Commun. Comput. Phys., 9 (2011), pp. 1257-1283. |
A Discrete Flux Scheme for Aerodynamic and Hydrodynamic Flows S. C. Fu ^{1*}, R. M. C. So ^{2}, W. W. F. Leung ^{3} 1 Mechanical Engineering Department, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong.2 Mechanical Engineering Department, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong; and School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907-2088, USA. 3 Mechanical Engineering Department and Research Institute of Innovative Products and Technologies, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. Received 31 October 2009; Accepted (in revised version) 24 November 2010 Available online 28 January 2011 doi:10.4208/cicp.311009.241110s Abstract The objective of this paper is to seek an alternative to the numerical simulation of the Navier-Stokes equations by a method similar to solving the BGK-type modeled lattice Boltzmann equation. The proposed method is valid for both gas and liquid flows. A discrete flux scheme (DFS) is used to derive the governing equations for two distribution functions; one for mass and another for thermal energy. These equations are derived by considering an infinitesimally small control volume with a velocity lattice representation for the distribution functions. The zero-order moment equation of the mass distribution function is used to recover the continuity equation, while the first-order moment equation recovers the linear momentum equation. The recovered equations are correct to the first order of the Knudsen number (Kn); thus, satisfying the continuum assumption. Similarly, the zero-order moment equation of the 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 DFS is equivalent to solving the lattice Boltzmann equation (LBE) with a BGK-type model and a specified equation of state. Thus formulated, the DFS can be used to simulate a variety of aerodynamic and hydrodynamic flows. Examples of classical aeroacoustics, compressible flow with shocks, incompressible isothermal and non-isothermal Couette flows, stratified flow in a cavity, and double diffusive flow inside a rectangle are used to demonstrate the validity and extent of the DFS. Very good to excellent agreement with known analytical and/or numerical solutions is obtained; thus lending evidence to the DFS approach as an alternative to solving the Navier-Stokes equations for fluid flow simulations. AMS subject classifications: 76M25, 76D05Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Aerodynamics, hydrodynamics, lattice Boltzmann equation. *Corresponding author. Email: mm.scfu@polyu.edu.hk (S. C. Fu), mmmcso@polyu.edu.hk (R. M. C. So), mmwleung@polyu.edu.hk (W. W. F. Leung) |