barenblatt g.i. scaling phenomena in fluid mechanics

These equations should be supplemented by the boundary conditions at the shock wave front: 07 uy — D = —poD 1.5 where p;,uy are the density and velocity just behind the shock front,

_—

G.I TAYLOR PROFESSOR OF FLUID MECHANICS

AT THE UNIVERSITY OF CAMBRIDGE FELLOW OF GONVILLE AND CAIUS COLLEGE, CAMBRIDGE

© Cambridge University Press 1994

First published 1994

Printed in Great Britain by George Over Ltd., Rugby

ISBN 0-521-46920-1 Paperback

AMBRIDGE

NIVERSITY PRESS

establish the new G.I Taylor Professorship of Fluid

Mechanics It may not be appropriate for me to go

The term ‘scaling’ denotes a seemingly very simple

thing: a power-law relationship between certain vari- ables y and x of the form

where A,a are constants Such relations often appear

in mathematical modelling of various phenomena, not only in physics, but also in biology, economics, and en-

gineering Scaling laws are not merely some special

simple cases of more general relations They never appear by accident Scaling laws always reveal a very important property of the phenomenon under consid- eration: its self-similarity Self-similar means repro- ducing itself on different time and space scales — I will

explain this later in more detail

I begin with one of the best examples of such self- similar phenomena: G.I Taylor’s analysis of a basic

intense shock wave propagates in the atmosphere and

the gas motion inside the shock wave can be considered

as adiabatic

In one of the worst and most alarming days of the

Battle of Britain, in the early autumn of 1940, Profes-

sor Geoffrey Ingram Taylor was invited to a business lunch at the Athenaeum by Professor George Thomson,

(the name ‘Maud’ had nothing to do with the fairy-tale:

it was the acronym for ‘Military Application of Uranium

Netonation’ t wac t hat ]

energy would be released by nuclear fission — the name

‘atomic bomb’ had not yet been used The question was:

c4

cial importance for the further development of events

Shortly before this conversation the confidential report

~ of G.B Kistyakovsky, the well known American expert

~~ in explosives, was received Kistyakovsky claimed that

even if the bomb exploded, its effect would be much less than was expected As R.W Clark writes in his instructive book (Clark, 1961) in the whole of Britain

there was only one man able to solve this problem — Professor G.I Taylor

To answer this question, GI] had to understand and

calculate the motion and pressure of the ambient gas after such an explosion It was clear to GI that after

a very short initial period (related as we now know to

who knew G.I Taylor

Figure 1 Very intense shock-wave propagates in quiescent

air (sketch)

the thermal wave propagation in quiescent air), a very intense shock wave would appear (Figure 1)

The motion was assumed spherically symmetric, that

is identical for all radi1 going out from the explosion

centre (This simplifying assumption received excel- lent confirmation in the first atomic test ) Hence the

following equations of motion inside the shock wave

had to be considered:

(1) the differential equation of conservation of mass

Op + 0,7? r2 pu = 0 (1.2)

where p is the gas density, u the gas velocity, t

the time, r the radial distance;

where p is the gas pressure; and finally

(3) the differential equation of conservation of en- ergy

p P\ _

Or (2) + ud, (2) =0 (1.4)

where + 1s the adiabatic Index, a constantprop- — ˆ

——— ertyofthe gas (y = 1.4 for air) Itwas cleartoGl[ ˆ

that at this early stage viscous effects could be

neglected and the gas motion could be considered

as adiabatic

These equations should be supplemented by the boundary conditions at the shock wave front:

07 (uy — D) = —poD (1.5)

where p;,uy are the density and velocity just

behind the shock front, D = dr; /dt is the shock front velocity, rs is the shock front radius, and

po is the initial gas density of ambient quiescent

alr;

(2) the condition of conservation of momentum

ps(us -D)? +pp=potpoD? (16)

shock front, and pp is the gas pressure of ambient

quiescent air; and

(3) the condition of conservation of energy

p)|_?—P/ „ 8=

p(r,0) = po; p(r,0) = po; uír,0) =0 (r >rọ),

p(r,0) = pi(r); plr,0) =p,(r); tuír,0) = u¿(r) (r < To),

Here ro is the initial radius of the shock wave, in fact

the radius where the shock-wave outstrips the thermal

wave; the constants pp and po denote, correspondingly, the initial pressure and density in the ambient air, and

