1.2. Relevant equations and conditions 5
Figure 1.1: Schematic diagram of a typical wave propagation
travel, are significant elements to be considered. Theoretically, all other parameters, such as wave-induced water velocities and accelerations, may be calculated using these values.
Two-dimensional coordinates in x and y are taken into consideration in Figure 1.1. On a surface wave, a crest is a point where the wave displacement is at its greatest. The smallest or lowest point in a wave cycle is called a trough. It is the polar opposite of a crest. The horizontal distance between two consecutive wave crests is the wavelength which is represented as L. Following is a list of the notations: h: water depth which is measured from the mean sea level to the sea-bed, η(x, t): vertical displacement above the mean water level, A= H2: amplitude where H is the overall amplitude of a wave, T: period of a wave, k = 2πL: wavenumber, and ω= 2πT : angular frequency of a wave.
Ocean waves that areirregular or seeminglyrandom are common. There are a variety of ways that regular waves (wave components) might combine to produce irregular ones.
A regular wave (wave component) is defined by a single frequency (wavelength) and amplitude (height of the wave). Wind waves and swells are examples of normal water waves. There are many instances of irregular waves such assurf beats,harbour resonance, seiche, andtsunamis. It is possible to classify water waves depending on their wavelength.
In terms of wave classification, there are two distinct sorts of waves: deep and shallow. When the wavelength of a water wave exceeds the depth of the water, it is considered shallow (or long wave). Tsunamis and tidal waves are examples of shallow water waves.
Deep water waves, on the other hand, have wavelengths that are shorter than the depth of the ocean (or short waves), e.g., rogue wave. There are waves that fall midway between shallow and deep, and they are called intermediate water waves. This relative depth of wave characteristics is defined byh/L, wherehdenotes constant depth. For shallow water wave, this ratio satisfies h/L < 201, for deep water waves, h/L > 12 and for intermediate waves, 201 < Lh < 12. A linearized theory of water waves is used when the wave amplitude is expected to be small in proportion to the theoretical wavelength. In general, waves in
1.2. Relevant equations and conditions 7 the ocean are nonlinear. However, the margin of error in most engineering scenarios is quite small even when linear theory is utilised instead of nonlinear theory. With small amplitude waves in mind, linear wave theory is used to describe the majority of water wave dynamics difficulties. The linearized water wave theory is used in tackling the problems in this thesis.
With a velocityU~, the equations for mass and momentum conservation in incompress- ible fluids with inviscid flow are as follows:
∇. ~U = 0, (1.1)
∂ ~U
∂t + (U .∇)~ U~ =~g− 1
ρ∇P, (1.2)
where ρ is the density of the water, ~g is the gravitational accelaration and P is the hydrodynamic pressure.
Irrotational flows are now considered which results in a scalar function Φ such that
U~ =±∇Φ. (1.3)
Mathematically, both positive and the negative signs are correct in (1.3). However, the gravity wave always flows from higher to lower potential which defines loss of energy.
Following this convention, the negative sign is being considered in all our works in this thesis. ThisΦis called the velocity potential. Substituting (1.3) in (1.1) gives the equation of continuity for a potential flow as
∇2Φ = 0, (1.4)
which is the Laplace’s equation.
Considering periodic waves with time harmonic motions, the velocity potential can be expressed in Cartesian coordinates(x, y, z)as
Φ(x, y, z, t) =Re[φ(x, y, z) exp (−iωt)], (1.5) with φ(x, y, z)being the spatial potential function and ‘Re’ the real part of a function.
Considering gravity to be acting in the negative z-direction, it gives
~g =∇(−gz). (1.6)
Then equation (1.2) gives
∇ ∂Φ
∂t +1
2∇Φ.∇Φ + P ρ +gz
= 0. (1.7)
After carrying out the integration with respect to the spatial variables in real time domain [0, t], it yields
∂Φ
∂t +1
2∇Φ.∇Φ + P
ρ +gz =f(t),
when the process starts from rest at t = 0 and f(t) is the integration constant.
Without loss of generality, f(t) can be combined with the potential Φ(x, y, z, t) which results in
∂Φ
∂t + 1
2∇Φ.∇Φ + P
ρ +gz = 0. (1.8)
This is called theBernoulli’s equationor it may be termed here as the dynamic bound- ary condition.
