Finite difference calculations of buoyant convection in an enclosure: verification of the nonlinear algorithm

Solutions are presented to nonlinear finite difference equations used to represent fire-driven buoyant convection in enclosures. The solutions depend upon the fact that these difference equations permit the decomposition of the discretized velocity field into solenoidal and irrotational components. The irrotational field is shown to satisfy a finite difference analog of Bernoulli's equation when the density is constant. This leads to a three-dimensional time-dependent solution to the difference equations. The solenoidal field is shown to possess steady-state two-dimensional solutions corresponding to a constant non-zero value of the discretized vorticity. The two solutions, together with results presented elsewhere describing finite difference approximations to linear internal waves in enclosures, have been used in the development and testing of the computer-based algorithms used to solve these equations. They have proved particularly useful in assessing the accuracy of finite difference approximations to the equations of inviscid fluid mechanics, as well as in debugging the computer codes implementing these algorithms.