Diffraction of strong shock waves by a sharp compressive corner

Ju1y, 1970.

· .

DIFFRACTION OF STRONG SHOCK WAVES

BY A SHARP COMPRESS IVE CORNER by

c.

K. Law

UTIAS Technica1 Note No.150 AFOSR 70-0767 TR

,

'

....

DIFFRACTION OF STRONG SHOCK WAVES BY A SHARP COMPRESSIVE CORNER

by

C. K. Law

Manuscript received Dec, 1969.

Ju1y, 1970. UTIAS Technica1 Note No. 150

y

SUMMARY

The diff~action of strong shock waves (2 <Ms <8) in gxygen, in dissociation equilibrium, by a sharp compressive corner (25 <

e

<600 ) has been investigated both theoretically and experimentally.

T~e

continuous curvature of the reflected shock wave R, for usual Mach reflection, changes when the pressure behind the bow shock gen-erated by the wedge exceeds that behind R. The appearance of a second triple shock system in irregular Mach reflection can be pre-dicted from usual gasdynamic considerations without requirements from chemical kinetics. The location of .the point where R kinks has also been predicted. The above predictions are supported by the experimental results. The relative velocity between the first and second triple poin~s can also be found, making i t possible to do a useful analysis,of conditions near the second triple point.

Diffraction patterns for different gases (0_{2 , N 2 ,} A and C02) were investigated. It was found that R changes curvature very early for all gases tested. However, the format ion of the second triple point is very dependent on the gas. In a monatomic gas, it is most difficult to exhibit. ~his effect, owing to the lower flow Mach number behind the moving incident shock wave.

An approximate solution was obtained for the first triple point trajectory angle as a function of incident shock Mach number and the wedge angle for dissociated oxygen. With the exception of small wedge angles and .strong incident shocks, the agreement with experiment is good. Some reasons for the disagreement are given.

An analysis was also made for oblique shock wave relations with imperfect gas effects (vibration and dissociation) including a two-shock confluence (regular reflection) and a three-shock confluence

ACKNOWLEDGEMENT.

This project was carried out under the direction of Dr. I. I. Glass, w~ose advice and support throughout the course of the work are .gratefully acknowledged.

Sincere appreciation is extended to Dr. A. K. Macpherson for his invaluable advice and d~scussions.

The stimulating talks with Dr. M. Bristow, Mr. O. Igra and Mr. N. K. Phung, both on the theoretical and experimental phase of the work, are acknowledged with thanks.

The financial support was provided jointly by the National Research Council of Canada, and by the Air Force Office of Scientific Research of the United States Air Force under Contract AF-AFOSR 68-1368A.

Chapter 1 2 3

4

5

6

7

8

Appendix TABLE OF CONTENTS NOTATION INTRODUCTION

PRESENT STATUS OF THE PROBLEM OBLIQUE SHOCK WAVE RELATIONS

OBLIQUE REFLECTION OF-SHOCK WAVES

DIFFRACTION OF SHOCK WAVES BY A CONCAVE CORNER - ANALYTICAL CONSIDERATION

DIFFRACTION OF A SHOCK WAVE BY A CONCAVE CORNER - EXPERIMENTAL RESULTS

DISCUSSION OF RESULTS CONCLUSIONS

REFERENCES

A - Thermodynamie Properties of a Symmetrie Diatomie Gas in Dissoeiation Equilibrium B - Solutions for a Three-Shock Confluence in

Perfect Gas

C ~ Numerieal Method of Solving a Set of Non-linear Equations

D - The Computer Program

E - Proof of

p

&

>F~ for a St rong Shoek Wave Figures 1 2

4

8

15 20 22 27 2~u

h P s U U' k Uk U _s B F I LD' LR M Mk,Mk I ,Mk M s P P_ji R R' R-B S S I T T'

V

NOTATION specific ent~alpy pressure fringe shift

velocity in region k, measured fn T or P frame velocity in region k, measured in T' frame

velocity in k region, measured in laboratory frame nonstationary flow velocity behind incident shock bow shock

function defined by Eq. 3-16

incident shock, or incident shock polar distances defined in Fig. 5-2

Mach stem

Mach number in region k, measured in T or P frame; in T' frame, and in the laboratory frame, respectively incident shock Mach number

contact point of incident shock with wedge surface in regula~~reflection

pressure ratio

reflected shock or reflected shock polar

additional shock in the second triple shock system the shock formed by Rand B

first slipstream second slipstream

temperature or first triple point second triple point

Greek Symbols

x,x'

first and second triple point path angle )' specific heat ratio

À wave length

CPk shock wave angle in region k

CPM

,CP

F _{shock wave angle of}

M

at T and foot of

M,

:r:espectively

p density

e

_{deflection angle in region k}

k

eM,e

F deflection angle of

M

at T and for T of

M

respectively

e

wedge angle

w

Sub s cr.ipts.

i , j state in front- of and behind a shock

n,t normal and tangential component of a velocity vector

0,1,2,3,

4,5,6

regions defined by Fig's.

2-1

and

5-2

,

.

....

Chapter 1: INTRODUCTION

When a planar shock wave propagating in a shock tube encounters a sharp compressive corner, two independent processes

take place simultaneously. The shock itself is reflected by the

wedge surface, whereas the non-stationary flow behind i t is deflected

by the wedge corner. We shall call the first process as shock

re-flection and the second as flow dere-flection and the overall phenomenon as shock wave diffraction.

The severity öf the diffraction depends on two factors: The strength of the incident shock Ms and how large an angle Bw

the wedge subtends. For each set of initial conditions each of the

flow deflection and shock reflection processes can assume various

configurations. The shock reflection process can be either of two

types: regular ~ Mack reflection. The flow deflection process is

achieved through an attached bow shock, a detached bow shock, or simply

a continuous subsonic turning. Since shock diffraction is the result

of the interaction between these two processes, the overall flow

field can assume an even larger number possible configurations. The

resultant configuration, however, will usually be self-similar with

respect to time. It is th us of interest to investigate the limit

when one configuration undergoes transition to another, as weIl as the flow field associated with each type of configuration.

With the strength of the incident shock continuously in-creasing, a perfect gasdynamical behaviour tend to be less and less

accurate. The effects of the internal degrees of excitation, the

chemical reactions of dissociation and ionization, aS weIl as the

interparticle (van der Waal) forces all cause deviations in the dynamic and thermodynamic parameters of the flow fr om their ideal values.

The complexity as ·well as the accuracy of the analysis dep end on how

many "non-ideal" effects we take into account, and whether these

can be justified. In the present report we based the analysis on

a symmetric diatomic gas in dissociation equilibrium. (App.A).

Any subsequent use of the term "imperfect" will imply the above model.

In doing a theoretical analysis of the problem of a

strong-shock wave diffracted by a sharp compressive corner, one will be confronted by three major difficulties: the nonuniformity of the overall flow field, the nonlinearity of the equations of motion,

and the inclusion of imperfect gas effects. Thus in the present

analysis several assumptions and simplifications were made. These

have been strengthened by the experimental results. Numerical and

graphical solutions were used in place of analyticalones. This

choice is not only simpIer, but is also found to be necessary.

