Online Library → Kurt Otto Friedrichs → Formation and decay of shock waves → online text (page 1 of 2)

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Up^.

MW YORK UNIVERSITY

tBfflUTEOFM

y^ iqJ 9S WÂ«vwÂ«y Mx*. *â– * Y by passing

through each point x = ^ on the line t = the forward charac-

teristic with the slope -rr = u(^) + c(^), and let u and c be

_

v 'We do not admit here the equality sign; the subsequent

results remain valid, however, also for a point of contin-

uity if the derivatives -râ€” and -r^ become infinite in an

appropriate manner.

-6-

constant along it. We set accordingly

(2)

= {u(^) + c(^) t + ^ , u = u(^), c = c(^)

Owing to the compressive character of the discontinuity

we have u Q + c Q < u^ + c-, , and hence the two wave regions in

the (x,t) -plane obtained from the two parts x > and x <

overlap. The shock line is now to be drawn in the common

region in an appropriate way such that the mutilated wave

regions fit together.

d)(o)

â€” *

Pig. 3

Two forward facing simple waves

separated by shock.

4. How to determine the path of the shock is best seen

in case the state of the gas ahead of the shock front is con Â«

s tant : u = u = 0, c = c for x > 0, initially, and hence

also later on. Relation (1) then entails

(3) u = (1 - ,e?)(u + c - c Q ), c = c Q + ^(u + c - c Q ).

The relationship between shock velocity U and gas velocity u

behind the shock is exactly given by

-7-

(4) U - (1 - y|)(U - C^/U) ,

see Manual III>Art36(4$, or using (3), by

/c -\ u+c n _ U c o

(5) - 1 _ -

o o

Since the velocity behind the shock front is a given function

u(^) of the initial parameter ? we can determine D from (3)

or (5) as a function U(^) of X . Thus we have for the shock

te) w-'W â€¢

We now consider the quantity t along the shock line as

function of Â£ and express x as function of ? by (2).

Differentiation of equation (2) with respect to "5 yields

â–

or

(V) Ju-u-c|Â£- Ju. + c.( t â– 1 .

{' â€” â€¢ ]S\- {Â»,â™¦â€¢>}

This is an ordinary linear differential equation, the "shock

differential equation", for t as function of J , which yields

a unique solution by virtue of the initial condition t =

for T = 0. Thus the propagation of the shock is determined.

Introducing the function

(8)

/^ d(m-c)

?=? -/'Â»

with any convenient constant of integration, the solution can

be written in the form

-8-

Insertion of this relation into (2) yields x as function of

? â€¢ Thus the path of the shock front is given in parametric

representation.

For the numerical treatment various simplifications are

useful to evaluate the integrals. Chandrasekhar, who has

developed the method in connection with a special case, uses

the quantity U instead of F as parameter. We found it more

convenient to employ

u+c-c^

(10) 5-= â€” râ€” 2

C

as parameter and to set

(11) U = c Q |l +Â£6- + Â£ s 2 } ,

a formula which agrees with (5) up to terms of third order

as is easily verified (cf. Appendix AI). (Note that the

dependence of U/c on (u + c)/c Q through (5) or (11) and the

dependence of Â£ on U/c is independent of yÂ») Inserting (11)

into (8) we find

(12) < = 21og|- 5 , â€¢"* = (ij?) 2

whence by (9),

(13 > 1 Â« 8( ^) 2 rÂ° sÂ±s^ .

(See Appendix AI.)

5. In the two problems of formation and decay mentioned

in the introduction, we just have the situation that the state

ahead of the shock front is constant. Hence these two problems

can be treated by the method outlined. The results for these

two problems are derived in Appendices All and AIII. We give

here a brief account.

