Initial decay of flow properties of planar, cylindrical and spherical blast waves

INITIAL DECAY OF FLOW PROPERTJES

OF PLANAR, CYLINDRJCAL AND SPHERICAL BLAST WAVES

October, 1983

by

TECHNISCHE

HOGESCHOOL DElFT

LUCHTVAART-E RUlfn:-vAARTTECHNJEK

BiEUOTHEE

Kluyverweg 1

- DELFT

H. S. I. Sadek and

J. J.

Gott1ieb

2.

JAN.

1984

UTIAS Technica1 Note No. 244

eN ISSN 0082-5263

•

INITIAL DECAY OF FLOW PROPERTIES

OF PLANAR, CYLINDRICAL AND SPHERICAL BLAST WAVES

by

H. S. I. Sadek and J. J. Gott1ieb

Submitted March, 1983

•

Acknowledgements

We want to express our gratitude to Dr. J. M. Dewey and Mr. D. J.

McMillin of the University of Victoria for providing much detailed experi-mental data on TNT and propane-oxygen explosions for convenient utilization in the present research report .

We would also like to thank Mr. N. H. Ethridge of the United States Army Ballistic Laboratory for his interest, encouragement and additional experimental data on ANFO explosions.

It is also our pleasure to acknowledge Professor I. I. Glass o·f UTIAS for his interest in our work and for his helpful suggesticns.

Our appreciation is extended to Mr. R. L. Deschambault, for his operation of his microcomputer and printer terminal to produce the manuscript for this report.

The financial support for the present research was provided by the Defence Research Establishment Suffield and the Natural Science and

Engineer-ing Council of Canada, and this support is acknowledged with many thanks. It made the M.A.Sc. thesis research of the first author possible, on which this report is based.

Abstract

Analytical expressions are presented for the initial decay of all major flow properties just behind planar, cylindrical, and spherical shock-wave fronts whose trajectories are known as a function of either di stance versus time or shock overpressure versus distance. These expressions give the time and/or distance derivatives of the flow properties not only along constant time and distance lines but also along positive and negative characteristic lines and a fluid-particle path. Conventional continuity, momentum and energy equations for the nonstationary motion of an inviscid, non-heat conducting, compressible gas are used in their derivation, along with the equation of state of a perfect gas. All analytical expressions are validated by comparing

results from the present work to those obtained indirectly from known

self-similar solutions for planar, cylindrical and spherical shock-wave flows generated both by a sudden energy release and by a moving piston. Furthermore, time derivatives of pressure and flow velocity are compared to experimental data from trinitrotoluene (TNT), pentolite, ammonium nitrate-fuel oil (ANFO) and propane-oxygen explosions, and good agreement is obtained. For this latter work it was necessary to review and compile available experiment al data from

the previously mentioned explosions. This collection of data with new curve fits for the shock-front trajectories are an important part of the present work.

1.0 2.0 Table of Contents Acknowledgement Abstract Table of Contents Notation INTRODUCTION

DATA FOR SURFACE EXPLOSIONS 2.1 Past Explosion Trials 2.2 Blast-Wave Sealing Laws 2.3 Curve-Fits for the Shock 2.4 Explosion Data for TNT 2.5 Explosion Data for ANFO

IN AIR

Trajectories

2.6 Explosion Data for Propane-Oxygen Mixture

2.7 Comparison of Results from Different Explosions

3.0 ANALYSIS AND ANALYTICAL RESULTS FOR THE INITIAL DECAY RATES 4.0 VALIDATION OF THE INITIAL DECAY RATE EXPRESSIONS

4.1 Sheet-, Line- and Point-Source Explosions

4.2 Shock Flows in Front of Moving Planar, Cylindrical and Spherical Pistons

Page 1.1. 1.1.1. 1.V v 1 3 3 5 6 8 11 11 12 13 19 20 23 5.0 GRAPHICAL RESULTS FOR THE INITIAL DECAY RATES 24 6.0 COMPARISON OF ANALYTICAL AND EXPERIMENTAL INITIAL DECAY RATES 28

6.1 TNT Explosions 6.2 ANFO Explosion 6.3 Propane-Oxygen Explosion 6.4 Pentolite Explosions 7.0 CONCLUDING REMARKS 8.0 REFERENCES Tables Figures

Appendix A: ASYMPTOTIC PROPERTIES OF THE CURVE FIT AND INITIAL DECAY RATES

1.V 28 29 29 29 30 30

•

Notation

a speed of sound

Cv specific heat at constant volume for a perfect gas E energy release in an explosion

I positive-overpressure impulse

K parameter defined by the integral ~n Eq. 4.9 L positive-overpressure length

Mlocal flow Mach number (u/a) n exponent in Eq. 2.8 N p q r R s T~ P

T~

c-t ti U v W Wo

geometric constant which is equal to 0, 1 and 2 for planar, cylindrical and spherical geometry, respectively

static pressure

dynamic pressure (pu2/2) distance or radius

radius of an explosive charge gas constant

scaled shock radius (rs/rc)

characteristic decay distance of a general flow variabie ~ along a constant time line

entropy

Sachs's scaling factor for distance Sachs's scaling factor for time

static temperature, positive-overpressure duration

characteristic decay time of a general flow variabie ~ along a constant distance line

characteristic decay time of a general flow variabie ~ along a particle path

characteristic decay time of a general flow variabie ~ along positive or negative characteristic lines

time

flow or particle velocity

velocity of the shock front, shock Mach number speed of an expanding piston

mass of an explosive charge in kilograms standard TNT mass of 1 kilogram

z scaling parameter, independent variable ln(r/t) in Eq. 4.16

a denotes (y

+ 1)

I

(y - 1) ai constant coefficients

S denotes (y - 1) 12y

Si constant coefficients

y ratio of the specific heats for a perfect gas

"6~ sudden change in a general flow property ~ across the shock- or blast-wave front

6ps peak overpressure of a shock- or blast-wave front

Z;; denotes the nondimensional variab1e u tir

n

denotes the nondimensional variab1e a2 t2/r2

8 ratio of kinetic energy to internal energy

p density

~ geometric factor used to illustrate blast-wave scaling ~ general flow variable

Subcripts

o denotes standard atmospheric conditions

1 denotes ambient conditions in the quiescent gas ahead of the shock wave

p denotes conditions of an expanding piston s denotes the shock front

*

denotes dimensi9nal variables Special Symbols

Is indicates that differentiation 1S taken along the shock line

lp indicates that differentiation is taken along a particle path

Ic± indicates that differentiation is taken along a positive or negative characteristic line

1.0 INTRODUCTION

Consider the simple case of a shock or blast wave moving with a known velocity or Mach number in a perfect gas. Let the initial conditions such as the pressure, density and temperature of the quiescent gas ahead of the wave be known. Then all flow properties including, for example, the pressure, density,

temperature and flow velocity just behind the shock front can be calculated readily, by using the well-known Rankine-Hugoniot relations (Ref. 1). If the shock velocity is initially unknown but the shock-front trajectory is known as a function of distance and time, then an extra simple step is required to determine the shock velocity from the trajectory information. The procedure for calculating the properties just behind planar, cylindrical and spherical shock fronts is weIl known, and it has been used frequently since the original shock equations were derived independently by Rankine in 1870 (Ref. 2) and Hugoniot in 1889 (Ref. 3).

