Transcript Gas Dynamics ESA 341 Bab 4

Gas Dynamics
ESA 341
Bab 4
Dr Kamarul Arifin B. Ahmad
PPK Aeroangkasa
Oblique shock wave
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Introduction
Control volume and symbols
Equation of motion
Relation between mach number(M) and
deflection and shock wave angles ( and )
Ratio of flow properties
Mach number relations
Relation of  and 
Introduction
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Definition

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A compression shock wave occurs that is inclined at an angle of
the flow
Still represent a sudden, almost discontinuous change in
fluid properties
We will be focused on the 2D straight oblique shock
wave.
A symmetrical wedge
A concave corner
Control volume and symbols
Vt1
Vn1
Vt2
Vn2

V1
Downstream
flow angle
V2


1

2
  2  1
y
T2
T1
x
1
P2
P1
y
Upstream flow
angle
2
x
Equations of motion
Continuity equation
 2Vn 2  1Vn1  0
( P1  1Vn1 )  ( P2   2Vn 2 )  0
2
Momentum Equation
Energy equation
2
2
2



Vn1
Vn 2 
 h1 
   h2 
0

 

2
2

 

Relation between mach number(M) and deflection and
shock wave angles ( and )
Vt  V1 cos
Vn1  V1 cos
Vt  V2 cos(   )
Vt1
Vn1
  -

V1
y
Vn2
Vn 2  V2 sin(   )
Mt 
Vt Vt cos

 M 1 cos
a
a
M n1  M 1 sin 
M t  M 2 cos(   )
M n 2  M 2 sin(   )
Ratio of flow properties
P2 2M 12 sin 2     1

P1
 1

2
  1M 12 sin 2 

1 2    1M 12 sin 2 
Upstream flow
angle

   1 2 2   2
1
M 1 sin   
M 12 sin 2   1

2
T2 
 1


T1
   12  2 2

 M 1 sin 


2


1


  1 2 2 
M 1 sin  
P02  2


P01 1    1 M 2 sin 2  
1
2


T02
1
T01
 /( 1)




1


2



1
2
2


M sin  
   1 1
  1 
1

  2  1
1 /( 1)
y
x
Mach number relations
Replacing M1sin for M1 and M2sin (-) for M2
M 2 sin   
  1M12 sin2   2
2M12 sin2     1
Relation of  and 
tan 
Vn1
Vt1
tan    
Vt1
Vn 2
Vt 2
Vn1  2

Vn 2 1
tan    1 2    1M12 sin 2 


X
  1M12 sin 2 
tan
2
   1 

M 12

cot  tan 
 1
 2 2
 2  M 1 sin   1 

V1
Vn1
y

Vn2
  0 when:
Normal
shock
  90
M1 sin   1, or
 
0
Mach
wave

Mach Wave
 -

Physical phenomena associated with the oblique
shock wave
1. For any given upstream Mach number M1, there is a maximum
deflection angle, max. If the the physical geometry is such that > max,
then the shock will be detached.
Physical phenomena associated with the oblique shock
wave
2)For any given < max, there will be two straight oblique
solutions for a given upstream Mach number. For
example, for M1=2.0 and =150, then from the graph, 
can be equal either 45.3 or 79.80. The smaller  is called
the weak shock solution, and the larger is called the
strong shock solution.

Physical phenomena associated with the oblique
shock wave
•This may sometimes be more conveniently plotted as:
Physical phenomena associated with the oblique shock
wave
3) For attached shocks with a fixed deflection angle, as the upstream
Mach number M1 increases, the wave angle  decreases, and the
shock wave becomes stronger. Or, when M1 decreases, the wave
angle increases, and the shock becomes weaker.
=200
M1=2.0
=200
M1=5.0
Mn1=1.60
Mn1=2.49
P2/P1=2.82
P2/P1=7.07
Physical phenomena associated with the oblique
shock wave
4)For attached shocks with fixed upstream Mach number,
as the deflection angle increases, the wave angle 
increases, and the shock becomes stronger. However,
when > max, the shock wave will be detached.
=100
M1=2.0
=200
M1=2.0
Mn1=1.26
Mn1=1.6
P2/P1=1.69
P2/P1=2.8
Oblique-shock reflections
Oblique-shock reflections
1.
2.
3.
4.
5.
cont.
For a given M1 and 1, find 1.
Find M2 and P2/P1.
Since 2 = 1, use M2 to find 2.
Find M3 and P3/P2.
Finally:
P3  P1
1
2
1
 1- 
P3 P2
P2 P1
Oblique-shock Application
Application
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Oblique shocks desirable on
supersonic intakes to reduce
total pressure losses.
Group Exercises 5
1) Consider a supersonic flow with a Mach number M = 2,
with a static pressure p = 105 Pa, and a static
temperature T = 288K. The flow is deflected at a
compression corner through 20o. Calculate the Mach
number, the static pressure, the temperature, the
stagnation pressure and the stagnation temperature
behind the resulting oblique shock wave.
2) Consider a supersonic flow with M = 2, p = 1 atm, and T
= 288K. The flow is deflected at a compression corner
through 20o. Calculate M, p, T, po and To behind the
resulting oblique shock wave.