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The shape
of the blast wave: studies of the Friedlander equation
John M. Dewey
Dewey McMillin & Associates, 1741 Feltham Road, Victoria, BC V8N 2A4, Canada
Professor Emeritus, Dept.
Physics & Astronomy, University of Victoria, BC Canada
ABSTRACT
The time-histories of pressure and other physical properties of centred blast waves, have a characteristic shape. Friedlander (1946) suggested a simple relationship, now known as the Friedlander equation, which accurately describes this shape over a wide range of shock strengths. Introduction of an additional coefficient allows the modified Friedlander equation to describe blast wave time histories over an even wider range.
This paper explores the ranges over which the modified and un-modified Friedlander equations can be used to describe the various physical properties of blast waves. It also shows the application of these equations to blast wave radius-profiles at various times after detonation.
A number of the unique characteristics
of the Friedlander equation are explored. It has long been suggested
that the simplicity of the equation indicates a fundamental relationship
with the explosive process. This possibility is investigated,
and an explanation is presented, relating the Friedlander equation to
the spherical piston path that drives the blast wave from a centered
explosion.
The Friedlander Equation
The time histories of hydrostatic overpressure and several other physical properties of centered blast waves, as measured by gauges at fixed locations, have a characteristic shape. At the arrival of the primary shock there is a virtually instantaneous increase in pressure, followed by a modified exponential-type decay such that after a time, t+, the pressure returns to the ambient value and moves into the negative phase. Friedlander (1946) suggested that this characteristic shape could be described by an equation of the form
, ( 1 )
where P is the overpressure at a fixed location, PS is the peak overpressure immediately behind the primary shock, t is the time after the arrival of the primary shock at that location and t+ is the positive duration. This simple two-coefficient relationship has long been known to be an excellent descriptor of overpressure time-histories produced by a variety of explosives over a wide range of peak overpressures.
This is illustrated in Fig.
1, which shows, in red, the overpressure time history measured using
a piezo-electric transducer at a distance of 523.5 m from the centre
of a surface-burst, hemispherical, 2.205 ktonne ammonium nitrate –
fuel oil explosion (MINOR UNCLE) (Slater et al, 1995).
This gauge record has been chosen for illustrative purposes because
independent measurements were also made of the density-time and the
total-pressure time-histories at the same location. It is rare,
if not unique, to have such independent gauge measurements of several
physical properties at the same location in the same blast wave.
Figure 1.
Measured pressure time history, in red, at a distance of 523.5 m from the centre of a surface-burst, hemispherical, 2.205 ktonne ammonium nitrate – fuel oil explosion (Slater et al, 1995). The black curve is the least squares fit of these data to the Friedlander equation (1).
Fig. 1 clearly illustrates the excellent way in which the Friedlander equation can be used to describe the pressure time-history, at that pressure level, and is the preferred way of determining the peak pressure and the positive duration from a noisy gauge signal.
The density and total (pitot)
pressure time-histories were also measured at the same location, and
from the hydrostatic and total pressures the dynamic pressure (½ρu2)
time-history was calculated. Each of these time histories are also fitted
by least squares to the Friedlander equation, and the results shown
in Figs 2, 3 and 4. The manner in which the Friedlander equation
is able to describe these time histories, using only two coefficients,
PS and t+, is typical
for this strength of blast wave produced by a variety of high explosives,
both air- and surface-burst.
Figure 2.
Density time-history, in red, measured by Slater et al (1995) using a β absorption gauge. The black curve is the least squares fit of these data to the Friedlander equation (1).
Figure 3.
Total (Pitot) pressure, in
red, measured by Slater et al (1995). The black curve is
the least squares fit of these data to the Friedlander equation (1).
Figure 4.
Dynamic pressure (½ρu2), in red, calculated from the hydrostatic and total pressures measured by Slater et al (1995). The black curve is the least squares fit of these data to the Friedlander equation (1).
The Modified Friedlander equation
At peak overpressures above about one atmosphere, the Friedlander equation is no longer able to describe accurately the pressure time-histories, and it is necessary to introduce another coefficient, α, in a modified version of the equation, viz.
