* Communicated by J. Okabe.
--- V2
19 dV2
dt (1.2)
far the energy-containingeddies in their initial period or
decay [2] §6.1. NamelY,
.Faculty WbMT
Dept. of Marine Technology Mekelweg 2, 2628 CD DelftThe Netherlands
Reports of Research Institute for Applied Mechanics Vol. VIII, No. 30, 1960
ON THE THEORY OF HOMOGENEOUS,
.AXISYMMETRIC TURBULENCE (IV)*
By Michio 0111I
Summing up typical results of the existing observations concerning the decay of homogeneous axisymmetric turbulence, a tentative theory, which aims at an explanation of the general trend, is presented on the basis or some bold simplifications and assumptions. There are, in particular, dis-tinguished two substages of the decay: an interacting stage' and a 'locally isotropic stage '. Essentially, the interacting stage is characterized by a strong tendency of the smaller eddies to local isotropy, while the locally isotropic stage is characterized bya double structure composed of the trans-verse part at smaller numbers and the isotropic part at larger wave-numbers.
1. Introduction. A tendency of turbulence to a homogeneous and isotropic state has been long known. This seems not surprising, considering that turbulence is a
statistical system composed of a very large number of constituents, i.e., 'eddies'
through interaction, of whatever kind,
between eddies, the initialdistribution would fade away as time elapses, and in the limiting state we have no reason to suppose any preferred position or, direction so long as no external agent exists.
The circumstances are somewhat analogous to the molecular kinetictheory of gases,
but to be remembered the analogy is not so complete in many important respects.
Especially, there is a considerable difference of characterizing relaxation times. For gas molecules of mean free path / and mean speed c, the characteristic time r4 may be represented by r4 =11c, whereas for eddies of mean size L and speed V we have=- LI V as their characteristic time.
Now, with the aid of the Reynoldsnumber -=; LV I v---LV Ic it follows
x R c2
where c is roughly equal to the sonic speed, showing that under ordinary
condi-tions (c>V
, R>1) the eddy relaxations are taking place much slower than themolecular ones: In addition, turbulence is a dissipative system.
Experiments on the decay of grid-turbulence have established the essential result:.
'
20 M. OHM
within the characteristic time of these eddies, a considerable portion of energy is dissipated. These facts suggest that in the case of turbulence a non-uniform transi-tory state is not so simply negligible as in the molecular case. In fact recent
ex-perimental work, e.g. [5] [17] [18], tends to show highly persistent structures of the energy-containing eddies or a weak tendency of turbulence to a uniform state. See also [19] § 3.8.
However, the characteristic time T. defined above is considered to be de-pendent on the scale L of eddies. Noting that
T. --= (VIL)--' (1.3)
is a rough measure of the velocity derivatives and hence of a magnitude of the vorticity which will increase as L decreases, will be small enough for smaller eddies. Thus, the fine structure of turbulence has a possibility
to settle into a
statistically uniform state even when the over-all structure is non uniform. This is nothing but the basic principle of 'local isotropy hypothesis' which has been partly supported by experiments [2] Chap. 6.
For conciseness' sake, let us confine ourselves to a non-isotropic state, as-suming the homogeneity of the field from the beginning. A tendency to isotropy
in a motion of homogeneous but non-isotropic turbulence may be ascribed to the
action of pressure forces, and a qualitative explanation has been given by G.K.
Batchelor [1] and [2] § 5.2. It is essentially a rough use of Bernoulli's theorem,
where the release of the pressure head must be isotropic while the contributions from the velocity head is non-isotropic. But a complete analytical theory has not
been given so far, nor it seems to be readily obtainable.
At any rate, we have now to do with the directional transfer of energy
be-tween different velocity components, and the problem will make its appearance in the simplest context when the turbulence has axisymmetry, or the simplest case
of non-isotropy. After the previous paper of the author [11] [12] [13]
(referred to
hereafter as 1, II, III, respectively) on the basic kinematics and formal dynamics of homogeneous axisymmetric turbulence, we shall turn to this problem in
what follows, referring to a few experimental results published in these several years.
2. Experiments concerning axisymmetric turbulence.
Even in the ordinary
wind-tunnel turbulence behind a standard grid, the motion associated with the lowest
wave-numbers is known to be non-isotropic throughout the whole history of its decay, but except in the final period, where thelargest eddies predominate, the non-isotropy is usually masked by the isotropic distribution of the bulk of the energy
[4]. A field of turbulence which is distinctly non-isotropic, on the other hand, can
be produced in practice by imposing a uniform distortion upon such a nearly iso-tropic turbulence behind a grid. Two examples of axisymmetric distortion should be mentioned here :
(1) Passage through a fine wire gauze
A wire gauze of extremely fine mesh (about 1 mm or less) placed at right
angles to the main stream has an effect of reducing the axial component (u11) of
down-THEORY OF AXIS YMMETRIC TURBULENCE (IV) 21 stream side the turbulence becomes axisymmetric about the direction of mean flow,
and in particular 11112 < u1.2 when the incident turbulence is isotropic. The theory
of this gauze-effect has been refined by G. I. Taylor & Batchelor [16], and then
checked by A. A. Townsend [17] experimentally. This method of obtaining axisym-metric turbulence is fairly simple, but an appreciable degree of non-isotropy is nec-essarily accompanied by a large attenuation of the absolute energy level.
