199  P a g e
DIFFRACTION OF OBLIQUE SHOCK WAVE PAST A BEND OF SMALL ANGLE
R.S. Srivastava
Formerly of Defence Science Centre, New Delhi, (India)
ABSTRACT
Lighthill considered the diffraction of normal shock wave past a small bend of angle . Srivastava and Srivastava and Chopra extended the work of Lighthill to the diffraction of oblique shocks (consisting of incident and reflected shock wave). In the present investigation the curvature of the reflected diffracted shock and vorticity distribution of a particle over the reflected diffracted shock have been obtained
Keyword: Curvature, Diffraction, Reflection, Vorticity.
I. INTRODUCTION
Lighthill (1949) considered the diffraction of a normal shock wave parallel to a wall and meeting a corner where the wall turns through a small angle . The analogous problem for a plane shock wave hitting the wall obliquely together with the associated reflected shock has been considered by Srivastava (1968) and Srivastava and Chopra (1970). Srivastava (1968) has developed the mathematical theory of diffraction when the relative outflow from the reflected shock wave before diffraction is subsonic and sonic. Srivastava and Chopra (1970) completed the theory when the relative outflow behind the reflected shock wave before diffraction is supersonic.
Reference may be made to the book by Srivastava (1994). This paper gives new results concerning curvature of the diffracted reflected shock and vorticity distribution over the same.
II. MATHEMATICAL FORMULATION
In Figure1, the velocity, pressure, density and sound speed are denoted by subscripts 0 ahead of incident shock wave, 1 in the intermediate region and 2 behind the reflected shock wave. U in the figure denotes the velocity of intersection of the incident and reflected shock and is the angle of bend. _{0} is the angle of incidence and _{2} is the angle of reflection. The RankineHugoniot equations across the incident and reflected shock for
1 . 4
( being the ratio of specific heats) are given as follows (Srivastava 1995, 2003)Fig.1 Schematic drawing for oblique shock configuration passing over a small bend before diffraction
Across the incident shock
0 2 2
2 0 0
1
sin 1 sin
6 5
U U a
q  (1)
sin 7
6
5
_{0}^{2}0 2 2 0 1
U a
p
 (2)
0 2 2
2 0 0 1
sin 1 5
6
U
a ^{ (3) }
0 0
0
p
a
Across the reflected shock:
^{} ^{} ^{} _{} ^{} ^{} ^{} ^{} ^{} _{} ^{}
_{2}1
* 2 1 1
* 1
2
1
6 5
q U q a
U q
q
 (4)
_{}
6 7
5
^{2} _{1}^{2}1
* 1 2
q a U
p
 (5)
1
^{2}* 2 1 1
2
5
1 6
q U
a
 (6)
Where
q
_{2} q
_{2}sin
_{2}, q
_{1} q
_{1}cos
_{0}
_{2}
1 1 1
2
*
sin ,
a ^{p} U
U
Also from Figure1, we have
201  P a g e
2 , cos
cos
_{2} _{2} _{0} _{2}1
q q
Let the pressure, density, velocity and entropy behind the reflected diffracted shock be denoted by
2 2 2
, , q
p
and S_{2}. Using small perturbation theory and conical field transformations, the flow equations can be linearized and they yield a single second order differential equation in p, namely2 2 2 2
1 y
p x
p y
y p x x p y y
x x
 (7)where
t a y Y t a
t q x X
q a
p p p
2 2
2 2
2 2
2
2
, ,
u v
q
q
_{2}
_{2}1 ,
,  (8)
It may be pointed out that in Figure1 and 2 for X,Y coordinates the origin is at the corner and in Figure2 for x,y coordinates the origin lies on the original wall produced. The origin with respect to x,y coordinates is
, 0
2 2
a
q
and the coordinates of the point of intersection of incident and reflected shock is
, 0
2 2
a q
U
.The characteristics of the differential equation (7) are tangents to the unit circle
x
^{2} y
^{2} 1 ,
consequently the region of disturbance will be enclosed by the arc of the unit circlex
^{2} y
^{2} 1 ,
reflected diffracted shock and the wall.A schematic drawing after diffraction is shown in Figure2. The problem is to determine the curvature and vorticity over diffracted shock as indicated in Figure2.
