Oblique Shock Calculator
Weak Shock Solution Properties
What Is an Oblique Shock Calculator?
An Oblique Shock Calculator solves the relationship between upstream Mach number, flow deflection angle, shock angle, and gas properties. It models a supersonic flow that turns through a compression corner and forms an angled shock wave.
The calculator uses the entered Mach number, deflection angle, and specific heat ratio to find the attached weak shock solution. It then applies normal-shock equations to the velocity component perpendicular to the wave. The result shows how Mach number, static pressure, temperature, density, and stagnation pressure change across the shock.
This tool is designed for quick calculations and technical study. It can identify a Mach wave at zero deflection, calculate an attached weak oblique shock, or report that the shock is detached when no attached solution is found.
How the Oblique Shock Calculator Formula Works
The calculator first computes the Mach angle, which is the lowest possible wave angle for the entered upstream Mach number.
It then searches between the Mach angle and 90 degrees for roots of the coded theta-beta-M relationship:
Here, M₁ is upstream Mach number, θ is flow deflection angle, β is wave angle, and γ is the specific heat ratio. The first root found is used as the weak shock solution. A second root, when found, is reported only in the aerodynamic notes as the strong solution.
The static property ratios come from the normal component of upstream Mach number:
For example, use air with γ = 1.4, M₁ = 2.5, and θ = 15 degrees. The calculator finds β = 36.94 degrees and M₁n = about 1.503. It displays M₂ = 1.874, p₂/p₁ = 2.4675, ρ₂/ρ₁ = 1.8665, T₂/T₁ = 1.3220, and p₀₂/p₀₁ = 0.9290.
How to Use the Oblique Shock Calculator: Step by Step
- Select a Gas Type. Available choices are air or another diatomic gas, carbon dioxide, helium or another monatomic gas, methane, and a custom specific heat ratio.
- If you select Custom γ, enter a specific heat ratio strictly greater than 1 and strictly less than 2.
- Enter the Upstream Mach Number (M₁). The calculation requires a value strictly greater than 1 because the modeled incoming flow must be supersonic.
- Enter the Deflection Angle (θ) in degrees. Use zero for an unturned flow and a positive value for a compression turn.
- Select Calculate. The tool searches for the weak attached shock solution and displays the calculated flow properties.
- Select Reset to restore air as the selected gas, clear the numeric fields, hide the custom gamma input, and remove the displayed results.
The wave angle is shown in degrees. Downstream Mach number is displayed to three decimal places. Static pressure, temperature, density, and stagnation pressure ratios are displayed to four decimal places. Each ratio compares the downstream value with the upstream value. For example, p₂/p₁ greater than 1 means static pressure increased across the shock.
How to Read Oblique Shock Calculator Results
Wave angle and downstream Mach number
The wave angle β measures the angle between the shock wave and the upstream flow direction. For an attached weak shock, β is greater than the Mach angle and less than 90 degrees. The downstream Mach number shows how much the flow slows after crossing the wave.
Static and stagnation property ratios
| Output | Meaning |
|---|---|
| p₂/p₁ | Downstream static pressure divided by upstream static pressure |
| T₂/T₁ | Downstream static temperature divided by upstream static temperature |
| ρ₂/ρ₁ | Downstream density divided by upstream density |
| p₀₂/p₀₁ | Downstream stagnation pressure divided by upstream stagnation pressure |
A stagnation pressure ratio below 1 represents total pressure loss. When that ratio is below 0.95, the calculator adds a note showing the loss as a percentage. That percentage is calculated as one minus the stagnation pressure ratio, multiplied by 100.
Zero deflection and detached shocks
At zero deflection, the code reports the Mach angle instead of a finite-strength shock. It keeps M₂ equal to M₁ and sets every property ratio to 1.0000. This represents unchanged properties across an infinitesimally weak Mach wave.
If the root search finds no attached solution, the wave angle displays as “Detached.” All other numerical outputs display N/A. The tool does not calculate the curved bow-shock structure or its downstream property field.
Assumptions and limitations
The calculation assumes steady, adiabatic flow with no external work and a calorically perfect gas. This means γ stays constant. Real gases can have temperature-dependent properties. The tool also focuses on the weak solution and does not model viscosity, heat transfer, three-dimensional effects, boundary layers, shock interactions, or chemical reactions.
The input field displays a deflection range up to 90 degrees, but the calculation script directly rejects only negative angles. Use physically meaningful compression angles. The numerical solver searches in fixed angular increments before applying bisection, so results near a solution limit may depend on the root search finding a sign change.
Frequently Asked Questions
What is an oblique shock wave?
An oblique shock wave is an angled compression wave in supersonic flow. It can form when the flow turns into itself at a wedge, ramp, inlet, or leading edge. Across the wave, static pressure, temperature, and density rise, while Mach number and stagnation pressure generally decrease.
How do I calculate the oblique shock angle?
Enter upstream Mach number, deflection angle, and the gas specific heat ratio. The calculator solves its theta-beta-M equation numerically between the Mach angle and 90 degrees. It uses the first root as the weak shock angle and may identify a second root as the strong solution.
What is the difference between weak and strong oblique shocks?
The weak solution has the smaller wave angle and is the solution used for the displayed property calculations. The strong solution has a larger wave angle. When the code detects that second root, it mentions the angle in the aerodynamic analysis but does not display separate strong-shock property ratios.
Why does the calculator say the shock is detached?
The calculator reports a detached shock when its numerical search cannot find an attached weak solution for the entered Mach number, deflection angle, and γ. It then displays N/A for downstream properties because the coded attached-shock equations do not model the curved and non-uniform flow behind a detached bow shock.
What specific heat ratio should I use?
Use the gas option that matches your problem. The calculator assigns γ = 1.400 for air or a diatomic gas, 1.289 for carbon dioxide, 1.667 for helium or a monatomic gas, and 1.299 for methane. A custom value must be strictly between 1 and 2.
What happens when the deflection angle is zero?
At zero deflection, the calculator reports an infinitesimally weak Mach wave rather than a finite shock. The wave angle equals arcsine of 1 divided by M₁. Downstream Mach number remains equal to upstream Mach number, and all static and stagnation property ratios remain exactly 1.0000.
How accurate is the Oblique Shock Calculator?
The calculator follows the equations and numerical search coded into the tool. Its results are suited to ideal compressible-flow calculations under the stated assumptions. Real aerodynamic results may differ because of variable gas properties, viscosity, boundary layers, geometry, heat transfer, shock interactions, measurement error, and other effects not included here.