Turbulence Modeling in Computational Fluid Dynamics (CFD) Jun Shao, Shanti Bhushan, Tao Xing and Fred Stern

Transcription

1 Turbulence Modelng n Computatonal Flud Dynamcs (CFD) Jun Shao, Shant Bhushan, Tao Xng and Fred Stern

2 Outlne 1. Characterstcs of turbulence. Approaches to predctng turbulent flows 3. Reynolds averagng 4. RANS equatons and unknowns 5. The Reynolds-Stress Equatons 6. The closure problem of turbulence 7. Characterstcs t of wall-bound turbulent t flows 8. Turbulence models and ranges of applcablty RANS 7.. LES/DES 7.3. DNS 9. Example: dffuser

3 Characterstcs of turbulence Randomness and fluctuaton: u = U + u Nonlnearty: Reynolds stresses from the nonlnear convectve terms Dffuson: enhanced dffuson of momentum, energy etc. Vortcty/eddes/energy cascade: vortex stretchng 3

4 Characterstcs of turbulence Dsspaton: occurs at smallest scales Three-dmensonal: fluctuatons are always 3D Coherent structures: responsble for a large part of the mxng A broad range of length and tme scales: makng DNS very dffcult 4

5 Approaches to predctng turbulent flows AFD, EFD and CFD: AFD: No analytcal solutons exst EFD: Expensve, tme-consumng, and sometmes mpossble (e.g. fluctuatng pressure wthn a flow) CFD: Promsng, the need for turbulence modelng Another classfcaton scheme for the approaches The use of correlatons: CD = f ( Re) Integral equatons: reduce PDE to ODE for smple cases One-pont closure: RANS equatons + turbulent models Two-pont closure: rarely used, FFT of Two-pont equatons LES: solve for large eddes whle model small eddes DNS: solve NS equatons drectly wthout any model 5

6 Deeper nsghts on RANS/URANS/LES RANS URANS LES 6

7 Reynolds Averagng Tme averagng: for statonary turbulence Spatal averagng: for homogenous turbulence Ensemble averagng: for any turbulence Phase averagng: for turbulence wth perodc moton 7

8 RANS equatons and unknowns RANS equaton U U P ρ + U S uu t ρ = + x x x μ ρ ( ) RANS equaton n conservatve form U P + + = + t x x x ( UU ) ( ) u u S ρ ρ μ Numbers of unknowns and equatons ( ) Unknowns: 10 = P (1) + U (3) + uu (6) Equatons: 4 = Contnuty (1) + Momentum (3) 8

9 The Reynolds-Stress Equaton Dervaton: Takng moments of the NS equaton. Multply the NS equaton by a fluctuatng property and tme average the product. Usng ths procedure, one can derve a dfferental equaton for the Reynolds-stress tensor. τ τ U U u u + Uk = τk τ k + ν t x x x x x k k k u u τ p p ν + uu u k ρ x ρ x xk xk NEW Equatons: 6 = 6 equaton for the Reynolds stress tensor NEW Unknowns: = u u ν 6 x x k k unkowns u u p p + 6 ρ x ρ x unkowns k u u u 10 k k unkowns 9

10 The closure problem of turbulence Because of the non-lnearty of the Naver-Stokes equaton, as we take hgher and hgher moments, we generate addtonal unknowns at each level. In essence, Reynolds averagng s a brutal smplfcaton that loses much of the nformaton contaned n the Naver-Stokes equaton. The functon of turbulence modelng s to devse approxmatons for the unknown correlatons n terms of flow propertes that are known so that a suffcent number of equatons exst. In makng such approxmatons, we close the system. 10

11 Characterstcs of Wall-Bound Turbulent Flows The turbulent boundary layer (zeropressure gradent) has unversal velocty dstrbuton near the wall (nner-layer) (Clauser 1951) u + yu + + = u = τ = y : y < u ν 5 τ u = ln( y ) : k 4 + = 0.3 :10 > y 30 u τ > y + 30 The velocty defect when plotted vs. y/δ collapse on a sngle curve (outer-layer) u u y = 9.6(1 ) u δ τ 11

