to
, and such flows are certainly turbulent. In this chapter, we
restrict ourselves to Rayleigh numbers of 
. At such Rayleigh
numbers, the flow is laminar.
Two-dimensional flow in a square cavity is considered, where the motion is driven by cooling of the fluid at one vertical wall. The other three boundaries of the cavity are insulating. Natural convection in a square box is a standard test problem for CFD codes.
The problem that originally motivated this study was an idealised treatment of the natural convection of molten steel in a large vat; however, the present study has merit in its own right as a natural convection problem at temperatures close to freezing. At the true Rayleigh number for molten steel, it would be necessary to incorporate a turbulence model, and to take better account of the approach to freezing. This would involve a much more complicated model than is assumed here.
Even so, as you will see shortly, this problem involves a complicated set of coupled equations and requires substantial algorithmic design and Fasttalk programming. The computer time required to solve this 2D transient convection problem is a big increase on any example treated so far in this Tutorial Guide. The natural convection problem is solved using an implicit timestepping procedure. The region near the freezing point is modelled using a temperature-dependent viscosity and specific heat, and the simulations are carried out only as far as the early stages of mushy zone formation.
The key parameters in the problem are the Rayleigh number Ra and the
Prandtl number Pr. The Rayleigh number describes the importance of
buoyancy forces which drive convection compared to the two diffusive
processes (heat, momentum) which act to stabilise convection. Rayleigh
numbers for the real situation might be as large as to
, and such flows are certainly turbulent. In this chapter, we
restrict ourselves to Rayleigh numbers of . At such Rayleigh
numbers, the flow is laminar.