Normal and oblique shock waves are critical phenomena in . These abrupt changes in flow properties occur when supersonic flow encounters obstructions or changes direction. Understanding shock waves is essential for analyzing and designing supersonic vehicles and propulsion systems.
Shock waves cause sudden increases in pressure, temperature, and density while decreasing velocity. The describe these changes mathematically. Oblique shocks, inclined to the flow direction, are weaker than normal shocks and allow downstream flow to remain supersonic.
Properties of normal shock waves
Normal shock waves are thin regions where flow properties change abruptly
Occur when supersonic flow encounters an obstruction or a sharp change in flow direction
Characterized by a discontinuous increase in pressure, temperature, and density across the shock
Pressure ratio across shock
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Pressure increases significantly across a
Pressure ratio depends on the upstream
Higher upstream Mach numbers result in larger pressure ratios across the shock
Pressure ratio can be calculated using the Rankine-Hugoniot relations
Temperature ratio across shock
Temperature also increases across a normal shock wave
Temperature ratio is a function of the upstream Mach number
Higher upstream Mach numbers lead to higher temperature ratios
Temperature increase is due to the conversion of kinetic energy into thermal energy
Density ratio across shock
Density increases across a normal shock wave
Density ratio is related to the pressure and temperature ratios
Can be calculated using the equation of state for an ideal gas
is necessary to satisfy conservation of mass
Mach number change
Flow velocity decreases across a normal shock wave
Upstream Mach number is always supersonic (M > 1)
Downstream Mach number is always subsonic (M < 1)
Mach number decrease is due to the increase in speed of sound across the shock
Entropy increase
Entropy increases across a normal shock wave
Entropy increase is irreversible and indicates a loss of available energy
Amount of entropy increase depends on the strength of the shock (upstream Mach number)
Entropy increase is a measure of the irreversibility of the shock process
Rankine-Hugoniot relations
Set of equations that describe the relationship between flow properties across a shock wave
Derived from the conservation laws of mass, momentum, and energy
Used to calculate the downstream flow properties given the upstream conditions and shock strength
Conservation of mass
Mass flow rate is conserved across a normal shock wave
Product of density and velocity must be equal on both sides of the shock
Equation: ρ1u1=ρ2u2, where ρ is density and u is velocity
Subscripts 1 and 2 denote upstream and downstream conditions, respectively
Conservation of momentum
Momentum is conserved across a normal shock wave
Sum of pressure and momentum flux must be equal on both sides of the shock
Equation: p1+ρ1u12=p2+ρ2u22, where p is pressure
Pressure increase across the shock balances the decrease in momentum flux
Conservation of energy
Energy is conserved across a normal shock wave
Total enthalpy (sum of static enthalpy and kinetic energy) is constant across the shock
Equation: h1+21u12=h2+21u22, where h is specific enthalpy
Kinetic energy is converted into thermal energy (static enthalpy) across the shock
Normal shock in ideal gas
Normal shock waves in an ideal gas exhibit specific behavior and properties
Ideal gas assumption simplifies the analysis and allows for closed-form solutions
Upstream and downstream states
Upstream (pre-shock) state is characterized by high Mach number, low pressure, low temperature, and low density
Downstream (post-shock) state has low Mach number, high pressure, high temperature, and high density
Ratio of downstream to upstream properties depends on the upstream Mach number
Property ratios increase with increasing upstream Mach number
Mach number limits
Normal shock waves can only occur in supersonic flow (upstream Mach number > 1)
Downstream Mach number is always subsonic (< 1) for a normal shock in an ideal gas
Maximum downstream Mach number is limited to 1, which occurs for an infinitely strong shock
Minimum upstream Mach number for a normal shock is 1, corresponding to a weak shock
Stagnation pressure ratio
Stagnation pressure (total pressure) decreases across a normal shock wave
(downstream to upstream) is always less than 1
Stagnation pressure ratio decreases with increasing upstream Mach number
Stagnation pressure loss is a measure of the irreversibility of the shock process
Maximum entropy increase
Entropy increase across a normal shock wave has a maximum value
occurs at a specific upstream Mach number (approximately 1.245 for air)
Upstream Mach numbers above or below this value result in lower entropy increases
Maximum entropy increase is an important consideration in the design of supersonic diffusers
