The shape of two-dimensional turbulent wakes in density stratified fluids

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THE SHAPE OF TWO-DIMENSIONAL

TURBULENT WAKES

IN DENSITY-STRATIFIED FLUIDS

by

Roy Hayden Monroe, Jr.

and

C. C. Mei

HYDRODYNAMICS LABORATORY REPORT NO. 110

Lab. v. Scheepsbo

Technische Hog(

Delft

Prepared under Contract No. Nonr-1841(59)

Office of Naval Research

U.S. Department of the Navy

Washington, D.C.

DEPARTME

OF

CIVIL

ENGINEER!

SCHOOL OF ENGINEE MASSACHUSETTS INSTITUTE OF Cambridge, Massachusetts

hool

1

T68-4

HYDRODYNAMICS LABORATORY

Department of Civil Engineering

Massachusetts Institute of Technology

Cambridge, Massachusetts

THE SHAPE OF TWO-DIMENSIONAL TURBULENT

WAKES IN DENSITY-STRATIFIED FLUIDS

by

Roy Hayden Monroe, Jr. and

C. C. Mei

June 1968

Report No. 110

Prepared under Contract No. Nonr-1841(59) Office of Naval Research

U.S. Department of the Navy

ABSTRACT

The results of an analytical and experimental investigation of the rate of growth of the zone of turbulent mixing behind a

two-dimen-sional circular cylinder are presented. The analytical phase of the

study formulated a dynamical model for a parcel of fluid subjected to

inertia, turbulent damping, and gravity restoring forces.

The experiments were conducted in a tank 120" long and 4.5" wide which was filled to a depth of 18" with a fluid having a linear density

gradient. _{The cylinder was towed at mid-depth by a variable speed}

motor-pulley system. A record of wake growth behind the cylinder was

made by means of motion pictures of an aluminum pigment _{tracer in the}

fluid. The Reynolds number ranged from 1300 to 3500 and densimetric

Froude numbers varied from 5 to 20.

The density stratification was found to exert a strong inhibiting

force on the wake growth. Using the dynamical model to correlate

exper-imental data, the mixing length in the turbulent zone was found to

decrease with increasing density stratification.

ACKNOWLEDGEMENTS

This study was carried out under the sponsorship of the Office of Naval Research under Contract Nonr-1841-59. _{The project is supervised}

by Dr. A. T. Ippen, Ford Professor of Engineering, and administered _by

the

M.I.T. Division of

_{Sponsored Research under designation DSR 78228.}

The authors are indebted to Mr. A. K. M. Wang, Research _Assistant, who assisted in the early phase of the investigation. Sincere thanks

are also extended to Dr. Seelye Martin of the Department of Meteorology at M.I.T. for freely giving of his time and advice.

TABLE OF CONTENTS

Page

ABSTRACT ft

ACKNOWLEDGMENTS

TABLE OF CONTENTS iii

I. INTRODUCTION 1

II. LITERATURE REVIEW 4

The Wake in an Homogeneous Fluid 4

The Wake in a Non-Homogeneous Fluid 10

III. THEORY ₁₆

A. _{The Mathematical Model 16}

1. The Wake in an Homogeneous Fluid ₁₆

2. _{The Wake in a Density-Stratified Fluid 18}

B. The Solution ₂₁

1. Qualitative Solution in the Phase Plane ₂₁

2. Quantitative Solution ₂₄

IV. _{EXPERIMENTAL APPARATUS AND PROCEDURE}

27

A. _{ExperimenLal Apparatus}

27

1. _{The Towing Tank}

27

2. _{The Two-Dimensional Cylinder}

29

3. The Towing System ₂₉

4. Mixing Equipment ₂₉

5. _{Timing Devices}

30

6. The Tracer Material ₃₃

7. _{Photographic Equipment}

33

8. _{Density Measuring Equipment}

34

B. _{Experimental Procedure}

36

1. _{Preparation of the Linear Density Gradient}

36

2. The Experiment ₃₆

3. Data Analysis ₃₇

V. _{PRESENTATION AND DISCUSSION OF RESULTS}

40

Summary of Experiments ₄₀

Wake Width Variation ₄₀

A.

B.

A. B.

TABLE OF CONTENTS (Conic')

C. 'Correlation Of Experimental Data with Theory

D. Effect of Density Stratification on Vortex, Region

Page

56

63

VI. CONCLUSIONS 68

VII. REFERENCES 70

VIII. APPENDICIES .A71

A. Definition of Symbols A-1

B. List of Figures B-1

C. List of Tables C-1

CHAPTER I

INTRODUCTION

A two-dimensional object traveling in an homogeneous fluid

gener-ates a region of disturbance in the otherwise still medium. The

struc-ture of this disturbed area or wake is almost always turbulent in

practical situations and increases in width with distance behind the

moving body due to turbulent diffusion of momentum. Consider now the

same body traveling in a fluid with a vertical density gradient. In

this case, the density stratification provides a force inhibiting the

turbulent diffusion at the edges of the wake. The effect of gravity

is to return globules of fluid to their own density level and thereby

to suppress the lateral spread of the wake. It is also possible that

as the turbulent energy is dissipated, gravity will become the dominant

force and growth of the wake region will cease altogether. In fact, as

gravity begins to return the globules of fluid in the wake to their own

density level, it might be expected that the wake width would not only

cease expanding, but actually decrease in extent.

It was the purpose of the investigation described herein to study

this wake "collapse" phenomena. _{Underwater vessels are subject to a}

thermal density gradient when traveling near the thermocline of the

ocean where the density gradient is a maximum. Therefore, a significant

deviation from the expected behavior of the ordinary turbulent wake in

an homogeneous fluid would be of interest both for the possible effects

on the craft itself and for the effects on the surrounding marine

envi-ronment. _{A better understanding of such regions and their nature may}

-1-also be of value to geophysical interests as it pertains to the

strati-fied atmosphere.

In the experiments, a two-dimensional circular cylinder with its

axis aligned normal to the direction of motion was towed through a

laboratory tank which contained salt water with a linear density

gradi-ent. The density variation was attained by carefully placing several layers of different proportions of fresh and salt water in the tank and

letting molecular diffusion smooth the profile before the passage of

the cylinder. Variation of wake width with distance behind the cylinder

was recorded by a 16 millimeter motion picture camera for three

differ-ent density gradidiffer-ents and a range of velocities. A series of

homogene-ous wakes were also recorded for comparison.

The study revealed a significant alteration in the growth of the

wake in a stratified fluid. The variation in wake width behind the

body was found to depend strongly on the density gradient and the

veloc-ity of the body. A mathematical model for predicting qualitatively the

variation of wake width was also developed. A result of comparing the

theoretical model with experimental data was that the density

stratifi-cation tends to reduce the mixing length in the turbulent zone.

Chapter II presents an historical background for the general

problem of turbulent wake flows. Both the development for the wake in

an homogeneous fluid and the previous findings for stratified wake flows

are discussed. Chapter III presents the theoretical considerations

which were utilized in formulating a mathematical model of the problem.

