NRL Memorandum Report 4333

Simulation of Complex Shock Reflections fromWedges in Inert and Reactive Gaseous Mixtures

D. BoOK, J. BoRwS, E. ORAN. AND M. PICONE

Laboratory for Computational Physics

S. ZALESAK

Geophysical and Plasma Dynamics

Plabma Physics Division

•nd .

A. KUHL

OR& Assmlates-P.O. Box 9685

4640 .4d*ra4y W.zy"Marina del Ray, Ca1•ornia 90291

Septcmber 30, 1980

This work was sup.porled by tho Defense Nucl-at Asency under subtask Y99QAXSG601work unit 22, tand work unit ifle, "01I Flux Cornvet rai•.ojt Codý,*

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JNIMULATION OF COMPLEX.§HOCK REFLECTIONS Interim report on aFRM I ES- continuing NRL problem.

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*Present address: R & D Associat-.s, P.O. Box 96954640 Admiralty Wray, Marina del Rey CA 90291

**NRL/NRC Postdoctoral Research Associate (continues)

I$. KEY 11O0110 (Co~IAllnu on tow*#*#.I wIi H099444 VOW I&Aa.I by. klocb ift'06

DetoraL ions Mach shockShock waves WedgesAir blast Induction time

'0. ABSTRACT (ContInu. n •o,.v,* old* Ii "Otem, 0 ~, 10iyt, by block etMb*.l)"The Flux-Corrected Transport (FCT) technique for solving fluidequations reduces numerical diffusion, permitting calculationswith Reynolds numbers considerably in excess of the cell Roy-nolds rumber. Recent advances in FeCT, including a multidimen-sional flux limiter and a dynamic adaptive rezone, are illus-trated in the problem of transient reflections of planar shocksfrom wedges in inert and reactive media. Abstract (continues)

DD , . 1473 0oDION OF I NOV .I IS O ,.LETI -5,1 0102.LF 4144 601 i SECURITY CLASSIPICATIOM oN 0 ,, T ,AI 0^0 ',46, , . '"0

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18. Supplementary Notes (Continued)

Geophysical and Plasma Dynamics Plasma Physics Division

This work was supported by the Defense Nuclear Agency under subtaskY99QAWSG601, work unit 22, and work unit title, "BlI Flux Corrective Trans-port Code."

20. Abstract (Continued)

-ýResults are obtained with high resolution which are in quantitativeagreement with experiments.

I:

SECURITY CLAMPICAY|ON 0",s415 PAOe(Uh..1 Dart u.tred)

CONTENTS

1. ADVANCES IN FCT TECHNIQUES .......... ........ .... . 1

2. SHOCK REFLECTIONS IN AIR ...................... ... 2

3. DETONATIONS ................... ....................... 3

4. CONCLUSIONS ................. .................... . . 4

REFERENCES .. ................... . . . . .. .. .. .... 4

iNi

SIMULATION OF COMPLEX SHOCK REFLECTIONS

FROM WEDGES IN INERT AND REACTIVE GASEOUS MIXTURES

1. Advances in FCT Techniques

In this paper we describe new adaptations of the Flux-Corrected Transport (FCT)algorithms developed by Boris and Book (1973, 1976) for solving fluid equations, anddiscuss their application to multidimensional shock reflections in inert and reactinggaseous-mixtures. In particular, we consider planar constant-velocity shocks reflectingfrom wedges. Under certain circumstances double Mach stems are formed. Historicallythese have proven to be difficult to calculate with high accuracy, although many schemeshave been available to analyze compressible flow on a computationally discretized mesh:the method of charactetistics, spline techniques, Glimm-type random choice schemes, andfinite element, finite difference, and spectral methods. We believe that the calcula-tional difficulties experienced on this problem were the result of excessive numericaldiffusion, especially in the region of the contact surface.

