ANALYTIC MODEL AND NUMERICAL SIMULATION OF SHOCK WAVE ... · analytic model and numerical simulation of shock wave propaoati--etc jan 81 j a schmitt. r e lottero, h j goodman unclassified

ANALYTIC MODEL AND NUMERICAL SIMULATION OF SHOCK WAVE PROPAOATI--ETCJAN 81 J A SCHMITT. R E LOTTERO, H J GOODMAN

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TECHNICAL REPORT ARBRL-TR-U5

ANALYTIC MODEL AND NUMERICAL SIMULATION OF

o SHOCK WAVE PROPAGATION INTOA REENTRANT CORNER

SJ.A. Schmit DTICH. I. Goodman

APR 8 EI01U

January lS1 B

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or speamrt" activity is prohibited.Mii1tio.&I cogies of this report may be obtainedfrom the National Technical Information SarvieU.S. Department of Commerce, Sprinfield, Vixgials22161.

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Shock Wave Propagation into a Reentrant Corner ______________

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7. AUTWO N l CONTRACT OR IUANT NUUMIIre

J.A. SchmittR.E. Lottero

P. 0 , ORSANIZATM W AN AOo 10. 0 ot %9oW, Ppmoo TAMBallistic Research Laboratory A Vulation oa

ATTN: DRDAR-BLBAberdeen Proving Ground, nD 21005 RDthE oL161102A43Il. CONTROLL1N OPFICEWAO AND ADDES IL REORT DATEUS Amy Armament Research Development Command January 1981Ballistic Research Laboratory A NMUNWER O PACKSAMTN: DRDAR-BL, APG, ND 21005 113iL MONITORIO AGENCY ANME G ADOMODEIEW mE h CWm*M O IL ECURITY CLASL ( We rapo")

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WS SUPPLEMNUTARY NOTES

1S. KEY WORM (C n &me enome "I Inese emv &W "M04b or .. amde*"Shock Tube Experiments ShocksAnalytic Model HydrocodeDORF Stair-StepReentrant CornerMach Reflection

AWUn ACr - - __ m N - U i md SuWW IV numaanalytical model is developed and numerical simulations are performed for

shock wave propagation into a reentrant corner. The analytical and mmericalresults are compared with shock tube experiments performed in a 1000 reentrantcorner with a moderate strength (P/P 0 a 2.37, shock overpressure 137.9 kPa)

incident shock and a weak (PI/P 0 a 1.12, shock overpressure 12.41 kPa) incident

shock. The analytical model provides an exact solution of the flow field if

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shock diffraction does not occur within the reentrant corner; it also providesan approximation to the peak vertex pressure which is comparable with experi-mental results if diffraction does occur. Both regular and Mach reflections aremodeled. When the corner has finite length, an estimate of the duration of thepeak vertex pressure is given. The mmerical simulations are performed with theulerian hydrodynamic computer code DORF. In the DORF calculations, the reen-

trant corner is formed by a rigid smooth wall and a rigid stair-step wall. Adetailed discussion of the stair-step construction and comparisons of thepressure distributions along these walls are included. Furthermore, theanalytical and numerical pressure results for an infinitely long reentrantcorner are compared.

.

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TABLE OF CONTENTS

Page

LIST OF TABLES. .. .. . ... . .. .. .. .. .. .5

LIST OF FIGURES . .. .. .. .. .. .. .. .. .. .6

1. INTRODUCTION. .. .. .. ..... . .... ...... 11

2. EXPERIMENTS . .. .. .. .. .. .. .. .. .. . .15I

3. ANALYTIC MODEL. .. .. .... ..... ....... .. 17

3.1 Theory for an Infinite Corner. .. .. . ... . .. 17

3.2 Examples . . . . .. .. .. .. .. .. .. .. .27

3.3 Peak Pressure Duration for a Finite Corner . . .. 34

4. DORF CODE SIMUJLATION OF AN INFINITE CORNER .. ..... 40

4.1 DORF Code Description .. .. ... . .... .... 40

4.2 Examples . . . . . . . . . . .. .. .. .. .. .43

4.3 Stair-Step Approximation . . .. .. .. .. .. .44

4.4 Comparison of the Pressure Along the Smoothand Stair-Step Walls .. .. .. .. .. ..... 53

S. COMPARISON OF PRESSURE VALUES FROM EXPERIMENTS,ANALYSES AND NUMERICAL SIMULATIONS. .. .... ... 76

5.1 Experimental, Analytical and Numerical PeakPressure Values at the Vertex .. .. .... . .. 76

5.2 Experimental and Analytical Peak Pressure

Duration Values . .. .. .. .. .. .. .. . .79

5.3 Analytical and Numerical Pressure ResultsWithin an Infinite Corner. .. .. .... . ... 80

ACKIOWLEDQENTS . . . . .... .. . ... .. .. .. .91

REFERENCES . . . . . . . . .. .. .. .. . .. . . . . 92

3

TABLE OF CONTENTS (Continued)

Page

APPENDIX A. DERIVATION OF THE OBLIQUE SHOCKRELATIONS ...... ................ 95

APPENDIX B. EXPLICIT FORMULAS FOR OBLIQUE

SHOCK CALCULATIONS ............. .. 99

LIST OF SYMBOLS ...... .................... .. 101

DISTRIBUTION LIST ...... .. ................... 103

4

LIST OF TABLES

Table Page

I Summary of Experimental Shots ................ .. 18

2 Regional Flow Properties for Shot 2in Laboratory Coordinates .... ............. ... 31

3 Regional Flow Properties for Shot 1in Laboratory Coordinates .... ............. ... 37

4 DORF's Initial Values for Shot 2 ...... .......... 45

S DORF's Initial Values for Shot I ... ........... .. 46

6 Peak Pressure Duration Data From Experimentsand Analytic Model ......... .... .......... .. 79

Aooession For

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LIST OF FIGURES

Figure Page

I Schematic of Incident Shock in a Concave Cornerof Infinite Width ....... .................... .12

2 Schematic of the Experimental Shock Tube Model .......... 16

3 Regular Reflection in a Neighborhood of theReflection Point Q .. ................... 21

4 Mach Reflection in a Neighborhood of theTriple Point Z ....... ..................... .24

5 Criterion for Determining the Presence of Regularand Mach Reflection (Adapted from Reference 14) ....... 28

6 Flow Field Before the Incident Shock Reaches theVertex for Shot 2 ............ . . . . . . . . .. 29

7 Flow Field Shortly After the Incident ShockReaches the Vertex for Shot 2 ..... .............. 32

8 Flow Field Before the Incident Shock Reaches the

Vertex for Shot 1 ....... .................... .35

9 Flow Field Shortly After the Incident Shock

Reaches the Vertex for Shot 1 .... .............. .36

10 Rarefaction Wave Propagation Along a Finite Wall

(a) Before and (b) After the Incident ShockReaches the Vertex ...... ................... .39

11 Initial Flow Field for DORF Simulations ... .......... .47

12 Reentrant Corner Construction .... ............... .. 48

13 Stair-Step Wall Constructions Used in AWFL-HULL9Code Study. ......... ....................... 51

14 Detailed Portion of Computational Domain Nearthe Vertex ......... ....................... 54

15 Comparison of Wall Pressure Values for Shot 2at t = 0 .......... ........................ 56

16 Comparison of Wall Pressure Values for Shot 2at t =69.74ps ......... ..................... 57

6

LIST OF FIGURES (Continued)

Figure Page

17 Comparison of Wall Pressure Values for Shot 2at t - 114.3 us ....... ..................... 58

18 Comparison of Wall Pressure Values for Shot 2at 183.6 us ....... ....................... 60

19 Comparison of Wall Pressure Values for Shot 2at t - 321.4 us ...... .................... .61

20 Comparison of Wall Pressure Values for Shot 1at t = 0 ........ ........................ 62

21 Comparison of Wall Pressure Values for Shot 1at t = 91.09 Us ....... .. .................... 63

22 Comparison of Wall Pressure Values for Shot 1

at t = 142.4 us ........ ................. . E4

23 Comparison of Wall Pressure Values for Shot 1at t = 240.7 us ....... ..................... 65

24 Comparison of Wall Pressure Values for Shot 1at t = 465.6 Us ....... ..................... 66

25 Comparison of Wall Pressure Histories at a Oneand a Half Cell Distance From the Vertexfor Shot 2 ....... ........................ 68

26 Comparison of Wall Pressure Histories at a Twoand a Half Cell Distance From the Vertex

for Shot 2 . . .................... 69

27 Comparison of Wall Pressure Histories at a Threeand a Half Cell Distance From the Vertex

for Shot 2 ....... ........................ 7028 Comparison of Wall Pressure Histories at a Four

and a Half Cell Distance From the Vertex

for Shot 2 ....... ........................ 71

29 Comparison of Wall Pressure Histories at a Oneand a Half Cell Distance From the Vertex

for Shot 1 ....... ........................ 72

7


Figur!e Page

30 Comparison of Wall Pressure Histories at Twoand a Half Cell Distance From the Vertex for Shot 1. .. 73

31 Comparison of Wall Pressure Histories at Threeand a Half Cell Distance From the Vertex for Shot 1. .. 74

32 Comparison of Wall Pressure Histories at Fourand a Half Cell Distance From the Vertex for Shot 1. .. 75

33 Comparison of Experimental, Analytical and NumericalValues of the Peak Vertex Pressure for Shot 2 ......... 77

34 Comparison of Experimental, Analytical and NumericalValues of the Peak Vertex Pressure for Shot 1 ..... .... 78

35 Comparison of the Analytic and DORF PressureHistories at Cell 1 (5 = along the smoothboundary) for Shot 2 ...... .................. .81

36 Comparison of the Analytic and DORF PressureHistories at Cell 4 (35 - along the smoothboundary) for Shot 2 ...... .................. .82

37 Comparison of the Analytic and DORF PressureHistories at Cell 30 (SS.1 =r along the axisof symmetry) for Shot 2 ...... ................ .83

38 Comparison of the Analytic and DORF PressureHistories at Cell 31 (off the axis of symmetry,55.9 rm from vertex) for Shot.2 ............... .84

39 Comparison of the Analytic and DORF PressureHistories at Cell 1 (5 = along the smoothboundary) For Shot I ...... .................. .86

40 Comparison of the Analytic and DORF PressureHistories at Cell 4 (35 m along the smoothboundary) for Shot I ....... .................. 87

41 Comparison of the Analytic and DORF PressureHistories at Cell 30 (SS.1 a along the axisof symmetry) for Shot 1. ................ 88

8


Figure Page

42 Comparison of the Analytic and DORF PressureHistories at Cell 31 (off the axis of symlmetry

55.9 mm from vertex) for Shot 1. .. .. ..... ... 89

9

I. INTRODUCTION

When a shock wave propagates into a concave corner, it is re-flected one or more times from the walls forming the corner. Uponreaching the corner, the direction of the shock propagation is reversed,one or more additional reflections may occur, and, in general, the lastreflected shock is diffracted. These multiple reflections cause signifi-cant increases in the pressure along the walls as compared with a singlenormal reflection. Therefore, such corners may be susceptible to damagefrom blast waves that might otherwise cause little or no damage elsewhereon a structure. Examples of such corners are the wing/body junctions ofaircraft and helicopters.

If the propagation direction of the incident shock lies in thecross-sectional plane of a reentrant corner and if the corner's widthis "large enough", then a two-dimensional model can be applied near themid-plane of the corner. See Figure 1. The purposes of this reportare: (1) to develop an analytical model for the two-dimensional shockwave propagation into a reentrant corner which can determine the peakpressure and its duration at the vertex of a corner of a general angleand for an arbitrary incident strength shock; (2) to simulate the above

phenomenon numerically using the DORF hydrodynamic computer code1 ; and(3) to validate the results of (1) and (2) by comparing them with shocktube experiments as well as with each other. By comparing the resultsfrom the analytic model, the numerical calculations, and the experiments,it is possible to quantify the capabilities and limitations of each.This is of particular importance relative to the numerical calculationswhich, in principle, can simulate more complex flows than any analyticmodel and can provide more complete information than any experiment.However, the reliability of a code in predicting a particular type offlow field must first be established. This report provides a partialevaluation of the DORF code as a tool for simulating shock wave propaga-tion in a reentrant corner. The pressure profile difference along a smoothboundary versus a stair-step boundary is discussed.

Section 2 of this report describes the shock tube experiments whichwere performed at the ARRADCOM Ballistic Research Laboratory (BRL).

The analytic model is described in detail in Section 3. The mathe-matical problem corresponding to Figure I with the additional assumptionof infinitely long walls has been solved analytically in several special

cases. Lighthill2 considered an arbitrary strength shock propagatinginto a corner with an vertex angle 2a which deviated only slightly from

1. Johnson.W.E., "Code Correlation Study", AFWL-TR-70-144, US AirForce Weapons Laboratory, Kirtland Air Force Base, NM (April 1971).

2. Lighthill, M.J., "The Diffraction of Blast II", Proc. Roy. Soc.,Series A, Vol. 198, pp 554-65, 1950.

11

_ _ I_

PLANE OF SYMMETRY

INCIDENTSHOCK

Figure 1. Schematic of Incident Shock in a Concave Corner of InfiniteWidth.

12

1800. Keller and Blank3 considered weak shock waves (acoustic waves)

propagating into any corner. Later, Keller4 considered the specialcases where no diffractions of the regular reflected shock waves occur

and determined the exact solutions by algebraic means. Schniffan et al.

