Unit 2: Shocks and its applications

I. Orientation — Shock-wave fundamentals

A shock wave is an extremely thin compression region across which pressure, temperature, density, entropy, and velocity change almost discontinuously. It develops when finite-amplitude pressure disturbances coalesce in a compressible flow, particularly when the upstream Mach number exceeds unity.

• Governing principle: A shock is treated as a discontinuity satisfying conservation of mass, momentum, and energy.
• Continuum assumption: The physical thickness is only a few molecular mean free paths, so engineering analysis neglects its internal structure.
• Flow model: Unless stated otherwise, the gas is calorically perfect, with constant specific heats and no external heat or shaft work.
• Mach number: The local Mach number is
  
  TEXT

    M = V/a,       a = √(γRT)

  
  where (M) is Mach number, (V) is flow speed, (a) is acoustic speed, (\gamma) is the specific-heat ratio, (R) is the gas constant, and (T) is absolute temperature.
• Compression and irreversibility: A physical shock raises static pressure, density, temperature, and entropy while reducing stagnation pressure.
• Stagnation temperature: For an adiabatic shock with no work, stagnation enthalpy and stagnation temperature remain constant.
• Directional distinction:
  • A normal shock is perpendicular to the upstream flow.
  • An oblique shock is inclined to the upstream flow and turns it toward itself.

II. Normal Shock Waves — One-dimensional compression discontinuities

A. Development of normal shocks

A normal shock develops when compression disturbances in supersonic flow converge into a nearly discontinuous wave normal to the local velocity.

• Wave steepening: Compression raises temperature and acoustic speed, so the rear portions of a finite compression wave travel faster than its front portions and eventually overtake them.
• Coalescence: Numerous weak compression waves combine into one finite-strength shock; expansion waves behave oppositely and spread into a continuous expansion fan.
• Supersonic requirement: In the shock-fixed frame, the normal upstream Mach number must satisfy (M_1>1), while the downstream state has (M_2<1).
• Physical changes:
  • Static quantities increase: (p_2>p_1), (\rho_2>\rho_1), and (T_2>T_1).
  • Velocity and Mach number decrease: (V_2<V_1) and (M_2<1).
  • Entropy increases: (s_2>s_1).
• Limiting behavior: As (M_1\rightarrow1), shock strength and entropy rise approach zero; stronger upstream Mach numbers produce larger pressure and stagnation-pressure losses.
• Typical formation: Normal shocks occur in over-compressed intakes, converging–diverging nozzles operating away from design conditions, and supersonic ducts subjected to excessive back pressure.

B. Governing equations

The normal-shock relations follow from applying steady control-volume conservation laws across the wave.

• Mass conservation:
  
  TEXT

    ρ₁V₁ = ρ₂V₂ = j

  
  where (\rho) is density, (V) is velocity normal to the shock, (j) is mass flux per unit area, and subscripts 1 and 2 denote upstream and downstream states.
• Momentum conservation:
  
  TEXT

    p₁ + ρ₁V₁² = p₂ + ρ₂V₂²

  
  where (p) is static pressure; viscous stresses are negligible outside the thin internal shock layer.
• Energy conservation:
  
  TEXT

    h₁ + V₁²/2 = h₂ + V₂²/2

  
  where (h) is specific enthalpy. For a perfect gas, (h=c_pT), with (c_p) denoting constant-pressure specific heat.
• Perfect-gas closure:
  
  TEXT

    p = ρRT,       γ = cₚ/cᵥ

  
  where (c_v) is constant-volume specific heat.
• Property ratios:
  
  TEXT

    p₂/p₁ = 1 + [2γ/(γ+1)](M₁²−1)
  
    ρ₂/ρ₁ = [(γ+1)M₁²]/[(γ−1)M₁²+2]
  
    T₂/T₁ = (p₂/p₁)/(ρ₂/ρ₁)

• Downstream Mach number:
  
  TEXT

    M₂² = [1 + (γ−1)M₁²/2] / [γM₁² − (γ−1)/2]

• Entropy condition: Although the conservation equations admit compression and expansion discontinuities mathematically, the second law permits only the compression shock because (\Delta s=s_2-s_1>0).

