Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 2, pp. 563-575, Warsaw 2012

50th Anniversary of JTAM

SIMILARITY SOLUTIONS FOR A CYLINDRICAL SHOCK

WAVE IN A ROTATIONAL AXISYMMETRIC GAS FLOW

J.P. Vishwakarama

Prerana Pathak

D.D.U. Gorakhpur University, Department of Mathematics and Statistics, Gorakhpur, India

e-mail: jpv univgkp@yahoo.com; preranapathak18@gmail.com

Similarity solutions are obtained for an adiabatic flowbehind a cylindri-
cal shockwavepropagating in a rotational axisymmetric flowof aperfect
gas, in which initial velocity and density are functions of the distance
from the axis of symmetry. The initial medium is considered to have a
variable azimuthal velocity component in addition to the variable axial
velocity. Initial velocities and density are assumed to obey power laws.
Distributions of the fluid velocities, density, pressure and vorticity com-
ponents are obtained in the flow-field behind the shock front. Effects of
variable initial velocities anddensity and thevariationof the shock-Mach
number are investigated.

Key words: shock wave, axisymmetric flow, rotating medium, adiabatic
flow

1. Introduction

The formulation of self-similar problems and examples describing adiabatic
motion of non-rotating gas models of stars, are considered by Sedov (1959),
Zel’dovich andRaizer (1967), Lee andChen (1968) and Summers (1975). Ro-
tation of stars significantly affects theprocess takingplace in their outer layers.
Therefore, question connectedwith the explosions in rotating gas atmospheres
are of definite astrophysical interest.Chaturani (1970) studied thepropagation
of cylindrical shockwaves through a gas having solid body rotation, and obta-
ined the solutions by a similarity method adopted by Sakurai (1956). Nath et
al. (1991) obtained the similarity solutions for the flow behind spherical shock
waves propagating in a non-uniform rotating interplanetary atmosphere with
increasing energy. Recently, Vishwakarma andVishwakarma (2007) andVish-
wakarma et al. (2007) obtained the similarity solution formagnetogasdynamic



564 J.P. Vishwakarama, P. Pathak

cylindrical shock waves propagating in a rotating medium which is a perfect
gas with variable density or a non-ideal gas with constant density. In all of
theworksmentioned above, the ambientmedium is supposed to have only one
component of velocity, that is the azimuthal component.
In the present work, we obtained the self-similar solutions for the flow be-

hinda cylindrical shockwave propagating in a rotational axisymmetric perfect
gas flow which has a variable azimuthal fluid velocity together with a varia-
ble axial fluid velocity (Levin and Skopina, 2004). The shock-Mach number
is not infinite, but has a finite value. The fluid velocities and the density in
the ambientmediumare assumed to obey power laws. It is expected that such
fluid velocity anddensitymay occur in the atmosphere of rotating planets and
stars.
Distribution of the velocities, density, pressure and vorticity components

in the flow-field behind the shock are obtained. The effects of change in the
indexof variation of initial velocity of themedium, the indexof variation of the
initial density and the variation of the shock-Mack number are investigated.

2. Basic equations and boundary conditions

The fundamental equations governing an unsteady adiabatic axisymmetric
rotational flow of a perfect gas, in which heat conduction and viscous stress
are negligible, are (Levin and Skopina, 2004)

∂ρ

∂t
+u
∂ρ

∂r
+ρ
∂u

∂r
+
ρu

r
=0

∂u

∂t
+u
∂u

∂r
+
1

ρ

∂p

∂r
−
v2

r
=0

∂v

∂t
+u
∂v

∂r
+
uv

r
=0

∂w

∂t
+u
∂w

∂r
=0

∂

∂t

( p

ργ

)

+u
∂

∂r

( p

ργ

)

=0

(2.1)

where ρ, p are density and pressure; u, v and w are the radial, azimuthal and
axial components of the fluid velocity q in the cylindrical coordinates (r,θ,z);
t is time and γ is the ratio of the specific heats of the gas. Also

v= rK (2.2)

where K is the angular velocity of the medium at the radial distance r from
the axis of symmetry. In this case, the vorticity vector ζ = 1/2 Curl q has
the components



