On internal gravity waves generated by submerged disturbances

TECHNICAL REPORT 117-13

ON INTERNAL GRAVITY WAVES GENERATED BY SUBMERGED DISTURBANCES

By

K. K. Wong August 1968

DISTRIBUTION OF THIS DOCUMENT IS UNLIMITED

Prepared Under

Office of Naval Research Department of the Navy Contract No. Nonr-331-9(OO)

TABLE OF CONTENTS

Pa g e

INTRODUCTION i

FORMULATION OF THE PROBLEM 2

FEATURES OF THE WAVE PATTERN 3

THE REGULAR WAVE PATTERN 8

TEEE-DIMENSIONAL EFFECTS 12

THE WAVES GENERATED BY A TWO-DIMENSIONAL WEDGE 15

-ii-LIST OF FIGURES

Figure 1 - Wave Number of Stationary Plane Waves in a Uniform Stream

Figure 2 - Velocity of Propagation of Wave Packets of Cylindrical System

Figure

3 -

Position of Wave Packets of Different _Pl One Unit of Time After Generation

Figure - Source Distribution Representing Wedge Beneath

Surface of Large Density Discontinuity Figure

5 -

Velocity Perturbation Upstream of Wedge Figure 6

-

Velocity Perturbation Downstream of Wedge

NOTATION

A Complex wave amplitude

d Depth of submergence of wedge

d* = dN/U

F1 Function defined by Equation [)9] F2 Function defined by Equation [60]

G Group velocity

H Heaviside function

G, G

Components of group velocity in Cartesian

y

coordinates

G , G0 Components of group velocity in plane polar

r

coordinates

GR,GO,G Components of group velocity in cylindrical polar coordinates

k Radial component of wave number in plane polar coordina tes

K Radial component of wave number in cylindrical polar coordinates

Half-length of wedge Source strength

Vaisala frequency

Zero or positive integer Source distributions

Fourier transform of q, q1 and q2 respectively Radial coordinate in cylindrical polar coordinates Radial coordinate in plane polar coordinates

Parameter in Laplace transform Time m N n q, q1, q2 q1, q2 R r s t

-iv-U, u1, u2 Horizontal velocity components x, y, z Cartesian coordinates

y Also an axial coordinate in cylindrical polar

coordinate s

y* _{= yN/U}

a, /3, y Cartesian components of wave number

e Polar angle of wave number plane in plane polar

coordina tes

e Polar angle of wave number space in cylindrical

polar coordinates

T Half of wedge angle

Polar angle in cylindrical polar coordinates Polar angle in plane polar coordinates

INTRODUCTION

In a recent paper, Wu and Mel (1967) discussed the problem of gravity waves generated by a submerged two-dimensional

dis-turbance moving horizontally in a stratified fluid with a free surface using a linearized theory. In addition to the usual

free surface wave mode, internal waves that behave asymptotically like outgoing cylindrical waves were found on the downstream side of the disturbance. Experimental evidences (Long

1955,

Yih

1959),

however, indicate that unattenuated waves (blocking) can also exist upstream as well as downstream of the disturbance. Since Wu and Nei have only examined the vertical component of the perturbation velocity their analyses do not rule out the possi-bility that a linear theory can account for the existence of the blocking phenomenon; the reason for this is that the blocking perturbations would consist essentially of horizontal motions.

This report attempts to resolve this issue within the frame-work of a linearized theory incorporating the Eoussinesq and

Oseen approximations. The fluid will be taken to be unbounded and possess a constant Vaisala frequency. The concept of group velocity (Lighthill

1960, 1967)

will be used to clarify the physical basis of the solution. In addition to the two-dimen-sional problem, three-dimentwo-dimen-sional effects on blocking will be

considered. In the last section the internal waves generated by a two-dimensional wedge moving horizontally in a stratified fluid beneath a surface of large density discontinuity is

-2-FORMULATION OF THE PROBLEM

Consider the problem of a submerged disturbance moving with constant horizontal velocity of magnitude U in an infinite ex-panse of density-stratified, incompressible and inviscid fluid under gravity. The Vaisala frequency N of the fluid is assumed

to be constant. Choose a Cartesian coordinate system (x, y) with origin at the center of the disturbance and oriented in such a manner that gravity points in the negative z direction and the fluid appears to be moving in the positive direction.

Let us assume that the disturbance can be represented by a source distribution q(x,y). A large class of moving solid bodies can thus be represented, although the correspondence is not

simple since the body hape depends in a complex way upon the stratification of the ambient fluid and velocity.

Within the Boussinesq and Oseen approximations the linearized equation governing the horizontal component of the perturbation velocity, u, is 2 ) + N2 + N21 q x2 y2 x2 [1]

where t is the time. We focus our attention on this component of the velocity because we expect the perturbations far upstream and downstream of the disturbance, if there are any, to be es-sentially horizontal. For a three-dimensional disturbance the corresponding equation would be

+ N2

+ +

y2 z2

where z is the third space coordinate.

It is well known that problems of the sort being formulated are not determinate and it is necessary to have a way to select that particular solution corresponding to physical reality. Fol-lowing Wu and Mel

(1957)

we consider the corresponding transient problem in which the forcing function is switched on at t = O and take the large time limit of the resulting time-dependent

solution. The switching operation is accomplished by multiplying the source term by the Heaviside function H(t).

FEATURES OF THE WAVE PATTERN

Without integrating [1] directly we can obtain certain in-formation about its associated regular, steady wave pattern using the concept of group velocity. The regular wave pattern may be considered to be composed of plane wave components and, because the pattern is steady, only those plane waves that are

stationary with respect to the disturbance qualify to be

in-cluded. Packets of these waves are continuously generated

2

(+_.\

+N2

q

x)

through the interaction of the disturbance with the free stream. These wave packets are then swept downstream and, at the same time, they propagate with their own group velocities.

Let us, therefore, consider the behavior of plane waves in a stratified fluid stream to select those which qualify to be

in-cluded in the regular wave pattern. Two dimensional plane waves have the form

i(ax + f3y - wt)

u=A e

[31

where the components of the wave number vector

(a,/3)

and the frequency w are real numbers, and the amplitude A is a complex

number. They must satisfy the associated homogeneous Equation of

[i]. This leads to the following dispersion relation:

(w - Ua)2(a2 +

2)

N2a2

- O

which has two branches, viz.

w1

= Ua

-w2 = Ua +

a2+

2 Na

V2+ p2

Na

[k]

All wave numbers corresponding to Wi = O and w2 = O represent stationary plane waves, they are

a=0

for

w1=Oandw2=O

[7]

V+

-

-

for Wi = 0

[8]

-

U

and are shown in Figure 1 where a double line is used to indicate the double root at a = O. The regular wave pattern will be com-posed solely of th'ese stationary plane waves.

To obtain further information about the wave pattern we use the concept of group velocity. Let us consider first the system of plane waves corresponding to the double root at a = 0. They have wave numbers pointing in the vertical direction and

their group velocities are given by

where G and G denote components in the x and y directions

x y

respectively. Packets of these waves, therefore, propagate un-attenuated in the horizontal direction. Since the wave packets

G N , G

=0 for

w1=0

[9]

N

G =0 for

w2=0

_[lo]

=

-6-are swept downstream with velocity (u,o), waves corresponding to the branch w2 = O always trail behind the disturbance while those corresponding to w1 = 0 appear ahead of the disturbance if

K and behind if > . [hen = the group

velocity is equal in magnitude but opposite in direction to the free stream velocity. These wave packets, therefore, do not move away from the disturbance after their generation and thus their amplitudes grow with time. The linear theory is not adequate for describing the behavior of these wave components.

Next we consider the system of plane waves corresponding to

[6] -

the cylindrical system. It is convenient to introduce

polar coordinates defined as

a = k cos e, = k sin e

x = r cos , y = r sin

In

polar coordinates the group velocities (G ,G ₎ of the

corn-r

e

ponents of this system of waves are given by

G

_r

=0,

G =Usin6

8

Figure 2 shows the vector sum of the group velocity and the free stream velocity for various components of this system. It is seen that they all point in the direction downstream of the dis-turbance. Thus, this system of waves always trails behind the

disturbance. When e = ± the group velocity is exactly equal in magnitude but opposite in direction to the free stream ve-locity so that again the linear theory is invalidated.

The resulting wave, of course, also depends on the excitation produced by the disturbance. For the linear theory to be valid we need to restrict ourselves to those source distributions whose Fourier transforms vanishes at the critical points

a=O,

['3]

Let

q = q + q2 [lu]

and using a bar over a function to denote its Fourier transform we have

= + q2 [15]

The above requirement is satisfied if q1 vanishes at a = O and

q2 at a2 + - = O because the sum vanishes at the

U

critical points; its values at any other point, however, is arbitrary.