For planar, cylindrical and spherical shock or blast waves having known shock trajectories in terms of a function of distance (r) and time (t), one can determine not OlUy the shock velocity (dr/dt or U) at any point on the trajectory but also the corresponding shock decay rate or deceleration (d2r/dt2 or dUldt) and higher-order time derivatives. It has al ready been mentioned that the flow properties just behind the shock front can be obtained from the shock velocity. However, the additional knowledge regarding the shock deceler-ation permits one to calculate the decay rate or first time derivative of all flow properties just behind the shock front. Furthermore, higher-order time derivatives of all flow properties can be determined by making use of the corresponding higher-order time derivatives of the shock trajectory. This is neither obvious nor weIl known but should be accepted here for introductory purposes. It will be substantiated later in this report.

Although the preceding comments apply directly to the case when the shock trajectory is known as a function of distance and time, they also apply with only slight modification to a similar case when the shock-front

over-pressure is known as a function of distance or time. This is obvious when one realizes that an expression for the shock trajectory in terms of distance versus time can be rewritten, with the aid of only one Rankine-Hugoniot

relation, in terms of the shock overpressure versus distance, and conversely. Hence, the decay rates of all flow properties just behind the shock front can be determined from a shock trajectory known as a distance-time or overpressure-distance function.

For the case of a spherical explosion with a shock-front trajectory known in terms of the shock overpressure versus distance, Kirkwood and Brinkley in 1944 (Ref. 4), Theilheimer in 1950 (Ref. 5) and Dewey in 1952 (Ref. 6) first derived equivalent expressions for the overpressure decay with time just behind the shock front. This initial decay rate was expressed as a function of the shock overpressure and its first derivative with respect to distance.

Additionally, by assuming that the overpressure-time signature of the shock-wave flow was a simple exponential function with the initial shock amplitude and decay rate, they obtained a particular partial solution for the shock wave. This particular exponential representation of the overpressure signature is, of course, only approximate, being more accurate nearer to the shock front. No attempt appears to have been made by these early researchers to derive decay rates for the other flow properties just behind the shock front and obtain a more complete set of exponential signatures describing the shock wave.

early research, although the lat ter work was not published in the open literature or presented in laboratory reports. For example, in private communication, N. H. Ethridge has pointed out that F. B. PorzeI derived

expressions for the partial time and distance derivatives of the most important flow properties at the shock- or blast-wave front. Furthermore, Ethridge

derived equivalent expressions, independently of PorzeI, and compared predicted and measured results for TNT explosions in air. Although he presented these results orally at a meeting of the American Physical Society (see his abstract in the Bull. Amer. Phys. Soc. , Ser. 11, Vol. 14, p. 1098, 1969), they have never been published in written form.

The partial overpressure-time solution for a spherical shock or blast wave proved very useful in explosion tests conducted during the 1940s, 1950s and 1960s. The shock trajectory was of ten measured by using a series of pressure gauges to record shock overpressures and/or shock arrival times at known distances from the explosion center. Either or both types of shock-trajectory data could then be used to obtain the flow properties and over-pressure decay rate just behind the shock front. The decay rate information was of ten used for comparison with overpressure-gauge records, because it permitted an independent check of the initial decay rates of these records (Refs. 7 and 8). This check was more important prior to about 1970 than today, because early pressure gauges were of po or quality an.d of ten gave untrustworthy signatures.

A knowledge of the decay rates of all flow properties just behind a shock front can have additional applications. For example, in numerical

analyses for predicting shock- or blast-wave flow fields, this knowledge would permit the initial decay rates of flow properties to be matched to those based on theshock trajectory. Although this matching condition is not necessary for first-order numerical analyses, which require as a minimum condition that the Rankine-Hugoniot equations be satisfied across the shock front, it is required for second-order numerical work. In past numerical analyses of reconstructing the flow field of an explosion behind a known shock trajectory with the aid of one or more known overpressure signatures (Refs. 9 and 10), overpressure-decay information could have been employed advantageously to ensure that the shock-trajectory and overpressure-signature data were matched in a consistent manner at the shock front.

The main purpose of the present work is to present a complete set of analytical expressions for the initial decay of all major flow properties just behind planar, cylindrical and spherical shock- or blast-wave fronts whose trajectories are known either as a function of distance versus time or shock overpressure versus distance. These expressions give the time and/or di stance derivatives of the flow properties not only along constant time and distance lines but also along positive and negative characteristic lines and a fluid-particle path. Their derivation is based on the conventional continuity, momentum and energy equations for the nonstationary motion of an inviscid, non-heat conducting, compressible gas, as weIl as the equation of state for a perfect gas. This complete set of equations for the derivatives of the shock flow properties is given herein for the first time. The analytical results are presented in a concise form for easy utilization by others.

All analytical expressions given for the derivatives of the shock flow properties are validated by comparing results from the present work to those obtained indirectly from known self-similar solutions of planar, cylin-drical and spherical shock- or blast-wave flows. The first complete check

..

reported in this work is made with von Neumann's self-similar solution for the point-source explosion without counterpressure (Ref. 11). This solution is extended so that it also includes solutions for line- and sheet-source explo-sions, in order to make validation checks for these latter two cases. The second complete check is provided by Taylor's self-similar solution for the shock-wave flow in front of an expanding spherical piston (Ref. 12). This solution is also extended to include cylindrical and planar piston and shock motions, thereby providing additional validation checks.

A few comparisons of analytical and experimental results for the decay of the flow proper ties just behind a spherical shock front are also

included in the present work. Comparisons are made mainly for time derivatives of overpressure and flow velocity along constant distance lines and

fluid-particle paths, respectively. The experimental information was obtained from the literature for" past trinitrotoluene (TNT), ammonium nitrate-fuel oil

(ANFO), pentolite and propane-oxygen explosions. Note that extensive compari-sons of analytical and experimental data are not possible, simply because sufficient and/or accurate experimental data for the initial decay rat es are not available.

The analytical prediction of the derivatives of the shock flow

proper ties requires a detailed knowledge of the shock trajectory as a function of distance versus time or shock overpressure versus distance, as mentioned previously. When the present work was initiated it was assumed that experi-mental data and their empirical curve fits for shock trajectories from various explosions published in the literature were reasonably consistent and accurate. Since this was not necessarily true, as was discovered during the work, it was necessary to review in detail and compile available experimental data for shock traj ectorles from TNT, pentolite, ANFO and propane-oxygen explosions. The

resulting collection of shock-trajectory data, the new curve fits to this data, and a detailed discussion of both constitute an" important part of this work. This work is presented first in the next chapter.

