An implicit finite element solution of thermal flows at low Mach number

Thermal flows at low Mach numbers are a basic problem in combustion, environmental pollution prediction and atmospheric physics areas. Most of the existing schemes for solving this problem treat convection explicitly, which confines time step width due to the CFL condition. In this paper, based on the pseudo residual-free bubble approach [F. Brezzi, L.P. Franca, T.J.R. Hughes, A. Russo, b=∫gb=∫g, Methods Appl. Mech. Eng. 145 (1997) 329–339; T.J.R. Hughes, Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilised methods, Method. Appl. Mech. Eng. 127 (1995) 387–401], we introduce an implicit finite element scheme for the thermal flow problem. We firstly give a low Mach number asymptotics of compressible Navier–Stokes equations for the thermal flows and then derive the numerical scheme for them in detail. Three representative case studies are used to investigate and to test the numerical performances of the proposed scheme.