Water wave models and wave forces with shear currents

COASTAL AND OCEANOGRAPHIC ENGINEERING LABORATORY

University of Florida Gainesvilie, Florida

-Technica1 Report No. 20

WATER WAVE MODELS AND WAVE FORCES WrTH SHEAR CURRENTS

by

Robert

A.

Da1rymp1e

This research has been sponsored, in part, by the following participants of the joint industry project, "Wave Force Analysis and Design Procedure Development"

Amoco Production Company Cities Service Oil Company

Chevron Oil Field Research Company Continental Oil Company

Gulf Research

&

Development Company Mobil Research

&

Development Corporation Pennzoil United, Incorporated

Phillips Petroleum Company Placid Oil Company

Shell Oil Company Texaco Incorporated

Union Oil Company of California

ACKNOWLEDGE}lliNTS

The author would like to thank Dr. Robert G. Dean, who first intro-duced him to the field of coastal engineering and who has fostered his interest in this area of research.

The author would also like to acknowledge the participants in the joint industry project, "Wave Force Analysis and Design Procedure

Development," under the management of Dr. Frank Hsu of Amoco Production Company. This project, which has provided support for the author for the past two years, has introduced him to the ar duo us nature of wave force analysis, yet taught him the realities of offshore design, to which this report in part is directed.

Mrs. Heida Hudspeth must be thanked for her aid and assistance in translating the French language references of which there were many.

The author would further like to express his appreciátion to the Northeast Regional Data Center for providing the use of its IBM 370/65 on which all the computer work of this report has been carried out and, second, for its policy of funding unsponsored Departmental research. In this regard, also, Dr. O. H. Shemdin, Director of the Coastal and Oceanographic Engineering Laboratory, and Dr. James H. Schaub, Chairman of the Civil and Coastal Engineering Department, must be thanked for their liberal disbursements from these funds.

Finally, I must gratefully thank the typist of my final draft, Mrs. Edna Larrick, who has spent many arduous and long hours to ensure

that the author would meet his deadline, and, as the reader will note, has done an excellent job, particularly in view of the rough draft with

which she had to work. Further, Mrs. Jerline White must be thanked for doing the original translation of much of the handwritten draft.

Ms. Denise Frank did all of the drafting and lettering of the figures, for which the au thor thankfully acknowledges her.

TABLE OF CONTENTS

ACKNOWLEDmlENTS LIST OF TABLES LISlOF FIGURES

KEY TO syr-IBOLSAND ABBREVIATIONS ABSTRACT CHAPTER I. 11. III. IV. I~lRODUCTION

I-A. Statement of the problem I-B. Currents in Nature

l\IATHEl,lATICAL FOmrtJLATION

II-A. Governing Diffcrenti~l Equations in Cartesian Coordinates . . . • . . . . • . • . II-B. Transformed Governing Equations . • . . II-C. Boundary Conditions on the Fluid Domains

Si.IALLAMPLITUDE WP_VESON LI:-'"EAAA..:XDBILH."'EARCVRRE:-.-rS lIl-A. Linear Shear Current

III-B. Bilinear Shear Current

NOl';"LHa:ARWAVES ON A LINEAR SHEAR CURRENT

IV-A. Theoretical Development for Colinear Waves and Currents . . . .

IV-B. Modifications for Currents and Waves in Different Directions . . . • IV-C. The Theoretical Wave: Validi ty and

Comparisons IV-C.l. IV-C.2.

Analytical Validity ..•.•

The Change in the Length of Waves Due to Theiy Height and Shear Currents The Effect of Constant Vorticity on Wave Breaking . . . • The Effect of Vorticity on the IV-C.3. IV-C.4. Wave Profile v Page iii viii ix xii xviii 1 1 3 8 8 15 18 22 23 27 35 35 41 46 46 50 53 60

CHAPTER IV.

TABLE OF CO~7E~!S (Continued)

Page (Continued)

IV-C.5. The Effect of Constant Vorticity in

the Water particle Kinematics 62 IV-C.6. Superposition... 62 IV-D. The Measured Wave--Problem Solution Procedure 72 IV-E. The Measured Wave--Comparison with Irrotational

Theory . 75

V. NO?-;LlNEARWAVES ON A BILIXEAR CURRENT 78 V-A. The The:)retical Wave--problem Solution

Procedure . . . . • . . . . Thc Theoretical Wave--Comparison with

78 V-Eo

Pr-eviou s ModeIs 82

86 V-Co The Measured Wave--Proble::nSolution Procedures

VI. FINITE DIPFERENCE 1.10DELFOR WAVES ON ARBITRARY

VII. VIII. CURRENTS 89 90 92 100 100 101 VI-A. The Perturbation Procedure of Daubert

VI-B. The Application of Finite Differences VI-Co Applicaticns of the Model

VI-C.l. Uniform Current Case VI-C.2. Irrotational Results

VI-C.3. Comparison to Numerical Shear Current Model . . . •

VI-C.4. The One-Seventh Power Law

103 107

CURRENTS AND WAVE FORCES 113

113 115 Wave Project 11 Data

Currents and the Morison Equation

Currents Obtained from the Wave Project 11 Data . . . • . • VII-D. The Effect of Current Directionality VII-A.

VII-B. VII-Co

124

on Wave Forces 127

CONCLUSIONS AND RECOl\lME:t'rDATIONSFOR FURTHER RESEARCH 133 133 135 VIII-A. Conclusions

VIII-B. Recommendations

TABLE OF CONTENTS (Continued)

APPENDICES Page

I. PERTURBATION APPROACH TO WAVES ON ARBITRARY CURRENTS IN CARTESIAN COORDINATES, INCLUDING

A FIRST ORDER WKB SOLUTION . . . . • . . . • 138 11. PERTURBATION PROCEDURE FOLLOWING GOUYON

AND MOISEEV . . . 142

IIl. BREAKING WAVES ON A UNIFORM CURRENT 148

A

.

B.

Theoretical Treatment . . . . NumericaI Verification of the Breaking Index Curve

148 155

REFERENCES 159

Table IV-1 V-1 VI-1 VI-2 VI-3 LIST OF TABLES

Errors in Linear Shear Current Model Representation of the Case A nnd B Waves for LU

=

0.03 sec-1

o

Comparison of Linenr and Bilinear "Shear Current Models for a Constant Vorticity Case

Comparison of Stream Function Wave Theory and the Fini te Difference i\!odelImprovement . . .

Comparison of the Linear Shear Current Model and Finite Difference Representations of a Wave Propagating on an Opposing Shear Current for the Same ~~ nnd kh . . . • . . . •

'I

Results and Errors in the Finite Difference

Representation of a Wave Propagating on an OPPosing Turbulent Current . . . • . . . . • . . . • . . viii Page 49 83 102 106 112

LIST OF FIGURES Figure

1-1 Computed Dimensionless Current, U'

=u

_;j

;

I '

versus s

Dimensionless Elevation for Two Different Values of (T s/Tb) . • •- • • . • • • . • • II-l 111-1 1II-2 II1-3 IV-1 IV-2 1V-3 1V-4 IV-5 IV-6 IV-7 IV-8 1V-9

Domains of Definition foy the Different Governing Equations . . . • . . . . . Linear and Bilinear Approximations to Design Current

Profile . . . ., ...•. . . . .

Definition Sketch for a Wave on a Current, U, an Observer Moving with the Wave .

as seen by

Definition sketch for a Wave on a Bilinear Current, as seen by an Observer Moving with the Wave.