2.0 DATA FOR SURFACE EXPLOSIONS IN AIR 2.1 Past Explosion Trials

TIle sudden release of energy from an electrical discharge (spark, lightning), a focussed laser beam, a high-energy explosive charge or a nuclear detonation in a small volume of the atmosphere each produces an initially smal I region of gas at a very high temperature and pressure. The ensuing rapid

expansion of this high pressure gas quickly generates an outward moving shock or blast wave, as sketched in Fig. 1. Roughly one-half of the energy released remains in the heated region or hot fireball and roughly one-half appears in the blast wave. This is evident as an increase in pressure, temperature and density, along with a mass movement away from the fireball. Following the almost instantaneous rise in the flow properties at the front of the blast wave, an exponential-like decay toward ambient conditions occurs as shown in

Fig. 1.

During the 1940s and 1950s many nuclear-explosion trials were conducted both to facilitate the design and development of various types of nuclear weapons and to provide a nuclear-explosion environment to test military and civil structures and equipment (Ref. 13). With the advent of the test ban on nuclear explosions in the early 1960s, large charges of high-energy

explosives (up to 500 tons of TNT or its equivalent) were utilized to simulate a nuclear-blast environment for the continuation of such testing. These high-explosive trials involving surface bursts are continuing today, although

height-of-burst trials are now also being conducted. Furthermore, small-scale testing of structures and equipment is being done in small and large blast-wave simulators. Many results from high-explosive and blast-wave-simulator trials can be found in the past seven proceedings of the Military Applications of Blast Simulation Symposia (1967 to 1981).

A number of different high-explosive solid and gaseous explosion trials are worth introducing here, because data from these trials are used in the present study. During the early 1960s the United States, the United Kingdom and Canada participated in a 20-ton TNT hemispherical explosion trial

(Ref. 14), which was held at the Canadian Defence Research Establishment

Suffield (DRES), Ralston, Alberta. Shortly thereafter in July 1964 a 500-ton

TNT hemispherical charge was detonated on the ground at DRES (Ref. 15), and in April of 1965 an identical charge was detonated on Kahoolawe Island in the Pacific Ocean by the United States Navy (Ref. 15). The latter two trials were known as Snowball and Sailor Rat, respectively. In 1966 a hemispherical

balloon filled with a stoichiometric mixture of propane and oxygen, 38 m in diameter and having an energy release equivalent to about 20 tons of TNT, was

detonated at DRES. This explosion trial was one of a series cal led Distant

Plain to evaluate whether detonable gases were as good or better than high explosives for simulating a nuclear-blast environment (Refs. 16 and 17). In 1976 a high-explosive trial known as Dice Throw was conducted on the White Sands Missile Range in New Mexico under the auspices of the Defence Nuclear Agency of the United States (Refs. 18 and 19). The charge for event Dice Throw was constructed of 628 tons of ANFO, with an energy equivalent of about 500

tons of TNT, in the form of a 6.86-m-high by 9.08-m-diameter cylinder, capped with a 4.54-m-radius hemisphere.

The progression from using TNT as the explosive to the propane-oxygen mixture and finally to ANFO is worth mentioning. For an equivalent energy a propane-oxygen mixture is much cheaper than TNT, besides having the additional advantage of not providing severe anomalous jetting from the exploding charge.

However, propane-oxygen explosions were abandoned in favor of

ANFO,

which is

also much cheaper than TNT, because the problem with the possibility of a

pre-mature charge detonation was never satisfactorily resolved. Trials from the

time of event Dice Throw to the present, which are conducted about every one to

two years, have all used

ANFO

as the high-energy explosive.

Although the main goal of high-explosive trials was to simulate a nuclear-blast environment for testing military and civil structures and equipment, it was fully realized that another complementary goal was just as important. The physics of explosions had to be understood, and a good

capability had to be developed to predict the physical properties of blast

waves. Consequently, it was important to collect the best possible

measure-ments of the blast-wave flow properties. This measured data would not only help to understand the behavior of blast waves but could also provide a means of checking analytical and numerical predictions of blast-wave flows.

Measured data from explosion trials are rather limited because

instrumentation for field use was neither extensive nor highly developed in the 1950s, 1960s and 1970s. Time-of-arrival measurements of the blast-wave were normally made during each trial, so that the blast-wave trajectory was known as a function of distance and time (e.g., see Refs. 8 and 20). These measurements

were usually quite accurate and were extremely useful in giving the shock

velocity from which all other blast-front properties could be obtained by using the Ranking-Hugoniot relations and the ambient air properties, as mentioned in the introduction. In most field trials the overpressure signature with time was measured with piezoelectric or strain-type transducers (e.g., see Refs. 8 and 14). Prior to 1970 the measured overpressure histories were not very reliable because transducers were sensitive to temperature and vibration and cables introduced noise and capacitive problems. In a couple of past field trials, density signatures with time were also measured, with beta-radiation gauges specially designed for field and shock-tube studies (Refs. 21 and 22). Finally, high-speed photography was used in a few field trials to record the motion of smoke tracers (trails or puffs) as the blast wave moves outward

(Refs. 22 and 23). From the film one can then obtain the time of arrival of the blast-wave front at each smoke tracer and the path of each tracer with time behind the blast-wave front. Since the tracer path is essentially the

fluid-particle path for the small smoke particles used in the trial, the flow velocity behind the blast-wave front can also be obtained.

The measured data from the aforementioned explosions of different explosives, different sizes and also for different ambient pressures and temperatures (see table 1) are important in providing a check for analytical and numerical predictions in this and other work. Before such comparisons are made, however, it is most convenient to reduce the data by some standard means

to obtain a consistent data set for one energy yield at standard atmospheric conditions. This reduction of data is accomplished by using standard

blast-wave scaling laws.

·2.2. Blast-Wave Scaling Laws

The simplest scaling laws for spherical blast waves were formulated originally by Hopkinson in 1915 (Ref. 24). Hopkinson was fully aware that an explosion charge would produce a blast wave with a spatial variation in its flow properties, like that shown at time t in Fig. 2a. The peak overpressure of the shock at radius rs is denoted by ~Ps and the positive-overpressure

length is shown as L. He realized that a different sized charge by a geometric factor ~ for the same explosion would produce a blast wave of the same ampli-tude (~ps) and similar spatial shape at a different time ~t, but distributed spatially over a radius ~r, as shown in Fig. 2b. Hence, the shock has the same amplitude at radius ~rs and the positive-overpressure length is now ~L.

Furthermore, if the positive-overpressure duration at radius r was T for the blast wave from one explosive charge, it would become ~T at radius ~r for the blast wave from another charge of the same explosive but of a different size by the factor ~.

Hopkinson then showed that the spatial distribution for the

properties of blast waves from different sized charges of the same explosive should be identical when plotted as a function of some scaling parameter. If the scaling variable is the charge radius rc' then the scaled radius R should be r/rc. If the scaling variable is the charye weight W (or energy), then a scaling parameter z can be selected to be r/W /3.