Chapter IV gives a detailed description of the experimental method

employed in the study as well as describing the apparatus used. Chapter

-V presents and discusses the findings in the present study, and Chapter

If the further assumptions. are made that velocity in the y direction

u

is Very stall, and hence v is much smaller than u ; and. can be neglected, the equation of motion reduces to

au

ax

1 3T p ay CHAPTER II LITERATURE REVIEW

It is the purpose of this section to provide 4 general summary of

the work leading up to the present study., _{the presentation here of the}

classical analysis for an homogeneous fluid is meant to be an aid in

Understanding the basic problems involved in all wake flows. A review

of the results Of previous investigators tab() have worked with a non-='

homogeneous fluid is presented in the second half of the chapter,

A. The Wake in an Homogeneous Fluid

_

In all previous work, the starting point has been,

to

_{realize that,}

in

the two-dimensional case shown in Figure 1, the width, b 1 of the.

mixing region is small, compared to x and the transverse velocity

gradient Is large compared to the gradient in. the x. direction. This

classifies wake flows into the boundary layer category and for the two dimensional, Steady case, Frandt1(1) has shown the equations of motion.

And

continuity to be;

ar..1

+ v

'au 1

ax

T3-7-0

₍₂₎

(3)

, = =

(1)

-u =

-4

Definition Sketch of the Wake Width, b , for a

Moving Body in a Still Homogeneous Fluid

Figure 1

In order to integrate equation (3), it is necessary to relate the

shearing stress, T , to the flow parameters. _{Boussinesq was the first}

to assume a form for T _{for turbulent flow.}

He said the shearing

stress was composed of a viscous term plus _{a turbulent term.}

aTi-

au

= py +

pc

Ty-Here v is the kinematic viscosity and E _{is the apparent or eddy}

kinematic viscosity. In this relation, E _{becomes zero for laminar}

flow, and for turbulent flow _{E is much larger than}

V and the viscous

term is discarded.

For turbulent wake flow, the shearing stress is expressed as

p-(5)

is not a property of the fluid _{as is V , but instead depends on}

mean flow characteristics.

Several investigators have offered

semi-empirical relations relating eddy viscosity to mean velocity. _{The most}

commonly used of these methods _{are: (1)}

Prandtl's mixing length

con-(4)

of magnitude as u'

T = -p

Z

b

, the resulting form of the shear stress is

n2 Cf176:171

u?V' = px

dyldy

Prandtl(2) has further offered a kinematic argument as to the

variation of wake width with distance behind the cylinder. The basic

assumptions are: (1) the ratio of mixing length, 2, , to the wake

width, b , is a constant; (2) the rate of increase of b with time

is proportional to the transverse velocity v' ; and (3) the mean

value of the velocity gradient in the vertical direction is proportional

to where

u1 is the maximum velocity defect. Introducing the

ex-pression for v' which is analogous to (6), these assumptions become:

8 = constant (8)

(7)

cept, (2) G. I. Taylor's vorticity transfer theory, and (3) the

Von Kerman similarity hypothesis, which assumes turbulent fluctuations

are similar at all points in the field of flow.

Prandtl's mixing length is defined as that distance in the

trans-verse direction which must be covered by an agglomeration of fluid

particles traveling with its original mean velocity in order to make

the difference between its velocity and the velocity in the new level

equal to the mean transverse fluctuation in turbulent flow. This is expressed as:

1,,

z

dy (6)

where 2 = _{Prandtl's mixing length.}

Assuming the transverse fluctuating component v' is of the same order =

Dbn

v' ISU1 Dt dy b 1 = Db If the total derivative' '

is taken as U:12- , expression (9)

dx becomes

'1

For wake flows, the total momentum flux across a control surface,

taken at a sufficient distance to guarantee no pressure forces, is

equal to the drag on a symwetrical cylindrical body. If the height of

the body is h , the diameter bo , and the width of the wake region

b , the momentum flux is given by

PUulhb

and the drag by

1 2

D =

_2D

CPU hb

o

where

CD = drag coefficient and is a function of Reynolds number.

Equating (11) and (12) yields

ulcc Cpbo

2b

Introducing (13) into (10) and integrating:

1/2

b

((iCDbox)

CDbo

1 / 2

With the variation of wake width and velocity known _{from the}

pre-ceding kinematic argument, Sch1ichting(3) _{was able to obtain a}

similar-(9) (10) (12) dx U

Do

(15)

ity solution to equation (3) of the following form: = 1/10 13(xCpb0)1/2 N-1/2 u1 /10 1 -U 1818 CDbo,

In his work Schlichting defines wake width, b , as the distance from

the centerline of the wake region to the point where the wake velocity

equals the free stream velocity. From experimental results of

Schlichting and Reichardt(4), was determined to have a value of

0.18. Introducing this constant into (16) and defining b112 as the

distance from the centerline of the wake to the point at which

- 0.5 , the following relation is obtained:

[

2

, ' u 1.1v) L12 uv = -A (x) 0 (19) (20) b112 = 0.25(xCDbo) 1/2 (18)

In another approach to the problem Reichardt(4) has developed an

inductive theory of turbulence.

Basically,

his theory utilizes the

fact that the velocity profiles can be very well approximated by a

Gauss function. In formulating the problem, Reichardt presents two

basic equations. The first is a time average of the momentum equation

in the x _{direction for frictionless flow and the second is an}

empiri-cal momentum transfer law. The equations are:

(16)

'3/2j

(17)

b

where AW =

momentum transfer length. Realizing

(PIC.

in free turbulent flows and introducing (20) into (19) the' equation reduces to 2 ki / \ 32U2

- =

A(x) %)x 2 Dy

which is analogous to the heat conduction equation with a conductivity

which is a function of time. The solution to this equation is a form

of the Gauss function and therefore will give a velocity profile which

agrees with experimental results. The distribution of momentum in a

two-dimensional wake can be represented by

+ e-(y/b)2,

where

C1 and C2 are empirical constants and b denotes the width

of the mixing region. The function in (22) is a solution to equation

(21) if the momentum transfer length is of the following form:

b db

A(x) =

dx

In essence, the inductive theory makes it possible to obtain a solution

for the velocity profile in the wake without using the concept of a

mixing length once the variation in wake width is known.

All the solutions for the homogeneous wake have assumed that the

region under consideration was sufficiently far behind the body to

guarantee that the wake would be fully turbulent and that velocity

profiles would be similar. Just exactly what a "sufficient distance"

is has been open to question. A. A. Townsend(5) has carried out

exten-sive experiments on two dimensional wake flows behind _{a circular} cylin-der and reports similarity of mean velocities at greater than 80.

Jt'

However similarity of the turbulent velocities was not achieved until

500 body diameters downstream. Roshko(6), using circular cylinders,

determined that for Reynolds numbers greater than 300, all periodicity

in the wake has disappeared by 50 body diameters downstream.

Reichardt(4) and Schlichting(3) reported similarity of mean velocities

for = 50, and Rouse(7) found - 10 gave similar

pro-Do

Do

files. In earlier studies at M.I.T., Harty(8) obtained velocity

simi-larity for

Do

= 7.27 and Froebel(9) achieved good results for

CDbo

= 10.8 .

