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Unit 1: Concepts of compressible flow

ASE204 — Aerodynamics-Ii 6 min read

I. Orientation — Foundations of Compressible Aerodynamics

Compressible-flow theory studies fluids whose density changes appreciably during motion; it combines conservation laws with thermodynamic relations to connect pressure, density, temperature, velocity, and energy.

A. Defining Framework

The analysis rests on conservation of mass, momentum, and energy, supplemented by an equation of state.

  • Continuum assumption: Fluid properties such as pressure (p), density (\rho), and temperature (T) vary continuously in space and time.
  • Perfect-gas model: Air at ordinary aerodynamic temperatures is commonly represented by:
    TEXT
      p = ρRT
  γ = cp/cv

    Here (R) is the specific gas constant, (c_p) and (c_v) are specific heats at constant pressure and volume, and (\gamma) is their ratio. For air, (R\approx287\ \text{J kg}^{-1}\text{K}^{-1}) and (\gamma\approx1.4).
  • Calorically perfect gas: The specific heats are treated as constant, giving (h=c_pT), where (h) is specific enthalpy.
  • Compressibility criterion: Density variation is usually negligible below Mach number (M\approx0.3), but becomes increasingly important at higher speeds.
  • Flow assumptions: Particular results may additionally require steady, one-dimensional, inviscid, adiabatic, or isentropic motion.

II. Thermodynamic Basis — Isentropic Flow and Its Scope

A. Introduction to isentropic flow

Isentropic flow is an ideal adiabatic and reversible process in which entropy remains constant along a streamline.

  • Defining condition: For specific entropy (s),
    TEXT
      ds = 0

    Adiabatic flow requires no heat transfer, while reversibility excludes friction, shocks, and other dissipative effects.
  • Property relations: For a calorically perfect gas undergoing an isentropic process,
    TEXT
      p/ρ^γ = constant
  T2/T1 = (p2/p1)^((γ−1)/γ)
  ρ2/ρ1 = (p2/p1)^(1/γ)

    Subscripts 1 and 2 identify two states on the same isentropic path.
  • Aerodynamic use: Smooth flow through an ideal nozzle, away from boundary layers and shocks, can often be approximated as isentropic.
  • Limitation: A shock is adiabatic but not isentropic because entropy increases through the irreversible compression.

B. Scope of compressible flow

Compressible-flow analysis applies whenever pressure changes create density variations large enough to influence the motion.

  • Major applications: Examples include aircraft at transonic and supersonic speeds, rocket nozzles, gas turbines, compressors, high-speed wind tunnels, pipelines, and blast waves.
  • Relevant mechanisms: Compressibility governs choking, acoustic propagation, shock formation, expansion waves, and aerodynamic heating.
  • Speed dependence: Velocity alone is insufficient; the controlling measure is (M=V/a), comparing flow velocity (V) with local sound speed (a).
  • Other causes: Compressibility may matter at moderate velocity when pressure ratios are large, as in pneumatic systems and compressor passages.

III. Governing Laws — Conservation in Compressible Motion

A. Energy and momentum equations for compressible fluid flow

Energy and momentum equations determine how pressure, temperature, and velocity exchange and respond to external forces.

  • Steady-flow energy equation: For one inlet and one outlet,
    TEXT
      h1 + V1²/2 + gz1 + q = h2 + V2²/2 + gz2 + ws

    Here (V) is speed, (g) is gravitational acceleration, (z) is elevation, (q) is heat added per unit mass, and (w_s) is shaft work delivered per unit mass.
  • Adiabatic aerodynamic flow: With negligible elevation change and no shaft work,
    TEXT
      h + V²/2 = h0 = constant

    The stagnation enthalpy (h_0) includes static enthalpy and kinetic energy.
  • Momentum equation: For steady one-dimensional inviscid flow without body forces,
    TEXT
      dp + ρV dV = 0

    Thus acceleration generally accompanies a pressure decrease, while deceleration accompanies a pressure increase.
  • Mass conservation: The governing equations are closed by continuity:
    TEXT
      ρAV = ṁ = constant

    Here (A) is flow area and (\dot m) is mass-flow rate.
  • Integral momentum balance: The net pressure and external forces on a control volume equal the change in momentum flux, (\dot m(V_2-V_1)).

