Unit 6: High speed flow over airfoils and wings - Subjective Questions

Shock stall is the rapid loss of lift and increase in drag caused by shock-wave-induced boundary-layer separation on an airfoil.

• As the freestream Mach number approaches the critical Mach number , local airflow over the airfoil first becomes sonic and then supersonic.
• The local supersonic region terminates through a shock wave, across which pressure rises abruptly.
• This sudden pressure rise creates a strong adverse pressure gradient.
• The boundary layer loses momentum and may separate behind the shock.
• Separation reduces suction over the upper surface, causing a loss of lift, a large drag rise, buffet, and changes in pitching moment.

Shock stall differs from ordinary low-speed stall because it is initiated primarily by shock-boundary-layer interaction rather than solely by excessive angle of attack.

The principal consequences of shock stall are:

• Lift reduction: Shock-induced separation reduces upper-surface suction.
• Wave-drag increase: Pressure losses across the shock produce a rapid drag rise.
• Buffeting: Unsteady shock movement and separated flow create structural vibrations.
• Pitching-moment change: Movement of the pressure distribution may cause a nose-down tendency known as Mach tuck.
• Control deterioration: Separated flow can reduce elevator or aileron effectiveness.

Shock stall can be delayed by:

• Using a supercritical airfoil with a flattened upper surface and aft camber.
• Reducing airfoil thickness-to-chord ratio.
• Employing wing sweep to reduce the Mach-number component normal to the leading edge.
• Using vortex generators or boundary-layer control.
• Designing a smooth area distribution and avoiding severe surface curvature.
• Restricting angle of attack or Mach number through flight-control systems.

A supercritical airfoil is designed to delay drag divergence and weaken shock waves during transonic flight. Its main characteristics are:

• A relatively flat upper surface, which limits excessive local acceleration.
• A rounded leading edge that provides acceptable low-speed characteristics.
• Greater thickness than a conventional airfoil having the same critical Mach number.
• Pronounced aft camber near the trailing edge to recover the required lift.
• A relatively flat or mildly curved lower surface.

Aerodynamically, the airfoil forms a larger but weaker supersonic region over its upper surface. The terminating shock is weaker and generally located farther aft. This reduces wave drag and delays shock-induced boundary-layer separation. Supercritical airfoils therefore provide a higher drag-divergence Mach number, improved transonic efficiency, increased cruise speed, and sufficient structural depth for the wing.

Selection of a supercritical airfoil requires balancing aerodynamic, structural, and operational requirements:

• Design Mach number: The section must provide low wave drag near the intended cruise Mach number.
• Design lift coefficient: Its pressure distribution should be optimized for the aircraft's normal cruise lift coefficient.
• Thickness ratio: A thicker section provides structural depth and fuel volume but may increase wave drag.
• Shock strength and position: A weak, aft-positioned shock is preferred to reduce separation and drag.
• Pitching moment: Strong aft loading can produce an undesirable nose-down moment and greater trim drag.
• Off-design behavior: Acceptable performance is required during climb, maneuver, and low-speed operation.
• Boundary-layer behavior: The airfoil should tolerate surface roughness and avoid severe shock-induced separation.
• Manufacturing constraints: Surface contour accuracy is important because small shape errors can alter transonic pressure distributions.

Final selection is normally based on computational analysis, wind-tunnel testing, and multidisciplinary optimization.

For a two-dimensional, thin surface in a supersonic stream, linearized theory assumes small disturbance angles and an attached flow. Let the freestream Mach number be and define

The pressure coefficient is

The linearized velocity-potential equation gives a pressure perturbation proportional to the local surface slope. Consequently,

for a surface turning the flow toward itself and producing compression. For an expansion through the same small angle,

Thus:

• Compression surfaces have positive .
• Expansion surfaces have negative .
• Pressure disturbances become weaker as increases.

This relation is the basis of linearized calculations for lift, wave drag, and pitching moment of thin supersonic profiles.

Consider a thin, symmetric airfoil at angle of attack in a supersonic stream. Under linearized theory, the pressure coefficients due to incidence are approximately

where

The pressure difference is

Integrating over the chord gives the lift coefficient

with expressed in radians.

The lift-dependent wave drag is obtained by resolving the pressure force in the freestream direction. For a flat plate,

For a profile with finite thickness, an additional thickness-dependent wave-drag term appears. Therefore, in general,

where is drag due to lift and is wave drag due to thickness.

The aerodynamic pitching moment about a reference point is obtained by integrating the pressure loading along the chord:

with the sign set by the chosen moment convention. The centre of pressure is the point at which the resultant lift acts, so

For a thin flat plate at a small angle of attack in linearized supersonic flow, is uniform along the chord. Its resultant therefore acts at the mid-chord:

Hence, the pitching moment about mid-chord is zero, while the moment coefficient about the leading edge has magnitude

Unlike subsonic thin-airfoil theory, in which the aerodynamic centre is near the quarter-chord, the aerodynamic centre of a two-dimensional thin supersonic airfoil is approximately at the half-chord.

