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SECOND-ORDER TURBULENT MODELING APPLIED TO MOMENTUMLESS WAKES IN STRATIFIED FLUIDS

W. Stephen Lewellen Milton Teske, et al

Aeronautical Research Associates Df Princeton, Incorporated

Prepared for:

Defense Advanced Research Project Agency

November 1973

DISTRIBUTED BY:

Nattnal ftdaical WHMIMI Strvict

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I i

Sponsore a bj

Defense Advanced Research Projects Agency DARPA Cxder No, 1910

1

A.R.A.P. REPORT NO.' 206

SECOND-ORDER, TURBULENT MODELING

APPLIED TO MOMENTUiMLESS WAKES IN

STRATIFIED FLUIDS u

D D C

JAN 8 1914

B

by

W. S. Lewellen, M. Teske and Coleman duP. Do.ialdson

This work was supported by the

Defense Advanced Research Projects Agency of the Department of Defers

and was monitored by the Office of Waval Research under

Contract IJ00014-72-C-0413

The view.-? and conclusions contained In this document are those of the authors and should not be Interpreted as necessarily representing the official policies, either expressed or implied, of the Defense Advanced Research Projects Agency or the U. S. Government

f

Aeronautical Research AiSoCiatea of Princeton, Inc. 50 Washington Road, Princeton, New Jersey 085^0

609-452-2950

*

November 197i

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«ATIMG ACTIVITY fC«p«-«. «»*«> 12.. NEPORT SECURITY CLASSIPICATIOr I. ORICINAT

Aeronautical Research Associates of Princetor __l!£illassiried

50 Washington Road, Princeton, N. J. 085^0

3. REPORT TITIE

"Secona-Order, Turbulent Modeling Applied to Momentumless Wakes in Stratified Fluids"

4. DESCRIPTIVE NOTES (Typ* ol »port mnd mcluJiv« dmlom)

Interim i. AUTHORISI (Fml name, middle inmml. ;••• nmm»)

W. Stephen Lewellen Milton Teste

Coletnan duP. Donaldson

S. REPORT DATE

November 19 73 7«. TOTAL NO. OF PAGES

^7 •■■ CONTRACT OR GRANT NO.

»00014-72-0-0413 /UMJ b. PROJECT NO.

DARPA Order No. 1910

10. DISTRIBUTION STATEMENT

Tfc. NO. or REFS

18 9«. ORtGINATOR'S REPORT NUMBERS)

A.R.A.P. Report No, 206

•6. OTHER REPORT NOISI (Any other number» ihmi mey be memigned ihie leporl)

This document has been approved for public release and sale; its distribution is unlimited.

11. SUPPLEMENTARY NOTES

13. ABS1RACT

12. SPONSORING MILITARY ACTIVITY Defense Advanced Research Projects Agency

Tue technique of second-order closure for turbulence modeling Is used to treat the growth and decay of turbulence, and the generation of Internal waves behind a submarine moving at constant velocity through a stratified ocean. A computer mcv-cl based on the solution of five partial differential equatiras and approximating nine others is generated. Required model coefficients are obtained by comparison with data obtained from laboratory flows and the atmospheric surface layer. The model is verified by comparison with Naudascher's wake data In the limit of v .nishing stratification; with Wu's observations in the limit of starting with a uniformly mixec region in a itrcn 1 stratified fluid; and with Hartman and Lewis's analytical predictions in the limit of a small perturbation to the ambient linear density gradient and vanishing turbulence. Model runs are made demonstrating that the strength of the Internal waves generated by a decaying wake Increase with increasing initial potential energy (unless the initial Richardson number Ri « 1 ), or increasing initial kinetic energy (unless Ri » 1 ), or increasing ambient stratification gradient.

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DD FORM < NO» «5 1473

Security Classification

iMaMM^ -

mmmmimmKm

Security Classification

KEY «ONos

Submarine wakes

Turbulence modeling

Internal wave generation

Momentumless wakes

J Security Classification

•MM* __^B_HaatBa

sw-W— t« ^J

I I

I

1

I 1

TABLE OF CONTENT.

Summary

Nomenclature

Introduction

Th« Turbulence Model

Computmtional ho nemo

Numerical Results

Conclusions and Future Plans

References

~

m

i I

i i

ii

—^m^mmmm

mm mm

1

1 i

1

1

a,A,b

CD

D /

Fr

g

N

P

q

r

Re

Rl

3

U

Ui'UJ'Uk

U

v

xi'xJ'xk

A.

