Unit 6: High speed flow over airfoils and wings

I. Orientation — Compressible high-speed aerodynamics

High-speed aerodynamics concerns flows in which density changes and pressure disturbances propagate at finite acoustic speed. Mach number governs whether disturbances influence the upstream flow and whether shocks, expansion waves, wave drag, and aerodynamic heating become important.

A. Governing framework

The analysis commonly assumes steady, adiabatic flow of a calorically perfect gas, with corrections added for viscosity, boundary layers, and high-temperature effects.

• Mach number: The ratio of flow speed to local speed of sound determines the compressibility regime.
  
  TEXT

    M = V/a,             a = √(γRT)

  
  Here, (M) is Mach number, (V) is flow speed, (a) is sound speed, (\gamma) is the specific-heat ratio, (R) is the specific gas constant, and (T) is absolute temperature.
• Flow regimes: Subsonic flow is commonly (M<0.8), transonic flow approximately (0.8<M<1.2), supersonic flow (1.2<M<5), and hypersonic flow (M>5); boundaries are approximate because local Mach number differs from free-stream Mach number.
• Compression and expansion: Supersonic compression produces oblique shocks with increased pressure, temperature, density, and entropy; turning away from the flow produces isentropic Prandtl–Meyer expansion fans.
• Aerodynamic coefficients: Forces and moments are nondimensionalized using dynamic pressure.
  
  TEXT

    CL = L/(q∞S),   CD = D/(q∞S),   Cm = m/(q∞Sc),   q∞ = ½ρ∞V∞²

  
  (L), (D), and (m) are lift, drag, and pitching moment; (S) is reference area, (c) is reference chord, and (\rho\infty), (V\infty), and (q_\infty) are free-stream density, speed, and dynamic pressure.

II. Shock-Induced Separation — Loss of lift near transonic speed

A. Shock stall

Shock stall is the abrupt deterioration of aerodynamic performance caused when a shock strengthens sufficiently to separate the boundary layer.

• Formation: On a lifting airfoil at transonic speed, upper-surface flow may accelerate to (M>1) even while (M_\infty<1); it then returns toward subsonic speed through a shock.
• Adverse pressure gradient: Pressure rises sharply across the shock, forcing low-momentum boundary-layer fluid to decelerate and possibly reverse direction.
• Stall sequence:
  1. Increasing (M_\infty) or angle of attack moves and strengthens the upper-surface shock.
  2. Shock–boundary-layer interaction creates separation behind the shock.
  3. Lift falls, drag rises, and unsteady buffet may appear.
• Critical Mach number: (M_{\mathrm{cr}}) is the free-stream Mach number at which sonic flow first occurs somewhere on the airfoil; the drag-divergence Mach number is higher and marks rapid drag growth.
• Pitching effect: Rearward shock movement followed by separation changes the pressure distribution and can cause Mach tuck—a nose-down pitching tendency.
• Control measures: Thin sections, supercritical airfoils, boundary-layer control, vortex generators, and reduced local curvature weaken the shock or delay separation.

III. Supercritical Sections — Delaying transonic drag rise

A. Supercritical airfoil selections

Supercritical airfoils are selected to reduce upper-surface peak Mach number, weaken the terminating shock, and raise drag-divergence Mach number.

• Characteristic geometry: A flattened upper surface limits acceleration, a relatively thick mid-chord preserves structural depth, and aft camber with a cusped trailing-edge region restores lift.
• Selection parameters: The designer matches airfoil thickness ratio, design lift coefficient, Reynolds number, sweep angle, and cruise Mach number; transport sections commonly operate near (M_\infty=0.75)–(0.85).
• Pressure distribution: The upper surface develops an extended, nearly uniform supersonic-pressure plateau followed by a weaker shock than on a conventional airfoil.
• Advantages: Higher usable thickness, delayed wave-drag rise, improved buffet margin, and lower cruise drag permit greater range or lower fuel consumption.
• Trade-offs: Strong aft loading may increase hinge moments and pitching moment; performance away from the design Mach number and lift coefficient can deteriorate.
• Three-dimensional selection: Wing sweep reduces the Mach component normal to the leading edge:
  
  TEXT

    Mn = M∞ cos Λ

  
  (Mn) is normal Mach number, (M\infty) is free-stream Mach number, and (\Lambda) is sweep angle; airfoil sections must therefore be selected together with wing sweep and local chord.

