ReportoftheNationalResearchCenterfor

DisasterPrevention，No．5，March1971

551，465：551，511．6：532．517

AMode1ofTurb111ence：

Statistica1Treatmentof

Vortexan

Ensemb1eof

Fi1aments

By

Yukio

Fujinawa∬か〃舳肋B〃〃oゐ，Nα〃o〃α1Rθ8ωプ肋C舳才〃1oκ1）ゴsos加γPγω舳ガo〃

Abstract

Turbulence

ismodelizedasamotion

due

to

an

ensembleof

circularvortex丘1aments．By the loca1isotropic hypothesis of Ko1mogoroff this ensemb1e canbe thought of as a canonical ensemb1e of classica1statistica1mechanics．From1．

thisstatistics，formulation

tionandenergy

spectrumis

madetensors

infor

eva1uating

isotropic

thevelocity

turbu1enceof

corre1ation

func−

incompressib1enuid．

IntrOdl1Cti011

In most of practica1ly important phenomena of iuid dynamics，various physica1
q。。。titi。・…h・…1・・ity，P・・・・・・…dt・mp…t・・・・・…tb・d・t・・mi・・d。。iq。。1y，。・di・・t・・ti・g…d・mlyi・・p・・…dtim・，th・y・・…11・dt・・b・1・・t・

Turbulence

has

been

oneof

the

most

di伍cu1tandmostattractive

prob1ems

ini．iddy・。mi・…dw・h・・…y・t・…ti・f・・t・・yth…y・ft・・b・1・…lth・・gh．b。。tf。。tyy・…h・…1・p・・d・i・…L．P…dt1p・・p…dhi・mi・i・g−1・・gthth。。・y．Thi・th…y・・…p1・i・th…i・・…11・g・・ithmi・1・w・・…w・1l・・dhas

been

These

successfu11y

facts

show

used

that

in

its

variousproblems

physica1pictureofturbu1ence

due

is

consistent

withtoits

thesimp1icity．

rea1turbulent

且OW．

AnewscopeinturbulencetheoryhasbeenopenedwhenG．I．Tay1orintro−
duced

themethodofcorre1ation

function

in1935．This

theoryisarather

mathe−

m．ti。。1．pP・…h．W・・…bt・i・th・・q・・ti…f・…bit…y・・d・…1・・ity
corre1ation tensor and the introduction of some assumption in regard to tensor
of a certain order makes the prob1em c1osed and so1ved．E．Hopf s functiona1．q。。ti・・（E．H・pf，1952；E．H・pfθ1α1．，1953）…t・i・・th・i・f・・m・ti…f・・・…
1ation functions of a11orders and might be said to be the widest genera1izationof corre1ation fmction method．Though a functiona1equation is hard to be
so1ved，it may prove to be usefu1in the genera1discussion．Recent1y R．H．
Kraichnan（1960）ca1cu1ated the non1inear interaction term by the perturbation
method using graphica1technique and succeeded in obtaining the（一5／3）Powe「1awfor

isotropic turbu1ence in his latter paper（R．H．Kraichnan，1965）。
There have been many investigations of turbu1ence，but a1most a11of them
are concerned with the homogeneous and isotropic turbu1ence，excepti㎎W．y．
R．M．1k。。（1956）wh・d・・i・・dth・m・・…1・・itydi・t・ib・ti…fn・wb・tw…th・P。。。l1・1p1・t・…i㎎th・m・・im・mdi・・ip・ti・・hyp・th・・i・・T・1m・m…1W・C・

Meechamandothers（1963and1968）expanded
一89一

theve1ocity

ie1dbyWiener一Reportof

theNationa1ResearchCenterfor

DisasterPre▽ention，No．5，March1971

Hermite functiona1，and from the Navier−Stokes equation they got equations
satisfied with its expansion coe冊cients，and they successfu11y app1ied these to
theprob1emof

turbu1ence

decay

nearawa11．

Thus，the recent trend of research in turbu1ence is strongly mathematica1
and tends to discard the physica1entity．The present author thinks that a true
so1ution of turbulence can not be attained without the understanding of physica1
picture of turbulent motion．In the present paper an ensemb1e of vortex丘1aments
is adopted as a physica1entity which causes turbulence，and is treated by themethod of c1assica1statistica1mechanics．

2．Form皿1atiom
Asiswe11−known

invectorana1ysis，anarbitrary

vector五eld

as a sum of irrotationa1and rotationa1vectors．In iuidpart is usua11y the vorticityωwhich can be expressedaScanbeexpressedmechanics the rotationa1by the ve1ocity vector〃