5

considered a series of reentrant corner problems, most of which involved900 corners. Some of these corners had one finite length wall. Forcorners formed at non-right angles, they considered only regular re-flection within an infinitely long corner and used approximations to theoblique shock relations in order to obtain estimates of the resultingpressure field. In Section 3.1, an analytic model is presented and theextension of the model to simple Mach reflection is made. The only re-striction on the corner angle and shock strength is that complex anddouble Mach reflections do not occur within the corner. Under the as-sumptions of the analytic model, the flow field within the corner canbe analyzed as a cascading series of step shock reflections, except forpossibly the final reflected shock. The model predicts the propagationof all the shocks within the corner, determines the type of reflectionoccurring at each reflection point within the corner, and calculates thegas states and shock wave parameters associated with each reflection.The flow field resulting from a shock wave propagating into an infinitetwo-dimensional corner can be solved algebraically provided that the fi-nal reflected shock is not diffracted as shown by Keller. However, ifany shock is diffracted by either the leading edge of a finite corner orthe final reflection process, no exact analytic treatment is possible.In these cases, an approximate technique (numerical or analytic) must beused. The present analytic model provides an exact solution of the-.flowfield if no diffraction oecurs and an approximation of the peak vertexpressure which is comparable with experimental results if diffractionoccurs. The flow fields behind both weak and moderate strength inci-dent shock waves propagating into an infinitely long corner are calcu-lated in Section 3.2. When*he corner has finite length, the rarefactionwave generated at the leading edge of the corner propagates into thecorner and decreases the maximum vertex pressure. In Section 3.3, aformula is derived using the results of Section 3.1 to determine theduration of the peak ivertex pressure in a finite corner.

3. Keller, J.B. and lank, A., "Diffraction and Reflection of Pulsesby Wedges and Corners", Communs. Pure and Appl. Math., Vol. IV,No. 1, pp. 75-94, 1951.

4. Keller, J.B., "MUltiple Shock Reflection in Corners", Journal ofApplied Physics, Vol. 25, No. S, pp. 558-590, 1954.

S. Schniffman, T., Heyman, R.J., Sherman, A., and Weimer, D., "PressureMultiplication in Re-Entrant Corners", in Proceedings of the FirstShock Tube Symposium, Air Force Weapons Center Report No. SWR-Th-$7-2 (AD467-201), 1957.

13

The numerical simulation of a shock wave propagation into a infinitereentrant corner by the DORF hydrodynamic computer code is described inSection 4. The DORF code solves a finite difference representation ofthe two-dimensional Euler equations in an Eulerian computational grid.The continuum being modeled is divided into rectangular zones, or cells,each of which represents a given mass of either pure or mixed materialin a state of thermodynamic equilibrium. The DORF code can model non-responding cells in the computational flow field. These are used toconstruct interior reflecting boundaries and rigid structures. A de-scription of the DORF algorithm is given in Section 4.1.

The geometry of a general reentrant corner is not easily modeledwith the DORF code. It is necessary to use the two-dimensional Car-tesian configuration which has a 900 angle between the x and y axes.One or both of the axes must be modified to model a non-right angle atthe apex of a reentrant corner. Because the code is limited to theorthogonal Cartesian mesh, this modification can not be done by rotat-ing the axes. The only means in the current version of DORF by whichthis can be accomplished is by stacking non-responding cells along aboundary to approximate a corner with an angle other than 90 degrees.DORF does not have the ability to model fractional cells. Consequently,the modified boundary along which the non-responding cells are stackedis not smooth, but rather consists of discrete steps. Because thepressure distributions on the walls forming a reentrant corner are ofspecific interest, it is important to determine what effect this stair-stepping has on the wall loading.

The DORF code has previously been used with stair-stepping to model.6the sides of a munition magazine , a cross-section of which is a trap#-

zoid. The trapezoidal structure was modeled as a stair-stepped rampfacing a spherical incident shock wave, followed by a rectangle and an-other stair-stopped ramp. The geometry used in the simulation wascylindrically symmetric. The actual ramp angles are 26.2, measuredfrom the horizontal. The stair-step ramps use one cell per step, eachcell having an aspect ratio Ar/Az = 2. Two calculations, simulatingdifferent incident shock strengths, were performed. On the forward facingramp the calculations differ by .13% to +29% from the experimental values,on the top by -20% to 12%, and on the rearward facing ramp by +37 to+71%.

The technique of stair-stepping to model inclined surfaces has also

been used in computations with the HULL 7 hydrodynamic computer code at

6. Goodman, H.J., "Calculations of Pressure Over the Surface of a1/30th Scale Model Munition Magazine", ARBRL-TR-02153, US ArmyBallistic Research Laboratory, Aberdeen Proving Ground, MD (April1979). (AD #B037702L)

7. Fry, ?.A., Durrett, R.E., Ganong, G.P., Matuska, D.A., Stucker, M.D.,Chambers, B.S., Needham, C.E., and Westmoreland, C.D., "The HULLHydrodynamics Computer Code", APWL-TR-76-183, US Air Force WeaponsLaboratory, Kirtland Air Force Base, NN (Sentember IQ76).

14

I . .. . . . ..__ _ _

the US Air Force Weapons Laboratory (AFWL). One series of HULL compu-

tations was used to determine the air blast over a dam slope caused bya nuclear burst on a surface of a reservoir. Here, the stair-stepped

dam slope faces rearward relative to the on-coming shock. The resultsappear to be qualitatively reasonable. However, there are no experimentaldata with which to compare the results. Another study at AFWL attempts

to determine an optimum stair-step design9 . A discussion of the stair-step design for the reentrant corner problem as well as a critique of

the AFWL method is given in Section 4.3.

The two examples of shock wave propagation which are calculated inSection 3.2 using the analytic model are recomputed using the DORF code.The DORF input values are given in Section 4.2. A comparison of thepressure profiles along the different walls is made in Section 4.4.

The experimental measurements, the analytic calculations, and thenumerical computations are compared in Section 5. The peak pressure atthe vertex was determined by all three methods and the results are com-pared in Section 5.1. In the experiments, a rarefaction ware is gener-ated at the leading edge of the corner which ultimately decreases thevertex pressure. The values for the duration of the peak pressure mea-sured by the experiment and determined by the analytic theory are comparedin Section 5.2. The flow field within an infinite corner is modeled

by both the analytic theory and the DORF calculation. The pressurehistories at various stations within the corner are compared inSection 5.3.

The summary of the results and conclusions are presented in Section6.

2. EXPERIMENTS

A series of shock tube experiments were performed at the BRL inwhich a step shock propagaed in air perpendicularly along a shock tubewall into a corner having an vertex angle of 500. (See Figure 2.) Inreference to Figure 1, the experiments simulated a symmetrically placedshock wave into a reentrant corner with a vertex angle of 2a - 100*.The shock tube wall replaces the plane of symmetry. The wall forming

the corner is 0.166. long and is of sufficiently heavy constructionthat its response to the loading is negligible. Two pressure gageswere inserted in the shock tube wall. The gage at the vertex recorded

8. Fry, N.A., Needham, C.E., Stucker, H., Chambers, B.S., III, and

Ganong, G.P., "AFWL HULL Calculations of Air Blast Over a Dam Slope",AFWL-TR-76-154, US Air Force Weapons Laboratory, Kirtland Air Force

Base, NM (October 1976).9. Happ, H.J., III, Needham, C.E., and Lunn, P.W., "AWVL HULL Calcu-

lations of Square-Wave Shocks on a Ramp", AFL-TR-77-82, US Air

Force Weapons Laboratory, Kirtland Air Force Base, NN (July 1977).

15

PRESSURE GAGES

1 / 1 43 0

INCIDENTSHOCK WAVE

Figure 2. Schematic of the Experimental Shock Tube Model.

16

the pressure history. Another upstream gage determined the strength ofthe incident shock wave. The corner's width (2S4=n) was long enoughto minimize three-dimensional effects.

Oscilloscope pressure-tine records were obtained10 for several ex-periments. From these records the value of the peak overpressure at thevertex, its duration and the overpressure decay at the vertex could bedetermined. The more relevant experimental data are the pressure peakand its duration, because the rarefaction wave which causes the pressuredecay is not explicitly modeled by either the analytical model or thenumerical simulation. The estimated accuracy of the overpressure mea-

surements is 5%. The estimated error in the peak pressure durationmeasurements is less than 7 ps. A series of four experiments were per-formed. We shall consider only the weakest incident shock, hereafterreferred to as Shot 1, and the strongest incident shock, hereafterreferred to as Shot 2. The experimental values are summarized in Table1.

3. ANALYTIC MODEL

3.1 Theory for an Infinite Corner.

The analytic model is based on four assumptions:

a. The incident shock propagates with constant velocity and issymmetrically placed within the corner at its mid-plane. (See Figure1.) This hypothesis permits the two-dimensional analysis of a shockpropagating with its velocity vector parallel to the plane of symmetry(a rigid wall) into a corner which has an acute angle equal to thebisected angle of the physical corner. This assumption, of course, canbe ignored, if the incident shock front is already propagating perpen-dicularly to a wall. Because the incident shock velocity is constant,its propagation can be considered pseudostationary if a frame of refer-ence is attached to the shock.

b. The medium in which the shock wave propagates is a perfect gasand has negligible viscosity. The latter part of this assumption excludesthe formation of boundary layers along the wall, and enables us to treatshock waves as discontinuous surfaces. Following the derivation in

Thompson11 or Courant and Friedrichs12 (see APPENDIX A), the Jump conditions

10. Taylor, W.J., MODUr, BRL, Private communication.11. Thompson, P.A., Compressible Fluid Dynamics, McGraw-Hill Book Co.,

New York, 1972.12. Courant, R. and Friedrichs, K.O., Supersonic Flow and Shock Waves,

Vol. 1, Interscience Publishers, Inc., New York, 1948.

17

- Nn

*41

: A

-

4J4

.840

0

0 4.

lit.

across a planar discontinuity can be written as:

v- W) • n - P v 0, (3.1.1)

bb a aPb [('vb n] P)• a[(v a. w) n ). Va. rb' (3.1.2)

vb t - va • t O, (3.1.3)

2,, h.. 0. 5,,-•n, + ha n a n) w (b v- (3.1.4)

where p, v, P, h, w, n, t are the density, velocity, pressure, specificenthalpy, the shock wave velocity, the unit outer normal vector to theshock wave, and the unit tangential vector to the shock wave, respec-tively. The properties imediately ahead of the shock wave are denotedby the subscript a and those immediately behind by the subscript b. Theperfect gas assumption postulates an equation of state of the formh - yP/[(y-l)p] where y is the ratio of two constants (the specific heatat constant pressure cp and specific heat at constant volume c ). The

sound speed in a perfect gas is given by a - (yP/P) . Equations (3.1.1)-(3.1.4) are commonly known as the oblique shock relations and are validat any point Q on the shock. If the shock is not curved in the immediatevicinity of Q, then the flow is uniform in this neighborhood and theflow properties computed at Q by the oblique shock relations are alsovalid in this neighborhood.

c. The walls forming the corner are rigid and infinite. Theassumption of wall rigidity is reasonable if the time duration ofthe loading is short relative to the response time of the wall. Theinfinite extent of the walls eliminates the rarefaction wave which isgenerated at the leading edge of the corner and causes the curvature ofsome reflected shocks.

d. Only regular and simple Mach reflections occur within the cor-ner. This restriction is necessary because only these types of reflec-tions are modeled. The theory of the model identifies the type of shockreflection and permits one to carry out a corresponding analysis.

The initial conditions for the shock reflection analysis are theabsolute pressure P0 and temperature TO in the undisturbed medium, the

incident shock strength and the angle of the apex 2a. From the initial

19

pressure value and shock strength, the pressure behind the incident shockcan be computed. In a perfect gas, the initial density pO is given by

P0 a Po/(ToR*), where R* is the gas constant, and the initial enthalpy is

given by ho - P/ [(y-1)PO].

The theory for the regular reflection of a shock wave in a perfect

gas from a solid boundary is well known.1 3, 14 Consider a step shock waveI which is propagating with a constant velocity, is incident at point Qupon an infinite plane rigid wedge making an angle 0 with the horizontal,w

and causes a regular reflected shock R to arise from the wedge. If weattach a frame of reference to the point Q, the incident shock velocityis zero and the flow in region 0 toward I is parallel to the wedge sur-face. (See Figure 3.) We define the region upstream of I as region 0,downstream of I and upstream of R as region 1, and downstream of R asregion 2. The properties in regions 0, 1, and 2 are related in aneighborhood of the reflection point Q. While passing through the inci-dent shock at an angle of 40* 900 - 8w, the flow is deflected from its

original direction towards I by an angle e1 and its dynamic and thermo-

dynamic properties are changed. These properties are related by theoblique shock relations (3.1.1)-(3.1.4) in the neighborhood of point Q.In these circumstances, the oblique shock relations can be simplifiedby substituting

4 4 44 4ua va~ -w, 11b = vb - w

Ua 9 n -u 0 sin 0ub , n = u, sin (*0 - el) ,

ua * t u0 cos *0u - u 1 cos (0- el),

and can be rewritten using the equivalent (AlO) instead of (3.1.4) as:

plUl sin(#0 - 81) = o0uo sin #of (3.1.5)

P1 + pl[ul sin (#0 el)]2 a P0 + P0 [u 0 sin *012, (3.1.6)

u1 cos (#0 " 81) Z u 0 Cos 0' (3.1.7)

h I 0.5[u1 sin (#0- 01)]2 a h0 0.S[u0 sin ,0]2, (3.1.8)

13. Bleakney, W. and Taub, A.H., "Interaction of Shock Waves", Reviewof Modern Physics, 21, pp. 584-605, 1949.

14. Polachek, H. and Seeger, R., "Shock Wave Interactions" in Funda-uentals of Gas emnics, H.W. Emons, Ed., Princeton UniveFtWP'ress, pp. 494-504, 1958.

