On waves generated on the surface of an incompressible fluid by a shock wave

ON WAVES GENERATED ON TEE SUWACE OF AN INCOMPIVESSIl5LE F’LUID BY A SBOCK WAVE (0 VOLNAKB. VOZBUZBDAE1YKA NA POVEBKflNOSTI NESZBIMAEMOI ZBIDKOSTI UDABNOX VOLNOI) PMM Vo1.24,

No.2, B. N.

(Received

1960,

pp.

240-248

RUMIANTSEV (Moscow) 10

September

1959)

We consider the two- and three-dimensional problems of motion of a fluid resulting from pressure applied to its surface, neglecting the effect of gravity. We investigate a number of self-similar solutions. When a blast occurs above the surface of a fluid (Fig. 1). then after a certain time the shock wave reaches the fluid and interacts with it. To determine the motion of the fluid and the gas, it is necessary to solve the problem simultaneously in both domains. However, considering the ratio of the densities of the two media, we may, as a first approximation. assume that. displacements of the fluid do not influence the brings us motion of the gas, which we suppose known. Such a formulation to the problem of determining the motion of a fluid due to a pressure applied to its surface which varies according to a known law. The analogously-formulated linear problem for a compressible fluid is considered in the paper [l I but the author restricts himself to pressure This assumption is justified fields. We suppose the fluid incompressible. in the case of air and water for pressures resulting from shock waves and nof exceeding

22 kg/cm.

1. The two-dimensional dimensional state

problem.

of equilibrium

problem.

In view of the fact there

exists

We first that

a velocity

consider

the two-

the motion starts potential

from a

$ which satisfies

the equation (1.1) in the domain of

flow

and the boundary condition

~(2,

0; t) = po(z, t)

&q/lh=O

on the x-axis

on the solid

344

:

boundaries

(1.2) (1.3)

Waves generated

on an incompreroible

fluid

by a shock

345

wawe

QJ”_ Solid boundary

-0

F

z

Fluid FIG. 1.

Here n is the direction of the normal to the solid boundary. If we use the Cauchy integral and ignore second-order terms and gravity, then condition (1.2) can be represented in the form

-ao = po(z, at

t) on the x-axis

(1.4)

We restrict ourselves to the consideration of three cases: a) 'lhe pressure due to the shock wave is constant; b) 'Ihepressure due to the shock wave is an arbitrary function of x/t; and c) 'Ihecase of a cylindrical blast on the surface of the water. a) Let a shock wave perpendicular to the wall CO travel along that wall with velocity V (Fig. 2). At 0 the shock wave encounters the free surface of a fluid of density p1 and moves along that surface so that the point A travels in the direction of the positive x-axis with velocity V csc a. ch the segment AB the pressure may be assumed to be zero. On OA the pressure is determined from the solution of the gasdynamical problem which is supposed known [2,3 1. The problem under consideration has self-similar solutions since the flow depends only on the dimensionless combinations c = x/Vt, q = y/Vt and the angles a and 8. In dimensionless coordinates the relations (1.11, (1.3) and (1.4) take the form

aa, -_== 0 aq

on the solid boundary

If we go over to complex variables by putting w = CD+ iY, then the boundary conditions (1.6) become

(l.i) z = C$+ iq,

B.N.

Ruuiant:tu

He (w - Cd?.@ / dc) = a, (61)

for 0 < ;I < 1 f: = =f)

Rc(w--

for 1 < il< 30(iI-i=)

Cdw/dC) = 0

We introduce a new analytic function W - w - [dap /d(. tion the boundary conditions (1.6) t&e the form Rt?W = CL~ (c,)

for

Rf?W = 0

0 < cl< I,

f1.S)

For this func-

for 1< :I < 03

(1.9)

Finally we introduce the variable r t 5’ which varies in the lowerright quadrant where k = n/2/3. I.& R = W,(r 1. Using the reflection principle we continue the function W,G 1 into the lower-left quadrant. As a result of symnetry relative to the ql-axis, condition (1.7) is automatically satisfied. Qk can now determine II, frem Schwarz’s forstula 1

w’,=

--A \

a&+-

(1.10)