Sachs developed more general scaling laws for blast waves (Ref. 25), so that the scaling for different sized charges of the same explosives would include the effects of a different ambient pressure (Pl) and temperature (Tl) cr sound speed (al). He showed that dimensionless groups for pressure p, time

tand impulse I, that is, P/Pl' a t/(E/Pl)1/3 and a 1I/(E/Pl)1/3p1 , should be unique functions of the dimensioniess radius r/(E/pl)1/3, where E is the energy release of the explosive charge. Hopkinson's geometric scaling laws are

included in Sachs's more general results. Sachs's results can be interpreted in a manner similar to Hopkinson's. For example, the shock strength or ratio PS/Pl is the same for different sized charges of the same explosive in

different atmospheres at the nondimensional radius r/(E/pl)1/3 and time

a 1t/(E/Pl)1/3. Note that Sachs's scaling laws have been shown to be accurate for blast waves which are not too close to the exploding charge, where the pressure and temperature are sufficiently low that air behaves essentially like

.a perfect gas (Refs. 8, 26 and 27).

In the present study all experimental data for the shock trajectories from different explosions with different ambient conditions are scaled to the case of a hemispherical explosion of 1 kg of TNT or its equivalent in a

standard atmosphere (pc ~ 101.33 kPa, Tc

=

_{288.15 K, a o}

=

340.29 mis). This was achieved by using Sachs's scaling laws to reduce all measured distances and times by

(2.1) and

(2.2)

respectively, where Sr denotes the distance factor, St denotes the time factor, W is the charge mass in kilograms of TNT or its equivalent, Wo is the standard TNT mass of 1 kg, and the subscripts 1 and 0 designate ambient and standard conditions, respectively.

Measurements of the trajectory of the blast-wave or shock-wave front, in terms of the shock arrival times at fixed distances from the explosion

center, are given in the first two columns of tables 2 to 7 for the previously mentioned explosions of interest in this study. Each set of measurement has been scaled for presentation to the case of a hemispherical explosion of a I-kg charge of TNT or its equivalent in a standard atmosphere. The TNT equivalent for ANFO and propane-oxygen explosions, along with all values of the scaling factors which were used for scaling the data, are listed in table 1. The original values of the time and radius for each shock-front trajectory can be recovered by simply multiplying this scaled data by the appropriate values of Sr and St from table 1.

2.3 Curve Fits for Shock Trajectories

An accurate curve fit for the trajectory of the blast or shock front in terms of the radius versus time (or the overpressure versus distance) has many important advantages. The curve fit can, of course, be used directly to reproduce the shock-front trajectory quickly and conveniently in computational work. However, a major advantage of a curve fit over the original data is that

it can be differentiated easily for subsequent use. Shock-front properties can then be obtained conveniently by using the Rankine-Hugoniot relations and

decay-rate expressions to be given later in this work. Some past curve fits are first reviewed in this section, and a new and bet ter curve fit for the

'

"

shock trajectory is introduced.

The most common form of curve fit for the shock trajectory of a blast wave is

6p

+

+

+

_(2.3)

where ~Ps is the shock overpressure, rs denotes the shock radius and ai are suitable coefficients. For example, see Refs. 7, 14 and 28. This polynomial curve fit has generally been fairly useful in representing shock-trajectory data over a certain limited range of shock overpressure (e.g., 5 to 0.1 MPa). If higher overpressure nearer to the exploding charge are to be represented accurately by this equation, the lower limit of the range is then shifted to much higher pressures, and vice versa. Furthermore, at large radii the shock overpressure is forced to decay like r~l according to the linear acoustic decay law and not slightly more rapidly like r;1(ln{r s})-1/2 for the nonlinear

acoustic decay law (Refs. 29, 30 and 31).

For a blast wave from a pentolite explosion, Goodman (Ref. 32) used the curve fit

(2.4)

to represent his experimental data. In this expression Rs equals rs/rc, the

. shock radius rs normalized by the charge radius rc. Since this curve fit includes the far-field nonlinear acoustic decay, r~1(ln{rs})-1/2, it is supposed to cover the entire range of rs from rc to infinity, even though experimental data at large radii were unavailable to provide a check. Goodman showed that his curve fit (Eq. 2.4) represents shock-overpressure data for pentolite explosions over a much wider range of radius than Eq. 2.3.

For experimental data of the shock trajectory in terms of radius and time, no simple analytical expression could be found to give a satisfactorily accurate fit over the entire range of measurements. Consequently, two differ-ent expressions, each covering an adjacdiffer-ent part of the range, were used to describe the shock trajectory for TNT explosions. In the work of Ref. 14 the data for the shock arrival time versus distance were patched together by

for measuremenls made at small radii corresponding to shock overpressures greater than 350 kPa and

=

for measurements made at larger radii corresponding to lower overpressures. (2.5)

(2.6)

In a more recent work by Dewey and co-workers (Refs. 15 and 16), a curve fit for the shock trajectory for TNT and propane-oxygen explosions,

(2.7)

was used repeatedly. This curve fit was found to be very useful for the time range corresponding to moderate shock-front overpressures between 700 and 30 kPa.

Although each curve fit given by Eqs. 2.5 to 2.7 was very successful within its specified range (limited), it should not be used outside of this range. For example, Eqs. 2.6 and 2.7 have the correct form for the linear acoustic term at large times (~2tS) _{but not the exact result (alt S)' since} ~2

is not set equal to the ambient sound speed al. Consequently, the shock Mach number and other shock-front properties obtained from these curve fits become inaccurate and even meaningless when the shock Mach number (aîldrs/dts)

approaches its linear acoustic limit (~2/al)' which can be greater or less than unity depending on the value of the coefficient ~2. Furthermore, Eqs. 2.6 and

2.7 do not have the correct nonlinear acoustic behavior at large times. The dominant nonlinear term in both cases is ln(t s )' when it should have been

(ln{t s })I/2.

In the present study a new curve fit for the shock radius-time trajectories of blast waves from TNT, ANFO and propane-oxygen explosions was sought, which would be accurate over a wide range of shock overpressures below about 2 MPa and have the correct linear and nonlinear acoustic terms for the far field (large times). The curve fit

(2.8)

was found to be very successful. The coefficients ~i and the exponent n can be determined, essentially, by means of a least-square fit to available' experi-mental data (e.g., first two columns of tables 2 to 7). During this fitting procedure the coefficient al of the second term alt s is forced to be the ambient sound speed for the particular explosion in question, so that the correct linear acoustic term is present. The last term reduces to the correct nonlinear acoustic term at large times, that is, (ln{t s })1/2. It will also give the correct nonlinear acoustic decay for the shock overpressure at large times, that is, t~l(ln{ts})1/2 or r~l(ln{rs})1/2 , as shown in the first part of appendix A. Note that, as time increases and the blast wave becomes very weak (~Ps < 1 kPa) , viscous effects would become important in altering the decay of the shock front. Such low-amplitude blast waves are not of interest in the present study.

