The Enthalpy at the wall is the stagnation enthalpy modified with a term "pariethal" entalphy recovery factor r usually =0.85 before the term M^2/2

Hwall = h(t)*(1+r*(M^2/2))

where t is the static temperatute a h(t) static enthalpy

There is a brief but instructive chapter on high speed flow heat transfer in e.g. Kays/Crawford/Weigand: Convective Heat and Mass Transfer, McGraw-Hill, 2005. The common method is (as Mr. tournes writes), to use the stagnant enthalpy for the flow as the fluid condition, hence Tfluid=Tinfinite+r*V^2/(2*cp),where r is the recovery factor. Due to thermal conduction effects in the boundary layer the 'effective' stagnant enthalpy is somewhat lower than the adiabaticstagnant enthalpy, and the correction factor (recovery factor) is roughly r=sqrt(Pr) which becomes r=0,85 for air.

As already explained by Erling, modeling heat flux should imply accurate description of the boundary layer. A simple 1D approach can be used to describe the flow in the boundary layer to understand the heat flux see Colonna et al, and Armenise et al. You have three different contribution to the heat transfer that must be considered separately. 

First of all you have the conductive or Fourier heat transfer. It is proportional to the gradient of temperature at the surface and slip-relations must be considered at low density. Therefore you need a boundary condition on the surface. A common approach is to fix the temperature on the surface to derive the heat flux. In alternative, but this is more complex, you can have a heat removal rate from the surface and its heat capacity, so that you can calculate also the temperature. 

The other contribution comes from processes at the surfaces, i.e. the catalytic heat flux. If atoms recombine on the surface, or vibrational energy is accommodated on the surface, part or all of the chemical energy goes to the surface, contributing to the heat flux. To calculate this term you need accommodation coefficients for the surface reactions. This is not a simple task because of the missing of data and of models for the energy accommodation on the surface. Therefore two limiting cases are considered, the fully catalytic and non catalytic approximation, the first considering equilibrium composition at the wall temperature and the last considering null fluxes of species perpendicular to the surface. 

The third contribution comes from the radiation. This term is very difficult to calculate (see Surzhikov) and need an equation of transport for photons in the plasma, considering emission and absorption by the gas particle, a reflectivity coefficient for the surface. Usually the radiative flux is smaller than the Fourier contribution, but in some region, where the gas temperature is small, the radiative flux is the dominant contribution.

Hope it helps

The Enthalpy at the wall is the stagnation enthalpy modified with a term "pariethal" entalphy recovery factor r usually =0.85 before the term M^2/2

Hwall = h(t)*(1+r*(gamma-1))(M^2/2)): (sorry the gamma term was missing in my previous response)

where t is the static temperatute a h(t) static enthalpy

The following information may be useful for your research

Ref: Large scale computer modeling of aerodynamics of hypersonic vehicle composite constructions by Dimitrienko et al,. Third International Conference of 'High Perfomance computing, Ukraine, October, 2013.

In this paper, heat transfer studies were conducted for hypersonic bodies and computer software package SIGMA was used in this model to corelate the results. 

hpc-ua.org/hpc-ua-13/files/proceedings/18.pdf

You can follow this text book also

Computational Fluid Mechanics and Heat Transfer, Third Edition

By Richard H. Pletcher, John C. Tannehill, Dale Anderson

The ratio of static temperature to stagnation temperature is taken care of Mach number, irrespective of its value, whether it is less than one, or equal to one or more than one. 

T/To= 1 + (gaama-1}/2 M^2, This value is hidden in the methodology of software.

All other parameters like ratio of stagnation pressure to static pressure, stagnation density, etc will depend on Mach number

These values are given in Chapter- 'Isentropic Heat Flow' of the Text Book on Gas Dynamics 

Appendix B in "Optimal MODIFIED CONTINUOUS Galerkin CFD" (Wiley 2014, ISBN 978-1-119-94049-4) fully details a real gas weak form algorithm for hypersonic shock layer prediction.   Near wall resolution is absolutely critical - view Figure B.6 for cold and near stagnation wall temperature Dirichlet BC solutions.   The weak form "trick" for accurate flux computation at surfaces with fixed temperature is detailed at the end of Ch. 2.3 of the text.