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Compressibility Effects

Aerodynamic changes caused by air density variations at higher Mach numbers, typically becoming important above M≈0.3.

Mach Number (M)

Ratio of aircraft velocity to local speed of sound.

Speed of Sound Equation

a = √(γRT)

Critical Mach Number (Mcrit)

Lowest freestream Mach number at which any point on the aircraft first reaches Mach 1.

Drag Divergence Mach Number (MDD)

Mach number where drag begins increasing rapidly due to shock-wave formation.

Transonic Flow Regime

Flow containing both subsonic and supersonic regions, typically M≈0.8–1.2.

Wave Drag

Additional drag caused by shock waves in compressible flow.

Prandtl-Glauert Compressibility Correction

Corrects incompressible aerodynamic coefficients for subsonic compressible flow.

Prandtl-Glauert Beta (β)

β = √(1 − M²)

Prandtl-Glauert Lift Correction

CL = CL0/β

Prandtl-Glauert Pressure Coefficient Correction

Cp = Cp0/β

Effect of Increasing Mach Number on CL (Prandtl-Glauert)

CL increases as Mach number increases.

Limitation of Prandtl-Glauert Theory

Becomes inaccurate near transonic conditions.

Karman-Tsien Correction

Improved compressibility correction that remains more accurate at higher subsonic Mach numbers.

Reason Karman-Tsien Improves Accuracy

Accounts for nonlinear compressibility effects ignored by Prandtl-Glauert.

Empirical Estimate of Critical Mach Number

Mcrit ≈ MDD − 0.08

Shock Wave

Thin region where flow properties change abruptly.

Normal Shock

Shock wave perpendicular to flow direction.

Oblique Shock

Shock wave inclined relative to flow direction.

Effect of Shock Waves on Drag

Increase drag and can cause flow separation.

Buffet

Onset of flow separation and aerodynamic instability caused by shocks.

Why Swept Wings Delay Compressibility Effects

Only the velocity component normal to the leading edge contributes to compressibility.

Normal Mach Number

Mn = M cos(Λ)

Leading Edge Sweep Angle (Λ)

Angle between wing leading edge and a line perpendicular to the fuselage.

Effect of Sweep on Critical Mach Number

Increases effective critical Mach number.

Effective Mach Number for Swept Wing

Meff = M cos(Λ)

Advantage of Swept Wings at High Speed

Reduced wave drag and delayed shock formation.

Disadvantage of Swept Wings

Reduced lift-curve slope and poorer low-speed performance.

Lift Curve Slope

Rate of change of lift coefficient with angle of attack.

Lift Curve Slope of Swept Wing

Lower than an equivalent unswept wing.

Area Rule

Cross-sectional area should vary smoothly to minimize wave drag.

Purpose of Area Rule

Reduce transonic wave drag.

Delta Wing

Triangular wing with large sweep angle.

Typical Delta Wing Sweep

Usually greater than 50°.

Advantage of Delta Wings at High Speed

Excellent performance in transonic and supersonic flight.

Primary Lift Mechanism at Low AoA on Delta Wing

Potential flow lift.

Primary Lift Mechanism at High AoA on Delta Wing

Vortex lift.

Vortex Lift

Additional lift generated by stable leading-edge vortices.

Why Delta Wings Produce Vortex Lift

Leading-edge flow separates and rolls into strong vortices.

Benefit of Leading-Edge Vortices

Low-pressure cores increase lift.

Polhamus Leading Edge Suction Analogy

Method used to estimate vortex lift on delta wings.

Potential Flow Lift Component

Linear lift contribution predicted by attached-flow theory.

Vortex Lift Component

Nonlinear lift contribution from leading-edge vortices.

Total Delta Wing Lift

Potential lift + vortex lift.

Characteristic of Delta Wing CL vs α

Strongly nonlinear due to vortex lift.

Reason Delta Wings Can Fly at High AoA

Vortex lift delays stall.

Supercritical Airfoil

Airfoil designed to delay shock formation and reduce wave drag.

Key Feature of Supercritical Airfoil

Flattened upper surface.

Another Feature of Supercritical Airfoil

Highly cambered aft section.

Benefit of Supercritical Airfoils

Higher critical Mach number and lower wave drag.