Numerical analysis for turbulent buoyancy-driven flows shares many common topics with other computational fluid dynamics (CFD) fields. However, it has several special problems that must be dealt with. The major contribution of this thesis is the development of a new algorithm, SIMPLET, for buoyancy-driven flows. The essence of the SIMPLE method lies in its coupling between the momentum and continuity equations. Almost all the algorithms of the SIMPLE family are based on one precondition, that is, the corrected velocity is obtained from the corrected pressure only. However, in buoyancy-driven flows, there are two major forces driving the fluid movement: the force caused by the temperature gradients and the force caused by the pressure (including kinetic pressure) gradients. In this thesis, the effect of the temperature correction on the velocity correction is considered during the derivation of the pressure linked equation. A modification to the SIMPLE algorithm--SIMPLET--was proposed. The development of the SIMPLET is divided into two stages. The first version of SIMPLET was developed on the basis of the Boussinesq assumption. Since the temperature variation in the flow fields encountered in modern electronic equipment and other industrial facilities is large enough that the Boussinesq assumption is not acceptable, the second version of SIMPLET was developed to remove this restriction so that it can be used for general cases. Because large temperature variations invariably cause turbulence, the flows with appreciable length scales are nearly always turbulent. As a preview of the application of the SIMPLET algorithm to real industrial problems, this thesis investigates several cases of turbulent mixed convection flows in a cavity problem using the RNG turbulence model. The test cases show that the SIMPLET method can speed up the energy equation convergence rate because of its linkage between temperature and velocity. When the convergence rate of the energy equation becomes the determinant in reaching a solution, the advantage of the SIMPLET method will be prominent.