6.4 Normal and oblique shock waves

Normal and oblique shock waves are crucial phenomena in supersonic flow. They occur when flow encounters obstructions or changes direction, causing abrupt changes in pressure, density, and temperature. Understanding these shocks is vital for designing supersonic aircraft, engines, and wind tunnels.

Shock waves are governed by conservation laws and the Rankine-Hugoniot relations. Normal shocks slow flow to subsonic speeds, while oblique shocks allow supersonic flow downstream. Both types play key roles in supersonic aerodynamics, from thin airfoil theory to inlet design and blunt body aerodynamics.

Characteristics of normal shock waves

Formation in supersonic flow

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• Normal shock waves form when a supersonic flow encounters an obstruction or sudden change in flow direction
• Occur when the upstream Mach number is greater than 1, indicating supersonic flow
• Can be observed in supersonic wind tunnels, rocket nozzles, and supersonic aircraft inlet systems
• Represent a thin region where flow properties change abruptly

Discontinuous changes across shock

• Flow properties experience discontinuous changes across a normal shock wave
• Pressure, density, and temperature increase sharply while velocity decreases
• These changes occur over a very small distance, typically a few mean free paths of the gas molecules
• The discontinuity is a result of the gas molecules' inability to adapt gradually to the changing flow conditions

Increase in pressure, density, temperature

• Pressure increases significantly across a normal shock wave, often by several times the upstream value
• Density also increases, leading to a more compressed and denser flow downstream of the shock
• Temperature rises due to the conversion of kinetic energy into thermal energy across the shock
• The increase in these properties depends on the upstream Mach number and the specific heat ratio of the gas

Decrease in velocity to subsonic

• The velocity of the flow decreases abruptly across a normal shock wave
• Upstream supersonic flow is decelerated to subsonic speeds downstream of the shock
• This deceleration is necessary to satisfy the conservation of mass and momentum equations
• The Mach number downstream of a normal shock is always less than 1, indicating subsonic flow

Irreversible process and entropy rise

• The process of a normal shock wave is irreversible, meaning it cannot be reversed without additional energy input
• Entropy increases across the shock due to the irreversible conversion of kinetic energy into thermal energy
• This entropy rise is a measure of the loss of available energy in the flow
• The irreversible nature of normal shocks leads to losses in efficiency and performance in supersonic devices

Governing equations for normal shocks

Conservation of mass, momentum, energy

• The governing equations for normal shocks are based on the conservation of mass, momentum, and energy
• Conservation of mass: $\rho_1 u_1 = \rho_2 u_2$, where $\rho$ is density and $u$ is velocity
• Conservation of momentum: $p_1 + \rho_1 u_1^2 = p_2 + \rho_2 u_2^2$, where $p$ is pressure
• Conservation of energy: $h_1 + \frac{1}{2}u_1^2 = h_2 + \frac{1}{2}u_2^2$, where $h$ is specific enthalpy

Rankine-Hugoniot relations

• The Rankine-Hugoniot relations are derived from the conservation equations and relate the upstream and downstream properties of a normal shock
• These relations express the downstream properties as functions of the upstream Mach number and specific heat ratio
• Examples include the pressure ratio: $\frac{p_2}{p_1} = \frac{2\gamma M_1^2 - (\gamma - 1)}{\gamma + 1}$ and the density ratio: $\frac{\rho_2}{\rho_1} = \frac{(\gamma + 1)M_1^2}{(\gamma - 1)M_1^2 + 2}$

Mach number relations

• The upstream and downstream Mach numbers are related by the normal shock relations
• The downstream Mach number $M_2$ can be expressed as a function of the upstream Mach number $M_1$ and specific heat ratio $\gamma$: $M_2^2 = \frac{(\gamma - 1)M_1^2 + 2}{2\gamma M_1^2 - (\gamma - 1)}$
• As the upstream Mach number increases, the downstream Mach number approaches a limiting value of $\sqrt{\frac{\gamma - 1}{2\gamma}}$

