Computation of the nonlinear energy transfer in a narrow gravity wave spectrum with a method derived by Dungey and Hui

Delft University of Technology

Department of Civil Engineering

Fluid Mechanics Group

846374

Computation of the nonlinear energy

transfer in a narrow gravity wave

spectrum with a method derived by

Dungey and Hui

G.Ph. van Vledder

COMPUTATION OF THE NONLINEAR ENERGY TRANSFER IN A NARROW GRAVITY WAVE SPECTRUM WITH A

METHOD DERIVED BY DUNGEY AND HUI

G.Ph. van Vledder

August 19~4

Delft University of Technology

Department of Civil Engineering

Fluid Mechanics Group

CONTENTS 1 • 2. 2.1 2.2 2.3 2.4 3. 3.1 3.2 3.3 4. 4.1 4.2 4.3 4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.4.5 5. 5.1 5.2 6. INTRODUCTION

SUMMARY OF AND EXTENSION TO THE METHOD OF DUNGEY AND HUI The coupling coefficient G

Representation of the local action density spectrum Reduction to single integrations

Computation of the one-dimensional energy transfer rate

NUMERICAL ASPECTS

Numerical evaluation of the single integrations Computation of k and E 0 Fit procedure RESULTS Introduction Description of testcases Results of the fit procedure

Computations of nonlinear transfer Transformation of transfer rates Testcase

Case RC3 Case 15 Case 25

COMPUTATIONAL ASPECTS

Accuracy and required computertime The one-dimensional transfer rate

CONCLUSIONS LIST OF SYMBOLS LITERATURE APPENDICES Figures page 1 3 3 4 4 5 7 7 8 8 11 11 11 13 15 15 15 16 16 17 18 18 19 20 21 22 24

1. INTRODUCTION

Nonlinear wave-wave interactions play an important role in the evolution in time and space of the directional energy spectrum of wind-generated surface waves. Calculations of the nonlinear energy transfer showed that they tend to stabilize the spectral shape (Hasselmann et al. 1973).

A general perturbation theory for the nonlinear resonant interaction of free waves in a random sea was developed by Hasselmann (1962, 1963a). His result for the contribution of the nonlinear wave-wave interactions in deep water to the rate of change of the action density spectrum n(~,t) can be written as:

00

(1)

where n

1

=

n(~i,t), ~ the two-dimensional wave number, related to the frequency w by wi

(gj~ij)~,

and g the gravitational acceleration. The Dirac a-functions in (1) reflect the resonant conditions given by Phillips

(1960). The coupling coefficient G is a complex function of the wave numbers

~

₁

, ~

₂

, ~

₃

and ~

₄

• The action density spectrum n(~) is related to the wave number spectrumS(~) by n(~)

=

gS(~)/w. To date no non-trivial solution to equation (1) has been found.

Numerical solutions to (1) were obtained by Hasselmann (1963b), Webb (1978), Resio and Tracy (1982), Masuda (1980) and Hasselmann and Hasselmann (1981). However, these solutions are very time consuming and therefore not applicable for usage in wave prediction models. They can be used in the study of the effects of nonlinear interactions to the directional energy spectrum.

Longuet-Higgins (1976) showed that the coupling coefficient G between four nearly equal wave numbers~, ~

₂

, ~

₃

, ~

₄

is finite and not zero. This

result implies that the exchange of energy within the peak of the spectrum is of dominant importance. This theory was extended by Fox (1976) who showed that for a spectrum which is a delta spike with normal (Gaussian) structure the sixfold integral in (1) reduces to a single integration, reducing the computer

-1-time needed and placing the numerical error under control. Dungey and Hui (1979) extended the work of Longuet-Higgins and Fox to include the effect of the width of a narrow gravity wave spectrum on the nonlinear resonant

interactions. This is done by perturbing the coupling coefficient G for a narrow spectrum and retaining only first order terms (in spectral width). Dungey and Hui showed that for an action density spectrum which can be represented as a sum of Gaussian terms the integral (1) reduces to a set of single integrations. A disadvantage of this method is that the required computertime increases with the third power of the number of Gaussian terms used. An other disadvantage is that it can only represent spectra which have a symmetrical directional distribution.

