HYBRID LES-RANS: Inlet Boundary Conditions for Flows With Recirculation

Transcription

1 1 HYBRID LES-RANS: Inlet Boundary Conditions for Flows With Recirculation Lars Davidson Div. of Fluid Dynamics Dept. of Applied Mechanics Chalmers University of Technology, Göteborg, Sweden lada

2 1 HYBRID LES-RANS Near walls: a RANS one-eq. k model. In core region: a LES one-eq. k SGS model. wall replacements URANS y URANS LES y + ml x wall

3 2 MOMENTUM EQUATIONS The Navier-Stokes, time-averaged in the near-wall regions and filtered in the core region, reads ū i t + ) (ūi ū x j = βδ1i 1 p + [ (ν + ν j ρ x i x T ) ū ] i j x j ν T = ν t, y y ml ; ν T = ν sgs, y y ml

4 3 TURBULENCE MODEL Use one-equation model in both URANS region and LES region. k T t + (ū x j k T ) = [ (ν + ν j x T ) k ] T k 3/2 + P j x kt C T ε j l P kt = 2ν T Sij Sij, ν T = C k lk 1/2 T LES-region: k T = k sgs, ν T = ν sgs, l = = (δv ) 1/3 URANS-region: k T = k, ν T = ν t, l y, Chen-Patel model (AIAA J. 1988)

5 4 SYNTHESIZED ISOTROPIC TURBULENCE E(κ) κ e κ 1 N u t (x) = 2 κ n σ n = n=1 E(κ n ) û n cos(κ n x + ψ n )σ n E(κ) κ 5/3 κ n κ max N = 6, û n = E(κ n ) κ n, κ n κ û n, ψ n, σ n i : amplitude, phase, direction of Fourier mode n. κ e = 13π/(55L t ), L t = k 3/2 /ε [κ 1, κ max ] divided into N modes, κ max = π min{ x, y, z}, κ 1 = κ e /2

6 5 TIME SCALES M independent realizations u t (x) are created. Thus no time correlation. A time correlation is introduced by (U ) m = a(u ) m 1 + b(u t (x)) m, m = time step 1 autocorrelation B(τ).6 The autocorrelation B(τ) is prescribed by setting a = exp( t/t ), b = (1 a 2 ) 1/2.4.2 exp( τ/t ).8 B(τ) from (U ) m τ

7 6 MEAN+FLUCTUATIONS Inlet fluctuations are set as (U ) m, (V ) m, (W ) m. The streamwise fluctuations are superimposed to the mean profile taken from experiments or RANS simulations

8 7 FLOW OVER a BUMP: DESider CASE ONERA bump. Re h = δ in /H =.3 W/H = 1.67 in expts. Here only a slice: W slice /H =.61 Mesh: x/δ in =.41, z/δ in =.44. z + = 125 and z + = 116. L 1 =.34H replacements h =.46H U in not to scale H L 2 =.88H L = 7.6H

9 8 RESOLVED SHEAR STRESSES x=.67h.33h x= x=.33h frag replacements C p ( ) u = 1 τ in ( ) u = 2 τ in ( u τ )in =.5

10 9 x=.67h RESOLVED SHEAR STRESSES: ZOOM.33H ( ) u = 1 τ in ( u τ )in = 2 ( u τ )in =.5 ments Inlet located at = 1.2 C p

11 1 1.5 PRESSURE AND FRICTION COEFFICIENTS 4 x ts C p.5 C f ( ) u = 1 τ in 6 ( ) u = 2 τ in ( u τ )in =.5

12 11 VELOCITIES x=.67h.33h x= x=.33h 1.17H 1.5H 2.8H 2.42H ts H p ( ) u = 1 τ in C f ( ) u = 2 τ in ( u τ )in =.5

13 12 PRESSURE AND FRICTION COEF.: 2D 3D, ( u τ )in = x ts C p.5 C f D 3D

14 13 VELOCITIES: 2D 3D, ( u τ )in = 1 x=.67h.33h x= x=.33h 1.17H 1.5H 2.8H 2.42H ts H p C f 2D 3D

15 14 RESOLVED SHEAR STRESSES: 2D 3D, ( u τ )in = 1 x=.67h.33h x= x=.33h x=.5h.83h 1.17H ts H p C f 2D 3D

16 15 SPECTRA ( u τ )in = 1 Eww,z κz ts 1 3 Eww,z κz κ z H κ z H.4 y/h w rms

17 16 DISSIPATIONS: ε SGS,mean ε SGS = 1 =.67 =.33 x= = lacements 19 C f C p ε SGS = 2ν T s ij s ij, ε SGS,mean = 2 ν T s ij s ij, ε SGS = ε SGS ε SGS,mean ε SGS,mean : dissipation term in ū i ū i /2 eq. (mean flow) ε SGS : approx. dissipation term in ū iū i /2 eq. (resolved turb)

18 17 SGS DISSIPATIION v. WAVE NUNMBER E(κ) frag replacements E(κ) κ 5/3 ε SGS In reality the SGS dissipation does not occur only at the cut-off, but ε SGS (κ) κ c κ

19 18 ENERGY SPECTRA and DISCRETE FFTs Two-point correlation is the inverse FFT of the energy spectrum Q ww (ξ) = E ww (k) cos (2π(k 1)ξ) k=1 ε zz = 2ν( w / z) 2 can be obtained from (z homogeneous dir.) ( w ) 2 ε zz = 2ν = 2ν 2 Q ww (ξ) N z ξ 2 = 2ν k 2 E ww (k) ξ= Not satisified in FV because the / ξ is not exact. Instead, form a DFT of w / z and then a Power Density Spectra. Then indeed ε zz = N k=1 ŵ k w ˆ k = P DS( w / z) where ŵ are Fourier coefficients of w / z k=1

20 19 DISSIPATION SPECTRA = 1, (y y wall )/H = ν T P DS(1) = ( ν T ū / y) 2 = = 1, y/h =.34 ν T P DS(1) = ( ν T ū / y) 2 = 1.1 nts κ z H κ z H ν T k 2 E ww (κ z ) ν T P DS( w / z) ν T P DS( u / z) ν T P DS( ū/ y)

21 2 DISSIPATION SPECTRA: CHANNEL FLOW Re τ = 4 y + = 34 (URANS region) y + = 25 (LES region) nts P DS(ε 1/2 SGS ) κ z κ z y + match = 125 ν SGS P DS( w z ) ν SGS k 2 ze ww (κ z ) ν SGS P DS( ū y )

22 z y δ =.5H x Inlet B.C. Dahlström & Davidson[?] H L 1 3.2H L 2 W outlet B.C. 21

23 22 PRESSURE COEFFICIENT, SHEAR STRESSES.5 x = 3H 2H H ts C p ( ) u = 1 τ in ( ) u = 2 τ in ( u τ )in =.5

24 23 ( 2 u τ )in = SURFACE STREAMTRACES 2 1. = z/h z/h ents 2 z/h x C f ents

25 24 CONCLUSIONS Reasonable inlet fluctuations are important, their exact timeand lengthscale not very important ( ) = 1 seems to be good u τ in If amplitude too small, too small resolved fluctuations are created In LES region SGS dissipation fairly evenly distributed at all wavenumbers In URANS region SGS dissipation is largest at low wavenumbers