I--_;

-1 Since in our case a1 = const, W has the form

To find the function &/df = t “C- it + iv), where u and tr are the horizontal and vertical velocity cmponents respectively, we must integrate the equation

After integration

‘IIEfget

(1.12) For k = 1, this integral

becomes &I =-z

,4 In& k-

If k is an integer greater than unity we can get the folltig formula: flU “a Ii -I)‘; ill (I -;- :) - lll(1 - ;I] _I$( -z-“-h!--1

general (1.14)

Stvtr

on an incorprs*tiblr

generated

where C, are integration

andk=

CllC

a 1+4+2tan-‘t--i:+4 --==----a & ( du:

a

v==--xi

+

(1--Cfe~,(ltc+f2)

Lln

I

2/3-(arc tg 2+

-

347

(1.15)

+

(1 - P)*

F)

tan-”

We now compute the velocity and (1.16) imply (u -i_ iz&

mave

for k = 2

2

dr.

by a *hock

to be determined from the condition of 3weget

constants

equilibrixsn at 00. Fork=2

flaid

+ xi] for

at the origin.

= UE 1(

i Q$)

k = 3

(l.lG)

Equations (1.131, (1.151,

(1.17)

=aa(1-iig?)

‘Lexus,the velocity at the origin is directed along the solid boundary. From (1.131, (1.15) and (1.16) we see that the horizontal velocity u is equal to ac in the high-pressure region and to zero in the region of zero pressure. lhe graphs of the dependence of the vertical velocity u1 = a - ’ Im da/d{ on [1 for k = 1, 2, 3 are sh
-G 6 - L3 FIG.

3.

lhe singularities at the origin for k = 1 and at 6 = 1 for all of k reflect the shortcomings of the linear theory, We compute the displacement S, of the particles

values

of the fluid

i

81 =

v

s

(u f ir) tft

(1.18)

0 If we express u and v in tern of 5 and caqute (1.18) we obtain the particle-displacement field for the fluid. We carry out the investigation only for points on the surface of the fluid. Clearly, the horizontal

348

BJV.

Ruriantseu

displacementis equal to acVt in the regionof high pressure and equal to zero in the region of zero pressure. We considerthe verticaldisplacementS*. Integrating(1.18)and introducingthe dimensionlessdistanceS = Sr/a~ Vt we get

The graphs of the functionss(c,) correspondingto these formulasare shown in Fig. 4. We note that for k > 1 the displacementfield has no singularities, b) When computingthe velocity field we assumedthat the pressurebehind the shock wave was constant.'lhisassumptionis valid for large and small incidenceanglesa (cf. Fig. a), i.e. for a = 0 and u J n/2. For intermediatevalues of a this is only approximatelycorrect (thus, it is shown in t]41 that when the incidenceangle of the shock wave is small, the increasedpressure is concentratedin a narrow region adjaining the shock wave). For a more accuratesolutionthe functional(cl)

FIG.

FIG.

4.

5.

which gives the distributionof pressureon the surfaceof the liquid must be determinede~ri~ntally, or it can be assumedto be known from the solutionof the gasdynamicalproblem of the reflectionof a shock wave from an arbitraryangle.We assume the functional(C1) to he known and we representit as a power series in (I ;, n,(iIf == 2 &n 71=* ~stituting (1.21)in fl.10)we get

(1.21)

Waves

generated

on

an

incorprcssible

fluid

by

a shock

mave

349

(1.22) Substituting

(1.22)

(1.11)

in

and differentiating

under the integral

sign we find dll. -Z-=---i d,

#l$ '

(1.23)

s

The constant of integration is determined from the condition of equiFor arbitrary integer values of k the inner integral librium at infinity. Thus, computation of the velocity field leads to elementary functions. for an arbitrary function al(&) reduces to quadratures. By way of example we consider one of the simplest distribution of pressure a,([,) = az[12, a2 = const. part of the pressure is concentrated in the vicinity Computing the integrals we get

cases of non-uniform Here, the greater of the shock wave.