The phenomena of flow deflection and shock reflection are

matched and the possibilities for the occurrence of various overall

configurations are disQ~saed. One interesting configuration is the

occurrence of a "curvature discontinuity" between the bow shock and

the reflected shock of a Mach reflection structure. For more

in-tense incident shock strengths an additional shock also issues from

the dependence of shock diffraction configuration on the incident

shock strength, the wedge angle, and the type of gas

u,~e'd.

The

results and their comparisons with theoretical predictions are

presented in Chapter

7.

Chapter 2.

PRESENT STATUS OF THE PROBLEM

During the early phase of investigation ot the shock

~if­

fraction problem, three possible configurations (Fig. 2-la, b, c)

were observed experimentally.

Since the solutionl tor regular

re-flection is relatively simple, subsequent theoretical and experimental

efforts were placed on the process of Mach reflection.

Although

Mach ref1ection was first noticed by E. Mach as early al the lalt

century, no work was done on it until 1940's.

Polachek and Seeger (Ref.l), von Neumann (Rets. 2,3), and

Bleakney and Taub (Refs. 4,5) have

exhaustively discussed the

ideal gas solutions of r.etlection and retraction (at a galeous

inter-face) ot the regular and Mach types.

These solutions, however, are

only lim1ted round the point ot contluence of the triple shock sy.tem

(the triple point).

.

Bargm nn (Ref.6) wa. able to obtain solution. tor the entire

flow

t i

ld by r

tricting hi. analysi. to weak Ihock. at nearly

gl ncin

incid ne • Lighthill (Retl.

7,8)

obt ined the pre •• ure

dil-tribut10n on the rigid boundary tor incident hockl with arbitrary

s t r n th di

f f r Cl

t d b Y

&

I ma 11 wed g e a n 11

9

_{w , wh i eh c a n b e e i th e r}

cone&v

or Qonv x.

Tinl and Ludlotf (R

t.~)

uled a difterent approach

to obt in th pre

ure and d nlity ti ldl, alain for incident Ihockl

of arbitr3ry tr n th and at clancinl incidlneo.

Tht Ilipltr am Ihowl

up automatically a

a relult of th ir mathematical modtl.

L@~n

(R f.10) atarted IXp rimontal involtilation in Enlland.

St ady atat M

eh

r fllction wal form d whln ,ttaoh.d bew Ihoakl

from two id@ntic31 w d I plaold at oppo.ito aidol of a wind tUnntl

int@r&ct

with @3@n oth r.

~hoek

tube aeon btcamt popular and Mach rtfllation wal

obt3inêd by

refl@@tinl

an incidtnt Ihoak eff alelid wldSI.

The

confisur&tiefi~

thua obtain@d il patude atationafY, and honet growl

uniformly with

~@a~oet

t@ time.

It i , h@w vtr, u ually plrturbod

by

êorn@r

~isn&l§

And boundary lay@r inttr3ctiena

~. ~mitn'a

oxporlm@ntal werk (Rof.ll) WAa prob&bly tnt

~êv@ral

important ebairvati@na Wêrê mAdêl

1) Fep

rê~ular

rorlê@ti@n, tn@ory And êxpêrim@fit

A~r@@

WIlle

~)

Fgr

MACn

r@fl@@tion,

tnêery And @xp@rlmênt A8r@ê w@ll rer

atrens ahe@Ka.

Fer WêAk aneCKa

(F

I~ ~ O.~)

tn@ r@lulAr

l@fl@êtien

und@rseo. tr3naltien

toeMA~n

rofl@ctien in

&

entlnueu§ mAnnêr, wnlch ia

centrA~y

te tno tn@ery.

3)

Ma@h rêfl@@tien deoa net

atA~t

imm@diAtoly whon tno thter@tiQal

limit

ef

r@~u13r

rêfl@ctien la êxc@tdod.

I

...

In an effort to explain L. G. Smith's difficulty in (2),

Fletcher (Ref.12) further investigated Mach reflection for weak shocks. The original discrepancy was reconfirmed, but without any fruitful

explanation.

It was Kawamura and Saito (Ref.13), who, by analyzing the stability of the subsonic flow behind the reflected shock for weak shock case, concluded that the transition from regular to Mack re-flection is continuous for sufficiently weak shocks. This explains the first difficulty as to why there is a disagreement between theory and experiment for weak shocks . Thus the tripl e-shock theory is not applicable for weak shocks wï~b( Ms '<)1 {4j)·:(R~:C .. 25).

Jahn (Ref.14) did an extensive study of the problem of shock refraction at a gaseous interface, and found similar discrepancies as in L. G. Smith's shock reflection problem. A severe subsonic rare-faction region was also noticed behind the reflected shock near the gaseous interface . Since the range of its occurrence is between the theoretical and experimental extreme angles of incidence, he argued that t his is actually the transition regime between the theoretical termination of regular reflection and the onset of Mach reflection. This successfully explains L. G. Smith's difficulties in (3) .

White (Ref.15) did an interferometric study of Mach

re-flection. The density fields so obtained were in good agreement with the theories of Ting and Ludloff (Ref.9) and of LighthilI (Refs . 7 and 8). For strong shocks, i t was noticed that the slipstream curls at the wedge surface and the reflected shock breaks at a point where for a stronger shock, a new tripl e-shock system is formed (Fig. 2-ld,

e) •

Among other mathemat ical t echniques developed to solve the shock diffraction problem, t here are Thornhi l l ' s two-dimensional pseudo-stat ionary equations of motion (Ref.16) and Whitham's theory (Ref.17 ). Henderson (Refs . 18, 19) by employing the shock polar t echnique, has been able to explain many interesting configurations for shock r efraction. This graphical solution of shock polars has also been adopted in the present theoretical analysis.

The first triple point trajectory angl e

X

,

which is needed t o f ind the regions around the first triple point, is usually de-termined empiracally. Cl~tterham and Taub (Ref.20) gave a theo-retical solution as

X

=

e

_w + T (p~/p ) .L 0

where,

PiP

,

is the pressure ratio across the shock, and hence a functionOof the incident shock strength only. Equation 2-1

OX

de

_w -1 (2-1) T is implies (2-2) The above result which attempts to approximate the actual confi guTat ions by assuming that the curvature at the triple point of the Mach stem and reflected shock wave are continuous, fails

whenever the shock wave "topology around the triple point is not analytic" (Ref.29) and is confirmed by experiment. Whitham's theory (Ref.17) gives better agreement with L. G. Smith's experimental re-sults (Fig.4-9).

Gvozdeva and Predvoditeleva (Ref.21) initiated 1nveet1ga-~ibtisLofothè~diffr~et'Oi.t strong shock 'waves (Incident shock velocity or the order of 2000 m/sec) in N2 and CO 2 . Again they noticed the appearance of the second triple point and the curled up slipstream, as in White's experiment. A vibrational excitational model is assumed in their theoretical analysis that finds good agreement with the ex-periment. The property of self-similarly is found to be valid

ex-cept in the region near the wedge apex. they concluded that the break in the reflected shock curvature is due to the fact that the

re-flected ~hock and the bow shock lie on different sides of the triple point trajectory path, and hence a smooth patching is impossible. They further argued that since the reflected shock can never be sit-uated below the trajectory path when a

=

1.4 is used in analysis, the real cause for the curvature break is due to the internal physio-chemical processes, which lower the value of Y.

