13.4 Oblique Shock Waves and Expansion Waves

Oblique shock waves and expansion waves are crucial phenomena in supersonic flow. These waves occur when supersonic flow encounters changes in geometry, causing sudden or gradual shifts in flow properties. Understanding their behavior is key to analyzing and designing supersonic systems.

Calculating flow properties across oblique shocks and expansion waves involves specific relations and functions. The $\theta-\beta-M$ relation for oblique shocks and the Prandtl-Meyer function for expansion waves are essential tools. These concepts are fundamental for designing supersonic airfoils and nozzles.

Oblique Shock Waves

Oblique shocks vs expansion waves

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• Oblique shock waves form when a supersonic flow encounters a concave corner (compression ramp) causing a sudden change in flow properties across the shock
  • Increase in pressure, density, and temperature
  • Decrease in velocity
  • Deflect the flow towards the surface, increasing the flow angle
• Expansion waves form when a supersonic flow encounters a convex corner (expansion corner) causing a gradual change in flow properties across the wave
  • Decrease in pressure, density, and temperature
  • Increase in velocity
  • Deflect the flow away from the surface, decreasing the flow angle

Oblique shock flow calculations

• Oblique shock relations relate the upstream and downstream flow properties across an oblique shock wave depending on the upstream Mach number ($M_1$) and the shock wave angle ($\beta$)
• Calculate flow properties using the normal shock relations in conjunction with the oblique shock angle
  1. Determine the normal component of the upstream Mach number: $M_{1n} = M_1 \sin \beta$
  2. Calculate the downstream normal Mach number ($M_{2n}$) using the normal shock relations
  3. Determine the downstream Mach number: $M_2 = M_{2n} / \sin(\beta - \theta)$, where $\theta$ is the flow deflection angle
• Determine the wave angle using the $\theta-\beta-M$ relation: $\tan \theta = 2 \cot \beta \frac{M_1^2 \sin^2 \beta - 1}{M_1^2 (\gamma + \cos 2\beta) + 2}$
  • Solve for $\beta$ given the upstream Mach number ($M_1$) and the flow deflection angle ($\theta$)

Expansion Waves and Interactions

Expansion wave property determination

• Prandtl-Meyer function relates the upstream and downstream flow properties across an expansion wave depending on the upstream Mach number ($M_1$) and the flow deflection angle ($\theta$)
• Calculate flow properties using the Prandtl-Meyer function
  1. Determine the upstream Prandtl-Meyer function value: $\nu_1 = \sqrt{\frac{\gamma + 1}{\gamma - 1}} \tan^{-1} \sqrt{\frac{\gamma - 1}{\gamma + 1} (M_1^2 - 1)} - \tan^{-1} \sqrt{M_1^2 - 1}$
  2. Calculate the downstream Prandtl-Meyer function value: $\nu_2 = \nu_1 + \theta$
  3. Solve for the downstream Mach number ($M_2$) using the inverse Prandtl-Meyer function

Wave interactions with boundaries

• Reflection of oblique shock waves
  • Regular reflection: Incident and reflected shock waves meet at the surface, flow downstream of the reflection point is parallel to the surface
  • Mach reflection: Incident and reflected shock waves do not meet at the surface, a Mach stem forms perpendicular to the surface
• Reflection of expansion waves
  • Expansion waves reflect as expansion waves from a solid boundary
  • Flow downstream of the reflected expansion wave is parallel to the surface

Applications in supersonic design

• Supersonic airfoils designed to minimize drag and control the location of shock waves
  • Use a combination of compression surfaces (oblique shock waves) and expansion surfaces (expansion waves) to achieve the desired flow properties
  • Diamond airfoil with a sharp leading edge and a symmetric wedge shape
• Supersonic nozzles designed to accelerate a flow from subsonic to supersonic speeds
  • Use a converging-diverging geometry to create oblique shock waves and expansion waves
  • De Laval nozzle with a converging section followed by a diverging section