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NEW YORK lJ^aVERSmr IMM-NYU 167

WSTTTUTE CF MATl lET.IATICAL SOÂ£HC3tS

LIURARY

2S WmwV PLcc, New YoA 3, KX.

3

3

DECAYING SHOCKS

A Comparison of an Approximate Analytic Solution

With a Finite Difference Method

[^ Prepared under

Navy Contract N6ori-201 Task Order No. 1

\ By the

. Institute for Mathematics and Mechanics

Sj. New York University

October, 1947

PREFACE

The present report by Miss Anneli Leopold,

prepared under Navy Contract N6orl-201, Task

Order No. 1, supplements Dr. Friedrichs' paper

Formation and Decay of Shock Waves report

IMM-NYU 158. It discusses the range of validity

of the numerical approximation given in the

latter report for the treatment of decaying shock

waves.

Miss Leopold was greatly assisted in the

numerical computations by Mr. John Butler.

R. Courant

DaCAYINQ SHOCKS

A Comparison of an Approximate Analytic Solution

with a Finite Difference Method.

INTRODUCTION .

In report IMM-NYU 158, Formation and Decay of

Shock Waves by K. 0. Friedrichs, an approximate method

was developed to determine the behavior of a decaying

shock wave in a gas, i.e. the behavior of a shock wave

weakened by a rarefaction wave overtaking it. This

method is expected to yield rather accurate results In

the case of weak shocks; it is of interest to deter-

mine for what range of shock strengths its use may be

Justified, and beyond what shock strength a more exact

solution is required.

In order to test the validity of the "approximate"

method by comparison with an "accurate" result, a corre-

sponding problem for surface waves in shallow water is

treated first by the "approximate" method of IMM-NYU 158,

and then by a finite difference scheme applicable to

shocks of any strength. The accuracy of the solution

obtained by the latter scheme is limited only by the size

of the mesh used. The discussion of a problem concerning

water wave phenomena rather than gas dynamics proper Is

advantageous; for, while the differential equations are

of the same form in both cases, there are no entropy

changes in water so that the finite difference scheme

is simplified considerably.

- 2 -

For every shock strength considered, a comparison

of the results obtained by both methods determines the

accuracy of the "approximate" method. The shock strengths

treated here. I.e. the ratios of height h^ of the

moving water to the height of h^ of the still water

(see fig. 1), are: 1.2, 1.5, 1.8, 3.0 respectively. The

results show a very good agreement of the two methods of

^1

solution for all these cases, even for ^ = 3.0 which is

o

a stronger shock than those usually classed among weak

shocks.

The formulation of the physical problem Is as follows;

A shock wave - or bore as It Is also called - moves with

constant speed U Into still water of uniform depth h^.

The height of the water behind the shock Is h^^, and Its

velocity Is u^ (see fig. 1).

1.

a.

Figure 1

At a certain time t = 0, a vertical wall Is suddenly In-

serted into the moving water at a point x = 0, so that

a depression (or rarefaction) wave is created which will

catch up with the shock, since such disturbances are propa-

gated with a speed greater than that of the shock (see

pages 5 and 6). Thus the shock will gradually be weakened.

* Such a bore may be created by pushing a rigid vertical

plate through the water at constant speed.

- 3 -

Figures 2 and 3 describe the physical situation at two

Instances after the insertion of the wall.

I

Plp^ure 2

UqU

Figure 3

In section 1 of this report, the "approximate" method

will be discussed and applied to the problem as formulated

here. In section 2, the finite difference method is de-

scribed and used. The graphs II and III show the results

of both methods.

A noteworthy result of these considerations is that

strong shocks weaken very rapidly at first. This suggests

the following treatment of strong shocks: One uses the

finite difference method initially but finds, after rela-

tively few steps, that the shock has weakened sufficiently

to justify the use of the "approximate" method in con-

tinuing the solution. Thus the method of IMM-NYU 158 is

an extremely useful one even in cases where shocks are

strong.

- 4 -

Let g be the acceleration due to gravity, h the

depth of the water, x the distance from the wall and t

the time after the Insertion of the wall. Let x^ be

the distance between the wall and the discontinuity at

the time the wall is inserted. I.e. at t = 0. We, use

the following definitions in order to work with dimension-

less units:

u "Vgh = particle velocity.

