Conical Flow Calculator
Taylor–Maccoll over a sharp cone — the true surface state, not a wedge approximation, at any γ.
Results
| Quantity | Just behind the shock | On the cone surface |
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What this is computing
A cone is not a wedge. Behind a wedge the flow is uniform; behind a cone it keeps turning, because the flow has a third direction to spread into. Nothing closed-form survives that, so this page integrates the Taylor–Maccoll equation inward from the shock until the flow runs parallel to a surface, and that surface is the cone.
Velocities are divided by Vmax = √(2h₀), which folds the energy equation in and puts V′ in (0, 1). The surface condition is Vθ = 0 — the flow no longer crossing the cone — and finding the shock angle that lands on a given cone is a shooting problem, solved here by bisection on β.
The surface state is not the post-shock state
At M₁ = 2 on a 20° cone the flow leaves the shock at M = 1.693 and arrives at the surface at M = 1.568, still turning the whole way. Two different answers to "the Mach number behind the shock", and which one you want depends on what you are computing.
A cone is far weaker than a wedge
The same 20° at M₁ = 2 gives a shock at 37.80° on a cone against 53.42° on a wedge, and a surface pressure of 1.91 p₁ against 2.84 p₁. Treating a cone as a wedge overestimates the load by about half.
And it stays attached much longer
At M₁ = 2 a wedge detaches past 22.97°, while a cone holds an attached shock to 40.69° — nearly twice the angle. It is the reason supersonic bodies are conical rather than wedge-shaped, and it is not a small correction.