(2) Axisymmetric contraction of duct
If a uniform turbulent flow is accelerated by an axisymmetric contraction, the vorticity filaments are elongated axially and contracted transversely with the result that the mean square of axial vorticity (o)//2) exceeds the transverse one
(0)1_2). This causes a reduction in 14112 and an increase in u12, and thus
axisym-metric turbulence such that i.-47(2 < ui2 is realized again. The modern theory of this phenomenon is due to Batchelor [2] § 4.3 and also independently to H. S. Ribner
& M. Tucker [14]. Careful experimental tests were made by the later authors as well as by M. S. Uberoi [23].
Now, we are interested not in the processes of distortions themselves but in
the behavior of axisymmetric turbulence once produced in either of these ways. Although available data are only a few at present, it seems that they are revealing the essential points to some extent. Typical measurements are as follows :
(a) Townsend's measurements of turbulence behind gauzes
Townsend [17] used four kinds of gauzes of various resistance coefficients. His results of measuring the down-stream variations of turbulent intensities are re-produced in Figs. 1 (A)--(D), where the air-speed U was 620 cm/sec, the mesh-length M of the grid was 5.08 cm, and hence the grid Reynolds number RmUMII, amounted to 22,000. In every case, the gauze was placed 79 cm (-15.5 M)
down-stream from the grid)) Townsend also experimented with several other sets of U and M covering the range 12,-8,000 to 32,000, but corresponding decay curves such as Fig. I are not presented. Further, by measuring the attenuation factors
for the mean squares of the velocity derivatives, Townsend verified the relation :
(atill)2
(= e'
axi , say) 2
closely behind the gauze (the xi-axis is in the direction of the stream). As is well-known, (2.1) is a necessary condition for local isotropy in the sense that the small-scale components which make dominant contributions to the mean square velocity derivatives are practically isotropic.') Summing up, Townsend concluded that a
short period of comparatively rapid interchange of energy between the velocity
components was followed by the locally isotropic stage in which the return to
isotropy became distinctly slow.
(2.1)
Various experiments for grid turbulence with moderate Reynolds numbers have shown that a distance about 15M.20M is required .before the onset of initial period of
decay.
A corollary is that local isotropy is also conditioned by the isotropy of the vorticity or the directional equipartition of the dissipation.
1.2
0.4
3 and 4. (From M., S.. Uberoi [23]) * A _ (A) - k=2.94 , ,.. , (C) ,k=1.12
I...
A.... ..
\
. ...,... + ....- ' --- ...\
... S....* .'"...
4. ...I. 0,... +:
t
t
(B) CD) 11: .,i k=1.70) ,. ,Ip--0.47 ssti
... x, ... .,---
-\
+...4. + 114.-,- -,-..4.=t
4 -3 2 (i-z-",-//p),v2 ___
_ 20 30 contraction 40 50 x/IV1Fig. 2. Decay of axisymmetric turbulence in a contracting strea-rn.
Rm = 3,700. 4 : 1 contraction. Note that the ordinate is
proportional to (i)1I 2 and (L7/2)1/2 in contrast to Figs.
22 M. .OHJI
144 10. 20
t
x/A4 10 20Fig. 4. Decay of axisymmetric turbulence behind the. gauze.
Rm= 22,000.
////2 u±2, T42 = u//2
The position of the gauze is indicated by an arrow. (From A. A. Townsend [17])
1
(b) Uberoi's measurements of turbulence through contractions
In Fig. 2 the effect of contraction on nearly isotropic turbulence produced by a grid of M= 5:08cm is illustrated,, where U= 109.7 cm/sec (before
contrac-0 1.2 0 8 0.4 , -I-10 1, +
tion) so that contraction ratio was 4: 1 [23]. The contraction began at 21 M (=
107 cm) and ended at 31 M (-158 cm) behind the grid. Uberoi measured the var-iation of e', namely the approach to local isotropy, too (Fig. 3(A)). Figs. 3(B) and (C) show the results of similar measurements for other values of contraction ratios. It is seen that within therange of observation local isotropy is not attained.
Uberoi's another set of [ 24 1 extended far down-stream. He measured the var-iations of u//2,
iii
(Fg. 4)and of (01/2 / CO-12 (in place of (Fig. 5) this ratio be-comes unity for local isotropy) as well as the ratio of
direc-tional transfer of energy to the rate of viscous dissipation(Fig. 6). A remarkable fact is that
in early locally non-isotropic stage energy transfer is
signif-icant indicating a strong
in-teraction between the axial
and transverse velocity
com-ponents. On the contrary, a
less violent interaction and an
weaker tendency to isotropy
is peculiar to later locally iso-tropic stage.
