An implicit dual-time stepping spectral difference lattice Boltzmann method for simulation of viscous compressible flows on structured meshes

In this work, the spectral difference lattice Boltzmann method (SDLBM) is extended and applied for accurately computing two-dimensional viscous compressible flows on structured meshes. Here, the compressible form of the discrete Boltzmann-BGK equation with the Watari model is considered and the numerical solution of the resulting LB equation is obtained by using the spectral difference method. The main benefit of the use of the LB method in simulating compressible flows is that a same formulation can be applied to compute the inviscid and viscous portions of the flowfield. Note that the LB formulation for simulating the viscous flows is the same as that used for the inviscid ones, however, the wall boundary conditions are changed in a manner to consider the no-slip conditions. Here, the SDLBM is also extended to use curved-edge cells for properly representing curved wall boundaries. In addition, to enhance the solution of the SDLBM the time integration is efficiently performed by implementing an implicit dual-time stepping method which does not require a matrix inversion. Both steady and unsteady flows are simulated. Different test cases including the Couette flow, the viscous shock–vortex interaction, the flow over a circular cylinder and the flow over a NACA-0012 airfoil are simulated to assess the accuracy and robustness of the solution procedure proposed based on the SDLBM in computing steady and unsteady viscous compressible flows. The present results obtained by applying the SDLBM exhibit good agreement compared to the analytical and available high-order accurate solutions of the LB and Navier–Stokes equations.