In any Eulerian calculation, numerical diffusion arises because material which hasjust entered a computational cell, and is still near one boundary, becomes smeared overthý whole cell. FCr minimizes thilts effet. F(T algorithms can be constructed as aweighted average of a low-order and a high-"rder finite-difference scheme. If the tluidevuations are written In conservative form, both schemes are implemented using .trans-portive fluxvs. Each flux describes the transfer of mass (or some other extensivequantity) from one point to a nvighboring point. The procedure for assigning weightsInvolves limiting or "correcting" the fluxes at cortain points, Thi higher-order schemeIs used to the greatest possible extent, consistent with avoiding the introduction ofdispersive ripples (undershoots and overshoots). The weights for the tow-order schohpare chomsen to be Just sufficient to eliminate these ripples, thus asssuring the propertyof "Mollotonicity" or "positivity." The result is to algorithm which effoetively reducesto the higher-order scheme wherever the fluid properties change gradually. Near sharpdiscontinuities, howevert, enough diffusion is supplied to retain onoltonicity. AtahockU fronts this procedure automatically produecs the correct local viscous hoating.

The prototyp. soeond-order finite-difforeoee formula~~~I ft~ .ntaj( it 0,|) nJ+lP}| it) - =11

S t~c~ to CO + .4. + V .(

Illustrates the procedure. Here labels grid position, n d@notes tim level andf &.m .tl A& X and V are diotensionless adveetiot And diffusion coefficients,r ?cjoyJ writ t'ý % Iii(j_,+o)% where r is d "cl~ipping faetor0' measourion tht"

.Xtro diffusion added to Achieve positivity. ••to t, -0, the above schene is tecond-orderl in the vicinity of shocks c •, I and it effectively reduces to first .order.

.A nu.rieal diffUion ROynoluls Number (Cle) * L,/eAs can be deflned, whore I. Id theclarae.torfitbe so.au of a structure Ill tue ffo, Even the moso ateurate spertral Sir*-lations requiro setting e - I to kuarantee pooitivity lifnerly. lltis givvel rlse to theusuil definition of the numerical atuyaiods number, IL/As. Algforithms such aim sCT which

C'..

guarantee monotonicity nonlinearly can have average values Ile, 1 0-10 2 intro-

ducing much less overall dissipation and permitting calculations with effective Reynoldsnumbers such that Re .(Re)ND >>2L/6x.

Four advances in FCT techniques have enabled us to perform a series of shock anddetonation calculations with high accuracy. These techniques are easy to program, andthey have wide applicability to general quasi-linear hyperbolic equations (i.e., equa-tions describing continuum conservation laws).

The first of these, a generalization of FCT due to Zalesak (1979), removes the

necessity of tinestep splitting in multidimensional hydrodynamics. This reduces errorsassociated with time splitting in regions of the flow which are nearly incompressible.The second refers to the development of FCT algorithms in which the spatial derivativescan be approximated to arbitrarily high order (fourth, sixth, eighth, etc., or pseudo-spectral). These innovations, which relate to the transport algorithm itself, havebeen implemented in a two-dimensional hydrocode which utilizes the leafrog-trapezoidal(L-T) algorithm and is therefore dissipationless. Both complex and double Mach stemstructures are obtained (cf. Ben-Dor, 1978; Ben-Dor and Glass, 1978, 1979).

The third new technique, adaptive rezoning, is an extension to two dimensions ofthe dynamic rezoning employed in detailed one-dimensional reactive flow simulations byOran, et al. (1979). This concentrates needed spatial resolution in the vicinity ofmoving shocks, contact discontinuities and reactive surfaces. The technique is illus-trated with shock calculations using a time-split code (FAST2D). In air for HW5 ando -45, the results fall very close to the boundary between regular and Mach reflection.Two calculated wall pressures are in detailed agreement with the results of Bertrand(1972). The fourth technique is a generalization of the induction time approximationused in earlier flame, ignition and shock work (Oran et al, 1980a, b). This provides asimple, efficient, yet reasonably accurate global chemical kinti tics package to be usedit connection with these comprehensive two-dimensional hydrodynamics calculations.

Section 2 describes the results of calculations in which a planar shock is reflettedfrom a wedge in an inert gas. In Section 3 we present the results of calculations ofdetonations initiated by shock reflections in stoichiometric mixtures of It, in air atlow pressure. Section.4 summariaes our conclusions.