Thus far the presentation has assumed an unconfined fluid so that

the fluid continues to expand until all turbulent energy is dissipated.

However both in practical situations and in laboratory experiments,

the fluid is confined either by a free surface or a solid boundary. To

account for the effect of the free surface on the velocity profiles in

the wake, Harty(8) has developed an image method which gives good

results.

B. The Wake in a Non-Homogeneous Fluid

In the case of wake flows in a fluid with a density variation,

there will be a restoring force due to gravity on parcels of fluid

moved away from their original density level. There also will be

pres-ent a mixing process in the wake as a result of the turbulpres-ent diffusion

of mass.

In a first study at M.I.T., Prych, Harty, and Kennedy(10)

inves-tigated the effects of the confining boundaries and the initial density

interface on mixing in the turbulent wake of a two-dimensional flat

plate traveling at the interface of a two layered fluid system.

As-suming that mass transport in the x direction is primarily due to

the mean velocity and that mass transport in the y direction is due

to the turbulent velocity fluctuations, the two-dimensional,

concen-tration diffusion equation becomes

_a

I

I. 1

-where E = vertical diffusion coefficient.

With the assumption of similar concentration profiles, and by

super-imposing a double image system to account for the confining boundaries,

equation (24) was integrated to obtain an error function solution

which agreed well with experimental observation. Prych et al. measured

the drag on the plate and although the drag coefficient, CD , varied

from 1.60 to 2.03, there was no systematic variation in the drag

coef-ficient for the density ranges investigated (0 Aa < 0.0061). In

order to characterize the flows according to the density difference

between the two layers and the free stream velocity, a dimensionless

stratification parameter was defined as follows:

-b--b g

p 0

2

CDU

From somewhat limited data, it was concluded that a wake collapse does

indeed occur. The results of the study are presented in Figure 2.

From these results, it is seen that as the parameter J increases, the

effect of gravity becomes more important. _{This is reasonable since an}

increasing J _{value implies either increasing density difference (i.e.}

J

(24)'

Variation' of Wake Width with Distance Behind Plate for

Different Values of the Stratification Parameter J (Prych)

Figura 2

J=0.049

_LD

J=0.063

e J.0.128

() J=0.134

0 J.

0.218

/

--...

...

/

/

/ /

I

/

i /

//

II///

FM.----...---I0

20

30

40

/

increasing restoring force) or decreasing free stream velocity (i.e.

decreasing source of mixing energy). However, it would be noted here

that due to the experimental method, the data was very sparse, and it

would be difficult to base any final conclusions on their findings.

Froebel(9), using the same two-dimensional plate in a two layered

system as did Prych, measured the velocity defect in the wake. Assuming a velocity defect of Gaussian form:

u12

e-C(y/b)

u1max

The empirical constant, C , was found to have a value of 0.70. An

(8)

earlier study by Harty determined C = 0.69 for an homogeneous

fluid. The relation for homogeneous wake flows,

_{ulb =}

constant,

max

was also verified for the stratified case under consideration. _Froebel has also observed a retarding effect on the spread of the wake due to

the density stratification. _{His experimental results are compared to}

Schlichting's solution for an homogeneous fluid in Figure 3. It is again noted that the data is extremely limited.

The collapse of the three-dimensional wake region was first ob-served in the presence of a linear density gradient by _{Schooley and}

(11)

Stewart.

In their report they observed the generation of _internal

waves by the collapse. _{They used a small self-propelled body to}

gen-erate the wake region and measured vertical and horizontal _{growth by} injecting dye into the wake and recording the spread of _{the dye on}

film. _{The initial wake growth was found to be similar to that in a}

homogeneous fluid, but a maximum vertical wake width _{was reached at} about 20 body diameters behind the object followed by a vertical

b112

bo

2.5

2.0

1.5

1.0

10

20

1)0

Variation of Wake Ralf Width With

Distance (Kennedy and Froebel)

Figure 3

30

lapse. Associated with the vertical collapse, was an increased

expan-sion of the wake in the horizontal direction. The collapse was

com-plete at a distance of about 60 body diameters.

Ina similar study, Stockhausen, Clark, and Kennedy(12) observed

the three-dimensional wake generated by a self-propelled body. Their

investigation was carried out in the presence of two different linear

density gradients. The general wake characteristics were the same as

those reported by Schooley and Stewart and internal waves, initiated

both by the passage of the body and the collapse of the wake, were

observed. The principal difference in the two studies being that the

distance behind the body at which the wake attains a maximum vertical

width is greater for the Kennedy study than that of Schooley and

Stewart. This fact is explained by realizing that Kennedy was working

with a weaker density gradient than were Schooley and Stewart and thus

gravity does not predominate over turbulent diffusion as soon. What

was not explained, was the fact that maximum wake width, normalized

with respect to the body diameter, in each case was the same. Both

studies showed a strong convergence of surface particles at about the

position of the vertical wake collapse. _{However, Stockhausen, Clark,}

and Kennedy pointed out that a surface convergence could _{occur only if}

the wake intersected the surface.

-15-CHAPTER III

THEORY

The classical approach to the problem of the turbulent wake in an

homogeneous fluid is based on Prandtl's kinematic argument to obtain

the wake width variation and the concept of a mixing length to achieve

a solution for the velocity distribution. The problem can equally well

be solved using Prandtl's relation for b(x) and Reichardt's inductive

theory. Since Froebe1(9) has found that in a non-homogeneous fluid the

velocity profile is still of a normal function form, Reichardt's theory

could be applied to the wake in a density stratified medium if the wake

width variation were known. It is to determine this wake width

varia-tion that a mathematical model is constructed.

A. The Mathematical Model

The method used in constructing such a model will be to develop

first a dynamical argument that will yield the same results for b(x)

for the homogeneous case as that obtained by previous investigators.

Then an additional term will be added to account for the force due to

gravity. However, it must be emphasized that the theory really is

exactly Prandtl's kinematic argument written in an alternate form, such

that a different physical meaning can be attached and a generalization

to the stratified case is easily obtained.

1. The Wake in a Homogeneous Fluid

Consider a parcel of turbulent fluid in the wake behind a

two-dimensional bluff body. In a homogeneous medium, this eddy or ball of

which can be expressed in a Lagrangian frame of reference as:

D2C

Dt2

In this term, refers to the vertical displacement of the parcel of

fluid from horizontal centerline of the body.

A second force acting on a turbulent eddy in homogeneous wake

flows is that due to turbulent damping. The form of this term is not

so easily arrived at. It is reasonable to hypothesize that turbulent

damping is related in some power of the velocity of the fluid eddy. This term can then be expressed in a Lagrangian form as:

a

ip4

n

Dt)

where

a

is an empirical constant and n is to be determined.

By combining terms (27) and (28), the dynamic model for the homo-geneous case is complete:

122._L + (2L.1

0

2 .)