. (2)
(It may be noted that some authors use β/t+ in place of α in the modified Friedlander equation. If the same data are fitted to the two equations, it will be found that, and that the fitted coefficients PS and t+ are not identical in the two cases. It is recommended that the form of (2) be used as the modified Friedlander equation.)
Fig. 5 shows an attempt to fit the Friedlander equation to the pressure time-history, derived from the AirBlast1
data base for the free-field
explosion of 1 kg TNT at a distance of 1.3m (3.7 atm). The green
curve shows the same data fitted to the modified Friedlander equation (2).
The modified Friedlander equation appears to provide a good description
of overpressure time-histories up to a peak hydrostatic overpressure
level of about 7 atm, and is probably the best method of obtaining the
peak pressure and positive duration from a noisy gauge signal, as illustrated
in Fig. 5.
Figure 5
. The data points, in red, show the hydrostatic overpressure time history derived from AirBlast at 1.3 m from a free-field 1 kg TNT explosion. The black curve is the Friedlander (1) fit to those data. The green curve is the modified Friedlander (2) fit to the same data.
Some properties of the friedlander equation
In addition to its value as a descriptor of the pressure time-history of the blast wave from a free-field or surface-burst explosion in the region below one atmosphere peak hydrostatic overpressure, the Friedlander equation has some other interesting and valuable properties. The impulse during the positive phase can be found by integrating the Friedlander equation over that period, viz.
. (3)
The result of this simple relationship shows excellent agreement with the positive impulse calculated by the numerical integration of gauge signals.
It may be of interest to note that the total impulse of a Friedlander blast wave is zero, viz.
, (4)
so that the impulse of the positive phase equals that of the negative phase. In practice, this does not occur because of the arrival, in the negative phase, of the secondary and subsequent shocks.
Thornhill (1959) points out another interesting property of the Friedlander equation. The relaxation time, t*, of an exponential decay is defined as the time for the property to decay to 1/e of its peak value. Appling this in the Friedlander equation gives
. (5)
Therefore,
, (6)
and the solution of (6) is
. (7)
This can be a useful relationship when interpreting a noisy gauge signal from which it is difficult to make an accurate measurement of the positive duration. The relaxation time, t*, may be easier to measure, and can be used in (7) to calculate t+. Similarly, by combining (3) and (7), the positive impulse can be calculated from
. (8)
Thornhill (1956) suggests that the valid region for application of the Friedlander equation can be extended above the peak hydrostatic overpressure level of one atmosphere by replacing t+ in the Friedlander equation (1) by 2.31t* from (7), but this is not correct because the two equations are algebraically identical.
The time of the minimum pressure in the negative phase of a blast wave that conforms to the Friedlander equation can be found by differentiating the equation and equating it to zero. This gives
, (9)
which gives the minimum overpressure in the negative phase as
. (10)
Blast wave distance profiles
The pressure time-history, measured at a fixed location in a blast wave, is a somewhat inadequate descriptor of the wave in that it is an Eulerian measurement of a Lagrangian process. In a uniform, non-decaying wave the time-history of a process can be used to describe the wave profile, but this is not the case for a decaying wave such as a blast wave. The wave profile of a blast wave, namely the variation of a physical property with distance at a fixed time, can be measured by photogrammetric techniques such as the particle trajectory analysis (PTA) method described by Dewey (1971) and Dewey & McMillin (1981, 1987). This technique was use to construct the AirBlast data base, from which blast wave distance profiles can be obtained.
Fig. 6 shows, in red, the hydrostatic overpressure wave profile, obtained from AirBlast, at 2 ms after the free-field detonation of 1 kg TNT. The black curve is the Friedlander fit to these data, where the Friedlander equation as been rewritten in terms of distance rather than time, viz.
, (11)
where r = RS – R, RS is the radius of the primary shock, R the radius of any other point in the wave profile, and r+ is the distance behind the primary shock where the overpressure, P, first returns to its ambient value.
Figure 6.
Hydrostatic overpressure wave profile at 2 ms after a free-field detonation of 1kg TNT. The red points are derived from AirBlast, and the black curve is the least squares fit to the Friedlander equation (11).
It will be noted that the peak
overpressure is above 1 atm, the upper limit of the Friedlander equation
for time histories. In the case of wave profiles, the Friedlander
equation is valid for peak overpressures above 1 atm, but not below.