IV. Supersonic Profile Loads — Linearized thin-airfoil results

A. Lift, drag, pitching moment and centre of pressure for supersonic profiles

Supersonic profile forces arise from pressure changes generated independently by local surface turning because disturbances remain inside Mach waves.

• Pressure coefficient: For a small flow-deflection angle (\theta), linearized theory gives:
  
  TEXT

    Cp = 2θ/β,             β = √(M∞² − 1)

  
  (Cp) is pressure coefficient, (\theta) is local turning angle in radians, (\beta) is the supersonic parameter, and (M\infty>1).
• Lift: A symmetric thin profile at small angle of attack (\alpha) has:
  
  TEXT

    CL = 4α/β

  
  Thus lift-curve slope decreases as Mach number increases; (\alpha) must be in radians.
• Wave drag: For a symmetric double-wedge airfoil of thickness ratio (t/c):
  
  TEXT

    CD,w = (4/β)[α² + (t/c)²]

  
  (C_{D,w}) is wave-drag coefficient, (t) is maximum thickness, and (c) is chord; thickness creates wave drag even at zero lift.
• Pitching moment: Linear theory gives a symmetric flat plate an aerodynamic centre at (c/2), unlike the subsonic quarter-chord location. With nose-up moment positive:
  
  TEXT

    Cm,LE = −CL/2

  
  (C_{m,LE}) is pitching-moment coefficient about the leading edge.
• Centre of pressure: For the uniform pressure difference predicted on a flat plate, (x_{cp}=c/2); camber and nonlinear shock effects can shift this position.

V. Transonic Wave-Drag Control — Longitudinal area shaping

A. Transonic area rule

The transonic area rule states that wave drag depends primarily on the smoothness of the aircraft’s total cross-sectional area distribution along its length.

• Equivalent body: At each longitudinal station, fuselage, wing, nacelle, and other exposed areas are combined into one cross-sectional area (A(x)), where (x) is distance from the nose.
• Drag mechanism: Rapid changes in (A(x)), particularly large curvature in the area distribution, create strong pressure waves and increased wave drag near Mach 1.
• Waisting: Narrowing the fuselage where wing area is added produces the “coke-bottle” shape, smoothing (A(x)) without reducing total wing area.
• Ideal tendency: A Sears–Haack-like smooth distribution gives low wave drag for prescribed length and volume, although practical aircraft require cockpits, engines, payload volume, and control surfaces.
• Application: Component placement is important: nacelles or stores can be staggered longitudinally so their area additions do not coincide with the wing’s maximum contribution.
• Limitation: The rule predicts transonic wave-drag trends but does not replace detailed analysis of lift, viscosity, local shocks, or three-dimensional interference.

VI. Supersonic Airfoil Geometry — Controlled compression and expansion

A. Airfoils for supersonic flows

Supersonic airfoils use thin, sharp-edged geometry to minimize flow turning and the resulting wave drag.

• Double-wedge section: Straight panels generate attached oblique shocks at compression corners and expansion fans at convex corners; its simple geometry suits analytical treatment.
• Biconvex section: Smooth convex surfaces distribute compression more gradually, potentially reducing peak pressure changes, but are harder to manufacture and analyze exactly.
• Thinness: Wave drag varies approximately with ((t/c)^2) in linear theory, encouraging low thickness-to-chord ratios.
• Leading edge: A sharp leading edge can support an attached shock when the wedge angle is below the attachment limit; a blunt edge produces a detached bow shock and higher drag.
• Trailing edge: Upper and lower streams must adjust toward a compatible downstream pressure, producing shocks, expansions, or a trailing wake.
• Practical compromise: Structural depth, fuel volume, stiffness, heat resistance, and leading-edge durability prevent sections from being arbitrarily thin or sharp.

VII. Finite-Turning Analysis — Nonlinear wave construction

A. Shock expansion theory

Shock expansion theory calculates inviscid supersonic flow over piecewise-straight surfaces by applying oblique-shock relations at compression corners and Prandtl–Meyer relations at expansion corners.