ω＝▽X〃，

where▽denotes
Now，infast

beinthecaseof

gradient

Huid

when

theinfinite（ゴ．θ。〃（oo）＝O（R■冊），〃＞3），the

expressedby

thevorticity＝

whereoperator．

incompressible，μ，2are

thedistribution∂H∂G■， o＝∂ひ ∂2tends
attozerosu冊ciently

anypoint

inspace

canas

∂F

∂∬∂G

∂F

一， 〃＝一，∂2 ∂∂∂〃orthogona1coordinates（瓦…）一÷／（ξiowve1ocityζ

and〃＝（〃，o，〃）．Andζ）・τ・・

whereκis the distance between the point（ ，μ，2）and the point（，ひ ，z ）atwhich vo1ume element of integra1dτ is situated andω＝（ξ，η，ζ）（Lamb，1932）．So

we can determine the velocity fie1d of incompressib1e iuid at1east in princip1e
if we know the distribution of vorticity in total space．But the introductionof assumption as to the distribution of vorticity at every point in space is as
di冊・・1t・・th・t・f・・1・・itydi・t・ib・ti・… d… pP・…ht・th・t。。b．1。。。。
from the standpoint of vorticity has no merit，and on1y introducing comp1exity．
So we do not enter into the vorticity distribution at any point in space，and

we consider vortex丘1aments．0utside a vortex i1ament there exists ve1ocity
potentia1，and the potential caused by an ensemble of vortex刮aments becomes
the sum of ihdividua1potentia1s if they are not overIapping，and the prob1em
is treated as a1inear one．0f course，the fundamental feature of turbulence・…i・t・i・it・…1i・…i・t・…ti・…
byany

techniqueof1inearization．In

dthi・…li・…ity・・…

fact

vortexfi1aments

tb・…1．d．dinteract

witheachother and are bom or dead at every instant and may be in a stationary state．
But as the Lagrangian approach of Navier−Stokes equation gives the linear ap−
proximation higher by an order than that of Eulerian equation（Pierson，1962），
the1inearization of turbu1ent ield making use of vortex ilaments is expected togive a better model than that from Eu1erian description．
In the present paper，the interaction of vortex i1aments is assumed to be so−90一

AModelofTurbulence−Y．Fujinawa
weak that it only exchanges energy and does not create a new fi1ament nor de・
struct other vortex fi1aments．In another paper we may see the effect of crea−tionandsaidtoAsis

we11−knownendsatany

point

faceofvation

annihi1ationbea

mode1ofproperty

ofvortex

fie1d．So

turbu1ence

usingavortex

inhydrodynamics，ain

space，but

formsboundary．Of

course，if

weofvortex，this

so1id

c1osedtakeassertionvortexcurve

account

wi11not

orof

this

approachmay

wel1befi1ament．

i1ament

neither

begins

has

ends

the

the

app1y．And

bothproperty

as

atof

we

nor

sur−

non−conser−

considerisotropic turbu1ence in this paper，aI1the vortex i1aments in the region
be assumed to be c1osed curve．Since the isotropic turbu1ence exists inon1ythemaythewe11
sma11

region compared with the scale of ie1d，1arger vortex fi1aments with an end at
the surface are thought to have an effect on that sma11region just1ike a meannOW．

Now we assume a11the vortex創aments to be c1osed curves，but the c1osed
curves have an ininite number of forms and for simp1icity we must introduce

some assumption as to the form of c1osed vortex i1aments．First，we assume
that a11the vortex丘1aments are simp1e ones and do not entang1e themse1ves．
This statement is justiied by the decomposabi1ity of entang1ed vortex fi1aments
into a number of simp1e curves of i1aments（Moffat，1969）．Second1y，we adopt
fora c1osed curve its simplest form，a circ1e．This restriction may not give any
serious modification and essentia1characteristics wi11be retained．

3．

Stati8tics

Asisdescribedabove，isotropic

turbu1ence

isthought

to

be

an

ensemb1eof

a great number of circu1ar vortex i1aments．Behaviors of a system of many
composites are treated by statistica1mechanics．Consider a re1ative1y sma11region
of

iuid

movingattheve1ocity

of

mean

iow，where

theisotropy

isa

good

ap−

proximation．In this region there are many vortex刮aments which interact witheach other，with the mean刊ow and with1arger vortex i1aments，and then there
may be created a condition of statistica1equi1ibrium of stationary state．Thesestatements may be justiied by the Ko1mogoroff s loca1isotropy hypothesis andthe interaction may we11be thought to be so weak that we shou1d only consider
theexchangeofenergy

ofasystemofvortex

i1aments

inthesmall

regioncon−cerned．Such a system is ca11ed in c1assica1statistica1mechanics a canonica1ensemb1e，and the probabi1ity of the system in the energy range ofτto T＋dT
is proportiona1toθ一阿，whereβis a constant determined by the abso1ute temper−
ature of the extema1丘eld having a very1arge degree of freedom compared withthesystem concerned．
Now the energy T of an ensemble of vortex i1aments is・一㍍閉∬d811㍗・（・）whereρis the density of the iuid，11｛the strength ofゴーth vortex，and〜denotes
the distance between the point of1ine integra1e1ement d曲of6−th vortex andthat of d8j ofプーth vortex．Using this formula，we can ca1cu1ate any mean value
as an expected va1ue．The constantβis here unknown，but may depend only on
the representative quantity of the iow and can be determined by the comparison
with experimenta1results．Phase space is composed of the vector in space coordi一一g1一ReportoftheNational

