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The Taylor-Maccoll equations below govern the gas dynamics around a supersonic, axially oriented cone. Figure 13.19 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.

  1. 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: $$ \tan(\delta) = 2 \cot(\theta_s) \frac{M_1^2\sin^2(\theta_s)-1}{M_1^2 \left[\gamma+\cos(2\theta_s) \right]+2}$$

  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?

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  • $\begingroup$ Your equation is not linear and that implies that you don't have guarantee of a unique solution. Even if you have two boundary conditions. $\endgroup$
    – nicoguaro
    May 13, 2023 at 14:27

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The traditional way of doing this was to start from the shock with a free stream Mach number of M and an arbitrary shock angle, then march inward until you encountered radial flow at some value of $\theta$ and then announce that you just solved the cone-flow flow problem for $(M,\theta)$. This was all back in the day when scientific computing was performed on desk calculators and was the method used to produce Zdenek Kopal's two volumes of "MIT Cone Flow Tables" (MIT Tech Rept No 1, 1947) which I still remember using. If these did not show the value of $\theta$ that you wanted, we all knew how to interpolate by hand. As soon as digital computers became available, NASA produced NASA SP 3004, J. L. Sims, Tables for supersonic flow around right circular cones at zero angle of attack, 1964, still available on the internet. Once you have even a modest computer available, the two-point boundary condition is not a problem. You just iterate to find the correct shock angle.

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