Continuity Equation for Compressible Flow

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2 Continuity Equation for Compressible Flow Velocity potential irrotational steady compressible

3 Momentum (Euler) Equation for Compressible Flow Euler's equation isentropic

4 velocity potential equation for steady, irrotational, isentropic compressible flow

5 Velocity Potential Equation for Compressible Flow a can be readily expressed in terms of φ as follows. nonlinear partial differential equation finite-difference numerical techniques Once φ is known, all the other flow variables can be obtained as: T γ γ p γ 1 2 γ 1 ρ γ 1 2 γ 1 T 0 = (1 + M 2 ) p 0 = (1 + 2 M ) = (1 + ρ 0 2 M )

6 Velocity Potential Equation for Compressible Flow nonlinear partial differential equation finite-difference numerical techniques Velocity Potential Equation For Incompressible Flow Laplace s equation is a second-order linear partial differential equation. If Φ 1, Φ 2, Φ 3,, Φ n represent n separate solutions of Laplace s equation, thenφ =Φ 1 +Φ 2 +Φ 3 + +Φ n is also a solution of Laplace s equation. Therefore, the solution of a complex flow are usually in the form of a sum of elementary flow solutions. linear partial differential equation Linear algebra analytical or numerical techniques

7 THE LINEARIZED VELOCITY POTENTIAL EQUATION uˆ vˆ << 1 << 1

8 (11.12)

9 freestream local

10 uˆ V uˆ V vˆ ~ 0.1 << 1; V 2 vˆ ~ 0.01 <<< 1; 2 2 V 2 ~ 0.1 << 1 ~ 0.01<<< < 1 M 0 < M 2 for 0 M 2 < 0.64 for 1.2 M < < 1 M < < M < 25 5

11 not valid for thick bodies and for large angles of attack not for transonic flow (0.8 < M < 1.2) or hypersonic flow (M> 5).

12 Pressure Coefficient C p

13 Linearized Form of Pressure Coefficient C p For an adiabatic flow of a calorically perfect gas

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16 Boundary Conditions θ At infinity At the body surface: The flow-tangency condition holds. Let θ be the angle between the tangent to the surface and the freestream.

17 PRANDTL-GLAUERT COMPRESSIBILITY CORRECTION Compressibility corrections for 0.3<M<0.7

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22 compressibility correction Prandtl-Glauert rule As early as 1922, Prandtl in his lectures at Gottingen, and first formally published by the British aerodynamicist, Hermann Glauert, in 1928.

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24 IMPROVED COMPRESSIBILITY CORRECTIONS

25 CRITICAL MACH NUMBER Linearized theory does not apply to the transonic flow regime, 0.8 < M <1.2. Transonic flow is highly nonlinear. Consider an airfoil in a low-speed flow, say, with M = 0.3, as sketched in Fig. a. In the expansion over the top surface of the airfoil, the local flow Mach number M increases. Let point A represent the location on the airfoil surface where the pressure is a minimum, hence where M is a maximum. Let us say this maximum is M A = Now assume that we gradually increase the freestream Mach number. As M increases, M A also increases. For example, if M is increased to M = 0.5, the maximum local value of M will be 0.772, as shown in Fig. b.

26 CRITICAL MACH NUMBER Linearized theory does not apply to the transonic flow regime, 0.8 < M <1.2. Transonic flow is highly nonlinear. Let us continue to increase M until we achieve just the right value such that the local Mach number at the minimum pressure point equals 1, i.e., such that M A = 1.0, as shown in Fig. c. When this happens, the freestream Mach number M is called the critical Mach number, denoted by M cr. By definition, the critical Mach number is that freestream Mach number at which sonic flow is first achieved on the airfoil surface. In Fig. c, M cr = One of the most important problems in highspeed aerodynamics is the determination of the critical Mach number of a given airfoil, because at values of M slightly above M cr the airfoil experiences a dramatic increase in drag coefficient.

27 Estimation of M cr C = f ( M, M p, A A )

28 Estimation of Mcr

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30 Thin airfoil Thick airfoil

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38 THE SOUND BARRIER C d = C d,0 2 1 M MACH NUMBER

39 DRAG-DIVERGENCE MACH NUMBER According to Prandtl- Glauert rule, C d becomes infinite at M = 1, C d = C d,0 2 1 M Point c is the critical Mach number. As we very carefully increase M slightly above M cr, (point d) a finite region of supersonic flow appears on the airfoil. At point e, the value of M at which this sudden increase in drag starts is defined as the drag-divergence Mach number. Beyond the drag-divergence Mach number, the drag coefficient can become very large, typically increasing by a factor of 10 or more.

40 point a~b point c point d point e drag-divergence Mach number

41 The Bell XS-l-the first rocket-propelled manned aircraft to fly faster than sound, October 14, Since 1945, research in transonic aerodynamics has focused on reducing the large drag rise. Instead of living with a factor of 10 increase in drag at Mach 1, can we reduce it to a factor of 2 or 3? This is the subject of the remaining sections of this chapter.

42 Reducing Drag at Transonic and Supersonic Flow 1. Thin airfoil section

43 Reducing Drag at Transonic and Supersonic Flow 2. Swept wings Adolf Busemann (1901~1986) The typical swept wing aircraft, F-86 fighter

44 Sweep Wing Obviously, it is desirable to reduce the Mach number of the flow over the airfoil section. A long time ago, it was discovered that the flow could be "fooled" by simply sweeping the wing.

45 Swept Wing Aircrafts F-86 Sabre fighter F-100 Super Sabre fighter

46 Richard Whitcomb (1921~2009)

47 THE WHITCOMB AREA RULE 1950 F-102 fighter

48 Area-Ruled Aircraft YF-102A

49 Area-Ruled Aircraft F-106

50 Area-Ruled Aircrafts F-104 fighter F-5 fighter B-1B bomber F-16 fighter

51 Reducing Drag at Transonic and Supersonic Flow 3. Supercritical airfoil section

52 Standard NACA 64 series airfoils with different thickness for high speed research in 1949 Shock wave move downstream as Mach number increases. A large region of separation flow downstream of shock wave for M=0.94, 0.87 and The separation flow is the primary reason for the increase in the drag near the M=1.0. Discontinuity pressure increase across a shock wave creates a strong adverse pressure gradient on airfoil surface and this adverse pressure gradient causes flow separation.

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55 Computational Fluid Dynamics (CFD)

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60 Blended Wing Body (BWB) Aircraft Design

61 Blended Wing Body (BWB) Aircraft Design Area-Ruled Aircraft

62 Computational Fluid Dynamics (CFD) National Transonic Facility (NTF)