HO CHI MINH CITY NATIONAL UNIVERSITY UNIVERSITY OF POLYTECHNIC THE FACULTY OF MECHANICAL ENGINEERING THE PFIEV PROGRAMGraduate thesisSTUDY 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 importanttechnology in the field of the coastal and ocean engineering. To date most laboratory testingof floating or bottom-mounted structures and studies of beach profiles and other relatedphenomena have utilized wave tanks, which are usually characterized as long, narrowenclosures with a wavemaker of some kind at one end; however, circular beaches have beenproposed for littoral drift studies and a spiral wavemaker has been used. For all of thesetests, the wavemaker is very important. The wave motion that it induces and its powerrequirements can be determined reasonably well from linear wave theory. Wavemakers are, in fact, more ubiquitous than one would expect. Earthquakeexcitation of the seafloor or human-made structures causes waves which can be estimatedby wavemaker theory; in fact, the loading on the structures can be determined. Any movingbody 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 varioustypes of structures and vessels, including models of ships, offshore platforms, and otherbodies. The theory of water waves has attracted scientists in fluid mechanics and appliedmathematics for at least one and a half centuries and has been a source of intriguing - andoften difficult - mathematical problems. Apart from being important in various branches ofengineering and applied sciences, many water-wave phenomena happen in everydayexperience. Waves generated by ships in rivers and waves generated by wind or earthquakesin oceans are probably the most familiar examples. The mathematical theory of water wavesconsists of the equations of fluid mechanics, the concepts of wave propagation, and thecritically important role of boundary conditions. The results obtained from theory may givesome explanation of a natural phenomenon or provide a description that can be testedwhenever an expanse of water is at hand: a river or pond, the ocean, or simply thehousehold bath or sink. However, obtaining a thorough understanding of the relevantphysical mechanisms presents fluid dynamicists and applied mathematicians with a greatchallenge In practical use, two types of wavemakers which use a paddle with two type of movingto produce water wave are the most popular. They are so-called piston-type and flap-typewavemaker.
Figure 1.3. Edinburgh Designs Ltd’s flap type wavemakerFigure 1.1. Some DHI Group’s wavemakers
Figure 1.2. An engineering testing tank with the use of wavemakerII. WAVEMAKER THEORY FOR PLANE WAVES PRODUCED BY A PADDLEAssumptions: 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 2.11. BOUNDARY VALUE PROBLEMS In formulating the small-amplitude water wave problem, it is useful to review, in verygeneral terms, the structure of boundary value problems, of which the present problem ofinterest is an example. Numerous classical problems of physics and most analyticalproblems in engineering may be posed as boundary value problems; however, in somedevelopments, this may not be apparent. The formulation of a boundary value problem is simply the expression in mathematicalterms of the physical situation such that a unique solution exists. This generally consists offirst establishing a region of interest and specifying a differential equation that must besatisfied within the region. Often, there are an infinite number of solutions to the differentialequation and the remaining task is selecting the one or more solutions that are relevant tothe physical problem under investigation. This selection is effected through the boundaryconditions, 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 velocitypotential 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 watersurface, certain physical conditions must be satisfied by the fluid velocities. Theseconditions on the water particle kinematics are called kinematic boundary conditions. Atany 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 derivedfrom the equation which describes the surface that constitutes the boundary. Any fixed ormoving 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 totalderivative 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)whereThis condition requires that the component of the fluid velocity normal to the surface berelated 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 (horizontalbottom) for a two-dimensional case where the origin is located at the still water level andrepresents the depth. If the bottom is impermeable, we expect that , as the bottomdoes 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 . Thekinematic boundary condition at the free surface is (2.5)whereCarrying the dot product yields (2.6)Dynamic Free Surface Boundary Condition A distinguishing feature of fixed (in space) surfaces is that they can support pressurevariations. However, surfaces that are “free”, such as the air-water interface, cannot supportvariations in pressure (neglecting surface tension) across the interface and hence mustrespond in order to maintain the pressure as uniform. A second boundary condition, termeda dynamic boundary condition, is thus required on any free surface or interface, to prescribethe pressure distribution pressures on this boundary. As the dynamic free surface boundary condition is a requirement that the pressure onthe 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 displacementof the paddle may be described as , the kinematic boundary condition iswhereor, 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 AHORIZONTAL BOTTOMSolution of the Laplace equationA convenient method for solving some linear partial differential equation is calledseparation of variables. For our case (2.9)For waves that are periodic in time, we can specify . The velocity potentialnow 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 dependsonly on . If we consider a variation in in Eq. (2.11) holding constant, the second termcould conceivably vary, whereas the first term could not. This would give a nonzero sum inEq. (2.11) and thus the equation would not be satisfied. The only way that the equationwould 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 wewill permit the separation constant k to have an imaginary value in this problem and ingeneral 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 forreal, , 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 heightwith distance, which may be valid for cases of waves propagating with damping or wavegrowth 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 EquationsRealImaginaryLinearization of dynamics free surface boundary conditionThe Bernoulli equation must be satisfied on , which is a priori unknown. Aconvenient 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 Taylorseries.where on .Now for infinitesimally small waves, is small, and therefore it is assumed that velocitiesand 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 asThe resulting linear dynamic free surface boundary condition relates the instantaneousdisplacement 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 conditionUsing the Taylor series expansion to relate the boundary condition at the unknownelevation, to , we haveAgain retaining only the terms that are linear in our small parameters, , , and , andrecalling that is not a function of , the linearized kinematic free surface boundarycondition results: (2.14a)or
(2.14b)3. APPLICATION TO THE PLANE WAVE PRODUCED BY A PADDLELet’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 requirethat 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 byneglecting the second term on the left-hand side.