Similarity solutions for a cylindrical shock wave... 565

ζr =0 ζθ =−
1

2

∂w

∂r
ζz =

1

2r

∂rv

∂r
(2.3)

The above system of equations should be supplemented with an equation
of state. Perfect gas behaviour of the medium is assumed, so that

p=ΓρT Um =CvT =
p

ρ(γ−1)
(2.4)

where Γ is the gas constant, T – temperature, Um – internal energy per unit
mass of the gas and Cv =Γ/(γ−1) is the specific heat at constant volume.
We assume that a cylindrical shock wave is propagating outwards from

the axis of symmetry in a perfect gas with variable density, which has zero
radial velocity, variable azimuthal velocity and variable axial velocity. Jump
conductions across the moving shock are

u1 =(1−β)Ṙ v1 = v0 w1 =w0

ρ1 =
ρ0
β

p1 = p0+(1−β)ρ0Ṙ
2

(2.5)

where Ṙ = dR/dt denotes the shock velocity, R is the shock radius and
subscripts 0 and 1 refer to the values just ahead and just behind the shock,
respectively. The quantity β is given by the equation

β=
γ−1+2M−2

γ+1
(2.6)

where M, the shock-Machnumber referred to the speedof sound in theperfect
gas
√

γp0/ρ0, is given by

M =
Ṙ
√

γp0
ρ0

(2.7)

Following Levin andSkopina (2004), we obtain the jumpconditions for the
components of the vorticity vector across the shock as

ζθ1 =
1

β
ζθ0 ζz1 =

1

β
ζz0 (2.8)

Ahead of the shock, the velocity components and the density are assumed
to vary as

u0 =0 v0 =BR
b

w0 =AR
a ρ0 =DR

d
(2.9)

where a, b, d,A,B,D are constants.



566 J.P. Vishwakarama, P. Pathak

Therefore, from (2.1)

p0 =
B2D

2b+d
R2b+d 2b+d> 0 (2.10)

Ahead of the shock, the components of the vorticity vector, therefore, vary
as

ζr0 =0 ζθ0 =−
aARa−1

2
ζz0 =

(b+1)BRb−1

2
(2.11)

From equations (2.2) and (2.9), we find that the initial angular velocity
varies as

K0 =BR
b−1 (2.12)

It decreases as the distance from the axis increases, if b−1< 0.
The total energy E of the flow-field behind the shock is not constant, but

assumed to be time dependent and varying as (Rogers, 1958; Freeman, 1968;
Director and Dabora, 1977)

E=E0t
s s­ 0 (2.13)

where E0 and s are constants. Thepositive values of s correspond to the class
in which the total energy increases with time. This increase can be achieved
by the pressure exerted on the fluidby an expanding surface (a contact surface
or a piston). This surfacemay be, physically, the surface of a stellar corona or
condensed explosives or a diaphragm containing a very high-pressure driver
gas. By sudden expansion of the stellar corona or the detonation products or
the driver gas into the ambient gas, a shock wave is produced in the ambient
gas. The shocked gas is separated from this expanding surface, which is a
contact discontinuity. This contact surface acts as a ’piston’ for the shock
wave. Thus the flow is headed by a shock front and has an expanding surface
as the inner boundary.

3. Self-similarity transformations

We introduce the following similarity transformations to reduce the equations
ofmotion into ordinary differential equations (Vishwakarma andYadav, 2003)

u= ṘU(η) v= ṘV (η) w= ṘW(η)

ρ= ρ0g(η) p= ρ0Ṙ
2P(η)

(3.1)



Similarity solutions for a cylindrical shock wave... 567

where U,V ,W , g and P are functions of the non-dimensional variable η only,
and

η=
r

R
(3.2)

The total energy of the perfect gas behind the shock is given by

E =2π

R
∫

Rp

[1

2
ρ(u2+v2+w2)+

p

γ−1

]

r dr=E0t
s (3.3)

where Rp is the radius of the inner expanding surface. Applying similarity
transformations (3.1) to (3.2) in relation (3.3), we find that themotion of the
shock front is given by the equation