NOMENCLATURE

model constants

drag coefficient of generating body

diameter of generating body

wake Froude number = r.N/U

gravitational acceleration

Brunt-Väisälä frequency ■ [-(g/p) 3p /J»3*

pressure

square root of twice the turbulent kinetic energy

wake radius

initial wake radius

Reynolds number = ur./v

Richardson number of turbulence = rfl^/q2

i m model constant

velocity departure In free stream direction normalized by U

Cartesian velocity components

free stream uniform velocity

horizontal velocity normalized by U

model constant

vertical velocity normalized by U

Cartesian coordinates

coordinate in free stream direction

coordinate in horizontal, vertical direction normalized by initial wake radius

normalized plotting coordinate ■ cy, cz (c = 1.45 in all figures except Fig. 3 and 6 where c - 2.75)

ill

■■ - m

mmmmmmm

I i i

i

i i

x

X

A

v

IT

P

P

Po

f

supencripts

coefl .clent of laminar diffusion for density perturbation

mlcroscale of turbulent dissipation

nacroscale of turbulent model

kinematic viscosity

perturbation pressure = p -i / gp dz

density

normalized perturbation density ■ (p - p )/(r.dp /bz)

ambient fluid density

stream function = / vdz = - I wdy

denotes time average

denotes fluctuation about the mean valuo

subscript

m denotes maximum value

iv

•' ' mm wmm

i-I

I

I

i

1

I

I I I I

1, INTRODUCTION

A submarine moving at constant speed through a stratified oceer

leaves behind a turbulent wak.e containing potential and kinetic

energy. The build-up of potential energy Is caused by the co-mixing

of heavier density fluid above the lighter density fluid. The

kinetic energy Is composed principally of the turbulent contribution.

In the Initial stages of the wake,the turbulent kinetic energy

dominates the potential energy and the wake spreads. This spreading

in turn 'ncreases the potential energy, until at some point gravita-

tional forces are able to overcome the turbulence and T"-)rce a collapse

of the vertical spread of the wake.

The wake collapse has been studied both experimentally (RftfS. i--

and theoretically (Refs. 4-10) to gain a qualitative understanding of

the phenomenon. The theoretical studies have dealt primarily with

idealized, invlscld collapse of an initially mixed region, the excep-

tion being Ko's analysis (Ref. 9) which attempts to include full

coupling between the dynamics of the turbulence and the collapse,

albeit in a simplified, integral approach.

The present analysis uses the techniques of second-order clotr.re

for turbulence modeling (Ref. 11) to treat the dynamic coupling

between the turbulence and the stratification in some detail. We

are concerned here with the growth and decay of the turbulence and

the generation of internal waves, but not with the propagation of

these waves to distances far outside the turbulent wake.

The turbulence model is discussed in Section 2. The computaiK . U

scheme for dividing the wake into regions governed by different physi-

cal phenomena is discussed in Section 3. Numerical results are

presented In Section 4.

am

^■1 —■ "—" •" U-lJl.lHilll

I 1 I I I I I i I I

2-1

2. THE TURBULENCE MODEL

The time-averaged equations of motion for a Boussinesq fluid

whose motion consists of a mean and a i'luctuating part may be

written as

la.

Du]

Dt

= 0

' o i

&±p

(i)

(2)

Whether the density variation is caused by a temperature variation

or a variation in composition, when the ambient density gradient

is constant the diffusion equation for the perturbation density P - P(

may be written as

C(p-P0) 3 , ^p-p,

Dt

eki'p '

dx . I 3x / " dx .

0,P, - u

1 dx (3)

Starting from the equations of motion for the fluctuating compon-

ents of velocity and density,It is possible to derive exact equa-

tions for the Reynolds stress ccrrelation u.'uI and the correia-

tions Involving the p' densily fluctuation. These may be

written as

BÖJÜT DT-^

^ (üjquj)

i ^p'u,'

PT ^X .

dp' i]

^o 1 + p_ \ 3x , 3x. / + v

32u'u r 1

dx ?

du? du' (4)

i Mwm -"W mmmm ijia

1 1 Du-p'

i • i, i

dx ,

du. %-

1 ••

Dt Vj Ujp' »j 5:p

- vp' o uJ

6xi - /cu'

1 ä -P;^(P'P')

2-2

(5;

Dp«' jh dx. dx. Jp jx, dxT J M ""J

.■-

Second-order turbulence nodellng involves deriving vor scae-

tlmes assuming) relationships between the third-order nut Wilt Mil appearing In Eqs. (4) - (6) and the derivatives of lowe:—order correlations. We choose to model the third-order correlations as (Ref. 11)

Dissipation teacost

du.1 du' r -2 2 1 (7)

I I I I I

d2u^ ^. j— + KU^ iL-^- = ^=7 (V + «) U'p1

dx'

with

dx'

2K dal&J.= £•

X2- A* Ta bAq/v)

(6

(9)

1 .«PW.OT II nmmimw m^m ■^

..

