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J. S. A tr . In st. M in . M eta l/., vol. 91, no. 4.A pr. 1 99 1. p p. 1 37 -1 44 .

The propagation of shock waves in a complextunnel system by B.W . SKEW S. and W .R. LAW .

SYNOPSIST his paper d escribes an inve stiga tio n into the b ehav iou r of a sho ck W 8Y 8p ro pag atin g thro ug h a sy stem of b ranch edducts such as can occur In an underground m ining complex after an explosion. A simple numerical model thatp red icts flo w pro perties w ithin th e d ucts w as evaluated by use of p ressu re m easurem ents and sch lleren ph otog rap hyof the transient flow w ithin the ducts. The results returned by the m odel com pared w ell w ith the m easured values,a nd v erif ie d th e m od el's a bility to p ro vid e a n a pp ro xim ate e stim ate o f s ho ck -w av e b eh av io ur In p ro pa ga tio n p ro blemsof this type. The experimental results revealed the establishm ent of complex flow pattems In the ducts after thea rrival of the Inciden t sh ock w ave. T he av erag e pressure Inside the test section rose to app ro xim ately do ub le th epressure behind the Incident shock. This pressure acts as a source to drive shock waves Into other sections ofthe duct system . It Is shown that the series of shock waves emerging from the test section Into an adjacent tunnelc oa le sc e to fo rm a n e ve n s tro ng er wav e, a nd th at th e a tte nu atio n o f t he wav e th ro ug h this typical tunnel geometryI s s ur pr is in gly sma ll.SAM~V A T TIN GH ie rd ie r efe ra at b es kr yf 'n ondersoek na die gedrag van 'n s ko kg olf w at in 'n ste ls el v an v erta kte to nn els v oo rtg ep la ntword, soos kan gebeur In 'n ondergrondse myn kompleks as gevolg van 'n o ntp lo ffin g. 'n E en vo ud lg e n um erie semod el om d ie v lo el-e ie ns ka pp e b ln ne d ie to nn els te vo ors pe l is g e6 va lu ee r d eu r v an d ru km etin g e n s ch lle re n-fo to gra fievan die vloei in die tonne Is gebruik te maak. D ie resultate wat d.m.v. die numeriese model verkry la, het gunstlgvergelyk met die waardes wat deur meting verkry Is en het bewys dat die model oor die verrnoi besklk om 'naanduiding te gee van die gedrag van skokgolwe wanneer hulle voortgeplant word, in sltuasles van hlerdle soort.Die eksperlment se resultate het die ontstaan van komplekse vloeipatrone In die tonnels na die aankoms van dieinvalskok getoon. D ie gem iddelde druk in die toetssegm ent het tot ongeveer dubbel die druk agter hlerdle invalskokgestyg. Hlerdle druk het die gevolg dat skokgolwe In ander dele van die tonnelstelsel Ingedryf w ord. D ie skokgolw eko mb lne er o m 'n sterker golf te vorrn, en die verswakklng van die skokgolf In so 'n tip le se to nn els la te em la v er ba se ndklein.

IntroductionK now ledge about the beh avio ur of shocks in branchedducts has im portant applications in m any engineeringfields, including the transmission of oil and gas inpipelines, the inlets and exhausts of reciprocatingm achines, the design of buildings in w hich to store explo-sives and, conversely, the design of structures to protectpeople from explosions. In the m ining industry, the studyof the behaviour of shock w aves in a system of branchedducts can serve as a basis for the prediction and controlof explosive fronts in mine tunnels. The geom etry of atunnel system can be a dom inant factor in determ iningthe m agnitude of the pressure increase associated w ithan u nd erg ro un d e xp lo sio n.In 1959, at the President Steyn Gold M ine, such anaccidental explosion caused a stron g shock w ave to m ovedow n the m ain haulage tow ards a stope and its series ofcross-cuts.. The inter-connecting tunnel system at theend of the haulage is indicated in Fig. 1. The severity ofthe damage caused by this blast on either side of thissystem suggested unexpected behaviour by the shockw aves in negotiating corners, branches, and dead ends.The exp er imenta l te st s ec tio n u sed in th e wo rk d es crib edin the present paper w as an idealized geom etric m odelof the portion of the m ine referred to above. T he invest-igation w as a direct attem pt to get an understanding, bo thby experim ent and through num erical m odelling, of the. U nive rsity o f the W itw atersran d, P.O . W its, T ransv aal.@ The South A frican Institute of M ining and M etallurgy, 1991. SAIS SN 0 03 8- 22 3X /3 .0 0 + 0 .0 0. P ap er re ce iv ed 4 th S ep temb er, 1 99 0.