20

L _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _

___________________

INCIDENT SHOCK REGION 0

REFLECTED R o EGSHOCK VELOCITY

REGION 2

Figure 3. Regular Reflection in a Neighborhood of the Reflection PointQ

21

where we define ui = 1*iI. Equations (3.1.S)-(3.1.8) represent a system

of four nonlinear algebraic equations with four unknowns, u0 O. u1

and 61, because h, - yP1 /[(y-l)P 1 ]. The solution of this system is

obtained easily, and the explicit formulas for the unknowns are derivedin the APPENDIX B. These formulas determine the flow in region 1 andare independent of the type of reflection occurring at the point Q.

The flow deflection across the incident shock causes the flow inregion 1 in the neighborhood of Q to approach the reflected shock oblique-ly at an angle #I' While passing through the reflected shock R, the flow

is deflected towards it by an angle 02 from its region 1 trajectory and

its dynamic and thermodynamic properties are altered. These propertiesare related by the oblique shock relations (3.1.1)-(3.1.4) in the

neighborhood of Q. In this case, the velocities are-I + 40 4 .ua V n = u1 sin l, b n a u2 sin (41 - 62),

+4 0 41_ua t a uI Cos 41 ub • t a u 2 cos (# 192).

In order that the resulting flow in region 2 adjacent to the wall beparallel to the wall in the neighborhood of Q, the deflection anglesmust be equal, that is, 62 a e,. In this framework both 01 and 62 are

positive angles and the difference in deflection direction is incorporatedin the formulation. The oblique shock relations for the reflectedshock R can be written in the form:

P2u2 sin (1 - 01) - p1 u sin #1, (3.1.9)

u2 cos (1 - 01) z u I cos 41, (3.1.10)

P2 + P2 [u 2 sin (1- 01)] = P1 P[Ul sin #1] 2, (3.1.11)

h 2 + O.S[u 2 sin (#1- 0 61)2 h I O.S [u I sin 1] 2 . (3.1.12)

In this system of equations, pl, uls P1, hl and 0 I e known. Thus,

equations (3.1.9)-(3.1.12) represent four nonlinear equations in P2 P

P2. u2, and *l when the enthalpy h2 is expressed in terms P2 and P2.

22

The solution to this system cannot be obtained by explicit formulas.

Instead the solution can be obtained nuerically by using the ItEL

subroutine ZSYSTM. (ZSYSTh solves a system of N simultaneous nonlin-

ear equations in N unknowns by using Brown's technique1 6.)

The described method determines the entire flow field in theneighborhood of Q in a shock-fixed coordinate system from the giveninitial conditions. Performing a simple transformation of the veloc-ities gives the solution in the laboratory coordinate system. Further-more, this flow configuration can be verified experimentally for a classof incident shock strengths and wall angles. For large incident anglesI and/or strong shocks, the equation system may not have a solution.

such cases a so-called "Mach reflection" takes place.

The theory of single Mach reflections from a solid boundary is dis-cussed in References 14, 17, and 18. We extend this theory to includethe case of an incident shock propagating into a nonquiescent region.Consider a step shock wave I which is incident upon a plane rigid wallthat makes an angle 0w with the horizontal, and which causes a Mach re-

flection to arise on the wall. The gas velocity ahead of the shock waveis towards I and is parallel to the wedge. The frame of reference isattached to the triple point Z. (See Figure 4.) The incident shock I,reflected shock R, and the Mach stem M emanate from Z as does the slip-line. The trajectory of Z is a straight line with angle X between it andthe wall surface. The region upstream of I and M is denoted by region0, upstream of R and downstream of I by region 1, downstream of R byregion 2, and downstream of M by region 3. The slipline divides regions2 and 3 which have equal pressures and equal flow directions with differ-ent speeds in this reference frame. We correlate the properties in thesefour regions in the imediate vicinity of the triple point. In thisshock-fixed coordinate system, the incident shock velocity is zero andthe gas velocity in region 0 relative to the wall velocity is parallelto the wall surface. The portion of the flow in region 0 hich passesthrough I makes an angle

#0 = tan v + w tan(ew+X) (3.1.13)Oy w

15. International Mathematical and Statistical Libraries, Inc., INSLLibrary 3, Edition 6, IMSL, Houston, Texas, 1977.

16. Brown, K.M., "A Quadratically Convergent Newton-Like Method BasedUpon Gaussian Elimination", SIAN Journal of Numerical Analysis,Vol. 6, No. 4, pp. 560-569, 1969.

17. Law, C.K., "Diffraction of Strong Shock Waves by a Sharp Compress-ive Corner", University of Toronto Institute for Aerospace StudiesTechnical Note No. ISO, 1970.

18. Ben Dor, G., "Regions and Transitions of Nonstationary ObliqueShock Wave Diffractions in Perfect and Imperfect Gases", Universityof Toronto Institute for Aerospace Studies Technical Report No. 232,1978.

23

INCIDENT SHOC uO

REGION 1 #

REFLECTED R RGOS HOC IC1

TRIPLE POINT-*/1 lt

Figure 4. Mach Reflection in a Neighborhood of the Triple Point Z.

24

with I where vOx, VOy , and w are the x and y components of the gas

velocity in region 0 and the incident shock speed in the laboratorycoordinate system, respectively. The resulting flow is then similar tothat described in the regular reflection case except that now the flowin region.2 need not be parallel to the wall surface. Instead, the flowvelocity u2 relative to the velocity of the wall must be parallel to the

wall surface. With the assumption that the incident and reflected shocksare straight line shocks (i.e. not curved) at least in a neighborhood ofZ, the oblique shock relations which now relate uniform flow propertiesin regions 0, 1 and 2 are:

Plu1 sin (#- 0 1) = pouo sin #0, (3.1.14)

2 2

PI p1°[Ul sin 01e)] 2 = P0 + 0[0sin *0] 2 , (3.1.15)

u 1 co - 01) * Uo cos 0 (3.1.16)

h I + 0.S[u I sin ( 0 1)] = h0 + 0.5[u 0 sin 0, (3.1.17)

Pu sin ) " = Plu1 sin #IV (3.1.18)

P2 + P2 [u 2 sin (1 1 " 02)] 2= PI + P1 [u1 sin 1i 2 , (3.1.19)

u cos C 1 (3.1.20)2 1 2 11

h 0.5[u2 sin (fI - 0 2)]= hl 0.5[u sin i] 2 (3.1.21)

The portion of the flow in region 0 which passes through the Mach stemmakes an angle *M with M. In general, the Mach stem will be a curved

shock, so M will vary with position along the Mach stem. While passing

through the Mach stem, the flow is deflected from its original directionby an angle e3 and its dynamic and thermodynamic properties are changed.

If the Mach stem is straight in a neighborhood of Z, the flow is uni-form in that vicinity. These properties are related by the obliqueshock relations (3.1.1)-(3.1.4). For the Mach stem the oblique shockrelations can be simplified with

ua • n a u0 sin #M' ub , n a u3 sin (#M - e3)1

ua , t - U0 Cos *MS ub 0 t - 9 3);

25

and rewritten as:

P3U3 sin N - 83) = PoUo sin OM, (3.1.22)

u3 cos (OM - 3) U0 cos , (3.1.23)

P3 + P3 [U3 sin (IM - 83)] = P0 + o[Uo sin m]2, (3.1.24)

h3 + 0.5[u 3 sin ( I - 03)]2 = h0 + O.. [u0 sin OM]2 . (3.1.25)

Furthermore, the flow fields in regions 2 and 3 are related across theslipline because equal pressures exist and the same flow direction occursnear the slipline:

P3 = P 21 (3.1.26)

03 = 01 02. (3.1.27)

For the special case where X = 0, the triple point Z attaches to the walland the slipline and region 3 are nonexistent. If one allows 83 = 0, equa-

tions (3.1.13)-(3.1.21) and equation (3.1.27) reduce to the regular re-flection case. For X 0 0, equations (3.1.13)-(3.1.27) represent 15equations in 16 unknowns x, *0' U0, Pl' u, 0l' *1' 0 2' P21 P2' u 2

ON, 03 3' P3, and u3 when the initial conditions (Po' Pl' TO' ew' Vox,

and v oy) are given. The perfect gas relations p = P/(TR*) and h =

yP/[(y-1)p] are assumed. In order to obtain the missing 16th equation,the entire Mach stem is assumed to be a straight line shock. Experiments

have shown that except for strong diffractions, the Mach stem is only17

slightly curved . Therefore this assumption does not introduce grosserrors. It is equivalant to the assumption of a uniform flow field aboutthe Mach stem. Because the flow adjacent to the wall must remain parallelto the wall's surface after passing through the Mach stem in the labora-tory coordinate system, the Mach stem must intersect the wall at 900.Consequently, we have the following geometric relation

--00 * + w (3.1.28)

along the entire Mach stem. Equations (3.1.13)-(3.1.28) form a systemof 16 nonlinear equations in 16 unknowns which determines the flow fieldin the neighborhood of Z when the initial conditions in Region 0 and theincident shock I are given.

26

For nonstationary flow, the criteria for distinguishing between thesundry types of reflections are contained in References 14 and 18. Re-flection occurs if the flow behind the incident shock is non-subsonic inthe shock-fixed coordinate system. Figure S delineates the regions ofregular reflection (bottom section) from Mach reflections (top section)in the angle of incidence - inverse shock strength plane. The curvelabeled *e is the limiting curve above which regular reflection istheoretically impossible. The curve labeled + is the limiting curve

below which the past history can not affect the reflection process. Theexperimental points indicate the smallest incident angle at which Machreflection has been observed. The termination of Mach reflection occurswhen the Mach number in the shock-fixed coordinate system of region 2is equal to or greater than one.

The implementation of the regular and Mach reflection theories toform the analytic model for shock wave propagation into a reentrantcorner is best illustrated by examples. Such examples are given in Sec-tion 3.2. In the next subsection the theory is used to compute the twoshock tube experiments described in Section 2.

3.2. Examples.

In the experiments, the walls forming the corners do not have in-finite extent. This limits the model's predictions to finite times afterthe incident shock reaches the vertex. To obtain the complete historyof the vertex pressure, a hydrodynamic computer code simulation of theentire experiment must be performed.

Consider the incident step shock from Shot 2 with P1/P0 = 2.3699

which propagates into an infinite reentrant corner with vertex angle 500.(See Figure 6.) From the geometry, the incident angle is 0 = 400. The

medium is assumed to have constant specific heats cv = 714.0 J/(kg-K)

and cp = 1001. J/(kg.K), and a gas constant R = 287.03 J/(kg.K).

The initial conditions in region 0 are P0 = 100.66 kPa, To = 295.48 K,

and zero gas velocity in the laboratory coordinate system with itsorigin at the vertex. Consider a point Q on the corner wall at which theincident shock impinges. If we make a Galilean transformation at Q, wecan apply the formulas in Appendix B. In the shock-fixed coordinate

3system, we compute p1 = 2.1561 kg/m , u1 = 667.35 m/s, T1 = 385.45 K,

81 - 1.208* and aI W 393.84 m/s. Because the flow is supersonic in

region 1, reflection occurs at Q. The point (#0 = 40* and P1/P0 =

0.4168) in Figure 5 falls below the # e curve, and thus regular reflec-

tion occurs at Q. Solving the four equations governing regular reflec-tion, equations (3.1.9)-(3.1.12), in the neighborhood of Q with ZSYSTM

27

~8O__*Experimental points 0

z ~ MACH REFLECTION

z4 0 -0O REGULAR REFLECTIONLU

020 IZ 0 0.2 0.4 0.6 0.8 1.0

INVERSE SHOCK STRENGTH P0/P,

Figure S. Criterion for Determining the Presence of Regular and MachReflection (Adapted from Reference 14).

28

P1 :236.55 ItPo REGION 0WT1

REGION 1

REGION 2P 0 10 0. 66 kPa

P2 s521.08 kPa 0

Figure 6. Flow Field Before the Incident Shock Reaches the Vertex forShot 2.

29

3we obtain p 2 u 3.7132 kg/m, u2 * 488.09 n/s, P2 u 521.08 kPa, and

*1=56.893*. (These results were obtained with ZSYSTM termination param-

eters EPS = 10 "1 0 , NSIG - 13 and II14AX - 100.) From geometric con-

siderations, the angle of reflection is 41.685 °. With the assumption ofan infinite corner, rarefaction waves do not exist within the corner,and the incident and reflected shock remain straight. Thus, the gasproperties behind these shocks are uniform and the values of the flowproperties calculated in the neighborhood of Q are those behind the en-tire extent of the shocks. Upon transforming back to the laboratorycoordinates, we obtain the configuration depicted in Figure 6. The gasproperties in regions 0, 1 and 2 are summarized in Table 2. (The veloc-ities are denoted by u in the shock-fixed coordinate system and by v inthe laboratory coordinate system.) The speeds of the incident and firstreflected shocks are denoted by wI and wRl respectively. The velocities

in the laboratory coordinates in regions 1 and 2 are parallel to theplane of symetry and the corner wall, respectively. The angle of in-cidence is not equal to the angle of reflection and this one reflectionprocess has already increased the pressure near the wall by a factor of5.18.

