Generation and Study of Wave Kinematics of Deep water and Intermediate water Ocean Waves
Smitha Pai B1a,Veera Pandin1,b , Bhaskar E1,C, Goutham M.A1,d, S.L.Pinjare1,e
1 a*Department of ECE, Nitte Meenakshi InstituteofTechnology,Bangalore
1b
Centre for NanoScience and Engineering,Indian Institute of Science (IISc),Bangalore
1 c Department of Mechanical, Nitte Meenakshi InstituteofTechnology,Bangalore
1d
Department of ECE ,Adichunchanagiri Institute of Technology,Chikmagalur
1eDepartment of ECE, Nitte Meenakshi InstituteofTechnology,Bangalore
1a* [email protected], 1,b [email protected] , 1,C bhaskar263e@gmail.
, 1,d [email protected] , , 1,e [email protected]
Abstract
In the ocean, waves are produced by variety of forces. Based on the criteria of origin, water depth and apparent shape, the ocean waves are classified as deep water, intermediate and shallow water waves. In this paper an attempt is made to simulate the ocean conditions by designing the wave generator which is housed in a wave flume of 58cm long, 44cm wide and 11cm depth. The wave generator which is used here to generate the wave is a flap type structure. The structure is hinged at both the ends of the flume and was driven by a DC motor to generate sinusoidal deep water wave with relative water depth d/L (ratio of water depth and wavelength of the generated wave) is of 2.5, which is greater than 0.5 and intermediate water wave with relative water depth in the range of 0.05<d/L<0.5.
This experimental set up is used to find the wave parameters such as wavelength, wave height, time period and wave celerity of both waves. The frequency of the generated, deep water and intermediate water waves are found to be in the range of 3Hz to 10 Hz. The values of wave parameter which were found experimentally are compared with that of values obtained by applying wave theory and are found to be closely matching. Emphasis is given to the selection of proper kinematic model and wave model based on Airys theory, which is outlined for the evaluation of wave kinematics such as velocity potential, wave particle displacement, particle velocity, particle acceleration and dynamic pressure exerted by the wave. Subsequently an attempt is made to analyse the behavior of wave kinematics for various water depths varying from SWL (Surface Water Level) till the sea bed. At the end, wave kinematic was applied in Morison equation for the evaluation of forces exerted on offshore structure and dynamic response of the structure is analysed for the same.
Keywords: Airys theory, Deep water, Intermediate water, Morison equation, Wave Kinematics.
1. Introduction
Ocean waves are random in nature and can be represented by superposing a number of known sinusoidal waves of different amplitude and frequency. The ocean waves can be classified based on three parameters. (i). As per apparent shape (ii). As per water depth
and (iii). As per Origin. The ocean waves are produced by variety of forces [1]. In the ocean the deep water waves are produced or generated due to metrological forces. These metrological forces are mainly because of wind, variation in temperature, pressure, salinity and density. Astronomical forces produce the intermediate and shallow water waves. Tides and tsunamis are shallow water waves and they are due to astronomical forces and earthquakes respectively [2].
1.1 Deep Water Waves:
In the case of deep water waves (d>0.5L where„d‟ is depth of water and „L‟ is the wavelength of the wave) the energy of the wave does not have any effect or impact on the bottom of the sea bed. The deep water waves change in to breaking waves as shown in Fig. 1, when they move in to shallow water. The water particle orbit gets flattened, since water particles drag along the bottom when the energy of the waves comes in contact with the ocean floor [3].
1.2 Intermediate Waves (Transitional Waves)
Intermediate waves occur if the ratio of d and L lies in the range of 0.05 ≤ d/L≤0.5. At this instant, the water movement of particles on the surface transitions from swells to steeper waves called peaking waves there will be generation of peaking waves as shown in Fig1. In case of intermediate waves, the friction will happen between the deeper part of the wave and bottom particles, which results in faster movement of the top of the wave compared to deeper part of the wave. This process results in a wave whose front surface becomes steeper when compared to back surface [3],[4].
1.3 Shallow water waves
Shallow water waves are produced when they satisfy the following criteria [4].
i. When the relative water depth d/L is less than 0.05.
ii. When the crest of the wave occurs at an angle less than 120 degrees.
iii. Height of the wave should be more than one-seventh of the wavelength (H>1/7L) and
iv. The height of the wave is more than three-fourths of the water depth (H>3/4d).
Fig. 1. Nature of different types of waves
1.4 Wave Kinematic Model
The deep water waves are progressive waves which are sinusoidal in nature and Airy‟s theory is feasible to explain the phenomenon. Airy‟s wave theory is applicable only if the wave height is smaller when compared with the wavelength of the wave produced and the water depth [5].