ResearchCenterfor

DisasterPrevention，No．

natesr｛of5，March1971

thecenterofvortex

fi1ament，the unit vectorωnor−
ma1to the plane of vortex，thed52ρ2〃

ε23radius尺1ofθ

グ21己andthevortex丘1ament．

Now we address ourse1ves to theprob1em of evaluation of energy

ε124S1

circ1estrength八of

ε22
ε13

夕θ

of

21asystem

inducedθ

bycorre1ation（andvelocitythem

andpotentia1

theve1ocity

function．

θ114．
Fig．1．

Orthogona1coordinatesof

twovortex

Calc皿1ation

We rewrite the equation（1）i1a−aSmentS．

r＝ρ
4π Σ八rハj，
峠ゴ

where
乃・一／／㌣芦We calculateτ12，since from this form otherτ在j can be easi1yinferred．We
takeθ｛1andθ勿on the surface normal toωsuch that three unit vectors2、、，θ、，，

ωconstructanixedspace．Weintheorthogona1system，andcantake201，θ02，θ03as3

3

ω＝Σ舳θ・j，

伽＝Σ伽θ。j，

ト1

whereρ幻and伽aredirection

θoj，respectively．Line

orthogona1vectors

write

ゴ＝1

cosines

e1ement

betweenθ、｛and2．j，andd81canbewritteninbetween②、andthe（伽，ω，伽）systemasd81＝（一1モ1sinθdθ，1モ1cosθdθ，0）．

Using

a

transformation

formu1ain

avectoranalysis3

仙＝α1＋Σρ〃！，
ゴ＝1whereα＝（α1，α2，α3）denotes

system

andin

this

the

positioncaseα＝n，we

vectorofanoriginofanew

coordinategetd・1・d・・＝朋・dθdg（ん1・i・θ・i・ダλ1・・i・θ…ρ
一■421COSθSin

g＋λ22COSθCOSg），

wh…伊d…t・・th…g1・b・tw…θ・1・・dth・p・・iti・・…t…f1i・・i・t・g。。1e1ement

d82，andんjtheimer

productof▽ectorθ1，andθ、j，ク．θ．
んj＝舳・吻．

IfweWegetdenote「1−r2＝r12，一g2一（2）

A

Model

Turbulence＿Y． Fujinawa

ofγ、、＝lr1，12＋R12＋見2＋2R1（C11cosθ十C12sinθ）一2凪（C21cos

g＋C22sin？）

一2R1R、（λ、。…θ…叶ん・…θ・i・ψ十ん1・i・θ…叶ん・・i・θ・i・伊）・

where
Cむ＝r12・吻j．

If

weassumethat

twovotexes

areenoughapartsoas

tobe（■貴：1）3一・・

wecaneva1uate

the1ineintegra1to

such

an（3）

approximationas

乃・一（W1，lI呂［・（舳・一舳）一1，l13／・11（・2山一・22ん1）
一・12（・21Ar・2・ん1）／l

Now

using

theformu1aof（五．〃）（B．0），whereλ×B

vectordenotesana1ysis，（五xB）．（0x〃）＝（λ．0）（B・一0）一thevectorproductof

two▽ectors五，B，weget

λ1工ん2−A2ん1＝θ11・ω

andC11（C21λ22−C22A21）一C12（C21λ12−C22A11）＝一（θ13・r12）（吻3・r12）十1r1212（θ13・223）・

and自na11y

weget

theτ12asτ1・一（服・）11、土131一（舳・）十1、…■・（m・1・）（・1lω）／

0f

course，thisform

does

notdependonthearbitrary

quantitiesθ11，θ12，伽。。dθ、，，b・td・p・・d… th・・t…gth・f…ti・ity・・dth…1・ti・…p…ti・…dmutual direction．Thus the tota1energy of a system of▽ortex i1aments isτ一篶M鮒I，1■十（舳）・■、l13（舳）（舳）／Next，weca1cu1ate