The pseudo-steady flow of Figure 6 remains unchanged until theincident shock reaches the apex. At that instant only the first re-flected shock remains (only regions 1 and 2). This shock continues topropagate along the plane of symmetry with an angle of 8.315' and aspeed of 3637.0 m/s. With an inverse strength of P1/P2 * .4578, regu-

lar reflection occurs at any reflection point Q' according to Figure 5.(See Figure 7.) Because the flow properties are already calculated inregions 1 and 2, only the flow in region 3 must be calculated. We makea Galilean transformation at Q. In this shock-fixed coordinate system,the velocity magnitudes are u1 3 3865.52 m/s and u2 = 3838.63 m/s in

and the flow deflection angle across the shock is 3.464e . Solving thefour equations governing regular reflection, equations (3.1.9)-(3.1.12),in the neighborhood of point Q1 with identical ZSYSTM termination param-

eters as before, we obtain 3 6.0394 kg/m3, u3 = 3808.8 m/s, P = 1.045

MPa, and *2 = 9.0690. From geometric considerations, the angle of reflec-

tion is 5.605° . With the infinite corner assumption, the second reflectedshock remains straight and the gas properties behind the shock are uniform.Thus, the values of the flow properties calculated in the neighborhood of

are those behind the entire extent of the second reflected shock. Upontransforming back to the laboratory coordinates, we obtain the flow fieldin Figure 7 with respect to regions 1, 2 and 3. The gas properties ofregion 3 are given in Table 2. The gas velocity in region 3 is parallelto the plane of symetry. The speed of the second reflected shock isdenoted by wR2.

30

Table 2. Regional Flow Properties for Shot 2 in Laboratory Coordinates.

Region 0 Region 1 Region 2

P0 - 100.66 kPa P1 - 238.5 kPa P2 a 521.08 kPa

PO * 1.1869 kg/n3 P 1 = 2.1561 kg/n P2 a 3.7132 kg/n

3

To - 295.48 K T1 a 385.4 K T2 = 488.91 K

a0 m 344.82 ./s a1 - 393.84 n/s a2 = 443.56 n/s

V 0x - 0 U/s vlx - 228.52 m/s V2x - 194.62 m/s

Vy - 0 U/s V - 0 m/s V = 231.94 m/sO n/ "ly 2y

WI 508.36 rn/s w RI 525.96 rn/s wR2 *355.24 rn/s

Region 3 Region 4* Region S*

P3 1.0447 MPa P4 - 1.8913 MPa P z 1.8913 MPa

3 3 3

P3 - 6.0394 kg/m3 P4 - 9.1668 kg/m3 p 5 = 8.7707 kg/m

3

T3 - 602.65 K T4 = 718.80 K TS = 751.26 K

a3 = 492.46 m/s a4 = 537.83 m/s a5 - 549.84 m/s

V3x ' 171.8 m/s V4x - -33.492 m/s V sx -101.90 m/s

v 3y a 0 /s v4y - -75.358 m/s v s= -121.42 m/s

WR3 - 479.713 m/s ' 2 - 497.18 m/s

*These values are valid only in the neighborhood of the triple point.

31

P2 z521.08 kPa 1.4 A~

P5 mi.S91MAPa

Figure 7. Flow Field Shortly After the Incident Shock Reaches theVertex for Shot 2.

32

The second reflected shock will subsequently impinge on the cornerwall at an angle of 55.605. With an inverse shock strength ofP2/P3 - 0.4988, Mach reflection occurs at the wall according to Figure

S. Aligning the second reflected shock with the schematic in Figure 4,the incident shock speed becomes 355.24 a/s and the gas velocity com-ponents in regions 0 and I are vx * - 249.84 m/s, vo, * - 171.03 r/s,

Vlx - - 16.781 m/s, and v ly - 170.98 r/s. For Mach reflections, a

Galilean transformation is made at the triple point Z. The velocityof the triple point depends on an unknown X of the configuration.Thus, in the shock-fixed coordinate system for Mach reflections, thevalues corresponding to #0Uo' Ul, 1 are not known, even though in

the laboratory coordinate system they are known. Because the thermo-dynamic properties are independent of the coordinate system, the valuescorresponding to Po, Pl, poi pl are known as is the wall angle 8 w - 900

- S5.60S5. In reference to Section 3.1, we have now IS equations and15 unknowns (P1 is now known). It was found by the first author that

the solution of these equations is more simply obtained by the follow-ing procedure: (a) guess an initial value of X, XG, and compute the

corresponding # from equations (3.1.13) and (3.1.28); (b) solve anMG

appropriate subset of the equations corresponding to equations (3.1.13)-(3.1.27) for *ll, P2, P29 u2 *-Mc, e., 03' P3 and u3 with the sub-

c

routine ZSYSTM; (c) iterate on X until ifM - * M is zero within aG c

given tolerance. Following this technique (tolerance u 10-4 ), thesolution of the flow field in the neighborhood of Z in the shock-fixedcoordinate system is

x a 11.083% 81 a 13.710% u0 - 805.86 m/s, P2 - 9.1668 kg/3,3

0= 48.664,' e 2 4.608, u1 - 649.33 a/s, P3 a 8.7707 kg/a ,

#M = 83.0599 e3 9.1010, u2 - 434.86 m/s, P2 = P3 = 1.8913 MPa.

01 M 80.807,' u3 = 352.39 m/s,

Table 2 lists the values of the flow variables in the laboratorycoordinate system. From geometric considerations the angle betweenthe incident shock (actually second reflected shock) and the reflectedshock in the Mach configuration is 64.239". If we extend this Machreflected shock in a straight line to the plane of symmetry, the in-cident angle at the point of intersection " I is 110.1S60 (an obtuseangle). Thus, no more reflection can occur. This reflected shockmust then impinge at Q"I at a right angle to the plane of symmetryso that the gas flow in the neighborhood of Q" is parallel to theplane of symmetry. Consequently, the reflected shock must be curvedto satisfy the required angles at points Z and Q/1, and the flow

33

properties in region 4 are not uniform. See Figure 7. The values ofthe flow properties in region 4 must be obtained by a hydrodynamiccomputer code simulation. However, an approximation of the vertex pres-sure value can be obtained by taking as the vertex value the pressurevalue calculated in the intersection of region 4 and the neighborhoodof Z. Although obviously incorrect, the resulting pressure value givesa peak pressure value comparable to those of experiments while retain-ing the simplicity of the model. If the incident angle at Q, wereacute, Q I would be another reflection point and the method of analysiswould have continued. If the incident angle at Q/J were 90', no furtherreflection would occur and the final shock would not be diffracted. Insuch a case, the reflected shock would remain straight, the gas proper-ties in region 4 would be uniform, and the method would give exact val-ues of the flow field within the entire infinite reentrant corner.

The Mach stem ZQI" is assumed to be straight and intersects thewall at 90' according to the discussion in Section 3.1. Its speed isdenoted by wR21 . The velocity in region S is parallel to the wall. The

pressure near the wall behind the Mach stem has increased by a factorof 18.8. A curved slipline separates regions 4 and S which has identicalpressure values across it. The velocities in regions 4 and S relativeto the unsteady motion of the slipline are parallel to the slipline.Because region 4 is nonuniform, the computed values are valid only in aneighborhood of the triple point and care must be used in any extrapola-tion. The speed of the Mach reflected shock wR3 is also correct only

in the neighborhood of Z.

Shot 1 is analyzed in a similar fashion. Conceptually, the onlydifference in the analyses is that at the last reflection point, regu-lar reflection occurs instead of Mach reflection. As before, the lastshock wave is diffracted and the pressure at the final reflection pointis taken as the vertex pressure value. The flow fields shortly beforeand after the incident shock reaches the vertex are shown in Figures 8and 9. The regional flow properties in laboratory coordinates aresummarized in Table 3.

3.3. Peak Pressure Duration for a Finite Corner.

To determine an analytic approximation for the duration of thepeak pressure at the vertex of a finite corner, we use the concepts devel-oped in Reference 19. The velocities of the incident shock wave and ofthe rarefaction wave generated at the leading edge of the finite cornerare calculated from the values computed in Section 3.1. Using theknown length of the corner wall, the time from peak pressure devel-opment (arrival of the incident shock at the vertex) to the instant therarefaction wave reaches the vertex can be computed.

19. Smith, L.G., "Photographic Investigation of the Reflection of PlaneShocks in Air", NDRC Report No. A-350, OSRD Report No. 6271, 194S.

34

I.-

P :113.20IPa REGION 0

REGION I

REGION 2 P0 " 100.80kPa

40

Figure 8. Flow Field Before the Incident Shock Reaches the Vertexfor Shot 1.

35

0000

10000

101-10

,.o1 00 0 U

0h

,o , 0

z U

oc1000 0

cqd6

c4..

ale

364

Table 3. Regional Flow Properties for Shot I in Laboratory Coordinates.

Region 0 Region 1 Region 2

P0 - 100.80 kPa P1 = 113.20 kPa P2 = 126.91 kPa

o = 1.1887 kg/ 3 Pl - 1.2912 kg/ 3 P2 = 1.4008 kg/n3

TO = 295.44 K TI m 305.45 K T2 = 315.64 K

a0 W 344.80 r/s aI = 350.60 n/s a2 = 356.40 m/s

VOx = 0 3/s Vlix = 28.79 M/3 V2x = 23.84 n/s

V y = 0 M/s Vly v 0 /s v2y = 28.40 m/s

wI a 362.51 m/s WRI = 363.39 3/s WR2 = 342.23 n/s

Region 3 Region 4*

P3 a 141.9 kPa P4 = 165.44 kPa

03 - 1.5175 kg/m P4 = 1.6922 kg/K3

T3 = 32S.98 K T4 = 340.62 K

a3 = 362.19 n/s a4 = 370.23 m/s

V3x = 19.2 n/s V4x = -15.95 n/s

V3y . 0 rn/s v4y = -19.01 ./s

WR3 = 370.10 n/s

* These values are valid only in the neighborhood of the last reflection

point.

37

The incident shock wave travels at a speed of wl/sin 0 along the

wall and traverses it in tw (dw sin #o)/wl, where dw is the length of

the wall. (See Figure lOa.) The rarefaction wave travels at a speedequal to the algebraic sum of the sound speed and the gas particlespeed. In tw seconds, the rarefaction wave in region 2 travels a dis-

tance dr - (a2 + v2)tw, because the direction of the gas velocity is

toward the vertex. The remaining distance that the rarefaction wavemust travel to reach the vertex is

dp = dw dr

(3.3.1)

= d[1 2v2 sin

The time that the rarefaction wave takes to traverse the distance d isPthe duration of the peak apex pressure. The velocity of the rarefactionwave depends on the gas properties in regions 2, 4 and 5 for theconfiguration depicted in Figure 7 and the gas properties in regions 2,3 and 4 for that given in Figure 9. Because the extent of region 3 at Q//near the wall is small (see Figure 9) we neglect its influence on thespeed of the rarefaction wave. The rarefaction wave continues to propa-gate into region 2 until it meets the final shock traveling at speed wF

along the wall from the vertex. (See Figure lOb.) The time at whichthis occurs is

dt* - a (3.3.2)a2+v2+WF

The time to traverse the remaining distance from their meeting point to thevertex is

+ wF t*t -- (3.3.3)aF-vF

where a F and v F are the sound speed and gas velocity in the final region,

respectively. For aF and wF we choose the well-defined gas properties

inmediately behind the last shock at the wall. These values are notaccurate for the whole region, and therefore, the calculation providesonly an approximation to the duration of the peak pressure. Addingequations (3.3.2) and (3.3.3), the approximation to the duration of thepeak pressure is

38

C144

LU 0

44

0,b4

cc x

90 uCu

*14

39.

t+ t+I + F d. (3.3.4)a F-VF a 2+vz1wFj

In both experimental Shots 1 and 2, the distance dw is 0.166 m and

angle f0 is 40°. For Shot 2, we use the appropriate values from Table

2 with w = wR2iaF = a5, and v = v5 in equations (3.3.1) and (3.3.4)

and compute the duration of the peak pressure to be 17.06 us. For Shot1 we use the appropriate values from Table 3 with wF = wf3, aF =a 4 and

vF = v4 to obtain from equations (3.3.1) and (3.3.4) the peak pressure

duration of 135.78 us.

4. DORF CODE SIMUIATION OF AN INFINITE CORNER

4.1 DORF Code Description.

The DORF hydrodynamic computer code is a two-dimensional, Eulerian,explicit code which solves the equations for conservation of mass,

2-P + • (p) ( , (4.1.1)a t

conservation of momentum,

+p ) + .(p ) = - P, (4.1.2)

at

and conservation of total energy,

a (pE + [V - -p _) *(P _V) (4.1.3)t v. (.E

together with an equation of state of the form

P = P(p,I), (4.1.4)

and a sound speed equation of the form

a = a(p,P). (4.1.5)

For the computations described in this report, the fluid istaken to be inviscid air and is assumed to be polytropic, so equation

(4.1.4) becomes

P = PI(y-l), (4.1.6)

with y = 1.402. The particular sound speed relation is ia (4.1.7)

40

The momentum equation (4.1.2) is derived from the Navier-Stokes equation

EEL . (P). pg- P .at

+ 2( .A)4+X(VWX), (4.1.8)

by assuming the fluid to be inviscid (v - X a 0) and the gravitational

acceleration * 0 0. The energy equation (4.1.3) is derived from

3(pE) + • (pE )] p • gat

* • { T - P A * + • -) (4.1.9)

by letting the heat transfer coefficient T - 0, and using the previousassumptions of zero viscosity and gravity.

The DORF code uses an explicit numerical method. The time step fora given sweep through the computational mesh is computed by sweepingthrough the entire mesh and computing, on a cell by cell basis, theminimum cell dimension

d = nmin (bx, Ay), (4.1.10)

the time step based on the sound speed in the celldi

At a n 4..1

the time step based on the x direction particle speed

A aAX (4.1. 12)

X

and the time step based on the y direction particle speed

At. y . (4.1.13)y v

y

Once these values are computed, a candidate time step based on the givencell is computed by

41

Atk min(Atat Atx t y). (4.1.14)

This process is repeated for each cell in the flow field with the finalvalue for the time step to be used during the next computational cyclebeing

At= min[AtlAt29 ... Atk, ...Atkm ax], (4.1.1S)

where n is the Courant-Friedrichs-Lewy stability factor.

