Modeling Turbulent Flows

Transcription

1 Modelng Turbulent Flows Introductory FLUENT Tranng 2006 ANSYS, Inc. All rghts reserved.

2 2006 ANSYS, Inc. All rghts reserved. 6-2 What s Turbulence? Unsteady, rregular (aperodc) moton n whch transported quanttes (mass, momentum, scalar speces) fluctuate n tme and space Identfable swrlng patterns characterze turbulent eddes. Enhanced mxng (matter, momentum, energy, etc.) results Flud propertes and velocty exhbt random varatons Statstcal averagng results n accountable, turbulence related transport mechansms. Ths characterstc allows for turbulence modelng. Contans a wde range of turbulent eddy szes (scales spectrum). The sze/velocty of large eddes s on the order of mean flow. Large eddes derve energy from the mean flow Energy s transferred from larger eddes to smaller eddes In the smallest eddes, turbulent energy s converted to nternal energy by vscous dsspaton.

3 Is the Flow Turbulent? External Flows Re 500,000 x Re 20,000 d Internal Flows Re 2,300 d h along a surface around an obstacle where ρu L Re L = µ L = x d, d,etc., h Other factors such as free-stream turbulence, surface condtons, and dsturbances may cause transton to turbulence at lower Reynolds numbers Natural Convecton Ra 9 10 where Pr 3 β g L T Ra = ν α ν µc p Pr = = α k β g L s the Raylegh number 2006 ANSYS, Inc. All rghts reserved. 6-3 = ρ 2 C p µ k 3 T s the Prandtl number

4 2006 ANSYS, Inc. All rghts reserved. 6-4 Turbulent Flow Structures Small structures Large structures Energy Cascade Rchardson (1922)

5 2006 ANSYS, Inc. All rghts reserved. 6-5 Overvew of Computatonal Approaches Reynolds-Averaged Naver-Stokes (RANS) models Solve ensemble-averaged (or tme-averaged) Naver-Stokes equatons All turbulent length scales are modeled n RANS. The most wdely used approach for calculatng ndustral flows. Large Eddy Smulaton (LES) Solves the spatally averaged N-S equatons. Large eddes are drectly resolved, but eddes smaller than the mesh are modeled. Less expensve than DNS, but the amount of computatonal resources and efforts are stll too large for most practcal applcatons. Drect Numercal Smulaton (DNS) Theoretcally, all turbulent flows can be smulated by numercally solvng the full Naver-Stokes equatons. Resolves the whole spectrum of scales. No modelng s requred. But the cost s too prohbtve! Not practcal for ndustral flows - DNS s not avalable n Fluent. There s not yet a sngle, practcal turbulence model that can relably predct all turbulent flows wth suffcent accuracy.

6 2006 ANSYS, Inc. All rghts reserved. 6-6 Turbulence Models Avalable n FLUENT RANS based models One-Equaton Models Spalart-Allmaras Two-Equaton Models Standard k ε RNG k ε Realzable k ε Standard k ω SST k ω Reynolds Stress Model Detached Eddy Smulaton Large Eddy Smulaton Increase n Computatonal Cost Per Iteraton

7 2006 ANSYS, Inc. All rghts reserved. 6-7 RANS Modelng Tme Averagng Ensemble (tme) averagng may be used to extract the mean flow propertes from the nstantaneous ones: u ( n ( ) ) x, t = lm u ( x, t) u Instantaneous component N 1 N N = n 1 ( x, t) = u ( x, t) + u ( x t), Tme-average component Fluctuatng component u ( x,t) u ( x,t) u ( x,t) Example: Fully-Developed Turbulent Ppe Flow Velocty Profle The Reynolds-averaged momentum equatons are as follows u u p u R u ρ + k + t x = + µ k x R = ρu u The Reynolds stresses are addtonal unknowns ntroduced by the averagng procedure, hence they must be modeled (related to the averaged flow quanttes) n order to close the system of governng equatons. (Reynolds stress tensor)