E0t
s

2πJ
=R2Ṙ2ρ0 (3.4)

where

J =

1
∫

ηp

[1

2
g(U2+V 2+W2)+

P

γ−1

]

η dη (3.5)

ηp being the value of η at the inner expanding surface.
For similarity solutions, the shock-Mach number M (whichoccurs in shock

conditions (3.11)) must be a constant parameter. Therefore, from (2.7)

Ṙ=QRb (3.6)

where Q is a constant. From equation (3.6), on integration

R= [(1−b)Q]
1

1−bt
1

1−b (3.7)

Also, from equation (3.4), we get

R=
( E0
2πJ

)

1

2b+d+2
( 1

Q2d

)

1

2b+d+2
t

s
2b+d+2 (3.8)

Comparing equations (3.7) and (3.8), we get

2b+d= s(1− b)−2 (3.9)

From equations (2.9), (2.10), (3.6) and (2.7), we obtain

B

Q
=
1

M

√

2b+d

γ
(3.10)



568 J.P. Vishwakarama, P. Pathak

Then, shock conditions (2.5) are transformed into

U(1)= 1−β V (1)=
1

M

√

2b+d

γ
W(1)=

A

Q

g(1)=
1

β
P(1)= 1

γM2
+1−β

(3.11)

where a= b.
The condition to be satisfied at the inner boundary surface is that the

velocity of the fluid is equal to the velocity of inner boundary itself. This
kinematic condition from (3.1) and (3.2) can be written as

U(ηp)= ηp (3.12)

Using transformations (3.1), equations of motion (2.1) take the form

− (η−U)g′+gU ′+g
(

d+
U

η

)

=0

− (η−U)U ′+ bU+
P ′

g
−
V 2

η
=0

− (η−U)V ′+V
(

b+
U

η

)

=0

− (η−U)W ′+bW =0

− (η−U)P ′+
(η−U)Pγg′

g
+Pd+2bP −Pγd=0

(3.13)

where the prime denotes differentiation with respect to η. From equations
(3.13), we have

U ′ =
1

γP − (η−U)

[v2g(η−U)

η
−
PγU

η
− b(η−U)Ug−Pd−2bP

]

V ′ =
V

η−U

(

b+
U

η

)

W ′ =
bW

η−U
(3.14)

g′ =
g

η−U

[

U ′+
(

d+
U

η

)]

P ′ =
V ′g

η
− bUg+(η−U)gU ′

Also, applying the similarity transformations on equations (2.3), we obtain the
non-dimensional components of the vorticity vector

ℓr =
ζr

Ṙ/R
ℓθ =

ζθ

Ṙ/R
ℓz =

ζz

Ṙ/R



Similarity solutions for a cylindrical shock wave... 569

in the flow-field behind the shock as

ℓr =0 ℓθ =−
1

2

bW

η−U
ℓz =

V

2

(1

η
+
b+U/η

η−U

)

(3.15)

Ordinary differential equations (3.14) with boundary conditions (3.11) can
now be numerically integrated to obtain the solution for the flow behind the
shock surface.

4. Results and discussion

Similarity considerations lead to the following relations among the constants
a, b, d and s

a= b 2b+d= s(1− b)−2 2d+d> 0 (4.1)

Then the following three cases may exist.

(i) The increasing velocity shock (b> 0),

(ii) the constant velocity shock (b=0), and

(iii) the decreasing velocity shock (b< 0).

Therefore, for the purpose of numerical calculations, we choose the follo-
wing three sets of values of the constants

(i) b=0.5, d=−0.5, s=5;

(ii) b=0, d=0.5, s=5/2;

(iii) b=−0.5, d=1.5, s=5/3.

The solutions to differential equations (3.14) with boundary conditions
(3.11) depend upon five constant parameters γ, M, A/Q, b and d. Numeri-
cal integration of these differential equations is performed to obtain the non-
dimensional variables U, V , W , g, P, by using the Runge-Kutta method of
the order four, for γ = 1.4; M2 = 5, 10; A/Q = 0.1; b = 0.5, 0, −0.5 and
d=−0.5, 0.5, 1.5.

