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The Formation of a Blast Wave by a

very Intense Explosion

Part I and Part II

Sir Geoffrey Ingram Taylor (* 7 March 1886, † 27 June 1975)

Proceedings of the Royal Society A

1950

THE ROYAL SOCIETY

6-9 Carlton House Terrace, London SW1Y 5AG

2

Contents

1 The formation of a blast wave by a very intense explosion I. Theoretical discussion

1.1 Summary and Introduction . . . . . . . . . . . . . . . . . . . . . .

1.2 Shock-Wave Conditions . . . . . . . . . . . . . . . . . . . . . . . .

1.3 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4 Numerical Solution for γ = 1.4 . . . . . . . . . . . . . . . . . . . .

1.5 Approximate Formulae . . . . . . . . . . . . . . . . . . . . . . . .

1.6 Blast Wave Expressed in Terms of The Energy of the Explosion

1.7 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.8 Velocity of Air and Shock Wave . . . . . . . . . . . . . . . . . . . .

1.9 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.10 Heat Energy Left in the Air After it has returned to Atmospheric

Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.11 Comparison with high Explosives . . . . . . . . . . . . . . . . . .

1.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 The formation of a blast wave by a very intense explosion. II. The

atomic explosion of 1945

2.1 Summary and Introduction . . . . . . . . . . . . . . . . . . . . . .

2.2 Comparison with Photographic Records of the First Atomic Explosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3 Calculation of the Energy Released by the Explosion . . . . . . .

2.4 Energy of the First Atomic Explosion in New Mexico . . . . . . .

2.5 An Alternative Possibility . . . . . . . . . . . . . . . . . . . . . . .

2.6 Some Dynamical Features of the Atomic Explosion . . . . . . . .

2.7 The initial rate of rise of air from the seat of the explosion . . . .

2.8 Distribution of Air Density after the Explosion . . . . . . . . . .

2.9 Calculation of the Rate of Rise of the Heated Air . . . . . . . . .

2.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

4

7

8

9

10

14

15

15

16

17

21

25

27

27

28

30

32

36

37

38

43

47

48

1 The formation of a blast wave by a very

intense explosion I. Theoretical discussion

1.1 Summary and Introduction

This paper was written early in 1941 and circulated to the Civil Defence Research Committee of the Ministry of Home Security in June of that year. The

present writer had been told that it might be possible to produce a bomb in

which a very large amount of energy would be released by nuclear fission-the

name atomic bomb had not then been used-and the work here described

represents his first attempt to form an idea of what mechanical effects might

be expected if such an explosion could occur. In the then common explosive bomb mechanical effects were produced by the sudden generation of a

large amount of gas at a high temperature in a confined space. The practical

question which required an answer was: Would similar effects be produced

if energy could be released in a highly concentrated form unaccompanied by

the generation of gas? This paper has now been declassified, and though it

has been superseded by more complete calculations, it seems appropriate to

publish it as it was first written, without alteration, except for the omission of

a few lines, the addition of this summary, and a comparison with some more

recent experimental work, so that the writings of later workers in this field

may be appreciated.

An ideal problem is here discussed. A finite amount of energy is suddenly

released in an infinitely concentrated form. The motion and pressure of the

surrounding air is calculated. It is found that a spherical shock wave is propagated outwards whose radius R is related to the time t since the explosion

started by the equation

2

1

−1

R = S(γ)t 5 E 5 ρ0 5 ,

where ρ0 is the atmospheric density, E is the energy released and S(γ) a calculated function of γ, the ratio of the specific heats of air.

The effect of the explosion is to force most of the air within the shock front

into a thin shell just inside that front. As the front expands, the maximum

pressure decreases till, at about 10 atm., the analysis ceases to be accurate.

At 20 atm. 45 % of the energy has been degraded into heat which is not

available for doing work and used up in expanding against atmospheric pres-

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

sure. This leads to the prediction that an atomic bomb would be only half as

efficient, as a blast-producer, as a high explosive releasing the same amount

of energy.

In the ideal problem the maximum pressure is proportional to R−3 , and comparison with the measured pressures near high explosives, in the range of

radii where the two might be expected to be comparable, shows that these

conclusions are borne out by experiment.

The propagation and decay of a blast wave in air has been studied for the case

when the maximum excess over atmospheric pressure does not exceed 2 atm.

At great distances R from the explosion centre the pressure excess decays as

in a sound wave proportionally to R−1 . At points nearer to the centre it decays

more rapidly than R−1 . When the excess pressure is 0.5 atm., for instance,

a logarithmic plot shows that it varies as R−1.9 . When the excess pressure

is 1.5 atm. the decay is proportional to R−2.8 . It is difficult to analyse blast

waves in air at points near the explosion centre because the initial shock

wave raises the entropy of the air it traverses by an amount which depends

on the intensity of the shock wave. The passage of a spherical shock wave,

therefore, leaves the air in a state in which the entropy decreases radially so

that after its passage, when the air has returned to atmospheric pressure,

the air temperature decreases with increasing distance from the site of the

explosion. For this reason, the density is not a single-valued function of the

pressure in a blast wave. After the passage of the blast wave, the relationship between pressure and density for any given particle of air is simply the

adiabatic one corresponding with the entropy with which that particle was

endowed by the shock wave during its passage past it. For this reason, it is

in general necessary to use a form of analysis in which the initial position of

each particle is retained as one of the variables. This introduces great complexity and, in general, solutions can only be derived by using step-by-step

numerical integration. On the other hand, the great simplicity which has

been introduced into two analogous problems, namely, the spherical detonation wave (Taylor 1950) and the air wave surrounding a uniformly expanding

sphere (Taylor 1946), by assuming that the disturbance is similar at all times,

merely increasing its linear dimensions with increasing time from initiation,

gives encouragement to an attempt to apply similar principles to the blast

wave produced by a very intense explosion in a very small volume.

It is clear that the type of similarity which proved to be possible in the two

above mentioned problems cannot apply to a blast wave because in the latter

case the intensity must decrease with increasing distance while the total energy remains constant. In the former, the energy associated with the motion

increased proportionally to the cube of the radius while the pressure and

velocity at corresponding points was independent of time. The appropriate

similarity assumptions for an expanding blast wave of constant total energy

5

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

are

pressure, p/p0 = y = R−3 f1 ,

density, ρ/ρ0 = ψ ,

3

radial velocity, u = R 2 φ1 ,

(1.1)

(1.2)

(1.3)

where R is the radius of the shock wave forming the outer edge of the disturbance, p0 and ρ0 are the pressure and density of the undisturbed atmosphere.

If r is the radial co-ordinate, η = r/R and f1 , φ1 and ψ are functions of η. It

is found that these assumptions are consistent with the equations of motion

and continuity and with the equation of state of a perfect gas.

The equation of motion is

∂u

∂u

p0 ∂y

+u

=−

.

∂t

∂r

ρ ∂r

(1.4)

Substituting from (1.1), (1.2) and (1.3) in (1.4) and writing f10 , φ01 for

3

p0 f10

0

− 52 dR

−4

0

φ1 + ηφ1 · R

+ R · φ1 φ1 +

= 0.

−

2

dt

ρ0 ψ

∂

f, ∂φ,

∂η 1 ∂η 1

(1.5)

This can be satisfied if

3

dR

= AR− 2 ,

dt

(1.6)

where A is a constant, and

3

p0 f10

0

−A ·

φ1 + ηφ1 + φ1 φ01 +

= 0.

2

ρ0 ψ

(1.7)

The equation of continuity is

∂ρ

∂ρ

+u +ρ·

∂t

∂r

∂u 2u

+

∂r

r

= 0.

Substituting from (1.1), (1.2), (1.3) and (1.6), (1.8) becomes

2

0

0

0

−Aηψ + ψ φ1 + ψ · φ1 + φ1 = 0 .

η

The equation of state for a perfect gas is

∂

∂

+u

· pρ−γ = 0 .

∂t

∂r

(1.8)

(1.9)

(1.10)

6

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

where γ is the ratio of specific heats.

Substituting from (1.1), (1.2), (1.3) and (1.6), (1.10) becomes

A · (3f1 + ηf10 ) +

rf1 0

ψ · (−Aη + φ1 ) − φ1 f10 = 0 .

