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HIGH-SPEED AERODYNAMICSMACE 31321

Lecture 7Oblique Shock Waves II

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OBJECTIVES OF THIS LECTURE

• To understand the inherent difference between the flow around a 2D wedge and a 3D conical object

• To acquire a understanding of the nature of a detached shock wave in front of a blunt object

• To acquire a understanding of what happens when an oblique shock wave impinges on a solid wall or intersects with another wave

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OBLIQUE SHOCK WAVE

When a supersonic flow is “turning into itself”, an oblique shock wave forms.

Across the wave, M decreases DISCONTINOUSLY, whereas p, ρ and T increases DISCONTINOUSLY.

Non-isentropic process: ∆s>0.

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SEQUENCE OF STEPS1. Find wave angle β from θ-β-M chart at

intersection of M1 and θ – interpolate if necessary

2. Find Mn,1 using3. Find flow property ratios across shock from

Normal Shock Property Table4. Find M2 using5. Find flow properties downstream of the shock

from the property ratios and the given upstream flow conditions, bearing in mind that

βsin11, MM n =

( )θβ −=

sin2,

2nM

M

1,02,01,02,0 ppandTT ≠=

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SUPERSONIC FLOW OVER WEDGES & CONES

Note: The shock property table provides the information for 2D flows ONLY.

2D body shape: uniform flow downstream

parallel to the wedge surface

3D body shape: 3D relieving effect, weaker

shock with a lower wave angle

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DETACHED SHOCK WAVE• Detached shock wave in front of a 2D blunt body

The detachment distance δ depends on M1 and the size and shape of the body.

θ−β−Μ diagram

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SHOCK INTERACTION AND REFLECTION• An oblique shock may impinge a solid surface or

intersect with another wave resulting in a complex shock wave system.

• Important implication to the design of supersonic aeroplanes, rocket engines and missiles.

X-15 hypersonic research vehicle built in 1950s

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SHOCK INTERACTION AND REFLECTION• Regular reflection of a shock wave from solid surface

A reflected shock is formed so as to turn the flow into the direction of the surface B

The strength of the reflected shock is weaker than the incident shock since M2 < M1.

The properties of the reflected shock are uniquely defined by M1 and θ.

β2

β2 = Φ + θ

The wave angle β is the angle between the shock wave and the incoming flow direction.

Φ

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SHOCK INTERACTION AND REFLECTION• Mach reflection

If M2 is smaller than the minimum Mach number for the required deflection angle, a straight reflected shock can not be formed. A normal shock is formed and a Mach reflection wave are produced.

Mβ

θ

M1M2

Normal shock

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SHOCK INTERACTION AND REFLECTION• Intersection of right- and left- running shock waves

The direction of velocity is parallel to the slip line.

p4 = p4’

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SHOCK INTERACTION AND REFLECTION• Intersection of two left-running shock waves

A stronger shock is formed due to merge of two shock waves.

A weak reflected wave is formed to ensure the flow is region 4 and 5 will be in the same direction.

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QUESTION• Consider an oblique shock wave generated by a

compression corner with a 10o deflection angle. The oblique shock wave subsequently impinges on a straight wall opposite the compression corner. Calculate the angle of the reflected shock wave, Φ, related to the straight wall and the Mach number, pressure, temperature behind the reflected shock wave.

• Answer: 17.3o, 2.55, 4.62atm, 458.01K)

M1=3.6p1=1atmT1=15oC

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SOLUTION• From the θ-β-M chart, for M1=3.6 and θ=10o, β1=24o.

Hence• From the normal shock table

• Hence

• From the θ-β-M chart, for M2=2.96 and θ=10o, β2=27.3o.Hence

464.124sin6.3sin 111, =×== on MM β

294.1,32.2,7157.01

2

1

22, ===

TT

ppMn

( ) ( ) 96.21024sin

7157.0sin 1

2,2 =

−=

−=

θβnM

M

β2Φ

Φ = β2 − θ = 27.3−10=17.3ο

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SOLUTION• Also

• From the Normal Shock Table,

358.13.27sin96.2sin 222, =×== on MM β

atmppp

ppp 62.4132.2991.11

1

2

2

33 =××==

229.1,991.1,7572.02

3

2

33, ===

TT

ppMn

( ) ( ) 55.2103.27sin

7572.0sin 2

3,3 =

−=

−=

θβnM

M

KTTT

TTT 01.458288294.1229.11

1

2

2

33 =××==

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REFERENCES

• In “Fundamentals of Aerodynamics” by Andersons, 2nd edition.–§9.3 –§9.4–§9.5