E is the energy, concentrated initially in the sphere

of radius rp The functions p;(r),p;(r),u,(r) give the distributions of density, pressure and velocity inside the initial shock wave respectively

Non-specialists should not despair when looking at these equations Indeed, this primary mathematical

model is so complicated that until now nobody has been

able to treat it completely analytically Moreover, the problem statement presented above is, in some sense,

incomplete because nobody knew then or knows now

distributed inside the initial shock wave

GI however was astute His ability to deal with un-

solvable problems, by aparently minor adjustment con-——

tive mathematics, was remarkable And here also GI

took several steps of crucial importance which allowed him to obtain the solution which was needed in a simple

and effective form In addition, this solution allowed

him to overcome the lack of knowledge of details of the

initial distributions of the gas density, pressure and

velocity

These steps were as follows:

(1) G.I Taylor replaced the problem by an ‘ideal’ one

As he has written (Taylor 1941; 1950a; 1963), this ideal problem is the following:

A finite amount of energy is suddenly released

in an infinitely concentrated form.’ This means

ss‘ that ro is taken equal to zero, that is the explo-

^ ™ 29 s afr wey Cs ms ava ™ cr ava oo "1O ˆ = re fs ^ ^^ arr 7

of energy It is clear that neglecting the initial

radius of the shock wave ro is allowable (if at all!)

only when the motion is considered at a stage

when the shock front radius r; is much larger

than ro If the initial shock-wave radius is taken

equal to zero, then the initial distributions of the air density, pressure and velocity inside the initial shock wave disappear from the problem

statement

(2) At the same time, GI restricted himself to con-

sideration of the motion at the stage when the maximum pressure of the moving gas reached

at the shock-wave front is large, much larger than the pressure pp in the ambient gas This

_ allowed him to neglect the terms involving the — -

(1.6), (1.7) at the shock wave

r7

The question was: what are the quantities on which

the shock wave radius r+ depends? In the original ‘non-

ideal’ problem they are obviously

E, PO; t, T0;P0; 7

The units for measuring these quantities In the c.g.s

system of units are

gem?

[E] — s2 › [Po] — — l = S, [ro] = cm, [po] — em s2 ` g

whlle + 1s a dimensionless number We shaill see later

how important it was that GI neglected the last two quantities ro and po replacing the problem by an ideal one

GI’s next step can be represented in the following way He constructed the quantity

/

\ po /

which is measured in units of length This means that

if we replace cm by another unit of length: m, mm,

um, km , or in general by cm divided by L, where L is

The quantity J depends in principle on the same

quantities as rs, and this dependence can be repre- sented, neglecting rp and po, as

where F is a certain function The arguments rp and

po were neglected, a step of crucial importance The argument 7 is an abstract numerical constant The first

three arguments of F have independent dimensions

This means, in particular, that the time t is measured in

time units, i.e seconds, or s/T where T is an arbitrary

positive abstract number Units of time are absent in the dimensions of the first two arguments Therefore,

by varying the number T we will vary the numerical

value of t while leaving the values of J_and two other

I likewise does not depend on po Similarly J does not

depend on S: by varying the unit of length we vary

, bu e value of J remains invariable us, the

function F’ is nothing but a constant depending on the value of 7, and so the famous G.I Taylor scaling law for -

the radius of the shock wave is obtained,

Later, GI’s processing (Taylor 1950b; 1963) of the

photographs taken by J.E Mack of the first atomic ex- plosion in New Mexico in July 1945 confirmed this scal-

ing law (Figures 2 and 3) — a well-deserved triumph of

GI’s intuition We see how important it was to neglect

the arguments ro and po, the initial radius of the shock

stead of the two arguments r and ¢ in the system

Figure 3 Logarithmic plot of the fire ball radius, showing

that re * is proportional to the time ¢ (Taylor 1950b, 1963)