Fig.2 Schematic drawing for oblique shock configuration passing over a small bend after Diffraction
Following Srivastava and Chopra (1970) the equation of the straight portion of the reflected shock after diffraction is given by
cot
2y k
x
^{where}2 2
a q
k U
 (9)The equation of the diffracted shock may therefore may be written as
y f y
k
x cot
_{2}
 (10)Where
^{f} ^{y}
^{ is small }From (10) we obtain
cot
_{2}1
f y dx
dy
 (11)and
_{2}
^{3}2 2
cot
y f
y f dx
y d
The curvature
is therefore given by y f
dx dy
dx y
d
^{3} _{2}32 2
2 2
sin 1
 (12)by neglecting higher order terms of
f y
.Following Srivastava (1968) and Srivastava and Chopra (1970), on the reflected diffracted shock we have the relation
B A G y f y y
p
2
2  (13)
where
G y q
_{2}sin
_{2}cos
_{2} a
_{2}y
 (14)In (13), B_{2}, A_{2} are constants.
From the equation (13) and (12) we have
y
p y
G A
B
^{3} _{2}2 2
1 sin
^{ (15) }Equation (15) can be put in the form
dy
x x
p y
G A B
1 1 2 3 2
2
1 sin
 (16)This is the mathematical expression giving the curvature of the reflected diffracted shock wave for all the three cases (when the relative outflow is supersonic, sonic and subsonic). In the present paper we take the case when the relative outflow is sonic.
We have the relation on the diffracted shock (Srivastava, 1968) as
cos
_{2} sin
_{2}tan θ
K y
^{cos} ^{sin} ^{tan} ^{θ}
sin
_{2}
_{2}
_{2}
 (17)where
K K Z
Z
1 θ 1
tan
_{2}2
1 cot
_{2}1
2
2
Z
Z
 (18)203  P a g e
2
1
sin
cos 1
h Z
z
 (19)From (18) and (19), y can be expressed in the form
cos 1 1 sin
1 cos
1 sin
cos cos
sin
21 1 2
2 2
2 1 1 2
2 2
2 2
2
z h
z y h
^{ (20) }
cos 1 1 sin
cos 1 1 sin
cos cos
sin
21 1 2
2 2
2
1 1 2
2 2
2 2
2
x h
x h
 (21)
on the real axis From (21) we obtain
1
^{2} ^{2}1 2
2 2 2
1 1 1 2
2
1
cos
sin
1 1
cot cos 4
x x h
x h dx
dy
^{ (22) }The values for the calculation are
00009 . 1
, 29 . 31
, 91 . 39
, 0
2 0 2 2
0 0
1
0
a
q U p
p
^{(Sonic)}For these calculations Bleakney and Taub (1949) paper has been used.
For these values Srivastava (1968) gives
^{} _{} _{} ^{}
^{}
cos 1
1 1
2 75
. 0 1
58272 . 0 4 5
. 0 1
31151 . 0 2
25 . 0 1
09153 . 0 4 1 2
2 1 0 1
0 1 1
1 1
1 12
1 1
1
x x x
x x D C x x
x e x x x
p
xwhere _{1} _{1}
1 1
1
1
tan z
t x x x
p y
p
^{}
 (23)
^{0} ^{.} ^{73114} ^{0} ^{.} ^{5010} ^{0} ^{.} ^{26885} ^{0} ^{.} ^{03039} ^{tan}
^{2}^{tan} ^{θ}
^{1}^{2}^{θ} ^{0} ^{.} ^{11300} ^{0} ^{.} ^{0215} ^{tan} ^{0} ^{.} ^{3685} ^{θ} ^{tan} ^{0} ^{.} ^{00930} ^{θ} ^{0} ^{.} ^{02617} ^{tan} ^{θ} ^{tan}
^{2}^{θ}
3 2
1
1
x p y
p
 (24)
^{1} ^{1}
θ cot
tan
_{2}2
2
Z
Z
 (25)
and
2
1
sin
1 cos Z
h
z
 (26)dx
1dy
is known from (22) andx
1p
is known from (23). Substituting these values in (16), curvature
^{ is }known.