R ET URN A IR WAY

HAULAGE

Fig. 1-The geometry of the m ine tunnel

behaviour of shock w aves as they m ove th,:ough such atunnel system . It is a simplified model and does notattempt to reproduce all the factors involved in theoriginal accident. It does not, for exam ple, take acco unto f b la st p ro paga tio n a cr os s th e in te rc onne ctin g c ro ss -c utsbetw een the haulage and the return airw ay, situated tothe left of the system shown in Fig. 1.T he o bje cts o f th e in ves tig atio n were two-fo ld : firs tly ,using schlieren photography and high-speed pressurem easurem ent, to investigate the behaviour of a shockw ave propagating through a system of branched ductsand, secondly, to evaluate an approxim ate num ericalm odel to predict shock-w ave M ach num bers, from whichpressures, tem peratures, and gas velocities w ithin thesystem boun daries can be calculated.

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EquipmentThe basic requirem ents of the laboratory apparatusw ere to generate a shock wave under controlled condi-tio ns, an d to fe ed it in to th e tes t sec tio n, whe re its s tren gthc ould be mea su re d at se ve ra l lo ca tion s. S chlie re n p ho to -graphs of the shock profiles and other flow phenom enaw ere also required. A plane incident w ave w ith uniformflow behind it was desirable, so that all the transversew aves and tw o-dim ensional effects observed w ere a con-sequence of the system of branches and were not con-founded by w aves that could arise further upstream .A conventional shock tube w as designed and built togenerate the shock w ave. T he tube w as 6 m long (a driversection 1 m in length and an expansion section of 5 m),and had a constant square cross-section of 20 by 20 mm.S eparating these tw o sections w as a thin diaphragm m adefrom cellulose acetate sheet. T he driver section could bepressurized to 1,5 M Pa. The expansion section rem ain-ed at atmospheric pressure (approximately 87 kPa).D ia ph ra gm bu rs ting was ac comp lish ed thro ug h a n atu ra lp re ssure b urs t o r e le ctric ally . This la tte r meth od in vo lv edpainting a 'bow tie'-shaped pattern on the surface of thediaphragm w ith conductive spray paint. T he diaphragmwas then installed in such a w ay that a suitable voltagecould be sw itched across it. T he m axim um resistance atthe centre of the pattern caused a sm all hole to be burntin the centre of the diaphragm once the switch had beenclosed. T he diaphragm , w hich w as highly pressurized attha t sta ge , sh atte re d, res ultin g in a p lan e s hoc k wav e mov -ing dow n the expansion section.The test section, or m odel, consisted essentially of avertically m ounted cylindrical 'slice' of m ild steel, ofdiameter 300 mm, into which the ducts (20 by 20 mm incross-section) had been m achined. It could be used in tw om odes: in pressure m easurem ent, for w hich a pressureplate containing transducers w as mounted on one sidew ith the other side blanked off, and in schlieren photo-g rap hy , where op tic ally cle ar g las s p late s w ere mou nte don both sides. Fig. 2 show s the test section, in schlierenmode, installed at the end of the shock tube. For thepressure m easurem ent, tw o-transducers (PCB 113A 21)w ere m ounted in ports along the expansion section of the138 APRIL 1" 1 JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY

F ig . 2 - T he te st s ec tio n in sta ll-ed at the end of the shock tube

tube, and another five in ports on the pressure plate posi-tioned to coincide w ith points of interest in the ducts. T hetransducers were linked to a computer-based data-a cq uisitio n sy stem, whic h allowe d p re ssu re -tim e d ata tobe recorded and analysed.S ch liere n p ho tog ra phy was ac hiev ed by u se o f an a rg on-jet light source, w ith a flash of 0,3 p,Sduration linked toa delay unit calibrated in m icroseconds and triggered byone of the transducers m ounted in the expansion sectionof the shock tube. Two 250 mm parabolic mirrors anda knife-edge and lens system produced the schlieren ef-fect, which w as recorded either on positive/ negativePolaroid instant film or 35 mm colour film . T he opticald en sity o f a sc hlie re n im ag e is p ro po rtio na l to th e d ens itygradients w ithin the gas, and is thus an ideal m ethod forthe v isua li za tion o f compress ib le gas dynamic phenomena.