8 2006 ANSYS, Inc. All rghts reserved. 6-8 The Closure Problem The RANS models can be closed n one of the followng ways (1) Eddy Vscosty Models (va the Boussnesq hypothess) R = ρu u = µ T u u + Boussnesq hypothess Reynolds stresses are modeled usng an eddy (or turbulent) vscosty, µ T. The hypothess s reasonable for smple turbulent shear flows: boundary layers, round ets, mxng layers, channel flows, etc. 2 3 µ T u k k δ 2 ρk δ 3 (2) Reynolds-Stress Models (va transport equatons for Reynolds stresses) Modelng s stll requred for many terms n the transport equatons. RSM s more advantageous n complex 3D turbulent flows wth large streamlne curvature and swrl, but the model s more complex, computatonally ntensve, more dffcult to converge than eddy vscosty models.

9 2006 ANSYS, Inc. All rghts reserved. 6-9 Calculatng Turbulent Vscosty Based on dmensonal analyss, µ T can be determned from a turbulence tme scale (or velocty scale) and a length scale. Turbulent knetc energy [L 2 /T 2 ] Turbulence dsspaton rate [L 2 /T 3 ] Specfc dsspaton rate [1/T] Each turbulence model calculates µ T dfferently. Spalart-Allmaras: Solves a transport equaton for a modfed turbulent vscosty. Standard k ε, RNG k ε, Realzable k ε Solves transport equatons for k and ε. Standard k ω, SST k ω Solves transport equatons for k and ω. k = u u 2 ε = ν u ( u + u ) ω = ε k µ µ µ T T T = f ~ ( ν) ρk = f ε ρk = f ω 2

10 2006 ANSYS, Inc. All rghts reserved The Spalart-Allmaras Model Spalart-Allmaras s a low-cost RANS model solvng a transport equaton for a modfed eddy vscosty. When n modfed form, the eddy vscosty s easy to resolve near the wall. Manly ntended for aerodynamc/turbomachnery applcatons wth mld separaton, such as supersonc/transonc flows over arfols, boundary-layer flows, etc. Embodes a relatvely new class of one-equaton models where t s not necessary to calculate a length scale related to the local shear layer thckness. Desgned specfcally for aerospace applcatons nvolvng wall-bounded flows. Has been shown to gve good results for boundary layers subected to adverse pressure gradents. Ganng popularty for turbomachnery applcatons. Ths model s stll relatvely new. No clam s made regardng ts applcablty to all types of complex engneerng flows. Cannot be reled upon to predct the decay of homogeneous, sotropc turbulence.

11 2006 ANSYS, Inc. All rghts reserved The k ε Turbulence Models Standard k ε (SKE) model The most wdely-used engneerng turbulence model for ndustral applcatons Robust and reasonably accurate Contans submodels for compressblty, buoyancy, combuston, etc. Lmtatons The ε equaton contans a term whch cannot be calculated at the wall. Therefore, wall functons must be used. Generally performs poorly for flows wth strong separaton, large streamlne curvature, and large pressure gradent. Renormalzaton group (RNG) k ε model Constants n the k ε equatons are derved usng renormalzaton group theory. Contans the followng submodels Dfferental vscosty model to account for low Re effects Analytcally derved algebrac formula for turbulent Prandtl / Schmdt number Swrl modfcaton Performs better than SKE for more complex shear flows, and flows wth hgh stran rates, swrl, and separaton.

12 2006 ANSYS, Inc. All rghts reserved The k ε Turbulence Models Realzable k ε (RKE) model The term realzable means that the model satsfes certan mathematcal constrants on the Reynolds stresses, consstent wth the physcs of turbulent flows. Postvty of normal stresses: u u > Schwarz nequalty for Reynolds shear stresses: ( u u ) u u Nether the standard k ε model nor the RNG k ε model s realzable. Benefts: More accurately predcts the spreadng rate of both planar and round ets. Also lkely to provde superor performance for flows nvolvng rotaton, boundary layers under strong adverse pressure gradents, separaton, and recrculaton.

