Supersonic Wings

P M V Subbarao Professor

Mechanical Engineering DepartmentI I T Delhi

An appropriate combination of Shocks & Expansion Waves…

Supersonic Flow Over Flat Plates at Angle of Attack

Review: Oblique Shock Wave Angle

tan 2 tan M1 sin

2 1

tan2 2 M12 cos 2

2 M1

2 sin2 1 tan 2 M1

2 cos 2

Prandtl-Meyer Expansion Waves

<0 .. We get an expansion wave (Prandtl-Meyer)

(M 2 ) (M1) (M ) 1

1tan 1 1

1M 2 1

tan 1 M 2 1

• Compare to Flat Plate

CL

pl

p

pu

p

2

M2

cos CD = 0

FpLift

Drag

Wings At Zero Angle of Attack

• Subsonic Wing in Subsonic Flow • Subsonic Wing in Supersonic Flow

M < 1

M > 1

• Supersonic Wing in Supersonic Flow

M > 1

M > 1

• Supersonic Wing in Subsonic Flow

Flow Separation

• Wings that work well sub-sonically generally Don’t work well supersonically, and vice-versa

Supersonic Airfoils• A leading edge in Supersonic Flow has a finite maximum wedge angle at which the oblique shock wave remains attached

=1.4

=1.1

=1.3

=1.4=1.3

=1.2

=1.1

=1.05

• Beyond that angle shock wave becomes detached from leading edge

Supersonic Flow Over an Airfoil

• Normal Shock wave formed off the front of a blunt leading causes significant drag

=1.1

=1.3Detached shock wave

Localized normal shock wave

Supersonic Airfoils

=1.1

=1.3

• To eliminate this leading edge drag caused by detached bow wave Supersonic wings are typically quite sharp at the leading edge

• Design feature allows oblique wave to attach to the leading edge eliminating the area of high pressure ahead of the wing.

• Double wedge or “diamond” Airfoil section

Supersonic Airfoils : Positive Angle of Attack

Dull Oblique Shock

Intense Oblique Shock

2

6

1 4

53

Supersonic Airfoils : Positive Angle of Attack

=1.1

=1.3

• A supersonic airfoil at positive angle of attack :• A dull shock at the top leading edge.•An intense shock at the bottom. • The airflow over the top of the wing is now faster.• Further acceleration through the expansion fans. • The Expansion fan on the top is more intense than the one on the bottom. • Combined result is faster flow and lower pressure on the top of the airfoil.

• We already have all of the tools we need to analyze the flow on this wing

Supersonic Airfoils : Negative Angle of Attack

=1.1

=1.3

•When supersonic airfoil is at negative angle of attack at the top leading edge there is a expansion fan and oblique shock at the bottom. • Result is the airflow over the top of the wing is now faster.• Airflow will also be accelerated through the expansion fans on both sides.• Result is much faster flow on top surface and therefore lower pressure on the top of the airfoil.

Supersonic Flow on Finite Thickness Wings at zero

• Symmetrical Diamond-wedge airfoil, zero angle of attack

Drag 2b p2l sin() p3l sin() sin()t / 2

l

Dragb p2

p3 t p2 > p1

Supersonic Wave Drag

• Finite Wings in Supersonic Flow have drag .. Even at zero angle of attack and no lift and no viscosity…. “wave drag”

• Wave Drag coefficient is proportional to thickness ratio (t/c)

• Supersonic flow over wings … induced drag (drag due to lift) + viscous drag + wave drag

CD wave

Drag

bcq_

p2 p3 2

pM 2

t

c

Symmetric Double-wedge Airfoil … Drag

CD wave

Drag

bcq_

p2 p3 2

pM 2

t

c

Thickness ratio

• Look at mach number Effect on wave drag

• Mach Number tends to suppress wave drag

Increasing mach

Thickness ratio

Induced drag

+=

Wave drag

• How About The effect of angle of attack on drag

Increasing t/c

Mach constant

Total drag

+=

Lift Coefficient Climbs Almost Linearly with

The effect of angle of attack on Lift

+=

• For Inviscid flow Supersonic Lift to drag ratio almost infinite for very thin airfoil

t/c = 0.035

• But airfoils do not fly in inviscid flows

+=

t/c = 0.035

• But contribute significantly to reduce the performance of supersonic wings

• Friction effects have small effect on Nozzle flow or flow in “large“ducts”

• Problem with sharp leading edges is poor performance in subsonic flight.

• Lead to very high stall speeds, poor subsonic handling qualities, and poor take off and landing performance for conventional aircraft

Disadvantages of Sharp Edged Wings

Wing Sweep Reduces Wave Drag

• One way to augment the performance of supersonic aircraft is with wing sweep …

• Lowers the speed of flowNormal to the wing …

• Decreasing the strengthOf the oblique shock wave

• Result is a Decrease in waveDrag and enhanced L/D

Geometrical Description of Wing Sweep

Equivalent 2-D Flow on Swept Wing

• Freestream Mach number resolved into 3 componentsi) vertical to wing … ii) in plane of wing, but tangent to leading edgeiii) in plane of wing, but normal to leading edge

i)M vert M sinii)M || M cos sinii)M M cos cos

• Equivalent Mach Number normal to leading edge

M eq M2 M vert

2 M sin 2 M cos cos 2

M 1 cos2 cos2 1 sin2 M 1 sin2 cos2

• Equivalent angle of attack normal to leading edge

tan eq M vert

M

M sin

M cos cos

tan cos

• Equivalent chord and span• Chord is shortened

• Span is lengthened

ceq c cos

beq b

cos

• Equivalent 2-D Lift Coefficient

CL eq

L2

pM eq2c cos b

cos

L

2

pM eq2cb

L2

pM2cb 1 sin2 cos2

CL

1 sin2 cos2

• Equivalent 2-D Drag Coefficient

CD eq

D / cos2

pM eq2c cos b

cos

D / cos

2

pM eq2cb

D / cos2

pM2cb 1 sin2 cos2

CD / cos

1 sin2 cos2

• Solve for CL, CD, L/D

CL CL eq1 sin2 cos2

CD CD eqcos 1 sin2 cos2

L

D

L

D

eq

cos

• Unswept Wing

CL: 0.205CD: 0.3606L/D: 5.68441

• 30 Swept Wing

CL: 0.2533 CD: 0.03909L/D: 6.4799

• WOW! … 14% IMPROVEMENT IN PERFORMANCE

F-14 Tomcat

The F-14's wing sweep can be varied between 20 and 68° in flight, and is automatically controlled by an air data computer. This maintains the wing sweep to give the optimum lift/drag ratio as the Mach number varies. The system can be manually overridden by the pilot if necessary. When the aircraft is parked, the wings can be swept to 75°, where they overlap the tail to save space on tight carrier decks.