Streamlines under a Wave with No Current Present Streamlines under a Wave Propagating Against a Linear Shear Current .. ...• .•.•.•...

Plan View of Streamlines at the Surf ace of Waves Propagating to the Right as seen by Observer Moving wi th the Speed of the waves . . . • . . . . Wave Length as a Functiön of Vorticity ior a Deep Water Wave

.

.

.

. .

. .

.

.

.

.

.

. .

.

.

. .

.

Wave Length as a Function of Vorticity for a Shallow Water Wave . . . • . . . . Breaking Index Curve with Experimental and Field Data Breaking Index Curve for Waves Propagating on Uniform Current, U . • . . . • . • • . .

o

Example of Triple-crestèd Wave in Shallow Water on a Linear Shear Current

Dimensionless Crest Elevation, ~ IH, as a Function of c

Vorticity for Case A ~nd Case B ...• ix Page 6 16 22 24 28 42 43 47 51 52 55 57 59 61

LIST OF FIGURES (Continueel) Figure

IV-I0 Change in Total Horizontal Velocity with Dimensionless Distance and Vorticity, -w , Evaluated at Mid-depth

o

for Case A Wave . . . • . . . .

IV-ll Change in Total Horizontal Acceleration with Dimin-sionless Distance and Vorticity, -w , Evaluated at Mid-depth for Case A Wave ... 0••••••

IV-12 Change in Horizontal Velocity with Dimensionless Distance and Vorticity, -w , Evaluated at Mid-depth

o

for Case B Wave . . • • . . . • • • . . • . . • . . IV-13 Change in Total Horizontal Acceleration with Dimen

-sionless Distance and Vorticity, -w , Evaluated at

Mid-depth for Case B Wave . . . . .0. • . • •

IV-14 Comparison of Horizontal Velocity Profiles by Linear Shear Current Model and Superposition Using Stream Function Wave Theory: Case A . . . • • . IV-15 Comparison of Horizontal Velocity Profiles by Linear

Shear Current Mod eI and Superposi tion Using Stream Function Wave Theory: Case B .

IV-16 Comparison of Horizontal Velocity Profiles on an

Opposing Uniform Current by Stream Function Theory

(U

=

-5.12 fps) and by Superposition ...

o

IV-17 .Measured Wave Fit by Stream Function Wave Theory

and Linear and Bil inear Shear Current Mod e.ls, with

Total Horizontal Ve~ocity Profiles Predicted under

Wave Crest ..•••...•.•..•.••...

v-t

Comparison of Linear and Bilinear Horizontal Velocity Profiles under the Crest of the Case A Wave for 3 fps Current at Still Water Level (y=0)

VI-l Schematic of Finite Difference Grid Used for the Solution of the Dureil-Jacotin Equation ..• VI-2 Number of Iterations Required for Convergence of

Irrotational Finite Difference Model as a Function of Relaxation Parameter,

8

...

VI-3 Comparison of Horizontal Velocity Profiles under the

Wave Crest Obtained Using Finite Difference Model and Stream Function Wave Theory . . . • . .

x Page 63 64 65 66 68 69 70 76 85 94 97 104

Figure VI-4

LIST OF FIGURES (Continued)

Page

Comparison of Streamlines Generated by Nineteenth Order Numerical Shear Current Mod el, and the Fini te

Diff erence Model . • . . . • . 108

VI-Sa Linear Shear Current Velocity Profile in the Absence

VI-Sb

of a Wave . 109

Comparisons of Velocity Profiles under Wave Crest

for Linear Shear Current . • . . . . 109 VI-Ga One-Seventh Power Law Current Velocity Profile in

the Absence of a Wave 111

VI-6b Velocity Profile under Wave Crest for One-Seventh

VII-1 VII-2 VII-3 VII-4 VII-5 Power Law Current 111

Drag Coefficients Obtained by Dean and Aagaard (1970) by

Least Squares Procedure Using Wave Project 11 Data 118

Inertia Coefficients Obtained by Dean and Aagaard (1970)

by Least Squares Procedure Using Wave Project 11 Data 119

Percentage Error in Drag Coefficient as a Function of

Relative Current Velocity (Linear Analysis) ... 121

Wave and Current Forces on a 3.71 ft Pile at Crest and Trough Positions as a Function of Elevation for

a Uniform Current . . . 129

Computed Wave Direction at Each Dynamometer Elevation

for 69 Wave Project 11 Waves ...•.. 132

A-IlI-1 Dimensionless Celerity for Breaking Waves from the Stream Function Wave Tables (Dean (1972» for Various

Dimensionless Water Depths .. 150

A-III-2 Dimensionless Stopping Current for Nonlinear Waves 153

A-III-3 Breaking Index Curve for Waves Propagating on Uniform

Current, Uo' and Dimensionless Crest Elevations, ~c/H 154 A-III-4 Stability Parameter, u IC, versus Wave Height for

Different Order StreamcFunction Wave Theory and

Uniform Opposing Current . . . . • • . . . . 157

a n,p A

A

n A

A

b C I C I CD C g ~ C o

C'

o

KEY TO SY1ffiOLS~~~ ABBREV1AT10NS

Daubert perturbation coefficient

sum for determining drag and inertia coefficients

constant in stream function expression

perturbation coefficient, Equation 11-8

projected area per foot of elevation

Daubert perturbation coefficient .

amplitude of interfacial wave

sum for determining drag and inertia coefficients

constant in stream function expression

sum for determining drag and inertia coefficients

wave celerity

wave celerity apparent to observer moving with current

drag coefficient

apparent drag coefficient group velocity

inertia coefficient

Airy wave theory deep water celerity

the Capparent to observer moving with current

o

a Bernoulli constant

d dl D DFSBC g h R

RB

H max depth of interface

sum for determining drag and inertia coefficient constant in stream function expression

dynamic free surf ace boundary condition

sum for determining drag and inertia coefficients total error, sum _{of El and E2}

mean squared error in the Bernoulli constant mean squared error in the free surf ace fit

mean squared error in the interfacial horizont al velocity mean squared error in the interfacial vertical velocity

vorticity function

sum for determining drag and inertia coefficients feet per second

vorticity function divided by H measured wave force

predicted wave force

acceleration due to gravity

terms in perturbation expansion, Appendix I

water depth wave height

breaking wave height

wave height associated with maximum steepness

i ... i I j ... j k KFSBC L L max L o L' o M

-1\1 n N NN

o

p psf p

grid index in ~ direction; summation index

unit vector

number of discrete points

grid index in x direction; iteration index

unit vector

wave number

kinematic free surface boundary condition

elevation above the bottom of the interface; summation index

wave length

apparent wave length

wave length associated with maximum.steepness

Airy theory wave length in deep water

apparent L to observer roovingwith current

o

number of grid points in

*

direct ion; measured Sarpkaya superposition parameter

grid index in x direction; coefficient number

number of grid points minus one in x direct ion

wave theory order parameter

objective function

pressure; iteration index

pounds per square foot

predicted

-q Q Q SOR t T u u c A U

u

U max

u

o I U

u

v

v

w

w

horizont al velocity vector Bernoulli constant

average Bernoulli constant successive overelaxation

time

wave period

wave-induced horizontal velocity

value of horizontal velocity at wave crest value of horizontal velocity at wave trough maximum horizontal velocity, Airy theory value of u at interface

horizo~tal current velocjty

maximum value of U

uniform current component

velocity increments to U due to vorticity

o

dimensionless horizont al current velocity mean horizontal current velocity

wave induced vertical velocity value of v at interface

vertical current velocity

volume per foot of elevation

integration variabIe

1ateral velocity componerrt

x X(n) X(x) D X(n) y z

z

~'

y è.x

Cartesian coordinate in wave direct ion

stream function constant

separation variabIe in Gouyon-Moiseev perturbation

Stream Function wave theory value of X(n)

