Calculator Inputs
Select a supersonic analysis mode, enter gas properties, and compute state changes, shock behavior, expansion effects, or nozzle branch solutions.
Example Data Table
| Case | Inputs | Typical Output Insight |
|---|---|---|
| Isentropic nozzle state | γ = 1.4, M = 2.0 | Useful for pressure, temperature, density, area ratio, and Mach angle estimates. |
| Normal shock | γ = 1.4, M₁ = 2.5 | Shows downstream Mach drop and strong static pressure rise. |
| Oblique shock | γ = 1.4, M₁ = 3.0, β = 35° | Calculates deflection angle and post-shock flow state. |
| Expansion fan | γ = 1.4, M₁ = 2.0, turn = 12° | Finds increased Mach number and reduced static pressure. |
| Area–Mach inversion | γ = 1.4, A/A* = 2.5 | Returns both subsonic and supersonic nozzle branch solutions. |
Formula Used
1) Isentropic relations
For a perfect gas:
T/T₀ = 1 / [1 + (γ−1)M²/2]
P/P₀ = [1 + (γ−1)M²/2]−γ/(γ−1)
ρ/ρ₀ = [1 + (γ−1)M²/2]−1/(γ−1)
A/A* = (1/M)[(2/(γ+1))(1 + (γ−1)M²/2)](γ+1)/(2(γ−1))
2) Normal shock relations
M₂² = [1 + (γ−1)M₁²/2] / [γM₁² − (γ−1)/2]
P₂/P₁ = 1 + [2γ/(γ+1)](M₁²−1)
ρ₂/ρ₁ = [(γ+1)M₁²] / [(γ−1)M₁² + 2]
T₂/T₁ = (P₂/P₁) / (ρ₂/ρ₁)
3) Oblique shock relation
tan θ = 2 cot β [(M₁² sin²β − 1) / (M₁²(γ + cos 2β) + 2)]
4) Prandtl–Meyer function
ν(M) = √[(γ+1)/(γ−1)] arctan √[((γ−1)/(γ+1))(M²−1)] − arctan √(M²−1)
5) Area–Mach inversion
The area relation is solved numerically because one area ratio generally maps to two Mach solutions: one subsonic and one supersonic.
How to Use This Calculator
Step 1: Choose the analysis mode that matches your problem: isentropic, normal shock, oblique shock, expansion fan, or area–Mach inversion.
Step 2: Enter the gas specific heat ratio γ. For air, 1.4 is a common default.
Step 3: Provide the main flow input such as Mach number, shock angle, turn angle, or area ratio.
Step 4: Optionally enter reference static pressure, temperature, and density to recover actual state values in engineering units.
Step 5: Press Submit. The result block appears above the form, directly below the header.
Step 6: Review summary cards, the detailed output table, and the Plotly graph. Export the results as CSV or PDF when needed.
FAQs
1) What does this calculator solve?
It solves isentropic flow states, normal shocks, oblique shocks, Prandtl–Meyer expansions, and nozzle area–Mach inversions for compressible perfect-gas flow.
2) Why is γ important?
The specific heat ratio controls compressibility behavior. It changes pressure, temperature, density, shock strength, and expansion results across all modes.
3) Can I use it for air?
Yes. Air is commonly modeled with γ = 1.4 in many aerospace and laboratory calculations, especially when the perfect-gas assumption is reasonable.
4) What is the difference between normal and oblique shocks?
A normal shock stands perpendicular to the flow and causes a larger Mach drop. An oblique shock is angled and usually produces a smaller downstream Mach reduction.
5) Why are there two area–Mach solutions?
For an area ratio above one, the nozzle equation can produce both a subsonic and a supersonic Mach number. The physical branch depends on choking and boundary conditions.
6) Does total temperature change across a shock?
For an adiabatic perfect-gas shock with no shaft work, total temperature stays constant. Total pressure drops because shocks are irreversible.
7) What if my oblique shock input fails?
The selected deflection angle may exceed the attached-shock limit, or the shock angle may be below the Mach angle. Reduce the angle and try again.
8) Are the exports based on the computed results?
Yes. The CSV and PDF buttons export the currently displayed result table, making it easier to document design checks and classroom work.