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

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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

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Oblique shock wave geometry Given : Find : Given : Find : or

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Galilean Invariance : 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 The tangential component of the flow velocity is preserved. Superposition of uniform velocity does not change static variables.

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

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and For a calorically perfect gas Special casenormal shock Note ： changes across a normal shock wave the functions of M 1 only changes across an oblique shock wave the functions of M 1 & Same algebra as applied to the normal shock equction

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and relation

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For =1.4 (transparancy or Handout)

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Note : 1. For any given M 1 ， there is a maximum deflection angle Ifno solution exists for a straight oblique shock wave shock is curved & detached, 2. If strong shock solution (large ) M 2 is subsonic weak shock solution (small ) M 2 is supersonic except for a small region near, there are two values of β for a given M 1

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3. 4. For a fixed 5. For a fixed M 1 and Shock detached Ex 4.1 (weak shock solution) →Finally, there is a M 1 below which no solutions are possible →shock detached

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4.3 Supersonic Flow over Wedges and Cones The flow streamlines behind the shock are straight and parallel to the wedge surface. The pressure on the surface of the wedge is constant = P 2 Straight oblique shocks Ex 4.4 Ex 4.5 Ex4.6 3-D flow, P s P 2. Streamlines are curved. 3-D relieving effect. Weaker shock wave than a wedge of the same, P 2,, T 2 are lower Integration (Taylor & Maccoll’s solution, ch 10)

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

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

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Important properties of the shock polar 1.For a given deflection angle, there are 2 intersection points D&B (strong shock solution) (weak shock solution) 2. tangent to the shock polar the maximum lefleation angle for a given For no oblique shock solution 4. Shock wave angle 5. The shock polars for different mach numbers. 3. Point E & A represent flow with no deflection Mach line normal shock solution

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

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

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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. Wave interaction Shock wave – a solid boundary Shock – shock Shock – expansion Shock – free boundaries Expansion – expansion

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

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diagram for sec 4.5

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4.7 Intersection of Shocks of Opposite Families and have (the same direction. In general they differ in magnitude. ) 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 Dividing streamline EF (slip line) If coupletely sysmuetric no slip line

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

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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 intersection

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

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4.9 Mach Reflection for M 2 Flow parallel to the upper wall & subsonic ( for ) A regular reflection is not possible A straight oblique shock Much reflection

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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 M 1. CFD is needed

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

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4.12 Prandtl – Meyer Expansion Waves Expansion waves are the antithesis of shock waves Some qualitative aspects : 1.M 2 >M 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. Centered expansion fan

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5. 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.

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

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

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procedures of calculating a Prandtl-Meyer expansion wave 1. from Table A.5 for the given M M 2 from Table A.5 4. the expansion is isentropic are constant through the wave

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