Following small amplitude of the motion, the linearised dynamic boundary condition (1.8) at free surface z =η becomes
∂Φ
∂t + P
ρ +gη= 0. (1.9)
The kinematic condition is given as the free surface as a material boundary for which a particle on the free surface always remains on the free surface and hence the total derivative of the free surface is zero. In reality, particles are in continuous motion; even due to Brownian motion, a particle will not stay on the free surface. A free surface is actually an idealization; it is a collection of many particles that form a surface. Therefore, the condition at the free surface z =η(x, y, t)is given by
∂Φ
∂z = ∂η
∂t + ∂η
∂x
∂Φ
∂x +∂η
∂y
∂Φ
∂y. (1.10)
The linearised kinematic condition (1.10) ultimately takes the form
∂Φ
∂z = ∂η
∂t at z = 0. (1.11)
Following linear water wave theory as mentioned earlier, the boundary value problem is governed by
∇2φ = 0. (1.12)
Assuming hydrodynamic pressure to be 0 at the free surface, and combining (1.9) and
1.2. Relevant equations and conditions 9
(1.11) gives us the mean free surface condition atz = 0 as
∂φ
∂z −Kφ= 0, (1.13)
whereK = ω2 g .
We define a no flow boundary condition for a rigid structure as follows:
∂φ
∂n = 0 on S, (1.14)
whereS is surface boundary of the structure andn is the outward normal of the surface.
For an uneven impermeable rigid bottomz =−h(x, y), the boundary condition reduces to
∂φ
∂z +∇x,yh· ∇x,yφ = 0, (1.15)
where∇x,y = ∂
∂x, ∂
∂y
.
For a constant bottom, it reduces into
∂φ
∂z = 0 at z =−h. (1.16)
The same principle may now be used to two layers of an immiscible fluid with mean free surface at z = 0, impermeable bottom at z = −h and linear interface at z = −hI. Considering time harmonic motion of the fluid properties where the syntax remains the same, we add subscripts 1 and 2 to denote the upper and lower layer fluid features, respectively. Hence, the potentials satisfy the governing equation as follows:
∇2φ1 = 0, for −hI ≤z ≤0, (1.17)
∇2φ2 = 0, for −h≤z≤ −hI. (1.18) The mean free surface condition at the upper layer gives
∂φ1
∂z −Kφ1 = 0. (1.19)
The impermeable bottom condition at the lower layer gives
∂φ2
∂z = 0. (1.20)
Assuming linear interface along the two-layer fluid with elevationη2(x, y, t), the continuity
of velocity results in
∂Φ1
∂z = ∂Φ2
∂z = ∂η2
∂t . (1.21)
Further, the continuity of momentum from linearised Bernoulli’s equation gives another boundary condition as follows:
P1 =P2,
=⇒ ρ1
∂Φ1
∂t −ρ2
∂Φ2
∂t =g(ρ2−ρ1)η2. (1.22) For time harmonic motions in a two-layer fluid, the conditions (1.21) and (1.22) become
∂φ1
∂z = ∂φ2
∂z , (1.23)
ρ1 ∂φ1
∂z −Kφ1
=ρ2 ∂φ2
∂z −Kφ2
. (1.24)
1.2.1 Solution for the potential in a two-layer fluid
For a two-layer fluid, we consider harmonic motion in the y-direction as well as in time.
Hence, the velocity potential Φj(x, y, z, t) for j = 1,2 in each fluid layer can be written as Φj(x, y, z, t) = Re
φj(x, z)ei(kyy−ωt)
where ky = k1sinθ with k1 being the incident wavenumber.
For the fluid region, the analytical solution is found by using the method of separa- tion of variables. Substituting Φj in equations (1.17) and (1.18), the Laplace’s equation gets transformed into a modified Helmholtz equation and subsequently, by applying equa- tions (1.20), (1.21) and (1.23), the spatial potentials can be written as
φ1(x, z) = e±iqxsinhk(h−hI)(Ksinhkz+kcoshkz)
KcoshkhI −ksinhkhI , (1.25) φ2(x, z) = e±iqxcoshk(z+h), (1.26) where q = p
k2−ky2 and (±) sign stands for the direction of the waves. Wavenumber k can be found from the dispersion relation, by using equations (1.25) and (1.26) in equation (1.24), as
K2(ρ+ cothkhIcothk(h−hI))−kK(cothkhI+ cothk(h−hI)) +k2(1−ρ) = 0. (1.27) This equation has two pairs of symmetric real roots k = ±kn, for n = 1,2, denoting the progressive wavemodes and an infinite number of symmetric purely imaginary roots k = ±kn =±itn for n = 3,4, . . . , where tn is a real number (See Appendix B), denoting
1.3. Fluid flow through porous media 11