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Gas DynamicsESA 341Bab 4
Dr Kamarul Arifin B. Ahmad
PPK Aeroangkasa
Oblique shock wave
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 Definition
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 wedgeA concave corner
Control volume and symbols
y
x
V1
Vt1 Vn1
V2
Vn2
Vt2
P1
P2
T1
T2
1
2
Downstream flow angle
y
x
12
1
Upstream flow angle
2
Equations of motion
Continuity equation
Momentum Equation
Energy equation 022
0)()(
0
22
2
21
1
2222
2111
1122
nn
nn
nn
Vh
Vh
VPVP
VV
Relation between mach number(M) and deflection and shock wave angles ( and )
)sin(
)cos(
cos
cos
22
2
11
1
VV
VV
VV
VV
n
t
n
t
y
V1
Vt1Vn1
Vn2
-
)sin(
)cos(
sin
coscos
22
2
11
1
MM
MM
MM
Ma
V
a
VM
n
t
n
ttt
Ratio of flow properties
1
1
1sin
1
21
sin2
11
sin2
1
sin12
1
1sin1
2sin
2
11
sin12
sin1
1
1sin2
01
02
)1/(1
221
)1/(
221
221
01
02
221
2
221
221
1
2
221
221
1
2
221
1
2
T
T
MM
M
P
P
M
MM
T
T
M
M
M
P
P
y
x
12
1
Upstream flow angle
Mach number relations
12sin21M2
22sin21M1
sin2M
Replacing M1sin for M1 and M2sin (-) for M2
Relation of and
1
2
2
1
2
2
1
1
tan
tan
n
n
t
n
t
n
V
V
V
V
V
V
X
M
M
221
221
2
1
sin1
sin12
tan
tan
y
V1
Vt1Vn1
Vn2
-
11sin2
1tancot
221
21
M
M
0 when:
orM ,1sin
90
1
0Normalshock
Machwave
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.
Stro
ng
shoc
k
Weak shock
•This may sometimes be more conveniently plotted as:
Physical phenomena associated with the oblique shock wave
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
=53.30
M1=2.0
Mn1=1.60
P2/P1=2.82
=200
=29.90
M1=5.0
Mn1=2.49
P2/P1=7.07
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.
Physical phenomena associated with the oblique shock wave
=100=39.2
0
M1=2.0
Mn1=1.26
P2/P1=1.69
=200
=530
M1=2.0
Mn1=1.6
P2/P1=2.8
Oblique-shock reflections
Oblique-shock reflections cont.
1. For a given M1 and 1, find 1.2. Find M2 and P2/P1.3. Since 2 = 1, use M2 to find 2.4. Find M3 and P3/P2.5. Finally:
1
2
2
313 P
P
P
PPP
1
1
2
1-
Oblique-shock Application
Application 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.