ψ

(1.11)

The equations (1.7), (1.9) and (1.11) may be reduced to a non-dimensional

form by substituting

f = f1 · a2 /A2 ,

φ = φ1 /A ,

(1.12)

(1.13)

where a is the velocity of sound in air so that a2 = γp0 /ρ0 . The resulting equations, which contain only one parameter, namely, γ, are

1 f0 3

− φ,

γψ

2

0

0

ψ

φ + 2φ/η

=

ψ

η−φ

φ0 (η − φ) =

3f + ηf 0 +

γψ 0

f (−η + φ) − φf 0 = 0 .

ψ

(1.7a)

(1.9a)

(1.11a)

Eliminating ψ 0 from (1.11a) by means of (1.7a) and (1.9a) the equation for

calculating f 0 when f , φ, ψ and η are given is

1

2

0

2

f (η − φ) − f /ψ = f · −3η + φ 3 + γ − 2γφ /η .

(1.14)

2

When f 0 has been found from (1.14), φ0 can be calculated from (1.7a) and

hence ψ 0 from (1.9a). Thus if for any value of η, f , φ and ψ are known their

values can be computed step-by-step for other values of η.

1.2 Shock-Wave Conditions

The conditions at the shock wave η = 1 are given by the Rankine-Hugoniot

relations which may be reduced to the form

7

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

ρ1 γ − 1 + (γ + 1)y1

=

,

ρ0 γ + 1 + (γ − 1)y1

U2

1

· {γ − 1 + (γ + 1)y1 } ,

=

2

a

2γ

u1

2(y1 − 1)

=

,

U

γ − 1 + (γ + 1)y1

(1.15)

(1.16)

(1.17)

where p1 , u1 and y1 represent the values of ρ, u and y immediately behind the

shock wave and U = dR/dT is the radial velocity of the shock wave.

These conditions cannot be satisfied consistently with the similarity assumptions represented by (1.1), (1.2) and (1.3). On the other hand, when y1 is large

so that the pressure is high, compared with atmospheric pressure, (1.15),

(1.16) and (1.17) assume the approximate asymptotic forms

ρ1 γ + 1

=

,

ρ0 γ − 1

2γ

U2

=

y1 ,

2

a

γ+1

u1

2

=

.

U

γ+1

(1.15a)

(1.16a)

(1.17a)

These approximate boundary conditions are consistent with (1.1), (1.2), (1.3)

and (1.6); in fact (1.15a) yields, for the conditions at η = 1,

ψ=

γ+1

,

γ−1

(1.15b)

f=

2γ

,

γ+1

(1.16b)

φ=

2

.

γ+1

(1.17b)

(1.16a) yields

And (1.17a) yields

1.3 Energy

The total energy E of the disturbance may be regarded as consisting of two

parts, the kinetic energy

8

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

R

Z

K.E. = 4π

0

1 2 2

ρu r dr ,

2

and the heat energy

Z

H.E. = 4π

0

R

pr2

.

γ−1

In terms of the variables f , φ, ψ and η

Z 1

Z 1

p0

1

2 2

2

2

ψφ η dη +

f η dη

,

ρ0 ·

·

E = 4πA ·

2

a2 · (γ − 1) 0

0

or since p0 = a2 ρ0 /γ, E = Bρ0 A2 , where B is a function of γ only whose value is

Z 1

Z 1

4π

2 2

B = 2π

ψφ η dη +

·

f η 2 dη .

(1.18)

γ · (γ − 1) 0

0

Since the two integrals in (1.18) are both functions of γ only it seems that for

a given value of γ, A2 is simply proportional to E /ρ0 .

1.4 Numerical Solution for γ = 1.4

When γ = 1.4 the boundary values of f , φ and ψ at η = 1 are from (1.15a),

(1.16a), (1.17a), 76 , 56 and 6. Values of f , φ and ψ were calculated from η = 1.0

to η1 = 0.5, using intervals of 0.02 in η. Starting each step with values of f 0 ,

φ0 , ψ 0 , f , φ and ψ found in previous steps, values of f 0 , φ0 and ψ 0 at the end of

the interval were predicted by assuming that the previous two values form a

0

geometrical progression with the predicted one; thus the (s + 1)th term, fs+1

in

0

0

0 2

0

a series of values of f was taken as fs+1 = (fs ) /fs−1 . With this assumed

value

0

the mean value of f 0 in the sth interval

was taken as 12 fs+1

+ fs0 and the in

0

0

0

crement in f was taken as (0.02) 21 (fs+1

+ fs0 ). The values of fs+1

, φ0s+1 and ψs+1

were then calculated from formulae (1.14), (1.7a) and (1.9a). If they differed

appreciably from the predicted values, a second approximation was worked

0

out, replacing the estimated values of fs+1

by this new calculated value. In

the early stages of the calculation near η = 1 two or three approximations

were made, but in the later stages the estimated value was so close to the

calculated one that the value of f 0 calculated in this first approximation was

used directly in the next stage.

The results are given in table 1.1 and are shown in the curves of figure 1.1.

These curves and also table 1.1 show three striking features: (a) the φ curve

rapidly settles down to a curve which is very nearly a straight line through

the origin, (b) the density curve ψ rapidly approaches the axis ψ = 0, in fact at

9

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

η = 0.5 the density is only 0.007 of the density of the undisturbed atmosphere,

(c) the pressure becomes practically constant and equal to 0.436/1.167 =

0.37 of the maximum pressure. These facts suggest that the solution tends

to a limiting form as η decreases in which φ = cη, φ0 = c = constant, f = 0.436,

0

f 0 , ψ and ψ 0 become small. Substituting for γ1 fψ from (1.7a), (1.14) becomes

f0

3

1

2γφ2

2

0

· (η − φ) = γφ (η − φ) + γφ − 3η + 3 + γ φ −

.

(1.19)

f

2

2

η

Dividing by η − φ (1.19) becomes

f0

2γφ

· (η − φ) = γφ0 − 3 +

.

f

η

(1.20)

If the left-hand side which contains f 0 /f be neglected the approximate solution of (1.20) for which φ vanishes at η = 0 is

(1.21)

φ = η/γ .

The line φ = η/γ is shown in figure 1.1. It will be seen that the points

calculated by the step-by-step method nearly run into this line. The difference

appears to be due to the accumulation of errors in calculation.

1.5 Approximate Formulae

The fact that the φ curve seems to leave the straight line φ = η/γ rather rapidly

after remaining close to it over the range η = 0 to η = 0.5 suggests that an

approximate set of formulae might be found assuming

φ = η/γ + α · η n ,

(1.22)

where n is a positive number which may be expected to be more than, say, 3

or 4. If this formula applies at η = 1,

2

1

+α=

γ

γ+1

or α =

γ−1

;

γ · (γ + 1)

(1.23)

inserting φ = η/γ + αη n , φ0 = 1/γ + nα · η n−1 in (1.20), the value f 0 /f at η = 1 is

f 0 /f = αγ · (n + 2)(γ + 1)/(γ − 1). From (1.14) and (2.15), (2.16), (2.17) the true

2 +7γ−3

value of f 0 /f at η = 1 is 2γ γ−1

. Equating these two forms,

n=

7γ − 1

.