3. Interpretation of the ex-perimental results. As is seen
from the various curves in
§ 2, the behavior of axisym-metric turbulence must depend more or less on the initial
con-ditions and the external pa-rameters. An well-organized quantitative description of these dependencies will not be possible until that much more systematic experimental data have been accumulated. Nev-ertheless, ignoring the details for the time being, the general trend may be abstracted in the following way :
(1) When the turbulence is locally non-isotropic, the
re-THEORY OF AXIS YMMETRIC TURBULENCE (IV) 23
measurements for 4: 1 contraction and I?, 8 4 2 10 10 r 9 1 8 10 20 20 16 1 16 12 8
-.0-
-Cr 4 .0.-1 = 13,200 4 3 15 4 3 2 2 C-10 20xim
30 40 (C)Fig. 3. Variations of X and e' (defined as (2.1)) 0 in a contracting stream. Rw=3,700. Contractions are 4 :1 9 :1 and 16: 1
respecti-vely. Arrows indicate the end of contraction
(From M. S. Uberoi [23] ) xim30 40 (B) 4 2
-
0- 44-N- --W.' 20 30 40 50 x/ii/ (A) 8 6 4 2 ,It e 6
2 0 -1 5 1 0 0 5 2 0-10' 20 40
Fig. 4. Decay of axisymmetric turbulence in a con-tracting stream. Rm = 13,200. 4 :1 contraction.
u/12 14,2.
142/3-(From M. S. (Iberoi [24])
20 40 60
M
Fig. 5. Tendencies of axisymmetric turbulence to isotropy and local isotropy. Rm= 13,200 4: 1
contraction. Ou 2/,,,,2 ,
(From M. S. Uberoi [241)
40
Fig. 6. Energy transfer between the axial and the transverse components. Rm = 13,200. 4 : 1
contraction.
0 (rate of transfer)/(rate of viscous loss) for grizi X (rate of gain)/(rate of viscous loss) for tint.
(From M. S. Uberoi[24])
turn to isotropy (the
equiparti-tion of energy) is fairly rapid,
being characterized by a strong
interaction between the velocity components.
But local isotropy (the equi-partition of dissipation) realizes
at a somewhat faster rate than
the equipartition of energy.
Once the locally isotropic structure has been established, the tendency to isotropy becomes considerably slow.
These suggest that two typical substages of the decay of
axisym-metric turbulence may be
distin-guished in the initial period.3) For
convenience, let us call them an
'interacting stage' and a 'locally
isotropic stage ', respectively. In the former stage intense activity of the smaller eddies will play an es-sential part,
while in the
latter stage a double structure of the en-ergy-containing eddies will be im-portant. Before entering into these particulars, we shall add a fewre-marks in favor of this point of
view.
(a) In Figs. 1(A) and (B), there
is observed an abnormal increase in un2 far down-stream. According
to Townsend [17], this effect ap-pears only when an appreciable level of turbulence is incident on the gauze of high resistance and may be attributed to either con-vection or instability of the mean
velocity pattern.
Thus we had
better to suppose that such aphe-nomenon is not representative of
3) The final period of the decay of
axisymmetric turbulence has been considered in Ill ; § 5 and § 6.
24 M. OHJI 1.5 10 0.5 4 3 2 _o_ 1 X 60 20
10
THEORY OF A XISYMMETRIC TURBULENCE (Iv) 25
mogeneous turbulence and so we shall omit this anomaly from the present con-siderations.
(b) In Townsend's experiments the approach to local isotropy seems to have been
very rapid. And yet we have no doubt that a considerable amount of energy was
redistributed among the velocity components within a short period. For, compared with Taylor-Batchelor's theoretical predictions, where the nonlinear inertial effects are disregarded at all, the observed degree of non-isotropy of transmitted turbulence was significantly less (Fig. 7), although the reduction in total energy was predicted
0.5
Fig. 7. Attenuation factors 77/i (for ui,2) and 77J_ (for u,2). theoretical, mean experimental. The groupings of experimental points are roughly represented by line segments or enclosed areas.
(From A. A. Townsend [17])
by the theory with adequate accuracy (Fig. 8). In Uberoi's experiments, on the other hand, the return to local isotropy was much slower (Figs. 3 and 4). Such a difference originates probably in the dissimilar initial conditions.
(c) To get a rough idea about the amount of energy interchange between the ve-locity components, a crude estimation will be made from Townsend's measurements
of the attenuation factors (Figs. 7 and 8). Let denote 77il = UT/2/u02, 7h
=
and 77 = 1412/31l02 -F 277 )/ 3, where u02 stands for the mean square velocity component in the isotropic turbulence far upstream the gauze, and let use thesu-perscripts th and ex to designate their theoretical and experimental values. Now, our present assumptions are : ( i ) the theory describes the earliest
state of the
transmitted turbulence correctly, while the experimental points show the results ofinteraction, (ii) the duration of the interacting stage is so short that viscous loss
can be neglected, (iii) 77" = 77"(=77, say), (iv) the process of interaction does not
2 3 4 -5
=
=
26
it
0 5
M. OHJ1
1 2 3 4
Fig S. Attenuation factor 77
la (for
u2).The groupings of experimental points are roughly represented by line segments or enclosed areas.
