Study on design of 2d ocean wave maker

HO CHI MINH CITY NATIONAL UNIVERSITY

UNIVERSITY OF POLYTECHNIC
THE FACULTY OF MECHANICAL ENGINEERING

THE PFIEV PROGRAM

Graduate thesis

STUDY ON DESIGN OF 2D OCEAN WAVE MAKER

Advisor: Assoc Prof PhD Nguyễn Tấn Tiến
Done by: Nguyễn Hồng Quân
Student code: 20402050

Ho Chi Minh City 2009

I. INTRODUCTION
The wave generation using a wave maker in a test basin has then become an important
technology in the field of the coastal and ocean engineering. To date most laboratory testing
of floating or bottom-mounted structures and studies of beach profiles and other related
phenomena have utilized wave tanks, which are usually characterized as long, narrow
enclosures with a wavemaker of some kind at one end; however, circular beaches have been
proposed for littoral drift studies and a spiral wavemaker has been used. For all of these
tests, the wavemaker is very important. The wave motion that it induces and its power
requirements can be determined reasonably well from linear wave theory.

Wavemakers are, in fact, more ubiquitous than one would expect. Earthquake
excitation of the seafloor or human-made structures causes waves which can be estimated
by wavemaker theory; in fact, the loading on the structures can be determined. Any moving
body in a fluid with a free surface will produce waves: ducks, boats, and so on.
Wavemakers are also used in experimental wave basins to measure wave effects on various
types of structures and vessels, including models of ships, offshore platforms, and other
bodies.

The theory of water waves has attracted scientists in fluid mechanics and applied
mathematics for at least one and a half centuries and has been a source of intriguing - and
often difficult - mathematical problems. Apart from being important in various branches of
engineering and applied sciences, many water-wave phenomena happen in everyday
experience. Waves generated by ships in rivers and waves generated by wind or earthquakes
in oceans are probably the most familiar examples. The mathematical theory of water waves
consists of the equations of fluid mechanics, the concepts of wave propagation, and the
critically important role of boundary conditions. The results obtained from theory may give
some explanation of a natural phenomenon or provide a description that can be tested
whenever an expanse of water is at hand: a river or pond, the ocean, or simply the
household bath or sink. However, obtaining a thorough understanding of the relevant
physical mechanisms presents fluid dynamicists and applied mathematicians with a great
challenge

In practical use, two types of wavemakers which use a paddle with two type of moving
to produce water wave are the most popular. They are so-called piston-type and flap-type
wavemaker.

Figure 1.1. A piston-type wavemaker (HR Wallingford Ltd)

Figure 1.2. HR Wallingford's multi-element wavemakers

Figure 1.3. Edinburgh Designs Ltd’s flap type wavemaker

Figure 1.1. Some DHI Group’s wavemakers

Figure 1.2. An engineering testing tank with the use of wavemaker

II. WAVEMAKER THEORY FOR PLANE WAVES PRODUCED BY A PADDLE
[1]

Assumptions:

1) The fluid is incompressible, irrotational. The water displaced by the wavemaker
should be equal to the crest volume of the propagating wave form (Figure 2.1).
2) The paddle moves with small amplitude and the wave height is small.
3) The water wave propagates in direction which tends to infinity. By using a wave
absorber at the other end of the basin, we can consider it equivalent.

Figure 2.1

1. BOUNDARY VALUE PROBLEMS

In formulating the small-amplitude water wave problem, it is useful to review, in very
general terms, the structure of boundary value problems, of which the present problem of
interest is an example. Numerous classical problems of physics and most analytical
problems in engineering may be posed as boundary value problems; however, in some
developments, this may not be apparent.

The formulation of a boundary value problem is simply the expression in mathematical
terms of the physical situation such that a unique solution exists. This generally consists of
first establishing a region of interest and specifying a differential equation that must be
satisfied within the region. Often, there are an infinite number of solutions to the differential
equation and the remaining task is selecting the one or more solutions that are relevant to
the physical problem under investigation. This selection is effected through the boundary
conditions, that is, rejecting those solutions that are not compatible with these conditions.

Figure 2.2

For the geometry depicted in Figure 1.1, the governing equation for the velocity
potential is the Laplace equation:
(2.1)

Kinematics boundary conditions

At any boundary, whether it is fixed, such as the bottom, or free, such as the water
surface, certain physical conditions must be satisfied by the fluid velocities. These
conditions on the water particle kinematics are called kinematic boundary conditions. At
any surface or fluid interface, it is clear that there must be no flow across the interface;
otherwise, there would be no interface.

