Similarity solutions for turbulent jets and plumes in a rotating fluid

Similarity Solutions for Turbulent Jets and Plumes in a

Rotating Fluid

T. J. PEDLEY

The Johns Hopkins University, Baltimore, Md. (Manúscript received 19 Jurie 1967)

ABSTRACT

Similarity solutions are derhîed for a turbulent jet (constant momentum flux) and a turbulent plume (constant buoyancy flux) directed along the axis of rotation of a body of fluid in soli&body rotation, on the assumption that the Rossby number is small The jet rachus is found to mcrease with axial coordinate

z as z115, and the plume radius as z317. A dètailed discussion of the order of magnitude of the Reynolds stress

terms is made necessary by the fact that there is, o p,iori, more than one velocity scale in the flow. There is some evidence that the given solutions.do not represent real flows, but this question cannot be answered fullyuntil certain problems in hydrodynamic stability are solved. An experiment is therefore suggested.

No. 5, September, 1968, pp. 789-795

tal. v Sckeepsbouwkunle

Techmsche Hogescheol

Deift

1. I.nfroductión become

Herbert (1965) has presented a similarity solution

for a steady axisymmetric laminar jet in a rotating

fluid. A jet here is defined as that flow which results from a constant point source of linear momentum,

directed in this case along the axis of an ambient solid--body rotation. (of angular velocity ). f Vo is a scale

for the swirl velocity V relative to the basic rotation, at a .given axial distance z from the source, and if ô characterizes the jet width at the san-ìe, value of z, then Herbert's solution is valid for all values of z for

which the. narrow jet assumption,

ö/z«1,

(1)

is valid añd the Rossby number,

V0!íò«1, (2)

is small. The second condition means that his solution

is valid as long as the ambient rotation dominates the

motion, since it turns out that the axial velocity W

(scale W0) has the same order of magnitude as 'V0, and the radial velocity U (sçale U0) has a smaller order of magnitude. When the Rossby number is small, inertia

forces are negligible, and the motion is a balance

be-tween viscous and Coriolis forces. If the 'similarity so-lution is written in the. form

ôz', W=zmf(r/ô),

V=z'g(r/ô), (3)

where r is thé radial coordinate in an (r,,z) cylindrical

polar coordinate system, then in Herbert's solution

the exponents take the values

n=, l=m=-1.

Conditions (1) and (2), by dimensional analysis, then

/M\1/v\l

z»()

and z»((),

\vu/ \çJJ

respectively, where M is the (constant) axial flux of linear momentum, and y is the kinematic viscosity. Thus, Herbert's solution is valid for large enough z. A plume is defined as the motion resulting from a constant point source of buoyancy (heat) in the field

of gravity, and it can be shown that there is no similarity

solution for a laminar plume in a rotating fluid (with

the heat source situated on the vertical axis of rotation)

corresponding to Herbert's jet solution; in this case the assumption of small Rossby number, eliminating inertia forces, is not sufficient, since the motion must

be a balance of three separate effectsviscous, Coriolis

and buoyancy forces.1n this paper a similarity

solu-tion is presented for the turbulent analogue of Hèrbert's

laminar jet, and a corresponding solution is found for

a turbulent plume.

2. Equations of motion and scale analysis

It will be assumed that the Reynolds number of the

mean flow in either case is large enough for the principle of Reynolds number similarity to h ld_in other words,

that the viscous terms may be neglected in the mean equations of motion. In addition, for the plume case, it will be assumed that the Prandtl number is not too

'small, so that molecular diffusiön of heat 'is negligible. It will also be assumed that the fluid is incompressible,

of density PO, which (in the plume case) leads to the Boussinesq approxmiation that density variations are

important only in the buoyancy force, not in the inertia

forces. Let the components of velocity relative. a1

ARCHIEF

Réprinted from JOURNAL OF ATMOSPHERIC SCIENCES, Vol. 25,

American Meteorological Society

790

JOURNAL OF THE ATMOSPHERIC

fra

z)ot.&: 4with the ambient flUid

at constanc a gula velb y' about the vertical z

V±i W-w) where a capital letter

h quantity, and small letters denote ctations from that mean.Define the modified

pressuriU

'''

P+p = gz,

where p' is the actual pressure, and the b'uoyancy

0+0= gp/po,

where p is the difference in density between thefluid

at the general point considered arid the ambient flUid, andg the gravitational accejeratiön. The thean motion

will be assumed to be axisymmetric and steady, so

that 8/â

and a/ai of all mean quantities are zero.