γ2 − 1

(1.24)

10

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

Figure 1.1

– · – curves f and ψ (step-by-step calculation); − − + − −, curve f (approximate

formulae). In the other curves the small dots represent the steps of the calculations, the larger symbols represent approximate formulae for: 4, curve φ; ,

curve φ = η/γ; •, curve ψ

11

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

Table 1.1

Step-by-step calculation for γ = 1.4

η

f

φ

ψ

1.00

1.167

0.833

6.000

0.98

0.949

0.798

4.000

0.96

0.808

0.767

2.808

0.94

0.711

0.737

2.052

0.92

0.643

0.711

1.534

0.90

0.593

0.687

1.177

0.88

0.556

0.665

0.919

0.86

0.528

0.644

0.727

0.84

0.507

0.625

0.578

0.82

0.491

0.607

0.462

0.80

0.478

0.590

0.370

0.78

0.468

0.573

0.297

0.76

0.461

0.557

0.239

0.74

0.455

0.542

0.191

0.72

0.450

0.527

0.152

0.70

0.447

0.513

0.120

0.68

0.444

0.498

0.095

0.66

0.442

0.484

0.074

0.64

0.440

0.470

0.058

0.62

0.439

0.456

0.044

0.60

0.438

0.443

0.034

0.58

0.438

0.428

0.026

0.56

0.437

0.415

0.019

0.54

0.437

0.402

0.014

0.52

0.437

0.389

0.010

0.50

0.436

0.375

0.007

12

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

The values of α and n have now been determined to give the correct values

of f 0 /f ; φ and φ0 at η = 1, ψ 0 is determined by (1.9a) so that all the six correct

values of f , φ, ψ, f 0 , φ0 , ψ 0 are consistent with (1.22) at η = 1. Substituting for

φ from (1.22) in (1.20),

f0

(n + 2)αγ 2 · η n−2

=

.

(1.25)

f

γ − 1 − γα · η n−1

The integral of (1.25) which gives the correct value of f at η = 1 is

2γ 2 + 7γ − 3

γ + 1 η n−1

2γ

−

· log

−

log f = log

γ+1

7−γ

γ

γ

(1.26)

At η = 0.5 this gives f = 0.457 when γ = 1.4. The value calculated by the

step-by-step integration is 0.436, a difference of 5 %.

The approximate form for ψ might be found by inserting the approximate

forms for φ and φ0 in (1.9a). Thus

Z 1

γ+1

3 + (n + 2)αγ · η n−1

log ψ = log

dη .

(1.27)

−

γ−1

(γ − 1)η − αγ · η n

η

Integrating this and substituting for α from (1.23),

γ+1

3

(γ + 5)

γ + 1 − η n−1

log ψ = log

+

· log η − 2

· log

.

γ−1

γ−1

7−γ

γ

(1.28)

When η is small, this formula gives

ψ = D · η 3/(γ−1) ,

(1.29)

where

log D = log

γ+1

γ−1

(γ + 5)

−2

· log

7−γ

γ+1

γ

.

(1.30)

When γ = 1.4 (1.30) gives D = 1.76 so that

ψ = 1.76 · η 7.5 .

(1.29a)

At η = 0.5 this gives ψ = 0.0097; the step-by-step calculation gives ψ = 0.0073.

At η = 0.8 formula (1.28) gives ψ = 0.387, while table 1.1 gives ψ = 0.370. At

η = 0.9 formula (1.28) gives ψ = 1.24, while the step-by-step solution gives 1.18.

Some points calculated by the approximate formulae are shown in figure 1.1.

In the central region of the disturbance the density decreases proportionally

to r3/(γ−1) ; the fact that the pressure is nearly constant there means that the

temperature increases proportionally to r−3/(γ−1) . At first sight, it might be

supposed that these very high temperatures involve a high concentration of

13

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

Table 1.2

Approximate calculation for γ = 1.666

η

f

φ

ψ

1.00

1.250

0.750

4.00

0.95

0.892

0.680

2.30

0.90

0.694

0.620

1.14

0.80

0.519

0.519

0.63

0.70

0.425

0.445

0.29

0.50

0.379

0.300

0.05

0.00

0.344

0.000

0.00

energy near the centre. This is not the case, however, for the energy per unit

volume of a gas is simply p/(γ −1) so that the distribution of energy is uniform.

Values of f , φ and ψ for γ = 53 calculated by the approximate formulae are

given in table 1.2.

1.6 Blast Wave Expressed in Terms of The Energy of

the Explosion

It has been seen in (1.18) that E /ρ0 A2 is a function of γ only. Evaluating

the integrals in (1.18) for the case for γ = 1.4, and using the step-by-step

calculations, it is found that

Z 1

Z 1

2 2

η φ ψdη = 0.185 and

η 2 f dη = 0.187 .

0

0

The kinetic energy of the disturbance is therefore

K.E. = 2π(0.185)ρ0 A2 = 1.164 · ρ0 A2 ,

(1.31)

while the heat energy is

H.E. =

4π

(0.187)ρ0 A2 = 4.196 · ρ0 A2 ;

(1.4)(0.4)

(1.32)

the total energy is therefore

E = 5.36 · ρ0 A2 .

(1.33)

14

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

1.7 Pressure

The pressure p at any point is

p = p0 R−3 f

2

A2

−3 ρ0 A

= 0.133 · R−3 E f .

=

R

f

a2

γ

(1.34)

The maximum pressure at any distance corresponds with f = 1.166 at r = R.

This is therefore

pmax. = 0.155 · R−3 E .

(1.35)

1.8 Velocity of Air and Shock Wave

The velocity u of the gas at any point is

3

3

1

1

u = R− 2 Aφ = R− 2 E 2 (Bρ0 )− 2 φ .

(1.36)

The velocity of radial expansion of the disturbance is, from (1.6),

3

3

1

1

dR

= AR− 2 = R− 2 E 2 (Bρ0 )− 2 ,

dt

so that, if t is the time since the beginning of the explosion,

1

1

5 1

1

2 5

t = R 2 (Bρ0 ) 2 E − 2 = 0.926 · R 2 ρ02 E − 2 ,

5

(1.37)

(1.38)

when γ = 1.4.

The formulae (1.34) to (1.38) show some interesting features. Though the

pressure wave is conveyed outwards entirely by the air the magnitude of the

pressure depends only on E R−3 and not on the atmospheric density ρ0 . The

1

time scale, however, is proportional to ρ02 . It is of interest to calculate the

pressure-time relationship for a fixed point, i.e. the pressure to which a fixed

object would be subjected as the blast wave passed over it. If t0 is the time

since initiation taken for the wave to reach radius R0 the pressure at time t

at radius R0 is given by

3

p

R0

f

=

·

,

(1.39)

p1

R

[f ]η=1

where p1 is the pressure in the shock wave as it passed over radius R0 at the

time t0 , R is the radius of the shock wave at time t and η = R0 /R. [f ]η=1 is the

maximum value of f , namely, 1.166 when γ = 1.4. η is related to t/t0 through

(1.38) so that

15

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

Figure 1.2

Pressure-time curve at a fixed point

η = (t0 /t)

2

5

and

p

1

=

p1

[f ]η=1

65

t0

[f ]η=(t0 /t)2/5 .

t

(1.40)

Values of p/p1 calculated by (1.40) for γ = 1.4 are shown in figure 1.2.

1.9 Temperature

The temperature T at any point is related to the pressure and density by the

relationship

pρ0

1.33 · E R−3

T

=

=

,

T0

p0 ρ

p0 ψ

when γ = 1.4 .

(1.41)

Since f tends to a uniform value 0.436 in the central region (r < 12 R) and ψ

tends to the value ψ = 1.76η 7.5 , T tends to the value

16

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

T

E R−3 (0.133)(0.436) −7.5

T0

=

η

= 0.033 · E R−3 η 7.5 .

T0

p0

1.76

p0

(1.42)

Thus the temperature near the centre is very high; for instance, when the

wave has expanded to such a distance that the pressure in the central region

is reduced to atmospheric pressure, p0 = (0.133)(0.436)E R−3 , then (1.42) gives

T /T0 = η −7.5 /1.76 and at η = 0.5, η −7.5 = 0.181 so that T /T0 = 103. If T0 = 273◦ ,

T = 27, 000◦ . The temperature left behind by the blast wave is therefore very

high, but the energy density is not high because the density of the gas is

correspondingly low.

1.10 Heat Energy Left in the Air After it has returned to

Atmospheric Pressure

The energy available for doing mechanical work is less than the total heat

energy of the air. The heated air left behind by the shock wave can in fact only

do mechanical work by expanding down to atmospheric pressure, whereas

to convert the whole of the heat energy into mechanical work by adiabatic

expansion, the air would have to be expanded to an infinite extent until the

pressure was zero. After the blast wave has been propagated away and the

air has returned to atmospheric pressure it is left at a temperature T1 which

is greater than T0 , the atmospheric temperature. The energy required to raise

the temperature of air from T0 to T1 is therefore left in the atmosphere in a

form in which it is not available for doing mechanical work directly on the

surrounding atmosphere. This energy, the total amount of which will be denoted by E1 , is wasted as a blast-wave producer.