XX^ = X,

U Ygh = shock velocity.

"1^0

= ygh so

0^

= ^ and

^o

\'

= t.

Wr

Our initial conditions are:

h(x) = h, for < X < 1. h(x) = h^ for at > 1

1 o

u(x,0) = for X > 1

The problem is to find the strength, velocity and position

of the shock, as functions of time.

- 5 -

Section I Â« Approximate method (applicable to weak shocks) .

If P Is the density and p the pressure of a gas and

V the velocity relative to the shock, then the mechanical

shock conditions for a gas are:

) =i [ i^ + 2]

* This assumption is consistent with the "approximate"

theory of report IMM-NYU 158, because the change In the

Riemann invariant across the shock appears only in third

and higher order terms.

- 8 -

In order to find the shock curve U = f[x( 5 ),'t(l )],

a differential equation for T( 5 ) is set up and solved.

We have now, on the left side of the shock,

i-=-i- = u( S) + c(M or X = [u( !>) + c(S-)] ^ + Â§ ;

differentiating.

dx

dt . -rr. '

[u( 5 ) + c( S )] ^ + t[u ( 5 ) + c ( 5 )) + 1 ,

where the prime denotes differentiation with respect to 5 â€¢

But

dx _ dx dt n ^^

d5 " dt dj - " dl

u II =hx{l )} || + "t [u (5) + c (S)] + 1

(11) or ^[y.(^(l)^ci$))]=:-tU\l)^c\l))-^l

n

Equation (S() is an ordinary differential equation in t( 1 ).

Setting r ^4^LllÂ£-^7Ti *i3 = ^ (3> ), the soluti

/ u-[u(S)+c(5)] -^

I/O

on

(12)

(13)

t( S ) = e

- i(M

e-I(5)

U-.(u+c) ^^5

\

>

x( 5 ) = 5 + (u + c)t

/

yields a pair of parametric equations for the shock curve.

In order to obtain a simple explicit expression for

"t( S), an expansion of the quantities u and U in terms

of the parameter cr(5) =u(!>) + c(S) - 1 (similar to

- 9 -

that described In IMM-NYTJ 158) Is made; terms up to second

order are kept and equations (12) and (13) become:

(14)

(15)

1(5) = k

ff^(36/5 - ef

_ o''^(36/5 - a^)"^

-1

:(5 ) = 5 + (OT + 1) t( !))

where S = 5(0) = u(0) + c(0) - 1 = u^ + c. - 1 .

O XI

In terms of the parameter 5 , these equations can be written

In the form ^

-2 1

r

(16)

t( !>) = k

<

(41/5

(41/5 - 8)(^

x(5) = 3 + -^ t(!,)

- â€¢^4^)(3 - 1)

K^ - 1)

- 1

where s = u. + c | or. In terms of the original coordinates

X and t.

t(3) = k

(41/5 - -^-^)(s - 1)

(41/5 - s) (jL^ - 1)

(17)

x(5 ) = ^^ t(3)

which is convenient for numerical work. Equations (17)

were used to compute the shock curves represented on the

graph by broken curves.

It Is interesting that only a part of the depression

wave interacts with the shock; the remaining part leaves

the shock unaffected. In the physical situation, the active

- 10 -

portion of the depression wave Is the forward moving part

whose height Is greater than h (see figure 5). In the

X, t-plane, the active region Is that In which the straight

characteristics satisfy the Inequality -^ = u + c > 1

(see figure 6). This behavior may be deduced by examining

equations (17). When ^^'^ = 1, x and t (the coordinates

of the point of Intersection of the shock with the charac-

teristic where 5 = J> = k - ^) become Infinite. Insert-

ing this value of

i+J

In (9 ) and (10 ), one has:

u( 3*) = 0, c( 5*) :* 1

This illustrates that only the forward moving part of the

water affects the shock (see figures 5 and 6), and that

it alone decays the shock for an infinltaly long time.