TheoryThe theory is based on a m ethod proposed by H eilig2for analysing shock m ovem ent through branched ducts,e xte nd ed to c as es w ith n on -symmetric a nd obtu se ly ang le dbranches. It is an approximate method that is beingevaluated in parallel w ith high-resolution num ericals ch emes , wh ic h a re compa ra tiv ely exp en siv e in compute rtime and are not reported on here.The aim of the method is to predict, independentlyfrom experim ental data, the intensity of the shocks in thebranches of a junction as a function of the intensity ofthe incident shock and geometry of the junction. Theprocedure is illustrated below for a sim ple 450 junctionas shown in Fig. 3, and can be summarized as follows.

(1) The shape of the diffracted shock at the instant w henit reaches apex R can be calculated by W hitham 'stheorY of shock dynam ics.(2) The assum ption is then m ade that the flux of energyow ing to a curved part-shock spanning the inlet toa branch is the sam e as that for w hen the part-shockhas sm oothed out into a plane shock in the branch.

W hitham 's theory is an approxim ate treatm ent thatgives inform ation on the shape of the diffracting shock.

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Fig. 3-A n exam ple to show the application of H elllg 's m ethodThe sho ck po sition s a t different tim es an d th e o rtho go naltrajectories of th is se t, term ed rays, fo rm an o rth og on almesh. The propagation of the shock between twoneighbouring rays can be treated as if the rays w ere solidw alls, L e. the ray tube s a re reg arded a s ch ann els. C hesterand Chisnell4 developed a 'channel form ula', given byA = A(M), in w hich the cross-sectional area A is adecreasing function of the M ach num ber M of th e sh ockmoving down the channel. W hitham 's theory does nottreat the afterflow region of the diffracted shock wavesince the particles can be considered to follow a ray onlyin the neighbourhood of the m oving shock w ave. It alsoignores the effect of this perturbed region on the diffra-tion geometry. The accuracy of the method has beenevaluated b y Sk ew s5.For the present illustrative case (Fig. 4), W hitham 'stheory enables the goemetry of the shock diffractedaround the inlet corner L of the junction to be calculatedat a fixed time for a given primary mach number, Moand a corner angle, 'Y . I n this diffraction process w ith'Y = 45, a sim ple w av e originating at L spreads ou t onthe plane shock and causes its curvature betw een Po an dPz, The angle e of a ray with the x-axis d ec re ase s in th efan from zero at Po to a negative final value at its endPz. Likewise, the shock m ach number, M, decreasesfrom Mo at Po to Mz at Pz. The diffrac ted sho ck leav in gthe fan at Pz is p la ne a nd mee ts th e w all p erp en dic ula rly .For all Mo, the las t charac te ri st ic , LPz, n ev er re ac he s th ewall, Le. the expansion fan ends within the flow field.The f ir st charac te ri st ic , LPo, is inclined to the x-axis atang les no t greater than 23 ,9 for all prim ary sh ock -w aveintensities.A fter transform ation onto C artesian coordinates, the

Mo Po

M"'--

Fig. 4-D lffractlon of a plane shock w ave around a corner

an gle e of ray inclination is related to th e shock Machnumbe r b y th e in te gra le(M) = I:o :~) , """"""""""""""""" (1 )

where W(M) is obtained fromW(M) = ../0,5 X (M z - 1) X K(M). (2 )

K(M) is the Chester function, and comes from theChe ste r-C hisn ell c ha nn el fo rmu la ,K(M) = 2 X[(1 + k~ 1 X

1 ~ILZ)(2 xlL+ 1 +M-~ rl,. (3)w here the variable IL is defined by

l = (k - 1) X Mz + 2 , " (4 )2 X k X Mz - (k - 1)an d k is the ratio of specific heats. The slope IL(M) ofea ch ch ara cteristic in the linear characteristic fan (this ILis different from the IL used by Chester) is given by

