The principal result obtained in this report is a generalization of Taylor's formula for a simple eddy. The discussion of the properties of the eddy indicates that there is a slight analogy between the theory of eddies in a viscous fluid and the quantum theory of radiation. Another exact solution of the equations of motion of viscous fluid yields a result which reminds one of the well-known condition for instability in the case of a horizontally stratified atmosphere

Bateman, H.

Bateman, H

NASA Technical Reports Server

.l
. .
REPORT No. 144
.—
THE DECAY OF A SIMPLE EDDY
BY
E BATEMAN
Cdfoda Instituted Technology
817
l
https://ntrs.nasa.gov/search.jsp?R=19930091209 2020-06-17T04:31:22+00:00Z

REPORT NO. 144.
THE DECAY OF A SIMPLE EDDY.
By H. BAT~.
.
JNTBODUCTION.
This subject, which has been studied reeently by G. I. Taylor 1 and EL A. Webb’ after
some initial investigations by Lxx+ is of considerable mathematical interest, as the theory
depends upon an exaot solution of the equations of motion of an ineomptible viscous fluid.
Sines very few exaot solutions of these equations ti known: it seems worth vrhile to study
in detail the one whioh describes the behavior of a simple eddy and to find, if possible, some
further solutions of the equations. The results of this study are herein set forth for publication
by the National Advisory Committee for Aeronautics.
SUMM.ABY.
The prinoipcd result obtained in this paper is a generalization of Taylor’s formula for a
simple eddy. The dkussion of the properties of the eddy indioates that there is a alight l
analogy between the theory of eddies in a viscous fluid and the quantum theory of radiation.
Another exact solution of the equations of motion of a viscous fluid yields a result vdieh
reminds one of the well-known condition for instability in the case of a horizontally stratified
atmosphere.
SOLUTION OF EQUATIONS OF TWO-DIMENSIONAL MOTION.
The equations of two-dimemional motion of an incompressl~e viscous fluid may be
written in the form
where U is the did vdooity, T the transveme vekity, p the pressure, P the density, and P
the kinematic viscosity. The variables (r, O) are cylindrical polar coordinates, t is the time,
and (1?, e) are the radial and transverse components of the esternal force on unit mass of
the fluid.
If e=0 and p, U, Y are independent of O,eqtitions (2) and (3) give
(4)
----
..—
.
.- .
.
—-
.
..—
..-
,-
820 REPORTNATIONAL ADVISORYCOMMITTEE FoR AERONAUTICS.
The last equation givea r U= K, where K is a fundion of t only. Equation (4) now becomes
(6)
Whan E is constant ah intereating solution of the partial differential equation is
V-B~e~ (7)
where s=: and B is a cunstant. Whens- O there is no radial vekeity and we obtain Taylor’s
solution.
In the gener~ cw” our solution repreaautaan eddy with a source or sink at the cen@l i. c.,
along the ~ it thus givea a very simple representation of the type of motion considered by
Webb and used as a model of a ease in whioh the deoay of an eddy arises from both dynamic
and visooue OS=.
The same solution may also be US-&to reprekmt an eddy whose center moves with uniform ‘-””
velooity, provided that U and V are interpret@ as radial and traneveme velocities relative b
the moving center.”” ..-
It shouId be notimd that V is a maximum when
.
()P-=4S4 8+; =2t (lC+P). (8) -
l
If, following Taylor, we dehe the radius of the gddy at time t as the radius of the ring of
maximum velocity, it appears that this radius is proportional to the equam root of K+ P. The
oonatants K and v are thus of equal importance as far as the rate of inoreasein size of the eddy
is ooncerned.
The distribution of pressure may be inferred from equation (1). Putting R-O, we have
(9)
The pressure thus increases from the renter outward.”
The kinetio energy of the fluid maybe rega@ed ~ made up of energy of radial motion and
energy of trmaverse motion. The former is infinite whm there is a sourw or aink at the origin,
but the latter is always finite. “Its value is in fact
.
(10)
Tho total energy of transverse motion at time t is thus represented by an expr~ion of
type $, whatever be the real vfdue ofs, and so the law of decay is always the same.
If we regard the tie of. .rnaximum velocity as”a boundary separating the inside of the “
vortex from the ‘outside, we find that the energy of transverse motion inaide tho vortex is always
a eonatant fraotion of the tital amount. This frmtion ~ is given by
When s-O, F==l-~ and is ks than ~m k “thii oaS6we have also
(II)
(12)
