Turbulence in wall-bounded flows

Transcription

1 Turbulence in wall-bounded flows Sergio Pirozzoli Matteo Bernardini Paolo Orlandi Dipartimento di Ingegneria Meccanica ed Aerospaziale Sapienza Universita di Roma HPC methods for Computational Fluid Dynamics and Astrophysics November , Bologna

2 Acknowledgments All simulations carried out using PRACE resources PRACE 3rd call PRACE 5th call PRACE 7th call PRACE 10th call PRACE 12th call Thanks to CINECA staff!

3 Why wall turbulence? Flow over solid surfaces most often turbulent Turbulent flow large increase of drag and heat transfer as compared to laminar state Drag and heat transfer prediction critical for efficient design of aircraft, cars, pipelines,... Practical impact: 1% drag reduction of A340 would result in fuel savings of about 100,000 Euros per aircraft per year Robust physical insight definitely needed

4 Wall turbulence: how it looks like Side view of turbulent flow over smooth horizontal surface (Re τ 4000)

5 Wall turbulence: how it looks like Wall-parallel view of turbulent flow over smooth horizontal surface (Re τ 4000)

6 Wall turbulence: a cartoon U 0 δ δ v Two velocity scales 1. U 0 (outer scale) 2. u τ = (τ w /ρ) 1/2 (inner scale) Two length scales 1. δ (outer scale) 2. δ v = ν/u τ (inner scale) Ratio of outer to inner length scales is an important parameter Re τ = δu τ /ν = δ/δ v

7 Wall turbulence: what we know (Approximately) universal near-wall behavior in inner scales, i.e. u + := u/u τ = f(y + ), y + := y/δ v Logarithmic meso-layer of mean velocity (Prandtl 25) u + = 1/k log y + + C, k = 0.41, C 5 Logarithmic layer of wall-parallel velocity variances (Townsend 76) u 2 /u 2 τ = A log y/δ + B The latter implies that peak velocity variances grow as log Re τ

8 Wall turbulence: what we don t know High Re τ outer eddies become more energetic Inner-outer layer interactions become more important Corrections to classical representation Asymptotic limit as Re τ? u + = f(y +, Re τ ) Subject of several recent reviews (Jimenez 2013, Smits et al. 2011, Marusic et al. 2010)

9 Why use supercomputers? Experiments limited in near-wall resolution, and difficulty to get clean data (e.g. derivatives of flow statistics) Revert to Direct Numerical Simulation (DNS), i.e. solve full Navier-Stokes equations in 3D unsteady version by resolving all scales of motion However, DNS of wall turbulence hampered by stringent computational requirements As δ v, simulating flows at high Re τ requires about (δ/ ) 3 = Re 3 τ points (in fact, N pts Re 11/4 τ ) Some effects (e.g. logarithmic scaling) only manifest themselves as Re τ O( ) points are required

10 DNS of wall turbulence State-of-the-art 10 4 Boundary layer Poiseuille Couette Pipe Reτ Year

11 DNS of wall turbulence Projection 10 4 Boundary layer Poiseuille Couette Pipe Reτ Re τ 10 4 likely to yield asymptotic wall turbulence (Jimenez 2012) Year

12 Outline DNS of incompressible wall-bounded flows (M = 0) Poiseuille flow up to Reτ 4000 Couette flow up to Reτ 1000 Passive scalars in Poiseuille flow Channel flow with unstable stratification DNS of compressible wall-bounded flows: see next talk

13 Governing equations Navier-Stokes-Boussinesq model u j x j = 0, u i t + u iu j x j = p x i + βgθδ i2 + Πδ i1 + ν 2 u i x j x j θ t + θu j x j = α 2 θ x j x j Controlling parameters Convection: bulk Reynolds number Re b = 2hu b /ν Buoyancy: bulk Rayleigh number Ra = βg θ(2h) 3 /(αν) Buoyancy/convection: bulk Richardson number Ri b = Ra/Re 2 b = 2βg θh/u 2 b Pr = ν/α 1

14 Numerical method Orlandi 2000 Projection method with direct Poisson solver based on Fourier expansions (Kim & Moin 87) Second-order approximation of space derivatives on staggered mesh (Harlow & Welch 65) Discrete conservation of total kinetic energy and scalar variance Implicit treatment of wall-normal viscous terms Third-order low-storage Runge-Kutta time stepping by A. Wray Pencil decomposition for efficient parallel implementation

