Abstract
Turbulent compressible flows are traditionally simulated using explicit time integrators applied to discretized versions of the Navier-Stokes equations. However, the associated Courant-Friedrichs-Lewy condition severely restricts the maximum time-step size. Exploiting the Lagrangian nature of the Boltzmann equation's material derivative, we now introduce a feasible three-dimensional semi-Lagrangian lattice Boltzmann method (SLLBM), which circumvents this restriction. While many lattice Boltzmann methods for compressible flows were restricted to two dimensions due to the enormous number of discrete velocities in three dimensions, the SLLBM uses only 45 discrete velocities. Based on compressible Taylor-Green vortex simulations we show that the new method accurately captures shocks or shocklets as well as turbulence in 3D without utilizing additional filtering or stabilizing techniques other than the filtering introduced by the interpolation, even when the time-step sizes are up to two orders of magnitude larger compared to simulations in the literature. Our new method therefore enables researchers to study compressible turbulent flows by a fully explicit scheme, whose range of admissible time-step sizes is dictated by physics rather than spatial discretization.
Highlights
One major challenge in fluid dynamics is the study of compressible turbulent flows, involving intrinsic as well as variable density compressibility effects [1,2,3,4,5,6,7]
We explore the capabilities of the semi-Lagrangian lattice Boltzmann method (SLLBM) for three-dimensional compressible flows
The SLLBM for three-dimensional compressible flows is a viable alternative to other solvers
Summary
One major challenge in fluid dynamics is the study of compressible turbulent flows, involving intrinsic as well as variable density compressibility effects [1,2,3,4,5,6,7]. Numerical simulations have become an indispensable tool to understand their physics, and many studies exploring compressible turbulent flows have been conducted using highorder compact finite difference, optimized dispersion-relation preserving schemes [19,24,25,26,27,28,29,30] for the spatial derivatives, often combined with low-dispersion-dissipation Runge-Kutta schemes for time-integration [19,31,32] These methods provide accurate results, the time steps are generally small [33], because of the methods’ Eulerian time derivatives, which describe how the variables of interest pass through fixed locations in the field.
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