A Discrete Flux Scheme for Aerodynamic and Hydrodynamic Flows (Communications in Computational Physics, 2020)
The objective of this paper is to seek an alternative to the numerical simulationof the Navier-Stokes equations by a method similar to solving the BGK-typemodeled lattice Boltzmann equation. The proposed method is valid for both gas andliquid flows. A discrete flux scheme (DFS) is used to derive the governing equationsfor two distribution functions; one for mass and another for thermal energy. Theseequations are derived by considering an infinitesimally small control volume with avelocity lattice representation for the distribution functions. The zero-order momentequation of the mass distribution function is used to recover the continuity equation,while the first-order moment equation recovers the linear momentum equation. Therecovered equations are correct to the first order of the Knudsen number (Kn); thus,satisfying the continuum assumption. Similarly, the zero-order moment equation ofthe thermal energy distribution function is used to recover the thermal energy equation.For aerodynamic flows, it is shown that the finite difference solution of the DFSis equivalent to solving the lattice Boltzmann equation (LBE) with a BGK-type modeland a specified equation of state. Thus formulated, the DFS can be used to simulate avariety of aerodynamic and hydrodynamic flows. Examples of classical aeroacoustics,compressible flow with shocks, incompressible isothermal and non-isothermal Couetteflows, stratified flow in a cavity, and double diffusive flow inside a rectangle are usedto demonstrate the validity and extent of the DFS. Very good to excellent agreementwith known analytical and/or numerical solutions is obtained; thus lending evidenceto the DFS approach as an alternative to solving the Navier-Stokes equations for fluidflow simulations.