Figure 5

â€¢â–ºX

Figure 6

- 11 -

Suppose that. Instead of moving the wall with such a

velocity that cavitation occurs (i.e. the case in which

c varies from c. to across the rarefaction wave),

the wall Is given a constant velocity w greater than

u. - 2c. hut less than u,. Then the rarefaction wave

Is narrowed and a constant state appears to the left of

the variable zone. The particle velocity in this con-

stant state is just the wall velocity w. If w < 0,

nothing essentially different happens, because then, the

rarefaction wave is still wide enough to Include the

"active" portion which decays the shock for an infinitely

long time. If however, < w < u., there is an interaction

which ends after a finite time during which the shock de-

cays from one constant shock to a weaker constant shock,

(see figures 7 and 8). The final constant shock is, of

course, constant only within the approximation considered

here.

UalL

Bzfoitdsceiy

h,

I

Uoa

L

T

i_

h.

Figure 7

- 12 -

Figure 8

In order to Investigate at which time a shock has

decreased from h. to an arbitrary height

2 ^

h = c ,(h < h < h^), one uses the equation

t [S (c*)] = k

(17/5 - e")(s - 1)

(41/5 - s)(c* - 1)

.2

for t as a function of c = h â€¢

(t = k represents the time at which the decay begins )>

Graph I represents the height as a function of

time in case of the following initial shock strengths:

2 ^1

c!' = Tr= = 1.2 , 1.5 , 1.8 , 3.0 , respectively.

Q

It Is apparent that the height decreases rapidly at first

and then approaches Its asympotlc value slowly.

7o t

- 13 -

Sactlon II. Finite difference method.

The differential equations describing the flow In

this problem are:

N

(1)

u. + uu + 2cc =

t X X

2c. + cu + 2uc =

t X X

The corresponding characteristic equations are

(2) ^ = u*o

u + 2c = const.

\

along C characteristics

(3)

11 = '"- =

u - 2c = const

along C characteristics

Replacing the derivative ^ In (2) by the difference

quotient, one obtains, for any Interior point of the net

shown In figure 9:

J^~- )s.s

Figure 9

- 14 -

iA\ ^Ik " ^1-1, k _ 1 râ€ž ,, ,

- X

(5) Ik 1,^-1 = I [T + T ]

hk \,k-l '^ ^^ ^*^ ^

where S^^^ = ^x^^ + c^^^ and T^^^ = u^^^ - c^^^

The slBiultaneoua solution of (4) and (5) yields:

^, ^

WSTTTUTE CF MATl lET.IATICAL SOÂ£HC3tS

LIURARY

2S WmwV PLcc, New YoA 3, KX.

3

3

DECAYING SHOCKS

A Comparison of an Approximate Analytic Solution

With a Finite Difference Method

[^ Prepared under

Navy Contract N6ori-201 Task Order No. 1

\ By the

. Institute for Mathematics and Mechanics

Sj. New York University

October, 1947

PREFACE

The present report by Miss Anneli Leopold,

prepared under Navy Contract N6orl-201, Task

Order No. 1, supplements Dr. Friedrichs' paper

Formation and Decay of Shock Waves report

IMM-NYU 158. It discusses the range of validity

of the numerical approximation given in the

latter report for the treatment of decaying shock

waves.

Miss Leopold was greatly assisted in the

numerical computations by Mr. John Butler.

R. Courant

DaCAYINQ SHOCKS

A Comparison of an Approximate Analytic Solution

with a Finite Difference Method.

INTRODUCTION .

In report IMM-NYU 158, Formation and Decay of

Shock Waves by K. 0. Friedrichs, an approximate method

was developed to determine the behavior of a decaying

shock wave in a gas, i.e. the behavior of a shock wave

weakened by a rarefaction wave overtaking it. This

method is expected to yield rather accurate results In

the case of weak shocks; it is of interest to deter-

mine for what range of shock strengths its use may be

Justified, and beyond what shock strength a more exact

solution is required.

In order to test the validity of the "approximate"

method by comparison with an "accurate" result, a corre-

sponding problem for surface waves in shallow water is

treated first by the "approximate" method of IMM-NYU 158,

and then by a finite difference scheme applicable to

shocks of any strength. The accuracy of the solution

obtained by the latter scheme is limited only by the size

of the mesh used. The discussion of a problem concerning

water wave phenomena rather than gas dynamics proper Is

advantageous; for, while the differential equations are

of the same form in both cases, there are no entropy

changes in water so that the finite difference scheme

is simplified considerably.

