Lars Davidson. Inlet boundary conditions for Hybrid LES-RANS

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1 Lars Davidson Inlet boundary conditions for Hybrid LES-RANS Div. of Fluid Dynamics Dept. of Applied Mechanics Chalmers University of Technology, Göteborg, Sweden lada

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

4 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 SYNTHESIZED ISOTROPIC TURBULENCE κ 3 E(κ) PSfrag replacements κ e κ 1 ements θ n ϕ n κ n da κ 2 E(κ n ) E(κ) κ 5/3 κ n κ max κ 1 u t (x) = 2 κ κ n σ n = κ n N û n cos(κ n x+ψ n )σ n, N = 15, û n = E(κ n ) κ n, κ n = κ n n=1 κ e = 13π/(55L t ), L t = k 3/2 /ε [κ 1, κ max ] divided into N modes, κ max = At high κ, E(κ) is a function of ν π min{ x, y, z}, κ 1 = κ e /2

6 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 PSfrag replacements.2 exp( τ/t ).8 B(τ) from (U ) m τ

7 CHANNEL WITH INLET-OUTLET Inlet fluctuations are set as (U ) m, (V ) m, (W ) m. The streamwise fluctuations are superimposed to the mean profile U + in = y + y ln(y + ) 5 < y + < 3 1 κ ln(y+ ) + B y + 3 where κ =.4 and B = 5.2

8 ts DOMAIN, Re τ = u τ δ/ν = 2 (Re b 8 ) 1.4δ 5.3δ 22.9δ inlet 2δ y x outlet 24δ (x, y, z) cells. z max = 6.3δ, x + 785, z δ/ z 5, δ/ x 2.5 The location of the matching plane at y =.75 (y + = 15)

9 TEST CASES Different inlet turbulent length and time scales have been used Using k = 4, ε = 7 (scaled with u τ and δ), the baseline scales were set to: Length scale: L t = k 3/2 /ε = L 1 =.11δ Time scale: τ t = 4k/ε = T 1 =.22δ/u τ

10 3 Time scale T 1 and length scale L U + ents y uv + PSfrag replacements +: U + = circles: DNS at Re τ = 5 solid lines: x/δ = 1.4 dashed lines: x/δ = 5.3 dash-dotted lines: x/δ = y/h

11 3 Time scale T 1 and length scale 2L U + ents y uv + PSfrag replacements +: U + = circles: DNS at Re τ = 5 solid lines: x/δ = 1.4 dashed lines: x/δ = 5.3 dash-dotted lines: x/δ = y/h

12 3 Time scale T 1 and length scale L t = 25.2 U + ents y uv + PSfrag replacements +: U + = circles: DNS at Re τ = 5 solid lines: x/δ = 1.4 dashed lines: x/δ = 5.3 dash-dotted lines: x/δ = y/h

13 3 Time scale τ = and length scale L U + ents y uv + PSfrag replacements +: U + = circles: DNS at Re τ = 5 solid lines: x/δ = 1.4 dashed lines: x/δ = 5.3 dash-dotted lines: x/δ = y/h

14 3 Time scale.25t 1 and length scale L U + ents y uv + PSfrag replacements +: U + = circles: DNS at Re τ = 5 solid lines: x/δ = 1.4 dashed lines: x/δ = 5.3 dash-dotted lines: x/δ = y/h

15 1.5 FRICTION VELOCITIES 1 u τ.95.9 replacements.85 T 1, L 1 T 1, 2L 1 :.25T 1, L x/δ : T =, L 1 +: T 1, L = : T 1, L 1 forcing

16 FORCING FLUCTUATIONS ADDED AT THE INTERFACE Object: to trig the momentum equations into resolving largescale turbulence interface u f, v f, w f LES region ents URANS region y y + ml wall x

17 INLET BOUNDARY CONDITIONS vs. FORCING Inlet U b (y) PSfrag replacements u (y, t) ts u (x i, t) LES region U b (x i, t) y x URANS region

18 FULLY DEVELOPED CHANNEL FLOW (periodic in x) 3 U + ents uv + PSfrag replacements y y/h +: U + = 1.4 ln y no forcing forcing (isotropic fluctuations)

19 T 1, L 1 FRICTION VELOCITIES, LONG CHANNEL forcing (isotropic fluctuations) u τ replacements no forcing x/δ

20 T 1, L 1 FRICTION VELOCITIES, X-LONG CHANNEL forcing (isotropic fluctuations) u τ.95.9 eplacements.85 no forcing x/δ

21 TWO-POINT CORRELATIONS. Time scale T 1. B(ζ) = 1/(z max w 2 rms) z max u (z)u (z ζ)dz. 1 Inlet 1 x = 22.9δ B(ζ) ents.5 PSfrag replacements B(ζ) ζ/δ ζ/δ DNS L t = L 1 2L 1

22 AUTO CORRELATIONS. Length scale L 1. B(τ) = 1/(T u 2 rms) T u (t)u (t τ)dt. Time scale.25t 1 Time scale T B(τ ) ents.6.4 PSfrag replacements B(τ ) τ = τu τ,in /δ τ = τu τ,in /δ x = 22.7δ, y + = 37 inlet DNS, y + = 37 x = 22.7δ, y + = 84

23 CONCLUSIONS Synthesized isotropic turbulence fluctuations have been used for prescribing fluctuating inlet velocities. Length scale L 1 and time scale T prescribed independently It has been shown that both L 1 and T are essential Far downstream the inlet, the standard hybrid LES-RANS forgets the inlet turbulence Hybrid LES-RANS with forcing conditions are needed for long channels Forcing conditions similar to inlet boundary conditions of fluctuating velocities. The object is to trig the equations into resolving turbulence.

24 REFERENCES Davidson, L., Dahlström, S., Hybrid LES-RANS: An approach to make LES applicable at high Reynolds number, International Journal of Computational Fluid Dynamics (to appear), 25. Davidson, L. and Billson, M., Hybrid LES/RANS: Using Synthesized Turbulence for Forcing at the Interface (submitted), International Journal of Heat and Fluid Flow, 25. Davidson, L. and Billson, M., Hybrid LES/RANS: Using Synthesized Turbulence for Forcing at the Interface, ECCOMAS 24, P. Neittaanmäki, T. Rossi, S. Korotov, E. Oñate and J. Périaux and D. Knörzer (eds), July 24-28, Finland, 24.

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