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Unit 3: Expansion waves and flow over nozzles

ASE204 — Aerodynamics-Ii 7 min read

I. Orientation — Supersonic Flow Foundations

Expansion waves and nozzle-flow regimes follow from compressible-flow conservation laws and the hyperbolic character of steady supersonic flow. When a supersonic stream turns away from itself at a convex corner, pressure disturbances form a continuous Prandtl–Meyer expansion fan; when nozzle exit pressure differs from ambient pressure, additional waves adjust the jet.

A. Governing Properties and Assumptions

The standard analysis treats the gas as calorically perfect and the smooth expansion as isentropic.

  • Flow model: The flow is steady, two-dimensional, inviscid, and adiabatic unless stated otherwise.
  • Perfect-gas relation: Pressure (p), density (\rho), and temperature (T) obey
    TEXT
      p = ρRT

    where (R) is the specific gas constant.
  • Mach number: Compressibility effects are measured by
    TEXT
      M = V/a,      a = √(γRT)

    where (V) is flow velocity, (a) is local sound speed, and (\gamma) is the specific-heat ratio.
  • Stagnation quantities: For adiabatic flow without shaft work, stagnation temperature (T_0) remains constant:
    TEXT
      T₀/T = 1 + [(γ − 1)/2]M²
  • Isentropic relations: Across a smooth expansion,
    TEXT
      p₀/p = {1 + [(γ − 1)/2]M²}^[γ/(γ − 1)]
  ρ₀/ρ = {1 + [(γ − 1)/2]M²}^[1/(γ − 1)]

    where (p_0) and (\rho_0) are stagnation pressure and density.
  • Characteristic propagation: In supersonic flow, infinitesimal disturbances travel along Mach lines inclined at the Mach angle
    TEXT
      μ = sin⁻¹(1/M)

    where (\mu) is measured from the local velocity direction.
  • Expansion trend: An isentropic expansion increases (M) and (V), while decreasing (p), (T), (\rho), and (\mu); stagnation pressure remains unchanged.

II. Prandtl–Meyer Expansion — Turning Around a Convex Corner

A. Prandtl-Meyer expansion flow and related problems

A Prandtl–Meyer expansion is a continuous fan of infinitesimal Mach waves that turns a supersonic stream through a convex angle while accelerating it isentropically.

  • Geometric condition: Expansion occurs when a wall turns away from the flow. A concave turn instead compresses the stream and normally produces compression waves or an oblique shock.
  • Fan structure: The upstream Mach line is the fan’s leading characteristic, and the downstream Mach line is its trailing characteristic.
    • Leading-line direction relative to a fixed datum: (\theta_1+\mu_1).
    • Trailing-line direction: (\theta_2+\mu_2), with signs adjusted to the turning orientation.
    • Inside the fan, velocity direction (\theta), Mach number (M), and Mach angle (\mu) vary continuously.
  • Prandtl–Meyer function: For (M\geq1),
    TEXT
      ν(M) = √[(γ + 1)/(γ − 1)]
         tan⁻¹{√[(γ − 1)/(γ + 1) · (M² − 1)]}
         − tan⁻¹{√(M² − 1)}

    where (\nu) is the Prandtl–Meyer angle in radians or degrees, used consistently, and (\gamma) is the specific-heat ratio.
  • Turning relation: For an expansion from state 1 to state 2,
    TEXT
      θ = ν(M₂) − ν(M₁)

    where (\theta) is the positive turning angle, (M_1) is upstream Mach number, and (M_2>M_1) is downstream Mach number.
  • Pressure calculation: Once (M_2) is found, static-pressure change follows from equal stagnation pressure:
    TEXT
      p₂/p₁ =
  {[1 + (γ − 1)M₁²/2] / [1 + (γ − 1)M₂²/2]}^[γ/(γ − 1)]