IV. Velocity Scales and Thermodynamic States

A. Reference velocities

Reference velocities provide physically meaningful scales for comparing compressible flows.

  • Local sound speed: The scale (a=\sqrt{\gamma RT}) depends on local temperature and is used in the ordinary Mach number.
  • Critical sound speed: The value (a^*) is the sound speed at the sonic state corresponding to the same stagnation condition:
    TEXT
      a* = √[2γRT0/(γ+1)]

    Here (T_0) is stagnation temperature; an asterisk denotes a critical or sonic property.
  • Maximum theoretical velocity: If all stagnation enthalpy becomes kinetic energy,
    TEXT
      Vmax = √(2cpT0)

    This limiting velocity corresponds ideally to static temperature approaching zero.
  • Alternative velocity parameter: The ratio (M^=V/a^) is useful because (a^*) remains fixed along an adiabatic streamline when (T_0) is constant.

B. Stagnation states

A stagnation state is obtained when a moving fluid is brought to rest adiabatically and isentropically.

  • Energy conversion: For a perfect gas,
    TEXT
      T0 = T + V²/(2cp)

    Static kinetic energy is converted into internal energy, raising temperature from (T) to (T_0).
  • Isentropic stagnation relations:
    TEXT
      T0/T = 1 + (γ−1)M²/2
  p0/p = [1 + (γ−1)M²/2]^(γ/(γ−1))
  ρ0/ρ = [1 + (γ−1)M²/2]^(1/(γ−1))

    Quantities (p_0) and (\rho_0) are stagnation pressure and density.
  • Physical distinction: Stagnation temperature remains constant in adiabatic flow without work, including across a shock, but stagnation pressure decreases across irreversible disturbances.
  • Measurement: A Pitot probe approximates stagnation pressure at its forward-facing opening; supersonic measurements require accounting for the shock ahead of the probe.

C. Critical states

A critical state is the local sonic condition (M=1) associated with a given stagnation state.

  • Critical ratios: Setting (M=1) in the stagnation relations gives:
    TEXT
      T*/T0 = 2/(γ+1)
  p*/p0 = [2/(γ+1)]^(γ/(γ−1))
  ρ*/ρ0 = [2/(γ+1)]^(1/(γ−1))
  • Air values: For (\gamma=1.4), (T^/T_0\approx0.833), (p^/p_0\approx0.528), and (\rho^*/\rho_0\approx0.634).
  • Choking: In a converging nozzle, the minimum available area can reach (M=1); further reduction of downstream pressure then cannot increase mass flow for fixed upstream stagnation conditions.
  • Area significance: The critical area (A^*) is the area at which a one-dimensional isentropic stream would be sonic.

V. Acoustic and Mach Parameters

A. Velocity of sound

The velocity of sound is the propagation speed of an infinitesimal pressure disturbance relative to the fluid.

  • Thermodynamic definition:
    TEXT
      a² = (∂p/∂ρ)s

    The derivative is evaluated at constant entropy because weak acoustic disturbances are nearly reversible and adiabatic.
  • Perfect-gas result:
    TEXT
      a = √(γp/ρ) = √(γRT)

    Sound speed therefore depends mainly on absolute temperature, not directly on pressure.
  • Air example: At (T=288\ \text{K}), (a\approx\sqrt{1.4(287)(288)}\approx340\ \text{m s}^{-1}).
  • Propagation direction: In one-dimensional flow, acoustic signals travel at (V+a) downstream and (V-a) upstream relative to a stationary observer.

B. Mach number

Mach number measures flow speed relative to the local speed of sound and determines the propagation of pressure information.

  • Definition:
    TEXT
      M = V/a

    Values (M<1), (M=1), and (M>1) denote subsonic, sonic, and supersonic local flow.
  • Physical meaning: At (M<1), disturbances can travel upstream because (V-a<0); at (M>1), both acoustic characteristics are swept downstream.
  • Thermodynamic influence: Since (a\propto\sqrt{T}), the same velocity can correspond to different Mach numbers at different temperatures.
  • Density-change estimate: For small pressure disturbances, increasing Mach number strengthens the coupling between velocity, pressure, and density changes.

