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Supersonic theory around a perfect cone can be written in a much simpler form than the fully continuity, momentum and energy equations. This is due to the symmetry of the problem, where the 5 coupled differential equations can be simplified to 1 ordinary differential equation and solved using boundary conditions. From this one ODE, the rest of the flow properties can be reconstructed.
The crucial equation (derived in section 13.6) is of the radial flow velocity between the cone surface and the oblique shock, is is below.
because of the symmetry of the problem, the flow properties along ANY ray from the cone vertex has to have constant values. This greatly simplifies all the equations.
Equation 13.78 is the key equation that must be solved. Non-dimensionalize by Vmax = sqrt(h0) where h0 is the total enthalpy and you get:
where eqn 13.80 is only a function of gamma and theta.
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