The mathematical expression for the kinematic boundary condition may be derived
from the equation which describes the surface that constitutes the boundary. Any fixed or
moving surface can be expressed in terms of a mathematical expression of the form
. If the surface vanes with time, as would the water surface, then the total
derivative of the surface with respect to time would be zero on the surface. In other words,
if we move with the surface, it does not change.

(2.2a)

or

(2.2b)

where the unit vector normal to the surface has been introduced as

Rearranging the kinematic boundary condition results:

(2.3)

where

This condition requires that the component of the fluid velocity normal to the surface be
related to the local velocity of the surface. If the surface does not change with time, then
; that is, the velocity component normal to the surface is zero.

The Bottom Boundary Condition (BBC)

The lower boundary of our region of interest is described as (horizontal
bottom) for a two-dimensional case where the origin is located at the still water level and
represents the depth. If the bottom is impermeable, we expect that , as the bottom
does not move with time.

The surface equation for the bottom is . Therefore

(2.4)

Kinematic Free Surface Boundary Condition (KFSBC)

The free surface of a wave can be described a s, where
is the displacement of the free surface about the horizontal plane . The
kinematic boundary condition at the free surface is

(2.5)

where

Carrying the dot product yields

(2.6)

Dynamic Free Surface Boundary Condition

A distinguishing feature of fixed (in space) surfaces is that they can support pressure
variations. However, surfaces that are “free”, such as the air-water interface, cannot support
variations in pressure (neglecting surface tension) across the interface and hence must
respond in order to maintain the pressure as uniform. A second boundary condition, termed
a dynamic boundary condition, is thus required on any free surface or interface, to prescribe
the pressure distribution pressures on this boundary.

As the dynamic free surface boundary condition is a requirement that the pressure on
the free surface be uniform along the wave form, the Bernoulli equation with
is applied on the free surface .

(2.7)

where is a constant and usually taken as gage pressure, .

Lateral Boundary Conditions

Consider a vertical paddle acting as a wavemaker in a wave tank. If the displacement
of the paddle may be described as , the kinematic boundary condition is

where

or, carrying out the dot product,

(2.8)

which, of course, require that the fluid particles at the moving wall follow the wall.

2. SOLUTION TO LINEARIZED WATER WAVE BOUNDARY VALUE PROBLEM FOR A
HORIZONTAL BOTTOM

Solution of the Laplace equation

A convenient method for solving some linear partial differential equation is called
separation of variables. For our case

(2.9)
For waves that are periodic in time, we can specify . The velocity potential
now takes the form

(2.10)

Substituting into the Laplace equation and dividing by gives us

(2.11)

Clearly, the first term of this equation depends on alone, while the second term depends
only on . If we consider a variation in in Eq. (2.11) holding constant, the second term
could conceivably vary, whereas the first term could not. This would give a nonzero sum in
Eq. (2.11) and thus the equation would not be satisfied. The only way that the equation
would hold is if each term is equal to the same constant except for a sign difference, that is,

(2.12a)

(2.12b)

The fact that we have assigned a minus constant to the term is not of importance, as we
will permit the separation constant k to have an imaginary value in this problem and in
general the separation constant can be complex.

Equations (2.12) are now ordinary differential equations and may be solved separately.
Three possible cases may now be examined depending on the nature of ; these are for
real, , and a pure imaginary number. Table 2.1 lists the separate cases. (Note that if
consisted of both a real and an imaginary part, this could imply a change of wave height
with distance, which may be valid for cases of waves propagating with damping or wave
growth by wind)

Table 2.1 Possible Solutions to the Laplace Equation, Based on Separation of Variables

Character of k, the Ordinary Differential
Solutions
Separation Constant Equations

Real

Imaginary

Linearization of dynamics free surface boundary condition

The Bernoulli equation must be satisfied on , which is a priori unknown. A
convenient method used to evaluate the condition, then, is to evaluate it on by

expanding the value of the condition at (a known location) by the truncated Taylor
series.

where on .