C. Critical Mach number

Critical Mach number is the freestream Mach number at which some point on an aerodynamic body first reaches local (M=1).

  • Local acceleration: Flow over an aerofoil can accelerate above the freestream speed, so the local sonic condition may occur while (M\infty<1), where (M\infty) is freestream Mach number.
  • Pressure connection: The surface point having the most negative pressure coefficient usually reaches sonic speed first.
  • Design dependence: Aerofoil thickness, curvature, angle of attack, and sweep influence the critical Mach number.
  • Consequence: Above the critical Mach number, a local supersonic region may terminate in a shock, producing wave drag and possible boundary-layer separation.
  • Distinction: This aerodynamic definition differs from a critical state, which simply means a local condition of (M=1).

VI. Wave Phenomena — Disturbance Propagation

A. Types of waves

Compressible flows transmit pressure changes through weak or finite waves whose form depends on disturbance strength and flow direction.

  • Mach waves: Infinitesimal disturbances in supersonic flow produce weak waves across which property changes are negligibly small.
  • Compression waves: Finite compression waves converge and steepen because higher-pressure portions propagate faster, eventually forming shocks.
  • Shock waves:
    • Normal shock: Stands perpendicular to the upstream flow and changes supersonic flow to subsonic flow.
    • Oblique shock: Inclines to the flow and turns it toward itself while reducing Mach number.
  • Expansion waves: A supersonic stream turning away from itself expands through a continuous Prandtl–Meyer fan; pressure and temperature fall while Mach number rises.
  • Contact surface: Pressure and normal velocity remain continuous, although density, temperature, or tangential velocity may change.

B. Mach cones

A Mach cone is the three-dimensional envelope of sound disturbances emitted by an object moving supersonically.

  • Formation: During time (t), the object moves a distance (Vt), while a disturbance emitted earlier expands through radius (at).
  • Envelope: Because (V>a), successive spherical wavefronts cannot travel ahead of the object and become tangent to a cone.
  • Information region: Disturbances affect points inside or on the cone but not points outside it.
  • Two-dimensional analogue: In planar flow, the cone appears as two Mach lines inclined at the Mach angle.
  • Observable effect: Coalesced compression disturbances may produce a sonic boom when their pressure signature reaches an observer.

C. Mach angle

The Mach angle is the half-angle between a supersonic flow direction and its Mach line or cone surface.

  • Geometric relation:
    TEXT
      sin μ = at/(Vt) = 1/M
  μ = sin⁻¹(1/M)

    Here (\mu) is the Mach angle and (t) is the elapsed propagation time.
  • Mach-number effect: At (M=1), (\mu=90^\circ); at (M=2), (\mu=30^\circ); as (M) increases, the cone narrows.
  • Domain: A real Mach angle exists only for (M\ge1); subsonic disturbances spread upstream and do not form a Mach cone.
  • Use: Mach lines indicate characteristic directions along which infinitesimal disturbances and compatibility information propagate.

VII. Compressibility Classification — Flow Regimes

A. Effect of Mach number on compressibility flow regimes

Mach number controls the importance of density variation and the qualitative structure of aerodynamic flow.

  • Incompressible regime, (M<0.3): Density changes are typically below about five percent, so constant-density equations are usually adequate.
  • Subsonic regime, (0.3<M<0.8): Compressibility corrections become important, but the flow remains locally subsonic and pressure disturbances propagate upstream.
  • Transonic regime, (0.8<M<1.2): Subsonic and supersonic regions coexist; shocks, rapid drag rise, and shock-induced separation may occur.
  • Supersonic regime, (1.2<M<5): Disturbances are confined by Mach lines, and compression occurs through oblique or normal shocks while expansion occurs through fans.
  • Hypersonic regime, (M>5): Strong shocks generate very high temperatures, making viscous interaction, aerodynamic heating, chemical dissociation, and real-gas effects increasingly significant.
  • Boundary dependence: These numerical limits are approximate; local Mach number, body geometry, temperature, and disturbance strength determine the actual behavior.
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