IL(M) = tan[e(M ) + m(M)], " (5 )1where m(M) = arctan [M X W(M)] .." (6 )

T he value m(M) is analogous to the Mach angle insteady supersonic flow . W ith the line elem ent ds(M)betw een tw o rays, the equations for the space variablesx an d y of the shock at time taredx = - sine X ds (7 )

an ddy = cose x ds. , (8)Finally, the initial values are

Xo = Mo x 01an d

Mo

Yo = IL(MJ x Mo x 01 .T he factor proportional to tim e, 01,indicates that thesh apes o f the diffra cted w av efron ts at d iffe ren t tim es aregeom etrically sim ilar w ith L as centre. T he integrationof equations (1), (2), (7), and (8) for a specific prim aryM ach num ber and wall angle can then be perform ed. Thevalue M1 comes from IL(M1) = 0, and M z from e(Mz)

= - 'Y.T hese geom etric 'boundary conditions', as theycan be called, w ill differ from every junction geom etry.T he second item in the theory m entioned above is thatrelated to energy conservation. As shown in Fig. 3, thediffracted shock splits up into two part-shocks, each ofwhich enters its own branch. The part-shock wave isassumed to smooth out and become plane. (This doesoccur in practice through the mechanism of multipleM ach reflections.) It is further assum ed that the energyflux of a part-shock is equal to that of its sm oothed outsta te. In M ach reflec tio n, pa rt o f th e reflected sh ock w aveovertakes the initial shock w ave from behind, its energybeing added back to the wave.Nevertheless, there are several energy losses notJOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY 13 1PRIL 1811

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included in the assumption based on an ideal gas, e.g.the energy lost by w all friction, heat conduction, or realg as e ffe cts. A ls o d ire ctly n eg le cte d in th e ap pro xim atio nis the energy of the part of the reflected shock that m ovesaw ay from the initial shock, the energy associated w iththe expansion w ave m oving upstream , and the energy ofany secondary shocks in the branches. It is assum ed thattheir energy fluxes should be low com pared w ith that ofthe dominant transmitted and deflected shocks. For atw o-dim ensional junction with depth 'T and usingR ankine-H ugoniot's equations w ith 1,4 as the ratio ofspecific heats, the energy assum ption m entioned aboveyields the termZ

E(M) = ~ x (M3 - 4 x M + 18zXM )x s x 'T.(9)6 M + 5M is the M ach num ber on a part of the shock w ith lengths , w hile E (M ) is th e energy increase in unit tim e no rm al-ized by the velocity of sound and the pressure ahead ofthe shock. That it holds true is the main assum ption onw hich this num erical m ethod is based.W ith reference to Fig. 3, the follow ing conservationequations hold in the main branch of the junction andin the side branch respectively :rM oJM(E(M) x ds + E(Mo) x (b - yo> = E(MT) X b (10)rM lJMzE(M) xds+E(Mz} xd= E(MD) x b, """ (11)where b = Xl X sin 'Y

d = IYz - xz l /.J2"for 'Y = 45.The plane portions of the total diffracted shock w ithth e le ng th s b - Yo an d d w ere described above in con-nection w ith the expansion fan. Equations (10) and (11)

are then solved for MT an d MD, these being the shockM ach num bers of the sm oothed-out deflected shock inthe m ain branch and the side branch respectively.B ased on this theoretical concept, all the junctions inthe m ine test section could be analysed. A ll that variesare the boundary conditions relating to the particulargeom etry of each branch.R es ults a nd D is cu ss io nFigs. 5 to 8 show schlieren photographs of the flowa s th e sh oc k-w av e s ystem p ro pa ga te s th ro ug h th e v ario usducts. These photographs were taken for an incidentshock wave w ith a Mach number of 1,38. The time in-

terval betw een the pictures is approxim ately 32 I-I-S .Fig. 5 a is the shape of the w ave predicted by W hitham ' stheory, and Fig.5b is the corresponding schlierenp ho to gra ph . T he c orre la tio n b etw een th e p re dic te d s ho ckgeom etry and that ob tained experim entally is good. A lsovisible in the photograph (and neglected by the theory)are tw o expansion w aves that em anate from each of thecorners. As the flow behind the incident shock wave issubsonic for the relatively low incident shock strengthte ste d, th ese w av es mov e u pstre am in a d ire ctio n o pp os iteto that in w hich the incident shock is m oving, the frontof the w ave being an acoustic pulse travelling at the localso nic v elo city . T he y c arry a 'm ess ag e' to th e ap pro ac hin gflow that an expansion has just occurred. W henever theshock expands in the rem ainder of the system , furtherexpansion w aves w ill be generated in a sim ilar m anner.