.2. Shock Reflections in AirThe utility of these advanced L vthods has been:det.onstrated by applying them

ii.to troe.stent reflections of plaolar shocks from wedges for variousa shock strengths H andWedge-40910s 0 - For HOnACtMlOWiR a dtch numbers greater than about 2.5 and wedge""angles betwoenW2O and 50 degrees, double tMach stems can develop. Numerical schemspreviously used for _this.problem reproduce qualitativtly the wave structure and shape,but have difficulty making accurate predictionu of flow details such as density contours(a conclusion drawn by Sen-bar and Glass, 197t) even itt the single Miach stem case. Toour knowledge, successful calculations of the double Match stem case have not vet been Yr

*ii publ ished. in this paper. we discuss the series of calculationm s arised in'gble I.

Open bouwidary condittons are used on thle left, right, and top edges of tel m, sh, 1.i.e., densityv pressure and velocity are set equal to their pre-shock or post-shlockvalues, depending on whether the incident shock front, has passed that point'. Refltrttingconditions are iposed on the bottomt of the mesh, which corresponds to the wedge surface.Examples of the calculated density contours and wave strueture for the double and complexHack reflecton. cases are shown ti Fig. 1. The incident sho.k, I, the contact surface,.CS, and the first and second Mach stes,. ft and I• , are Indicated. In Fig.. Ia. %ote in

a.!rtticular the foward curl of the contact surfacl near the vall and the mall region .(4 by 7 mesh points) of high-doesity g•s just to the left of th-, point, where the contact.surface Impacts the aill& The latter causes a second peak i • st, essune and density

• - dtistribution on the wall, as shoum in Fig. Ie. The accuracy .n W.ag.atuultionas t• o ben"verified by comparilon with experimental density distributions along the vail, as shown.1in Fig. 2, and with experimntal pressure m.Asure.mnts (Bertrand, 1972). NOte that FCT

.rovides idequate resolution of the key surfaces (conltact surfere and second 11401 stem)" ili~ regia" •is. mull As S by ces. "'. .

2 1'-[.

S• a I

Two additional cases were calculated with larger values of e (Fig. 3). As thewedge angle increases, the Mach stem develops tiore slowly, being separated from thewedge by a triple-point angle of only one or two degrees. [The triple-point angle ais the angle subtended by the Mach stem as viewed from the end of the wedge at whichthe shock was first incident (Fig. 4.] To reduce the size of the mesh needed, it isnecessary to calculate in the frame of the Mach stem and to rezone. For these calcula-tions we employed the time-split code FAST2D (with a 150 x 50 mesh), which incorporatesan automatic continuous ("sliding zone") regridding procedure (Oran et al., 1979). Forthe small 0 cases discussed above, where regridding is necessary, FAST2D yielded re-suits very similar to those obtained with the L-T code. The cases with wedge anglesof 440 and 46.50 constitute a severe test of the numerical algorithm because of thesmall triple-point angle. Because a is approximately equal to 2.8* and 1.5%, respec-"tively, considerable spatial resolution and a large amount of running time are usuallynecessary to get accurate flow fields. Figure 4 illustrates our adaptive rezone tech-"nique on a grid of 60 x 40 cells with varying cell dimensions, 6x, 6y. This methodrequires one-fifth the number of cells required in a uniform grid calculation. Auniform region consisting of the smallest cells covers part of the incident shockfront, the Mach stems, and the reflected shock structure. Outside this finely griddedregion, we have transition zones in which the cell dimensions increase smoothly to theirmaximum values, 10 6X and 106ymOn*