The energy so wasted at any stage of the disturbance can be calculated by

finding the temperature T1 to which each element of the blast wave would be

reduced if it were expanded adiabatically to atmospheric pressure. If T is the

temperature of an element of the blast wave

T

=

T1

p

p0

(γ−1)/γ

=

A2 −3

fR

a2

(γ−1)/γ

.

Also

T

pρ0

f A2 −3

=

=

R ,

T0

p0 ρ

ψ a2

17

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

hence

f 1/γ

T1

=

·

T0

ψ

A2 R−3

a2

1/γ

.

(1.43)

The total heat energy per unit mass of air at temperature T1 is

T1

T1 p0

· (gas constant) =

.

γ−1

(γ − 1)ρ0 T0

The increase in heat energy per unit mass over that which

contained be the air

p0

T1

fore the passage of the disturbance is therefore (γ−1)(ρ0 ) · T0 − 1 . The increase

p0 ψ

T1

per unit volume of gas within the disturbed sphere is therefore (γ−1) · T0 − 1 .

Hence from (1.43) the total energy wasted when the sphere has expanded to

radius R is

)

2 −3 1/γ

Z 1(

AR

1/γ

3 p0

·

f ·

− ψ η 2 dη .

(1.44)

E1 = 4πR

2

γ−1 0

a

This expression may conveniently be reduced to non-dimensional form by

dividing by the total energy E of the explosion which is related to A by the

formula (1.18). After inserting a2 /γ for p0 /ρ0 , this gives

#

" 2 −3 1/γ Z 1

Z 1

4π

a2

E1

AR

=

·

·

·

f 1/γ η 2 dη −

ψη 2 dη . (1.45)

E

B(γ − 1) · γ

A2 R−3

a2

0

0

4

πρ0 R3

3

is the total mass of air in the sphere of radius R.

R1

This is also 4πR3 ρ0 · 0 ψη 2 dη, so that

Z 1

1

ψη 2 dη = .

3

0

(1.46)

The quantity A2 R−3 /a2 is related to the maximum pressure p1 at the shock

wave by the equation

y1 =

p1

A2 R−3

=

[f ]n=1 ,

p0

a2

where y1 is the pressure in the shock wave expressed in atmospheres. (1.46)

therefore reduces to

"

#

1/γ

Z 1

E1

4π

f

1

1/γ

2

.

(1.47)

=

[f ]η=1 · y1 ·

η dη −

E

B(γ − 1) · y1

fη=1

3

0

18

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

Table 1.3

y1 (atm. at shock 10,000

wave)

1,000

100

50

20

10∗

5∗

E1 /E (proportion

of energy wasted)

0.069

0.132

0.240

0.281

0.325

0.333∗

0.28∗

(E1 + E2 )/E

0.096

0.189

0.337

0.393

0.455

–

–

∗

Formulae inaccurate when y1 < 10.

For γ = 1.4 numerical integration gives

Z

B = 5.36 see (1.18) , fn=1 = 1.166 ,

1

f 1/γ η 2 dη = 0.219 .

0

(1.47) reduces therefore to

i

1 h

E1

1/1.4

=

· 0.958 · y1

− 1.63 .

E

y1

(1.48)

Some values of E1 /E are given in the second line of table 1.3.

It is clear that E1 /E must continually increase as R increases, and y1 decreases because the contribution to E1 due to the air enclosed in the shockwave surface when its radius is R2 , say, remains unchanged when this air

subsequently expands. A further positive contribution to E1 is made by each

subsequent layer of air included within the disturbance. The fact that formula

(1.48) gives a value of E1 /E which increases till y1 is reduced to 10 and then

subsequently decreases is due to the inaccuracy of the approximate boundary

conditions (1.15a), (1.16a) and (1.17a), which are used to replace the true

boundary conditions (1.15), (1.16) and (1.17).

When γ = 1.4 and y1 = 10 the true value of ρ1 /ρ0 is 3.8 instead of 6.0 as is

assumed, the true value of U 2 /a2 is 8.7 instead of 8.6 and the true value of

u1 /U is 0.74 instead of 0.83.

When γ = 5 the errors are much larger, namely, ρ1 /ρ0 is 2.8 instead of 6.0,

U 2 /a2 is 4.4 instead of 4.3, and u1 /U is 0.64 instead of 0.83. The proportion

of the energy wasted, namely, E1 /E , is shown as a function of y1 in figure 1.3,

y1 being plotted on a logarithmic scale.

It will be seen that the limiting value of E1 /E is certainly greater than 0.32, its

value for y1 = 20. It is not possible to find out how much greater without tracing the development of the blast wave using laborious step-by-step methods

for values of η, less than, say, 10 or 20.

19

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

Figure 1.3

Heavy line, (E1 + E2 )/E ; thin line E1 /E . (E1 + E2 )/E is the proportion of the initial

energy, which is no longer available for doing work in propagation. E2 /E is the

work done by heated air expanding against atmospheric pressure (see note

added October 1949 (p. 172))

20

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

1.11 Comparison with high Explosives

The range within which any comparison between the foregoing theory and the

blast waves close to actual high explosives can be made is severely limited.

In the first place the condition that the initial disturbance is so concentrated

that the mass of the material in which the energy is originally concentrated

is small compared with the mass of the air involved in the disturbance at

any time, limits the comparable condition during a real explosion to one in

which the whole mass of air involved is several times that of the explosive. In

the second place the modified form of the shock-wave condition used in the

analysis is only nearly correct when the rise in pressure at the shock-wave

front is several -say at least 5 or 10- atmospheres. In a real explosive this

limits the range of radii of shock wave over which comparison could be made

to narrow limits. Thus with l0 lb. of C.E.∗ the radius R at which the weight

of explosive is equal to that of the air in the blast wave is 3 ft., while at 3.8 ft.

the air is only double the weight of the explosive. The pressure in the blast

wave at a radius of 6 ft. was found to be 9 atm., while at 8 ft. it was about

5 atm. It seems, therefore, that in this case the range in which approximate

agreement with the present theory could be expected only extends from 3.8

to 6 ft. from the l0 lb. charge.

Taking the energy released on exploding C.E. to be 0.95 kcal./g. the energy released when 10 lb. is exploded is 1.8 · 1014 ergs. If this energy had

been released instantaneously at a point as in the foregoing calculations the

maximum pressure at distance R given by (1.35) is

pmax. R3 = (0.155)(1.80 · 1014 ) = 2.79 · 1013 ergs.

(1.49)

Expressed in terms of atmospheres pmax. is identical with y1 . If R is expressed

in feet, (1.49) becomes

y1 R 3 =

2.79 · 1013

= 9.9 · 102 .

(30.45)3 · 106

(1.50)

The line representing this relationship on a logarithmic scale is shown in

figure 1.4. Though no suitable pressure measurements have been made, the

maximum pressure in the blast from l0 lb. of C.E. has been found indirectly

by observing the velocity of expansion of the luminous zone and, at greater

radii, the blast-wave front. These values taken from a curve given in a report

on some experiments made by the Road Research Laboratory are given in

table 1.4. The observed values of U in ft./sec. given in column 2 of this table

and the values of y1 (in atmospheres) found from the shock-wave formulae

are given in column 3, where they are described as observed values though

they were not observed directly. The “observed” values are shown in figure

21

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

Figure 1.4

Blast pressures near 10 lb. charge of C.E. compared with calculated blast pressures due to instantaneous release of energy of 10 lb. C.E. at a point. The

numbers against the points on the curve give distances in feet

1.4. The values of y1 calculated from (1.50) are given in column 4.

Though the observed values are higher than those calculated, it will be noticed that in the range of radii 3.8 to 6 ft., in which comparison can be made,

the observed curve is nearly parallel to the theoretical line y1 R3 = 990. In this

range, therefore, the intensity of the shock wave varies nearly as the inverse

cube of the distance from the explosion. The fact that the observed values are

about twice as great as those calculated on the assumption that the energy

is emitted instantaneously at a point may perhaps be due to the fact that the

measurements used in table 1.4 correspond with conditions on the central

plane perpendicular to the axis of symmetry of the cylindrical charge used.

