Numerical study of unsteady flows in large air-driven blast simulators with single and multiple drivers

Kluyve eg 1 - 2629 HS DELft'

7 AU •

NUMERICAL STUDY OF UNSTEADY FLOWS

IN LARGE AIR-DRIVEN BLAST SIMULATORS

WITH SINGLE AND MULTIPLE DRIVERS

March 1990

by

G. L. Petrini and J. J. Gottlieb

UTIAS Report No. 334 eN ISSN 0082-5255

NUMERICAL STUDY OF UNSTEADY FLOWS

IN LARGE AIR-DRIVEN BLAST SIMULATORS

WITH SINGLE AND MULTIPLE DRIVERS

by

G. L. Petrini and J. J. GottHeb

Subrnitted January 1990

March 1990

©Institute for Aerospace Studies

UTIAS Report No. 334 eN ISSN 0082-5255

.

-Abstract

Large air-driven blast simulators have multiple drivers connected to a

single channel with a test section of 50 m2 _{in cross-section and larger where}

full-scale or large military test objects are subjected to simulated blast waves. However, previous numerical analyses of unsteady blast flows in such facil-ities have been done with the multiple drivers combined into a single com-posite driver with an area variation based on the sum of the multiple-driver areas. This study presents numerical results for the first time for the case when the unsteady flows in these multiple drivers are treated separately and then all joined together with a specially developed junction cell to continue the unsteady flow solution throughout the channel. These results are com-pared to numeri cal solutions obtained for equivalent blast simulators with single composite, conical and curved drivers, in order to identify and illus-trate important differences in the two types of solutions, explain the origin and behaviour of many secondary and tertiary shock and rarefaction waves superposed on the sirnulated blast wave, and assess the advantages of ob-taining fuIl rnultiple-driver solutions with the correct secondary and tertiary flow features.

•

Acknowledgements

The financial assistance received from the Natural Sciences and Engi-neering Research Council of Canada under operating grant number

OGPIN-013 is much appreciated.

Table of Contents

Page Title Page ... i Abstract ... ii Acknowledgements ... 111 Table of Contents ... IV List of Symbols ... V 1.0 INTRODUCTION ... 1 1.1 Background Information ... 1

1.2 Effects of Driver Geometry on Simulated Blast Waves ... 2

1.3 Numerical Models for Blast-Simulator Flows ... 4

1.4 Objectives of the Present Study ... 5

2.0 ANALYSIS FOR BLAST-SIMULATOR FLOWS ... 5

2.1 One-Dimensional Equations for Blast-Simulator Flows ... 6

2.2 Initial and Boundary Conditions for Blast-Simulator Flows ... 7

2.3 Numerical Solution Procedure for Blast-Simulator Flows ... 8

2.4 Junction Cell Connecting Multiple Driver Flows to Channel Flow .. 9

3.0 BLAST SIMULATOR MODEL GEOMETRIES ... 11

4.0 NODE SELECTION FOR NUMERICAL MODEL ... 13

5.0 VALIDATION OF THE JUNCTION CELL ... 15

6.0 NUMERICAL RESULTS AND DISCUSSION ... 16

6.1 Blast Simulator with a Conical Driver ... 16

6.2 Blast Simulator with a Curved Driver ... 18

6.3 Blast Simulator with a Composite Driver ... 19

6.4 Blast Simulator with Multiple Drivers ... 20

6.5 Comparison of Simulated Blast Waves from Different Drivers ... 20

6.6 Effects of Awall on Simulated Blast Waves ... 21

7.0 CONCLUSIONS ... 21

8.0 REFERENCES ... 22 Figures

List of Symbols

Alphanumeric Symbols

a speed of sound of the gas

A

duet cross-seetional area

Ao

duet cross-sectional area at

x

=

0

AL duet cross-sectional area at

x

= L

Cp

gas specifi.c heat at constant pressure

Cv

gas specific heat at constant volume

D duet diameter

e total gas energy per unit volume

L length of an area change

M flow Mach number

p gas pressure

R

gas constant

S gas state

t time

f:1t time interval equal to tn - tn-1

T gas temperature

u gas flow velocity

x distance

f:1x width of the junction cen

x normalized distance in a duct area change Greek Symbols

ï specific heat ratio

(Cp/Cv)

p gas density

Subscripts

center center of the junction cen

chan channel

J

ph

flow into the left side of the junetion cen

left left side of junction cen

max maximum value

right right side of junction cell

wall wall of left side of junction cell Superscripts

cone curve

n

for a conical driver geometry for a curved driver geometry time level index

1.0 INTRODUCTION

1.1 Background Information

World War 11 stimulated the development and use of new weaponry based on nu-clear fission and fusion. After the war these new bombs were tested both in the Earth 's atmosphere and Earth's crust, in order to gather information about fission and fusion explosions and develop more advanced weapons [Glasstone, 1957]. At the same time their destructive potential to both military and civilian equipment and structures was being assessed. The ban on atmospheric nuclear testing in 1961 made the nuclear re-search community explorevarious methods of simulating the neutron, thermal and blast-wave environments of a nuclear explosion for testing purposes (survivability of military and civilian equipment and structures). Since the 1960s, lots of small-scale testing and some large-scale experiments were conducted using relatively large quantities of chemi-cal explosives-20 to 500 tons of trinitrotoluene (TNT) and 100 to 1000 tons of ANFO (ammonium nitrate and fuel oil mixture). Although these blast-wave simulations from detonating chemical explosives at the Earth's surface could be conducted more frequently and less expensively than the corresponding nuclear tests, they were still relatively in-frequent and expensive for extensive military testing purposes, and furthermore they generally did not provide sufficiently long duration blast waves for full-scale tests. From the late 1950s to the present time, in addition to the chemical explosion testing, there has been a continual development of specialized shock tubes in the form of blast sim-ulators. These facilities not only provide an alternative to both nuclear and chemical explosion testing, but they also provide a less expensive and more repetitive means of producing good quality simulated blast waves, which is achieved in a more laboratory like environment.

Typical signatures of a blast wave from either a nuclear or chemical explosion detonated far from the Earth's surface or next to the Earth's surface are depicted in figure 1. The expanding fireball in figure la produces an outwards travelling blast wave which decays with increasing distance (on its way to the test site). At the test site the temporal signatures of overpressure, density, temperature and flow velo city generally have an exponential-like shape, as illustrated in figure 1b, and the amplitude and positive-phase duration depend on the explosion yield and the distance from the explosion center.

The response or survivability of military equipment and structures depends on the amplitude, duration and shape of these blast signatures [Glasstone, 1957]. Survivability should be assessed at a very early stage in the prototype or pre-production stages of product development, in order to select proper materials and finalize designs that are most blast resistant. The design criteria and choice of materials should be fully confirmed by relatively inexpensive small-scale model tests with blast waves and/or more costly full-scale blast test ing.