?-.:

Pressure strain lerins

"307 £1(. p V

- S fu'u' -^^) (10)

- A* U.p' 11)

Diffusion terms

ukuiuj

BL n. U' ■ Po 1

squj ^uj^ äujuT

du] u'

(12)

(13)

u-u.p. . " vcqA Sx

äujpl (14)

i Ujp. - vcqA i-

j (15)

I ll

I I I

^ " vc^ hq H rpi (16)

Reasonable values for the coefficients a, b and were

determined in Refs. 11 and 12 as 2.5, 0.125 and 0.3, respectively,

oy examining the turbulent distributions in free jets and wall shear

layers. The coefficients A and s which influence the effect of

stratification were assigned the values of 0.75 and I.Ö.respectively,

in Ref. 13 to obtain good agreement with the turbulence distributlor.i;

observed in the atmospheric surface layer. With this choice of modei

coefficients, model predictions for the mean velocity gradient, man

temperature gradient, Richardson number, rms vertical velocity

ima^mm mwmm

1 I

I I I I I

2-4

fluctuations, rms temperature fluctuations and horizontal heat flux

agreed favorably with surface layer experimental observations over

the complete range of stability condltionip. The values cited above

A and s will be used in the for the coefficient ,

present analysis.

Equations (1) through (6) represent a formidable set for the

general three-dimensional wake. To keep the problem manageable

for our relatively small computer (a Digital Scientific Corp, META-4),

we have simplified the equations in certain regions of the downstrean:

development of the wake. Our approximation consists of neglecting the

convectlve and diffusion terms in the equations obtained by substitutl:-

Eqs. (7) through (16) into (4) through (6) to form a set of algebraic

relationships between the turbulent correlations and the mean flow

derivatives, but retaining convection and diffusion of the turbulence

by carrying the equation for q2 = üjü? . Equations (4) through (6)

are replaced by

^ 2 uiuj dx s.-

- 2 — U-p' + P^ lr 3X - [«•*+ *> Hj] - ^vq

^ü. bu. bu. u>p« " " UK 33C " uJuk giT - % -S~ " S

U'p« (17)

i P,

~im'£öxi>~W'viii*u 0 = ^1%-^^-%^-^^

(ie)

(19)

l I

0 = - 2 ^F" % - 28^ iT- (20

MMH ■■■

2-t

..

.

«•

Note that the function f has been added to Eq. (18) to allow Kq. (i

to be added without making the system overdetcrmined. The term is

added in such a way that the Influence of the corveeclve and dlffusiui

terms in Eq. (17) are distributed equally between the three components

of q' Also, the diffusion term in Eq, (l?) is somewhat simplified

from that which would be obtained for ujuj from Eq. (M.

The set of equations (1), (2), (3), (17), (18), (19) and (20) make

up what we will call the quasi-equxlibrlum (QE) approximation for

second-order closure. It is closely related to what Melior and Herring

(Ref. 14) refer to as mean turbulent änergy closure. It reduces to the

same number of differential equations but retains more of the c#U|>llng

between the components of the Reynolds stress through the algebraic

relations, Eqs. (IS) - (20),

Figure 1 gives a comparison of the QE results with that for the

full set for the case of an unstratified, axisymmetric wake. Results

lor tne full set are taken from Ref. 12 from an axisymmetric calcula-

tion while the 9ß results are taken from a 3-D, stratified calculati;,;;

In the limit of negligible stratification. The q distributions ae;-- -

quite closely while the mean velocity departure from free stream ■hows abuut a 10^ difference wnen it has been allowed to decay an order 01

magnitude from the Initial conditions. The great simplification m<le

possible by the QE approximation appears to be ample justification tar

t! is slight loss in accuracy. Comparisons of the two methods have

liao been made in calculations of the planetary boundary layer which

snow relatively little JUference between the two, but these results

Will not be shown tu. re.

The scale lengtr: A In Eqs. (17) - (20) should in fact be iolved

by its own aiffere.-.t lal equation, enabling us to determine the A

/aflation across the wakt. is the solution proceeds downstream. The

„qaation for A would be similar to the equation for q , Eq. (17*

We choose not to du tnis, however, since the scale equation has not

yet become an integral part of the second-order turbulence model we

mmm ■^MMU INI 111—It

2-6

K>

I I I I !

gOlxZX/^n ^OlXjb 4ZOIX0M

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O nJ O ^

(M to P ii ;<

WO) <H

+» BQ , --• rt\r■

S co X 9 ta w

£-,.£ —

p C 3

O TJ IJ - cr0

4-5

CO 4J rH

ra

U

■3

a o

ce « o o o

•ri tQ

CO a o c: ■■- •H a;