(From A. A. Townsend [17])
cause any appreciable change in the structure of the larger eddies.
Then, if a is
a fraction of energy transferred between the velocity components, and is thatcontained within the isotropic range, a little consideration leads
3,Y17 = 2( _17X) , (3.1)
and
nix (1-13) nil' + fin 721' or 19
771'
Making use of the mean experimental curves, the following table is obtained :
(k: resistance coefficient of the gauze)
It may be roughly concluded that about 15% of the total energy is rapidly inter-changed and about 5096 of the total energy, distributed over the higher
wave-number range, settles into a state of equipartition. Indeed, we cannot attach too (3.2) 7111; 7711 '7`
a
1 0.194 0 543 0.299 0.491 0.427 0.08 0.45 2 0.046 0.345 0.149 0.293 0.245 0.14 0.52 3 0.020 0.251 0.102 0.210 0.174 0.16 0.53 4 0.019 0.199 0.075 0.171 0.139 0.13 0.47 5 0.022 0.166 0.067 0.144 0.118 0.13 0.47=
+
THEORY OF AXIS YMMETRIC TURBULENCE
(w)
27great significance to these figures, but it seems permissible to suppose local isotropy includes a considerable fraction of the energy-containing eddies at these Reynolds numbers ( R11---0( 104) )
To be remembered in this connexion, the existence of a unique statistical state of the energy-containing eddies has been known for a wide class of
grid-turbulence [2] § 7.2. For instance, R. W. Stewart & Townsend [15] have
meas-ured the two lateral velocity correlations for various grid shapes and found that about 80 % of the total energy is approximately independent of the initial
condi-tions. A similar tendency of the small-scale components to be independent of the
initial conditions was also observed by H. Tsuji [20] [21] for turbulence behind two grids.
Up to the present, various experimental tests of local isotropy have been made by many authors. Their common result is that when the Reynolds number is mod-erately large, local isotropy can be practically justified for most of turbulence
ex-cept near the wall. Lack of local isotropy is presumably associated
with a high
rate of mean shear which reinforces the vorticity in a preferred direction. In aturbulent flow without mean shear, therefore, a locally non-isotropic structure is not
persistent by the nature of the case.
Thus, we shall be safe to consider that the locally isotropic stage is an
es-sential part for the decay of homogeneous non-isotropic turbulence. A slow return
to isotropy during that stage, being in agreement with the result of Townsend's
another experiment on the turbulence passed through a non-axisymmetric distorting duct [18] , seems to be typical of decaying boundary-free turbulent motions.
4. A tentative theory for interacting stage. Let us start here with the dynamical
equation of the form : a
a, (8± - 8B) + 2in-2 (61
611) t- Ery = 0,1; (6.17), where EN(K,I) and i(r, t) are the axial and the transverse spectra of
energy, respectively, and t) represents inertial interactions between them (cf.
I; (6.15b)).
In principle, must be expressed as a certain multiple integralmanifesting complicated links between different wave-number vectors, and the equa-tion like (4. 1) is only the first member of an infinite chain of equations which connect the spectrum functions of successive orders in turn. The state of affairs
is similar to, but even more troublesome than, that in isotropic turbulence, for we
are now concerned with the cascade process and the directional _transfer of energy simultaneously. in the present paper we shall be contented with some intuitive
considerations of avoiding this difficulty as simply as possible.
First, we assume that the interaction function t) depends, in effect, upon the quantity 45 (A-, t) = 5_,_-5,,4) on the one hand, andupon the characte-ristic time T .(A- , I) of eddies associated with the wave-numbermagnitude
on the
other, i.e.,
(4.1)
This equals to -V(, t) defined in I; (4.11 b). Weassume it positive (i.e., ki2<11i.2) hereafter for definiteness.
.(d)
28 M. OHJI
E, (jr, 1) (A
;T).