The profiles of U, V , W , g and P are shown in Figs. 1 to 5. Also, the
non-zero and non-dimensional components ℓθ and ℓz of the vorticity vector



570 J.P. Vishwakarama, P. Pathak

Fig. 1. Variation of the non-dimensional radial velocity U in the flow-field behind
the shock front

Fig. 2. Variation of the non-dimensional azimuthal velocity V in the flow-field
behind the shock front

Fig. 3. Variation of the non-dimensional axial velocity W in the flow-field behind
the shock front



Similarity solutions for a cylindrical shock wave... 571

Fig. 4. Variation of the non-dimensional density g in the flow-field behind the shock
front

Fig. 5. Variation of the non-dimensional pressure P in the flow-field behind the
shock front

Fig. 6. Variation of the non-dimensional azimuthal component of vorticity ℓθ in the
flow-field behind the shock front



572 J.P. Vishwakarama, P. Pathak

Fig. 7. Variation of the non-dimensional axial component of vorticity ℓz in the
flow-field behind the shock front

are plotted inFigs.6 and 7, respectively. Values of ηp (the position of the inner
expanding surface) are shown in table 1 for different cases.

Figures 1 and 2 show that the non-dimensional radial velocity U incre-
ases from the shock front to the inner expanding surface, whereas the non-
dimensional azimuthal velocity V decreases. Figure 3 shows that the non-
dimensional axial velocity W increases, remains constant or decreases from
the shock front to the inner surface according as b < 0, b = 0 or b > 0, i.e.
according as the shock velocity (or initial azimuthal fluid velocity) is decre-
asing remaining constant or increasing with the shock radius. Figures 4 and 5
show that the non-dimensional density g and pressure P increase or decrease
from the shock front to the inner expanding surface according as b > 0 or
b< 0 (i.e. according as the shock velocity is increasing or decreasing with the
shock radius). Figure 6 shows that the non-dimensional azimuthal component
of vorticity ℓθ increases behind the shock front and tends to infinity at the
inner surface, remains zero or decreases behind the shock front and tends to
negative infinity at the inner surface according as b< 0, b=0 or b> 0. Figu-
re 7 shows that the non-dimensional axial component of vorticity ℓz increases
behind the shock front and tends to infinity within a narrow region at the
inner surface.

From Table 1 and Figs. 1 to 7, it is found that the effects of the increase
in the value of M2 are

• to increase ηp, i.e. to decrease thedistance of the inner expanding surface
from the shock front. Physically, it means that the gas behind the shock
is compressed, i.e. the shock strength is increased (see Table 1);



Similarity solutions for a cylindrical shock wave... 573

• to increase the radial velocity U and to decrease the azimuthal veloci-
ty V at any point in the flow-field behind the shock (see Figs. 1 and 2);

• todecrease (or increase) theaxial velocity W when the indexof variation
of the initial azimuthal velocity b has a positive (or negative) value (see
Fig.3); and

• to increase density g and pressure P when b is positive (see Figs.4
and 5).

Table 1. Position of the inner expanding surface ηp for γ =1.4, A/Q=0.1
and various values of M2, b and d

M2 b d ηp

0.5 −0.5 0.8676
5 0 0.5 0.8538
−0.5 1.5 0.8419

0.5 −0.5 0.9018
10 0 0.5 0.8917

−0.5 1.5 0.8831

The effects of the increase in the value of the index of variation of the
initial azimuthal velocity b of the medium are

• to decrease the distance of the inner expanding surface from the shock
front, i.e. to increase the shock strength (see Table 1);

• to increase the radial velocity U and to decrease the azimuthal velo-
city V and axial velocity W at any point in the flow-field behind the
shock (see Figs.1 to 3);

• to increase thedensity g and thepressure P at anypoint in theflow-field
behind the shock (see Figs. 4 and 5);

• to decrease ℓθ and to increase ℓz behind the shock (see Figs. 6 and 7);
and

• to increase the shock velocity (see equation (3.16)).