15 Numerical discretization Incompressible NS equations u j x j = 0, u i t + u iu j x j = p x i + ν 2 u i x j x j Projection method to satisfy divergence-free constraint Semi-implicit discretization in time v v n = N n+1/2 + 1 t 2 (L(v ) + L(v n )) Gp n (1) In general, v is not divergence-free For that purpose, a correction step is implemented with ϕ TBD v n+1 v t = Gϕ (2)

16 Numerical discretization Take discrete divergence of previous equations Discrete Poisson equation G Gϕ = G (v n+1 v ) t = G v t Relation between ϕ and pressure can be derived by combining steps (1) and (2) p n+1 = p n + ϕ Lϕ The latter relation is used to feed step (1)

17 Numerical discretization Computational issues Inviscid 1D case Assume D = C2 v n+1 j v j t = Dϕ j D Dϕ = Dv t ϕ j 2 2ϕ j + ϕ j+2 = h v j+1 v j 1 2 t The stencil (j 2, j, j + 2) only contains odd or even indices decoupling of discrete pressure field

18 Numerical discretization Avoiding odd-even decoupling Staggered grid arrangement (MAC method, Harlow & Welch 65) v i,j+1/2 u i 1/2,j p i,j u i+1/2,j y v i,j 1/2 [ ] u 2 x i+1/2,j [ ] uv y i+1/2,j [ ] 2 u 2 x 2 i+1/2,j [ ] p x i+1/2,j x = u2 i+1,j u2 i,j + O( x 2 ) x [ ] (uv)i+1/2,j+1/2 (uv) i+1/2,j 1/2 = y + O( y 2 ) = u i+3/2,j 2u i+1/2,j + u i 1/2,j x 2 + O( x 2 ) = p i+1,j p i,j x + O( x 2 )

19 Numerical discretization Avoiding odd-even decoupling v i,j+1/2 Interpolations are needed u i+1,j = 0.5 ( u i+1/2,j + u i+3/2,j ) u i 1/2,j p i,j u i+1/2,j y (uv) i+1/2,j+1/2 = [ u i+1/2,j + u i+1/2,j+1 ] [v i+1,j+1/2 + v i,j+1/2 ] /4 v i,j 1/2 x Connection with skew-symmetric form discrete energy conservation [ ] u 2 x i+1/2,j ( ui+1/2,j + u i+3/2,j ) 2 ( ui 1/2,j + u i+1/2,j ) 2 u 2 j+3/2 u2 j 1/2 = 4 x 1 u 2 2 x + u u x 4 x + u i+1/2 u i+3/2,j u i 1/2,j 2 x

20 Numerical discretization Solution of Poisson equation for ϕ G Gϕ = G v / t In general requires iterative solvers impractical for DNS Direct solvers available in the presence of two periodic/symmetric directions (Kim & Moin 87) Fourier expansion in homogeneous directions (say x, z) ϕ(x, y, z) = ˆϕ l,m (y)e iklx e ikmz k l = 2πl/L x, k m = 2πm/L z Set f = G v / t l,m ˆϕ l,m ( kx2 + k z2 ) + Dy2 ˆϕ l,m = ˆf l,m kx2, k z2 are MWN for the second space derivative, e.g. k x2 = 2(1 cos(k l x))/ x 2, k z2 = 2(1 cos(k m z))/ z 2 Standard tridiagonal system in y direction must be solved l, m

21 Accuracy Linear analysis not necessarily the ultimate word Simple exercise: consider Burgers vortex model Representative for small-scale vortex tubes (Jimenez 93) u θ (x) 1 ξ (1 e ξ2) ξ = x/r b, r b = 3.94η We evaluate a single derivative as in MAC method u 2 j+3/2 u2 j 1/2 4 x + u i+1/2 u i+3/2,j u i 1/2,j 2 x 1 u 2 2 x + u u x

22 Accuracy Error analysis L 2 error norm E nd order 4th order 6th order 8th order C nd order 4th order 6th order 8th order h/η Practical DNS of wall turbulence use typical grid spacings in the wall-parallel direction x/η 5, z/η 3 FD formulas still far from reaching the asymptotic convergence limit The additional computational cost may not be justified MAC scheme successfully used for DNS of many wall-bounded flows E 2