    Here (p_1) and (p_2) are upstream and downstream static pressures.
  • Velocity change: Since (T_0) is constant, the reduction in static enthalpy supplies the increase in kinetic energy:
    TEXT
      V₂² − V₁² = 2cₚ(T₁ − T₂)

    where (c_p) is specific heat at constant pressure.
  • Related-problem procedure:
    1. Confirm that the incoming flow is supersonic and the corner is convex.
    2. Evaluate (\nu_1=\nu(M_1)).
    3. Set (\nu_2=\nu_1+\theta).
    4. Invert the Prandtl–Meyer function to obtain (M_2).
    5. Apply isentropic relations for (p_2), (T_2), and (\rho_2).
  • Worked example: For air with (\gamma=1.4), (M_1=2.0), and (\theta=10^\circ), (\nu_1\approx26.38^\circ). Thus (\nu_2\approx36.38^\circ), giving (M_2\approx2.38). The pressure ratio is approximately
    TEXT
      p₂/p₁ ≈ 0.55

    confirming substantial pressure reduction and Mach-number increase.

B. Applications and Limitations

Prandtl–Meyer theory provides the local states and wave directions for ideal supersonic expansions.

  • Applications: It is used for convex aerofoil surfaces, nozzle contours, supersonic inlets, jet boundaries, and method-of-characteristics calculations.
  • Entropy behavior: Unlike a finite shock, the ideal fan is isentropic because it consists of infinitesimally weak expansion waves.
  • No expansion shock: A finite expansion shock would violate the entropy condition for ordinary gases; expansion disturbances therefore spread into a fan.
  • Practical limitations: Boundary layers, viscosity, heat transfer, three-dimensionality, condensation, and variable specific heats cause departures from perfect-gas predictions.

III. Nozzle Operating Regimes — Pressure Matching and Jet Adjustment

A. Under-expanded and over-expanded nozzles

A convergent–divergent nozzle is correctly expanded only when its supersonic exit pressure equals the surrounding back pressure.

  • Reference pressures: Let (p_e) be nozzle exit static pressure and (p_b) be ambient or back pressure.
  • Design condition:
    TEXT
      pₑ = p_b

    The jet leaves without requiring external compression or expansion waves.
  • Area–Mach relation: For quasi-one-dimensional isentropic flow,
    TEXT
      A/A* = (1/M)
  { [2/(γ + 1)] [1 + (γ − 1)M²/2] }^[(γ + 1)/(2(γ − 1))]

    where (A) is local area and (A^*) is the sonic critical area. The supersonic branch applies in the diverging section after choking.

  1. Under-expanded nozzle: Here (p_e>p_b), so the exit flow has not expanded enough inside the nozzle.

    • Initial adjustment: Prandtl–Meyer fans originate at the nozzle lip, turning the boundary outward and reducing jet pressure.
    • Jet pattern: Expansion waves reflect from the free boundary as compression waves; repeated adjustments create shock cells.
    • Strong mismatch: A highly under-expanded jet may contain barrel shocks and a Mach disk, a near-normal shock that causes a large pressure rise and Mach-number reduction.
    • Jet shape: The plume initially expands beyond the nozzle exit area.

  2. Over-expanded nozzle: Here (p_e<p_b), so the internal expansion has reduced pressure below ambient.

    • Initial adjustment: Compression waves or oblique shocks form at the nozzle lip, turning the jet inward and raising its pressure.
    • Moderate mismatch: External oblique shocks can restore approximate pressure compatibility while the internal flow remains attached.
    • Severe mismatch: An adverse pressure gradient can separate the boundary layer inside the divergent section, producing losses and unsteady side loads.
    • Jet shape: The plume initially contracts downstream of the exit.
  • Thrust connection: Ideal one-dimensional nozzle thrust is
    TEXT
      F = ṁVₑ + (pₑ − p_b)Aₑ

    where (F) is thrust, (\dot m) is mass-flow rate, (V_e) is exit velocity, and (A_e) is exit area. The second term is pressure thrust.
  • Altitude dependence: A fixed-area nozzle may be over-expanded near sea level, correctly expanded at its design altitude, and under-expanded at higher altitude because atmospheric pressure decreases.

B. Operational Significance and Limitations

Nozzle classification predicts wave structure, efficiency, and mechanical loading.

