An Overview of Computational Fluid Dynamics

Transcription

1 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

2 Outlne CFD? What s that? Modelng flud flow and assocated processes Geometry and grd selecton Equatons Numercal methods Soluton technques multphase Eample Secondary clarfer

3 Computatonal Flud Dynamcs (CFD) CFD s the scence of determnng a soluton to flud flow through space and tme. CFD models nclude a descrpton of the flow geometry, a set of coupled dfferental equatons descrbng the physcs and chemstry of the flow, boundary and ntal condtons, and a structured mesh of ponts at whch these equatons are solved The equatons of moton are solved by a fnte dfference, fnte element, or fnte volume technque.

4 Role of CFD Researchers, engneers, and educators can use CFD as a Desgn Tool Evaluate desgn alternatves Retroft alternatves Troubleshootng Tool Eplan strange flow phenomena Eplan poor process performance A tool to better understand the process dynamcs

5 Impact of alternatve ppng confguraton on UV reactor performance

6 Mcrobal transport n UV Reactor Desgn A Desgn B

7 Chemcal Mng n a flow through Reactor nvolvng chlorne and ammona

8 Result of Flow Feld and Chlorne decay

9 Modelng Grease nterceptor

10 Velocty Modelng Grease Interceptors Ol concentraton

11 Modelng Spatal Varaton n Floc Sze n Reactors Aal Flow Impeller Radal Flow Impeller

12 Common and tradtonal uses of CFD? Aerospace Bomedcal F18 Store Separaton Temperature and natural convecton currents n the eye followng laser heatng. Automotve

13 Common and tradtonal uses of CFD? Chemcal Processng Polymerzaton reactor vessel - predcton of flow separaton and resdence tme effects. Hydraulcs HVAC Streamlnes for workstaton ventlaton

14 Ducoste s Areas of Research Interest Dsnfecton Chemcal dsnfectants UV Bologcal nutrent removal Anaerobc dgeston Secondary Clarfers Chemcal mng Flocculaton DAF Bosystems/bonformatcs Advance odaton processes Fat, ol, grease, removal and roots control Wetlands Boremedaton

15 Modelng flud flow and assocated processes Modelng ncludes: Geometry, doman, grd specfcaton Governng equatons Intal and boundary condtons Selecton of models for dfferent applcatons

16 Geometry, doman, grd specfcaton Smple geometres can be easly created by few geometrc parameters (e.g. crcular ppe) Comple geometres must be created by the partal dfferental equatons or mportng the database of the geometry (e.g. Multlamp UV reactor, comple MBR confguraton) nto commercal software Doman: sze and shape Typcal approaches Geometry appromaton CAD/CAE ntegraton: use of ndustry standards such as Parasold, ACIS, STEP, or IGES, etc. The three coordnates: Cartesan system (,y,z), cylndrcal system (r, θ, z), and sphercal system (r, θ, Φ)

17 Modelng (coordnates) z Cartesan z Cylndrcal z Sphercal (,y,z) (r,θ,z) φ (r,θ,φ) z y θ r y θ r y Coordnates should be chosen for a better resoluton of the geometry (e.g. cylndrcal for crcular ppe).

18 Grd Specfcaton Grds can ether be structured or unstructured. Depends upon type of dscretzaton scheme and applcaton Scheme Fnte dfferences: structured Fnte volume or fnte element: structured or unstructured Applcaton Unstructured grds useful for comple geometres Unstructured grds permt automatc adaptve refnement based on the pressure gradent, or regons nterested (FLUENT, Comsol, AEA, Flow 3D, etc.) The grd selecton/dstrbuton can mpact the accuracy of the flow feld/turbulence and scalar varable feld.

19 unstructured j j structured

20 Grd Specfcaton It s mportant to make sure that the flow doman s properly dscretzed wth an approprate grd dstrbuton. The accuracy of a soluton and ts assocated cost due to computng tme and hardware are dependent n part on the grd densty. Some cost savngs can be acheved by optmzng the grd or cell dstrbuton wthn the flow doman (.e., makng the cell sze non-unform). For non-unform grd densty, the cells are fner n areas where large varatons or gradents occur from pont to pont and coarser n regons wth relatvely small varatons n the solved varables.

21 Grd Specfcaton The user must stll decde on the grd densty that s needed to capture the specfc process and flow characterstcs and produce a stable numercal (also known as a grd ndependent) soluton. A stable numercal soluton s acheved when solved varables at all dscrete cell locatons do not sgnfcantly change (.e., less than 0.1 percent) wth ncreasng grd densty. The user should utlze some knd of grd checker to safeguard aganst dstorted cells (.e., long thn cells). Grd cell dstorton wll lead to a poorly converged soluton.

22 Governng equatons Governng equatons (2D n Cartesan coordnates) 1. Naver-Stokes Equatons 2. Contnuty equaton 3. Equaton of state g y u u p y u v u u t u ρ μ ρ ρ ρ = g y y v v y p y v v v u t v ρ μ ρ ρ ρ = ( ) ( ) ( ) = z w y v u t ρ ρ ρ ρ p ρrt = Convecton Pressure gradent Vscous terms

23 = 0 U + = j j j j j u u U v P U U ρ 1 ε σ + + = j j j t k t U U U v k v k U k C U U U v k C v U j j j t ε ε ε σ ε ε + + = j j j t j k U U v u u δ = = + c u C C v C U t C j j j j j j ρ ρ ρ ) ( Pr ) ( T T C C u c μ ρ ) ( Pr =

24 Governng equatons Based on the physcs of the fluds phenomena, CFD can be dstngushed nto dfferent categores usng dfferent crtera Vscous vs. nvscd Eternal flow or nternal flow Turbulent vs. lamnar Incompressble vs. compressble Sngle- vs. mult-phase Thermal/densty effects Free-surface flow and surface tenson Chemcal reactons and combuston and so on.

25 Turbulent Flow Above a certan range of Reynolds No., a Lamnar Flow wll become unstable and turn turbulent Most of the envronmental flows we deal wth are turbulent Turbulence Characterstcs : General randomness n pressure and velocty Turbulence n 3-D Turbulence contans a range of Eddy Szes or scales of moton Eddy Flud Partcles movng laterally and longtudnally Eddy dea helps us thnk about stretchng, turnng,.

26 Turbulent Flow (Eddes Acton) a b c d

27 Reynolds Averagng In turbulent flows, there are velocty fluctuatons n all drectons u 2 u 3 u 1 u 1, u 2, u 3 Total Velocty

28 Reynolds Averagng Snce u1, u2, u3 are all non-zero we have to keep all three Naver Stokes Equatons. Represent Velocty as : u u u = U 1 = U = U u ' 1 + u + u ' 2 ' 3 where U mean veloctes and u ' Random Fluctuatng Veloctes Lets take a 1 T t0 + T t0 u { 1 dt tme average of = 1 T t0 + T t0 U dt + 1 u 1 t0 + T 1 ' u dt T 1 t u1= U1 ' u1= 0 Smlary u ' 2 = u ' 3 = 0

29 Reynolds Averagng The eample above demonstrates that tme averagng may reduce the complety of our equatons Reynolds Average Contnuty Equatons : = = + + = T t t T t t T t t dt u U dt u U dt u U u Substtutng u u ) ( T 1 ) ( T 1 ) ( T 1 0 and tme averagng:, u, 0 u u u Assumes 0 ' ' '

30 Reynolds Stresses The Reynolds average of Naver Stokes : problem. How many unknowns? How many equatons? a s Ths ) u ρu ( U μ P U ρu t U ρ ρ ρ ρ u U u Reynold Stresses ' j ' j j j 2 k k ' ' = + + = + =

31 Turbulence Closure Problem Most Common Approach : Eddy Vscosty τ Does unts U t How do we descrbe v ν ν j t = ρu + U ' k u ' j match? U = ρν Property of Property of k t = U 1 ρ j P t Flow Flud where? ν t + (ν + ν = Turbulent t 2 U ) j j eddy vscosty Second-moment (more sophstcated, accurate, and substantally more costly n terms of computng)

32 Turbulence Closure Problem Most Common Approach : Eddy Vscosty three Eddy vscosty model categores are zero-equaton, one-equaton, and two-equaton models most wdely used are standard k-ε, RNG k-ε and Chen Km k-ε, and Realzable k-ε Second-moment (more sophstcated, accurate, and substantally more costly n terms of computng) the turbulent flues of momentum are not descrbed by smple consttutve relatonshps, but are now descrbed by ther own transport equatons

33 Two Equaton Turbulent Model The standard k-ε s the most popular two-equaton turbulence model and avalable n every commercal software ε ν σ ν + + = + j j j t k t U U U k k U t k k c U U U k c U t j j j t e t ε ν ε ε σ ν ε ε + + = + ε ν μ 2 k t = C The standard k-ε s the most popular two-equaton turbulence model and avalable n every commercal software. Model constants are: = 0.09 μ C 0 σ k = 1. = 1.3 e σ 44 c 1 = 1. c 2 = 1.92

34 Two Equaton Turbulent Model The RNG k- ε s derved from mathematcal methods employng comple space-tme Fourer transforms where the turbulence beng modeled s governed by the long-tme large scale behavor of the system The energy equaton contans an addtonal producton term. Ths new term changes the value of C 1 n the standard k- ε turbulence model and reduces the level of dsspaton through these addtonal terms Cμη (1 η / η ) 1+ βη 3 * 0 1/ 2 c 2 = c2 + η = ( 2E E ) 3 j j σ = 0.72 = k σ Cμ = c 1 = c 2 = β = η 0 = e k ε

35 Two Equaton Turbulent Model The Chen Km k- ε model ntroduces an addtonal tme scale term to mprove the dynamc response of the standard k-ε model (Chen and Km, 1987). The addtonal source term s ntroduced nto the ε equaton. Ths etra term represents the energy transfer rate from large scale to smallscale turbulence controlled by the producton range tme scale and the dsspaton range tme scale. S = ε c 4 k ( ) 2 E j E j The model constants for Chen Km k-ε are: c 4 =.25 e = 1.15 σ k = σ c 1 = c 2 = 1. 9

36 Two Equaton Turbulent Model The Realzable k-ε ncorporates a tme-scale realzablty (T t ) and an addtonal source term (E) n the ε equaton for mproved performance n flows contanng averse pressure gradents (.e., flow separaton) (Goldberg et al., 1998). The revsed energy dsspaton equaton s as follows: = + t j j j t e t T E k c U U U k c U t ε ν ε ε σ ν ε ε ε ν μ μ 2 k f C t = ( ) ε ξ k T t 1 ma 1, = 2 2 τ νε ξ C k = E T t V A E ε ρ Ψ = ( ) ( ) 4 1/ 2 1/, ma νε k V = = Ψ 0, ma j j k τ ε τ k = { } 1 ma 1, 1 1 = ξ μ μ t t R R A e e f νε 2 k R t =

37 Two Equaton Turbulent Model The values of the constants for the realzable k-ε model are: C μ = 0.09 σ k = 1.0 c 1 = 1.44 A E = 0.3 C τ = 2 σ e = 1.3 c 2 = 1.92 A μ = 0.01 The constants used n these k-ε models are the default values typcally used n commercal CFD software. It s possble to fne-tune the model constants to acheve better agreement between model predctons and epermental results. However, snce the velocty and turbulence data s typcally not avalable for most reactors at the dfferent operatng condtons, ths method of tweakng results s dscouraged and default values are recommended.

38 Turbulence Model Selecton, Advantages and Dsadvantages Model Advantages Dsadvantages Two-Equaton (standard k-ε, RNG k-ε, Chen Km k-ε and Realzable k-ε) Smplest turbulence models that only requre ntal/boundary condtons Most wdely tested models that have performed reasonably well for many types of flow condtons Have been found to poorly perform for the followng cases: some unconfned flows flows wth curved boundary layers or swrlng flows rotatng flows fully developed flows n noncrcular ducts Second moment (RSTM/ASTM) Most general of tradtonal turbulence models that only requre ntal/boundary condtons More accurate representaton of Reynolds stresses that have better characterze the turbulent flow propertes n wall jets flows wth curved boundary layers or swrlng flows rotatng flows fully developed flows n noncrcular ducts Sgnfcantly hgher computng cost Has not been wdely tested compared to the two equaton models Stll has flow condtons that t has been found to behave poorly: as-symmetrc jets unconfned recrculatng flows

39 Turbulent Flow measurements wthn a UV Reactor (Model Comparsons) 2 flow measurement locatons were performed usng DPIV

40 Turbulent Flow measurements wthn a UV Reactor (Velocty Profle)

41 Turbulent Flow measurements wthn a UV Reactor (Knetc energy Profle)

42 Boundary Condtons y o Slp-free: u=0,v=? Slp-free: u=?,v=0 No-slp walls: u=0,v=0 Inlet r o v=0, dp/dr=0,du/dr=0 Outlet

43 Boundary Condtons Boundary condtons n turbulence modelng nvolve specfcatons for nlet and outlet condtons as well as wall boundary condtons. For nlet condtons, the average mean velocty normal to the nlet plane must be specfed. All tangental veloctes are set to zero. The turbulent knetc energy and energy dsspaton rate nlet condtons are defned as follows: k nlet = (I U) 2 ε nlet = k nlet 1.5 /(0.1 D) where I s the turbulence ntensty whose value ranges from , U s the normal average velocty at the nlet, and D s a characterstc dmenson. U s calculated as the flow rate dvded by the cross sectonal area of the nlet plane. D s typcally taken as the nlet ppe dameter.

44 Boundary Condtons For the outlet condtons: Gradents of all varables are zero n the flow drecton wth the ecepton of the pressure. The pressure s set to the eternal pressure or zero. The outlet boundary should be placed far from any flow obstructons so that the flow occurs n one drecton and no recrculatng flow regons are produced at part of the outlet boundary. If a recrculatng regon does develop at the outlet boundary, the outlet regon should be placed further downstream by etendng the flow doman wth a straght secton.