Weynants (Ref.22) used even stronger shocks (M

9

=

5.2 to

11.7) in oxygen and obtained some very clear pictures (F1g. 7-lt). It was further noticed that for the very intense cases the Mach stem is quite severely distorted.

To sum up, on the theoretical side the individual process of shock reflection and flow deflection is each quite well under-stood. The overall flow field as a result of the interaction between the two processes is still unexplained except for the case of very weak incident shocks . Some speculative analysis were made to explain the peculiar configurations obtained through experiment, but none seems satisfact ory. On the experimental side the information we have is scant (especi all y for strong shocks) and not systematic enough to shed light on the theory. This was the status of the strong shock diffraction problem when the present project started.

Chapter 3: OBLIQUE SHOCK WAVE RELATIONS

Consider a standing normal shock in the laboratory frame. Let the pre-and post-shock states be denoted by i and j respectively, and n the normal velocity components . The equations of motion

connecting the two states are given by

Mass U. P.

=

_{U jn Pj} 3-1 1n 1 Moment um p₁ . + p_{1 1n} ,U. 2

₌

_iJ + _{P jU jn} 2 3-2 j Energy h. + ~ U. 2 1 1n

=

hj + ~ Ujn 2 3-3

The method of solution for the above set of equations can be found in Ref.23.

.

...

For an observer travelling with a velo city Ut in a direc-tion parallel to the shock front he will notice both the incoming and outgoing flows are now inclined to the shock front. Clearly their velocities in this new frame of reference will now be

U _i

=

(U. 2 _~n + U 2) and U _{t j}

=

(U _jn 2 + U_t 2 ) respectively.

(Fig.3-1). The three equations of motion now can be transformed by expressing U. and U

j in terms of U., U.,

ep .,

and (~. as:

~n n ~ J ~ J

Uin = U.§in epi ~

3-4

Ujn = Uj~in(epi

-

~ : )

3-5

The continuity of the tangential velocity requires

Ut = Ui öosepi = Uj60s (epi - 9_j ) 3-6

Thus the conservation laws of Eqs. 3-1 to 3-3 can now be expressed as:

3-7

+ P. U. 2

. 2ep

+ P U. 2 !::jin 2 (ep._9 ) P. Énn .

=

_Pj

~ ~ ~ ~ j J ~ j 3-8 h i + ~ U. 2 /3,in_~ 2epi

=

_hj + ~U j 2g;in 2 ( epi -9 j )

3-9

Dividing Eq.3-7 by Eq.3-6, we have an alternate form for the con-tinuity of the tangential velocity;

By assuming a of temperature P i h. ~ p. t an

ep.

= P j t an

(ep. -

9 j ) ~ ~ ~

dissociation equLHbri um model,p and pressure only (Appendix A) , = p. ( P. , T ).

.

,

_Pj

=

P. (p j , T j )

~ ~ ~ J

= _{h. (Pi' Ti) ;} h

=

_{hj (p j , T . )}

~ j _J

3-10 and hare functions i . e ;

3-11

Thus in the systems of equations Eqs. 3-7 to 3-10, we have eight variables, namely p., T., ·U., Pj' T., uj,ep., 9 .. Therefore by

~ ~ ~ J ~ J

specifying any four of the above eight quantities the set of the four nonlinear equations can be solved simultaneously. This does not mean that the solution is unique nor does i t guarante-e its existan~e at all (see Ref. 18 and Appendix B for details). When a multiplicity of roots occurs, the choice of the physically likely solution can of ten be chosen from considerations of the boundary

conditions.

Generally such an oblique shock is generated when a uniform supersonic flow of gas with velocity Ui is deflected by a sharp concave

corner with angle 9. For suitable angles of 9 a straight oblique

shock is formed

Whi~h

makes an angle 9i with thJ oncoming flow (Fig.

3-1) .

For an ideal gas Eqs. 3-7 to 3-10 can be solved analytically

and a closed-form expression can be obtained. The method of solution

can be found in text books on gasdynamics (Ref.24). For example, the

relations between the pressure ratio P

ji, ~i and 9 j a r e simply: P .. J). tan~ . J 1 + 2cotJ:l; . ). 3-12 2 M. (y+cos2~.)+2 ). ). 3-13

fue graph of P . versus

e

for constant M 's for a diatomic gas with

j). j i

y= 1.4 is plotted on Fig.3-2. This is known as the pressure ratio vs.

deflection angle shock polar. Each curve represents the locus of all

the oblique shocks possible pertaining to an-- oncoming flow M ..

).

Several points are worth mentioning .~~~ every

e

_j there exists

two possible oblique shocks with the appropriate P ji 'Se The one with

the smaller P . is called the weak solution, whereas the one with the

larger P ji ,

i~).known

as the strong solution. Physically only the weak

solution usually exists. (It can be argued t,h(att every real wedge has a

blunt leading edge where a detached shock occurs and the st rong shock

quickly decays to the weak shock case). The state j behind the straight

oblique bow shock is uniform and supersonic. For 9

=

0, the weak

solution represents a Mach line with P ji

=

1,

wherea~

the strong solution

is the case of a normal shock.

For a given Mi and y, there is a .maximum deflection angle,

(9 j) lm'à~, beyond which no solution exists. _{( 9j )} :ma·:x: as a function of

Mi and y can be found by setting 09 j/~i

=

°

in Eq. 3-13. A

transcen-dantal equation for (~i)mar is obtained, viz:

M. 2sin[

2(~.)

] ). ). max

M.2sin2(~.)

-1 ). ). rnax 2M.2sin

[2(~.)

] ). ). max + --~---~~~---M.

2(y+cos[2(~.)

]-)+2 ). ). max 2

=

0 sin[2(~. ) ] ). max 3-14

(~i)ma~ can be found by solving Eq. 3-14. Putting (~i)~i~= into Eq.

3-13, ~~j) max can be found.

(~j)m'ax ' versus Mi for y

=

1.1, 1.4 and 5/3 are plotted on

,

..

Fig. 3-3. It is evident that gases with a low Y (polyatomic ) gases)

have the largest (8_{j )mai}Y '

When

e

>( 8 )max , th ere is no solution for the straight

ob-lique shock

rela~ion.j

What happens physically is that region jeeases

to be a uniform flow, signals now propagate upstream. To accommodate

the more severe deflection, the portion of the previously straight

attached shock at the ·corner curves and beao~es more normal to M .. This process continues until i t is normal to M. corresponding to~a

specific wedge angle (8 j) . Further increase ~of

e

will force this

o j

shock to detach from the corner and propagate upstream to increase

the effective shock strength with respect to M. in order to achieve

the more severe compression. Therefore, the ~bow shock is straight

a t t ached, curved a t t ached, and curve d det ache d in the range s 8

j

<

(e.)

. J

( 8.) . and

J 0 ' 8 j

>

(8 j )0 . respectively.

The solution of Eqs. 3-7 to 3-10 is exp~cted to be more

com-plex for an imperfect gas. Although by specifying any four of the

eight variables the set of four nonlinear equations is solvable, a good

choice of the given parameters can greatly simplify the method of

solution and P., T;, U. and

P.

were chosen in this case.

~ ... ~ J

By eliminating the terms Ui sin CPi and U j Sin(4) i - 8

j ), Eqs.