Once the time step is established, the DORF code proceeds to compute

phase I, which is the Lagrangian phase. The flux term [. (pvv)] inequation (4.1.2) is temporarily dropped, leaving

= - v . (4.1.16)

which is used to compute the fluid acceleration caused by the pressure

gradient between two adjacent flow field cells. The flux term [ (pe)]in equation (4.1.3) is also dropped, leaving

I • (F;), (4.1.17)at

which is used to compute the work done at the flow field cell boundaries.In the Lagrangian phase, the continuity equation (4.1.1) becomes

S0 .(4.1.18)atSubstituting equation (4.1.18) into equations (4.1.16) and (4.1.17), we

obtain = P, (4.1.19)

and

P W .(PV) (4.1.20)

respectively. If we take the specific kinetic energy equation (the dot

product of equation (4.1.19) with V) and subtract it from equation(4.1.20), the equation for the specific internal energy for the Lagrangianphase is obtained. The updated Lagrangian phase values of the momentumand the specific internal energy are then computed using central spatialdifferences and forward time differences for the corresponding deriva-tives in the momentum equation (4.1.19) and the specific internal energyequation.

42

The initial values in the Eulerian or fluxing phase are the updatedLagrangian phase values. The governing equations in this phase are thecontinuity equation (4.1.1), the mmentum conservation equation (4.1.2)

without the stress term - P,

!sea -" (J'wp), (4.1.21)at

and the total energy conservation equation (4.1.3) without the work rate

term - - (Fi),

at(pe = _(pev) (4.1.22)at

The DORF code is set up so that the grid origin is at the lower leftcorner of the grid. The cell indexing increases monotonically whenmoving either upward from the origin or to the right from the origin,as do the spatial coordinates being simulated. The DORF code uses thedonor cell fluxing method to update the mass, momentum, and specifictotal energy values. The updated value of the specific internal energyis computed by subtracting .the specific kinetic energy (obtained fromthe updated velocities) from the updated specific total energy values.These post-Eulerian phase values are taken to be the values at the endof a given time step. Within the Eulerian phase, a logical constraintexists so that a given cell cannot flux out more material than it con-tains. The donor cell fluxing method is first order accurate. Thusthe entire accuracy of the solution is first order. A more detaileddiscussion of the DORF code is found in Reference 5.

4.2. Examples.

The DORF code was used to simulate the gas flow within an infinitelylong reentrant corner for the case depicted in Figure 1 with 2a=100.An infinite reentrant corner was chosen, because the corresponding gas flowis considerably simpler than the flow within a finite corner, and because acomparison of the results of DORF and the analytic model was intended. Thesimplification which is made possible by the existence of a plane of symmet-ry within the 1000 corner was not exploited in the numerical simulation.

For a non-right angle corner, one of the straight walls forming thereentrant corner within the framework of the DORF code must be approximatedby a stair-step. Consequently, the simulation of an 1000 corner with itsaxis of symetry and its more separated walls is preferable to the simula-tion of a 50 corner. The quantification of the effects of stair-steps isa primary objective of this numerical computation. A discussion of thedifferent types of stair-step approximations is given in Section 4.3.

43

Using the DORF code we simulate the two experiments that were alsoanalyzed by the analytic model. The initial conditions used by DORFare those obtained from the analytic model corresponding to the gasflow configuration before the incident shock reaches the vertex. ForShot 2 the initial conditions correspond to those depicted in Figure 6and for Shot 1 those depicted in Figure S. Because the corner's orien-tation for the computer calculation is different from that for the ana-lytic calculation, and because the full 100* reentrant corner is ana-lyzed, the direction of the velocity vectors must be modified. Figure11 shows the initial flow field for the 100* corner based on the 50*corner analysis. Table 4 lists the initial values of density, specificinternal energy (I-c vT) and velocity components for Shot 2 while Table

5 lists the initial values for Shot 1. The walls forming the reentrantcorner (the bottom boundary and the 10* wedge emanating from the topleft corner in Figure 11) are reflective boundaries. The top and rightboundaries are transmissive inflow boundaries for the present problems.However, no special boundary routine was incorporated into DORF tosimulate a steady inflow. Instead, in order to avoid erroneous bound-ary signals from propagating into the portion of the flow field ofinterest, the top and right boundaries were placed sufficiently far up-stream of the vertex of the reentrant corner and the incident shock.It was estimated that any waves from the boundaries theoretically travelless than 10% of the distance along the wall toward the vertex duringthe simulated time. The actual speed of such boundary signals in thecomputational domain can be monitored during the calculation, becauseregion 2 near the transmissive boundary and along the smooth horizontalwall should remain uniform until the reflected shock emanating from thevertex reaches this region.

For the sample calculations, we chose the stability factor n = 0.4and did not activate the artificial viscosity option. The mesh size andthe stair-step approximation are discussed in the next section. Theresults of the DORF simulation are stated in Sections 4.4, S.1, S.2 and5.3.

4.3. Stair-Step Approximation.

The computational domain is a two-dimensional Cartesian grid forminga 1000 reentrant corner. The bottom boundary is a smooth reflectingboundary representing one side of the reentrant corner. (See Figure 12.)The left boundary is also a reflecting boundary, but it is along thisboundary that the non-responding flow field cells are stacked to constructthe other side of the reentrant corner. As may be seen in Figure 12,the actual reentrant corner side along the left boundary is approximatedby a series of discrete steps. If a single computational cell per stepwere used with its diagonal parallel to the reentrant wall the cellwould have a cell aspect ratio of

44

Table 4. DORF'S initial ValUeS for Shot 2.

Resion 0 Rtesion 1

Po - 1.89k/ - 2.1561 k/

10 a210.97 ks I~ I 275.21 ks

vx X0-0.U/s v 1 -146.89 u/

v -0 /S v -175.06 rn/s

Ratios 2 ad e 41.685*

02 a P21 a 3.7132 ks/rn3

1 2 a 12/ - 349.08 WIks

vx2 ' -302.78 rn/s

v 2a 0 U/s

v * -225.05 rn/s

4S4

Table S. DORF's Initial Values for Shot 1.

Region 0 Region 1

P0 = 1.1887 kg/n3 *1 0 1.2912

I 0 = 210.94 kJ/kg I I - 218.09 kJ/kg

Vo= 0Oa/s Vxl - -18.506 u/sv X0 O0U/s vxi,-1.6 lvyo = 0 U/s Vyl - -22.054 r/s

Regions 2 and 2' = 40.116"

P2 ' 02/ = 1.4008 kg/m3

12 = 12/ = 225.37 k/kg

Vx2 - 37.079 ls

v = 0 m/s

Vx2/ = 6.4387 r/s

Vy2/ = - 36.516 a/s

46

I

0 TRANSMISSIVEBOUNDARIES

REGION 2'

REFLECTIVEBOUNDARY REGION I

REGION 0 REGION 2

Y o

X k-REFLECTIVE BOUNDARY

Figure 11. Initial Flow Field for DORF Simulations.

47

ACTUAL REENTRANTCORNER SIDE

REENTCTANT CORNERRAPE

X-AXIS

Figure 12. Reentrant Corner Construction.

48

A~y = Y= 1 5.7 (4.3.1)

ASx ax tano (4..1

where ASy, ASx are the dimensions of a step, and Ay and Ax are the di-mensions of a cell, in the y and x directions, respectively. A largeaspect ratio such as this can cause numerical instabilities in DORF.This stair-step construction can be modified by dividing the single cellinto several cells. For the case of ASy > ASx, the values of ASx,ASy and Ay are given by:

ASx a Ax, (4.3.2)

and ASx

Ay= ASY/ (I t 1) (4.3.4)

where j is the greatest integer function. In our case 6 - 100, Ax =10, and Ay is calculated via equations (4.3.2)-(4.3.4) as 9.4521 mm.The steps are designed so that each complete step has six cells in they-direction and one cell in the x-direction. Furthermore, the stair-step wall is constructed so that the line representing the actual, smoothreentrant corner wall bisects both the horizontal and vertical sectionsof each step. The computational grid contains 60 equal cells in the x-direction and 60 equal cells in the y-direction.

A study comparing the effects of constructing a stair-step wallwith more than one square cell per step as opposed to a single highaspect ratio cell per step for shocks propagating parallel to a coor-

dinate axis has been performed at the Air Force Weapons Laboratory . Theconclusion reached in that report is that the better way to construct theapproximated wall is by using high aspect ratio cells like those thatwould be generated by equation (4.3.1). This has intuitive appeal, andmay be quite valid for codes which can accurately compute with high as-pect ratio cells and for cases where the incident angle between the walland the shock is not near 45. For a wall angle of 45", this methodreduces to using a single square cell per step. With single cell steps,a normal shock reflection occurs at every cell when the shock reachesthe wall, which does not occur on the physically straight wall. For ahigh aspect ratio cell whose long side is along the direction of travelof the shock, this normal reflection will be weaker than for a squarecell. In any case, the numerical simulation may not represent the physi-cal flow because of the many small normal shock/surface interactions.

49

Figure 13 shows a partial reconstruction of the grids used for two

AFWL 9 problems designated by the ediors 19.6026 (hereafter referred toas Problem A) and 19.8029 (hereafter referred to as Problem 5). ProblemB uses high aspect ratio cells with Ax - 3 = and Ay a 0.$29 am. Adiagonal dram from the cell lem left coer to the cell upper rightcorner is at a 10" alo with respect to the lower bmdary of the cell.This construction provides a pod qpximatin of the 100 wall angle.Using this constructiom a W mpaslem oeom at me and of eech cell,a normal reflection occurs at the ether eud, ad the comted cell-center values represemt m averaging between these two extremes. Theabsolute atmospheric pressur for this cemputatios is 101.13 kPa and theshock overpressure is 544.6 kPa. The omputed peek for Problem 3 is423 kPa, which compares very well with the predicted value of 427 kPataken from Figure 3.766 (f Neferemce 20. Problem A uses steps construct-ed of square cells, 3 - . i sie, with the steps being one cell highand five cells wide. Ike azi --- t does not approximate a 10* wallbut rather approximates -a 11.31 sll. If these cells were constructedas prescribed by equaties (4.3.2)-(4.3.4), this discrepancy would berectified. The five flew field cells comprising say given step inProblem A represent a variety of cemiltioms. The forward-most flow fieldcell along a step, for exmle cell "A" in Figure 13, is essentially thefirst cell around a 90 expansiom cerer after a normal reflection. Thelast flow field cell on a step, for example cell "r' in Figure 13, isthe last flow field cell prior to a mm-responding wall against which theshock undergoes a normal ref lectiou. The intervening three flow fieldcells provide a transition of umnmown accuracy between the two extremes.Problem A shows a peak reflected overpressure of 0.758 NPa, 77% above

the predicted20 value. However, the flow field cell used in Reference9 to give the peak pressure value is the same type as cell "B" inFigure 13. This higher pressure is the result of the interaction ofthe shock with a one-cell-high normal reflecting surface. It would bemore informative to see the overpressure histories for all five flowfield cells on a step, particularly the third, or middle, cell. Thethird cell has its center at the geometric center of the step, Just asthe cell center for the high aspect ratio cell is also the geometriccenter of the single cell step in Problem B. A comparison between thethird cell in Problem A and the high aspect ratio cell in Problem Bmight be more meaningful.

Another comparison in Reference 9 is between a HULL computation and

the corresponding experiment21 for a step shock (P1/P0 - 27.3) impinging

on a 40* wall. In that example a Mach reflection develops on the wall.

20. Glasstone, S. and Dolan, P.J., '"he Effects of Nuclear Weapons",Department of the Army Pamphlet No. 50-3, Headquarters, Departmentof the Army (March 1977).

21. Bertrand, B.P., "Measurement of Pressure in Mach Reflection ofStrong Shock Waves in a Shock Tube", BRL-MR-2196, US ArmyBallistic Research Laboratory, Aberdeen Proving Ground, ND4(June 1972). (AD #746613)

so

........ ............ _

-1~ccn

z

Lu 0I .- U.

a-Il I-

zLU

z'"

00

~t 0.. 4c0

0J 0

o -o

.4

41

Id;

As before, the incident shock is propagating parallel to the horizontalcoordinate axis. The grid used for the HULL calculation is constructedof nearly square cells (Ax - 3mm, Ay - 2.5173im) in the region of thewall. Each cell on the wall surface constitutes a stair-step. Thediagonal of each cell is parallel to the physical wall. As the wallangle approaches 45" both methods studied in the APWL report became thesame. Thus, for this case the procedure recommended by AFWL would bethe same as the one used in the DORF simulations. The maximm differ-ence between HULL's results and the experimental peak and late time pres-sures is 11.S%. However, the structures of the pressure histories donot compare well. The pressure plateaus seen experimentally are notevident in the calculations and the computed times between the initialshock arrival and peak pressure for points well up the wall also do notagree. Thus, the HULL calculation for this problem shows inconsistentsimulations of the physical phenomena.

The stair-step formulation used in this report is quite similar to

the multiple-square-cell step construction described in the AFWL 9 report.However, the physical phenomena considered in this report differ fromthose flows simulated by AFWL because the incident shocks do not impingenormally on the stair-step (see Figure 13) but rather obliquely (seeFigure 12), and the incident shock strengths considered in this reportare smaller (by 7S%-SO%) than those in the AFVL study. This reportconsiders both regular and Mach reflections, while the AFWL report isonly concerned with MACH reflections. Furthermore, this report dealswith the DORF code while the AiWL report documents the HULL code re-sults. Thus the AFWL study is informative, not definitive.