<i>

Si

•H Ct hO

rr1

0) I

•H to aJ 3— n<

CO

O O

id ?iS +j bi) co

tu M c « OJ CO

•o

tu

O nl to rs •H O" U a) aJ Q.rH t: rH O !3

a»

4-> H Ö «5 a> a> H 2

o rJ w o -Q

--' to «J

I

I I 1 I 1 1

I 1 I I I I I I

are ualng. toftead, m cnoost to use a A geared to tte brt^dLh

the q ta^ment Hate region. For our analysis here and in ttM

free Jet exatniiution (Ref. 12), we assume that A is proportional

to the distance from tnc radius of the Mxlmwi q2 (typically

r = 0) out to the radius where the value of q2 has dropped to l/k

its maximum value. Comparison with the axisymmetrlc flows In the

near-wake region yield a proportionality factor of 0.2.

The A that results from this algebraic relationship now

expands or collapses depending upon the turbulence within the wake.

Sine •,when collapse occurs, the scale in the vertical z direction

can be greatly distorted from the norizontal y scale; we have

chosen to compute separate scales along the y and z axes. These

resulting Ay'r, and Az'5 then go directly into their approprla'.

diffusion terms in me q''^ equation. The QE approximation ana

the dissipation terra -2bq3/A use only one scale length. Since w

anticipate that the collapse will yield an ellipse-like wake region,

we have chost-n the eqaitlon for A as

A » üLf A + A y ü

Tnus, when Ay » ^ for complete collapse, A - <2AZ and enable«

A to track the details of the vertical collapse. Following Ref. 13

we also impose an upper limit on the vertical scale

A, Ri q^ ^m

max JL. o

^ o az

I where Rl* is a critical Richardson number found from the atin®»pherl

I surface iay.:r to be Rl

■ mm Ma -

.,

-

COr-iFUTATlüNAL SCHEME

A typical turbulent wate i.n a stratified medium can be divided

into regions governed by different physical phenomena. At the aft

end of the body, the flow is dominated by the details 01 the body

shape and mode of propulsion (this problem will not be discusser.

here). A short distance downstream of the body in the near watee,

the kinetic energy of the mean flow is converted into turbulent

kinetic energy. As long as the potential energy in the near nake

is much smaller than the kinetic energy, which should be true as

long as

?,& op _ O a -c' 3 7^

U I.e., -^^r=Fr «CD (-

it should be possible to !g..oi't the effect of stratificatio;. Ln th region. Thus if there ar» no asymmetries introduced by the bed;, , near wake may be treated S8 an axisymmeti ic now. Equation'., (2)

4) were integrated for austratIfled, axisymmetric flow In Kef. i ^he model demonstx'ated the strong influence of net mean motnen- a:

the development of the wa.^i-, with the model predictions verified ;. comparisons with existing data for a wake behind a self-propoil. I

and for a wake with significant mean momentum.

The near-wakt .region should remain valid as long as the turoul« Is strong enough to dominate the stratification, i.e., us long u

-?- ^r - Ri « 1

I I • -\

Although the strat iriccition does not inriuence tne turbuiexu-e

significantly In the near-waKe region, the density distribution is

determined by the turbulence. Thus, Vor stratified flow it is

necessary to carry the additional Eqs, (3), (5) and (6). Howevt.:-.

as long as the Initial conditions contain only the streaming

velocity, it is possible to reduce Eq. (2) to a scalar equation

for u . Assuming that diffusion in tne free stream direction Is

negligible compared with that normal to the free stream, that the

pressure gradient in the free stream direction is zero, and that

the mean flow departure from the free stream is small, we can redu» •

Eqs. (2) and (3) to:

°u _ ±_ v2 _ <W. _ 3u'v' 3x Re 3z c5y v J

dx m ¥ äz dy

where ,, N2 N2 T2 » i-« + 2

dy dz

If we accept the quasl-equillorium approximation for the torhulenc«

discussed In the last section, then the set is completed by tatür«g

pi = _ ^-vrp- - 2 IPV^^ - 2 ir^-|H

1 '(^^A [ ? Rex

mM ^M

I I 1 1 1 I 1 I

3-3

together with th« ulgebraic x-eiations given In Eqs (o) - {20). m

these equations ,iengttis have beeri normalized with respect to "-he

initial wake radius r. ; velocities with the free streatu velocity,

U ; and tne perturbation density with - r. 3h /dz . The dimeasion-

less parameters are defined as Re = U r /v , a Reynolds number;

ft» - [( - grj^ äp0/^l)/p U^]"2 , a Froude number; and Pr mK/v , the

Prandtl number. The algebraic relations between the second-order

correlations bring In no additional parameters so the variables are

a function only of Re, Pr and Fr plus the initial conditions on q ,

u , and p . The diffusion terms In the turbulent energy equation

have been permitted to be more anls.:tropic in Eq. (27) than the form

In Eq. (if). This form permits a larger influence of the stratifica-

tion, but we have not baen able TO firmly establish which form is

more accurate yet.

