The Taylor-Maccoll equations below govern the gas dynamics around a supersonic, axially oriented cone.
Using the notation from Anderson's Fundamentals of Aerodynamics, which uses a dimensionless variable $V^\prime$ which can be found with:
$$V^\prime = \left[ \frac{2}{(\gamma -1) M^2} + 1 \right]^{- \frac{1}{2}}, \\ V_r^\prime = V^\prime \cos(\theta_s - \delta), \\ V_\theta^\prime = V^\prime \sin(\theta_s - \delta)$$
$$\frac{\gamma - 1}{2} \left[ 1 - (V_r^{\prime})^2 - \left( \frac{dV_r^{\prime}}{d \theta} \right)^2 \right] \left[2V_r^{\prime} + \frac{dV_r^{\prime}}{d \theta} \cot(\theta) + \frac{d^2 V_r^{\prime}}{d \theta^2} \right] - \frac{dV_r^{\prime}}{d \theta} \left[ V_r^{\prime} \frac{dV_r^{\prime}}{d \theta} + \frac{dV_r^{\prime}}{d \theta} \frac{d^2 V_r^{\prime}}{d \theta^2}\right] = 0$$
Obviously, as this is a 2nd-order ODE, we need two independent boundary conditions for a unique solution.
Shock angle $\theta_s, \, M_1$ are assumed to begin with. As a result, we use the standard oblique shock, and normal shock relations (changing the variables to match the diagram above), as well as the definitions for $V^\prime$ to find $V_r^\prime$ directly behind the shock: $$ \tan(\delta) = 2 \cot(\theta_s) \frac{M_1^2\sin^2(\theta_s)-1}{M_1^2 \left[\gamma+\cos(2\theta_s) \right]+2}$$
Cone Surface is impenetrable. The normal component of flow velocity where $\theta = \theta_c$ is zero.
$$V_\theta^\prime(\theta = \theta_c) = \frac{dV_r^\prime}{d \theta} = 0$$
My main problem lies in the simultaneous usage of these two boundary conditions. Anderson's text explains that a numerical solution could be done by:
"Using the above value of $V_r^\prime$ directly behind the shock as a boundary value, solve for $V_r^\prime$ numerically in steps of $\theta$, marching away from the shock. Here, the flow field is divided into incremental angles $\theta$, as sketched in Figure 13.19. The ordinary differential equation can be solved at each$\theta$ using any standard numerical solution technique.
At each increment in $\theta$, the value of $V_\theta^\prime$ is calculated. At some value of $\theta$, namely $\theta = \theta_c$, we will find $V_\theta^\prime = 0$. The normal component of velocity at an impermeable surface is zero. Hence, when $V_\theta^\prime = 0$ at $\theta = \theta_c$ then $\theta_c$ must represent the surface of the particular cone that supports the shock wave of given wave angle $\theta_s$ at the given Mach number $M_\infty$ as assumed previously. That is, the cone angle compatible with $M_\infty$ and $\theta_s$ is $\theta_c$. The value of $V_r^\prime$ at $\theta_c$ gives the Mach number along the cone surface."
Overall, I'm not sure how a numerical technique can be implemented in this situation, as the two boundary conditions are not "on the same side". How can I "march away" from the shock if I don't have two boundary conditions there as well?
