Turbulence Models and Their Application to Complex Flows R. H. Nichols University of Alabama at Birmingham


 Garry Elliott
 1 years ago
 Views:
Transcription
1 Turbulence Models and Ther Applcaton to Complex Flows R. H. Nchols Unversty of Alabama at Brmngham Revson 4.01
2 CONTENTS Page 1.0 Introducton 1.1 An Introducton to Turbulent Flow Transton to Turbulent Flow Statstcal Concepts for Turbulent Flow Analyss Turbulent Length and Tme Scales Vortcty and Stran Tensors Classfcaton of Turbulence Models 115 References Classcal Turbulence Closure Methodologes.1 Reynolds Averagng 1. Favre Averagng 4.3 Boussnesq Approxmaton 5 References Turbulent Boundary Layer and Shear Layer Theory 3.1 Boundary Layer Theory Shear Layer Theory 37 References Algebrac Turbulence Models 4.1 Inner Eddy Vscosty Model CebecSmth Model BaldwnLomax Model Wake and Jet Model Algebrac Model Shortcomngs Grd Senstvty for a Flat Plate wth Adabatc Walls Grd Senstvty for an Axsymmetrc Bump Grd Senstvty for a Flat Plate wth Heat Transfer Grd Senstvty for a Nozzle wth Heat Transfer Summary 41 BaldwnLomax Applcaton Hnts 4 References OneEquaton Turbulence Models 5.1 Theory SpalartAllmaras Model Rotaton/Streamlne Curvature Correctons Grd Senstvty for a Flat Plate wth Adabatc Walls Grd Senstvty for an Axsymmetrc Bump Grd Senstvty for a Flat Plate wth Heat Transfer Grd Senstvty for a Nozzle wth Heat Transfer Summary 50 SpalartAllmaras Applcaton Hnts 50
3 References TwoEquaton Turbulence Models 6.1 Theory Tradtonal kε Models kω Models SST Model RNG Model Numercal Implementaton Compressblty Correcton for Shear Layers Intalzng Turbulence Values for a Gven Profle Shape Rotaton and Curvature Correcton Grd Senstvty for a Flat Plate wth Adabatc Walls Grd Senstvty for an Axsymmetrc Bump Grd Senstvty for a Flat Plate wth Heat Transfer Grd Senstvty for a Nozzle wth Heat Transfer Summary 69 TwoEquaton Model Applcaton Hnts 630 References Reynolds and Algebrac Stress Models 7.1 Reynolds Stress Models Algebrac Stress Models Grd Senstvty for a Flat Plate wth Adabatc Walls Grd Senstvty for an Axsymmetrc Bump 71 RSM and ASM Model Applcaton Hnts 718 References Large Eddy Smulaton 8.1 The Flterng Operaton Dervaton of the LES Equatons Smagornsk Model Dynamc Smagornsk Model kequaton Model Inflow Turbulence Boundary Condton Other LES References Spatal Mxng Layer Example 810 References Hybrd RANS/LES Turbulence Models 9.1 Theory Crcular Cylnder WICS Bay Delayed Detached Eddy Smulaton (DDES) Summary 930 Hybrd RANS/LES Applcaton Hnts 931
4 References Wall Functon Boundary Condtons 10.1 Theory Grd Senstvty for a Flat Plate wth Adabatc Walls Grd Senstvty for an Axsymmetrc Bump Grd Senstvty for a Flat Plate wth Heat Transfer Grd Senstvty for a Nozzle wth Heat Transfer Wall Functon Applcaton Hnts 100 References Boundary Layer Transton Smulaton 11.1 Transton Models Based on Stablty Theory Transton Models wth Specfed Transton Onset Transton Models wth Onset Predcton Capablty Transton Models wth Onset Predcton Capablty References 1114
5 Introducton Some basc knowledge of turbulence and an understandng of how turbulence models are developed can help provde nsght nto choosng and applyng these models to obtan reasonable engneerng smulatons of turbulent flows. Ths effort s drected at producton users of Computatonal Flud Dynamcs (CFD) and attempts to provde the basc nformaton requred to choose and use currently avalable turbulence modelng technques. The lmtatons of each modelng technque and applcaton tps are also provded. It s beleved that the NaverStokes equatons can be used to fully descrbe turbulent flows, but current lmtatons n computatonal horsepower have made the drect soluton of the NaverStokes equatons mpractcal for all but very smple flows at low Reynolds numbers. Ths s because current computers do not allow for the resoluton of the wde range of length and tme scales assocated wth turbulence. Many complex flud dynamc applcatons are drected at determnng tmeaveraged quanttes, and hence t s desrable to fnd a means to obtan these mean quanttes short of solvng the full unsteady NaverStokes equatons for all of the length and tme scales assocated wth the turbulence. The quest for the ultmate turbulence model has been ongong for nearly a century now. Early turbulence models were emprcally derved algebrac relatons. As computers developed and numercal smulaton evolved dfferental equaton based transport type turbulence models became the turbulence smulaton methodology of choce. It should always be remembered that transport models are emprcally calbrated. The use of transport type turbulence models has become standard practce for most engneerng applcatons. Many current researchers are now solvng the unsteady NaverStokes equatons for largescale, or grd realzed, turbulence and modelng the smaller, or subgrd, turbulent scales that cannot be captured on the computatonal grd. Although the ultmate generalpurpose turbulence model has yet to be developed, turbulence modelng has matured to the pont that reasonably accurate results can be obtaned for a wde range of engneerng applcatons wth the current class of computers. As computer technology contnues to mprove both the role and the form of turbulence models wll contnue to evolve. 1.1 An ntroducton to Turbulent Flow An understandng of what consttutes turbulent flow s requred before proceedng to dscuss turbulence modelng. Turbulence can be parameterzed by several nondmensonal quanttes. The most often used s Reynolds number. Reynolds number represents the rato of nertal forces to vscous forces. The vscous forces domnate at low Reynolds numbers and dsturbances are damped rapdly. These dsturbances begn to amplfy as Reynolds number s ncreased and eventually transton nto fully turbulent flows. Launder 1 gves the followng
6 1 defnton for turbulent flow: At moderate Reynolds numbers the restranng effects of vscosty are too weak to prevent small, random dsturbances n a shear flow from amplfyng. The dsturbances grow, become nonlnear and nteract wth neghborng dsturbances. Ths mutual nteracton leads to a tanglng of vortcty flaments. Eventually the flow reaches a chaotc, nonrepeatng form descrbable only n statstcal terms. Ths s turbulent flow. Turbulence s a threedmensonal unsteady vscous phenomenon that occurs at hgh Reynolds number. Turbulence s not a flud property, but s a property of the flow tself. Turbulent flow can be hghly nonlnear and s random n nature. Turbulent dsturbances can be thought of as a seres of threedmensonal eddes of dfferent szes that are n constant nteracton wth each other. Ths model of turbulence s Lagrangan n nature, wth these turbulent flow structures beng transported downstream by the mean flow. These structures exst for a lmted amount of tme before they are dsspated away by molecular vscosty. An example of the flud structures that appear n a turbulent flow can be seen n the wake of a crcular cylnder n Fg. 1.1a and the boundary layer n Fg. 1.1b. Fgure 1.1a Turbulent structures n the wake of a crcular cylnder.
7 13 Fgure 1.1b Turbulent structures n a boundary layer. Turbulence s dffusve n nature. The rates of mass and momentum transfer are much hgher n a turbulent flow than n a lamnar flow. Ths dffusveness can be a hghly desrable property for enhancng the mxng of physcal and/or chemcal propertes wthn a flow. The dffusve nature of turbulence also causes boundary layers and shear layers to become much thcker than ther lamnar counterparts, whch s often an undesrable result of turbulent flow. A generc turbulent knetc energy spectrum that ndcates how the turbulent energy s parttoned among the varous sze eddes s shown n Fg 1.. Energy Producton Inertal Range Dsspaton Log(Energy) Log(Wave number) Fgure 1. Turbulent energy spectrum. The wave number s nversely proportonal to the turbulent length scale. It can be seen n Fg 1. that most of the turbulent knetc energy s n the large turbulent length scales (or low wave numbers). The large scale eddes take knetc energy from the mean flow n the energy producton regon of the turbulent spectrum. Smaller eddes feed off of the larger energy producng eddes n the nertal range. Here the turbulence s essentally n equlbrum and the transfer of
8 14 energy by nertal forces s the domnant process. At the smallscale end of the turbulent spectrum the eddes become so small that vscous dsspaton converts the knetc energy nto heat. As mentoned earler, t s generally accepted that the NaverStokes equatons can be used to drectly smulate turbulent flows. Ths type of flud modelng s called Drect Numercal Smulaton (DNS). Drect numercal smulaton of the turbulent energy cascade usng the NaverStokes equatons requres that the grd spacng be smaller than the smallest turbulent length scale. For a typcal fullscale arcraft, the smallest length scale of the turbulence may be of the order of 106 tmes smaller than the arcraft reference length. More than grd ponts would be requred to dscretze ths example. Snce engneerng problems of ths sze are mpractcal at ths tme, some model of the turbulence s requred so that engneerng answers may be obtaned for largescale flud problems. For many engneerng applcatons the length and tme scales of the turbulence are much smaller than the length and tme scales of the problem of nterest. Ths tends to smplfy the modelng of turbulence and remove the need to treat the unsteady aspects of the turbulent flow snce they occur at tme scales much smaller than those of nterest. When the turbulent and problem scales become of the same sze the modelng becomes more dffcult. 1. Transton to Turbulent Flow Lamnarturbulent transton s an extraordnarly complcated process. The followng s a quck summary of the process. More detal can be found n Schlchtng. The process begns by transformng external dsturbances nto nternal nstablty oscllatons n the boundary layer or shear layer. A lamnar boundary layer or shear layer s susceptble to dsturbances from both the free stream and the body surface. The ablty of external dsturbances to penetrate the boundary layer or shear layer and then be amplfed s defned as the receptvty of the boundary layer. The free stream dsturbances may nclude acoustc waves, partcles n the flow, transported vortcty from upstream n the flow, and pressure, densty, or temperature fluctuatons. Surface dsturbances nclude the roughness of the body or moton of the body. Many of these dsturbances eventually get damped by vscosty. The transton process starts when a dsturbance n the boundary layer or shear layer s no longer damped but gets amplfed. Ths results n the formaton TollmenSchlchtng waves whch are the frst mode nstablty of the flow. Whle these waves travel downstream, three dmensonal waves and vortces begn to develop. At certan ponts n the boundary layer, small rregularly shaped turbulent spots wll occur and wll be convected n a wedge shaped regon as shown n the Fgure.1.3. These spots, known as Emmons spots, appear at random locatons on the plates at rregular tme ntervals. As these spots move downstream, they grow and eventually fuse wth each other to encompass the entre boundary layer or shear layer. Ths process eventually leads to fully developed turbulent flow.
9 15 Lamnar Transtonal Turbulent Fgure 1.3 Growth of turbulent spots n a flat plate boundary layer. The transton mechansm mentoned above s known as natural transton. Flows wth strong free stream dsturbances can bypass the mechansm of formaton of TollmenSchlchtng waves and form turbulent spots drectly from the nfluence of the free stream dsturbances. Ths s called as bypass turbulence 3. For compressble flows, most notably flows wth a Mach number of. or greater, Mack 4,5 showed that multple modes of nstablty exst. For flows between M=. and M=4.5, the frst mode of the nstablty, known as the vscous nstablty, s the most unstable. As the Mach number ncreases beyond 4.5, the second mode, known as the nvscd nstablty, becomes the most unstable mode. Ths second mode ncludes acoustcal dsturbances that are characterzed by very large fluctuatons n pressure and temperature. Boundary layer stablty theory seeks to predct the receptvty of the boundary layer n terms the crtcal Reynolds number for a flow. The crtcal Reynolds number s defned as the Reynolds number at whch the dsturbances n a lamnar boundary layer begn to amplfy and the flow becomes turbulent. Accordng to ths theory, the flud moton s decomposed nto a tmeaveraged mean flow component and a fluctuatng component. U = U u (1.1) Here the overbar sgnfes a tme averaged quantty and the prme s used for perturbaton quanttes. If the quanttes n the NaverStokes equatons are decomposed as n Eq. 1.1 and the mean flow quanttes are subtracted out, the twodmensonal ncompressble NaverStokes equatons become
10 16 u v x y u u du 1 p U v = ν ( u ) t x dy ρ x v v 1 p U = ν v t x ρ y (1.) Eq. 1. has three equatons and three unknown dsturbances u, v, and p. If a perodc dsturbance s assumed then the soluton can be expressed as a Fourer seres. Ths soluton s lnear, so solutons may be summed up. Note ths s only vald for very small ampltudes for the dsturbances. For twodmensonal flow, the streamfuncton can be defned as Ψ(x,y,t) = Φ(y)e (α xβ t) (1.3) Where α = π s the wave number and λ s the wavelength of the dsturbance λ and β = β r β, where β r = πf ( f s the dsturbance frequency) and β s the amplfcaton factor for the dsturbance. The rato of β/α s β βr β c = = = cr c α α (1.4) Here c r s the propagaton velocty n the xdrecton and s known as the phase velocty. The magnary part, c, s the degree of dampng f c s postve and s the degree of amplfcaton f c s negatve. Substtutng the expressons from Eq. 1.3 nto Eq. 1. φ ( αx βt ) u = = φ ( y) e y φ ( αx βt ) v = = αφ ( y) e x (1.5) 4 ( U c)( φ α φ) U φ = ( φ α φ α φ) α Re (1.6) Eq. 1.6 s called the OrrSommerfeld equaton. The equaton s shown n nondmensonal form. The terms on the left sde represent the nerta terms. The terms on the rght sde represent the vscous terms from the NaverStokes
11 17 Uδ equatons. Here the Reynolds number (Re) s defne as Re= ν boundary layer thckness. where δ s the Solvng the OrrSommerfeld equatons frst requres a steady state fully lamnar soluton of the NaverStokes equatons. The pressure and velocty from the lamnar soluton are used as nputs to the OrrSommerfeld equatons. An example soluton of the OrrSomerfeld s shown n Fg The vertcal axs s the rato of the shear layer thckness to the wavelength of the dsturbance. The horzontal axs s the Reynolds number based on the shear layer thckness. Re Fgure 1.4 Example soluton to the OrrSommerfeld equaton. The curve shown n Fg. 1.4 s the neutral stablty curve. Dsturbances are amplfed n the crosshatched regon, whle they are damped n the regon outsde the curve. The Reynolds number at whch dsturbances are frst amplfed s called the crtcal Reynolds number (R crt ). For Reynolds numbers below R crt, dsturbances are damped. As the Reynolds number ncreases above R crt, almost all dsturbances are amplfed and turbulent flow occurs. The effect of pressure gradent s shown n Fg The 0 curve represents no pressure gradent, the 5 curve represents an adverse pressure gradent, and the 4 curve represents a favorable pressure gradent. An adverse pressure gradent causes transton to occur more rapdly, whle a favorable pressure gradent delays the onset of transton.
12 18 Re Fgure 1.5 Effect of pressure gradent on boundary layer stablty. Transton locatons n boundary layer flows are generally characterzed by two parameters: Re x and Re θ. Re x s the Reynolds number based on dstance from the leadng edge. Re θ s the Reynolds number based on the momentum thckness defned as θ y= δ = y= 0 ρu ρ u e e u 1 dy u (1.7) e Here δ s the boundary layer heght and the subscrpt e denotes the edge of the boundary layer. Fg. 1.6 shows the transton process on the skn frcton on a flat plate. The skn frcton changes by almost an order of magntude durng the transton process.
13 19 Fgure 1.6 Skn frcton along a flat plate. 1.3 Statstcal Concepts for Turbulent Flow Analyss Several statstcal moments are used to descrbe turbulent flows. Ths secton wll revew those most often assocated wth modelng turbulence. These statstcs are usually performed on ensembles of dscrete observatons n both expermental and computatonal analyss of turbulent flows. The frst moment s the mean or average and s defned as u = 1 n n u = 1 (1.8) Here the average s taken of n dscrete observatons. The hgher moments are defned relatve to the mean. The second moment, or varance, s defned as 1 n = u = n = 1 ( ) u u σ (1.9) The varance provdes nformaton on the spread of the data away from the mean. The thrd moment, or skewness, s gven by S 3 u = 3 σ = σ 1 n 3 n = 1 ( ) 3 u u (1.10)
14 110 Skewness gves the amount of tme the sgnal s above or below the mean. The fourth moment, or kurtoss, s defned by K 4 u = 4 σ = σ 1 n 4 n = 1 ( ) 4 u u (1.11) Kurtoss gves the amount of tme the sgnal s away from the mean. A smple example of these moments can be constructed usng sne waves. Four cases are defned as πt Case1: sn 1000 πt Case : sn 1000 πt πt Case3: sn sn ,000 πt πt Case 4 : sn sn (1.1) The four sgnals are shown n Fg Fgure 1.7 Example sgnals for statstcal analyss. The statstcal moments for these four cases are shown n Table 1.1.
15 111 Case Mean σ S K Table 1.1 Statstcal moments for four test sgnals. Sgnals that are symmetrc about the mean such as Case 1 and 3 have a zero mean and skewness. Case 4 s the only sgnal that produces a nonzero skewness. Tme ensembles must be statstcally statonary for the moments to have any meanng. A statstcally statonary flow s defned as a flow n whch statstcal parameters do not depend on the nterval n tme used to evaluate them. An example pressure sgnal s gven n Fgure 1.8. Wndow Wndow 4 Wndow 1 Wndow 3 Fgure 1.8 Unsteady pressure measurement. It s very dffcult to determne f the sgnal s statonary by vsual nspecton. Four overlappng data wndows of 4096 samples each are used to assess whether ths flow s statonary. The overlapped wndowng technque s effectve f at least two
16 11 of the wndows are completely ndependent as s the case for Wndow 1 and Wndow 4. The number of wndows and the number of samples n each wndow s generally determned by the number of total samples avalable n the sgnal. Fourer transforms are useful for transformng the sgnal between the tme doman and the frequency doman. The Fourer transform of the four data wndows and the average of the four transforms are shown n Fgure 1.9. Fgure 1.9 Fourer transform of pressure sgnal. Sound pressure level s proportonal to the log of the varance of the pressure sgnal. The Fourer transforms for the ndvdual wndows show a large amount of dsagreement. Ths s due to the dscreet nature of the transform and the lmted number of samples n each wndow. The statstcal moments and the frequency of the maxmum spectral peak for each of the data wndows and the average of the wndows are shown n Table 1.. The maxmum error n Table 1. s defned as the error between the ndvdual data wndows and the wndow average. Ths sgnal s reasonably statonary for the moments examned. It should be noted that demonstratng that a sgnal s statonary wth respect to a gven moment does not assure that the sgnal s statonary for hgher moments. The wndowng technque can be appled to both expermental and computatonal data. Ths technque can be used to assess convergence for unsteady computatons 6.
17 113 Average Pressure σ S K Frequency of Peak SPL (hz) Wndow Wndow Wndow Wndow Wndow Average Max Error 0.66% 1.71% 14.3% 4.18% 3.88% Table 1. Statstcs from pressure sgnal data wndows. 1.4 Turbulent Length and Tme Scales It s useful to quantfy the range of length and tme scales assocated wth a turbulent flow. These estmates of the maxmum and mnmum eddy length and tme scales can be used n selectng the approprate computatonal grd spacng and tme step for a gven problem. Turbulent scales are defned n terms of the turbulent knetc energy (k), turbulent dsspaton (ε), and the knematc vscosty (ν). The turbulent knetc energy and turbulent dsspaton wll be derved n Chap.. These two quanttes are usually assocated wth the larger scales of turbulence. From dmensonal analyss, the large scale, or turbulence producng, eddes (see Fg. 1.) have length scales characterzed by 3 k L = (1.13) ε The large scale eddes have tme scales of the order of k Τ = (1.14) ε The smallest scale, or dsspatve, eddes (the Kolmogorov scale) have length and tme scales gven by
18 ν λ = (1.15) ε 1 ν τ = (1.16) ε The rato of the smallest to largest eddy length and tme scales s then λ L 3 4 = R t (1.17) τ = Τ 1 R t (1.18) where R t s the turbulent Reynolds number (k /(νε)). These relatonshps ndcate that the range of turbulent scales may span orders of magntude for hgh Reynolds number flows. Snce these turbulent length scales are much smaller than the physcal scales (.e. wng chord, channel heght) assocated wth a flow of nterest, t s easy to see that large amounts of grd ponts would be requred to fully smulate a hgh Reynolds number turbulent flow. 1.5 Vortcty and Stran Tensors Two tensors that are often used n the development of turbulence models are the vortcty and stran tensors. The vortcty tensor s defned as Ω = 1 u x u x (1.19) The trace of the vortcty tensor (the sum of the dagonal elements) s equal to zero snce Ω 11 =Ω =Ω 33 =0. The largest magntudes of vortcty typcally occur near the wall n a boundary layer and n the core of a vortex. The stran tensor s defned as S = 1 u x u x (1.0) The trace of the stran s U r, whch s zero for steady ncompressble flows. The largest magntudes of stran also occur near the wall n a boundary layer. The magntude of the stran s almost zero n the core of a vortex. For typcal boundary layers and shear layers the magntude of the vortcty s equal to the
19 115 magntude of the stran snce only one of the cross dervatves wll domnate both tensors. 1.6 Classfcaton of Turbulence Models Turbulence models can be classfed by what turbulent scales they choose to model and what scales they choose to smulate by solvng the unsteady Naver Stokes equatons on a computatonal grd. The tradtonal Reynolds Averaged NaverStokes (RANS) use a tme averagng process to remove the necessty of smulatng all of the scales of the turbulence spectrum. The RANS approach uses one length scale to characterze the entre turbulent spectrum, as shown n Fg The use of a sngle length scale places a tremendous burden on the turbulence modeler snce t can be dffcult to fnd one length scale that s approprate for all cases. When ths can be accomplshed the flow can be treated as a steady flow snce all the unsteadness s assumed to occur at scales below the computatonal grd sze and are handled by the turbulence model. Ths allows for the use of tme marchng numercal algorthms wth large numercal dsspaton snce the obect of the calculaton s to dsspate all wavelengths (whether they are of physcal or numercal orgn) of the unsteady flow durng the convergence process. Numercal algorthms that converge to a steady state soluton rapdly may not be suted for unsteady flow smulatons because these algorthms contan large amounts of numercal dsspaton and may overdamp the actual flow. The RANS approach s dscussed n more detal n Chapter. RANS Hybrd RANS/LES Log(Energy) LES DNS Log(Wave number) Fgure 1.10 Regon of the turbulent energy spectrum modeled by dfferent turbulence model approaches.
20 116 Drect numercal smulaton (DNS) 7 attempts to smulate all of the scales of turbulence and model nothng. The grd resoluton and the maxmum allowable tme step for a DNS calculaton must be small enough to capture the Kolmogorov scales (Eq and 1.16) of the turbulent flow. The nverse relatonshp of these scales wth Reynolds number shown n Eq and 1.18 ndcate that the grds must become ncreasngly fne as Reynolds number s ncreased. A DNS soluton s nherently unsteady, so t must be run for long perods to assure that the resultng soluton s statstcally statonary and free from the ntal condtons provded to start the computaton. The numercal algorthm used n the DNS soluton process must have very low numercal dsspaton n order to allow all the wavelengths of the turbulent flow to naturally persst. The numercal dsspaton must be much smaller than the molecular vscosty or t wll manfest tself as an ncrease n the effectve molecular vscosty, and subsequently as a decrease n the effectve Reynolds number 8. Most of the hgher order numercal algorthms currently used n DNS are only vald on computatonal grds that contan unform spacng. Grd stretchng tends to reduce the order of the numercal algorthm and ncrease the numercal dsspaton of the algorthm. The requrement of nearly unform grds wth small cell szes leads to extremely large numercal grds even for small Reynolds numbers. DNS computatons also requre a model for the nflow free stream dsturbances. For channel flows, the nflow turbulence can be obtaned by recyclng the outflow turbulence. To date DNS calculatons have only been performed for low Reynolds numbers and smple geometres. Large Eddy Smulaton (LES) s an attempt to move beyond DNS by modelng only the smallest turbulent scales n a problem 9,10,11 as can be seen n Fg LES wll be dscussed n detal n Chapter 7. The smaller turbulent scales are more nearly sotropc, so they may be modeled wth farly smple turbulence models. Hence a smple turbulence model s used to smulate the subgrd turbulence (turbulent scales that can not be realzed on the computatonal grd) and the NaverStokes equatons are solved for the remanng scales. Ths requres a spatal flterng of the turbulence spectrum. Lke DNS, LES solutons are unsteady, and care must be taken to choose a numercal algorthm wth low numercal dsspaton or a decrease n effectve Reynolds number wll be seen n the calculaton. LES solutons must also be run a large number of tme steps to elmnate startng transents and to allow the soluton to become statstcally statonary. Whle LES allows for smulatng much hgher Reynolds numbers than DNS for the same computatonal resources, t s stll not capable of smulatng flght Reynolds numbers wth reasonable engneerng turnaround tmes. LES has been found useful for many lower Reynolds number applcatons such as combuston calculatons where the chemcal reactons are drven by the unsteady turbulent mxng of the fuel and the oxdzer. Recently a new class of turbulence models has arsen for unsteady hgh Reynolds flows. These turbulence models are called hybrd RANS/LES models and wll be dscussed n detal n Chapter 8. These hybrd turbulence models are extensons of the LES models n whch modfed RANS turbulence models are
21 117 used as subgrd models for the NaverStokes equatons. The goal for most applcatons s to use RANS to calculate the boundary layer where extremely small turbulent scales are present and use an LES lke model to smulate the smaller turbulent scales away from the body. Hence the large scale turbulent structures away from the body are smulated by the unsteady NaverStokes equatons. These models serve as a brdge between tradtonal RANS models and LES models as shown n Fg Ths allows for coarser grds than LES snce the RANStype subgrd model s vald for nonsotropc scales of turbulence. These models requre some sort of spatal flterng to determne the local value for the subgrd turbulent vscosty. Hybrd RANS/LES models do not need the extremely low numercal dsspaton requred by DNS and LES and hence can be robust and easy to mplement. Chapter 1 References: 1. Launder, B. E., An Introducton to the Modelng of Turbulence, VKI Lecture Seres , March 181, Schlchtng, H., BoundaryLayer Theory, McGrawHll, Inc., Morkovn, M., "Bypass  Transton Research Issues and Phlosophy," n Instabltes and Turbulence n Engneerng Flows, Netherlands: Kluwer Academc Publshers, 1993, pp Mack, L., "Lnear Stablty Theory and the Problem of Supersonc Boundary Layer Transton," AIAA Journal, Vol. 13, Mar. 1975, pp Mack, L., "Stablty of Axsymmetrc Boundary Layers on Sharp Cones at Hypersonc Mach Numbers," AIAA , Nchols, R., Comparson of Hybrd RANS/LES Turbulence Models for a Crcular Cylnder and a Cavty, AIAA Journal, Vol. 44, No. 6, June 006, pp Mansour, N., The Use of Drect Numercal Smulaton Data n Turbulence Modelng, AIAA9101, Jan Mon, P., Numercal and Physcal Issues n Large Eddy Smulaton of Turbulent Flow, JSME Internatonal Journal, 1998, Seres B, Vol. 41, No., pp Breuer, M., Large Eddy Smulaton of the Subcrtcal Flow past a Crcular Cylnder: Numercal and Modelng Aspects, Internatonal Journal of Numercal Methods n Fluds, 1998, Vol. 8, pp Rod, W., Large Eddy Smulaton of the Flows past Bluff Bodes, n: Launder, B. and Sandham, N. (eds) Closure Strateges for Turbulent and Transtonal Flows, Cambrdge Unversty Press, Cambrdge, Menon, S. A Numercal Study of Secondary Fuel Inecton Technques for Actve Control of Combuston Instablty n a Ramet, AIAA , January 199.
22 1.0 Classcal Turbulence Closure Methodologes The drect numercal soluton of the NaverStokes equatons for hgh Reynolds numbers flow requres a grd fne enough to capture all of the turbulent length scales and a tme step small enough to capture the entre turbulent spectrum. The computatonal requrements for such calculatons are well beyond the current state of computers. Obvously, some approxmatons to the Naver Stokes equatons are requred to allow practcal engneerng solutons to hgh Reynolds number flow problems. Two averagng technques used to smplfy the NaverStokes equatons are Reynolds and Favre averagng. Both of these approaches wll be dscussed n ths chapter. Both of these approaches results n tmeaveraged correlaton terms that must be modeled to close the equaton set. Two other methods of closure are large eddy smulaton (LES) and hybrd RANS/LES modelng. Both of these technques use spatal flterng to decde what turbulent scales should be modeled and what scales can be smulated. LES wll be dscussed n detal n Chapter 8. The hybrd approach wll be dscussed n Chapter 9..1 Reynolds Averagng Osborne Reynolds 1 suggested that for a statstcally statonary flow (see Chapter 1) the varables n the NaverStokes equatons could be decomposed nto tmeaveraged and turbulentfluctuaton terms. The velocty can then be wrtten as U ( x, t) = U ( x) u ( x, t) (.1) where the average velocty U s defned as 1 U = T T T Udt (.) and the tme average of the fluctuatng velocty u s 1 T 1 u = = u dt T T T T T ( U U ) dt = 0 (.3) Note that the tme scale of the ntegraton T must be much larger than the turbulent tme scales dscussed n Chapter 1 or the flow wll not be statstcally statonary. In the Reynolds averagng process the unsteady behavor of the turbulent flow s replaced by steadystate correlatons terms that must then be calbrated wth turbulent flow measurements. Ths has the advantage of sgnfcantly reducng
23  the computer tme requrements for obtanng engneerng qualty values of steady state varables. It s often dffcult to obtan emprcal correlatons that are vald for a wde range of turbulent flows, and hence the Reynolds Averaged NaverStokes (RANS) turbulence models must often be tuned to a partcular class of flows. Assumng that fluctuatons n densty ( ρ ), vscosty ( μ ), and thermal conductvty ( T κ ) are neglgble (whch s often not the case for hgh speed and reactng flows) and applyng Reynolds averagng to the NaverStokes equatons yelds ( ) 0 = U x t ρ ρ (.4) ( ) ( ) x x P u u x U U x t U = σ ρ ρ ρ (.5) ( ) ( ) ( ) = T x T x u x U x u u u u u U e u x H U x t E κ σ σ ρ ρ ρ ρ ρ 1 (.6) where = kk S S δ μ σ 3 1 (.7) = x U x U S 1 (.8) k U U e E = 1 (.9) ρ P E H = (.10) 1 u u k = (.11)
24 3 Note that the turbulent knetc energy (k) has been ncluded n the defnton of total energy ( E ). The Reynolds averagng has ntroduced new unknown correlatons k, u, u u e, u u u, and u σ. When the velocty correlaton uu s multpled by the densty ρ t represents the transport of momentum due to the fluctuatng (.e. turbulent) moton of the flud. The term ρ u u s the transport of x momentum n the drecton of x (or vce versa). Ths term acts as a stress on the flud and s therefore called turbulent or Reynolds stress. In most regons of a turbulent flow, the turbulent stresses are much larger than ther lamnar counter parts U ν. The Reynolds stress s a x symmetrc tensor. The trace of the stress tensor s two tmes the turbulent knetc energy, and s coordnate system ndependent. The off dagonal terms of the Reynolds stresses are coordnate system dependent. The veloctyenergy correlaton ρ u e can be thought of as a turbulent heat flux. The correlaton represents turbulent transport or dffuson. The turbulent dsspaton. ρ u u u trple u σ term s the Closure of the equaton set requres correlatons be developed for these new terms. These terms are often dffcult to measure expermentally. The turbulent dsspaton, uσ, ncludes correlatons of the perturbed velocty and spatal dervatves of the perturbed velocty. Ths quantty cannot be measured expermentally. Transport equatons can be developed for the turbulent stresses and turbulent dsspaton to close the system. Ths approach to closure s called Reynolds Stress Modelng (RSM) and wll be dscussed n Chapter 6. A smplfed approach to modelng the turbulent stresses n whch the RSM equatons are replaced wth algebrac relatonshps s called Algebrac Stress Modelng (ASM) and wll also be dscussed n Chapter 6. The Reynolds stresses must meet certan constrants to be physcally plausble. Schumann ntroduced the realzablty constrant. Ths constrant specfes that all the component energes of the turbulent knetc energy (the dagonal terms of the Reynolds stress tensor) reman nonnegatve and all offdagonal components of the Reynolds stress tensor satsfy Schwartz s nequalty. Ths can be wrtten as u u 0 (.1) u u u u u u 0 (.13)
25 4 ( ) ( ) u u u u u u u u u u u u u u u u u u u u u u u u u u u u (.14). Favre Averagng Reynolds averagng for compressble flows produces a very complcated set of equatons because the fluctuatons of densty, pressure, and temperature must also be accounted for. Favre 3 averagng or Favre 3 flterng has been used to smplfy the equatons set. Favre averagng s a densty weghted averagng process n space. The Favre flter can be defned for any varable as ), ( ) ( ~ ), ( t x u x U t x U = (.15) where () () () ρ ρ ρ ρ U dt t dt t U t U T T T T = = ~ (.16) Note that the over bar sgnfes a tme averaged quantty, and the tlde represents a Favre fltered quantty. Other auxlary relatons nclude U U U u ρ ρ ρ ρ = = = ~ ~ 0 (.17) Thus, usng Favre averagng, U u U t x U t x ~ ~ ), ( ), ( ρ ρ ρ ρ = = (.18) The equvalent expresson usng Reynolds averagng would yeld u U t x U t x = ρ ρ ρ ), ( ), ( (.19) Note that under these defntons the Reynolds average perturbaton 0 = u, but the Favre averaged perturbaton 0 u. Applyng ths flterng operaton to the NaverStokes equatons, and assumng that the flterng commutes wth the dervatve operaton, yelds
26 5 ρ t x ~ ( ρu ) = 0 (.0) ~ ~ ~ P ( ρu U ) ( ρu u ) = ( σ σ ) ρu ~ t x x x x ~ ρe t x x ~ ~ ( ρu H ) x ~ ρu e U ρuu ~ ( ) ( ) T T U ~ ~ σ uσ U σ κ T κ T x x x x 1 ρu uu ~ = (.1) (.) For almost all flows ~ >> σ, and hence the σ ~ σ T T T Smlarly, for most flows >>, and hence the x x terms are usually neglected. x terms are usually neglected. The Favre averaged equatons are smlar n appearance to the ncompressble Reynolds averaged NaverStokes equatons, but the correlatons are not the same. Favre averagng should only be used n flows wth large densty fluctuatons such as n combuston or hypersonc flow. In other cases the smpler Reynolds averagng should be used..3 Boussnesq Approxmaton Boussnesq 4 suggested that the turbulent stresses can be treated n an analogous form to the vscous stresses n lamnar flows. The Reynolds stresses are modeled as U U ρuu = μt ρkδ (.3) x x 3 The Boussnesq approxmaton states that the Reynolds stresses are proportonal to the local mean flow stran rate U U. The eddy vscosty x x (μ t ) s the proportonalty factor for the Boussnesq approxmaton. Ths s sometmes called a local equlbrum assumpton snce the turbulent stresses are assumed to be proportonal to the mean flow stran rates. The last term n Eq..3 was added so that the normal stresses would sum to the turbulent knetc energy (k). The Boussnesq approxmaton reduces the turbulence modelng process from fndng the sx turbulent stress components as dscussed above to determnng an approprate value for the eddy vscosty.