(a )

.....

(b )

F ig. 5 -(a) N um erical pred ictio n of a shock-w ave p ro file(b ) T he c orresp on din g sch liere n p ho to grap hIn F ig. 6 (30 I-I-Sater), the sh ock h as expand ed in to th efirst ju nction. T he oval-shaped w ave is th e reflection offthe apex. T his reflection continues to grow in size untilit is itself reflected off the side w alls, form ing secondaryand tertiary reflections that weaken and eventuallydissipate in the turbulence of the system .S om e 60 I-I-Sater, as show n in Fig. 7, the shock m ovesin to the secon d junctio n. A strong vortex , visible as a darkm ass of rotating fluid, form s as the shock negotiates theobtuse corner. A n expansion w ave is again generated,

moving in a direction opposite to that of the incidentshock. The shock w ave reflecting off the far w all of thisjunction appears w hite in the photograph since the den-sity gradient is in the opposite direction to that of thein cid en t w av e.A t th is s ta ge o f th e flow, a n in te re stin g fe atu re d ev elo psin the inlet junction attached to the low er corner of theentrance duct and inclined to the axis of this duct ata pp ro xim ately 2 0 . T his s tro ng d en sity g ra die nt is 'n ea rlystationary w ith respect to the junction. Furtherm ore, aph otograph taken 321-1-sater, F ig. 8, shows tw o of thesefeatu res, lying parallel to each other and still em beddedin the inlet junction. W hile practically all the otherfeatures visible in the schlieren photographs can bee xp lain ed in te rm s o f re fle ctio ns, e xp an sio ns , o r v ortic es,these features require further study. A t this stage of the

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Fig. 6- The Initial w ave reflection In the firstJunction

Fig. 7-W ave Interactions In the secondJunction

Fig. a-Flow features In the entrance region

process, the flow separation at the sharp corners of theentrance junction has developed into a turbulent patchspanning the full w idth of the duct. This turbulencequickly grows in magnitude until it engulfs the wholesystem, making individual shocks very difficult todistinguish.

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Fig. 9 shows a trace from the pressure transducerlocated in the low er branch of the first junction. T he firstpressure rise is due to the passage of the initial shockw ave. T his is follow ed by a short tim e delay as this w avetravels to the end of the duct, is reflected, and propagatesback over the transducer, giving a further pressure in-crease. This reflected wave results in a pressure on theback w all that is equivalent to tw ice that behind the inci-dent shock, w hich has itself approxim ately doubled thepressure ahead of it. The strength of the reflected w aveis determ ined by the boundary condition that the flowat the end face of the duct m ust have zero velocity. Sincethis reflected w ave m ust bring flow at nearly 100C andmoving at over 600 km lh to rest, the pressure ratio acrossit is understandably high.

LUa:::>enenLUa:a. .

4 =>y0 800 1600 (uSec)40 0 3200

TIMEFig . 9 -P artlcu lar p ressu re ch arac teristics o f reflectio nThe subsequent fluctuations of pressure can be cor-related w ith the w ave patterns in the schlieren photo-graphs. The general trend of pressure is, however, ofmo re s ig nific an ce . F ig . 1 0 shows th e r esults o bta in ed fromthree of the transducers, superim posed. T he transducerlocation w ithin the ducts is not im portant, since all thetransducers, over the long tim e base represented in thistrace, show the sam e general behaviour. It is noted thatby about 3,6 ms after the shock wave first enters thesystem , the average pressure inside the duct system has

risen to approxim ately double the pressure behind theoriginal shock. The interaction of all the w aves in all thebranches of the test section thus reaches a generalm axim um after the incident shock first arrives, produc-ing the general doubling of pressure over that behind theincident w ave. T his high-pressure source can then act todrive a shock down the exit leg of the duct.T he basis of the num erical m odel, as presented earlier,involves a m ajor explicit approxim ation-nam ely, theneglect of all reflected w aves. Judged from the schlierenphotographs, this w ould seem to be an invalid approx-imation, yet it depends on the energy density of thesereflected w aves in com parison w ith that of the incidentshock. T he results obtained from the program generallycompa red wel l w ith th e sho ck -wave Mach numbers d eriv edfrom the pressure traces. Fig. 11 shows the num erical

LUa:::>enenLUa:a. .

