Abstract
Current approaches to Direct Numerical Simulation (DNS) are computationally quite expensive for most realistic scientific and engineering applications of Fluid Dynamics such as automobiles or atmospheric flows. The Lattice Boltzmann Method (LBM), with its simplified kinetic descriptions, has emerged as an important tool for simulating hydrodynamics. In a heterogeneous computing environment, it is often preferred due to its flexibility and better parallel scaling. However, direct simulation of realistic applications, without the use of turbulence models, remains a distant dream even with highly efficient methods such as LBM. In LBM, a fictitious lattice with suitable isotropy in the velocity space is considered to recover Navier-Stokes hydrodynamics in macroscopic limit. The same lattice is mapped onto a cartesian grid for spatial discretization of the kinetic equation. In this paper, we present an inverted argument of the LBM, by making spatial discretization as the central theme. We argue that the optimal spatial discretization for LBM is a Body Centered Cubic (BCC) arrangement of grid points. We illustrate an order-of-magnitude gain in efficiency for LBM and thus a significant progress towards feasibility of DNS for realistic flows.
Highlights
Improving the accuracy of LBM, while keeping its parallel efficiency and large time-stepping intact, is an important challenge
A fundamental problem with such approaches is that the local accuracy of the method remains unchanged and optimization is done only with respect to the global distribution of grid points
For decaying turbulence in a periodic geometry, local grid refinements cannot improve the accuracy of LBM
Summary
We begin with the observation that the introduction of the stair-case geometry, as done in LBM, is similar to the generation of a Wigner-Seitz cell for a given lattice structure[29] One can expect that RD3Q27 should show enhanced accuracy for finite Knudsen flows as compared to D3Q2738 This set-up provides a good indication of the convergence of the discrete velocity model towards the Boltzmann equation. Where β is the discrete relaxation time Notice this is similar to D3Q27 model, if we choose Δt =Δx, the advection happens from one lattice point to another and the method does not require any spatial interpolations. We recall that the BCC grid in 3D has an alternate interpretation in terms of replica of simple cubic grids displaced from each other by Δx/2 in each direction where Δx is the grid spacing
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