13 2006 ANSYS, Inc. All rghts reserved The k ω Turbulence Models The k ω famly of turbulence models have ganed popularty manly because: The model equatons do not contan terms whch are undefned at the wall,.e. they can be ntegrated to the wall wthout usng wall functons. They are accurate and robust for a wde range of boundary layer flows wth pressure gradent. FLUENT offers two varetes of k ω models. Standard k ω (SKW) model Most wdely adopted n the aerospace and turbo-machnery communtes. Several sub-models/optons of k ω: compressblty effects, transtonal flows and shear-flow correctons. Shear Stress Transport k ω (SSTKW) model (Menter, 1994) The SST k ω model uses a blendng functon to gradually transton from the standard k ω model near the wall to a hgh Reynolds number verson of the k ε model n the outer porton of the boundary layer. Contans a modfed turbulent vscosty formulaton to account for the transport effects of the prncpal turbulent shear stress.

14 2006 ANSYS, Inc. All rghts reserved Large Eddy Smulatonn Large Eddy Smulaton (LES) LES has been most successful for hgh-end applcatons where the RANS models fal to meet the needs. For example: Combuston Mxng External Aerodynamcs (flows around bluff bodes) Implementatons n FLUENT: Subgrd scale (SGS) turbulent models: Smagornsky-Llly model Wall-Adaptng Local Eddy-Vscosty (WALE) Dynamc Smagornsky-Llly model Dynamc Knetc Energy Transport Detached eddy smulaton (DES) model LES s applcable to all combuston models n FLUENT Basc statstcal tools are avalable: Tme averaged and RMS values of soluton varables, bult-n fast Fourer transform (FFT). Before runnng LES, consult gudelnes n the Best Practces For LES (contanng advce for meshng, subgrd model, numercs, BCs, and more)

15 2006 ANSYS, Inc. All rghts reserved Law of the Wall and Near-Wall Treatments Dmensonless velocty data from a wde varety of turbulent duct and boundary-layer flows are shown here: τw U τ = ρ + yu u τ y = u + = ν U Wall shear stress where y s the normal dstance from the wall τ For equlbrum turbulent boundary layers, walladacent cells n the log-law regon have known velocty and wall shear stress data

16 2006 ANSYS, Inc. All rghts reserved Wall Boundary Condtons The k ε famly and RSM models are not vald n the near-wall regon, whereas Spalart-Allmaras and k ω models are vald all the way to the wall (provded the mesh s suffcently fne). To work around ths, we can take one of two approaches. Wall Functon Approach Standard wall functon method s to take advantage of the fact that (for equlbrum turbulent boundary layers), a log-law correlaton can supply the requred wall boundary condtons (as llustrated n the prevous slde). Non-equlbrum wall functon method attempts to mprove the results for flows wth hgher pressure gradents, separatons, reattachment and stagnaton. Smlar laws are also constructed for the energy and speces equatons. Beneft: Wall functons allow the use of a relatvely coarse mesh n the near-wall regon. Enhanced Wall Treatment Opton Combnes a blended law-of-the wall and a two-layer zonal model. Sutable for low-re flows or flows wth complex near-wall phenomena Turbulence models are modfed for the nner layer. Generally requres a fne near-wall mesh capable of resolvng the vscous sublayer (at least 10 cells wthn the nner layer ) outer layer nner layer

17 2006 ANSYS, Inc. All rghts reserved Placement of The Frst Grd Pont For standard or non-equlbrum wall functons, each wall-adacent cell + centrod should be located wthn the log-law layer For enhanced wall treatment (EWT), each wall-adacent cell centrod should + be located wthn the vscous sublayer 1 EWT can automatcally accommodate cells placed n the log-law layer How to estmate the sze of wall-adacent cells before creatng the grd: y + y + p uτ y p ν p = y p = ν uτ y p τ ρ y p w U τ = = U e C 2 f The skn frcton coeffcent can be estmated from emprcal correlatons: Flat plate: C C f Duct: 1 4 Re f Re L 2 Dh Use postprocessng tools (XY plot or contour plot) to to double check the nearwall grd placement after the flow pattern has been establshed.