This limitation of the theory is entirely due to finite amplitude effects and has nothing to do with the Eoussinesq ap-proximation. Following an argument similar to the above, one can

2/

V+ U

and

2'

-8-easily show that elimination of the Boussinesq approximation fails to remove this limitation.

THE REGULAR WAVE PATTERN

In the previous section we have obtained a broad picture of the regular wave pattern, here we will derive it analytically. Referring to [1] let us split u into two components u1 and u2

such that

u1 =

+N2

2 u2 x2 2 ( t + U + N2 q1H(t) [16] ( + 2 + N2

22

(+u_)

x2 y2

ui = -2

hm

(271)

s- O

i u2 = - him 2 (271) I m u -

hm

27r2 s-4 O

s O

-co - co

It is noted that we have split the total source distribution into two components q and q2 as discussed in the previous sec-tion where j, the Fourier transform of q1 vanishes at a = O and

q2 at a2 + - = o.

u2

Using the method of Fourier-Laplace transforms we obtain the following integral representations of the steady state solu-tions of [16] and [17]: co a[(s+iaU)2+ N2] -

i(ax+y)

dad3 e (s+iaU)2(a2+ 2)+ N2a2 [18] co i(ax + /3y) dade a[ (s+iaU)2 + N2] q2 e (s+iaU)2(a2+ 2) + N2a2 [19]

Let us first consider u1 and rewrite the integral in polar coordinate form as follows:

co

deJ

dk cose[s+ikU cose)2+ N211(k cos ,k sin e)X (s + ikU cos e)2 + N2cos2 e

ikr cos

(ê_ct)

ikr cos (e+)

N i

+ I

de N sin2G - (N N r cos

(e+)

q1 cos

e,

sin e J e

for x > O,and for x < O we have

rcos(G-i Re

[

Nsin2e- (N

N u1

=

-2w dG U cos e q1 ces

e,

sin G

)

e

-10-Since vanishes at cos G = O the integrand has simple poles at

is

U ces

e - u

and only their residues contribute to the regular wave pattern. Thus, retaining only the regular wave part, u1 becomes

Using the method of stationary phase we obtain the following asymptotic form for

u1:

[21]

N sin2ê - IN N i r cos(G-) de _q cos G, sin G ) e Re u1 = 27r

Re u1 - - H(x)

V

27Fr

.2

sine

- (N N q1 cas sin e 7T

[2]

Thus u1 represents an outgoing cylindrical wave system trailing behind the disturbance.

Next we consider u2. Since q2 vanishes at a2 + 2 O U2

the contributions from the integral to the regular wave pattern come from the residues of the simple poles at

a = is

(u+N)(u-N)

Retaining only the regular wave part, u2 becomes N/U

iy

U2 = Re

_U-N

e o 00

iy

- H(x)

ReÍ

d

N22(O,)

e 7F _J /32U2-N2 O

Therefore u2 does not depend on x except for the Heaviside

function which separates fore and aft behaviour, thus it repre-sents unattenuated waves and accounts for the phenomenon of blocking.

-12-From

[26]

it is noted that disturbances whose Fourier transforms vanish at a = O cannot excite the fore and aft unat-tenuated wave system and, therefore, no blocking can occur. These source distributions satisfy the condition

f

q(x,y) dx = 0

_[27]

-that is, for all y the sources are exactly balanced by the sinks. In particular, a source distribution that is antisymmetric in x would satisfy this condition.

It is also noted that the unattenuated waves ahead of the disturbance have small wave numbers. Thus a disturbance whose vertical extent £ is small compared with U/N can excite only very weak forward unattenuated waves because the Fourier trans-form of the disturbance will be small in the low wave number

range.

By integrating u2 over vertical sections upstream and down-stream of the disturbance it is noted that one half of the

total discharge from the source distribution flows upstream and the other half flows downstream.

THREE-DIMENSIONAL EFFECTS

We have seen above that fore and aft unattenuated waves can be excited by two-dimensional disturbances moving horizontally in a stratified fluid. For a disturbance having a finite span-wise extension, however, this wave system attenuates due to lateral spreading and therefore, no blocking can occur.

The equation corresponding to [LI] for the three-dimensional jase is

(w-Ua)2(a2+

2)

-

N2 (ct2+

2)

= 0

_[28]

which also has two branches, viz.

