Introducing a new kinetic model which admits an H-theorem for simulating the nearly incompressible fluid flows

Abstract

We present a new kinetic model, for simulating the nearly incompressible fluid flows. The model uses constant speed particles which are free to move in all directions. In the new model, the Maxwellian is replaced by a fractional form, while the Navier–Stokes and continuity equations are recovered. It is known that the conventional lattice Boltzmann model (LBM) with polynomial equilibrium distribution function cannot admit an H-theorem (Wagner, 1998) [9]. In the present work, we show that the new model admits an H-theorem and the numerical schemes which stem from the new model are more stable than the conventional LBM. For the streaming stage, two different approaches, namely the lattice based and the discontinuous Galerkin based schemes, are introduced. The former is more cost effective than the conventional lattice Boltzmann models while it maintains their simplicity. The latter is a higher order spectral element method, using which one can employ non-uniform triangular elements with high accuracy and geometrical flexibility. Unlike the conventional LBM, the new model is derived from continuous relations. Hence, it does not require a symmetric discrete velocity model. The accuracy and stability of the model have been verified, by simulating three benchmark problems. The plane Couette flow, lid driven square cavity, and flow around an impulsively started cylinder have been simulated. The results of the present work are in excellent agreement with the exact solutions and with the results reported by others.