We have investigated the accuracy of the numerical simulation by comparing tile* results with experimental data (Bertrand et al, 1972). Because the cases 0 * 44'

and 46.5" are so similar, we will discuss only the latter. The computed value of aif,'. •or U • 46.5" is approximately 2.5' for a real-air equation of state and approximately3.2' Vor an ideal gas with y - 1.35. Both agree with the measurements to within tileexperimental errors, - 2Z. In Fig. 5, we compare the calculated (using a real-airequation of state) and experimental values of the pressure at the surface of the wedgefor 0 - 46.5. The agreement in the shapes of the pressure curve is striking•-•.hevaIuet- of the lower pressure peak, corresponding to the Mich stem, are nearly Identical.Tieh calculated value of the second pressure peak is 117 lower than the vxperimentalvalue and Is thus within experimental uncertainty. Figure 6 shows the history of thepressure on the wedge calculated for an ideol gas with ' - 1,.35. We note that the curveshave much the same shape as for the real-air simulationst however, the first pressurepeak its again 11%? lower than the experimental value. Figure I shows that the doublettach re-flet ion shock -structure is well rosolved In the himulation.

3-•. ttonat ionsWe havo also conside-red analogous shock reflections in reactive gases (stelehio-

metric mixitures of It hI air) at low pressure (0.1 atm). The induetion time hypothesisO(e.g., tran et at, 1980) represents the chemistry through ai composite proeoss, in which

-rvadlitants begln to combine into combustion products only after a finite time has elapsed.The rate at which thl energy-releasing reactions proceed depentds upon a single paraeitvr,thle indution, time. This ti turn ti a funtiota of the local thtrodyamic vr.i4bles.

Figure 8 ihows the time ••volopmwnt of a detonation Initiated by a weak r•fletinog"shock. . The incident shock was chogen so that th1 pressre behind It is too low to causmedetonation to take place within thte tine of the calculation. As with the caleulationof. Section 2, we have used open boundary conditions at the gides and top of thie system.

, the sequence of sIx pressure contour plotsi traves the evolution of a detotnation waveinitiated by couplesx Ma.h rflec tion at the surfave of the wedge. Figure 84 show§ the

* presure contours; corre•sponding to an ineident sthock with * 2V and MI 4.0 vhicihhas Just begunto reflect. The I•ieh stem is in itilly ton •rli to be resolved. In."Pig. 11 the ,aeh stem becomes dise.rnible, but as yet no apparent reaetinn htas occurred.Sy fwrae (c) tile vttterial his begun to ignite at a poitioln lwlt,l belhind th.e-urrentlocgation of thie 4Ach stem. .MWiao the Mtach stem passed that position the prcsstturv

- itt cvase heated the mixture siufflenatloy to cause Ignition after a short indu•t•io,time charavteristie of the li '-att mixture. in frams (d) and Ce), at later time,the pressure tat the H•ch stt, cofntfinus to grow, leading to ,tortor charatteristItiitduetion timwe for material between the Bttah stem and the oritgintl ignitiott point.

S huN we seo ge the inLited- rglion acchelrate along the wedge ourfaue tow4ard the ;airh •tom.ecausem, re eneorgy ti baltng relewaed as the.burniLngp cootitiuk, the boundary. o the

- 3

ignited region also accelerates in the direction of the reflected shock front, com-pressing and heating the material into which the burned gases expand. In the lastframe the burn front has overtaken the reflected shock and Mach stem, as we see fromthe decrease in the separation of the pressure contours near both locations. A stabledetonation pattern has not yet emerged, however. This is evident from the bending outof the Mach stem and the lower density of contours between the Mach stem and the re-flected shock/detonation front.

We anticipate that shock tube experiments will confirm this wave structure andthat such reactive flow calculations will be extremely useful in quantitatively ex-plaining the experimentally observed multicell structure of detonations (Oppenheim 1970).

4. Conclusions

Our calculations of complex and double Mach reflection are in close agreementwith measurements for shocks reflecting in air from wedges. Because of the accuracyand speed of FCT algorithms and the effectiveness of adaptive rezonintg, the calculationsare accurate and economical even when the Mach stem develops very slowly. All of theimportant features (location of surfaces of discontinuity, pressure loading on thewedge surface, density contours) are correctly predicted. The results do not dependsensitively on whetlter the I.-T or FAST2D code is used. Of the advances discussed inSection l. multidimensional flux limiting and the adaptive regridding technique seemto be the most efficacious for reflections in nonreactive media. We conclude that FLCTalgorithms reduce numerical diffusion dratmitically, assuring qualitative improvementsin accuracy. We believe that to achieve comparable accuracy and efficiency, otherhydrocodvs must employ similar nonlinear algorithms and rezoning techniques.