The mean values of the Navier-Stokes equations now

become

la

ô V2+

- _[r(U2±u2)]+_[UW+uw]

rôr

Oz r (4) a,.

i:ô

O

- _[*2(U

V+)]+_{VW+vw]±2Tj 0,

r2ôr .9z

lar

_-i 'Or

_i

OP

---i r(UW+uw) +I W2+w2 ¡+0=--,

rôrL

J

.9zL

J

'where a bar over a product denotes the mean value

that product. We also have, the 'mean, equation

continuity, '

lô

rO,. Oz

and that of heat convection,

10

a

- [r(u®+uO)i+[W0±w0]0. ''

(8)

rär

In the case of a jet, where: O is identically zero, (6) may be multiplied by 2r',and' integrated from 'r = O

tor=ootogive

'

j 2r(W2+w2±P)rdr

I:2ir(ti W±uw)r= Ô,

dzJo :

since=U=0 on r=0, and it

is assumed that âll perturbations to' 'the ambient rotation tend to er'o at large radius in particùlar, both W and w tend 'to zero,

.nd the radial mass flow 2irr (U-fu) is at most a finite

SCIENCES

1kiJtÀ

- VOLUME 25

constant. Thus, this integral constraint becomes

2-ir f (W2+±P)rdrM,

(9)

where M i the constant axial momentum flux. Simi--' larly, in the case of a plume, (8) may be integrated to

give

2rf

(W0+)rdr=Q,

(10)

where Q is the constant vertical flux of buoyancy.

The only approximations made so far involve the

neglect of molecular diffusive processes, and use of the Botissine'sq approximation. Further impliñcation of

the equations of motion can be achieved by an

order-of-magnitude analysis based on a boundary-layer type

approximation, m which the radial length scale ô of

any flow under' consideratión "is assuThed to be much smaller-than the axial length-scale i (equal to z), i.e.,

ô/kKl. - ,-

(li)

If the orders Of ñiagñitude of the quantities U, V; W, -P; O are denoted by U0, V0, Wo, P0; 0o, then (6)

teils us that - ' ' ' ' ' -

-Üo= Wóô/l<Wo.

'

(12)

The quahtities whoSe Orders of m'agnitiide re4uirè sOme discussion are' the Reynolds stress terms, u2, etc. It will also be necessary to consider' how the

z-de-peñdences of Such quantities iii a fimnilarity solution are

related to those of the mean velocities, given by

&-pressions like (2) above. In a free turbulent shear flow,

like a jet or a plume, where the turbulence is

main-tained by the mean shear, it is generally assumed that

the properties of the turbulence, in a region whose dimensions are small compared with the length scale of variations of the mean flow, depend' on the local nature of the mean flow. -Thus, in a nonrotating jet-,

say, the magnitudes' of the quantities, u2, ,

tç., .t a

given- z, depend only on those of the quantities, U, W,

etc., at thatvalue of z. In nonrotating jets or plumes,

'this. assumption is easily applied, for there is only one dimensional parameter of the motion, M or Q,

respec-tively Thus, thmensional analysis shows that the jet (or plume) -width ô must be directly proportional to the height z above the soUrce, j.c;, ôz, or n=-1,in(3)

above. Because of this, U0 has the same z-dependence

as Wo [from (12), since l=z], say, Now:there is no

swirl, so all velocity scales of the motion mst vary as

z", and this includes the turbulent fluctuations.' Hénce,

, forinstance, varies äs Z2m. We have not yet, how-ever, answered the question of what order of magnitude

to assume for the- Reynolds stress terms. It may be

first argued that ail components of the velocity fluctua-fions have tJ ame order of' magnitude as each' oth'er

of of

C since turbulence is essentially a threedimensional

phenomenon), and second, on dimensional grounds, that this common order of magnitude is that of the

maximum variation of the largest mean velocity

com-ponent as r variesin other words, that in the

ilon-rotating jet añd plume cases, , etc., all have order

f magnitude W02. An alternative approach is the semi-empirical one of Townsend (1956) in which he writes

down the mean equation of motion (6), say, and

determines what order of magnitude the quantity must be in order that the dominant Reynolds tress

term [1/r8/8r(ri)] should balance the largest of

the other terms, and should hot be left identically zerO because it was assumed to be too large. If the Reynolds

stress terms were negligible in all the equations, the motion would not be turbulent. In the case of