45 Boundary Condtons For the wall boundary condtons: No slp condton (.e., veloctes equal to zero) s appled to all sold surfaces. At very small dstances near the sold wall, a vscous sublayer ests followed by an ntermedate layer and turbulent core. In the vscous sub-layer, the flow s nfluenced by vscous forces and does not depend on free stream turbulent parameters. The velocty n ths sub-layer only depends on the dstance normal to the wall, flud densty, vscosty, and the wall shear stress.

46 Boundary Condtons For the wall boundary condtons: The ntermedate sub-layer s brdged by utlzng emprcal wall functons to provde near-wall boundary condtons for the mean-flow and turbulence transport equatons. The purpose of these emprcal functons s to connect the wall shear stress to the dependent varables at the near-wall grd node. Ths grd node must le outsde ths sub-layer and resde n the fully-turbulent zone. There are two types of wall functon provded n commercal CFD codes: a) equlbrum log-law wall functons and non-equlbrum log- law wall functons.

47 Boundary Condtons For the wall boundary condtons: The non-equlbrum log- law wall functons should be used when the turbulent transport of heat, and also speces, at a reattachment pont s requred. The equlbrum wall functons are those approprate to a near wall layer n local equlbrum. The velocty, turbulent knetc energy, and energy dsspaton rate n the equlbrum wall functons are as follows: u + + / uτ = Ln( Ey ) /κ k = u τ / C μ ε = Cμ k /( κy )

48 Boundary Condtons For the wall boundary condtons: u+ s the absolute value of the resultant velocty parallel to the wall at the frst grd node, u t s the resultant frcton velocty ( u τ = τ w / ρ ), Y s the normal dstance of the frst grd pont from the wall, y + + s the dmensonless wall dstance ( y = uτ Y /υ ), C m s a constant based on the two-equaton turbulence model selecton, κ s the von Karman constant and E s a roughness parameter. κ = 0.41 and E = 8.6, whch s approprate for smooth walls.

49 Boundary Condtons For the wall boundary condtons: These wall functons are known as the wall logarthmc law and should only be used when the y+ value range between 30 and 500. Devaton from ths y+ value range may cause poor predcton of the pressure drop through the reactor. If the y+ value falls outsde ths range, grd refnement must be performed to correct for these devated values. In some commercal CFD codes, the refnement procedure can occur automatcally durng the model eecuton.

50 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA The soluton technque and convergence crtera are essental components of the numercal process to ensure that a stable converged soluton has been acheved. There are many equatons that must be solved n order to determne the local value of the specfc varable. In addton to beng numerous, these sets of equatons are often non-lnear and strongly coupled. The soluton procedure nvolves an teratve technque, where the purpose s to reduce the mbalance between the left and rght hand sdes of every equaton so that the magntude of the dfference neglgble.

51 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA The method for solvng the equatons descrbng the target varables wll be based on the type of commercal CFD code and the method used to dscretze these varables (.e., fnte element, fnte dfference, and fnte volume). FDM-smplest and requres structured mesh FEM-smlar to FD but requres nterpolaton functons FVM-FD method drectly assocated wth conservaton laws Focus on solvers assocated wth fnte volume technque snce t s used by several commercal CFD codes There are several teratve solvers that are avalable n commercal CFD codes. These solvers can be dvded nto two groups: coupled solvers and sequental or segregated solvers.

52 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA For the coupled solver approach, dscretzed equatons are placed nto one large global matr equaton system that wll be solved wth a specfc numercal approach. For the segregated approach, no global matr s bult. As an alternatve, the dscretzed equatons for each flow varable are assembled nto smaller equaton systems where they are solved teratvely. In the coupled solver category, methods that are typcally found n commercal codes nclude but are not lmted to the followng: Jacob, Gauss Sedel, Successve Over-relaaton (SOR), and Stone.

53 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA The Jacob method s consdered the smplest teratve approach that solves the system of equatons from a pont by pont method. The Jacob technque has been found to lead to slow convergence due to a reduced lnk between grd cells from a slow propagaton of change through the flow doman. Ths slow propagaton can be useful when dfferent varables are tghtly coupled and the value of solved varable n a cell s more dependent on other varables n the same cell than on the value of the varable n neghborng cells. However, the Jacob method s not recommended for 3-D problems.

54 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA The Gauss Sedel approach s smlar to the Jacob method but utlzes an updated teratve value for a specfc varable as soon as t becomes avalable. It s twce as fast as the Jacob method due to a reducton n the storage for unknowns. SOR dffers from the Gauss Sedel approach by multplyng the change n the varable from successve teratons wth a constant called an over-relaaton factor. It reverts back to the Gauss Sedel approach when the over-relaaton constant s one. Determnng the correct value for ths over-relaaton constant s mportant to the rate of convergence. An optmzed value can be determned but not wthout sgnfcant computatonal cost. SOR s consdered the most wdely used method for solvng non-lnear sets of smultaneous equatons.