Our c.alculations ini reactive gas mixtures show that detonations tend to beginwhere a secondary pressure peak aris•es as the slip s.urfaee •ipproaches the wedge.w " ecausv oef the f(iite vinduction time in our klnittleg model, the detonation begins+somewhat behind this pressure peak. 1he high reoloattion our caleulations achieve"en1ables us to follow multiple reflectiono ,.id Is capatblv of providing quantitat ive""r eittl tons of dvtottit Ion phenea.

11ert attd, RI. P., Mcatment cj Pressure. in Mih Rvflvc•tio of Strong Shock Wavoiti it i,•"Shok Tubv. 1tl 1i.t iR vootorch lUtboratorle•o rey IRp t -l-1 (s 6l 72W...

* it-lbor, U., kev.tons +rod TransItits of nstgt tioinarv Ohlliqte V •hock•*we __ tmntti .er(Qet ndilmp1erfct0rt UaMe, UTIAS l•eport 4.3, August 1978, 6,1 p1)4#019 4 apetptedicks.

Wiot-tkr, tC.., ad Cillas1. I I., 'Noostatiottary Oblique Shock Reflectiont - Actual Igo-pyetikc simd hNumorteal- fiaples" AMM J. 16, i114 (1978).

lo.-K•r, G., aid |Ultsd, 1. 1., "I- Iti atd houlwatrivt* of Notlstiottsotry Oblique.Shockwdve- Rofleetion"t4 1.-tlatomc Cirs," ji Fluid Mkeh. 91, 4&9 (1979).

- erig, J. P. Attd lkD, D. *1,.. !Flux-Correated Transport, I. IMIASMA, A-Fluid Tratt•portAlgorithm tMat totk:¾, J. tootm. PhVo. 11. -38 (0973),

Sonies, J . P. tind Book ,v -L ".4lution 0 Cont inuitY f41o tItto by th lte thId f Flus-Lorro.ted Trdusport,` i "iods lit Co" tVol. lIt, 11

(Acadetmicimi l'bs n. o or,17) e lo S~on, .3. P., Pu-oreo rtAS Muardit Ittfb. • SKI.

-- : - teI4. A. K., lntroduhetito to ~dvnamicsots , ttti tlitlelal .,"ie for-S. •hI. Sc+io~ese-tourite+ mtand lveeturem So. , Springer-Verldg, pp 2.-'3 t1970).

Orwti K. 4., oVoun•l T. R..,. and lorim, J. V., "Appliedttion of tim-Dlependent NCulNoethodt4 to the 0t#griptieol of Uiletive Shock!+1 6 eventtonth eVqo*h• (uteroat ionol)

* t ~a~tit~ inP. I(h Coabuittion Ittittitutce, Pittittaungli, 14ýw, S. dr .3+ J. P. loris, T, R, Voung, I'. hurnk. H. i alnigtan, Mnd M. Pleaem,, Nulutiedl

• SImultlotin of DetontatIon in ftl-atr on l-u i flIttres. ir -iedrtt mothte l.$t+t$y ,ii ..•he~rn tmlotal) on 1 sthtln, ti Comsttbtbool Intiltute, Pllttmburgh, PA,"IYii• In pro, Sl"ouAt.ti"11 of % Ii-T' .t, Detktjtoio. totrodaltiobt o lthe hto u tt

:hrPiliter Noel'" ,NotI. if r4mona Report n% 080h (1i1 preos). "Z.Is"4k. S. T. "Full+ .......... n a "Por t-5 t: Wag.. fo +-: :