Dt Dt

It is now necessary to solve (29) subject to the initial _conditions (27)

at t = 0 (30)

and compare the solution to Schlichting's similarity solution. This yields

n = 3 (a) 0.2P a - _(b)

a2cUb

Do

(31)

(29)

The mathematical model for the homogeneous case is then:

3

0.2p

,R2CdUbo, Dt

is the empirical constant defined by Schlichting as the ratio

be-tween Prandtl's mixing length and the wake width. For the homogeneous

case, 8 = 0.18 .

2. The Wake in a Density-Stratified Fluid

In the presence of a density gradient, the inertia force and the turbulent damping force will still be present and retain their same

form. However, an additional force will now be present due to the

effect of gravity. In order to formulate this restoring term, consider

the particle of fluid shown in Figure 4. If this particle is displaced,

due to turbulent diffusion, a distance E from its original density

level, the gravity force, FR , acting on the particle will be:

FR = g[p(y) - p(y

+ C)]

(33)

Displacement of a Parcel of Fluid from its Original Density Level

Figure 4

= 0 (32)

ORIGINAL DENSITY LEVEL

-The density at the new level,

y + E

, can be expanded in a Taylor's

series, and if all terms in the expansion of order E2 and greater are neglected, the expression in (33) becomes:

dp

FR

-g -

dy

Here _{is taken to be the density gradient in the undisturbed fluid.}

dy

By adding (34) to the model for homogeneous wake flows, _{an equaLLon is}

obtained which will be valid for density-stratified wake flows.

2 D /-) 2 Dt 3

EDt)f-la

agldyl

= 0 (35)

In this case, an additional empirical _constant a has been added. In

the stable case and with the coordinate _{system shown in Figure 1, the}

density gradient will have a negative sign and this has been accounted

for by changing the sign in front of the _{restoring force term and using}

absolute value notation on the density gradient.

Using the transformation U _{to convert (35) to an}

Dt dx

Eulerian frame of reference, the equation becomes:

( 34)

a

Pk-1 = 0 ₍₃₆₎

p dy

It must be emphasized here that since the actual motion of a turbulent

globule of fluid is highly random due _{to collisions with other globules,}

the trajectory given by equation (36) should be regarded as meaningful

only in some statistical sense. _{However, since only the wake width,}

which can be defined by the maximum _{excursion of such trajectories, is}

desired, the precise statistical sense is unimportant.

d2E 0.2 dx 3

149

4-2 dx ,a2Cpbo,

+

Ub

-Equation (36) is now complete and could be solved in terms of the

parameters of the coefficients. This however would yield a different

solution for each different set of parameters. In order to obtain one

solution which will be valid for all values of density stratification,

cylinder diameter, and drag coefficient, a new pair of variables will

be defined,

Where

bo is the cylinder diameter which non-dimensionalizes the

origi-nal variables and X and B are scale factors which will be chosen

such that the coefficients of the non-dimensionalized equation are all

unity. Substituting transformations (37) into equation (36) the

fol-lowing equation is obtained:

3

d2Z

1C-1

^2

dx

And the new variables Z and are defined as in (37) where:

2 1 dp

gbo p dy

U2

With equation (38), it is now possible to obtain one solution which will

= 0 (38) (C) (37) X B =

-(aX)-112 CD (a) (b) (39) (aX) II 0.2 boX boB x

be applicable regardless of the flow parameters.

B. The Solution

It is now possible to investigate what type of solution may be

achieved from the mathematical model as given in (38).

1. Qualitative Solution in the Phase Plane

As a means of getting a qualitative idea of the solution, the

dZ

substitution p = yields:

dx

ap_ p3 + Z

dZ

It is now possible to plot the solution curve in the phase (p vs. Z)

plane by noting:

Along the line p3 -Z , the slope is zero.

Along the Z axis, the slope is infinite.

Along the p axis, as p + , the slope becomes infinite,

In the first quadrant, p and Z are both positive and

therefore the slope must always be negative.

In the third quadrant, p and Z are both negative and the

slope will always be negative.

In the second quadrant, p is positive and Z is negative.

Therefore, above the line p3 = -Z , the slope will be

negative and below that line, the slope will be positive.

In the fourth quadrant, p is negative and Z is positive.

3

Above the line p = -Z , the slope is always positive and

below the line, the slope is negative.

With this behavior in mind, it is possible to construct the curve in

(40)

i)

li)

the phase plane which has the correct slopes. This curve is shown in

Figure 5. _{The solution in Figure 5 starts at}

_{p = co}

_{and Z = 0}

because of the initial conditions of the problem. Referring to equation

(38) it is seen that for the stratified case, initially the inertia force and the turbulent damping force are the dominant terms since Z 0. _{Therefore it is expected that the initial behavior for the}

stratified wake would be qualitatively the same _{as for the homogeneous}

wake.

The Phase Plane Solution Figure 5

Using the solution shown in Figure 5 and the definition of the

variable p , it is possible to construct a qualitative solution to

the problem in a Z - x axis system. _{In doing this, the following}

points should be noted:

-Z A

dZ

i) At X = 0 , Z = 0 and the slope is infinite.

dx

dZ .

The slope p = is zero at values of Z which alternate

dx

in sign, but decrease in magnitude.

The displacement distance, Z , is zero between each zero of

dZ

P=

dx

With these characteristics of the solution, it is easy to plot the

behavior of Z versus x qualitatively. The solution is shown in

Figure 6(a). Z A

Qualitative Solution to Equation (38) for a Single Eddy

WAKE BOUNDARY

Composite of Several Eddies Originating at Different Points

Qualitative Solution to Equation (38) Figure 6

a)

The fact that Z is seen to oscillate about the x axis should not be

too surprising since the model for the problem was one for an individual

homogeneous eddy displaced from its own density level. Therefore as increases,, the gravity term dominates and tends to return the parcel

of fluid to its original level. In the process of reaching the

equi-librium position, the eddy oscillates about that position. The wake

of course does not behave this way. However, the wake may be imagined

to be composed of infinitely many eddies each having statistically

identical trajectories originated at different points behind the body.

Hence a reasonable approximation to the wake boundary is the envelope

formed by the extreme of the trajectories of all eddies. This picture

suggests that the wake width reaches a constant value equal to the

height of the first peak of all trajectories as shown in Figure 6(b).

2, _{Quantitative Solution}

The behavior of the solution is known from the phase plane

anal-ysis, but without a quantitative solution to equation (38), it is not

possible to correlate experimental results with theory. Therefore, in

an attempt to get such a solution, an infinite series method was used.

In choosing a form for the series, the primary requirement was that the

leading term should be of a parabolic nature. In this manner, for small

values of X , the solution will approach that of the homogeneous case.

The general form of the solution was then assumed to be:

^2 An Z = x _{[Co + Clx +} C2x +... + Cnx +...]

^1/2

x Cn

n= 0

-24-(41)

Differentiating (41) once yields:

dZ 1 "(n - 1/2)

E (n

+

2) Cnx

dx n = 0

Differentiating (42) once yields:

d2 1 1 "(n - 3/2)

(n

+

2)(n - 2) Cnx ^

dx2 n = 0

Expressions (41), (42), and (43) can now be substituted into equation

(38). _{The resulting expression will in general only be equal to zero}

if the coefficients of each power of x are equal to zero. This

re-quirement yields successive recursion relations to determine all the

coefficients,

Cn . The development of the expressions for the coeffi-cients is presented in Appendix D.