Stagnation pressure ratio vs Mach number

• The stagnation pressure ratio across a normal shock is a function of the upstream Mach number and specific heat ratio
• Stagnation pressure decreases across a normal shock, indicating a loss in the flow's ability to do work
• The stagnation pressure ratio is given by: $\frac{p_{02}}{p_{01}} = \left[\frac{(\gamma + 1)M_1^2}{2 + (\gamma - 1)M_1^2}\right]^{\frac{\gamma}{\gamma - 1}} \left[\frac{\gamma + 1}{2\gamma M_1^2 - (\gamma - 1)}\right]^{\frac{1}{\gamma - 1}}$
• As the upstream Mach number increases, the stagnation pressure ratio decreases, indicating greater losses

Oblique shock waves

Formation by flow deflection

• Oblique shock waves form when a supersonic flow encounters a sharp corner or a wedge-shaped object
• The flow is deflected by an angle $\theta$, causing an oblique shock to form at an angle $\beta$ relative to the upstream flow direction
• The shock angle $\beta$ depends on the upstream Mach number $M_1$ and the deflection angle $\theta$
• Oblique shocks are more common in practical applications than normal shocks due to the gradual flow deflection

Oblique vs normal shock waves

• Oblique shocks differ from normal shocks in several ways:
  • Oblique shocks form at an angle to the flow, while normal shocks are perpendicular to the flow
  • The flow downstream of an oblique shock remains supersonic, while it becomes subsonic after a normal shock
  • The changes in flow properties across an oblique shock are less severe than those across a normal shock
• However, both types of shocks satisfy the same conservation equations and share some qualitative similarities

Shock angle and deflection angle

• The shock angle $\beta$ and deflection angle $\theta$ are related by the $\theta$-$\beta$-$M$ relation: $\tan \theta = 2 \cot \beta \frac{M_1^2 \sin^2 \beta - 1}{M_1^2 (\gamma + \cos 2\beta) + 2}$
• For a given upstream Mach number and deflection angle, there are two possible shock angles satisfying the relation
• These two solutions correspond to the weak and strong shock cases
• The shock angle increases with increasing deflection angle and upstream Mach number

Weak vs strong shock solutions

• For a given upstream Mach number and deflection angle, there are two possible oblique shock solutions: weak and strong shocks
• Weak shocks have a smaller shock angle and a lower pressure ratio than strong shocks
• The downstream Mach number is higher for weak shocks, and the flow remains supersonic
• Strong shocks have a larger shock angle, higher pressure ratio, and lower downstream Mach number
• In most practical cases, weak shocks are observed as they are more stable and have lower losses

Oblique shock properties

Downstream Mach number and pressure

• The downstream Mach number $M_2$ and pressure $p_2$ can be calculated using the oblique shock relations
• These relations are similar to the normal shock relations but include the shock angle $\beta$
• The downstream Mach number is given by: $M_2^2 \sin^2 (\beta - \theta) = \frac{1 + \frac{\gamma - 1}{2}M_1^2 \sin^2 \beta}{\gamma M_1^2 \sin^2 \beta - \frac{\gamma - 1}{2}}$
• The pressure ratio across an oblique shock is: $\frac{p_2}{p_1} = 1 + \frac{2\gamma}{\gamma + 1}(M_1^2 \sin^2 \beta - 1)$

Pressure, density, temperature ratios

• The ratios of pressure, density, and temperature across an oblique shock can be expressed in terms of the upstream Mach number and shock angle
• Pressure ratio: $\frac{p_2}{p_1} = 1 + \frac{2\gamma}{\gamma + 1}(M_1^2 \sin^2 \beta - 1)$
• Density ratio: $\frac{\rho_2}{\rho_1} = \frac{(\gamma + 1)M_1^2 \sin^2 \beta}{2 + (\gamma - 1)M_1^2 \sin^2 \beta}$
• Temperature ratio: $\frac{T_2}{T_1} = \frac{p_2}{p_1} \frac{\rho_1}{\rho_2}$
• These ratios increase with increasing shock angle and upstream Mach number

Supersonic flow downstream of shock

• Unlike normal shocks, the flow downstream of an oblique shock remains supersonic
• The downstream Mach number $M_2$ is always greater than 1 for weak shock solutions
• This allows for the possibility of multiple oblique shocks in succession, such as in supersonic inlet designs
• The downstream flow is deflected by an angle $\theta$ relative to the upstream flow direction