1.5 Airy’s Wave Equation
The concept of two-dimensional ideal flow is used to derive the Airy wave. Fig.2 shows a progressive sinusoidal wave of wavelength L, height H and period T on a undisturbed water of depth d defined in the region, -∞≤x≤∞ and -d≤z≤η.
Fig.2. Sinusoidal Progressive Wave
Let η be the surface elevation variation with time, the reference being still water line.
η = H/2 cos{2π(X/L-t/T)}---(1)
where X is the meaured distance in meters along horizontal axis and t being time in secs , indicating η as a function of both space and time.
The derivation of small amplitude and finite amplitude wave theories can be deduced by obtaining velocity potential ϕ(x,y,z) associated with the progressive wave.The velocity potential can be accomplished with the help of following three equations and two boundary
conditions i,e Dynamic free surface boundary condition and Kinematic boundary condition[6].
Laplace equation being the governing equation: ---(2)
Continuity equation: + =0--- (3)
Bernoullis equation: - u2+v2+w2)+ +gz=0---(4)
where u,v,w are particle velocities in x,y and z directions respectively. The solution of
Velocity Potential „ϕ(x,y,z)‟ = ---(5)and
η= a ) ---(6) where a=H/2 , H being wave height, g=accceleration due to gravity, 𝜎= angular velocity, k= wavenumber=2π/L , L is the wavelength of the wave and d=depth of water. In a fluid medium, when wave varies sinusoidally, then for each elevation η of a wave, the velocity potential varies in cosine form.The magnitude of the velocity potential is maximum at the water surface and it becomes minimum at the sea
bed, as the magnitude of the velocity potential varies in the form of hyperbolic cosine form.Velocity potential is a function of wavelength, waveheight and wave period. It is a main source for all wave kinematics such as, particle velocity components, particle displacement, acceleration , pressure and energypossessed.
2.0 Laboratory Experimental Setup for wave generation
The main objective of this experimental setup is to generate deep water and intermediate water waves and also to study the nature of generated waves in two dimensions. This setup as shown in Fig.3 provides the demonstration of the properties of waves and wave propagation phenomenon with the help of optical properties of waves. The experimental set up to generate water waves involves the components such as (i). Flume (ii). Wave Maker (iii). DC Motor (iv). Power Supply (v). Optical Source. The flume is a shallow tank with 58cm length, 44cm width and 11 cm height, with transparent bottom and sides to visualize generated waves. The depth of water in the tank or flume is adjusted to generate the required waves. To generate the deep water wave and intermediate water wave , the depth of the water in the flume is adjusted to 80mm and 20mm respectively so that the relative water depth d/L of deep water wave is >0.5 and intermediate water wave lies in the range of 0.05≤d/L≤0.5.
The wave maker used to generate the wave is flap type structure made up of soft fiber material, with 3cm width and 33.5cm length, which is hinged at both the ends of the flume. It is driven by a simple DC motor. The speed of the DC motor and hence the frequency of the wave is adjusted by varying the applied dc voltage to the motor with the help of regulated DC power supply. This experimental set up was able to generate uniform plane waves as shown in Fig.4, in both cases (deep water and intermediate water waves) in the range of 3 Hz to 10Hz.
Fig.3. Experimental set up for wave generation
Flume Wave
Maker DC Supply
Optical
Source
The flume is illuminated from top with the help of lamp and while light rays pass through water in the flume, the pattern of generated water waves on the water surface show up as shadows on the screen i, e the white paper underneath the flume as shown in Fig.4. To find the frequency of the generated wave, the video recording of the experiment is carried out for „n‟ seconds and later it was played in slow motion. The numbers of waves passing a point in „n‟ seconds are counted during the run in slow motion and it was divided by „n‟
to find the frequency.Similarly to calculate the wavelength, measured the length of the number of waves and then devided it by the number of waves. Celerity of the wave is found by using the formula, i.e, Wave Speed= Frequency*Wavelength.
Fig.4. Uniform Plane wave Generation
In both the cases, generated uniform plane wave in the low frequency range and corresponding wavelength, wave height,celerity are calculate and they are indicated in the table1& table 2.