= Ua

+ N

a2+

,2

2+

_{2+ y2}

In polar cylindrical coordinates defined as

a

= K cos B; y = K sin S; = x = R cos ; z = R sin ; y = y r _are K Wi = UK cos 5 - N

_[32]

V2,2

K

-t-g

0)2 = UK cos B + N K K2 +

[301

[311

[331

Ua -

N

2+ y2

_[29]

a2-'- 2+ ,2

-1k-Therefore, the stationary plane waves are

K = O

K2+ 2

cos e =

VK2+2

cos e

-The system of plane waves given by

[3k]

corresponds to the wave system that causes blocking in the two-dimensional case. Our

primary interest is to see the three-dimensional effects on the blocking phenomenon and thus we shall concentrate our attention on this system.

Members of this system have z = constant planes as surfaces of constant phase and the components in cylindrical coordinates of their group velocities are

GR__

N , G

=0, G

=Oforci1 =0

O y

N

,G0 =0, Gy=Oforci2 =0

It is seen that those corresponding to ci1 = O have energy flow-ing radially inwards and are, therefore, not excited by a

lo-calized disturbance.

for

ci1 =

O and w2 = 0

[3k]

for cii = o cos e > 0

[35]

The wave packets, while propagating with their own group velocities are, at the same time, being washed downstream. Fig-ure 3 shows the plane view of the position of wave packets of different one unit of time after their generation by the

dis-turbance which is localized at the origin of the coordinate

sys-tem, Wave packets with > N/U trail behind the disturbance and are restricted to a sector with a half included angle of

.-i

N

..

. N

sin wriile packets with appears in front as well as behind the body.

THE WAVES GENERATED BY A TWO-DIMENSIONAL WEDGE

In this last section we would like to apply some of the theoretical results developed in the previous sections to a specific problem.

As we have seen, the Fourier transform of the source dis-tribution that represents the disturbance must vanish at the

critical points given by Equation _£131 for, otherwise, waves that have group velocities equal but opposite to the main stream

would be excited. The linear theory is not adequate for treating these particular wave components. One of the source distribu-tions that satisfies this requirement consists of two line sources of constant strength m arranged as shown in Figure k where 2

denote the length of the source lines and 2d is the distance separating them. It will later be shown that d has to be

-16-d = 7r(n + ) U/N

[39]

where n is zero or any positive integer. Taking

m=2UtanT

['FO]

where T 15 supposed to be small, then according to the thin body theory, the body corresponding to each of the source lines is a wedge of angle 2T and length 22 followed by a stern of thickness

22 tan T. If both T and £ tan T are small then,aside from the vicinity of the stagnation point and the turning point at the

shoulders,the perturbations are small and we should expect the linear theory to be reasonably adequate. Furthermore, from the governing equations and the symmetry of the disturbance,it is

clear that the flow is symmetric about the x-axis and the flow in the lower half plane may be considered to be produced by a wedge moving horizontally beneath a wall or a surface of large density

dis continuity.

The Fourier transform of this source distribution is

q = +m OOS d sin a2

a

[hl]

At a = O

therefore d has to satisfy Equation

_[391

in order that (O,/3) vanishes at the critical points.

can now be written as the sum of i and 2 so that ¿j

vanishes at a = O and q2 at a2 + 32 - N2/U2 = O as follows

- 4mU (Nd . IN2

q1

= M sin e cos ) sin sin

sin aß 4mU (Nd

)

. (Mt

q2 = 4m cos 13d cos cos e sin sin

a

Nsine

U U

The outgoing wave system trailing behind the wedge is ob-tained by substituting equation [45] into equation [24] to give

u1 H(x)2m / 2U tan (Nd 'Nt (Nr \

VTN

cos cos t)sin (-j_sin

)

cos

-We shall not get into the details of this wave system beyond mentioning the fact that the wave is not singular at = ±

i.e., along the y axis,in spite of the factor tan , because

cos (Nd cos ct/U) vanishes there.

To obtain the fore and aft unattenuated wave system we substitute equation

[6]

into equation

[26].