The velocity of propagation of the luminous zone is greater on this plane and

on the axis of symmetry than in other radial directions so that the pressures

deduced in column 3 of table 1.4 are greater than the mean pressures at the

corresponding radii.

On the other hand, it has been seen that by the time the maximum pressure

has fallen to 20 atm., 32 % of the energy has been left behind in the neighbourhood of the concentrated explosive source, raising the air temperature

there to very high values. The burnt gases of a real high explosive are at a

very much lower temperature even while they are at the high pressure of the

detonation wave. Their temperature is still lower when they have expanded

22

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

Table 1.4

Calculated values y1 from equation (1.50)

y1 (atm.)

R (ft.) U (ft./sec.) observed

with C.E.

y1 calculated for

concentrated

explosion by (1.50)

8

2350

6.2

.

6

3100

9.3

4.6

5

3800

14.0

7.9

4

4820

22.6

15.5

3

6200

37.5

.

2

8540

71.8

.

range of

comparison

Table 1.5

Pressure y1 p0 at distance R from explosion of weight W of T.N.T. - R.D.X. mixture

1

1

R/W 3 (ft./lb. 3 )

U (thousands ft./sec.)

y1 (atm.)

1

3

0.5

1.0

1.5

2.0

2.5

3.0

3.5

14.3

11.0

8.4

6.6

5.1

4.0

3.3

68.3

42.0

25.1

15.4

10.5

16.2

21.5

27.0

32.4

37.8

198

4

1

3

R/E · 10 (cm./ergs. )

5.39

117

10.8

adiabatically to atmospheric pressure, so that little heat energy is left in them.

To this extent, therefore, a real high explosive may be expected to be more efficient as a blast producer than the theoretical infinitely concentrated source

here considered.

Note added, October 1949. The data on which the comparison was based

between the pressures deduced by theory and those observed near detonating explosives were obtained in 1940. More recent data obtained at the Road

Research Laboratory using a mixture of the two explosives R.D.X. and T.N.T.

have been given by Dr. Marley. These are given in table 1.5, which shows

1

the values of U observed for various values of R/W 3 . R the distance from the

explosive is expressed in feet and W its weight in pounds. The third line in

table 1.5 shows the result of deducing y1 from U using γ = 1.4 in (1.16) and

a = l, l00 ft./sec. in (1.16).

For comparison with the concentrated point-source explosion, the value of

23

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

1

1

RE − 3 expressed in cm. (erg.)− 3 is found by multiplying the figures in line 1,

1

1

−3

table 1.5, by 30.41 ·

. The first factor converts ft. (lb.)− 3

1 = 1.078 · 10

7

(454) 3

(1,200·4.2·10 ) 3

1

to cm. (g.)− 3 , and the second replaces 1 g. by the equivalent energy released

1

by this explosive, namely, 1,200 cal. The values of RE − 3 are given in line 4,

1

table 1.5. In figure 1.5 values of log10 y1 are plotted against log10 RE − 3 , and

the theoretical values for a point source of the same energy as the chemical

explosive are plotted in the same diagram. Comparing figures 1.4 and 1.5

it seems that the more recent shock-wave velocity results are qualitatively

similar to the older ones in their relation to the point-source theory. The

range of values of y1 for which comparison between theory and observation

might be significant, is marked in figure 1.5.

It will be seen that the chemical explosive is a more efficient blast producer

than a point source of the same energy. The ratio of the pressures in the

range of comparison is about 3 to 1. This is more than might be expected in

view of the calculation of E1 /E as a function of y1 which is given in table 1.3.

E1 is the heat energy which, is unavailable for doing mechanical work after

expanding to pressure p0 . Of the remaining energy, E − E1 a part E2 is used in

doing work against atmospheric pressure during the expansion of the heated

air. The remaining energy, namely, E − E1 − E2 , is available for propagating

the blast wave.

To find E2 , the work done byunit volume

of the gas at radius ηR in expanding

T1 p

to atmospheric pressure is T p0 − 1 p0 . From (1.43)

T1

=

T

A2 −3

R f

a2

(1−γ)/γ

and

p

A2 R−3

=f

,

p0

a2

hence

E2 = 4πR p0 ·

3

Z

1

0

but

A2

a2 R3

=

y1

,

(f )η=1

2

−3 A

R f 2 − 1 η 2 dη ,

a

(1.51)

so that

4πR3 p0

E2

=

·

E

E

"

y1

fη=1

1/γ Z

·

0

1

f 1/γ η 2 dη −

Z

1

#

η 2 dη .

(1.52)

0

The first integral has already been calculated and found to be 0.219 when

γ = 1.4 (see (1.47) and (1.48)). Substituting for pmax. from (1.35),

"

#

(1−γ)/γ

E2

0.219 · y1

1

= 4π · (0.155) ·

−

.

(1.53)

E

(1.166)1/γ

3y1

24

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

Values of (E2 + E1 )/E have been added as a third line in table 1.3 and a corresponding curve to figure 1.3.

1.12 References

Taylor, Sir Geoffrey 1946 Proc. Roy. Soc. A, 186, 273.

Taylor, Sir Geoffrey 1950 Proc. Roy. Soc. A, 201, 175.

25

1 The formation of a blast wave by a very intense explosion I. Theoretical

discussion

Figure 1.5

Blast pressures near a chemical explosive (R.D.X. + T.N.T.) compared with theoretical pressure for concentrated explosion with same release of energy. Heavy

line (upper part) is taken from shock-wave velocity measurements. Heavy line

(lower part) is from piezo-electric crystals. Thin line, y1 = 0.155 · E /(p0 R3 ). The

figures against the points represent the ratio of the mass of the air within the

shock wave to the mass of the explosive

26

2 The formation of a blast wave by a very

intense explosion. II. The atomic explosion

of 1945

2.1 Summary and Introduction

Photographs by J.E. Mack of the first atomic explosion in New Mexico were

measured, and the radius, R, of the luminous globe or “ball of fire” which

spread out from the centre was determined for a large range of values of t,

the time measured from the start of the explosion. The relationship predicted

5

in chapter 1, namely, that R 2 would be proportional to t, is surprisingly

5

accurately verified over a range from R = 20 to 185 m. The value of R 2 t−1 so

found was used in conjunction with the formulae of chapter 1 to estimate the

energy E , which was generated in the explosion. The amount of this estimate

depends on what value is assumed for γ, the ratio of the specific heats of air.

Two estimates are given in terms of the number of tons of the chemical

explosive T.N.T. which would release the same energy. The first is probably

the more accurate and is 16,800 tons. The second, which is 23,700 tons,

probably overestimates the energy, but is included to show the amount of

error which might be expected if the effect of radiation were neglected and

that of high temperature on the specific heat of air were taken into account.

Reasons are given for believing that these two effects neutralize one another.

After the explosion, a hemispherical volume of very hot gas is left behind and

Mack’s photographs were used to measure the velocity of rise of the glowing

centre of the heated volume. This velocity was found to be 35 m./sec.

Until the hot air suffers turbulent mixing with the surrounding cold air, it may

be expected to rise like a large bubble in water. The radius of the “equivalent

bubble” is calculated

√ and found to be 293 m. The vertical velocity of a bubble

of this radius is 23 · g 29, 300 or 35.7 m./sec. The agreement with the measured

value, 35 m./sec., is better than the nature of the measurements permits one

to expect.

2 The formation of a blast wave by a very intense explosion. II. The atomic

explosion of 1945

2.2 Comparison with Photographic Records of the

First Atomic Explosion

Two years ago, some motion picture records by Mack (1947) of the first atomic

explosion in New Mexico were declassified. These pictures show not only the

shape of the luminous globe which rapidly spread out from the detonation

centre, but also gave the time, t, of each exposure after the instant of initiation. On each series of photographs, a scale is also marked so that the

rate of expansion of the globe, or “ball of fire”, can be found. Two series of

declassified photographs are shown in figure 2.5, plate 7.

