Unit 2: Shocks and its applications - Subjective Questions

A normal shock wave is a very thin compression region positioned perpendicular to the direction of flow, across which the flow changes abruptly from supersonic to subsonic conditions.

Development of a normal shock:

• Weak pressure disturbances travel at the local speed of sound relative to the fluid.
• In supersonic flow, disturbances generated by a compressing surface cannot propagate upstream.
• The compression waves converge and coalesce into a narrow discontinuity called a shock wave.
• Across the shock, pressure, temperature, density, and entropy increase abruptly.
• Velocity and Mach number decrease, with and for a stationary normal shock.
• The process is adiabatic but irreversible, so stagnation temperature remains constant while stagnation pressure decreases.

The shock thickness is only a few molecular mean free paths, so it is usually treated as a discontinuity.

For a steady, one-dimensional, adiabatic flow without shaft work, the equations across a stationary normal shock are:

1. Conservation of mass:

2. Conservation of momentum:

3. Conservation of energy:

For a calorically perfect gas, , giving

The perfect-gas equation is

These equations show that mass, momentum, and total enthalpy are conserved. However, entropy increases because the shock is irreversible. Consequently, stagnation pressure decreases even though stagnation temperature remains constant.

Applying conservation of mass, momentum, and energy to a perfect gas gives the normal-shock relations in terms of upstream Mach number and specific-heat ratio .

Pressure ratio:

Density ratio:

Since continuity requires ,

Temperature ratio:

Downstream Mach number:

For , these relations give , , , , and .

The Hugoniot equation relates the thermodynamic states on the two sides of a shock. The steady-flow energy equation is

Using continuity with specific volume ,

so

The momentum equation gives

therefore

Substituting into the energy equation produces

Since , this may also be written as

This is the Rankine-Hugoniot relation.

Physical significance:

• It specifies all possible downstream equilibrium states compatible with conservation laws for a given upstream state.
• Its curve on a pressure-specific-volume diagram is called the Hugoniot curve.
• The physically admissible compression-shock branch satisfies , , and .
• It is applicable even when the thermodynamic behavior is not represented by a calorically perfect-gas model.

Across a normal shock, the static properties change abruptly:

• Static pressure: increases, so .
• Static temperature: increases, so .
• Density: increases, so .
• Velocity: decreases, so .
• Mach number: changes from to .
• Entropy: increases, so .

Because the shock is adiabatic and no external work is done,

For a calorically perfect gas, this implies

However, irreversibility causes a loss in stagnation pressure:

Its ratio is

Thus, a shock converts organized kinetic energy into internal energy and reduces the capacity of the flow to perform useful work.

Stationary normal shock:

• It remains fixed relative to the duct, nozzle, or observer.
• The upstream fluid moves supersonically toward the shock.
• In the shock-fixed frame, and .
• It commonly occurs in supersonic diffusers, overexpanded nozzles, and wind tunnels.

Moving normal shock:

• It propagates through a fluid with shock speed relative to a stationary observer.
• Conditions must be analyzed using velocities relative to the shock.
• If the laboratory-frame fluid velocities are and , the shock-fixed velocities are
  
  with signs selected consistently.
• The normal-shock conservation equations apply in the shock-fixed reference frame.
• It appears in shock tubes, explosions, blast waves, and rapid valve or piston motions.

Therefore, the governing physics is the same in both cases; the principal difference is the choice of reference frame.

A moving shock is most conveniently analyzed in a reference frame attached to the shock. Let be the shock speed and let and be the laboratory-frame fluid velocities before and after the shock. Define relative velocities by

The conservation equations in the shock-fixed frame are:

• Mass:
  

• Momentum:
  

• Energy:
  

The relative upstream Mach number is

The stationary normal-shock relations can then be used with . After calculating , the downstream laboratory velocity is recovered from

This reference-frame transformation is essential because it converts an unsteady laboratory-frame problem into a steady shock-fixed problem.

Applications of stationary shocks:

• Supersonic air intakes: shocks compress and decelerate air before it enters a compressor or combustor.
• Convergent-divergent nozzles: a normal shock may stand in the divergent portion when back pressure is above the design value.
• Supersonic wind tunnels: normal shocks occur during tunnel starting or in diffuser systems.
• Supersonic diffusers: shocks help convert kinetic energy into pressure, although they cause stagnation-pressure loss.

Applications of moving shocks:

• Shock tubes: a moving shock produces well-defined high-pressure and high-temperature test conditions.
• Blast and explosion analysis: moving shocks describe rapidly propagating pressure fronts.
• Piston-driven compression: sudden piston motion generates a shock that travels into the gas.
• Internal-combustion and detonation systems: transient compression waves may steepen into shocks.