ments．Outside
ve1ocity

athe

vortexve1ocity

fi1ament

potentia1φ（r）andbepotentia1due

theve1ocity

toanie1d（4）ensemb1e

of

can

described

bevortexi1a−bythewritten

1（・）一￡lc葦3・…

where3denotes

the ang1e between the vector rLr and the surface norma1，and〆th．p・・iti・・…t…f…f…i・t・g・・1・1・m・・tdS1（L・mb119321§150）・A・。。1。。ityp・t・・ti・1i・・1i・…q…tity，w・・・…1・・1・t・th…1・・ityp・t・・ti・1d・・

toan

ensemb1e

asthesumofindividua1potentia1φ｛inducedby

individua1vortex

i1ament，φ＝Σφ1・

Just

asin

the

previous

ca1cu1ationforenergy，wecan

ca1cu1ate

theve1ocityP．t。。ti・1d・・t…i…1・・…t・・i1・m・・t・f・t…gth＾i・th・・pP…im・ti・・（1ξll）3一・aS

一g3一（・）

R・p・・t・fth・N・ti…1R・・・…hC・・t・・f・・Di…t・・P。。。、。ti。。，N。．

5，March1971φ。（、）一「lR。・（「・1剛4

1剛3

where1「｛＝「1一「・（6）N・ww・・・…mp・t…1・・ity・・…1・ti・・f…ti・・i・th。。。m．m．m。。。。
th・…1・・ti…f・・…t・d・・1・・i・・t・・i・ti・・1m・・h・・i・・．V・1。。i・。c．rre1，ti。、

functi・・f・・tw…m・・Rり（ト・）一州・。（・）…b…1・・1・t・d。。f．11．ws，whe、、
th・・…1・idb・・d…t・・th…p・・t・d。。1。。．
R。、（卜、・）一∂φ（「）1φ（〆）

∂伽∂
：∂仙∂

∂

！

∂．、φ（r）φ（〆），！

andφ（r）φ（〆）＝Σφ血（r）φβ（〆）α，β＝Σφ皿（r）φ皿（〆）十Σφ血（r）φβ（〆）
血

＝

wh…

αキβ

φ1（・）φ1（〆）十〃（W−1）φ1（・）φ2（〆），i・th…mb…f…t・・m・m・・t・．E・・1・・ti…fthi．i．t．g。。1。。dth．

discussion

wi11begiven

inthenextpaper．

I己eferences1︶

Ho・f・E・（1952）：St・ti・・i・・1h・d・・m・・h・・i・…df…ti…1・・1・・1・・．〃α〃．吻ε．ル、1．，2︶

1，87＿123．

H・pf・E…dTi・・lE・W・（1953）：O・・・…i・・p・・i・1・・1・ti・…fφ一。q。。・i。。。f．t．ti。。i。。13︶hydromechanics．∫。Rα〃〃1θ肋．λ伽1．，2，587−591．

K「ai・h…lR・H（1960）：D…mi…f…1i・・・・・…h・・ti・…t・m・．∫肋肋．伽．，2，4︶

124＿148．

K「aich・…R・H・（1965）：L・・・…i・・hi・t・…1・・・・・・・・…im・・i・・f・・…b・1。。。。．物。．

5︶

1『Z〃ゴ∂8，8，575＿598．

Ma1k…W・V・R・（1956）O・・1i…f・・h・・…f…b・1・・t・h…i・w．∫肋州θ。ん．，6︶

1，521＿539．

7︶
8︶Im・m・…T・・M…h・m，W・C…dSi・g・1，A．（1963）：Symb．li。。。1。。1。。。fth．Wi。。。。
process and Wiener−Hermite functiona1s．∫．肋肋．P妙∫．，6，695−706．
Mee・h・m・W・C…dJ・…D・T・（1968）：U…fth・Wi・・…H・・mi・…。・・。i。。f。。。。、rl．
norma1turbu1ence．∫．〃〃〃〃12c〃．，32，225−249．

9︶Lamb，H．（1932）：助〃oφ伽〃6s．Cambridge

Univ．Press，London．

Pie「so・，W・J・（1962）：P・・t・・b・ti・・…1・・i・・f・h・N・・i・・一S・・k…。・・ti…i．L。。。。。。ia，

f・・mwith・・1・・t・d1i・・・…1・ti…．∫．Gω伽・．肋．，67．3151−3160．
10）Mo耐・t・H・K・（1969）：Th・d・・・…f㎞・t・・d・・…f・…1・d…t・・li…．〃1。舳；θ。ん．，
35，117＿129．

（Manuscript

一94

received1ODecember1969）

A

Modelof

Turbu1ence−Y．Fujinawa

乱流の一つのモデル：うず糸の集合の統計的処理
藤縄幸雄
国立防災科学技術セソター平塚支所
乱流を数多くの円形のうず糸の集まりとみなす．局所等方性の仮定から非圧縮性流体の等方性乱流においてこのうず糸の集合が標準集団として扱えることを用いて，乱流場の速度棉関，エネルギースペクトルテソソルを算出する定式化を行なった．

一95一