55 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA The Stone solver (also known as the strongly mplct procedure) s desgned to solve a system of algebrac equatons that were developed from the dscretzaton of partal dfferental equatons as n flud mechancs problems. The man dea behnd Stone s approach s that the convergence can be mproved by the addton of cell values at the corners n the teratve soluton process. The Stone approach usually converges n a smaller number of teratons than SOR mentoned earler.

56 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA For ncompressble flud flow we are n the busness of solvng: 0 u u u = + + j j 2 t k k U ) ν (ν P ρ 1 U U t U + + = + If the flow s compressble: The contnuty equaton can be used to compute densty. Temperature follows from the enthalpy equaton. Pressure can then be calculated from the equaton of state p=p(ρ,t). What to do wth the pressure feld whch s a scalar quantty wth no other equaton to compute ts value

57 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA So-called pressure-velocty couplng algorthms are used to derve equatons for the pressure from the momentum equatons and the contnuty equaton. The most commonly used algorthm s the SIMPLE (Sem- Implct Method for Pressure-Lnked Equatons). An algebrac equaton for the pressure correcton p s derved, n a form smlar to the equatons derved for the convecton-dffuson equatons: a P p' = a p' + b' Each teraton, the pressure feld s updated by applyng the pressure correcton. The source term b s the contnuty mbalance. The other coeffcents depend on the mesh and the flow feld. nb nb

58 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA SIMPLE: It s based on the premse that flud flows from regons wth hgh pressure to low pressure. Start wth an ntal pressure feld. Look at a cell. If contnuty s not satsfed because there s more mass flowng nto that cell than out of the cell, the pressure n that cell compared to the neghborng cells must be too low. Thus the pressure n that cell must be ncreased relatve to the neghborng cells. The reverse s true for cells where more mass flows out than n. Repeat ths process teratvely for all cells.

59 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA SIMPLE s the default algorthm n most commercal fnte volume codes. Improved versons are: SIMPLER (SIMPLE Revsed). SIMPLEC (SIMPLE Consstent). PISO (Pressure Implct wth Splttng of Operators). All these algorthms can speed up convergence because they allow for the use of larger underrelaaton factors than SIMPLE. The dfferences are n speed and stablty.

60 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA The segregated solver approach conssts of decouplng the nonlnear equatons descrbng velocty, pressure, and turbulence usng ether a pressure projecton (PP) or pressure correcton (PC) technque PC s essentally the same as SIMPLE, PP s smlar to SIMPLER, Of the two methods, the PP segregated approach was the most effcent when evaluated based on CPU tme and storage. Typcally, the segregated solver needs less CPU storage and sgnfcantly less computaton tme per teraton.

61 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA The conjugate gradent (CG) approach s based on the dea that the soluton to a system of non-lnear equatons can be determned n a smultaneous manner as oppose to a sequental manner as done n the Jacob, SOR, or Stone approach. It s a method used tradtonally to mnmze a functon usng the functon s gradent and can be consdered a type of segregated solver. The CG approach may produce more rapd convergence than Jacob, SOR, or Stone approaches.

62 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA The selecton of the most approprate solver wll depend on the followng tems: 1) sze of the problem, 2) amount of computng resources (.e., avalable RAM), 3) problem physcs, and 4) specfc program capabltes. For large three dmensonal problems, solvers based on the segregated approach should be used due to ts reduced computng resource requrement.

63 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA In convertng the equatons of moton nto algebrac equatons, dscretzaton schemes must be used to descrbe a varable n one cell as a functon of the varable n nearby cells. The accuracy and stablty of the fnal soluton wll depend n part on the selecton of the dscretzaton scheme. There are several dscretzaton methods used by commercal CFD codes. These dscretzaton methods dffer n how the varable s calculated on the cell surface. The dscretzaton schemes can be dvded nto order of accuracy (.e., 1st, 2nd, or hgher). 1st and 2nd order accurate dscretzaton methods are called upwnd-dfferencng scheme (UDS) and central dfferencng scheme (CDS).

64 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA Both UDS and CDS have been shown to be numercally stable wth CDS dsplayng only a hgher degree of accuracy than UDS. UDS has been shown to suffer from numercal dffuson (.e., addtonal transport due only to numercal formulaton and not flud flow) whle CDS s only approprate for low cell Peclet (Pe) number condtons (.e., Pe < 2). When dealng wth 1st and 2nd order dscretzaton schemes, the value of the Pe number becomes mportant n determnng the nfluence of upstream computatonal cells on the current cell. At hgher Pe values, the CDS calculaton becomes unstable. Some CFD codes utlze a Hybrd scheme, whch uses CDS for cells that satsfy the Pe number restrcton and uses UDS for cells where Pe 2.

