In this paper, lattice Boltzmann method and large eddy simulation are adopted to
simulate the laminar and turbulent Poiseuille flow through square duct. The shear
or friction Reynolds number based on the duct width are 10 for the laminar
flow, whereas are 360 for the turbulent flow. We validate the governing equations
and the boundary conditions by simulating laminar Poiseuille duct flow. The results
of the laminar flow show good agreement compared with analytic solution. For the
turbulent Poiseuille square duct flow at Re = 360, our simulation is able to capture
the turbulence quantities by observing the mean streamwise velocity profile along
the wall bisector compared with the direct numerical simulation data and the law
of the wall. Then the simulation result of turbulence intensities and the Reynolds
stress variation along the wall bisector is also shown in present research.

1 Introduction
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature survey . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Lattice Boltzmann method . . . . . . . . . . . . . . . . . . 3
1.2.2 Multiple-relaxation-time lattice Boltzmann models . . . . . . 3
1.2.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . 4
1.2.4 Large eddy simulations . . . . . . . . . . . . . . . . . . .. 5
1.2.5 Turbulent Poiseuille flow . . . . . . . . . . . . . . . . . . 6
1.3 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Methodology
2.1 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . 7
2.2 The BGK and the low-Mach-number approximations . . .. . . . . . 9
2.2.1 The BGK approximation . . . . . . . . . . . . . . . . . . . . 9
2.2.2 The low-Mach-number approximation . . . . . . . . . . . . . . 11
2.3 Discretization of the BGK equation . . . . . . . . . . . . . . .12
2.3.1 Spatial discretization . . . . . . . . . . . . . . . . . . . .12
2.3.2 Temporal discretization . . . . . . . . . . . . . . . . . . . 14
2.4 The multiple-relaxation-time model . . . . . . . . . . . . . . .16
2.5 Large Eddy Simulation . . . . . . . . . . . . . . . . . . . . . 18
2.5.1 The filtering operation . . . . . . . . . . . . . . . . . . . 18
2.5.2 The filtered Navier-Stokes equations . . . . . . . . . . . . .19
2.5.3 The lattice Boltzmann Subgrid-scale model . . . . . . . . . . 19
3 Numerical algorithm
3.1 Simulation procedure . . . . . . . . . . . . . . . . . . . . . .22
3.2 The external forcing term . . . . . . . . . . . . . . . . . . . 23
3.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . 23
3.4 Parallel algorithm . . . . . . . . . . . . . . . . . . . . . . .24
4 Numerical results
4.1 3D laminar Poiseuille flow . . . . . . . . . . . . . . . . . . .26
4.2 3D turbulent Poiseuille flow . . . . . . . . . . . . . . . . . .27
5 Conclusions

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