The transonic area rule states that the wave drag of an aircraft near sonic speed depends strongly on the smoothness of its total cross-sectional area distribution along the longitudinal axis, including the fuselage, wings, nacelles, and other components.

A sudden increase in area, such as that caused by the wing-fuselage junction, creates strong pressure disturbances and shock waves. To compensate, the fuselage may be narrowed near the wing, producing a characteristic waist or coke-bottle shape.

The desired area distribution is smooth and is often compared with that of a low-wave-drag body such as the Sears-Haack body. A smooth distribution:

• Reduces abrupt flow acceleration and compression.
• Weakens shock waves.
• Decreases pressure losses across the shocks.
• Lowers transonic wave drag and delays drag divergence.

The rule concerns the total aircraft area distribution, not merely the shape of an isolated fuselage or wing.

Application of the transonic area rule involves calculating the aircraft's equivalent cross-sectional area at a sequence of longitudinal stations. The areas of the fuselage, wing, nacelles, pylons, and empennage are combined.

The designer then modifies the configuration to make the resulting area curve smooth. Typical measures include:

• Narrowing the fuselage near the wing.
• Increasing fuselage area gradually before and after the wing.
• Carefully locating engines, nacelles, and external stores.
• Blending wing-body junctions with suitable fairings.
• Staggering components so that their maximum areas do not occur at the same station.
• Using computational fluid dynamics to evaluate Mach-plane or equivalent-area distributions.

These changes reduce peaks in the rate of area change, weaken transonic shocks, and lower wave drag. Structural volume, cabin layout, fuel capacity, stability, and manufacturing requirements must also be considered, so the theoretically ideal area distribution is rarely achieved exactly.

• Flat plate: It has minimum thickness wave drag in ideal theory and is useful as a reference profile. It has negligible internal volume and poor structural practicality.
• Single wedge: It produces an attached oblique shock on its compression side and an expansion on the opposite side. It is simple but generally has unsuitable rear-surface pressure recovery.
• Double-wedge or diamond airfoil: Compression occurs at the leading edges, while expansions or recompressions occur at the corners and trailing edges. It provides structural thickness but produces thickness wave drag.
• Biconvex airfoil: Its smoothly curved surfaces distribute flow turning more gradually than a double wedge. It may provide improved practical performance but is more difficult to analyze exactly.

A good supersonic airfoil is generally thin, has a sharp leading edge, and avoids large flow-deflection angles. Selection depends on design Mach number, required lift, structural depth, heating, manufacturing, and off-design performance.

Shock-expansion theory evaluates inviscid supersonic flow over piecewise straight surfaces by treating compression turns as oblique shocks and expansion turns as Prandtl-Meyer fans.

The procedure is:

1. Divide the upper and lower surfaces into straight panels.
2. Determine the flow-turning angle at each corner.
3. For a compression corner, use the oblique-shock relation to determine the shock angle, downstream Mach number, and pressure rise.
4. For an expansion corner, apply the Prandtl-Meyer function to determine the downstream Mach number and pressure decrease.
5. Carry the downstream state from one panel to the next.
6. Calculate the pressure coefficient on every panel.
7. Resolve and integrate panel pressure forces to obtain lift, wave drag, and pitching moment.

The method captures finite shock and expansion effects more accurately than linearized theory. It assumes steady, inviscid, two-dimensional flow with attached shocks and does not directly predict viscous separation or boundary-layer losses.

At a compression corner, the flow is turned through an oblique shock. The turning angle , shock angle , and upstream Mach number satisfy

The normal component is used with normal-shock relations to obtain the pressure rise and downstream Mach number.

At an expansion corner, the flow turns through a centered Prandtl-Meyer fan. The turning angle is

where

Flow through an expansion is isentropic: Mach number increases while static pressure and temperature decrease. Across an oblique shock, pressure and entropy increase while the Mach number decreases.

For a symmetric double-wedge airfoil at zero angle of attack:

• The upper and lower leading surfaces turn the flow toward the body, creating attached oblique shocks.
• At the maximum-thickness corners, the surfaces turn away from the flow, generating Prandtl-Meyer expansion fans.
• Additional waves form near the trailing edge to make the upper and lower streams compatible with the downstream flow.
• Symmetry makes the upper and lower pressure distributions equal, so .
• The pressure forces on the inclined panels have rearward components, producing thickness wave drag.

At a positive angle of attack, the lower leading surface produces stronger compression, while the upper surface experiences weaker compression or an expansion. The resulting pressure difference generates lift. The asymmetric pressure forces also produce lift-dependent wave drag and a pitching moment. Pressures on all panels are found sequentially from oblique-shock and Prandtl-Meyer relations, after which their normal and axial components are summed.

A disturbance in supersonic flow is confined within a Mach cone whose half-angle is

This limited propagation of disturbances strongly affects supersonic wings. Wing sweep reduces the Mach-number component normal to the leading edge:

where is the sweep angle.

A leading edge is classified as:

• Supersonic leading edge: The component normal to the leading edge is supersonic, generally . Disturbances cannot propagate upstream along the wing, and attached leading-edge waves may form.
• Subsonic leading edge: The normal component is subsonic, generally . Pressure disturbances can communicate along the span, and leading-edge suction may occur.