Consequently, blast simulators should have the capability of producing good quality long-duration simulated blast signatures in a contained environment in which nearly full-scale or full-sized equipment can be tested extensively and hopefully economically. This approach has stimulated the development of specialized and sophisticated blast simulators of large size. These blast simulators need to be very large in order to accommodate actual test objects such as tanks, jeeps, trucks, personnel carriers, tactical or fighter aircraft, and modest sized helicopters and bomber aircraft. A typicallarge blast-wave simulator

must be capable of producing a simulated waveform with the correct decaying shape, a peak overpressure ranging from 0 to 400 kPa, and positive overpressure durations in the range from 0.05 s to 3 s.

In the late 1950s and through the 1960s the largest blast simulators were explosive driven. One example of such an early facility is sketched in figure 2. The explosive charge in the form of a heli cal coil of explosive is placed in the smallest end of the facility (called the driver) and detonated, in order to create high-pressure combustion products which expand down the channel of increasing area and produce a simulated blast wave [Clare, 1979; Leys, 1983]. This wave decays during its propagation along the expanding channel to the test section. Such explosive facilities generally produce high amplitude blast waves with a short positive-phase duration, and considerable effort is needed to obtain more suitable long duration waves.

The trend from the early 1970s to the present time has been in the direct ion of developing large blast simulators that are driven with high-pressure air (instead of explo-sives), which have the capability of producing good quality simulated blast waves with a well-controlled amplitude and long durations. These types of facilities avoid the hazards of using high explosives and the expense of involving munitions and safety officers in tests. However, blast simulators using high-pressure air drivers have to be sophisticated in order to produce good-quality simulated blast waves with a wide range of amplitudes and durations. One example of a large air-blast simulator is sketched in figure 3. This simulator has seven independent drivers with separate adjustable internallengths, seven diaphragms to break simultaneously, seven expansion ducts to the single channel that

contains the test section with a cross-sectional area of 70 m2

, and a specially designed

active reflection eliminator with a horizontal set of louvers which rotate during the

pas-sage of the simulated blast wave [Crosnier

&

Monzac, 1977; Gratias

&

Monzac, 1981;

Cadet

&

Monzac, 1981]. Note that a similar sized facility has recently been constructed

in Germany [Klubert, 1987], which has. numerous identical high-pressure air tanks as

drivers, bursting diaphragms to initiate the flow, a mountain tunnel for the channel, and a foam plug as a reflected-wave eliminator (RWE) and environment al noise suppressor. Furthermore, a larger blast simulator with many separate drivers, controlled flow rate

valves instead of diaphragms, a channel cross-sectional area of about 165 m2 _{and an active}

RWE is planned for construction in the United States.

1.2 Effects of Driver Geometry on Simulated Blast Waves

The geometry of the driver greatly affects the quality of the simulated blast wave at the test section in the channel, in terms of obtaining the decaying or exponential-like profiles of overpressure, density and flow velo city typical of explosion generated blast waves. This will become dear during the following discussion of the shapes of simulated blast waves in different types of air-driven shock tubes and blast simulators.

Consider first the case of the conventional constant-area shock tube [Ferri, 1961;

Wright, 1961; Zucrow

&

Hoffman, 1976]. The geometry of the shock tube, the

time-distance diagram of the wave motion and the time histories of overpressure, density,

temperature and flow velocity at the test section are illustrated in figure 4. It is obvious

that the decaying profiles typical of blast waves are not readily obtainable with such

and the density and temperature contain an extra discontinuity (contact surface). The amplitude and duration of the positive phase can be controlled individually by changing the initial pressure of the compressed air in the driver and the length of the driver, but the flat-topped nature of the initial part of the signature remains (unless the wave is permitted to propagate to very large distances from the driver). Hence, the conventional constant-area shock tube is not suitable as a blast simulator.

However, flat-top shocks are useful for other studies. To increase the duration of the flat top and delay the arrival of the contact surface at the test section, this section must be located far from both the driver and open end of the channel (making the channel extremely long for the case of long-duration waves). The long channel beyond the test section could be eliminated if the channel was terminated with an active RWE which controls the channel exit area such that reflected waves are insignificant.

The conventional shock tube can be converted into a much better blast-wave sim-ulator by inserting a set of spatially distributed perforated plates in the driver [Reichen-bach, 1966; Zhang & Gottlieb, 1986], or by installing a perforated cone inside the driver [Gottlieb

&

Funk, 1981; Gottlieb, Igra

&

Saito, 1983; Zhang

&

Gottlieb, 1986]. This modification to the internal geometry is illustrated in figure 5. In each case the flow out of the driver is restricted by the perforations and their associated pressure losses, thereby resulting in a decaying wave travelling along the channel. Predicted and mea-sured overpressure signatures from a small shock tube with perforated plates are shown in figure 6 for interest [Zhang

&

Gottlieb, 1986]. Although the simulation is of good quality, perforated plates and a perforated cone cannot easily be scaled up for use in large blast simulators, since a large single driver with perforated plates attachments and diaphragm assembly cannot easily resist the high-pressure induced wall stresses, and this normally results in a cumbersome and expensive pressure vessel design of the driver.

Another approach to producing a simulated blast wave in a constant-area channel, of a better quality than that obtained in a conventional shock tube, is to alter the driver geometry [Crosnier, Monzac

&

Cantaloube, 1972; Amann, 1977; Gratias

&

Monzac, 1981; Gottlieb

&

Funk, 1981]. One of the first geometrical modifications to the driver was to essentially keep the driver constant in area but insert a throat (with the diaphragm) between the driver and channel (see figure 7a). This produces a wave with a small leading spike and a following decaying overpressure in the form of a set of descending flat-topped steps. Another modification was to make the driver conical, with and without a constricting throat at the diaphragm location (figures 7b and 7

c).

This geometric modification was important in producing a more smoothly decaying wave with a much better exponential-like shape. Although these are fairly good simulated blast waves, the driver and diaphragm assembly of such shock tubes cannot easily be scaled up to a sufficiently large size suitable for full-scale blast simulators. The driver diameter and length would be of the order of 15 mand 50 m, respectively. Again, a large single driver and a diaphragm assembly which can resist the high-pressure induced wall stresses would be very cumbersome and expensive to build.

One option of overcoming the shortcomings of a large single driver and its di-aphragm assembly is to use multiple drivers and didi-aphragms of much smaller size. This approach was embarked on by the French researchers in building the Gramat blast sim-ulation facility (figure 3) [Crosnier

&

Monzac, 1977; Gratias

&

Monzac, 1981; Cadet

&

Monzac, 1981]. In their facility, the higher amplitude and longer duration simulated blast wave are obtained by using larger initial driver-air pressures and increasing the lengths of the drivers. In addition, the seven independent drivers are normally set at different lengths to control the shape of the blast wave, thereby obtaining an exponential-like waveform. Seven drivers permit fairly good control of the decaying blast wave, and the use of an active RWE helps considerably in increasing the duration of this wave.