18 2006 ANSYS, Inc. All rghts reserved Near-Wall Modelng: Recommended Strategy Use standard or non-equlbrum wall functons for most hgh Reynolds number applcatons (Re > 10 6 ) for whch you cannot afford to resolve the vscous sublayer. There s lttle gan from resolvng the vscous sublayer. The choce of core turbulence model s more mportant. Use non-equlbrum wall functons for mldly separatng, reattachng, or mpngng flows. You may consder usng enhanced wall treatment f: The characterstc Reynolds number s low or f near wall characterstcs need to be resolved. The physcs and near-wall mesh of the case s such that y + s lkely to vary sgnfcantly over a wde porton of the wall regon. Try to make the mesh ether coarse or fne enough to avod placng the wall-adacent cells n the buffer layer (5 < y + < 30).

19 2006 ANSYS, Inc. All rghts reserved Inlet and Outlet Boundary Condtons When turbulent flow enters a doman at nlets or outlets (backflow), boundary condtons for k, ε, ω and/or must be specfed, dependng on whch turbulence model has been selected u u Four methods for drectly or ndrectly specfyng turbulence parameters: Explctly nput k, ε, ω,or Ths s the only method that allows for profle defnton. See user s gude for the correct scalng relatonshps among them. Turbulence ntensty and length scale Length scale s related to sze of large eddes that contan most of energy. For boundary layer flows: l 0.4 δ 99 For flows downstream of grd: l openng sze Turbulence ntensty and hydraulc dameter Ideally suted for nternal (duct and ppe) flows Turbulence ntensty and turbulent vscosty rato For external flows: 1 < µ t /µ < 10 u 1 2 k Turbulence ntensty depends on upstream condtons: I = < 20% U U 3 Stochastc nlet boundary condtons for LES and RANS can be generated by usng spectral syntheszer or vortex method

20 2006 ANSYS, Inc. All rghts reserved GUI for Turbulence Models Defne Models Vscous Invscd, Lamnar, or Turbulent Turbulence Model Optons Defne Boundary Condtons Near Wall Treatments Addtonal Optons

21 2006 ANSYS, Inc. All rghts reserved Example #1 Turbulent Flow Past a Blunt Plate Reynolds-Stress model ( exact ) Standard k-ε model Contour plots of turbulent knetc energy (TKE) The Standard k ε model s known to gve spurously large TKE on the font face of the plate

22 2006 ANSYS, Inc. All rghts reserved Example #1 Turbulent Flow Past a Blunt Plate Predcted separaton bubble: Skn Frcton Coeffcent Standard k-ε (SKE) Realzable k-ε (RKE) SKE severely underpredcts the sze of the separaton bubble, whle RKE model predcts the sze exactly. Expermentally observed reattachment pont s at x/d = 4.7

23 2006 ANSYS, Inc. All rghts reserved Example #2 Turbulent Flow n a Cyclone 0.1 m 40,000-cell hexahedral mesh Hgh-order upwnd scheme was used m Computed usng SKE, RNG, RKE and RSM (second moment closure) models wth the standard wall functons 0.97 m U n = 20 m/s Represents hghly swrlng flows (W max = 1.8 U n ) 0.2 m

24 2006 ANSYS, Inc. All rghts reserved Example #2 Turbulent Flow n a Cyclone Tangental velocty profle predctons at 0.41 m below the vortex fnder

25 2006 ANSYS, Inc. All rghts reserved Example #3 Flow Past a Square Cylnder (LES) Dynamc Smagornsky Drag Coeffcent 2.28 Strouhal Number (Re H = 22,000) Dynamc TKE Exp.(Lyn et al., 1992) Iso-Contours of Instantaneous Vortcty Magntude Tme-averaged streamwse velocty along the wake centerlne C L spectrum

26 2006 ANSYS, Inc. All rghts reserved Example #3 Flow Past a Square Cylnder (LES) Streamwse mean velocty along the wake centerlne Streamwse normal stress along the wake centerlne

27 2006 ANSYS, Inc. All rghts reserved Summary: Turbulence Modelng Gudelnes Successful turbulence modelng requres engneerng udgment of: Flow physcs Computer resources avalable Proect requrements Accuracy Turnaround tme Near-wall treatments Modelng procedure Calculate characterstc Re and determne whether the flow s turbulent. Estmate wall-adacent cell centrod y + before generatng the mesh. Begn wth SKE (standard k-ε) and change to RKE, RNG, SKW, SST or V 2 F f needed. Check the tables n the appendx as a startng gude. Use RSM for hghly swrlng, 3-D, rotatng flows. Use wall functons for wall boundary condtons except for the low-re flows and/or flows wth complex near-wall physcs.