This is illustrated in Fig. 7.
Figure 7.
Hydrostatic overpressure wave profile at 10 ms after a free-field detonation of 1kg TNT. The red points are derived from AirBlast. The black and green curves are, respectively, the least squares fits to the Friedlander and modified Friedlander equations
Wave profiles with peak overpressures below 1 atm can no longer be described by the Friedlander equation, but are well described by the modified equation, written as
, (12)
where r and r+ are defined above, and PS, α and r+ are the fitted coefficients.
discussion
Several authors, including Thornhill (1956), noting that the simple Friedlander equation, with only two fitted coefficients, is such an excellent descriptor of the blast waves produced by a variety of free-field and surface-burst explosives, have asked if there may not be a more fundamental relationship between the physical processes of an explosion and this equation, i.e. that it is not merely an empirical relationship.
Taylor (1946) suggested that
any centered blast wave can be produced by a spherical piston with an
appropriate radius time trajectory. That trajectory can be determined
experimentally by high speed photogrammetry of a flow tracer, i.e.
a smoke puff or trail established close to an explosive source shortly
before initiation. This technique has been used extensively during
the past fifty years to reconstruct the blast wave properties produced
by high explosives ranging from a few kilograms to several kilotons.
Fig. 8 shows the particle trajectory used to reconstruct the blast wave
from a hemispherical surface-burst explosion of 2,445 tons of ANFO (Operation
MISERS GOLD).
Figure 8.
The upper curve is the trajectory of a smoke tracer originally formed at about 150 m from the center of a 2,445 ton ANFO explosion. The points on the curve were measured and used as the spherical piston path to reconstruct the physical properties of the blast wave.
The piston-path points in Fig.
8 have been rotated and fitted by least squares to the Friedlander equation,
as shown in Fig. 9. It is now clear that the spherical piston
which drives a centered blast wave has the form of the Friedlander equation,
and thus, it should not be unexpected that the resulting wave profiles
and time histories also have that form. Future work will investigate
the form to be expected from an expanding pressurized sphere.
Figure 9.
Points from the spherical piston path of MISERS GOLD fitted by least squares to the Friedlander equation.
In order to obtain a suitable orientation for the points shown in Fig. 9, the origin of the piston path was selected as the radius and time when the primary shock first reached the smoke tracer, and the resulting radii were subtracted, arbitrarily, from 80 m. The “ambient” line was determined as an additional coefficient in the Friedlander fit, which then appears as
, (13)
where R is the inverse radius defined above, and R0, Rmax and t+ are the fitted coefficients.
acknowledgements
The author gratefully acknowledges Doug McMillin, who first observed the Friedlander form of the spherical-piston path.
REFERENCES
Dewey, J.M., 1971, The properties of blast waves obtained from an analysis of the particle trajectories, Proc. Roy. Soc., A 324, 275-299.
Dewey, J. M. & McMillin, D. J., 1981, An Analysis of the Particle Trajectories in Spherical Blast Waves reflected from Real and Ideal Surfaces, Can. J. Phys., 59, 10, pp 1380-1390.
Dewey, J. M. & McMillin, D. J., 1987, Numerical Reconstruction of the Flow Field in Experimentally Observed Mach Stem Blast Waves, Proc. 10th Int. Symp. Military Appl. Blast Simulation, MABS 10, Vol. 1, pp 287-299.
Freidlander, F. G, 1946, The diffraction of sound pulses. I. Diffraction by a semi-infinite plate, Proc. Roy. Soc. Lond. A, 186, 322-344.
Slater, J. E., Boechler, D. E. and Edgar, R. C., DRES Measurement of free-field airblast, Minor Uncle Symp. Rept., Defense Nuclear Agency, POR 7453-4, Vol. 4, 2, 1-98.
Taylor, G. I., 1946, The Air Wave Surrounding An Expanding Sphere, Proc. Roy. Soc. Lond., A, 186, 273-292.
Thornhill,
C. K., 1959, The shape of a spherical blast wave, Armament Research
and Development Establishment (ARDE) Memo. (B) 41/59, HMSO, London.
1 AirBlast is an interactive data base of blast wave properties provided by Dewey McMillin & Associates.
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