1.3 Numerical Models for Blast-Simulator Flows

The prediction of the unsteady flows in shock tubes and blast-wave simulators with constant-area duet segments, area changes, a rapidly opening diaphragm, passive and aetive RWEs, friction and heat transfer between the duet walls and flowing air, and mass flow losses through openings in the side walls, is a not an easy task. However, this is becoming more standard by using recent numerical modelling and modern computational

techniques [Mark, 1981; Gottlieb, Igra

&

Saito, 1983; Zhang

&

Gottlieb, 1986; Opalka

& Mark, 1986]. In almost all cases, the standard one-dimensional Euler's equations

with inhomogeneous or source terms are solved in either finite-difference or finite-volume form, and in one case the equations were solved in Lagrangian form [Crowley, 1967]. One-dimensional unsteady flow solutions are normally suflicient to describe the primary flow

features fairly accurately. If the blast simulator flows have significant two-dimensional

characteristics, then the facility is basically inadequate because a good simulator should produce essentially one-dimensional unsteady blast-wave flows. Note that there are a few cases in which the two-dimensional equations of motion have been solved numericaIly, in order to investigate secondary flow characteristics or simulator designs. One solution worth mentioning is that of Hisley [1987].

Previous modelling, analyses and computations were performed for the case of shock tubes and blast-wave simulators, and these simulators were modelled as single ducts. In each case there was only one driver linked to one channel. No modelling and computations have been do ne for blast simulators which have multiple and independent drivers of different lengths, which is a charaeteristic of the French facility (figure 3). All previous work performed for the French and similar facilities have merged the multiple drivers into a single driver of supposedly equivalent geometrical characteristics. The method of combining multiple drivers into a single composite driver is illustrated in figure 8a and

8b. The area of the composite driver at a given distance from the diaphragm is simply

the sum of the areas of the multiple drivers which occur at this same location. Since the multiple drivers have different lengths, this leads to a series of discrete area changes. Furthermore, the area of the composite throat is equal to the sum of the areas of the multiple throats, and the diaphragm remains at the same location.

If the multiple drivers are sufliciently small in diameter and thereby also numerous,

then the area changes in the composite driver would each be correspondingly smaIl, numerous and weIl distributed. In this case the area variation of the composite driver would become quite smooth. The limiting case of a driver with a smooth area variation is depicted in figure 8c.

The main reasons why the blast simulator with multiple drivers have been modelled as a single composite driver are (a) simplicity in modelling and analysis, (b) reduction in computational time, and (c) unavailabilty of ajunetion cell for combining all individu al

flows from the multiple drivers and introducing this resulting flow into the channel. Furthermore, the flow predictions obtained by means of the model for the single composite driver were deemed to be in reasonable agreement with limited experiment al data, and this reduced the interest in furthering research on this topic.

1.4 Objectives of the Present Study

It has become increasingly important in recent times to build large air-blast

simula-tors which are fairly easy to operate and also capable of producing simulated blast waves of good quality. Correspondingly, it has also become important to accurately model the unsteady flows in these blast simulators, in order to provide better design data and mini-mi ze expensive prototype testing. One can weIl afford better numerical modelling in lieu of doing experiment al testing with large prototypes.

With advanced numeri cal methods available today, combined with modern powerful computers, the extra numerical modelling and computational efforts of handling the multiple drivers separately are not considered to be serious drawbacks in achieving better accuracy in the predietion of flows in such blast simulators. As a consequence, the primary objectives of this study can be summarized as follows:

• develop an appropriate model, analysis and computational program to predict unsteady flows in actual blast-wave simulators with multiple drivers of various

lengths (figure Sa), with the flow in each driver considered independently, and

• compare numeri cal results from the multiple driver case (figllre Sa) to those from

the simpier single composite driver with a nonsmooth area variation (figure Sb)

and also to those from the composite driver with the smooth area variation (figure Sc), in order to assess the relative differences in the simulated blast waves as a result of the three different driver geometries.

The methods by which these objeetives are achieved and most of the numeri cal results culminating in this report's conclusions are presented herein.

2.0 ANALYSIS FOR BLAST-SIMULATOR FLOWS

The analysis required to solve unsteady flows in blast-wave simulators with single and multiple drivers linked to a single channel is presented in this chapter. The relatively straightforward analysis for solving unsteady flows in a single duet with area changes is given first, and the new analysis for a special junction cell which combines the multiple flows from multiple drivers into one flow which enters the single channel is presented at the end of this chapter.

The unsteady flows in the blast-wave simulators considered in this study are treated as one-dimensional, because actual facilities are designed to pro duce blast waves with flows moving in one direction, at least ahead of the target in the test section. Furthermore, the driver and channel ducts are relatively long compared to their diameter and the flows are essentially one-dimensional. Hence, one-dimensional computations give reasonable results for the average flow properties across the duct. Finally, a two-dimensional analysis of the unsteady flow in a blast simulator, in order to perform a fairly extensive study of the flow differences owing to single and multiple drivers, would be prohibitively expensive.

2.1 One-Dimensional Equations for Blast-Simulator Flows

The three equations of motion (continuity, moment urn and energy) in partial differ-ential form for describing fairly general one-dimensional unsteady flows of a compressible gas in pipes or duets with area changes can be expressed in weak conservation form as [Shapiro, 1953; Rudinger, 1969; Groth & Gottlieb, 1988]

a

a

at lp]

+

ax [pul

-a

a

.

at [pul

+

ax [pu2

+

p] 1 dA -[pul A dx' _[pu2]

~ ~:'

a

a

1 dA

at[e]

+

ax[u(e+ p)] = -[u(e+p)]Adx'

(1)

(2)

(3)

where p, p, e, u, A, x and t denote the pressure, density, total energy per unit volume, flow velocity, duct area, distance and time, respectively. For a gas which is thermally perfect (i.e., p

=

pRT) and also calorically perfect (i.e., constant specific heats Cv

=

-t:l

Rand

Cp

=

:f!:ïR), the energy can be expressed as e

=

pCvT

+

~pu2

=

"Y~IP

+

~pU2, where R,

Tand I denote the gas constant, temperature and specific heat ratio, respectively. Friction and heat transfer between the gas and duct walls have been neglected in this study, because they are relatively small for flows in large blast simulators. In addition, pressure losses due to flow turbulence generation and dissipation have been negleeted (e.g., from flows passing over broken diaphragm rernnants that protrude into the duct, and flows passing through steep area reductions and enlargements).

The three previous partial differential equations and the state equation (p

=

pRT)

become a closed set for the three dependent variables p, pand u only if the duet area is

known or specified as a funetion of distance. In this study, all of the area changes in the multiple drivers of the blast simulator (figure

8a),

and also the composite driver case with nonsmooth area changes (figure 8b), are taken to be continuous and smooth, although some of these area transitions are very short. The duct diameter, area and derivative of area are given by the expressions

D(x)

=

V;A(X) ,

(4)

(5)

d~ix)

=

~

A(x)

In

V~:

sin (

,,~)

,

(6)

where Ao is the initial duet area at i

=

0, AL is the final duet area at i

=

L, L

=

Xl - Xo

is the length of the area change, and i is the local distance variabIe for the area change (i.e., 0 ~ i ~ Land :î:

=

X - xo). Note that this type of smooth area change starts and ends with the derivative dA/di equal to zero, and (l/A)dA/d:î: is a symmetric function with its maximum or minimum value occurring at the center of the area change.

