This formula calculates the wall temperature of a hypersonic vehicle based on the freestream velocity and temperature. It is crucial for determining thermal loads and designing the thermal protection systems required for vehicles moving at hypersonic speeds.

Symbols:

• Tw​: Wall temperature (K)
• Te​: Freestream temperature (K)
• Ve: Freestream velocity (m/s)
• cp: Specific heat at constant pressure (J/kg·K) 

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This formula calculates the boundary layer thickness in hypersonic flows. The boundary layer is a thin layer of air near the surface of a vehicle where the flow velocity changes from zero to the freestream velocity. Understanding boundary layer thickness is critical for predicting drag and heat transfer in hypersonic vehicles.

• $\delta$: Boundary layer thickness (m)
• C: Constant depending on flow conditions (dimensionless)
• μ: Dynamic viscosity (Pa·s)
• ρ: Freestream density (kg/m³)
• U​: Freestream velocity (m/s)

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This formula calculates the ratio of specific heat in hypersonic flows. The specific heat ratio is a key parameter in a compressible flow, influencing shock wave behaviour and thermodynamic properties of gases at high speeds.

Symbols:

• γ: Ratio of specific heats (dimensionless)
• Cp​: Specific heat at constant pressure (J/kg·K)
• Cv: Specific heat at constant volume (J/kg·K) 

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This formula calculates the dynamic pressure in a hypersonic flow. Dynamic pressure is a key parameter in determining the aerodynamic forces on a body moving through a fluid, especially at high Mach numbers.

Symbols:

• q1​: Dynamic pressure
• γ: Ratio of specific heats (dimensionless)
• p1​: Freestream static pressure (Pascals)
• M1​: Freestream Mach number (dimensionless) 

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This formula calculates the Stanton number in hypersonic flows, which is a measure of the heat transfer rate relative to the convective heat transport. It is a critical parameter in analysing thermal protection systems for hypersonic vehicles.

• Ch​: Stanton number (dimensionless)
• qw: Heat flux at the wall (W/m²)
• ρ: Freestream density (kg/m³)
• V: Freestream velocity (m/s)
• h0​: Total enthalpy (J/kg)
• hw​: Wall enthalpy (J/kg) 

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This formula calculates the heat transfer rate to a surface in hypersonic flows, which is essential for understanding thermal loads on the surfaces of high-speed vehicles. High-speed flows generate significant heat due to air compression, and this equation helps in managing thermal protection.

Symbols:

• q: Heat transfer rate per unit area
• ρ: Freestream density (kg/m³)
• V: Freestream velocity (m/s)
• Ch​: Stanton number (dimensionless) 

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This calculator evaluates the deflection angle for Prandtl-Meyer expansion waves in hypersonic flows. It uses an approximate relation, which becomes more accurate as the Mach numbers (M1 and M2​) increase. The formula helps determine the change in the direction of flow due to expansion, which is vital in high-speed aerodynamics where flow expansion waves play a significant role in reducing pressure and temperature downstream of the wave. The flow through an expansion wave is isentropic, making this relation important for designing efficient nozzles and airfoils in hypersonic regimes.

Symbols:

• θ: Flow deflection angle (radians)
• M1​: Freestream Mach number before the expansion wave (dimensionless)
• M2: Mach number after the expansion wave (dimensionless)
• γ: Ratio of specific heats (dimensionless)

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This calculator approximates the shock wave angle for slender wedges in hypersonic flow. For vehicles with sharp, slender geometries, understanding the shock wave angle is essential for predicting the location of shock waves and their effect on vehicle performance. This relation helps in designing hypersonic bodies with minimized drag and optimized shock wave behavior for efficient high-speed flight.

Symbols:

• β: Shock wave angle (radians)
• θ: Flow deflection angle (radians)
• γ: Ratio of specific heats (dimensionless)

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This calculator evaluates the pressure coefficient for hypersonic flow behind an oblique shock. The pressure coefficient is used to analyze pressure distribution around hypersonic vehicles, influencing drag, lift, and overall aerodynamic performance.

• Cp​: Pressure coefficient (dimensionless)
• γ: Ratio of specific heats (dimensionless)
• M1: Freestream Mach number (dimensionless)
• p2/p1: Pressure ratio (dimensionless)                 

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This calculator computes the velocity component perpendicular to the flow after a shock in hypersonic conditions. This component is key to predicting flow redirection and determining how shocks interact with the vehicle’s surfaces during high-speed flight.

• v2​: Velocity component perpendicular to the flow after the shock (m/s)
• V1​: Freestream velocity (m/s)
• M1: Freestream Mach number (dimensionless)
• γ: Ratio of specific heats (dimensionless)
• β: Shock wave angle (radians)               

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