However, it will be shown in the present report that it is possible to extend the method of Dungey and Hui for spectra with non-symmetrical directional distribution.

In this report the method of Dungey and Hui will be summarized and extended to include the effect of skewness of the directional distribution. Additional information will be given about the derivation and computation of the single integrations mentioned above. The method of Dungey and Hui is applied to a number of theoretical spectra. The results of these computations will then be compared with the results of exact computations by Sell and Hasselmann (1972) and Hasselmann and Hasselmann (1981). Finally, some comments will be made about computational aspects of the method of Dungey and Hui.

-2-X (6b)

N(,bi) ( 6c)

2

E t (6d)

andcS() is the DiraccS-function.

2.2 Representation of the local action density spectrum

Dungey and Hui assumed that the local action density spectrum can be expressed as a sum of s Gaussian terms:

N(,b) s

I

j=l 2 2 R . exp[-~P (1--U.) - Q.l-J] J r J J . (7)

where R., P.,

Q.

> 0 for j=1, ... ,s. The parameter U. is used to shift the

J J J J

position of the peak of a Gaussian term along the k -axis. Using expression

X

(7) it is only possibly to represent action density spectra which are symmetric round the !--axis and thus in 8. When (7) is used with only one

Gaussian term (s

=

1), the action density spectrum is also symmetric round the line A

=

u

1• Contours of this spectrum will consist of concentric ellipses. Using more than one Gaussian term it is possible to represent action density spectra which can be asymmetric in k -direction.

X

With a small extension of (7) it is possible to represent spectra which may also be asymmetric in k -direction. We now assume that the local action

y

density spectrum can be written as

N(L)

..

(8)

where R., P . , Q . > 0 for j

=

1, •.• , s.

J J J

The parameters U. and V. are used to shift the position of the peak of a

J J

Gaussian term relative to the wave number k . _-o

2.3 Reduction to single integrations

In the following we assume that the local action density spectrum may be expressed as a sum of s Gaussian terms, namely

-4-Using the single integration expressions developed for 1₁, _{1 2 , 13 and 14 ,}

it is possible to perform the ~l integration of the equations (13) (see Appendix C).

So for action density spectra of the form (9), the local one-dimensional energy transfer rate may be expressed as a set of single integrations to order

2

E by

(14)

-6-3. NUMERICAL ASPECTS

3.1 Numerical evaluation of the single interactions To obtain numerical values of the integrals I

1, I2, I3 and I4, standard numerical integration routines can be used. Advantages of these methods are

that they are simple to use and that the accuracy can be controlled.

Disadvantages of these methods are that the number of points needed to achieve a certain accuracy can become very great. Further, most of these routines work as a black box.

Tests were made to gain insight in the computertime and the number of function evaluations needed to compute the integrals to a certain accuracy. It became clear very soon, that the available standard routines are very time consuming. Therefore a numerical integration method was developed which can control the numerical accuracy and the number of function evaluations. This resulted in a numerical integration method of the Gauss-Lobatto type, which will be briefly explained in the following.

Numerical integration routines can nearly always be written in the form:

b / f(x)dx a n

L

wi f(xi) i=l

where the n values w. are the weights to be given to the n functional values

1

f(x.). The points x. are usually equally spaced so there is no choice in the

1 1

selection of these points. If the x. are not fixed and if we place no other

1

restrictions on them, it follows that there are 2n undetermined parameters, which might apparently suffice to define a polynomial of degree 2n-l. Formulas which are determined in such a way, with unequal spacing of the xi, are of the Gauss-type. An n-point formula of the Gauss-type will integrate poly-nomials up to degree 2n-l exactly. (Carnahan, Luther and Wilkes, 1969). Integration formulas of the Gauss-type which have pre-assigned points at the beginning and end of the integration interval are called Gauss-Lobatto

formulas.

The numerical accuracy of an integration method can be controlled as follows. The first step in such a method is to compute an approximate value of the