In the last case of the composite driver with the limiting smooth shape (figure 8e), the driver shape will sometimes be taken as conical and at other times as curved. For the conical driver, let the cone have a length, maximum diameter and maximum area denoted by Lcone, D~:: and A~::, respectively. Then the driver diameter, area and first derivative of this area can be specified as a function of distance by

(7) A(x) = Acone X ( ) 2 maz Lcone ' (8) dA(x) =

~

A(x) , dx x (9)

where the distance x is measured from the apex of the cone.

For the case of the curved driver, let the length, maximum diameter and maximum area be denoted similarly by Lcurlle Dcurlle _{' m a z} and Acurlle _{maz ,} respectively Then the driver _•

diameter, area and first derivative of this area can be specified as a function of distance by

D() _X

₌

D curve· ( 1rX ) _{maz SIn 2Lcone '} (10) A(x) = Acurlle SIn • 2 ( 1rX )

maz 2Lcurlle' (11)

dA(x) . 1r cos(1rxj2Lcurlle)

=

--A(x) ,

dx Lcurlle sin( 1rX j2Lcurlle) (12) where the distance x is again measured from the driver apex. Values for L, Dmaz and

Amaz for both the conical and curved drivers will be given later in this report.

2.2 Initial and Boundary Conditions for Blast-Simulator Flows

The governing one-dimensional equations for nonstationary gas flows in ducts, pre-sented earlier as equations 1, 2 and 3, are a system of nonlinear inhomogeneous hyperbolic partial differential equations. The solution of this system of equations depends on both the initial and boundary conditions. The initial and boundary conditions are equally important in obtaining an accurate numerical solution for the unsteady flow field in a blast simulator.

The prescription of initial conditions at some initial time (e.g., t = 0) for the un-steady flow in blast simulators is straightforward. At the initial time a diaphragm sepa-rates the high-pressure quiescent air in the driver from the quiescent atmospheric pressure air in the channel, and the temperatures of the air in both the driver and channel are normally atmospheric. When the diaphragm is broken, the wave motion begins and a simulated blast wave is eventually produced at the test section as time progresses. Hence, the initial conditions (just before the diaphragm is broken), can easily be specified at each node of the numeri cal grid in the driver and channel. In the case of the driver nodes the specified pressure is higher than atmospheric, the temperature and sound speed are atmospheric, and the flow velocity is zero. In the case of the channel nodes, the specified

pressure, temperature and sound speed are all atmospheric and the flow velocity is also zero.

The prescription of boundary conditions at both ends of the numerical grid (and also at duet ends) is easy to define but not always simple to implement. There are essentially three types of boundary conditions associated with shock tubes and blast simulators--closed, nonrefleeting and open-duet ends. All of these boundary conditions are implemented in terms of aetual flow properties (not their derivati ves ), and therefore these are called Dirichlet boundary conditions. For the simplest case of the closed end occurring in both single and multiple drivers, the flow velocity at this stationary boundary (wall) is zero. By using known flow properties from the node next to the boundary, the flow properties at the boundary node are determined at the same time level by means

of a fairly simple Riemann problem, and the details can be found elsewhere [Zhang &

Gottlieb, 1986; Groth & Gottlieb, 1988].

The channel of the blast simulator in this study is considered as ending with a perfect RWE, or the channel is assumed to be infinitely long. In this case a nonrefleeting or perfectly-transmissive boundary condition is applied at the channel end of the nu mer-ical grid. By using known flow properties from the node next to the boundary, the flow properties at the boundary node are determined at the same time level by means of a Riemann problem. Since this Riemann problem is trivial because it contains no waves, the flow properties at the boundary node are simply equal to those from other node inside

the numeri cal grid [Zhang & Gottlieb, 1986; appendix A of Groth & Gottlieb, 1988].

For the case of the multiple drivers (figure 8a), the end of each diverging duet seetion

terminates abruptly in the large channel. The outflows and inflows at the end of each of these sections are considered herein to be very similar to the outflow or inflow occurring for a duet ending in the atmosphere. Hence, an open-duet-end boundary condition is applied at the end of eaeh diverging duet section. This is done by using the known flow conditions at the node next to the boundary node and an effeetive "atmospherie" pressure and temperature of the flow at the entrance of the ehannel (special junction eell which connects the multiple drivers to the channel), in order to obtain the flow conditions at the boundary node. The details of this open-duet-end boundary condition

based on a specialized Riemann problem are presented elsewhere [Zhang & Gottlieb, 1986;

appendix A of Groth & Gottlieb, 1988], and the method of obtaining a particular effeetive

atmospheric pressure and temperature required in appling th is boundary condition is described at the end of this chapter.

2.3 Numerical Solution Procedure for Blast-Simulator Flows

Euler's equations with inhomogeneous terms (Eqs.I-3) are nonlinear and hyper-bolie partial differential equations, and the solutions for blast simulators contains many discontinuites (shocks and contact surfaces). The approach taken in this report to solve these equations is the random-choice method (RCM). The RCM is based on the solution of Riemann problems with quasirandom sampling, and it us es an operator splitting

tech-nique to include inhomogeneous terms. It is a rather unconventional finite-volume and

explicit method of solution, which preserves the sharpness of discontinuities in a natural manner without smearing and Gibb's phenomena (numerical overshoots and undershoots

- - - ,

at disco.ntinuities). This is achieved with no. extra effo.rt, and no. explicit artificial visCo.sity

and implicit numerical viscosity are required.

The reaso.ns fo.r selecting the ReM fo.r so.lving the blast-simulato.r flo.WS in this study are: (a) simple metho.d to. apply to. so.lve o.ne-dimensio.nal unsteady flo.ws, (b) preserves disco.ntinuities in a natural way, (c) has go.o.d accuracy, (d) uses variabie no.de spacing and Io.cal time stepping which significantly reduces co.mputatio.nal time. The ReM emplo.yed in this study inco.rpo.rates all o.f the mo.st recent impro.vements, including a very efficient Riemann so.lver with an effective quasirando.m sampling pro.cedure and a no.nstaggered gridding system with variabie no. de spacing and Io.cal-time stepping. Further details o.f this numerical metho.d can be fo.und in recent repo.rts by Go.ttlieb & Anderso.n, 1987; Gro.th & Go.ttlieb, 1987; Go.ttlieb, 1988; Go.ttlieb & Gro.th, 1988].