12 Turbulent models (RANS) Boussnesq eddy-vscosty approxmaton Algebrac (zero-equaton) equaton) models Mxng length Cebec-Smth Model Baldwn-Lomax Model One-equaton models Baldwn-Barth model Spalart-Allmaras model Two-equaton models k-εε model k-ω model Four-equaton (vf) models Reynolds-stress (seven-equaton) models 1

13 Turbulent models (RANS) Boussnesq eddy-vscosty approxmaton U uu ν = U + δ k 3 T x x Dmensonal analyss shows: ν T = CqL μ, where q s a turbulence velocty scale and L s a turbulence length 1 scale. Usually q = k where k = uu s the turbulent knetc energy. Models that do not provde a length scale are called ncomplete. 13

14 Turbulent models (0-eqn RANS) Mxng length model: ν = l T mx du dy Assume l = αδ ( x) for free shear flow, then mx α=0.180 for far wake α=0.071 for mxng layer α=0.098 for plane et α=0.080 for round et Comments: Relable only for free shear flows wth dfferent α values Not applcable to wall-bounded flows 14

15 ν T Turbulent models (0-eqn RANS) ν T, = ν To Inner layer: Outer layer: Cebec-Smth Model (Two-layer model) y y y > ν T y m m ym y ν T = ν To Where s the smallest value of for U V = lmx + y x y A0 lmx = κ y 1 e * T e v Kleb ( ; ) ν = αu δ F y δ Closure coeffcents: κ = α dp dx α = A = y ρu 6 1 τ y δ y; δ = * δ v = 1 UU δ 0 ( ) FKleb Comments: Three key modfcatons to mxng length model; Applcable to wall-bounded flows; Applcable to D flows only; Not relable for separated flows; *,, U dffcult to determne n some cases; δδ v e 1 ( ) e dy 15

16 ν T ν T, = ν To Inner layer: Outer layer: Turbulent models (0-eqn RANS) Baldwn-Lomax Model (Two-layer model) y y y > ν T y m Closure coeffcents: Comments: m ym y ν T = ν To Where s the smallest value of for + + y A0 = lmx ω lmx = κ y 1 e ( ; ) ν = αc F F y y C T0 cp wake Kleb max Kleb 6 y FKleb ( y; ymax CKleb ) = ymax C Kleb κ = 0.40 α = Applcable to 3D flows; Not relable for separated flows; * No need to determne 1 δδ,, U v A + = C = 1.6 C = 0.3 C = 1 cp Kleb V U W V U W 1 ω = + + x y y z z x e wk 16

17 Turbulent models (1-eqn RANS) Baldwn-Barth model Knematc eddy vscosty : ν = C ν R T μ TDD 1 Turbulence Reynolds number : ( νr T) ( νr T) ( νr T) 1 ν ( ν R T ) T + U = ( Cεf Cε1) νrt P + ( ν + νt σε) t x x x σ x x Closure coeffcents and auxlary relatons: C ε = 1. C ε =.0 C μ = U U U Uk Uk P ν = T + x x x 3 xk xk + 1 C A + = A = 10 ( C C ) ε k k μ = 041 ε1 κ = 0.41 ε σ κ y A0 D = 1 e ε k y A D = 1 e + + C 1 C ε ε1 1 1 D + + y A D y A f = DD 1 DD 1 e e Cε Cε κ y DD A 0 A 1 k 17

18 Turbulent models (1-eqn RANS) Knematc eddy vscosty : Eddy vscosty equaton: Spalart-Allmaras model ν = ν f T v1 ν 1 c b1 U ν cb 1S cw 1f ν ν w ( ν ν ) ν + = ν ν t x d σ xk xk σ xk xk Closure coeffcents and auxlary relatons: c b1 = c b = 0.6 c v1 = 7.1 σ = 3 c w = 0.3 c =.0 κ = 0.41 w f = 3 χ v1 3 3 χ + cv 1 f v ν ν c b1 w1 ( + c ) 1 b c = + κ σ ν = + r = S κ d χ = g r c 6 w ( r r ) g 16 χ = 1 1+ c 6 w3 ν 1+ χ f fw = g 6 6 v1 S S f g + c w33 κ d = + v S = Ω Ω 18