65 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA The two approaches used to reduce numercal dffuson nclude refnng the grd densty or utlzng hgher order dscretzaton schemes. Whle refnng the grd densty may be a smple soluton, t s generally not recommended snce sgnfcant refnement would be necessary to acheve small reducton n numercal dffuson. The preferred approach to reducng numercal dffuson s to apply hgher order dscretzaton schemes.

66 Non-Lnear Schemes Scheme B(r) Notes SMART ma(0,mn(2r, 0.75r+0.25, 4)) Gaskell & Lau [1988]): bounded QUICK, pecewse lnear H-QUICK 2 (r+ r )/(r+3) Waterson & Deconnck [1995]; harmonc based on QUICK, smooth UMIST ma( 0, mn(2r, r, r, 2) ) CHARM r(3r+1)/(r+1) 2 for r > 0; B(r) = 0. for r <= 0 Len & Leschzner [1994]; bounded QUICK, pecewse lnear Zhou [1995]; bounded QUICK, smooth MUSCL ma( 0, mn( 2r, r, 2) ) van Leer [1979]; bounded Fromm Van-Leer harmonc (r+ r )/(r+1) bounded Fromm OSPRE 3 (r 2 +r)/{2.(r 2 +r+1)} Waterson & Deconnck [1995]; bounded Fromm van Albada [1982] (r 2 +r)/(r 2 +1) van Albada [1982]; bounded Fromm Superbee ma( 0, mn(2r,1), mn(r,2) ) Roe [1986], Hrsch [1990] Mnmod ma( 0, mn(r, 1) ) Roe [1986], Hrsch [1990] H-CUS 1.5 (r+ r )/(r+2) Waterson & Deconnck [1995]; bounded CUS Koren ma( 0, mn( 2r, 2r/3+1/3, 2) ) Koren [1993]; bounded CUS

67 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA The non-lnear schemes n the prevous table are used along wth Equaton 1 below, whch calculates the cell-face value of the convected varable. φ f = φ c + 0.5B(r) (φ c - φ u ) φf s the varable across the control-volume face f, φc s the varable value at the cell center, φu represents the upstream value, and φd represents the downstream value. B(r) s termed a lmter functon, and the gradent rato r, whch s defned as: r = (φ d - φ c )/( φ c -φ u )

68 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA The lnear schemes can be compactly epressed as follows: UDS upwnd dfference B(r) = 0 CDS central dfference B(r) = r B(r) = 0.5*{ (1+K)*r + (1-K) } where LUS lnear-upwnd K = -1 QUICK quadratc upwnd K = 0.5 FROMM Fromm's upwnd scheme K = 0 CUS cubc upwnd scheme K =

69 Performance of Schemes Smulaton of a scalar varable downstream of a pont source n a stream of unform velocty as shown. The two ponts to eamne are: Does the band spread unduly? Do the values le wthn the range from 10.5 to 0?

70 Eact Upwnd DS Central DS Quck

71 LUS FROMM CUS

72 SMART KOREN HQUICK Quck

73 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA Numercal convergence s also mportant for the stablty of the model and conssts of reducng the error assocated wth performng guesses for the varables n the flow doman. The convergence of the numercal soluton can be based on two requrements: the magntude of the resduals the dfference n the values for solved varables between successve teratons.

74 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA Resduals are essentally mbalances caused by the guesses made at each teraton. In order to understand the concept of resduals, an eample s presented n terms of steady, snglephase flow. The speces transport equaton (constant densty, ncompressble flow) s gven by: c c + ( uc ) = D + S t Here c s the concentraton of the chemcal speces and D s the dffuson coeffcent. S s a source term. We wll dscretze ths equaton (convert t to a solveable algebrac form) for the smple flow feld shown on the rght, assumng steady state condtons. c W c N c P c E c S

75 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA The balance over the control volume s gven by: Auc A u c + Av c Avc = e e e w w w n n n s s s dc dc dc dc DA DA + DA DA + S e w n s p d e d w dy dy n s Ths contans values at the faces, whch need to be determned from nterpolaton from the values at the cell centers. j,y,v,,u c N A n,c n c W c P A e,c e A w,c w A s,c s c S c E Notaton A, A, A, A : areas of the faces c w n e s, c, c, c : concentratons at the faces w n e s c, c, c, c : concentratons at the cell centers W N E S u, u, u, u, v, v, v, v : veloctes at the faces w n e s w n e s u, u, u, u, v, v, v, v : veloctes at the cell centers W N E S W N E S S P : source n cell P D: dffuson coeffcent

76 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA The smplest way to determne the values at the faces s by usng frst order upwnd dfferencng. Here, let s assume that the value at the face s equal to the value n the center of the cell upstream of the face. Usng that method results n: Au c A u c + A v c Av c = DA ( c c )/ δ e P P w W W n P P s S S e E P e DA ( c c )/ δ + DA ( c c )/ δ y DA ( c c )/ δ y + S w P W w n N P n s P S s P Ths equaton can then be rearranged to provde an epresson for the concentraton at the center of cell P as a functon of the concentratons n the surroundng cells, the flow feld, and the grd.