  • Efficiency: Correct expansion maximizes useful axial momentum for the specified stagnation state and ambient pressure.
  • Shock losses: Compression through shocks increases entropy and reduces stagnation pressure, unlike Prandtl–Meyer expansion.
  • Separation risk: Strong over-expansion is especially important in rocket nozzles because asymmetric separation can impose damaging lateral forces.
  • Model limitation: Real plume structure is viscous, turbulent, three-dimensional, and often chemically reacting, so one-dimensional theory predicts mean states rather than every local feature.

IV. Turning Limits — Finite Expansion Capacity

A. Maximum turning angle

The maximum ideal expansion turn is reached asymptotically when the downstream Mach number tends to infinity.

  • Limiting Prandtl–Meyer angle:
    TEXT
      ν_max = (π/2)[√((γ + 1)/(γ − 1)) − 1]

    where (\nu_{\max}) is in radians.
  • Available turn from a given state:
    TEXT
      θ_max = ν_max − ν(M₁)

    where (\theta_{\max}) is the largest mathematical turn from upstream Mach number (M_1).
  • Air value: For (\gamma=1.4),
    TEXT
      ν_max ≈ 130.45°

    This is the limiting turn from the sonic state (M=1); a flow already at (M_1>1) has less remaining turning capacity.
  • Physical meaning: At the limit, static pressure and temperature approach zero in the ideal model while Mach number approaches infinity; no real nozzle reaches this state.
  • Dependence on gas properties: Lower (\gamma) gives a larger (\nu_{\max}), so the available angular expansion depends on thermodynamic behavior.

B. Practical Significance

The mathematical limit bounds ideal calculations but does not by itself establish a practical nozzle contour.

  • Geometric constraint: If a proposed turn requires (\nu(M2)>\nu{\max}), no finite downstream Mach number satisfies the Prandtl–Meyer relation.
  • Real-flow constraint: Viscous effects, finite temperature, boundary-layer behavior, and nozzle length impose useful turning limits far below the asymptotic value.
  • Design implication: Large turns are distributed gradually in contoured nozzles to control characteristic interactions and avoid undesirable nonuniformity.

V. Flow-Field Classification — Characteristic Interactions

A. Simple and non-simple regions

Simple and non-simple regions distinguish whether one family or both families of characteristics carry spatially varying flow information.

  1. Simple region: One family of characteristics consists of straight, parallel lines, while flow properties vary through waves of the other family.

    • Expansion fan: A centered Prandtl–Meyer fan is a simple region because characteristics of one family radiate from the corner.
    • Compatibility relations: For two-dimensional, steady, irrotational supersonic flow,
      TEXT
           K₊ = θ + ν = constant along C₊
     K₋ = θ − ν = constant along C₋

      where (C+) and (C-) are the two characteristic families and (K+), (K-) are Riemann invariants.
    • Property determination: Knowing one invariant from the uniform adjacent flow allows (\theta) and (\nu), hence (M), to be found along the varying family.

  2. Non-simple region: Characteristics from both families interact, and neither invariant is globally fixed throughout the region.

    • Formation: Such regions occur where expansion fans intersect, waves reflect from walls or symmetry lines, or nozzle characteristics converge and cross.
    • Spatial variation: Both (\theta) and (M) vary in two characteristic directions, so a single Prandtl–Meyer turning relation is insufficient.
    • Solution method: The method of characteristics determines an interior point from intersecting (C+) and (C-) lines by applying both compatibility equations.

B. Analytical Significance

The classification determines how much information is required to construct a supersonic flow field.

  • Simple-region advantage: A wall turn or known uniform state can generate the complete local solution with one invariant held constant.
  • Non-simple-region requirement: Data from both characteristic families are necessary, commonly obtained through a numerical characteristic mesh.
  • Nozzle application: Contoured supersonic nozzles combine simple expansion regions with non-simple wave-cancellation regions to produce nearly uniform, parallel exit flow.
  • Validity boundary: The characteristic relations assume continuous supersonic flow; shocks require separate jump conditions, including mass, momentum, energy, and entropy change.
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