19 Turbulent models (1-eqn RANS) Comments on one-equaton models: 1.One-equaton models based on turbulence knetc energy are ncomplete as they relate the turbulence length scales to some typcal flow dmenson. They are rarely used..one-equaton models based on an equaton for the eddy vscosty are complete such as Baldwn-Barth model and Spalart-Allmaras model. 3.They crcumvent the need to specfy a dsspaton length by expressng the decay, or dsspaton, of the eddy vscosty n terms of spatal gradents. 4 Spalart-Allmaras model can predcts better results than Baldwn-Barth B model, and much better results for separated flow than Baldwn-Barth model and algebrac models. 5 Also most of DES smulatons are based on the Spalart-Allmaras model. 19

20 ,,,, k-ε model: k-ω model: Turbulent models (-eqn RANS) ν T = C k μ ε k k U k + U = τ ε + ( ν + νt σk) t x x x x ε U U ε C ε τ C ε + = + ( ν + ν σ ) ε t x k x k x x ε1 ε T ε C ε 1 = C ε 19 = σ = 13 = 1.9 C μ 0.09 σ = ν T = k ω k σ ε ( Ck μ ) 3 ω = ε l = C k ε k k U * * k + U = τ β kω+ ( ν + σ ν T) t x x x x ω U U ω ω ( T) ω + = α τ βω + ν + σν t x k x x x 13 α = β = β 5 f 0 β * β * f 1 * = χ * β 0 σ = β σ = β 0 = ω f 15 β = χω 1, χk 0 ΩΩ ksk χ = * 9 1 k ω ω = = * ( ) 3 β 0 = f β * χ χ k k 3 βω χ 0 k > ω x x χk * 1 ε = β ωk l = k ω μ 0

21 Turbulent models (-eqn RANS) Comments on two-equaton models: 1. Two-equaton models are complete;. k-ε and k-ω models are the most wdely used twoequaton models and a lot of versons exst. For example, a popular varant of k-ω model ntroduced by Menter has, been used n our research code CFDSHIP-IOWA. There are also a lot of low-reynolds-number versons wth dfferent dampng functons. 3. k-ω model shows better results than k-ε model for flows wth adverse pressure gradent and separated flows as well as better numercal stablty. 1

22 ,,,, vf-kε model: Turbulent models (4-eqn RANS) ν T = Cμ v T C 1 v = kf ε + ( ν + νt ) 3 Dv v v Dt k x x P k L f f = C T k k Dk ν t k Dε ε ε ν t ε = P ε + ν + = Cε1 P Cε + ν + Dt x σ k x Dt k k x σ ε x C 10 1 = 0.4 C = 0.3 C L = 0.3 C η = 70 C ε = 1.9 σ ε = Cε 1 = ( CLd L) k ν k ν T = max,6 L= max C, L C η ε ε ε ε 1 n n ν = C k v ω vf-kω model: T C k v Dv ε v Dt k x x T μ = kf 6v + ( ν + νt ) Dk U * ν t k = τ β ωk + ν + Dt x x σ k x * α = v 1 v P k T k 3 T k k L f = ( C1 1) 5 C 1 n ω ω U v ν t ω = α τ βω + ν + σω D Dt k x k x x β = 0.09 β = 340 σ k = 1.0 σ ω = k ν 3 3 k ν T = max,6 L= max CL, C η ε ε ε ε

23 Turbulent models (4-eqn RANS), Comments on four-equaton models: 1. Fortwo-equaton models a maor problem s that t s hard to specfy the proper condtons to be appled near walls.. Durbn suggested that the problem s that the Reynolds number s low near a wall and that the mpermeablty condton (zero normal velocty) s far more mportant. That s the motvaton for the equaton for the normal velocty fluctuaton. 3. It was found that the model also requred a dampng functon f, hence the name vf model. 4. They appear to gve mproved results at essentally the same cost as the k-ε and k-ω models especally for separated flows. Hopefully the vf-kω model can have better numercal stablty than vf-kε model as ther counterparts behave n two-equaton models. 3