Cartesian coordinate in vertical direction

Cartesian coordinate in lateral direct ion

roughness lengths of Reid

Moiseev shear current

angle between horizontal velocity vector and wave direct ion

transformation variabIe in Appendix 11

vorticity proportionality constant

incremental distance in x direct ion

6~ incremental distance in

*

direction € perturbation parameter

function in perturbation form of stream function; displacement of interface from mean elevation

n

displacement of free surface from still water level

n

crest displacement c

n

M measured value of

n

np

predicted value of

n

8 angle between current direction and wave direction

8 angle between current direction and reference axis c

p

w

o

angle between wave direction and reference axis relaxation parameter

shallow water breaking parameter Lagrange multipliers

angle for which total horizontal velocity is zero 3.1415927....

density of water

angular frequency of wave

mean-square error in predicted versus measured force

bottom and surface shear stress, respectively

function in Daubert perturbation procedure

stream function

Regions I and 11 stream functions for bilinear shear current

free surface streamline

wave component of stream function

separation variabie in Gouyon-~~iseev perturbation constant negative vorticity

constant negative vorticities in Regions land 11

current component in Daubert solution

two-dimensional Laplacian operator in Cartesian coordinates.

ABSTRACT

Water wave modeIs, incorporating shear currents, are developed

for linear and nonlinear waves. The first model assumes a constant

vorticity over the depth of the fluid; the case for a wave propagating

on a linear shear current. For greater generality, a second model is

presented which assumes that the fluid is composed of two layers, each

with a different, but constant, vorticity. The nonlinear solutions

require the use of a nurnerical perturbation procedure. The last wave

model, using a finite difference approach, generates waves propagating

on arbitrary vorticity distributions.

Examples of the effect of the vorticity on the waves are

presented. Further, the use of two of these models in the analysis

of actual measured wave data is shown.

Wave force measurement programs conducted for the purposes of

obtaining drag and inertia coefficients on structures are affected by

the presence of currents, and the biases introduced into these

coefficients by neglecting the currents are investigated via small

amplitude wave theory. Further, an approximate procedure for de

ter-mining currents for measured wave force data is presented, as weIl as

some results from Wave Project 11 dàta, which were obtained during

Hurricane Carla in the Gulf of Mexico.

CHAPTER I

INTRODUCTI ON

I-

A

.

Statement of the problem

The engineer who must design offshore structures faces a

difficult task, in large part due to the presence at sea of storm waves and currents; bath of wh Lch are not adequately predictabie. At present, most offshore design of drilling rigs and oil storage tanks is done on the design wave concept; that is, the forces pre-dicted on the structure are associated with a wave of given height and period in the design water depth--a wave which has been determined to be the maximum wave likely (by some criterion) to be experienced by the structure during its lifetime. Quite often, the presence of

oceanic or wind-driven currents is neglected in the design, while rarely would this quiescent condition exist. Indeed in all cases

. 1

1

whe.ce large waves are present, for example, during a hurricane, the high velocity winds that are responsible for generating the design

waves would also generatc currents, with magnitudes of approximately 3% of the wind speed at the water surface.

The presence of these currents causes a significant change in the magnitudes of the forces exerted on the structure, even if the current velocities are small relative to the wave-induced veloeities, For example, if the maximum horizontal velocity due to the design wave

, I

!

I

2

is 16 fps, then the presence of a 2 fps current would inCrease the drag force on the structure by over 25%! It is obvious, then, that rational

offshore design must include the effect of currents.

A primary reason for the lack of a good model for waves

propagating on currents is the formidable mathemat~cal problem that

results from an analytical treatment. Formal work on the mathematical

description of waves began with Gerstner in 1802, and since that time,

despite the numerous researchers who have attacked the problem, most

of the work has dealt with the simpier, but by no means easy, problems

of waves propagating in quiescent water or on a current which has a

Uniform velocity over depth. The goal of this research has been p

ri-marily to provide wave models which would provide the design engineer

with the capability to include reasonable current profiles in his wave

force design technique.

The problem of waves on currents has been attacked here in

several different ways. The first wave models are based on currents

which possess constant vorticity, i.e., the velocity profiles are

linear over-depth. These models are developed, in the beginning, using

small amplitude wave theory assumptions, and then, more generally,

based on computer generation of nonlinear waves. The last wave model

presented generates waves on arbitrary current distributions over

depth. This method is based on an iterative finite difference scheme.

The actual prediction of wave forces on structures involves

the use of the Morison equation which relates the water partiele

3

force exerted on the structure through the use of two empirical

coef-ficients: the drag, CD' and the inertia, CM' coefficients. These

coefficients have been obtained by many researchers based on different

experimental techniques and they either all disagree or the scatter in

their data is quite large. However, in almost all.of the studies for

these coefficients, the presence of an ambient current has been

ignored, with resulting (but unknown) biases in the final values of

In the last chapter, using smal1 amplitude wave theory,

the effect of this neglect of currents is demonstrated to be important.

Finally, the directionality of the current in relation to the

direct ion of the wave is discussed, particularly in application to

wave forces on a pile-supported structure. It is shown there that the

force loading on a pile due to waves and currents moving in different

directions varies in strength and in direction over the wetteti vertical length of the pile.

I-B. Currents in Xature

In order to discuss realistic models for wave on currents, it is first necessary to investigate the types of currents that w0uld occur in design situations. Currents may be classified into three major subdivisions: those caused by density differences in the fluid, those caused by tidal forces and finally those caused by wind. The first class of currents is of interest mainly to oceanographers, rather than ocean engineers, as the currents encountered in design situations on the Continental shelves are generally induced by tides or wind

4

Currents generated by tides are usually small, except where

the water is either very shallow or near the coastline, where lateral

boundaries constrict the flow. The velocity profiles occurring with

these flows would be similar to those observed in open channel flows,

that is, the profile would be logarithmic, conforming to the turbulent

velocity profiles first described analytically by Prandtl.

Wind-drivcn currents, which are created whenever wind blows

over water, can be very important during the occurrence of storms

and, for design situations, would probably be thc major curren~con

-sidered. The first analytical work on currents due to wind was done

by Ekman, who developed the Ekman spiral. This spiral velocity law

predicts that at the surf ace of infinitely deep water the current flows

at 45° to the right (in the Northern Hemisphere) of the wind direction,

and, with incr~asing depth into the fluid, the current velocity vector

decreases exponentially in magnitude, but continues to rotate in

a clockwise direction. _{In shallow water, the angular deviation of the}

current vector is suppressed by the presence of the bottom (see, for

instance, Neumann and Pierson (1966) for a good exposition of the

subject) .

The effect of land masses complicates the current distribution

cansiderably. In the extreme case of lateral constraints on the cur

-rents, laboratory studies have been conducted in closed basins to

measure the velocity profile of the current (see, for example, Baines

and Knapp (1965), Tickner (1961) and Francis (1953, 1959)). _{In most}

5

tank, resulting in a setup of fluid (the storm surge) at the downwind

end and a fast forward current at the surface and return current near

the bottom. Reid (1957), using the Prandtl mixing length theory with

a mixing length that increased from both the bottom and the top,

reaching a maximum at mid-depth, derived current velocity profil es

which are dependent on the ratio of the surface shear stress, ~_s, to

the bottom shear stress, Tb. The fit of these profiles to measured

laboratory data is reasonably good for closed channel data with

~S/Tb

=

-0.097. This profile is sho~n in Figure I-I. For an

open-ended tank, with no induced surf ace slope, the steady state solution

obtained by Reid is also shown in Figure I-I for ~s/Tb

=

1. It is

expected that this type of current profile would be very often

encoun-tered in design situations; however, due to the presence of the waves,

it seems that the mixing length assumptions of Reid may not be totally

valid.

