The Lattice Boltzmann Method for Laminar and Turbulent Channel Flows

Transcription

1 The Lattice Boltzmann Method for Laminar and Turbulent Channel Flows Vanja Zecevic, Michael Kirkpatrick and Steven Armfield Department of Aerospace Mechanical & Mechatronic Engineering The University of Sydney Australia

2 Introduction We aim to develop a high performance, highly scalable parallel code for the simulation of fundamental turbulent flows. In this work, we use the lattice Boltzmann method to simulate laminar and turbulent channel flows. This method allows an efficient implmentation on highly parallel architectures, in this case graphics processing units (GPUs). It is also suitable for more coarse parallelization across multiple processors or GPUs. One drawback of the method is that non uniform grids are difficult to implement. We compare a relatively high resolution uniform grid using our code to results published by Moser et. al. [3] using a spectral method with the grid refined at the walls. Our results aproach Mosers results as the grid is refined.

3 LBM - Summary Tracks particle populations f i (x,y) Time advancement: collision step + streaming step. f i = ρ and f i c ia = ρu a conserved. Artificially compressible. Review by Benzi, Succi & Vergassola [1]. Bhatnagar Gross Krook collision operator.

4 Domain f y Solid boundary y x Periodic boundary z Simplest BC s are bounce-back. We use the Guo BC [2].

5 Hardware & Technologies Hardware CPU intel E8600 GPU 2 GTX295 RAM 4 GB 1066 MHz DDR3 PSU 1500 W Mobo GA-X48 Chipset intel X48 PCI-e 2 16 PCI-e 2.0 Technologies OS Debian Lenny AMD64 Compiler gcc, nvcc libs lcudart, pthreads GPU nforce 200 PCI-E bridge RAM Hardware Schematic north bridge CPU V-RAM

6 Parallel Implementation GPU uses fine grained data-parallel approach. Many execution units (eg. 240) and even more threads (millions). Synchronization over the entire domain cripples performance. [4] All communications in LBM are nearest neighbor and data dependence requires two synchronizations per time step. Conjugate Gradient solvers require multiple synchronizations and whole domain communications per solver iteration. Allows us to get 92% parallel efficiency when scaling from one to four GPU s.

7 Domain Decomposition GPU 2 A boundary A B C C boundary A C write A boundary B write C boundary synchronize read A & C boundaries

8 Laminar Channel Flow - Parameters 1 Define the following dimensionless quantities, l char = δ u char = u max t char = l char u char l d = l lu l char (1) u d = u lu u char (2) t d = t lu t char (3) The following are constants. t lu = 1 x lu = 1 (4) c s = speed of sound = 1 3 (5)

9 Laminar Channel Flow - Parameters 2 The following equations can be used as constraints, t d = 1lt = u char (1lt) = u max t char l char δ (1lt) (6) x d = 1lu = 1 (1lu) (7) l char δ Ma = u max = t d c s c s x d (8) Re = u maxδ ν (9) For a laminar channel flow we can estimate: u max = fδ2 2νρ (10)

10 Laminar Channel Flow - Parameters 3 8 free variables are t d, x d, δ, Ma, ν, f, u max and Re 5 equations linking these variables. Usually we would specify t d, x d and Re, however it is more helpful to specify Ma. The Mach number plays a similar role to the finite element CFL number. CFL = t luu max x lu = u max (11) Ma = u max c s = CFL c s = 3 CFL (12) We use Ma = 0.1 for our turbulent simulation which is somewhat equivalent to, CFL = 1 3 Ma = 0.06 (13)

11 Laminar Channel Flow - Results Error vs Ma (Ny = 160) 1.50E 04 y = x R² = Error Error 1.00E E E E E E E E E E E E+00 0E+00 5E 05 1E 04 2E 04 2E 04 3E 04 dx^2 Ma Error vs dx^2 (Ma=0 Extrapolation) 1.40E 03 y = x e-06 R² = Error compared to the Poiseuille profile. 2nd order in space and 1st order in Mach number.

12 Turbulent Channel Flow - Parameters 1 Compared to spectral method of Moser et. al. [3] fδ = τ w = u z µ u τ = fδ ρ Re τ = u τδ ν = δ ν τw ρ = fδ ρ (14) = 180 (15) N y l y N x l x N z l z N tot LBM π 176 π LBM π 176 π LBM π 212 π LBM π 246 π Moser π π

13 Turbulent Channel Flow - Parameters 2 Initial condition = curl noise. D3Q15 had instability and aliasing.

14 Turbulent Channel Flow - Onset of Turbulence Friction Velocity vs Time friction velocity LBM LBM t (turnover times)

15 Turbulent Channel Flow - Mean Profile u mean 10 u mean LBM - 92 LBM LBM LBM Moser et al. 4 2 LBM - 92 LBM LBM LBM Moser et al y+ Log law region and viscous sublayer both apparent y+

16 Turbulent Channel Flow - TKE LBM - 92 LBM LBM LBM Moser et al. 4 LBM - 92 LBM LBM LBM Moser et al. 3 3 TKE TKE y+ Height of peaks is a good indicator of agreement y+

17 Turbulent Channel Flow - Energy Spectrum Energy density LBM LBM pi 1E 06 Moser et al. -5/3 Power Law 1E Wave number Dotted line shows power law expected in inertial range. Nyquist frequency in Mosers results has a wave number 32. 2π stream-wise dimension appears to be too small with significant energy in the largest wavelength.

18 Performance LBM appears compute bound. High performance. Good scaling. Performance Theoretical Updates per s GFLOPS (DP) Gbps

19 Conclusion Convergence was verified using laminar flow. The LBM was successful in simulating a fully turbulent flow. Found interesting turbulent channel flow results comparing a fine square grid with a locally refined stretched grid. High performance and good parallel scaling. Research will continue on to thermal turbulent flows using a Boussinesq approximation.

20 References [1] R. Benzi, S. Succi, and M. Vergassola. The lattice boltzmann-equation - theory and applications. Phys. Rep., 222(3): , Dec [2] Z. L. Guo, C. G. Zheng, and B. C. Shi. An extrapolation method for boundary conditions in lattice boltzmann method. Phys. Fluids, 14(6): , Jun [3] R. D. Moser, J. Kim, and N. N. Mansour. Direct numerical simulation of turbulent channel flow up to Re τ =590. Phys. Fluids, 11(4): , Apr [4] NVIDIA. NVIDIA CUDA C Programming Best Practices Guide, 2009.