Table 1. Wave parameters of generated deep water waves
Voltage(V)
Frequency(H
z) Amplitude(mm)
Experimental Value of
Wavelength (mt)
Theoritical value of Wavelength(mt)
Relative Water Depth(d/L)
4 2.9 5 0.192 0.185 0.43
5 3.95 4.54 0.101 0.099 0.79
6 5 4.1 0.068 0.066 1.17
7 6.05 3.55 0.044 0.042 1.81
8 7.08 3.1 0.033 0.031 2.58
9 8.05 2.52 0.025 0.024 3.33
Wave Make r
Flume
Wave
Image
Table 2. Wave parameters of generated intermediate water waves
Voltage(V)
Frequency(H
z) Amplitude(mm)
Experimental Value of
Wavelength (mt)
Theoritical value of Wavelength(mt)
Relative Water Depth(d/L)
4 1.8 6.4 0.2 0.31 0.1
5 2.55 6 0.1 0.23 0.2
6 3.55 5.6 0.066 0.1 0.3
7 4.2 5.1 0.05 0.077 0.4
8 5.05 4.5 0.04 0.052 0.5
9 5.8 3.9 0.033 0.043 0.6
The celerity or the speed of the wave in the direction of propagation is given by the equation and since from the above equation we get
--- (7)
Celerity in case of deep water conditions is only a function of T i,e = ---(8) and =1.56 ---(9). Celerity in the case of intermediate wave is a function of both, time and water depth [7]. Our experimental results are matching with the results of theritical calculation.For eg, For 7.05 Hz frequency of deep water wave, the obtained wavelength and celerity using relevent formulae are 3.11 cm and 0.2341m/sec, whearas experimentally found results are 3.33cm and 0.2347m/sec.
3. Progressive wave behavior and impact on Fluid kinematics
To evaluate the force exerted by wave on the offshore structure, it is essential to know the fluid kinematics i,e, the particle velocity and acceleration[8].
3.1 Water particle velocity components under Progressive waves
The Velocity components of wave at any water depth z are orbital velocity or horizontal water particle velocity and Vertical water particle velocity, both being the functions of space and time [9].
The orbital or horizontal water particle velocity component u and vertical fluid particle velocity component „w‟ is given by
) --- (10)
…..(11)
Since velocity potential varies along the depth of the water, the velocity components also vary along the depth of the water. For a given z, orbital velocity component varies sinusoidally, while verticle particle velocity is out of phase with the wave variation (η) and horizontal particle variation.
In the case of deep water waves, it is found that the maximum horizontal particle velocity equals the maximum verticle velocity for any given distance from the surface as horizontal water particle displcement and verticle particle displcement is same thereby
taking the path of circular orbit, whereas in intermediate water waves they are not equal as water particle displacement takes elliptical orbit[9].
For the generated water wave of frequency 5Hz,8mm wave height, Wavelength of 0.068 m in the case of deep water wave and 9mm wave height with wavelength 0.04 m of intermediate wave, the particle velocity components at various water depth are studied and is shown in Fig5.
Fig. 5a. Deep Water Wave of Height 8mm at 5 Hz frequency
Fig.5b. Variation of Maximum Horizontal and Vertical Particle Velocity with respect to different water depth in case of deep water waves
Fig.5c. Variation of Maximum Horizontal and Verticle Particle Velocity with respect to different water depth in case of intermediate water waves Fig.5. Behavior of Particle Velocity components at various water depth
4. Water particle displacement under Progressive waves
As the waves pass through water, the energy associated with the wave sets the particles of water into orbital motion. In the case of deep water waves, it is observed that near the surface, the water particles movement follows a path of circular orbit, whose diameter approximately matches with the wave height. As we go down deeper in the water, the energy of the wave and the orbital diameter decreases. It is found that when the depth of water becomes equal to half the wavelength (d=1/2L) or less than that, the wave energy will not have any impact on the water [10]. In case of intermediate water waves, the water particle displacement movement takes the path of elliptical orbit, and it is observed that both horizontal particle displacement and vertical particle displacement reduces along the depth of the water [9][10]. For the generated water wave of frequency 5Hz,8mm wave height, Wavelength of 0.068 m in the case of deep water wave and 9mm wave height with wavelength 0.04 m of intermediate wave, the particle velocity components at various water depth are studied and is shown in Fig6.
Fig.6a. Water particle Displacement of deep water wave at surface level i,e z=0 and z= 100% (d/L)
Fig.6. Behaviour of Particle displacement of intermediate water wave
5. Pressure distribution of progressive waves
By knowing the velocity potential of a progressive wave, we can compute the pressure distribution of a wave with the help of linearised Bernoulli equation. Two pressure components associated with the waves are static and dynamic pressure. These components vary along the depth of water.
i.e. -γz---(12)
In the above equation the first component is the dynamic pressure component and the second component is static pressure component. Here the dynamic pressure component reduces as we go towards the seabed from still water line as shown in Fig.7.
