Propagation of Oblique Shock Waves in Troposphere

Propagation of oblique shock waves in the troposphere

By V. B . Sin g h

Terminal Ballistics Research LahoraJlry, Chandigarh AND

Pr e m Ku m a r

Indian Institute of Technology, Delhi {^Received 20 November 1970—Revised 21 April 1971)

Propagation o f the shock waves, when propagated at a constant angle to the vertical, in the troposphere, is discussed. In a special case, the results o f au earlier paper nve deduced.

In t r o d u c t io n

Propagation o f shock waves in the earth’s atmosphere in the vortical direction was discussed by Singh (19G9), in which the model o f Mitra (1952) was used witli minor changes. It was found that the shock velocity increases as the shock front propagates in the vertically upward direction. In the isotliei'inal part o f the at­

mosphere, the increase in shock velocity is smaller, but larger in layers whore t-ho temperature decreases. At a height o f 78 44 km the fluid velocity behind the ejiock front becomes greater than the escape velocity. In this paper, wo generalize the idea o f previous work for oblique shock waves. Propagation o f shock wave IS considered only in the troposphere. Curvature of the earth is riegleciod, so that density, pressure and temperature of the atmosphere are varying in planes parallel to the surface ol the earth. We consider the component o f the shock front making angle 0 to the vertical, so that the planes of variation o f thermody­

namical parameters are inclined at an angle 0 to the shock surface which is assumed to be plane

It is found that the rate o f increase o f shock velocity decreases as 0 increases, being the maximum in vertical direction. In figure 2, the variation o f shock velocity is shown for 0 = 0, tt/G, tt/4 as the shock projiagates in the troposphere.

In figure 3 the shock envelope at different interv^als o f time is drawm.

Fo r m u l a t io n o f t h e p r o b l e m

If we take the point o f explosion as origin and denote it by 0 , r^ the radius vector, 0 the angle made by the radius vector to the vertical and g^^ the acceleration due to gravity at a distance from the origin, then

yo=

(

^0+^0 cos

whore gg the value of g^ at the surface o f the earth, and Rq the radius of the earth 564

We assume that there is an explosion at the point 0 , which is on the surface of the eaith. Shock wave created by the explosion propagates in all directions.

We consider the component o f the shock which propagates along the radius vector

?*() and inclined at an angle 0 to the vertical. For simplicity we assume that the shock front is plane.

Let ps, Tg, and Pq, pressure, density and absolute temperature at the surface of the earth and at a radial distance r^, and u, U, p, p, r, K, t, and f/ bo the dimensionless fluid velocity, shock velocity, gas pressure, density, radius vector, earth s radius, time, and acceleration due to gravity^ given by ISingh (1069) in equation (4). Then equations o f motion in terms o f dimensionless para­

meters are given by

Propagation of oblique shock waves in the troposphere

t+ ⁴ ;+ 's -»

dll , du , I dp dt ^ d r ^ p dr

B

cos 0)²cos 0 — 0 (

2

[u equilibrium conditions the variation of pressure in the atmosphere is given by _ _ ___ \ ® cos 0 .

P ~ ' ■

dp _ _ I R Y ^

P \ cos 0 / T dr (3)

wliere T is the dimensionless absolute temperature such that at the surface o f the earth

p, = RpsTa (4)

R being the gas constant. Here we have assumed that there is no wind in the troposphere

When shock created at point 0 roaches the distance jump conditions across the shock front at r = ^ are, (Pai, 1959, Singh 1909)

p.i{u^-U) = - p U P i - P =

^ _ y . _ 2 r - 1 p

y Pzj^\

r-1 P 2

(6⁾

where pg, p^, w, are the values o f p, p, u behind the shock front and y being adi- batic gas constant. Quantities beliind the shock front are functions o f i and t whereas, ahead the shock front they are functions o f ^ alone, i being the shock position. It is assumed that fluid velocity u in front of the shock is zero. Solving relations (6) for Wg, pg, p^ in terms o f Mach number M, defined as

M ^U K ypIp)^

we got

Pi(i, n_ (y + l)p lf»

9(M)

(6)

where

f{M ) ^ I 2 i f a - Z i : l |^ - n 1 7 g{M) = { 2 + ( y - l ) J / 2 }

c 2 = y p j p

(7)

To visualize the problem we must know four unlmovm variables u.j,, p,, p..aml M in terms o f p , p To relate these four variables wo have throe jump conditions (6). One extra relation between these four variables is required, which we get fiorn the rule deviced by Whitham (1958). According to the rule, we write equa­

tions o f motion just behind the shock front along the positive characteristic axis In this equation wo substitute the expressions for u^, Pg, po from (6).

Equation o f motion along the positive characteristic axis, just behind the shock front is

^Pt+PiC^‘U^+ Pf-K- { ----? --- \*oos 0 df = 0

«2+c* \ ii+ f cos

/

whoro f is the shock position. Using relations (0) in (8) and after some aimpli- fications we get,

Jyi p c

, / R y { y + l ) ‘ M‘ h(M) WB 6d(

\ n + $ cos 0 I ‘c={2(Jlf“- l ) + V^t^ } '

where, _ J y f{^ )

V g{M)

(0)

(10)

Relation (9) gives the variation of M, if p and c are known. I f the variation of absolute temperature is known, the pressure can be found from (3) and thus M can be calculated from (9),

Propagation of oblique shock waves in the troposphere

Wo discuss below the x^arialion of M as the shook propagates in the tropos­

phere. As the height of troposphere is very small compared to the radius of the earth, our assumption, that the earth’s surface round the location is plane, is justified.