23 Accuracy FD vs. spectral Plane channel at Re τ = 2000 Lines: second-order finite differences (Bernardini et al. 14); symbols: pseudo-spectral method (Hoyas & Jimenez 06) u/uτ u 2 i /u2 τ y Max difference about < 2% in turbulence intensity y +

24 Parallel implementation Slab vs. pencil decomposition Operations in x z plane can be carried out locally Operations in y (e.g. tridiagonal matrix inversion in Poisson solver) are global, requiring data transposition Main limitation: given N 3 mesh, max N processors can be used E.g points can be (inefficiently) handled on 1000 MPI processes Number of halo cells gets higher as # of cores increases

25 Parallel implementation Pencil decomposition The limit on the number of processes is now N p = N 2 Size of the halo cells on every core decreases with increasing # cores Example of work flow: solution of Poisson equation 1. transpose (a) (b), then take real FT in z 2. transpose (b) (c), then take complex FT in x 3. transpose (c) (a), then solve tridiagonal system in y Fourier coefficients 4. Transpose (a) (c), then take inverse FT in x 5. Transpose (c) (b), then take inverse FT in z to get physical solution 6. Transpose (b) (a) to have solution in original arrangement

26 Parallel scalability Weak scaling

27 DNS of Poiseuille flow DNS setup Flow case Line style Re b Re τ N x N y N z x + z + T u τ /h CH1 Dashed CH2 Dash-dot CH3 Dash-dot-dot CH4 Solid Computational box L x = 6πh, L z = 2πh dp/dx 2 h

28 DNS of Poiseuille flow DNS at Re τ = 4000: contour lines of u low-speed and high-speed

29 DNS of Poiseuille flow Mean velocity u/uτ CH1 CH2 CH3 CH y +

30 DNS of Poiseuille flow Mean velocity Comparison with experiments u/uτ CH1 CH2 CH3 CH4 Shultz & Flack y +

31 DNS of Poiseuille flow Mean velocity Overlap layer u/uτ y + Log-law fit u + = log y + /0.386 Recent estimates yield k 0.37, C = 3.7 (Nagib PoF 2008)

32 DNS of Poiseuille flow Diagnostic function Ξ = y + du + /dy + 3 Ξ 2 1 CH2 CH3 CH4 No range with flat behavior y +

33 DNS of Poiseuille flow Diagnostic function Ξ = y + du + /dy + Generalized log law 3 Ξ 2 1 CH2 CH3 CH4 Modified log law y + Refined overlap arguments (Afzal 1976) suggest Ξ = 1 k + αy h + β Re τ We get k = 0.41, α = 1.15, β = 180 (close to Jimenez & Moser 2007)

34 DNS of Poiseuille flow Velocity variances Horizontal components u 2 /u 2 τ 6 4 w 2 /u 2 τ y y +

35 DNS of Poiseuille flow Velocity variances Horizontal components u 2 /u 2 τ 6 4 w 2 /u 2 τ y + Experiments by Shultz & Flack (2013) y +

36 DNS of Poiseuille flow Velocity variances Horizontal components u 2 /u 2 τ 6 4 w 2 /u 2 τ y y + Townsend attached-eddy hypothesis: u 2 i /u2 τ B i A i log(y/h) Typically quoted values A , B We find A , B

37 DNS of Poiseuille flow Reynolds stress peaks Trend with Re τ u 2 pk/u 2 pk(ch1) U V W UV Re τ Clear logarithmic growth of u and w Sub-logarithmic growth of v

38 DNS of Poiseuille flow Spectral densities (pre-multiplied) u, inner scaling CH1 CH2 CH3 CH4

39 DNS of Poiseuille flow Production excess over dissipation y + (P + ε + ) Equilibrium hypothesis clearly violated at sufficiently high Re τ y +

40 DNS of Couette flow DNS setup Flow case Line style Re c Re τ N x N y N z x + z + T u τ /h C1 Dashed C2 Dash-dot C3 Dash-dot-dot C4 Solid Computational box L x = 18πh, L z = 6πh U 2 h