= DECAY OF A SIXPLE EDDY. 321
where C is a funotion of t. It should be noticed that the pressure does not become a minimum
.
when &F is a nuurimum but is a minimum when r-O.
.
The angular momentum of the eddy is
Jff=~2zrfk#d7
r’=2rpB *~e-%
(13)
The angular mormmtum remains oonstant, as we should expeot from dynamicai considerations.
The angular momentum within the eddy is a constant fraction j of the whole, where
(M)
Whens -O, f=- 1 – ~ e-~and is quite amsll; thus most of the angular mcngentum is outside the
vortex.
The oimdation in a ring is 2mV, and this is a mxximum when
~;t-u+l. (16)
It iseasily seen that the cmaximum circulation deoays according to a law of type ~, while the
maximumvelooity deoays acoording to a law of type ~.
The sngular velocity
(16)
.—
isa maximum wheu
P
42- 8“ (17)
The maximumangdar velooi~ deoays acoording h a law of type -
n_.; (18)
where E is a oonstant.
VISCOUSFLUIDMOTIONANIITHE QUANTUM THEOItY OF R4DL4TION.
It follows from the Iast rault that tie total energy of transverse motion is proportional
to the *um angular velooity. -ti result is slightly mmlogous b the qu@um law in the
theory of radiation. Punming the analogy we shall attempt to oompare oritioal values of the
angular momentum in Bohr’s atomic theory with oritical valuca of the Reynolds number ~L
in the theory of viscous fluid motiom It should be noticed in the &et place that ang&
momentum has the same dimensions as (a Reynolds number) multiplied by (a mass and a lshw
matio viscosity). The msss required for our amdogy maybe the mass of m ehwtron d the
vkcosity the viscosity of the ether.
It should be notioed also that we mu actually find inetanms of viscous fluid motion whioh
exist only for oertain ptioulsr wlues of a constant.
.
.-
322 REPORT NATIONAL ADVISORY COWIIT~.E FOR AERONAUTIC%
Let us consid6r, for instance, a type of eddy motion in which the transveme velocity V
a’v
is zaro for r-O and r-00 while some derivative w is fite and difTerentfrom zero for r= 0.
An expression of type.(7) evidantly satisdes the conditiom when 28+1 I=% hence a SOIU-
tion of the required t~e exieta whenavar 28+1 isa positive integer. Moroover, wo get into
difkltica when we try to satisfy the conditions by means of a solution of (6) of type .
for this does not generally become zero when rs oo. Whens ievmy large, for i.netanoe,T“is prac-
tically equal to B $“ To illustrate the way in which a sohtion of the required typo cm be
derived from the infinite series, let us cotidkr the CMOn= 1, m= 2. We may then writo “
{
I=B; 17..:i&+(1_811(2 _81(;ty-.”” m.]..]
.
~~tiplying V by 6– k, where k is a positive integar and making s~k, wo obtain a solution of the
required type. Webb sssuom that for differ~t edc@ of tha ~me ra~~ a, the radial velocity
at r - a is proportional ta the”strengtli. This means ,t@t
(21)
Where A is a nondimensional constant, henoe 6 is a Reynolds number and n solution of our
problem exists only for certain oritiml values of a Reynolds number.
ANO!I’EEE ME?I’EOD OE SOLUTION.
bother exact solution of tie equations of variable motion of an incompressible viscous
fluid may be obtained as follows: -
Let us andeavor to satisfy the equation
(22)
by expressions of type
lb - W-kvcm qx, v= -“tp+ sin (p ($43)
where k, g, and h are conetante. These expressions
tinuity -
:+g=o
We iind that equations (22) are satisfied if
evidemtly satisfy the equation of Colk
(24)
THE DEOAY OF A SIhU?LE EDDY. 828
These equations can be satisfied either by putting k-g or by putting k= r (P –~). The fornm
case is of no interest, for it corresponds to a well-known case of irrotatiomd motion.
In the latter case we have
(25)
where (? is a constant, P being supposed to be also constant. When k< g the motion is dying
down while when k >q the ruot!on is becoming ”morc violent. When k-=q the motion is steady
and irrotational.
Ifat anyinetant wefi thevahteof p at y-0, we futd thet tithe case of growing orun- .
a(39stable motion tie pressure gradient ~ .O is huger than in the steady case when k==q. In
---
the case of a decaying motion the presmxregradient is less than in the steady case. This result
may be compared with the well-known theorem that a horizontally stratified atmosphere is
unstable when the Iapse rate of temperature is higher than the adiabatic value.
bother point worth noticing is that
.-
Hence whau the pressure over y=O is ftxed, the sign of the vortioity is dil%ent according as
k is greater than or leas than g, i. e., according as the motion is growing or decaying.
.