Equations (23) to (27) cease to be a valid approximation for

the wake when the potenl Lai enepgy due to the perturbed density

increases to ehe point where It is the same order of magnitude as

the kinetic energy of the turbulence, i.e., wher. Rj > 0.1 . Then

the effect of the Induced vertical velocity on the convection terms

can no longer be neglected. To continue the wake calculation! into

the collapse region,ii is then necessary to Include the two transverse

momentum equations from Eq. (2).

In the case of a momentumless wa>lte, the mean velocity departure

from free stream decay« mucn more rapidly then the rms turbulent

velocity so that with, density ■tratlfication typical of the ocean (Brunt-Valsala period of th« order uf on«:-half bear) the mean velocity

departure becomes negligible before Ri reaches 0.1. Therefore, for

many cases of interest we cm set u _ o and drop the axial momentum

equation in the collapse region. The set of equations to be solved

In this region within the quasi-equlllbrlua approximation now becomes

- ■ -

3-5

In the overlap region, which should occur as long as Ri immediately behind the body is sufficiently smull, either set of equations may be useo. If there Is no overlap, such as suay occur in the wake

behind a low drag body moving slowly through a steep stratification,

then our present computational scheme is not valid. Future modifica- tions to our computer model will provide for the slmultaneoua solution

of all three moment urn equations so that this more general case can be

analyzed. Figure 2 gives a comparison between tue results of a Phase (l) and (II) computation for a case with a large overlap region. No

significant differe ues are observed in the two solutions until after Ri = 0.07 at .04 B.''. (Br unt-Välsälä periods) after wake generation.

Of course. Phase (I) represents a much more efficient computational scheme in the overlap region.

We have all c! the Information needed ^o solve the equations of

Interest for our' Phase (I) or ^ ' . ' regions. We begin our calcula- tions in the near-walce region by nssuming initial conditions fcr the

turbulence, tne perturbation ülenslty and mean u profile. The Phase (l) regier is pemltted to ouild to the overlap region, at which

time we Introduce zne i jraal velocities v and w and the pressure

TT to proceed through the collapse Phase (11). To reduce redundant calculations, we consider only a quarter-plane of the wake cross-

section, thus requiring us to impose aj propriate sy: ;:,etry conditions along both axes.

Since our primary purpose is to calculate the wate dynamics rather than the full wave pattern between the wake and the surface, we impose

an artificial porous liner a few radii from the turbulent wake; this liner serves to absorb any waves emitted from the wake without reflect-

ing them back to the walce. In this way we are Justified in requiring all the variables -► 0 as r ■• " . Thli absorption of the waves is

implemented by adding the damping terms - kw and - kv to the right- hand sides of Eqs, (28) and i y 1, respectively, and the term - v dk/dy

- w dk/dz to the pressure Sq, (32). The damping factor k is zero

I»--. m-am

■=-6

B.V.

Figure 2. Ccmparison of Phase (I) and (II) calculations (Initial conditions - Fr = 228., q2 .00436, um - .00236, pa « 1.» V • « » 0.) m

P.E. !-'io • +oo

J-. g(. -,0);dYdz .

mm^mmm mmm ■■■MHMMM

..

.. 3-7

-:

out to a few wato radii and then Increases smoothly to absorb any waves

A: the wake dimension grom», the position of the liner also moves out

at a proportional rate.

Figure 3 is a plot of density contours for two different positions

of the absorbing liner. Although the contours ax^e quite different

outside the inner liner, there is little difference in the region of

Interest around the turbulent portion of the wake Itself. A detailed

comparison of these two cases shows that the average numerical differ-

ence at the points inside the Inner liner Is less than 5* of the

maximum local v^lue.

The differential equations discussed above are solved numerically

by replacing them with rorwird-time-centered-space finite difference

equations (Ref. L7). Since the equations involve two space directions

y and z and a tlme-lifce direction x , we choose to solve them by

the Alternating Dtr«ction Implicit (AD1) method (Ref. 18). The

implicit technlq;.- pennlta stable solution development with a

contrcllec Lncreast in ■er step. The mesh in the and directions Is variable, with mesh spacing Ay and Az minimizing

curvature change:, from point to point.