The effects of an increase in the value of the index of variation of the
density d are

• to increase the distance of the inner expanding surface from the shock
front (see Table 1);

• todecrease the radial velocityUand to increase theazimuthal velocity V
and the axial velocity W at any point in the flow-field behind the shock
front (see Figs. 1 to 3);



574 J.P. Vishwakarama, P. Pathak

• to decrease the density g and pressure P at any point in the flow-field
behind the shock (see Figs. 4 and 5); and

• to increase the vorticity components ℓθ and to decrease ℓz (see Figs.6
and 7).

References

1. Chaturani P., 1970, Strong cylindrical shocks in a rotating gas, Appl. Sci.
Res., 23, 1, 197-211

2. DirectorM.N.,DaboraE.K., 1977,Anexperiment investigationofvariable
energy blast waves,Acta. Astronaut., 4, 391-407

3. Freeman R.A., 1968, Variable energy blast waves, J. Phys., D1, 1697-1710

4. LeeT.S., ChenT., 1968,Hydromagnetic interplanetary shockwaves,Planet.
Space Sci., 16, 1483-1502

5. Levin V.A., SkopinaG.A., 2004,Detonationwave propagation in rotational
gas flows, J. Appl. Mech. Tech. Phys., 45, 457-460

6. Nath O., Ojha S.N., Takhar H.S., 1991, A study of stellar point explosion
in a self-gravitating radiativemagnetohydrodynamicmedium,Astrophys. Space
Sci., 183, 135-145

7. Rogers M.H., 1958, Similarity flows behind strong shock waves, Quart. J.
Mech. Appl. Math., 11, 411-422

8. Sakurai A., 1956, Propagation of spherical shock waves in stars, J. Fluid
Mech., 1, 436-453

9. Sedov L.I., 1959, Similarity and Dimensional Methods in Mechanics, Acade-
mic Press, London

10. Summers D., 1975, An idealised models of a magnetohydrodynamics spheri-
cal blast waves applied to a flare produced shock in the solar wind, Astron.
Astrophys., 45, 151-158

11. VishwakarmaJ.P.,MauryaA.K., SinghK.K., 2007,Self-similar adiabatic
flowheaded by amagnetogasdynamic cylindrical shockwave in a rotating non-
ideal gas,Geophys. Astrophys. Fluid Dyn., 101, 155-168

12. VishwakarmaJ.P.,VishwakarmaS., 2007,Magnetogasdynamiccylindrical
shockwaves in a rotating gaswith variable density, Int. J. Appl. Mech. Engng.,
12, 283-297

13. Vishwakarma J.P., Yadav A.K., 2003, Self-similar analytical solutions for
blast waves in inhomogeneous atmospheres with frozen-in-magnetic field,Eur.
Phys. J.,B 34, 247-253



Similarity solutions for a cylindrical shock wave... 575

14. Zel’dovich Ya.B., Raizer Yu.P., 1967, Physics of Shock Waves and High
Temperature Hydrodynamic Phenomena, II, Academic Press

Rozwiązania automorficzne dla cylindrycznej fali uderzeniowej przy

rotacyjnym osiowo-symetrycznym przepływie gazu

Streszczenie

Wpracyprzedstawiono rozwiązaniaautomorficzne otrzymanedla adiabatycznego
przepływu czynnika za cylindryczną fala uderzeniową rozchodzącą się w rotacyjnym,
osiowo-symetrycznym opływie gazu doskonałego, którego prędkość początkowa oraz
gęstość są funkcjami odległości od osi symetrii. W stanie początkowym, analizowa-
ny czynnik posiada oprócz zmiennej składowej osiowej prędkości dodatkowo zmienną
składowąazymutalną.Założono, że prędkościpoczątkowe i gęstośćopisują funkcje po-
tęgowe. Rozkłady prędkości płynu, gęstość, ciśnienie oraz składowe wirowości otrzy-
manowpolu przepływuza frontem fali uderzeniowej. Zbadano takżewpływprędkości
początkowych oraz gęstości na zmiany wartości liczbyMacha.

Manuscript received April 8, 2011; accepted for print October 26, 2011