English

The decay of a simple eddy

Caltech Authors - Main

REPORT No. 144 
THE DECAY OF A SlllPLE EDDY 
BY 
H. BATEl\.lil~ 
California Institute of Technology 
• 
817 
• 
~. 
REPORT No. 144. 
THE DECAY OF A 811\IPLE EDDY. 
By B. B.6.TKlliM. 
INTRODUCTION. 
This subject, which has been studied recently by G. I. Taylor 1 and H. A. Webbs after 
some initial investigations by Lees, is of considerable :ma.th.ematical interest, as the theory 
depends upon an exact solution of the equations of motion of an incompressible viscous .fluid. 
Since very few exa.ct solutions of these equations are known,1 it seems worth while to study 
in detail the one which describes the behavior of a simple eddy and to find, if possible, some 
further solutions of the equations. The results of this study are herein set forth for publication 
by the National .Advisory Committee for .Aeronautics. 
smDIARY. 
The principal result obtained in this paper is a generalization of Taylor's formula for & 
simple eddy. The discussion of the properties of the eddy indicates that there is a slight 
analogy between the theory of eddies in a viscous fluid and the quantum theory of radiation. 
Another exact solution of the equations of motion of a viscous fluid yields a result which 
reminds one of the well-known condition for instability in the case of a horizontally stratified 
atmosphere. 
SOLUTION OF EQUATIONS OP TWO-DIMENSIONAL MOTION. 
The equations of two-dimensional motion of an incompressible viscous fluid may be 
written in the form 
a_l.!+ ua u +£°au_ VJ -n-!.~+r{z,s l7 +!.al!+! cas u -2i• -u)} (1) 
at ar r as r p ar c;rt" ,. ~ r' ~ 8 
ar + ua v +_yav + uv ,,. 0 _!.~+r~v +!.av+! ("v +2a u _ v)l (2) at Fr" r N ,. P roe l or r C>r r ~ 2>B I 
C> ar ~~m+~-o w 
where U is the radial velocity, V the transverse velocity, p the pressure, p the density, and r 
the kinematic viscosity. The variables (r, 8) are cylindrical polar coordinat-es, t is the time, 
and (R, 0) are the radial and tre.nsverse components of the ext-ernal force on unit mass of 
the fluid. 
If e-0 and p, l7, V are independent of B, eqwitions (2) and (3) give 
2> ar(rU)-0 
t Bdd.111. A.d't'flOQ' Cammlttae b' .Asmaut1m. B.. .I: lC. No. 198, DsmaJJer, 1918. 
I B. &r: !ii:. No. ml, Kay, 1918. 
(4) 
(5) 
For mme C1111e1 of hm-"bnen!msl lta:sdy :matlm- a. B. lettecy, PhD. llq. t. 211 cm.&>. p. -&51; a. L Taylar, Proc. BoJ'. Soc. A.. (1.917), 
p. 99; and IAmb's Hydrod,rnamlm. 
819 
• 
820 REPORT iunoNAL ADVISORY COHJ.tt!."!EB FOB AEBONAU'l'ICS. 
The last equation gives r U~ K, where K is a. function of t only. Equation (4) now becomes 
~~("~ _g\(~Y+I\ 
'" • ar r) C>r . r} (6) 
When K is constant a.iJ. interesting solution of the partial differential equation is 
(7) 
where 1--:; and Bis a constant. When s-0 there is.no radial velocity and we obtain Taylor's 
solution. 
In the generaj case our solution represents an ed«!y with a source or sink at. t.he ccn~, i.e., 
along the axis; it thus gives a. very simple representation of the t.ype of motion considered by 
Webb and used as a model of a case in which the decay of a.n eddy arises from both dynamic 
and viscous causes~· · 
The same solution may also be used t.o represent an eddy whose center moves with uniform 
velocity, provided ~t U and V are interprete4 as radial and tra.nsverse velocities relative to 
the moving center. 