4

F.a: (

w w -

""l 0 0 0 0 i 0 0

oo

U) 0 CL

Z

4

4J U) U

w' w

0p.44

""4O

DOUBLE MACH REFLECTION30

° t00I -i i 1104 50' ý K M

15 140 130 Ial 110 100 90 so To 60 50 40 30 20 10 0CELL. (cma

REDUCED DENSITY CONTOURS (M:6S.9,9ws2O*,yzl.35) 393.85ps30

I' P/nL, I P)

-~20 Gul &M

IL

• WALL DENSITY, PRESSURE 393.85 p•s200 is.. ... 40

to 'S lis .... 41

650 L1210 MM4- 1

0. ,

.50 i40 130 W20 1s 100 90 00 70 60 50 40 30 20 10 0CELL,* (cm)

P~tg. 1 -(a) Wave structure and density contours for double Mach,- reflection from a wedge; (b) reduced density contours (o/0^) for• complex Mach reflection with levels chosen to ag~ree with ti~ose

.. of IDen-Dor and Glass (1978); (€) corresponding pressure and den-'i...sity profiles on the wedge, plotted against cell number.

f.:. 6

S• • ... - . . ... ..L D N ITY, PRESSURE mblil 39 1 i m nmikli IB ro y/ i As b ~ ii

17-- Ms 6.9

Ow :20*161.35

15 --

i• i4 -14

13

12

9-

wJ

-7

4

V 5.

-CALCULATED VALUES.. .. EXPERIMENTAL VALUES

"(BEN-DOR, 1978)

140 130 120 10 100 90 80 70 60 50 40 30 20 10 0CELL,t (mm)

Fig. 2 Comparison of the calculated density profile on thewedge (Fig. l(b)) with measure" values (ion-Dor and Glass, 1978.

7

RR

60

usw

S IDMR (0)

40

20

MACH NO-.o n e

note~ eg~ ~ relect on sk ci4tts tjgO s0 ~ e nd do~~bl ac'10o

t X . C t Wo f s hopckIO Y m e I )m p l 1 ef O t o de t a C h ed X u st -

fig.b3 TYPOOCn inlk oenoex v yrgla ~ eO'nTeI aflectiu res8

30 COARSE ZONES 30 FINE ZONES

14-- .•12

r!! . . . . . "-10- 15 COARSE

2• :/1 ZONES

REFLECTED

0 4 a 12 1S 20 24X (cm) -

Fig. 4 - Gridding for complex shock reflection problems withFAST2D code. Shown are innident, reflected and Mach shocks"(solid lines) and slip surfaces (dotted lines) for incidentshock coming from the left.

• -

STEP 601 t - 126usec STEP 1501 t 232pssec E

UAU

i!.U

cc645 4.48~

5"• Uj ° .".

'U

IL262 1.74 W

0 o_ L

33 63

POSITION (cml POSITION (cm)

Sii itwl 430 PSI0.lOp SIGIoIYPSATO 1 3.13I

P11 S30.3 PlI& 10 tU

a 43j

Fig. 5 -Upper and lower diagrams show pressure in PSIA on thewedge as a function of position for o1 u 46.5*(real air equa-tion of state) at two times in the si ulation and as a functionof time at two stations in the experiment, respectively.

10

46.50 WEDGE ANGLE-IDEAL GAS EQUATION OF STATE

167Sosec ]4.0

S436 - -3.0

29 - 2.0oU

-1.014161.

50 40 3D 20 10 0POSITION (cm)

Fig. 6 - Structure of the calculated shock (8 - 46.5", M- 5.15,1.35) as a function of position for severh1 times.

ii

40-1 44

2 2

30• 3

2390 350IW

CELL INUMBERFig. 7 - Plot of pr~essure ye distance along the trajector'y 1-2-

3-4 shown in the inset, which •.ntersects all surfaces of dis-

~1I continuity norma1ly.

1 12

0 0

SU•4

k0

0 k

04

0 d

4) 00U4Jg 4A

4J

13

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Analytic Services, Inc. Denver, University of400 Army-Navy Drive Colorado SeminaryArlington, VA 22202 Denver Research Institute

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Teledyne Brown EngineeringCummings Research ParkHuntsville, AL 35807

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