28 Appendx 2006 ANSYS, Inc. All rghts reserved.

29 2006 ANSYS, Inc. All rghts reserved RANS Turbulence Model Descrptons Model Spalart Allmaras Standard k ε RNG k ε Realzable k ε Standard k ω SST k ω Reynolds Stress Descrpton A sngle transport equaton model solvng drectly for a modfed turbulent vscosty. Desgned specfcally for aerospace applcatons nvolvng wall-bounded flows on a fne near-wall mesh. FLUENT s mplementaton allows the use of coarser meshes. Opton to nclude stran rate n k producton term mproves predctons of vortcal flows. The baselne two-transport-equaton model solvng for k and ε. Ths s the default k ε model. Coeffcents are emprcally derved; vald for fully turbulent flows only. Optons to account for vscous heatng, buoyancy, and compressblty are shared wth other k ε models. A varant of the standard k ε model. Equatons and coeffcents are analytcally derved. Sgnfcant changes n the ε equaton mproves the ablty to model hghly straned flows. Addtonal optons ad n predctng swrlng and low Reynolds number flows. A varant of the standard k ε model. Its realzablty stems from changes that allow certan mathematcal constrants to be obeyed whch ultmately mproves the performance of ths model. A two-transport-equaton model solvng for k and ω, the specfc dsspaton rate (ε / k) based on Wlcox (1998). Ths s the default k ω model. Demonstrates superor performance for wall-bounded and low Reynolds number flows. Shows potental for predctng transton. Optons account for transtonal, free shear, and compressble flows. A varant of the standard k ω model. Combnes the orgnal Wlcox model for use near walls and the standard k ε model away from walls usng a blendng functon. Also lmts turbulent vscosty to guarantee that τ T ~ k. The transton and shearng optons are borrowed from standard k ω. No opton to nclude compressblty. Reynolds stresses are solved drectly usng transport equatons, avodng sotropc vscosty assumpton of other models. Use for hghly swrlng flows. Quadratc pressure-stran opton mproves performance for many basc shear flows.

30 2006 ANSYS, Inc. All rghts reserved RANS Turbulence Model Behavor and Usage Model Spalart-Allmaras Standard k ε RNG k ε Realzable k ε Standard k ω SST k ω Reynolds Stress Behavor and Usage Economcal for large meshes. Performs poorly for 3D flows, free shear flows, flows wth strong separaton. Sutable for mldly complex (quas-2d) external/nternal flows and boundary layer flows under pressure gradent (e.g. arfols, wngs, arplane fuselages, mssles, shp hulls). Robust. Wdely used despte the known lmtatons of the model. Performs poorly for complex flows nvolvng severe pressure gradent, separaton, strong streamlne curvature. Sutable for ntal teratons, ntal screenng of alternatve desgns, and parametrc studes. Sutable for complex shear flows nvolvng rapd stran, moderate swrl, vortces, and locally transtonal flows (e.g. boundary layer separaton, massve separaton, and vortex sheddng behnd bluff bodes, stall n wde-angle dffusers, room ventlaton). Offers largely the same benefts and has smlar applcatons as RNG. Possbly more accurate and easer to converge than RNG. Superor performance for wall-bounded boundary layer, free shear, and low Reynolds number flows. Sutable for complex boundary layer flows under adverse pressure gradent and separaton (external aerodynamcs and turbomachnery). Can be used for transtonal flows (though tends to predct early transton). Separaton s typcally predcted to be excessve and early. Offers smlar benefts as standard k ω. Dependency on wall dstance makes ths less sutable for free shear flows. Physcally the most sound RANS model. Avods sotropc eddy vscosty assumpton. More CPU tme and memory requred. Tougher to converge due to close couplng of equatons. Sutable for complex 3D flows wth strong streamlne curvature, strong swrl/rotaton (e.g. curved duct, rotatng flow passages, swrl combustors wth very large nlet swrl, cyclones).

