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Supersonic Aerodynamics
Methods and Applications
Outline
• Introduction to Supersonic Flow
• Governing Equations
• Numerical Methods
• Aerodynamic Design
• Applications
Introduction to Supersonic Flow
• What does “Supersonic” mean ?– The term Supersonic implies that the flow is faster
than the speed of sound in the entirely regions concerned.
• Motivations– High speed civil transport (HSCT)
– Military fighters with supersonic cruise
• Shock wave• Entirely supersonic
Characteristics of Supersonic Flow
Bow shock on supersonic wedge airfoil, M = 1.7
Bow shock on supersonic wedge airfoil
• Subsonic Leading Edge• Supersonic Leading Edge
Characteristics of 3D Supersonic Flow
Spanwise pressure distributions for two different cases.
Influence on aerodynamic center
Governing Equations
• N-S Equations
• Euler Equations
• Full Potential Equation
• Prandtl-Glauert Equation
0)1( 2 =++− ∞zzyyxxM φφφ
Prandtl-Glauert Equation
• The flow is assumed to be steady, inviscid and irrotational.
• The flow is assumed to be essentially free stream, with only small perturbations.
• If the flow is entirely supersonic, then the right-hand-side of the transonic small disturbance equation can also be neglected, and the governing equation we must solve reduces to the linear form.
Prandtl-Glauert EquationPrandtl-Glauert Equation
The right-hand-side of the transonic small disturbance equation can be neglected
• Second order, linear partial differential equation• Not difficult to solve in this form• In the form of wave equations
Numerical Methods
• Panel Methods for Prandtl-Glauert Equation– Woodward, F. A.: Analysis and Design of Wing-Body
Combinations at Subsonic and Supersonic Speeds. J. Aircraft., Vol. 5, No. 6, pp. 528-534, (1968).
• Finite Difference Methods– See chapter 8
• Finite Volume Method– MGAERO
– CFD-FASTRAN
Aerodynamic Design• Supersonic flight is a challenge. We can get some insight
from the range equation:
• At supersonic speed the main problem is drag !
– Wing configuration
– Area-rule
– Slender body theory
• Control the shift of aerodynamic center
Wing Configuration
• In three dimensions, the bow shock is in reality a cone in shape (a Mach cone) as it extends back from the nose of the airplane.
• The Mach cone becomes increasingly swept back with increasing Mach numbers
• As long as the wing is swept back behind the Mach cone, there is subsonic flow over most of the wing and relatively low drag.
Wing Configuration
• A delta wing has the advantage of a large sweep angle but also greater wing area than a simple swept wing to compensate for the loss of lift usually experienced in sweepback .
Wing Configuration• At still higher supersonic Mach numbers, the Mach cone may
approach the leading edge of even a highly swept delta wing.
• This condition causes the total drag to increase rapidly and, infact, a straight wing (no sweep) becomes preferable.
Wing Configuration• Arrow Wings
– Arrow wing is a class of planforms for which exact linear theory solutions are available.
Wing Configuration
Drag of arrow wing
Modified arrow wing
Modified arrow wing
Donald D. Baals, A. Warner Robins and Roy V. Haris, Jr., “Aerodynamic Design Integration of Supersonic Aircraft,” Journal of Aircraft, Vol. 7, No. 5, Nov-Dec. 1970.pp. 385-394.
Area Rule
supersonic wave drag may be minimized by using area ruling.
Slender Body Theory
• The connection between the drag coefficient and fineness ratio:
• The wave drag integral: ( See chapter 5 )
Slender Body Theory• The von Kármán Ogive
– The shape that produces minimum wave drag for a specified base area and length, according to slender body theory.
Slender Body Theory• The Sears-Haack Body
– This is the minimum wave drag shape for a given length and volume according to slender body theory.
– The body is closed at both ends and has a very slightly blunted nose, and is symmetric about the mid-point.
Control the Shift of Aerodynamic Center
• Modifying arrow wing
• Modifying delta wing– See Concord configuration
Application• There have actually been only a few truly supersonic airplanes.
This means airplanes that can cruise supersonically.
• Before the F-22, classic “supersonic” fighters used brute force
(afterburners) and had extremely limited duration.
F-22
The small number of actual “cruisers”:- B-58 - SR-71
- Concord - XB-70
- F-22
B-58
XB-70 SR-71
Application: Concord
use delta modified wing and control the shift of
aerodynamic center
Application: Low-aspect-ratio wing
F-104
High speed civil transport
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