3-

0 3,2 6,4TIME

9,6 (mSec)2,8

Fig. 10-G eneral pressure characteristics of the systemestim ate of the M ach num ber in the low er branch of thefirst jun ctio n c ompa re d w ith th e ex pe rim en tal v alu e, b othplotted against the inlet M ach num ber to the system . Thelargest recorded error in this branch w as 4,8 per cent. Inthat case, the error is a sm oothly increasing function ofM ach num ber and could be extracted to generate an em -pirical factor, itself a function of M ach num ber, w hichcould be subtracted from further estim ates to producea m odified estim ate w ith negligible error. W hether thiswould apply over a wider range of incident Machnum bers w ould need to be established. C alculations forthe second junction (both branches) predicted themeasured values to within 10 per cent. This lower ac-curacy is due to the curvature of the wave incident onthe junction, w hereas the analysis assum es a plane w ave.Since the output shock strength for one lim b of the ductacts as an input to the next junction, the errors wouldtend to accum ulate unless there w as sufficient distancebetw een junctions to ensure the developm ent of a planewave.F ig. 12 is a num erical estim ate of the tim e delay, alongw ith the m easured value, both again plotted against theinlet M ach num ber to the system . The tim e delay, m en-tioned earlier, w as the tim e taken for a shock to pass overa transducer in a closed duct, reflect off the end-wall,and return as a reflected wave. This particular resultproduced a m axim um recorded error of only 2,4 per cent.T ypical errors (i.e. the difference betw een the m easuredand estim ated results) w ere generally in the range 2 to6 per cent. The measured and numerically estimatedresults follow the sam e general trend, indicating that theapproximate theoretical model is of value. W hile theresults presented here were obtained on a smalllaboratory-scale rig, the num erical procedure can beapplied to any branched duct system for estimates ofshock strengths. N um erical m odels are currently underdevelopm ent that w ill predict fuller details of the w holefl ow fi eld .In th e o rig in al in ves tig atio n, it w as n oted th at s ub sta n-tia l b la st e ffe cts were evid en t in th e re tu rn a irway, p ara lle lto the haulage in w hich the explosion had occurred. It isthus of interest to establish the strength and behaviour of

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1,6.s::g 1,5CO.c~ 1,4.QQ)-= 1,3.SFig . 11 -Numerlca l estimate of the Mach number 2E 1,2:Jc:.s::aJ 1,1::E

0,4

11>. Measured 0 N ume ri ca l1

1,1 1,2 1,3 1,4 1,5 1,6In le t M ach n um be r

0,35 11>. Measured 0 N um er ic al 1rn"Cc:8 0,3)~.E.s 0,25>-o~ 0,2Q)Ei= 0,15

I> .F ig . 1 2-N um eric al e stim ate of the tim e d ela y

1,2 1,25 1,3 1,35 1,4 1,45 1,50,1 1 1,05 1,1 1,15In le t M ac h n um ber

the waves that are fed into this tunnel from the develop-m ent end. In order to investigate this in the laboratory,an extension tube, I m in length, w as added to the exhaustoutlet of the test section. This tube had the sam e cross-section and wall roughness as the inlet tube. From thesc hlie re n p ho to gra ph s g iv en p re vio usly , it is e vid en t th ata series of waves will pass into this duct from the testsection. E ighteen pressure transducer ports w ere fittedalong the length of this tube in order to track the m otionof these w aves.Fig. 13 shows the pressure traces from the six centraltransducers for a test with an inlet M ach num ber of 1,6.T he traces are arbitrarily separated on the pressure axisfor clarity, w ith the record for the transducer closest tothe test section at the top. T hree distinct pressure jum psare noted on this trace that identify three distinct shockwaves em erging from the test section, the second wavebeing the strongest. It should be noted that the absolutemagnitudes of these jumps should not be comparedd ir ec tly b etw een t ra ce s b ecau se o f t he d iff er in g tr an sducercalibrations.The tim e intervals between the waves on successivetraces d ecrease for transd ucer po sition s furthe r from th etest section , L e. th e w ave s are o vertak in g each other. T hefirst shock w ave em erging from the test section is over-tak en by the seco nd at the transd ucer po sition co rrespo n-ding to the second-lowest trace on Fig. 13, therebyresulting in a stronger leading wave. This overtaking

1,5 1,55

wa::JCl)Cl)wa:a.

