Simple lattice Boltzmann model for simulating flows with shock wave

Abstract

We propose a lattice Boltzmann model for compressible Euler equations. The numerical examples show that the model can be used to simulate shock wave and contact discontinuity. The results are compared with those obtained by traditional methods.

Finite Difference Lattice Boltzmann Method for Compressible Thermal Fluids

“…Consequently, he obtained an expression for given by D n 2=D n where n is the number of extra degrees of freedom. The finite difference LBM based on this proposal is numerically stable and represents an improvement over those recently proposed in [24,25]. Therefore, it is possible to extend the BGK/BE to simulate thermal fluid flows in the subsonic to supersonic range without any numerical instability [22].…”

Progress in particle-based multiscale and hybrid methods for flow applications

“…They applied their scheme to the 1D and 2D Sod shock tube problem as done by Guangwu et al (1999). Despite the absence of tabulated data in the work of Joshi et al (2010), their scheme qualitatively exhibited the same behaviour as the pure LBM, see Guangwu et al (1999). Compared to two solutions obtained from the pure Euler equation making use of the Godunov and Flux Vector Splitting (FVS) scheme, their solution compared well to the Euler results with spurious oscillation and numerical dispersion still present.…”

“…Compressible effects are difficult to model since the small number of discrete velocities in the speed model allows only small temperature variations. Successful applications have been presented, for example, by Guangwu et al (1999), who simulated Sod's and Lax's shock tube problem. Despite following the trend of the reference solution, spurious oscillation, as well as numerical dispersion, was present.…”

“…Joshi et al (2010) constructed a hybrid LBM in conjunction with a finite volumebased Euler solver where the primitive variables at the cell centres were obtained from the Euler equations, while the inter-cell fluxes were approximated by the LBM through a modified equilibrium distribution function to account for thermal effects. They applied their scheme to the 1D and 2D Sod shock tube problem as done by Guangwu et al (1999). Despite the absence of tabulated data in the work of Joshi et al (2010), their scheme qualitatively exhibited the same behaviour as the pure LBM, see Guangwu et al (1999).…”

“…To incorporate heat transfer and to compute the temperature field, as stated previously, the equilibrium distribution function needs to be modified and higher-order speed models should be used, i.e those that take their immediate neighbours and their neighbours into account. Various models have been introduced to account for the energy and thus for the temperature (Guangwu et al 1999;Kataoka and Tsutahara 2004a, b;Shi et al 2001) and are reviewed in Kun (2008). Mezrhab et al (2004) discussed the thermal models available in the Lattice Boltzmann framework and concluded that these are still premature and need further development.…”