In the wave produced by a pis ;on moving with constant

-9-

1 2

acceleration , x =Â« bt , a shock develops at the time

"" â„¢^" â€” """" ~~â„¢ â€” ~ ~" â€” " â€” â„¢ c c

t s> = (1 - -a) â€” Â» at the position x,o = (1 -zt) -r~ Â» (see

Pig, 1). The path of the shock is (cf. Appendix All), given

by

r 3 b(t-t f )~

(14) x - x = c \{t - t ) + 2-a +

I 2

8(1-^)

9 b 2 (t-t^,) 3

128 (l-^) 53 "7^"

J-

One observes that the acceleration of the shock front is

initially about 3/4 of the value b of the acceleration of

the piston, but then increases.

6. For the decaying shock wave (cf â€¢ Appendix AIII) we

obtain more striking results. Suppose the piston is first

moved with a velocity u-, and stopped at a time t R at

x = Xrj = u-,t R , (see Pig, 2), A shock starts moving with

constant speed at t = from the piston; it is overtaken by

a rarefaction wave starting at the time t R when the piston is

arrested. At the time t = t-, of overtaking the interaction

begins and the theory of Sec, 3 and 4 is to be applied from

then on. The motion of the decaying shock is found to be

given by

(15) t = t R + (t x - t R ) k 2 (^) 2 , 6 1 > S> ,

x = Xp + (1 + â€¢

It is remarkable that up to the order considered the shock

depends only on the initial distribution of CJ = u + c and

on the local value c ^ of c but not explicitly on the value

of y.

These formulas can be used to determine the formation

of a shock in a compression wave which is produced when a

piston is pushed into the gas beginning with acceleration

zero. This will be explained in detail in Appendix AVI.

10. The decay of steady two-dimensional shock waves

can be treated in much the same way as the decay of moving

one-dimensional shock waves has been treated. The reason

is that for steady two-dimensional waves, just as for one-

dimensional unsteady waves, the transition relations for

shocks and simple waves are the same up to terms of second

order.

Consider a backward-facing simple wave that is forward

inclined. Suppose further that the incoming supersonic flow

with the constant velocity u=q > c , v = is turned into

J ^o o

a flow with velocity u = q cos Â©, v = q sin 0, (see Pig. 7).

The strength of this wave may be characterized by the angle

Â© and the changes of all quantities across the wave may be

expanded in powers of 0. Similarly, consider a backward-

facing forward-Inclined shock front, (see Pig. 8) across

which the flow with velocity (q ,0) is turned into a flow

with velocity u = q cos 9, v = q sin Â©. Then again the changes

of all quantities can be expanded in powers of Â©. This ex-

pansion now agrees with the expansion for the simple wave up

to terms of second order, (see Appendix BII). For this reason

-17-

we may in second approximation consider the flow behind the

shock as if it had resulted through a simple wave.

Pi

â€¢

Backward-facing forward-

inclined simple wave in-

dicated through a set of

straight Mach lines.

Pig. 8

Backward-facing forward-

inclined shock. Forward-

inclined Mach lines are

shown on both sides.

This fact may also be expressed as follows: consider

in the (u,v) -plane the shock polar and the epicycloid through

the same point (q ,0) employing the same value for the critical

speed c v ,, (see Manual III, 35, and IV, 61, 69) â€ž Then these

two curves agree up to second order in the sense that their

representations in terms of the parameter Â© agree up to second

order. If, therefore, the flow velocity u,v behind a shock

with a certain angle of flow direction, represented by a

point on the shock polar, is replaced by a velocity with the

same angle represented by a point on the epicycloid, then

the shock relations for u and v are satisfied up to terms of

second order in the shock strength. The sound speed c is

2 2 2 2 2

determined from Bernoulli's law juQ. + (1 - /*)c = câ€ž and

p and p are functions of c, if we assume the entropy to be the

same on both sides.

Suppose now a supersonic -low along a straight wall with

-18-

a bump is to be determined, (see Fig. 9). We may, for ex-

ample, consider the upper part of an airfoil cross section

as such a bump. We assume that the incoming flow is super-

Pig, 9

Plow along a straight wall with a

bump. The resulting shock and for-

ward-facing Mach lines are shown.

sonic M = Q /c > 1> that the profile of the bump begins

Up^.

