6.1 Introduction

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6.1 Introduction
CH.6 OBLIQUE SHOCK WAVES 6.1 Introduction 1.
Definition of Oblique Shock - a straight
compression shock wave inclined at an angle to
the upstream flow direction - In general, the
oblique shocks produce a change in flow direction
as indicated in Fig. 6.1. (p119) 2.
Occurrence 1 external flow - due to the
presence of wedge in a supersonic flow - due
to the presence of concave corner in a supersonic
flow 2 internal flow - in supersonic flow
through an over-expanded nozzle
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3. Distinction 1 2, 3 dimensional shock 1)
2-dimensional shock due to the presence of wedge
etc. 2) 3-dimensional shock due to the
presence of cone etc. 2 attached, detached
shock 1) attached oblique shock
straight line for a given 2)detached shock
curved shock " where
deflection angle 4. Momentum Consideration 1
statement The oblique shock relations can be
deduced from the normal shock relations by noting
that the oblique shock can produce no momentum
change parallel to the plane in which it lies.
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2 proof 1) control volume (see p120 Fig. 6.2)
2) Because there is no momentum change parallel
to the shock, must equal .
3) flow normal to an oblique shock wave (see p120
Fig. 6.3) All the properties of oblique shocks
can be obtained by modification and manipulation
of the normal shock relations provided that angle
of the shock relative to the upstream flow
is known.
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6.2 Oblique Shock Wave Relations
6.2 Oblique Shock Wave Relations 1./ Basic
assumptions frictionless surface steady
2-dimensional planar adiabatic flow no
external work, negligible effect of body
forces 2./ Governing Equations (see 1
p124-125) 1. control volume (T p121 Fig.
6.4, 1 p124 Fig. 6.3) - unit area parallel to
the oblique shock wave - shock wave angle
deflection angle or turning angle or wedge
angle (change in flow direction induced by
the shock wave) 2. continuity equation

(6.1)
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3. momentum equation 1 normal momentum
equation
(6.1) 2
tangential momentum equation 4. energy
equation (Eq. (6.3), (6.4))
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If eqs. (6.1), (6.2), and (6.4) are compared
with the equations derived for normal shock waves
it will be seen that they are identical in all
respects except that and replace and
respectively. 5. Rankine-Hugoniot Relations for
Oblique Shock Waves
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6. Relations between the Changes across the Shock
Wave and the Upstream Mach Number 1 geometric
relation 2 Eqs. (6.1), (6.2), (6.4) becomes
These are, of course, again identical to
those used to study normal shocks, except that
occurs in place of and occurs
in place of .
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• Hence, if in normal shock relations is replaced
  by and by
  the following relations for oblique shocks are
  obtained using equations given in Ch.5.
• 3 Relations in terms of Upstream Mach Number
  and Wave Angle , Turning Angles

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• 7. Limit Values of
• 1) for normal shock ---gt for
  oblique shock
• 2) for normal shock ---gt for
  oblique shock Hence, for an oblique shock
  wave, can be greater than or less than 1.
• 3)
• The minimum value that can have is,
  therefore, i.e., the minimum shock wave
  angle is the Mach angle. When the shock has this
  angle, Eq. (6.10) shows that is equal to
  1, i.e., the shock wave is a Mach wave.
• The maximum value that can have is, of course,
  
• , the wave then being a normal shock wave.
  Hence
• 
  (6.15)
• 8. Relation between and (see p125
  Fig. 6.6)

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1 formula

(6.19) 2 meaning of eq. (6.19) (see
p124 Fig. 6.5, Fig. 6.6) 1) The turning angle ,
is zero when and also when
is equal to 1, i.e. , when
normal shock
and Mach wave
Thus an oblique shock lies between a normal shock
and a Mach wave. In both of these two limiting
cases, there is no turning of the flow. Between
these two limits reaches a maximum. 2) The
normal shock limit and Mach wave limit on the
oblique shock at a given value of are given by
the intercepts of the curves with the vertical
axis at (see p125 Fig. 6.8)
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• 3 value of ( maximum turning
  angle) for a given
• 1) derivation p124
• 2) variation of maximum turning angle with
  upstream Mach number for (see p126 Fig.
  6.8)
• 4 Remarks
• -For flow over bodies involving greater angles
  than this, a detached shock occurs. (see p126
  Fig. 6.10)
• It should also be noted that as increases,
  increases so that if a body involving a given
  turning angle,accelerates from a low to a high
  Mach number, the shock can be detached at the low
  Mach numbers and become attached at the higher
  Mach numbers.
• 9. Strong and Weak (non-strong) Shocks
• 1 two possible solutions for a value
• If is less than , there are two
  possible solutions,