31 2006 ANSYS, Inc. All rghts reserved The Spalart-Allmaras Turbulence Model A low-cost RANS model solvng an equaton for the modfed eddy vscosty, ~ ν 2 D ~ ν ν ~ ν ( ) ν ~ = G µ + ρ ~ ν + C ρ b2 Yν + S~ ν Dt Eddy vscosty s obtaned from µ = ρ ~ ν The varaton of very near the wall s easer to resolve than k and ε. t f v 1 ~ ν 3 ( ν / ν) 3 3 ( ~ ν / ν) + C Manly ntended for aerodynamc/turbomachnery applcatons wth mld separaton, such as supersonc/transonc flows over arfols, boundary-layer flows, etc. f v1 = ~ v1

32 2006 ANSYS, Inc. All rghts reserved RANS Models - Standard k ε (SKE) Model Transport equatons for k and ε: where C D Dt D Dt ( ρε) k t ( ρk) = µ + + G ρε = µ µ + σ t ε µ = k ε µ σ k ε 0.09, Cε 1 = 1.44, Cε2 = 1.92, σ = 1.0, σ = C e1 ε k G k k ρc ε2 ε k 2 The most wdely-used engneerng turbulence model for ndustral applcatons Robust and reasonably accurate; t has many sub-models for compressblty, buoyancy, and combuston, etc. Performs poorly for flows wth strong separaton, large streamlne curvature, and hgh pressure gradent.

33 2006 ANSYS, Inc. All rghts reserved ρ Dk Dt Dω ω ρ = α τ Dt k RANS Models k ω Models = τ u k µ t = α ρ ω µ t k ρβ f k ω + µ + β σk u µ ω 2 t ρβ f ω + µ + β σω specfc dsspaton rate, ω ε ω k 1 τ Belongs to the general 2-equaton EVM famly. Fluent 6 supports the standard k ω model by Wlcox (1998), and Menter s SST k ω model (1994). k ω models have ganed popularty manly because: Can be ntegrated to the wall wthout usng any dampng functons Accurate and robust for a wde range of boundary layer flows wth pressure gradent Most wdely adopted n the aerospace and turbo-machnery communtes. Several sub-models/optons of k ω: compressblty effects, transtonal flows and shear-flow correctons.

34 2006 ANSYS, Inc. All rghts reserved RANS Models Reynolds Stress Model (RSM) t ( ) ( ) T ρu u + ρuk u u = P + F + D + Φ ε k Stress producton Rotaton producton Attempts to address the defcences of the EVM. Turbulent Dsspaton dffuson Pressure Stran Modelng requred for these terms RSM s the most physcally sound model: ansotropy, hstory effects and transport of Reynolds stresses are drectly accounted for. RSM requres substantally more modelng for the governng equatons (the pressure-stran s most crtcal and dffcult one among them). But RSM s more costly and dffcult to converge than the 2-equaton models. Most sutable for complex 3-D flows wth strong streamlne curvature, swrl and rotaton.

35 2006 ANSYS, Inc. All rghts reserved Standard Wall Functons Standard Wall Functons Momentum boundary condton based on Launder-Spauldng law-of-thewall: y for y < y 1/4 1/2 1/4 1/2 ν U Cµ k where U P P ρc µ k = P y U = 1 2 y = ln( E y ) for y > yν Uτ µ κ Smlar wall functons apply for energy and speces. Addtonal formulas account for k, ε, and ρu u. Less relable when flow departs from condtons assumed n ther dervaton. Severe p or hghly non-equlbrum near-wall flows, hgh transpraton or body forces, low Re or hghly 3D flows p

36 2006 ANSYS, Inc. All rghts reserved Standard Wall Functons Energy T = 1 Prt ln κ ρc 1 4 µ 2q k 2 2 [ Pr U + ( Pr Pr ) U ] ρprc µ P 1 ( Ey ) + P + Pr y + ln( E y ) 1 2 t P t k 2q c U κ for y for y < > y y t t P = 9.24 Pr Pr t exp Pr Pr t Speces Y Sc Sc y 1 ln κ ( E y ) = t for y > yc + Pc for y < y c

37 2006 ANSYS, Inc. All rghts reserved Non-Equlbrum Wall Functons Non-equlbrum wall functons Standard wall functons are modfed to account for stronger pressure gradents and non-equlbrum flows. Useful for mldly separatng, reattachng, or mpngng flows. Less relable for hgh transpraton or body forces, low Re or hghly 3D flows. The standard and non-equlbrum wall functons are optons for all of the k ε models as well as the Reynolds stress model.