MW YORK UNIVERSITY

tBfflUTEOFM

y^ iqJ 9S WÂ«vwÂ«y Mx*. *â– * Y by passing

through each point x = ^ on the line t = the forward charac-

teristic with the slope -rr = u(^) + c(^), and let u and c be

_

v 'We do not admit here the equality sign; the subsequent

results remain valid, however, also for a point of contin-

uity if the derivatives -râ€” and -r^ become infinite in an

appropriate manner.

-6-

constant along it. We set accordingly

(2)

= {u(^) + c(^) t + ^ , u = u(^), c = c(^)

Owing to the compressive character of the discontinuity

we have u Q + c Q < u^ + c-, , and hence the two wave regions in

the (x,t) -plane obtained from the two parts x > and x <

overlap. The shock line is now to be drawn in the common

region in an appropriate way such that the mutilated wave

regions fit together.

d)(o)

â€” *

Pig. 3

Two forward facing simple waves

separated by shock.

4. How to determine the path of the shock is best seen

in case the state of the gas ahead of the shock front is con Â«

s tant : u = u = 0, c = c for x > 0, initially, and hence

also later on. Relation (1) then entails

(3) u = (1 - ,e?)(u + c - c Q ), c = c Q + ^(u + c - c Q ).

The relationship between shock velocity U and gas velocity u

behind the shock is exactly given by

-7-

(4) U - (1 - y|)(U - C^/U) ,

see Manual III>Art36(4$, or using (3), by

/c -\ u+c n _ U c o

(5) - 1 _ -

o o

Since the velocity behind the shock front is a given function

u(^) of the initial parameter ? we can determine D from (3)

or (5) as a function U(^) of X . Thus we have for the shock

te) w-'W â€¢

We now consider the quantity t along the shock line as

function of Â£ and express x as function of ? by (2).

Differentiation of equation (2) with respect to "5 yields

â–

or

(V) Ju-u-c|Â£- Ju. + c.( t â– 1 .

{' â€” â€¢ ]S\- {Â»,â™¦â€¢>}

This is an ordinary linear differential equation, the "shock

differential equation", for t as function of J , which yields

a unique solution by virtue of the initial condition t =

for T = 0. Thus the propagation of the shock is determined.

Introducing the function

(8)

/^ d(m-c)

?=? -/'Â»

with any convenient constant of integration, the solution can

be written in the form

-8-

Insertion of this relation into (2) yields x as function of

? â€¢ Thus the path of the shock front is given in parametric

representation.

For the numerical treatment various simplifications are

useful to evaluate the integrals. Chandrasekhar, who has

developed the method in connection with a special case, uses

the quantity U instead of F as parameter. We found it more

convenient to employ

u+c-c^

(10) 5-= â€” râ€” 2

C

as parameter and to set

(11) U = c Q |l +Â£6- + Â£ s 2 } ,

a formula which agrees with (5) up to terms of third order

as is easily verified (cf. Appendix AI). (Note that the

dependence of U/c on (u + c)/c Q through (5) or (11) and the

dependence of Â£ on U/c is independent of yÂ») Inserting (11)

into (8) we find

(12) < = 21og|- 5 , â€¢"* = (ij?) 2

whence by (9),

(13 > 1 Â« 8( ^) 2 rÂ° sÂ±s^ .

(See Appendix AI.)

5. In the two problems of formation and decay mentioned

in the introduction, we just have the situation that the state

ahead of the shock front is constant. Hence these two problems

can be treated by the method outlined. The results for these

two problems are derived in Appendices All and AIII. We give

here a brief account.

In the wave produced by a pis ;on moving with constant

-9-

1 2

acceleration , x =Â« bt , a shock develops at the time

"" â„¢^" â€” """" ~~â„¢ â€” ~ ~" â€” " â€” â„¢ c c

t s> = (1 - -a) â€” Â» at the position x,o = (1 -zt) -r~ Â» (see

Pig, 1). The path of the shock is (cf. Appendix All), given

by

r 3 b(t-t f )~

(14) x - x = c \{t - t ) + 2-a +

I 2

8(1-^)

9 b 2 (t-t^,) 3

128 (l-^) 53 "7^"

J-

One observes that the acceleration of the shock front is

initially about 3/4 of the value b of the acceleration of

the piston, but then increases.