It should be noticed that Vis a maximum when 
(8) 
If, following Taylor, we define the radius of the ¢ciy at time t as the radius of the ring of 
maxim.um. velocity, it appears that this radius is proportional t.o the square root of K+,,. Tho 
constants K and .,, a.re thus of equal importance a.s far as the rate of incroaso in size of the eddy 
is concemod. 
The distribution of pressure may be inferred from equation (1). Putting R-0, wo have 
1 ~ Kl 1""+1 _,. 
- --+.&-e bi. 
P r ,a "t"+' . (9) 
The pressure thus increases from the center outward.· 
The kinetic .energy of the fluid may be regarded ~ m&de up of energy of radial motion and 
energy of transverse motion. The former is infinite whEIIl there is a source or sink at the origin, 
but the latter is always :finite. "Its value is in fact 
T- fu"' 2rrdr (~P VS) 
jr'f+I "' 
-rp.& Jo "t"+' e-w dr . 
1 (B)I f'° 
-2 rp t (211)11...S Jo t-f ri+l~~. (10) 
Tho total energy of transverse motion at time tis thus represented by an expression of 
type:, whatever be the real value of s, and so the la.w of decay is always the same. 
If we regard the circle .of. ;muimum. velocity as" & boundary separating the inside of the 
vortex from t.b.e·outside, we find that the energy of transverse motion inside tho vortex is always 
a constant fraction of the total amount. This fraction Fis given by 
F J: • e-f ~ ~- Jo11+1,-e ~1 d~. 
When s-0, F-1-~ and is less than j. In "this oaae we have. also 
P._ 0-.8111 e-£, 
p f:I 
{11) 
(12) 
TRK DECAY OF A. SlllPLE EDDY. 821 
where 0 is a function of t. It should be noticed that the pressure docs not become a. minimum 
he 1 T1\I" • b"" .. h w n 2P t - JS a mA.XJmum U11 JS a ID.lWID.Um w en r-0. 
The angular momentum of the eddy is 
.Jl ... Jo• 2rrdrZpr V 
""'kpB rr;:: 4-E, dr 
-rpB (4J,)1+1Jo ... 4-r r-+1 dt. (13) 
The angular momentum remains constant1 as we should expect from dynamical considerations. 
The angular momentum within the eddy is a constant fraction! of the whole, where 
ifo"' e-rr-.iar-- J:•+te-rr-.iar. (14) 
When a-O,f=-1-; ri and is quite small; thus most of the angular mOID.entum. is out.side the 
vortex. 
The circulation in a ring is 2rr V, and this is a maximum when 
(15) 
It is easily seen that. the ma.ximum circulation decays according to & law of type f, while the 
maximum velocity decays according to & la.w of type£· 
The angular velocity 
is & maximum when 
v ,.s. .. 
n-- -=B- 4-iii r t"fl 
r 4.t_,_ 
The maximum angular velocity decays according to a law of type 
E a.u.--r 
where E is a constant. 
VISCOUS FLUID MOTION AND THE QUANTUM THEORY OF RADIATION. 
(16) 
(17) 
(18) 
It follows from the last result that the total energy of transverse motion is proportional 
to the maximum angular velocity. ~result is slightly analogous to the quq.ntum law in the 
theory of radiation. Pursuing the analogy we shaJ.l att-empt to compare critical values of the 
a.ngular momentum in Bohr's atomic theory with critical values of the Reynolds number VL 
" in the theory of viscous fluid motion. It should be noticed in the first place that angular 
momentum has the same dimensions as (a Reynolds number) multiplied by (a mass and a kine-
matic viscosity). The mass required for our e.nalogy may be the mass of &n electron and the 
viscosity the viscosity of the ether. 