(1.2}-(1.4), GI obtained one single argument r/r; in

his solution and so was able to reduce the problem

of solving partial differential equations to that of the solution of a set of ordinary differential equations The

solution was sufficiently simple that he himself was able to make all necessary numerical computations In

particular, he showed that the constant C in the scaling

Friday 27th June 1941 The great American mathe-

matician, J von Neumann, also involved in the atomic

~~ problem, submitted his paper (von Neumann 1941; see — ˆ

12

the solution in closed form Later, the solution of this

problem was obtained in the Soviet Union by L.I Sedov

(Sedov 1946, 1959) who also found the energy integral

Therefore tt hut; ‘dered al ;

called the Taylor-von Neumann—Sedov solution

blatt 1979, Barenblatt et al 1980) was as follows: look

at the fire ball surface Radiation losses, dust, moisture

- all these effects lead to energy losses at the shock- wave front Apparently these losses are not large, in view of the excellent coincidence of the data with GI’s

the shock But nevertheless, as a matter of principle,

let us adopt a simple model to account for the energy

oy L3 - n mn“ ˆ

e e e

_ self-similarity Namely, assume that the rate of loss —

of energy at the front, per unit mass per unit time, g, 1S

proportional to the temperature at the front:

(2.1)

We can represent an arbitrary negative value of the constant K as

(xì — 1) — 1)

where 1 < +; < + The differential equations (1.2)-

1.4) 4 bing t] ion inside the shocl

nt] | lo the shocl Front Hi

tions (1.5), (1.6) and the initial condition (1.8) The

only difference is in the energy condition (1.7) at the

~ shock wave front, where the energy loss (2.1) should be -

en in unt The trick 1 is condition has the same form as (1.7), but instead of 7 there appears the effective adiabatic index 7, < y For 7; = y we

obtain the G.I Taylor solution For 7, < y we are

seemingly able to do exactly as GI did for y, = +, and consider the ideal problem of concentrated explosion The only difference in the solution should then be that the quantities involved should depend additionally on

an argument y, Therefore, apparently, we should ob-

our basic assumption concerning the losses, because

—_ “=4mj{-p(up-D)KP, dt Pf

The first factor on the right-hand side is the current

shock-wave area, the second is the mass flux across the shock wave, the third is the rate of energy losses Thus,

only the trivial solution, when the energy € is equal to zero, can satisfy both requirements This trivial solu-

The contradiction obtained shows that the solution to

the ‘ideal’ problem in GI’s sense does not exist for 7, < ^

reproduced numerically,thereby checking the numeri-

cal procedure However, for 7; < 7 the situation was different (Figure 5)

For each 7, a straight line, 1.e a scaling law to which

the solution rapidly converged, was again obtained

However, the slope of this straight line was different from the value predicted by dimensional analysis It depended on both 7; and ¥, but not on the initial condi-

Figure 5 Numerical calculations of the modified problem

of the very intense explosion showed that for 7, < +7 there exists a scaling law for the shock-wave radius rs However,

problem with finite ro Therefore the relation for r,

cannot be represented, generally speaking, in the form

of a simple scaling law (1.12) Instead, the same type

of consideration leads us to the relation

and now the problem statement is fixed and no further choices are possible: we cannot require the function F' not to depend on the first argument n = ro/(Et?/po)s When solving the ideal problem of a concentrated ex- plosion, GI put 7 = 0 and the required function F then depends on 7 alone We tried to do the same However, long ago classics coined an eternal truth: ‘Quod licet Jovi, non licet bovi.’ (What is allowed to Jupiter is not

allowed to a bull.) Thus, for 7; < + the solution of the

ideal problem does not exist, and for the following very

power-law asymptotics, viz

lem o£a concentrated explosion, the function ?'(?, +, +1)

is equal to zero, and we obtain an empty relationr; = 0 However, we do not need in fact the solution of the

ideal problem What is really needed is intermediate -

asymptotics: that is, asymptotics valid for times and

distances at which the influence of fine details of initial and/or boundary conditions is lost, although the system

is still far from an ultimate equilibrium

This concept of intermediate asymptotics has an 1m-

portant general significance, and not only in mathemat-

ical physics For instance, this concept is always used

in our perception of visual art (think of Van Gogh!)