2.4 Junctio.n CeU Co.nnecting Multiple Driver FIo.WS to. Channel FIo.W

The blast simulato.r sho.wn previo.usly in figure 8a has multiple drivers o.f different lengths. Each driver has a co.nverging duct sectio.n to. the thro.at where the diaphragm is Io.cated, fo.llo.wed by a diverging duet sectio.n which ends abruptly in the channel. Af ter the diaphragm is bro.ken and the flo.W fro.m each driver to. the channel is established, the pressure decay characteristics will eventually be different in each driver during the generatio.n o.f the simulated blast wave at the test sectio.n, because the multiple driver lengths and vo.lumes differ. Hence, the flo.WS fro.m these drivers and at the exits o.f the diverging duet sectio.ns will also. eventually differ. In fact, at late times when the different pressures in the drivers are sufficiently Io.w, so.me flo.WS can be leaving and o.thers entering the di verging duct ends and drivers.

In arealistic numerical predietio.n o.f the simulated blast-wave flo.WS in a blast simulato.r with multiple drivers, the flo.WS fro.m these drivers sho.uld be co.nsidered as independent and also. co.nnected to. the single flo.W at the entrance o.f the channel. The unsteady flo.W in each driver with its co.nverging and diverging duct sectio.ns can be predieted numerically by the ReM, pro.vided that the initial co.nditio.ns are specified and

bo.undary co.nditio.ns at both ends o.f this single duet flo.W can be implemented. This is similarly true fo.r the co.nstant-area channel. The flo.WS in the multiple drivers and the flo.W in the channel are intimately conneeted where the diverging duct seetio.ns end abruptly in the channel. As a co.nsequence, a special junetio.n cell is required to. link and pro.vide co.ntinuity between the driver and channel flo.ws.

The special junctio.n cell is sketched in figure 9. The flo.WS into. the ceU from the multiple diverging duet seetio.ns are sho.wn o.n the left-hand side, and the resulting flo.W fro.m the cell to. the channel are sho.wn o.n the right side. The states Sj-left, Seenter and Srigltt o.f the air flo.WS at the cell entrance, center and exit defined in this study by the

pressure, density and flo.W velo.city, where subscript j deno.tes the flo.W fro.m the

ph

driver. Fo.r example, the center state Seenter deno.tes the flo.W pro.perties Peenten Peenter and Ueenter'

No.te that the state Seenter deno.tes the average values o.f the flo.W pro.perties o.f the junetio.n

cell at a given time; Srigltt deno.tes the average flo.W pro.perties at the right junctio.n cell

bo.undary, fo.r a ceU starting at the Io.catio.n o.f Seenter and ending at Io.catio.n Seitan; and

similarly fo.r Schan'

The o.ne-dimensio.nal flo.WS fro.m the drivers separate fro.m the duet exit wall o.n entering the cell, because the to.tal duct exit area is generally smaller than the channel

area. This also means that part of the cell area on the left si de is a wall and has no flow

entering or leaving, but there is a wall pressure denoted by Pw41l that generally differs from

the panter' The flows entering and leaving the eell are assumed to thoroughly mix inside

the ceU, and the flow at the right side of the junction cen is assumed one-dimensional (uniform across the channel).

The method of updating the state Scenter of the flow properties at the center of the

junction cen at each new time interval in the computational process is now described.

One can start with the partial differential equations (Eqs.1-3) and integrate around the junction cen in space and time [Godunov, 1959] or else the integral form of the same equations can be written in terms of mass, momentum and energy flux balances (control volume approach). The fin al results for the updated density, momentum and energy are given by

n n-l

~t

[()

~

( )

Aj-lelt ]

P center

=

P center - A pU right - L..J pU j-lelt - A '

~x j=l chlln

(13)

(pu): ... " -

(pu):;;;:" -

~:

[(PU

2

+

p),;,ht

~

( 2

)

A j -lelt AWllll] - L..J pU

+

P j-lelt - A - PWllll-A ' j=l chlln chlln (14) ( 1 1

2)

n

( 1 1

2

)n-l

- - p + - p U = - - p + - p U , - 1 2 center , - 1 . 2 center

~t

[(

"y . 1

3)

~

(

"y

3)

A,;_lelt ]

- - - - P U + - p U - L..J - - P U + p U - - ,

~X , - 1 2 right j = l ' - 1 j-lelt A chlln

(15)

at the new time level n in terms of the previous time level n - 1. Note that the time step

.6.t = tn - tn-1 and the width of the junction cen is denoted by .6.x.

The change in density from time level n - 1 to n in Eq.13 is a function of the difference in mass fluxes at the left and right sides of the junction cell. The first term in the square bracket is the mass flux leaving the cen and entering the channel, whereas the second term is a sum of the mass fluxes entering the junction cen from the seven drivers

and their expansion sections. Each flux includes its corresponding area ratio A j-lelt / A chlln

because the area of the flow entering the junction een differs from the area of the junction ceU (or channel). The change in energy between time levels, given by Eq.14, has a similar structure.

The change in momentum from time level n - 1 to n in Eq. 15 is not only a function of the difference in momentum fluxes at the left and right si des of the cen, but it is also a function of the pressures at both sides of the cell. This dep enden ce leads to the terms

pu2

+

P at the left and right si des of the ceU, and the additional term PWllllAwllll/ A chlln

because the left side of the junction cen consists partIy of a wan throughwhich there is

Once the density, mömenum and energy updates have been done, one can easily calculate the individual flow properties for flow velocity and pressure. The flow veloc-ity foUows directly from the updated densveloc-ity and momenum, and the pressure is then obtained from the updated energy, flow velocity and density.

The update of the center state Scenter requires a knowledge of the states on the left

and right sides of the ceU-Sj•/eft and Srighh respectively. Each individual state Sj-Ieft is

obtained by means of a special Riemann problem for a duet ending in a special atmo-sphere, as explained earlier in section 2.2 for boundary conditions. In order to apply this boundary condition at each duet exit to the junction ceU, one needs a representative atmospheric pressure near the junetion's left waU for the case of a flow from the driver into the junction ceU, and alternatively a representative atmospheric pressure and sound speed (or temperature) near the junction's left wall for the opposite case of a flow from

the junetion cell to the driver. For the case of flow into the junction cell, the representa

-tive pressure at the junetion's left waU is simply set equal to the statie pressure for the center of the cello In other words, the separated jet flow that enters the junction ceIl is

assumed to enter an environment with a pressure Pwall

=

Pcenten and flow losses from the

junction's left boundary to the junction cell's center are ignored. For the opposite case of a flow out of the junction ceU the representative pressure and sound speed are taken as the stagnation values of the state at the junction ceIl's center. Hence, the flow is driven from an assumed quiescent atmosphere into the duct leading to the driver by stagnation conditions given by

..::J....