The analysis reveals that all odd coefficients disappear and _the

solution through

C6 is as follows:

= X1/2[1 - 0.133X2 - _{0.0088X4 + 0.000065X6]}

(44)

This solution is shown in Figure 7. _{Since the first peak represents}

the maximum wake width of the envelope, _{this is the crucial point in}

the wake profile and the above series solution _{is sufficiently accurate}

in predicting where this occurs.

-1 I

0.5

1 1

1.0

X

Infinite Series Solution to Equation (36)

Figure 7 1 1

WAKE WIDTH

1 I 1 I 1.5

2.0

CHAPTER IV

EXPERIMENTAL APPARATUS AND PROCEDURE

Previous experimental studies of the two-dimensional wake problem

in a stratified medium yielded suggestive results on the variation of

wake width. However, the data upon which these results were based,

was rather sparse because of the great labor involved in the

experi-ments. _{For this reason, it was thought to be necessary to utilize a}

new approach in obtaining data on the wake growth. The present study

utilized a visual technique for recording variation of the wake width

with distance behind the body. A movie camera running at an extremely

slow frame rate took time exposure pictures of the motion of a tracer

in the fluid. _{The wake growth in this manner was recorded continuously}

and afforded a more complete set of data for _{analysis than had been}

obtained before. _{The visual technique also allowed an observer to}

actually see what occurred in the wake region _{and gave some additional}

justification for interpreting the data. _{Another step toward a more}

practical and easily controllable situation was the use of a linear density gradient rather than a two layered system.

A. _{Experimental Aparatus}

1. The Towing Tank

The experiments were conducted in a metal frame towing tank with

glass walls and bottom. The tank was 120" long and 4.5" wide and was

filled with fluid to a depth of 18". _{The distance that the cylinder}

was towed in the tank varied somewhat from run to _{run, but was generally}

VARIABLE SPEED MOTOR

120"

LIGHT SOURCE

@

Schematic Sketch of the Experimental Setup

Figure 8

PRO P0

r.

TIONING

TANK

_FRESH

_WATER

_SALT

WATER

in- 16 MM MOVIE

_

scaling of the pictures taken. A schematic sketch of the tank is given in Figure 8 and photographs are shown in Figures 9 and 10.

The Two-Dimensional Circular Cylinder

The towed body was a circular plexiglass cylinder 0.5" in diameter

and 4-3/8" long. There was 1/16" clearance at each end of the cylinder.

The cylinder was towed by a braided stainless steel seizing wire 1/32" in diameter which was attached at the center of the body by means of a

screw on each side of the cylinder. The cylinder was prevented from

oscillating either in the horizontal plane or the vertical plane by means of two support wires of the same braided stainless steel material

as the tow line. _{These wires were tensioned and passed through the}

cylinder at a distance of 7/16" from each end.

The Towing System

The two-dimensional body was towed through the tank by means of a

motor-pulley system. _{The power was supplied by a 1/8 horsepower,} 43 RPM Bodine speed reducer motor which drove a shaft through a gear

connection. _{This shaft was part of the pulley system and the tJW line}

consisted of one continuous wire which was driven by the shaft. The

speed of the motor was controlled by means of a rheostat. The general

arrangement is illustrated in Figure 8 and a photograph of the motor connection is presented in Figure 11.

Mixing Equipment

In order to achieve the linear density gradient desired in the

experiments, eighteen different layers of fluid, each of one inch depth,

were placed in the tank. _{Each successive layer was less dense than the}

2,

preceeding layer by 1/18 of the density difference between top and

bot-tom layers. To achieve this, a proportioning tank, which, when filled

to a depth of 17 centimeters had sufficient volume to fill a one inch

layer in the towing tank, was used to mix proportionately quantities of

fresh and salt water. The first layer would be 17 centimeters of the

given salt solution. The second layer then was 16 centimeters salt

solution and one centimeter of fresh water and so on until the

eight-eenth layer was 17 centimeters of fresh water. The mixture was then

drained into the towing tank by gravity. The arrangement of the fresh

water, salt water and proportioning tanks is given in Figure 8.

5. Timing Devices

Two electric stopclocks were utilized in the experiments. One

was synchronized with the motor so that it was possible to determine

the time necessary to travel the length of one run. This clock could be read to the nearest 0.05 of a second. The average velocity calcu-lated from this time might be expected to be somewhat in error due to

including acceleration and deceleration of the cylinder. However, a

check on the accuracy of the stopclock was made by using a manual

stop-watch over the central section of the tank only and there was found to

be, on the average, less than one percent error in the stopclock

read-ing.

A second stopclock, accurate to 0.01 seconds was recorded on the

film to note at what spacing in time the pictures were being taken.

1"

The Experimental Apparatus Figure 9

7;1-477 -

'7-7-R.;

The Experimental Apparatus

Figure 10 -31-1

cP=n

-you 1 5.

1

t.,

r

ciP.

The ,Driving Mechanism

Figure 11

4,

-32-,

p.

The Camera and Drive Mechanism

Figure 12

The Tracer Material

To follow the motion of the fluid in the wake of the cylinder,

Alcoa aluminum pigment number 606 was placed in the tank. The tracer

was wetted with acetone prior to placement in order to prevent the

pig-ment from adhering to the surface. The pigment was so fine that even

after 24 hours, there was still some tracer in suspension. Yet, when

illuminated, each particle was clearly visible. The acetone and

alumi-num pigment was placed in the tank by means of a hypodermic syringe and

20" long needle. The diameter of the needle was 1/16" so there was a

minimum of disturbance of the stratification in placing the tracer.

Photographic Equipment

The movements of the tracer were recorded by means of a commercial

16 mm Bell and Howell movie camera placed in front of the tank. The

camera was driven by an external motor connected through the hand crank

mechanism. _{The 1/70 horsepower, 2.9 RPM Bodine speed reducer motor was}

necessary in order to achieve the extremely slow frame rate desired.

This drive system produced a frame rate of one frame per second which

provided an exposure time of 0.57 seconds. The slow frame speed and

long exposure time were necessary to obtain a definition of the wake

region which would be identifiable on the film. With a 0.57 second

exposure time, the wake region appeared as a blurred region surrounded

by individual tracer particles where no motion was present.

The tracer particles were illuminated by means of a 650 watt high

intensity iodine crystal lamp which was placed beneath the tank and

shone up through the glass bottom. _{The one limitation this system had}

the significance of this problem, a plane light source was borrowed. A

lens system collimated the light into a sheet approximately 1/4" thick

throughout the entire depth of the tank. The plane light was used for

all of the homogeneous runs and for two of the stratified runs. There

was no significant variation in the results found using the plane light

as compared to the light source which illuminated the entire tank.

In addition to the light illuminating the tracer, a stroboscopic

light source was used to stop the clock which was recorded on film.