(non-rotating) wake flow, this indeed gives that ii has the same order of magnitude as (Wmn_Wrnjn)2, but in

the jet case, it shows that

must have order of

magnitude W028/l, not W02, in order that the necessary

balance of Reynolds stress with inertia terms may be achieved. Since /l is small, this means that the

turbu-lent intensity (the. order of magnitude of iii', etc.) is

small compared with the square of the mean velocity,

which is observed to be the case experimentally. It is such an empirical approach that we shall, therefore,

employ.

In the case of rotating flows, however, neither of the

above results concerning- z-depéndence and order of

magnitude of the turbulent terms is obvióusly going

to be valid. First ofi all, there is a. second dimensional

parameter ) in the problem, so that dimensional con-siderations no longer demaiid 'thatöxz. Thus, U0 will in general not have the same z-dependence as W0; in any case, there is no a priori reason to suppose that

V0 has the same z-dependence as Wo. Thus, there isno

uniform z-dependence for the mean velocities, and it is not immediately possible to deduce what z-dependence the. turbulent fluctuations will have. Secondly, since it is well known that lotation inhibits radial flow, might

it not therefore alsó inhibit the radial componentu of the turbulent fluctuations, so that u is of smaller order

of magnitude than y or w, and the turbulence is not

fully three-dimensional? This seems to be a possibility,

but there is a question as to whether such a motion

properly be described as turbulence; should we not treat

it as a separate phenomenon, applying to it new tech-,

niques? Bearing these difficulties in mind, we now pro-ceed to make certain assumptions, hopefully plausible, but ultimately only verifiable by experiment. The main

assumption will be that the turbulence is fUlly

three-dimensional, the three components of the velocity

fluctuatiöns being essentially interdependent, and hence of the same order of magnitude (Uo) and zLdependence

as each other. If the radial component should be

in-hibited, then the turbulence as a whole is inhibited. The

relationship of u with, say, W0 remains to be

deter-mined by Townsend's Semi-empirical method. Secondly,

the z-dependence of the fluctuating velocities will,, as in

the non-rotating case, be assumed to be thesame as that of Wo, i.e., ztm. This is self-consistent; in fact, be-cause in both the cases under consideration Vo and W0 turn out to have the same z-dependence, as in Herbert's laminar jet.

Let us denote the common order of magnitude of

all quantities like u2, , etc., by u02, and of those like

uO by u000. The order of magnitude of the terms in (4)(6) and (8), using (12) to eliminate Uo, may now be written, in order, as follows:

W0ô u02 W02ö UO2* . Vo _Uo2

p

-- 2V0 (4a)

12 _a j2

ô

V0W0 u02 V0W0 U0i* _12Wy5

a

i i

'-

i (Sa)

UO2 lVo2 UO2* _{go po}

a i,

- (6a)

woe0 Woo 14000*

i ô i (8a)

The boundary layer approximation &«i shows that

the fourth tem (with asterisks) in eachrow is negligible

compared with the second terni, although Townsend

pgins out (p. 87) that it is sometimes

necessary to retain the fourth term for an accurate description of the flow1 This hs no ffect on the similarity scheme when

ô _{z, but would render it invalid in other circumstances,'}

so we shall ignore it as the order of magnitude analysis

suggests. Secondly, ye notice that if the rotation is to

interact significantly with the mean axial flow, it must

do so through the pressure terms in Eqs. (4) and (6). Thus, the pressure term in (6) cannot be negligible, and hence from (6a), Po is at least of order Wo2; this has the result that the pressure term in (4a) is much largerthan the inertia terms, which may therefore be

neglected. Finally, we make the small Rossby number assumption, i W02 i vo

«1.