(4.2)Then, as a general matter of fact, the value of SE, is expected to be the larger, the larger is z c' and the smaller is T The simplest form of (4.2) with this pro-perty is
= 4c
46(.,1)
t)
T(tr, t)
c being a disposable numerical factor. Thus the equation (4.1) reads
[3 at + 2 (v
+
t-)146,
= o.Now, if I/(ff, t) is the characteristic speed for eddies of the size proportional
to we have
1 1
T(,, t)
A.v (ff, X V(r, t)Or
T,,--1 = A-2N(K ,t) ; N (K , t) V (K , K. (4.5)
The function N (K , t) is a quantity of (velocity) x (length), and has a meaning
a-nalogous to the molecular kinematic viscosity v. After the manner of W. Heisenberg [61 [71, let us regard it as an effective eddy kinematic viscosity due to the motion of the eddies smaller than K-1, on assuming
N , t) =
rf
,/5(,,,
, (4.6)where r is a constant, and 5 should be understood. A still further simplification is to assume that N is independent of the decay time t, namely N
(K , t) N (K), in view of the square root sign in the integrand of (4.6). Under
these assumptions and simplifications, the equation (4.4) can be integrated to give
(,)
(K, t) = C (K)expE-2vK2(1 ti){1+ cN }i (4.7)
in which t, is a time origin and C (,) stands for the initial profile z/F_;: (ff ,t1). If,
therefore, the eddy viscosity or a kind of 'Reynolds number' cN(K)I v is known together with C(K), we can find the energy difference through
.M(0=
(u1L2u112) =f 6 t)2 (4.8)
But considering the crude nature of the present arguments, we have to proceed here in a qualitative way.
It is known that when the Reynolds number of turbulence is large, there may exist a higher wave-number range where the energy spectrum falls monotonously with and where the direct effects of viscosity are negligible. Within this 'inertial range ', two extreme cases should be noticed :
(4.3) (4.4) , t) .
d/
=
THEORY OF AXIS YMMETRIC TURBULENCE (IV) 29( .(i)
5 )
In this case, we have from (4.5) and (4.6)N (4.9)
and so V (1C)---qcN (r)-10.. When 8 (r) decreases at a slower rate than r'r, V(lc)
must increase with which is contradictory to empirical knowledges. ,5 In this case, we have
N (r),--40 (4.10)
When 5 (r) decreases at a faster rate than r-3;T, (lc) must increase with A-,
which is again contradictory to empirical knowledges. Consequently, it follows
that
N (IC)"'/CP *i 1 <p < 2,,, (4.11)
for the inertial range. Of course, such an inverse power law as (4.11) cannot be
authorized elsewhere, but since N is, at any rate, a certain monotonously decreasing
function of r (provided that (4.6) is assumed), we shall make a very bold approx-imation of substituting (4.11) into (4.7) at once without regard to the magnitude of lc. Then (4.7) reads,
t) = C (r) exp, 2
(v/r2' ar2-P)(t-11)11, (4.7')where a arid p are considered as constants. At large Reynolds numbers, the
viscosi-ty is unimportant, and the above expression still reduces to
.4 6 (K, t:) = C(x)exp [-2alcg(tt1)] 0 <q < 1,. (4.7") This shows that 4'5. falls off more or less exponentially with r and t,, representing
a tendency to local isotropy. The lower wave-number range becomes more and more important as t increases, and in view of the fact that C (K) = Cur4 -I- 0(,e5 In r) (Ill; (3.10)) we have in the limit of t--)co
dE(t)Cof
letexp [-2a(tti)x,i] clff =(-5
[2a(t t1)1 5/ul0 q q
(4,12)
T' (x) being the gamma function.. Notwithstanding an apparent resemblance to the theory of the final period of decay (ii!; p. 263), however, the asymptotic law (4.12) is likely to be of little use because the underlying assumptions must be no longer acceptable for large values of (tt1).. 'We are suggested at most some
in-verse power law
E t Ell Ep(t (4.13)
to represent the general trend as a whole. An appropriate value of m 0) re-mains undetermined here, but perhaps it does not exceed 51q.
On the other hand, referring to the experimental results described in § 2 (cf. Figs. 1 and 4), it is not far from being the truth to assume another inverse power law
2E1 ± E0 (4.14)1
for the decay of total energy. 'The decay law of this type is familiar in the theory of isotropic turbulence, where n 1 is widely adopted [2] § 7.1. Recently Tsuji [22],
(o)-/c',
(K)" ;=
(2
(t
-30 Nit OHM
has criticized such a linear decay law both theoretically and experimentally. Al-though the similar circumstances must exist in the axisymmetric case too, we shall
not enter into their details now.
Then, from (4.13) and (4.14), we obtain
E,
- [E0(tt0)-"
E1(1-11)-"1,and (4.15)
E // = 31
[Eo (t to)-" 2E1(tt1)-1],
respectively. As is stated above, we have no theoretical means to determine m and n at present. But to observe the general tendency, for the time being, an
illustra-tive plot of (4.15) is made in Fig. 9 with m=5/2 and n=1 tentaillustra-tively, on
assum-ing t1=0, where EWE() and to are parameters. These curves resemble to the ex-perimental ones (Figs. 3 and 4) for initial time intervals, 5) and especially a re-markable feature is that if is initially large, the .E11-curve has a maximum while E, -curve is monotonous, showing an excessive transfer of energy between the velocity components in an early period. Though our present choice of the power indices in and n is quite arbitrary, it seems that the inequality :
5
=2.55) Note that each curve is extended far beyond the proper limit of the interacting stage.
