Kurt Otto Friedrichs.

Formation and decay of shock waves online

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5



Up^.



MW YORK UNIVERSITY
tBfflUTEOFM

y^ iqJ 9S W«vw«y Mx*. *■* 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 -r— and -r^ become infinite in an
appropriate manner.



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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)



— *



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 «
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 •

We now consider the quantity t along the shock line as
function of £ and express x as function of ? by (2).
Differentiation of equation (2) with respect to "5 yields



or

(V) Ju-u-c|£- Ju. + c.( t ■ 1 .



{' — • ]S\- {»,♦•>}



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)



?=? -/'»



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

? • 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-= — r— 2

C

as parameter and to set

(11) U = c Q |l +£6- + £ 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 £ on U/c is independent of y») Inserting (11)
into (8) we find

(12) < = 21og|- 5 , •"* = (ij?) 2

whence by (9),

(13 > 1 « 8( ^) 2 r° s±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



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1 2

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

"" ™^" — """" ~~™ — ~ ~" — " — ™ c c

t s> = (1 - -a) — » at the position x,o = (1 -zt) -r~ » (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 • 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 + •

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 ©, v = q sin 0, (see Pig. 7).
The strength of this wave may be characterized by the angle
© 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 ©. Then again the changes
of all quantities can be expanded in powers of ©. This ex-
pansion now agrees with the expansion for the simple wave up
to terms of second order, (see Appendix BII). For this reason



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we may in second approximation consider the flow behind the
shock as if it had resulted through a simple wave.




Pi







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) „ Then these
two curves agree up to second order in the sense that their
representations in terms of the parameter © 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 = c„ 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


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Online LibraryKurt Otto FriedrichsFormation and decay of shock waves → online text (page 1 of 2)