(13)

Thus, the mean flow inertia terms in (4a) and (5a)

are negligible compared with the Coriolis terms, and

(5a) then, gives (using Townsend's semi-empirical

argument)

uoi=1Woa2/i. (14)

Now in (6a) both the pressure term and the Reynolds

stress term must be importänt, for reasons already

given, so that uihg (14) we have

Po= luo'/a oWo

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JOURNAL OF THE ATMOSPHERIC SCIENCES

VOLUME 25

Thus, from (4a) we have

(15)

so that

Po=zôWo»Wo2. Again in (4a), we have

u,2/& ÇÌW0&/1 2V0ô/l«lVo.

Thus, the only terms left

in Eqs.

(4)-(6) are,

respectively,

2V=ôP/Or

(16)

-(r)+2ZU0,

(17)

r'ar

--()+®= --.

la

ap

rar

as (18)

Eqs. (16) and (18) may then be cróss-difierentiated to eliminate P, giving

a

aria

-I

--(2lV)=-I --()+O

(19)

ôz

ôrLrôr

J

Thus, the reduced equations of motion, finally, are

(17) and (19).' The above order of magnitude analysis

reduces the integral constraint (9) for the jet to the

form

2ir

Prdr-ilt.

(20)

In the plume case, the heat equation (8) is useful for the generation of similarity solutions only

in its

integral form (10), but it is nevertheless instructive

to consider the order of magnitude f ternis in it. By

Townsend's semi-empirical argument, (8a) gives the

relation

UoO0WOØO

o -

-1

since, in this case, the turbulent diffusion term must

be of the same order as the term describing convection

by the mean flow; in the momentum equations, the Reynolds stress had to balance the Coriolis forces,

an order of magnitude larger than the mean flow inertia

'The scale analysis is really a rationalization of the fact that,

in the flows considered, there is a balance between Coriolis forces

and Reynolds stresses, leading to Eqs. (17) and (19) and the solutions of Section 3 The same solutions infact result if we assume that u is small compared with y and w (u'wVo/2ö), although more of the Reynolds stress terms are importantin that

case.

(2!)

terms. We note that (21) implies

so the integïal constraint (10) becomes

27rf W8rdrrQ

(22)

3. The sinlilarity solutions

In both the jet case and the plume case we assume

séif-preserving forms fOr the mean flow quantities, with

s-dependence of the form (3) extended by the above

arguments to

öz"; W,z";

Vozk;

eoza

(23) w, üi, etc.,, z2m From (12) we have

U, zm+ni.

so that for both terms in (17) to have the same z-de pendence we require

2mn=m+n-1.

(24)

Similarly., fòr the Coriolis term in (19) to balance the

Reynolds stress term, which it must in both the jet

and the plume case, wè require

k-1=2m-2n,

which, combined with (24); gives

k=m=2n-1.

(25)

a. The jet case ()0)

Here a third relation between the exponents in (22)

is provided by the integral constraint (20). From (16), the z-dependence of P is shown to be

Pz",

so that (20), requiring M to be independentof z, gives

k+3n= 0,

Combining this with (25), we finally obtain

n=*, k=ma.

(26)

Since there are two dimensional parameters M and fi in the problem, dimensional analysis yields unique

dimensional forms for the various mean flow variables.

From the continuity equation (7), we may define a

mean Stokes st.rearn function 'I'such that

i i

U=-û'I'/ôz, W= ---ô'I'/âr,

so the mean flow variables may be written as follows: I'= M41 r315s-1/5f (n) [whence W= M21512"5z316f'(n)/n, etc.]

V Í.1'2'5z3I5g(n)

uw= M4/512'5z°'5gi(n) uii= i1415í215z615g2 (n) where = M"51J215z'5r

and the functions f,

g, gi, g2 are nondimensional

functions of the nondimensional similarity variable n.

The equations to be satisfied by these, functións are

2d

d

frOm (17), and

5d

dn

-n(nf) =

(n2g2)

2

-r,2g±flgí'gi = O

from (19), with the integral constraint (20) giving

2lrfgn2dn

_-1;

these, however, will only be of use if the forms of the functions gi and g2, say, are known, in which case f and

g, can be predicted. The important result is the fact

that a similarity solution exists, of the form (27).

