Supersonic Airfoils Calculator

Wave pattern and surface pressures

Drawn to scale, freestream horizontal

Oblique shock Prandtl–Meyer fan Mach wave (no turning) Arrows: pressure pushing on each face, length ∝ p/p_∞; red p > p_∞, blue p < p_∞

Result

Second line in each tile: linear (Ackeret) theory for the same case and its difference from shock-expansion theory.

The three cases side by side

Same M_∞, α and γ; each case keeps its own geometry. Click a row to switch.

Surface-by-surface summary

F_i = p_i ℓ_i is the absolute-pressure force per unit span on face i, acting normal to the face and into the body. ΔL′ and ΔD′ are its components perpendicular and parallel to the freestream. The p_∞ parts cancel around the closed contour, so summing absolute forces and summing gauge forces give the same L′ and D′.

Worked solution

March the flow over each face

Worked solution

Pressures → forces → coefficients

Resolve the pressure forces in body axes first: N′ normal to the chord, A′ along it.

Both surfaces have length c and are parallel to the chord, so the plate carries no axial force:

Each face has length ℓ = c/cos δ, with ℓ cos δ = c and ℓ sin δ = t/2. The base of height t carries p_b:

With face lengths ℓ_f (front) and ℓ_r (rear), ℓ_fcos ε₁ = x_m and ℓ_fsin ε₁ = ℓ_rsin ε₂ = t/2:

Then rotate by α into the wind axes and divide by , with :

Sweep in angle of attack

Shock-expansion theory Linear (Ackeret) theory Shaded: no valid shock-expansion solution (shock detached or flow behind a front shock subsonic)

Reference

Relations used

Turning toward the flow: oblique shock

For a compression of θ, find the weak-shock angle β from the θ–β–M relation, then use normal-shock relations on the normal component.

If θ exceeds θ_max(M₁, γ), no attached solution exists. The shock stands off the nose and this method no longer applies.

Turning away from the flow: Prandtl–Meyer expansion

An expansion is isentropic. Add the turning angle to the Prandtl–Meyer function and invert for M₂.

Fan edges sit at the local Mach angle, μ = sin⁻¹(1/M), measured from the local flow direction.

Pressure coefficient

Linear (Ackeret) theory check

For small angles every surface obeys , with θ positive for compression. Integrating over the surfaces gives

The base term uses the same prescribed base pressure as shock-expansion theory; it is zero for p_b = p_∞.

For the symmetric diamond (x_m = c/2) the bracket becomes α² + (t/c)².

Thickness adds drag but no lift in linear theory. The center of pressure of the lift sits at mid-chord, so for all three cases.

Assumptions and limits

Two-dimensional, inviscid, calorically perfect gas: no skin friction, so c_d here is wave drag (plus base drag if p_b ≠ p_∞).
The leading-edge shocks must be attached. A detached bow shock invalidates the method.
Each face has uniform pressure. On the diamond, the flow on each front face must stay supersonic so the shoulder turn is a Prandtl–Meyer fan.
The wedge base pressure is an input, not a prediction.
Trailing-edge waves and the slip line form downstream of the body. They do not change any surface pressure and are not needed for the forces.
α is positive nose-up. c_m is positive nose-up.

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