for

li=

1

(1.24)

for

k= 2

(1.23)

Figure 5 give s the distributions of vertical (solid lines) and horizontal (broken lines) velocities computed from (1.24) and, for comparison, the corresponding curves in the case of uniform distribution of pressure. Substituting (1.24) ment of the surface

S=

and (1.25)

in (1.18)

we get the vertical

i(ln~+r,1*1~~-_‘ta~-lC,-~(:--.‘)-_

ln*$)

for

2:,)

for

0

Ii == 2

‘Ihe graphs of these functions appear in Fig. and (1.25) it is easy to compute the horizontal the same in both cases and has the value s’ = S,’ / Vt = fziE(1 -

displace-

0 < Cl < 1,

(1.2i)

4. Using Formulas (1.24) displacement S’. S’ is

S'=O

for

~
c) We assume that a cylindrical region of pressure is propagated ward from the point 0. Ihe pressure on the surface of the fluid is

out-

B.N. Rsriantrctt

350

supposedto vary in the mauner correspondingto a strong cylindrical blast with energy (l/2)B. In this case the flow dependson the parametersyS PI* ps8 E, x, Y, t, where y is the ratio of specificheats for the gas, pr is the densityof the fluid and pz is the density of the gas. Accordingtodimsnsiou theory the flow will depend on the dimensionless parameters

We shall restrictourselvesto the most iqmtaut case of a fluid of infinitedepth. &uation (1.5) holds in the 1-r half-plme occupiedby the fluid with the velocitypotentiald,given by E In dismsiouless coordinatesthe boundarycondition(1.4) on the surface takes the form (1.28) where h(e) is a knowu functionES 1 which describesthe distributionof pressurein dimensionlesscoordinatesin case of a strong blast and p2 is the pressurebehind a strong shock wave. If we introducethe coqlex variablesu=@+i41r, z- e+iu, theu condition(1.28)becomes Re(z$)=

Re(Zg)

-cl(E) for [Ej
= 0

for f$/>

1

(I .q

'lhus,to determinethe fuuctionzdw/dz it is necessaryto solve the Dirichletproblem.Using the Schwarz integralwe find 1

dw

1

(1.30)

dz=s;iz

This formla describesthe velocityfield.To get an approximateidea of this field we take a,(t) = u, a constant,which *lies an error only in the vicinityof the shock wave. Ihen (l.30)yields

Using the approximatefomla (1.31)it is easy to coaqmtethe displacementfield.The result is FIO. 6. =%Pl 4s= -~,@,t',

= - i(i$ :

in s-2)

(1.32)

llaves

generated

The graph of the vertical

displacements

face of the fluid based on this zontal displacements we get S’=n,-

1-

fluid

on an incompressible

E?

formula

by a shock

of the particles

is given

5’=

for I f I < 1,

in Fig.

Using

the approximate

of the sur6. For hori-

for ICI>1

0

We now compute the energy E, blast.

351

wave

acquired by the fluid formula (1.31) we obtain

as a result

of the

(1.33)

Ihus,

the magnitude of the energy

al to the ratio 2. tions

Ihe

of the densities

to the fluid

three-dimensional problem. Without

we consider

half-space

the problem of motion of a fluid

under pressure

half-space ing axial ordinates

imparted

the velocity

due to a point potential

changing which fills

in the gas.

the Laplace

on the surface

our assumpthe lower

In the lower

equation.

expressed

in polar

Assmco-

for Z = 0

PI (P9 4

at

blast

satisfies

symaetry the condition takes the form a9 --=

is proportion-

of the two media.