38 2006 ANSYS, Inc. All rghts reserved Enhanced Wall Treatment Enhanced wall functons Momentum boundary condton based on + Γ Γ a blended law-of-the-wall (Kader). u = e ulam + uturb e Smlar blended wall functons apply for energy, speces, and ω. Kader s form for blendng allows for ncorporaton of addtonal physcs. Pressure gradent effects Thermal (ncludng compressblty) effects Two-layer zonal model A blended two-layer model s used to determne near-wall ε feld. Doman s dvded nto vscosty-affected (near-wall) regon and turbulent core regon. ρ y k Based on the wall-dstance-based turbulent Reynolds number: Re y µ Zonng s dynamc and soluton adaptve. Hgh Re turbulence model used n outer layer. Smple turbulence model used n nner layer. Solutons for ε and µ T n each regon are blended: λε ( µ t ) outer + ( 1 λε )( µ t ) nner The Enhanced Wall Treatment opton s avalable for the k ε and RSM models (EWT s the sole treatment for Spalart Allmaras and k ω models).

39 2006 ANSYS, Inc. All rghts reserved Two-Layer Zonal Model The two regons are demarcated on a cell-by-cell bass: Turbulent core regon Re y > 200 Vscosty affected regon Re < 200 y y s the dstance to the nearest wall Zonng s dynamc and soluton adaptve Re y ρ y µ k

40 2006 ANSYS, Inc. All rghts reserved Turbulent Heat Transfer The Reynolds averagng of the energy equaton produces an addtonal term u t Analogous to the Reynolds stresses, ths s the turbulent heat flux term. An sotropc turbulent dffusvty s assumed: u t = ν Turbulent dffusvty s usually related to eddy vscosty va a turbulent Prandtl number (modfable by the users): Pr t ν = ν t T T T Smlar treatment s applcable to other turbulent scalar transport equatons.

41 2006 ANSYS, Inc. All rghts reserved Menter s SST k ω Model Background Many people, ncludng Menter (1994), have noted that: The k ω model has many good attrbutes and performs much better than k ε models for boundary layer flows. Wlcox orgnal k ω model s overly senstve to the free stream value of ω, whle the k ε model s not prone to such problem. Most two-equaton models, ncludng k ε models, over predct turbulent stresses n wake (velocty-defect) regons, whch leads to poor performance of the models for boundary layers under adverse pressure gradent and separated flows.

42 2006 ANSYS, Inc. All rghts reserved Menter s SST k ω Model Man Components The SST k ω model conssts of Zonal (blended) k ω / k ε equatons (to address tem 1 and 2 n the prevous slde) Clppng of turbulent vscosty so that turbulent stress stay wthn what s dctated by the structural smlarty constant. (Bradshaw, 1967) - addresses tem 3 n the prevous slde Outer layer (wake and outward) Inner layer (sub-layer, log-layer) k ω model transformed from standard k ε model ε = k l 3 2 Modfed Wlcox ε k ω model Wlcox orgnal k-ω model Wall

43 2006 ANSYS, Inc. All rghts reserved Menter s SST k ω Model Blended equatons The resultng blended equatons are: ρ Dω Dt = γ ν t τ Dk u * ρ = τ β k ρω + Dt u βρω 2 + µ µ + σ T ω µ T µ + σk ω + ρ k ( 1 F ) σ 2 1 ω2 1 ω k ω ( F ) φ ; φ = β, σ, σ γ φ = F φ +, ω k Wall

44 2006 ANSYS, Inc. All rghts reserved u ( x, t) = u ( x, t) + u ( x t), Large Eddy Smulaton (LES) N-S equaton u uu 1 p + = t ρ + u ν Flter, Instantaneous component Resolved Scale Subgrd Scale Fltered N-S equaton u u u + = 1 t ρ Spectrum of turbulent eddes n the Naver-Stokes equatons s fltered: The flter s a functon of grd sze Eddes smaller than the grd sze are removed and modeled by a subgrd scale (SGS) model. Larger eddes are drectly solved numercally by the fltered transent N-S equaton p + u ν τ τ = ρ ( u u u u ) (Subgrd scale Turbulent stress)

45 2006 ANSYS, Inc. All rghts reserved LES n FLUENT LES has been most successful for hgh-end applcatons where the RANS models fal to meet the needs. For example: Combuston Mxng External Aerodynamcs (flows around bluff bodes) Implementatons n FLUENT: Sub-grd scale (SGS) turbulent models: Smagornsky-Llly model WALE model Dynamc Smagornsky-Llly model Dynamc knetc energy transport model Detached eddy smulaton (DES) model LES s applcable to all combuston models n FLUENT Basc statstcal tools are avalable: Tme averaged and RMS values of soluton varables, bult-n fast Fourer transform (FFT). Before runnng LES, consult gudelnes n the Best Practces For LES (contanng advce for meshng, subgrd model, numercs, BCs, and more)

46 2006 ANSYS, Inc. All rghts reserved Detached Eddy Smulaton (DES) Motvaton For hgh-re wall bounded flows, LES becomes prohbtvely expensve to resolve the near-wall regon Usng RANS n near-wall regons would sgnfcantly mtgate the mesh resoluton requrement RANS/LES hybrd model based on the Spalart-Allmaras turbulence model: D ~ ν = C Dt b1 ~ S ~ ν C d w1 f w ~ ν d σ ( µ + ρ ν) +... One-equaton SGS turbulence model In equlbrum, t reduces to an algebrac model. DES s a practcal alternatve to LES for hgh-reynolds number flows n external aerodynamc applcatons ~ ν ( d, C ) = mn w DES ~ ν ~

47 2006 ANSYS, Inc. All rghts reserved V 2 F Turbulence Model A model developed by Paul Durbn s group at Stanford Unversty. Durbn suggests that the wall-normal fluctuatons v 2 are responsble for the near-wall dampng of the eddy vscosty Requres two addtonal transport equatons for v 2 and a relaxaton functon f to be solved together wth k and ε. Eddy vscosty model s ν ~ v 2 T nstead of ν T ~ k T T V 2 F shows promsng results for many 3D, low Re, boundary layer flows. For example, mproved predctons for heat transfer n et mpngement and separated flows, where k ε models fal. But V 2 F s stll an eddy vscosty model and thus the same lmtatons stll apply. V 2 F s an embedded add-on functonalty n FLUENT whch requres a separate lcense from Cascade Technologes (

48 2006 ANSYS, Inc. All rghts reserved Stochastc Inlet Velocty Boundary Condton It s often mportant to specfy realstc turbulent nflow velocty BC for accurate predcton of the downstream flow: Dfferent types of nlet boundary condtons for LES No perturbatons Turbulent fluctuatons are not present at the nlet. Vortex method Turbulence s mmcked by usng the velocty feld nduced by many quas-random pont-vortces on the nlet surface. The vortex method uses turbulence quanttes as nput values (smlar to those used for RANS-based models). Spectral syntheszer u ( x, t) = u ( x) + u ( x t), Instantaneous Tme Averaged Coherent + Random Able to synthesze ansotropc, nhomogeneous turbulence from RANS results (k ε, k ω, and RSM felds). Can be used for RANS/LES zonal hybrd approach

49 2006 ANSYS, Inc. All rghts reserved Intal Velocty Feld for LES/DES Intal condton for velocty feld does not affect statstcally statonary solutons However, startng LES wth a realstc turbulent velocty feld can substantally shorten the smulaton tme to get to statstcally statonary state The spectral syntheszer can be used to supermpose turbulent velocty on top of the mean velocty feld Uses steady-state RANS (k ε, k ω, RSM, etc.) solutons as nputs to the spectral syntheszer Accessble va a TUI command: /solve/ntalze/nt-nstantaneous-vel