Indeed, we have to look at paintings from distances

great enough not to see the brush-strokes, but at the same time small enough to enjoy not only the painting

as a whole but also its important details (Remem-

ber also Gulliver’s Travels by Jonathan Swift (1992)

Gulliver’s impressions of the fine details of the skin of

a giant Brobdingnag beauty, who had the custom of

putting him upon her breast, are especially instructive

from this viewpoint It is clear from his description that her admirers restricted themselves to an interme-

diate asymptotic perception of her!) As far as I know

the concept of intermediate asymptotics was formally introduced into mathematical physics by Ya B Zel- dovich and myself (Barenblatt & Zeldovich 1971, 1972;

The relation (2.6), valid for ry large compared with

ro, but at the same time ry small enough to ensure

and density of the shock-wave at the front are valid:

Here

Dp-! _ 21-8) rs (2.10)

dt 5 t

is the speed of the shock-wave front

Also, the pressure, velocity and density inside the shock wave are given by the relations (1.15), but the

‘scales’ ry, ps, uz and py are different They are deter- mined by the scaling laws (2.7) and (2.8)

In GI’s case 7; = 7, the exponent @ was equal to zero

dinary differential equations satisfying the appropriate

boundary conditions does not exist However there is

an exceptional value of @ for which such a solution does

exist We thus obtain a nonlinear eigenvalue problem, whose numerical solution is rather simple and whose

results coincide with those obtained by numerical inte- gration of the system of partial differential equations mentioned above

Dimensional analysis:

Intermediate asymptotics

Thus, we have achieved a self-similar solution to the

solution for a very intense explosion The important difference is that this solution contains time exponents

which are continuous functions of the adiabatic indices

y and y,, and are obtained by solving an eigenvalue problem rather than from simple dimensional consid-

erations as in GI’s case 7; = y In particular, the gov- erning parameter for these solutions is not the energy

E, but the quantity

5S

A= Er,” (3.1)

having the ‘anomalous’ dimensions M L?† 1“5T~2, Self-

similar solutions with such anomalous dimensions—

which cannot be obtained by dimensional analysis but require the solution of a nonlinear eigenvalue problem—

began to appear in fluid mechanics in the middle of

the 1940s The names and works of the first pathfind- ers, Guderley (1942), von Weizsacker (1954), Zeldovich

21

(1956) (see also Zeldovich & Raizer 1966, 1967) should

be mentioned here

Let us look at the problem from the group-theoretical

viewpoint In GI’s case, 7; = ¥, it is possible to replace

where B < 11s an arbitrary positive number, and noth- ing will change in the asymptotic solution (1.12), (1.14), (1.15) In other words this solution is invariant with

respect to a transformation group (3.2) and B is the group parameter The case 7, < 7 is different: the

asymptotic solution is invariant with respect to the

more complicated transformation group

Ty = BTo, FT = DT, Tr = DT

p; = B“pr, u;= Brus, p =p

This i mm] le of what ¡ led t :

ill be ile to give here a general defini- tion of this important concept as it is now understood (Barenblatt 1993b) The derivation of the self-similar

— form of the solution to G.I Taylor’s ideal problem was

essentially based on dimensional analysis (for details

see Barenblatt 1987.) This is in fact a sequence of sim- ple rules related to the fundamental covariance prin- ciple: all physical relationships can be represented in

a form that is equally valid for all observers Let us consider a certain physical relationship

mensions This means that by choosing appropriate

fundamental units it is possible to vary the values of

Q,, ,@, arbitrarily and independently:

đi = Aja), ., @, = Agag, (3.5) where A), ., Ax are arbitrary positive constants At the same time the dimensions of a, 6), ., b,, can be

where [7] is the symbol for the dimension of quantity 7,

introduced by James Clerk Maxwell Therefore, by the very definition of dimensions under the transformation

(3.5) of the parameters with independent dimensions, the values of a, b;, ., 6,, transform as

the covariance principle, the relationship (3.4) can be

represented as a relation between the dimensionless

where the invariants have the form

¬ Qa

an ay: ay

b, YW; = = (¡ = Ì, , m) (3.9)

Ất cớ,

Returning in (3.8) to the previous function f and its

arguments, we find that every function f entering a physical relationship possesses the fundamental prop- erty of generalized homogeneity:

Let-us-now consider the reduced form (3.8) of the —

physical relationship (3.4) What happens if in (3.8) certain dimensionless parameters IT,,; , ., Im, cor-

nomenon we always neglect the factors which are con- sidered to be non-essential and the main argument is

very often that the relevant dimensionless parameter

is small If there exists a finite, non-zero limit of the function ®(II,, , Tj, Wi4i, , Um) when M14), ., UT, tend to zero, then, for sufficiently small values of IT; +, , IIlm, the function ® can indeed be replaced with