Pwall - Peenter

[1

+

,~1 Mc~nter]

'")'-1 (16)

and

1

[ ,-1

2

]2

awall = acenter 1

+

-2-Mcenter . (17)

Although this procedure is not unduly sophisticated for a fairly complex flow problem, it should still give good results because it does contain the correct physical principles and flow trends. For example, in the case when the flow through the driver throat area is choked and the flow into the junction cell is not changing rapidly with time, any reasonably low pressure in the junction ceIl will give essentially correct fluxes of mass, momentum and energy. This choked condition normally occurs for a significant portion of the flow time for large-sized blast simulators with high-pressure drivers.

The update of the center state Scenter also requires a knowledge of the state on the

right side of the cell-Sright, as mentioned earlier. This state is obtained by solving a

standard Riemann problem between the two adjacent states Scenter and Schan. Then the

solution of the state at the" center of this Riemann problem yields values for Pright, Pright

and Uright, and these flow proper ties can then be used to calculate the required fluxes for

updating the center state Scenter of the junetion cello

3.0 BLAST SIMULATOR MODEL GEOMETRIES

The three different geometrical models for the blast simulators of importance to this study have already been introduced in figure 8, but this was do ne using schematic diagrams. The actual dimensions of these three blast-simulator models are detailed in this chapter, because they will be needed later for the numerical computations.

In each case the channel has a constant cross-seetional diameter and area of 9.44 m

and 70 m2_{, respectively, and the length is 105 m. The channel includes a test section and}

terminates with a RWE. The center of this test seetion is located at a distance of 37.5 m from the RWE or the open end of the channel. The RWE operation is considered ideal or perfectly functioning-in other words there are no reflected waves from the channel end which propagate upstream and disrupt the simulated blast flow properties in the test section. (Alternatively, this channel can be considered as having an infinitely long extension.) The type of RWE is not important in this study, because the end of the channel is modelled numerically as a nonreflecting boundary (or as a totally transmissive boundary).

In the case of the multiple drivers depieted in figure 8a, there are seven drivers

and each one has a constant cross-seetional diameter and area of 1.33 mand 1.39 m2

,

respectively. The variations of diameter, area and the funetion (1/

A)dA/dx

with axial

distance along the driver and first part of the channel are shown graphically in figure 10 (to scale). The longest driver (shown in this figure) has a constant-area length of 20.0 m, and the three remaining pairs (not shown) have constant-area lengths of 16.5, 12.0 and 6.50 m. Each of these drivers has one end that is closed, and the other has a 0.8-m-long converging duct seetion to a short constant-area throat with a length, diameter

ánd area of 0.25m, 0.665m and 0.347m2

, respeetively. The diaphragm is located at the

center of the short throat seetion. From the end of the throat seetion to the channel there is a diverging duct seetion which has a length of 2.4 m, and it terminates with a

short constant-area seetion of length, diameter and area of 0.25 m, 1.90 mand 2.84 m2,

respectively. Note that the tot al area of all seven divergent duet exits at the channel

ent rance is 19.85 m2_{, which is markedly less than the channel cross-sectional area of}

70 m2 _{(by a factor of 3.5). The sum of the volumes of the seven drivers of different}

lengths, from their closed end to their diaphragm, is 129.5 m3_.

In the case of the composite driver sketched in figure 8b, the four sections of

con-stant cross-sectional areas are 1.39, 4.17, 6.95 and 9.73 m2

, and their equivalent

diame-ters are 1.33, 2.30, 2.97 and 3.52 m, respeetively. The variations of diameter, area and

(1/ A)dA/dx

with axial distance are shown graphically in figure 11. These constant-area sections are connected by smooth area transitions with additionallengths of 0.5, 0.3 and 0.23 m (from the smallest to the largest constant area). The convergent section of the

driver to the throat section has a length of 0.8 m, as in the multiple driver .case, and

it also ends with a short constant-area throat of 0.25 m long, composite area of 2.43 m2

and equivalent diameter of 1. 76 m. The overalliength from the closed end to the start of the convergent section is 20 m, and these area transitions do not increase the total driver length to the diaphragm. The diaphragm is again located at the center of the throat section. The divergent seetion to the channel is 2.65 m long (equivalent to 2.4+0.25 m from the multiple driver case). This divergent seetion ends with a composite area equal

to 19.85 m2 _{and the equivalent diameter is 5.03 m. The total driver volume is again}

129.5m3_.

For the last case of the composite driver with the smooth area variation, shown in figure 8c, the shape of the driver can be either conical or curved. For the conical

figure 12. The equations for these variations are D = Dcone

(=-)

mcz L ' (IS)

A -

A~:: (~r,

(19) 1 dA 2

- - - -

,

x Adx (20)

where the distance x is measured from the apex of the cone. For our specific case of a

conicallength of 20 mand a total driver volume of 129.5.m3_{, D~:: equals 4.S4 m and A~::}

equals lS.4m2• For the curved driver the variations of diameter, area and (lJA)dAJdx

with axial distance are shown in figure 13. The equations for these variations are

D _ Dcurve . _{- mar sm}

n (7rX)

_{2L '} (21)

A _ A curve •

2n ( 7rX)

- mar sm 2L' (22)

!.

dA

=

n7r cot

(7rX)

A dx L 2L' (23)

where the distance x is again measured from the apex. For our specific case of a curved

driver length of 20 m, nis equal to 0.60, and for the same total driver volume of 129.5 m3

,

D':,..u;;e equals 3.64 m and A~;;e equals 10.39 m2_•

4.0 NODE SELECTION FOR NUMERICAL MODEL

To obtain accurate solutions to Euler's equations of motion for blast simulators in

this study with the first-order

RCM,

a sufficient number of nodes must be used in the

nu-merical solution procedure. This is, to some degree, a process based on trial and error and

previous experience with the

RCM.

For better accuracy, no des must be concentrated at

locations where spatial gradients in the flow properties are large. This normally occurs at discontinuities (shock, contact surface), steep rarefaction and compression waves, and in large area changes. Therefore, in this study, nodes are concentrated in the driver seetion and in the area changes of the blast simulator. This chapter is a brief discussion of the node seleetion procedure to determine the final number of no des for our blast-simulator computations, in order to achieve sufficiently accurate numeri cal results. Furthermore, a method of filtering or smoothing the numerical results is also given.

Two special cases of blast-simulator geometries are considered for this numerical

study of node seleetion. In the first case the driver consists only of one constant-area

duet of length, diameter and area of 12.S m, 3.52 mand 9.73 m2_{, respectively. This driver}

diameter and area correspond directly to the largest diameter and area of the composite

driver shown in figure 11, and the convergent-divergent throat seetion and channel are

identical to those of this composite driver. The second case is that of a multiple driver

blast simulator shown in figure Sa and 10, with only one difference in th at the drivers

are all of the same length of 12.S m. Note that the total volume of the driver is 129.5 m3

For the single driver case the total number of no des considered was 706 and 1880. Without giving all of the details, no des were concentrated in area changes by a factor of about four with respect to the constant area sections. On changing from 706 to 1880 nodes, the node densities everywhere were changed by the same factor. The corresponding multiple driver case had a total number of no des of 1880, considering only one driver, the convergent-divergent section and the channel. When the other six independent drivers are included, the total number of nodes increases to 4190.

