Continuity Equation for Compressible Flow

Velocity potential irrotational

steady compressible Momentum (Euler) Equation for Compressible Flow

Euler's equation

isentropic velocity potential equation for steady, irrotational, isentropic compressible flow Velocity Potential Equation for Compressible Flow

nonlinear partial differential equation a can be readily expressed in terms of φ as follows. finite-difference numerical techniques

Once φ is known, all the other flow variables can be obtained as:

−γ −1 T γ −1 2 −1 p γ −1 ρ γ −1 2 γ − = (1+ M ) = (1+ M 2 )γ −1 = (1+ M ) 1 ρ 2 T0 2 p0 2 0 Velocity Potential Equation for Compressible Flow

nonlinear partial differential equation

finite-difference numerical techniques

Velocity Potential Equation For Incompressible Flow linear partial Laplace’s equation is a second-order linear differential Φ Φ Φ partial differential equation. If 1, 2, 3, equation … , Φn represent n separate solutions of Laplace’s equation, thenΦ =Φ1+Φ2+Φ3+… +Φn is also a solution of Laplace’s Linear algebra equation. Therefore, the solution of a analytical or complex flow are usually in the form of a numerical sum of elementary flow solutions. techniques THE LINEARIZED VELOCITY POTENTIAL EQUATION

uˆ <<1 vˆ <<1 (11.12) freestream local uˆ vˆ ~ 0.1<<1; ~ 0.1<<1 V∞ V∞ uˆ 2 vˆ2 <<< <<< 2 ~ 0.01 1; 2 ~ 0.01 1 V∞ V∞

2 0.32 < 1− M ∞ <1

2 0 < M ∞ < 0.64 for 0 ≤ M ∞ ≤ 0.8

2 0.44 < 1− M ∞ < 24

2 1.44 < M ∞ < 25 for 1.2 ≤ M ∞ ≤ 5 not valid for thick bodies and for not for transonic flow (0.8 < M < 1.2) large angles of attack or hypersonic flow (M> 5). Pressure Coefficient C p Linearized Form of Pressure Coefficient C p For an adiabatic flow of a calorically perfect gas

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. PRANDTL-GLAUERT COMPRESSIBILITY CORRECTION

Compressibility corrections for 0.3

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.

IMPROVED COMPRESSIBILITY CORRECTIONS 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 MA = 0.435. • Now assume that we gradually increase the freestream Mach number. As M∞ increases, MA 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. 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 MA = 1.0, as shown in Fig. c. • When this happens, the freestream Mach number M ∞ is called the critical Mach number, denoted by Mcr. 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, Mcr = 0.61. • One of the most important problems in high- speed aerodynamics is the determination of the critical Mach number of a given airfoil, because at values of M ∞ slightly above Mcr the airfoil experiences a dramatic increase in drag coefficient. Estimation of Mcr

Cp,A = f (M ∞ , M A ) Estimation of Mcr

Thin airfoil

Thick airfoil

THE SOUND BARRIER

C C = d ,0 d 2 1− M ∞

MACH NUMBER DRAG-DIVERGENCE MACH NUMBER

According to Prandtl- Glauert rule, Cd becomes infinite at M∞ = 1, C C = d ,0 d 2 1− M ∞

• Point c is the critical Mach number. As we very carefully increase M∞ slightly above Mcr, (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. point a~b point c

point d

point e drag-divergence Mach number The Bell XS-l-the first rocket-propelled manned aircraft to fly faster than sound, October 14, 1947. 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. Reducing Drag at Transonic and Supersonic Flow

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

2. Swept wings

Adolf Busemann (1901~1986)

The typical swept wing aircraft , F-86 fighter 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. Swept Wing Aircrafts

F-86 Sabre fighter F-100 Super Sabre fighter Richard Whitcomb (1921~2009) THE WHITCOMB AREA RULE 1950

F-102 fighter Area-Ruled Aircraft YF-102A Area-Ruled Aircraft F-106 Area-Ruled Aircrafts

F-104 fighter F-5 fighter

B-1B bomber F-16 fighter Reducing Drag at Transonic and Supersonic Flow

3. Supercritical airfoil section 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 0.79. • 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.

Computational Fluid Dynamics (CFD)

Blended Wing Body (BWB) Aircraft Design Blended Wing Body (BWB) Aircraft Design

Area-Ruled Aircraft Computational Fluid Dynamics (CFD)

National Transonic Facility (NTF)