RaNS.1 RaNS + PDE Turbulence Closure Models

Transcription

1 RaNS.1 RaNS + PDE Turbulence Closure Models Time-averaged Navier-Sokes PDE closure models one-equaion: k wih l specified υ, Re ranspor wo-equaion: k, ε ranspor k,ω ranspor Turbulen kineic energy ranspor D : ( ) = k k U k 1 1 E L k + U i υ +ε+ uuu + p u 0 j x ij x x x i i j ρ = j j j j j υ ui ui dissipaion : ε δ 3/ / l jk 3 xj x k k - 1 i Re sress : δ + U U k υ S, S = ij ij ij ij + 3 xj xi 1 1 υ k closure model: uuu i i j+ pu j - ρ σ x k j j

2 RaNS. RaNS + Single Turbulence Closure Models Prandl one-equaion model selecs ε 3/ l assumed proporional o mixing lengh Baldwin-Barh Re ranspor PDE Spalar-Allmaras υ ranspor PDE = C k / l D eddy viscosiy: υ C υre DD (le Re R) μ 1 ( ) υr υr ( υr) 1/ 1 υ ( υr) ranspor: L ( υr) = + U υ+ υ σ C( RP) 0 j ε υ + = x x σ x x j j ε k k closure: 7 coeffeciens + 3 funcions ( Wilcox, pg.110 ) eddy viscosiy : υ = υf υ 1 υ υ 1 υ υ υ ranspor : L ( υ) = + U ( υ υ) C C υs C 0 j + + = x σ x x 1 x 3 d j k k k closure : 8 coeffeciens + 3 funcions (Wilcox, pg.111)

3 RaNS.3 RaNS + Two PDE Turbulence Closure Models Fundamenal variables for RaNS+ PDE closure models Kolmogorov: ω dissipaion rae, D ( -1) υ k/ ω, l k 1/ / ω, ε kω Chou: υ k/ ε, l k 3/ / ε Roa : υ k/ l, ε k3/ / l Kolmogorov, Speziale, W ilcox, Peng k-ω closure L ( k) = k k U i k + U + β kω ( υ+ σ υ ) = 0 j x ij x x x j j j j L ω ω ω U i ω j j j closure: υ = k/ ω, consans α, β, β, σ, σ, β= β f, β = β f ( ) = ω + U α ( ) 0 j ij + β ω υ + συ = x k x x x j β 0 β 1, χ k χ ΩΩ ij jk S f ω ki,,, 0, ω -3 k χ f ω = = χ χ β ω β + χ > k k k 1+80χ ω ( β 0 ω) x j x j χ k 1 1 definiions: ε = β kω, Ω Ui, j U j, i, S U ij ij i, j + U j, i

4 RaNS.4 RaNS + Two PDE Turbulence Closure Models Jones & Launder, Launder & Sharma k-ε closure L( k) and L( ε) wih momens modelled already covered u exac ( ) i L ε υ ( L ( u )) = 0 x x i j j eqn.(4.45) in Wilcox Yakho and Orzag renormalizaion group (RNG) model L ( k ) and L ( ε) remain in " sandard " form C λ 3 (1 λ / λ0 ) μ k C C +, λ S S ε ε 1 3 ji ij + βλ ε all model consans slighly alered Two PDE closure model coefficien deerminaion compaibiliy wih BL similariy resoluion, decaying homogeneous isoropic urbulence

5 RaNS.5 Similariy Soluions for RaNS + PDE Closures Uni-direcional flow primiive PDEs and similariy soluion forms

6 RaNS.6 Similariy Soluions for RaNS +PDE Closures Insering self-similar forms ino DP x (u), η y/x u 1 d j du D P ( η): L ( U) V - η N = S U x u η ηdη dη where : N( η) ransformed eddy viscosiy erm V ( η) ransverse momenum erm S ( ) sreamwise convecion erm u η Similariy form for k-ω and k-ε closure models