77 Rearrangng the prevous equaton results n: c ( A v + Au + DA / δ + DA / δ y + DA / δ + DA / δ y ) = c ( A u + DA / δ ) + P n P e P w w n p e e s s W w W w w Ths equaton can now be smplfed to: c ( DA / δ y ) + N n n c ( DA / δ ) + E e e c ( Av + DA / δ y ) + S S s S s s P ac = a c + ac + ac + ac + b P P W W N N E E S S = anbcnb nb + b Here nb refers to the neghborng cells. The coeffcents a nb and b wll be dfferent for every cell n the doman at every teraton. The speces concentraton feld can be calculated by recalculatng c P teratvely

78 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA ap and an are the convecton-dffuson coeffcents obtaned from the UDS: φp s the cell-average value of φ stored at the cell center; and the summaton s over the adjacent nodes n. The resdual Rp at a specfc locaton s then computed as follows R p = a p φ p - (a n φ n ) - S p - B p B p s the deferred-correcton source term. The goal of the teraton process s to mnmze the value of R p wth successve teratons. The process of convergence conssts of summng the absolute resdual sources over the whole soluton doman as Resdual = R p

79 SOLUTION TECHNIQUE AND CONVERGENCE CRITERIA Overall Procedure: (put convergence resdual plot here) Spot Values Resdual error

80 Multphase Flow PHOENICS

81 Multphase Flow Multphase flow descrbes Flow wth dfferent phases(.e. sold, lqud or gas). Flow wth dfferent chemcal substances but same phase (.e. lqudlqud lke ol-water). Prmary and secondary phases One of the phases s consdered contnuous (prmary) and others (secondary) are consdered to be dspersed wthn the contnuous phase. A dameter has to be assgned for each secondary phase to calculate ts nteracton (drag) wth the prmary phase (ecept for VOF model). Dlute phase vs. Dense phase; Refers to the volume fracton of secondary phase(s)

82 Multphase Flow Multphase flow can be dvded nto the followng regmes: Partcle flow: Dscrete sold partcles n a contnuous flud Bubbly flow: Dscrete gaseous or flud bubbles n a contnuous flud Droplet flow: Dscrete flud droplets n a contnuous gas Slug flow: Large bubbles (nearly fllng cross-secton) n a contnuous flud Stratfed/free-surface flow: Immscble fluds separated by a clearly-defned nterface

83 Lagrangan Multphase Model Good for modelng partcles, droplets, or bubbles dspersed at volume fractons less than 10%) n contnuous flud phase Computes trajectores of partcle streams n contnuous phase. Can allow for heat, mass, and momentum transfer between dspersed and contnuous phases. Neglects partcle-partcle nteracton.

84 Lagrangan Multphase Model

85 Couplng Between Phases One-Way Couplng Flud phase nfluences partculate phase va drag and turbulence transfer. Partculate phase have no nfluence on the gas phase. Two-Way Couplng Flud phase nfluences partculate phase va drag and turbulence transfer. Partculate phase nfluences flud phase va source terms of mass, momentum, and energy.

86 Partcle Trajectory (Lagrangan Models) the flud phase s treated separately from the partcle phase and are coupled through nterphase source terms n the partcle momentum equaton: du p, m p = D p, ( u u p ) + m g = 1,2,3, p dt D p = 0.5 ρ A C u u C D = ( Re ) p D p where subscrpt represents the projecton values of a varable n dmensonal as, mp s the partcle mass, Dp, s a drag functon, u s the carrer phase nstantaneous velocty, up, s the partcle velocty, and g s the gravtatonal acceleraton. The source terms on the rght hand sde are drag force and gravtatonal force. Other forces as needed can be ncluded (.e., lft, vrtual mass) 24 Re Re

87 Euleran Multphase Models (IPSA) Approprate for modelng gas-lqud or lqud-lqud or sold-lqud flows where: Phases m or separate Large range of bubble/droplet/sold volume fractons Inapproprate for modelng stratfed or free-surface flows. Two approaches: Interphase Slp Algorthm (IPSA) Algebrac Slp Model (ASM)

88 Euleran Multphase Models (IPSA) Each phase s regarded as havng ts own dstnct velocty components. Phase veloctes are lnked by nterphase momentum transfer - droplet drag, flm surface frcton etc. Each phase may have ts own temperature, enthalpy, and mass fracton of chemcal speces. Phase temperatures are lnked by nterphase heat transfer. Phase concentratons are lnked by nterphase mass transfer. Each phase can be characterzed by a 'fragment sze'. droplet or bubble dameter, flm thckness or volume/surface area. Phase 'fragment szes' are nfluenced by mass transfer, coalescence, dsrupton, stretchng etc. Each phase may have ts own pressure - surface tenson rases the pressure nsde bubbles, and nterpartcle forces prevent tght packng, by rasng pressure.