On the reflected diffracted shock combining (17) and (18) we have
^{1}
2
sin
^{2}2
2
Z Z y
^{ (27) }From (19) when
Z , z
_{1} 1
^{ and when }Z 0 z ,
_{1}
From (27)
1 , 1
2
sin y
_{2} Z z
^{1}
^{and }
_{1}
2
, 0 2 0
sin y Z z
. The variation of2
2sin
y
on the diffracted shock is therefore0 2 sin
_{2}
y
(Point of intersection of diffracted shock and wall)to
1
2 sin
_{2}
y
(Point of intersection of diffracted shock and unit circle).We give below the table showing
,
(
being the curvature and is the bend) on the diffracted shock.Table1
2
2sin
y
0 0.30 0.4 0.6 0.8 1
∞2 . 6 10
^{55}0 . 09 10
^{30} 0 . 105 0 . 000534
^{0 }The perturbed velocities uand v behind the diffracted shock are given by
2 2 1 2 1
2 2
2
2 2 1 2 2 2 2
2 2
2
1 θ cos
sin 2
sin sin
1 sin θ sin
6 cos 5
V q a
U y
f
V y a
f q
y f y a y f q a u

(28)
2 2 1 2 2 1
2 1 2 2
2
2 2 1 2 2 2 2
2 2
2
1 sin sin 2
cos sin
1 cos sin sin 6 cos
5
V q a
V U a
y f
V y a
f q
y f y a y f q a v
205  P a g e
U ^{sin} ^{}
2q
1^{sin} ^{}
V
 (29)We have
(Vorticity)
y u x v 2
1
 (30)We find from (29) that
0
x
v
as all the terms on the right hand side of (29) are function of y. Therefore using(30) we have
y
u 2
1
2 2 1 2 1
2 2
2
2 2
2 2 2 1 2 2 2
1 cos sin 2
sin sin
sin cos 1
1 sin 12
5
V q a
U
q y V a
y a q f
2 2 1 2 1
2 2
2 2
2 2
2 2 2 1 2
2 2
1 cos sin 2
sin 6 sin
5
sin cos sin 1
6 5 2
1
V q a
q U
q y V a
a y q
f
y q B
a A y
f
_{2} _{2} _{2}2
2
sin 2 cos sin
2 sin 2
1
^{ (31) }In (31),
A
^{ and }B
are constants.From (12)
sin
^{3}
_{2}
f y  (32)From (31) and (32) we have
y q B
a
A
_{2} _{2} _{2}2 2
2
3
sin 2 cos sin
2 sin sin
2
1
We give below the table showing
( is the Vorticity and is the bend) over the diffracted shock.Table2
2
2sin
y
0 0.3 0.4 0.6 0.8 1
∞2 . 52 10
^{55}0 . 8 10
^{30} 0 . 07557 0 . 000297
^{0 }III. NUMERICAL CALCULATION 3.1 Curvature
The computed curvature for the data indicated in the paper is shown in Table1. The table shows that the curvature takes an infinite value at the point of intersection of the oblique diffracted shock and the wall. The curvature then falls and takes the negative values while approaching the point of intersection of oblique diffracted shock and Mach circle and finally it becomes zero at the intersection of Mach circle and reflected diffracted shock.
3.2 Vorticity
The computed vorticity distribution is shown in Table2. Here also we find that vorticity takes an infinite value at the point of intersection of diffracted shock and the wall. The vorticity then falls and take negative values while approaching the point of intersection of shock and Mach circle and finally it becomes zero at the intersection of shock and Mach circle.
IV. CONCLUSION
Development of theory concerning normal shock diffraction has been completely analyzed. But in the case of oblique shock diffraction (consisting of incident and reflected shocks) investigations are lacking. From that point, the work presented in the paper concerning curvature and vorticity assumes importance. It is expected that more work would flow after this work.
REFERENCES
[1]. Lighthill, M.J. The diffraction of blast1, Proc. Roy. Soc. A, 198,454470 (1949).
[2]. Srivastava, R.S. Diffraction of blast wave for the oblique case. British Aeronautical Research Council, Current Paper No. 1008 (1968).
[3]. Srivastava, R.S. and Chopra, M.G. Diffraction of blast wave for the oblique case, J. Fluid Mech 40, 4, 821831 (1970).
[4]. Srivastava, R.S. Interaction of Shock Waves, Kluwer Academic Publishers (1994).
[5]. Srivastava, R.S. On the starting point of curvature for reflected diffracted shock wave (subsonic and sonic cases), Shock Waves, 13, 323326 (2003).
[6].
Srivastava, R.S. Starting point of curvature for reflected diffracted shock wave, AIAAJ33, 2230 2231 (1995).
[7].