• Compression corner: The shock angle (\beta_s) is found from the (\theta)-(\beta_s)-(M) relation for upstream Mach number and turning angle; pressure and Mach number then follow from normal-shock relations.
• Expansion corner: An isentropic fan changes the Prandtl–Meyer function:
  
  TEXT

    ν(M2) − ν(M1) = θ

  
  (\nu) is the Prandtl–Meyer angle, (M_1) and (M_2) are upstream and downstream Mach numbers, and (\theta) is the expansion turning angle.
• Procedure:
  1. Trace upper and lower surfaces from the leading edge.
  2. Update Mach number and pressure after every shock or expansion.
  3. Convert panel pressures into normal and axial force components.
  4. Integrate forces and moments over the profile.
• Strength: Unlike linear theory, it captures finite turning, entropy increase through shocks, and nonlinear pressure changes.
• Limitations: It neglects viscosity, boundary-layer displacement, shock separation, rounded corners, and wave interactions not represented by the assumed panel sequence.

VIII. Three-Dimensional Supersonic Lifting Surfaces

A. Supersonic wings

Supersonic-wing performance depends on planform, sweep, aspect ratio, thickness, and the position of edges relative to the Mach cone.

• Mach angle: Disturbances propagate along Mach lines:
  
  TEXT

    μ = sin⁻¹(1/M∞)

  
  (\mu) is Mach angle and (M_\infty) is supersonic free-stream Mach number.
• Edge classification: A leading edge inside the Mach cone is subsonic; one outside it is supersonic. This distinction controls whether spanwise pressure communication is possible.
• Delta wings: High sweep reduces normal Mach number, provides structural volume, and gives a long root chord with manageable thickness ratio.
• Aspect ratio: Low-aspect-ratio wings reduce supersonic wave drag and structural bending loads, but usually have poorer low-speed lift-to-drag ratio.
• Lift distribution: Pressure disturbances are confined by Mach lines, so tips influence only limited regions rather than the entire span.
• Control and stability: Rearward aerodynamic-centre movement at supersonic speed affects trim; all-moving tails are often used because ordinary elevators may lose effectiveness across shocks.

IX. Aircraft-Level Integration — Balancing mission requirements

A. Design considerations for supersonic aircraft

Supersonic aircraft design balances wave drag, propulsion, stability, structure, heating, low-speed operation, and environmental constraints.

• Drag control: Slender bodies, thin swept wings, smooth area distribution, and careful component integration reduce wave and interference drag.
• Propulsion matching: Variable-geometry or mixed-compression inlets decelerate air through controlled shocks before it reaches the compressor; poor shock positioning can cause inlet unstart.
• Trim and control: Centre-of-pressure movement between subsonic and supersonic regimes requires sufficient tail authority, fuel transfer, or aerodynamic balancing.
• Structure: Thin wings face bending and torsional-stiffness penalties; titanium, high-temperature aluminium alloys, and composites provide strength with controlled mass.
• Operational envelope: Take-off and landing require high lift despite low aspect ratio and high sweep, motivating flaps, leading-edge devices, or vortex lift.
• Environmental effects: Sonic-boom intensity depends on aircraft volume, lift, altitude, and pressure-wave shaping; airport noise and emissions also constrain propulsion choices.
• Safety margins: Flutter, shock buffet, inlet stability, control reversal, and thermal expansion must be assessed across Mach number and altitude.

X. Thermal Effects — Conversion of kinetic energy into heat

A. Aerodynamic heating

Aerodynamic heating results from compression and viscous dissipation as high-speed air is brought toward rest near the aircraft surface.

• Stagnation temperature: For a calorically perfect gas in adiabatic flow:
  
  TEXT

    T0/T∞ = 1 + [(γ − 1)/2]M∞²

  
  (T0) is stagnation temperature, (T\infty) is free-stream static temperature, (\gamma) is specific-heat ratio, and (M_\infty) is free-stream Mach number.
• Critical locations: Nose tips, wing leading edges, inlet lips, and shock-impingement regions experience the highest heat flux because of strong compression and thin boundary layers.
• Geometry effect: A small nose radius intensifies local heating, while a blunter nose moves the shock away and spreads heat over a larger area, though wave drag may increase.
• Boundary-layer effect: Turbulent boundary layers generally transfer more heat than laminar ones because mixing brings high-energy fluid closer to the wall.
• Consequences: Heating reduces material strength, causes thermal expansion and stress, changes structural clearances, and can damage coatings, seals, electronics, or fuel systems.
• Thermal protection: Designers use heat-resistant alloys, insulation, ceramic coatings, heat sinks, active cooling, or ablative materials according to speed and exposure duration.