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Equations for a Boussinesq model describing transient buoyant convection driven by a heat source in a rectangular enclosure are presented and solved by finite difference methods. Gravity is allowed to have an arbitrary direction relative to the enclosure so that the enclosure is inclined to horizontal. Computational results for three-dimensional dissipation-free flows and for two-dimensional flows with and without dissipation are presented. The hydrodynamics is based directly on the time-dependent Euler or Navier-Stokes equations. No turbulence model or other empirical parameters are introduced. The previous algorithms had been verified by comparisons with exact solutions to the equations in simple, special cases, and overall predictions of the model when the viscosity and thermal conductivity are zero have been compared with experimental results. The use of Lagrangian particle tracking allows one to visualize the flow patterns. The effects of a fire-induced flow in a corridor, and a stair well (or escalator) are examined.