For nearshore water motions due to extreme wind systems,

numerous studies have been made using averaged (over depth) currents.

Bretschneider (191:>7)developed al"analytic ·technique for calculating

the average currents during the passage of a hurricane. Recently,

more sophisticated models have been made possible by the use of

numer-ical methods in conjunction with computers. The disadvantage of these

models has been that, although they predict storm surge elevations

quite weIl for regions of complex bathymetry, the average velocities

are very sensitive to the.choice of friction factors, and thus the

-- 0

-

~

-+--:--+----1 ~ o o o

o

o o I'(') 11 11 NO

\_~

r-...

;;~

~~

I~

~ ...Q 11 -"" - 0 ~---+----~~~

-___

....- I~ ,_--~r_--_r----~---~r_--~N

-1---

=

----I'----

==

~~==o

\1

~

1

,_ 1'+---

<,'rf' 11 -- ... I~

-~---~

--

-r

--

--

r

---~~

~

-+

~

.

----

+-

---+----

,_

--

~

---~o

==

r---~~--+---~-~

-

--

--

--

--

-~---+---

-+--

--'_----

r-

---+----,_----~---+----

,_

----~Q

- - 1

-

--

--

--

--

--

--

-- -0 N - , -I

-

--

--

--

--

--

-- • ! -- -0 - Ol'(') 1

- r-o <TI CD 11 6 L1...

o

Cl) :::> Cl) 0:: W

>

11 f-Z W 0:: 0:: :::> (.) Cl) Cl)

w

_J

z

o

Cl)

z

w

~ Cl Cl W

....

:::> Cl.

::E

o

(.) H W 0:: :::> (!) u,

7

obtained through these means, the distribution of the current over depth would still be unknown.

..

In conclusion, the following statements may be made vis-a-vis currents: (1) currents probably flow in directions other than the wind direct ion , except where the flow is constrained by lateral boundaries or the water is very shallow, (2) the velocity profile of the current varies with elevation in a marmer which depends on the agent that caused the current, the bathymetry of tbe area and also the characteristics of the bottom, and (3) present knowleàge regarding currents in nature is inadequate for offshore design.

·CHAPTER 11

~~THEfi~TICAL FOR1ruLATION

II-A. Governing Differential Equations in Cartesian Coordina~es

Historically the problem of waves propagating on currents has been approached through several different methods. In each of these, however, there have been corrunonassumptions. First, the motion of the fluid is assumed two-dimensional in the plane of the direction of the wavesand current. This implies co-linearity between the directions of the waves and the current and also that the waves are long-crested. Second, the water is assumed to be incompressible. This is quite a good assumption for water and also guarantees the existence of a stream function,~. Denoting x and y as the horizontal and vertical coordinate directions, respectively, this condition may be expressed as

(U+u) + (V+v) = 0

x y (II-l)

where (U,V)

=

horizont al and vertical velocity components of the current

(u,v) = horizontal and vertical velocity components of the wave motion.

The subscripts denote partial differentiation. The third assumption is that the form of the current profile is weIl established, and,

henee, the effect of viseosity over a short time is negligible.

9

The wave problem may now be formulated as a boundary value problem, consisting of a differential equation which governs the motion of the fluicl,and boundary conditions which must be satisfied on the boundary of thE:fluid domain, which comprises one wave.

The governing differential eq~ations for the assumed inviscid fluid motion are those of Euler (Lamb (1945».

(U+u)t + (Us-u)(Utu) x + (V+v) (Uju)y = --1 p (II-2) 1 = -_ p p -y g .

...

Here p is the pressure, and p is the density of the fluid. The total stream function is now substituted into thes~ equations as follows: U+u

=

-

*

_y (II-3) V+v

=

~

x Therefore, (II-4)

Eliminating the pressure terms in these equations by cross-differentia-tion yields the following nonlinear partial differential equation:

(II-S)

In this equation, the two-dimensional Laplacian of

*

is interpreted

10

readily seen. The vorticity, f, is defined

f

=

(V+v) - (U+1J)

x y

I ,I, = <y2,lr

=

'l'x+x 'ryy T (II-6)

Therefore, the equation is interpreted as requiring that the vorticity

óf a moving fluid element be conserved.

In the derivation of Equation 1I-5.it was not assumed that the

motion was steady in time. If the coordinates x and y are translated

with the speed of the wave, C, and the wave is assumed of constant form,

then the problem may be rendered time independent, and the governing

equation may be written as

(II-7)

Note that now the total stream function must be redefined as

U+u - C

=

-1jr ;

y V+v = 1jrx •

If a periodic wave form, moving with celerity, C, is assumed

to exist, than a regular perturbation technique can be used to obtain

a series solution. The stream function is assurned to be of the follow

-ing form, where the current is steady in time.

1jr(x,y)=-JY(U(y/)-C)dyl +

o

€n A

C

(y)eink(x-ct).

n n (II-8)

The € represents a small quantity of the order of the wave slope,

H

I

L,

where H is the height of the wave and L is the wave length.

Substituting 1jr into Equation 11-7 and grouping terms of the same order

of E, the following equations result for the

C

(y) functions.

11

(U

yy + n

2 2)

k

ç

U-C n

(II-9)

where g is a compl icated function consisting of the

ç

(1:;;;m:;;;n-1) .1

. n m

The difficulty with this form of the equations is that no gener al solu-tion exists for this second order partial different;ial equation wi th a variable coefficient.

For n

=

1, gl

=

0, the first order equation may be wri tten as

(II-lO)

This equation is the Rayleigh equation, used in the theory of inviscid hydrodynamic stability; it is also knO\VTIas the inviscid form of the

or-r-vsommer ïeld equatLon , This linear (in €) form of the equation,

which is valid for infinitesimal waves, has been used by numerous investigators, generally using simplifieà models -:torthe current. Abdullah (1949) and Thompson (1949) studied wave motion on exponential

and linear currents, respectively. Diesel (1950) a150 studied the waves on a linear current, using this equation. Burns (1953) studied 10nJ waves on a current, enabling him to simplify Equation 11-10 by

. 2

allowlng k to go to zero. Hunt (1955) used this linear equation for 1

a more realistic current, one representable by a (7)power law, thus extending the results of Lighthill (in Burns (1953» from stationary waves to progressive waves. Recently Shemdin (1972) used this equation

in conjunction with the same equation for the air motion above the wave,

1

The development of the perturbation equations is given in Appendix 1.

12

to calculate the phase speed of a wave in the presence of wind and current. An approximate solution of this equation for an arbitrary current distribution valid for long waves can be obtained by the WKB method, as is developed in Appendix I.

Equation 11-7, in its full nonlinearity, was attacked by a Frobenius solution by Hunt, attempting to find a solution in terms of

a power series in y. His approach, which utilized a solitary wave, is not readily usabIe.

Using the same equation, Sun-Tsao (1959)showed that an irro-tational wave can only exist on a current which varies vertically with depth either in a uniform or linear manner. This may be stated in the form of a theorem.

Theorem: 1rrotational waves,which propagate without change in form, can exist only on flows which vary lin-early with depth.