3-7 to 3-9 are reduced to:

o

_3-15

Che trivial solution is l-P_ji = 0, or Pi

=

_{Pj ' the} solution for a Mach l ine . This is discarded due to the ~resence of a finite 8

j in our

problem, Since P.

=

P. (P. ,T.); h. - h. (P. ,T, ) are known from p.,

~ ~ ~ ~ ;I. ~ ~ ~ ~

Ti_ Pj is given, '-;',:" therefore Pj =' Pj(Pj,T_j )

=

_{Pj(T.),h j}

=

hj(Pj,T.)

=

J . J

hj(T

j ); F is a function of Tj only. Thus F

=

F(T.)

=

[p.-p.J[l + P .. ] -2p.(h.-h.)

J 1 J J~ J ~ J

o

3-16

Equation 3-16 must be solved through iteration for T (Ref.23 and

Appendix C). Once Tj is known, P ., h. can also be

f~und.

The rest of

J J

the quantities are subsequently determined by:

-1

cp

.

=

sin ~ p.p .. (B .. =-l) 1 Jl -J~ 2 -' P.U. (p .. -l) 1 ~ Jl -1 8. =

cp

.

-

tan [P .. tan

cp.]

J ~ Jl ~ U. U. sin

cp.

1 1 P .. s int

cp.

-

8 . ) 3-17 3-18 3-19

For oxygen at Pi = 15 mm of Hg and Ti = 3000K, various oblique shock relations are plotted on Figs. 3-5 to 3-7. By comparing the Pji vs . 8_j curve on constant M~ for ideal (Fig.3-2) and imperfect gas

cal-culat~on, we observe that ~8j) ~ax is generally larger in the imperfect gas case. This is further illustrated in Fig. 3-3 where (8:1) ma::ic vs. Mi

are plotted for both ideal gases and imperfect oxygen. Fof very low

Mi's the oxygen curve coincides with = 1.4 curve. As Mi increases i t deviates towards the ,= 1.1 curve, hence " is capable of sustaining a much larger (8 j ) ~~i~ For an imperfect gas the lowering of , is due to

internal degrees of excitation and dissociation effects. This is

dis-cussed in Appendix A.

In shock tubes the freestream flow Mi discussed above is

generated by the nonstationary flow behind the incident shock Ms' Thus

tor an ideal gas Mi is dependent on Ms through

M

2 1 + ,-I M 2 )

M,

s 2 s ~ 2(,-1) (,M 2 +1) 2

(

,M

2

-

₂

,-I ) s 2 (M -1) 5 (,+1)2 M s 3-20 s

For an imperfect _{gas Mi,Pi,T i are found by knowing Ms and the} state parameters ahead of i t. Two nonlinear equations in two unknowns Pi' ,Ti are solved simultaneously to give the solution (Ref.23). We have p otted (8_{j )} max vs. M both for ideal gas with,= 1.1,1.4 and 5/3, as weIl as for imperfect

~xygen

witp the undisturbed gas ahead of Ms having Po

=

15 Torr and Ta = 300 o K. (Fig. 3-8) . The oxygen curve exhi bi ts a peak at Ms '~~ 13.5, 'implying tha t wi th in c reas ing inc i den t sho ck strength,

the maximum deflection angle where a straight attached bow shock is

s t i l l possible, first increases, th en decreases. (The crossing of the Ms = 14, 15 curves on Fig. 3-11 is also due to this). This behaviour is due to the fact that the imperfect specific heat ~atio j~ 4a~,· the êtate behind t he incidènt, shock ipa~,'diatomic' gas .fïrst decreases from 1.4 to approximately 1.1, but then i t will increase again to a value approaching 5/3" as the gas is fully dissociated. _{Thus for a certain Ms and 8 j , an} increase in M might cause an already attached shock to detach, pro-viding an ill~strative example of the influence of chemical processes to the gasdynamical behaviour of an imperfect gas. Curves for other pr'operties, P jo' T. , and

a.

_i across such an oblique shock formed by the nonstationary flowJ€ehind an incident shock are shown on Figs. 3-9 to 3-11. These may be compared with the previous Figs. 3-5 to 3-7. res-pectively.

Chapter 4. OBLIQUE REFLECTION OF SHOCK WAVES

Consider a normal incident shock I (Fig. 4-1) colli ding with a rigid surface inclined at an angle 8_{w to the direction of propagation}

of I. Denote the states ahead and behind I by 0 and 1, respectively. Let us now move from the laboratory frame to a frame attached to the reflection point P where I meets the rigid wedge surface. The flow

in

surface and approaches I wi th veloci ty U

=

_{Us Sec 8 w at an angle} cp

=

o 0

900~w. Af ter passing I, the flow UI, in state 1, is deflected by the

shock through an angle 8₁ from the original direction of U.

(De-flection angles are positive when measured counter clockwi~e from the

original flow direction) . This deflection causes UI to approach the

wedge surface obliquely at an angle 8

1 (Fig.4-l). The situation we

have now is j~st the flow deflection process discussed in the previous

chapter. Here the equivalent freestream flow is UI' the deflection

angle 8_{1 , and the wedge apex is the point} P. Thus depending on the appropriate magnitudes of UI' _{8 1 ,} PI and Tl' we can have either an attached shock, a detached shock or just a subsonic turning (that is, a grazing incidence case where the flow is effectively subsonic be-hind a normal shock wave) with respect to the "corner"p. We will first discuss the case when an "attached shock" is possible or the case of regular reflection.

R= gular Re fle cti on

The regular reflection process is illustrated in Figs. 4-1 and 4-2 .

It is seen that incident shock wave

(I)

and the reflected shock wave

(R) can be treated fr om a stationaryfrane of reference with respect

to P by superposing the velocity vector UO. The required boundary

c_{ondition is that UI be deflected by an angle 8 2}

= -

8

1 , such that U2

is again parallel to the wedge surface ~ ~reating land R each separately,

and by using the oblique shock wave relations Eqs. 3-7 to 3-10, the equations of motion for regular reflection can be written as

For'I:

For R:

Po + Po U₀2sin2

CPo

=

1>1 + Plu:)..2sin2(cpo -8 1)

h +

~

U 2sin2 cp

=

hl +

~u12sin2(cp

-8 1)

o 0 0 0

fue boundary condition is 8

2

= -

81 4-1 4-2 4-3 4-4

4-5

4-6

4-7

4-8

4-9

Among the 13 variables defining Eqs. 4-1 to 4-9, (i.e., Po'

PI' P 2' T 0 ' Tl' T 2' U 0 ' UI' U 2' cp 0 '

ck

_{l ,} 8 l ' 8 2 ), i f f 0 u r 0 f th e mar e

known the rest can be determined. The four given parameters are usually

taken as Po' To ' 8_w

=

90 -

CPo

and Us

=

uosan

CPo

=

UoSos 8

w ' namely, the

undisturbed pressure and temperature, the incident shock velocity, and the wedge angle.

Even for an ideal gas case the solution of Eqs. 4-1 to 4-9 is

quite involved. We therefore have to resort to graphical solutions of

pressure ratio vs. deflection angle shock polars.

