Abstract
We present a multi-scale lattice Boltzmann scheme, which adaptively refines particles' velocity space. Different velocity sets of lower and higher order are consistently and efficiently coupled, allowing us to use the higher-order model only when and where needed. This includes regions of high Mach or high Knudsen numbers. The coupling procedure of discrete velocity sets consists of either a projection of the higher-order populations onto the lower-order lattice or lifting of the lower-order populations to the higher-order velocity space. Both lifting and projection are local operations, which enable a flexible adaptive velocity set. The proposed scheme is formulated for both a static and an optimal, co-moving reference frame, in the spirit of the recently introduced Particles on Demand method. The multi-scale scheme is validated with an advection of an athermal vortex and in a jet flow setup. The performance of the proposed scheme is further investigated in the shock structure problem and a high-Knudsen-number Couette flow, typical examples of highly non-equilibrium flows in which the order of the velocity set plays a decisive role. The results demonstrate that the proposed multi-scale scheme can operate accurately, with flexibility in terms of the underlying models and with reduced computational requirements.
Highlights
With its roots in kinetic theory, the lattice Boltzmann Method (LBM) describes fluid flow via the propagation and collision of discretized particle distribution functions, which are associated with a set of discrete velocities and constructed to recover the macroscopic Navier-Stokes equations (NSE) in the hydrodynamic limit
LBM has matured to a competitive alternative to conventional numerical solvers, with a vast range of applications including compressible flows [1], complex moving geometries [2], multiphase flows [3,4] and rarefied gas dynamics [5], to mention a few
The motivation for using different velocity sets in these cases stems from the errors of the standard D2Q9 lattice as the flow velocity increases, and the refinement criterion between the higher- and lower-order velocity sets is based on a Mach number threshold
Summary
With its roots in kinetic theory, the lattice Boltzmann Method (LBM) describes fluid flow via the propagation and collision of discretized particle distribution functions (populations), which are associated with a set of discrete velocities and constructed to recover the macroscopic Navier-Stokes equations (NSE) in the hydrodynamic limit. The most common LBM models use so-called standard lattices such as the D2Q9 or the D3Q27 in two or three dimensions (D = 2, 3) with Q = 9 and Q = 27 discrete velocities, respectively While this is mainly due to their simplicity and efficiency, the limited number of speeds puts severe restrictions on their range of validity. The recently proposed Particles on Demand (PonD) method [20,21] eliminates the Galilean invariance errors of the standard lattices from the outset by representing the populations in a co-moving reference frame Note that while both of these approaches get by with minimal velocity sets, another alternative to lift the aforementioned constraints is the use of multispeed lattices. For illustration of the coupling procedure, we use a dual population, multispeed LBM model with variable Prandtl number and adiabatic exponent as proposed in [1] for high-Mach regions.
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