24 Turbulent models (7-eqn RANS) Some of the most noteworthy types of applcatons for whch models based on the Boussnesq approxmaton fal are: 1. Flows wth sudden changes n mean stran rate. Flow over curved surfaces 3. Flow n ducts wth secondary motons 4Fl 4. Flow n rotatng tt fluds fld 5. Three-dmensonal flows 6. Flows wth boundary-layer separaton In Reynolds-stress models, the equatons for the Reynolds stress tensor are modeled and solved along wth the ε-equaton: τ τ U U u u u p u p τ + Uk = τ k τ k + ν ν + u u u k t xk xk xk xk x k ρ x ρ x xk xk 4

25 Turbulent models (7-eqn RANS) These are some versons of Reynolds stress models: 1. LRR rapd pressure-stran model. Lumley pressure-stran model 3. SSG pressure-stran model 4. Wlcox stress- model Comments on Reynolds stress models: 1.Reynolds stress models requre the soluton of seven addtonal PDEs and those equaton are even harder to solve than the two-equaton models.. Although Reynolds stress models have greater potental to represent turbulent flow more correctly, ther success so far has been moderate. 3. There s a lot of current research n ths feld, and new models are often proposed. Whch model s best for whch knd of flow s not clear due to the fact that n many attempts to answer ths queston numercal errors were too large to allow clear conclusons to be reached. 5

26 Turbulent models (LES) Large scale motons are generally much more energetc than the small scale ones. The sze and strength of large scale motons make them to be the most effectve transporters of the conserved propertes. LES treats the large eddes more exactly than the small ones may make sense LES s 3D, tme dependent and expensve but much less costly than DNS. 6

27 Turbulent models (LES, flterng) LES needs a velocty feld that contans only the large scale components of the total feld, whch s acheved by flterng the velocty feld (Leonard, 1974) ( x) = G( x ξ ) u ( ξ ) dξ u G(x-ξ) s the flter kernel, s a localzed functon, whch ncludes a Gaussan, a box flter (a smple local average) and a cutoff (a flter whch elmnates all Fourer coeffcents belongng to wavenumbers above a cutoff) Each flter has a length scale assocated wth t, Δ. Eddes of sze large than Δ are large eddes whle those 1 G( x ξ ) = Δ 0 smaller than Δ are small eddes and need to be modeled. f x ξ otherwse Box or top-hat flter Δ exp πδ 1 γ ( ) γ x ξ G x ξ = G( x ξ ) Gaussan flter Δ sn = kc = π c Δ k c ( kc ( x ξ )) ( x ξ ) Cutoff flter 7

28 Turbulent models (LES, Governng Equatons) Fltered Naver-Stokes equatons (constant densty, ncompressble): ( ρu ) ρ x = 0 ( ρuρ u ) ( ρuρ u ) u p u u t Note that: + x = x u u u u Introducng Subgrd-scale Reynolds Stress S + x = ρ μ x + x ( u u u u ) The flter wdth Δ> grd sze h The models used to approxmate the SGS Reynolds τ stress are called subgrd-scale (SGS) or subflter-scale models. 8

29 Turbulent models (LES, Smagornsky model) The earlest and most commonly used subgrd scale model s one proposed by Smagornsky (1963), whch s an eddy vscosty model. As the ncreased transport and dsspaton are due to the vscosty n lamnar flow, t seems reasonable to assume that 1 u u τ μ S S S, τ kk δ, = μt + = t 3 x x S μt Eddy vscosty μ t = C S ρδ C 0. Stran rate of the large scale or resolved feld S Model constant Drawbacks: 1. Cs s not constant t. Changes of Cs are requred n all shear flows 3. Need to be reduced near the wall. 4. Not accurate for complex and/or hgher Reynolds number flows. S 9

30 Turbulent models (LES, Scale-smlarty model and Dynamc model) Dynamc model: 1. fltered LES solutons can be fltered agan usng a flter broader than the prevous flter to obtan a very large scale feld.. An effectve subgrd-scale feld can be obtaned by subtracton of the two felds. 3. model parameter can then be computed. Advantages: 1. model parameter computed at every spatal grd pont and every tme step from the results of LES. Self-consstent subgrd-scale model 3. Automatcally change the parameter near the wall and n shear flows Dsadvantages: backscatter (eddy vscosty<0) may cause nstablty. 30