Proof: Assume

*

=

JY(

U(y

/

)-C)dyl

+

0/

o

where ~ represents the stream function of the wave motion and where "72

0/

=0, that is, the wave motion is

irrotational. Substituting into Equation 11-7,

~

(f

Y U(y', dy,)

=

0

x yyy

o

or U(y)

=

0, which corresponds only to a linear or yy

13

The physical meaning of Equation 11-7 is that the gradient of vorticity is perpendicular to the streamlines of the flow and, therefore, the vorticity is a constant along a streamline. 1ntegrating along a streamline, then, yields Poisson's equation,

(IJ-l1)

This equation is in general difficult to solve due to the appearance of the dependent variabie in the vorticity function, f(*), and as the domain of defini tion is unkriown; principally because the water surf ace elevation is unknown a priori. However, for simple forms of f(W), solutions are readily obtainable.

For the case of f(W)

=

0, the Laplace equation results, which would hold when the motion of the fluid is irrotational. This case, corresponding to either a quiescent fluid or to water moving with a uniform (over depth) current, has been treated quite fully in clas -sical memoirs. G. B. Airy (1845)first solved for the linear case of small amplitude waves, now called the Airy wave theory, and later stokes (1847)developed a series solution for finite height waves, taking into consideration the nonlinearitiés which appear, not in the governing equation, but in the boundary conditions, which will be discussed in the next section. More recently, Chappelear (1961) and Dean (1965~have developed numerical perturbation procedures for solving the Laplace equation for finite height waves to any order, that is, including as many series terms in the solution as desired. Chappelear developed his procedure by using the velocity potential, which involved considerable complexity, while Dean used the stream

14

function represt::ntationwhich reduced the amount of effort to achieve

a solution to the necessary degree of accur.acy,but which also satisfied

the boundary conditions better. Dean's Stream Function theory, which

can treat waves on a uniform (over depth) current, will be expanded in

the later chapters to waves on linear shear currents.

For the vorticity, f(~), equal to a constant, say,

-w

o (l1-12)

corresponding to a current which varies linearly with depth, Sun-Tsao

(1959)has obtained a linear solution corresponding to that of Biesel

and also a series solution to third order in wave slope. Using the

Stream Function approach, Sun-Tsao's results will be extended to any

arbitrary order in Chapter IV. F0r f(l~)equal to ± y~2, exponerrttal

and sLnusoLdal types of currents may be treated. Wehausen (1965)gives

the general linear solution for these rotational waves, and Eliasson

and Engsl.urd (1972)show an approach to nonlinear waves on a hyperbolic

sine varying current.

One other form of the governing differential equation in

Cartesian coordinates has been de!ived by Freeman and Johnson (1970),

who used a coordinate stretching perturbation procedure to develop

a rotational version of the Cnoidal wave theory for shallow water.

Their resulting equation was the Korteweg-de Vries equation, often

15

ll-B. Transformed Governing Equations

Three approaches to the problem of waves on currents have utilized transformed equations with good success. The first was that of Dubreil-Jaeotin (1934), whieh was later applied to Daubert (1961) to periodic waves and Brooke-Benjamin (1962) to solitary waves.

~Thle.Dubreil-Jaeotin, in a treatise extending the convergenee proofs of Levi-Civita (1925) and Struik (1926) for irrotational waves to waves with arbitrary vorticity distributions, utilized a transformation of variables. The governing equation, as previously presented, is based on • as the dependent variabie and x and y, the independent variables, that is, ~ as a funetion of x and y. Dubreil-Jacotir. posed the problem with Y as a function of x and ~'. Her transformat ion resul ts by using the stream function definition Equations 11-3 written as

(I1-13)

and transforming Equation 11-6. The resulting nonlinear second order equation is

(II-14)

Despite the obviously more difficult governing equation, the domain of the problem has been transformed into a rectangle, with a base the Iength of the wave and a height equal to the value of the surface streamline,

*

~,

rather than the wave-shaped region for the poisson equation (Equation 11-11). See Figure 11-1.

o/(x,y) a) Stream Function Approach, Equation

rr -

11 16 Y(X,o/) Equation n-14

x

u (x,o/) v (x,o/) b) Dubreil - Jacotin Approach, Equation TI - 14 c) Gouyon Approach Equations

TI-15,

Ir - 16

FIGUR

E

TI-I

DO

MA

l

N

S

OF

D

E

FI

N

I

T

ION

FOR

THE

DIF

FERE

NT

GOV

ERN

I

N

G

EQUATI

ON

S

A furthcr advantage is gained as the vorticity distribution, f(V), is now specified as a function of an independent variabie. Dubrei l-Jacotin showed that for a given wave number, k, water depth, h, a wave height parameter, and an f(V), there is a rotational wave which

exists and is unique. Dubreil-Jacotin did not, however, offer a sol u-tion to her equation, and Daubert, using a regular perturbation

technique, developed linear through third order solutions to this equation. His procedure, based on perturbing the uniform current, will be discussed iurther in Chapter VI.

Geuyon (1961) used a von Mises transiormation of the continuity equation (Equation 11-1) and the rotationality condition (Equation 11-6), arriving at the following equations ior u(x,.) and v(x,.) which, in this case, represent the total veloeities:

17

= 0 (Il-15)

= f(1jt) . (II-16)

The domain of the u and v is the same as that of Dubreil-Jacotin,

as shown in Figure 11-1. Gouyon, using a perturbation procedure, was able to give a general convergence proof for waves on arbitrary

.

.

currents, incorporating the results of Levi-Civita and Struik.

Moiseev (1960)improved on Gouyon's procedure by, instead of perturbing

the uniform flow, -C, using the steady current represented by

derivable from Equation 11-16, when the wave is not present, i. e.,

v

=

O. Using nonlinear integral equations, Mo i.seev obtained a general solution for Gouyon's equations. A perturbation procedure, utilizing these equations with 1biseev's steady current is developed in

Appendix 11. Germain (1963),using a coordinate stretching pe

rturba-tion procedure, developed long wave solutions to the equations of Gouyon.

The problem has also been solved by perturbation methoès in

Lagrangian coordinates by Kravtchenko and Daubert (1957). They devel-oped a third order theory for waves with closed water partiele

tra-jectories. Their work extended Gerstner's classic work (1802)on a

rotational wave theory, yet also results in a vorticity in the fluid

opposite to that expected in nature. Their theory predicts no net

transport of the fluid particles, contrary to what is observed

18

II-C. Boundary Conditions on the Fluid Domains

For each domain of definition of the wave problem, boundary conditions must be prescribed to fully specify the mathematical problem.

These boundary conditions are quite similar for all the techniques and

will be developed here.

Thc first two conditions are relatively straightforward.

First, no flow is allowed into thc bottom, which is assumed horizontal. The na flow bottom boundary condition is specified as

v

+ v

=

0 at the bottom. (ll-18)

This condition is not generally true in nature as the presence of the wave may induce flow into the usually permeable bottom. However, the effect of bottom percolation, while important over long distances in reducing wave energy, is locnlly assumed insignificant. Next, as a wave solution is desired, periodicity is imposed on the motion. For example,

u(x)

=

u(x + L) (I1-19)

where L is the wave length.

At the free surface, denoted as ~(x), two conditions are generally specified, as the location of the free surface is not known

a priori, in contrast to the bottom which is fixed. One boundary

con-dition, the dynamic free surface boundary condition, is necessary to

specify it as a constant pressure surface and the seconct condition, the kinematic free surface boundary condition, requires that this sur-face be a streamline.

