Numerical simulation of a low-Mach-number flow with a large temperature variation

Compressibility is important in a low-Mach-number flow with a large temperature variation. However, it is well known that time-dependent compressible flow schemes become ineffective at low Mach numbers. This ineffectiveness occurs because a wide disparity exists between the time scales associated with convection and the propagation of acoustic waves. For this reason, scale analysis modifies the compressible equations in order to remove acoustic waves from them. The pressure is divided into the thermodynamic and dynamic parts. The density variation caused by the variation in the dynamic part is neglected. The scale analysis shows the conditions under which the modified equations are applicable. Examination of small-amplitude waves show that the modified equations contain internal gravity waves (buoyant effects), while they exclude acoustic waves. Similarly to the modified equations, the Boussinesq equations are derived under a further condition. A finite difference scheme integrates the modified equations. The scheme is essentially the MAC method. First, thermal convection of a Boussinesq fluid in a square cavity is simulated in order to validate the calculation method. The result is in good agreement with a benchmark solution. Second, thermal convection with a large temperature variation is simulated in a vertical pipe furnace used for the heat treatment of semiconductor wafers. An axisymmetric steady flow is obtained.