Knowing p~ , _{To ' U and assuming a value of PI' the}

correspond-• . - _. _ , 0

lng

CPo'

~, knd äll thé downstream properties can be determined by the

method discussed in Chapter 3 (Eqs.3-15 to 3-19). Thus a shock polar I

can be constructed in a step-by-step fashion in going from PlO

=

1 to

a PlO corresponding to the normal shock wave value of Uo' Among

all the ~IS, there is a particular one corresponding to the given ~ .

The point on the polar represents state 1. Again we choose the weakgr

solution with the lower PlO as the physically meaningful root.

With state 1 completely known, a second polar R representing all the possible reflected shocks can be constructed with point 1 as

the origin. To use the same coordinate. as the I-polar, P

21 has to be

multiplied by PlO and the deflection angle for R has to be subtracted

from 8 (Fig. 4-2b). Since 8

1 = -~., the solution for the regular

reflection is the point P

20 where the R-polar intercepts the

P/Po

axis.

The Transition Regime

Regular reflection does not always occur. When 8_{w is}

sufficie-ntly small, U is more normal to I . . This results in a weaker U

l and a

larger 8

1 , bo~h of which tend to induce R to detach from P. The

term-ination of regular reflection can be predicted from the shock polar

diagram ..

Let the strength of the incident shock increase, bu~ keep

the Mach number U of the incoming flow constant. This means only the

incident angle ch~nges. Thus point 1, the origin of the R-po~ar, is

continuously being displaced to a larger 8 value. The strong and weak

solutions on R will eventually degenerate to one point. Upon further

displacement, the R-polar ceases to intercept the pip axis, i~pl~ing

that no solution exists. 0

For an ideal gas the limit for the termination of regular

reflection (Ref.4) is expressible as a sixth degree polynomial ~e­

lating pressure ratio PlO and wedge angle 8_{w for a given} 1, viz:

,

a,nd 6 W (PlO' 8w'y)

=

L

i=O A -K(y + 1) 0 2 A4 = 1) [2-K(y-l)] Al A

₃

=

A

₅

=

0 A 2 A6 (Y+l)P lO+ 1) (Y-l)P_lO+ x

=

tan8_w

o

= 1-2K. 1) .y 4 1) 4-10

,

_, y-l y+l K - (1)-1~t2-(y+l) (1)-1)]

The polynomial W has been solved for y

=

1.1, 1.4, 5/3 with the results presented on Fig.4-3.

When the limit for regular reflection is exceeded, shock R does not "detach" fr om P instantly. Analogous to the flow deflection problem, the portion of R at P will curve first until i t is normal to U

l , then i t will advance into Ulo The fact that there is a finite transition region between the termination of regular reflection and the ons et of Mach reflection explains the experimental result that Mach reflection occurs a few degrees less in 8w beyond the theoretical

curve for the termination of regular reflection.

The transition from regular to Mach reflection can be viewed as if the reflected shock R detaches from P at a position rand moves to s (see Fig. 4-2c). The lower part of R collides with the existing incident shock I. The lower part of I that has been hit becomes inten-sified and is known as the Mach stem through the interaction. This intensification causes the stem to protrude forward into state O. Thus we now have three shocks, i.e. the undisturbed part of land R, and the intensified portion of I. The three shocks are all confluent at one point where the interaction started. This is the (first) triple point T. (Fig.4-4).

Mach Reflection

It was noted above. that the Mach stem moves forward in such reflections and the configurations is not stationary with respect to the incident shock wave I. The moment this configuration starts to develop occurs when the incident shock first hits the wedge corner. As the reflection process is similar in both regular and Mach

re-flections, a line joining the wedge corner to any point on the config-uration should remain at a constant angle with respect to the wedge surface at all times. One such point of particular significance is the triple point T. We shall call the angle between the wedge surface

and the linear trajectory path traced out by T as X(Fig.4-4).

Consider another reference frame (Fig.4-4) with Tand I at rest. The gas in state 0 now approaches I with a velocity

U

=

U csc cp at an incident angle cp

=

900 - (8 + X).

o s 0 0 w

For the flow to reach from 0 to region bounded by Rand M, the process operates in two ways in the immediate neighbourhood of T.

The gas above the triple point trajectory has to pass through two shocks land R, whereas the gas below i t only has to pass through one shock M.

From stability consideration the gas must be compressed to the same

pressure and moves along a common line S issuing from T. This line is

a slipstream, contact surface or entropy layer. We denote the region

above and below S to be 2 and 3 respectively.

To formulate the equations of motion for this triple~shock

system, we again treat I, Rand M individually using the oblique shock

solutions. They are then relate"d to each other by imposing the

approp-riate boundary conditions. Although experimentally i t is observed that state 3, and sometimes state 2, are not uniform, i t is assumed that

the equations of motion are s t i l l applicable in the immediate

neigh-bourhood of T. Therefore for the incident shock I we have:

P ta~ = Pltan(~ -8₁)

o 0 0

P U _{0 0 0} sin~

=

PIUlsin(~ ₀ -8₁)

Po + _PoUo2Sin2

~o

p~

+

Plul?Sin2(~o-81)

ho +

~

U_o2Siri2

~

o

= hl

+

~ u12sin2(~o-81)

Similarly for reflected shock R:

Plta~l

=

P2tan(~1- _{82 )}

PlUlsi~l

=

P2u2sin(~1-82)

PI +

PIU{sin2~1

P2 +

P2u22sin2(~1-82)

hl +

~ U12~1 =

_h2 +

~ u22sin?(~1-82)

For the Mach stem M:

potan ~M

=

P3tan(~M-83) poUosi~M = P3U3sin(~M-83) Po +

pouo2Sin2~M

=

P3 +

P3u32sin2{~M-83)

1 2 1 2. 2( ) ho + 2 _Uo ~M

=

h3 + 2 U₃ Sln ~M-83 4-11 4-12 4-13 4-14 4-15 4-16 4-17 4-18 4-19 4-20 4-21 4-22

where, ~M is the incident angle for U

o at the Mach stem in the vicinity of the triple point T.

The boundary conditions are:

There are eighteen variables in the set of 14 equations 4-23 4-24

..

Eqs. 4-11 to4-24, namely Po' PI' P2'

U3 ' cfJ_{o '}

C/i'

~, _{81 ,8 2 ,8 3 , Thus tne}

any four of the above 18 quantities.

p~, T , Tl' T_{2 ,} T_{3 , U , UI' U2 '}

problgm is soIvable b? knowing

For an ideal gas the above system of equations was found to

be reducible to a single polynomial of degree 10 (Ref.18). With the

pressure ratio P~O as the polynomial variabIe, the polynomial

coefficie-hts 'were taken tö be functions of y, M ,and PlO' This will be

dis-cussed in more detail in Appendix B. 0

To: solve Eqs. 4-11 to 4-24 for an imperfect gas, we generally

use p , T

,CfJ

and U as the given parameters.

· 0 0 0 0

The construction of shock polars for Mach reflection is

essen-tially the same as for regular reflection. Here, however, since we are

in a different reference frame, the incident velocity is U= U_{s sec(8 w} +X)

instead of U

=

_{U sec 8w '} From the boundary requirement o? Eqs. 4-23

o s

to 4-24, the solution occurs where polar I intersects polar R (Fig.

4-4b). Among the 10 possible solutions, there is only one physically

realistic root (Appendix

B).