[II
111
I
II
5 15 25 5 15 25 15 25
I; in arbitrary scale
Fig. 9. Theoretical curves for the decay of axisymmetric turbulence in the interacting stage. t4,2>un2
in 5/2, n =1. a stands for E1/E0. (cf. equation (4.15)) + 4.= a=0.5 a=2 3 2 4
THEORY OF AXIS YMMETRIC TURBULENCE
(w)
31in > n
(4.16)is essential to explain the observed rapid interchange of energy during the
inter-acting stage. In short, the interacting stage is characterized by the fact that the decay of total energy is taking place at a slower rate than the reduction ofenergy difference due to violent activity of smaller eddies in non-isotropic state.
To end this section, a question may be asked : when does the interacting stage actually end? It is not easy to answer this, but previous remarks § 3 (c) and
(d) suggest that when about a half of total energy has redistributed in an isotropic
manner, the turbulence will settle into a rather stable structure which is typical of
the locally isotropic stage.
5. A tentative theory for locally isotropic stage. The basic relation characteriz-ing the locally isotropic stage is isotropy of the mean square velocity derivatives or
vorticity (like (2.1)) as well as distinct non-isotropy of theenergy distribution. In
other words, within the energy-containing range, the smaller eddies are practically isotropic, and the larger non-isotropic eddies make only negligible contributions to the vorticity. Hence, the dissipation in the non-isotropic range is also negligible
on the one hand, and in view of (1.3) the characteristic time of the non-isotropic eddies must be large enough on the other. Under these conditions, we have simply
0 for the isotropic range, but is so small for the non-isotropic range
that such a simple assumption as (4.3), or more generally (4.2), becomes inadequate to represent the inertial effects.
However, we have to remember a slow approach to isotropy in that period.
From this empirical fact, it is natural to suppose that the scales of smaller isotrop-ic eddies and of larger non-isotropisotrop-ic eddies are effectively separated. If, on the contrary, these two groups of eddies were considerably overlapping in wave-number space, the observed stability of the energy-containing eddies could not be expected
as a result of their mutual interactions.
Let us again confine ourselves to the case of oblate axisymmetry
in conformity with experimental conditions (§ 2 (1) and (2)).
In III;
§ 7 theauthor has examined some dynamical aspects of the transverse turbulence,6) and
has pointed out that during a period which is short compared with its characteristic time the energy of transverse turbulence is approximately frozen within the trans-verse plane in spite of the presence of energy-transferring pressure forces. Taking these points into account, we shall make the following assumptions :
The larger-scale components of the energy-containing eddies are practical-ly in the state of transverse turbulence (referred to as the transverse part), and the dissipation can be neglected there.
The smaller-scale components of the energy-containing eddiesare isotropic (referred to as the isotropic part), and because of the difference in their effec-tive sizes, multiplicaeffec-tive cross interactions between the transverse and the iso-tropic parts are negligibly small.
The characteristic time for the transverse part is large in comparison
6) Transverse turbulence is defined as a special type of axisymmetric turbulence in which
no axial components of velocities are present. I; § 8.
=
(rir2>ii,72)
with the time scale of the over-all decay.
In consequence of these assumptions, it may be permissible to consider the decay of each part separately. This is the leading idea of the proposed 'double structure
model ', which is now to be formulated in a little more analytical form. (a) Decay of the transverse part
Omitting the viscosity term (assumption (1)), the basic dynamical equation
for transverse turbulence I; (8.5b) reads
a
Y, t) 1"3 (tc, it, t) , (5.1)
at K2
or through integration over the surface of sphere of radius
at Et (K,
+
t) 0,2 11
respectively (cf. ; (8.4a)). F_), is the K-spectrum of energy distributed over the
transverse part, and stands for inertial modulations. Since, however, the details of is not known at present, we shall assume most simply the self-preserving decay, a customary notion in the theory of isotropic turbulence. Then, what kind of self-preservation should be applied here?
It is worth noting that the decay of transverse part will depend on the behav-ior of the largest eddies for which we have in general Et(,, t) J(1)0 -1- 0(i,-5 In K) (III; (3.10)). According to Batchelor & I. Proudman [3] and also III, the
co-effcient J is not necessarily invariant contrary to the past belief. But, as has been
pointed out in III;
5 7, the older predictions about the large-scale structure oftransverse turbulence is still available in an approximate sense provided that its characteristic time is sufficiently large. In the present model, this is exactly the case (assumption (3)). Thus, ignoring the time variation of J as before, we are suggested to postulate a self-preserving law of the form :
t) JK,40(xL) ;
J
const. , (5.4)for the transverse part, where L is its representative scale and 0 is a certain
di-mensionless function of one variable.