Since the exposure time was 0.57 seconds, the hand on the clock would

have been blurred if it were illuminated all the time. Therefore, the

stroboscopic light was adjusted so that it illuminated the clock once

every second. The particular light used was a Strobotac type 1531-A

manufactured by the General Radio Company, Concord, Massachusetts.

A black velvet backdrop was used as background for the pictures

because of its high absorbtion of light.

8. Density Measuring Equipment

To check on the presence of a

linear density gradient, samples

of the stratified fluid were removed before each run by means of a

hypo-dermic syringe and needle. The density was measured with an hydrometer

and the excellent density profiles achieved are shown in Figure 13.

The temperature profile was also taken before each run. The variation

in temperature within the tank for any one run was found to be small.

0

0.1

I

6

0.2

0.3

0.4

0.5

0.6

a_

0.7

0

cc

0.8:11' I

I-Q ,a9

a

9

I

9' 5A

.-0 58

650

o 5E.

a 4A

z 4B

A 4C

L

40

6, 4E

thS

A. _

1.0295

1.0495

1.0695

1.0895

,grn

(CT/

Density

_{Stratification for Series 4 and}

Figure 13 35

1.0095

5

0-5D

-o

-o

1.2

_-o

1.3

1.4

B. Experimental Procedure

Preparation of the Linear Density Gradient

The method of achieving a linear density gradient has been

ex-plained in the section on mixing equipment. Briefly, the tank was

filled to a depth of 18 inches by placing 18 layers of fluid of

de-creasing density. The proper reduction in density of each layer was

achieved by mixing proportionately fresh and salt water in a tank which

was then drained into the towing tank. The fluid entering the towing

tank does so at a very slow rate and is admitted at both ends of the

tank through a fibrous packing material so that its energy is dissipated

as much as possible to reduce mixing. The presence of the individual

layers was visible and a sharp interface was present between each layer.

The tank was filled very slowly and the average time to fill the tank

was just over eight hours. The filled tank then was let stand

over-night so that the density profile could be smoothed by molecular

dif-fusion. The average time between completion of stratification and the

run was seventeen hours.

Experiment

The standard procedure for making an individual run included the

following steps. First, the density gradient was checked by withdrawing

samples at five different depths and measuring density with a

hydrom-eter. On occasion a complete density profile would be taken with

sam-ples being withdrawn from as many as ten different depths. The rather

consistant linear profile generated in this manner is shown in Figure

13. Figure 13 shows all the density samples taken for two of the three

stratifications investigated. Temperature profiles were taken at the

same time as density samples, but as was mentioned before, the maximum

variation in temperature in one run was only 1.7°C. Once the density

samples and temperature readings were taken, the tracer was placed in

the tank by means of a hypodermic syringe and needle. The initial

posi-tion of the cylinder was recorded and when moposi-tion in the fluid ceased,

the lights and camera motor were turned on. The motor driving the

cylinder and the clocks was started and when the cylinder reached the

other end of the tank, the motor and camera were stopped and the final position of the cylinder was recorded along with the stopclock _time.

3. Data Analysis

In order to calculate the characterizing _{parameters of the flow,}

average values of density, p , and dynamic viscosity, p , were

com-puted. The average value of density was arrived _{at by taking the}

meas-ured points on the density profile and averaging them. To arrive at an

average p _{for each run, the average temperature was computed and the}

dynamic viscosity corresponding to this average temperature was

selec-ted. _{With these quantities, the values of densimetric Froude number} and Reynolds number were calculated.

The wake width, b , was defined as the vertical distance from

the horizontal centerline of the body _{to the point where there was no}

motion of the tracer. _{To obtain a record of wake width variation with}

distance behind the body, the film was projected onto a screen, and with the velocity of the cylinder known and the time since the passage of the cylinder known from the stopclock _{in the picture, it was possible}

to determine how far behind the body each _{point in the picture was.} The picture was projected to full scale by matching a grid on the

screen, which was the same size as the one on the wall of the towing

tank, with the grid in the film. Some difficulty was encountered in

taking the wake width measurements. First, the edge of the wake is

necessarily not sharply defined due to the fact that motion of the

tracer particles does not completely cease at some point, but rather

dies out gradually. Therefore the definition of the wake width was

open to some subjectivity. Secondly, the wake is not in reality a

steady state phenomena. The occurrence of internal gravity waves along

with the normally ragged wake boundary make a scatter of data points

inevitable. In fact in the cases in which the cylinder velocity is slow

enough that the same point behind the body appears in more than one

frame, there can be a substantial difference in observed wake width at

the same distance behind the body. This is observed especially in the

case of the homogeneous wake. Because of this sizable scatter of data,

it was necessary to draw banding curves on each side of the data points

and then average these banding curves to obtain an average curve through

the data.

One other difficulty in taking data from the film was that, for

the Reynolds number range investigated (1300 < IF < 3500), there was

a vortex region immediately behind the body. This region continued for

approximately twenty to thirty body diameters behind the body before

becoming a truly turbulent wake. For this reason, the data taken for

the early part of the wake is not considered as representing the

tur-bulent region,

Once the average curves were drawn, relations (37) and (39) were

employed in order to determine values of the empirical parameters a

theoretical curve with the maximum point on the average experimental

curve. Once a and (3 were determined for each run, the average

ex-perimental curve was converted into the dimensionless variables x and

Z and plotted with the theoretical curve. Then the variation of a

and (3 with densimetric Froude number and Reynolds number was

CHAPTER V

PRESENTATION AND DISCUSSION OF RESULTS

Summary of Experiments

Fourteen different runs including three different stratifications

were analyzed. Also three homogeneous runs were analyzed for

compari-son. _{These runs covered a range of Reynolds numbers from 1300 to 3500}

and densimetric Frounde numbers from 5 to 20. Table 1 gives a

compre-hensive summary of all runs analyzed.

Wake Width Variation

As was discussed in Chapter IV, a record of wake width variation

behind the body was obtained by projecting a movie film which recorded

the motion of the tracer particles full scale onto a screen and

meas-uring the wake region on the screen. A typical series of these pictures

is given in Figure 14. Figure 14(a) is a homogeneous run while 14(b)

is of a density stratified case. The difference qualitatively in the

growth of the wake region is quite obvious from these pictures. The

presence of a vortex street is apparent in both series of pictures, but

in the homogeneous case, the wake does continue to expand after it

be-comes turbulent. However, in Figure 14(b), after the vortex region,

there appears to be little, if any, expansion of the wake region. There

also does not seem to be any very appreciable decrease in wake width in

the stratified case.

It should be noted here, that the boundaries of the wake region

are extremely irregular. This is especially true for the homogeneous

case. This property is in accordance with Townsend,s(13) model of the

-40-A.