- 2 -

For every shock strength considered, a comparison

of the results obtained by both methods determines the

accuracy of the "approximate" method. The shock strengths

treated here. I.e. the ratios of height h^ of the

moving water to the height of h^ of the still water

(see fig. 1), are: 1.2, 1.5, 1.8, 3.0 respectively. The

results show a very good agreement of the two methods of

^1

solution for all these cases, even for ^ = 3.0 which is

o

a stronger shock than those usually classed among weak

shocks.

The formulation of the physical problem Is as follows;

A shock wave - or bore as It Is also called - moves with

constant speed U Into still water of uniform depth h^.

The height of the water behind the shock Is h^^, and Its

velocity Is u^ (see fig. 1).

1.

a.

Figure 1

At a certain time t = 0, a vertical wall Is suddenly In-

serted into the moving water at a point x = 0, so that

a depression (or rarefaction) wave is created which will

catch up with the shock, since such disturbances are propa-

gated with a speed greater than that of the shock (see

pages 5 and 6). Thus the shock will gradually be weakened.

* Such a bore may be created by pushing a rigid vertical

plate through the water at constant speed.

- 3 -

Figures 2 and 3 describe the physical situation at two

Instances after the insertion of the wall.

I

Plp^ure 2

UqU

Figure 3

In section 1 of this report, the "approximate" method

will be discussed and applied to the problem as formulated

here. In section 2, the finite difference method is de-

scribed and used. The graphs II and III show the results

of both methods.

A noteworthy result of these considerations is that

strong shocks weaken very rapidly at first. This suggests

the following treatment of strong shocks: One uses the

finite difference method initially but finds, after rela-

tively few steps, that the shock has weakened sufficiently

to justify the use of the "approximate" method in con-

tinuing the solution. Thus the method of IMM-NYU 158 is

an extremely useful one even in cases where shocks are

strong.

- 4 -

Let g be the acceleration due to gravity, h the

depth of the water, x the distance from the wall and t

the time after the Insertion of the wall. Let x^ be

the distance between the wall and the discontinuity at

the time the wall is inserted. I.e. at t = 0. We, use

the following definitions in order to work with dimension-

less units:

u "Vgh = particle velocity.

XX^ = X,

U Ygh = shock velocity.

"1^0

= ygh so

0^

= ^ and

^o

\'

= t.

Wr

Our initial conditions are:

h(x) = h, for < X < 1. h(x) = h^ for at > 1

1 o

u(x,0) = for X > 1

The problem is to find the strength, velocity and position

of the shock, as functions of time.

- 5 -

Section I Â« Approximate method (applicable to weak shocks) .

If P Is the density and p the pressure of a gas and

V the velocity relative to the shock, then the mechanical

shock conditions for a gas are:

) =i [ i^ + 2]

* This assumption is consistent with the "approximate"

theory of report IMM-NYU 158, because the change In the

Riemann invariant across the shock appears only in third

and higher order terms.

- 8 -

In order to find the shock curve U = f[x( 5 ),'t(l )],

a differential equation for T( 5 ) is set up and solved.

We have now, on the left side of the shock,

i-=-i- = u( S) + c(M or X = [u( !>) + c(S-)] ^ + Â§ ;

differentiating.

dx

dt . -rr. '

[u( 5 ) + c( S )] ^ + t[u ( 5 ) + c ( 5 )) + 1 ,

where the prime denotes differentiation with respect to 5 â€¢

But

dx _ dx dt n ^^

d5 " dt dj - " dl

u II =hx{l )} || + "t [u (5) + c (S)] + 1

(11) or ^[y.(^(l)^ci$))]=:-tU\l)^c\l))-^l

n

Equation (S() is an ordinary differential equation in t( 1 ).

Setting r ^4^LllÂ£-^7Ti *i3 = ^ (3> ), the soluti

/ u-[u(S)+c(5)] -^

I/O

on

(12)

(13)

t( S ) = e

- i(M

e-I(5)

U-.(u+c) ^^5

\

>

x( 5 ) = 5 + (u + c)t

/

yields a pair of parametric equations for the shock curve.