Since the Mach stem is curved, state 3 is not uniform and in the hodograph plane i t corresponds to a segment of polar I (see Fig.

4-4b). As flows adjacent to the wedge surface must be moving parallel

to it, the incident angle for U at the foot of the Mach stem must be

900 in the laboratory frame. TRus in the triple-point (T) frame, this

angle, say

1J

F, will a~sume a value

1J

F = 90

0

-

X.

Thus the segment QT

corresponding to state 3 on polar I extends from the triple shock solution point to a 8

F corresponding to cfJF. (Some authors (Ref.13)

have errorneously mapped state 3 from the solution point to the pip

axis, corresponding to a cfJ

F

=

90

0

) . 0

The validity of the triple-shock theory has been verified for moderately strong shocks (Ref.ll) where chemical kinetic effects are

still insignificant. It is thus necessary to check if the same

con-clusion s t i l l holds for intense shock-wave diffraction.

The triple-point region in the experimental picture (Fig.7-lt) from Ref.22 was reconstructed from the triple shock theory by knowing

M = 11.7, p = 14 Torr, T ~. 3000K, 8 = 150 , and X = 110 • A

s . 0 0 w

dissociation equilibrium model was assumed for the oxygen gas used in

the experiment. The agreement between the theoretical and experiment al

configurations is very good (Fig.4-5) .

Determination of Triple-Point Trajectory Angle

X

It was shown that by knowing p , T , U , cfJ , the equations of

motion for a triple point confluence arg so~vab~e. °In the laboratory,

however, a set of more realistic initial parameters are p , T , as

before, since they enable us to take into account imperfegt g~s effects,

and U , 8 which specify the severity of diffraction. It is

unfor-tunat~ tha~ a direct functional dependence of U ,

1J,

on U and 8

By introducing an additional parameter X, the triple point

trajeetory angle, U ,~ can then be expressed as:

o 0

U

=

U sec(e + X)

o s w

4-25

~o

4-26

This means instead of

18,

we now have

19

variables (i.e.

U

,~ being

replaced by U , e and X) defining the set of

14

equationsOEqs~

4-11

to

4-24.

Thu~ wewnow need to specify

5

parameters, say U , e ,X,P ,

s w 0

T , as given values. This is quite undesirable since X is not known

o

before an actual experimental picture of the configuration is taken. An alternative way of solving the problem is to find an

additional independent equation. Thus with

15

equations and

19

variables,

U , e , p , T are sufficient to define the problem.

s w 0 0

It has been observed experimentally that except for strong

diffractions, the Mach stem is only slightly curved (Fig.

4-4).

As

at the foot of the Mach stem ~F =

90

0 - X, we therefore assume

approxi-mately that ~M = ~F' or

4-27

as the required additional independent equation.

The major assumption made in Eq.

4-27

is that the Mach stem

at the triple point be normal to the wedge surface such that ~M = ~F.

The Mach stem need not be straight between the triple point and the surface regions.

To solve the

15

equations Eqs.

4-11

to

4-24,

and

4-27

obviously a tedious task. However, i f we can expressX as

X = X(Po,To,Us,e_w)

is

4-28

the shock polar technique can again be applied to solve the problem graphically.

A systematic procedure for a graphical solution of

X= X (M , e ) wi th Po' T held constant was developed. The method

s w 0

of solution is as

follows:-Choose any M , then with p a n d T known, a shock polar I can

o 0 0

be drawn.

(2) Choose a spectrum of ~ 's at equally spaeed intervals (say

30

0 ,

32

0 ,

34

0 , etc) and thu~ (X + e ) =

90

0 - ~ (say

60

0 ,

58

0 ,

56

0 ,

w 0

etc) . The shock Mach number Ms is also found as M = M sin ~ .

(3) For each ~o' we can raise a second polar R at the corresponding

e l '

(4) The intersection of land R (see Fig.4-4) gives the solution. From the intersection point of

e-=l'

~M is determined. Then using Eq. 4-27 the angle X is foünd as X

=

900 - ~M'

(5) With a different M , repeat (1) through (4) with the same

~ IS. 0

( 6 )

o

Thus for n values of M IS, we have n points of (M , X) for each X +

e .

Thus X vs~ M can be plotted on cons~ant

(x +

e

)Isw as shown in Fi~.4-6. Using this graph X can be

determYned for any (M

,e ),

at the constant p , T as follows.

s w 0 0

At the specified M , each of the ( X +

e )

curve has its X as the ordinate value~ Thus a set of [X ,

e

W

=

(X + 8 ) - X] can be obtained. Thus X can be plotted agains~ 8 on th~ constant Mand X can then be determined for a partYcular 8 •

s

w

Su c h a g rap h o f

J

vs. M f 0 r con sta n t (X + 8 ) f 0 r 0 xy gen at p = 15 Torr and T = 300 K has Been plotted in Fig. ~-6. The

o8rresponding grapRs of X vs.

e

on constant Mand X vs. M on

constant

e

have also been plot~ed in Figs. 4-7 and 4-8. Ffgure 4-8 also has aWsummary of the present experimental results. It is seen that reasonably good agreement exists for wedge angles 8 of 400 and 350 over the entire range 2

<

M

<

7, but not so for small~r

e .

A comparison was made with the e~perimental results of L. G. SmYth in air (Ref.ll). The agreement is very close except at very large wedge angles. The theoretical curve of Whitam (Ref.17) was also plot-ted. It seems present analytical results gives improved agreement

(Fig.4-9). Comparisons and discussions of these curves are given in Chapter 7.

Chapter 5. DIFFRACTION OF SHOCK WAVES BY A CONCAVE CORNER -ANALYTICAL CONSIDERATION

With the aid of our previous discussion on the processes of flow deflection and shock reflection, we are now in a position to analyze the overall configuration of a normal shock wave diffracted by a sharp compressive corner in a shock tube. Several shock diff-raction structures have been observed so faro At moderate shock strength (say M

<

2.5) the configurations are those shown in Fig. 2-la, band C. sIn the case of Mach reflection, the reflected shock

R, starting from the triple point T, assumes acontinous (smooth) curvature until it finally terminates either at the wedge corner, for supersonic flow over the wedge, or on the shock tube wall ahead of i t , for subsonic flow over the wedge. When the incident shock strength increases, implying a more severe diffraction, the

curva-ture of the reflected shock in Mach reflection reverses discontinuously at a point TI (Fig.2-1d). Between Tand TI the shock is straight.

We call this portion R. The remaining portion of the shock will be called B. The combined structure of Rand B is called the R-B

shock. When the incident shock strength is further increased, an additional shock RI issues fr om TI behind R. (Fig.2-le). There is

now a second triple shock system, with T' as the triple point, and R,

R' and B as the appropriate "incident", "reflected" and "Mach" shocks.

A second slipstream S' was observed by White in one interferogram

(Ref.15) and in many photographs in the present work (see Fig. 7.1).

In contrast to the first triple-shock system, we have not established

that this second triple-shock system can be solved by the triple-shock

theory discussed in Appendix B, as it is not certain that all the

waves emanating from T' are straight and if additional waves are

llL,L

required in the matching process.

In the following an explanation is

offered for the irregular Mach reflections shown in

Fig~2.l,

and

possibly set a limit as to when one configuration undergoes a

transi-tion to the other.