Three sets of coefficients for Eq. 2.8 for curve fits to experimental shock-trajectory data for TNT, ANFO and propane-oxygen explosions from tables

2 to 7 (first two columns) are listed for convenient reference in table 8. Additional details concerning these curve fits and an assessment of how well they represent the shock-trajectory data are presented in the next three sections.

2.4

Explosion Data for TNT

numerous different distances from the explosion center were taken from Ref. 14 for the 20-ton TNT explosion trial. Three different sets of data, obtained by the American, British and Canadian teams, were combined and scaled for the case of a surface explosion in a standard atmosphere for a I-kg hemispherical charge of TNT, by using the appropriate sealing factors given in table 1. The result-ing 360 time-radius pairs are presented in the first two columns of table 2. The value of the coefficient a_q was fixed initially, and a least-squares method was then employed to find values for al and a₃ of the equation rs

=

al + _{alt s} + a

3(Zn{1

+

a tS})1/2 that was fitted to data at large distances or low

over-pressures

(1

_{PS /PI} < 0.29). With al'a.₃ and aq fixed, Eq. 2.8 was then fitted

to all the data by using a least-square method with a weighting factor of r , to obtain values for the remaining coefficient a₂ and exponent n. Finally, the process was repeated by judiciously selecting different values of a_q until a good curve fit was obtained, based on a subjective assessment. The

coefficients and exponent for the curve fit for the scaled 20-ton TNT results are given in table 8.

The capability of the curve fit (Eq. 2.8) to represent the

experimental data is demonstrated in two different ways. Firstly, shock radii obtained from the curve fit for each of the measured shock arrival times are listed in the third column of table 2 for comparison with the actual shock radii in the second column. The percentage error between the curve-fit and actual radii are also shown in the fourth column. Secondly, the curve-fit results and measured data are compared also in the time-distance plot given in Fig. 3. From these tabulated and graphical comparisons it is evident that the curve fit is an excellent representation of the measured data over the range of radii from 1 to 14 m, corresponding to a range of shock overpressure ratios

(PS-PI)/PI from 14 to 0.08. Note that the shock Mach number U obtained from

aïldrs/dts from the curve fit and the corresponding shock properties including the nondimensional flow velocity uS/al' sound speed aS/al' density PS/PI and overpressure (PS-PI)/PI from the Rankine-Hugoniot relations are tabulated in the fifth to nineth columns for interest and convenient reference.

Measured shock times of arrival of the blast-wave front at many different distances from the explosion center for two different SOO-ton TNT surface explosion trials called Snowball and Sailor Rat (Ref. 15) were scaled for the case of a surface explosion in a standard atmosphere for a I-kg

hemisherical charge of TNT. The scaling factors that were used are given in table 1, and the scaled data are listed in the first two columns of table 3 for Snowbali and table 4 for Sailor Rat. Measured flow velocities (us) at the blast-wave front are shown also in the third column of tables 3 and 4. These were obtained from an analysis of the motion of smoke tracers in the blast-wave flow that was recorded by high-speed photography.

The capability of the curve fit (Eq. 2.8), with coefficients obtained previously from the scaled 20-ton TNT data, to represent the scaled experi-mental data from the two SOO-ton TNT explosion trials is now demonstrated.

Firstly, shock radii obtained from the curve fit for measured shock arrival times are listed in the fourth column of tables 3 and 4, for comparison with the actual radii in the second column. The percentage error between the curve-fit and actual radii are also given in the fifth column of each tabie.'

Furthermore, the curve-fit results and experimental data for both explosions are compared in the time-distance plot given in Fig. 4. These tabulated and graphical comparisons show that the curve fit is a very good representation of these two sets of data over the combined range from 1 m to 5.5 m, corresponding to a range of shock overpressure ratios of 12 to 0.3. Secondly, flow

veloc-ities for the blast-wave front from the curve fit, along with the percentage error from the measured values, are listed in the sixth and seventh columns of tables 3 and 4. Curve-fit and measured flow velocities are compared also in Fig. 5. These tabulated and graphical are in good agreement, illustrating that the curve fit for the shock trajectory can be used with confidence to obtain the shock-front flow velocity and very likely other shock-front flow properties as weIl.

The fact that only one curve with one set of coefficients is needed to represent the scaled data from the 20-ton and two 500-ton TNT explosions illustrate both the value of Sachs's scaling laws and the good consistency of the scaled experimental data. Perfect scaling of experimental results should not be expected, however, since Sachs's scaling laws do not include real and viscous gas effects and explosions on soft and hard ground would obviously produce slightly different shock trajectories. A slightly better curve fit to each set of data for Snowball and Sailor Hat could have been achieved by

slightly readjusting the coefficients and exponent in Eq. 2.8. This was deemed unnecessary, however, partly because of the good agreement already achieved and partly because of the desire to have only one curve fit with one set of

coefficients to represent all scaled TNT data.

In addition to the good agreement between the curve-fit results and available experimental data for TNT explosions, it is of much interest to show that the curve-fit results are also in good agreement with widely quoted

numerical results of both Brode (Ref. 33) and Lehto and Larson (Ref. 34), and also those compiled in tabular form in Baker's book (Ref. 8, page 156). Brode, as weIl as Lehto and Larson, solved the nonstationary, one-dimensional,

inviscid. gasdynamic equations of motion for the explosion of a bare spherical TNT charge in real air. Their initial conditions were those of a spherical detonation wave in the TNT charge, according to the description of Taylor

(Ref. 35). Brode computed the flow field, in terms of Sachs's sealing

parameters for radius and time, and obtained a solution out to a radius where the blast-wave peak overpressure ratio (PS-Pl)/Pl was as small as 0.06. Lehto and Larson's solution technique was very similar but they continued the

solution to a much larger radius where the overpressure ratio was only 0.0001, since they were able to reduce the smearing effect of artificial viscosity on the blast-wave front. The tabulated results in Baker's book, nondimension-alized in terms of Sachs's sealing parameters, are essentially a smoothed combination of Goodman's measured pentolite data near the charge (Ref. 32), some old TNT data at moderate distances from the charge, and Lehto and Larson's numerically calculated data at larger distances from the charge.

Results from the curve fit, Brode, and Baker are compared in Fig. 6, in terms of the shock overpressure ratio versus distance, for the case of a I-kg TNT surface explosion in a standard atmosphere. In order to present the results of both Brode and Baker as a function of radius from a I-kg surface 6 charge, instead of nondimensional radius r s /(E/pl)l/3, a TNT energy of 4.5x10 J/kg and a reflection factor of 2 were used. Note that the curve-fit results are shown as solid line with a dotted extension for shock overpressure ratios higher than about 20, where it is no longer a good fit to TNT data.

