Oblique Shock at a Corner

What is modelled

Supersonic flow cannot turn smoothly: when a wall turns into the stream, a straight oblique shock forms from the corner. Across it the component of velocity normal to the shock drops like a normal shock while the tangential component is unchanged, so the stream leaves parallel to the new wall. Mach number, shock angle and turning angle are tied by the theta-beta-M relation, which has a weak and a strong solution up to a largest turning angle.

Equations

• tan θ = 2 cot β (M² sin²β − 1) / (M² (γ + cos 2β) + 2).
• Normal Mach number Mn1 = M sin β, pressure ratio 1 + 2γ (Mn1² − 1) / (γ + 1), density ratio (γ + 1) Mn1² / ((γ − 1) Mn1² + 2).
• M2 = Mn2 / sin(β − θ); Prandtl-Meyer dν = √(M² − 1) dM / ((1 + (γ − 1) M² / 2) M).

Checks run in the page

• Momentum and energy residuals and flow turning from the solved shock angle.
• The weak-disturbance limit equals the Mach angle.
• Total-pressure ratio by two routes.
• Prandtl-Meyer function by numerical integration.

Honest simplifications

A calorically perfect gas, a straight shock from a sharp corner in steady inviscid flow, no boundary layer, no shock-boundary-layer interaction and no real-gas effects at high temperature. The detached bow shock is only sketched. The drifting dots are a cartoon of the particle paths, not a simulation of the flow field. The 3D page is a wave visualisation; this page is the textbook shock calculation.

• Press Mach 5 wedge and see the shock hug the wall.
• Press Strong family: same corner, much steeper shock, subsonic flow behind it.
• Press Detached bow shock and raise the Mach number to see the attached solution come back.