7 RaNS.7 Similariy Soluions for RaNS +PDE Closures Similariy closure parameers / funcions Key disincion beween k-ω and k-ε closure models is χ ω χ = f ( Ω Ω S ) Ω Ω S + Ω Ω S for -D flows ω ij jk ki U 1 U 0 U 1 U 1 U 0 x y x r 0 0 ry, 1 U V 1 U V D: S = 0 Axi: S = 0 Boh: Ω = 1 U ij ij y y 0 0 ij r r ry, V r χ ω 1 U U V 1 U U V 1 U V - + =0 χ - + ω 4 y x y 4 r x r 4 y r

8 RaNS.8 RaNS + PDE Farfield Similariy Soluions Numerical similariy soluions for farfield urbulen flows wake mixing layer

9 RaNS.9 RaNS + PDE Farfield Similariy Soluions Numerical similariy soluions for farfield urbulen flows plane je round je

10 RaNS.10 RaNS +k-ω Soluion for Turbulen Channel Flow Perurbaion mached-region soluions, k-ω model k ω closure Re = 13,750 H o DNS (Mansour) P + U = υ / u 4 xy y ε + = υε/ u4

11 RaNS.11 RaNS + PDE Closure Model Near Wall BCs PDE closure models mus address nearfield low Re region BCs recall BL similariy U+ u/ u = κ -1log( y+ E) + C y+ u y/ υ producion = dissipaion in L ( k) υ -1 = κ ε = (κ ) -1 = / μ ω = (κ ) yu y u k u C y u 3 Modificaions for pressure gradien, surface roughness dp P+ υ ρu 3 dx 5 C 8.5 in log law

12 RaNS.1 RaNS + PDE Closure Model Low Re Effecs TS applied o DM + DP x for flucuaing velociies near surface u fx( x,z,) y + O( y ) 3 v fy( x,z,) y + O( y ) as y 0 w fz( x,z,) y + O( y ) Limiing behavior for ime-averaged properies 1 k ( f + f ) y + O( y3), ε υ ( f + f ) y+ O( y), ( f f ) y3+ O( y4) x z x z xy x y Implemenaed by modificaions o PDE closure models for BLs ε = ε +ε 0 υ f k μ μ = C / ε

13 RaNS.13 RaNS +PDE Low Re Closure Modificaions Correlaion Re numbers : Re T = k /ε, R y = k 1/ y/ν, y + = u y/ν

14 RaNS.14 RaNS + PDE Low Re Closure Modificaions Examining low Re model asympoic behavior for k-ε as y 0: k y ε/ k υ / y y 3, y 4 for Lam-Bremhors xy BCs a y =0: k =0 = ε k 1/ k ε=ε +ε, ε= f, 0 0 y y k ε= υ y for Lam-Bremhors ε = 0 y ransiion o k-ω: ε= C kω = β0kω μ k + = k / u

15 RaNS.15 RaNS + Low Re k-ω Closure Model Low Re k - ω closure model idenical excep for α * υ = α k / ω T 0 α 1 Closure model consans now funcions of urbulen Re s β = 9 f β /15, σ = 1/ = σ σ =β /3, α = 1/9 R = 8 β R = 6 k R =.95 ω

16 RaNS.16 RaNS + k-ω Low Re Soluion for Channel Flow k ωlow Re closure Re = 13,750 H o DNS (Mansour) P + U = υ / xy u y ε = υε/ u + 4 4

17 RaNS.17 Summary: RaNS + PDE Turbulence Closure Time-averaged incompressible NS + PDE closure models single PDE : Prandl mixing lengh wih L ( k) Baldwin-Barh: L (Re ) closure consans, funcions Spalar-Allmaras: L ( υ ) dual PDE : Kolmogrov,...: L ( k), L (ω), υ = k /ω Jones Launder,...: L( k), L( ε ), υ = C k / ε Yakho,...,RNG: C ε modified for Uni-direcional farfield flows, similariy soluion characerizaion μ S ij L ω(, Ω ) admis vorex rollup for -D axisymmeric flow S ij ij improved agreemen for all farfield cases accepable agreemen for channel flow Uni-direcional nearfield flows, low Re modificaions low Re models address wall damping asympoically L ( S ) ij ω(,α ) provides necessary modificaions for k-ω improved agreemen for k, P, ε nearwall disribuions