Brode's calculated curve for the shock overpressure lies slightly below both the curve-fit results and Baker's tabulated data. Hence, Brode's prediction of the shock overpressure is slightly lower than the experimental data for TNT explosions. It can be also be seen that Baker's tabulated shock-overpressure data are virtually indistinguishable from the curve-fit

results below a shock overpressure ratio of about 10. For shock overpressure ratios from 0.1 to 0.0001 (not shown in Fig. 6) this good agreement continues between the curve-fit results and Baker's data (which is now really Lehto and Larson's numerical predictions). In this overpressure range (0.1 to 0.0001) the difference between the curve-fit and Lehto and Larson's results is typically less than 1%, with a maximum of 2%. Such good agreement in this range was unexpected, since the curve-fit coefficients were obtained soley by using the scaled shock trajectory from the 20-ton TNT surface explosion trial, for which the lowest shock overpressure ratio was only 0.08. The agreement, however, is very satisfying. It adds much confidence that the linear and nonlinear far-field terms in the curve fit expression to the shock trajectory

(Eq. 2.8) are essentially correct and the curve fit can be used to obtain accurate values of the shock-front flow properties.

2.5 Explosion Data for ANFO

Experimental data for the time of arrival of the blast-wave front at many distances from the explosion center were taken from Refs. 18 gnd 19 for the 628-ton ANFO explosion trial. This set of data was scaled for the case of a surface explosion in a standard atmosphere for a I-kg hemispherical charge of TNT, by using the scaling factors given in table 1. The resulting 80 time-radius pairs are listed in the first two columns of table 5. Measured over-pressures are shown alongside these pairs in the third column.

Equation 2.8 was fitted to this scaled data to obtain the coef-ficients ai and exponent n in a manner similar to that for TNT, with one exception. The coefficients a₃ and a₄ were now kept equal to the previous values for TNT, since it was assumed that the far-field term should remain the same if 628 tons of ANFO and 500 tons of TNT would produce equivalent blast waves away from the charge. The coefficients and exponent for the curve fit for the scaled 628-ton ANFO results are given in table 8.

Shock radii obtained from the curve fit are compared to the actual radii in both table 5 and Fig. 7, in the same manner as for TNT. These

tabulated and graphical data show that the curve fit represents the data quite weIl over the range of radii from 1 to 23 m, corresponding to a range of shock overpressure ratios from 20 to 0.05. Shock overpressures obtained from the curve fit are also compared to the measured values in both table 5 and Fig. 8. The curve fit represents the data quite weIl over most of the range of shock overpressure ratios from 400 to 0.3, but it is obviously below the data in the small range of overpressure ratios from 0.3 to 0.05.

It is not clear why the measured shock overpressures are higher than the curve-fit results at large radii. This implies that 628 tons of ANFO

produces a stronger blast wave than 500 tons of TNT and their assumed far-field equivalency is not necessarily correct. Furthermore, the measured data at large radii are few in number, and the scatter in the data implies that the measurements may have been inaccurate. Consequently, the curve-fit coef-ficients were not adjusted to obtain a better curve fit to this measured data at large radii.

2.6 Explosion Data for a Propane-Oxygen Mixture

blast-wave front at many different distances from the explosion center were obtained from Ref. 16 and 17 and directly from the authors for the 38-m-diameter hemispherical propane-oxygen explosion trial. One set of data was obtained with many mechanical-type shock-front detectors, and the other was obtained with high-speed photography. Bath sets of data were kept separately and scaled for the case of a surface explosion in a standard atmosphere for a

I-kg hemispherical charge of TNT, by using the sealing factors given in table 1. The scaled sets of data for the shock radius and time are given in the first two columns of table 6 for the case of the mechanical detectors and in table 7 for the case of high-speed photography. Measured flow velocities at th.e blast-wave front are shown also in table 7. These were obtained from an analysis of the motion of smoke tracers in the blast-wave flow that was

recorded by high-speed photography. Note that the measured data in table 7 are presented in groups of fours. In each grouping the first to fourth measured value corresponds to high-speed photography measurements for smoke tracers

taken at angles of 5, 10, 15 and 20 degrees above ground level, respectively, for which the vertex of the measured angle is at the explosion center.

Equation 2.8 was fitted to this scaled data to obtain the coef-ficients ai and exponent n in a manner similar to that for TNT, with one exception. The coefficients a₃ and a₄ were kept equal to the previous values for TNT, since it was assumed that the far-field term should remain the same if the 38-m-diameter hemisphere of propane-oxygen mixture and 20-tons of TNT would produce equivalent blast waves away from the charge. The coefficients and exponent for the scaled propane-oxygen results are given in table 8.

Shock radii obtained from the curve fit are compared to the actual radii both in table 6 and Fig. 9 for the case of the mechanical detectors and both in table 7 and Fig.

la

for the case of the high-speed photography. These

tabulated and graphical results illutrate that the curve fit represents the data very weIl over the range of radii from 0.8 to 33 m, corresponding to a range of shock overpressure ratios from

la

to 0.02. Shock flow velocities obtained from the curve fit are also compared to the measured values (high-speed photography) in both table 7 and Fig. 11. The curve fit also represents this data fairly weIl, although it is slightly low in the center portion around a scaled shock radius of 3 m, corresponding to a shock overpressure ratio of 1. Note that the reason for the fairly large scatter in the measured flow-velocity data given in Fig. 11 is not known.

2.7 Comparison of Results from different Explosions

. The curve-fit results for TNT, ANFO and propane-oxygen explosions are now compared, not only for interest but also to illustrate that the scaled results from different explosives may be similar qualitatively but in general are different quantitatively.

Shock trajectories from the curve fit (Eq. 2.8) for TNT, ANFO and propane-oxygen explosions are compared first in Fig. 12, where all results are given in scaled form for a I-kg TNT equivalent surface explosion in a standard atmosphere. These results show that the three trajectories differ fairly significantly at early times (or small radii), but come closer together at later times.

Eq. 2.8.

The behavior of the shock trajectory at later t~mes can be found from

., 1/2

.~

0.7200 m, al

=

0.3403 mIs and a₃

=

1.290 m from table 8 for both TNT and

propane-oxygen explosions, their shock trajectories have the same asymptote and therefore merge at later times. For ANFO, however, one coefficient is

different, that is, al

=

1.000. Hence, the shock trajeetory for ANFO has a slightly different asymptote which is 0.2800 m ahead of the other two

trajectories at large times, and it therefore does not merge exactly with the other two tra~ectories. At large times or radii, however, this slight

difference is negligible.

Curve-fit results for the shock overpressure versus distance for TNT, ANFO and propane-oxygen explosions are finally compared in Fig. 13. These results show that the shock-overpressure curves differ fairly significantly at small radii, but come closer together at large radii. This might have been anticipated from the previous results. The TNT and ANFO curves merge slowly as the radius increases and the overpressure ratio diminishes. Although the

propane-oxygen curve crosses the TNT and ANFO curves and then lies above them, it will eventually merge with them (not shown). This must occur because the three curves have the same asymptote 2ya3/(y+1)alts(ln{a4ts}) 1/2, as mentioned previously and also shown in detail in the first part of appendix A. The

propane-oxygen curve takes longer to merge with the TNT and ANFO curves because the exponent n for the propane-oxygen curve (n

=

0.48) is smaller than that for both the TNT and ANFO curves (n

=

1.00).

