Abstract
A double-distribution-function based lattice Boltzmann method (DDF-LBM) is proposed for the simulation of polyatomic gases in the supersonic regime. The model relies on a numerical equilibrium that has been extensively used by discrete velocity methods since the late 1990s. Here, it is extended to reproduce an arbitrary number of moments of the Maxwell–Boltzmann distribution. These extensions to the standard 5-constraint (mass, momentum and energy) approach lead to the correct simulation of thermal, compressible flows with only 39 discrete velocities in 3D. The stability of this BGK-LBM is reinforced by relying on Knudsen-number-dependent relaxation times that are computed analytically. Hence, high Reynolds-number, supersonic flows can be simulated in an efficient and elegant manner. While the 1D Riemann problem shows the ability of the proposed approach to handle discontinuities in the zero-viscosity limit, the simulation of the supersonic flow past a NACA0012 aerofoil confirms the excellent behaviour of this model in a low-viscosity and supersonic regime. The flow past a sphere is further simulated to investigate the 3D behaviour of our model in the low-viscosity supersonic regime. The proposed model is shown to be substantially more efficient than the previous 5-moment D3Q343 DDF-LBM for both CPU and GPU architectures. It then opens up a whole new world of compressible flow applications that can be realistically tackled with a purely LB approach.This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.
Highlights
The lattice Boltzmann method (LBM) is a popular numerical scheme capable of computing solutions to the Boltzmann equation (BE) in a regime of small deviations from the local equilibrium state [1,2]
We extend the above compressible LBM by deriving exponential discrete equilibria designed to recover an arbitrary number of the Maxwellian moments, with particular emphasis on the 13-moment approach
A new type of quadrature free LBM was introduced in the context of compressible flow simulation
Summary
The lattice Boltzmann method (LBM) is a popular numerical scheme capable of computing solutions to the Boltzmann equation (BE) in a regime of small deviations from the local equilibrium state [1,2]. Simulations are stable for relatively low values of the relaxation time (τf = 0.57, Pr = 1), even with the standard BGK operator, while LBMs based on a polynomial discrete equilibrium usually require a large number of discrete velocities (e.g. 81 in 2D [73]), more robust collision models [16,17] or numerical discretizations [86], in order to achieve similar results. The Knudsen number based relaxation times drastically improve the stability of the present model, allowing the simulation of the 1D Riemann problem in the zero-viscosity limit (τf = 0.5, Pr = 1), with an accuracy that competes with LBMs coupled with shock capturing techniques [87] The behaviour of this kinetic sensor is investigated in more detail in figure 6. This is very promising considering the fact that a relatively coarse mesh and the very simple halfway bounce back rule were used
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