Any type of stair-stepping represents a potentially serious problem.For example, along the bottom boundary of the grid, the incident angleis #B = 40° everywhere, as desired. However, along the left boundary,

the incident angle is *L = 500 because of the stair-stepping. Moreover,

at every sixth cell along that side there is a local reentrant corner,having OL = S0 and u 40*. Each of these discrete steps disturbs

the flow field.

A better solution to the problem of forming a stair-step wall wouldbe to add the capability of including partial rigid cells in a flow

field. Such a capability has been successfully implemented22 in the

SAMS code23 which belongs to the same family of hydrodynamic computercodes as do DORF and HULL. Another alternative is to turn to recentlydeveloped techniques for generating body conforming coordinate systems.

22. Coleman, M., ARRADCOM, BRL, Private communication.23. Traci, R.M., Fan, J.L., and Liu, C.Y., "A Numerical Method for the

Simulation of Muzzle Gas Flows With Fixed and Moving Boundaries",BRL Contract Report No. 161, June 1974. (AD #784144)

S2

However, irrespective of the possible alternatives to stair-stepping astraight boundary, a purpose of the present study is to quantify thedifference in the flow field in the vicinity of a stepped reflectingboundary as opposed to that near a smooth reflecting boundary.

4.4 Comparison of the Pressure Along the Smooth and Stair-StepBoundaries.

The pressure values for Shots I and 2 along both the smooth andstair-step walls are compared in two ways: the spatial pressure pro-files are given along the walls at five specific times and then thepressure histories at four selected pairs of wall positions. Thepressure values are cell-centered. The cells chosen for the comparisonare adjacent to each wall for example, cells 1,2,3,4,5,... along thesmooth wall and cells 1,21,3J,4,S/,... along the stair-step wall asshown in Figure 14. Therefore, only one stair-step corner is initiallydownstream of the incident shock. The distance from the vertex of thecorner to a given point along either wall is denoted by d. The initialposition of the shock is along the straight line between the points(O.158m, 0.0m) and (0.09m, 0.0567m) in the computational domain.

An inconsistency in the reflected pressure values at the first re-flection point arises in the DORF simulation due to the stair-step approxi-mation. For example, consider the simulation of Shot 2. Because theactual incident angle of the incident shock along the stair-step is 500,Mach reflection is predicted at the first reflection point (see Figure 5)rather than regular reflection as on the smooth boundary (incident angle40*). Even when the same type of reflection is predicted, the pressurebehind the reflected shock along the stair-step wall is different fromthe value along the smooth wall, because the angles of incidence of theshock are different. This inconsistency is inevitable given the con-straints of the present DORF capabilities. The discrepancies betweenthe stair-step and smooth boundary pressures caused by this inconsistaen-cy may be lessened somewhat, because the conditions imposed in region 2'are identical to those in region 2.

The variations of the absolute pressure values for the numericalsimulations of Shots 1 and 2 along the smooth and stair-step walls arecompared at the initial time (t=0), at a time immediately before theincident shock reaches the vertex, at the time when the maximum pressuredevelops at the vertex, and at early and late times after the final shockleaves the vertex. The spatial extent of the comparison is O.18m fromthe vertex along each boundary. Figures 15-19 show the histories ofthe computed wall pressures for Shot 2; Figures 20-24 describe thosefor Shot 1.

53

.15 - - - -

.10 - - PHYSICAL PLANE

SOF SYMMETRY--"

.05INITIAL POSITION_OF INCIDENT SHOCK -

00 .05 .10 .15 .20 .25 .30

X-AXIS (M)

Figure 14. Detailed Portion of Computational Domain Near the Vertex.

54

Figure 15 compares the initial pressure values along the walls, asprescribed by the initial conditions. The transition from the ambientpressure to the pressure behind the reflected incident shock is smearedout over 3 to 4 cells. These initial curves are not identical, becausethe uniform mesh spacings along the x-coordinate are not equal to thosealong the y-coordinate.

At t = 69.74 ps (Figure 16), the incident shock has theoreticallypassed the stair-step corner but has not reached the vertex. However,the calculated pressure has begun to rise due to the shock smearing incell 1. Along the smooth boundary, the pressure beyond d = 0.045mis within 4% of the theoretical value of 521.08 kPa. For d > .10m, thepressure is within 1%. Along the stair-step boundary, three pressurepeaks corresponding to the three stair-step corners are present. Thefirst pressure peak at d = 0.0331m is substantially larger than theremaining two. For the first corner, the incident shock propagatinginto this small "reentrant corner" is reflected, with an accompanyingincrease in pressure. For the other corners, the initial gas velocitiesin region 2' are such that the air is partially stagnated in thesecorners which causes a lesser pressure rise than the shock reflectionin the first corner. Although the pressures along the stair-stepboundary do oscillate around the corresponding pressures on the smoothboundary, the character of the flow along the stair-step boundary ismarkedly different from that along the smooth boundary. The amplitudeof the oscillations is between 20% and 61% of the smooth boundary valuesin this example. The last two pressure peaks in Figure 16 do not occurexactly at the corner position because they reached their maxima pre-viously at the corners and are now decreasing and propagating from thecorners.

At t = 114.3 Vs (Figure 17), the pressure at the vertex (in cell 1)reaches a maximum value of 1.643 MPa. The incident shock has reachedthe vertex and the reflected shock is ready to propagate from the vertex.The pressure curve along the smooth boundary approaches the theoreticalfar-field value without any oscillations and with only a slight under-shoot. Along the stair-step wall the effects of the three corners areapparent. The pressure in the first corner has decreased slightly andhas propagated along the neighboring smooth section of the stair-stepwalls. The areas of increased pressure near the second and thirdcorners have enlarged and the pressure values have decreased from thoseshown in Figure 16. The pressure values in cells 2' and 3' are greaterthan those along the corresponding section of the smooth boundary. Thepercent differences between the pressure values are 49.7% and 25.1%,respectively. The reason for these differences is that regular reflec-tion occurs along the smooth boundary and Mach reflection occurs alongthe stair-step boundary, due to the different angles of incidence.

55

usa

zI

44

OL 4.

ft.0

IL

cog

I.-

-0!

.4

A

C144

S6

1.0STAIR- STEP WALL

-SMOOTH WALLPOSITION OFSTAIR-STEP CORNERS

uJ

0.n 0.2..1

DISTANCE FROM APEX (n

Figure 16. Comparison of Wall Pressure Values for Shot 2 at ta69.74is.

57

1.6

)A \ _STAIR STEP WALL-SMOOTH WALL*POSITION OF

STAIR-STEP CORNERS

1.2

'U~1.o

06

OA

0.2

0 a I a I a 1 I0 0.04 0.06 Q32 0.16

DISTANCE FROM APEX (in

Figure 17. Comparison of wall Pressure Values for Shot 2 at ta114.3uas.

58

Figure 18 gives the pressure profiles after the final reflectedshock has left the vertex. At this early time, the magnitudes of thepressures differ by nearly 25% along the wall before the first stair-step corner. The first corner, which is just downstream of the pressureovershoot, changes the slope of pressure decay along the stair-stepboundary. The effects of the second and third corners can still be seen.Along the smooth wall the pressure decays without oscillations from itsovershoot to the far field pressure value.

Figure 19 shows the pressure profiles at a later time, t = 321.4us. The position of the pressure peak is 0.085m from the vertex alongthe smooth waill and 0.0995m along the stair-step wall. The percent differ-ence in locations is 17%. Both pressure profiles are oscillatory be-hind the peaks. However, the pressure values for d < 0.45 a are within5% of one another. Near the peaks, the pressure values differ greatly.Ahead of the peaks, the previously described behavior again occurs.Hence, only for "sufficiently" large distances behind the pressure peaksare the pressure values along the smooth and stair-step walls comparable.The beginning of the equilibration of the pressure values in a smallregion near the vertex is the physical reason for the 5% agreement.

Figures 20-24 show the comparison between the absolute pressurevalues along the smooth and stair-step walls for the lower pressure ShotI. This series of figures is analogous to Figures 15-19 for Shot 2. InFigures 20-24, the pressure scale is more refined and thus the percentpressure differences are significantly smaller than those in Figures 15-19. Figure 20 gives the comparison of the initial pressure profiles.The characteristics are identical to those in Figure 15. At t - 91.09 us,(Figure 21) the incident shock is between the vertex and the first stair-step corner. The pressure in the vertex cell has began to rise due toshock smearing. Behind the incident shock overshoot (d > .055m), thecalculated pressures along the smooth boundary are within 2.4% of thetheoretical value. Along the stair-step boundary, a pressure peakoccurs near each of the three corners. The sam phenomena which arediscussed in relation to Figure 16 are present again. The cornereffects are not so severe, because the initial flow field is weaker.The calculated pressure values along the stair-step boundary are differ-ent by at most 5.3% from the calculated smooth boundary values. Theseresults are significantly better than those depicted in Figure 16. Thepressure profiles at t - 142.4tis (Figure 22) show that the incident shockhas reached the vertex. The numerically caused pressure rise along thesmooth boundary at 0.065 a deviates by less than 2% from the theoreticalvalue. The effects of the three corners along the stair-step wall areapparent. The pressure peak at the first corner has decreased from thatin Figure 21, and those at the second and third corners have expandedalong the smooth sections of the stair-step walls. The maximm devia-tion of the stair-step wall profile from the smooth wall profile is 9.2%and occurs at the first corner. The differences between the pressureprofiles at the second and third cell centers are 1.2% and 2.3% respec-tively. This significant improvement over the calculation for Shot 2

59

1.86-

1.6

--'-STAIR-STEP WALL1.4 - SMOOTH WAL

POSITION OFt STAIR-STEP CORNER

1.2 V

U)

-0

0.4

0.2

0 004 0.08 0.12 036DISTANCE FROM APEX (m)

Figure 18. Comparison of Wall Pressure Values for Shot 2 at 183.6 us.

60

2.0

--- STAIR-STEP WALL-- SMOOTH WALL

POSITION OFSTAIR-STEP CORNERS

16

1LA

ul-! I- i

0.0 .

0.6

0-

0 004 0.0 012 0.16DISTANCE FROM APEX (mf

Figure 19. Comparison of wall Pressure Values for Shot 2 at t=321.4ps.

61

150

140 -- STAIR -STEP WALL

-SMOOTH WALL

POSITION OFSTAIR -STEP CORNERS

a

w130..0 . . . T-I

u,

mV20 1w If

011o0

101 I THEORETICALI-POSITION OF

REFLECTION

0 D4U 0.12 0.16DISTANCE FROM APEX (m)

Figure 20. comparison of Wall Pressure Values for Shot I at t o.

62

-- STAIR -STEP WALL

is0 SMOOTH WALLPSITION OFSTAR-SEPCORNERS

a

U,

09130

0in

0 0.04 0m08 0.12 0.16DISTANCE FROM APEX (in)

Figure 21. Comparison of Wall Pressure values for Shot 1 at t=9 l.O9 lis.

63

160

STAIR-STEP WALL150 - SMOOTH WALL

POSITION OFSTAI R- STEP CORNERS

-140a

LU

b __ , ,-_ .0\/

9120

do

0 004 06 0.12 0.16

DISTANCE FROM APEX Ia)

Figure 22. Comparison of Wall Pressure Values for Shot I at tul42.4ps.

64

170

160-- STAIR- STEP WALL-SMOOTH WALL

STAIR- STEP CORNERS150

of140

1-130-

0 /

12T

110

OL AMBIENT

10 004 0.06 0.12 0.16DISTANCE FROM APEX (in

Figure 23. Comparison of Wall Pressure Values for Shot 1 at t=240.7us.,

65

170

160 1

150 9

B .

± n

,<

U)

~130

0

120 STAIR-STEP WALLSMOOTH WALLPOSITION OFSTAIR-STEP CORNERS

110

AMBIENT

1O0i I ,, I i I a I0 0.04 0.08 0.12 0.16

DISTANCE FROM APEX (m)

Figure 24. Comparison of Wall Pressure Values for Shot I at t=465.6us.

66

is attributed to the fact that regular reflections occur on both wallsand the incident shock is a very weak shock. Figure 23 shows that thepressure peaks have moved away from the vertex at t - 240.7us. Thepositions of the largest peaks are very close and their magnitudes dif-fer by approximately 1%. The maximum difference between the pressureprofiles is less than 6%. As in Figure 18, the smooth wall profileahead of the maximm pressure is smooth while the stair-step profile con-tains oscillations. In Figure 24 the pressure peaks along both wallsoccur approximately at 0.105m from the vertex. The profiles oscillatebehind their pressure peaks and differ from each other by at most 6%from the vertex to their peak. Thus, the pressures for Shot 1 equili-brate faster behind the final reflected shock than for Shot 2.

Figures 25-32 compare the pressure histories at corresponding posi-tions [(2,2'), (3,3'), (4,4') and (5,5)] in Figure 14. Because themesh increment in the x-direction (10 m) is not equal to the incre-ment in the y-direction (9.4521m) the cell-centers at which the pres-sures are recorded are not equidistant from the vertex. The distancesfrom the vertex to the cell-centers are: 15mm for cell 2 and 14.18 mmfor cell 2', 25 m for cell 3 and 23.63 - for cell 3', 3S=m for cell4 and 33.08 as for cell 41, and 4S =m for cell 5 and 42.53 mm forcell 5'. Consequently, the pressure comparisons show small phase dif-

ferences. In particular, the initial pressure rise along the smoothwall precedes that along the stair-step wall because the position ofthe station along the smooth wall is closer to the initial position ofthe incident shock than the corresponding cell-centered positions alongthe stair-step wall. Furthermore, the final reflected shock returnsto the stair-step wall cell position before it reaches the correspondingsmooth wall cell position. Consequently, the crossover of the solidand dashed curves during the final pressure rise in Figures 25-32 isas expected. The duration of the initial pressure plateau decreaseswith successive comparisons of cells 2 and 2', 3 and 3, 4 and 4', andS and 5 because the time for the incident shock to reach the appro-priate cell-center decreases with increasing cell number.