Moving normal shock waves
Normal shock waves can be either stationary or moving relative to an observer
Moving shocks introduce additional complexity in the analysis of flow properties
Stationary vs moving shocks
Stationary shocks are fixed in space and the flow moves through the shock
Moving shocks propagate through a stationary or moving fluid
Reference frame can be changed to convert a moving shock into a stationary shock and vice versa
Flow properties across the shock are the same in both reference frames
Shock velocity relative to flow
Velocity of a moving shock wave is superimposed on the flow velocity
Upstream and downstream velocities relative to the shock are different
Relative velocity upstream of the shock is supersonic, while downstream is subsonic
Shock velocity can be determined using the Rankine-Hugoniot relations
Shock propagation in ducts
Moving normal shocks can propagate through ducts or channels
Shock propagation is influenced by the duct geometry and flow conditions
Shock velocity in a duct is affected by the change in cross-sectional area
Converging ducts accelerate the shock, while diverging ducts decelerate it
Oblique shock waves
Oblique shock waves are inclined at an angle to the flow direction
Occur when a supersonic flow encounters a concave corner or a compression ramp
Oblique vs normal shock waves
Oblique shocks have a non-zero angle with respect to the flow direction, while normal shocks are perpendicular
Flow downstream of an oblique shock remains supersonic, while it becomes subsonic after a normal shock
Oblique shocks are weaker than normal shocks for the same upstream Mach number
Oblique shocks can be attached to the surface or detached, depending on the flow conditions
Shock wave angle
Angle between the and the upstream flow direction
Shock wave angle depends on the upstream Mach number and the of the surface
Increases with increasing upstream Mach number and deflection angle
Can be calculated using the theta-beta-Mach relation
Deflection angle
Angle through which the flow is turned by the oblique shock wave
Deflection angle is determined by the geometry of the surface (ramp angle or corner angle)
Maximum deflection angle exists for a given upstream Mach number, beyond which the shock becomes detached
Deflection angle is related to the shock wave angle through the theta-beta-Mach relation
Weak vs strong solutions
For a given upstream Mach number and deflection angle, there are two possible shock wave angles
has a smaller shock wave angle and a higher downstream Mach number
has a larger shock wave angle and a lower downstream Mach number
Weak shock solution is usually observed in practice, unless the flow is highly disturbed or the deflection angle is large
Oblique shock relations
Flow properties across an oblique shock wave can be calculated using
Relations are derived from the Rankine-Hugoniot equations and the geometry of the oblique shock
Pressure ratio across oblique shock
Pressure increases across an oblique shock wave
Pressure ratio depends on the upstream Mach number and the shock wave angle
Pressure ratio increases with increasing Mach number and shock wave angle
Can be calculated using the oblique shock pressure ratio equation
Density ratio across oblique shock
Density also increases across an oblique shock wave
Density ratio is related to the pressure ratio and the upstream Mach number
Can be calculated using the oblique shock density ratio equation
Density ratio is always greater than 1
Temperature ratio across oblique shock
Temperature increases across an oblique shock wave
Temperature ratio depends on the upstream Mach number and the shock wave angle
Can be calculated using the oblique shock temperature ratio equation
Temperature ratio is always greater than 1
Downstream Mach number
Mach number downstream of an oblique shock wave is lower than the upstream Mach number
Downstream Mach number depends on the upstream Mach number, shock wave angle, and deflection angle
Can be calculated using the oblique shock Mach number equation
Downstream Mach number is always supersonic for an attached oblique shock
Supersonic flow over wedges
Wedges are simple geometries that produce oblique shock waves in supersonic flow
Wedge flow is a fundamental problem in compressible aerodynamics
Attached vs detached shocks
Oblique shock wave can be attached to the wedge apex or detached from it
Attached shock occurs when the deflection angle is less than the maximum deflection angle for the given Mach number
Detached shock occurs when the deflection angle exceeds the maximum deflection angle
Detached shock is curved and stands off from the wedge apex
Wedge angle for attached shock
Wedge angle is the angle between the wedge surface and the freestream direction
For an attached shock, the wedge angle is equal to the deflection angle
Maximum wedge angle for an attached shock depends on the freestream Mach number