The validity of this solution is restricted by the

various assumptions which have been made concerning

order of mangitude. Assuming that för a typical point

in the jet the functions f, g, etc., are of order 1 when

is of order 1, then the conditions ö/l«1 and Wo/ô

= V0/2ô«1 imply

&

Vo fM\"5

-=--=(---} «1,

i 2& \Ç22z4/

or in other words, the solution is valid when

z»(MÌ-2).

(28)

Far downstream from the source, then, the similarity

solution (27) should be valid. b. The plume case

Here e is nonzero, and the e term must be of the

same order of magnitude as the other two terms in

(19). Hence

a n = k

1, and so

a 3n- 2

from (24). The final relation between the exponents is

given by the integral comstraint (22), which yields

m+a+2ñ=0.

(27)

Hence,

n=3/7, k=m=-1/7, a=-5/7.

The final dimensional form for this similarity

solu-tiòn is 'I'= Q4'7cr517z517f(,) [whence W= -Q2"77z117f'(n)/n etc.] V=Q2I711"s'7g(n) = Q3'7l5'7s517h (n)

i=

üii= 7t2217z2"7g2(n) = Q°'7'7s°'7g3(ij) where now, n = Q"7[2'7z317r

The resulting ordinary differential equations for the functions f, g, etc. may easily be derived as in the jet

case.

The conditions &/l«1 and V0/7â«1 this time lead to the requirement ô

V0 fQ'"7

-=--=(--I «i,

¡ 7ô \1i3z4/ or

z» (Qr3)

(30)

for the solution to be valid.

Note that neither of these similarity solutiöns could

have been obtained by assuming a uniform eddy

viscosity N, independent of z. If we wanted to think

in terms of eddy viscosity, we would have to assume a

z-dependence for N, so that in (6), for instance, we

could write

13

1(92W 10W\

+--),

(31)

r(9r \3r2

r är/

(neglecting â2W/8z2 by the boundary layer

approxima-tion ô«l); if N z we must have ß= m+n, which

takes the value -2/5 in the jet case, and +2/7 in the

plume case. Similarly, if we assumed a z- dependence K z7 for the eddy heat diffusivity (in the plume case),

the reduced form of (8) shows that

= m+2n- i

=-2/7. If we went further, and asserted that these

eddy diffusivities were functions only of z (N=Nizß, K=K1z, where N1, K1 aré constants), which is often

a fairly good assumption for free turbulent shear flows [see, for example, Townsend (1956),p. 107], then the

functions gi, g, g3 in the similarity forms (27) and (29)

may in either case be expressed in terms of f, g, h,

respectively, and in terms of nondimensional forms of

the constants N1, K1. This is one way of closing the equations. In the jet case, the final set of

nondimen-sional ordinary differential equations would be the same

(29)

as for Herbert's laminar jet, and hence the. mean

velocities would have the same profiles as in the laminar

case. In the plume case, the final set of equations is

nonlinear [because convection terms are importänt in the heat equation (8)1, of the seventh order, and con-tains one nondimensional parameter (a sort of eddy

Prandtl number). Thus, although it

is possible to integrate the equations numerically, the rather arbitrary

assumption that eddy diffusivities are independent of

radius,

together with complete ignorance of

their magnitude, means that the value of the-results would not justify the considerable effort required to obtain them.

4. Discussion

The danger inherent in a search for similarity solu-tions in fluid dynamics is that they can often be found but rarely confirmed, either by experiment br by more detailed analysis. Sometimes, indeed, a closer

examina-tiòn of the problem reveals that the flow which the similarity solution purports to describe cannot exist.

So in the present instance there is doubt as to whether jets and plumes in a rotating fluid can become or remain

turbulent at low Rossby numbers, owing to the in-hibition of turbulence by rotation. In an attempt to

resolve this doubt we must examine the criteria for the existence of free turbulent shear flows; If the jet, say,

is generated in an initially laminar way, then it will become turbulent only if the initial flow is unstable

to small disturbances. If, on the other hand, the initial flow is fully turbulent, as it may well be in an experi-ment, then the problem is to decide if it will remain so.

Now this is certainly related to the linear stability

problem; but even if that indicates stability, we cannot

say that the flow will become laminar unless we can

show that no energy will be transfèrred from the mean flow to any disturbance, however convoluted and

non-linear it might be. Thus, the first big problem, about which almost nothing is known,, is the investigation

of the exact relationship between the stability of a

laminar flow and the laminarization of its turbulent

counterpart.