These photographs show that the ball of fire assumes at first the form of a

rough sphere, but that its surface rapidly becomes smooth. The atomic explosive was fired at a height of 100 ft. above the ground and the bottom of the

ball of fire reached the ground in less than 1msec. The impact on the ground

does not appear to have disturbed the conditions in the upper half of the

globe which continued to expand as a nearly perfect luminous hemisphere

bounded by a sharp edge which must be taken as a shock wave. This stage of

the expansion is shown in figure 2.6, plate 8 which corresponds with t = 15

msec. When the radius R of the ball of fire reached about 130m., the intensity

of the light was less at the outer surface than in the interior. At later times,

the luminosity spread more slowly and became less sharply defined, but a

sharp-edged dark sphere can be seen moving ahead of the luminosity. This

must be regarded as showing the position of the shock wave when it ceases

to be luminous. This stage is shown in figure 2.7, plate 9, taken at t = 127

msec. It will be seen that the edge of the luminous area is no longer sharp.

The measurements given in column 3 of table 2.1 were made partly from

photographs in Mack (1947), partly from some clearer glossy prints of the

same photographs kindly sent to me by Dr. N. E. Bradbury, Director of Los

Alamos Laboratory and partly from some declassified photographs lent me by

the Ministry of Supply. The times given in column 2 of table 2.1 are taken

directly from the photographs. To compare these measurements with the

analysis given in chapter 1 of this paper, equation (1.38) was used. It will be

seen that if the ball of fire grows in the way contemplated in my theoretical

5

analysis, R 2 will be found to be proportional to t. To find out how far this

prediction was verified, the logarithmic plot of 52 log R against log t shown in

figure 2.1 was made. The values from which the points were plotted are given

in table 2.1. It will be seen that the points lie close to the 45◦ line which is

drawn in figure 2.1. This line represents the relation

5

log10 R − log10 t = 11.915 .

2

(2.1)

28

2 The formation of a blast wave by a very intense explosion. II. The atomic

explosion of 1945

Table 2.1

Radius R of blast wave at time t after the explosion

authority

5

2

t (msec.)

R (m.)

log10 t

0.10

11.1

4.0

3.045

7.613

0.24

19.9

4.380

3.298

8.244

0.38

25.4

4.580

3.405

8.512

0.52

28.8

4.716

3.458

8.646

0.66

31.9

4.820

3.504

8.759

0.80

34.2

4.903

3.535

8.836

0.94

36.3

4.973

3.560

8.901

strip of declassified

photographs lent

by Ministry of Sup-

ply

1.08

38.9

3.033

3.590

8.976

1.22

41.0

3.086

3.613

9.032

1.36

42.8

3.134

3.631

9.079

1.50

44.4

3.176

3.647

9.119

1.65

46.0

3.217

3.663

9.157

1.79

46.9

3.257

3.672

9.179

1.93

48.7

3.286

3.688

9.220

3.26

59.0

3.513

3.771

9.427

3.53

61.1

3.548

3.786

9.466

3.80

62.9

3.580

3.798

9.496

4.07

64.3

3.610

3.809

9.521

4.34

65.6

3.637

3.817

9.543

4.61

67.3

3.688

3.828

9.570

106.5

2.176

4.027

10.068

130.0

2.398

4.114

10.285

145.0

2.531

4.161

10.403

175.0

2.724

4.243

10.607

185.0

2.792

4.267

10.668

strip of small

images MDDC221

strip of small

images from

MDDC221

large single

photographs

MDDC221

15.0

25.0

34.0

53.0

62.0

log10 R

log10 R

29

2 The formation of a blast wave by a very intense explosion. II. The atomic

explosion of 1945

The ball of fire did therefore expand very closely in accordance with the theoretical prediction made more than four years before the explosion took place.

This is surprising, because in those calculations it was assumed that air

behaves as though γ, the ratio of the specific heats, is constant at all temperatures; an assumption which is certainly not true.

At room temperatures γ = 1.40 in air, but at high temperatures γ is reduced

owing to the absorption of energy in the form of vibrations which increases

Cv . At very high temperatures, γ may be increased owing to dissociation. On

the other hand, the existence of very intense radiation from the center and

absorption in the outer regions may be expected to raise the apparent value

of γ. The fact that the observed value of R5 t−2 is so nearly constant through

the whole range of radii covered by the photographs of the ball of fire suggests

that these effects may neutralize one another, leaving the whole system to

behave as though γ has an effective value identical with that which it has

when none of them are important, namely, 1.40.

2.3 Calculation of the Energy Released by the

Explosion

The straight line in figure 2.1 correspond with

R5 t−2 = 6.67 · 102 (cm.)5 sec.−2 .

The energy, E , is then from equation (1.18) of chapter 1

4π

2

I2 ,

E = ρ0 A · 2πI1 +

γ · (γ − 1)

(2.2)

(2.3)

where

1

Z

ψφ2 η 2 dη

I1 =

(2.4)

0

and

Z

I2 =

1

f η 2 dη ,

(2.5)

0

where f , φ and ψ are non-dimensional quantities proportional to pressure,

velocity and density which are defined in chapter 1.

A is found by integrating equation (1.6) of chapter 1, so that

30

2 The formation of a blast wave by a very intense explosion. II. The atomic

explosion of 1945

Figure 2.1

5

Logarithmic plot showing that R 2 is proportional to t

31

2 The formation of a blast wave by a very intense explosion. II. The atomic

explosion of 1945

2 5

A = R 2 t−1 .

5

(2.6)

Writing

4

4π

K=

· 2πI1 +

I2 ,

25

γ · (γ − 1)

E = Kρ0 R5 t−2 .

(2.7)

(2.8)

It was shown in chapter 1 that I1 and I2 are functions of γ only. For γ =

1.40, their values were found to be I1 = 0.185, I2 = 0.187, using step-by-step

methods for integrating the equations connecting f , φ and ψ. Using the

approximate formulae (1.22) to (1.30) of chapter 1, values of f , φ and ψ were

calculated in chapter 1 for γ = 1.667 and are there given in table 1.2 of chapter

1. Further calculations have now been made for γ = 1.20 and γ = 1.30 using

the approximate formulae. The results for γ = 1.30 are given in table 2.2 and

are shown in figure 2.2. The corresponding values of I1 , I2 and K are given in

table 2.3; K is shown as a function of γ in figure 2.3.

2.4 Energy of the First Atomic Explosion in New

Mexico

Having determined R5 t−2 and assuming that ρ0 may be taken as 1.25 · 10−3

g./cm.3 , the figures in table 2.3 were used to determine E from equations

(2.8) and (2.2). Different values are found for different assumed values of

γ. These are given, expressed in ergs, in line 5 of table 2.3. It has become

customary to describe large explosions by stating the weight of T.N.T. which

would liberate the same amount of energy. Taking 1 g. of T.N.T. as liberating

1,000 calories or 4.18 · 1010 ergs, 1 ton will liberate 4.25 · 1016 ergs. The T.N.T.

equivalents found by dividing the figures given in line 5 of table 2.3 by 4.25·1016

are given in line 6.

It will be seen that if γ = 1.40, the T.N.T. equivalent of the energy of the

New Mexico explosion, or more strictly that part of the energy which was not

radiated outside the ball of fire, was 16,800 tons.

32

2 The formation of a blast wave by a very intense explosion. II. The atomic

explosion of 1945

Table 2.2

Calculation for γ = 1.30 using formulae 1.22 to 1.30 of chapter 1

η

f

φ

ψ

T /T1

1.0

1.130

0.869

7.667

1.00

0.98

0.896

0.833

4.534

1.34

0.96

0.772

0.801

2.989

1.75

0.94

0.666

0.772

2.067

2.19

0.92

0.606

0.745

1.454

2.73

0.90

0.563

0.721

1.058

3.61

0.85

0.499

0.669

0.509

6.76

0.80

0.468

0.623

0.255

12.5

0.75

0.453

0.580

0.128

24.0

0.70

0.445

0.540

0.063

48.0

0.65

0.441

0.501

0.029

103

0.60

0.439

0.462

0.013

229

0.55

0.438

0.423

0.006

.

0.50

0.438

0.386

0.002

.

0.45

0.438

0.347

0.001

.