In order to obtain a simple explicit expression for

"t( S), an expansion of the quantities u and U in terms

of the parameter cr(5) =u(!>) + c(S) - 1 (similar to

- 9 -

that described In IMM-NYTJ 158) Is made; terms up to second

order are kept and equations (12) and (13) become:

(14)

(15)

1(5) = k

ff^(36/5 - ef

_ o''^(36/5 - a^)"^

-1

:(5 ) = 5 + (OT + 1) t( !))

where S = 5(0) = u(0) + c(0) - 1 = u^ + c. - 1 .

O XI

In terms of the parameter 5 , these equations can be written

In the form ^

-2 1

r

(16)

t( !>) = k

<

(41/5

(41/5 - 8)(^

x(5) = 3 + -^ t(!,)

- â€¢^4^)(3 - 1)

K^ - 1)

- 1

where s = u. + c | or. In terms of the original coordinates

X and t.

t(3) = k

(41/5 - -^-^)(s - 1)

(41/5 - s) (jL^ - 1)

(17)

x(5 ) = ^^ t(3)

which is convenient for numerical work. Equations (17)

were used to compute the shock curves represented on the

graph by broken curves.

It Is interesting that only a part of the depression

wave interacts with the shock; the remaining part leaves

the shock unaffected. In the physical situation, the active

- 10 -

portion of the depression wave Is the forward moving part

whose height Is greater than h (see figure 5). In the

X, t-plane, the active region Is that In which the straight

characteristics satisfy the Inequality -^ = u + c > 1

(see figure 6). This behavior may be deduced by examining

equations (17). When ^^'^ = 1, x and t (the coordinates

of the point of Intersection of the shock with the charac-

teristic where 5 = J> = k - ^) become Infinite. Insert-

ing this value of

i+J

In (9 ) and (10 ), one has:

u( 3*) = 0, c( 5*) :* 1

This illustrates that only the forward moving part of the

water affects the shock (see figures 5 and 6), and that

it alone decays the shock for an infinltaly long time.

Figure 5

â€¢â–ºX

Figure 6

- 11 -

Suppose that. Instead of moving the wall with such a

velocity that cavitation occurs (i.e. the case in which

c varies from c. to across the rarefaction wave),

the wall Is given a constant velocity w greater than

u. - 2c. hut less than u,. Then the rarefaction wave

Is narrowed and a constant state appears to the left of

the variable zone. The particle velocity in this con-

stant state is just the wall velocity w. If w < 0,

nothing essentially different happens, because then, the

rarefaction wave is still wide enough to Include the

"active" portion which decays the shock for an infinitely

long time. If however, < w < u., there is an interaction

which ends after a finite time during which the shock de-

cays from one constant shock to a weaker constant shock,

(see figures 7 and 8). The final constant shock is, of

course, constant only within the approximation considered

here.

UalL

Bzfoitdsceiy

h,

I

Uoa

L

T

i_

h.

Figure 7

- 12 -

Figure 8

In order to Investigate at which time a shock has

decreased from h. to an arbitrary height

2 ^

h = c ,(h < h < h^), one uses the equation

t [S (c*)] = k

(17/5 - e")(s - 1)

(41/5 - s)(c* - 1)

.2

for t as a function of c = h â€¢

(t = k represents the time at which the decay begins )>

Graph I represents the height as a function of

time in case of the following initial shock strengths:

2 ^1

c!' = Tr= = 1.2 , 1.5 , 1.8 , 3.0 , respectively.

Q

It Is apparent that the height decreases rapidly at first

and then approaches Its asympotlc value slowly.

7o t

- 13 -

Sactlon II. Finite difference method.

The differential equations describing the flow In

this problem are:

N

(1)

u. + uu + 2cc =

t X X

2c. + cu + 2uc =

t X X

The corresponding characteristic equations are

(2) ^ = u*o

u + 2c = const.

\

along C characteristics

(3)

11 = '"- =

u - 2c = const

along C characteristics

Replacing the derivative ^ In (2) by the difference

quotient, one obtains, for any Interior point of the net

shown In figure 9:

J^~- )s.s

Figure 9

- 14 -

iA\ ^Ik " ^1-1, k _ 1 râ€ž ,, ,

- X

(5) Ik 1,^-1 = I [T + T ]

hk \,k-l '^ ^^ ^*^ ^

where S^^^ = ^x^^ + c^^^ and T^^^ = u^^^ - c^^^

The slBiultaneoua solution of (4) and (5) yields:

^, ^

1

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