Mixed convection boundary layers with reverse flows

What is it about?

The other limiting case where the flow is dominated mainly by free convection is considered in the present study. First, an asymptotic approximation is developed in order to describe the stagnation point flow. Because of the upstream influence originating from the coexistence of ascending and descending flow in the opposite mixed convection region, a direct integration of the boundary layer equations fails to converge. In order to resolve this difficulty, the boundary layer equations are separately integrated on each side of the curve along which the longitudinal component of the velocity vanishes. It is however assumed, in accordance with our approach of weakly perturbed free convection boundary layer flow, that the boundary layer thickness remains a well defined function along the surface and the first approximation to the external flow therefore remains the classical potential flow past the geometrical body.

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Why is it important?

A free convection boundary layer perturbed by a transverse stream around a heated circular cylinder is investigated. Whereas obvious Cauchy conditions, deduced by symmetry of the governing equations, are associated with the symmetric mixed convection problem, the asymmetric case has no trivial Cauchy conditions because of the downstream–upstream interaction occurring in the opposite mixed convection domain. In the limiting case of small forced convection effects, a semi-analytical solution is found near the free convection stagnation point. This solution allows us to retrieve approximate Cauchy conditions and accounts for the particular local structure of the flow. The latter is then extended outside the stagnation zone by solving the whole problem by a direct second order numerical method in the favorable mixed convection domain. This procedure however fails to converge in the opposite mixed convection domain. In order to avoid numerical divergence owing to the existence of reverse flow, an iterative method which takes into account the flow direction is implemented. The procedure works primarily by seeking the boundary curve separating the upstream and downstream flows. It is based upon a variational method which consists of calculating the extremal of the jumps of the radial derivatives of the temperature and the angular velocity component along the free curve. In this way, the equilibrium state is reached with only a few iterations.