Sh o c k w a v e s i n t h e t r o f o s p h e e e

In the troposphere, which extends iipto a height of 10 kilometers above the equator, the gases are in adiabatic equilibrium fn this region the variation of pressure and the equation of state in terms of dimensionloss parameters p, p.

can be writtes as

1 dp ( It V n

— = ■— ( --- ^ ) cos 6 p dr \ cos 0 f

p =

Combining (11) and (12) we get after some simplifications 7 - ! = - / ____

^ \ R-Vr eoa d j

(1 1)

(12

(13)

Integrating (13) from the surface of the earth, whore r = 0, p = \ to the oiiiTent point r, where density is p, wo get,

- - [i y - i ^ r c o B g (14^)

^ L y cos 0 J

From this we can easily get pressure p and temperature T as follows y _ l R r cos d I 7—1y

— [ i —Zir — —

~y ^4*rcos0 1 ’

T = [ i - Z z l

L r i J + r o o s ^ J

(Ub)

(14c)

Substituting the values of p and c at r = | from equations (14) and (17) in equation (9) after some simplifications we got

2 z

_ _ _ _ K,(M) f if e o s e 1 L ^ y cos B I

(15)

where

K^(M) = K y ( U ) + y ^

(16a)

U M ) -

I [ y + y ? ! y M % ( M)

... (15b)

K (M) = (16c)

In figure ], we have drawn the value o f K^(M) versus M, lor y — 1.4. It is found that the variation o f K^(M) is small for M ^ S. as compared to the varia­

tions o f M.

fitAC'/ KJMiJf/t A

Figure 1. Variation o f K^J^M) versus M .

From figure 1, it ie seen that K^{M) increases sharply as M increases from 1 to 3, having values 0 and 0.5150 at i f = 1 and 3, respectively. When J f increases from 3 onward, it’s variation becomes negligible. We take the value of M at the surface o f earth as equal to 4. Thus if wo take K^{M), which no doubt is a function of i f , as a constant and evaluate i f from equation (15), the error is less than 2% This error decreases as i f increases. This fact can easily-be seen from figure 1, where the curve showing the variation of i i3( if) , becomes a straight line as M becomes greater than 4.

Thus we neglect the variations in K^{M) in the process o f integration o f the equation (15). Integrating (15) and taking K^(M) as a constant during integra­

tion, wo get,

M = i f . [ 1 - ^

L r 5+f ⁰ j

^{— ^}

where Ms is the values o f M at | = 0. From (16), (14) and the definition o f M, we get shock velocity U, given by,

Propagation of oblique shock waves^ in the troposphere

u = \ l- t z l

*• y is+f oos e J

Relations (16) and (17) give the variation of Mach number and the shock velocity as the shook propagates along the radial distance at an angle 0 to tho vertical.

In figure 2, we have drawn the variation o f shock velocity versus dimensionless

fJAD/Al DIS-'AI^U

F i g u r i ' 2 V a r i a t i o n o f p h o o k v e l o c i i v U oersits t h o r a d i a l d iK ta n c o

shock position f for 6^ = 0, njQ, ;r/4. It is seen that the rate o f increase o f shock velocity decreases with the increase of 0

Equation (17) can be written as.

A f = f 1 - t - - ' ) / *

L y R+f

6 J

where

(18)

iL = u dl

From (18) we can compute the distance ^ at a particular time t. In figure 3, we have drawn the shook envelop at different time intervals after the explosion.

All the figures are drawn in the dimensionless parameters.

OONOLUSION

In figure 2 we have plotted the variation of shock velocity in non-dimen- aional form. Here we have used the following data to nondimensionalize. gr, — 981 cm/aec2, p, = 10« dyiies/cm^, = 1 ,3 x 1 0 “ ® gm/om®, a 1276.3x10“ ® cm“ ^ From computations we find that the shook velocity at a distance 9.4 km is 1.807,

Figure 3. Positiorx ol' shock front at dimensionless times 0.02, 0 1, 0.2 the origin being the point o f explosion.

1.783, 1.712, 1.614 km/sec for 6 — 0,tt/12, tt/6 tt/4, respectively. Tt is seen that the rate o f increase o f shock velocity is less for larger values o f 0.

Ill liguro 3 we have drawn the shock envelope at time intervals of 0.5665, 2.8275, 5.6550 seconds, respectively. It is shown that in a specific time the distance covered by the shock wave decreases as 6 increases from 0 to tt/2, being maximum for 6 — 0. In all the above calculations, the initial shock velocity is taken to be 1.313 km/soc

AoILN OW LBDG E M B N T

One o f us (VPS) is thankful to Col. G. S. Sawhney, Director, Terminal Ballistic Research Laboratory Chandigarh, for his permission to publish the work. Thanks arc due to the referee for very useful comments.

Re f e b b n o b

Mifcra S. K. 1962 U pper A tm osphere, Asiatic Society, Calcutta.

Pai S. I 1959 Introduction to the T heory o f Compressible Flow , D . Van Nostrand Co., N.Y.

Smgh V. P. 1969 In d ia n J. P h ys. 43, 619.

Whitham G. B. 1958 J. F lu id M ech. 4. 337.