# The Role of Buoyancy Induced Instability in Transpirational Cooling Applications
###### Abstract  
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  
experiments. Other computational studies have been performed that quantify cooling efficacy versus inlet conditions, thermal properties, and boundary conditions [10; 11]. All previous studies have either analytically, experimentally, or numerically, investigated transriprational cooling in geometric configurations where the hot surface is on the top or side and the cold surface is on the bottom or side. However, if the geometry is such that the hot surface and cold surface are on the bottom and top, respectively, then Benard-type convection can ensue resulting in a mixed convection phenomena. This type of arrangement occurs on the bottom wall of the aircraft or the top wall of the combustion chamber. In these cases, an understanding of the buoyancy-induced instability mechanisms need to be described and understood. The tools of linear stability analysis provide a pathway for this understanding. A similar geometric and flow configuration to that occurring in transriprational cooling applications, albeit with no base flow, is the Horton-Rogers-Lapwood (HRL) problem. The HRL problem consists of a buoyancy-induced instability in porous media due to heating from below [12; 13]. The HRL problem is extensively described and characterized in [PERSON] and [PERSON] [14]. A pioneering variation of the HRL problem where base flow is perpendicular to the base temperature gradient was studied by [PERSON] [15]. Several variations of the HRL and Prats problems have been investigated by [PERSON] and [PERSON] [16; 17; 18]. The variation of the HRL problem most relevant to the transriprational cooling problem at hand is with a base flow parallel in the direction of the base temperature gradient, i.e., vertical base flow. This problem was first studied by [PERSON] [19] on a semi-infinite (vertical) porous domain. [PERSON] then characterized stability in a finite domain-of both width and height-against through-flow velocities of a small magnitude [20]. Initially, a small negative through-flow has a slight destabilizing effect which transitions to a stabilizing effect as velocity is increased. This effect is exaggerated as the domain width becomes smaller. Next, [PERSON] and [PERSON] investigated the stability theory on a finite domain for both strong and weak through-flows [21]. They found that mass discharge in the positive vertical direction destabilizes while the opposite was true for negative through-flow. The previous works provide much insight to buoyancy-induced instabilities occurring in porous media applications such as transriprational cooling.