Sweep can therefore reduce wave drag, but excessive sweep may reduce lift-curve slope, worsen low-speed handling, increase structural weight, and promote spanwise flow.

Finite supersonic wings are influenced by Mach-wave geometry, aspect ratio, sweep, taper, and leading-edge condition. Their lift is generally accompanied by both lift-dependent wave drag and vortex-induced drag.

Delta wings have:

• Large leading-edge sweep and low aspect ratio.
• Low wave drag at supersonic speed when the leading edges lie within the Mach cone.
• High structural volume and stiffness.
• Vortex-generated lift at high angles of attack.
• Relatively high induced drag during low-speed flight.

Highly swept conventional wings have:

• A reduced normal Mach-number component.
• Better compatibility with moderate supersonic or transonic cruise.
• Potentially better low-speed efficiency than very low-aspect-ratio delta wings.
• Greater susceptibility to spanwise flow, tip stall, and aeroelastic deformation.

The optimum planform depends on cruise Mach number, range, maneuverability, runway performance, structural requirements, and internal volume.

For a flat plate at a small angle of attack , linearized supersonic theory gives the upper- and lower-surface pressure coefficients as

Therefore, the pressure difference is

Because this difference is uniform over the chord, integration gives

Thus, the two-dimensional lift-curve slope is

The predicted lift-curve slope decreases as the supersonic Mach number increases. The expression is not reliable very close to because linearized theory becomes singular there and transonic nonlinear effects dominate.

A supersonic aircraft must balance several competing requirements:

• Wave drag: Thin wings, slender bodies, sharp leading edges, and smooth area distributions reduce shock strength.
• Lift-to-drag ratio: The wing planform, sweep, aspect ratio, camber, and cruise lift coefficient must be optimized.
• Propulsion integration: Inlets must slow the air efficiently while controlling shocks and avoiding unstart.
• Stability and trim: The aerodynamic centre shifts aft as the aircraft accelerates, potentially causing Mach tuck and trim drag.
• Low-speed performance: Thin, highly swept wings often require high-lift devices, vortex lift, or high take-off speeds.
• Structural design: Slender configurations must resist bending, torsion, fatigue, and aeroelastic effects.
• Aerodynamic heating: Materials and systems must tolerate elevated skin and stagnation temperatures.
• Sonic boom: Volume and lift distributions should be shaped to reduce ground overpressure.
• Operational factors: Range, payload, runway length, noise, maintainability, and cost constrain the aerodynamic optimum.

During acceleration through the transonic regime, local shocks form and move over the aircraft. This alters the pressure distribution and generally moves the wing's aerodynamic centre aft—from approximately the quarter-chord in subsonic flow toward the half-chord in ideal two-dimensional supersonic flow.

Important effects include:

• Mach tuck: The aft movement of the resultant aerodynamic force creates an increased nose-down pitching moment.
• Trim drag: Greater tail force or control deflection may be required to maintain equilibrium.
• Control effectiveness changes: Shocks and separated flow may reduce hinge effectiveness or cause buffet.
• Control reversal risk: Aeroelastic wing twisting can oppose the intended control input.
• Directional effects: Shock interactions with the vertical tail can alter directional stability.

Design solutions include an all-moving tailplane, powered controls, suitable centre-of-gravity management, fuel transfer, adequate tail volume, aeroelastic tailoring, and automatic flight-control compensation.

Aerodynamic heating occurs because kinetic energy is converted into internal energy as high-speed air is decelerated in the boundary layer and near stagnation regions. Compression across shock waves also raises the gas temperature.

For steady adiabatic flow with no shaft work, the total enthalpy is constant:

For a calorically perfect gas, , so

Using

and

we obtain

Thus,

Actual wall temperature is usually below because of incomplete recovery and heat transfer, but stagnation regions can approach high recovery temperatures. Heating increases strongly with speed and can cause thermal expansion, strength loss, oxidation, and equipment-temperature problems.

Aerodynamic heating is nonuniform. The highest heating rates generally occur at stagnation points, sharp leading edges, nose regions, control-surface gaps, and locations where shocks interact with the boundary layer.

Its effects include:

• Reduction of material strength and stiffness.
• Thermal expansion and distortion of the airframe.
• Thermal stresses caused by temperature gradients.
• Damage to coatings, seals, electronics, fuel systems, and transparencies.
• Changes in structural clearances and surface contours.
• Increased oxidation and fatigue.

Mitigation methods include:

• Selecting heat-resistant aluminum, titanium, steel, nickel alloys, or composites.
• Using insulation, heat sinks, thermal barriers, and high-emissivity coatings.
• Providing expansion joints and allowing for thermal deformation.
• Increasing leading-edge radius where wave-drag and heating requirements permit.
• Employing active cooling for severe conditions.
• Placing sensitive equipment away from hot regions.
• Limiting Mach number or high-speed exposure time through the flight profile.

Thermal protection must be integrated with aerodynamic, structural, mass, and maintenance considerations.