Fig.7. Dynamic Pressure Variation at z=0 and z=d due to intermediate water
6. Wave loads on Structures
The precise evaluation of force exerted by waves on structure is difficult to obtain as waves are nonlinear, unsteady and exhibit non uniform flow condition, for which dynamic effects will be non-deterministic in nature. Morison equation is used for the formulation of the wave forces on a pile which is fixed on the sea bed. The application of Morison equation is feasible if D⁄L<0.2 and if H⁄D>1 where D is the diameter of the pile [11].
The force exerted on the structure in such case is the summation of drag force and inertial force, which are dependent on orbital particle velocity and particle acceleration respectively. The considered offshore structure here, is a cylindrical structure of 3mm diameter and 25 mm length. For this taken dimension D⁄L=0.075 and H⁄D=3, hence Morison equation is applied.
Based on the structure dimension and wave characteristics of intermediate wave the Keulegan –Carpenter number (KC=Umax/D T) and scattering parameter (Ka= πD/L) are calculated and they are found to be 9.42 and 0.2355. For the obtained values of Kc and Ka, the wave force regime indicates that inertial force dominates the drag force [12].
The total force in such a case is the summation of orbital particle velocity dependent drag force and particle acceleration dependent inertial force.
Inertial force (per unit length) is 𝞀 ---(13) and Drag Force (per unit length) is = 𝞀D‖u‖u---(14) where CD and Cm are smoothly varying functions of the Keulegan-Carpenter number, u is instantaneous particle velocity and is water particle acceleration.
Fig. 8a. Phase Variation of Inertial Force at Z=0%of Fig.8b. Phase variation of Drag Force at Z=0% of Wavelelength (L0) Wavelength (L0)
Fig.8c. Phase Variation of Inertial Force at Z=100% of Fig.8d. Phase Variation of Drag Force at Z=100% of Wavelength(Lo) Wavelength (Lo )
Fig.8e. Variation of Total Wave Force(Total inertial Force+Total Drag Force) in deep water wave with respect to the phase angle
Fig.8. Wave load force components at various water Depth
7.0 Conclusion
In this paper, an attempt is made to produce progressive (sinusoidal) deep water and intermediate water wave as per the criteria of apparent shape, water depth and origin. The water surface elevations are measured and with the help of this obtained information, the water wave kinematics was derived by using Airy‟s theory. Airys theory was feasible to apply as generated wave amplitude was found to be small. Expressions for the wave speed, wavelength, time period and wave height obtained experimentally are found to be in line with the results obtained by solving kinematics equations of Airys classical theory.
The effect of wave forces on offshore structures, the effect of generated low frequency pressure waves on the underwater acoustic sensor, both in the coastlines and oceans to a great extent are governed by the local distribution of water particle velocities and acceleration. Hence, in coastal and ocean engineering, wave kinematic has become a central field of study and in this paper an attempt is made to simulate ocean conditions in the laboratory by generating deep water, intermediate water waves followed by the study of behavior of wave kinematics.
Acknowledgment
Authors are thankful to Department of Mechanical Engineering (NMIT) and
Center for Nano Science and Engineering (CENSE), IISC, Bangalore, for providing the facilities with valuable guidance.
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Smitha Prabhu, is working as Associate Professor at NMIT. She has obtained her M.Tech from VTU University in VLSI Design& Embedded Systems. She is currently pursuing her Ph.D from VTU University. She has around 7 research publications. Her research interest includes VLSI Design and MEMS.
Veera Pandi N, BE, is working as Senior Facility Technologist at Centre for Nanoscience and Engineering, Indian Institute of Science, Bangalore. He has one patent and around 10 research publications. His research interests in Design and Development of MEMS based sensors.
BHASKAR.E is a faculty of Mechanical department of Nitte Meenakshi Institute of Technology. He has more than 10 years of industrial experience.
Executed and presented various Aero modeling projects at national level.
Dr. Goutham M A is Professor & Head of ECE, AIT. He obtained his Ph.D from Jadhavpur University.He has20 years of research and teaching experience. He has published over 20 research papers in national and international journals. His current research interest includes VLSI Design, MEMS and Signal Processing.
Dr. Sampatrao L. Pinjare is Senior Professor of ECE, NMIT. He obtained his Ph.D. from IIT Madras.He has 39 years of research and teaching experience. He has published over 50 research papers in national and international journals. His current research interest includes VLSI Design, MEMS and Nanotechnology.