0.40

0.438

0.309

0.000

.

Table 2.3

Calculated constants used in determining the Energy E of the explosion with a

range of assumed values for γ

γ

1.20

1.30

1.40

1.667

I1

0.259

0.221

0.185

0.123

I2

0.175

0.183

0.187

0.201

K

1.727

1.167

0.856

0.487

14.4

9.74

7.14

4.06

34, 000

22, 900

16, 800

9, 500

E · 1020 erg.

T.N.T. equivalent (tons)

33

2 The formation of a blast wave by a very intense explosion. II. The atomic

explosion of 1945

Figure 2.2

Distribution of radial velocity φ, pressure f , density ψ and temperature T /T1 for

γ = 1.30 expressed in non-dimensional form

34

2 The formation of a blast wave by a very intense explosion. II. The atomic

explosion of 1945

Figure 2.3

Variation of K with γ

35

2 The formation of a blast wave by a very intense explosion. II. The atomic

explosion of 1945

2.5 An Alternative Possibility

If the effect of radiation, which cannot be estimated, is disregarded, a mean

value of γ might be taken which is appropriate to the temperature calculated

5

to correspond with the mean value of R in the range over which R 2 t−1 is nearly

constant. The least value of R which lies near the line in figure 2.1 is R = 20

m. and the greatest is R = 185 m. The mean value is therefore approximately

100 m. It will be found that the value of γ appropriate to the temperature

behind the shock wave at 100 m. is about 1.3. The pressure at any point is

from equations (1.1), (1.6) and (1.12) of chapter 1

p = p0 R−3 f1 = p0 R−3

A2

4 ρ0 −3

R f (R5 f −2 ) .

f=

2

a

25 γ

For γ = 1.3, the value of f behind the detonation front is 1.13, so that

p = 9.3 · 1022 ρ0 R−3

The temperature T1 at that point is given by

p ρ0

T1

=

,

T0

ρ p0

where T0 is the undisturbed atmospheric temperature. When

γ = 1.3, ρ0 /ρ = 1/ψ = 1/7.667,

Thus

and if ρ0 = 0.00125, p/p0 = 7.43 · 1013 R−3 .

T1

= 0.97 · 1013 R−3 .

T0

(2.9)

At R = 100 m., T1 /T0 = 9.7, so that if T0 = 15◦ C = 288◦ K, T1 = 2,800◦ K.

Values of Cp at temperatures up to 5,000◦ K have been calculated for nitrogen

by Johnston & Davis (1934) and for oxygen by Johnston & Walker (1935).

At 2,800◦ K, Cp is given for nitrogen as 8.82 and for oxygen as 9.43, so that

for air Cp = 8.92. Since there is little dissociation at that temperature, it

seems that Cv = Cp − R = 6.92 and γ = 8.92

= 1.29. Thus the use of γ = 1.30 for

6.92

calculating the temperature at R = 100 m. is justified when effects of radiation

are neglected.

Using the curve, figure 2.3, it seems that the value of K appropriate to γ =

1.29 is 1.21. Using ρ0 = 0.00125 and R5 t−2 = 6.67 · 1023 , equation (2.8) gives

E = 1.01 · 1021 ergs and the T.N.T. equivalent is 23,700 tons.

Of these two alternative estimates, it seems that the first, namely, 16,800

tons, is the more likely to be accurate.

36

2 The formation of a blast wave by a very intense explosion. II. The atomic

explosion of 1945

2.6 Some Dynamical Features of the Atomic

Explosion

It will be seen in figure 1.1 of chapter 1 that if γ = 1.4 the air density is a maximum at the shock wave front where it reaches six times atmospheric density.

Within the shock wave, the density falls rapidly till at a radius of about 0.6 · R

it is nearly zero. Within the radius 0.6 · R the gas has a radial velocity which

is proportional to the distance from the centre, a very high temperature, and

a uniform pressure about 0.43 time the maximum pressure.

The maximum pressure at the shock front is found by inserting the value of

E from line 5 of table 1.3 in the formula (1.35) of chapter 1. The pressure

-expressed in atmospheres- is

p

= 0.155R−3 (7.14 · 1020 ) · 106 = 1.11 · 1014 R−3 .

p0

(2.10)

At R = 30 m. this is 4,100 atm., or 27 tons/sq.in. At R = 100 m. the pressure

would be 1 ton/sq.in. These pressures are much less than would act on rigid

bodies exposed to such blasts, but the pressures on obstacles depend on their

shape so that no general statement can be made on this subject.

The temperature rises rapidly as the centre is approached, in fact the ratio

T /T1 , T1 being the temperature just inside the shock wave, is equal to

f

ψη=1

p1 ρ 1

p p0 ρ1 ρ0

=

=

·

.

(2.11)

p ρ

p0 p1 ρ0 ρ

fη=1

ψ

The values of T /T1 for γ = 1.3, namely

table 2.2, and are shown in figure 2.2.

7.66f

,

1.13ψ

are given in the fifth column of

37

2 The formation of a blast wave by a very intense explosion. II. The atomic

explosion of 1945

2.7 The initial rate of rise of air from the seat of the

explosion

When the shock wave had passed away from the ball of fire, it left a cloud

of very hot air, which then rapidly rose. Mack (1947, p. 37) gives a rough

picture of the process in a series of diagrams representing the outlines of

the boundary of the heated region, so far as his photographs could define

them, at successive times from t = 0.1 to t = 15.0 sec. It is not possible to

know exactly what these outlines represent, though in the later numbers of

the series they seem to show the limits of the region to which dust thrown off

from the ground and sucked into the ascending column of air has penetrated.

This dust rapidly expands into a roughly spherical shape owing, to turbulent

diffusion or convection currents in the central region. The radius of the outer

edge of the glowing region is not the same in all photographs taken nearly

simultaneously, but the height of its apparent centre seems to be consistent

when photographs taken simultaneously from different places are compared.

The heights, h, of the top of the illuminated column and their radii, b, were determined, so far as was possible, from Mack’s published photographs. These

rough measurements are given in table 2.4. The height of the centre of the

glowing area is taken as h − b, and points corresponding with those given

in table 2.4 are plotted in figure 2.4. It will be seen that the centre of the

glowing volume seems to rise at a regular rate. The line drawn in figure 2.4

corresponds with a vertical velocity of

U = 35 m./sec.

(2.12)

Table 2.4

Height, h, and radius, b, of the glowing region from 3 12 to 15 sec. after the

explosion

authority

Mack MDDC221,

sketches on p. 37

photographs on p. 38

t (sec.)

3.5

8.0

10.0

15.0

14.8

b

(radius)

(m.)

160

240

300

360

550

h

(height

of top)

375

688

810

1060

1200

h−b

(height

of centre)

215

448

510

700

650

38

2 The formation of a blast wave by a very intense explosion. II. The atomic

explosion of 1945

Figure 2.4

Height of centre of glowing region from 3 12 to 15 sec. after the explosion.

from diagram p. 37, from photograph p. 38 of MDDC221

39

2 The formation of a blast wave by a very intense explosion. II. The atomic

explosion of 1945

Figure 2.5

Succession of photographs of the “ball of fire” from t = 0.10 msec. to 1.93 msec.

40

2 The formation of a blast wave by a very intense explosion. II. The atomic explosion of 1945

Figure 2.6

The ball of fire at t = 15 msec., showing the sharpness of its edge

41

2 The formation of a blast wave by a very intense explosion. II. The atomic explosion of 1945

Figure 2.7

The glowing volume t = 127 msec. showing an indefinite edge and hemispherical shape

42

2 The formation of a blast wave by a very intense explosion. II. The atomic

explosion of 1945

2.8 Distribution of Air Density after the Explosion

To give a dynamical description of the rise of hot air from the seat of the

explosion it is first necessary to know the distribution of density immediately

after the shock wave has passed away and left hot air at atmospheric

pressure. Formulae are given in chapter 1 for the temperature ultimately

attained by air, which passed through the shock wave when its pressure

was y1 p0 , but the position at which this air comes to rest when atmospheric

pressure was attained was not discussed. If the ground had not obstructed

the blast wave, the distribution of temperature would evidently be spherical.