Now for infinitesimally small waves, is small, and therefore it is assumed that velocities
and pressures are small; thus any products of these variables are very small: , but
, or . If we neglect these small terms, the Bernoulli equation is written as

The resulting linear dynamic free surface boundary condition relates the instantaneous
displacement of the free surface to the time rate of change of the velocity potential,

Since by our definition will have a zero spatial and temporal mean, , thus

(2.13)

Linearization of kinematics free surface boundary condition

Using the Taylor series expansion to relate the boundary condition at the unknown
elevation, to , we have

Again retaining only the terms that are linear in our small parameters, , , and , and
recalling that is not a function of , the linearized kinematic free surface boundary
condition results:

(2.14a)

or

(2.14b)

3. APPLICATION TO THE PLANE WAVE PRODUCED BY A PADDLE

Let’s recall boundary conditions:

The linearized form of the dynamic and kinematics free surface boundary conditions:

(2.15)

(2.16)

The bottom boundary condition is the usual no-flow condition

(2.17)

To the lateral boundary condition, in the positive direction, as becomes large, we require
that the waves be outwardly propagating, imposing the radiation boundary condition
(Sommerfield, 1964). At , a kinematics condition must be satisfied on the wavemaker.
If is the stroke of the wavemaker, its horizontal displacement is described as

(2.18)
where is the wavemaker frequency.

The function that describes the surface of the wavemaker is

(2.19)

The general kinematics boundary condition is

(2.20)

where and . Substituting for yields

(2.21)

For small displacement S(z) and small velocities, we can linearize this equation by
neglecting the second term on the left-hand side.

As at the free surface, it is convenient to express the condition at the moving lateral
boundary in terms of its mean position, . To do this we expand the condition in a
truncated Taylor series

(2.22)

Clearly, only the first term in the expansion is linear in and , the others are
dropped, as they are assumed to be very small. Therefore, the final lateral boundary
condition is

(2.23)

Now that the boundary value problem is specified, all the possible solutions to the
Laplace equation are examined as possible solutions to the determine those that satisfy the
boundary conditions. Referring back to Table 2.1, the following general velocity potential,
which satisfies the bottom boundary conditions, is presented.

(2.24)

When using wavemaker to generate water wave, in addition to the desired progressive
wave, there exist several evanescent standing waves. The subscripts on indicate that that
portion of is associated with a progressive ( ) or a standing wave ( ). For the wavemaker
problem, must be zero, as there is no uniform flow possible through the wavemaker and
can be set to zero without affecting the velocity field. The remaining terms must satisfy the
two linearized free surface boundary condition, made up of both conditions. This condition
is

(2.25)

which can be obtained by eliminating the free surface from Eqs. (2.15) and (2.16).
Substituting our assumed solution into this condition yields

(2.26)
and

(2.27)

The first equation is the dispersion relationship for progressive waves, while the second
relationship, which relates to the frequency of the wavemaker, determines the wave
numbers for standing waves with amplitudes that decrease exponentially with distance from
the wavemaker. Rewriting the two equations as

(2.28)

The solutions to these equations can be shown in graphical form (see Figure 2.3 and 2.4).

The first equation has only one solution or equivalently one value of for given values of
and whilst there are clearly an infinite number of solutions to the second equation and
all are possible. It means that there exist one progressive wave and countless standing wave.
Each solution will be denoted as , where is an integer. The final form for the
boundary value problem is proposed as

(2.29)

The first term ( ) represents a progressive wave, made by
the wavemaker, while the second series of waves (
) are standing waves which decay away from the wavemaker.

Figure 2.3

Figure 2.4

To determine how rapidly the exponential standing waves decrease in the direction, let us
examine the first term in the series, which decays the least rapidly. The quantity ,
from Figure 2.3, must be greater than , but for conservative reason, say ,
therefore, the decay of standing wave height is greater than . For , ,
for , it is equal to 0.009. Therefore, the first term in the series is virtually negligible
two to three water depths away from the wavemaker.

For a complete solution, and need to be determined. These are evaluated by
the lateral boundary condition at the wavemaker.

or

(2.30)
Now we have a function of equal to a series of trigonometric functions of on the right-
hand side, similar to the situation for the Fourier series. In fact, the set of functions,
form a complete harmonic series of
orthogonal functions and thus any continuous function can be expanded in terms of them
(Sturm-Liouville theory). Therefore, to find , the equation above is multiplied by
and integrated from to . Due to the orthogonality property of these
functions there is no contribution from the series terms and therefore

(2.31)

Multiply Eq. (2.17) by and integrating over depth yields

(2.32)

Depending on the functional form of , the coefficients are readily obtained. For the
simple cases of piston and flap wavemaker, the are specified as

(2.33)

Thus

(2.34)

(2.35)

The wave height for the progressive wave is determined by evaluating far from the
wavemaker.