6. For the decaying shock wave (cf â€¢ Appendix AIII) we

obtain more striking results. Suppose the piston is first

moved with a velocity u-, and stopped at a time t R at

x = Xrj = u-,t R , (see Pig, 2), A shock starts moving with

constant speed at t = from the piston; it is overtaken by

a rarefaction wave starting at the time t R when the piston is

arrested. At the time t = t-, of overtaking the interaction

begins and the theory of Sec, 3 and 4 is to be applied from

then on. The motion of the decaying shock is found to be

given by

(15) t = t R + (t x - t R ) k 2 (^) 2 , 6 1 > S> ,

x = Xp + (1 + â€¢

It is remarkable that up to the order considered the shock

depends only on the initial distribution of CJ = u + c and

on the local value c ^ of c but not explicitly on the value

of y.

These formulas can be used to determine the formation

of a shock in a compression wave which is produced when a

piston is pushed into the gas beginning with acceleration

zero. This will be explained in detail in Appendix AVI.

10. The decay of steady two-dimensional shock waves

can be treated in much the same way as the decay of moving

one-dimensional shock waves has been treated. The reason

is that for steady two-dimensional waves, just as for one-

dimensional unsteady waves, the transition relations for

shocks and simple waves are the same up to terms of second

order.

Consider a backward-facing simple wave that is forward

inclined. Suppose further that the incoming supersonic flow

with the constant velocity u=q > c , v = is turned into

J ^o o

a flow with velocity u = q cos Â©, v = q sin 0, (see Pig. 7).

The strength of this wave may be characterized by the angle

Â© and the changes of all quantities across the wave may be

expanded in powers of 0. Similarly, consider a backward-

facing forward-Inclined shock front, (see Pig. 8) across

which the flow with velocity (q ,0) is turned into a flow

with velocity u = q cos 9, v = q sin Â©. Then again the changes

of all quantities can be expanded in powers of Â©. This ex-

pansion now agrees with the expansion for the simple wave up

to terms of second order, (see Appendix BII). For this reason

-17-

we may in second approximation consider the flow behind the

shock as if it had resulted through a simple wave.

Pi

â€¢

Backward-facing forward-

inclined simple wave in-

dicated through a set of

straight Mach lines.

Pig. 8

Backward-facing forward-

inclined shock. Forward-

inclined Mach lines are

shown on both sides.

This fact may also be expressed as follows: consider

in the (u,v) -plane the shock polar and the epicycloid through

the same point (q ,0) employing the same value for the critical

speed c v ,, (see Manual III, 35, and IV, 61, 69) â€ž Then these

two curves agree up to second order in the sense that their

representations in terms of the parameter Â© agree up to second

order. If, therefore, the flow velocity u,v behind a shock

with a certain angle of flow direction, represented by a

point on the shock polar, is replaced by a velocity with the

same angle represented by a point on the epicycloid, then

the shock relations for u and v are satisfied up to terms of

second order in the shock strength. The sound speed c is

2 2 2 2 2

determined from Bernoulli's law juQ. + (1 - /*)c = câ€ž and

p and p are functions of c, if we assume the entropy to be the

same on both sides.

Suppose now a supersonic -low along a straight wall with

-18-

a bump is to be determined, (see Fig. 9). We may, for ex-

ample, consider the upper part of an airfoil cross section

as such a bump. We assume that the incoming flow is super-

Pig, 9

Plow along a straight wall with a

bump. The resulting shock and for-

ward-facing Mach lines are shown.

sonic M = Q /c > 1> that the profile of the bump begins

1 2

Online Library → Kurt Otto Friedrichs → Formation and decay of shock waves → online text (page 1 of 2)