(2.1)

where pl@a, t) is a known fuuction. Let Q@, z; t) = d+/dt. Obviously, this function must also satisfy Laplace’s equation. Consider the zero order Ha&e1

transform ipo of Q UY(5,

Z; t) =

j r@((r,

z; l)J,(Er)

dr

IJ Multiplying

both sides

grating

with

respect

clusion

that Q” satisfies

of the Laplace

suitable

by rJO(&)

and inte-

at the well-kuonn

con-

the equation A5!!$42@0

whose solution

equation

to r from 0 to ODwe arrive

in cases when 9+

=1() 0 as z + - m has the form

a0 = A (5, t) $2 hkltiplying z = 0 in the

formula

for

(2.1) by rJ,,([r), integrating latter relation we find A@, t) the Hankel transform we get

with respect to r and putting and then using the inversion

(2.2) Formula (2.2)

gives

the pressure

field

in a fluid

due to au arbitrary

352

B.N.

surface pressure p1 (p,

t).

We now consider special

a) Let the function p1 (p, PI =

a1

Runiantseu

have the form

t)

for p < lit,

cases.

p1 == 0

for

p > Vt

(al,

V = const)

‘Ihis corresponds to a circular region of constant pressure which is propagated in all directions with constant velocity. In this formulation the problem is one with self-similar solutions, all flow parameters depending on the dimensionless quantities z1=

In dimensionless variables

the velocity

fi

potential

takes the form

‘p tp. 2; q = fJ2@ (p1, 21) The

dimensionless form of Equation (2.2)

We now expand Qb(p,, z,) first two terms @ fP1,

is

in powers of zr but restrict

211 =

@, (PI) -I- 21 @l (Plf -t

ourselves to the

*. *

(2.4)

Substituting (2.4) in (2.31, expanding the right side of (2.3) in a series of powers of zl, and equating coefficients, we obtain the following expressions for the terms of order zero and one: (2.5)

From (2.5) we find the horizontal surface of the fluid dQlo u=dpl=a From (2.6)

for p1 < 1,

velocity

of the particles

u=o

we find the corresponding vertical

for p1> 1

of the

(2.7)

velocity

‘Ihe arbitrary constant is chosen using the condition of equilibriws at infinity. The graph of vertical velocity based on this formula is given in Fig. ? (the dotted curve). We now compute the displacasmsmtof

Waves generated

the points distance

on an incompressible

on the surface

S = S,n’/

of the fluid.

fluid

by a shock

We denote by 5’ the dimensionless

Vt 2 a, where S, is the dimensional

quantity.

Further,

we denote by S1snd S* the horizontal and vertical displacements. (2.8) and (1.18) we get for the horizontal displacement S’=a ‘lhe

for

graph of vertical

as the dotted

b) We now assume that as would result fluid

S’=

p1 < I,

displacement

curve in Fig.

blast

‘Ihe velocity

potential function

The dimensionless

imparted

self-similar

1 and (1.18)

appears

%=:

po(p,

t 1 is such

0 on the surface

to the gas.

In this

with the flow depending

2

’

(~~pz~/~~/6

c$ is expressed

in the following

form of

p1>

of pressure

at the point

P

PIz (~~p2jbt%

for

based on (2.81

the distribution

with E/2 the amOunt of energy

dimensionless

0

Using

8.

from a strong

tion the problem is likewise dimensionless combinations

353

vave

’

7*

of the formulaon the

HfE Pl

in terms of the corresponding manner:

EQuation (2.2 ) becomes in this

case

1

a, EJ0

cw

KPd eF;zls rJo (Er) h (r) drdE 0

a=--Here h(r)

is a known function

8 PZ 5 l-l + 1) Pl which gives

the distribution

of pressure

for a strong blast I5 I. Its graph appears in Fig. 9. When h(r) is a complicated function, the inner integral is not expressed in terms of elementary

functions, As can be seen from Fig.

9, h(r)

be well approximated by putting b = 0.366. lhen (2.9) becomes

(

am

*+2Pl3p-z17jg

aat )

can

h(r)

=

=

co =

ab

s

e’*‘Jo EPI) JIG)

0

dS

(Z.l(J)

354

B.N.