The hypothesis like (5.4) has been formerly proposed by T. von Karman and C. C. Lin [8] [9] for isotropic turbulence, yielding the decay law (energy) cc t -10 originally due to A. N. Kolmogoroff [10] . To recapitulate the analysis, the
equa-tion (5.2) is integrated from K = 0 to ic = i.e.,
a
f
(r, t) dx -F S1(e, t) = 0, (5.5) where Et0,, 0'ft
i(1/22)
and 32 M. OHJI (5.2) (5.3) e(K=
t)=
5,(ic,=
=
where
(e, t)
(A-, t)Now, from dimensional considerations, self-preservation requires
St (i,7,1) = J312 L-'712 (A -L) (5.6)
and, since cb and are functions of L alone, (5.5) reduces to an ordinary
dif-ferential equation
Jv2L-312V, (x') 5L ddLt x4cb (x) dx = 0 (5.7)
. o on putting irL = x and ,,'L = x', from which it follows
dL
PI' L-3/2 cc L dt or L cc . (5.8)
The decay law of energy is therefore
co
=
5 t, t) d
-L5 1 x40 (x)dx cc i-"17.
0 o
This type of decay law, being based on the 'permanence of big eddies ', is essentially connected with self-preservation of smaller wave-numbers, and hence
does not properly reflect the behavior of large wave-numbers. By this reason it has been open to criticism when applied to isotropic turbulence (see e.g. [15] ), but
in the present case such a difficulty seems to be not so objectionable because of large characteristic time and large scale of the transverse part.
To be noted further, since the direct effects of viscosity are quite unimportant
within the transverse part, there must exist an escape of energy into the isotropic
part caused by the coupling process localized at intermediate wave-numbers. Al-though the two parts, the transverse and the isotropic, are decaying in their own manners respectively, they are not supposed to be utterly decoupled, in the sense that the latter acts as a sink of energy for the former.
(b) Decay of the isotropic part
For the over-all decay of the isotropic part, the behavior of wave-numbers
at its lower end is assumed unimportant. The appropriate law of self-preservation
in this case is of the form :
= ox(d) (5.10)
where is the energy spectrum of the isotropic part, and its representative scale 1 together with 5o are functions of time. This is what is called the quasi-equilib-rium hypothesis [2] § 7.3, from which a similar analysis as (5.4)-(5.9) yields (see also [15] )
/ cc tu= (5.11)
THEORY OF AXIS YMMETRIC TURBULENCE
(w)
33and
34 M. OHJ1
.,
Et = J-
5 t (A-,, t)IdK cc I-112. ,(5.12)o
In short, the essential point is that notwithstanding the presence of the transverse
part, the isotropic part behaves as if it were alone because of rapid self-adjustment characterizing the locally isotropic structure.
Combining, now (5.9) and (5.12), we have in. general
Ej
Et + Et --=EL,(t
t2)-1, + r=1.0b7, 1and,. . (5.13)
--ar- E2
for a proper choice of the time orgin, E2, E3 and t2 being certain constants.
The-oretical decay curve for E, and Ell are shown in Fig. 10, and the variations of EilEtt are in Fig. 11, for several values of the parameters E2/Ea and t2. It will be
seen that an approach to isotropy is distinctly slow in the locally isotropic stage.
No direct comparison with the experimental curves li,ke Figs. 1 and 4 is, however,
intended there, in view of the qualitative nature of the present theory.
Lastly, it must be remembered that the locally isotropic stage is conditioned by
L > 1
(5.14) 12--= t: = 0 b=-0.5 H G=0.5, , b= 0.5 b=1 li=11 b =11 . . " 1 1,=-2 b=2 h=2 I'
--.'---"'--7---L--- -1,5 25 5 15 25 5 1,5 25 t; in arbitrary .scaleFig.. 10. Theoretical curves for the decay of axisymmetric
turbulence in the locally isotropic
stage.72
b stands for E2/E3. (cf. equation (5.13))E3
(t
-5
20 10 -2.5 0.5 2 = 0.5 I
h2
b (1 5 35THEORY OF AXISYMMETRIC TURBULENCE (1V)
2.0
2
10
5 15 25 35 45
t; arbitrary scale
Fig. 11. Theoretical curves showing the variation of e =_
/
iii72 in the locally isotropic stage. (cf. equation (5.13))in our notation (assumption (2)). According to (5.8) and (5.11), / increases at a faster rate than L with t, and so the condition (5.14) would be violated more
and more as time increases, showing the limitation of the double structure model. But, since at the same time this is accompanied by a decrease in turbulent velocities, the final period will eventually set in, where the degree of non-isotropy is
practical-ly constant (cf. III; § 5 and § 6).
6. Summary and acknowledgements. Typical results of the existing observations concerning the decay of homogeneous axisymmetric turbulence can be summedup as follows
In the very early stage, the turbulence is locally non-isotropic and return
to isotropy is fairly rapid.
But local isotropy realizes at a somewhat faster rate than true isotropy.
Once the locally isotropic structure has been established, tendency to
isotropy becomes considerably slow.
Correspondingly, two substages of the decay of axisymmetric turbulence are distinguished : an 'interacting stage ', and a 'locally isotropic stage '.
3.0
2.0
10
:
M.. OHJI
The interacting stage is characterized by an intense activity of the smaller
eddies in non-isotropic state. A tentative theory for the interacting stage yields
2E_L+ Ell Eo (t
to)'
andE Ell = E1(t
showing a rapid interchange of energy between Ei and EH (Fig. 9).