27 6 The eddy vscosty s a property of the flow feld and not a property of the flud. The eddy vscosty can vary sgnfcantly form pont to pont wthn the flow, and also from flow to flow. The eddy vscosty concept was developed assumng a relatonshp or analogy exsts between molecular moton and turbulent moton. Although turbulent eddes can be thought of as dscrete lumps of flud that collde and exchange energy, they are not rgd bodes and ther mean free paths are not small compared to the eddy sze, and hence turbulence does not really satsfy the constrants of knetc gas theory. In spte of the theoretcal weakness of the eddy vscosty concept, t does produce reasonable results for a large number of flows. The eddy vscosty concept has been successful n the predcton of flows n whch the shear stress ( τ = ρ u u ) s the turbulent stress of greatest mportance. Ths class of flows ncludes smple shear layers and attached boundary layers. The eddy vscosty s ntroduced as a scalar n Eq..3 so that the eddy vscosty s the same for all stress components. The assumpton of sotropc eddy vscosty s a smplfcaton that lmts the performance of ths class of turbulence models n complex flows. In a drect analogy to the turbulent momentum transport defned by the Boussnesq approxmaton (Eq..3), the turbulent heat or mass transport s often assumed to be proportonal to the gradent of the transported quantty ϑ uϑ = Γ (.4) x where Γ s the turbulent dffusvty of heat or mass. Lke the eddy vscosty, Γ s not a property of the flud but s a functon of the flow. Reynolds analogy between heat or mass transport and momentum transport suggests that Γ s proportonal to the eddy vscosty ν t Γ = (.5) σ t where σ t s the turbulent Prandtl or Schmdt number. Experments have shown that the σ t vares lttle throughout a flow feld and hence t s usually taken as a constant. Usng the Boussnesq approxmaton and the assumpton of a constant turbulent Prandtl number, the Reynolds averaged NaverStokes equatons (Eq..4,.5, and.6) can be wrtten as ρ t x ( ρu ) = 0 (.6)
28 7 ρu t x ( ρu U ) P = x σ x (.7) ρe t x ( ) ( ) μ μ t T ρu = H Uσ x x Pr Prt x (.8) where σ 1 ( μ μt ) S Skkδ (.9) 3 = S 1 U = x U x (.30) 1 E = e U U k (.31) P H = E (.3) ρ where k s the turbulent knetc energy. The turbulent knetc energy s often gnored n the defnton of total energy (Eq..31). Under the assumpton of a constant turbulent Prandtl number (Pr t ), the equaton set s closed once the eddy vscosty s defned. Some common approaches to defnng eddy vscosty models are dscussed n Chapters 3, 4, and 5. Chapter References: 1. Reynolds, O., On the Dynamcal Theory of Incompressble Vscous Fluds and the Determnaton of the Crteron, Phlosophcal Transactons of the Royal Socety of London, Seres A, Vol. 186, pp , Schumann, U., Realzablty of Reynolds Stress Turbulence Models, Physcs of Fluds, Vol. 0, pp , Favre, A., Equatons Fondanmentales des Fluds a Masse Varable en Ecoulements Turbulents, Pages 478 of Favre, A., Kovasznay, L., Dumas, R., Gavglo, J., and Coantc, M. (eds.), La Turbulence en Mcanque de Fluds, CNRS, Pars, 196. Englsh edton: The Mechancs of Turbulence, Gordon and Breach, New York, Boussnesq, J., Theore de l ecoulement Turbllonnant et Tumultueux des Lqudes Dans les lts Rectlgnes a Grande Secton, GaultherVllars, 1897.
29 Turbulent Boundary Layer and Shear Layer Theory A bref revew of smple theory for turbulent boundary layers and shear layers wll ad n the understandng of the development and calbraton of turbulence models. Most of these relatonshps are based on emprcal correlatons of the shape of the velocty profle across the boundary layer or shear layer. Ths shape can be used n conuncton wth the NaverStokes equatons to back out relatonshps for the turbulent stresses or the eddy vscosty. 3.1 Boundary Layer Theory Turbulent boundary layers are usually descrbed n terms of several nondmensonal parameters. The boundary layer thckness, δ, s the dstance from the wall at whch vscous effects become neglgble and represents the edge of the boundary layer. Two ntegral parameters across the velocty profle are the * dsplacement thckness δ and the momentum thckness θ. * ρu δ = 1 dy eu (3.1) 0 ρ e ρu u θ = dy eu 1 e u (3.) ρ 0 e The ntegraton s performed normal to the wall and the subscrpt e s used to denote the edge of the boundary layer at y=δ. The dsplacement thckness s a measure of the ncreased thckness of a body due to the velocty defect of the boundary layer. The momentum thckness s the dstance that, when multpled by the square of the freestream velocty, equals the ntegral of the momentum defect across the boundary layer. If a smple power law velocty profle as shown n Eq. 3.3 s assumed and the flow s ncompressble, the relatonshps n Eqs can be obtaned from approxmatons of the NaverStokes equatons. u ue δ x = y = δ 1/5 x 1/ Re (3.3) (3.4) * δ x = Re 1/5 x (3.5)
30 3 θ x = Re 1/5 x C τ = = f w 1 1/5 ρ eue Re x (3.6) (3.7) C f s called the skn frcton. The dstance from the leadng edge n the streamwse drecton s gven by x. Both the boundary layer growth rate and the skn frcton decrease as the Reynolds number ncreases. Reynolds analogy can be used to develop a relatonshp for heat transfer. Reynolds analogy says that the rato of the shear stress to the heat transfer s a constant near the wall. Thus the Nusselt number can be defned as hx C f Nu x = = Re x Pr (3.8) k Here h s the heat transfer coeffcent, k s the thermal conductvty, and Pr s the Prandtl number. Ths relatonshp s ndependent of the equaton used to determne skn frcton. The power law relatonshp s not extremely accurate, but s useful for developng some useful turbulent boundary layer relatonshps. A more accurate relatonshp for skn frcton for adabatc ncompressble flow on a flat plate s gven by Whte and Chrstoph 1 C f 0.4 = (3.9) ln ( Re ) x Ths relatonshp s only applcable n the turbulent regon of a boundary layer, and does not apply n the lamnar or transtonal regons of the boundary layer. The skn frcton s affected by a number of parameters, ncludng pressure gradent, surface roughness, compressblty, and surface heat transfer. Adverse pressure gradents cause the skn frcton to be reduced as the boundary layer s pushed toward separaton. Boundary layer separaton occurs when the skn frcton becomes negatve. Hgh values of skn frcton are an ndcaton of a very stable (.e. dffcult to separate) boundary layer. Unfortunately hgh values of skn frcton also equate to hgh values of vscous drag. The effect of both compressblty and heat transfer on the flat plate turbulent skn frcton are shown n Fg Both a hot wall and compressblty tend to reduce the skn frcton over the ncompressble adabatc value.
31 33 Fgure 3.1 Effect of compressblty and heat transfer on the skn frcton on a flat plate. As mentoned above, the power law relatonshp n Eq. 3.3 s not very accurate. Better approxmatons of the velocty profle shape are generally wrtten n terms of the parameters u and y defned as u = u u τ (3.10) y ρ wu y = τ μ w (3.11) and u τ s the frcton velocty u τ w τ = (3.1) ρ w The subscrpt w denotes the value at the wall and τ w s the wall shear stress defned by
32 34 u τ w = μ (3.13) y w The boundary layer velocty profle can be dvded nto four regons. The ncompressble velocty profle n each of the subregons regons of the nner regon shown n Fg. 3. s gven by Lamnar sublayer 0 < y < 5 u = y (3.14) Buffer layer 5 < y < 30 u = 5 ln y (3.15) Log layer 30 < y < 1000 u = κ 1 ln y B (3.16) The values of κ, the von Karmen constant, and B are often debated, but are generally accepted to be 0.4 and 5.5 respectvely. The log layer s also called the law of the wall. Inner Regon Outer Regon Sublayer Buffer Log Layer Wake Layer Fgure 3. Boundary layer regons. The y value where the profle transtons from the nner to the outer profle vares wth the Reynolds number and the pressure gradent. The outer regon s much more senstve to pressure gradent. Clauser s equlbrum parameter β s often used to characterze the pressure gradent.
33 35 * δ β = τ w p e x (3.17) Coles 3 ntroduced the wake functon W gven by y π y W = sn δ δ (3.18) The velocty profle n the outer regon s gven by u 1 = ln y κ Π y B W κ δ (3.19) where Π s gven by ( 0.5) Π = 0.8 β (3.0) Spaldng 4 gven by proposed a composte form for the ncompressble velocty profle y = u e κb e ΠW e κu 1 κu ( κu ) ( κu ) 6 3 (3.1) Whte and Chrstoph 1 gve a law of the wall velocty profle that ncludes the effects of compressblty, heat transfer, and pressure gradent. sn 1 Γu Θ = sn Q 1 1/ Γu0 Θ Γ Q κ φ 1 φ 0 1 ( φ φ ) ln 0 φ 1 φ0 1 (3.) Where Q = ( Θ 4Γ) 1/ and = ( 1 αy ) 1/ φ. The values of y 0 and u 0 are taken as 6 and 10 respectvely. The parameters Θ, Γ, and α represent the effects of heat transfer, compressblty, and pressure gradent respectvely. qwμw Θ = (3.3) T k ρ u w w w τ ruτ Γ = (3.4) c p T w
34 36 μw pe α = (3.5) ρ τ u x w w τ T Here q w s the wall heat transfer ( kw ), μ w s the wall vscosty, T w s the wall y w temperature, ρ w s the wall densty, r s the recovery factor (normally taken as the Prandtl number to the one thrd power), k w s the wall thermal conductvty, and c p s the specfc heat at constant pressure. It s not obvous what effect each of the parameters defned n Eqs has on the velocty profle. Fg. 3.3 shows the effect of an adverse pressure gradent and compressblty has on the velocty profle. Increasng α Increasng Γ Fgure 3.3 Effect of adverse pressure gradent and compressblty on boundary layer profle shape. The boundary layer thckens and the skn frcton decreases as the pressure gradent s ncreased. Compressblty also causes the skn frcton to decrease. Heat transfer effects are shown n Fg. 3.4.
35 37 Increasng Θ (Cold Wall) Decreasng Θ (Hot Wall) Fgure 3.4 Effect of wall heat transfer on boundary layer profle shape. A cold wall (wall temperature less than the adabatc wall temperature) creates a thnner boundary layer and ncreases the skn frcton. A hot wall (wall temperature greater than the adabatc wall temperature) thckens the boundary layer and decreases the skn frcton. The temperature dstrbuton wthn the nner part of the boundary layer boundary layer s gven by the CroccoBusemann equaton ( 1 Θu Γ( ) ) T = Tw u (3.6) where Θ s defned n Eq. 3.3 and Γ s defned n Eq For adabatc wall cases, β =0 and the CroccoBusemann equaton reduces to T = T w 1 ( γ 1) r u (3.7) The pressure s assumed to be constant n the normal drecton from the wall n the nner part of the boundary layer. Densty dstrbutons can be defned based on the temperature dstrbuton and the equaton of state. 3. Shear Layer Theory A free shear layer s always ntated from a surface of some knd. The boundary layer profle remans for a short perod. If no external pressure gradent s present, the shear layer wll eventually become selfsmlar. A selfsmlar profle s one n whch the profle shape remans unchanged as you move downstream f
36 38 the profle s defned n terms of smlarty varables. Chapman and Korst 5 suggested the smlarty varable for twodmensonal shear layers σy η = (3.8) x where x s the downstream dstance from the orgn of the shear layer, y s the normal dstance across the shear layer (y=0 denotes the center of the shear layer, and σ s the shear layer spread rate parameter. Brown and Roshko 6 suggested usng y y = (3.9) * y 0.5 δ ω where the vortcty thckness δ ω s defned as U U = u y 1 δ ω (3.9) max Here U 1 s the velocty at the hgh speed edge of the shear layer and U s the velocty at the low speed edge of the shear layer. Chapman and Korst 5 suggested that the velocty profle was gven by * u u = = 0.51 [ erf ( η) ] (3.30) U U 1 where erf s the error functon. Ths profle shape s vald for both lamnar and turbulent shear layers. Sammy and Ellot 7 obtaned Laser Doppler Velocmeter (LDV) data on a shear layer. Measurements were made at several downstream locatons between the tralng edge of the spltter plate and a staton 10 mm downstream of the spltter plate. The flow parameters are gven n Table 3.1. Eq s plotted wth data n Fg T 0, K P 01, kpa M 1 M M c U 1, m/sec U /U 1 ρ /ρ 1 δ 1, mm Table 3.1 Flow parameters for the spatal mxnglayer case
37 39 The shear layer thckness s gven by Fgure 3.5 Shear layer velocty profle. x b = (3.31) σ The shear layer spread parameter (σ) s affected by compressblty. The accepted value for subsonc flow ssung nto quescent ar s σ=11. Ths value wll ncrease as compressblty effects become greater, and wll cause the shear layer to become thnner. Expermental values 8 for σ for twodmensonal ets ssung nto quescent ar are shown n Fg Fgure 3.6 Shear layer spread parameter for et ssung nto quescent ar.
38 310 Subsonc spread parameter expermental results 8 as a functon of velocty rato are shown n Fg Fgure 3.7 Shear layer spread parameter for subsonc ets. Ths ndcates that the shear layer wll become thnner as the rato of the veloctes of the two streams ncreases. Chapter 3 References: 1. Whte, F. M. and Chrstoph, G. H., A Smple New Analyss of Compressble Turbulent TwoDmensonal Skn Frcton Under Arbtrary Condtons, AFFDLTR , Feb Clauser, F. J., Turbulent Boundary Layers n Adverse Pressure Gradents, Journal of the Aeronautcal Scences, Vol. 1, 1954, pp Coles, D. The Law of the Wake n the Turbulent Boundary Layer, Journal of Flud Mechancs, Vol. 1, Part, July 1956, pp Spaldng, D. B., A Sngle Formula for the Law of the Wall, Journal of Appled Mechancs, Vol. 8, No. 3, 1961, pp Chapman, A. and Korst, H., Free Jet Boundary wth Consderaton of Intal Boundary Layer, Proceedngs of the Second U. S. Natonal Congress of Appled Mechancs, The Amercan Socety of Mechancal Engneers, New York, 1954, pp Brown. G. and Roshko, A., On Densty Effects and Large Structure n Turbulent Mxng Layers, Journal of Flud Mechancs, Vol. 64, No. 4, 1974, pp Sammy, M. and Ellot, G., Effects of Compressblty on the Characterstcs of Free Shear Layers, AIAA Journal, Vol. 8, No. 3, 1990, pp
39 Brch, S. and Eggers, J., A Crtcal Revew of Expermental Data for Developed Free Turbulent Shear Layers, n Free Turbulent Shear Flows, NASASP31, 197, pp
40 Inner Eddy Vscosty Model 4.0 Algebrac Turbulence Models Most of the earlest turbulence models were based on Prandtl s mxng length hypothess. Prandtl 1 suggested that the eddy vscosty could be represented by u μ t = ρl m (4.1) y The Prandtl mxng length relates the eddy vscosty to the local mean velocty gradent. The secret to successful applcatons of the mxng length hypothess s to fnd some general method of defnng the mxng length. Most algebrac models dvde the boundary layer nto an nner and outer regon as descrbed n Chapter 3. The nner layer ncludes the vscous sublayer, the buffer layer, and part of the fully turbulent log regon. The outer layer ncludes the remanng part of the log layer and the wake regon. The eddy vscosty n the nner layer follows Prandtls form and s gven by ( ) = ρl Ω μ (4.) t nner m where the mxng length L m s gven by L m y κ y 1 exp (4.3) A = where y s the dstance normal to the wall, κ s the von Karmen constant, and the vortcty Ω s defned as u v Ω = (4.4) y x 4. CebecSmth Model CebecSmth suggested that the outer eddy vscosty be expressed as * ( μ ) ραu δ t outer = (4.5) e Here α s usually assgned a value of for flows where the Reynolds number based on momentum thckness (Re θ ) s greater than 5000, δ * s the dsplacement thckness, and u e s the velocty at the edge of the boundary layer. The fnal eddy vscosty s
41 4 t [( μ ) ( μ ) ] μ = mn, (4.6) t nner t outer Ths model s farly smple, but t requres knowledge of the condtons at the edge of the boundary layer and the boundary layer thckness. These quanttes are not always easy to calculate n complcated flows wth a NaverStokes code snce t s often dffcult to defne where the boundary layer edge actually occurs. 4.3 BaldwnLomax Model BaldwnLomax 3 developed a form of the outer eddy vscosty that dd not requre knowledge of the condtons at the edge of the boundary layer. Ths model has become qute popular for CFD applcatons. The eddy vscosty n the outer layer s defned as ( t ) ρkccpfwakefkleb μ = (4.7) outer where F wake contans the mxng length term and F kleb s the Klebanoff ntermttency factor. These terms are defned as F { y F, C y U F } = wake mn max max wk max dff / (4.8) max The quanttes F max and y max are taken from the maxmum of the F functon defned as y F( y) = y Ω 1 exp (4.9) A and U dff s gven by ( U V ) max ( U V ) mn U dff = (4.10) The F functon s calculated along a lne normal to the wall. The F functon for a typcal attached boundary layer s shown n Fg. 4.1.
42 43 y (F max,y max ) F Fgure 4.1 The F functon (Eq. 4.9) for an attached boundary layer. The Klebanoff ntermttency factor 4 s used to reduct the eddy vscosty to zero at the outer edge of the boundary layer. The Klebanoff ntermttency factor s gven by 1 6 C kleb y F ( ) kleb y = (4.11) max y The fnal eddy vscosty s t [( μ ) ( μ ) ] μ = mn, (4.1) t nner t outer The constants used n the BaldwnLomax model are A = 6, Ccp = 1.6, Ckleb = 0.3, Cwk = 1.0, κ = 0.4, K = (4.13) Algebrac models work well for flows that can be characterzed by a sngle length scale such as attached boundary layers or smple shear layers. The Baldwn Lomax model determnes the approprate mxng length from the locaton of the peak n the F functon. For smple flows there wll only be one peak n the F functon. In more complcated flows wth multple shear layers such as separated boundary layers or wall ets the F functon wll contan multple peaks. When multple peaks are present t becomes dffcult to automatcally choose the proper peak to use, f a sngle peak can be used to model the turbulent flow. Degan and Schff 5 recommended a procedure to automatcally select the frst sgnfcant peak n the F functon. Ths modfcaton to the search procedure for the peak has been shown to mprove the performance of the BaldwnLomax for hgh angleofattack flows wth crossflow separaton.
43 Wake and Jet Model The BaldwnLomax model has been modfed for use n wakes and ets (Ref. 6). Frst, U dff n Eq s redefned as dff ( U V ) max ( U V ) y max U = (4.14) The exponental term n the defnton of F(y) (Eq. 4.9) s set to zero yeldng F ( y) = y Ω (4.15) The F wake functon (Eq. 4.8) s redefned to be U dff F wake = ymax Fmax = Ω (4.16) max Ω max The y max locaton n Eq s defned to be the locaton where Ω Fnally the Klebanoff ntermttency factor F kleb (Eq. 4.11) s rewrtten as max occurs. 1 6 y y max F kleb = Ckleb (4.17) U dff max Ω Ths wake formulaton wll work for ndvdual smple shear layers. If multple shear layers are present, the user must partton the flow and apply the turbulence model to each shear layer ndvdually. Ths can easly become mpractcal for complcated geometres. 4.5 Algebrac Model Shortcomngs The flow n a twodmensonal channel wth a crcular arc bump contracton can be used to demonstrate two shortcomngs of the BaldwnLomax turbulence models. These shortcomngs are the model s tendency to swtch F max peaks and the lack of any transport terms n the turbulence model. Fg. 4. shows the geometry and the eddy vscosty contours for both the BaldwnLomax and SpalartAllmaras turbulence models. The eddy vscosty predcted by the BaldwnLomax s seen to reduce tself almost to zero at the start of the bump. The SpalartAllmaras oneequaton transport turbulence model (dscussed n the next secton) predcts a much smoother dstrbuton of eddy vscosty on the bump. The reason for the anomaly n the BaldwnLomax model can be seen n
44 45 Fg The locaton of the F max peak moves very close to the wall at the start of the bump (x=0) and rses back to ts orgnal level at the bump tralng edge (x=1). The relatve magntude of the F max peak s also shown f Fg. 4.3, and s seen to ncrease n the regon of the bump. The eddy vscosty s proportonal to the product of the dstance of the peak off the wall and the magntude of the peak, and s seen to decrease n the regon above the bump because the F max peak has moved very near the wall. The sudden swtchng of the F max peak causes the turbulence level to be greatly reduced n the favorable pressure gradent regon near the bump leadng edge. If the BaldwnLomax model ncluded transport terms, then the hgher upstream values of eddy vscosty would be transported downstream at the begnnng of the bump, and the BaldwnLomax predcted eddy vscosty dstrbuton would not demonstrate the dscontnuous behavor t shows here. Fgure 4. Eddy vscosty contours for a crcular arc bump n a twodmensonal channel for two dfferent turbulence models.
45 46 Fgure 4.3 Dstance above the wall of the F max peak and the F max value for a twodmensonal channel wth a crcular arc bump. 4.6 Grd Senstvty for a Flat Plate wth Adabatc Wall The ntal wall spacng of the computatonal grd and the grdstretchng rato (the rato of the change n grd spacng normal to the wall) can affect the accuracy of the BaldwnLomax model. Fgure 4.4 shows the senstvty of the skn frcton to ntal wall spacng for a flat plate. The grdstretchng rato was 1. for all these cases. The plots nclude the theoretcal skn frcton curves of Whte and of Spaldng.
46 47 Fgure 4.4 Flat plate skn frcton predctons for the BaldwnLomax turbulence model for varyng ntal wall grd pont spacngs. The calculated boundary layer s seen to become fully turbulent around a length Reynolds number (Rex) of 1x10 6. The results for y =0. and y =1 are vrtually dentcal ndcatng a grd ndependent soluton. The y =5 soluton shows some small dvergence from the y =1 soluton at the lower length Reynolds numbers whle the y =10 soluton shows large dfferences from the other solutons. Predcted velocty profles for the flat plate boundary layer for varous ntal wall grd pont spacngs are shown n Fg The velocty profle shows lttle effect of the ntal spacng for all but the y =10 profle. All of the profles but the y =10 profle are n good agreement wth the theoretcal profle from Spaldng. Note that the theoretcal profle does not nclude the lawofthe wake (see Chapter 3), and hence the predcted profles dverge from the theoretcal profle n the wake regon of the boundary layer. The predcted eddy vscosty for varous ntal wall spacngs s shown n Fg. 4.6.
47 48 Fgure 4.5 Flat plate boundary layer profles predcted by the BaldwnLomax turbulence model for varyng ntal wall grd pont spacngs. Fgure 4.6 Eddy vscosty dstrbuton predcted by the BaldwnLomax turbulence model for varyng grd ntal wall spacngs. Here agan t s seen that the y =0. and the y =1.0 results are almost dentcal. The y =5 and y =10 results show the solutons are no longer grd ndependent at larger wall spacngs.
48 49 The effect of grd stretchng rato on skn frcton for a flat plate s shown n Fg All of these solutons used an ntal wall spacng of y =1. Fgure 4.7 The effect of grd stretchng rato on the skn frcton for a flat plate boundary layer usng the BaldwnLomax turbulence model. There seems to be very lttle effect of grd stretchng for ths case ndcatng that the ntal wall spacng s the more crtcal parameter for skn frcton predctons for flat plates wth the BaldwnLomax turbulence model. Ths s also the case for the velocty profle, as seen n Fg The eddy vscosty does change as the stretchng rato ncreases as shown n Fg It s nterestng to note that a wde range of eddy vscosty dstrbutons have lttle effect on skn frcton and the velocty profle for a flat plate boundary layer. Ths nsenstvty to the absolute eddy vscosty level s probably a maor reason why the eddy vscosty concept has worked so well n practce.
49 410 Fgure 4.8 The effect of grd stretchng rato on the velocty profle for a flat plate boundary layer usng the BaldwnLomax turbulence model. Fgure 4.9 The effect of grd stretchng rato on the eddy vscosty dstrbuton for a flat plate boundary layer usng the BaldwnLomax turbulence model. 4.7 Grd Senstvty for Axsymmetrc Bump A second example of the grd senstvty of the BaldwnLomax turbulence model that ncludes a pressure gradent s the NASA Ames transonc axsymmetrc
50 411 bump experment (Ref. 7). The geometry s shown n Fg The model conssted of a sharplpped hollow cylnder wth a 15. cm outer surface dameter. The bump was a crcular arc 0.3 cm long and 1.9 cm hgh that begns 60.3 cm downstream of the cylnder leadng edge. The upstream ntersecton of the bump and cylnder was fared wth a crcular arc. The test was run at a Mach number of and a chord Reynolds number of.67x10 6. M=0.875 Shock Recrculaton D=15. cm c=0.3 cm h=1.9 cm Fgure 4.10 Geometry for the transonc axsymmetrc bump. The effect of ntal grd spacng on the pressure coeffcent dstrbuton along the bump s shown n Fg The stretchng rato was 1. for these cases. The pressure coeffcent seems to be relatvely nsenstve to the ntal grd spacng, wth the y =10 and y =0 curves beng slghtly dsplaced from the other curves. The velocty dstrbuton at the aft uncton of the bump and the cylnder (x/c=1) s shown n Fg The y =0 soluton predct a larger velocty n the reverse flow regon than the other solutons. Grd stretchng effects on the pressure coeffcent dstrbuton along the bump s shown n Fg The ntal grd spacng was 1. for these cases. The pressure dstrbuton coeffcent changes slghtly as the grdstretchng rato s ncreased to 1.5. The soluton n the separated regon dffers greatly for a grdstretchng raton of.0. The effect on the velocty dstrbuton at x/c=1 s shown n Fg As wth ncreasng ntal grd spacng, ncreasng the grd spacng ncreases the sze and the magntude of the separated flow regon.
51 41 Fgure 4.11 The effect of ntal grd wall spacng on the pressure coeffcent for the axsymmetrc bump usng the BaldwnLomax turbulence model. Fgure 4.1 The effect of ntal grd wall spacng on the velocty dstrbuton at x/c=1 for the axsymmetrc bump usng the BaldwnLomax turbulence model.
52 413 Fgure 4.13 The effect of grd stretchng on the pressure coeffcent for the axsymmetrc bump usng the BaldwnLomax turbulence model. Fgure 4.14 The effect of grd stretchng on the velocty profle at x/c=1 for the axsymmetrc bump usng the BaldwnLomax turbulence model. The results for the Ames axsymmetrc bump ndcate that the grdstretchng rato s a crtcal parameter when pressure gradents are present n the flow. The stretchng rato should probably be kept between 1. and 1.3 to assure that the grd can capture the pressure gradent effects. Ths s true n both structured and unstructured grds.
53 Grd Senstvty for a Flat Plate wth Heat Transfer Calculatng heat transfer accurately can be more dffcult than predctng skn frcton. Ths can be seen n the subsonc flat plate example when the wall temperature s specfed to be 1.5 tmes the freestream temperature. The senstvty of the skn frcton and heat transfer result wth varyng ntal grd wall spacng s shown n Fg and Fg The grd stretchng rato was fxed at 1. for these results. Both the skn frcton and heat transfer seem to be relatvely nsenstve to the wall spacng for wall spacngs less than y=5 when no pressure gradent s present. Fgure 4.15 The effect of wall spacng on the skn frcton on a flat plate wth heat transfer usng the BaldwnLomax turbulence model.
54 415 Fgure 4.16 The effect of wall spacng on the heat transfer (Stanton number) on a flat plate usng the BaldwnLomax turbulence model. The velocty and temperature profles for a length Reynolds number (Re x ) of 1.0x10 7 are shown n Fg and 4.18 respectvely. Both the velocty and temperature profles are relatvely nsenstve to wall spacng for ths model when no pressure gradent s present. Fg The effect of wall spacng on the velocty profle on a flat plate wth heat transfer usng the BaldwnLomax turbulence model.
55 416 Fg The effect of wall spacng on the temperature profle on a flat plate wth heat transfer usng the BaldwnLomax turbulence model. Grd stretchng effects on skn frcton and heat transfer predctons are shown n Fg and 4.0. The effect of grd stretchng on the velocty and temperature profles s shown n Fg. 3. and 3.3 respectvely. The ntal wall spacng was held at y=0.1 for these calculatons. The results reach a grd ndependent result for a stretchng rato of less than 1.3. Fgure 4.19 The effect of grd stretchng on the skn frcton on a flat plate wth heat transfer usng the BaldwnLomax turbulence model.
56 417 Fgure 4.0 The effect of grd stretchng on heat transfer (Stanton number) on a flat plate usng the BaldwnLomax turbulence model. Fgure 4.1 The effect of grd stretchng on the velocty profle on a flat plate wth heat transfer usng the BaldwnLomax turbulence model.
