# Implementing pressure inlet boundary condition

I am interested in implementing pressure inlet boundary conditions for the 2D compressible Euler equations.

My equation of state is an ideal gas ($p=\rho R T$) which is thermally prefect but not calorically perfect, i.e. $c_p\equiv c_p(T)$ is a non-constant functions of $T$ alone.

The boundary condition specifies total pressure $p_t$, total density $\rho_t$, and angle of attack $\alpha$ on the boundary. I would like to know how to compute a consistent state at the boundary assuming an isentropic and adiabatic process.

I see from the ANSYS Fluent manual, that in case of a perfect gas one does the following:

• Static pressure on the boundary $p_b$ is set equal to the static pressure in the adjacent cell $p_i$
• Mach number on the boundary is computed from the isentropic relation $$p_t = p_b\left(1+ \frac{\gamma-1}{2}M^2\right)^{\gamma/(\gamma-1)}$$
• Static temperature on the boundary $T_b$ is computed from the isentropic relation $$T_t = T_b\left(1+\frac{\gamma-1}{2}M^2\right)$$

Any references on implementing this kind of boundary condition in case of a ideal gas which is solely thermally perfect is highly appreciated.

My first approach was

• set the mach number on the boundary equal to that of the adjacent cell
• solve the adiabatic process equation $$h_t = h(T) + \frac{1}{2}M^2 c(T)^2$$ for $T$ to determine the temperature on the boundary
• Solve $s(T,p) = s(T_t,p_t)$ for $p$ to determine the pressure on the boundary

However, this approach leads to a flow which differs significantly from what I see in ANSYS. In fact, the computed mach numbers are much lower.