MAE 3293 · Fundamentals of Aerodynamics · High-speed flow

Compressible Aerodynamics Calculator

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Supersonic Airfoils

Shock-expansion theory for three airfoils of increasing complexity. Pick a case, then change the freestream Mach number, the angle of attack, or the geometry. Every step below updates: the wave on each surface, the surface pressures, the force on each face, and finally cl, cd and cm.

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

Fi = pi ℓi 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:

\[ N' = (p_2-p_1)\,c,\qquad A' = 0 \]

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 pb:

\[ N' = (p_2-p_1)\,c,\qquad A' = (p_1+p_2)\,\frac{t}{2} - p_b\,t \]

With face lengths ℓf (front) and ℓr (rear), ℓfcos ε1 = xm and ℓfsin ε1 = ℓrsin ε2 = t/2:

\[ N' = (p_3-p_1)\,x_m + (p_4-p_2)\,(c-x_m) \]
\[ A' = (p_1+p_3-p_2-p_4)\,\frac{t}{2} \]

Then rotate by α into the wind axes and divide by \(q_\infty c\), with \(q_\infty=\tfrac12\rho_\infty V_\infty^2=\tfrac{\gamma}{2}p_\infty M_\infty^2\):

\[ L' = N'\cos\alpha - A'\sin\alpha,\qquad D' = N'\sin\alpha + A'\cos\alpha \]
\[ c_l=\frac{L'}{q_\infty c},\qquad c_d=\frac{D'}{q_\infty c},\qquad c_{m,\mathrm{LE}}=\frac{M'_{\mathrm{LE}}}{q_\infty c^2} \]

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.

\[ \tan\theta = 2\cot\beta\,\frac{M_1^2\sin^2\beta-1}{M_1^2(\gamma+\cos2\beta)+2} \]
\[ M_{n,1}=M_1\sin\beta,\qquad \frac{p_2}{p_1}=1+\frac{2\gamma}{\gamma+1}\left(M_{n,1}^2-1\right) \]
\[ M_{n,2}^2=\frac{1+\frac{\gamma-1}{2}M_{n,1}^2}{\gamma M_{n,1}^2-\frac{\gamma-1}{2}},\qquad M_2=\frac{M_{n,2}}{\sin(\beta-\theta)} \]

If θ exceeds θmax(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.

\[ \nu(M)=\sqrt{\tfrac{\gamma+1}{\gamma-1}}\tan^{-1}\!\sqrt{\tfrac{\gamma-1}{\gamma+1}(M^2-1)}-\tan^{-1}\!\sqrt{M^2-1} \]
\[ \nu(M_2)=\nu(M_1)+\theta,\qquad \frac{p_2}{p_1}=\left(\frac{1+\frac{\gamma-1}{2}M_1^2}{1+\frac{\gamma-1}{2}M_2^2}\right)^{\frac{\gamma}{\gamma-1}} \]

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

Pressure coefficient

\[ C_p=\frac{p-p_\infty}{q_\infty}=\frac{2}{\gamma M_\infty^2}\left(\frac{p}{p_\infty}-1\right) \]

Linear (Ackeret) theory check

For small angles every surface obeys \(C_p = 2\theta/\sqrt{M_\infty^2-1}\), with θ positive for compression. Integrating over the surfaces gives

\[ c_l=\frac{4\alpha}{\sqrt{M_\infty^2-1}},\qquad c_d=\frac{4\alpha^2}{\sqrt{M_\infty^2-1}} \]
\[ c_l=\frac{4\alpha}{\sqrt{M_\infty^2-1}},\qquad c_d=\frac{4}{\sqrt{M_\infty^2-1}}\left[\alpha^2+\left(\frac{t}{2c}\right)^2\right] - C_{p,b}\,\frac{t}{c} \]

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

\[ c_l=\frac{4\alpha}{\sqrt{M_\infty^2-1}},\qquad c_d=\frac{4}{\sqrt{M_\infty^2-1}}\left[\alpha^2+\frac{(t/c)^2}{4}\left(\frac{c}{x_m}+\frac{c}{c-x_m}\right)\right] \]

For the symmetric diamond (xm = 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 \(c_{m,c/4}=-\alpha/\sqrt{M_\infty^2-1}\) for all three cases.

Assumptions and limits

  • Two-dimensional, inviscid, calorically perfect gas: no skin friction, so cd here is wave drag (plus base drag if pb ≠ 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. cm is positive nose-up.

Face numbers: flat plate and wedge 1 = upper, 2 = lower, b = base; diamond 1 = upper front, 2 = upper rear, 3 = lower front, 4 = lower rear; ∞ = freestream.