For all blast-simulator computations given in this chapter, the initial driver pressure and temperature are 33 atm and 298 K, and the initial channel pressure and temperature are 1 atm and 298 K. For the first case of the single driver and a total number of no des equal to 706, normalized time histories of pressure, density, sound speed and flow velo city are presented in figure 14. These results exhibit fairly large numerical oscillations or noise, especially in the first half of the signatures. This is due mainly to having insufficient concentration of nodes in the area changes (the operator-splitting corrections for area changes are too large and therefore inaccurate). When the number of nodes is increased by a factor of 2.66 to 1880, this noise is reduced markedly, as illustrated in figure 15. For the multiple driver case with a corresponding number of nodes (1880), the noise level is slightly less, as shown in figure 16. Based on these numerical results, and ot hers using different numbers of nodes, it was concluded that 1880 no des were sufficient for providing good solutions in this study. In the case of longer drivers or channels the number of no des is correspondingly increased.

The advantage of using more nodes to achieve bet ter accuracy is offset by the increase in computational run times. Doubling the number of nodes halves the time step and almost quadruples the computer run time. This becomes a major obstacle for the multiple driver case, where each driver flow is computed independently, leading to a large total number of nodes. Note that for the case of 1880 nodes in the single driver case, the corresponding total number of nodes for the multiple driver case is much larger at 4190. With 1880 nodes, a typical

epu

time for the single driver case is 2 hours and that for a multiple driver case is 5 hours, when an Apollo 4500 computer without a floating-point accelerator is used.

In this study, numerically computed time histories for different driver geometries are compared by plotting them together on the same graph. In most cases these time histories are very similar and the small differences between various time histories are obscured byeven a small amount of numeri cal noise. To circumvent this problem, the computed time histories are filtered numerically before they are compared graphically.

The numeri cal filtering or smoothing of records is done in a piece-wise manner. Parts of each recored between selected discontinuities are filtered independently, so that these selected discontinuities are preserved (not smoothed out). Each part or fini te set of data is extrapolated beyond its two ends in an antisymmetric manner, and then the data is smoothed by the following algorithm:

j=+oo

Pi

=

(211"0"2)-1/2

L

j=-oo ( i _

i)2

Yiexp ~ , (24)

where

Yi

is the ith _{value of the noisy data,}

Pi

_{is the corresponding i}th _{value of the smoothed}

This root-mean-square value controls the size of the "filtering window" or the "degree of smoothing" .

Some numerically filtered time histories are shown in figures 17 and 18. These correspond directly to the unfiltered time histories shown previously in figures 14 and 15 respectively. One can see that the filtered results for the case of 706 no des have aphysical spikes, whereas those in figure 18 for the case of 1880 nodes are much smoother and, therefore, more aceeptable. Note that the accuracy of the results from the random-choice method cannot be improved by using the filtering proeess alone. However, the use of filtering to decrease the numeri cal noise is very helpful in facilitating the comparison of more than one set of time histories.

5.0 VALIDATION OF THE JUNCTION CELL

The junction ceU developed earlier in this study represents a new approach to solving multidriver blast-simulator problems. Therefore, the validity of this ceU should be assessed before it is used for predicting flows in such blast simulators. This assessment is the subject of this chapter.

In order to assess the validity of the junction ceU, a proper test problem related to blast simulators needs to be defined. Consider two blast-wave simulators, one with a single driver and one with seven multiple drivers. The only differences between the single and multiple drivers are that the multiple drivers are all identical and have a constant area and volume of 1.39 m2 and 18.5 m3, which are exactly one-seventh that of the single

driver (9.73 m2

, 129.5m3). The length of each multiple driver is equal to that of the single

driver (with a constant-area section 12.8 m long). The convergent-divergent duct sections from each multiple driver is also one-seventh the corresponding area of the single driver case, and these sections are the same length. For both cases the channel is the same area of 70 m2 _{and length of 105 m, and the seven multiple driver expansions exactly match}

the area of the channel (Awall

=

0 in Eq. 14). These two blast simulators were selected

because they will produce identical blast flows. However, in the single driver case the flow can be computed fairly easily without using a junction ceU, whereas in the other case of multiple drivers the computations require the use of a junction cello Since the flows are theoreticaUy identical, any differences in the flow computations should be due directly to inadequacies of the junction cell.

In

other words, the goodness of agreement of the two flow computations should provide an assessment of the validity of the junction ceU.

The flow computations were performed for both the single and multiple driver cases and compared for four different initial driver pressures of 5, 33, 71 and 160 atm, an initial channel pressure of 1 atm, and initial driver and channel temperatures of 298 K.

These conditions produce simulated blast waves with amplitudes of approximately 1.09, 1.56, 2.2, 3.2 atm, which are refered to as very low, low, medium and high amplitude simulated blast waves (or shocks). Comparisons of computed time histories at the test section for these four simulated blast-wave cases, with and without the junction ceU, are presented in figures 19 to 22. The agreement between the two different computations with and without a junction eeU is very good. Therefore, the validity of the junction ceU is considered as demonstrated, and this has been done for a wide range of simulated blast-wave amplitudes and signature features.

6.0 NUMERICAL RESULTS AND DISCUSSION

The numerical results for the blast simulators with conical, curved, composite and multiple drivers are presented and discussed in this chapter. These results consist of both time histories of the flow properties at the test section in the channel and spatial distributions of some of these flow properties in the driver and channel. The results for

blast simulators with conical and curved. drivers are presented first, in order to illustrate

the details of the wave motion occurring inside the driver and channel, since these wave patterns are the simplest and easiest to understand. Then the more complex wave pat-terns in blast simulators with composite and multiple drivers are presented. This chapter ends with a comparison of the numeri cal results from the blast simulators with different drivers and an assessment of the importance of modelling flows in blast simulators with multiple drivers as independent flows rather than combining them into one flow as in the composite driver case.

6.1 Blast Simulator with a Conical Driver

Time histories of the flow properties in the test section of the blast simulator with

the conical driver (figure 12) are presented in figures 23, 24 and 25, for the cases of

simulated blast waves with low, medium and high nominal overpressure amplitudes of

0.6, 1.2 and 2.2 atm, respectively. These flow properties include the pressure, density,

sound speed and flow velocity. All of the flow properties for the low amplitude case and the pressure and flow velocity of the medium and high ampitude case exhibit the basic decaying signature of a blast wave. The decay of the density and sound speed for the medium and high amplitude waves is interrupted when the contact surface (C) arrives at the test section, whereas for the low amplitude case the contact surface does not reach

the test section and cause a disruption . .

