Ch4 Oblique Shock and Expansion Waves 4.1 Introduction Supersonic flow over a corner.

4.2 Oblique Shock Relations …Mach angle (stronger disturbances) A Mach wave is a limiting case for oblique shocks. i.e. infinitely weak oblique shock

Oblique shock wave geometry Given : Find : or

Galilean Invariance : The tangential component of the flow velocity is preserved. Superposition of uniform velocity does not change static variables. Continuity eq : Momentum eq : parallel to the shock The tangential component of the flow velocity is preserved across an oblique shock wave Normal to the shock

Energy eq : The changes across an oblique shock wave are governed by the normal component of the free-stream velocity.

Same algebra as applied to the normal shock equction For a calorically perfect gas and Special case normal shock Note：changes across a normal shock wave the functions of M1 only changes across an oblique shock wave the functions of M1 &

and relation

For =1.4 (transparancy or Handout)

Note : 1. For any given M1 ，there is a maximum deflection angle If no solution exists for a straight oblique shock wave shock is curved & detached, 2. If strong shock solution (large ) M2 is subsonic weak shock solution (small ) M2 is supersonic except for a small region near , there are two values of β for a given M1

3. (weak shock solution) 4. For a fixed →Finally, there is a M1 below which no solutions are possible →shock detached 5. For a fixed M1 and Shock detached Ex 4.1

4.3 Supersonic Flow over Wedges and Cones Straight oblique shocks 3-D flow, Ps P2. Streamlines are curved. 3-D relieving effect. Weaker shock wave than a wedge of the same , P2, , T2 are lower Integration (Taylor & Maccoll’s solution, ch 10) The flow streamlines behind the shock are straight and parallel to the wedge surface. The pressure on the surface of the wedge is constant = P2 Ex 4.4 Ex 4.5 Ex4.6

4.4 Shock Polar –graphical explanations c.f Point A in the hodograph plane represents the entire flowfield of region 1 in the physical plane.

Shock polar Increases to (stronger shock) Locus of all possible velocities behind the oblique shock Nondimensionalize Vx and Vy by a* (Sec 3.4, a*1=a*2 adiabatic ) Shock polar of all possible for a given

Important properties of the shock polar For a given deflection angle , there are 2 intersection points D&B (strong shock solution) (weak shock solution) tangent to the shock polarthe maximum lefleation angle for a given For no oblique shock solution 3. Point E & A represent flow with no deflection Mach line normal shock solution 4. Shock wave angle 5. The shock polars for different mach numbers.

ref：1. Ferri, Antonio, “Elements of Aerodynamics of Supersonic Flows” , 1949. 2. Shapiro, A.H., ”The Dynamics and Thermodynamics of Compressible Fluid Flow”, 1953.

4.5 Regular Reflection from a Solid Boundary (i.e. the reflected shock wave is not specularly reflected) Ex 4.7

4.6 Pressure – Deflection Diagrams -locus of all possible static pressure behind an oblique shock wave as a function deflection angle for given upstream conditions. Shock wave – a solid boundary Shock – shock Shock – expansion Shock – free boundaries Expansion – expansion Wave interaction

(+) (-) (downward consider negative) Left-running Wave : When standing at a point on the waves and looking “downstream”, you see the wave running-off towards your left.

diagram for sec 4.5

4.7 Intersection of Shocks of Opposite Families C&D:refracted shocks (maybe expansion waves) Assume shock A is stronger than shock B a streamline going through the shock system A&C experience or a different entropy change than a streamline going through the shock system B&D 1. 2. and have (the same direction. In general they differ in magnitude. ) Dividing streamline EF (slip line) If coupletely sysmuetric no slip line

Assume and are known & are known if solution if Assume another

4.8 Intersection of Shocks of the same family Will Mach wave emanate from A & C intersect the shock ? Point A supersonic intersection Point C Subsonic

(or expansion wave) A left running shock intersects another left running shock

4.9 Mach Reflection ( for ) ( for ) A straight oblique shock A regular reflection is not possible Much reflection Flow parallel to the upper wall & subsonic for M2

4.10 Detached Shock Wave in Front of a Blunt Body From a to e , the curved shock goes through all possible oblique shock conditions for M1. CFD is needed

4.11 Three – Dimensional Shock Wave Immediately behind the shock at point A Inside the shock layer , non – uniform variation.

4.12 Prandtl – Meyer Expansion Waves Expansion waves are the antithesis of shock waves Centered expansion fan Some qualitative aspects : M2>M1 2. 3. The expansion fan is a continuous expansion region. Composed of an infinite number of Mach waves. Forward Mach line : Rearward Mach line : 4. Streamlines through an expansion wave are smooth curved lines.

i.e. The expansion is isentropic. ( Mach wave) Consider the infinitesimal changes across a very weak wave. (essentially a Mach wave) An infinitesimally small flow deflection.

…tangential component is preserved. as …governing differential equation for prandtl-Meyer flow general relation holds for perfect, chemically reacting gases real gases.

Specializing to a calorically perfect gas --- for calorically perfect gas table A.5 for Have the same reference point

procedures of calculating a Prandtl-Meyer expansion wave from Table A.5 for the given M1 2. M2 from Table A.5 the expansion is isentropic are constant through the wave