As at the free surface, it is convenient to express the condition at the moving lateralboundary in terms of its mean position, . To do this we expand the condition in atruncated Taylor series (2.22) Clearly, only the first term in the expansion is linear in and , the others aredropped, as they are assumed to be very small. Therefore, the final lateral boundarycondition is (2.23) Now that the boundary value problem is specified, all the possible solutions to theLaplace equation are examined as possible solutions to the determine those that satisfy theboundary 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 progressivewave, there exist several evanescent standing waves. The subscripts on indicate that thatportion of is associated with a progressive ( ) or a standing wave ( ). For the wavemakerproblem, must be zero, as there is no uniform flow possible through the wavemaker andcan be set to zero without affecting the velocity field. The remaining terms must satisfy thetwo linearized free surface boundary condition, made up of both conditions. This conditionis (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 secondrelationship, which relates to the frequency of the wavemaker, determines the wavenumbers for standing waves with amplitudes that decrease exponentially with distance fromthe 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 andall 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 theboundary value problem is proposed as (2.29)The first term ( ) represents a progressive wave, made bythe wavemaker, while the second series of waves ( ) are standing waves which decay away from the wavemaker.
Figure 2.3 Figure 2.4To determine how rapidly the exponential standing waves decrease in the direction, let usexamine 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 negligibletwo to three water depths away from the wavemaker. For a complete solution, and need to be determined. These are evaluated bythe 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 oforthogonal 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 thesefunctions 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 thesimple 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 thewavemaker. (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 fluidflux 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 MODELINGThis 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 angularfrequency ) and the basin’s depth .The wave numbers of the progressive wave and standing waves are computed from thedispersion equations (2.26) and (2.27)where the first equation has a unique root while the second has numerous roots but wejust need to find some smallest roots, say 3, because the standing wave’s amplitudesdecrease exponentially with .These equations is non-linear, and the solutions can be achieved with a programminglanguage’s function, e.g. the Python’s function scipy.optimize.fsolve() which is awrapper around MINPACK’s hybrd and hybrj algorithms.Such a numerical root finding subroutine need a starting estimate. For we can make useof the approximation suggested by Fenton and McKee (1989) for the wave length:where and .For , three starting estimates of three roots , , can be deduced fromFigure 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 ofthe standing waves. We knew that the standing wave amplitudes decreasing exponentially with and the distance . So we can determine the effectiverange relied on the first standing wave component, say and . This range is definedso that the ratio of the standing wave’s amplitude to that of the progressive is less than athreshold, usually 0,001.Denote . Following is the binary algorithm to search thenearest distance beyond which the standing wave is negligible within the increasingsequence 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 found3. Computing the elevation of water waveThe computation of progressive wave’s elevation at numerous positionsat several continuous instants is massive because itneeds evaluation of many trigonometric functions. It can be lightened with followingiterative computational method .The elevation at i-th time step:Introduce in companion with as:Denote ,The elevation at (i+1)-th time step:Denote ,
ThusThe elevation component caused by standing wavesis only considerable in the range which we determined earlier and can be computed with themethod above.The final elevation is the sum of those of those waves.IV. MODELING’S RESULTFollowing 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 typeWave number 5.765 5.765Wave length 1.09 1.09Wavemaker’s stroke 5.84 11.05 19.6 19.6Standing wave’s wave numbers 41.24 41.24 62.402 62.402Standing wave effective range 17 39
Following is the plots of the wave elevations. We can see in the case of flap typewavemaker, the affection of standing waves is more remarkable than of piston typewavemaker.Figure 4.1. The total elevation of waves generated by piston type wavemakerFigure 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 wavemakerFigure 4.4. The elevation of standing wave in case of flap type wavemakerConclusion: In the case of shallow water, the piston type wavemaker is more effective thanthe 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 Dean & Dalrymble, Water wave mechanics for engineers and scientists, WordScientific, 1991 Ben T. Nohara, A Survey of the Generation of Ocean Waves in a Test Basin