In practical designs, shocks may be necessary for compression, but their strength is controlled to limit entropy generation, stagnation-pressure loss, heating, vibration, and structural loading.

An oblique shock wave is a compression shock inclined at an angle to the upstream velocity. It is commonly generated when a supersonic flow is turned toward itself through a compression angle .

Differences from a normal shock:

• A normal shock is perpendicular to the upstream flow, whereas an oblique shock is inclined to it.
• Across an oblique shock, only the velocity component normal to the shock is compressed.
• The tangential velocity component remains unchanged:
  
• The normal component follows normal-shock relations using
  
• Flow direction changes by across an oblique shock, whereas it does not change across a normal shock.
• The downstream flow behind a weak oblique shock may remain supersonic.
• For the same upstream Mach number, an oblique shock usually causes less stagnation-pressure loss than a normal shock.

A normal shock is the limiting case of an oblique shock when .

Resolve the upstream velocity into components normal and tangential to a shock inclined at angle :

The normal component satisfies the normal-shock relations, while the tangential component is unchanged. Combining continuity, momentum, and the geometric relation between upstream and downstream velocities gives

Here, is the upstream Mach number, is the flow-deflection angle, and is the shock angle.

Characteristics of the solutions:

• The minimum possible shock angle is the Mach angle
  
• For , the equation generally has two solutions.
• The weak solution has smaller , smaller property changes, and usually a supersonic downstream flow.
• The strong solution has larger and usually produces subsonic downstream flow.
• At , the weak and strong solutions merge.
• If , an attached straight shock is impossible and a detached bow shock forms.

The oblique shock is analyzed by applying normal-shock relations to the velocity component normal to the shock.

Procedure:

1. Determine the shock angle from the -- relation.
2. Calculate the upstream normal Mach number:
   
3. Calculate the downstream normal Mach number:
   
4. Since the downstream velocity makes an angle with the shock,
   

The property ratios follow by replacing in the normal-shock equations with :

The tangential velocity is unchanged, but the normal velocity decreases. Consequently, the flow is deflected while its pressure, density, temperature, and entropy increase.

For a specified and a deflection angle below , the -- relation generally gives two shock angles.

Weak-shock solution:

• Has the smaller shock angle .
• Has a smaller normal Mach component .
• Produces smaller increases in pressure, density, temperature, and entropy.
• Causes a smaller stagnation-pressure loss.
• Usually leaves the downstream flow supersonic.

Strong-shock solution:

• Has the larger shock angle, closer to .
• Has a larger value of .
• Produces greater compression and entropy generation.
• Usually leaves the downstream flow subsonic.

The weak solution is normally observed because it introduces the smaller disturbance and lower irreversible loss. The strong solution may occur when imposed downstream pressure conditions require subsonic flow or when confinement and back pressure force the stronger branch.

When an incident oblique shock reaches a solid wall, the flow behind it is generally directed toward the wall. Since flow cannot penetrate the wall, a reflected shock forms and turns the flow back until it becomes parallel to the wall.

Regular-reflection process:

• The incident shock compresses and deflects the upstream flow toward the wall.
• At the reflection point, a reflected oblique shock is generated.
• The reflected shock supplies an opposite flow deflection.
• Downstream of the reflected shock, the normal velocity at the wall is zero.
• Pressure and temperature increase across both shocks.
• Stagnation pressure decreases across each shock because of irreversibility.

The reflected-shock angle is determined by applying the -- relation to the Mach number and flow direction behind the incident shock. If the required turning cannot be achieved by an attached reflected shock, regular reflection is replaced by Mach reflection.

Regular reflection occurs when an incident shock and a reflected shock meet directly at a wall. The reflected shock turns the post-incident-shock flow until it is parallel to the wall.

Mach reflection occurs when a simple reflected shock cannot satisfy the required wall boundary condition. Its structure consists of:

• An incident shock.
• A reflected shock.
• A nearly normal shock segment called the Mach stem.
• A triple point where the three shocks meet.
• A slip line extending downstream from the triple point.

Across the slip line, pressure and flow direction are compatible, but velocity, density, temperature, and entropy may differ.

Compared with regular reflection, Mach reflection generally occurs for larger shock strength, larger incidence or wedge angle, or conditions for which the reflected-shock solution cannot remain attached. The Mach stem produces stronger compression and can generate large local pressure and thermal loads near the wall.

Oblique shocks are described as belonging to the same family when they turn the flow successively in the same rotational sense. When two such shocks intersect or occur in sequence:

• The first shock compresses and turns the flow.
• The second shock acts on the already compressed flow and causes further turning in the same direction.
• Pressure, temperature, density, and entropy rise across each shock.
• Mach number and stagnation pressure decrease progressively.
• The downstream state is found by applying the oblique-shock relations successively, using the output from the first shock as the input to the second.

If the total required turning is excessive, an attached-shock pattern may not be possible. The interaction can then produce a detached shock, a Mach stem, or a more complex reflected-shock structure. Distributing a fixed compression through several weaker oblique shocks generally causes less stagnation-pressure loss than achieving the same compression through one strong shock.

Shocks of opposite families turn the flow in opposite directions. When they intersect, each shock enters a flow region modified by the other, so the shocks generally change strength and direction after the interaction.

For compatibility downstream of the interaction:

• The final streams must have equal static pressure at their common boundary.
• Their velocity directions must be parallel to that boundary.
• Their velocity magnitudes, densities, temperatures, and entropy levels need not be equal.

The unequal thermodynamic histories of the streams mean that a single uniform downstream state may not be possible. A slip line then extends from the interaction point and separates the two streams.

The interaction is analyzed using shock polars. The intersection of the appropriate shock-polar branches provides a compatible downstream pressure and flow direction. If no suitable intersection exists, a more complex pattern involving reflected shocks or a Mach stem may be required.

A slip line is a contact discontinuity separating two streams that have different velocities or thermodynamic properties but are in mechanical compatibility. It commonly originates at a triple point in Mach reflection or at an oblique-shock interaction.

For an ideal, inviscid slip line:

• Static pressure is continuous:
  
• The normal velocity is continuous and is zero relative to the line:
  
• The flows on both sides are tangent to the slip line.
• Tangential velocities may differ:
  
• Density, temperature, entropy, Mach number, and stagnation pressure may also differ.
• No mass crosses the slip line in the idealized model.

Because tangential velocity is discontinuous, a slip line represents a vortex sheet. In a real viscous flow, it develops into a finite-thickness shear layer and may become unstable and mix downstream.

A shock polar is a graphical locus of all possible downstream velocity states, pressure ratios, or flow-deflection states obtainable from a given upstream supersonic state through an oblique shock.

Main features:

• Every point on the polar corresponds to a possible shock angle.
• The weak and strong branches represent the two solutions of the -- relation.
• The maximum-deflection point corresponds to .
• The limiting normal-shock state occurs when .

Uses:

• For shock reflection, a second polar is constructed using the state behind the incident shock. Its intersection with the wall-direction requirement determines the reflected-shock state.
• For two-shock interaction, polars based on the two incoming states are compared.
• A polar intersection identifies downstream states with equal pressure and compatible flow direction.
• If velocity magnitude or density differs at the compatible state, the two regions are separated by a slip line.
• Failure to obtain a required intersection indicates that regular interaction is impossible and a Mach-reflection-type structure may form.

A shock produces a rapid pressure rise. When it meets a viscous boundary layer, this rise acts as a strong adverse pressure gradient on the low-momentum fluid close to the wall.

Interaction mechanism:

• The boundary layer thickens as it approaches the shock.
• If the pressure rise is sufficiently strong, wall shear stress falls to zero and then reverses.
• The boundary layer separates upstream of the nominal shock location.
• The separated shear layer deflects the outer supersonic flow and generates compression waves or a separation shock.
• A recirculation bubble may form between separation and reattachment.
• A reattachment shock may occur as the shear layer returns to the wall.
• The original shock can split or become curved, creating a complex shock system.

The interaction is influenced by Mach number, Reynolds number, shock strength, boundary-layer thickness, wall temperature, and whether the boundary layer is laminar or turbulent.

Types of interaction:

• Normal shock interacting with a boundary layer in an internal passage.
• Oblique-shock impingement on a wall boundary layer.
• Compression-corner interaction.
• Shock interaction with laminar, transitional, or turbulent boundary layers.

Consequences:

• Boundary-layer thickening and possible flow separation.
• Increased drag and stagnation-pressure loss.
• Reduced intake pressure recovery and possible inlet unstart.
• Distortion of downstream velocity and pressure profiles.
• High local heating and pressure loads.
• Unsteady shock motion, buffet, vibration, and fatigue.
• Reduced control-surface effectiveness.

Control methods:

• Use multiple weak shocks instead of one strong shock.
• Apply boundary-layer bleed through slots or porous surfaces.
• Use suction, vortex generators, or micro-ramps to energize near-wall flow.
• Optimize ramp angle, surface contour, and shock location.
• Use passive cavities or active flow-control devices where appropriate.

Effective control seeks to reduce the adverse pressure gradient or increase near-wall momentum, thereby limiting separation and unsteadiness.