An extended B′ formulation for ablating-surface boundary conditions

The B^′ formulation can be understood as a mass and energy conservation formalism at a reacting singular surface. In hypersonics applications, it is typically used to compute the chemical equilibrium properties of gaseous mixtures at ablating surfaces, and to estimate the recession velocity of the interface. In the first half of the paper, we derive the B^′ formulation to emphasize first principles. In particular, while we eventually specialize to the commonly considered case of chemical equilibrium boundary layers that satisfy the heat and mass transfer analogy, we first derive a general interface jump condition that lets us highlight all the underlying assumptions of the well-known B^′ equations. This procedure helps elucidate the nature of the B^′ formalism and it also allows us to straightforwardly extend the original formulation. Specifically, when applied at the interface between a porous material and a boundary layer (as in thermal protection systems applications), the original formulation assumes unidirectional advective transport of gaseous species from the porous material to the boundary layer (i.e., blowing). However, under conditions that may appear in hypersonic flight or in ground-based wind tunnels, boundary layer gases can enter the porous material due to a favorable pressure gradient. We show that this scenario can be easily handled via a straightforward modification to the B^′ formalism, and we demonstrate via examples that accounting for gas entering the material can impact the predicted recession velocity of ablating surfaces. In order to facilitate the implementation of the extended B^′ formulation in existing material response codes, we present a short algorithm in section 5 and we also refer readers to a GitHub repository where the scripts used to generate the modified B^′ tables are publicly available.