C. Hugoniot equation

The Hugoniot equation identifies all thermodynamic states that can be connected to a specified initial state by an adiabatic shock.

• Energy form:
  
  TEXT

    e₂ − e₁ = ½(p₁+p₂)(v₁−v₂)

  
  where (e) is specific internal energy and (v=1/\rho) is specific volume.
• Derivation basis: Eliminating velocities between mass, momentum, and energy equations produces the relation without requiring shock speed explicitly.
• Hugoniot curve: On a pressure–specific-volume diagram, the equation defines the shock adiabat through the known state ((p_1,v_1)).
• Rayleigh line:
  
  TEXT

    p₂ − p₁ = −j²(v₂−v₁)

  
  Its intersection with the Hugoniot curve determines the possible downstream state for mass flux (j).
• Physical branch: A compression shock lies at (p_2>p_1) and (v_2<v_1); the corresponding entropy is greater downstream.
• Distinction from isentropic compression: The Hugoniot curve represents irreversible shock states, not a constant-entropy path. The difference becomes significant as shock strength increases.

D. Stationary and moving normal shock waves and applications

A normal shock may be stationary relative to a duct or moving relative to the observer, but the same jump equations apply in a frame attached to the shock.

1. Stationary normal shock:

   • Reference frame: Shock speed is zero, so the velocities in the jump relations are ordinary flow velocities.
   • Nozzle application: A shock may stand in the diverging section of a converging–diverging nozzle when back pressure exceeds the design value.
   • Intake application: A terminal normal shock in a supersonic inlet converts the captured stream to subsonic flow before a compressor, but causes substantial stagnation-pressure loss.
   • Duct effect: Downstream static pressure rises sharply, while the reduced stagnation pressure limits pressure recovery and thrust.

2. Moving normal shock:

   • Relative velocity:
     
     TEXT

          w₁ = u₁−Uₛ,       w₂ = u₂−Uₛ

     
     where (u) is laboratory-frame gas velocity, (U_s) is shock velocity, and (w) is velocity measured in the shock-fixed frame.
   • Analysis: Replace (V_1,V_2) in the stationary equations by the relative velocities (w_1,w_2), then transform the result back to the laboratory frame.
   • Applications: Moving shocks occur in shock tubes, explosions, piston-driven compression, compressor surges, and pressure waves travelling through pipelines.
   • Piston mechanism: A rapidly advancing piston launches compression waves that merge into a moving shock; gas behind the shock moves in the piston’s direction.

III. Oblique-Shock Systems — Compression with flow turning

A. Oblique shock

An oblique shock is an inclined compression wave that reduces the normal velocity component while leaving the tangential component unchanged in inviscid flow.

• Geometry: The shock angle (\beta) is measured between the upstream velocity and shock, while the deflection angle (\theta) is the turning of the flow.
• Normal component:
  
  TEXT

    Mₙ₁ = M₁ sinβ

  
  where (M{n1}) is the upstream Mach number normal to the shock. Normal-shock relations apply using (M{n1}).
• Turning relation:
  
  TEXT

    tanθ = 2cotβ [(M₁²sin²β−1)/(M₁²(γ+cos2β)+2)]

• Downstream Mach number:
  
  TEXT

    M₂ = Mₙ₂/sin(β−θ)

  
  where (M_{n2}) is obtained from the normal-shock Mach-number equation.
• Two solutions: For an attached shock below the maximum turning angle, the weak solution usually leaves the downstream flow supersonic, whereas the strong solution commonly makes it subsonic.
• Attachment limit: If (\theta) exceeds the maximum allowed value for (M_1) and (\gamma), the shock detaches and forms a curved bow shock.
• Applications: Oblique shocks provide staged compression in supersonic intakes and occur over wedges, cones, aircraft surfaces, and compression ramps.