4.

5

200 600 (~Sec)0000TIME

Fig. 13-Pressure distributions In the outlet duct

p hen om en on is readily exp lained . T he reg io n be tw een thetwo w aves is reasonably uniform , and a shock w ave hasthe property that the flow into it relative to the wave issu person ic, and the ou tflo w is subso nic. T hu s, the seco ndw av e m ov es sup erso nically w ith respect to th e g as ah eadof it, which is itself subsonic with respect to the secondwave. The two waves must thus eventually m erge.

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2,22

1,81,61,4

'"": 1,2j"ew::!! 01 3=

0,60,40,20

0 0,2 0,4 0,6DISTANCE meters

Fig. 14 is a wave diagram (or x, t plot) of the wavemotion in the outlet duct. All three of the shock waveshave m erged into a single w ave before the end of the ductis reached. It is noted that the locus of the waves on thew ave diagram are nearly linear, and the w aves thus pro-pagate at approxim ately constant velocity. T hese velo-cities can thus be determ ined from the slope of the lines,from which the M ach num bers, and hence tem peraturesand pressures, can be calculated. F or the case show n, theM ach num ber at the inlet to the test section w as 1,6, cor-responding to a pressure ratio of 2,82. The first shockemerging from the test section had a M ach number of1,14 and a pressure ratio of 1,35, whereas the finalmerged wave was moving at M ach 1,41 and a pressurera tio o f 2 , 1 5. T he comp le x tu nn el sy stem b etw een th e in le tand the outlet ducts thus had a net effect of reducing thepeak pressure behind the blast by only 24 per cent forthe case tested. The effects of different inlet wavestrengths and other geom etries have not been tested, andit is not inconceivable, if area changes occur, that w aveamplifica tio n may re su lt.ConclusionA m odel study of gas-dynam ic effects resulting froma shock w ave propagating through a com plex system ofb ran ch ed d uc ts s howe d a n umber o f in teres tin g fea tu re s.

. T he flow is com plex, but can be understood w hen theind iv id ua l w av es a re tra ck ed o n a se rie s o f in div id ua lschlieren photographs taken at short tim e intervals.

Fig. 14-W ave diagram for the outlet duct

0, 8

. A first approximation of the strength of the wavespropagating in the various lim bs can be m ade by useo f s imple n umerica l pro ce du re s.. R eflection boundaries in the tunnel com plex result ina general increase of pressure in the cavity, whichresults in the transmission of shock-induced highpressures to the connecting tunnels.. W aves em erging into adjacent tunnels tend to becom eplane and overtake each other, thereby resulting instronger w aves further along the tunnel.T he study explored only the w ave behaviour in a singletunnel geom etry, and over a lim ited range of inlet shock-wave M ach numbers. The effects of wall roughness on

w ave attenuation w ere not explored, nor w ere the effectsof the initial wave-pressure profile. A substantiallyextended series of tests would be needed in order toad dre ss th ese fa cto rs .References

I. SMOLENIEC, S. Report on blast wave effects on the site of accidentat President Stern G old M ine on 24-12-1959. Johannesburg, U niver-sity of the W itwatersrand, 1962.

2. HEILlG , W .H. Propagation of shock waves in various branchedducts. Proceedings of the Tenth International Shock Tube Sym -

posium. K yoto, 1975.3. WHITHAM, G.B. A new approach to problems of shock dynamics.

J. Fluid M ech., vol. 2. 1957. pp. 145-165.4. CHESTER, W ., and CHISNELL, R.F. The motion of a shock wave ina channel . J . F lu id M e ch ., vol . 2 . 1957. p .286 .5. SK EWS,B .W . T he shape of a diffracting shock w ave. J. F lu idMech., v ol. 2 9, p t 2 . 1%7 . p p. 2 86 -3 04 .

144 APRIL 1111 JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY