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Supersonic flows are characterized by speeds greater than the speed of sound in a medium. Understanding the critical parameters, such as Mach number and shockwave formation, is essential for analyzing and designing high-speed aerodynamic systems.
Mach Number in Supersonic Flows
The Mach number is a dimensionless quantity representing the ratio of an object’s speed to the local speed of sound. It is calculated as M = v / a, where v is the flow velocity and a is the speed of sound.
In supersonic flows, the Mach number exceeds 1.0. As the Mach number increases, flow properties such as pressure, temperature, and density change significantly, affecting the behavior of the flow around objects.
Shockwave Formation
Shockwaves are abrupt discontinuities in the flow where properties like pressure, temperature, and density increase sharply. They form when the flow speed exceeds the local speed of sound, causing compression waves to coalesce into a shockwave.
Shockwaves can be classified as oblique or normal, depending on their orientation relative to the flow. Normal shockwaves directly face the flow and cause a sudden decrease in Mach number, while oblique shockwaves are inclined and result in a change in flow direction.
Calculating Critical Parameters
Determining the Mach number at which shockwaves form involves analyzing flow conditions and applying the shock relations. The critical Mach number is the minimum Mach number at which a shockwave appears in a given flow configuration.
Key equations include the normal shock relations, which relate upstream and downstream flow properties. For example, the pressure ratio across a normal shock is given by:
P₂ / P₁ = 1 + 2γ / (γ + 1) * (M₁² – 1)
where γ is the specific heat ratio and M₁ is the upstream Mach number.
- Flow velocity
- Speed of sound
- Pressure and temperature conditions
- Shockwave angles