Mach angle and Mach cone

• The Mach angle $\mu$ is the angle between the flow direction and the Mach wave, which is a weak disturbance that propagates at the speed of sound relative to the flow
• The Mach angle is related to the Mach number by: $\sin \mu = \frac{1}{M}$
• The Mach cone is a conical surface formed by the Mach waves emanating from a point source moving at supersonic speeds
• The half-angle of the Mach cone is equal to the Mach angle $\mu$
• Oblique shocks always lie within the Mach cone of the upstream flow

Oblique shock applications

Supersonic thin airfoil theory

• Supersonic thin airfoil theory is used to analyze the aerodynamics of thin, sharp-edged airfoils at supersonic speeds
• The theory assumes that the flow disturbances caused by the airfoil are small and can be treated as oblique shocks
• The lift and drag coefficients of the airfoil can be calculated using the oblique shock relations and the airfoil geometry
• Supersonic thin airfoil theory is useful for designing supersonic aircraft wings and control surfaces

Supersonic inlet design

• Supersonic inlets are used to decelerate and compress the airflow before it enters the engine of a supersonic aircraft
• The inlet design often involves a series of oblique shocks followed by a normal shock to efficiently slow down the flow
• The goal is to minimize the total pressure loss and flow distortion while providing sufficient pressure recovery
• Careful design of the inlet geometry and shock system is crucial for optimal performance

Shock-expansion theory

• Shock-expansion theory is a method for analyzing the flow over supersonic airfoils and bodies with sharp corners
• The theory combines oblique shock relations with Prandtl-Meyer expansion waves to calculate the flow properties and aerodynamic forces
• The flow is assumed to undergo an oblique shock at the leading edge, followed by an isentropic expansion around the corner
• Shock-expansion theory is more accurate than supersonic thin airfoil theory for airfoils with finite thickness and sharp corners

Shock-boundary layer interaction

• Shock-boundary layer interaction occurs when an oblique shock impinges on a viscous boundary layer
• The interaction can cause flow separation, unsteadiness, and increased heat transfer
• The adverse pressure gradient imposed by the shock can lead to boundary layer thickening, separation bubbles, or shock-induced turbulence
• Understanding and controlling shock-boundary layer interactions is important for designing efficient supersonic inlets, control surfaces, and nozzles

Detached and bow shocks

Blunt body shock formation

• Detached and bow shocks form when a supersonic flow encounters a blunt body, such as a spherical nose cone or a rounded leading edge
• Unlike sharp-edged bodies, blunt bodies cannot sustain an attached oblique shock due to the high shock angle required
• Instead, a detached curved shock forms ahead of the body, called a bow shock
• The bow shock is nearly normal to the freestream flow at the centerline and becomes more oblique away from the centerline

Subsonic region behind bow shock

• The flow immediately behind a bow shock is subsonic, as the shock is nearly normal to the flow at the centerline
• This subsonic region is called the shock layer and is characterized by high pressure, density, and temperature
• The thickness of the shock layer depends on the body geometry and the freestream Mach number
• The subsonic flow in the shock layer gradually accelerates and becomes supersonic as it moves away from the centerline

Shock standoff distance

• The shock standoff distance is the distance between the bow shock and the blunt body surface at the centerline
• The standoff distance depends on the body geometry, freestream Mach number, and specific heat ratio
• For a spherical body, the standoff distance can be estimated using empirical correlations or numerical methods
• A larger standoff distance indicates a thicker shock layer and higher drag on the body

Entropy layer and flow separation

• The flow in the shock layer has a higher entropy than the freestream flow due to the irreversible nature of the bow shock
• This high-entropy flow forms a thin region called the entropy layer near the body surface
• The entropy layer is characterized by low velocity, high temperature, and high density compared to the inviscid flow outside the layer
• The adverse pressure gradient and the entropy gradient can cause flow separation and recirculation near the body surface
• Controlling flow separation is crucial for designing efficient blunt body geometries, such as reentry vehicles and supersonic parachutes