Reversal in Curvature of the R-B Shock

Consider a time

T

=

0

(Fig.5-la) when the incident shock has

just arrived at the wedge corner.

The shock itself and the nonstationary

flow behind it have velocities U and Ü

_l

respectively, with U

>

0l'

For better viaualization, at thil instant "colour" all the ga=

mole-cules ahead of I blue and those behind it red, and they maintain that

colour later regardless of their state for identification purposes

only.

At a later time 5T (Fig.5-lb) the incident shock I would have

gone a distance LR • U 5T

along the shock tube measured from the wedge

corner.

Por a moment focus on the gas

~olecules

in the upper part

ot

the shock tube.

They have not yet felt the presence of the wedge

re-presented by the shock B, which is assumed to stop at the demarcation

line between the red and blue molecules for descriptive purposes only.

Since I propagates much taster than the flow behind it (the red

mole-cules), there will now be a layer

ot

blue molecules between land the

red molecules.

_{The line of demarcation is at a distance LD • 015T trom}

the corner.

Now consider the gas in the lower part of the shock tube,

along the wedge.

The incident shock I is now compressing the b.lue gas

(at rest) ahead of it on an inclined surtace, or from one-dimensional

considerations, I is now propagating in a converging ohannel.

To

meet thi. new boundary condition the incident .hook ha. to become

stronger, through a .hock refleotion prooess.

A. only blut moleoulel

have part1aipated in thil prooe •• , the relulting Ihook-r.fl.ction

contiguration conli.t. only of blut mol.cul...

prom th. prop.rty of

self-.imilarity the .hock .truatur. grow. uniformly in tim..

In

partiaular, th. r.tlected .hook R weuld have grewn a l.ngth

~R

during

e

T

terminating at the demarcation line.

Th. red molecule. Itart.d te pa •• ev.r th. w.d,. at

T·O.

Ir

O~ i~ ~uper~oniQ,

a bew Ihook re.ult..

From .imilar aon.id.ration.

a.

a~cve

we ccnclude that thi. flow d.tlection .truoture oon.i.t.

only of red moleculel.

~h.

tip

ot

the bow .heak B allo t.rminat ••

on th. 11ne of demarcatien.

For th • • ake of argument we let the

shock B be an attaahed eblique .hock wave (Fig.;-lb).

Figure '-lb, ef oour.e, i. phy.ically unreali.tic becau.e

ga.dynamic di.continuitie. do not terminate in the middle

ot

a flow

field.

An interactien mu.t take place between the •• two •• parat.

proce •••• with the re.ult that Rand B

a~.

bridg.d.

From the

int.r-ferograms for weak diffractions (M < 2.5 in oxygen) the bridging is s

very smooth. However, for stronger waves a kink develops between R and B. We denote the point where Reeases to be straight as T'.

Sinee regions 2 and 6 may be considered as being formed through two independent processes, their flow properties, in particular the

pressure, must be different. This pressure difference is transmitted with the loeal speed of sound between (2) and (6) with the result that the pressure field is adjusted from P6 to P2 in a continous manner. This adjustment zone forms the bridging part of the B-R shock. Thus the type of bridging depends on P2 and Ph and the pressure distribution between them. As the two processes of snock refleetion and flow de-flection are self-simi1ar, the configuration of shock diffraetion as a result of their interaction is also self-similar. As the bridging part of B is more normal to U (in the frame T) than R, the flow

pressure P6 below it must be fligher than P2· Thus P6> P2 is required for the formation of sueh a kink. For strong shocks, i t is shown in Appendix E that for an ideal gas P6 is usually greater than P2. How-ever, it wou1d be wel1 to confirm quantitatively the model given above.

Summarizing, several conditions are needed for the occurrence of a curvature reversal:

(1) The nonstationary flow

Ü

_l behind I m~st be supersonic so that a bow shock can be generated. When U

l is subsonic, there is no bow shock and R is continuously attenuated until it terminates on the shock tube wall.

(2) For P6>P2 (or p > P2 if B is detached, Fig.2':1) requires the incident shock t6 be sufficiently strong. The exact magnitude has not been found.

(3) The flow behind R must be supersonic with respect to the triple point T, i.e. M

2 >1. If M2< 1, a sonic disturbance will influ-ence all of reg10n .. 2. The adjustment of the pressure gradient will start at the triple point T. (Thus, the straight portion of R-B vanishes and we can say the point of curvature reversal eoincides wi th T). The requirement of M

2 > 1, however, is

auto-matically satisfied if condition (1), Wh1Ch requires a much stronger incident shock, is satisfied. For a )'

=

1.4 gas,

M

2> 1 occurs when M >1.45 (Ref.13); whereas M2> 1 occurs when

M > 2.:tI·C1~;::' ' .. ':.:1 .. s ;:: .1. It should be noted that for M

2 <1

r~gion 2 is nonunif~rm, having a different pressure from that calculated from the triple-shock theory and the orientation of R wi 11 al so change. Thi s i s prob ably the re as on for the discrepaney between the theoretical and experimental results for weak shock diffraction (Ref.ll).

From a consideration of the criteria (1) and (3) alone, it seems the onset of curvature reversa1 should occur for moderately strong shock waves, 2 <M < 3, and is verified by experiment.

Location of Point of Curvature Reversal reversal blue and of T I is and L D

=

An inspection of Fig. 5-lb indicates the point of curvature

T' is in the neighbourhood of the' line of demarcation of the

red molecules. Thus a reasonable estimate for the location

the location of the demarcation line. Noting LR

=

U 5T

- S

U 1 5T, we obtain

5-

1

It should be noted that the approximate value obtained ·from

Eq. 5.1 is always less than the actual measured value. (see Fig.

7-3).

The Relative Velocity of Tand Tl

From the assumption of self-similarity, point T' is receding

from T at a constant velocity V. From simple geometrical

considera-tion, (Fig.5-2 and Eq. 5-1), the magnitude of this velocity is found to be

where rf,

'1-'0' point T.

5-2

CPl and

e

are obtained from the analysis of the first triple

From self-similar consideration the path ,traced out by

every point on the Mach reflection structure, in particular T' must

be inclining at a constant angle X I with respect to the wedge surface.

An indirect way to look at this argument is that since T' is moving at a constant velocity with respect to T, but T itself has a constant velocity in the laboratory frame, therefore T' is also moving at a

constant velocity in the same frame. The angle X I between the linear

trajectory of T' and the wedge surface can again be found geometri-cally as follows:

U

tan(x +

e

)

w (1 - + ~ tan(x +

Ü~

·

e )

w

5-3

The relative velocity of T' with respect to T is important

in that if we want to do an analysis around T', we have to.work. in

a frame of reference where T' is at rest. This is precisely the same

reason why we have to transform from the laboratory frame to the first triple point T frame to do an analysis for the initial Mach

reflection. In both cases the equations of motion are thereby

changed from the nonstationary case with three independent variables x, y, t to the pseudo stationary case of two independent variables

x/t and y/t (Ref.16). To transform from the T to the T' frame, V has

to be substracted vectorially from all the velocities computed fr om

the T frame. All the scalar quantities remain invariant. The flow

in Fig. 5-3. Further discussions on obtaining state

4

and its relation to actual data is given in next section.