89 Euleran Multphase Models (IPSA) Phase contnuty regarded as the equatons governng the phase volume fractons. d(r ρ )/dt + dv( r ρ V - Γ r grad(r ) ) = ñ j Transent Convecton Phase Dffuson Mass Source r = phase volume fracton, m 3 /m 3 ρ = phase densty, kg/m 3 V = phase velocty vector, m/s Γ r = phase dffuson coeffcent, Ns/m 2 ñ j = net rate of mass enterng phase from phase j, kg/(m 3 s)

90 Euleran Multphase Models (IPSA) Phase conservaton equatons d(r ρ φ )/dt+dv( r ρ V φ -r Γ φ grad(φ )-φ Γ r grad(r ) ) = S +S p Transent Convecton Wthn-phase and Phase Dffuson Sources φ = any varable r = phase volume fracton, m 3 /m 3 ρ = phase densty, kg/m 3 V = phase velocty vector, m/s Γ φ = wthn-phase dffuson coeffcent, Ns/m 2 Γ r = phase dffuson coeffcent, Ns/m 2 S = wthn-phase volumetrc sources, kg φ/(m 3 s) S p = nterphase volumetrc sources, kg φ/(m 3 s)

91 Euleran Multphase Models (ASM) The mture model s a multphase flow model, partcularly well suted for suspensons, that s, mtures of sold partcles and lqud. Based on momentum balance and mass conservaton of each phase, the mture model uses the followng equatons: Here u denotes mture velocty (m/s), ρ mture densty (kg/m 3 ), p pressure (Pa), cd mass fracton of the sold phase (kg/kg). Furthermore, u slp s the relatve velocty between the two phases (m/s), τ Gm the sum of vscous and turbulent stress (kg/(m s 2 )), and g the gravty vector (m/s 2 ).

92 Euleran Multphase Models (ASM) The mture velocty (m/s) s defned as Where φ c and φ d denote the volume fractons of the lqud (contnuous) phase and the sold (dspersed) phase (m 3 /m 3 ), respectvely, u c the lqud-phase velocty (m/s), u d the soldphase velocty (m/s), ρ c the lqud-phase densty (kg/m 3 ), ρ d the sold-phase densty (kg/m 3 ), and ρ the mture densty (kg/m 3 ). The relatonshp between the veloctes of the two phases s defned by

93 Euleran Multphase Models (ASM) For the slp velocty, the Hadamard-Rybczynsk drag can be used for the slp velocty and s defned as: where d d denotes the dameter of the sold partcles (m). For the mture densty and knematc vscosty:

94 Multphase Models Choosng a Multphase Model Assess volume fracton Less than 10%: LPM (no aggregaton/breakup or partcle/partcle nteracton All volume fractons: Euleran approaches (IPSA: two phase only), (ASM: 3 or more phases) Aggregaton/breakup can be ncluded wth user defned subroutnes

95 CFD Process Geometry Physcs Mesh Solve Reports Post- Processng Select Geometry Heat Transfer ON/OFF Unstructured (automatc/ manual) Steady/ Unsteady Forces Report (lft/drag, shear stress, etc) Contours Geometry Parameters Compressble ON/OFF Structured (automatc/ manual) Iteratons/ Steps XY Plot Vectors Doman Shape and Sze Flow propertes Convergent Lmt Verfcaton Streamlnes Vscous Model Precsons (sngle/ double) Valdaton Boundary Condtons Numercal Scheme Intal Condtons

96 Eample (Secondary Clarfer)

97 Eample (Secondary Clarfer)

98 Eample (Secondary Clarfer) The Mture Model was used to solve ths problem. Physcal propertes of the two phases accordng to the followng table: Var ρ c η c ρ d d d g z φ ma v n v out φ n VALUE 1000 kg/m Pa s 1100 kg/m m m/s m/s m/s DESCRIPTION Lqud phase densty Lqud phase vscosty Sold phase densty Dameter of sold partcles (Constant) z-component of gravty vector Sold phase mamum packng concentraton Inlet velocty Outlet velocty Volume fracton of sold phase of ncomng sludge

99 Results (Secondary Clarfer) Velocty Contour

100 Results (Secondary Clarfer) Sludge Mass Contour

101 Results (Secondary Clarfer) Sludge Mass Contour