When needed, the missen equation (Eq. (32)) for the perturbation

pressure v Is solved by adding a time-like term - bv/bt to the

left-hand side of Eq. (32) and iterating by the ADI method to a steady

state solution (so that dw/dt ~ 0) for every step in q2 , p , v

and w . For the flr»t two Lrerstlon steps. At is given a value

proportional to the area of the wake (At = y z /4ü) and allowed v •'max max ' axiuweu to Increase or decrease thereaft««1 at a rate determined internally in

such a way as to try to aalntalB a change of z 10% with each iteration.

In practise our solution scheme takes a step ZiX in the main

variables (q^ , , , and u . or v and. w as the case dictates),

then the pressure is iterated for these new values. We find that

our procedure worKs ,ery well Cor Phase (I) (where no pressure

Iteration is required), but the Phase (II) pressure solution has

3-ö

2S..

20..

Figure 3. Comparison of density contours at one B.V. after waice generation for two positions of the porous liner Out- side the indicated dashed line the liner resistance increases from zero exponentially. — contours for the case of the larger liner. (Initial conditions - Fr = 1.15, qm= um = v w . 0 , ^ l.) J. .280 with contour for P/P» - +.5 m denoted by -1-; -.5 by -2-; ..1 by -3*| -.1 by -4-; +;01 by -5-; and -.01 by

1 I 3-9

i 1 [

I 1 I

caused the bulk, of our numerical problems. It seems that our ADI

method Is not an extremely viable one when steady state is sought

for 77 at every step Ax . Rather, we found that stability and

convergence were better controlled by limiting the number of pressure

iterations at each step; a maximum of five Iterations were permitted.

This serves to keep the pressure close to an accurate value, and in

turn keep v and M tnus p , accurate also. We monitor our accuracy by trying to keep the relative error

TT h j (Afrrdy dz

r 2 / F^dy dz < 0.001

I I

where Aw is the change In TT in the step At , and F is the

value of the right-hand side of Eq. (32). F has also been slightly modified by realizing that the term of most importance - 1/F bp/Sz

can be estimated for the next step size ^x and the pressure estimated ahead accordingly. Since the numerical technique is not exactly

divergence free, the extra term on the right side of Eq. (32) in the form bV-'v/dx is retained to make V-v" = 0 at the future Ax main step.

A typical run in Phase (I) takes about 3 hours on the Digital Scientific META-4 (12 min. on CDC 6600), while Phase (II) takes

about 12 hours on the META-4 (50 min.on CDC 6600) of which about

30^ of the time is involved In the pressure iteration loop.

I I I

MMhft

4-1

{

I I

4. NUMERICAL RESULTS

Considerable confidence in our turbulence model was gained by

the comparisons Ifch uhe unstratlfled wak.es in Ref. 12 and with

the stratified atmospheric shear layer in Rtf. 13. To gain some

confidence In our computer model's capability to predict the

generation of Internal waves, we made a check: with Hartman and

Lewis's linear, Inviscid solution (Ref. 5). Within our self-

imposed limit of 40 x 40 mesh points, we were unaule to maintain

good agreement in the iumedlate neighborhood of the discontinuity

at the edge of the homogeneously mixed region. However, results

within the center of the mixed region agree with the exact solution

to times greater than one B.V. Figure 4 is a comparison of the model

results with the linear analytical predictions for radii less than

00^ of the radius at which the discontinuity occurs. The bars on

the model results give the r.m.s. deviation of the value at

individual points about their mean.

Another check on the model is given in Fig. 5 by comparing with

the experimental results >f Wu (Ref. 3). Wu used a mechanical mixer

to produce a cylindrical region of homogeneously mixed fluid

surrounded by a strongly stratified fluid. His observations of

the width of the mixed region as a function of time after release

are plotted as gl^en In Ref. 10. The model prediction for growth

of the wake width for a case with lar^e initial Ri and an initial

aomogeneously mixed wake is given as the solid curve. The agreement

is similar to that obtained by the numerical models of Refs. 6 and 10.

The streamline patterns predicted for this two-dimensional flow at

times of 0.5, 1.0 and 1.5 B.V. after release are presented in Pig. 6.

At earlier times there is only one vortex in the quadrant but other

modes appear as time progresses. At t * 1.0 B.V. there are three

distinct vortices, while at 1.3 B.V. two of these of the same sign

have Joined and a fourth is forming along the vertical axis.

im. MM

LSLJ- i- . i„.^imm

*

I I 1 I

n-.

-I.Z1-

Flgure 4. Perturbation density and velocity variations as a function of time for an Invlscld, small amplitude linear perturbation In density within a cylinder of fluid. e(= 10-5) defined by Initial condition on p . ana- lytic solution given by Hartman and Lewis (Ref. 5)i J numerical model predictions with r.m.s. scatter In the Inner 00% of the radius of the cylinder as Indicated.