Denoting the vector quantities in the T' frame by "primes", the relations of the velocity vector in states 1 and 2 in the two frames are;

1

lUi I

= [

(IUllcos~l-

Ivl)2+ (

IUllsin~1)2

J2

~'

=

tan-l 1 -1 8 I

=

tan 2 lUl I sin

~l

Appearance of the Second Triple Shock System

5-4

5-5

5-6

5-7

5-8

In analyzing Mach reflection, we have restricted ourselves

to the vicinity of the triple point T. It is therefore worthwhile to

investigate what takes place around T'. In order to do so an

under-standing of the behaviour of the slipstream S is required. The slipstream is generated by the two flows U

2 and U3

~ gliding past each other. To analyze the present problem U₂ and U

3 have to be transformed to the laboratory frame where the wedge is at

rest. The transformation is simp1y to subtract U from U and U

vectorially. The re1ations between IU

2 I , I U3 I, °and I Ü2

f

and PÜ3 I

are;

When the flows meet the wedge surface they have to turn at

the surface to meet the boundary condition. If the flow veloeities

with respect to the wedge is subsonic, a continuous turning is

possi-ble. If the flow is supersonic, a bow shock wil1 be generated. (We

are indebted to Dr. L. G. Gvozdeva of the Institute of High

Tempera-ture, Moscow, U.S.S.R. for this suggestion). Thus when M

2

>

1, the generation of such a bow shock R' (see Fig. 2.1c) forms another

triple-shock system with the origina1 shocks Rand B. Therefore,

we conclude that the second triple-shock system appears when the

flow in region. 2 becomes supersonic with respect to the wedge.

shock theory of Chapter

4.

In that theory, once we are given the upstream conditions ahead of the "incident shock" R, the problem is solved with the orientations of RI and B uniquely determined (see Fig.

5.3). The regions around the triple point TI are all uniform. In

our present scheme RI is formed in order to satisfy a gasdynamic.

boundary condition. Moreover in actual fact (see Fig. 7.1k) the

re-gion behind R; even in the immediate neighbourhood of TI, is

non-uniform. The flows behind RI and B adjust themselves subsonically

until flow stability is achieved with a resultant second slipstream SI If the region around TI is describable by the triple-shock

theory, we have sufficient initial conditions (i.e. PI' Ti, Ui, ~i

from Eqs. 5-5 and 5-6) for a solution. The procedure is as follows

(see Fig. 5. 3): with the analytically determined value of

X

we can

analyze the region around T using the triple-shock theory, then

employing the foregoing transformation scheme between Tand TI, we

can further analyze TI again using the triple-shock theory. The

solution around TI is given by the intersection point

(4

1

,

5

1 ) of

shock polars RI and R in the TI frame. It is worth noting that the

R polar in the TI-frame has the same PIjp as the R polar in the TI

frame. The polars are translated throughOthe addition of the

velo-city vector V, which also changes the shape of the RI polar in the

TI-frame. That is, common flow quantities P2' and PI are.unchanged

by such a transformation. However, i t is unTortunate that the

sec-ond triple shock system is not well-behaved enough to take advantage of the above analysis.

Lastly, since the Mach stem M is nearly a normal shock

with respect to U

,U

will be subsonic (see point 3 in polar

solu-tion Fig.5.3) andowi11 re sult in a smooth turning at the wedge

sur-face.

Strong Shock Diffraction with Regular Reflection

Since for strong incident shock waves the pressure behind the reflected shock is much stronger for a regular reflection (Fig. 7.1b) than for a Mach reflection, P2 will generally be greater than Ps (Fig.2.1a), and a kinked shock wlll not occur in regular reflec-t1on.

Chapter

6:

Motivation.

DIFFRACTION OF A SHOCK WAVE BY A CONCAVE CORNER-EXPERIMENTAL RESULTS

Since several postulates were raised in the previous chap-ters concerning the diffraction of a strong shock wave by a wedge, i t was necessary to check their validity against experimental

re-sults. However, so far the experiment al information on this

sub-ject has been scant. The interferograms and density contours

ob-tained by White (Ref.15) were conducted at low shock Mach numbers

where chemical kinetic processes are s t i l l unimportant. Gvozdeva

et al (Ref.21) performed experiments up to M

=

8

in different gases,

but the information obtained was s t i l l inade~uate. Moreover, it

ex-perimental verification. In 1967·Weynants (Ref.22) took a few shadow-graphs of strong shocks (up to M = 11.7) diffracting over a 150 wedge in oxygen. The configurations a~peared to be complex and interesting enough to warrant further experimental investigation.

Instrumen tat ion

The UTIAS 4 x 7 combustion-driven hypersonic shock tube (Ref. 26) was used to generate the incidept shocks. A Mach-Zehnder inter-ferometer was used for recording the nonstationary process. The

light source was a Ruby rod laser. 'Simultaneous dual-frequency inter-ferograms were taken 'with wavelengths of 6943R and 3471R, respectively .

. Kodak RS Pan film sheets were found to yield the best results.

Wedges with e = 250 , 350 , 400 and 600 were used. The wedge was fastened to the lowlr wall· of the working section of the shock tube (Fig.6.1a). Although this arrapgement tends to introduce bound-ary layer interactions at the w~dge corner, i t was adopted owing to

simplicity in design and stability du~ing impact of a strong shock wave. The model (Fig.6-lb) used by Gvozdeva et al (Ref.21) elim~nates the

boundary layer interaction at the corner, but at the same time intro-duces a more serious gasdynamic problem. When the bow shock at the wedge corner becomes detached, the flow field beneath the wedge will influence that above, hence. disturb~ng the region of interest. A symmetric. wedge (Fig.6-lc) would el'iminate the above problem,. but.

this model occupies a larger volume, thereby generating greater dis-turban ces that could choke the flow ahd even generate hig~ tempera-tures to crack the window surfaces. Another model (Fig.6-ld), an

asymmetrical wedge with wedge angles c e ) l and (e )2 has the advantage of offering two sets of data[ (M , (e )lj), [(M ,

(if

)2)] for each .run. However, i t has the. combined di~adva~tages ofsmodeYs ~ and c.

The commercial bottled oxygen gas used was 99.9% pure. Experimental Set Up

For each ofthe 250 , 350 and 400 wedges, experiments were run for incident shock Mach numbers between approximately 1.9 to

8.

We did not venture to go beyond M =

8,

as in similar experiments:

(Ref.27) previously conducted_ wit~ the same facility, the .high quali ty optic-al windows were burned. The origin of such d!:J,mage is thought to be fr om the further compression of the gas behind the incident shock by the wedge. The rea~tivity of dissociated oxyge~ mayalso have contributed.

Whenever possible the initial pressure p was kept at 15 Torr. The initial temperature T bein'g' the labora~ory teinperature,

o 0 60

whLch was usually be.tween 22 C to 2 C • . The nea·r constancy of the initial temperature and p~essure' were ma~ntained to compare the experimentally determined X with the theoretical curve of Fi~.

4.6where X = X ( M , e , p ~T)=X(M,e )whenp = 15 Torra.nd

T = 3000_{K. s w 0 0 s w 0}

o

Oxygen was used for most of the experiments. Several 'runs in N