Q: How is the oblique shock angle calculated from deflection angle?

The θ-β-M relation connects deflection angle θ, shock wave angle β, and upstream Mach number M₁: tan θ = 2 cot β · (M₁²sin²β − 1) / (M₁²(γ + cos 2β) + 2). For given M₁ and θ, this transcendental equation is solved numerically for β (weak shock solution). The normal component M₁ sin β is then used with the normal shock relations to find M₂ and pressure ratio. Oblique shocks are weaker and more efficient than normal shocks.

Q: How do I use these relations in CFD simulations?

In CFD (OpenFOAM sonicFoam, Fluent density-based solver): use isentropic relations to set inlet total pressure P₀ and total temperature T₀ from the freestream Mach number. For supersonic inlets, the normal shock table gives the downstream conditions used as initial conditions. The A/A* ratio is used to size nozzle cross-sections in rocket and jet engine design. Always verify that your Mach number places the operating point on the correct branch (subsonic or supersonic) of the area-Mach curve.

Question 1

What are isentropic flow relations?

Accepted Answer

Isentropic flow relations describe adiabatic reversible (entropy-constant) compressible flow. From the energy equation and equation of state: T₀/T = 1 + (γ-1)/2·M², P₀/P = (T₀/T)^(γ/(γ-1)), ρ₀/ρ = (T₀/T)^(1/(γ-1)), where T₀, P₀, ρ₀ are stagnation (total) conditions. These relations are fundamental for converging-diverging nozzle design, supersonic wind tunnel calculations, and CFD boundary condition specification.

Question 2

What happens at a normal shock wave?

Accepted Answer

A normal shock wave occurs in supersonic flow (M>1) perpendicular to the flow direction. Across the shock, flow always decelerates to subsonic (M<1). The Rankine-Hugoniot relations give: M₂ = √[(M₁²(γ-1)+2)/(2γM₁²-(γ-1))], P₂/P₁ = (2γM₁²-(γ-1))/(γ+1). Since shocks are irreversible, total pressure P₀₂ < P₀₁ (entropy increases). Normal shocks appear in supersonic nozzles, inlet diffusers, and ahead of blunt bodies.

Question 3

How is the oblique shock angle calculated from deflection angle?

Accepted Answer

The θ-β-M relation connects deflection angle θ, shock wave angle β, and upstream Mach number M₁: tan θ = 2 cot β · (M₁²sin²β − 1) / (M₁²(γ + cos 2β) + 2). For given M₁ and θ, this transcendental equation is solved numerically for β (weak shock solution). The normal component M₁ sin β is then used with the normal shock relations to find M₂ and pressure ratio. Oblique shocks are weaker and more efficient than normal shocks.

Question 4

How do I use these relations in CFD simulations?

Accepted Answer

In CFD (OpenFOAM sonicFoam, Fluent density-based solver): use isentropic relations to set inlet total pressure P₀ and total temperature T₀ from the freestream Mach number. For supersonic inlets, the normal shock table gives the downstream conditions used as initial conditions. The A/A* ratio is used to size nozzle cross-sections in rocket and jet engine design. Always verify that your Mach number places the operating point on the correct branch (subsonic or supersonic) of the area-Mach curve.

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