Compressible Inviscid Flow 10 - Expansion Waves and Lift and Drag due to Pressure on Supersonic Aerofoils Flashcards
Properties of expansion wave
Lower density
Lower pressure
Flow properties change continuously through expansion fan
Similarities between oblique shock and expansion fan
Both deflect supersonic flow to direction parallel to wall
Define centred expansion wave
An expansion wave emanating from a sharp convex corner
Theory was worked out by Ludwig Prandtl and Theodore Meyer in 1907-08, so are commonly called Prandtl-Meyer expansion waves
Property changes across a Prandtl-Meyer expansion wave
Mach number and velocity increase (opposite to incompressible and subsonic)
Pressure and density decrease
What is an expansion fan?
Continuous expansion region that can be considered as an infinite number of Mach waves, each making the Mach angle with the local flow direction
Bounded upstream and downstream by two Mach waves with Mach angles
Isentropic as it takes place across a continuous succession of Mach waves and ds = 0 for each Mach wave
Lift and drag on a supersonic flat plate
On the top surface, an expansion wave occurs at the leading edge
p2<p1 can be calculated from expansion wave theory
On the bottom surface, an oblique shock wave occurs at the leading edge
p3>p1 can be calculated from oblique shock theory
Pressure force gives the exact solution for supersonic inviscid flow
Lift and drag on a supersonic wedge
Top and bottom surface are the same as a flat plate
Pressure force gives F2 and F3 (neglecting small contribution from the base)
Project F2 and F3 into lift and drag directions to calculate lift and drag
Can then calculate force coefficients
Shock-expansion theory for supersonic inviscid flow
Whenever there is an aerofoil made up of straight-line segments and the deflection angles are small enough so that no detached shock waves occur, the flow over the body goes through a series of oblique shock and expansion waves
Pressure distribution on the surface can be obtained exactly from either the shock or expansion wave theories
Effects of boundary layer (viscous effects) and skin friction are not included
Theory is useful as pressure forces are dominant for supersonic flow at a very high Reynolds number
Lift and drag on a diamond aerofoil
Supersonic flow over the airfoil is first compressed and deflected by the oblique shock wave at the leading edge
At mid-chord, the flow is expanded by twice the angle of deflection, creating an expansion wave
Pressures on opposite faces are uniform due to oblique shock and expansion wave theory
No lift due to symmetry
In the drag direction, pressure on front faces is larger than on back faces, resulting in finite drag
Wave drag for supersonic flow
Theories predict finite drag for 2D profiles
Contrasts results for 2D bodies in low-speed, incompressible flow, where drag was theoretically 0 with inviscid theory
In supersonic flow d’Alembert’s paradox does not occur
In a supersonic, inviscid flow, drag per unit span on a 2D body is finite - this new source of drag is called wave drag
Existence of wave drag is related to entropy increase and therefore loss of total pressure across oblique shock waves created by the airfoil
What do the force coefficients on an aerofoil due to supersonic inviscid flow depend on?
Aerofoil shape
Angle of attack
Freestream Mach number
Oblique shock boundary layer interaction
Due to viscous effects (no-slip boundary condition), BL develops on aircraft surfaces at supersonic speed
To account for viscous drag, either viscous flow equation (N-S) or BL equations need to be solved to calculate skin friction drag and BL effect on pressure
Complicated by interaction of shock wave with BL, invalidating BL assumption and inviscid flow theory due to shock induced flow separation
Effect of oblique shock boundary layer interaction
Major effect on pressure, shear stress and heat transfer distributions along the wall
Surface pressure starts to rise earlier than inviscid flow impinging point due to interaction, showing a plateau in separated flow region