If, however, we accept that the stability to small

disturbances of a laminar flow is sufficient to preclude

the existence of the corresponding turbulent flow, we

still have to solve the stability problem. A full investi-gation in our cases would be overwhelmingly difficult,

because of the complicated nature of the undisturbed

flow, involving Herbert's self-similar profiles in the jet

case, and quite unknown profiles in the plume case. We have only a few qualitative criteria which we can

use. Howard and Gupta (1962) showed that a sufficient

condition for a flow [0, V(r), W(r)J to bestable to

inviscid, axisymmetric disturbances is that

1 d -

1f dW\

- (r2V2)

₄

r3dr

4\dr

everywhere. Scaling in an obvióús way for our problem (jet or plume), this gives (approximately)

(Wó2

t' ç2)

But the low Rossby. number condit n imposes the

constraint W0/2&«1, so our flows are clearly stable tq iiiviscid, axisymmetric disturbances. The fact that they

fall so far within the region of stability is a strong

argument that the proposed turbulent flows do not exist. It is not conclusive, however, for nothing has been said of non-áxisyrnmetric disturbances, and it is well known that despite their intractability, it is these disturbances which are frequently the most unstable (cf. Batchelor and Gill,

1962, who showed that a

non-rotating jet is unstable only to non-axisymmetric disturbances).2 In any case, the túrbulence generated

at the source will not die out for. some distance

down-stream, and our similarity solutions could be valid

over a length of axis bounded above by this decay

distance, and below by the constraint (28) or (30).

Another -consideration to be remembered is that

vis-cósity will tend to stabilize the flow if the local Reynolds

number R,= Woo/v (where z' is the constant kinematic

viscosity), is not large enough. In the plume case, the

similarity solution (29) yields i(Q3Z2\1/7

2j ':.

which increases with z, and the stabilizing inuence of

viscosity decreases, as z increases. In the jet case,

(27) yields

-16.

i iM3'15

R=(

_-)

y\z2

which decreases with z, so that this effect, too, puts an upper bound on the value of z for which the

simi-larity solution may be valid.

- The two big questions raised in this section concern

the connection between sability of laminar flows and laminarization of turbulent flows, and the stability of a laminar swirling flow at low Rossby numbers to viscous, non-axisyrnmetric disturbançes. Until these

questions are answered, we cannot conclusively decide whether' the similarit solutions presented above do or

do nòt represent

experimentally realizable flows.

Professor B. R. Morton has indicated in a private

communication that he was able, in some qualitative

2 The author has recently demonstrated that axial shear will destabilize solid-body rotation at arbitrarily small Rossby num-bers,. in an inviscid fluid (Pedlèy, 196S). He has also solved the riscóus problem for flow in a rapidly rotating pipe, and found a surprisingly .sjll critical Reynolds number, again fôr small Rossby numbers. Thus, the "not unstable" argument against

the present solutiOns is demolished.

experiments, to generate a turbulent jet at low Rossby

numbers but not a turbulent plume. A controlled,

quantitative experiment is desirable, therefore, as it

would probably yield results before the above theoreti-cal questions are satisfactorily answered.

Acknowledgments. This work was supported partly by a Research Studentship from the Science Research Council of Great Britain (at Cambridge University) and

partly by the U. S. Office of Naval Research (at the

Johns Hopkins University) under contract number

Nonr 40t0(02).

REFERENCES

Batchelor, G. K., and A. E. Gill, 1962: Analysis of the stability of axisymmetric jets. J. Fluid Mec/i., 14, 529-551.

Herbert, D. M., 1965: A laminar jet in a rotating fluid. J. Fluid

Mech., 23, 65-75.

Howard, L. N., and A. S. Gupta, 1962: On the hydrodynamic and hydromagnetic stability of swirling flows. J. Fluid Mech.,

14, 463-476.

Pedley, T. J., 1968; On the instability of rapidly rotating shear flows to non-axisymmetric disturbances. J. Fluid Mech.,

31, 603-607.

Townsend, A. A., 1956: The Structure of Turbulent Sheer Flow.

Cambridge University Press, 315 pp.