It has been pointed out, however, that the shape of the upper half of the

blast wave has not been affected by the presence of the ground. The same

thing seems to be approximately true of the temperature distribution, for

Mack publishes a photograph showing the luminous volume at t = 0.127 sec.

when the shock wave had moved well away from the very hot area. This is

reproduced in figure 2.7, plate 9. It will be seen that the glowing air occupies

a nearly hemispherical volume, the bottom half of the sphere being below

the ground. It seems that it may be justifiable to assume that most of the

energy associated with the part of the blast wave, which strikes the ground

is absorbed there. In that case, we may neglect the effect of shock waves

reflected from the ground and consider the temperature distribution as being

that calculated in part I for an unobstructed wave. With this assumption, the

distribution of density will be calculated.

The following symbols will be used:

T0 , p0 , ρ0 , the temperature, pressure and density in the undisturbed air,

T , y1 p0 , ρ, R, the temperature, pressure, density and radius at the shock

wave,

T1 , ρ1 , r, the temperature, density and radius after the pressure has become

atmospheric.

From (1.43), chapter 1,

T

(γ−1)/γ

= y1

=

T1

A2 f R−3

α2

(γ−1)/γ

(2.13)

and

f A2 −3

T

=

R ,

T0

ψ α2

(2.14)

so that

T1

T T0

1

=

= ·

T0

T1 T

ψ

A2 f R−3

α2

,

(2.15)

43

2 The formation of a blast wave by a very intense explosion. II. The atomic

explosion of 1945

and since

A2

4E

,

=

2

α

25Kγp0

−3/γ

R

T1

=β ·

,

T0

r0

(2.16)

where

1

β= ·

ψ

4f

25Kγ

1/γ

(2.17)

,

and r0 is the length defined by

r03 = E /p0 .

(2.18)

r0 is introduced in order to make the equations non-dimensional, ψ and f

have their values at the shock wave front. When γ = 1.40, f = 1.167, ψ = 6.0;

K = 0.856 (see table 2.3), so that

(2.19)

β = 0.044

(2.16) may be written

γ

γ

3

T0

T0

R

γ

=β ·

= 0.1265 ·

r0

T1

T1

when

γ = 1.40 .

(2.20)

To find the values of r and T1 /T0 corresponding with a given value of y1 the

equation of continuity must be used. This is

ρ0 R2 dR = ρ1 r2 dr ,

(2.21)

and since T0 /T1 = ρ1 /ρ0 , (2.21) can be integrated after substitution from (2.20),

thus

3

γ−1

r

γ

T0

γ

=β ·

·

,

(2.22)

r0

γ−1

T1

and from (1.35) of chapter 1 and (2.20)

R

r0

3

0.155

=

= βγ ·

y1

T0

T1

γ

.

(2.23)

Eliminating T0 /T1 from (2.22) and (2.23) and using values appropriate to

γ = 1.40

44

2 The formation of a blast wave by a very intense explosion. II. The atomic

explosion of 1945

Table 2.5

Density ρ1 at radius r expressed as a proportion of ρ0 , the undisturbed density

y1 (atm.)

r/r0

ρ1 /ρ0

uncorrected

C

ρ1 /ρ0

corrected

R/r0

10

(0.360)

1.155

0.753

0.873

0.249

20

0.337

0.704

0.858

0.605

0.198

30

0.325

0.527

0.900

0.475

0.173

40

0.316

0.430

0.923

0.397

0.157

60

0.304

0.321

0.947

0.305

0.137

100

0.289

0.223

0.968

0.216

0.116

500

0.247

0.071

0.071

0.068

5000

0.200

0.013

0.013

0.031

r

r0

3

= 0.0906 · y1−0.2857

(2.24)

ρ0

T1

=

= 0.167 · y10.7143 .

ρ1

T0

(2.25)

and

Values of R/r0 , ρ1 /ρ0 , r/r0 calculated for a range of values of y, are given in

table 2.5.

It has been pointed out that T1 /T0 contains two factors, T /T0 and T1 /T . T /T0

represents the temperature change through the shock wave and is equal

γ−1

to ρ0 y1 /ρ. In calculating this the approximate expression ρρ0 = γ+1

was used

instead of the true value

ρ0

γ + 1 + (γ − 1)y1

=

.

ρ

γ − 1 + (γ + 1)y1

(2.26)

The proportional error in calculating T /T0 for a given values of y1 is therefore

equal to the proportional error in using (γ − 1)/(γ + 1) instead of the correct

expression for ρ0 /ρ. The second factor, T1 /T , which represents the reduction

in temperature for given expansion ratio is correct, so that correct values of

ρ/ρ0 for a given value of y1 can be found by multiplying the figures given in

45

2 The formation of a blast wave by a very intense explosion. II. The atomic

explosion of 1945

Figure 2.8

Distribution of density after the explosion. The upper scale corresponds with

calculations using the value of E given in table 2.3 for γ = 1.40

column 3 of table 2.5 by a correcting factor

C=

γ − 1 + (γ + 1)y1 γ − 1

.

·

γ + 1 + (γ − 1)y1 γ + 1

(2.27)

Values of C are given in column 4, table 2.5, and the corrected values of ρ1 /ρ0

in column 5. It must be pointed out that though the figures in column 5 are

correct, the values obtained for r/r0 when y1 is less than about 40 are subject

to an appreciable error owing to using approximate values of f , φ and ψ at

the shock front.

The variation of ρ1 /ρ0 with r/r0 is shown in figure 2.8. It will be seen that

the density is very small when r/r0 < 0.2, that it begins to rise steeply at

about r/r0 = 0.28, and that it has nearly attained atmospheric density when

r/r0 = 0.36.

46

2 The formation of a blast wave by a very intense explosion. II. The atomic

explosion of 1945

2.9 Calculation of the Rate of Rise of the Heated Air

Though it would be difficult to calculate the effect of buoyancy on a fluid

with the density distribution shown in figure 2.8, the rise of bubbles in water,

when the change from very light to heavy fluid is discontinuous, has been

studied. It has been shown both experimentally and theoretically (Davies &

Taylor 1950) that the vertical velocity U of a large bubble is related to a, the

radius of curvature of the top of the bubble, by the formula

2 √

· ga .

(2.28)

3

It seems worth while to compare the observed rate of rise of the air heated

by the New Mexico explosion with that of a large bubble in water, and for

that purpose it is necessary to decide on the radius of a sphere of zero

density which might be expected to be comparable with air having the density

distribution of figure 2.8. The simplest guess is to take the radius as the

value of r for which ρ/ρ0 = 12 ,and in figure 2.8 this corresponds with the

broken line for which

U=

r/r0 = 0.328 .

(2.29)

In the New Mexico explosion the best estimate of E was that corresponding

with the measured rate of expansion of the ball of fire, assuming γ = 1.40.

This is given in table 2.3, namely E = 7.14 · 1020 ergs.

Using this value and p0 = 106 dynes/sq.cm.

1

r0 = (7.14 · 1014 ) 3 = 8.9 · 104 = 894 m. .

(2.30)

The radius chosen for comparison with a bubble rising in water is therefore

r = 0.328 · 894 = 293 m. .

(2.31)

The predicted velocity of rise is therefore

2 p

· (981 · 2.93 · 104 ) = 35.7 m./sec.

(2.32)

3

Comparing this with the observed value of the vertical velocity of the centre

of the glowing volume, namely, 35 m./sec., it will be seen that the agreement

is better than the nature of the measurements would justify one in expecting.

A far less good agreement would justify a belief that the foregoing dynamical

picture of the course of events after the atomic explosion is essentially correct.

U=

47

2 The formation of a blast wave by a very intense explosion. II. The atomic

explosion of 1945

2.10 References

Davies, R. M. & Taylor, Sir G. 1950 Proc. Roy. Soc. A, 200, 375.

Johnston, H. L. & Davis, C. O. 1934 J. Amer. Chem. Soc. 56, 271.

Johnston, H. L. & Walker, M. K. 1935 J. Amer. Chem. Soc. 57, 682.

Mack, J. E. 1947 Semi-popular motion picture record of the Trinity explosion.

MDDC221. U.S. Atomic Energy Commission.

48

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