Figures 25-28 compare the pressure-time histories tor Shot 2. The"plateaus" of the solid pressure curves immediately after the first in-crease, which can be seen in Figures 26-28, are due to the differencein time between the arrival of the incident shock at a given positionand the return of the final reflected shock to that position. Asexpected, the duration of the pressure plateau increases as the distancefrom the vertex to the cell increases. These plateaus are missing in thedashed curves, because the stair-step corner prevents analogous be-havior. The deviations of the pressure values in the flow field awayfrom a stair-step corner can be significant when compared with a cor-responding position off a smooth wall. In Figure 25, the position ofcells 2 and 2' are within 5.5% of each other, and yet the maximum dif-ference in their pressure values before peak pressure occurs is 82% att - 0.122 as. The pressure history at the stair-step reentrant corner

67

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EE

EE

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d

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27

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(cell 41) is given as the dashed curve in Figure 27. The inter-action between the incident shock and this small reentrant corner re-sults in the pressure rise at t a 0.08 as. A 71% difference betweenthe dashed and solid curves exists at this time. In Figure 28, thefirst pressure rise of the dashed curve is due to the incident shock,the second increase is due to the propagation of the high pressure seenat cell 4' (Figure 27) along the smooth section of the stair-step wallto cell S° , aid the final rise is due to the arrival of the final re-flected shock propating from the vertex. In all the Figures 25-28, thesmaller magnitudes of the final pressure jumps and the steadier risealong the stair-step boundaries may account for the smaller pressureovershoots in the stair-step wall pressure curves than in those alongthe smooth wall. The differences between the final pressure valuesalong the smooth and stair-step walls decrease as a function of timedue to the equilibration of the pressures in the vertex region. This phe-nomenon is quite apparent in Figure 25.

Figures 29-32 compare the pressure-time histories at correspondingpositions along the smooth and stair-step walls for Shot 1. Figures 29-32 are analogous to Figures 25-28 and similar coments are applicable.The percentage differences are significantly smaller and the pressurescales are more refined for this numerical simulation than for Shot 2.In both Figures 29 and 30 the maxims deviation in the pressure values be-fore peak pressure occurs is less than 7%. The marked increase in thepressure values of the dashed curve at approximately t a .12 ms in Figure31 is due to the corner at cell 41. The maxims pressure deviation nearthis time is 14%. The numerically caused undershoot of the solid curvein Figure 32 at t a 0.13 ms is absent in the dashed curve because a com-pression wave caused by the high pressure region at cell 4' has traveledalong the smooth section of the stair-step wall to station 5'.

5. COMPARISON OF PRESSURE VALUES FROM EXPERIMENTS,ANALYSES, AND NUERICAL SIMULATIONS

5.1. Experimental, Analytical and Numerical Peak PressureValues at the Vertex.

The experimental pressure values at the vertex are compared with theanalytical and numerical results in Figures 33 and 34. In the experi-ment, a rarefaction wave originates at the leading edge of the wedge.propagates along the wall's surface and ultimately lowers the pressureat the vertex. This expansion wave is absent in the analytic model andthe DORF simulation because of their infinite wedge assumption. Con-sequently, the experimental peak pressure value is compared with thefinal pressure values from the analytical model and for corner cell I inthe DORF simulation (see Figure 14). In the DORF results pressure oscil-lations are produced by the numerical simulation of the shock waves. Asingle comparable pressure value was obtained from the pressure historyin cell 1 by a least squares fit to a constant.

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For Shot 2 (Figure 33), the peak experimental value is 1.8195 ±O.08S9 N4Pa, the analytic value is 1.8913 bWa, and the least squaresvalue from the DORF results is 1.4406 ?4Pa. The analytically computedpressure is 3.95 larger than the experimental value and is within theexperimental error. The DORF value differs by -20.82% from the experi-mental value and by -23.83% from the analytical value. The pressurevalue from the analytic model should be the largest, because the as-sumption of an inviscid gas does not allow any viscous effects. Vis-cous forces which dissipate the pressure are present in the experi-ment. Although the DORF simulation assumed an inviscid gas, the numeri-cal method in DORP introduces diffusion which is effectively an implicitartificial viscosity. In the determination of the type of reflectionwhich occurs at the first reflection point for Shot 2, the quantities#0 = 40o and Po/Pi a 0.4168 generated a point Just below the curve #cin Figure S. The close comparison of the analytic and experimentalvalues indicates that the past history of the flow did not affect thereflection process.

The comparison of the experimental value with both the analyticaland the DORF results for Shot 1 is given in Figure 34. The experimentalpressure peak value at the apex is 154.58 ± 2.69 kPa. The analyticalpressure value is 165.44 kPa. The least squares straight line fit ofthe DORF pressure values is 151.97 kPa. Consequently, the analyticpressure value is 7.03% larger than the experimental value, and theDORF value is 1.69% smaller. The DORF value is 8.14% smaller than theanalytic value. The relative position of the three values is the sameas in Figure 33. However, contrary to the results for Shot 2, the DORFresults for Shot I provide better approximations to the experimentalvalues than do the analytical values.

5.2. Experimental and Analytical Peak Pressure Duration Values.

A comparison of the experimental measurements with the analyticapproximations to the peak pressure duration for a finite length re-entrant corner can be made when we combine the peak pressure durationresults from Sections 2 and 3.3 into Table 6.

Table 6. Peak Pressure Duration Data From Experiments and AnalyticModel.

Designation Experimental AnalyticalValue Value

Shot 1 131±7 ps 135.78 us

Shot 2 22±5 us 17.06 us

79

...... ..

The analytical values are within the measurement errors for the experi-mental peak pressure duration values. The experimental value of 22 Psis an estmate of the duration, because the experimental pressure pulseon the oscilloscope was extremely short.

5.3 Analytical and Numerical Pressure Results Within an InfiniteCorner.

The characteristics of the pressure history will vary at differentpositions throughout the flow field according to the analytic model de-veloped in Section 3. For example, at a position along a wall forming %1000 corner, the ambient pressure is initially increased to the pressurebehind the first reflected shock (see Figures 6 and 8) and then by thereturning final reflected shock (see Figure 7 or 9). For a positionalong the plane of symmetry, the situation is different. The ambientpressure is initially increased by the incident shock (see Figure 6or 8), then by the first reflected shock's reflection and lastly by thefinal curved shock (see Figure 7 or 9). Because both the DORF calcula-tion and the analytic model simulate the gas flow within an infinitereentrant corner, their results are comparable. The comparisons aremade between the analytic results and the DORF data at cells 1, 4, 30,and 31 shown in Figure 14.

Figures 35-38 compare the DORF results (solid curve) with the ana-lytic calculations (broken curve) for Shot 2. The short duration of theanalytically determined pressure plateau behind the first reflectedshock in Figure 35 is the result of the 5 - distance from the cell-center of cell 1 to the vertex. This highly transient plateau is lost inthe numerical simulation probably because each shock is smuared overseveral cells. The final DORF least squares fitted pressure values areover 23% lower than the analytic values, and this trend persists, as isshown in Figures 36, 37 and 38. Figure 36 shows that at cell 4 the DORFvalues indicate the pressure plateau behind the first regular reflectedshock wave. However, the duration of this constant pressure value isconsiderably shorter than the analytically determined plateau. Awayfrom the edges of the pressure plateau, the DORF pressure values arewithin 9% of the analytic value. Because this same plateau exists fora longer period in cell 4 than in cell 1, a comparison of Figures 35and 36 indicates that highly transient results are less likely to beaccurately calculated by DORF, at least using the present computationalflow field grid, than those of a pseudo-steady or steady nature. Thepressure histories along the plane of symmetry (cell 30) and just offthe plane of symmetry (cell 31) are depicted in Figures 37 and 38,respectively. Because the speed of the final curved reflected shockalong the plane of symmetry is not known from the analytic model, thearrival time of that shock at any given position can not be determined.

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Hence, the initiation of the final pressure plateaus in Figures 37 and38 can not be identified. The constant pressure value behind the secondreflected shock in Figure 37 is only slightly suggested by the DORFresults. The discrepancy between the results is most likely due to theclose proximity of cell 30 to the stair-step corner combined with thedifferent incident angle along the stair-step wall. At cell 31 theeffect of each reflection which is predicted by the analytic study canbe seen in the DORF results. Although the arrival time of a given shockand the duration of the associated pressure plateau are not accurate, thedistinct constant pressure values associated with regular reflection arenear the analytic values. As before, the pressure values behind the Machreflection do not compare as well as do those behind regular reflection.Because the DORF code solves a finite difference approximation to theinviscid compressible flow equations rather than the simple equation ofthe analytic model, Figure 38 verifies the analytic model at least quali-tatively.

The pressure histories for Shot I calculated by the DORF code andthe analytic model are compared in Figures 39-42. These figures areanalogous to Figures 35-38 and many of the previously stated featuresare the same. The scaling used in Figures 39-42 is considerably differ-ent from that used in Figures 3S-38 and care must be taken in comparingthese sets. In Figure 36, the pressures from the DORF calculation andthe analytic model behind the first reflected shock seem to compare morefavorably than in Figure 40, at first glance. However, the opposite istrue. For Shot 1, the difference is 4.7% and for Shot 2 the differenceis 5.6%. Figure 39 shows the steady increase of the DORF pressure valuesat cell 1 from the ambient to the final peak value. The highly transientpressure plateau predicted by the analytic model is again not simulatedin the DORP calculation. As discussed in Section 5.1, the analyticvalue is 8% higher than the least squares fit of the DORF values. Thisdifference is approximately the same in Figures 40-42. Figures 40-42show the pressure histories farther down the smooth boundary, on the planeof symmetry, and off the plane of symmetry, respectively. The durationsof constant pressure other than those initially prescribed are not dis-cernable in these figures. The DORF values in the intermediate pressureplateaus of Figures 40 and 41 have maximum variations of 4.7% and 7%,respectively, from the analytic values. In Figures 36 and 37 (Shot 2),the corresponding percentages are 5.6% and 8.7%, respectively. Figure42 shows a monotonic rise in the pressure and no correspondence with thepressure plateaus.

6. CONCLUSIONS

Shock tube experiments which simulate symmetrically placed shockwaves propagating into a 100' reentrant corner were performed and thepressure histories at the corner's vertex are presented. These experi-ments are simulated by an analytic model and by the DORF hydrodynamic

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computer code. Two experiments having incident shock strengths of

P1/Po 2.3699 (Shot 2) and P1/P = 1.1231 (Shot 1) are simulated by both

the analytic model and numerical calculations.

A detailed description of the analytic model, which excludes realgas effects, is given. The sole restriction of the analytic study onthis corner's vertex angle and incident shock strength is that complex ordouble Mach reflections do not occur within the corner. Thus, the modelis widely applicable. Because a criterion for the type of reflection(regular and simple Mach) at each reflection point is given, the procedureindicates when the model is applicable. The mathematical techniquesused in the model are simple: Galilean transformations and a method tosolve a nonlinear system of algebraic equations. When the final reflectedshock is not diffracted, the model calculates the exact solution of theentire flow field within an infinite corner, or the exact solution ofthe flow field near the vertex in a finite corner until the rarefactionwave approaches the vertex. When the final reflected shock is diffracted,the model provides an estimate of the peak vertex pressure value which isshown to be within 7% of the experimental shock tube data. For a finitecorner, a formula is derived to estimate the duration of the peak pres-sure value at the vertex. The delineation of the corner into distinctregions by the model is verifiable by the DORF computer code simulations.Because of its predictive capabilities, the model could be used as anaid to experimental design and as a benchmark problem for hydrodynamiccomputer codes.

The ntmerical results are for a single computational mesh size.The effects of a different mesh size on the DORF results are not ad-dressed in this report. The quantitative results for Shot I are signi-ficantly better than for Shot 2. For example, the peak vertex pressurevalue computed by DORF was less than 2% smaller than the experimentalvalue for Shot 1 but close to 21% smaller for Shot 2. The primary fea-ture of the DORF simulation is the use of the rigid stair-step boundaryto form one of the infinite straight walls forming the 1000 reentrantcorner and a reflecting smooth boundary to form the other wall. Theinitial placement of the incident shock in the numerical simulation wassuch that the shock traverses a stair-step corner before reaching thevertex. This stair-step corner acts like a small reentrant corner.Therefore, the flows adjacent to the smooth and stair-step walls weresignificantly different from one another. The DORF code results quanti-tatively demonstrate the differences. The pressure profiles adjacent tothe stair-step wall and smooth wall for several times and the pressurehistories at corresponding positions along the walls were compared. Itwas found that the calculated pressures for Shot 2 at comparable smoothversus stair-step wall positions differ from one another by at least20% and at particular positions and times up to 82%. However, thesepercent differences were significantly lower for Shot 1, being at most9.2%. The comparisons showed that only after the final reflected shockpasses a given cell do the pressures in corresponding cells on the twowalls begin to equilibrate. The time needed for the convergence to the

90

same pressure value depends on the strength of the involved shocks.Consequently, the accuracy of any type of transient pressure calculationsis strongly dependent on shock strength and position, and also the wayin which the walls are formulated in the computational grid. Even forcells away from the stair-step corner, the pressure results may deviateby 80%. Hence, care must be taken in the interpretation and use ofpressure results from a calculation involving stair-stepping to simulatea smooth plane inclined at an angle to an oncoming shock. Alternativesto stair-stepping (e.g. the utilization of partial rigid cells to modelinclined surfaces or techniques for generating body conforming coordi-nate system) are highly recommended.

