The Role of Buoyancy Induced Instability in Transpirational Cooling Applications

Transpirational cooling is an effective thermal protection method in hypersonic vehicles. In order to properly manage the high heat load, an understanding of the convective flow regimes as well as the thermophysical properties of the working fluid are required. Often, the vehicle’s fuel is re-purposed as the coolant or working fluid that is passed through the porous media. If the geometry is such that the coolant is heated from below, buoyancy-induced instability can ensue resulting in a mixed convection phenomena. Transpirational cooling applications require a unique analysis which combines a Darcy–Forchheimer relationship for the momentum relation, a flowing base state which introduces non-negligible convective terms for the energy equation, and a novel consideration of a cubic density dependence on temperature. This latter feature is justified by fitting thermodynamic data for typical transpirational cooling conditions. A base state solution is provided and the onset of instability is investigated using linear stability analysis. The governing equations are solved utilizing multiple methods, comparing results from a combination of analytical solutions, finite difference, power series, and Chebyshev methods. Results demonstrate excellent consistency in predictions across these methods and indicate that including non-linear density effects promote a stabilizing effect. Finally, the effect of varying the net through-flow in the porous media is investigated.

THE INFLUENCE OF A CUBIC DENSITY LAW ON PATTERNED GROUND FORMATION

A theoretical analysis is presented for the formation of polygonal ground which consists of stone borders forming regular hexagons with soil centers. The model is based on the onset of convection in a saturated soil below which is a cold permafrost layer. Darcy’s law is adopted with zero fluid inertia and the density is assumed constant except in the buoyancy term. In this term we take the equation of state proposed by Merker et al.,5 this being the one which assumes the density has a cubic dependence on temperature. Theoretical predictions of the model, particularly those for width to depth of the patterned ground cell are reported in detail.