The shock overpressure curve for a nuclear explosion in real air is also given in Fig. 13 for comparison. This curve was taken from Ref. 37, and it has been scaled such that nuclear shock overpressures at large radii match the TNT curve. Hence, the equivalence is based on an explosion that produces the same shock overpressure not too near to the explosion center.

The results shown in Figs. 12 and 13 illustrate clearly that scaled explosion results for different explosives do not reduce to one unique curve. Hence, no single shock trajeetory can be used for all different types of

explosives. However, different explosives, each of a certain size or mass, do produce equivalent blast waves at relatively large distances from the explosion center, where their shock trajectories and amplitudes become almost indistin-guishable. For TNT, ANFO, propane-oxygen and nuclear explosions, this occurs at radii larger than about 5 m for a I-kg TNT equivalent charge, or for shock overpressure ratios smaller than about 0.6. This also corresponds to the case when the blast-wave front has engulfed a mass of air equal to about 20 times the mass of the charge. Consequently, the terms equivalent charges and equivalent explosions refer to the equivalent amount of different explosives that would produce the same blast wave at large radii. This is the reason why 500 tons of TNT are supposed to be equivalent to 628 tons of ANFO, and 20 tons of TNT are supposedly equivalent to a 38-m-diameter hemisphere of a stoichio-metrie mixture of propane and oxygen.

3.0 ANALYSIS AND ANALYTICAL RESULTS FOR THE INITIAL DECAY RATES

Consider the case of a planer, cylindrical or spherical shock or blast wave that moves into a quiescent gas with known conditions (e.g.,

pressure PI' density PI and sound speed al). Let the shock-front trajectory be specified as a function of distance or radius rand time t. In functional notation, therefore,

and the velocity U of the shock front is simp1y

drl

dt s •

u

= _(3.2)

The vertica1 1ine with the subcript s indicates that the differentiation is to be taken a10ng the shock trajeetory. The shock trajeetory r(t) can be given by

Eq. 2.8, for example, which is a curve fit to experimenta1 shock time-of-arriva1 data.

In the preceding equations the symbols r, tand U denote nondimen-sional shock distance r*/Sr' time t*/St and velocity U*/a 1 , respective1y. In this report the dimensiona1 variables are identified by the subcript *, the dimensiona1 ambient or reference conditions ahead of the shock or blast wave are denoted by the subcript 1, and Sr and St are sca1ing factors for radius and

time. Furthermore, the shock distance r, time tand other flow properties will

not have a subcript s, as they had in the previous sections, because this notation is too cumbersome here.

The gas is assumed to be both thermally and calorical1y perfect for

the present work. Consequent1y, once the instantaneous shock velocity or shock

Mach number U is known, the flow properties just behind but at the shock front

can be obtained by using the following Rankine-Hugoniot equations (Refs. 1, 2

and 3): p 1 + (U2 - 0/0.8, (3.3) p (1 + o.p)/(a+p), (3.4) a

=

[p(o. + p)/(1 + o.p)]1/2

=

[p/p]1/2, _(3.5) u (p - 0/[y28 (1 + o. p)]1/2 (3.6) s (y - 0- 1 Zn(p/pY) • (3.7)

In these expressions the nondimensiona1 shock pressure p, density p~ sound

speed a, flow velocity u and entropy s are P*/P1' P*/P1' a*/a 1, u*/a 1 and (s* - sl)/R, respectively, where R is the gas constant. The symbol Y denotes the specific heat ratio, a equals (y+l)/(y-l) and 8 equa1s (y-l)/2y. Note that

all other shock properties 1ike nondimensiona1 temperature T

=

a 2, flow Mach

number M

=

u/a and dynamic pressure q

=

pu 2/2 can be obtained direct1y from

Eq. 3.3 to 3. 7.

If the shock-front trajeetory is known as a function of shock pressure versus distance, that is,

p = per) , (3.8)

instead of distance versus time, the above procedure for obtaining the shock properties is then altered, but on1y slight1y. Once the shock pressure is

obtained at a certain distance from Eq. 3.8, the corresponding shock properties

fo11ow direct1y from Eqs. 3.4 to 3.7. Furthermore, when Eqs. 3.2 and 3.3 are

combined to give

t

J

[8 (1

+

a p)] -1/2

dr,

(3.9)

,

.

the shock trajectory is then obtained as a function of distance and time. Time and distance derivatives of the flow properties along the shock trajectory are needed in subsequent work. The first derivatives with respect to time follow directly from Eqs. 3.3 to 3.7 by differentiation, and they are listed below. = dP

!

dt s 2

u

dU! 0'.

S

dt s (3.10) 0'. - _{P d}P_! (3.11) 0'. + P dt s

d

P

!

dt S 2 dP! 0'.+2p-O'.a (3.12) 2 a (1 + 0'. p) dt s

dal

dt s

dU!

dt s u (0'. P + 0'. + 2) dP! 2 (p - 1) (1 + 0'. p) dt s (3.l3)

dS!

dt S 1 (1 Y(O'. - p») dP! Y - 1

P -

P (0'. + p) dt s (3.14)

The first derivatives with respect to distance are not listed, because they follow directly from

d\jJ! dr S

(3.15)

whp.re the variabie \jJ is one of the flow properties. Higher-order time and distance derivatives can also be obtained readily by further differentiation.

It has been demonstrated that the flow properties just behind the

shock front, along with their first time and distance derivatives along the shock trajectory, can be obtained fairly easily when the shock trajectory is known as a function of distance versus time or overpressure versus distance. This shock-front information is needed in the following analysis for the decay rates of flow properties just behind the shock front.

The flow properties of a shock or blast wave are governed by weil known partial differential equations for mass, momentum and energy. For a nonstationary, one-dimensional, inviscid flow of a gas these equations are

(Re f s. 1 and 3 6) :

+

dU

at

+

dS

at

+

dU u

-ar

as

u -dr dU

Par

+

1

ap

+

y P

ar

o ,

Npu r

o ,

o ,

(3.16) (3.17) (3.18)

where N takes the value of 0, 1 and 2 for planer, cylindrical and spherical flows, respectively. Additional relations are required to close the above set of equations. In the present work the gas is assumed to be both thermally and calorically perfect. Consequently, the nondimensional equation of state and

the nondimensional sound-speed relation are p pT = and

=

_y

1 rdp)

_{dP s} p Y exp [ C y - 1) s] T = 1:. p (3.19) (3.20)

When Eq. 3.20 is differentiated with respect to both time and distance,

da a dp a dP

at

2 p

at

2

pat'

(3.21)

and

da a dp a dP

dr 2 p dr 2

p

dr ' (3.22)

are obtained for future reference.

Equations 3.16 to 3.20 apply across and behind the assumed discon-tinuous shock or blast front. Just behind but at the shock front the partial derivatives with respect to time and distance of a general flow property ~ are related fo the time derivative of the property ~ taken along the shock

trajectory, according to the mathematical equation

d~\

dt s

+

u

d~ •

dr (3.23)

Since U is the shock velocity and not the flow velocity, this should not be confused with the substantial, material or particle derivative.