Let us consider first the forward wake. For x < O we have

mßN _'±ij F1 TrU UJ w h. er e yN dN F1 '

= Fl(y*,d*) = cas d* cos

P

t-* *

*

*

*

*

= - sin d sin y Cin(y -d ) - Cin(y + d )

L

+ cos y* [Si(y*_d*) Si(y*+ d*)

in Figure 5 we have plotted F1 as a function of y for the case where d* _{= /2} [so] For x > O we have m N U2

-

rU (F1

-

TrF2)

where

F2 (y*, - sin d* cos y*[sgn(d* + y*) + sgn

[51]

In Figure

6,

the expression (F1 - 7TF2) is plotted against y*

for d* given by Equation [50].

It is noted that for the case where the fore and aft unat-tenuated perturbation profiles depend only on the volume of fluid displaced by the wedge (i.e. mß) but not on the shape of the

wedge, we can arbitrarily vary m and L respectively so long as we keep mL constant. The cylindrical wave system, however, would not stay the same.

-20-REFERENCES

Lighthfll, M. J, Trans. Royal Soc., Vol.

252A, pp 397-k30. 1960.

Lighthill, M. J., J. FluId Mech.,

Vol.27,

_pp

725-752, 1967.

Long, R. R., Tellus, Vol.7, pp

341-357, 1955.

Wu, T. Y., Mel, C. C., Phys. of Fluids, Vol.10. pp

482-486, 1967.

Yih. C. S., J. Geophys, Res,

Vol.64,

_{pp 2219-2223., 1959.}

B

n/U

FIGURE 1 - WAVE NUMBERS OF STATIONARY PLANE WAVES

U SIN e U COS e U SIN 8 U u cos e U 13 U SIN e U COS e U u cos e

FIGURE 2 - VELOCITY OF PROPAGATION OF WAVE PACKETS OF

CYLINDRICAL SYSTEM

U SIN e

U

y

SIN1 N

U

N/lW 1(31 >N/U

FIGURE 3 - POSITION OF WAVE PACKETS OF DIFFERENT 1131 ONE UNIT OF TIME AFTER GENERATION

U MAIN STREAM X = -y IMAGE y =d LI NE SOURCE STRENGTH m

FIGURE 4 - SOURCE DISTRIBUTION REPRESENTING WEDGE BENEATH SURFACE OF

LARGE DENSITY DISCONTINUITY

X = +

y = -d

o -1T - 2ir - 5ir/2 -3ir -7 u/2 -4 o F1 irUu2/mN FIGURE 5

- VELOCITY PERTURBATION UPSTREAM OF WEDGE

-3

-lt - 3ir/2 - 2it -5ir/2 - - 7ir/2 2 -4 -3 -2

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(Security cIaaefcati ot ttfle, body of abstract and indexing annotation murt be entered en the overall report is class,fied)

t. ORIGINATIN G ACTIVITY (Co!porate author)

HYDRONAUTICS, Incorporated, Pindell School Road, Howard County, Laurel, Maryland

2e. REPORT SECURITY C LASSIFICATION UNCLASSIFIED

2b GROUP

3. REPORT 'ITLE

ON INTERNAL GRAVITY WAVES GENERATED BY SUBMERGED DISTURBANCES 4. DESCRIPTIVE NOTES (Type of report and inclusive dates)

Technical Report

5. AUTHOR(S) (Laat n&ne, first namo initial)

Worig, K. K.

6. REPORT DATE

August 1968 7a. TOTAL NO. OF PAGES₃₁ 7b. NO. OF REF5₅

ea. CONTRACT OR GRANT NO.

Nonr-33.9(OO), NR

062-266

b. PROJECT NO.

c.

d

9.. ORIGINATORS REPORT NUMBER(S)

Technical Report 117-13

9b. OTHER RPORT NO(S) (Any other number. that may be a..ied

thia report)

IO. AV A IL ABILITY/LIMITATION NOTICES

DISTRIBUTION OF THIS DOCUMENT IS UNLIMITED

11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

Office of Naval Research Department of the Navy 13. ABSTRACT

The problem of gravity waves generated by a disturbance moving horizontally in a stratified fluid with a free surface has been con-sidered recently by experiments and theory. The linear theory,

which considered only the vertical component of the perturbation velocity, found internal waves only on the downstream side of the

disturbance. The experiments, however, have disclosed the presence of upstream unattenuated waves (blocking). This report attempts to resolve this issue within the framework of a linearized theory in-corporating the Boussinesq and Oseen approximations, and accounting for the blocking effects as essentially horizontal motions. The fluid is assumed unbounded with a constant Vaisala frequency, and the concept of group velocity is used to clarify the physical basis of the solution.