19

For determining the first of these condltions, it is necessary

to develop the Bernoulli equation for rotational flow, with the motion

rendered stationary by moving the coordinates with speed, C. The Euler

equations written in terms of the stream function are

~y~yx _*

*

1

-

= -

_

_Px x yy p -~y~xx +

~ V

1

-P Py-g xxy (I1-20) (I1-21)

lf Equation 11-11 is used, these equations may be rewritten as

Qx

=

*

_xf(~)/g 2 2

*

+

*

Q =

*

fq)/g where Q=_E_+2 __ y + Y y _Y pg 2g (I 1-22) (I1-23)

and Q is defined as the Bernoulli constant. li Equation 11-22 is multiplied by

-

y

and Equation II-23 by .~, arid they ar c added

y x

together, the result is

-

*

_{y x}Q + ~_xQ_y = 0 (I1-24)

or Q is a constant along a streamline. This can be stated as Q = Q(*) only. The Bernoulli equation is then stated as

1jT2 + ~2

_

p + x y

= + y .

pg 2g (II-25)

Therefore, substituting back into Equation 11-22, and using the chain rule,

=

*

f(~)/g

x (II-26)

20 1ntegrating, ~r 1

J

I I Q(~) = - f(~ )d~ + Cl g 0

cr

1-27)

TI1erefore,for two-dimensional steady flow, the Bernoulli constant may be specified to within a constant if f(~) is known for each stream

-line. As an example of the Bernoulli equation, if f(V) was equal to

V

constant, -w , then Q(~)

=

-w -

+ Cl' and-the Bernoulli equation, valid

o og

throughout the flow field, would be written,

p pg 2 (U+u-C) + --- + y + 2g 2 + v Ulo~ g (11-28)

---

=

For application as the free surface boundary condition, specifying

that the free surface be a constant pressure surface (that is,

atmospheric pressure) , the Bernoulli equation is rendered into the

following form along the free surface, ~.

2

11

+ (U+u-C) 2g 2 + v

-= Q (II-29)

The final boundary condition to be imposed requires that the free surf ace be a streamline. This may be stated as

(U+u-C)

nx

=

v on

11 •

(II-30)

This condition is s2tisfied exactly by all of the forms of the problem

in the previous section; however, the utility of the condition is in

prescribing the value of the free surface streamline and, therefore,

an alternate free surf ace condition is imposed. "I'hLs condition requires that the presence of the wave does not change the location of the mean

21

sea level; that is, the amount of fluid in the prescribed domain is

the same with or without a wave. This mean sea level constraint is specified as

1 L

L

J

T](x)dx = O.

o

CHAPTER 111

SMALL N,lPLITL'DE WAVES ON LlNEAR AAm BILI:r.'EAR CURRENTS

The simplest approxlmation to an ambient current velocity

profile is a straight linei that is,

U(y)

=

w

(y+h) + U ,

o 0 (III-l)

where

w

is related to a constant vorticity throughout the fluid and

o

U' is the magnitude of a uniform (over depth) component of the current.

o

A better fit is achieved if the eurrent ean be assumed to be repre

-sented by two straight lines, as shown in Figure III-lc.

(

a

)

Design Current Profile (b) Lineor Approximotion (c) Bilinear Approxi mation FIGURE

:m: -

I

L!NEAR

CURRENT

ANO

BI

LI

N

EAR

APPROXIMATION TO DESIGN PROFILE

23

In this chapter, small amplitude wave models are developed for these linear approximation profiles. In the succeeding chapters, nonlinear,

or finite amplitude wave theories will be developed for design situa

-tions where the wave heights are large. lIl-A. Linear Shear Current

The governing differential equation for this case is

-

w

o 011-2)

which is valid in the region

o

<x <L, -h <y <0. On the boundaries

of this region, the four boundary conditions of Chapter 11 are imposed:

no flow through the bottom, the motion is periodic in space, and the two free surface boundary conditions must be satisfied, but only to

the first order i~ wave slope, as will be seen. .Adefinition sketch

of the wave is ShO\V11in Figure 111-2 to illustrate the notatLou,

As the governing differential equation is linear, a solution

of the following form is assumed.

~(x,y) = -(1)

o

2 (y+h)

2 -(U -C)yo + *<x,y) (III-3) where'W is the stream function associated with the irrotational wave motion. The free surfaee displacement, ~(x),is assumed to be

1')(x)

=

H cos kx .

2 (III-4)

The kinematic free surface boundary condition (KFSBC),

Equation 11-30, whieh is nonlinear due to the presenee of the product

term, (U+u-C) 1'),

x x is simplified by the assumption of a small amp

x

u

J

>

j__L__-o

>-24

z

UJ 0::: W

o

Cl) LL. Cl) :I: Cl

U

W

~

..>

~::>

~

Cl)

a-:

Z

z

w

OW:I:

-

et::

~

~

gs

Z

U::c

-

~ LL. _

~~

3:

C\J I ~ W ct:

::>

(!) LL.

25

that all terms involving H/L to powers greater than one are relatively small and therefore may be neglected. Thus, the KFSBC may be

expressed as

(U+U1-C)~

=

v

= ~

o x x on y

=

°

(111-5)

where U1 == Woh, the velocity increment to the uniform flow due to the shear curr-cnt at the still water level. Note that this form of the KFSBC is only valid for small waves.

The dynamic free surface boundary condition (DFSBC), Equation 11-29, must also be linearized. Equation 11-29 applies strictlyon y

= ~;

however, in this problem the domain extends upwards only to

y

=

0, and therefore analytic continuation in the form of a Taylor series is used, similar to the procedure of Stokes (1847), to extend the

domajn to the free surface, ~. lherefore, the D~SBC 15 written as

11

+

=

Q (1 II-G)

where the superscripts mean that the bracketed terms are evaluated on y

=

0. Linearizintrwith respect to H/L, and retaining only the no

n-constant terms, the linearized dynamic free surface boundary condition

results. (U + U1 - C)W

11

0 y = g+ CU +U1 - C) U o y on y

=

0 . (III-7)

Differentiating with respect to x and substituting into Equation 111-5 yields

2

(U +U - C)

Y

=

y

Cg + (U +U1- C) U ) on y

=

0,

o 1 x-y x 0 y

26

This is the linearized combined free surf ace boundary condition which is applicable for any small amplitude wave problem regardless of the currer..t profile, U(y), provided Uo+U1 is interpreted as the value of the ambient current at the still water level, y

=

O. To specialize the condition to the problem of linear shear current, U =(J) and

y 0

U =(J) hare introduced. -1 0

The final solution, first derived by Thompson (1949) in a different manner, and then later by Biesel (1950) and Sun-Tsao (1959), is ~(x,y)

=

- (U -C) y - (J) o 0 2 (y+h) 2 H (Uo+Ul-C) + -2 sinh k(h+y) sin kx sinh kh (IIl-9)

with the dispersion relationship (obtained from the combined linear free surface boundary condition) which must be satisfied by k, the wave number (k

=

2niL) ,

(IIl-IO)

This equation is nonlinear in k, or alternatively, L, and can be solved by iteration. This will be deferred until Chapter IV, when a compar-ison of nonlinear and linear mode~s is made. For the case of zero vorticity, i.e., (J) =0, both *(x,y) and the dispersion relationship

o

reduce to that for irrotational waves on a uniform flow (cf., Lamb (1945), page 375).

The presence of a linear shear current causes a change in the dynamic pressure under the wave, when compared to irrotational flow.

