High-order semi-Lagrangian kinetic scheme for compressible turbulence

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.

Cubature rules for weakly and fully compressible off-lattice Boltzmann methods

“…• Last, for three-dimensional compressible flows, a recently introduced D3Q45 velocity set was applied [48], based on a cubature rule by Konyaev [49,50]. This off-lattice velocity set was compared in an off-lattice Boltzmann simulation to the state of the art on-lattice counterparts D3Q103 and D3V107 applied to compressible on-lattice Boltzmann simulations.…”

“…The nominator C can be either set to C = Ma 2 or to C = 1; this work made use of the latter case. Table 4 [48]. Figure 8 depicts the mean kinetic energy over time for a resolution of 256 3 obtained by the velocity sets D3Q103, D3V107 and by the SLLBM D3Q45 simulation.…”

Prediction of Moments in the Particles on Demand Method for LBM

“…Interpolation of the off grid values leads to the loss of the conservation properties of the numerical scheme. In the Semi-Lagrangian methods [51][52][53][54], the populations are streamed from the off-grid positions. In the context of LBM, this kind of issue is resolved by resolving spatial discretization with finite-difference [55][56][57][58], finite-volume [59], finite element [60], Taylor expansion and least squares approximation [61] methods.…”

“…The optimal multidimensional cubatures are not used in most LBM simulations since their points can't be scaled to match the grid nodes, and interpolation of the off-grid values is often avoided. But, in PonD, there is nothing to loose since the streaming positions are already off-grid, and the search of optimal high-order cubatures has become relevant again [54]. Obtaining the explicit gauge transfer expression even for the smallest 3D velocity sets is a difficult task, and the matrix inversion at each lattice node at each time step is a costly operation.…”

Particles on Demand for flows with strong discontinuities

“…While originally conceived for standard velocity sets in order to gain geometrical flexibility [26][27][28], off-lattice propagation schemes received some traction more recently in the context of compressible flows. Interesting realizations for compressible flows at moderate Mach numbers using a semi-Lagrangian advection have recently been reported in [29][30][31]. Another class of numerical schemes are finite volume methods, such as the discrete unified gas kinetic scheme (DUGKS) [32,33].…”