(2.36)

Substituting for , we can find the ratio of wave height to stroke as

(2.37)

(2.38)

The power required to generate these water can be obtained by determining the energy fluid
flux away from the wavemaker
(2.39)

where is the total average energy per unit surface area
(2.40)

is the phase velocity, and with is the group velocity:

(2.41)

Figure 2.4

III. MATHEMATICAL MODELING
This modeling problem consists of:

- With the given wave height, period of the desired water wave, calculating the wave
numbers, the wave length and the wavemaker’s stroke to generate that wave.
- Finding the distance beyond which the evanescent standing wave can be neglected.
- Calculating the elevation of the water wave at several positions, counting the
standing waves at positions where they are significant.

1. Computing the wave numbers:

The given parameters: the desired wave height: , the period (hence, the angular
frequency ) and the basin’s depth .

The wave numbers of the progressive wave and standing waves are computed from the
dispersion equations (2.26) and (2.27)

where the first equation has a unique root while the second has numerous roots but we
just need to find some smallest roots, say 3, because the standing wave’s amplitudes
decrease exponentially with .

These equations is non-linear, and the solutions can be achieved with a programming
language’s function, e.g. the Python’s function scipy.optimize.fsolve() which is a
wrapper around MINPACK’s hybrd and hybrj algorithms.

Such a numerical root finding subroutine need a starting estimate. For we can make use
of the approximation suggested by Fenton and McKee (1989) for the wave length:

where and .

For , three starting estimates of three roots , , can be deduced from
Figure 2.3. They are, respectively, , , .

The wavemaker’s stroke can be calculated using (2.37) and (2.38).

2. Finding the range in which the standing waves is considerable

In order to choose a relevant length for the basin, we must determine the effective range of
the standing waves. We knew that the standing wave amplitudes
decreasing exponentially with and the distance . So we can determine the effective
range relied on the first standing wave component, say and . This range is defined
so that the ratio of the standing wave’s amplitude to that of the progressive is less than a
threshold, usually 0,001.
Denote . Following is the binary algorithm to search the
nearest distance beyond which the standing wave is negligible within the increasing
sequence of discrete positions .

1. Begin with , ,
2. At , calculate
3. If , let , then – ; if , let , then
– ; and if , break the iteration loop.
4. Return to step 2 and so on, until , the iteration stops with xp is found

3. Computing the elevation of water wave

The computation of progressive wave’s elevation at numerous positions
at several continuous instants is massive because it
needs evaluation of many trigonometric functions. It can be lightened with following
iterative computational method [2].

The elevation at i-th time step:

Introduce in companion with as:

Denote ,

The elevation at (i+1)-th time step:

Denote ,

Thus

The elevation component caused by standing waves

is only considerable in the range which we determined earlier and can be computed with the
method above.

The final elevation is the sum of those of those waves.

IV. MODELING’S RESULT
Following is the result for the modeling of water wave with wave height , period
, generated by piston type and flap type wavemaker in a basin with the water level of
height.

Piston type Flap type
Wave number 5.765 5.765
Wave length 1.09 1.09
Wavemaker’s stroke 5.84 11.05
19.6 19.6
Standing wave’s wave numbers 41.24 41.24
62.402 62.402
Standing wave effective range 17 39

Following is the plots of the wave elevations. We can see in the case of flap type
wavemaker, the affection of standing waves is more remarkable than of piston type
wavemaker.

Figure 4.1. The total elevation of waves generated by piston type wavemaker

Figure 4.2. The elevation of standing wave in case of piston type wavemaker

Figure 4.3. The total elevation of waves generated by flap type wavemaker

Figure 4.4. The elevation of standing wave in case of flap type wavemaker

Conclusion: In the case of shallow water, the piston type wavemaker is more effective than
the flap type.

In addition, we have a comparison between piston type and flap type wavemaker:

Piston type:

- Advantage: Shorter wavemaker stroke, less affection of evanescent standing wave.

- Disadvantage: In deep water, this type wastes more energy to move the lower water
layers.

Flap type:

- Advantage: In deep water, this type doesn’t waste energy to move the lower water
layers as piston type.
- Disadvantage: Longer wavemaker stroke, more affection of evanescent standing
wave.

Reference
[1] Dean & Dalrymble, Water wave mechanics for engineers and scientists, Word
Scientific, 1991

[2] Ben T. Nohara, A Survey of the Generation of Ocean Waves in a Test Basin

Study on design of 2d ocean wave maker

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