M* _*«_

ms^B^^POB* '• ""

10

»0 -I

01 01

Log (B.V.)

Figure 5. Horizontal wake spread as a function of time aft generation, (initial conditions - Fr er

m m 1.15, 0' Pm = !•) model predic-

tions, V* experimental observations of Wu (Ref. 3)

--

.mmmmmzm w4im£.^msmmmmm^^^mtiiBi

4-4

*m - •128

1.5 B.V.

^n ' •178 1,0 B.V.

Y

^ - .298

0.5 B.V.

Figure 6. Streamline patterns for the same conditions as Fig. 5. Contours for ^/^m as denoted in Fig. 3

iJi ■ H--

I J 1

I

»

H-5 j

For our case of a momenturnless waKe the principal parameters are i

the Froude nurrber Fr « [-(g/r)äp /äz]~2U/D , the Initial level of

turbulence q /U . the initial density profile in the wake, and the

Reynolds number of the turbulence ( q Vv). 1t. For a submarine wake

the most important single parameter is a combination of the first two,

a Richardson number for the stratified turbulent wake (Eq. (23)). As

seen in the previous section when Rl « 1 right behind the body, the

wake will grow for some distance behind the body before the wake

collapses. Then the initial density distribution is relatively

unimportant since the turbulent entrainment of fluid in the growth

region will lead to a density distribution at the point where collapse

begins which is independent of the initial distribution. The Reynolds

number is not influential as long as it is large enough so tnat qA/v

remains much larger than one through the Initial collapse. The only

parameter we will consider in detail herein is Ri.

If Naudascher's (Ref. 15) values for turbulent intensities behind

a self-propelled bod,y ax'e used as a star-ting point (although (iran (Ref. l6y

indicates that his values are about a factor of two higher than would be

obtained behind ct streamlined body driven by an efficient propeller), then

a body with a radius of five meters moving through a stratified fluid 2 -5 -2 with N = 10 Vsec at a velocity of five meters/sec would have _2

.Ri ^10 at a station 20 diameters downstream of the body. Figure 7

is a plot of some of the wake variables as a function of distance dowr:-

.itrtam of the body or.equivalently, time after generation for a run in

walch we start with a homogeneously mixed density and Naudascher's

values of turbulence at x/D ^ 20 . Up to x/D = 200 where Ri a 1,

the results for q are the same as obtained in Ref. 12 for an unstrali-

fled wake; the denslt-. perturbation Is being reduced as the wake entrains

•uore fluid; and v uu] w are much less than q , At larger distances

ijwnstream there is a significant difference In the shape of the wake

although the maximum q shows little difference. Although v and w

build to maximum velocities at approximately 0.5 B.V., they remain

smaller than q . Figures fa, 9 and 10 show the contours of p and q^

__.. _«-^^___

■"■ " ■' '■ ' -»T^iBl^i^^PW^WW^lBI im

4-6

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-3.5 ..

-4

r^w7^ -3

•- o

-+-

m—!T— LOG CB.VO 1

t \-

l 2 3 LOG (x/D-20)

H 4

Figure 7. Maximum values of q , p , v and w as a functjon of time after generation, (initial conditions - Fr = 228 qm = -GMtf, v ^ w . 0 ?„, - 1.) Equivalent distance m behind a Naudascher body Is also indicated.

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4-9

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1.0 B.V.

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fm - .00128

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Figure 10. Normalized streamline patterns at different times after generation for the same conditions as Fig. 7. Contour notation for f/Tp as in Fig. 3.

-

i I I i 1

4-10

and the streamline patterns of the radiating waves at 0.0, 0.5, 1.0

and 1.5 B.V. periods after the wake generation.

When the stratification is increased the picture changes as seen

In Figs. 11 to 14, Stratification oecomes important a shorter

distance behind the body, before tue wake has grown to as large a

radius as previously, and the strength of the streamline patterns

produced by the collapse is increased. As a function of x/'D , q

and p are the same for both runs until x/D - o? (a 0.1 B.V. on

Fig. 11) at which point the stronger stratification forces p and

q to be decreased at a faster rate as v and ■ Increase. In the

region of the principal collapse (0.1 to 1 B.V.) v and w scale

fairly closely with N , as seen In rigs. 7 ana 11, and the stream-

line patterns are quit« similar as seen In Pigs. 10 and 14.

When the Initial Richardson number is decreased by increasing

the intensity of the turbulence, the results shown in Figs. 13 to 18

are obtained. The Influence of ptratificatton is delayed until the

potential energy car: build to a large:- value. Consequently, a

stronger streamline pattern Is generated by the collapse although

its occurrence is delayed further downstream.