41 DNS of Couette flow DNS at Re τ = 1000: contour lines of u low-speed and high-speed

42 DNS of Couette flow DNS at Re τ = 1000: contour lines of u low-speedand high-speed

43 DNS of Couette flow Mean velocity u/uτ y +

44 DNS of Couette flow Mean velocity u/uτ y + Log-law fit u + = log y + /0.41

45 DNS of Couette flow Diagnostic function Ξ = y + du + /dy Ξ 2.5 k = No apparent tendency to log law! y +

46 DNS of Couette flow Velocity variances Horizontal components u 2 /u 2 τ w 2 /u 2 τ y y + Logarithmic decrease of w Emergence of outer peak of u

47 DNS of Couette flow Reynolds stress peaks Trend with Re τ 1.4 ui 2 pk /u i 2 pk (C1) Re τ Clear logarithmic growth of u and w No growth of v

48 DNS of Couette flow Spectral densities (pre-multiplied) Inner scaling C1 C2 C3 C4

49 DNS of Couette flow Production excess over dissipation y + (P + ε + ) Equilibrium hypothesis clearly violated at sufficiently high Re τ y +

50 DNS of Couette flow Coherent part of stresses (< u > u) 2 /u 2 τ < w > 2 /u 2 τ < v > 2 /u 2 τ y + < u >< v >/u 2 τ y/h Rollers responsible for significant fraction of Reynolds stresses

51 DNS of unstably stratified flows Computational set-up θ 2 y Π θ Q 2h L x u θ 1 > θ 2 Q L z Channel boundary layer Confined domain No rotation No roughness Imposed θ

52 Effects of unstable stratification Streets of cumulus clouds 2 4 km

53 Flow cases Reb Ra

54 Selection of computational mesh Grid selected so as to simultaneously satisfy Spacing restriction for Rayleigh-Bénard convection (Shishkina et al. 2010) Grid spacing restrictions for forced convection x + z For all simulations, /η 3 throughout Box size is a sensitive issue (reminiscent of Couette flow)... We eventually went for L x = 16h, L z = 8h

55 Flow cases Flow case Re b Ra Ri b h/l Re τ Nu C f N x N y N z RUN Ra9 Re NA RUN Ra9 Re E RUN Ra8 Re NA RUN Ra8 Re E RUN Ra8 Re E RUN Ra8 Re E RUN Ra8 Re E RUN Ra7 Re NA RUN Ra7 Re E RUN Ra7 Re E RUN Ra7 Re E RUN Ra7 Re3.5 LA E RUN Ra7 Re3.5 SM E RUN Ra7 Re3.5 NA E RUN Ra7 Re E RUN Ra7 Re E RUN Ra6 Re NA RUN Ra6 Re E RUN Ra6 Re E RUN Ra6 Re E RUN Ra6 Re E RUN Ra6 Re E RUN Ra6 Re E RUN Ra5 Re E RUN Ra5 Re E RUN Ra4 Re E RUN Ra4 Re E RUN Ra0 Re E RUN Ra0 Re E RUN Ra0 Re E

56 Flow cases DNS at Ri = 1: contour lines of θ cold and hot

57 Spectra Channel centerplane v Ez(v)(εν 5 ) 1/ k z 5/3 u Êz(u) v k zη θ λ z/h Êz(v) Êz(θ) Ri= Ri=100 Ri=10 Ri=1 Ri=0.1 Ri=0.01 Ri= λ z/h λ v = λ w = 2λ u λ z/h

58 Spectra Near-wall plane (y = y P ) u 10 0 v 10 0 kzêz(u) Ri b kzêz(v) h w λ z/h θ λ z/h kzêz(w) kzêz(θ) Ri b = Ri b =100 Ri b =10 Ri b =1 Ri b =0.1 Ri b =0.01 Ri b = λ z/h λ z/h

59 Free convection (Ri b ) Expectations Prandtl (32): the typical vertical velocity scale of buoyant plumes is vp = (βgqy) 1/3 the associated temperature scale is θp = Q 2/3 (βgy) 1/3 Prt const. Consequences Scaling of mean velocity and temperature θ(y) θ 0 θ τ = 3B θ [ (y/l) 1/3 (y 0 /L) 1/3[ u(y) u 0 u τ = 3B u [ (y/l) 1/3 (y 0 /L) 1/3] Velocity fluctuations to scale as v 2 /u 2 τ (y/l) 2/3 Temperature fluctuations to scale as θ 2 /θ 2 τ (y/l) 2/3

