On the Computation of Some External or Partially Enclosed Natural Convection Flows

We address the issue of the computation of natural convection in external or partially enclosed domains, such as natural convection flow in vertical open channels or along heated vertical plates which raises the question of the proper choice of artificial boundary conditions and of their numerical implementation. To this aim we perform a singular value decomposition of the discrete Stokes operator that arises from the discretization of the governing Navier-Stokes equations, using the classical staggered grid formulation. We show that some choices of boundary conditions lead to an increase of the kernel of this operator making the solution of the full nonlinear equations indefinite. This arises in particular when Neumann type boundary conditions are imposed on the velocity component normal to the main inlet and outlet boundaries. The existence of one or more velocity-pressure combinations solutions of the homogeneous Stokes operator can be used to derive a superposition algorithm in which a linear combination of these modes is added to a particular solution of the nonlinear equations in order to satisfy given constraints such as one or more pressure conditions for instance. Sample calculations are performed to demonstrate the effectiveness of this methodology, which has many applications in other configurations such as pipe flows with several outlets or free plane jets for instance.