The maximum potential energy pw unit length of the wake occurs

when the local Richardson number Is of order pet. Figure 19 shows

how this maximum value varies as a function of the initial Richardson

number over the range of initial Richardson numbers expected to be of

interest for a subriarlne (10 < Hi < I) when the initial distribu-

tions of density and t irbulence are held constant. Sensitivity to

variations In *ric distril .'i^... of the Lnit Lai turbulence have not

been computed "cut I". ! ;-.. ct«d that these would have little influ-

ence on the a-.; Mien --. ■»».. ÜI rig. 19 as long J.S the initial

Kinetic energy *»~i-- : I ;. I •-. results are sensitive to variations

in the Initial ratio o: potent Lai to kinetic energy. Two curves

obtained by starting with different Initial density profiles are

included on this curve to demonstrate this variation. As long as

Ri is small enough, the initial density distribution is unimportant

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LOG CB-VO

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Figure 11 ( Maximum values of q , p , v and w as a function of time after generation. Initial conditions are the same as Fig. 7 except Fr is decreased by y/TÖ.

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^m = .00162

1.0 B.V.

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10,

Tp = .00286 m

0.5 B.V.

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Figure 14. Normalized streamline patterns at different times after generation for the same conditions as Fig. 11 Contour notation per Fig. 3.

■

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I I I :

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4-15

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-2.5 ..

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Figure 15. LOG (x/0-20)

Maximum values of q , p , v and w as a function of time after generation. Initial conditions are the same as Fig. 7 exceot q2 is increased by ten. ' m

---

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MI...II^IJB.BjlinHI^^WWL,__. . ...l^. il!l!JBlil,.:|il_t*J.i^^ ^^■iPSIBI^pili m^mmmmmmsm^. .. _....__., .Jl J._..

4-18

10^

^ = .00172

1.5 B.V.

z

V'm - .00157

10,

::

fm = .00234

0.5 B.V.

Figure 18. Normal .'.zed streamline patterns at different times after generation for the same conditions as Fig. 15. Contour notation per Fig. 3.

wtmmmmmam*^.

4-19

m

01 rH nj cd

•a o C

0) u

<U 0)

x: 0)

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l 4-20

since the »afee will continue t.o grow and develop its appropriate

density distribuLlon prl®r to collapse. When RIQ > I, collapse

is imminent and the initial density distribution is a dominant

factor. Since in this case the Initial potential energy is the

maximum value of potential energy it is independent of Rio .

The limiting asymptote for zero initial potential energy varies

approximately with the -0.0 power of the initial Richardson number.

A summary curve showing the distortion of the wake as a function

of time after generation is presented In Fig,. 20 for several of the

points on Fig, 19. The experimental variation as reported by

Schooley and Stewart (Ref. 1) is liso Included. A detailed match

of Schooley - Stewart 's run h^s not been made yet since their initial

Rl ~- 0.1 , where it is necessary to also know the initial density

distribution In the Make. It appears that a reasonable match could

be made since their results fall between trie Hi - ü.04o3 and 0.465

model results.

>.^ ■ ■ ■ ■■

^P^PliiBPBgflr ^mimfmm mmmmmimmm&!wmmmmm^mmmmKmt

4-21

A) Ri ».00465

B) Ri -.0465

C) Ri = 000465

D) Ri =0.465 x) Schooley - Stewart dato

(Ref. I)

.2..

Ill I I I I

.5 -!7S

B.V.

"tig I.!

Figure 20. Wake distortion as a function of time for the four different points Indicated on Fig. 19.

iM

tmm ■ ■ " ■ Ll1- •"■ ■ ■—

5-?-

We also recommend that experimental laboratory data be ehosei. by the contract monitor so thai detailed comparisons between our

model ppedictlona and observations can b- made ror a stratified wake similar to that expected behind a submarine.

..

::

i

I I I I I

I I

6-2

14. Mellor, G. L. and H. J. Herring: "A Survey of the Mean Turbulent Field Closure Models," AIAA J. U, No. 5, 1973, PP. 590-599.

15. Naudascher, E.: "Plow In the Wake of Self-Propelled Bodies and Related Sources of Turbulence," J. Fluid Mechanics 22, 1965» pp. 625-656.

16. Gran, R. L.: "An Experiment on the Wake of a Slender Propeller- Driven Body," TRW Report 20086-6OO6-RU-00, June 1973.

17. Roache, P. J.: Computational Fluid I>v mics, Hermosa Publishers (1972).

18. Carnahan, B,, H. A. Luther, and J. 0. 'likes: Applied Numerical Methods, Wiley Publishers (1969), PP. 452-453.

-

' -