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• i.e., two possible values for , for a
  given and
• .(see p127 Fig. 6.11)
• 2 classification
• 1) strong shock larger
  dotted line in Fig. 6.6
• 2) weak shock smaller
• 3 experimental results
• Experimentally, it is found that for a given
  and in external flows the shock angle, ,
  is usually that corresponding to the weak or non
  strong shock solutions.
• - Under some circumstance, the conditions
  downstream of the shock may cause the strong
  shock solution to exist in part of the flow. In
  the event of no other information being
  available, the non-strong shock solution should
  be used.
• 4 physical meaning of
• 1) meaning (physical interpretation)

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2) remarks if shock wave Mach
wave greater ---gt greater discontinuity
intensity of shock 5 general relation of
1) for both case 2) strong shock
weak shock 6 Occurrence of weak shock
and strong shock 1) whether weak or strong
shock f (boundary condition)
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2) weak shock typically occurs in external
aerodynamic flows Of the two choice for ,
it is an experimental fact that the one
corresponding to the weak shock usually
occurs. 3) strong shock The strong shock wave
occurs if the downstream pressure is sufficiently
high. The high downstream pressure may occur in
flows in wind tunnels, in engine inlets, or in
other ducts. 10. Characteristics of the Oblique
Shock Wave 1 Reason for the deflection of
stream direction velocity component So
is deflected from the direction of ,
i.e., fluid stream is deflected toward the
oblique shock wave.
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• 2 Distinction between Mach wave and shock wave
  by normal velocity component
• 1) Mach wave ( shock wave of zero intensity)
  
• 2) shock wave
• 3 deflection angle
• 1) formula
• 2) application
• applicable to conical shock as well as plane
  shock
• valid only for
• 3) case of
• a) (Mach angle)
  Mach wave
• b) normal shock
  wave
• 4

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• 1)
• 2) if
• the basic relation previously presented are
  not applicable.
• 5 in 2, 3 dimensional shock wave
• 2-dimensional shock ( plane shock) angle
  of wedge
• angle of concave corner
• 2) 3- dimensional shock
• angle of cone
• in this case streamlines after the conical
  shock must be curved
• in order that the 3-dimensional continuity eq. be
  satisfied.
• 6 corresponding to
• ( maximum flow deflection angle for a given
  )

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Example 6.1 ltSol.gt
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6.3 Reflection of Oblique Shock Waves
6.3 Reflection of Oblique Shock Waves 1.
Reflection of an Oblique Shock Wave from a Plane
Wall (see p129 Fig. 6.12) 2. Wall Pressure
Distribution near Point of Oblique Shock Wave
Reflection in Ideal Case (see p131 Fig.
6.13) 3.Wall Pressure Distribution near Point of
Oblique Shock Wave Reflection in Real Case (see
p131 Fig. 6.14) --gtboundary layer separation
during shock wave-boundary layer interaction
(p132 Fig. 6.15) 4. Mach Reflection (p132-133
Fig. 6.16, 6.17, 6.18 ) 5. No Reflection of
Wave (p135 Fig. 6.19) (Neutralization,
Cancellation or Absorption of an Oblique Shock
Wave)
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Example 6.2
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ltSol.gt
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Example 6.3 ltSol.gt
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Example 6.4 ltSol.gt
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6.4 Interaction of Oblique Shock Waves

• 6.4 Interaction of Oblique Shock Waves
• 1. Intersection of Multiple Left-Running Oblique
  Shock Waves
• (p136 Fig. 6.20)
• 2. Intersection of Oblique Shock Waves on a
  Curved Wall
• (p137 Fig. 6.21)
• 3. Intersection of Right- and Left-Running
  Oblique Shock Waves
• (p138 Fig. 6.23)

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Example 6.5
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ltSol.gt 1 Region 2
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• Turning angle produced by the oblique shock wave
  between region2 and 42
• Turning angle produced by the oblique shock wave
  between region2 and 42
• Find the value of that makes
• 5 Repeating Procedure
• region 2 region 42
• region 2 region 42

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6.5 Conical Shock Waves
6.5 Conical Shock Waves 1. Schematic
Representation (p142 Fig. 6.27, 28 p143
Fig. 6.29, 30) 2. Comparison with Wedge Flow
- In this case, both and are
small -gt attached shock - wedge
2-dimensional flow cone 3-dimensional
flow - strength of shock wave wedge case
gt cone case for the same and - flow
deflection angle after shock wedge
deflection angle wedge half angle cone
deflection angle lt cone half angle because of
3-dimensional effect
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• - surface pressure
• wedge surface pressure static pressure
  behind the shock
• cone surface pressure const lt wedge surface
  pressure
• cone surface Mach number gt wedge surface Mach
  number
• 6.6 Concluding Remarks

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