31 Turbulent models (DES) Massvely separated flows at hgh Re usually nvolve both large and small scale vortcal structures and very thn turbulent boundary layer near the wall RANS approaches are effcent nsde the boundary layer but predct very excessve dffuson n the separated regons LES s accurate n the separated regons but s unaffordable for resolvng thn near-wall turbulent t boundary layers at ndustral Reynolds numbers Motvaton for DES: combnaton of LES and RANS. RANS nsde d attached boundary layer and LES n the separated regons 31

32 DES Formulaton Modfcaton to RANS models was straghtforward by substtutng the length scale d w, whch s the dstance to the closest wall, wth the new DES length scale, l defned as: Δ= max( δ, δ, δ ) l = mn( d, C Δ) w DES x y z where CDES s the DES constant, t Δ s the grd spacng and s based on the largest dmensons of the local grd cell, and δx, δy, δz are the grd spacng n x, y and z coordnates respectvely Applyng the above modfcaton wll result n S-A Based, standard k-ε (or k-ω) based and Menter s SST based DES models, etc. 1 * ρβ ω ρ ω lk ω = k ( βω) = l = mn( l, C Δ) k * 3 DRANS = k = k / lk k 3 D k DES ρ l k ω DES 3

33 Resolved/Modeled/Total Reynolds stress (DES) TKE Modeled Resolved Total Modeled Reynolds stress: ' u v ' u = ν t y + v x 33

34 Resolved/Modeled/Total Reynolds stress (URANS) Modeled Total Resolved 34

35 Deep nsght nto RANS/DES (Pont oscllaton on free surface wth power spectral analyss) EFD DES URANS -5/3 HZ 35

36 Drect Numercal Smulaton (DNS) DNS s to solve the Naver-Stokes equaton drectly wthout averagng or approxmaton other than numercal dscretzatons whose errors can be estmated (V&V) and controlled. The doman of DNS must be at least as large as the physcal doman or the largest turbulent eddy (scale L) The sze of the grd must be no larger than a vscously determned scale, Kolmogoroff scale, η The number of grd ponts n each drecton must be at least L/ η The computatonal t cost s proportonal to Re 3/ 4 L ( 0.01Re) 3 4 Provde detaled nformaton on flow feld Due to the computatonal cost, DNS s more lkely to be a research tool, not a desgn tool. 36

37 Examples (Dffuser) Asymmetrc dffuser wth separaton s a good test case for turbulence models. A nlet channel was added at the dffuser nlet to generate fully developed velocty profle Boundary layer n the lower dffuser wall wll separate due to the adverse pressure gradent. Results shown next nclude comparsons between Vf and k-ε LES smulaton of ths geometry can be found n: M. Fatca, H. J. Kaltenbach, and R. Mttal, Valdaton of LES n a Plan Asymmetrc Dffuser, center for turbulence research, annual research brefs,

38 Mean velocty predcted by Vf agreed very well wth EFD data, partcular the separaton regon s captured. K-ε model fals to predct the separaton caused by adverse pressure gradent. Examples (Dffuser) 38

39 Examples (Dffuser) vf k-ε TKE predcted by Vf agreed better wth EFD data than k- ε model, partcular the asymmetrc dstrbuton. Rght column s for the skn frcton coeffcent on the lower wall, from whch the separaton and reattachment pont can be found. x/h 39

40 References 1. J. H. Ferzger, M. Perc, Computatonal Methods for Flud Dynamcs, 3 rd edton, Sprnger, 00.. D. C. Wlcox, Turbulence Modelng for CFD, S.B.Pope, Turbulent Flows, Cambrdge, P. A. Durbn and B. A. Pettersson Ref, Statstcal Theory and Modelng for Turbulent Flows, John Wley & Sons, LTD, P.A. Davdson, Turbulence: An Introducton for Scentsts and Engneers, Oxford,