Run dy _

(cm')

cm/sec 2B 2C 2F 2G 2H 4A 4B 4C 4E SA 5B 5C 5D 5E

Summary of Experimental Runs

Table 1

I°

b j max bo b bomax a .00159 12.44 7.85 1630 1.70 80 .077 .0139 .00159 21.95 13.83 2877 2.75 50 .158 .111 .00164 10.30 6.39 1350 1.70 60 .089 .0164 .00168 15.94 9.78 2091 2.67 80 .122 .0216 .00148 24.87 16.26 3220 2.65 60 .138 .105 .00201 9.24 5.19 1322 1.60 50 .092 .0155 .00197 12.98 7.36 1830 1.88 50 .109 .0312 .00198 17.50 9.89 2489 2.70 70 .131 .0288 .00191 25.24 14.52 3515 3.38 120 .125 .0213 .00106 9.81 7.58 1323 2.80 70 .136 .0168 .00101 14.14 11.19 1819 2.30 70 .112 .0366 .00106 17.19 13.29 2138 2.50 50 .143 .101 .00113 21.03 15.74 2817 3.50 60 .183 .100 .00107 24.90 19.15 3243 3.22 160 .103 .0208 p

-42-amal11111=10... rormac

-4411e--

-414111r-- .-,

0071101710111i.

-rrr 7 0

St

(a) Homogeneous Run (b) Stratified Run 4C

= 2700

11:?= 2489

F1=9.9

Typical Film Record of Wake

Width Variation

4

scale)

32 Figure 14

wake region in a homogeneous fluid. That is, that the bounding surfaces of the wake are moved by the convective action of a system of large

eddies and these contorted surfaces may in some places approach the

central plane of the flow. It is this action in the homogeneous case

that leads to a wide scatter of data points. The ragged edges of the

wake are substantially reduced in the stratified case, due primarily to

the retarding effect of gravity on the spreading of the eddies into a

region of a different density level, However, in the density stratified

case, another difficulty was present. The passage of the body

gener-ated a field of internal gravity waves which also caused a scatter of

data points _{In addition to the irregularity of the boundary, some}

subjectivity is involved in deciding exactly where the edge of the wake

is. _{This difficulty is not as apparent from the small scale pictures}

as it is when the film is projected to full scale.

The experimental record of variation of wake width behind the body

is shown in Figures 15(a) through 15(h). These figures include the data

points from which the scatter mentioned above is seen, The method of

obtaining one average curve through the data points was to draw banding

curves on each side of the data points and then average these curves to

get one curve representing the wake growth. Figures 15(a) through 15(h)

show the data, the banding _{curves, and the final average curve. From} these figures, the scatter in the homogeneous case is seen to be

sub-stantially greater.

It should be noted that because of the _{vortex street region}

pres-ent in all runs, the early part of the wake is actually _{not turbulent.} The end of the vortex region for each _{run is denoted by the solid circle}

I I I I I 1 1 I 1

Run 3B (Homogenous Fluid)

_{---Banding Curves}

0Experimental Observation

Average Curve

End of Vortex Region

0

HR=1470

07

b0=-1.27 cm.

,,-/ co

/ 00

4o

eoo

o

0

0

0

.00' .00 Figure 15(a)

o

o

0

0

CD

00

/

_/

I I I I I I I 1 1 I I 1

o0

10

20 30 40 50 60 70 80 90 100 120

NORMALIZED DISTANCE BEHIND CYLINDER, 170

Experimental Variation of Wake Width

with Distance Behind Cylinder

9

7

4

9

J3128

7

6

5

4

NI

3

4

.2

2

0

---Banding Curves

0 Experimental Observation

Run 3C (Homogenous Fluid)

o

_

o

10 20 30 40 50 60 70 80 90

NORMALIZED DISTANCE BEHIND CYLINDER, x

bo

Experimental Variation of Wake Width

with Distance Behind Cylinder

Figure 15(b)

Average Curve

End of Vortex Region

...0...

_ ...0

.0p.

0

IR

1940

-...0

....-0 Co

bo =1.27cm.

°*--0

0°0

00°-..

0

°

60,

0

."

,...

0

0

0

,..,

...

---,.., 0 __... .

..----1001 110 1120 130 140 1150

[III

I

0

I I I I I I I I I I

9

I28

6

5

a

4

NJ

3

2

2

01/11/11 I

1111

Run 3D (Homogeneous Fluid)

--Banding Curves

0

Experimental Observation

Average Curve

End of Vortex Region

fR= 2470

bo =1.27 cm.

I I I 1 I I I I I I I I I I

o0

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

NORMALIZED DISTANCE BEHIND CYLINDER,

bo

Experimental Variation of Wake Width

with Distance Behind Cylinder

Figure 15(c)

00

0

/0

II

I I

III

If

0

10 20 30 40 50 60 70 80 90 100 110 120

130 140

NORMALIZED DISTANCE BEHIND

CYLINDER,

Experimental Variation of Wake Width

with Distance Behind Cylinder

Figure 15(d)

H

LU

CD3

Nuil

cl

0

2

₀

2

0

.11

Banding

Average

0

Curves

°Experimental Point

Curve

'End of Vortex Region

MMD .11. ..ONM

0

0

/o

oe O...

Run 2B

_{Run 2C}

0 0

0 0

0

0

Run 21F

o

c"'"11.'

fR =1630

fid=7.85

= =

y

_,-,etrbarerear-ca=

fR= 2877

ffd=13.83

77

0 0 0

..

0

217

2

0 ...m.a=

=NM

j fR=I350

ff-d=6.39 If

0

III

0

0 0

o Experimental Observation

End of Vortex Region

_ _ ___o___0

3

--0

0

... -"'"...._,COO

0o...,

."

d'...

2)"9.0.".."Wcp

0

n

° 0,

(lb

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0

...

2

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02

C)0..._

21

saa

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cc) I

Run 2G

fR = 2091

Fd =9.78

i-ve

0 0

_

--

---.-0'0'0-...-...

-____

u..1

3

*.--.

0

00_

--/

0

00

0

B

00

oo

0o

o00°00o0

0

2

/

_/

0

.---_____

_

_ ---0_______

I 1

nn..0

''''''

0:."."-' ... Lli

)(:

Run 2H

FR =3220

Fd =16.26

rn

N

/

.4

2

2

....0..1,1

.".... .... . . . ."'. '''

--,...

Ct

I

Run

flA

I FR

=I1322

Iffd=5.191I

I I I

0," r

I

0

10 20 30 40 50 60 70 80 90 100 110 120 130 140

NORMALIZED DISTANCE BEHIND CYLINDER, r)T(0

Experimental Variation of Wake Width

with Distance Behind Cylinder

.012

0

3

2

LL.1

0

0

L.L.J

3

2

0

00°

0

.D.C6

Run 4B

FR=I830

d736

0"40 =MN.

Run 4C

fR =2489

Fid =9.89

2

/ 0

/

_/

ot Expi rn

eritental Observation

End iof Vortex Region

0 Ir

0

10

20 30 40 50 60 70 80 90

100 110 120 130 140

NORMALIZED DISTANCE BEHIND

CYLINDER, g(-(-)

Run 4E

IR= 3515

Experimental Variation of Wake Width

with Distance Behind Cylinder

Figure 15(f)

T I

300

ImMO =Ewen

1rd =14.52

Gomm ammo .rmwa

0

0.0.01) 1111)..._

0IO

... -ow. .... 0 '''''' .. .. . ''. 0 - 0 ' '' '''' ' -.09. ' o.

o

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--/

At° <0101

,--0-0

0

---P

u

/

i

..---.." -- MEMO INIMI/ °/1IM 4101=11 /MINIM

/

0 6ID ...-./.0.. ./ I]

i

o (:),.

