Normal shock waves and oblique shocks are crucial concepts in compressible fluid flow. They occur when supersonic flow encounters obstacles, causing sudden changes in flow properties like pressure, temperature, and velocity.

These phenomena play a vital role in designing supersonic aircraft, engines, and wind tunnels. Understanding their behavior and relationships helps engineers optimize performance and efficiency in high-speed applications.

Normal Shock Waves in Compressible Flow

Formation and Characteristics

• Normal shock waves form in supersonic flow when the flow encounters an obstruction or sudden change in flow direction, causing the flow to decelerate abruptly to subsonic speeds
• Across a normal shock, there is a discontinuous change in flow properties
  • Sudden increase in pressure, density, temperature, and entropy
  • Flow velocity decreases
• The Mach number upstream of a normal shock is always greater than 1 (supersonic), and downstream of the shock, it is always less than 1 (subsonic)
• The strength of a normal shock depends on the upstream Mach number
  • Higher Mach numbers result in stronger shocks and more significant changes in flow properties
• Normal shocks are highly irreversible and non-isentropic
  • Cause an increase in entropy and a loss of total pressure across the shock
• The thickness of a normal shock is on the order of a few mean free paths of the gas molecules (extremely thin region compared to the overall flow field)

Examples

• A supersonic aircraft creates a normal shock wave in front of its engine inlet to decelerate the flow to subsonic speeds before entering the engine
• In a supersonic wind tunnel, a normal shock forms when the flow encounters a model or a sudden change in the test section geometry

Normal Shock Relations

Equations and Calculations

• The normal shock relations are a set of equations that relate the flow properties upstream and downstream of a normal shock, based on conservation of mass, momentum, and energy
• The upstream Mach number ($M_1$) is the primary input for the normal shock relations, along with the specific heat ratio ($\gamma$) of the gas
• The static pressure ratio ($p_2/p_1$) across a normal shock is given by: $\frac{p_2}{p_1} = \frac{2\gamma M_1^2 - (\gamma-1)}{\gamma+1}$
• The static temperature ratio ($T_2/T_1$) across a normal shock is given by: $\frac{T_2}{T_1} = \frac{(2\gamma M_1^2 - (\gamma-1)) * ((\gamma-1)M_1^2 + 2)}{(\gamma+1)^2 * M_1^2}$
• The density ratio ($\rho_2/\rho_1$) across a normal shock is given by: $\frac{\rho_2}{\rho_1} = \frac{(\gamma+1)M_1^2}{(\gamma-1)M_1^2 + 2}$

Downstream Mach Number and Total Pressure Ratio

• The downstream Mach number ($M_2$) can be calculated using: $M_2^2 = \frac{(\gamma-1)M_1^2 + 2}{2\gamma M_1^2 - (\gamma-1)}$
• The total pressure ratio ($p_{02}/p_{01}$) across a normal shock is given by: $\frac{p_{02}}{p_{01}} = \left(\frac{(\gamma+1)M_1^2}{2 + (\gamma-1)M_1^2}\right)^{\frac{\gamma}{\gamma-1}} \left(\frac{\gamma+1}{2\gamma M_1^2 - (\gamma-1)}\right)^{\frac{1}{\gamma-1}}$

Examples

• For a normal shock in air ($\gamma = 1.4$) with an upstream Mach number of 2.0, the static pressure ratio ($p_2/p_1$) is 4.5, and the downstream Mach number ($M_2$) is 0.58
• In a supersonic nozzle, a normal shock can form if the back pressure is higher than the design value, leading to a sudden decrease in the exit velocity and an increase in pressure

Oblique Shock Waves in Supersonic Flow

Formation and Behavior

• Oblique shock waves form when a supersonic flow encounters a wedge or a compression corner, causing the flow to turn and compress
• The angle between the upstream flow direction and the shock wave is called the shock angle ($\beta$)
  • Depends on the upstream Mach number ($M_1$) and the deflection angle ($\theta$) of the flow
• For a given upstream Mach number, there exists a maximum deflection angle ($\theta_{max}$) beyond which an attached oblique shock cannot form
  • The shock becomes detached, forming a bow shock
• The downstream flow behind an oblique shock remains supersonic, with a reduced Mach number ($M_2$) and a change in flow direction
• The strength of an oblique shock increases with increasing deflection angle and upstream Mach number, leading to larger changes in flow properties across the shock

Classification and Limiting Cases

• Oblique shocks can be classified as weak or strong
  • Weak shocks have a smaller shock angle and a higher downstream Mach number compared to strong shocks
• In the limiting case where the deflection angle approaches zero, an oblique shock becomes a Mach wave
  • A weak, isentropic compression wave that does not change the flow properties

Examples

• Supersonic aircraft wings are designed with a swept-back angle to generate oblique shocks and reduce wave drag
• In supersonic inlet design, a series of oblique shocks are used to decelerate the flow and increase pressure before entering the engine

Oblique Shock Relations

Equations and Calculations

• The oblique shock relations are a set of equations that relate the flow properties upstream and downstream of an oblique shock, based on the conservation of mass, momentum, and energy, as well as the geometry of the shock
• The primary inputs for the oblique shock relations are the upstream Mach number ($M_1$), the deflection angle ($\theta$), and the specific heat ratio ($\gamma$) of the gas
• The $\theta$-$\beta$-$M$ relation is used to determine the shock angle ($\beta$) for a given deflection angle ($\theta$) and upstream Mach number ($M_1$). This relation is given by: $\tan(\theta) = 2 * \cot(\beta) * \frac{M_1^2 * \sin^2(\beta) - 1}{M_1^2 * (\gamma + \cos(2\beta)) + 2}$
• The downstream Mach number ($M_2$) can be calculated using: $M_2^2 * \sin^2(\beta-\theta) = \frac{M_1^2 * \sin^2(\beta) + (2/(\gamma-1))}{(2\gamma/(\gamma-1)) * M_1^2 * \sin^2(\beta) - 1}$
• The static pressure ratio ($p_2/p_1$) across an oblique shock is given by: $\frac{p_2}{p_1} = 1 + \frac{2\gamma}{\gamma+1} * (M_1^2 * \sin^2(\beta) - 1)$

Temperature, Density, and Total Pressure Ratios

• The static temperature ratio ($T_2/T_1$) and density ratio ($\rho_2/\rho_1$) can be calculated using the same equations as for normal shocks, replacing $M_1$ with $M_1 \sin(\beta)$
• The total pressure ratio ($p_{02}/p_{01}$) across an oblique shock is given by: $\frac{p_{02}}{p_{01}} = \left(\frac{p_2}{p_1}\right)^{\frac{\gamma}{\gamma-1}} * \left(\frac{(\gamma+1)M_1^2 * \sin^2(\beta)}{2 + (\gamma-1)M_1^2 \sin^2(\beta)}\right)^{\frac{1}{\gamma-1}}$

Examples

• For a 10-degree wedge in a Mach 2.5 flow of air ($\gamma = 1.4$), the shock angle ($\beta$) is approximately 32 degrees, and the downstream Mach number ($M_2$) is 1.87
• In a supersonic wind tunnel, an oblique shock generated by a wedge model can be used to study the effects of shock-boundary layer interaction on aerodynamic performance