60 Flow statistics Free convection: power-law diagnostics θ d log(θcl θ)/d log y u d log u 2 /d log y Nu y/h Nu y/h KH θ d log θ 2 /d log y v d log v 2 /d log y Nu y/h Ra=10 6 Ra=10 7 Ra=10 8 Ra= Nu y/h Kader & Yaglom 90: u should depend on h-sized eddies (y/l) 2/3

61 Forced convection u u/uτ Re b = Re b =10 4 Re b = θ (θw θ)/θτ yu τ /ν k k θ yu τ /ν

62 Mixed convection Mean profiles θ (θ θcl)/ θ Ri b Ri b = Ri b =100 Ri b =10 Ri b =1 Ri b =0.1 Ri b =0.01 Ri b =0.001 Ri b =0 u u/ucl Ri b Ri b = 100 v y/h θ y/h v 21/2 /uτ Ri b θ 21/2 /θτ Ri b Ri b y/h y/h

63 Flow statistics Fluxes and correlations Ri b 1 Ri f Pr t Ri b yu τ/ν yu τ/ν 0.8 Ri b 0.8 Cvθ Cuθ Ri b =100 Ri b =10 Ri b =1 Ri b =0.1 Ri b =0.01 Ri b =0.001 Ri b =0 Ri b yu τ/ν Ri f = βgv θ u v du/dy, yu τ/ν Pr t = ν t α t = u v v θ dθ/dy du/dy

64 Monin-Obukhov similarity Obukhov 46, Monin & Obukhov 54: single length scale encapsulating effect of shear and buoyancy Flow fully characterized by u τ, L L = u3 τ Qβg Constancy of total stress and total heat flux u v τ w, v θ Q Universal representation ( ) y dθ y θ τ dy = ϕ h, L u 2 1/2 ( ) ( ) i y = ϕ i, θ 21/2 y = ϕ θ, u τ L θ τ L ( ) y du y u τ dy = ϕ m L u θ Asymptotic behavior discussed by Kader & Yaglom 90 Q = ϕ uθ ( y L )

65 Monin-Obukhov similarity Parametrizations Unstable stratification: Businger 71, Dyer 74 ϕ h = 1 k θ (1 + γ h y/l) α h, ϕ m = 1 k (1 + γ my/l) αm, Typical choice (Paulson 70): k = k θ = 0.35, γ m = γ h = 16, α h = 1/2, α m = 1/4 Quantity k DNS γ DNS α DNS α KY α BDP ϕ h /3-1/2 ϕ m /3-1/4 ϕ /3 / ϕ /3 1/3 ϕ θ /3 / ϕ uθ /3 / KY Kader & Yaglom 90, BDP Businger-Dyer-Panofsky Account for deviations from alleged free-convection scaling Widely used as wall models for LES of atmospheric BL (e.g. Khanna & Brasseur 97, 98)

66 Monin-Obukhov similarity DNS data θ 10 1 u 10 1 y/θτ dθ/dy 10 0 y/uτ du/dy y/l y/l Solid: KY, Dashed: BDP; Chained: DNS fit

67 Monin-Obukhov similarity DNS data u 10 1 v 10 1 u 21/2 /uτ 10 0 v 21/2 /uτ y/l y/l Solid: KY, Dashed: BDP; Chained: DNS fit

68 Monin-Obukhov similarity DNS data θ 10 1 u θ 10 1 θ 21/2 /θτ 10 0 u θ /Q y/l y/l Solid: KY, Dashed: BDP; Chained: DNS fit

69 Summary DNS of canonical flows starting to approach Re τ typical of asymptotic wall turbulence Near log layer found for mean velocity Significant flow-dependent deviations: Re τ limit unclear Logarithmic growth of velocity variances confirmed: unbounded growth in wall units? Rollers form in wide variety of flow conditions (0.01 Ri b 100) Aspect ratio 2 at least Strong modulation of near-wall region Prandtl scaling for natural convection does not show up Monin-Obukhov scaling applies to bulk flow, but not to individual profiles Significant deviations in light-wind regime DNS data available at