Rumiantscv

As in the previousexamplewe obtain a differentialequation for 4 and aI by substituting(2.4) in (2.101,expandingthe integralin a series and equatingthe coefficientsof like powers of z,. Solving these differential-equat&s we get for the horizontalvelocityu = d@,&, on the surfaceof the fluid ab

(‘Z---

.'p;/l

for

p1<

24 = 0

I,

for

p1>

1

and for the verticalvelocityu1 = QIn2/crb. F1

2‘1 = -

*

(3-PTP

[5

(1 + PT

eossydp+

!’ 1 -pa

1 i_ pa

\ m-tin

-dp+L p

p%

I

0

(2.11)

0

The arbitraryconstantC is determinedfrom the conditionof finite energy impartedto the fluid.We find that C = 0. 'Ihegraph of the vertical velocityu1 appearsas the solid curve in Fig. 7. -al We introducethe dimensionlessdis-at tance s = f

s1

c) _;

FIG.

--I’+,

8.

'lhesolid curve in Fig. 8 is the graph of the dimensionlessvertical displacementobtainedby nuaericalintegrationusing (2.11) and (1.18). 'lhegraph in Fig. 9 shows that the line h(r) = b deviatesstrongly from the curve in the vicinityof the shock wave where on a small interval the pressure takes on large values.One cau take this fact into

9, 10

u FIG.

9.

considerationand so make the theoryjust developedmore precise;that is, we can assuae that the functionh,(r) = h(r) - b is of the form 1

hl (r) = cE (r -

I),

c =

I’ 0

[h(r)

-

b] dr

Favor

where 6(r -

generated

on an incorprcsriblc

fluid

by a shock

355

uave

1) is the Dirac delta function.

Let @ be the velocity potential the linearity of the problem

of the required motion. In view of

@=q)+@~ where Qa stands for the solution of the flow problem with uuifoxm pressure end ‘$ for the velocity potential of a flow due to a ringlike pressure zone. Since $, was computed earlier, it remains to determine $j. TO this end we substitute

h,(r)

in (2.9)

and obtain

(2.92)

In au analogous manuer we obtain the following horizontal and vertical velocities:

expressions for the

U&= 0

Using Formulas (2.13) and (1,181 we obtain the corresponding vertical displacement. Its graph (multiplied by a factor of b/c) appears as the dotted curve in Fig. 8. In case of a point blast it is Possible to compute the energy E1 imparted to the fluid. Using (2.111 we obtain, after comPuti.ngthe integral, the following formula

In conclusion I wish to thank N.N. Moiseev for valuable interest in this paper*

advice and

BIBLltOGRAPliY 1.

Bagdoev, A. G., Issledovanie zadachf o pronitanii davleniia v glub’ azhimaemoi zhidkosti (Investigation of the problem Qf Penetration of pressure into a compressible fluid). YJJtII. Yosk. univcrsitcta No. 3, 1957,

2.

Ludloff,

H.F., Aerodi~avika vtaryunykh voln. Y cb. atatti * Problcry *ekhanikis(Aerodyna*icr of Blast Waves; in the collection of articles aProblcvs of Mechanics*, (edited by R+ Mite5 and T. Karran) Izd-vo

inostr.

lit-rs,

1955.

356

3.

B.N.

Rumiantseu

Ludloff, H.F. and Friedman, H.B., of blast around finite corners,

4.

Ryzhov, O.S. and Khristianovich, bykh udarnykh voln (On nonlinear PMMVol. 22, No. 5, 1956.

5.

Sedov,

L. I.,

Similarity

Metody and

podobiia

Dimension

i in

Aerodynamics Vol. 22,

JAS

of blast -diffraction No. 1, pp. 27-34, 1955.

S.A.. 0 nelineinom otrazhenii slareflection of weak shock waves).

raznernosti

Mechanics).

u nekhanike

Gostekhizdat.

(Methods

of

1957.

Translated

by

A.S.