Each flow property signature is headed by the primary shock (Sp), and secondary

shocks (S and Sa) and secondary rarefaction waves (R) are superposed on this basic

decaying waveform. In almost all cases these secondary waves appear in pairs, in the

form of a shock followed irnrnediately by a rarefaction wave, and also conversely. This

frequently gives the pair the appearance of a spike. Furthermore, although these time histories do not illustrate the process, these coupled waves continually interact as they propagate along the driver and channel, and their amplitudes are reduced as aresult

of the interaction. If the propagation continues sufficiently far, one of the waves will

disappear during the interaction, and the dominant wave will survive as a single wave. For all three cases of low, medium and high amplitude blast waves, these secondary

waves (with the exception of the shock Sa in the medium and high amplitude cases) have

a basic sequence of two shock-rarefaction wave pairs followed by a rarefaction wave. This

triplet of waves (SR, SR and R) appears repeated once or twice until the waves are too

small in amplitude to be recognized, as can be seen in figures 23-25. The origin of the

primary shock in all three figures is obviously due to the initial difference in pressure across the diaphragm, before it is broken. The origins of wave pairs, the triplet sequence,

and the shock Sa are not so obvious, and these will now be explained, starting with the

wave pairs and triplet sequence.

After the diaphragm is broken a rarefaction wave moves toward the origin, spreads out, and continually reflects from the conical area reduction, as shown in the spatial

distributions of pressure for the driver in figure 26. At the tail of this rarefaction wave

(R) an inward moving shock forms, becoming stronger as it approaches the origin. (This

RS wave combination is a common feature of most explosions, and every rarefaction wave that approaches the origin must eventually be followed by a shock.) On reflection from the origin this shock is followed immediately by a rarefaction wave, leading to an SR

wave combination, and this combination decays as it moves outwards into the expanding area of the cone and towards the convergent-divergent section of the blast simulator. The relative strength of these waves will determine which one is dominant and survives with increasing distance, and the driver flow geometry will dictate the type of wave pairs and their relative amplitudes ..

In this first reflection case, the shock is dominant and the SR pair interacts with the convergent-divergent section to produce a reflected SR combination. This new SR

combination moves towards the origin and reflects as another SR combination. The resulting sequence of reflections from both ends of the driver produces an oscillatory wave behaviour inside the driver, as shown in figure 26, and the amplitude of these combined waves decreases with time. The overall effect of this oscillatory wave behaviour is to reduce in an unsteady manner the pressure in the driver, during the loss of driver air through the throat to the channel.

In some cases the rarefaction wave in the reflected SR combination from the origin is dominant and eliminates or nearly eliminates the shock within the driver, resulting in a surviving rarefaction wave. This expansion wave then interacts with the convergent-divergent section and pro duces a reflected expansion wave in the driver. However, this reflected rarefaction wave always results in an RS wave combination on approaching the origin, and then this leads to an SR wave combination on reflection from the origin.

Each interaction in the driver of a rarefaction wave

R,

or shock and rarefaction wave combination (SR), with the convergent-divergent section also results in a transmitted wave or wave combination of the same type in the channel. These transmitted waves in the channel are shown in the spatial distributions of pressure for the channel in figure 27. Following the primary shock Sp, one can identify the triplet sequence of SR, SR and

R in these spatial distributions. As mentioned earlier, when the transmission is only a rarefaction wave the rarefaction wave was initially dominant in the SR wave pair reflected from the driver origin. The sequence of waves in each pair and which wave survives within the pair is directly due to the driver geometry, and only somewhat affected by the initial driver pressure. This helps explains the origins of the secondary waves that are superposed on the decaying signature of the simulated blast wave, their occurrence and order within the pair, and the occurrence of the triplet sequence in the time histories and spatial distributions of the flow properties.

Before leaving figures 26 and 27, it is worth noting a few interesting features of the flow in this blast simulator with a conical driver. Firstly, the primary shock is initially spiked on emerging from the convergent-divergent area section. However, this spike quickly decays and becomes unnoticeable as the shock propagates along the channel. (This is a common feature of shocks passing through large area expansions.) Secondly, on breaking the diaphragm the primary shock rapidly establishes choked flow at the throat, supersonic flow downstream in the expansion section, and an upstream-facing shock wave which terminates the supersonic flow. As the pressure in the driver diminishes with time

this upstream-facing shock slowly moves forward and the choked flow at the throat and the supersonic flow eventually end. For sufliciently small pressures in the driver, the flow is unchoked and subsonic, and the upstream-facing shock does not exist. Thirdly, there is a contact surface between the original expanded air from the driver and the shock-heated air from the channel, which cannot be readily identified in the spatial distributions of pressure in figure 28. However, this contact surface (C) can easily be identified by the large spike in the spatial distributions of density presented in figure 28. When the first two SR wave pairs pass through this contact surface, from the cold to the hot side, small reflected shocks or actually an SR wave combi nat ion are produced. Two of these are

identified and labelled s in figure 27.

_,

In contrast to the case of the low amplitude shock, the case of the medium amplitude shock introduces some new features. In the time histories of the flow properties presented earlier in figure 24, a shock labelled

Sa

was notieed. The origin of this shock is obvious from spatial distributions, and these are presented for both the pressure and density in figures 29 and 30 respectively. In the medium amplitude case the driver pressure is initially sufliciently high that the upstream-facing shock is swept downstream by the supersonic flow. This shock is swept through the area expansion and down the channel (by about 10 m). However, as the driver pressure diminishes, the strength of the supersonic flow decreases and the upstream-facing shock reverses its motion and moves back into the area change, wh ere it then slowly moves forwards and eventually dies out. The interaction of this shock with the area change on its return produces a reflected compression wave which steepens into an outward moving shock, which is labelled

Sa

in figures 24, 29 and 30. An additional feature of the medium amplitude shock case is that higher driver pressure with more initial air in the driver causes the contact surface to be swept farther downstream in the channel before slowing and coming to rest. The contact surface in this case is swept downstream past the test section, which is fairly obvious from the results of figures 24 and 30.

The shock

Sa

also exists in the time histories of the case of the high amplitude shock, as seen earl ier in figure 25. lts origin is the same as in the medium amplitude shock case. However, because of the high initial driver pressure, the upstream-facing shock is swept much farther down the channel, as shown in the spatial distributions of figure 31. It thereby returns to the area change much later in time and the subsequent reflected compression wave and shock

Sa

occurs much later in the time histories.

6.2 Blast Simulator with a Curved Driver

Time histories of the pressure, density, sound speed and flow velocity at the test section of the blast simulator with the conical driver (figure 13) are presented in figures 32, 33 and 34, for the cases of simulated blast waves with low, medium and high nominal overpressure amplitudes of 0.6, 1.2 and 2.2 atm, respectively. These time histories exhibit a dominant primary shock, the overall decay of a blast wave, and secondary shock and rarefaction waves, very similar to the previous case of the conical driver. However, there are a number of small but interesting differences. For instanee, the initial decay just after the primary shock is more humped or less rapid. In addition, the wave pairs in this case are initially all of the type SR, followed by a couple of expansion waves R, and these differences are due only to a change in driver geometry (from conical to curved).