Sun-Tsao (1960) showed that, even in the absence of gravity, there is a restoring force to cause wave propagation. To examine the pressure

27

under these waves, the Bernoulli constant in Equation 11-28 can be evaluated at the free surface. In this case, Cl is found (to the first order in RIL) to be

- p

At any location in the fluid, then, the pressure can be found from the Bernoulli equation by applying i~ between the free surface and

,

the elevation at which the pressure is desired. Carrying out the mathe-matics, the pressure at an elevation (y) is given by the following

equation p(y) - pgy + p 'Tl cosh k(h+y)

u

(U +U ,...C) 1 0 1 )

J

+ h- (yk cosh k(h-sy ) - sinh k(h-s y ) . sinh kh \ (Ill-II)

III-B. I3ilinearShear Current

To represent the ambient current with a bilinear current profile, as shown in Figure lIl-Ic, it is necessary to separate the

domain of the wave into two separate regions, land 11, each with a different, but constant, vorticity. This is shown in Figure 111-3

For each region, Poisson's equation is specified as

o

:5: x :5: L; -h<y< -d (IlI-12)

N :::::>

+

:::::>

+

0 0 0 :::> :::::> :::::> X ~ 0:: ~ _.J

<!

1.LJ

_>

:::x:

Cl::

_w

Cf)

z

CO

0

₀

L1J

>-~ CD

?:

z

w

<t

L1JCf) L1J 0:: _Cf) ~

Ft

H ~

0 <t

~ ....J IJ... c: c:

.

2

0

::r:

..

w

0'1 0'1

...

:c

Q) Q) U

Z

l-a:

a:

...

_w

w

Cl:: ~ _0:: Cf) _:::::>

:c

U

...

z

3:

0

Cl::

...

<t

(.!') L1J

Z

_Z

Z

IJ... _...J

>

L1J

-

0

0 al

:ë

>-

0

r<>

~.

w

0:: :::::>

5:2

IJ... 28

29'

At the free surface of the water, the same linear free surface con di-tions given in Section III-A have to be satisfied by the stream func

-tion in the upper region, VII' At the bottom, the no-flow condition

must be met by Î' Further, at the interface of the two regions , I'

y = - d + " ~ - d, wh ere , represents the displ acement of the interface from its mean position due to the presence of the wave, the pressure must be the same on both sides. If it is specified that uI

=

uIl and vI

=

vII on the tnterface streamline, the pressure will be a constant, as can be readily seen fro~ the Bernoulli equation (Equation 11-25).

Therefore,

*

I

=

y

II Y y W

=

.1, on y

₌

- d+S I "II x x (III-14) (III-15)

Also the interface must be a streamline; therefore a kinematic

condition must also be prescribed (from Equation 111-5) = (U +U1-C) ,

o x on y = (III-16)

Next, it is assumeu that the form of the stream function for each region is as follows:

2

,1, (x y) = - (U -C)y - U (hy +y /2) + D sinh k(h+y) sin kx

'1'1' 0 1 (h - d) U2 y2 ,_'1'11'1, (x y)

=

-

(U +U +U -C)y - 2d + (A sinh ky + B cosh ky) sin kx . 0 1 2 (III-17) (III-18)

30

Here Uo is the eurrent velocity at the bottom, Ui is the veloeity

increment .at y= -d, and U2 is the veloei ty inerement at the still water level, By definition,

U

2/d

=

wIl; Ul/t

=

WI' Further it is assumed that the free surfaee, ~, may be written as

11

=

H/2 sin kx (IIl-19)

and

C

as

C

-= b/2 sin kx , (IIl-20)

From Equation lIl-S, the line2rizeàKFS3C yields

(IlI-21) From Equation 111-7, the linearized DFSBC yields H/2 (I11-22) and eombining the two relates B to A, (III-23) At the interface, Equation 111-5 must be satisfied

Dk sinh kCh-d) eos kx

=

Dk sinh kt eos kx

(III-24)

or

D

=

(Uo+U1-C)b

31

AIso, the vertical velocity must be continuous across the interface,

~II x

=

~I

x

on y

= -

d+

ç

(III-26)

(-A sinh kd + B cosh kd)k cos kx

=

Dk sinh k(h-d) cos kx or

- A sinh kd + B cosh kd ~ D sinh kt .

Finally, the horizontal velocity is forced to be continuous across the

interface, or

-

~

_I Y

= -

*

II

Y on y (III-27) U1b (U C) + U + sin kx - Dk cosh kt sin kx 0- 1 2(h-d) or Ulb - Dk cosh kt = 2(h-d) U b 2 _ Hk cosh kd + Bk sinh kl,. 2d

Substituting for b from Equation 111-25

[ ((h-d)Ui -

ct

U2)((sinU+Uh kt-C)) - k cosh kt

J

D

o 1

= - Ak cosh kd + Bk sinh kd. (IlI-28)

In this equation, the coefficient D in turn can be expressed in terms

of A and B from Equation 111-26, and B in terms of A from Equation

32

(IIl-29)

The resulting solvability condition, then, becomes independent of A,

which, after some algebra, is

tanh

k

(lIl-30)

This dispersion condition relates the depths of the two regions and

the current profile to the wave length of the wave propagating on the

current. This result is a generalization of the results of G. I. Taylor

(1955) who, while investigating the case of wave breaking by bubble

breakwaters in infinitely deep water, solved the case for Uo=UI

=

0,

.that is, the case for a shear current only in the upper layer. Further,

it is an extension of the earlier work of Thompson (1949) who treated

the special case of a linear shear layer in the lower regions and

a uniform velocity in the upper 1ayersi U

=

U

=

0. Binnie and

o 2

Cloughley (1971) solved the problem of stationary waves on a similar

current, that is, further specializing to the case where C

=

0. All of the previous special cases may be found from Equation 111-30. Further, this condition will reduce to the condition for a single linear shear

33

in other words, letting Region I expand to the surface. Also, by

allowing the vorticity in the lower layer to approach infinity as t goes to zero, that is, allowing the lower Region I to collapse onto the bottom, the case for a uniform (over depth) current results, as given by Lamb, i.e.,

(lII-31)

From Equations 111-21, 111-22, and 111-27, the stream functions

tand

t

can be written in terms of the wave height, H. I II 2 UI (hy+ (~)) (U0-C)y - (h _ d) + Hr

- L

(

U

+U +U -C) k cosh kd 2 0 1 2 ~_I (g+ (U +U1+U2-C) (U2/d)) ] o sinh kd sLnhkïh-y ) sin kx, -h<y<-d keU +U1+Ur,-C) o ... (III-32) 2 u2y t (x y) = -(U +U +U -C)y ---II' 0 1 2 2d (III-33)

which, with the solvability condition, Equation 111-30, completely

specifies the problem to within an arbitrary constant, the wave height.

It is possible, by dividing the ~luid into numerous regions,

to fit a current profile with any number of straight line segments.

34

procedure, except that stream functions of the form of Equation·III-18

would exist in each region, with the coefficients determined by the

interface at the top of the region. The mathematics would get extremely

CHAPTER IV

NONLI}j'"EARWAVES ox A LI]\"EARSHEAR CURRETI

IV-A. Theoretical Development for

Col inear \'iavesand Curren ts

In design, it is desirabie to specify a wave on an assu.aed

linear current profile by the following characteristics: wave height,

H, wave period, T, water dept h ,'h, aridthe current parameters, which are the constant vorticity, -'J) , and a uniform current component, U. For

o 0

the linear wave models in the last chapter, this was possible; however, these waves were restricted to small heights, and, in fact, the th~ories

pre~ict waves which possess su~face profiles,

n,

which va~y in space as the trigonometric sine or eosine. These waves are, therefore, not entirely realistic as waves in nature have peaked crests and flat troughs. For more realistic waves it is necessary to use higher order

wave theory to describe large waves. Chappelear (1961) and Dean (1965a) have developed numerical perturbation procedures for computer calcula-tion of irrotational waves of aoy arbitrary order. Of the two methods,

Dean's procedure is more efficient in that there is only one set of

unknown coefficients. As it involves the use of the stream function, the theory can be modified to accon~odate currents with constant

vor-ticity. It has been shown in Sun-Tsao's theorem in Chapter 11 that the Stream Function wave theory cannot be used for more general

vor-ticity distributions.

