30 supersonic aerodynamics
TRANSCRIPT
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Supersonic Aerodynamics
Methods and Applications
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Outline
• Introduction to Supersonic Flow
• Governing Equations
• Numerical Methods
• Aerodynamic Design
• Applications
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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
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• Shock wave• Entirely supersonic
Characteristics of Supersonic Flow
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Bow shock on supersonic wedge airfoil, M = 1.7
Bow shock on supersonic wedge airfoil
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• Subsonic Leading Edge• Supersonic Leading Edge
Characteristics of 3D Supersonic Flow
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Spanwise pressure distributions for two different cases.
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Influence on aerodynamic center
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Governing Equations
• N-S Equations
• Euler Equations
• Full Potential Equation
• Prandtl-Glauert Equation
0)1( 2 =++− ∞zzyyxxM φφφ
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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.
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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
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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
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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
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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.
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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 .
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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.
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Wing Configuration• Arrow Wings
– Arrow wing is a class of planforms for which exact linear theory solutions are available.
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Wing Configuration
Drag of arrow wing
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Modified arrow wing
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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.
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Area Rule
supersonic wave drag may be minimized by using area ruling.
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Slender Body Theory
• The connection between the drag coefficient and fineness ratio:
• The wave drag integral: ( See chapter 5 )
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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.
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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.
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Control the Shift of Aerodynamic Center
• Modifying arrow wing
• Modifying delta wing– See Concord configuration
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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
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The small number of actual “cruisers”:- B-58 - SR-71
- Concord - XB-70
- F-22
B-58
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XB-70 SR-71
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Application: Concord
use delta modified wing and control the shift of
aerodynamic center
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Application: Low-aspect-ratio wing
F-104
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High speed civil transport