The locally isotropic stage is. characterized by a double structure composed Of the transverse part at smaller wave-numbers and the isotropic part at larger wave-numbers. These two parts are assumed to decay in their own manners re-spectively. Theory predicts
= E2 (t -
t2)- E3f-lo7,and
E11 E2 (t
showing a slow approach to isotropy (Figs. 10, 11).
It should be noted that E1> EN has been assumed throughout these con= siderations.
The proposed theory aims at an explanation of the general trend only. A
complete theory must wait on future work..
The author would like to express his hearty thanks to Professor Hikoji
Ya-mada of Kyushu University (now of Kyoto University) and to Professor Isao Imai of Tokyo University for their continual encouragements in the course of this work. He is also grateful to Professor Kazunori Kitajima of Kyushu University for advice. and discussion.,
36
n>0
THEORY OF AX/SYMMETRIC TURBULENCE (IV) 37
References
[ 1 ] Batchelor, G. K., The theory of axisymmetric turbulence. Proc. Roy. Soc. A, 186 (1946) 480.
[ 21 Batchelor, G. K., The theory of homogeneous turbulence. Cambridge Univ. Press (1953).
[ 3 ] Batchelor, G. K. & Proudman, 1., The large-scale structure of homogene-ous turbulence. Philos. Trans. A, 248 (1956) 369.
[4] Batchelor, G. K. & Stewart, R. W., Anisotropy of the spectrum of turbu-fence at small wave-numbers. Quart. J. Mech. App. Math. 3 (1950) 1. [ 5 ] Grant, H. L. & Nisbet, 1. C. T., The inhomogeneity of grid turbulence.
J. Fluid. Mech. 2 (:957) 263.
[ 6 ] Heisenberg, W., Zur statistischen Theorie der Turbulenz. Z. Phys. 124
(1948) 628.
[ 7 ] Heisenberg, W., On the theory of statistical and isotropic turbulence. Proc. Roy. Soc. A, 195 (1948) 402.
[8] von Karman, T. & Lin, C. C., On the concept of similarity in the theory of isotropic turbulence. Rev. Mod. Phys. 21 (1949) 516.
[ 9 ] von Karman, T. & Lin, C. C., On the statistical theory of isotropic
tur-bulence. Advances in applied mechanics. Academic Press Inc. vol. 2
(1951) I.
Kolmogoroff, A. N., On degeneration of isotropic turbulence in an in-compressible viscous liquid. C. R. Acad. Sci. URSS. 32 (1941) 16. Ohji, M., On the theory of homogeneous axisymmetric turbulence. Rep. Res. Inst. App!. Mech. (this journal) 6 (1958) 63. (referred to as I). Ohji, M., On the theory of homogeneous axisymmetric turbulence (II). Rep. Res. Inst. App!. Mech. 6 (1958) 153. (referred to as II).
Ohji, M., On the theory of homogeneous axisymmetric turbulence (III). Rep. .Res. Inst. App!. Mech. 7 (1959) 259. (referred to as III).
Ribner, H. S. & Tucker, M., Spectrum of turbulence in a contracting stream. N.A.C.A. Rep. no. 1113 (1955).
Stewart, R. W. & Townsend, A. A., Similarity and self-preservation in
isotropic turbulence. Philos. Trans. A, 243 (1951) 358.
Taylor, G. I. & Batchelor, G. K., The effect of wire gauze on small dis-turbances in a uniform stream. Quart. J. Mech. App!. Math. 2 (1949) 1. Townsend, A. A., The passage of turbulence through wire gauzes. Quart. J. Mech. App!. Math. 4 (1951) 308.
Townsend, A. A., The uniform distortion of homogeneous turbulence. Quart. J. Mech. App!. Math. 7 (1954) 104.
Townsend, A. A., The structure of turbulent shear flow. Cambridge Univ. Press (1956).
Tsuji, H., Experimental studies on the characteristics of isotropic turbu-lence behind two grids. J. Phys. Soc. Japan, 10 (1955) 578.
Tsuji, 1-I., Experimental studies on the spectrum of isotropic turbulence
behind two grids. J. Phys. Soc. Japan, 11 (1956) 1096.
Tsuji, H., A contribution to the energy decay law of isotropic turbulence in thc initial period. Rep. Aero. Res. Inst. Tokyo Univ. on. 345 (1959). Uberoi, M. S., Effect of wind-tunnel contraction on free-stream turbulence. J. Aero. Sc!. 23 (1956) 754. 112) 117) [181 1191 [20] ,1211 122] ][23]
38 M. OHJI
[241 Uberoi, M. S., Equipartition of energy and local isotropy in turbulent
flows. J. App!. Phys. 28 (1957) 1165.
(Received December 1, 1960)
(Note added in proof) In III we have defined as e=u/12/u12 (III; (5.13)), while in the present paper e=u,2/14772is to be understood throughout. Thus both 'e' are reciprocal