36

A disadvantage of the Stream Function procedure is that it does not converge on wave height, but rather a specified Bernoulli constant.

This requires that the procedure be used several times to generate a wave of desired height. Through use of a Lagrange multiplier

approach, this drawback can be eliminated and tue procedure for linear shear currents is developed below to converge on wave height.

The boundary value problem was presented in Chapter 11 and will be repeated here for convenience.

The governing differential equation (Equation 11-12) is

=-(.1);

o -h:::;y:S:'Tl;

In contrast to the small amplitude wave model, the fluid domain extends directiy up to the free surface, ï].

r.:e

botto:::boundary cor..dition(Equation 11-18) is

~_x = 0 on y = - h

and the dynamic free surface boundary condition (DFSBC) (Equation 11-29) is

'2 + J,2)

C

1

'!'

'Tl + _x--::-___::..y_

2g Q on y = 'Tl

and must be satisfied. The mean sea level constraint (Equation 11-31) is specified, also.

1 L

L

J

'Tl dx = 0 . o

1t is assumed that a stream function of the following form is a series representation of the exact solution to the boundary value

37

problem. It can be sbown readily tbat tbis stream function is an exact solution of tbe governing differential equation.

2 W (b+y) o (U-oC)y - --2--+ l','N+1

t

n::::2 2n(n-l)(h-sy) X(n) si

rul

_ _;__..:...;_.::...;:_

L cos 2TT(n-1)x L (IV-I)

Tbe form of tbe stream function is similar to tbat used in tbe preceding cbapter with tbe exception of the series terms on tbe right

-hand side, which for n

=

2, would be approximately the same as

Equation III-9. Tbe X(n) s (2::Çn::Ç}l'X+l),tbe stream function coeffi

-cients, are unknown a priori, and for computational convenience, tbe

otber unknown s , Land y(x,ï]) are defined as X(l) and X(l'.'N2).+ Tbe

constant }.")i, is the order of tbe wave tbeory, as tbere are NN terms

in the trigonometrie series for the stream function.

Tbe assumed form of ~(x,y) satisfies two of the boundary condi

-tions exactly; it is periodic in space over a distance, L, and tbe vertical velocity vanishes at the bottom. Tbe X(n)s, tben, must be

cbo~en such tbat tbe dynamic free surface boundary condition and the

mean sea level constraint are best satisfied.

A

measure of how weIl the DFSBC is satisfied can be prescribed

as

I

I: (Qi -

Q

)

2 i=1

(IV-2)

where El would be zero if tbe Bernoulli constant, Qi' evaluated at I

discrete points along the wave profile, was equal to the average

38

-Q

=

(IV-3)

The free surf ace of the wave, necessary for the determination of Q, is found from Equation IV~l by substi tuting y =

n,

and solving for

Tl.

Tl =

::\:;+1 + 2U1h .L: n=2 2n(n-l) (h+71) X(n) sinh ---~~~ L 1,. cos kX)2. (IV-4)

Note that

n

must, in fact, be determined by iteration due to the presence of

n

in the square root expression.

The solution for the X(n)s proceeds by first assuming a trial set of X(n)s. It is convenient to use the results of the linear model in Chapt~r IIl for the first two coefficients; that is, X(l)(=L) is

H (Uo+Ul-C)

found from Equation III-10, and X(2) =

'2

sinh kh by inspection of Equation 111-9. The starting coefficients are not critical in most cases; for instance, the values obtained from Airy's theory without a current can often be used. The remaining coefficients are set to zero. The value of El obtained by these original coefficients would be quite large, and a nonlinear perturbation procedure involving iter-ation similar to that used by Dean (1965a) and Chappelear (1961) is used to improve these X(n) values. The perturbation procedure can be specified as a nonlinear least squares procedure on El with linear con-straints to ensure the still water level is not changed and that the specified wave height is achieved. The procedure then involves

39

minimizing, wi th r-espoct to the X(n)s, the following objective function,

j+1 t 1

o ,a iteration j+ , using a Lagrange multiplier approach.

j+1

o

(IV-5)

There are two important points to note here about the objective

func-tion. First, the Lagrange multiplier approach allows the incorporation

of the zero still water level and the wave height constraints directly

into the minimization procedure, as opposed to the less direct method

of Dean. Second, as a result of sy~netry in the wave about its crest,

the problem need only be specified from 0 ~ x~L/2. The objective func-tion is quasi-linearized by expanding it in a first order Ta ylor series

involving the X(n)s at iteration j.

~-X+2

z

neL

eo

j) X'en) X(n) (IV-6)

,

where X (n) is an incremental change in X(n).

The expanded objective function is then minimized with respect

to the X(n)s and also with respect to the additional unknowns, ~ and

(oj+l)

=

0 for m

₌

1,NN+2

}

X(m) (IV-7) (Oj+l)

=

0 for 1- = 1,2 ~

40

The minimization with respect to the X(m)s, Equation IV-7, gives

equations of the form:

2À1 L/2 ( )

+

L

J

T]x(m) dx + r'2 (T](0» X(m) - (T](L/2» X(m)

o

(IV-8)

The evaluation of (Qi-Q)xCm) (= CQi)XCm) - QXCffi»)follows by

first evaluating CQi)X(m), using the chain rule of differentiation on

Equation II-29,

(IV-9)

Then, using the definition of Q, the value of Q. is obtained by

"X(m)

summation,

1 I

~(ffi)= I

r:

(Q.)

i=l 1. X(m)

Minimization with respect to

~ and ""2yields (IV-1Q) NN+2 L/2 . 2: .

x

'

(n)

f

n=l .0 1'1

x

(n) dx L/2

=

-

f

1'1 dx (IV-1l) o :NN+2 L: n=l

41

Combining these equations in matrix form results in an (NN+4) square

matrix which must be inverted for the X'(n)s, ~ and À2. The resulting

,

X (n)s are then added to the X(n)s at iteration j to give a better fit

to the DFSBC. Due to the nonlinearity in the problem, the procedure

must be iterated several times for adequately small values of El'

Once the best fit X(n)~ are determined, the values of parameters within

the wave are readily determined through the use of the stream iunction,

Equation IV-l.

As an example of the stream function solution for a given wave

height, Figures IV-1 and IV-2 depiet the·streamlines under a 2-foot-high

wave in 10 feet of water. The different values of the streamlines re

pre-sent the difference in the flow rates between the streamlines for the

two different cases, due to the addition of a current for the second case. The different values of the surface streamlines reflect, for the .

first wave, the mass transport of water associated with the wave, and,

for the second wave, the value of *(x,~) represents the mass transport of the wave minus the volume transport of the current in the opposite direct ion of the wave.

IV-B Modifications for Currents and Waves

in Different Directions

For a steady current flowing in a direct ion other than that of

the wave, it is still possible to formulate the problem with a stream

function despite the three-dimensional nature of the flow. Form~lly, a stream function arises through the continuity equation which in