Equations 3.16, 3.17, 3.18, 3.21 and 3.22, along with five more from Eq. 3.23 for ~ equal to p, P, a, u and s, comprise a set of ten algebraic equations in terms of ten partial time and distance derivatives of p, P, a, u and s. Cramer's rule can be applied to obtain explicit expressions for each partial derivative. The final results are listed below.

dp

at

au

at

=

+ yNpuU Cu - u) r (3.24)

[a2 _{+uCU-u)] dP \ + PudU\ + CY-1)PUdS\ + NpuUCU-u)}

dt s dt s y(U - u) dt s r

(3.25)

a2 - Cu - u) 2

Cy -l)aU dU\ _ Cy -1)a3U ds \ 2 dt s 2yCU - u) dt s [a2 +uCU-U)]dU\

+~dP\

dt s YP dt s a2 - Cu - u) 2 + Na2uU r + Cy - 1) NauU Cu - u) 2r (3.26) (3.27)

dS

at

dp dr dP dr da

ar

dU

ar

dS dr = u dSI

u -

u dt s (U _ ) dp I _{u dt}

+

_{YP dt} du I

+

yNpu (U - u) _r s s

(u-u) dPI

+

pdul

+

(Y-OPdSI

+

Npu(U-u)

dt s dt s y(U - u) dt s r a2 - (U - u) 2 (U _ U) da I

+

(y - 1) a du I dt s 2 dt s (U _ u) du I

+

_1 dp I dt s yp dt s 2 + Na u r (Y-1) a3ds l 2y(U - u) dt s

+

(Y - 1)Nau(U - u) 2r (3.28) (3.29) (3.30) (3.30 (3.32) (3.33)

These partial time and distance derivatives of the flow properties just behind the shock front are the initial time and distance decay rates of the shock proper ties along constant distance and time lines, respectively. These decay rates are expressed explicitly in terms of the shock properties and their derivatives along the shock trajeetory. These latter properties and their derivatives, which appear on the right-hand side of the expressions, all depend on the shock trajeetory, which can be specified by means of a curve fit to experimental data.

Once the partial time and distance derivatives of a general flow variable ~ are known, the initial time decay rate of ~ along aparticle path can then be obtained by employing the substantial derivative

d~l

dt p

d~

at

+

u d~ _dr • (3.34)

The vertical line with the subcript p signifies that the differentiation is to be taken along the partiele path. The resulting expressions for the shock properties along aparticle path are listed below.

a2 dp I

+

yp(U - u ) -dul

+

yNpu(U - u) 2

dpl dt s dt s r

dt p _{a 2 - (U - u) 2} (3.35)

a 2 dp

I

+

p(u - u) -dul

+

(y-1)p~1

+

Npu(U - u) 2

dPI = dt s dt s y dt s r

dal dt p dul dt p = 2 dal +

a dt s (y - l)a(U - u) dul _ (y - 1)a

3 _ds

I

2 dt s 2y dt s a2 - (U - u) 2 a2 - (U - u) 2 + (y - 1)Nau(U - u) 2 2r (3.37) (3.38) (3.39)

Note that the corresponding distance derivatives of the shock properties along aparticle path follow directly from

d1jJ1

dr p

=

(3.40)

and they are not listed here for brevity.

The time decay rate of the shock properties along both a positively sloped characteristic line (c+) and negatively sloped characteristic line (c-)

·can also be obtained in a manner similar to that for the partiele path. By using

dljll

dt c± =

a

Ijl

at

+

(u ± a)

aljl

_ar

(3.41)

and EqS. 3.24 to 3.33, the following final results are obtained.

dpl dt c±

dPI

dt c± dal dt c± dul dt c±

=

[a 2

+

a (U - u)]

~~

I

s [ a 2 + a (U - u) ] dda

I

t s

dul

+ yp(U-u+a) dt s a2 - (U - u) 2 + yNpu (U - u) (U - u + a) r

p(U-u+a) dul + (y-1)p(U-u+a) dsl dt s y(U - u) dt s Npu(U-u)(U-u+ a)

+

r[a2 - (U - u)2] (3.42) (3.43)

+ (y-1)a(U-u+a) dul _ (y-1)a3(u-u+a)~1

2 dt s 2y(U - u) dt s

+

+

(U - u + a) dp

I

yp dt s a2 - (U - u) 2 a2 - (U - u) 2 (y - 1) Nau (U - u) (U - u + a) 2r[a 2 - (U - u)2] (3.44) (3.45)

dS!

dt c± ± a

dS!

U - u dt S (3.46) Furthermore, by using

d\jJ!

dr c±

=

1

d\jJ!

u ± a dt c± ' (3.47)

the corresponding distance derivatives of the shock properties along positive and negative characteristic lines can be obtained.

Equations 3.24 to 3.47 are a fairly complete set of expressions for

the initial time and distance decay rates of most major shock properties (p,

P,

a, u, s) not only along constant time and distance lines but also along

positive and negative characteristic lines and a fluid particle path. The initial decay of other shock properties like the nondimensional temperature

(T = a2) , flow Mach number (M

=

u/a) and dynamic pressure (q = pu2/2) can also

be obtained from the present results. For example, the initial time decay rate

of the Mach number and dynamic pressure along aparticle path are

dM! 1 dU! M dal (3.48) dt p a dt p a dt p and dq

!

dt p pu -

dU!

dt p

+

.9. d P

_{! '}

P

dt p (3.49)

respectively. Finally, it is worth mentioning that higher-order derivatives of

the shock properties can also be obtained. However, these results are not of interest in the present work.

4.0 VALIDATION OF THE INITIAL DECAY RATE EXPRESSIONS

All of the analytical expressions for the initial decay rates of the shock flow properties along constant time and distance lines, positive and negative characteristic lines and a fluid-particle path were written into a computer program, so that calculated decay rates can now be obtained readily

for any given shock trajectory. In order to eliminate possible errors made

both in deriving the decay rate expressions and in writing the computer

program, a means of checking the computed decay rates is needed. It should be

obvious that one of the best methods of validating the decay rate expresions and checking the computer code is to use known, exact solution to planar,

cylindrical and spherical shock- or blast-wave flow problems. The shock

trajectory from the known solution can be used to calculate the initial decay

rates with the computer program. These computed decay rates can then be

compared to decay rates obtained directly or indirectly from the known flow-field solution j ust behind the shock traj ectory, thereby providing the desired verification.

Of the few known solutions to blast-wave flows that are of use for the present validation, the self-similar solutions of von Neumann for a point-souree explosion without counterpressure (Ref. 11) and Taylor for a shock-wave flow in front of an expanding spherical piston (Ref. 12) are the most useful. In each case the solution is extended to include planar and cylindrical geomet-ries, so that additional verifications for these geometries are also possible.