It should be noticed also that we can actually find instances of viscous fluid motion which 
exist only for oe?tain particular l"alues of & constant. 
~
822 BEPOR'.l' NATIONAL ADVISORY COYlUTTRE l!'OR AERONAUTI<.'S. 
Let us consider, for instance, a type or eddy motion in which the transverse velocity V 
is zero for r-0 and r- ao while some derivative%:~ is finite and different from zero for r-0. 
An expression of type. (7) evidently ea.tis.fies the conditions when 28 + 1-n, hence a solu-
tion of the required type exists whenever 28+ 1 is a positive integer. Moreover, wo got into 
diffic.ulties when we try to satisfy the conditions by JJ:i.ee.ns of a solution of (6) of type 
7 K;.{ m.. r m tm+lJ er\ } l- fm l- (l+n-28) (n+ar;t+ cr+-n-=--28) (a+n-2Bffn-+s> (n+5) ~i)...... <19> 
for this does not generally become zero when r-::-+ oo. . When 8 is very largo, for instance, r is prac-
tically eque.1 to B ~= · To illustrate the way in which a solution of the required typo can be 
derived from the infinite series, let us consider the csso n - 1, m- 2. We mo.y then write 
(20) 
Multiplying V by 8-k, where k ia a positive integer and making 8-+k, we obtain a solution of the 
required type. We}>b assumes that for rlifferent eddies of tho ~me radius a, the radial velocity 
at r-a is proportional to the strength. This means th&t 
K aV 
8 ... --x-2v v (21) 
Where X is a nondimensiona.l constant, hence 8 is o. Reynolds number and o. solution of our 
problem exists only for cert.a.in critical values of a Reynolds number. 
ANOTHER METHOD OJ!' SOLUTION. 
Another exact aolution of the equations of variable motion of a.n incompressible viscous 
fluid may be obtained as follows: 
Let us endeavor to satisfy the equation 
(22) 
by expressions of type 
u-ks>J-l:r cos qz, v- -·qs>J-l:r sin qz (28) 
where k, q, a.nd X a.re constants. These expressions evidently satisfy tho equation of con· 
tinuity - · 
e>u+ov 0 .::1. ~ Oy-
We :find that equations (22) are satisfied if 
_!. ~ -k { X- J1 (k'-</) } e>.t-l:f cos qz p~ . 
-~ g~- -!l {A- JI (k'-</)} eM-:-~' sin !li:-k</ at<M-tr> 
(24) 
THE DECAY OF A SIMPLE EDDY. 828 
These equations can. be satisfied either by putting "k-q or by putting >."", (J:I-</). The form.et 
case is of no interest, for it corresponds to a well-known case of irrota.tional motion. 
In the latter case we have 
l ... o-! <t t1tli-l:n 
p 2 (25) 
where 0 is a constant, p being supposed to be also constant. When k < q the motion i8 dying 
down while when k>q the motion is becoming.more violent. When 'k-q the motion is steady 
and irrotational. 
If at any instant we fix the value of pat y-0, we :find that in the case of growing or un-
stable motion the pressure gradient (~),-o is larger than in the steady case when k-q. In 
the case of a decaying motion the pressure gradient is less than in the ste&dy case. This result 
may be compared with the well-known theorem that a horizontally str&tifi.ed atmosphere is 
unstable when the lapse rate of t-em~rature is higher than the a.die.be.tic value. 
Another point worth noticing is that 
Clt1 Cl-u Clz - Cly - ("Ii'- <f) t1'-tr cos qz. 
Hence when the pressure over y = 0 is fixed, the sign of the vprticity is different according as 
Tc is greater than or less than q, i. e., according as the motion is growing or decaying. 


The decay of a simple eddy

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