Can be calculated using the theta-beta-Mach relation and the maximum deflection angle
Maximum deflection angle
Maximum angle through which the flow can be turned by an attached oblique shock
Depends on the upstream Mach number and the specific heat ratio of the gas
Increases with increasing Mach number
Flow cannot be turned by an angle greater than the maximum deflection angle without creating a detached shock
Reflection of oblique shocks
Oblique shock waves can reflect from solid surfaces or interact with other shock waves
Reflection patterns depend on the incident shock strength and the flow conditions
Regular vs Mach reflection
Regular reflection occurs when the incident and reflected shocks meet at the surface
Mach reflection occurs when the incident and reflected shocks meet above the surface, forming a Mach stem
Transition from regular to Mach reflection depends on the incident and the flow deflection angle
Mach reflection is more likely to occur for strong incident shocks and large deflection angles
Shock-shock interaction
Oblique shocks can interact with each other, resulting in complex flow patterns
Interaction can be between two oblique shocks or between an oblique shock and a normal shock
Shock-shock interaction can lead to the formation of a triple point, where three shocks meet
Flow properties and shock angles change discontinuously across the triple point
Shock polars
Graphical representation of the relationship between the flow deflection angle and the shock wave angle
Used to analyze and predict the behavior of oblique shocks and their interactions
Different branches of the shock polar correspond to different shock solutions (weak, strong, or detached)
Intersection of shock polars determines the flow conditions and shock angles in shock-shock interactions
Shock wave-boundary layer interaction
Interaction between shock waves and boundary layers can significantly affect the flow field
Shock-boundary layer interaction can lead to , unsteadiness, and increased drag
Shock-induced separation
Adverse pressure gradient imposed by a shock wave can cause the boundary layer to separate
Separation occurs when the boundary layer cannot overcome the pressure rise across the shock
Shock-induced separation can lead to the formation of a recirculation bubble and increased flow unsteadiness
Severity of separation depends on the shock strength, boundary layer state, and surface geometry
Lambda shock structure
Characteristic shock pattern that forms when a shock wave interacts with a boundary layer
Consists of a normal shock near the surface, followed by an oblique shock that merges with the incident shock
Lambda shock structure is associated with shock-induced separation and the formation of a recirculation bubble
Can occur in supersonic inlets, transonic airfoils, and other flow situations where shocks interact with boundary layers
Shock train in supersonic flow
Series of successive shock waves that form in a supersonic flow with a boundary layer
Shock train is caused by the interaction between the shocks and the boundary layer
Each shock in the train is weaker than the previous one, and the spacing between shocks decreases downstream
Shock trains can occur in supersonic diffusers, isolators, and other flow passages with adverse pressure gradients
Applications of shock waves
Shock waves have numerous applications in aerospace engineering and other fields
Understanding and controlling shock waves is crucial for the design and operation of supersonic vehicles and devices
Supersonic inlets
Inlets are used to decelerate and compress the flow before it enters the engine of a supersonic vehicle
Shock waves are employed in supersonic inlets to efficiently reduce the Mach number and increase the pressure
Inlet design must balance the conflicting requirements of high pressure recovery and low flow distortion
Shock wave-boundary layer interaction and shock stability are major challenges in supersonic inlet design
Shock tubes and tunnels
Shock tubes and tunnels are experimental facilities used to study shock waves and high-speed flows
Shock tube consists of a high-pressure driver section and a low-pressure driven section separated by a diaphragm
When the diaphragm is ruptured, a shock wave propagates into the driven section, followed by an expansion wave
Shock tunnels use the high-temperature, high-pressure flow behind the reflected shock to simulate hypersonic flight conditions
Shock wave lithotripsy
Medical application of shock waves for the non-invasive treatment of kidney stones and other calculi
Focused shock waves are generated outside the body and propagate through tissue to the target stone
Shock waves induce stress and fracture in the stone, leading to its fragmentation into smaller pieces
Fragmented stones can then be easily passed through the urinary tract or dissolved by the body's natural processes