57 418 Fgure 4. The effect of grd stretchng on the temperature profle on a flat plate wth heat transfer usng the BaldwnLomax turbulence model. 4.9 Grd Senstvty for a Nozzle wth Heat Transfer Flow through a supersonc nozzle wth a constant temperature wall can serve as a test case for evaluatng the performance of the turbulence model n the presence of strong pressure gradents. Back, Masser, and Ger 8 measured the wall pressure dstrbuton and heat transfer for a convergngdvergng nozzle wth a throat dameter of meters and an ext dameter of 0.17 meters. Hghpressure ar was heated by the nternal combuston of methanol and flowed along a cooled constant area duct wth a length of meters and a dameter of meters before enterng the nozzle. The nozzle geometry and boundary condtons are shown n Fg The gas could be treated as a calorcally perfect gas wth a ratoofspecfc heats (γ) of The nozzle ext Mach number was approxmately.5. The molecular vscosty and thermal conductvty were assumed to vary accordng to Sutherland s law. P t =150psa T t =1418 o R T wall =0.5T t Fgure 4.3 Nozzle geometry. Supersonc Outlet
58 419 The ntal wall spacng was vared wth a grd stretchng rato of 1. n the boundary layer. The grd spacng was held constant n the core of the nozzle. Comparsons of the predcted and measured pressure along the nozzle are shown n Fg. 4.4 for varyng ntal wall spacngs. The throat s located at x=0.091 meters. The pressure dstrbuton s seen to be nsenstve to the ntal wall spacng. The heat transfer at the wall s shown n Fg The results are n poor agreement wth the data and are qute senstve to the ntal wall spacng, especally for wall spacng greater than y=1. Fgure 4.4 The effect of wall spacng on the pressure dstrbuton for a supersonc nozzle wth heat transfer usng the BaldwnLomax turbulence model. Fgure 4.5 The effect of wall spacng on the wall heat transfer dstrbuton for a supersonc nozzle usng the BaldwnLomax turbulence model.
59 40 The poor performance of the BaldwnLomax model for ths case s due to the dffculty n choosng the proper F max peak as was descrbed n Secton 4.5. The F functon at the nozzle throat s shown n Fg Notce that multple peaks are present n the functon. It s extremely dffcult to predct whch, f any, of these three peaks s the proper value for ths case. The eddy vscosty dstrbuton at the nozzle throat s shown n Fg The SpallartAllmaras eddy vscosty dstrbuton s ncluded n Fg. 4.6 for comparson. The BaldwnLomax eddy vscosty s much lower than the SpalartAllmaras dstrbuton and s seen to cut off prematurely. Ths low value of eddy vscosty results n a low predcton of the wall heat transfer. R/Rthroat Fgure 4.6 The F functon at the nozzle throat.
60 Summary Fg. 4.7 The eddy vscosty dstrbuton at the nozzle throat. Algebrac turbulence models were popular n the 1970 s and 1980 s because of ther smplcty and robustness. As NaverStokes CFD applcatons became more complex n the 1990 s these models began to lose popularty because of accuracy lmtatons for flows that contan multple shear layers or boundary layer separaton. The dffcultes encountered when multple peaks occur n the F functon have been demonstrated. These models have also lost favor n unstructured grd applcatons snce they requre a velocty profle over multple grd ponts algned wth the flow for successful applcaton. The eddy vscosty predcted by an algebrac turbulence model s only a functon of the local velocty profle used to generate the F functon. Thus the eddy vscosty relates drectly to the local nstantaneous vortcty feld of that profle and cannot model the transport of turbulence by the flow. Ths makes these models a poor choce for unsteady flows and for flows where turbulent transport s mportant.
61 4 BaldwnLomax Applcaton Hnts 1. The BaldwnLomax model requres that the F functon be well defned. Ths normally requres that at least three ponts be located wthn the sublayer (y <10). The frst pont off the wall should be located about y <5 for pressure dstrbutons, y < to obtan reasonable skn frcton values, and y <0.5 for heat transfer. The grd stretchng normal to the wall should not exceed 1.3. Improved heat transfer results can be obtaned by usng a constant spacng for the frst three cells off the wall.. In order to reduce the probablty of fndng a second peak well off the wall, t s usually good to lmt the number of ponts over whch the F functon s calculated. 3. Care should be taken not to dvde vscous regons such as boundary layers when dvdng the computatonal doman for blocked or chmera applcatons snce the entre velocty profle s requred to properly defne the F max and U dff quanttes. Chapter 4 References: 1. Prandtl, L., Bercht uber Untersuchugen zur Ausgebldeten Turbulenz, Z. Angew Math., Meth., Vol. 5, pp Cebec, T, and Smth, A. M. O., Analyss of Turbulent Boundary Layers, Academc Press, New York, Baldwn, B. S. and Lomax, H., Thn Layer Approxmaton and Algebrac Model for Separated Turbulent Flows, AIAA , Jan Klabenoff, P. S., Characterstcs of Turbulence n a Boundary Layer wth Zero Pressure Gradent, NACA Rep. 147, Degan, D. and Schff, L. B., Computaton of Supersonc Vscous Flows Around Ponted Bodes at Large Incdence, AIAA , Jan Bunng, P. G., Jesperson, D. C., Pullam, T. H., Klopfer, G. H., Chan, W. C., Slotnk, J. P., Krst, S. E., and Renze, K. J., OVERFLOW User s Manual, Verson 1.8l, July Johnson, D. A., Predctons of Transonc Separated Flow wth an Eddy Vscosty/Reynolds Shear Stress Closure Model, AIAA , July Back, L. H., Masser, P. F., and Ger, H. L., Convectve Heat Transfer n a ConvergentDvergent Nozzle, Internatonal Journal of Heat Mass Transfer, Vol. 7, 1964, pp
62 OneEquaton Turbulence Models 5.1 Theory The general form for all transport turbulence models s smlar to the form of a speces equaton n reactng flows Z t U Z x 1 = σ x Z x ( ν ν ) P( Z) D( Z) t (5.1) where Z s a general turbulence varable. The terms on the left hand sde of Eq. 5.1 represent the convectve transport of Z. The frst term on the rght hand sde of Eq. 5.1 s the dffuson of Z assumng a gradentdffuson model. The functons P(Z) and D(Z) on the rght hand sde of Eq. 5.1 represent the producton and destructon of Z. The producton functon for turbulence models are usually functons of the eddy vscosty and the flud stran or vortcty. A functonal relatonshp between Z and the eddy vscosty s requred to complete the model. Several oneequaton transport models for turbulence have appeared over the years. The most popular oneequaton models have been the BaldwnBarth 1 model and the SpalartAllmaras model. The SpalartAllmaras model wll be dscussed n some detal here. 5. SpalartAllmaras Model The SpalartAllmaras turbulence model was derved usng emprcal relatonshps, dmensonal analyss, and Gallean nvarance. The goal was to produce a turbulent transport model that was fast, numercally stable, and reasonably accurate for both shear layers and boundary layers. The model uses a turbulence varable ν ~ that has the dmensons of vscosty. The model can be wrtten as ~ ν ~ ν 1 U ~ = b D t x σ [ (( ν ~ ν ) ~ ν ) ( ~ C ν ) ] P( ~ ν ) ( ν ) (5.) where the producton term s gven by ~ ( ~ ) = ~ ν P ν Cb 1ν Ω f v κ d (5.3) and the dsspaton s gven by D ~ ( ν ) = C w1 f w ~ ν d (5.4)
63 5 Here Ω s the magntude of the vortcty, d s the dstance to the nearest wall, and f v and f w are gven by f v χ = 1 (5.5) 1 χf v C w3 f w = g 6 6 (5.6) g Cw3 and the remanng functons are gven by f v1 3 χ = χ 3 C 3 v1 (5.7) ~ ν χ = (5.8) ν 6 ( r r) g = r Cw (5.9) ~ ν r = (5.10) ( Ωκ d ~ ν ) f v The remanng constants are gven n Table 5.1. C b1 C b σ κ C w1 C w C ν / C κ (1 C ) / σ b1 / b Table 5.1 Coeffcents for the SpalartAllmaras model. The eddy vscosty s defned as ν = ~ ν (5.11) t f v 1 The orgnal model also ncludes a boundary layer transton model. The trp functon can be used for smple geometres, but s dffcult to apply for complex geometres and s usually omtted. The turbulent transton locaton can also be fxed by settng the producton term to zero n the regon of the flow upstream of the desred transton locaton. The wall boundary condton s ~ ν = 0. At free stream boundares, ~ ν s set to a small number, generally ν/10. The turbulence varable ~ ν s usually lmted to be greater than the free stream value n the feld so that t wll not go negatve. The model wll selfntalze from ths small value of ~ ν n the freestream. The
64 53 numercal mplementaton for mplct algorthms s dscussed n detal n Ref.. Generally the convectve terms are treated wth upwnd dfferencng. The dffuson term s treated wth central dfferencng. The full source term acobans P( ~ ν ) D( ~ ν ) ( ~, ν ~ ) should be used n mplct formulatons. When the r functon ν becomes large, the f w functon becomes a constant, so r can be clpped at a value of 10 to avod numercal overflow problems. The SpalartAllmaras model suffers from a defcency common to transport type models n that t fals to predct the reducton n shear layer growth rate wth ncreasng et Mach number. The model wll predct the ncompressble shear layer growth rate regardless of the et Mach number. Pacorr and Sabetta 3 suggest a compressblty correcton for ths model. 5.3 Rotaton/Streamlne Curvature Correctons Conventonal lnear eddy vscosty models have dffculty predctng flows wth large system rotaton or streamlne curvature. The SpalartAllmaras model falls nto ths category. The vortcty reaches a local maxmum n the core of a vortex causng the eddy vscosty to ncrease rapdly. Ths leads to excessve dsspaton of the vortex core. To remedy ths problem the producton term (Eq. 5.3) be modfed by multplyng by a correcton factor 4. P ~ ν κ d ( ~ ν ) = C ~ ν Ω f F ( *) (5.1) b 1 v r1 r where F r1 (r*) s a correcton for rotatonal flow gven F r 1( r*) = [ 1 Cvor mn( 0, r * 1) ] (5.13) and ε r* = S (5.14) Ω ε where S s the magntude of the stran. The stran s defned as u S = 0. 5 x u x (5.15) The magntude of the stran s gven by S = S S (5.16)
65 54 The ε n Eq. (8) s a small number used as a threshold value so that the turbulence model returns to ts baselne form n areas of low vortcty and stran such as near free stream boundares. The ε value can be defned as U ref ε = 0.5 (5.17) Lref Where U ref and L ref are the reference velocty and length scale for the calculaton respectvely. Ths correcton form mples that curvature effects are a functon of the rato of the local stran magntude and vortcty magntude. The stran goes to zero n the core of a vortex whle the vortcty reaches a local maxmum. Thus Eq. (5.13) converts the producton term of the SA model nto a dsspaton term n the vortex core regon f C vor s greater than one. Ref. 4 recommended C vor =. The modfed form of the producton term has almost no effect n boundary layers and shear layers snce the magntude of the stran and the magntude of the vortcty are almost equal for these flows and the correcton term goes to zero. Ths correcton s generally referred to as the approxmate SpalartAllmaras rotatonal correcton (ASARC). Shur, et al. 5 ntroduced a correcton to the SpalartAllmaras turbulence model for rotatng and curved flows. The correcton s appled to the producton term n a manner smlar to the ASARC correcton. The correcton has the form F r1 r * 1 r * [ r3 r ] 1 ( ~ 1 r r )) = ( 1 c ) 1 c tan ( c ~ r ) c (5.18) where *, r1 r c ( ) ε mn S n ε mn S n Ω m ~ Ωk S k DS r = (5.19) 4 D Dt The remanng term D s defne as ( ) D = 0.5 S Ω (5.0) The recommended values for the constants are c r1 =1.0, c r =1.0 and c r3 =1.0. The Lagrangan dervatve of the stran tensor n Eq. (5.19) s ncluded to obtan the proper value for the curvature correcton. Ths term makes the correcton expensve and cumbersome to mplement. Some mplementatons of ths correcton gnore the contrbuton of the tme dervatve of the Lagrangan dervatve and the reference frame rotaton to smplfy the codng. Ths s referred as the SARC correcton.
66 55 An example of the standard SA, SARC, and ASARC models appled to flow n a uduct 6 s shown n Fg The dstance along the wall s denoted by s. The standard SA model underpredcts the pressure coeffcent and the skn frcton at the ext of the ubend. Both the SARC and ASARC models mprove the comparson wth data. Fgure 5.1. Pressure coeffcent and skn frcton on the outer wall of a uduct.. The standard SA model also overdamps the flow n the core of a vortex. Ths tends to cause vortces to be damped out prematurely n vortex domnated flows such as wng or rotor tp vortces. The producton term for the SA model (Eq. 5.3) s based on the vortcty magntude. The vortcty magntude reaches a local maxmum n a vortex core, hence the model tends to produce much more eddy vscosty than s requred to properly smulate the flow n the vortex core. The SARC and ASARC correcton terms are neglgble n smple shear layers and boundary layers and do not affect the model snce the magntude of the stran s approxmately equal to the magntude of the vortcty. The correcton term transforms the producton term nto a dsspaton term n the core of a vortex where the magntude of the stran s zero. Ths addtonal dsspaton tends to reduce the eddy vscosty n the core of the vortex and sgnfcantly mproves the smulaton of vortex domnated flows. The wng tp vortex on a NACA 001 wng 4 tp at an angleofattack of 10 o s shown n Fg. 5. for both the orgnal and modfed SA producton terms. The tangental velocty predcted by the modfed SA models s n much better agreement wth the data. Fg. 5.b also ncludes results for the hybrd RANS/LES Detached Eddy Smulaton (DES) models that are dscussed n Chapter 9.
67 56 Fgure 5.a Axal velocty for a NACA 001 wng tp vortex. Tangental Velocty Vortex Radus Fg. 5.b Vortex tangental velocty. 5.4 Grd Senstvty for a Flat Plate wth Adabatc Walls The ntal wall spacng of the computatonal grd and the grdstretchng rato can affect the accuracy of the SpalartAllmaras model. Fgure 5.3 shows the senstvty of the skn frcton to ntal wall spacng for a flat plate. The grdstretchng rato was 1. for all these cases. The plots nclude the theoretcal skn frcton curves of Whte and of Spaldng.
68 57 The boundary layer s seen to become fully turbulent around a length Reynolds number (Rex) of 1x10 6. The results for y =0. and y =1 are vrtually dentcal ndcatng a grd ndependent soluton. The y =5 soluton show some small dvergence from the y =1 soluton at the lower length Reynolds numbers whle the y =10 soluton shows large dfferences from the other solutons. Predcted velocty profles for the flat plate boundary layer for varous ntal wall grd pont spacngs are shown n Fg The velocty profle shows lttle effect of the ntal spacng for all but the y =10 profle. All of the profles but the y =10 profle are n good agreement wth the theoretcal profle from Spaldng. Note that the theoretcal profle does not nclude the lawofthe wake (see Chapter 3), and hence the predcted profles dverge from the theoretcal profle n the wake regon of the boundary layer. The predcted eddy vscosty for varous ntal wall spacngs s shown n Fg Fgure 5.3 Flat plate skn frcton predctons for the SpalartAllmaras turbulence model for varyng ntal wall grd pont spacngs.
69 58 Fgure 5.4 Flat plate boundary layer profles predcted by the SpalartAllmaras turbulence model for varyng ntal wall grd pont spacngs. Fgure 5.5 Eddy vscosty dstrbuton predcted by the SpalartAllmaras turbulence model for varyng grd ntal wall spacngs. Here agan t s seen that the y =0. and the y =1.0 results are almost dentcal. The y =5 and y =10 results show the solutons are no longer grd ndependent at larger wall spacngs. The effect of grd stretchng rato on skn frcton for a flat plate s shown n Fg All of these solutons used an ntal wall spacng of y =1.
70 59 There seems to be very lttle effect of grd stretchng for these cases ndcatng that the ntal wall spacng s the more crtcal parameter for skn frcton predctons for flat plates wth the SpalartAllmaras turbulence model. Ths s also the case for the velocty profle, as seen n Fg The eddy vscosty does change as the stretchng rato ncreases as shown n Fg It s nterestng to note that a wde range of eddy vscosty dstrbutons have lttle effect on skn frcton and the velocty profle for a flat plate boundary layer. Fgure 5.6 The effect of grd stretchng rato on the skn frcton for a flat plate boundary layer usng the SpalartAllmaras turbulence model. Fgure 5.7 The effect of grd stretchng rato on the velocty profle for a flat plate boundary layer usng the SpalartAllmaras turbulence model.
71 510 Fgure 5.8 The effect of grd stretchng rato on the eddy vscosty dstrbuton for a flat plate boundary layer usng the SpalartAllmaras turbulence model. 5.5 Grd Senstvty for and Axsymmetrc Bump A second example of the grd senstvty of the SpalartAllmaras turbulence model that ncludes a pressure gradent s the NASA Ames transonc axsymmetrc bump experment descrbed n the prevous chapter. The effect of ntal grd spacng on the pressure coeffcent dstrbuton along the bump s shown n Fg The stretchng rato was 1. for these cases. The pressure coeffcent seems to be relatvely nsenstve to the ntal grd spacng for y <10. The y =0 result shows a sgnfcant dfference from the other results n regons where a pressure gradent s present. The velocty dstrbuton at the aft uncton of the bump and the cylnder (x/c=1) s shown n Fg The y =0 soluton predct a larger velocty n the reverse flow regon than the other solutons. Grd stretchng effects on the pressure coeffcent dstrbuton along the bump s shown n Fg The ntal grd spacng was 1. for these cases. The pressure dstrbuton coeffcent changes slghtly as the grdstretchng rato s ncreased to 1.5. The soluton n the separated regon dffers greatly for a grdstretchng raton of.0. The effect on the velocty dstrbuton at x/c=1 s shown n Fg As wth ncreasng ntal grd spacng, ncreasng the grd spacng ncreases the sze and the magntude of the separated flow regon.
72 511 Fgure 5.9 The effect of ntal grd wall spacng on the pressure coeffcent for the axsymmetrc bump usng the SpalartAllmaras turbulence model. Fgure 5.10 The effect of ntal grd wall spacng on the velocty dstrbuton at x/c=1 for the axsymmetrc bump usng the SpalartAllmaras turbulence model.
73 51 Fgure 5.11 The effect of grd stretchng on the pressure coeffcent for the axsymmetrc bump usng the SpalartAllmaras turbulence model. Fgure 5.1 The effect of grd stretchng on the velocty profle at x/c=1 for the axsymmetrc bump usng the SpalartAllmaras turbulence model. As was seen wth the BaldwnLomax model, grd stretchng s a crtcal parameter for ths turbulence model when adverse pressure gradents are present. Care should be taken to keep the grdstretchng rato between 1. and 1.3.
74 Grd Senstvty for a Flat Plate wth Heat Transfer Calculatng heat transfer accurately can be more dffcult than predctng skn frcton. Ths can be seen n the subsonc flat plate example when the wall temperature s specfed to be 1.5 tmes the freestream temperature. The senstvty of the skn frcton and heat transfer result wth varyng ntal grd wall spacng s shown n Fg and Fg The grd stretchng rato was fxed at 1. for these results. Both the skn frcton and heat transfer seem to be relatvely nsenstve to the wall spacng for ntal wall spacngs less than y =1. Fgure 5.13 The effect of wall spacng on the skn frcton on a flat plate wth heat transfer usng the SpalartAllmaras turbulence model. The effect of wall spacng on the velocty and temperature profles for a length Reynolds number (Re x ) of 1.0x10 7 s shown n Fg and 5.16 respectvely. The profles are relatvely nsenstve to wall spacng for ntal wall spacngs less than y =1.0.
75 514 Fgure 5.14 The effect of wall spacng on the heat transfer (Stanton number) on a flat plate usng the SpalartAllmaras turbulence model. Fg The effect of wall spacng on the velocty profle for a flat plate wth heat transfer usng the SpalartAllmaras turbulence model.
76 515 Fg The effect of wall spacng on the temperature profle for a flat plate wth heat transfer usng the SpalartAllmaras turbulence model. Grd stretchng effects on skn frcton and heat transfer predctons are shown n Fg and The effect on velocty and temperature profles for a length Reynolds number (Re x ) of 1.0x10 7 are shown n Fg and 5.0. The ntal wall spacng was held at y =0.1 for these calculatons. The results reach a grd ndependent result for stretchng ratos of less than 1.3. Fgure 5.17 The effect of grd stretchng on the skn frcton on a flat plate wth heat transfer usng the SpalartAllmaras turbulence model.
77 516 Fgure 5.18 The effect of grd stretchng on heat transfer (Stanton number) on a flat plate usng the SpalartAllmaras turbulence model. Fgure 5.19 The effect of grd stretchng on the velocty profle on a flat plate wth heat transfer usng the SpalartAllmaras turbulence model.
78 517 Fgure 5.0 The effect of grd stretchng on the temperature profle on a flat plate wth heat transfer usng the SpalartAllmaras turbulence model. 5.7 Grd Senstvty for a Nozzle wth Heat Transfer Flow through a supersonc nozzle wth a constant temperature wall can serve as a test case for evaluatng the performance of the turbulence model n the presence of strong pressure gradents. Detals of the geometry and boundary condtons for the convergngdvergng supersonc nozzle are gven n Chapter 4. Hghpressure ar was heated by the nternal combuston of methanol and flowed along a cooled constant area duct before enterng the nozzle. The gas could be treated as a calorcally perfect gas wth a ratoofspecfc heats (γ) of The nozzle ext Mach number was.5. The molecular vscosty and thermal conductvty were assumed to vary accordng to Sutherland s law. The grd ntal wall spacng was vared and the grd stretchng rato was held at 1. n the boundary layer. A unform grd was used n the nozzle core. Predcted wall pressure dstrbuton results for varyng ntal wall spacngs are shown n Fg The pressure dstrbuton s seen to be nsenstve to the ntal wall spacng. Predcted wall heat transfer s shown n Fg. 5.. The results are somewhat senstve to the ntal wall spacng for values of y less than one. The predcted results dverge rapdly from the data for wall spacng greater than y =1.
79 518 Fgure 5.1 The effect of wall spacng on the pressure dstrbuton for a supersonc nozzle wth heat transfer usng the SpalartAllmaras turbulence model. Fgure 5. The effect of wall spacng on the wall heat transfer for a supersonc nozzle usng the SpalartAllmaras turbulence model. The grd stretchng rato was vared n the boundary layer whle the ntal grd spacng held at y =0.5. A unform grd was used n the nozzle core. Predcted wall pressure dstrbuton results for varyng ntal wall spacngs are shown n Fg The pressure dstrbuton s seen to be nsenstve to the grd stretchng
80 519 rato. Predcted wall heat transfer s shown n Fg The results are nsenstve to the grd stretchng ratos less than 1.3. The predcted results dverge slghtly from the data for grd stretchng ratos greater than 1.3. Fgure 5.3 The effect of grd stretchng rato on the pressure dstrbuton for a supersonc nozzle wth heat transfer usng the SpalartAllmaras turbulence model. Fgure 5.4 The effect of grd stretchng rato on the wall heat transfer for a supersonc nozzle usng the SpalartAllmaras turbulence model.
81 Summary The SpalartAllmaras turbulence model s a very stable and generally reasonably accurate model for a wde range of turbulent flows. The model s relatvely easy to mplement n both structured and unstructured NaverStokes codes. One drawback s the model requres the calculaton of the dstance to the nearest wall for all feld ponts. Ths can be an expensve computaton, especally for unstructured grd codes. The model has been used extensvely for threedmensonal geometres and s well documented. The model has been used wth some success for some unsteady flows. The example gven here are for subsonc and supersonc flows. An excellent valdaton source for hypersonc flows can be found n Ref. 7. SpalartAllmaras Applcaton Hnts 1. The frst pont off the wall should be located about y =1 to obtan reasonable skn frcton values and about y =0.5 for heat transfer. The grd stretchng normal to the wall should not exceed The eddy vscosty should be lmted so that t wll not run away n some complex applcatons. Generally a lmt of ν t /ν=00,000 s acceptable. 3. Care should be taken not to dvde vscous regons such as boundary layers when dvdng the computatonal doman for blocked or overset applcatons snce the model requres the dstance from the nearest wall. 4. Ths model tends to smear out threedmensonal vortcal flows. Rotaton and curvature correctons (SARC and ASARC) can sgnfcantly mprove the results. 5. The model can overdamp some unsteady flows. 6. The model contans no correctons for compressblty and wll overpredct the growth rate of hgh speed shear layers. Chapter 5 References: 1. Baldwn B. S. and Barth, T. J., A OneEquaton Turbulence Transport Model for Hgh Reynolds Number WallBounded Flows, AIAA , Jan Spalart, P. R. and Allmaras, S. R., A OneEquaton Turbulence Model for Aerodynamc Flows, AIAA , Jan Pacorr, R. and Sabetta, F., Compressblty Correcton for the Spalart Allmaras Model n FreeShear Flows, Journal of Spacecraft and Rockets, Vol. 40, No. 3, MayJune DaclesMaran, J., Zllac, G., Chow, J., and Bradshaw, P., Numercal/Expermental Study of a Wngtp Vortex n the Near Feld, AIAA Journal, Vol. 33, No. 9, September 1995, pp
82 Shur, M. L., Strelets, M. K., Travn, A. K., and Spalart, P. R., Turbulence Modelng n Rotatng and Curved Channels: Assessng the SpalartShur Correcton, AIAA Journal, Vol. 38, No. 5, May 000, pp Nchols, R., Algorthm and Turbulence Model Requrements for Smulatng Vortcal Flows, AIAA , Jan Roy, J. and Blottner, F., Methodology for Turbulence Model Valdaton: Applcaton to Hypersonc Flows, Journal of Spacecraft and Rockets, Vol. 40, No. 3, MayJune 003.
83 TwoEquaton Turbulence Models Twoequaton turbulence models have been used for almost ffty years. There are a tremendous number of these models n the lterature. Most of these models solve a transport equaton for turbulent knetc energy (k) and a second transport equaton that allows a turbulent length scale to be defned. The most common forms of the second transport equaton solve for turbulent dsspaton (ε) or turbulent specfc dsspaton (ω). Some of these twoequaton models are vald down to the wall (low Reynolds number models) and some are only vald outsde the nner regon of the boundary layer (hgh Reynolds number models). All of these models have ther own strengths and weaknesses. Several of the currently popular models are descrbed n ths chapter. 6.1 Theory A transport equaton for turbulent knetc energy can be derved from Naver Stokes equatons and has the form x u x u x U u u p u u u x x k U t k = ν ρ (6.1) The left hand sde of Eq. 6.1 contans the temporal change and convectve terms for the turbulent knetc energy. The frst term on the rght hand sde of Eq. 6.1 s the dffusve transport of the turbulent stresses (D k ). The second term on the rght hand sde of Eq. 6.1 s the producton of turbulent knetc energy (P k ). The last term on the rght hand sde of Eq. 6.1 represents the dsspaton of turbulent knetc energy (ε). All of the terms on the rght hand sde must be modeled. Applyng the Boussnesq approxmaton (Eq..) the producton term becomes t t t k x U k S S x U k x U S x U k x U x U x U P = = = ν ν ν (6.) The last term n Eq. 6. s zero for ncompressble flow snce the dvergence of the velocty ( x U ) s zero. Ths term s also neglected n many compressble flow applcatons snce t can cause numercal dffcultes near strong shocks and expansons. S s the stran rate defned as 1 = x u x u S (6.3) Kato and Launder 1 suggested redefnng the producton term as
84 6 P =υ S Ω (6.4) k t Where Ω s the vortcty defned as Ω = 1 u x u x (6.5) The purpose of ths modfcaton s to reduce the tendency of many twoequaton models to overpredct the turbulent producton n regons wth large normal stran,.e. regons wth strong acceleraton or deceleraton. Ths correcton s often used n stagnaton regons such as at the leadng edge of a wng or blade. The dffuson term can be modeled usng the gradentdffuson approxmaton as D k = x ν t k ν (6.6) σ k x where σ k can be thought of as a turbulent Schmdt number. Relatonshps for the eddy vscosty (ν t ) and the turbulent dsspaton (ε) are requred for closure. The square root of the turbulent knetc energy can be used to represent a velocty scale for the largescale turbulent moton. Usng ths n an eddy vscosty relatonshp yelds ν k L (6.7) t where L s a turbulent length scale. A relatonshp for the turbulent length scale of the large scale eddes s gven n Eq Usng ths defnton of the turbulent length scale n Eq. 6.7 produces k ν t = Cμ (6.8) ε where C μ s a constant. The system of equatons would be closed wth a relatonshp for the turbulent dsspaton (ε). The two features that dstngush dfferent two equaton models are the treatment of the wall and the form of the turbulent dsspaton equaton. It s possble to derve a transport equaton for turbulent dsspaton from the NaverStokes equatons, but the equaton contans fluctuatng correlaton terms that cannot be
85 63 easly modeled. Turbulent dsspaton cannot be measured drectly expermentally, but must be nferred from measurements of other correlatons. Because of the dffculty n dervng a turbulent dsspaton equaton based on physcs, turbulence modelers have tradtonally resorted to usng a transport equaton wth emprcally determned producton and dsspaton terms. The two most popular equatons used to close the turbulence model are based on the dsspaton ε and a turbulence varable ω defned by Wlcox 3 as ε ω = (6.9) C μ k The eddy vscosty for the Wlcox formulaton s k ν t = (6.10) ω 6. Tradtonal kε Models The standard form of the ε transport equaton s ε ε ν t ε ε ε U = C f Pk C f R t x x ν ε1 ε1 ε ε σ ε x k k (6.11) The producton and dsspaton terms of the dsspaton equaton are formed from the producton and dsspaton terms of the turbulent knetc energy equaton scaled by ε /k and multpled by emprcally determned constants and wall dampng functons (C ε1 f ε1 and C ε f ε ). An addtonal dampng term must be ncluded for the eddy vscosty n the kε equatons for applcatons near walls so that k and ε wll have the proper behavor n the near wall regon. The constants C ε1, C ε, and C μ are emprcally determned from comparsons wth expermental data. In grdgenerated turbulence the dffuson term and the producton term n Eq. 6.1 and Eq are neglgble and C ε can be determned from the measured rate of decay of the turbulent knetc energy k behnd the grd. The value of C ε was found to le between 1.8 and.0. For shearlayers n local equlbrum (producton=dsspaton), Eqs. 6.1 and 6.8 can be combned to produce u v C μ = (6.1) k Measurements n equlbrum flows suggest that C μ ~0.09. In the nearwall regon of the logarthmc regon of a boundary layer turbulent producton and dsspaton
86 64 are approxmately equal and the convecton of ε s neglgble. Under these assumptons Eq reduces to C κ = C (6.13) σ C ε1 ε ε μ Note that all of these emprcal constants are defned for near equlbrum condtons, and thus are not truly vald when producton and dsspaton are not of the same sze. Ths s often a source of numercal dffcultes when solvng these models and leads to the use of ad hoc relatonshps to control the behavor of the turbulence model for cases when producton s much greater than dsspaton. The general form of the eddy vscosty relatonshp for low Reynolds number models s k ν t = Cμ f μ (6.14) ε Many kε turbulence transport models exst. The coeffcents of some of the more popular are shown n Table 6.1. The wall dampng functons are shown n Table 6.. Model C ε1 C ε σ k σ ε C μ κ Standard Jones Launder Chen Spezale Table 6.1 Coeffcents for low Reynolds number kε models. Model f ε1 f ε f μ Standard Jones Launder ν k νν u 3.4 t 1 1.3exp( R ) exp ε t y ε y Chen 4 ν k 1 R y y ε t C 1. exp 1 ε exp ν k exp y ε y ( R ) t
87 65 Spezale 5 1 R 1. exp t 36 y * 1 exp y 1 tanh R t 70 Table 6. Wall dampng functons for low Reynolds number kε models. The last column of Table 6.1 also contans the von Karman constant (κ) calculated from Eq All of the models shown here produce κ values slghtly above the usually accepted value of wth the excepton of the Spezale model. The wall boundary condton for the turbulent knetc energy s k = 0 (6.15) wall Snce the turbulent knetc energy s zero at the wall, then the eddy vscosty s also zero at the wall. A wall boundary condton for the dsspaton (ε) can be found from the turbulent knetc energy equaton ε k wall = x ν x (6.16) Ths boundary condton can produce numercal stablty problems, so Spezale 4 suggested the boundary condton be replaced wth the form k ε wall = ν (6.17) x The turbulent knetc energy s usually assgned to the free stream turbulence level at free stream boundares f t s known. If the free stream turbulence level s not known, then the turbulent knetc energy s set to a small number at the free stream boundary. The eddy vscosty s also set to a small number at free stream boundares. ν t ν < 0.1 (6.18)
88 66 Once the free stream turbulent knetc energy and eddy vscosty are known, the free stream value of turbulent dsspaton may be calculated from the defnton of eddy vscosty. 6.3 kω Model Usng Wlcox s formulaton, a transport equaton for ω can be wrtten as ω U t ω = x x ν ω t κ ν β σ ω x σ ϖ C μ ω Pk k βω (6.19) The ω transport equaton s essentally the ε transport equaton (Eq. 6.11) wth the addton of a cross dffuson term of the form Cross Dffuson = ν t ε k ν σ ε x x (6.0) The kω models are derved for wall bounded flows and requre no addtonal wall dampng terms when used n boundary layer flows. The coeffcents for Eq are gven n Table 6.3. β σ ω C μ κ Table 6.3 Coeffcents for the kω model. The wall boundary condton for the turbulent knetc energy s also gven by Eq The defnton of ω (Eq. 6.9) suggests that ω should go to nfnty at the wall snce the turbulent knetc energy (k) s gong to zero. Wlcox, based on asymptotc arguments, suggested that ω be gven the value 60ν ω wall = (6.0) C μ ( Δy ) at the wall. Here Δy s the normal dstance to the frst pont off the wall. The followng free stream boundary condtons are recommended ν ν U = 0.001, ω 10, ν ω (6.1) t = k = L t The kω model has been shown to be senstve to the free stream value of ω chosen.
89 SST Model In practce, the kε models are generally more accurate n shear type flows and are well behaved n the far feld. The kω models are more accurate and much more numercally stable n the near wall regon. Recognzng that each model has ts strength and weakness and that the forms of the equatons are smlar, Menter 6 suggested a blended model that s a kω model near the wall and transtons to a kε model away from the wall. The model has the form k U t k x = x k x ( ν σ ν ) P C ωk k t k μ (6.) ω ω U = t x x σ ω k ω 1 1 ω x x ( F ) ( ν σ ν ) ω t ω ω γ Pk x k βω (6.3) The coeffcents are blended forms of the two baselne models. The blendng functon F 1 s defned as 4 ( ) F 1 = tanh arg 1 (6.4) where arg 1 mn max C k 500ν 4ρσ ω k,, ωy y ω CDkω y = μ (6.5) Here y s the normal dstance to the wall and CD kω s the postve porton of the crossdffuson term 1 k ω 0 CDk ω = max ρσ ω, 10 (6.6) ω x x The coeffcents σ k, σ ω, γ, and β n Eq. 6.3 are computed wth the general form ( F 1 ) φ = F (6.7) 1 φ 1 1 φ where the φ 1 corresponds to coeffcents from the kω model and φ corresponds to coeffcents of the kε model. The eddy vscosty s calculated from a k 1 ν t = (6.8) max( a1ω, ΩF )
90 68 where F s gven by ( ) F = tanh arg (6.9) and arg k 500ν = max, C μω y ω (6.30) Here Ω s the magntude of the vortcty vector. The frst term n the denomnator of Eq. 6.8 comes from the tradtonal defnton of eddy vscosty for a kω model (Eq. 6.10). The second term n the denomnator represents an attempt to mprove the performance of the model for adverse pressure gradents and comes from a one equaton turbulence modeled developed by Bradshaw 7 based on a relatonshp for shear stress n a boundary layer smlar to Eq. 6.1 v = a k (6.31) u 1 The second term n the denomnator of Eq. 6.8 reduces the eddy vscosty n the logarthmc and wake regon of the boundary layer when adverse pressure gradents are present. The use of the shear stress relatonshp of Eq n the defnton of the eddy vscosty (Eq. 6.8) has gven rse to callng ths model the shear stress transport (SST) model. The coeffcents used wth the SST model are gven Tables 6.4 and 6.5. Set 1 kω Set kε β σ k σ ω γ β 1 σ ω1κ C C ( ) μ C ε 1 C C 1 μ ε1 μ Table 6.4 SST model functons. C ε1 C ε C μ κ a Table 6.5 SST model coeffcents. The boundary condtons for SST model are the same as the kω model. The SST model s relatvely nsenstve to the free stream value of ω.
91 RNG Model One of the maor crtcsms of the kε model s that t s not derved from the NaverStokes equatons n any systematc fashon. Yakhot and Orszag 8 appled Renormalzaton Group (RNG) methods to derve the kε equatons. In ths approach, an expanson s made about an equlbrum state wth known Gaussan statstcs usng the correspondence prncple that the effects of the mean strans can be represented by a random force. Bands of hgh wave numbers (small scales) are systematcally removed and space s rescaled n a manner analogous to that used n phase transtons. The successve removal of larger scales ultmately leads to a kε model of the form k t U k x = x ν t k ν σ k x P k ε (6.3) ε ε ν t ε ε ε U = C Pk C R t x x ν ε1 ε σ ε x k k (6.33) where R s defned as ( 1 ζ / 4.38) ζ P R = kε (6.34) 3 ( ζ ) k ( S S ) 1/ k ζ = (6.35) ε The parameter ζ s the rato of the turbulent to mean stran tme scale. The remanng constants are gven n Table 6.6. C ε1 C ε C μ σ k σ e Table 6.6 RNG model coeffcents. The RNG model s smlar to the tradtonal kε model wth the excepton of the addtonal term R and the lower value of C ε. These two dfferences decrease both the rate of producton of turbulent knetc energy and the rate of dsspaton of ε, leadng to lower values of the eddy vscosty. In free shear flows, ζ (and thus R) s zero. The model presented s the hgh Reynolds number form. The addtonal terms requred for the low Reynolds form can be found n Ref. 9.
92 Numercal Implementaton The flux terms on the rght hand sde of Eqs. 6.1, 6.11, and 6.19 are usually treated wth upwnd dfferences. The dffuson terms are generally treated wth central dfferences. The turbulent knetc energy producton term P k s lmted to be less than 0ε n the turbulent knetc energy source term to keep the model from runnng away n complex flows. The maxmum value of eddy vscosty s also capped (normally < x10 5 ν) to lmt nonphyscal behavor of the model. The turbulent quanttes (k, ε, and ω) are lmted to be postve n the feld. The exact source term Jacoban term can be used for the ε or ω equatons for mplct computatons. Use of the exact source term for the Jacoban of the turbulent knetc energy equaton usually results n numercal nstablty. It s recommended that the turbulent knetc energy source term Jacoban be replaced wth TKE source term Jacoban = P k ε (6.36) k The two equatons need not be solved fully coupled, and can be lagged from the mean flow equatons. For tme accurate mplct solutons the turbulence equatons should be solved wthn a Newton teraton loop wth the mean flow equatons to assure that the soluton s locally converged at each tme step. 6.7 Compressblty Correcton for Shear Layers The standard kε and kω turbulence models wll over predct the mxng for hghspeed compressble shear layers. The models tend to predct the ncompressble growth rate for the shear layer rather than decreasng the growth rate as compressble effects ncrease. A common method of correctng the kε model for ths effect s gven n Ref. 10. The compressble turbulent knetc energy equaton s wrtten as ρk t ρu k x = x μ t k μ σ k x ρp k ρ ( ε ε ) p"d" c (6.37) where ε c = α M ε (6.38) 1 t M t s the turbulent Mach number defned as k M t = (6.39) γ RT
93 611 The last term n Eq represents the pressure dlataton. Ths term s modeled as p " d" α ρpk M t α 3ρεM t = (6.40) Based on drect numercal smulaton, the recommended values for the constants n Eqs and 6.40 are α =.0, α = 0.4, α 0. (6.41) = The compressblty correcton must also be ncluded n the SST model snce the SST model reduces to the standard kε model away from the wall. A method for ncorporatng the compressblty correcton on the SST model s gven n Ref. 11. Further correctons for hgh temperature et flows are descrbed n Ref. 1. The need for a compressblty correcton can be seen n predctons of the supersonc axsymmetrc et shear layer of Eggers 13. The et ext Mach number was. and the et ext statc pressure was matched to the quescent outer ar. Fg. 6.1 shows the predcted and measured axal velocty on the et centerlne. Fg. 6. contans velocty comparsons at varous downstream locatons n the et. Fgure 6.1. Centerlne axal velocty for the Ref. 1 et.
94 61 Fgure 6.a. Velocty profle at X/R=5 for the Ref. 1 et. Fgure 6.b. Velocty profle at X/R=50 for the Ref. 1 et.
95 613 Fgure 6.c. Velocty profle at X/R=100 for the Ref. 1 et. Wthout the compressblty correcton all of the turbulence models predct a faster growth rate than s seen n the expermental data. The SST model wth the compressblty correcton s seen to do a good ob for accountng for the compressblty effect. 6.8 Intalzng Turbulence Values for a Gven Profle Shape There s really no unque way to construct turbulence varables for a gven velocty profle shape for an nflow boundary condton for transport type turbulence models. Here are two smple approaches. Often a combnaton of these approaches s requred n real world applcatons. One approach s to specfy a turbulent ntensty and a turbulent length scale for a gven profle. The turbulent ntensty s specfed as a percentage (a) of the local tmeaveraged flud velocty u = au (6.4) The turbulent ntensty s normally less than twenty per cent of the mean flow velocty. A turbulent length scale (L) must then be specfed. Some typcal choces are shown n Table 6.7. Flow Boundary Plane Jet Round Jet Plane Ppe Layer Wake L 0.41δ 0.53δ 0.44δ 0.95δ 0.5D
96 614 Table 6.7 Typcal values for the turbulent length scale L. In Table 6.7, δ s defned as the dstance between ponts where the velocty dffers from the free stream velocty by one percent of the maxmum velocty dfference across the layer. For symmetrcal flows and boundary layers, δ s the dstance from the symmetry plane or wall to the one percent pont at the outer edge. D s the nternal ppe dameter. The turbulent knetc energy can be found from 3 3 k = u = ( au ) (6.43) The turbulent length scale s related to the eddy vscosty by the Kolmogorov Prandtl expresson μ = C ρ t μ k L (6.44) The tradtonal defnton of eddy vscosty for twoequaton turbulence models s gven by μ t = C μ k ρ ε (6.45) Thus the turbulent dsspaton s defned as k 3 / ε = (6.46) L Although ths method s relatvely smple to mplement, t s not very accurate for boundary layers and looses meanng for complex velocty profle shapes. The second method takes advantage of the turbulence equlbrum assumpton (turbulent producton = turbulent dsspaton). Ths requres an estmate of the eddy vscosty along the profle. The eddy vscosty profle can be obtaned from a smple algebrac turbulence model such as BaldwnLomax or a oneequaton turbulence model such as SpalartAllmaras. The turbulent dsspaton s then gven by ε = (6.47) P k Fnally, the turbulent knetc energy s obtaned from the defnton of eddy vscosty n Eq. (6.45)
97 615 k μtε = (6.48) ρ C μ Ths method works well for general applcatons and can be useful for startng kε models. These models often have dffcultes generatng eddy vscosty when a soluton s started from a unform free stream condton. 6.9 Rotaton and curvature correcton A rotatonal and curvature correcton for the SST model can also be formulated n a manner smlar to the SARC and ASARC correctons used n the SA oneequaton model. The correcton has a much smaller effect because the SST uses stran rather than vortcty n the defnton of the producton term. The producton term used n both the k and ω equatons (Eq. 6. and 6.3) s redefned as P k = μ S S F ( r*, ~ r ) (6.49) t r1 The correctons defned n Eq. (5.13) and Eq. (5.18) are used wth Eq. (6.49) to provde a general threedmensonal rotaton and curvature correcton for the SST model n ths study. The Eq. (5.13) and Eq. (5.18) correctons are lmted 0.0<F r1 <1.5 for the two equaton model applcaton. The SST model wth the Eq. (5.13) correcton s called the ASSTRC model and the SST model wth the Eq. (5.18) correcton s named the SSTRC model. An example of the standard SST, SSTRC, and ASSTRC models appled to flow n a uduct 13 s shown n Fg The dstance along the wall s denoted by s. The standard SST model slghtly underpredcts the pressure coeffcent and the skn frcton at the ext of the ubend. Both the SARC and ASARC models mprove the comparson wth data. Fgure 6.3. Pressure coeffcent and skn frcton on the outer wall of a uduct..
98 Grd Senstvty for a Flat Plate wth Adabatc Walls The ntal wall spacng of the computatonal grd and the grdstretchng rato can affect the accuracy of the SST model. Fgure 6.4 shows the senstvty of the skn frcton to ntal wall spacng for a flat plate. The grdstretchng rato was 1. for all these cases. The plots nclude the theoretcal skn frcton curves of Whte and of Spaldng. Fgure 6.4 Flat plate skn frcton predctons for the SST turbulence model for varyng ntal wall grd pont spacngs. The boundary layer s seen to become fully turbulent around a length Reynolds number (Re x ) of 1x10 6. The results for y =0. and y =1 are vrtually dentcal ndcatng a grd ndependent soluton. The y =5 soluton show some small dvergence from the y =1 soluton at the lower length Reynolds numbers whle the y =10 soluton shows large dfferences from the other solutons. The SST model tends to be more senstve to ntal grd wall spacng than the Spalart Allmaras or BaldwnLomax models. Predcted velocty profles for the flat plate boundary layer for varous ntal wall grd pont spacngs are shown n Fg The velocty profle shows lttle effect of the ntal spacng for all but the y =10 profle. All of the profles but the y =10 profle are n good agreement wth the theoretcal profle from Spaldng. Note that the theoretcal profle does not nclude the lawofthe wake (see Fg. 3.), and hence the predcted profles dverge from the theoretcal profle n the wake regon of the boundary layer. The predcted eddy vscosty for varous ntal wall spacngs s shown n Fg. 6.6.
99 617 Fgure 6.5 Flat plate boundary layer profles predcted by the SST turbulence model for varyng ntal wall grd pont spacngs. Fgure 6.6 Eddy vscosty dstrbuton predcted by the SST turbulence model for varyng grd ntal wall spacngs. Here agan t s seen that the y =0. and the y =1.0 results are almost dentcal. The y =5 and y =10 results show the solutons are no longer grd ndependent at larger wall spacngs.
100 618 The effect of grd stretchng rato on skn frcton for a flat plate s shown n Fg All of these solutons used an ntal wall spacng of y =1. Fgure 6.7 The effect of grd stretchng rato on the skn frcton for a flat plate boundary layer usng the SST turbulence model. There seems to be very lttle effect of grd stretchng for ths case ndcatng that the ntal wall spacng s the more crtcal parameter for skn frcton predctons for flat plates wth the SST turbulence model. Ths s also the case for the velocty profle, as seen n Fg The eddy vscosty does change as the stretchng rato ncreases as shown n Fg It s nterestng to note that a wde range of eddy vscosty dstrbutons have lttle effect on skn frcton and the velocty profle for a flat plate boundary layer.
101 619 Fgure 6.8 The effect of grd stretchng rato on the velocty profle for a flat plate boundary layer usng the SST turbulence model. Fgure 6.9 The effect of grd stretchng rato on the eddy vscosty dstrbuton for a flat plate boundary layer usng the SST turbulence model Grd Senstvty for an Axsymmetrc Bump A second example of the grd senstvty of the SST turbulence model that ncludes a pressure gradent s the NASA Ames transonc axsymmetrc bump experment descrbed n the BaldwnLomax Chapter 4. The effect of ntal grd
102 60 spacng on the pressure coeffcent dstrbuton along the bump s shown n Fg The stretchng rato was 1. for these cases. The pressure coeffcent seems to be relatvely nsenstve to the ntal grd spacng, wth the y =10 and y =0 curves beng slghtly dsplaced from the other curves. The velocty dstrbuton at the aft uncton of the bump and the cylnder (x/c=1) s shown n Fg The y =0 soluton predct a larger velocty n the reverse flow regon than the other solutons. Grd stretchng effects on the pressure coeffcent dstrbuton along the bump s shown n Fg The ntal grd spacng was 1. for these cases. The pressure dstrbuton coeffcent changes slghtly as the grdstretchng rato s ncreased to 1.5. The soluton n the separated regon dffers greatly for a grdstretchng raton of.0. The effect on the velocty dstrbuton at x/c=1 s shown n Fg As wth ncreasng ntal grd spacng, ncreasng the grd spacng ncreases the sze and the magntude of the separated flow regon. Fgure 6.10 The effect of ntal grd wall spacng on the pressure coeffcent for the axsymmetrc bump usng the SST turbulence model.
103 61 Fgure 6.11 The effect of ntal grd wall spacng on the velocty dstrbuton at x/c=1 for the axsymmetrc bump usng the SST turbulence model. Fgure 6.1 The effect of grd stretchng on the pressure coeffcent for the axsymmetrc bump usng the SST turbulence model.
104 6 Fgure 6.13 The effect of grd stretchng on the velocty profle at x/c=1 for the axsymmetrc bump usng the SST turbulence model. As was seen wth the BaldwnLomax model and the SpalartAllmaras model, grd stretchng s a crtcal parameter for ths turbulence model when adverse pressure gradents are present. Care should be taken to keep the grdstretchng rato between 1. and Grd Senstvty for a Flat Plate wth Heat Transfer Calculatng heat transfer accurately can be more dffcult than predctng skn frcton. Ths can be seen n the subsonc flat plate example when the wall temperature s specfed to be 1.5 tmes the freestream temperature. The senstvty of the skn frcton and heat transfer result wth varyng ntal grd wall spacng s shown n Fg and Fg The grd stretchng rato was fxed at 1. for these results. Both the skn frcton and heat transfer seem to be relatvely nsenstve to the wall spacng for wall spacngs less than y =0.1.
105 63 Fgure 6.14 The effect of wall spacng on the skn frcton on a flat plate wth heat transfer usng the SST turbulence model. Fgure 6.15 The effect of wall spacng on the heat transfer (Stanton number) on a flat plate usng the SST turbulence model. Profles of velocty and temperature at a length Reynolds number (Re x ) of 1x10 7 are shown n Fg and 6.17 respectvely. The results are relatvely nsenstve to the wall spacng for ntal wall spacngs of y >5.
106 64 Fgure 6.16 The effect of wall spacng on the velocty profle on a flat plate wth heat transfer usng the SST turbulence model. Fgure 6.17 The effect of wall spacng on the temperature profle on a flat plate wth heat transfer usng the SST turbulence model. Grd stretchng effects on skn frcton and heat transfer predctons are shown n Fg and The velocty and temperature profles for a length Reynolds number (Re x ) of 1x10 7 are shown n Fg. 6.0 and 6.1. The ntal wall spacng was held at y =0.1 for these calculatons. The results reach a grd ndependent result for stretchng ratos less than 1.3.
107 65 Fgure 6.18 The effect of grd stretchng on the skn frcton on a flat plate wth heat transfer usng the SST turbulence model. Fgure 6.19 The effect of grd stretchng on heat transfer (Stanton number) on a flat plate usng the SST turbulence model.
108 66 Fgure 6.0 The effect of grd stretchng on the velocty profle on a flat plate wth heat transfer usng the SST turbulence model. Fgure 6.1 The effect of grd stretchng on the temperature profle on a flat plate wth heat transfer usng the SST turbulence model Grd Senstvty for a Nozzle wth Heat Transfer Flow through a supersonc nozzle wth a constant temperature wall can serve as a test case for evaluatng the performance of the turbulence model n the presence of strong pressure gradents. Detals of the geometry and boundary
109 67 condtons for the convergngdvergng supersonc nozzle are gven n Chapter 4. Hghpressure ar was heated by the nternal combuston of methanol and flowed along a cooled constant area duct before enterng the nozzle. The gas could be treated as a calorcally perfect gas wth a ratoofspecfc heats (γ) of The nozzle ext Mach number was.5. The molecular vscosty and thermal conductvty were assumed to vary accordng to Sutherland s law. The grd ntal wall spacng was vared and the grd stretchng rato was held at 1. n the boundary layer. A unform grd was used n the nozzle core. Predcted wall pressure dstrbuton results for varyng ntal wall spacngs are shown n Fg. 6.. The pressure dstrbuton s seen to be nsenstve to the ntal wall spacng. Predcted wall heat transfer s shown n Fg The results are somewhat senstve to the ntal wall spacng for values of y less than one. The predcted results dverge rapdly from the data for wall spacng greater than y =1. Fgure 6. The effect of wall spacng on the pressure dstrbuton for a supersonc nozzle wth heat transfer usng the SST turbulence model.
110 68 Fgure 6.3 The effect of wall spacng on the wall heat transfer for a supersonc nozzle usng the SST turbulence model. The grd stretchng rato was vared n the boundary layer whle the ntal grd spacng held at y =0.5. A unform grd was used n the nozzle core. Predcted wall pressure dstrbuton results for varyng ntal wall spacngs are shown n Fg The pressure dstrbuton s seen to be nsenstve to the grd stretchng rato. Predcted wall heat transfer s shown n Fg The results are nsenstve to the grd stretchng ratos less than 1.3. The predcted results dverge slghtly from the data for grd stretchng ratos greater than 1.3.
111 69 Fgure 6.4 The effect of grd stretchng rato on the pressure dstrbuton for a supersonc nozzle wth heat transfer usng the SST turbulence model. Fgure 6.5 The effect of grd stretchng rato on the wall heat transfer for a supersonc nozzle usng the SST turbulence model Summary Twoequaton models have been the ndustry standard for many years. These models have been used for a wde varety of nternal and external flows. Many
112 630 varants are avalable that have been tuned for specfc classes of flows. These models are more dffcult to mplement than are the oneequaton models and can often be less numercally stable. For these reasons many code developers have avoded twoequaton models. The examples gven here are manly for subsonc and supersonc applcatons. An excellent summary of hypersonc valdaton of these models can be found n Ref. 15. TwoEquaton Model Applcaton Hnts 1. The frst pont off the wall should be located about y =1 to obtan reasonable skn frcton values and about y =0.5 for heat transfer. The grd stretchng normal to the wall should not exceed 1.3. A constant spacng should be used for the frst three cells off the wall for heat transfer calculatons.. The eddy vscosty should be lmted so that t wll not run away n some complex applcatons. Generally a lmt of ν t /ν=00,000 s acceptable. 3. Care should be taken not to dvde vscous regons such as boundary layers when dvdng the computatonal doman for blocked or overset applcatons snce the model requres the dstance from the nearest wall. 4. kε models are often dffcult to start. It s usually easer to come off the ground wth another turbulence model such as BaldwnLomax or SpalartAllmaras and swtch to the kε model after ν t /ν>1.0 n the computatonal doman. Ths can be done by frst calculatng the turbulent knetc energy producton P k wth Eq. 6.. Assumng the turbulence s n local equlbrum, then the dsspaton can be found from ε =P k. The turbulent knetc energy k can then be found from the defnton of eddy vscosty n Eq Ths model can overdamp some unsteady flows. 6. Compressblty correctons should be ncluded for hghspeed shear layer flows. Chapter 6 References: 1. Kato, M. and Launder, B. E., "The Modelng of Turbulent Flow Around Statonary and Vbratng Square Cylnders", Proc. 9th Symposum on Turbulent Shear Flows, Kyoto, August 1993, pp Jones, W. P., and Launder, B. E., The Calculaton of LowReynolds Number Phenomena wth a TwoEquaton Model of Turbulence, Int. J. of Heat and Mass Transfer, Vol. 16, 1973, pp
113 Wlcox, D. C., Reassessment of the ScaleDetermnng Equaton for advanced Turbulence Models, AIAA Journal, Vol. 6, No. 11, 1988, pp Chen, J. Y., Predctons of Channel BoundaryLayer Flows wth a Low ReynoldsNumber Turbulence Model, AIAA Journal, Vol. 0, Jan. 198, pp Spezale, C., Abd, R., and Anderson, E., A Crtcal Evaluaton of Two Equaton Models for Near Wall Turbulence, ICASE Report No , June Menter, F. R. and Rumsey, C. L., Assessment of TwoEquaton Turbulence Models for Transonc Flows, AIAA , June Bradshaw, P., Ferrss, D. H., and Atwell, N. P., Calculaton of Boundary Layer Development Usng the Turbulent Energy Equaton, Journal of Flud Mechancs, Vol. 8, 1967, pp Yakhot, V., Orszag, S., Thangam, S., Gatsk, T., and Spezale, C. G., Development of Turbulence Models for Shear Flows by a Double Expanson Technque, Physcs of Fluds A, Vol. 4, No. 7, 199, pp Barber, T., Cho, D., Nedungad, A., Orszag, S., Konstantnov, A., and Staroselsky, I., Advanced Turbulence Models for HghMach Number Wall Bounded Flows, AIAA , Jan Spezale, C., Abd, R., and Anderson, E., A Crtcal Evaluaton of Two Equaton Models for Near Wall Turbulence, ICASE Report No , June Suzen, Y. B. and Hoffmann, K. A., Investgaton of Supersonc Jet Exhaust Flow by One and TwoEquaton Turbulence Models, AIAA9803, Jan AbdolHamd, K. Pao, S. Massey, S., and Elmlgu, A., Temperature Corrected Turbulence Model for Hgh Temperature Jet Flow, AIAA , Jun Eggers, J. Velocty Profles and Eddy Vscosty Dstrbutons Downstream of a Mach. Nozzle Exhaustng to Quescent Ar, NASATN D3601, September Nchols, R., Algorthm and Turbulence Model Requrements for Smulatng Vortcal Flows, AIAA , Jan Roy, J. and Blottner, F., Methodology for Turbulence Model Valdaton: Applcaton to Hypersonc Flows, Journal of Spacecraft and Rockets, Vol. 40, No. 3, MayJune 003.
114 Reynolds and Algebrac Stress Models 7.1 Reynolds Stress Models One means of obtanng closure for the Reynolds or Favre averaged Naver Stokes equatons s to nvestgate hgher moments of the equatons themselves. Transport equatons for each of the Reynolds stresses can be derved from the momentum equatons. These equatons are called the second moment equatons. The second moment equatons can be obtaned by multplyng the momentum equatons by U and then Reynolds averagng the resultng equatons. The ncompressble form of the second moment equaton s uu uu U U U k = u uk uuk t xk xk xk p uu uk ν ( uu ) ( uδ k u δ k ) xk xk ρ (7.1) p u ρ x u x u ν x k u x k The frst term on the rght hand sde of Eq. 7.1 s the turbulent producton term. The second term on the rght hand sde s the dffuson of the Reynolds stresses. The thrd term on the rght sde s called the pressure stran. The pressure stran term redstrbutes energy among the turbulent stress components. The last term on the rght hand sde s the turbulent dsspaton. The balance of these source terms s crtcal to the proper smulaton of a turbulent flow. The second moment equatons produce a host of new unknowns and hgher order correlatons that must be modeled n order to close the system. These hgher order correlatons are dffcult, and n many cases mpossble, to determne from expermental measurements. Drect Numercal Smulaton (DNS) has been used to determne these quanttes n many cases. Unfortunately DNS s restrcted to low Reynolds numbers and smple geometres at the present tme, so the data base for determnng these hgher order correlatons, whether expermentally or numercally generated, s lmted. Eq. 7.1 may be rewrtten as u u t U k u u x k = P x k v t ( Dk ) ( Dk ) Π ε x k (7.) The producton term s defned usng the exact form
115 7 = k k k k x U u u x U u u P (7.3) Molecular dffuson also follows the exact defnton ( ) k v k u u x D = ν (7.4) There are several models suggested for the turbulent dffuson. Daly and Harlow 1 suggested = l l k s t k x u u u u k C D ε (7.5) Hanalc and Launder used the form = l l k l k l l k l s t k x u u u u x u u u u x u u u u k C D ε (7.6) Mellor and Herrng 3 suggests = k k k s t k x u u x u u x u u k C D ε (7.7) Many models for the pressurestran term have been developed snce the poneerng work of Launder, Reece, and Rod 4. A general form s gven by ( ) k k k k mn mn k k k k mn mn k k W b W b k C S b S b S b k C ks C b b b b C C b = Π δ δ ε (7.8) where the stress ansotropy tensor s gven by k u u b δ 3 1 = (7.9)
116 73 The stran rate tensor s defned as = x u x u S 1 (7.10) and the rotaton tensor s gven by = x u x u W 1 (7.11) If the turbulence s assumed to be locally sotropc (.e. the same amount of turbulent energy s dsspated by each of the turbulent energy components u ) then the dsspaton term n Eq. 7.1 can be reduced to ε εδ 3 = (7.1) where ε s the total rate of energy dsspaton. A transport equaton for the dsspaton must also be solved to close the system of equatons. The dsspaton equaton can be wrtten as ( ) t k k k k D x x x k C P k C x U t ε ε ε ε ε ν ε ε ε ε = 1 (7.13) The frst term on the rght hand sde of Eq s the producton of dsspaton. The second term represents the dsspaton of the dsspaton. The last two terms are the molecular and turbulent dffuson of dsspaton respectvely. The producton of dsspaton s gven by = x U u u P ε (7.14) The dsspaton equaton turbulent dffuson term s modeled by = l k t x u u k C D ε ε ε ε (7.15) Typcal values for the constants n the above equatons for the LaunderReece Rod 4 (LRR), GbsonLaunder 5 (GL) and SpezaleSarkerGatsk 6 (SSG) RSM models are
117 74 Model C 1 C 1 C C 3 C 4 C s C ε1 C ε C ε LRR GL SSG P/ε (b b ) 1/ Table 7.1 Coeffcents for three Reynolds stress models. A system of seven transport equatons (sx Reynolds stress components and one turbulent dsspaton equaton) must be solved at each tme step to provde the Reynolds stresses for the NaverStokes equatons. The above correlatons must be further modfed when appled n the presence of a wall, and hence many of these models utlze wall functons (see Chapter 10) to avod the complcatons and the addtonal grd ponts ntroduced by the presence of the wall. The Reynolds stresses produced by these models must meet certan constrants for the models to be applcable to a large range of problems wthout sgnfcant tunng. Two of the more mportant constrants are tensor nvarance and realzablty. Tensor nvarance requres the replaced terms to have the same tensor form as the orgnal terms. Ths wll nsure that the modeled terms transform properly n dfferent coordnate systems. Schumann 7 ntroduced the realzablty constrant. Ths constrant requres the equaton for the turbulent stresses to have the property that all the component energes of the turbulent knetc energy (the dagonal terms of the Reynolds stress tensor) reman nonnegatve and all offdagonal components of the Reynolds stress tensor satsfy Schwartz s nequalty. Ths can be wrtten as u u 0 (7.16) u u u u u u 0 (7.17) 1 u1u 1 uu u u 3 3 u u 3 u u 3 u1u ( u1u uu3 uu u1u3 ) 0 ( u1u u3u3 uu3 u1u3 ) (7.18) The modelng of the rght hand sde source terms n Eq. 7. can often cause numercal stablty problems wth the second moment closure models. The dffculty n dervng a turbulent dsspaton equaton was dscussed n Chap. 6 and s a source of weakness n ths approach of modelng turbulence as well a weakness n the applcaton of twoequaton turbulence models. Because of the numercal stablty problems, the extra computatonal tme requrements, and the lmted mprovement n soluton accuracy over lower order models (Ref. 8), second moment closure s not commonly used for many complex CFD problems
118 75 today. Some cases where second moment closures have shown mprovement over lower order models can be found n Ref Algebrac Stress Models Under certan assumptons the Reynolds stress transport equatons (Eq. 7.1, 7.) can be reduced to a system of algebrac equatons that requre knowledge of the turbulent knetc energy (k) and the turbulent dsspaton (ε). Ths class of turbulence models s called an Algebrac Stress Model (ASM). ASM models reduce the closure problem to solvng two transport equatons and a system of algebrac equatons. Ths s sgnfcantly faster than solvng the full Reynolds stress transport equatons set. The algebrac equatons nclude most of the models and assumptons that were used to solve the full equaton set. Rod 10 suggested that the transport of u u s proportonal to the transport of the turbulent knetc energy k, the proportonalty factor beng the rato u u /k (whch s not a constant). Ths yelds Du u Dt Dff ( u u ) = u u k Dk Dt Dff ( k ) = u u k ( Pr oducton ε ) (7.19) Equaton 7.19 s vald when the temporal and spatal change n u u /k s small compared wth the change of u u tself. The algebrac stress model of Rod s gven by ( 1 C ) 3 U U U u ul u ul δ uu ε x 3 l xl x u = δ u k (7.0) 3 U uu x C 1 1 ε A more recent algebrac stress model derved by Abd, Morrson, Gatsk, and Spezale 11 (AMGS) s gven by ρu u = μ S ρkδ S k δ α ε ( S W S W ) * t kk 4 k k k k 5 k α S ε k S k 1 3 S kl S kl δ (7.1) where
119 76 * μ t 6 6 ( 1 η ) 0.( η ξ ) 3 k = α1 ρ (7.) η 6η ξ 6ξ η ξ ε ( ) 1/ k η = S S (7.3) ε ( ) 1/ k ξ = W W (7.4) ε 4 α1 = C 3 α = 4 g g ( C ) α = ( C ) 4 α = 5 g 4 ( C ) α = ( C ) 3 3 g 3 g = 1 g 4 ( C / ) C (7.5) The remanng coeffcents for the model of Rod and AMGS are gven n Table 7.. Model C 1 C C 3 C 4 C 5 Rod AMGS Table 7. Coeffcents for the ASM models of Rod and AMGS Rod s model n Eq. 7.0 does not nclude the necessary terms to be vald near walls and requres wall functons (descrbed n Chapter 10). The AMGS model s vald down to the wall. The soluton obtaned from an ASM model s not ndependent of the choce of model used to provde k and ε. The pressure coeffcent on the Ames axsymmetrc bump for a free stream Mach number of s shown for the AMGS ASM model usng a kω and kε two equaton models for closure n Fg The velocty dstrbuton at the tralng edge of the bump s shown n Fg. 7.. Note the solutons are qute dfferent for the same ASM model.
120 77 Fgure 7.1 Pressure coeffcent dstrbuton for the axsymmetrc bump at M= Fgure 7. Velocty Dstrbuton at the tralng edge of the bump for the axsymmetrc bump at M= Grd Senstvty for a Flat Plate wth Adabatc Walls The ntal wall spacng of the computatonal grd and the grdstretchng rato can affect the accuracy any turbulence model. Fgure 7. shows the senstvty of the
121 78 skn frcton to ntal wall spacng for a flat plate for the SSG RSM model. Flat plate results for the AMGS ASM model are shown n Fg The grdstretchng rato was 1. for all these cases. The plots nclude the theoretcal skn frcton curve of Whte. The boundary layer s seen to become fully turbulent around a length Reynolds number (Re x ) of 1x10 6 for both models. These models are relatvely nsenstve to wall spacng below a y =10. Fgure 7.3 Flat plate skn frcton predctons for the SSG RSM turbulence model for varyng ntal wall grd pont spacngs.
122 79 Fgure 7.4 Flat plate skn frcton predctons for the AMGS ASM turbulence model for varyng ntal wall grd pont spacngs. Predcted velocty profles for the flat plate boundary layer for varous ntal wall grd pont spacngs are shown n Fg. 7.5 and 7.6. The velocty profle shows lttle effect of the ntal spacng for all but the y =10 profle for the ASM model. Fgure 7.5 Flat plate boundary layer profles predcted by the SSG RSM turbulence model for varyng ntal wall grd pont spacngs.
123 710 Fgure 7.6 Flat plate boundary layer profles predcted by the AMGS ASM turbulence model for varyng ntal wall grd pont spacngs. The effect of grd stretchng rato on skn frcton for a flat plate s shown n Fg. 7.7 and 7.8 All of these solutons used an ntal wall spacng of y =1. Fgure 7.7 The effect of grd stretchng rato on the skn frcton for a flat plate boundary layer usng the SSG RSM turbulence model.
124 711 Fgure 7.8 The effect of grd stretchng rato on the skn frcton for a flat plate boundary layer usng the AMGS ASM turbulence model. The RSM model shows lttle senstvty to grd stretchng below a raton of. The model has dffculty convergng as can be seen by the spkes n the soluton. The ASM soluton shows senstvty begnnng at a raton of 1.5. The boundary layer profle predcted by both of these models show the same senstvtes to skn frcton as the skn frcton. Ths can be seen n Fg.7.9 and Fg Fgure 7.9 The effect of grd stretchng rato on the velocty profle for a flat plate boundary layer usng the SSG RSM turbulence model.
125 71 Fgure 7.10 The effect of grd stretchng rato on the velocty profle for a flat plate boundary layer usng the AMGS ASM turbulence model. 7.4 Grd Senstvty for an Axsymmetrc Bump A second example of the grd senstvty of these turbulence models that ncludes a pressure gradent s the NASA Ames transonc axsymmetrc bump experment descrbed n the BaldwnLomax Chapter 4. The effect of ntal grd spacng on the pressure coeffcent dstrbuton along the bump for the RSM and ASM models s shown n Fg and Fg The stretchng rato was 1. for these cases. The pressure coeffcent seems to be relatvely nsenstve to the ntal grd spacng for the ASM model, wth the y =10 and y =0 curves beng slghtly dsplaced from the other curves. The velocty dstrbuton at the aft uncton of the bump and the cylnder (x/c=1) are shown n Fg and The trends are smlar to those seen wth the pressure coeffcent. Grd stretchng effects on the pressure coeffcent dstrbuton along the bump s shown n Fg and Fg The ntal grd spacng was 1. for these cases. Both models show senstvty to grd stretchng raton n the separated flow regon. The velocty profles shown n Fg and 7.18 show a senstvty to stretchng rato. The senstvty of the Reynolds shear stress predcted by the SSG RSM model to both ntal wall spacng and grd stretchng rato s shown n Fg and Fg The shear stress predcton s shown to be extremely senstve to stretchng rato.
126 713 Fgure 7.11 The effect of ntal grd wall spacng on the pressure coeffcent for the axsymmetrc bump usng the SSG RSM turbulence model. Fgure 7.1 The effect of ntal grd wall spacng on the pressure coeffcent for the axsymmetrc bump usng AMGS ASM turbulence model.
127 714 Fgure 7.13 The effect of ntal grd wall spacng on the velocty dstrbuton at x/c=1 for the axsymmetrc bump usng the SSG RSM turbulence model. Fgure 7.14 The effect of ntal grd wall spacng on the velocty dstrbuton at x/c=1 for the axsymmetrc bump usng the AMGS ASM turbulence model.
128 715 Fgure 7.15 The effect of grd stretchng on the pressure coeffcent for the axsymmetrc bump usng the SSG RMS turbulence model. Fgure 7.16 The effect of grd stretchng on the pressure coeffcent for the axsymmetrc bump usng the AMGS ASM turbulence model.
129 716 Fgure 7.17 The effect of grd stretchng rato on the velocty dstrbuton at x/c=1 for the axsymmetrc bump usng the SSG RSM turbulence model. Fgure 7.18 The effect of grd stretchng rato on the velocty dstrbuton at x/c=1 for the axsymmetrc bump usng the AMGS ASM turbulence model.
130 717 Fgure 7.19 The effect of ntal wall spacng on the shear stress dstrbuton at x/c=1 for the axsymmetrc bump usng the SSG RSM turbulence model. Fgure 7.0 The effect of grd stretchng rato on the shear stress dstrbuton at x/c=1 for the axsymmetrc bump usng the SSG RSM turbulence model.
131 718 RSM and ASM Model Applcaton Hnts 1. RSM models generally suffer from numercal stablty ssues. Ths s due to the complexty of the modeled terms. Ths s the prmary reason that these models are not regularly used n large scale applcatons.. ASM models are senstve to the transport equatons used to obtan k and ε. These models are generally less numercal stable than tradtonal twoequaton models. 3. The choce of the pressure stran term s crtcal to the performance of both RSM and ASM models. More complex pressure stran models are generally more accurate but less numercally stable. Chapter 7 References: 1. Daly, B. and Harlow, F., Transport Equatons of Turbulence, Physcs of Fluds, 1970, pp Hanalc, K. and Launder, B., Contrbuton Towards a ReynoldsStress Closure for LowReynoldsNumber Turbulence, Journal of Flud Mechancs, Vol. 74, 1976, pp Mellor, G. and Herrng, H., A Survey of the Mean Turbulent Feld Closure Models, AIAA Journal, Vol. 11, 1973, pp Launder, B., Reece, G., and Rod, W., Progress n the development of a ReynoldsStress Turbulence Closure, Journal of Flud Mechancs, 1975, Vol. 68, Part 3, pp Gbson, M. and Launder, B., On the Calculaton of Horzontal, Turbulent, Free Shear Flows under Gravtatonal Influence, Journal of Heat Transfer, Vol. 98, 1976, pp Spezale, C., Sarker, S., and Gbson, T., Modelng the PressureStran Correlaton of Turbulence: An Invarant Dynamcal Systems Approach, Journal of Flud Mechancs, Vol. 7, 1991, pp Schumann, U., Realzablty of Reynolds Stress Turbulence Models, Physcs of Fluds, Vol. 0, pp , Leschzner, M., Batten, P. and Craft, T., ReynoldsStress Modelng of Transonc Afterbody Flows, The Aeronautcal Journal, June 001, pp Nallasamy, M., Turbulence Models and Ther Applcatons to the Predcton of Internal Flows: A Revew, Computers and Fluds, Vol. 15, pp , Rod, W., Turbulence Models and Ther Applcaton n Hydraulcs A State of the Art Revew, Internatonal Assocaton for Hydraulc Research, nd Ed., Feb Abd, R., Morrson, J., Gatsk, T., and Spezale, C., Predcton of Aerodynamc Flows wth a New Explct Algebrac Stress Model, AIAA Journal, Vol. 34, No. 1, Dec
132 Large Eddy Smulaton Theoretcally the NaverStokes equatons can be used to smulate turbulent flows. The computatonal grd used n such a smulaton would have to be fne enough to allow the smallest turbulent length scales to be realzed and the computatonal tme step would have to be small enough to smulate the hghest frequences of the turbulent spectrum. In practce such computatons are prohbtvely expensve for hgh Reynolds numbers. The Reynolds averaged NaverStokes (RANS) equatons were derved assumng that all of the unsteadness due to the turbulent nature of the flow could be modeled wth emprcally derved correlatons. Ths reduces the tme and length scales that must be smulated, but also lmts the applcablty of the smulaton for unsteady flows. Large eddy smulatons (LES) were developed to extend the smulaton of unsteady flows beyond DNS. The desred result of an LES computaton s to obtan a DNS equvalent soluton for the largescale turbulence on a much coarser grd than s requred for DNS. An LES smulaton requres: 1. A grd fne enough to dscretze the small nearly sotropc scales of the turbulence. A low dsspaton numercal scheme 3. A flter functon to determne the dvson of the turbulent spectrum nto grd realzed and subgrd regons 4. A subgrd turbulence model A true LES smulaton s more than a hgh Reynolds number computaton run wthout a turbulence model. Although the resultng soluton from such a smulaton may resemble turbulent flow, the resultng soluton wll most lkely not represent an equvalent DNS soluton. 8.1 The Flterng Operaton The flterng operaton s crtcal to LES. Consder a flterng operaton wth a unform characterstc flter wdth Δ (whch mples sotropc grd elements). Leonard 1 defned the followng flter n physcal space φ ( x ) = G( x ξ ) φ( ξ ) dξ (8.1) Note that n ths chapter the over bar represents a fltered quantty, not a tme averaged quantty as n the prevous chapters. The flterng operaton s a spatal operaton as opposed to the Reynolds averagng operaton dscussed n Chapter that s a temporal operaton. The orgnal functon φ s then decomposed nto a fltered feld (or grd resolved) and a subgrd term φ = φ φ (8.)
133 8 The functon G defned n Eq. 8.1 s the flter functon. The flter functon may be any functon defned on an nfnte doman that satsfes the followng requrements (Ref. 1): 1. G ( ξ ) = G( ξ ). G ( ξ ) dξ = 1 n 3. G ( ξ ) 0 as ξ 0 such that all moments G ξ ) ξ dξ Δ Δ 4. G (ξ ) s small outsde, One mportant and useful feature of ths choce of flter s ( ( 0) n exst φ φ = x x (8.3) Three common flters are Spectral cutoff flter: πξ sn Δ G ( ξ ) = (8.4) πξ Δ Gaussan flter: = 1 C Cξ G ( ξ ) exp (8.5) Δ π Δ Box flter: 1 Δ f ξ < Δ G( ξ ) = (8.6) 0 otherwse The spectral cutoff flter s normally appled n spectral space as ˆ ϕ ( k) = Gˆ ( k) ˆ( ϕ k) (8.8)
134 83 π 1 f k < Gˆ ( ξ ) = Δ (8.9) 0 otherwse The constant C n the Gaussan flter (Eq. 8.5) s somewhat arbtrary, and values from to 6 have been used n practce. Note that the box flter (Eq. 8.6) s vald on both fnte and nfnte domans. For ths reason box flters are often used to relate DNS to LES for physcal space numercal schemes. There are advantages to usng spectral and pseudospectral soluton methodologes for LES. The frst advantage s that the approxmate feld, whch s dscrete n spectral space, s a fnte sum of contnuous functons n physcal space. Thus the approxmate feld, the spectral flter, and ther dervatves are contnuous functons n physcal space. The spectral flters defned n Eqs. 8.8 and 8.9 have the followng addtonal useful propertes φ = φ (8.10) φ = 0 (8.11) Ths says that flterng a grd resolved quantty yelds the orgnal grd resolved quantty and that the spectral flter of a subgrd quantty s equal to zero. Another advantage of the spectral approach s that dervatves yeld an exact value for the approxmate feld. In other words, f one s approxmatng a functon usng N Fourer modes, then as long as one has at least (N1) ponts n physcal space, t does not matter f one has 0 or 0 mllon ponts: the value of the dervatve at any gven pont s the same. Therefore one can take a flow feld generated wth a spectral DNS code, flter and coarsen the soluton, and produce a flow feld whch satsfes the governng equatons for LES. There are drawbacks to spectral flters. Frst, they have lmted applcablty n realstc flow stuatons because of boundary constrants. Also, spectral flters are nonpostve (.e. the flter functon s negatve at some ponts n space). The latter condton results n a subgrd stress tensor that does not satsfy the Reynolds stress realzablty condtons outlned n Chapter. Both Gaussan flters (Eq. 8.5) and box flters (Eq. 8.6) are postve functons, and thus the Reynolds stress realzablty condtons outlned n Chapter wll be satsfed on the resultng subgrd stress tensor. Unfortunately Eq and Eq are not vald for these flters. Furthermore, f a spatal numercal solver s to be used the flter must also be dscretzed. Therefore t s necessary to examne the dscretzed system that s solved n the actual code because dscrete systems
135 84 sometmes have dfferent propertes than the orgnal contnuous systems from whch they were derved. 8. Dervaton of the LES Equatons The Favre averaged or fltered NaverStokes equatons were derved n Chapter. The Favre flter can be defned for any varable as ρ ρφ φ = ~ (8.1) Note that the over bar sgnfes a spatally fltered quantty, and the tlde represents a Favre fltered, or grd resolved, quantty. Thus the Favre flter may be thought of as a densty weghted flter n space. Applyng ths flterng operaton to the NaverStokes equatons, and assumng that the flterng commutes wth the dervatve operaton, the LES equatons are 0 ~ = x u t ρ ρ (8.13) ( ) = u u u u x x x x p x u u t u ~ ~ ~ ~ ~ ~ ~ ~ ρ τ τ τ ρ ρ (8.14) ( ) ( ) ( ) = x T x T x x T x u u x x u pu pu x Eu Eu x u p E x t E ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ κ κ κ τ τ τ ρ ρ ρ (8.15) The resolved vscous stress tensor n Eqn takes the form = k k x u x u x u δ μ τ ~ 3 1 ~ ~ 1 ~ ~ (8.16) The vscosty and thermal conductvty s assumed to be calculated from the Favre averaged temperature (T ~ ). The total energy ( E ~ ) s defned as sgs k u u e E = ~ ~ 1 ~ ~ (8.17) The subgrd knetc energy, k sgs, s defned as the effect of the subgrd scales on the knetc energy of the resolved feld
136 85 sgs 1 ~ k = u u u~ u ~ (8.18) Fnally, the LES equaton of state s ~ p = ρrt (8.19) As wth Reynolds averagng, the flterng operaton produces terms that must be modeled n order to close the equaton set. The LES momentum equaton, Eqn. 8.14, contans two such terms. The frst, ( τ ~ τ ), represents the dfference x n the vscous terms between Favre and straght flterng. The second term, ~ ρ u u u~ u ~, arses from the convecton terms and has been observed to x behave much lke a vscous stress. Thus t s frequently redefned n terms of the subgrd stress tensor τ defned as sgs ~ sgs τ = ρ u u u~ u ~ (8.0) The LES energy equaton, Eq. 8.15, contans four subgrd terms. The frst term, ~ ρ Eu Eu ~~, arses from the convecton term. The second term s a x pressurevelocty correlaton term, ( pu pu~ ), whch s also related to the x convecton term. The thrd term, ( u τ u ~ ~ τ ), s a vscous subgrd term x smlar to the term n the LES momentum equaton. Ths term represents the transfer of energy due to subgrd vscous forces. The fourth term, ~ T T κ ~κ, s a heat flux subgrd term. The subgrd knetc energy, k sgs, x x x used n the defnton of the Favre fltered total energy n Eq. 8.7 must also be defned for closure. The subgrd stress tensor, sgs τ s usually treated n a smlar manner as the Boussnesq approxmaton for the RANS equatons (Chapter )
137 86 τ sgs ~ 1 ~ 3 3 sgs = ρν t S Skkδ ρk δ (8.1) In the above equaton, S ~ represents the resolved rate of stran tensor ~ S = 1 u~ x u~ x (8.) Note that a subgrd eddy vscosty, ν t, has been ntroduced n Eq The subgrd eddy vscosty accounts for the turbulence that cannot be resolved on the computatonal grd. Introducng the subgrd stress tensor nto Eq and assumng that the straght fltered stress tensor s equal to the Favre fltered stress tensor ( τ = ~ τ ) the LES momentum equaton becomes ρu~ t ρu~ u ~ x p = x ~ τ x τ x sgs (8.3) The LES energy equaton (Eq. 8.15) can be further smplfed as follows. Usng the defnton of total enthalpy p H = E (8.4) ρ two of the subgrd terms can be combned to form x ~ ( ) ~ ρ Eu ρeu ~~ pu pu~ = ρ Hu ρhu ~~ (8.5) x x Ths term can then be modeled wth an eddy dffuson model as x ~ ~ sgs H ρ Hu ρhu ~~ ceρ k Δ (8.6) x x where Δ s the local grd spacng and c e s an emprcal coeffcent. The thermal ~ T T conductvty term, κ ~κ, s neglected based on the assumpton that x x x the thermal conductvty s locally constant and that the Favre flterng s equvalent to the straght flterng. The vscous transport term s modeled as
138 87 x ( ) sgs k u τ u ~ τ μ x x ~ (8.7) Ths term s generally small and s often neglected n practce. Thus the modeled LES energy equaton may now be wrtten as ~ ρe t x x ~ k μ x ~ ( ρe p) sgs x u = ceρ k x ~ ~ T κ x sgs ~ u~ ~ H τ Δ x x (8.8) Inspecton of the modeled LES equatons (Eqs. 8.13, 8.3, and 8.8) shows that these equatons are qute smlar to the RANS equatons ncorporatng the Boussnesq approxmaton derved n Chapter. It should be noted that the tme averagng process used to derve the RANS equatons and the spatal flterng operatons used to derve the LES equatons are qute dfferent, and hence the terms n the two equaton sets are not the same. Three thngs are requred to close the modeled LES equatons: 1. The subgrd turbulent knetc energy k sgs. The subgrd eddy vscosty ν t 3. The emprcal coeffcent n the energy equaton, c e 8.3 Smagornsky Model As wth the RANS equatons, there are numerous models for the LES eddy vscosty. One of the earlest models for the LES eddy vscosty was proposed by Smagornsk and s gven by t 1 ~ ( C Δ ) ( S ~ S ) ν = (8.9) s g where C s s the Smagornsky coeffcent and Δ g s the local grd spacng. Ths model gnores the subgrd knetc energy k sgs. The Smagornsky model provdes a smple closure for the LES equatons and has been used effectvely for a number of applcatons. But, lke ts algebrac counterpart n the RANS regme, t has been shown to be lackng n smulatng complex turbulent flows. The Smagornsky model s not vald near walls, and a wall dampng term (Ref. 3) s often added to the model. The walldamped form s gven by
139 t 1 y / 5 ~ ~ ( C ( 1 ) ( ) sδ g e SS ν = (8.30) The coeffcent C s must be defned for ths model. In practce no unversal value of C s exst Dynamc Smagornsky Model Several nvestgators have attempted to dynamcally calculate C s based on equatng the hghest wave number resolved turbulent stresses wth the subgrd stress. One dynamc LES closure model outlned n Ref. 4 can be descrbed as follows. The subgrd stress tensor s wrtten as ( x, t) Δ S( u ~ ) S u ) sgs τ = C ( ~ (8.31) where C(x,t) s the Smagornsky coeffcent to be determned dynamcally and Δ s the grd spacng. For ths purpose a second spatal flter, called the testflter, of wdth larger than the grd flter s ntroduced. We choose the testflter scale Δ = Δ. Ths flter generates a second set of resolvablescale felds (denoted by ( )). Then, the dynamc SGS model of the τ at the grd flter and of the T at the testflter are wrtten as: τ T δ τ 3 δ T 3 kk kk = CΔ = Cβ = CΔ = Cα S S S S (8.3) A least squares method s developed to predct C(x,t) as: l ( α β ) ( α mn β mn )( α mn β mn ) 1 C( x, t) = (8.33) a kk wth l = l ( δ /3) l = Cα Cβ. 8.5 kequaton Model Another approach to closng the LES equatons s to ntroduce a transport equaton for the subgrd knetc energy, k sgs. The exact subgrd knetc energy equaton may be wrtten
140 89 ρk t x sgs x sgs ( ρk u~ ) ~ ( ρku ρku ~ ) p = u x sgs u~ τ τ x sgs u~ x u~ p u x τ x u~ τ x (8.34) Ths equaton contans several subgrd terms. The equaton s modeled n a smlar manner as the RANS turbulent knetc energy equaton. The modeled equaton s ρk t x sgs ~ sgs sgs ( ~ k u sgs ) ( k ) ρk u = ρν τ ρc 3 sgs x t x x ε Δ (8.35) where the terms on the rghthandsde are the dffuson, producton, and dsspaton of the subgrd knetc energy respectvely. An algebrac expresson for the dsspaton s generally used nstead of a transport equaton as n twoequaton RANS turbulence models. The turbulent vscosty s gven by sgs ν = k Δ (8.36) t c ν The LES energy equaton dsspaton coeffcent c ε and the eddy vscosty coeffcent c ν are determned dynamcally by equatng the hghest wave number resolved turbulent propertes wth the subgrd propertes. The dynamc process for determnng these coeffcents s descrbed n detal n Ref Inflow Turbulence Boundary Condton Flows that nclude a developng boundary layer at the nflow plane of a smulaton requre a boundary condton that ncludes the specfcaton of the nflow velocty fluctuatons. Soterou and Ghonem 6 show the development of an ncompressble mxng layer wth and wthout perturbaton at the nflow plane. Compared to the perturbed case, the development of the mxng layer wthout perturbaton s delayed sgnfcantly. Ths s because dsturbances wthn the boundary layer are amplfed and lead to the KelvnHelmholtz nstablty that eventually causes the shear layer to roll up. If these dsturbances are not present n the boundary layer, then the shear layer wll develop more slowly untl dsturbances are generated through mnute numercal errors. Several approaches have been taken to add the perturbatons to the nflow. The smplest approach s to add whte nose to the velocty at the nflow plane as was done by Comte et al 7. Ths approach s somewhat unphyscal n that the perturbatons have no correlaton n space or tme. A second approach s to add perturbatons at dscreet frequences as was done by Soterou and Ghonem 6. Ths can be effectve for studyng forced flows, but t must be remembered that turbulence s broadband and does not confne tself to dscrete frequences. A
141 810 thrd approach to ths problem s to attempt to reconstruct the unsteady flow at the nflow usng correlatons for the perturbatons. One example of ths can be found n Ref. 5, n whch a box of frozen turbulence s generated and saved. Ths box s scaled and appled as perturbatons at the nflow of the smulaton. 8.6 Other LES References LES methods are stll evolvng as researches nvestgate methods for smulatng the nearwall flow physcs and methods to mprove the performance of the subgrd models used n LES smulatons. Fureby, et al 8, provde a good summary of the LES methods currently beng employed and nvestgated. Ref. 9 ncludes a wde range of expermental test cases for valdaton of LES smulatons. 8.7 Spatal MxngLayer Example The mxng layer experment of Sammy and Ellot 8 can be used to demonstrate the behavor of LES models. The data s also ncluded n Ref. 10. The expermental setup had an upper supersonc stream and a lower subsonc stream mxng at a matched statc pressure. Laser Doppler Velocmeter (LDV) measurements were made at several downstream locatons between the tralng edge of the spltter plate and a staton 10 mm downstream of the spltter plate. The flow parameters are gven n Table 8.1. T 0, K P 01, kpa M 1 M M c U 1, m/sec U /U 1 ρ /ρ 1 δ 1, mm Table 8.1 Flow parameters for the spatal mxnglayer case The solutons were run for 30,000 tme steps to wash out ntal transents and then statstcs were taken over the next 60,000 tme steps. The tme step used was 5.0x108 seconds. The flow solver used for these calculatons was 4 th order n space and nd order n tme. The computatonal grd dmensons were 181 x 11 x 61. The nflow plane perturbatons were specfed usng the box of turbulence approach from Ref. 5. Further detals of the experment and the computatons can be found n Ref. 10. All of the turbulence models nvestgated predct roughly the same mxnglayer thckness for ths case as shown n Fg 8.1. The Hybrd kε model n Fg. 8.1 s the multscale hybrd RANS/LES turbulence model descrbed n Chapter 9. The DES model s the SpalartAllmaras DES hybrd model that s also descrbed n Chapter 9. The tradtonal RANS models (kε and SpalartAllmaras) have the worst agreement wth the expermental data.
142 811 Fgure 8.1 Mxnglayer thckness The streamwse velocty 10 mm downstream of the spltter plate s shown n Fg. 8.. Agan all of the models are n reasonable agreement wth the expermental data. Fgure 8. Nondmensonal streamwse velocty at x=10 mm
143 81 The streamwse turbulence ntensty, Reynolds stress, streamwse velocty fluctuaton skewness, and streamwse velocty fluctuaton flatness at x=10 mm are shown n Fg The performance of the tradtonal RANS models s seen to deterorate as the order of the turbulent velocty fluctuaton correlaton ncreases. The LES and hybrd RANS/LES models are all n reasonable agreement wth the expermental data. Fgure 8.3 Streamwse turbulence ntensty at x=10 mm Fgure 8.4 Reynolds stress profles at x=10 mm
144 813 Fgure 8.5 Stream wse velocty fluctuaton skewness at x=10 mm Fgure 8.6 Streamwse velocty fluctuaton flatness at x=10 mm. Chapter 8 References: 1. Leonard, A., Energy Cascade n LargeEddy Smulatons of Turbulent Flud Flows, Advances n Geophyscs, Vol. 18A, pp , Smagornsky, J. S., General Crculaton Experments wth the Prmtve Equatons I: The Basc Experment, Monthly Weather Revew, Vol. 91, pp , Pomell, U., Zang, T. A., Spezale, C. G., and Hussan, M. Y., Physcs of Fluds, Vol., pp. 57, Wasstho, B., Haselbacher, A., Naar, F.M., Taft, D., Balachandar, S. and Moser, R.D., Drect and Large Eddy Smulatons of Compressble Wall
145 814 Inecton Flows n Lamnar, Transtonal and Turbulent Regmes, AIAA , 38 th AIAA/ASME/SAE/ASEE Jont Propulson Conference and Exhbt, July 710, 00, Indanapols, IN. 5. Nelson, C. C., Smulatons of Spatally Resolved Compressble Turbulence Usng a Local Dynamc Subgrd Model, Ph.D. Thess, Georga Insttute of Technology, Soterou, M. and Ghonem, A., Effects of the FreeStream Densty Rato on Free and Forced Spatally Developng Shear Layers, Physcs of Fluds, Vol. 7, No. 8, 1995, pp Comte, P., Leseur, M., Laroche, H., and Normand, X., Numercal Smulatons of Turbulent Plane Shear Layers, n Turbulent Shear Flows 6, SprngerVelag, 1989, pp Fureby, C., Aln,N., Wkstrom, N., Menon, S., Svanstedt, N. and Persson, L., LargeEddy Smulaton of HghReynoldsNumber WallBounded Flows, AIAA Journal, Vol. 4, No. 3, March 004, pp A Selecton of Test Cases for the Valdaton of LargeEddy Smulatons of Turbulent Flows, AGARDAR345, Aprl Sammy, M. and Ellot, G., Effects of Compressblty on the Characterstcs of Free Shear Layers, AIAA Journal, Vol. 8, No. 3, 1990, pp Nelson, C. and Nchols, R., Evaluaton of Hybrd RANS/LES Turbulence Models Usng an LES Code, AIAA , June 003.
146 Hybrd RANS/LES Models 9.1 Theory Several nvestgators have noted a lmtaton wth RANS turbulence models when appled to unsteady flows. The turbulence models produce too much eddy vscosty and overdamp the unsteady moton of the flud. The problem s nherent n the constructon of the turbulence models because of the assumpton that all scales of the unsteady moton of the flud are to be captured and modeled by the turbulence model. One approach to overcomng the shortcomngs of RANS models appled to unsteady flows s to spatally flter the RANS turbulence models such that the eddy vscosty does not nclude the energy of grd resolved turbulent scales. The spatally fltered RANS turbulence model may be thought of as a subgrd model for very large turbulent eddes. Ths class of turbulence models has been called hybrd RANS/LES models because they ncorporate aspects of both forms of turbulence modelng. It s desrable that the spatal flter functon chosen not degrade the performance of the turbulence model when the largest turbulent scales present are below the resoluton of the grd as s often the case n current arcraft CFD applcatons. There are two possbltes for developng hybrd models from exstng RANS models. A frst approach would be to modfy the producton or dsspaton source terms for exstng turbulence model dfferental equatons to nclude addtonal terms to adust the local eddy vscosty so that t does not nclude the grd realzed contrbuton. Ths means that the turbulence quanttes normally transported by the RANS turbulence model wll be created or destroyed based on the local grd resoluton. A second approach would be to solve the exstng RANS turbulence model n the normal manner and then flter the results to determne the level of eddy vscosty that wll be used n the soluton of the NaverStokes equatons. Ths approach s somewhat smpler to develop snce t requres no tunng of the dfferental transport equatons and can be easly extended to dfferent RANS turbulence models. Both approaches have been appled to unsteady flows. One example of the frst form of a hybrd RANS/LES turbulence model based on the SpalartAllmaras oneequaton turbulence model can be found n Ref. 1. The standard Spalart Allmaras turbulence model contans a destructon term for eddy vscosty that s nversely proportonal to the square of the dstance from the wall (d). Ref. 1 suggests replacng the wall dstance (d) n the destructon term wth ( d ) d ~ = mn, C L (9.1) DES g
147 9 where C DES s a constant of O(1) and L g s a grd length scale defned by L g ( Δx, Δy Δz) = max, (9.) where Δx, Δy, and Δz are the local grd lengths. For twodmensonal or axsymmetrc problems, the out of plane drecton s smply gnored n defnng the local grd length. The modfed destructon term has the effect of decreasng the eddy vscosty n regons of tght grd spacng. Ths modfcaton causes the SpalartAllmaras RANS turbulence model to behave lke a Smagornsky LES turbulence model (see Chapter 8) when the grd spacng (L g ) s less than the dstance from the wall, whch s generally the case outsde of the boundary layer. Note that the transton from RANS to LES does not nclude any turbulent length scale dependence, but s solely a functon of the local grd spacng. Ref. 1 ntroduces the term DetachedEddy Smulaton (DES) to descrbe ths model. DES has been appled to a number of unsteady flow problems ncludng flow over a sphere, flow over a delta wng 3 and flow over an arcraft 4. A smlar modfcaton to an SST twoequaton model s gven by Strelets 5. In ths hybrd model the dsspaton term (ε) n the turbulent knetc energy equaton (k) s replaced by 3 ω k ε ε = = = (9.3) * β k mn( Lt, CDES Lg ) Lg mn 1.0, C DES Lt where 3 k k L t = = (9.4) * β ω ε ( 1 F1 ) C DESKE F C DESKW C DES 1 = (9.5) here ω s the specfc dsspaton, β * =0.09, C DESKE =0.61, and C DESKW =0.78. The turbulent dsspaton ε defned n Eq. 9.3 s effectvely ncreased when the grd sze length scale L g (Eq. 9.) s less than the turbulent length scale L t (Eq. 9.4). Ths causes the eddy vscosty to be reduced n these regons. Unlke the oneequaton SADES model, ths model does nclude a dependency on the local turbulent length scale. The turbulent length scale used n ths model s a functon of the fltered (subgrd) turbulent quanttes. Ths model behaves lke a k equaton LES subgrd model when the turbulent length scale s greater than the grd length scale, and the dsspaton equaton s decoupled from the knetc energy equaton n ths regon.
148 93 Nchols and Nelson 6 gve an example of the second approach for developng a hybrd RANS/LES turbulence model whch they have desgnated a multscale model. The method was mplemented n conuncton wth the SST twoequaton turbulence model and s termed a multscale model. The turbulent length scale used n ths effort s defned by t ( 6.0 ν / Ω, k / ) 3/ ε L = max (9.6) trans RANS RANS where ν trans s the unfltered (large scale) eddy vscosty and Ω s the local mean flow vortcty. Ths length scale s a mxture of the tradtonal turbulent scale 3/ defnton for twoequaton turbulence models ( k RANS /ε RANS ) and the defnton usually assocated wth algebrac turbulence models ((ν trans /Ω) 1/ ). The turbulent length scale defnton could be easly adapted to other types of turbulence models. The subgrd turbulent knetc energy s defned as k = k f (9.7) LES RANS d The dampng functon s defned as f d ( 1 tanh( ( Λ 0.5) ))/ = π (9.8) where 1.0 Λ = (9.9) 4 / 3 L t 1.0 L g where L g s defned n Eq. 9.. The eddy vscosty s then calculated from t trans d ( f ) 1 d ν tles ν = ν f (9.10) where the LES based subgrd eddy vscosty s gven by ν tles = mn( L g k LES, ν trans ) (9.11) The multscale hybrd model behaves lke a tradtonal SST model on the RANS end of the spectrum and transtons to a nonlnear kequaton model on the LES end of the spectrum. The transton functon n Eq was chosen to allow a smooth transton from the standard RANS turbulence model to the LES subgrd model. Ths hybrd RANS/LES approach can easly be extended to other RANS turbulence models wth lttle to no alteraton provdng that a length scale can be derved for that model and a value for the turbulent knetc energy can be approxmated by that model. The multscale hybrd model, lke the SSTDES
149 94 model, transtons from RANS to LES as a functon of the rato of the local turbulent length scale predcted by the RANS model and the local grd spacng rather than beng a functon of the grd spacng alone as s the case for the SpalartAllmaras DES model. Nelson and Nchols evaluated the SpalartAllmaras DES and the multscale hybrd turbulence models as subgrd turbulence models for LES applcatons n Ref. 7. The hybrd models were appled to a hghspeed shear layer usng a true LES flow solver. The results ndcate that the hybrd models perform as well as more complcated LES subgrd models for ths applcaton. Nchols 8 nvestgated these three hybrd RANS/LES models appled to a crcular cylnder and a generc weapons bay. These results are shown n the followng sectons. 9. Crcular Cylnder Unsteady threedmensonal calculatons were performed for the vortex sheddng from a crcular cylnder for M=0. and Re d =8x10 6. Three computatonal grds were used n the smulaton: Fne 401x01x01 Md 01x101x101 Coarse 101x51x51 The md and coarse level grds were constructed by removng every other pont from the fne and md level grds respectvely. The grds have a span of ten cylnder dameters. Perodc boundary condtons were appled at the sde planes of the grd. The ntal wall spacng was x104 dameters for the fne grd, whch corresponds to a y of about 0. Wall functon boundary condtons were used n the calculatons. Calculatons wth three dfferent tme steps were run on the md grd wth the multscale hybrd turbulence model to determne an acceptable tme step for ths smulaton. The results are shown n Fg The prmary sheddng frequency for ths example s ust less than 60 Hz (Strouhal number=0.8). These solutons correspond to 50, 00, and 800 tme steps per sheddng cycle. The results from the largest tme scale are qute dfferent from the smaller tme step results. The spectral peak predcted wth the large tme step occurs at a much lower frequency than the peak predcted usng the smaller tme steps. Based on these results all subsequent smulatons were performed usng the 9.0E4 second tme step wth 00 tme steps per sheddng cycle.
150 95 Fgure 9.1. Normal force power spectral densty for 3 dfferent tme steps usng the SSTMS model on the medum grd. The data n Ref. 9 was obtaned usng both ar (γ=1.4) and Freon (γ=1.13) as a test medum. Calculatons were performed on the medum grd wth the SSTMS hybrd model to assess the affect of test medum. The normal force spectral results are shown n Fg. 9.. Ths ndcates that the test medum s not an mportant factor at the condtons of ths study. All subsequent solutons were performed usng ar as the test medum. Fgure 9.. Normal force power spectral densty for ar and Freon usng the SST MS hybrd model.
151 96 The calculatons were run 10,000 teratons and the fnal 4096 tme steps were statstcally analyzed. The centerlne plane of the three grds s shown n Fg Fne Grd Medum Grd Coarse Grd Fgure 9.3. Cylnder grd centerlne. A comparson of the normal force power spectral densty for the three hybrd models, the SpalartAllmaras model, and the SST model for the medum grd are shown n Fg The hybrd models show smlar trends n that they all have a rather broad spectral peak at a Strouhal number of about 0.8. The two RANS models have a much narrower spectral peak at a smlar Strouhal number. The RANS models have much lower energy away from the peak. Ths s an ndcaton that the RANS models are provdng too much dampng of the unsteady solutons. Fgure 9.4. Normal force power spectral densty on the medum grd.
152 97 Instantaneous xvortcty sosurfaces computed wth the SSTDES hybrd turbulence model are shown n Fg. 9.5 for all three grds. The turbulent structure n the wake of the cylnder s clearly evdent n the fne grd soluton. The turbulent structure s reduced n the md grd soluton. There s very lttle structure present n the coarse grd soluton. Fne Grd Medum Grd Coarse Grd Fgure 9.5. Instantaneous vortcty sosurfaces colored by Mach number for the SSTDES hybrd turbulence model. Instantaneous contours of Mach number and eddy vscosty for the SADES, SSTDES, and SSTMS hybrd turbulence models computed on the three grd levels are shown n Fgs a. Fne b. Md c. Coarse Fgure 9.6. Instantaneous Mach number contours for the SADES hybrd turbulence model.
153 98 a. Fne b. Md c. Coarse Fgure 9.7. Instantaneous eddy vscosty contours for the SADES hybrd turbulence model. a. Fne b. Md c. Coarse Fgure 9.8. Instantaneous Mach number contours for the SSTDES hybrd turbulence model. a. Fne b. Md c. Coarse Fgure 9.9. Instantaneous eddy vscosty contours for the SSTDES hybrd turbulence model.
154 99 a. Fne b. Md c. Coarse Fgure Instantaneous Mach number contours for the SSTMS hybrd turbulence model. a. Fne b. Md c. Coarse Fgure Instantaneous eddy vscosty contours for the SSTMS hybrd turbulence model. All three hybrd turbulence models have smlar trends on the three grd levels. Sgnfcant turbulent structure can be seen n the fne grd solutons wth both largescale and smallscale turbulent structure present. The md level solutons have largescale structure present, but the smaller turbulent scales are absent. The md level grds also have structure that appears to be more perodc than does the fne grd soluton. The coarse level solutons show almost no turbulent structure and produce an almost steady state wake away from the cylnder. The level of eddy vscosty n the wake dffers for the three turbulent models. The eddy vscosty for the SADES model shown n Fg. 9.7 ndcates that ths model effectvely shuts off the eddy vscosty outsde the boundary layer. The eddy vscosty for the SSTDES and the SSTMS models s reduced as the grd s refned as would be expected for these models. The SSTDES model tends to predct hgher levels of eddy vscosty n the wake regon than does the SSTMS model. The SSTDES model also predcts hgher eddy vscostes along the
155 910 edges of the wake than ether the SADES or SSTMS models. Both the SST DES and SSTMS models are tendng toward a RANS type soluton n the far wake of the coarse grd soluton. The rato of the turbulent length scale to the local grd length scale for the SST MS model s shown n Fg. 9.1 for the three grd levels. a. Fne b. Md c. Coarse Fgure 9.1. Instantaneous rato of turbulent length scale to the grd length scale for the SSTMS hybrd turbulence model. The length scale rato seems to scale wth the grd refnement for the md to fne grd results, whch ndcates that the turbulent length scales predcted on the md and fne grds are smlar. The md level grd results n a length scale rato of greater than two for most of the wake regon. Ths s the desred performance trend wth grd refnement for hybrd turbulence models. The coarse grd results do not seem to contnue the scalng trend. The power spectral denstes (PSD) of the axal normal force coeffcent are shown n Fgs for the three hybrd turbulence models. The md and fne level grd results are n general agreement for the SADES hybrd model. The locaton and ntensty of the prmary spectral peak s stll varyng wth the SSTDES and SSTMS hybrd models. Ths may be due to the fact that the SSTDES and SSTMS hybrd models use a rato of the local turbulent length scale to the local grd length scale to determne the subgrd eddy vscosty, whle the SADES model only uses the local grd length scale. The coarse grd solutons are seen to be sgnfcantly dfferent and the energy s contaned n narrower peaks for all of the hybrd models.
156 911 Fgure Power spectral densty of the normal force coeffcent for the SA DES hybrd model. Fgure Power spectral densty of the normal force coeffcent for the SST DES hybrd model.
157 91 Fgure Power spectral densty of the normal force coeffcent for the SST MS hybrd model. The average ntegrated drag coeffcent (Cd), standard devaton of the lft coeffcent (σ(cl)), and the peak lft coeffcent Strouhal number (St) for the 3 hybrd models computed on the three grd levels are shown n Table 9.1. The md and fne level grd results are n general agreement wth each other and the expermental data 9,10,11. The results are also n reasonable agreement wth the Ref. 1 results obtaned at a Reynolds number of 3.0x10 6. Model Grd Average Cd σ(cl) St SA Coarse SST Coarse SADES Coarse SSTDES Coarse SSTMS Coarse SA Md SST Md SADES Md SSTDES Md SSTMS Md SADES Fne SSTDES Fne SSTMS Fne DATA Ref DATA Ref DATA Ref SADES Ref Table 9.1 Force coeffcent and Strouhal number predctons on the 3D crcular cylnder.
158 WICS Bay Unsteady NaverStokes calculatons were also performed for the WICS (Weapons Internal Carrage and Separaton) L/D=4.5 weapons bay 13 for M=0.95 and Re=.5x10 6 /ft. The wnd tunnel data were obtaned n the AEDC fourfoot transonc wnd tunnel. The weapons bay was 18 n. long, 4 n. wde, and 4 n. deep. The computatonal geometry was a flat plate that extended 15 n. upstream of the bay to match the experment and 5 n. downstream of the bay. The sdes of the computatonal grd extended 50 n. on ether sde of the bay centerlne. The full bay geometry was modeled usng wall functons. The wall spacng was chosen as n., whch corresponds to a y of 50 on the upstream plate. Ths wall spacng was used for all the grds n the grd refnement study. Grd resoluton effects were nvestgated by modfyng the bay grd and the grds n the vcnty of the bay grd. The grds upstream and to the sdes of the bay were not modfed. Detals of the grd systems and grd spacngs used n ths study are shown n Table 9.. The centerlne plane of the three bay grds s shown n Fgure Grd Total Ponts Bay Grd Dmensons Bay Grd Δx max Bay Grd Δy max Bay Grd Δz max Fne 1.8x x61x n. 0.1 n. 0.1 n. Medum 1.1x x41x n. 0. n. 0. n. Coarse 7.9x x31x n. 0.3 n. 0.3 n. Table 9.. Parameters for the grd refnement study. A tme step study was performed usng the SSTMS hybrd model and the medum grd set. Tme steps of 8.0x106 seconds to 8.0x105 seconds were evaluated. A tme step of 8.0x106 seconds corresponds to 60 tme steps per cycle for the prmary spectral mode of the cavty at 480 hz. The calculatons were run 000 steps to remove the ntal transents and all unsteady results were processed over the last 819 tme steps. The results are compared to data (Ref. 1) for the K16 and K18 transducer locatons n the bay n Fg K16 s located on the bay celng centerlne 0.75 nches from the back wall. K18 s located on the bay back wall centerlne 0.75 nches from the bay openng. The sound pressure levels predcted by the hybrd turbulence models ncludes only the contrbuton of the grd resolved turbulent scales. Subgrd turbulent energy, whch ncludes most of the energy n the boundary layer, s not ncluded. Thus the spectrum predcted usng the hybrd models should have lower energy at hgher frequences (.e. should have a more rapd roll off) than the expermental data.
159 914 Fgure Centerlne plane of the bay grds. Fgure 9.17a. Sound pressure level spectra at the K16 locaton for varyng tme step.
160 915 Fgure 9.17b. Sound pressure level spectra at the K18 locaton for varyng tme step. The agreement wth the measured spectra s seen to mprove at the hgher frequences as the tme step s reduced. The soluton wth the largest tme step predcts the prmary spectral peaks at too low a frequency. The soluton wth the smallest tme step provdes reasonable agreement for the frst four spectral peaks. The soluton usng 1.6x105 second tme step has reasonable agreement wth the frst three spectral peaks. All subsequent solutons were performed usng the 1.6x105 second tme step. A comparson of the sound pressure level for the three hybrd models, the SpalartAllmaras model, and the SST model for the medum grd are shown n Fg The hybrd models show smlar trends throughout the spectral range. The two RANS models predct the frst two spectral peaks, but are well below the data away from the peaks. Ths s an ndcaton that the RANS models are provdng too much dampng of the unsteady solutons.
161 916 Fgure 9.18a. Sound pressure level spectra at the K16 locaton for the medum grd. Fgure 18b. Sound pressure level spectra at the K18 locaton for the medum grd. Instantaneous Mach number and eddy vscosty contours on the bay centerlne are shown n Fgs
162 917 a. Fne b. Medum c. Coarse Fgure 9.19a. Instantaneous Mach number contours on the bay centerlne usng the SADES hybrd model. a. Fne b. Medum c. Coarse Fgure 9.19b. Instantaneous eddy vscosty contours on the bay centerlne usng the SADES hybrd model. a. Fne b. Medum c. Coarse Fgure 9.0a. Instantaneous Mach number contours on the bay centerlne usng the SSTDES hybrd model.
163 918 a. Fne b. Medum c. Coarse Fgure 9.0b. Instantaneous eddy vscosty contours on the bay centerlne usng the SSTDES hybrd model. a. Fne b. Medum c. Coarse Fgure 9.1a. Instantaneous Mach number contours on the bay centerlne usng the SSTMS hybrd model. a. Fne b. Medum c. Coarse Fgure 9.1b. Instantaneous eddy vscosty contours on the bay centerlne usng the SSTMS hybrd model. The turbulent structure shown n the Mach number contours s seen to dramatcally ncrease wth grd resoluton for all three hybrd models. The shear layer s seen to grow and then burst as t traverses the bay. The eddy vscosty decreases wth ncreasng grd resoluton as s expected for all three hybrd models. The SSTDES hybrd model has the largest levels of eddy vscosty, and the maxmum eddy vscosty s located n the shear layer. The rato of the
164 919 turbulence length scale to the grd length scale (L t /L g ) for the SSTMS model s shown n Fg. 9.. The contours scale wth the grd refnement as s expected for ths model. The tme averaged pressure coeffcent on the bay celng s shown n Fg. 9.3 for the three hybrd models. The sound pressure level on the bay celng s shown n Fg The solutons are smlar for the three models for both of these propertes. There s some senstvty to grd densty for the average pressure coeffcent on the downstream end of the celng. The medum and fne grds produce smlar results for the sound pressure. The coarse grd sound pressure s only slghtly dfferent from the fner grds. Ths ndcates that average pressure coeffcent and sound pressure level are not overly senstve to grd densty. Fne Medum Coarse Fgure 9.. Instantaneous rato of turbulent length scale to the grd length scale contours on the bay centerlne usng the SSTMS hybrd model.
165 90 Fgure 9.3a. Average pressure coeffcent on the WICS celng usng the SA DES model. Fgure 9.3b. Average pressure coeffcent on the WICS celng usng the SST DES model.
166 91 Fgure 9.3c. Average pressure coeffcent on the WICS celng usng the SST MS model. Fgure 9.4a. Sound pressure level on the WICS celng usng the SADES model.
167 9 Fgure 9.4b. Sound pressure level on the WICS celng usng the SSTDES model. Fgure 9.4c. Sound pressure level on the WICS celng usng the SSTMS model. The sound pressure level spectra at the K16 and K18 locatons for the three hybrd models are shown n Fgs The medum and fne grd spectra are n reasonable agreement wth the frst three spectral peaks for all three hybrd
168 93 models. The coarse grd spectra are not qute as good, but are not totally unacceptable. The fne grd solutons underpredct the prmary spectral peak, whle the coarse and medum grds generally overpredct the prmary spectral peak. Spectral results seem to be more senstve to grd densty than the tme averaged quanttes. Fgure 9.5a. Sound pressure level spectrum at the K16 locaton usng the SA DES model. Fgure 9.5b. Sound pressure level spectrum at the K18 locaton usng the SA DES model.
169 94 Fgure 9.6a. Sound pressure level spectrum at the K16 locaton usng the SST DES model. Fgure 9.6b. Sound pressure level spectrum at the K18 locaton usng the SST DES model.
170 95 Fgure 9.7a. Sound pressure level spectrum at the K16 locaton usng the SST MS model. Fgure 9.7b. Sound pressure level spectrum at the K18 locaton usng the SST MS model. 9.4 Delayed Detached Eddy Smulaton (DDES) Hybrd RANS/LES models were developed to provde a RANS soluton n the boundary layer along a body and to transton to LES away from the body. The
171 96 functons used to transton from RANS to LES n these hybrd models can transton too early and lead to solutons that are nether LES nor RANS. Ths usually occurs n small pockets of separated flow such as a shock nduced separaton on an arfol. If the grd s fne enough and the turbulent ntenstes are hgh enough n ths regon the hybrd model wll begn to reduce the eddy vscosty level. The reduced level may not be low enough to allow the flow to become unsteady. Recently a correcton has been proposed for the DES models to force the models to reman turbulent n the boundary layer. The new formulaton s called Delayed Detached Eddy Smulaton (DDES). The DDES 14 verson of the SA model modfes Eq. (9.1) to force the modfed wall dstance parameter to equal the dstance from the wall further nto the boundary layer. Ths delays the transton from RANS to LES. The new modfed dstance functon s gven by ~ d = d f d max, DES ( 0 d C Δ) (9.1) where f d and ([ 8r ] ) 3 (9.13) 1 tanh d r d ν t ν (9.14) Sκ d f d s desgned to be 1 n the LES regon where r d << 1, and 0 elsewhere. r d can be thought of as a rato of the turbulent length scale defned by ν t / S and the wall dstance d. Ths length scale rato wll work for smple turbulent flows, but may have dffculty n more complcated flows snce the eddy vscosty s a transported quantty whle the stran s a locally derved quantty. It s not dffcult to magne flows where a large value of eddy vscosty s transported nto a regon of relatvely small stran. Ths could cause the DDES model to transton from LES to RANS mode away from the body. A DDES verson of the SST model 15 can be derved by replacng L g n Eq. (9.3) wth a wall corrected grd length scale L gcor defned as L gcor where = L t f D ( L L ) t g (9.15)
172 97 f D 3 L t = 1 tanh 1.5 d (9.16) Here d s the dstance to the wall. The turbulent dsspaton n the turbulent knetc energy equaton s effectvely ncreased when the grd sze length scale L g s less than the turbulent length scale L t. The wall correcton ncreases the sze of L gcor near the wall and causes the turbulence model to reman n RANS mode n the near wall regon wthout regard to the local grd sze n the boundary layer. Note that the turbulent length scale used here s based on the rato of the transported turbulent quanttes k and ω only. The pressure dstrbuton for the transonc flow over a NACA wng test can be used to show the beneft of the DDES correcton. The freestream Mach number was 0.799, the angleofattack was.6 degrees, and the Reynolds number based on chord was 9x10 6. A 381x81 two dmensonal grd was used for ths study wth 301 ponts on the arfol. The pressure coeffcent on the arfol s shown n Fg All of the solutons were run n an unsteady mode, but all of the models produced steady state solutons for ths case. The SA and SST RANS model produce smlar results for ths case. The SA DES model moves the shock on the upper surface upstream and produces a weak shock on the lower surface. Ths s due to a decrease n eddy vscosty n the boundary layer. The SA DDES model produces a soluton that s closer to the SA model result. The SST DES model moves the shock well upstream on the upper surface and produces a shock on the lower surface. The SST DDES model produces a soluton smlar to the SA DDES model.
173 98 Fgure 9.8a Pressure coeffcent for a NACA 001 arfol for SA models. Fgure 9.8b Pressure coeffcent for a NACA 001 arfol for SST models. The SA DES and SA DDES model modfed wall dstance ( d ~ ) s shown n Fg. 9.9 for the separated flow regon behnd the shock at x/c=0.75 on the wng upper surface. The boundary layer edge s approxmately at z/c= Note ~ d = d that for the SA model. The SA DDES model s seen to delay the transton to LES to a much larger dstance from the wall than does the SA DES model. The effectve turbulent length scale used for transtonng the SST DES and SST DDES models at the same locaton s shown n Fg The SST DES s seen to transton very near the wall. The SST DDES modfed turbulent length scale moves the RANS to LES transton much further away from the wall.
174 99 Fgure 9.9 Wall dstance parameter for the SA, SA DES, and SA DDES turbulence models. L Fgure 9.30 Turbulent length scale for the SST, SST DES and SST DDES turbulence models.
175 Summary These smulatons ndcate a fundamental dffculty n verfcaton and valdaton for unsteady flows. As the grd s refned, smaller scale turbulent structures are resolved n the soluton. Ths process wll contnue untl the grd s refned below the Komologrov scale and all of the turbulent scales are grd resolved. The lmtng grd refnement case cannot be approached wth current computng hardware for most real world hgh Reynolds problems of nterest. Comparson of statstcal quanttes derved from the unsteady solutons can be made to assure that the mportant features of the unsteady flow have been captured wth a gven grd, but the nature of the statstcal analyss makes t dffcult to assess whether a true grd convergence has been acheved. The tme step study ndcated that about 00 tme steps per prmary sheddng cycle were requred for temporal accuracy wth the SSTMS hybrd model. Usng a larger tme step causes the prmary spectral peak to occur at a lower frequency than predcted wth the smaller tme step and than s seen n the data. The smulatons on the md and fne grd systems are n good general agreement ndcatng that a level of grd convergence can be acheved for the large turbulent scales. The turbulent length scale to grd length scale rato was greater than two n the wake regon of the cylnder and n the bay. Ths may serve as a rule of thumb for grd resoluton for hybrd model applcatons. Although the prmary sheddng frequency for the cylnder s approxmately rght for the coarse grd, the ntegrated forces and the spectral shape are qute dfferent from the md and fne grd solutons. Ths ndcates that these turbulence models can produce solutons that appear to capture the relevant physcs, but not capture the physcal detals of the flow. The coarse grd solutons were totally nonphyscal. The hybrd RANS/LES turbulence models are relatvely new and wll need to be exercsed for a wde varety of problems to determne ther accuracy before they become an accepted tool for flud modelers. They seem to offer much for unsteady flow applcatons, but ssues such as grd senstvty need to be further addressed. Hopefully more effort wll go nto these models n the near future so that they can be matured for use n everyday applcatons.
176 931 Hybrd RANS/LES Applcaton Hnts 1. These models are very new and have not been used extensvely. They should be used wth care and only when valdaton calculatons can be performed.. These models are for unsteady applcatons, and should not be used wth local tme steppng or wth other nontme accurate algorthms. 3. Turbulent flows are three dmensonal, and hence these models should be used only n 3D (see Ref. 17). 4. Because of the unsteady nature of these models, they may requre a large number of tmesteps to obtan a statstcally statonary soluton for analyss. 5. These models may be senstve to the computatonal mesh because the flter functon s nversely proportonal to the grd spacng. A ruleofthumb s that the rato of the turbulent scale to grd length scale should be greater than two n the regon of nterest when usng hybrd models. 6. As wth all unsteady applcatons, care should be taken to be sure the tme step s small enough to temporally capture the unsteady phenomena of nterest. 7. The models can produce solutons that are nether RANS or LES when appled to flows wth small pockets of flow separaton such as shock nduced separaton on an arfol. Chapter 9 References: 1. Spalart, P., Jou, WH. Strelets, M., and Allmaras, S. "Comments on the Feasblty of LES for Wngs and on a Hybrd RANS/LES Approach," Frst AFOSR Conference on DNS/LES, August 1997, C. Lu and Z. Lu edtors, Greyden Press, Columbus, Oho.. Constatnescu, G. and Squres, K., LES and DES Investgatons of the Turbulent Flow over a Sphere, AIAA , Jan Mtchell, A., Morton, S., and Forsythe, J., Analyss of Delta Wng Vortcal Substructures usng DetachedEddy Smulaton, AIAA , Jun Forsythe, J., Squres, K., Wurtzler, K., and Spalart, P, DetachedEddy Smulaton of Fghter Arcraft at Hgh Alpha, AIAA Strelets, M., "Detached Eddy Smulaton of Massvely Separated Flows," AIAA , Jan Nchols, R. and Nelson, C., Applcatons of Hybrd RANS/LES Turbulence Models, AIAA , Jan Nelson, C. and Nchols, R., Evaluaton of Hybrd RANS/LES Turbulence Models usng an LES Code, wth C. C. Nelson, AIAA , Jun. 003.
177 Nchols, R., Comparson of Hybrd RANS/LES Turbulence Models for a Crcular Cylnder and a Cavty, AIAA Journal, Vol. 44, No. 6, June 006, pp Jones, G. W., Cncotta, J. J., and Walker, R. W., Aerodynamc Forces on a Statonary and Oscllatng Crcular Cylnder at Hgh Reynolds Number, NASATRR300, October Roshko, A., Experments on the Flow Past a Crcular Cylnder at Very Hgh Reynolds Number, Journal of Flud Mechancs, Vol. 10, Part 3, May 1961, pp Schlchtng, H., Boundary Layer Theory, 7 th Edton, McGrawHll, New York, Travn, A., Shur, M., Strelets, M. and Spalart, P., DetachedEddy Smulatons Past a Crcular Cylnder, Flow, Turbulence and Combuston, Vol. 63, pp , Dx, R. E. and Bauer, R. C., Expermental and Theoretcal Study of Cavty Acoustcs, AEDCTR994, May Spalart, P. R., Deck, S., Shur, M. L., Squres, K. D., Strelets, M. Kh., and Travn, A., A New Verson of DetachedEddy Smulaton, Resstant to Ambguous Grd Denstes, Theor. Comput. Flud Dyn., Vol. 0, 006, pp Nchols, R. H., A Comparson of Hybrd RANS/LES Turbulence Models for a Generc Weapons Bay Wth and Wthout a Spoler, AIAA , Aug Maksymuk, C. M. and Pullam, T. H., Vscous Transonc Arfol Workshop Results Usng ARCD, AIAA Paper , Jan Shur, M., Spalart, P., Squres, K., Strelets, M., and Travn, A., Three Dmensonalty n ReynoldsAveraged NaverStokes Solutons around Two Dmensonal Geometres, AIAA Journal, Vol. 43, No. 6, June 005, pp
178 Wall Functon Boundary Condtons Turbulent transport models that are applcable all the way to the wall are called low Reynolds number turbulence models snce they are vald at low turbulent Reynolds numbers (ρk /(με)). Low Reynolds number models requre tght grd spacng near the wall to resolve the large gradents n velocty, temperature, and turbulent quanttes near the wall. Ths can lead to large grd requrements for 3D NaverStokes applcatons. The stablty of a numercal algorthm s lmted by the smallest cell sze n a grd, so the small cells near the wall can severely restrct the maxmum allowable tme step for a problem. The nearwall turbulence dampng terms are generally expressed n exponental functons. Ths can lead to dffcultes n convergng the numercal soluton n the near wall regon. Turbulence models that do not nclude wall correcton terms n the dfferental equatons are called hgh Reynolds number turbulence models. Hgh Reynolds number turbulence models rely on emprcally derved algebrac models of the nearwall regon of the boundary layer to provde boundary condton nformaton to the mean flow NaverStokes equatons at the frst pont off the wall. These emprcally derved relatonshps are called wall functons. Snce the hgh gradent regon near the wall s modeled wth these emprcal relatonshps, the frst pont off the all may be placed much farther away than wth low Reynolds number models. Ths reduces the number of ponts requred to dscretze a flow feld and ncreases the maxmum allowable tme step. Wall functons have been used by turbulence modelers for a number of years. Wall functons were a matter of necessty durng the years before computers began to mature, and are stll a convenent and effcent means of modelng many turbulent flows. Most wall functon mplementatons are based on emprcal relatons for ncompressble adabatc flows. Some mplementatons nclude a decoupled relaton for the temperature dstrbuton n the lower porton of the boundary layer that allows the wall functons to be appled to flows wth heat transfer at the wall. The wall functon boundary condton formulaton developed here s based on a coupled velocty and temperature boundary layer profle that s applcable to ncompressble flows, compressble flows, and flows wth heat transfer. The wall functons descrbed here are compatble wth most transport type turbulence models Theory There are sx fundamental assumptons used n the development of wall functon boundary condtons for compressble flows: 1. Analytcal expressons are avalable for the velocty and temperature profles n the lower part of the boundary layer.
179 10. Analytcal expressons are avalable for the turbulent transport varables at the frst pont off the wall. 3. The pressure s constant n the lower part of the boundary layer. u 4. The shear stress ( τ = τ w = ( μ μt ) ) s constant n the lower part of y the boundary layer. T 5. The heat transfer ( q = qw = ( k kt ) ) s constant n the lower part of y the boundary layer. 6. There are no chemcal reactons (.e. the chemstry s frozen) n the lower part of the boundary layer. Several emprcal relatonshps for the velocty n a boundary layer are avalable for use n wall functon boundary condtons. A typcal boundary layer velocty profle s shown n Fg. 3.. The smplest profle form s the adabatc ncompressble unversal lawofthewall ( y ) B 1 u = log κ (10.1) where u and y are defned n Eqs and The unversal lawofthewall s vald for the y range of about 30<y <1000. The velocty n the sublayer (0<y <10) s gven by u = y (10.) The sublayer must be blended wth the lawofthewall to cover the nterm regon. Spaldng 1 suggested a unfed form vald for the log layer and the sublayer as well as the transton regon. Ths form s gven by y = u e κ B e κ u 1 κ u ( κ u ) ( κ u ) 6 3 (10.3) κ u Note that the e B κ e term n Eq s a restatement of the ncompressble adabatc lawofthewall (Eq. 10.1). The constants κ and B are generally taken as 0.4 and 5.5 respectvely. The wall functon boundary condton can be tuned for such effects as surface roughness by tunng the κ and B constants (Ref. ). Compressblty and heat transfer effects can be ncluded by replacng the ncompressble adabatc lawofthewall term n Spaldng s equaton (Eq. 10.3) wth the outer velocty form of Whte and Chrstoph 3. Whte and Chrstoph s outer velocty s gven by
180 103 Γ Φ Γ = 0 ln sn 1 y y Q u κ β (10.4) where ( ) ( ) 3 1 / w w w w w w p Pr r B exp y 4 Q Q sn u k T q T c ru = = Γ = = = Γ = κ β β ϕ ρ μ β τ τ (10.5) The nondmensonal parameter Γ models compressblty effects and the parameter β models heat transfer effects. The outer velocty form of Whte and Chrstoph reduces to the standard lawofthewall (Eq. 10.1) for ncompressble adabatc flow. Eq can then be wrtten as ( ) ( ) = 6 u u u 1 e y u y 3 B Whte κ κ κ κ (10.6) where ( ) B Q u y Whte κ φ β κ Γ Γ = exp sn exp 1 (10.7) The temperature dstrbuton wthn the boundary layer s gven by the Crocco Busemann equaton ( ) ( ) w u u 1 T T Γ = β (10.8) For adabatc wall cases, β =0 and the CroccoBusemann equaton reduces to ( ) = w u r 1 1 T T γ (10.9)
181 104 The wall shear stress for adabatc flows s determned as follows: 1. Set the velocty at the wall to zero for nonmovng body problems or to the grd velocty for movng body problems.. Gven the velocty and temperature at the frst pont off the wall, solve Eq for the wall temperature (T w ). 3. Extrapolate the pressure from the frst pont off the wall and solve for the wall densty (ρ w ) usng the equaton of state. 4. Iteratvely solve Eq for the wall shear stress n the wall coordnate system. 5. Rotate the stress tensor nto the computatonal coordnate system and replace the vscous flux calculaton at the wall for cellcentered algorthms or at the half node for nodecentered algorthms. The wall shear stress and the heat transfer for constant temperature wall flows are determned by: 1. Set the velocty at the wall to zero for nonmovng body problems or to the grd velocty for movng body problems.. Extrapolate the pressure from the frst pont off the wall and solve for the wall densty usng the equaton of state wth the gven wall temperature. 3. Iteratvely solve Eq and Eq for the wall shear stress and heat transfer n the wall coordnate system. 4. Rotate the stress tensor and the heat transfer vector nto the computatonal coordnate system and replace the vscous flux calculaton at the wall for cellcentered algorthms or at the half node for nodecentered algorthms. One approach to ntroducng the wall functon corrected wall shear stress and heat transfer nto the calculaton of the vscous fluxes has been to calculate an effectve wall eddy vscosty so that the dscrete shear stress at the frst half pont off the wall gven by μ μ μ μ u = u 1 t1 t 1 τ 1.5 (10.10) y y1 yelds the correct value of the wall shear stress. Unfortunately ths approach ntroduces errors nto the energy equaton snce the slope of the temperature dervatve wll not be calculated correctly (.e. a separate effectve wall eddy vscosty would be requred for the temperature). Sondak and Pletcher 4 descrbe a procedure to perform a transformaton of the stresses for generalzed coordnate systems. Although ths transformaton s much more general than the effectve wall eddy vscosty approach and ntroduces no error n the energy equaton, t s also much more complex. In ther
182 105 transformaton the wall functon generated wall shear stress s assumed to be n the wall coordnate system,.e. a system algned wth velocty vector at the frst pont off the wall and the normal vector to the wall. Ths s not always the case for separated flows, but the assumpton seems to produce reasonable results even when the flow s separated. Another method of correctng the computatonal shear stress at the wall wth the wall functon value s to calculate the rato of the magntude of the computed wall shear stress to the wall functon value and then scalng the computed values by ths rato. Ths s equvalent to performng the transformaton of Sondak and Pletcher 4, and s much smpler to apply. The temperature dervatve at the wall may be treated n a smlar manner. Once the wall shear stress has been determned the turbulence transport varables must be determned at the frst pont off the wall. The eddy vscosty at the frst pont off the wall for ncompressble adabatc flow can be found usng the constant stress assumpton to be μt μ w = κ exp ( ) ( ) ( κ u ) κ B exp κ u 1 κ u (10.11) The equvalent form of the eddy vscosty for compressble flows wth heat transfer can also be found usng the constant stress assumpton. The eddy vscosty s gven by ( κu ) = (10.1) μ t ywhte μw 1 1 κexp( κb ) 1 κu μw y μw y where μ w 1 s the molecular vscosty at the frst pont off the wall and whte s y gven by ywhte κ Γ = y whte y Q [ 1 ( Γu β ) / Q ] 1 / (10.13) Eq reduces to Eq for adabatc ncompressble flows. The values of the transport model turbulence varables at the frst pont off the wall must also be defned. The values for the Spalart varable ν ~ s gven by ~ 4 ~3 μtν ν = μ C (10.14) t 3 v1 The turbulent knetc energy and turbulent dsspaton at the frst pont off the wall for kε models are gven by
183 106 k uτ = (10.15) C μ Cμ ρk ε = (10.16) μ t The turbulent knetc energy and specfc turbulent dsspaton at the frst pont off the wall for kω models are gven by 6μ w ω = (10.17) 0.075ρ w y ϖ o uτ = (10.18) C κy μ ϖ = (10.19) ϖ ϖ o ϖμ k = t (10.0) ρ The two level model for ω n Eqs was suggested by Veser, Esch, and Menter 5. Most upwnd flow solvers requre no further modfcatons than the ncluson of the corrected wall stress and heat transfer n the vscous flux calculatons. The smoothers used wth central dfference algorthms requre some addtonal modfcatons. The fourthorder smoother used to mantan numercal stablty wll produce excessve dsspaton at the wall when wall functons are employed because of the large velocty dfference between the wall and the frst pont off the wall. To prevent ths from occurrng, the smoother should be modfed as follows. The standard form of the explct fourthorder smoother n the ξ computatonal drecton s ( 4 ) [ Δ Λ ] Δt (10.1) ξ ψ where ξ ξ r Λ = Δ ( Jq) 1 ξ ξ ξ (10.) n
184 107 where Δt s the tme step, Δ ξ s the central dfference operator, and ψ (4) s the explct fourthorder smoothng coeffcent. The second dervatve term Λ ξ s normally set to zero at the wall for vscous calculatons. Ths term s also set to zero at the frst pont off the wall when wall functons are used. Ths effectvely turns the fourthorder smoothng off at the frst pont off the wall and reduces t sgnfcantly at the second pont off the wall. Once the wall shear stress s calculated usng wall functons, the y value of the frst pont off the wall s known. It s a smple matter to automatcally swtch between the wall functons and ntegratng to the wall based on the local value of y. The turbulence model chosen should contan the low Reynolds number terms requred for the model to be vald n the near wall regon f automatc swtchng s mplemented. Automatc swtchng offers advantages n complex confguratons snce computatonal grd ponts can be saved by usng the larger wall functon wall spacngs on nonessental portons of the geometry and ntegratng to the wall where hgh qualty skn frcton or heat transfer s requred. Automatc swtchng s also useful n grd sequencng or multgrd algorthms where the wall functons can provde an mproved estmate of the wall shear stresses and heat transfer on the coarse mesh solutons. Two examples of wall functon applcatons for the SpalartAllmaras and the SST turbulence models are ncluded here. Other example of wall functon applcatons for structured and unstructured grds are gven n Ref. 6 and Ref. 7 respectvely. 10. Grd Senstvty for a Flat Plate wth Adabatc Walls The predcted and theoretcal values for the skn frcton dstrbuton on an adabatc flat wth varyng ntal grd wall spacngs usng are shown n Fg for the SpalartAllmaras turbulence model and n Fg. 10. for the SST turbulence model. The y =1 results dd not use wall functons and are ncluded for reference. All the results shown here used a grdstretchng rato of 1.. The wall functon predctons of skn frcton are very smlar for both turbulence models. The models are n reasonable agreement for length Reynolds numbers (Re x ) above 1.0x10 6. Velocty profles for the two turbulence models are shown n Fg and Agan the y =1.0 results were not run wth wall functons and are ncluded for reference. The velocty profles are n reasonable agreement for all of the wall spacngs run here.
185 108 Fgure 10.1 Flat plate skn frcton predctons for the SpalartAllmaras turbulence model wth wall functons for varyng ntal wall grd pont spacngs. Fgure 10. Flat plate skn frcton predctons for the SST turbulence model wth wall functons for varyng ntal wall grd pont spacngs.
186 109 Fgure 10.3 Flat plate velocty profle predctons for the SpalartAllmaras turbulence model wth wall functons for varyng ntal wall grd pont spacngs. Fgure 10.4 Flat plate velocty profle predctons for the SST turbulence model wth wall functons for varyng ntal wall grd pont spacngs.
187 Grd Senstvty for an Axsymmetrc Bump A second applcaton of wall functons for adabatc flows wth pressure gradents s the Ames axsymmetrc bump descrbed n Chapter 5. Both the Spalart Allmaras and the SST models were run wth varyng ntal grd wall spacngs. Results for the pressure coeffcent for the SpalartAllmaras model and the SST model are shown n Fg and 10.6 respectvely. Results for the velocty dstrbuton at x/c=1.0 (the tralng edge of the bump) are shown for the Spalart Allmaras and the SST models n Fg and 10.8 respectvely. The y =1.0 results were not run wth wall functons and are ncluded for reference. For both the pressure coeffcent dstrbuton and the velocty profle at x/c=1.0 the results for each model are smlar. The results tend to start to dverge for y =00. The wall functons provde reasonable results n the shock nduce separated flow regon. Fgure 10.5 Pressure coeffcent predctons for the Ames axsymmetrc bump for the SpalartAllmaras turbulence model wth wall functons for varyng ntal wall grd pont spacngs.
188 1011 Fgure 10.6 Pressure coeffcent predctons for the Ames axsymmetrc bump for the SST turbulence model wth wall functons for varyng ntal wall grd pont spacngs. Fgure 10.7 Velocty dstrbuton at x/c=1.0 for the Ames axsymmetrc bump for the SpalartAllmaras turbulence model wth wall functons for varyng ntal wall grd pont spacngs.
189 101 Fgure 10.8 Velocty dstrbuton at x/c=1.0 for the Ames axsymmetrc bump for the SST turbulence model wth wall functons for varyng ntal wall grd pont spacngs Grd Senstvty for a Flat Plate wth Heat Transfer Calculatng heat transfer accurately can be more dffcult than predctng skn frcton. Ths can be seen n the subsonc flat plate example when the wall temperature s specfed to be 1.5 tmes the freestream temperature. The senstvty of the skn frcton and heat transfer result wth varyng ntal grd wall spacng s shown n Fg and Fg for the SpalartAllmaras model and n Fg and 10.1 for the SST model. The y =0.1 results were not run wth wall functons and are ncluded for comparson. The grd stretchng rato was fxed at 1. for these results. Both the skn frcton and heat transfer seem to be relatvely nsenstve to the wall spacng for the wall spacngs presented here.
190 1013 Fgure 10.9 The effect of wall spacng on the skn frcton on a flat plate wth heat transfer usng the SpalartAllmaras turbulence model. Fgure The effect of wall spacng on the heat transfer (Stanton number) on a flat plate usng the SpalartAllmaras turbulence model.
191 1014 Fgure The effect of wall spacng on the skn frcton on a flat plate wth heat transfer usng the SST turbulence model. Fgure 10.1 The effect of wall spacng on the heat transfer (Stanton number) on a flat plate usng the SST turbulence model.
192 1015 Profles of velocty and temperature at a length Reynolds number (Re x ) of 1x10 7 are shown n Fg and respectvely for the SpalartAllmaras model and n Fg and for the SST model. Agan the y =0.1 results were not run wth wall functons and are ncluded for comparson purposes. The wall functon results are n reasonable agreement for all wall spacngs nvestgated for both turbulence models. Fgure The effect of wall spacng on the velocty profle on a flat plate wth heat transfer usng the SpalartAllmaras turbulence model and wall functons.
193 1016 Fgure The effect of wall spacng on the temperature profle on a flat plate wth heat transfer usng the SpalartAllmaras turbulence model and wall functons. Fgure The effect of wall spacng on the velocty profle on a flat plate wth heat transfer usng the SST turbulence model and wall functons.
194 1017 Fgure The effect of wall spacng on the temperature profle on a flat plate wth heat transfer usng the SST turbulence model and wall functons Grd Senstvty for a Nozzle wth Heat Transfer Flow through a supersonc nozzle wth a constant temperature wall can serve as a test case for evaluatng the performance of the wall functon formulaton n the presence of strong pressure gradents. Detals of the geometry and boundary condtons for the convergngdvergng supersonc nozzle are gven n Chapter 5. Hghpressure ar was heated by the nternal combuston of methanol and flowed along a cooled constant area duct before enterng the nozzle. The gas could be treated as a calorcally perfect gas wth a ratoofspecfc heats (γ) of The nozzle ext Mach number was.5. The molecular vscosty and thermal conductvty were assumed to vary accordng to Sutherland s law. The grd stretchng rato used n ths study was 1.. Comparsons of the predcted and measured pressure along the nozzle are shown n Fg for the SpalartAllmaras model and n Fg for the SST model. The results for a wall spacng of y=0.1 were not run wth wall functons and are ncluded for reference. The pressure dstrbuton s seen to be nsenstve to the wall spacng. The wall heat transfer s shown n Fg for the SpalartAllmaras model and n Fg for the SST model. The SST results for heat transfer are much less senstve to the ntal wall spacng than are the SpalartAllmaras results. The heat transfer predctons are adequate for many applcatons even for an ntal wall spacng of y=100.
195 1018 Fgure The effect of wall spacng on the pressure dstrbuton for a supersonc nozzle wth heat transfer usng the SpalartAllmaras turbulence model and wall functons. Fgure The effect of wall spacng on the pressure dstrbuton for a supersonc nozzle wth heat transfer usng the SST turbulence model and wall functons.
196 1019 Fgure The effect of wall spacng on the heat transfer dstrbuton for a supersonc nozzle wth heat transfer usng the SpalartAllmaras turbulence model and wall functons. Fgure 10.0 The effect of wall spacng on the heat transfer dstrbuton for a supersonc nozzle wth heat transfer usng the SST turbulence model and wall functons.
197 100 Wall Functon Applcaton Hnts 1. The frst pont off the wall should not exceed y =100. A good rule of thumb s to place the frst pont off the wall at y =50.. Wall functons should not generally be used for calculatons requrng hghly accurate frcton drag or heat transfer. 3. Note that the computatonal doman for the turbulence varables does not nclude the frst pont off the wall when usng wall functons snce t s prescrbed by the boundary condton. 4. Post processng of terms that depend on the velocty or temperature gradent at the wall must nclude the wall functon n order to match the wall velocty or temperature gradent used n the calculaton. Chapter 10 References: 1. Whte, F. M., Vscous Flud Flow, McGrawHll, Inc., ISBN , Rod, W., Turbulence Models and Ther Applcaton n Hydraulcs A State of the Art Revew, Internatonal Assocaton for Hydraulc Research, nd Ed., Feb Whte, F. M. and Chrstoph, G. H., A Smple Analyss of Compressble Turbulent Twodmensonal Skn Frcton Under Arbtrary Condtons, AFFDLTR , Wrght Patterson AFB, OH, Feb Sondak, D. L. and Pletcher, R. H., Applcaton of Wall Functons to Generalzed Nonorthogonal Curvlnear Coordnate Systems, AIAA Journal, Vol. 33, No. 1, pp , January Veser, W., Esch, T., and Menter, F., Heat Transfer Predctons usng Advanced TwoEquaton Turbulence Models, CFX Technologcal Memorandum CFXVAL10/060, Nov Nchols, R. H., and Nelson, C. C., Wall Functon Boundary Condtons Includng Heat Transfer and Compressblty, AIAA Journal, Vol. 4, No. 6, pp , June Frnk, N. T., Assessment of an UnstructuredGrd Method for Predctng 3D Turbulent Vscous Flows, AIAA9609, Jan
198 Boundary Layer Transton Smulaton The transton of a boundary layer from lamnar to turbulent mpacts the characterstcs of a flow feld, but ts underlyng physcs has yet to be well understood. The lack of a valdated boundary layer transton smulaton capablty s a maor source of error for many applcatons, such as Reynolds number scalng of wnd tunnel results to flght, hypersonc flght, and hgh alttude turbne engne propulson. In spte of the large amount of turbulence transton data avalable, no emprcal method has been formulated that can relably predct transton for a varety of flght condtons and geometres. Though current efforts usng Drect Numercal Smulatons are nterestng and have shown promsng results, ther use n producton applcatons on real geometres s stll years away. One method for predctng the onset of transton s the use of the boundary layer stablty theory descrbed n Chapter 1. Ths methodology has a number of shortcomngs: 1) The method cannot be used for smulatng bypass transton. ) The method can only be used to predct the onset of transton and not for modelng the transton regon or the fully developed turbulent flow regon. There s no way to drectly use the results of ths method to obtan a soluton for the entre transtonal flow feld. 3) The method requres a hghly converged steadystate lamnar flow soluton to the NaverStokes equatons. Ths can be dffcult to obtan for many confguratons, especally f large scale flow separaton s present. Probably the most popular method for smulatng transton s the use of ReynoldsAveraged NaverStokes (RANS) CFD codes wth a low Reynolds number turbulence model. The RANS equatons are solved ether drectly or n conuncton wth some emprcal correlatons. The use of low Reynolds number RANS models has proven unrelable n predctng the change n skn frcton and heat transfer wthn the transton regon. No model of ths type performs satsfactorly under the nfluence of freestream turbulence ntensty and pressure gradents. It s extremely dffcult to obtan the correct locaton of the onset of transton wth ths class of models. Recently methods have been developed for smulatng boundary layer transton that utlze addtonal transport equatons for ntermttency or for a dsturbance knetc energy. These models rely heavly on emprcal functons to predct the onset and extent of the transton regon. Transport based transton models are of current nterest because they can be easly coupled to exstng transport equaton turbulence models and can be used on complex confguratons. Turbulence transton models have been broadly categorzed n two groups: models based on stablty theory and models not based on stablty theory. Models not based on stablty theory are further dvded nto two groups: models wth specfed transton onset and models wth onset predcton capablty.
199 Transton Models Based on Stablty Theory The e n method proposed by Smth and Gamberon 1 and Van Ingen, based on lnear stablty theory, s one of the most popular methods avalable for transton predcton. There are three steps n the applcaton of the e n method. The frst step nvolves the computng of the lamnar velocty and temperature profles at dfferent stream wse locatons for the gven flow. In the second step, the amplfcaton rates of the most unstable waves are calculated for each profle by usng the e n method. In the thrd step, these amplfcaton rates are used to calculate the transton locaton. There are several problems wth the e n method. The maor crtcsms that the e n method has receved s that t was developed based on the lnear stablty theory wth an assumpton that the flow s locally parallel. The value of the n factor for transton s not unversal and needs to be determned based on expermental data. Ths value vares from one wnd tunnel to the next. The lnear Parabolzed Stablty Equatons (PSE) method addresses the nonparallel effects neglected n the lnear stablty theory and assumes that the mean flow, ampltude functons and wave number (α) are dependent upon the stream wse dstance (x). A further development of the lnear PSE, known as the nonlnear PSE, ncorporates the nonlnear effects that have been neglected n the lnear stablty theory. Methods based on the stablty theory have one maor drawback  they need to track the growth of the dsturbance ampltude along a streamlne. Ths lmtaton poses a sgnfcant problem for threedmensonal flow smulatons where the streamlne drecton s not algned wth the grd. Couplng of such methods wth CFD codes requres an unrealstcally hgh grd densty to yeld the boundary layer data wth the requred level of accuracy. These methods also requre a wellconverged steadystate soluton, whch may not be obtanable for realworld problems nvolvng local flow separaton. The man advantage of these methods s that they gve the correct treatment of the surface curvature. Some dfferent technques have been employed to use these stabltybased methods more effcently 3. One method s to generate a database of the soluton of the lnear stablty equaton for dfferent velocty profles n advance. The local flow stablty can then be determned quckly based on the local velocty calculated from CFD codes. The valdty of these models s lmted to the range of velocty profles avalable n the database. 11. Transton Models Wth Specfed Transton Onset The transton regon models n ths secton are unable to predct the locaton of the transton. The transton locaton s determned from emprcal data or results from an e n computaton. The transton regon s modeled by modfyng exstng turbulence models. In Ref. 4, sx transton models were mplemented nto the commercal NaverStokes code GASP. These sx models were the Baldwn
200 113 Lomax model, the Wlcox kω model, the Schmdt and Patankar lowre kε model whch had a producton term modfed for modelng transton, the Warren, Harrs and Hasan oneequaton model, the algebrac transton model developed at ONERA/CERT, and the lnear combnaton transton model developed by Dey and Narasmha. These models were used to smulate hypersonc expermental cases that ncluded transton on a cone at Mach 6 5, a compresson ramp at Mach , and fve flared cone test cases at Mach ,8. Out of the fve flared cones used there were two wth favorable pressure gradents, two wth adverse gradents, and one wth a zero pressure gradent. 1) BaldwnLomax algebrac turbulence model: The BaldwnLomax model was used to predct the transton regon by turnng off the turbulence model for the lamnar regon by settng the eddy vscosty equal to zero and then ust turnng t on at the transton pont. It was found that n most of the cases ths model adequately predcted the peak heat transfer, but under predcted the transton length. ) Warren, Harrs and Hassan (WHH) oneequaton model: Ths model attempts to nclude the effect of second mode dsturbances n addton to the frst mode 9. The transtonal stress, ncorporatng both modes, s calculated by usng the followng formula for the eddy vscosty length scale ( l μ ) l μ t ( Γ)[ l l ] Γl = 1 (11.1) TS SM μ Where l TS and l SM are the length scale contrbutons from the frst and second modes, respectvely, whch are calculated from expermental correlatons. l t μ s the turbulent length scale, and s the ntermttency factor. The ntermttency of the flow s defned as the fracton of the tme that the flow s turbulent. The expresson for ntermttency used here was developed by Dhawan and Narasmha 10 and s gven by λ ( x ) x t Γ = 1 e (11.) Where x t s the locaton of transton onset, s the stream wse dstance between ponts where = 0.5 and = The applcablty of ths equaton has been confrmed for hypersonc flows. The model was used to smulate cases n whch the frst mode dsturbances domnate the transton process (M < 4) and cases n whch the second modes are domnant (M > 4) n Ref. 9. In all cases, the model performed satsfactorly. Ths model was later mplemented n GASP 4 and agan was found to be qute accurate. 3) Wlcox kω turbulence model 11,1 :
201 114 The low Re kω model developed by Wlcox was used to predct the transton regon. The predcton of the transton regon was obtaned by trppng the boundary layer at a gven pont by decreasng the value of the dsspaton so as to destablze the boundary layer and cause transton. The applcaton of ths model n Ref. 4 showed that t was not very easy to trp the boundary layer at the desred locaton due to the senstvty to the ntal condtons. Ths model predcted a short transton length and overpredcted the peak heat transfer for some cases. 4) Schmdt and Patankar producton term modfcatons 13 : Schmdt and Patankar have developed modfcatons to the producton term n the turbulent knetc energy (TKE) equaton of the Lam and Bremhorst kε model 14. These modfcatons lmted the producton of the knetc energy. For the use of ths model, a tral and error method was needed to make transton occur at the desred poston by varyng the nlet condtons. The results dd not compare well to the expermental cases n Ref. 4, and the method was found to be very senstve to the grd spacng near the wall. Due to the defects n ths model a few modfcatons were suggested n Ref. 4. Snce t was found that the model was dffcult to trgger turbulence transton, a spot wth hgh TKE was ntroduced nto the boundary layer. Ths spot then grew and caused transton to take place. In order to mprove the predcton of the length of the transton regon, an exponental functon was used for the maxmum allowable producton of TKE. These modfcatons mproved the results for some cases but gave worse results for other cases. 5) Algebrac transton model 15,16 : The algebrac transton regon model was developed at ONERA/CERT and s descrbed n Arnal 15,16. The form of the model of Snger et al. 17,18 was mplemented n Ref. 4. Ths model predcts transton by multplyng the eddy vscosty by a transton functon before addng t to the flud vscosty. Ths functon was found to be related to the momentum thckness growth. As a result, n test cases wth severe adverse pressure gradents, where the momentum thckness decreases, the model dd not produce transton. Theoretcally ths model should be compatble wth any turbulence model. However, t was found that ths model dd not perform well wth two equaton models. In Ref. 4 the model was used wth the BaldwnLomax model. Correctons to the calbraton of the transton functon for hgh speed flows were also suggested n Ref. 4. The new model predcted the cases tested better than the orgnal model. 6) Lnear combnaton transton model 10 : Ths model was developed by Dey and Narasmha 19 and s based on the concept that the transton flow s a combnaton of the lamnar and turbulent flow felds. The contrbuton from lamnar and turbulent values s proportoned based on the ntermttency factor developed by Dhawan and Narasmha 10 mentoned above. Ths model requres that a complete lamnar flow smulaton be run frst. Ths smulaton s followed by a turbulent one, wth the turbulent boundary layer
202 115 startng at the pont of transton. The model then uses these two solutons to generate the transtonal soluton. For example, the mean velocty (U) and skn frcton coeffcent (C f ) are gven by U C f ( Γ) U L ΓUT = 1 (11.3) ( Γ) C fl ΓC ft = 1 (11.4) In the above equatons the subscrpts L and T stand for values n the lamnar and turbulent boundary layers, respectvely. The peak heat transfer was not predcted n the test cases smulated n Ref. 4. The man dffculty n gettng accurate results wth ths model was that one of the modelng constants needs to be modfed from case to case to obtan good results. Many researchers such as Abd 0 have used the ntermttency functon from the lnear combnaton model as an algebrac transton regon functon to proporton the amount of the eddy vscosty added to the flud vscosty. The results usng ths method were found to be very smlar to the lnear combnaton model mentoned above, but there are some notceable dfferences 4. The transton length was always underpredcted. For the cases wth no pressure gradent and adverse pressure gradents, the heat transfer predcted at the end of transton and through the turbulent regon was sgnfcantly hgh. In addton to the above models, efforts have been conducted to modfy exstng turbulence models for turbulence transton. In Ref. 1, the performance of the Spalart Allmaras (SA) and the BaldwnBarth (BB) 3 oneequaton models and three twoequaton models for smulatng hypersonc transton were evaluated. The two equaton models assessed n Ref. 1 ncluded a low Re kε model wth the modfcatons of Nagano and Hshda 4, the hybrd kω model of Menter (SST) 5, and the Wlcox kω model 6. The Sanda Advanced Code for Compressble Aerothermodynamcs Research and Analyss (SACCARA) was used to evaluate these models n Ref. 1 usng two flow cases. The frst case was the flow over a flat plate at Mach 8 wth flow condtons correspondng to an alttude of 15 km, where a perfect gas was assumed. The second flow case consdered was the flow over a reentry flght vehcle at Mach 0 and an alttude of 4.4 km, where real gas effects need to be taken nto account. The method employed n Ref. 1 to specfy transton from lamnar to turbulent flow was as follows. The turbulence transport equatons were solved over the entre doman, wth a transton plane specfed by the user. Upstream of ths plane, the effectve vscosty was smply the lamnar value whereas at downstream the effectve vscosty was the sum of the lamnar and turbulent vscostes. An advantage of ths approach was that the turbulence transport equatons were solved over the whole doman, thus promotng turbulent behavor downstream of the transton plane. On the other hand, f the turbulence source terms were smply turned on after the transton plane, the turbulence model mght not transton to turbulent flow untl farther downstream, dependng on the free stream turbulence level. However, a dsadvantage of the approach was that a dscontnuty n the total
203 116 vscosty (lamnar plus turbulent) could occur at the transton plane. All models except the SA model and the low Re kε model predcted the transton at the correct locaton for the flat plate case at Mach 8. For these two models, the free stream turbulence values needed to be ncreased. All the models tested n Ref. 1 provded the correct skn frcton levels for ths case. For the reentry flght vehcle, the wall heat flux predcted by the SA model, the SST model, and the Wlcox kω model were found to be n reasonable agreement wth the expermental data. The BB and the Low Re kε model greatly overpredcted the heat flux n the turbulent regon Transton Models wth Onset Predcton Capablty These models not only smulate the characterstc of the transton regon, but also predct the onset of transton. 1) kζ turbulence/transton model: The kζ turbulence model 7 was used to study the effect of hgh dsturbance envronments (HIDE) on the transtonal smulatons carred out n conventonal hypersonc facltes. Snce HIDE cannot be descrbed by lnear stablty theory, a mnmum heat flux crteron was used to determne onset of transton. Ths s done by assumng ntal transton onset ponts and employng lnear nterpolaton for nteror ponts. After runnng a few teratons, the mnmum heat flux crteron s employed to fnd the locatons where the wall heat flux s a mnmum. The soluton s ndependent of the ntal guess as long as the ntal transton ponts are ahead of the actual locatons. Ths approach s smlar to the WHH model mentoned above. In ths case, the eddy vscosty was modfed usng the formula e ( Γ) μnt Γ t μ = 1 μ (11.5) μ nt s the eddy vscosty due to the nonturbulent fluctuatons, and can be calculated as μ = 0. 09ρ (11.6) nt kτ nt where τ nt s the nonturbulent tme scale. In addton, the dsspaton tme scale n the turbulent knetc energy equaton was also chosen as the combnaton of tme scales of turbulent and nonturbulent fluctuatons. The tme scale for calculatng the eddy vscosty and the dsspaton tme scale were derved for three dfferent transton mechansms: cross flow nstabltes, second mode nstabltes, and HIDE. These three mechansms were selected because they were beleved to be responsble for transton over 3D bodes n conventonal hypersonc wnd tunnels 8. The smulatons were performed on an ellptc cone at the Mach number of 7.93 and were compared wth expermental results. It was concluded from the results that HIDE had a hgher mpact on the transton mechansm than the other two mechansms. The man dsadvantage of ths model s that t does
204 117 not solve the nonturbulent fluctuatons usng transport equatons, whch lmts the flexblty of ths method. Also, ths model requres an ntal guess for the transton locaton. ) Papp and Dash model 9 : Papp and Dash proposed a concept analogous to that of the WHH model was used. The SSGZJ kε model developed by So et al. 30 was mplemented wth compressblty correctons for hypersonc flows. An addtonal transport equaton was solved for the nonturbulent fluctuatons. The nonturbulent fluctuatons ncluded the frst and secondmode mechansms. The locaton of the onset of transton 31 was sad to be the mnmum dstance along the surface for 1 ν lt Ret 1, where C μ s the turbulent vscosty coeffcent (0.09), ν s the flud Cμ ν knematc vscosty, and ν lt s the eddy vscosty due to the nonturbulent fluctuatons whch can be calculated as ν = C τ (11.6) lt μk l nt where τ nt s the vscosty tme scale obtaned for dfferent transton mechansms, and k l s the lamnar turbulent knetc energy, whch s obtaned from a transport equaton n the Papp and Dash model 9. Ths model was ncorporated nto a ReynoldsAveraged NaverStokes (RANS) flow solver by multplyng the turbulent eddy vscosty by the ntermttency before addng to the flud vscosty. In all cases smulated, the transton onset was properly obtaned. However, n some cases the peak n heat transfer was not reproduced correctly. Ths has been attrbuted to the algebrac nature of the ntermttency functon used. Ths s the bggest dsadvantage of the model. 3) Suzen and Huang model [3]: Ths model uses a transport equaton for the ntermttency factor. Ths equaton not only reproduces the ntermttency dstrbuton of Dhawan and Narasmha 10, but also gves a realstc varaton of the ntermttency n the crossstream drecton. The ntermttency transport equaton ncludes source terms from two dfferent models: the Steelant and Dck model 33 and the Cho and Chung model 34. The model s ncorporated nto the NaverStokes solvers by smply multplyng the eddy vscosty obtaned from the turbulence calculatons wth the ntermttency factor. The Menter s shear stress transport (SST 5 ) model was used to calculate the turbulent quanttes. The onset of transton was determned by comparng the local Reynolds number wth a transton onset Reynolds number (Re θ ) calculated usng the correlaton of Huang and Xong 35, where Re θ s a functon of the free stream turbulent ntensty and the acceleraton parameter. Ths model was tested for zero and varablepressure gradent flows wth dfferent free stream turbulence ntenstes. The numercal result showed good agreement wth the expermental data of Savll 36,37. Ths model s not a sngle
205 118 pont model snce t uses the free stream turbulence ntensty value to calculate the transton onset Reynolds number, whch requres global parameters. 4) Walters and Leylek model 38 : Ths model s based on the concept that bypass transton s caused by very hgh ampltude stream wse fluctuatons. These fluctuatons are very dfferent from turbulent fluctuatons. Mayle and Schulz 39 proposed a second knetc energy equaton to descrbe these fluctuatons. Ths knetc energy was called lamnar knetc energy k L. In the nearwall regon, the turbulent knetc energy (TKE, k T ) was splt nto smallscale energy and large scale energy. The smallscale energy (k T,s ) contrbutes drectly to the turbulence producton, and the largescale energy (k T,l ) contrbutes to the producton of lamnar knetc energy. These two energes can be calculated from the k T based on the turbulent length scale. The eddy vscostes based on both scales are calculated from the respectvescale knetc energes. For the onset of transton, a parameter s calculated from k T, the knematc vscosty and the wall dstance. When ths parameter exceeds a certan threshold, transton s assumed to start. The onset of transton s assocated wth the reducton of k L and the consequent ncrease of k T (ndcatng the breakdown of lamnar fluctuatons nto turbulence). Ths model was ncorporated nto a RANS flow solver for the calculaton of the total eddy vscosty and eddy thermal dffusvty to account for contrbutons from the smallscale as well as largescale turbulent knetc energes. For all test cases smulated, the model responded correctly to ncreases n the free stream ntensty. It yelded reasonable results for cases wth hgh pressure gradents and streamlne curvatures. Advantages of ths method are that t s very smple to mplement t nto the exstng CFD codes snce t s based on a RANS framework. Ths s a sngle pont transton model meanng that t requres only local nformaton, whch makes ths method easly applcable to unstructured and parallel computatons. The lowre kε models are typcally not calbrated for transton predcton, but provde the transton locaton as a byproduct of ther vscous sublayer formulaton. Snce ths transton model s developed based on the lowre kε model, the embedded vscous sublayer formulaton coupled wth the added transton predcton capablty cannot be calbrated ndependently. Hence, a change n the transton formulaton would affect the soluton n the fully turbulent regon. In addton, t s generally observed that these models are not flexble enough to suffcently cover the wde range of transton mechansms observed n realty 40,41. 5) Local Correlaton Based Transton Model (γ Re θ model or LCTM) 40,41 : Ths model s based on the vortcty Reynolds number. The vortcty Reynolds number s an extremely mportant parameter and s a local property and can be easly calculated n CFD codes. The maxmum value of the vortcty Reynolds number n a boundary layer profle s drectly proportonal to the momentum thckness Reynolds number. The vortcty Reynolds number s used n trggerng transton nstead of drectly usng the momentum thckness Reynolds number. Ths model solves a transport equaton for ntermttency and also a transport equaton for the Reynolds number based on the transton onset momentum
206 119 thckness. The frst transport equaton ncludes two terms that control producton. These are F length, a parameter whch controls the length of transton zone, and Re θc whch s the momentum thckness Reynolds number at the pont where the ntermttency starts to ncrease n the boundary layer. These two varables are calculated from emprcal functons of the transton momentum thckness Reynolds number ( Re θ t ). A second transport equaton s requred to solve for Re θ t and to nclude the nonlocal nfluence of the turbulence ntensty, whch vares wth the free stream turbulence knetc energy and the free stream velocty. In the case of flows wth boundary layer separaton, ths transton model s modfed so that the ntermttency s allowed to exceed the unty when the boundary layer separates. Ths event results n larger producton of knetc energy leadng to correct predcton of reattachment 41. Ths model s appled by modfyng the producton and destructon terms of the orgnal SST model usng the ntermttency. The model was valdated wth many complcated D and 3D confguratons. In all cases, good agreement wth the expermental data was obtaned. Ths model offers two man advantages: 1) t s based on local varables; ) t s very flexble and can be used for any mechansm as long as the emprcal correlaton can be formulated. However, the emprcal correlatons used wth ths model are propretary. 6) The model of Lodefer et al. 4 : Ths model s also based on the concept of pretranstonal fluctuatons smlar to the Walters and Leylek model. However, ths model uses the concept of ntermttency to descrbe the transton regon. The ntermttency equaton used was proposed by Steelant and Dck 43. The producton term of ths ntermttency equaton was modfed by multplyng t wth a new factor whch s used to locate the start of transton. Ths factor s zero before the start of transton and rapdly goes to unty after the onset pont. Smlar to LCTM, the vortcty Reynolds number s used n trggerng transton nstead of drectly usng the momentum thckness Reynolds number ( Re θ t ). Unlke LCTM, the equaton used to calculate the crtcal value of Re θ t for transton s calculated from the local free stream turbulence ntensty and not from a transport equaton. The emprcal correlaton used for Re does not nclude a pressure gradent term. The model s θ t ncorporated nto the SST model both by multplyng the eddy vscosty wth the ntermttency and by modfyng the producton terms of the k and ω equatons. These modfcatons are used to ensure that the turbulence quanttes have small nonzero values at the start of transton as n the concept of pretranstonal fluctuatons. The man dsadvantage of ths model s that t uses the free stream ntensty to determne the onset of transton, whch makes the model nonlocal unlke LCTM. 7) Model of Lan and Shyy 44 : Ths model was developed for smulaton of flow around the wng of a mcro ar vehcle (MAV). The approach used n ths model was to couple an
207 1110 ncompressble RANS solver wth the e n method. The kω model of Wlcox [11] was selected for modelng turbulence n the RANS solver. Ths couplng s accomplshed as follows. The computaton s started wth the soluton of the RANS equatons; however, the eddy vscosty s not added to the effectve vscosty. The boundary layer parameters requred for the soluton of the e n method are extracted from the NaverStokes solutons to evaluate the amplfcaton factor. Once the threshold value of the nfactor s reached, the flow s allowed to become turbulent by multplyng the eddy vscosty wth the ntermttency factor and addng t to the effectve vscosty. The ntermttency n ths case s calculated from an emprcal formula. The e n method employed n ths case s based on the assumptons that the ntal dsturbance s small and that the boundary layer s thn. 8) Model of Arthur and Atkn 3 : Ths method s based on lnear stablty theory (e n method) appled wthn a RANS framework. The overall process s as follows. The vscous flow over the confguraton of nterest s frst calculated wth an ntal guess of the transton onset locaton. A seres of pressure dstrbutons s extracted from the RANS soluton at dfferent lne ofsght postons across the span. These pressure dstrbutons are fed nto a boundary layer code to predct the boundary layer parameters wth great accuracy and fdelty. The stablty analyss, together wth some n factor crteron s conducted to yeld the transton locaton. Ths nformaton s then passed onto the RANS solver for further soluton. Ths process s contnued untl the transton locaton and the pressure dstrbuton are converged. For flows wth hgh pressure gradents, t was found that the predcted transton locaton can move upstream more easly than downstream durng the teraton process. Thus, for ths method t s essental that the ntal guess s downstream of the fnal, predcted transton locaton. The method does not have any ntermttency model to predct the nature of the regon of transton LCTM Appled to a Flat Plate The LCTM method solves two addtonal transport equatons for the ntermttency (γ) and the transton momentum thckness Reynolds number ( Re θ t ). The ntermttency equaton can be wrtten as ( ργ ) D Dt μt γ = Pγ Eγ μ (11.7) x σ f x The ntermttency equaton source term s gven by P 0.5 [ γf ] ( c γ ) = F c ρs (11.8) γ1 length a1 onset 1 e1
208 1111 where S s the stran rate. Ths term s desgned to be zero n the lamnar boundary layer and actve everywhere the local vortcty Reynolds number exceeds the local transton onset crtera. The vortcty Reynolds number s defned as Re ν ρy = Ω μ (11.9) where Ω s the vortcty magntude. The vortcty Reynolds number s assumed to be proportonal to the momentum thckness Reynolds number ( Re ) max ν Reθ = (11.10).193 F onset s defned as follows Re ν F onset 1 = (11.11).193Reθ c 4 ( max( F, F ), ) F = mn (11.1) onset onset1 onset1 3 RT F onset3 = max 1, 0 (11.13).5 F onset ( F F ) = max 3,0 (11.14) onset onset where R T s defned as (ρk)/(μω). The two functons, F length and Re θc must stll be defned. Re θc s the crtcal Reynolds number where the ntermttency frst starts to ncrease n the boundary layer. Ths occurs upstream of the transton Reynolds number, Re θt, because the turbulence must grow to a large enough level to trgger transton before any change n the lamnar profle can be seen. Hence Re θc s the locaton where turbulence starts to grow and Re θt s the locaton where the velocty profle frst departs from a lamnar profle. F length s an emprcal correlaton that controls the length of the transton regon. Both Re θc and F length are defned as functons of a second transport equaton varable The destructon/relamnarzaton source term s defned as ( c 1) a Ω turb e Re. E = c ρ γf γ (11.15) γ where θ t
209 R 4 T F = e (11.16) turb The constants for ths equaton are c e1 =1.0, c e =50.0, c a1 =.0, ca =0.06, and σ f =1.0. The transport equaton for Re θ t s D ( ρre ) θt = P σ ( μ μ ) Dt θt x θt t Re x θt (11.17) Outsde the boundary layer, the source term P θt s desgned to force the transported scalar Re θ t to match the local value of Re θt calculated from an emprcal correlaton. The source term s defned as P θt ( Re )( 1. F ) ρ = cθ t Re θt θt 0 θt (11.18) t Here t s a tme scale defned as 500μ t = (11.19) ρu The blendng functon F θt s defned as 4 y δ γ 1/ ce F = θ t mn max Fwakee, 1.0, 1.0 (11.0) 1.0 1/ ce where θ BL Reθ tμ = (11.1) ρu 15 δ BL = θ BL (11.) Ωy δ = 50 δ U BL (11.3) F wake e Reω 5 1x10 = (11.4)
210 1113 Re ω ρωy = (11.5) μ The F wake functon ensures that the wake functon s not actve n wake regons downstream of an arfol. The system of equatons s closed except for the emprcal relatonshps for F length, Re θc, and Re θt. The frst two functons are consdered propretary by the developers. The relatonshp for Re θt s gven by Re θt = Tu F Tu f ( Tu ) F( λ ), θ ( λ ), θ f Tu 1.3 Tu > 1.3 (11.6) Tu s the turbulence ntensty defned as Tu = k x100 U (11.7) where k s the turbulent knetc energy. F(λ θ ) s an emprcal relatonshp for the effect of pressure gradent gven by 3 1 ( ) ( 1.986λ ) = θ 13.66λθ λ λ θ θ Tu 35λθ ( 1 e ) e, Tu 1.5, λ 0 F θ (11.8) λθ > 0 where λ θ s defned as ρθ du λ θ = (11.9) U ds Note ths s only vald for 0.1<λ θ <0.1. For a zero pressure gradent case, λ θ =0 and F(λ θ )=1. The remanng two correlaton functons are assumed to be 4 ( Re 0.9) Reθ = Reθ (11.30) c t x θt e 1.5 F length 7 8.5x10 = (11.31) Re 3 θt The LCTM s used to predct the skn frcton on a low speed flat plate for free stream turbulence ntenstes of 0.3, 0.9, 3.3, and 6.5 percent n Fg The
211 1114 Model does an excellent ob of predctng the onset and length of the transton regon for all but the hghest free stream turbulence level. The Tu=6.5 case may need more grd refnement near the leadng edge of the plate to adequately resolve the transton process. Fgure 11.1 Skn frcton for a flat plate predcted usng LCTM. Chapter 11 References: 1. Smth, A. M. O., and Gamberon, N., Transton, Pressure Gradent and Stablty Theory, Douglas Arcraft Co. Rept. ES6388, El Segundo, Calforna, Van Ingen, J. L., A Suggested Sememprcal Method for the Calculaton of the Boundary Layer Transton Regon, Unversty of Technology, Department of Aero. Eng., Rep. UTH74, Delft, Arthur, M. T., and Atkn, C. J., Transton Modelng for Vscous Flow Predcton, AIAA , McKeel, S. A., Numercal Smulaton of the Transton Regon n Hypersonc Flow, PhD dssertaton, Blacksburg, Vrgna, Stanback, P., Fsher, M., and Wagner, R., Effects of WndTunnel Dsturbances on Hypersonc Boundary Transton, AIAA Paper 7181, Holden, M. and Chadwck, K., Studes of Lamnar, Transtonal, and Turbulent Hypersonc Flows Over Curved Compresson Surfaces, Tech. Rep , CalspanUB Research Center, Kmmel, R., 1993, Expermental Transton Zone Lengths n Pressure Gradent n Hypersonc Flow, Symposum on Transtonal and Turbulent Compressble Flows (Kral, L. and Zang, T., eds.), Vol. FED 151, pp , ASME Fluds Engneerng Conference, 1993.
FORCED CONVECTION HEAT TRANSFER IN A DOUBLE PIPE HEAT EXCHANGER
FORCED CONVECION HEA RANSFER IN A DOUBLE PIPE HEA EXCHANGER Dr. J. Mchael Doster Department of Nuclear Engneerng Box 7909 North Carolna State Unversty Ralegh, NC 276957909 Introducton he convectve heat
More informationDamage detection in composite laminates using cointap method
Damage detecton n composte lamnates usng contap method S.J. Km Korea Aerospace Research Insttute, 45 EoeunDong, YouseongGu, 35333 Daejeon, Republc of Korea yaeln@kar.re.kr 45 The contap test has the
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationwhere the coordinates are related to those in the old frame as follows.
Chapter 2  Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of noncoplanar vectors Scalar product
More information"Research Note" APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES *
Iranan Journal of Scence & Technology, Transacton B, Engneerng, ol. 30, No. B6, 789794 rnted n The Islamc Republc of Iran, 006 Shraz Unversty "Research Note" ALICATION OF CHARGE SIMULATION METHOD TO ELECTRIC
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationIntroduction. by a source term ( ) 242 / Vol. XXVIII, No. 2, AprilJune 2006 ABCM. Denise Maria V. Martinez et al
ρ ρ Dense Mara V. Martnez et al Dense Mara V. Martnez densevmartnez@yahoo.com.br Departamento de Matemátca Fundação Unversdade Federal do Ro Grande 9601900 Ro Grande, RS, Brazl Edth Beatrz C. Schettn
More informationIMPACT ANALYSIS OF A CELLULAR PHONE
4 th ASA & μeta Internatonal Conference IMPACT AALYSIS OF A CELLULAR PHOE We Lu, 2 Hongy L Bejng FEAonlne Engneerng Co.,Ltd. Bejng, Chna ABSTRACT Drop test smulaton plays an mportant role n nvestgatng
More informationCalculating the high frequency transmission line parameters of power cables
< ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager,
More informationRiskbased Fatigue Estimate of Deep Water Risers  Course Project for EM388F: Fracture Mechanics, Spring 2008
Rskbased Fatgue Estmate of Deep Water Rsers  Course Project for EM388F: Fracture Mechancs, Sprng 2008 Chen Sh Department of Cvl, Archtectural, and Envronmental Engneerng The Unversty of Texas at Austn
More informationMAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date
Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPPATBDClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller
More informationPSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 12
14 The Chsquared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationA hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm
Avalable onlne www.ocpr.com Journal of Chemcal and Pharmaceutcal Research, 2014, 6(7):18841889 Research Artcle ISSN : 09757384 CODEN(USA) : JCPRC5 A hybrd global optmzaton algorthm based on parallel
More informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationCan Auto Liability Insurance Purchases Signal Risk Attitude?
Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? ChuShu L Department of Internatonal Busness, Asa Unversty, Tawan ShengChang
More informationSimulating injection moulding of microfeatured components
Smulatng njecton mouldng of mcrofeatured components T. Tofteberg 1 * and E. Andreassen 1 1 SINTEF Materals and Chemstry, Oslo, Norway terje.tofteberg@sntef.no; erk.andreassen@sntef.no Numercal smulaton
More informationNonlinear data mapping by neural networks
Nonlnear data mappng by neural networks R.P.W. Dun Delft Unversty of Technology, Netherlands Abstract A revew s gven of the use of neural networks for nonlnear mappng of hgh dmensonal data on lower dmensonal
More informationTHE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES
The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes causeandeffect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationCommunication Networks II Contents
8 / 1  Communcaton Networs II (Görg)  www.comnets.unbremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP
More informationActuator forces in CFD: RANS and LES modeling in OpenFOAM
Home Search Collectons Journals About Contact us My IOPscence Actuator forces n CFD: RANS and LES modelng n OpenFOAM Ths content has been downloaded from IOPscence. Please scroll down to see the full text.
More informationCHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol
CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 /  Communcaton Networks II (Görg) SS20  www.comnets.unbremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More informationEffect of a spectrum of relaxation times on the capillary thinning of a filament of elastic liquid
J. NonNewtonan Flud Mech., 72 (1997) 31 53 Effect of a spectrum of relaxaton tmes on the capllary thnnng of a flament of elastc lqud V.M. Entov a, E.J. Hnch b, * a Laboratory of Appled Contnuum Mechancs,
More informationCharacterization of Assembly. Variation Analysis Methods. A Thesis. Presented to the. Department of Mechanical Engineering. Brigham Young University
Characterzaton of Assembly Varaton Analyss Methods A Thess Presented to the Department of Mechancal Engneerng Brgham Young Unversty In Partal Fulfllment of the Requrements for the Degree Master of Scence
More informationRotation Kinematics, Moment of Inertia, and Torque
Rotaton Knematcs, Moment of Inerta, and Torque Mathematcally, rotaton of a rgd body about a fxed axs s analogous to a lnear moton n one dmenson. Although the physcal quanttes nvolved n rotaton are qute
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationA frequency decomposition time domain model of broadband frequencydependent absorption: Model II
A frequenc decomposton tme doman model of broadband frequencdependent absorpton: Model II W. Chen Smula Research Laborator, P. O. Box. 134, 135 Lsaker, Norwa (1 Aprl ) (Proect collaborators: A. Bounam,
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMISP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationINVESTIGATION OF VEHICULAR USERS FAIRNESS IN CDMAHDR NETWORKS
21 22 September 2007, BULGARIA 119 Proceedngs of the Internatonal Conference on Informaton Technologes (InfoTech2007) 21 st 22 nd September 2007, Bulgara vol. 2 INVESTIGATION OF VEHICULAR USERS FAIRNESS
More informationProject Networks With MixedTime Constraints
Project Networs Wth MxedTme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa
More informationAutomated information technology for ionosphere monitoring of loworbit navigation satellite signals
Automated nformaton technology for onosphere montorng of loworbt navgaton satellte sgnals Alexander Romanov, Sergey Trusov and Alexey Romanov Federal State Untary Enterprse Russan Insttute of Space Devce
More informationCHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES
CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES In ths chapter, we wll learn how to descrbe the relatonshp between two quanttatve varables. Remember (from Chapter 2) that the terms quanttatve varable
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationSPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:
SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and
More informationCompaction of the diamond Ti 3 SiC 2 graded material by the high speed centrifugal compaction process
Archves of Materals Scence and Engneerng Volume 8 Issue November 7 Pages 67768 Internatonal Scentfc Journal publshed monthly as the organ of the Commttee of Materals Scence of the Polsh Academy of Scences
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a twostage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationCHAPTER 9 SECONDLAW ANALYSIS FOR A CONTROL VOLUME. blank
CHAPTER 9 SECONDLAW ANALYSIS FOR A CONTROL VOLUME blank SONNTAG/BORGNAKKE STUDY PROBLEM 91 9.1 An deal steam turbne A steam turbne receves 4 kg/s steam at 1 MPa 300 o C and there are two ext flows, 0.5
More informationThe Application of Fractional Brownian Motion in Option Pricing
Vol. 0, No. (05), pp. 738 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qngxn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com
More informationMolecular Dynamics Simulation of Ablation and Spallation of Gold irradiated by femtosecond laser
Molecular Dynamcs Smulaton of Ablaton and Spallaton of Gold rradated by femtosecond laser Bran J. Demaske, V.V. Zhakhovsky, Ivan I. Oleynk Department of Physcs, Unversty of South Florda Ablaton Spallaton
More informationSection 2 Introduction to Statistical Mechanics
Secton 2 Introducton to Statstcal Mechancs 2.1 Introducng entropy 2.1.1 Boltzmann s formula A very mportant thermodynamc concept s that of entropy S. Entropy s a functon of state, lke the nternal energy.
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More informationRESEARCH ON DUALSHAKER SINE VIBRATION CONTROL. Yaoqi FENG 1, Hanping QIU 1. China Academy of Space Technology (CAST) yaoqi.feng@yahoo.
ICSV4 Carns Australa 9 July, 007 RESEARCH ON DUALSHAKER SINE VIBRATION CONTROL Yaoq FENG, Hanpng QIU Dynamc Test Laboratory, BISEE Chna Academy of Space Technology (CAST) yaoq.feng@yahoo.com Abstract
More informationPassive Filters. References: Barbow (pp 265275), Hayes & Horowitz (pp 3260), Rizzoni (Chap. 6)
Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called
More information) of the Cell class is created containing information about events associated with the cell. Events are added to the Cell instance
Calbraton Method Instances of the Cell class (one nstance for each FMS cell) contan ADC raw data and methods assocated wth each partcular FMS cell. The calbraton method ncludes event selecton (Class Cell
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationForecasting the Direction and Strength of Stock Market Movement
Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract  Stock market s one of the most complcated systems
More informationSTANDING WAVE TUBE TECHNIQUES FOR MEASURING THE NORMAL INCIDENCE ABSORPTION COEFFICIENT: COMPARISON OF DIFFERENT EXPERIMENTAL SETUPS.
STADIG WAVE TUBE TECHIQUES FOR MEASURIG THE ORMAL ICIDECE ABSORPTIO COEFFICIET: COMPARISO OF DIFFERET EXPERIMETAL SETUPS. Angelo Farna (*), Patrzo Faust (**) (*) Dpart. d Ing. Industrale, Unverstà d Parma,
More informationCHAPTER 14 MORE ABOUT REGRESSION
CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp
More informationAn InterestOriented Network Evolution Mechanism for Online Communities
An InterestOrented Network Evoluton Mechansm for Onlne Communtes Cahong Sun and Xaopng Yang School of Informaton, Renmn Unversty of Chna, Bejng 100872, P.R. Chna {chsun,yang}@ruc.edu.cn Abstract. Onlne
More informationLinear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits
Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.
More informationExhaustive Regression. An Exploration of RegressionBased Data Mining Techniques Using Super Computation
Exhaustve Regresson An Exploraton of RegressonBased Data Mnng Technques Usng Super Computaton Antony Daves, Ph.D. Assocate Professor of Economcs Duquesne Unversty Pttsburgh, PA 58 Research Fellow The
More informationEE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN
EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson  3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson  6 Hrs.) Voltage
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 MultpleChoce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multplechoce questons. For each queston, only one of the answers s correct.
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More informationInequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More informationSIMULATION OF THERMAL AND CHEMICAL RELAXATION IN A POSTDISCHARGE AIR CORONA REACTOR
XVIII Internatonal Conference on Gas Dscharges and Ther Applcatons (GD 2010) Grefswald  Germany SIMULATION OF THERMAL AND CHEMICAL RELAXATION IN A POSTDISCHARGE AIR CORONA REACTOR M. Mezane, J.P. Sarrette,
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationAnalysis of Premium Liabilities for Australian Lines of Business
Summary of Analyss of Premum Labltes for Australan Lnes of Busness Emly Tao Honours Research Paper, The Unversty of Melbourne Emly Tao Acknowledgements I am grateful to the Australan Prudental Regulaton
More informationAn Empirical Study of Search Engine Advertising Effectiveness
An Emprcal Study of Search Engne Advertsng Effectveness Sanjog Msra, Smon School of Busness Unversty of Rochester Edeal Pnker, Smon School of Busness Unversty of Rochester Alan RmmKaufman, RmmKaufman
More information+ + +   This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationImplementation of Deutsch's Algorithm Using Mathcad
Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages  n "Machnes, Logc and Quantum Physcs"
More informationPortfolio Loss Distribution
Portfolo Loss Dstrbuton Rsky assets n loan ortfolo hghly llqud assets holdtomaturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment
More informationThe Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets
. The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely
More informationx f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60
BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true
More informationTime Domain simulation of PD Propagation in XLPE Cables Considering Frequency Dependent Parameters
Internatonal Journal of Smart Grd and Clean Energy Tme Doman smulaton of PD Propagaton n XLPE Cables Consderng Frequency Dependent Parameters We Zhang a, Jan He b, Ln Tan b, Xuejun Lv b, HongJe L a *
More informationInner core mantle gravitational locking and the superrotation of the inner core
Geophys. J. Int. (2010) 181, 806 817 do: 10.1111/j.1365246X.2010.04563.x Inner core mantle gravtatonal lockng and the superrotaton of the nner core Matheu Dumberry 1 and Jon Mound 2 1 Department of Physcs,
More information7.5. Present Value of an Annuity. Investigate
7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on
More informationAnalysis of Reactivity Induced Accident for Control Rods Ejection with Loss of Cooling
Analyss of Reactvty Induced Accdent for Control Rods Ejecton wth Loss of Coolng Hend Mohammed El Sayed Saad 1, Hesham Mohammed Mohammed Mansour 2 Wahab 1 1. Nuclear and Radologcal Regulatory Authorty,
More informationLaws of Electromagnetism
There are four laws of electromagnetsm: Laws of Electromagnetsm The law of BotSavart Ampere's law Force law Faraday's law magnetc feld generated by currents n wres the effect of a current on a loop of
More informationThe Analysis of Outliers in Statistical Data
THALES Project No. xxxx The Analyss of Outlers n Statstcal Data Research Team Chrysses Caron, Assocate Professor (P.I.) Vaslk Karot, Doctoral canddate Polychrons Economou, Chrstna Perrakou, Postgraduate
More informationTime Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University
Tme Seres Analyss n Studes of AGN Varablty Bradley M. Peterson The Oho State Unversty 1 Lnear Correlaton Degree to whch two parameters are lnearly correlated can be expressed n terms of the lnear correlaton
More informationForecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network
700 Proceedngs of the 8th Internatonal Conference on Innovaton & Management Forecastng the Demand of Emergency Supples: Based on the CBR Theory and BP Neural Network Fu Deqang, Lu Yun, L Changbng School
More informationAn Evaluation of the Extended Logistic, Simple Logistic, and Gompertz Models for Forecasting Short Lifecycle Products and Services
An Evaluaton of the Extended Logstc, Smple Logstc, and Gompertz Models for Forecastng Short Lfecycle Products and Servces Charles V. Trappey a,1, Hsnyng Wu b a Professor (Management Scence), Natonal Chao
More informationThe Analysis of Covariance. ERSH 8310 Keppel and Wickens Chapter 15
The Analyss of Covarance ERSH 830 Keppel and Wckens Chapter 5 Today s Class Intal Consderatons Covarance and Lnear Regresson The Lnear Regresson Equaton TheAnalyss of Covarance Assumptons Underlyng the
More informationIDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS
IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS Chrs Deeley* Last revsed: September 22, 200 * Chrs Deeley s a Senor Lecturer n the School of Accountng, Charles Sturt Unversty,
More informationFrequency Selective IQ Phase and IQ Amplitude Imbalance Adjustments for OFDM Direct Conversion Transmitters
Frequency Selectve IQ Phase and IQ Ampltude Imbalance Adjustments for OFDM Drect Converson ransmtters Edmund Coersmeer, Ernst Zelnsk Noka, Meesmannstrasse 103, 44807 Bochum, Germany edmund.coersmeer@noka.com,
More informationOnLine Fault Detection in Wind Turbine Transmission System using Adaptive Filter and Robust Statistical Features
OnLne Fault Detecton n Wnd Turbne Transmsson System usng Adaptve Flter and Robust Statstcal Features Ruoyu L Remote Dagnostcs Center SKF USA Inc. 3443 N. Sam Houston Pkwy., Houston TX 77086 Emal: ruoyu.l@skf.com
More informationJoe Pimbley, unpublished, 2005. Yield Curve Calculations
Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward
More informationLecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.
Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook
More informationErrorPropagation.nb 1. Error Propagation
ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then
More informationSection 5.4 Annuities, Present Value, and Amortization
Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today
More informationAn Overview of Computational Fluid Dynamics
An Overvew of Computatonal Flud Dynamcs Joel Ducoste Assocate Professor Department of Cvl, Constructon, and Envronmental Engneerng MBR Tranng Semnar Ghent Unversty July 15 17, 2008 Outlne CFD? What s that?
More informationStudy on CET4 Marks in China s Graded English Teaching
Study on CET4 Marks n Chna s Graded Englsh Teachng CHE We College of Foregn Studes, Shandong Insttute of Busness and Technology, P.R.Chna, 264005 Abstract: Ths paper deploys Logt model, and decomposes
More informationTopical Workshop for PhD students Adsorption and Diffusion in MOFs Institut für Nichtklassische Chemie, Germany, www.unileipzig.
Gas Separaton and Purfcaton Measurement of Breakthrough Curves Topcal Workshop for PhD students Adsorpton and Dffuson n MOFs Adsorpton on Surfaces / Separaton effects Useful features Thermodynamc effect
More informationSIMPLE LINEAR CORRELATION
SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.
More informationA machine vision approach for detecting and inspecting circular parts
A machne vson approach for detectng and nspectng crcular parts DuMng Tsa Machne Vson Lab. Department of Industral Engneerng and Management YuanZe Unversty, ChungL, Tawan, R.O.C. Emal: edmtsa@saturn.yzu.edu.tw
More informationThe Effect of Mean Stress on Damage Predictions for Spectral Loading of Fiberglass Composite Coupons 1
EWEA, Specal Topc Conference 24: The Scence of Makng Torque from the Wnd, Delft, Aprl 92, 24, pp. 546555. The Effect of Mean Stress on Damage Predctons for Spectral Loadng of Fberglass Composte Coupons
More informationThe eigenvalue derivatives of linear damped systems
Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by YeongJeu Sun Department of Electrcal Engneerng IShou Unversty Kaohsung, Tawan 840, R.O.C emal: yjsun@su.edu.tw
More informationAPPLICATION OF PROBE DATA COLLECTED VIA INFRARED BEACONS TO TRAFFIC MANEGEMENT
APPLICATION OF PROBE DATA COLLECTED VIA INFRARED BEACONS TO TRAFFIC MANEGEMENT Toshhko Oda (1), Kochro Iwaoka (2) (1), (2) Infrastructure Systems Busness Unt, Panasonc System Networks Co., Ltd. Saedocho
More informationComparison of Control Strategies for Shunt Active Power Filter under Different Load Conditions
Comparson of Control Strateges for Shunt Actve Power Flter under Dfferent Load Condtons Sanjay C. Patel 1, Tushar A. Patel 2 Lecturer, Electrcal Department, Government Polytechnc, alsad, Gujarat, Inda
More information1 Example 1: Axisaligned rectangles
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton
More informationViscosity of Solutions of Macromolecules
Vscosty of Solutons of Macromolecules When a lqud flows, whether through a tube or as the result of pourng from a vessel, layers of lqud slde over each other. The force f requred s drectly proportonal
More informationAn Enhanced SuperResolution System with Improved Image Registration, Automatic Image Selection, and Image Enhancement
An Enhanced SuperResoluton System wth Improved Image Regstraton, Automatc Image Selecton, and Image Enhancement YuChuan Kuo ( ), ChenYu Chen ( ), and ChouShann Fuh ( ) Department of Computer Scence
More informationImperial College London
F. Fang 1, C.C. Pan 1, I.M. Navon 2, M.D. Pggott 1, G.J. Gorman 1, P.A. Allson 1 and A.J.H. Goddard 1 1 Appled Modellng and Computaton Group Department of Earth Scence and Engneerng Imperal College London,
More informationTHE USE OF CFD TOOLS FOR INDOOR ENVIRONMENTAL DESIGN
Chen, Q. and Zha, Z. 24. "The use of CFD tools for ndoor envronmental desgn" Advanced Buldng Smulaton, Edted by A. Malkaw and G. Augenbroe, Spon Press, New York, pp. 94. THE USE OF CFD TOOLS FOR INDOOR
More informationCredit Limit Optimization (CLO) for Credit Cards
Credt Lmt Optmzaton (CLO) for Credt Cards Vay S. Desa CSCC IX, Ednburgh September 8, 2005 Copyrght 2003, SAS Insttute Inc. All rghts reserved. SAS Propretary Agenda Background Tradtonal approaches to credt
More information