Run 5A

FR =1323

Fd =7:58

/

/

0

v

ost"

*--°7";.;

o 0

0 0

o

0

-.0

o

-"

0 o

0o-

Run 5B

IR=1819

/1M=./

o

1Fd =7.58

0

V=.0 ema wilmo am.. ay 0 ...n. Iamb sw,. a-tz e....

0

0

0

0

00

"-. ''"''

./. .

fa".

0 . `a

1crirr...0elo

9

---... MMI /...g.

#

0

/ 0 0

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0

0 0 0

0 0

0...'.

0

0

=1 -MaM1 5*.m. 1

/

,..1=1 MM. ./M1.. . ../ . ...-"60... .0.... 1=M3

/

Run 5C

fR=2138

Ird=13.29

°

....---:,

i

.

0 Experimental Observation

End of Vortex Region

10

20 30 40 50 60 70 80 90 100 110 120

130 140

NORMALIZED DISTANCE BEHIND CYLINDER, 1);

Experimental Variation of Wake Width

with Distance Behind Cylinder

Figure 15(g)

3

2

_oo

3

0

o

3

2

4

1 0

3

_oo

/

°Experimental Observation

End of Vortex Region

/

_/

..-/

/c<1

...

/

... ''"''

o

/

/

/

01/

H

i

II

II

1-I I I I ...- 0-0

'''"

./.."

I

_/

0

00

o ....

0000

0

-Run

Run 5D

R2817

1Fd=15.74

o

-o--

---so-6

o o

...- ""o o

...

0.-0.-...

I...

...

/

.... *"..

/

ii

I ...

0 ...

o0

I

Run 5E

R=3243

Fd=19.15

I 1 1 I I

I

I I I 1 I I I

10 20 30 40 50 60 70 80 90

100 110 120 130 140

x

NORMALIZED DISTANCE BEHIND

CYLINDER, i:7-0

Experimental Variation of Wake Width

with Distance Behind Cylinder

figure 15(h)

...M. ..,.

IMED

mmim, omm 0.0 7:rdiows MO

°... 0. 0...0 ..c.7

07

MND amm MNIMM WOmM

0-0-0

a/MO. =1111.

/

0

The fact that the experimental results show no drastic collapse of the wake region, (i.e., the wake width does not decrease after a

maximum is reached), is a significant feature of all the runs.

How-ever, this should not be too surprising if the features of the

two-dimensional wake are considered. First the central core of the wake

is a region which is relatively well mixed. If the entire wake were

well mixed, there could be no collapse in the two-dimensional case.

This follows from the fact that in effect a well mixed wake region would

create a stable three layered system. The top and bottom layers having

the original linear density variation and the middle layer being

homo-geneous with a density equal to the mean density of the central region.

In reality the wake is not entirely well mixed, but rather consists of

a central core which is nearly homogeneous plus the interfacial region

between the core of the wake and the surrounding quiescent fluid. It

is in this interfacial zone that the effects of gravity will be

ob-served. Therefore, the only collapse that should be observed in the

two-dimensional case, consists

of

restoring these edge regions to an

equilibrium position near the central core. In the three-dimensional

case, the homogeneous core will seek its own density level in the

un-mixed fluid in the horizontal plane thus creating a true vertical

col-lapse.

From non-dimensionalizing equation (36), it is seen that the

sim-ilarity parameter, X , is the reciprocal of a densimetric Froude

num-ber squared. Where the densimetric Froude number is given by:

fF

-d 1 b 2

dy g o

It is of interest to consider what effect varying the densimetric Froude number will have on the wake behavior. As the velocity of the body

increases, the mixing in the wake region will be greater for a given

stratification and therefore the effect of gravity is less. In the

same manner, for a given velocity, as the density gradient becomes less,

the effect of gravity decreases. In both of these situations, the

densimetric Froude number increases. In fact as the densimetric Froude

number becomes infinite, the effect of gravity disappears and

homogene-ous wake flow is approached. Therefore it would be expected that as the

densimetric Froude number increases, the maximum wake width,

b*

bo I

(46)

would increase and the position, behind the body, at which the maximum

wake width occurs

A X ,

t

D b , = X am bo = °)

would also increase. Figures 16(a) and (b) present the variation of

bm and xm with dersimetric Froude number, While there is appreciable

scatter of the experimental points, the general trend is as expected.

There are four points in Figure 16(b) corresponding to runs 2C, 2H, 5C,

and 5D which do not follow the expected trend. The reason for this discrepency is not apparent at this time.

In order to illustrate the differences between wake growth in a

homogeneous fluid and in a density stratified fluid, Figures 17(a),

(b), and (c) present the experimental variation of wake width behind the

body for homogeneous and density stratified mediums at approximately the (47) -m,

2.5

o run 2B

6

run 4A

o run 5A

run 2C

run 4B

a run 5B

fun 2F

A

run 4C

ra run 5C

a run 2G

4run 4E

II

run 50

e run 2H

run 5E

0

10

Frd

Variation of Maximum Wake Width, with Densimetric Froude Number

Figure 16(a)

15

20

-5

4

0

run 4A

0

run 5A

A

run 48

0

run 5B

run 4C

run 5C

run 4E

r

run 5D

Cg

run 5E

0 0

0

GI 0

0

1 1

2.5

5

10

FFd

Variation of the Point Rebind the Cylinder at which the

Maximum Wake Width Occurs with Densimetric Froude Number

Figure 16(b) 15

20

X77.1

160

140

120

100

80

60

0

run 2B

0

run 2C

run 2F

Gi

run 2G

e run 2H

40

20

same Reynolds number. From these figures, the suppression of wake

ex-pansion in a density stratified medium is obvious. The early wake

growth cannot realistically be compared because of the presence of the

vortex region.

C. Correlation of Experimental Data with Theory

Table 2 shows values of (3 computed for the homogeneous runs

using equation (16) with a drag coefficient,

CD , of 0.97 (Rouse(14)).

The values of 3 so obtained are somewhat less than Schlichting's value

of 0.18. However, these coefficients were computed using values of

and -12- taken from the average curves of each run. Figure 18 shows

bo bo

the log-log plot of the data points without averaging as compared to

equation (16) with CD = 0.97 and (3 = 0.18. From this figure, the

data seem to cluster about the line representing equation (16). This

indicates a value of 8 equal to 0.18.

8 as Computed from Equation (16) with CD = 0.97

Table 2

In the density stratified fluid, expressions for the constants a

and may be obtained by utilizing relations (37) and (39) which yield

1/ 4 = (a),) Run Average $ 3B 0.176 3C 0.147 3D 0.163 (a) (48) 0.2 1/2 b* CDD