Supersonic Aerodynamics. Methods and Applications

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1 Supersonic Aerodynamics Methods and Applications

2 Outline Introduction to Supersonic Flow Governing Equations Numerical Methods Aerodynamic Design Applications

3 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

4 Characteristics of Supersonic Flow Shock wave Entirely supersonic

5 Bow shock on supersonic wedge airfoil Bow shock on supersonic wedge airfoil, M = 1.7

6 Characteristics of 3D Supersonic Flow Subsonic Leading Edge Supersonic Leading Edge

7 Spanwise pressure distributions for two different cases.

8 Influence on aerodynamic center

9 Governing Equations N-S Equations Euler Equations Full Potential Equation Prandtl-Glauert Equation ( 1 M 2 ) φ + φ + φ = xx yy zz 0

10 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-handside of the transonic small disturbance equation can also be neglected, and the governing equation we must solve reduces to the linear form.

11 Prandtl-Glauert Equation Prandtl-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

12 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 , (1968). Finite Difference Methods See chapter 8 Finite Volume Method MGAERO CFD-FASTRAN

13 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

14 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.

15 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.

16 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, in fact, a straight wing (no sweep) becomes preferable.

17 Arrow Wings Wing Configuration Arrow wing is a class of planforms for which exact linear theory solutions are available.

18 Wing Configuration Drag of arrow wing

19 Modified arrow wing

20 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 pp

21 supersonic wave drag may be minimized by using area ruling. Area Rule

22 Slender Body Theory The wave drag integral: ( See chapter 5 ) The connection between the drag coefficient and fineness ratio:

23 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.

24 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.

25 Control the Shift of Aerodynamic Center Modifying arrow wing Modifying delta wing See Concord configuration

26 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

27 The small number of actual cruisers : - B-58 - SR-71 - Concord - XB-70 -F-22 B-58

28 XB-70 SR-71

29 Application: Concord use delta modified wing and control the shift of aerodynamic center

30 Application: Low-aspect-ratio wing F-104

31 High speed civil transport

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