The pressure histories from the analytic model and DORF simulationsare compared along the smooth boundary, and on and off the plane ofsymmetry. Qualitatively, the agreement between the DORF and analyticresults for Shot 2 is much better than that for Shot 1. Some of thepressure plateaus predicted by the analytic model are also indicated bythe numerical results. In particular, each of the cascading constantpressure regions predicted by the analytic model can be seen in thenumerical calculation near the plane of symetry (see Figure 38). How-ever, the smearing of the shocks by the DORF code results in the distor-tion of some shock arrival times and in elimination of some pressureplateaus. The small amplitude oscillations in the computation for Shot1 combined with the numerical diffusion in the computational grid makeit difficult to conclusively identify a pressure plateau in the numeri-cal results. Quantitatively the agreement between the DORF and analyticresults for Shot 1 is better than that for Shot 2. For example, thenumerical pressure value behind the Mach reflection in Shot 2 differsfrom the analytic value by 24%, while the percent difference betweenthe numerical and analytical results for Shot 1 is at most 8%. In gen-eral, the results show that DORF models regular reflection significantlybetter than Mach reflection for the flow situations modeled and compu-tational grid used.

ACIOMLEDGMENT

The shock tube experiment was designed by Mr. William Taylor andthe experimentalists were Mr. Robert Peterson and Mr. Rodney Abrahams.The first author is indebted to Dr. Raymond Sedney for his advice duringthe course of this work and to Mr. William Buchheister for his programmingassistance. Dr. Joseph Santiago's assistance in the formulation of thecomputational flow field grid design is gratefully acknowledged.

91

REFERENCES

1. Johnson, W.E., "Code Correlation Study", APWL-TR-70-144, U.S. AirForce Weapons Laboratory, Kirtland Air Force Base, N (April 1971).

2. Lighthill, N.J., "IThe Diffraction of Blast II", Proc. Roy. Soc.,Series A, Vol. 198, pp 5S4-65, 1950.

3. Keller, J.B., and Blank, A., "Diffraction and Reflection of Pulsesby Wedges and Corners", Communs. Pure and Appl. Math., Vol. IV,No. 1, pp. 75-94, 1951.

4. Keller, J.B., "Multiple Shock Reflection in Corners", Journal ofApplied Physics, Vol. 25, No. S, pp. 558-590, 1954.

S. Schniffuan, T., Heyman, R.J., Sherman, A., and Weiner, D., "PressureMultiplication in Re-Entrant Corners", in Proceedings of the FirstShock Tube Symposium, Air Force Weapons Center Report No. SWR-Th-57-2 (AD467-201), 1957.

6. Goodman, H.J., "Calculations of Pressure Over the Surface of a1/30th Scale Model Munition Magazine", ARBRL-TR-021S3, US ArmyBallistic Research Laboratory, Aberdeen Proving Ground, MD (April1979). (AD #B037702L)

7. Fry, M.A., Durrett, R.2., Ganong, G.P., Matuska, D.A., Stucker, M.D.,Chambers, B.S., Needham, C.E., and Westmoreland, C.D., "The HULLHydrodynamics Computer Code", APWL-Th-76-183, US Air Force WeaponsLaboratory, Kirtland Air Force Base, NN (September 1976).

8. Fry, M.A.. Needham, C.E., Stucker, N., Chambers, B.S.,III, andGanong, G.P., "AFWL HULL Calculations of Air Blast Over a DamSlope", AFWL-TR-76-154, US Air Force Weapons Laboratory, KirtlandAir Force Base, NM (October 1976).

9. Happ, H.J., 1II, Needham, C.E., and Lunn, P.W., "AFWL HULL Calcu-lations of Square-Wave Shocks on a Ramp", APL-TR-77-82, US AirForce Weapons Laboratory, Kirtland Air Force Base, NN (July 1977).

10. Taylor, N.J., ARRADCOM, BRL, Private Communication.

11. Thompson, P.A., Compressible Fluid Dynamics, McGraw-Hill Book Co.,New York, 1972.

12. Courant, R., and Friedrichs, K.O., Supersonic Flow and Shock Waves,Vol. 1, Interscience Publishers, Inc., New York, 1948.

13. Bleakney, W.,and Taub, A.H., "Interaction of Shock Waves", Reviewof Modern Physics, 21, pp. 584-605, 1949.

92

14. Polachek, H., and Seeger, R., "Shock Wave Interactions" in Funda-mentals of Gas Dynamics, H.W. Emmons, Ed., Princeton UniversityPress, pp. 494-504, 1958.

15. International Mathematical and Statistical Libraries, Inc., IMSLLibrary 3, Edition 6, It.SL, Houston, Texas, 1977.

16. Brown, K.M., "A Quadratially Convergent Newton-Like Method BasedUpon Gaussian Elimination", SIAM Journal on Numerical Analysis,Vol. 6, No. 4, pp. 560-569, 1969.

17. Law, C.K., "Diffraction of Strong Shock Waves by a Sharp Compress-ive Corner", University of Toronto Institute for Aerospace StudiesTechnical Note No. 150, 1970.

18. Ben Dor, G., "Regions and Transitions of Nonstationary ObliqueShock Wave Diffractions in Perfect and Imperfect Gases", Universityof Toronto Institute for Aerospace Studies Technical Report No. 232,1978.

19. Smith, L.G., "Photographic Investigation of the Reflection of PlaneShocks in Air", NDRC Report No. A-350, OSRD Report No. 6271, 1945.

20. Glasstone, S., and Dolan, P.J., "The Effects of Nuclear Weapons",Department of the Army Pamphlet No. 50-3, Headquarters, Departmentof the Army (March 1977).

21. Bertrand, B.P.,"Measurement of Pressure in Mach Reflection of StrongShock Waves in a Shock Tube", BRL-MR-2196, US Army Ballistic ResearchLaboratory, Aberdeen Proving Ground, MD (June 1972). (AD #746613)

22. Coleman, M., ARRADCOM, BRL, Private Communication.

23. Traci, R.M., Fan, J.L., and Liu, C.Y., "A Numerical Method for theSimulation of Nuzzle Gas Flows With Fixed and Moving Boundaries",BRL Contract Report No. 161, June 1974. (AD #784144)

93

APPENDIX A

DERIVATION OF THE OBLIQUE SHOCK RELATIONS

Consider a planar discontinuity (a shock frat S propagating in an

inviscid fluid with velocity w. See Figure IA.

REGION "REGION a

Figure 1A. A Propagating Planar Discontinuity.

The outward unit normal vector and unit tangential vector at anypoint Q on S are n and t, respectively. The fluid properties immediately

ahead of the shock wave are denoted by the subscript a and those immedi-ately behind by the subscript b. The velocity in the laboratory coor-dinate system, density, pressure and specific internal energy of the

gas are v, p, P, and I, respectively. By applying a standard controlvolume argument around an arbitrary position of the discontinuity andby using the principles of conservation of mass, momentum, and total

11energy of the discontinuity, we obtain (see Thompson Section 7.2 or

Courant and Friedrichs12 Sectio, 5.4)

Pb(vbwj -"a a" • - 0, (Al)b " c, LwPb " P a a- b nA2)

95

- Bu

2 v!2

Ob T ) (Vb"') " ( a~'a 2 a- " (M3)

= a , (P a n) V b " (P b" (

respectively. To derive the normal momentum and tangential velocityequations of the oblique shock relations, we add to equation (A2) thequantity (-w) times equation (Al) and obtain

b( Vb- )(Vb - ) " - w - ia ) " = (P -Pb)i" (A4)

bvw(vw -n a a a a Pb)

The scalar product of equation (A4) with n yields the conservation ofnormal momentum:

r4 ~ 2 4.42 p 2P bLv bw a a ) " n] = (P aP b). (AS)

The scalar product of equation (A4) with the unit tangent vector at Qcombined with equation (Al) yields the continuity of tangential veloc-ity:

4. 40 4 4.vb • t = va t (A6)

The usual form of the conservation of energy equation for the

oblique shock relations is obtained by adding +Pb(Vb-W) •-4 4. _+ -41

P a( a-w) - n to both sides of equation (A3) and using equations (Al)

and (A4) with the definition of specific enthalpy h = I + (PIp). Theresult is written as

2 v 2b a(hb + _) - (ha + --) = w 0 b b a (A7)

Using equation (A6), we rewrite equation (A7) as

[hb ( r) - [ a + ( ) (V b..Va)" (AS)

Equations (Al), (AS), (A6) and (AS) are commonly known as the obliqueshock relations and are valid at any point Q on the discontinuity S.

96

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AD0A097 471 ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMAND ABERO -ETC F/62 1ANAL YTIC MODEL AND NUMERICAL S IMULATION OF SMOCK WAVE PROPAGATIETJAN AD J A SCHMITT, R E L 0TTERD H JGOODM AN

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MICROCOPY RESOLUTION TEST CHART

NATIONAL BUREAU OF STANOARDS-1963-A

For reference, we can rewrite equation (A7) as

hb 2 (V b b ~2b 2 *

or

1 2 1 2hb +2 (vb w) h (v a - w) 0. (A9)

Using equation (A6), equation (A9) reduces to

h [.- " ha [(a- w " nJ= 0. (AIO)

97

APPENDIX B

EXPLICIT FORMULAS FOR OBLIQUE SHOCK CALCULATIONS

We derive the explicit formulas from the equation system (3.1.5)through (3.1.8) as follows. We multiply equation (3.1.6) by the quantity

L._* .- , use equation (3.1.5) and obtain

_(p - Po) [ 11 + ] O.S[u2 sin2 *oo2 u12 sin 2 (*O . 0l)] . (B1)

Rewriting equation (3.1.8) in terms of pressure and density instead ofenthalpy, we have

-j P - PO 0.5 [u02 sin2 *0 _ u2 sin2(# _ - 1)] (B2)

We equate the left hand sides of equation (B1) and (B2). Multiplyingthe resulting equation by the ratio p1/P OP we obtain the density in

region 1 in terms of y, P1/P0 and 00:

P=01 (

1 o +- 1 + - (B3)

The ratio of equations (3.1.5) and (3.1.7) is

P1 tan (0O - el) = p0 tan 00. (B4)

From equation (B4), the deflection angle 01 can bd expressed as

01= O - tan "1 [00 tan 0)/Ol] • (BS)

The velocity magnitude in region 0 can be expressed in terms of thequantities u1, *O and 01 from equation (3.1.7)

u0 a u1 cos(#O - el)/(cos #O).. (B6)

99

____ _I -

Using equation (6), we can rewrite equation (3.1.6) as

(PI - P o)/coS (4o - 0) a " 2 UI tan (00 " e- + Po tan2 0]" (07)

Using equation (BS), we rewrite equation (37) as

U1 +1I - PO POj sin (0- i) (18)

By solving equations (13), (BS), (38), and (16) in order and by apply-ing the perfect gas relations hI a yP1/ [(y-l)p 1] , T Pl/(R-p,), and

a, . (yP1/Pl) , we obtain all the values of the flow variables in theintersection of region 1 and a neighborhood of the reflection point Q.

100

LIST OF SYMOLS

a - sound speed (n/s)

cp a specific heat at constant pressure (J/(kg.K))

cv - specific heat at constant volume (J/kg.K))

d - distance from the corner's apex (m)

dai n = minimm dimension of a cell (mn)

d Z distance defined by equation (3.3.1) (m)P

d - distance rarefraction wave travels in tw (a)

dw = length of wall forming a finite cormer (a)

E = specific total energy (J/kg)-s 2g = gravitational acceleration (m/s )

h - specific enthalpy (J/kg)

I = specific internal energy (J/kg)

n - unit outer normal vector

P = absolute pressure (Pa)

Q a point at which shock reflection occurs

R = gas constant (J/(kg. K))

S - surface of discontinuity

T = temperature (K)

t - unit tangential vector

t ,t a defined by equations (3.3.2) and (3.3.3), respectively (s)

1w = time taken for incident shock to traverse d. (s)

u - velocity vector in the shock-fixed coordinate system

v - velocity vector in laboratory coordinate system• V y. (a/Is)

w shock wave velocity vector (a/s)

Z - triple point in the Mach reflection configuration

A x a step size in the x direction (m)

AS a step size in the y direction (mn)

101

At - time stop selected by DORF code (s)

Ax a computational cell size in the x direction (m)

Ay a computational cell size in the y direction (m)

y w ratio of specific hosts a Cp/cv

8 a reentrant corner angle variation from 90" ()

n = Courant-Priedrichs-Lewy stability factor

ei - deflection angle of the incmng velocity as it passesthrough a shock into region i ( )

a w = angle a wall makes with the horizontal (0)

v,x a kinematic and bulk viscosity coefficient, respectively, (Pa-s)

p a density (kg/a 3 )

T - host transfer coefficient (w/(a.K))

#B a incident angle of initial shock along the smooth wall (0)

- the incident angle of the velocity in region i at the shock

front (0)

#L = incident angle of initial shock along the stair-stop wall (*)

x a trajectory angle of the triple point in Mach Reflections ()

Subscripts

a denotes gas flow variables ahead of the shock

b denotes gas flow variables behind the shock

F denotes the gas flow variables in the last region

i denotes region i, i1,2,3,4,S

0 denotes initial region or region 0

Operators

Ix - curl operator

I. * divergence operator

I = gradient operator

102

r .... . . ... -.. . . . . . . . . . . .

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