B. Reflection of flow

Shock reflection occurs when an incident oblique shock reaches a solid boundary or another constraint that requires a further change in flow direction.

1. Regular reflection:

   • Wall condition: The incident shock turns flow toward the wall; a reflected shock turns it back parallel to the wall, ensuring zero normal velocity at the surface.
   • Structure: Incident and reflected shocks meet directly at the wall.
   • Pressure effect: The gas undergoes two successive compressions, giving a greater final pressure and stagnation-pressure loss.

2. Mach reflection:

   • Formation: If regular reflection cannot satisfy the turning and pressure conditions, a nearly normal Mach stem forms near the wall.
   • Triple point: Incident shock, reflected shock, and Mach stem meet at one point, from which a slip line emerges.
   • Importance: Mach reflection produces high local pressures and heating in supersonic vehicles, blast propagation, and nozzle flows.

C. Interaction of oblique shock waves

Oblique-shock interaction redistributes pressure and flow direction when two or more shocks intersect.

• Same-family interaction: Shocks turning flow in the same direction may merge into a stronger shock, with the final state governed by the combined deflection.
• Opposite-family interaction: Shocks turning flow in opposite directions can intersect and generate transmitted shocks whose strengths satisfy downstream pressure and direction compatibility.
• Shock polar method: Each possible post-shock state is represented by pressure versus flow-deflection angle; intersections of shock polars identify compatible downstream states.
• Unequal downstream states: Interacting shocks can create regions with equal pressure and flow direction but different entropy, density, temperature, and tangential velocity.
• Engineering significance: Such interactions occur in multi-ramp intakes, supersonic nozzles, jet plumes, blade passages, and between shocks generated by nearby aircraft surfaces.

D. Slip line

A slip line is a contact discontinuity separating streams that have equal pressure and normal velocity but different tangential properties.

• Compatibility conditions:
  
  TEXT

    pA = pB,       Vn,A = Vn,B

  
  where (A) and (B) identify the two streams and (V_n) is velocity normal to the slip line.
• Permitted jumps: Tangential velocity, density, temperature, entropy, and Mach number may differ across the line.
• No mass crossing: The normal velocity relative to the slip line is zero, so fluid particles remain on their respective sides.
• Origin: Slip lines emerge from triple points in Mach reflection and from intersections where different shock sequences produce compatible pressure and direction but unequal entropy.
• Shear instability: The tangential-velocity jump forms a vortex sheet and may develop Kelvin–Helmholtz instability in real viscous flows.

IV. Viscous Shock Interaction — Coupling with near-wall flow

A. Shock-boundary layer interaction

Shock-boundary layer interaction occurs when the pressure rise across a shock imposes a severe adverse pressure gradient on a viscous boundary layer.

• Boundary-layer response: Low-momentum fluid near the wall decelerates more readily than the external stream and may reverse direction.
• Separation: If the pressure rise is sufficiently large, wall shear stress falls to zero and becomes negative, producing a separated recirculation region.
• Shock restructuring: Separation often replaces a single shock with a system of compression waves, separation and reattachment shocks, commonly forming a (\lambda)-shaped pattern.
• Boundary-layer influence:
  • A laminar layer generally separates under a smaller pressure rise.
  • A turbulent layer has greater near-wall momentum and resists separation more effectively, but produces higher skin friction.
• Consequences: Interaction increases total-pressure loss, aerodynamic drag, wall heat transfer, pressure fluctuations, and structural loading; unsteadiness may also cause inlet buzz.
• Applications: Critical examples include supersonic inlet ramps, transonic wings, turbine and compressor passages, rocket nozzles, control surfaces, and hypersonic thermal-protection regions.
• Control measures: Boundary-layer bleed, vortex generators, reduced compression angles, distributed compression, and careful shock positioning can weaken separation and improve pressure recovery.