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**Ch4 Oblique Shock and Expansion Waves**

4.1 Introduction Supersonic flow over a corner.

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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 : or

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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

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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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**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 &

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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 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

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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

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**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

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**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.

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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* adiabatic ) Shock polar of all possible for a given

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**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 polarthe 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 Shock wave angle 5. The shock polars for different mach numbers.

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**ref：1. Ferri, Antonio, “Elements of Aerodynamics of Supersonic Flows” , 1949.**

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

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(+) (-) (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.

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diagram for sec 4.5

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**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. and have (the same direction. In general they differ in magnitude. ) 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

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

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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 M1. 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 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.

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**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**

from Table A.5 for the given M1 2. M2 from Table A.5 the expansion is isentropic are constant through the wave

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