1 Module3: Waves in Supersonic Flow Lecture14: Waves in Supersonic Flow (Contd.)
Mach Reflection: The appearance of subsonic regions in the flow complicates the problem. The complications are also encountered in shock reflections, when they are too strong to give the simple or regular reflections. If M after the incident shock is lower than the detachment Mach number forθ, then no solution with simple oblique wave is possible. A three-shock Mach reflection appears that satisfies the downstream conditions. M >1 M 1 0 M <1 Slip stream A normal, or, nearly normal, shock that appears near the wall forms with the incident and reflected shocks a triple intersection point at O. Due to the difference in entropy on streamlines above and below the triple point, the streamline that extends downstream from the triple point is a slipstream. The nearly normal shock is termed shock stem. The subsonic region behind the shock stem makes a local description of the configuration impossible. The triple point solution that occurs in a particular problem and the location of the triple point are determined by the downstream conditions which influence the subsonic part of the flow. Shock-Expansion Theory
3 Oblique shock wave and simple isentropic wave relations can be used to analyze many -D supersonic flow problems, particularly for geometries with straight segments. (1) Diamond-section airfoil: Consider a diamond section or double-wedge section airfoil with semi-vertex angle ε. Assume the semi-vertex angle to be sufficiently smaller than θ max associated with the free stream Mach number M 1. An attached oblique shock appears at the nose that compresses the p 3 M 1, p 1 M 1 >1 ε t ε 4 p 1 p 4 flow to pressure p.on the straight portion, downstream of the shock the flow remains uniform at M. The centered expansion at the shoulder expands the flow to pressure p 3 and the trailing edge shock recompresses it to nearly the free stream pressure ( p4 p1 ). Hence, an overpressure acts on the forward face and an under-pressure acts on the rearward face. Since the pressure on the two straight portions is unequal, a drag force acts on the airfoil. This drag force is given by ( ) cosε ( ) D = p p t p p t per unit span 3 3 p 3 t is the section thickness at the shoulder. Pressure values p and p 3 can be obtained using the shock and expansion relations. This drag exists only in supersonic flow and is called supersonic wave drag.
4 () Flat plate at incidence: Consider a flat plate of chord c set at an angle of attack α. Due to no upstream influence, the streamlines ahead of the leading edge are straight and the upper surface flow is independent of lower surface. The flow on the upper surface turns at the nose through a centered expansion by the angle α whereas on the lower side the flow is turned through a compression angleα by an oblique shock. The reverse happens at the trailing edge. p 3 M 1 >1 M 1, p 1 α Slip stream 3 p 1 p From the uniform pressures on the two sides, the lift and drag forces are ( ) L= p3 p ccosα ( ) D= p3 p csinα The shock on the lower surface at the nose is weaker than the shock at the trailing edge on the upper surface (shock at higher Mach number). Hence, the increase in entropy for flow on the two sides is not same and consequently the streamline from the trailing edge is a slipstream inclined at a small angle relative to the free stream. (3) Curved airfoil section An attached shock forms at the nose. Subsequently, continuous expansion occurs along the surface. The flow leaves at the trailing edge through an oblique shock. For the shocks to be attached, it is required that nose and tail be wedge shaped with half angle less thanθ max. Since the flow over the curved wall varies continuously, no simple expression for lift and drag forces is obtained in this case.
5 If a larger portion of the flow field is considered, then the shocks and expansion waves will interact. The expansion fans attenuate the oblique shocks, making them weak and curved. At large distances they approach asymptotically the free-stream Mach lines. Due to the interaction the waves will reflect. The reflected wave system will alter the flow field. In shock-expansion theory, the reflected waves are neglected. For a diamond airfoil and a lifting flat plate, the reflected waves do not intercept the airfoil at all. Hence, the shock-expansion results are not affected.
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