Governing equations of water waves

Section 5.1 Governing equations of water waves

We shall consider the case of waves on the free surface of a two-dimensional potential fluid (hence inviscid, incompressible, and irrotational). The free surface bounds the water from above, and is assumed to be given by

\begin{equation*} y = \eta(x, t) \qquad -\infty < x < \infty. \end{equation*}

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Since the fluid satisfies the assumptions of potential flow, there exists a velocity potential $\bu = \nabla \phi$ where

\begin{equation*} \nabla^2 \phi = \pd{^2 \phi}{x^2} + \pd{^2\phi}{y^2} = 0 \qquad \text{in the fluid}. \end{equation*}

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What makes water-wave problems unique (and challenging) is that not only do we need to solve for the potential (and hence velocities) within the fluid, as given by Laplace’s equation above, but that we must do so in an a priori unknown domain, where the free-surface location $y = \eta(x, t)$ must also be solved in parallel. Key considerations, then, must relate to the precise boundary conditions to impose.

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Subsection 5.1.1 Boundary conditions

Subsubsection 5.1.1.1 Bottom boundary conditions

If we consider waves on a fluid of infinite (or deep) depth, then far below the free surface, we expect that the velocity tends to zero. Hence a suitable boundary condition is

\begin{equation} \bu = \nabla \phi \to 0 \quad \text{as $y \to -\infty$}.\tag{5.1.1} \end{equation}

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On the other hand, for the case of fluid bounded below by a flat bed, say at $y = -H\text{,}$ we have

\begin{equation} \bu \cdot \bn = 0 \Longrightarrow \pd{\phi}{y} = 0 \qquad \text{on $y = -H$}. \tag{5.1.2} \end{equation}

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Subsubsection 5.1.1.2 Dynamic boundary conditions

There are now two boundary conditions to consider at the free surface. The dynamic boundary condition corresponds to the requirement of satisfying a force balance at the free surface. At the free surface, the pressure within the liquid must be equal to the pressure within the air. This leads to the dynamic boundary condition that

\begin{equation} p = p_{\text{atm}} \qquad \text{at $y = \eta(x, t)$},\tag{5.1.3} \end{equation}

tv where $p_{\text{atm}}$ is the atmospheric pressure above the fluid (assumed to be constant).

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Typically, we need the above pressure condition expressed in terms of $\phi\text{,}$ and this suggests using Bernoulli’s equation. We require yet another version of Bernoilli’s equation from a previous chapter, this one for unsteady flow and posed in terms of a potential function. This involves returning to an earlier manipulation of the momentum equation giving (3.5.5):

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Theorem 5.1.1. Bernoulli’s equation for unsteady potential flow.

Bernoulli’s equation for unsteady potential flow states that

\begin{equation} \pd{\phi}{t} + \frac{1}{2}|\nabla\phi|^2 + \frac{p}{\rho} + \chi = F(t),\tag{5.1.4} \end{equation}

for an arbitrary function $F(t)\text{,}$ and where $\bg = -\nabla\chi$ is the conservative body force.

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Proof.

Returning to (3.5.5), we re-write in terms of the potential function and set the vorticity $\omega = \nabla \times \bu = 0\text{,}$ giving

\begin{equation*} \pd{}{t}(\nabla \phi) = - \nabla \left(\frac{1}{\rho}\nabla p + \frac{1}{2} |\nabla \phi|^2 + \chi\right). \end{equation*}

We can interchange the order of differentiation on the left

\begin{equation*} \nabla\left( \pd{\phi}{t}+\frac{1}{\rho}\nabla p + \frac{1}{2} |\nabla \phi|^2 + \chi\right) = 0. \end{equation*}

We now integrate each of the spatial components and this yields the desired result.

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Below, we will set the body force indeed to be gravity. Thus if we write

\begin{equation*} \chi = gy \Longrightarrow \bg = -\nabla \chi = \Bigl[0, -g\Bigr], \end{equation*}

and gravity is then directed in the negative $y$-direction.

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We now evaluate Bernoulli’s equation (5.1.4) on the free surface, $y = \eta\text{.}$ This gives

\begin{equation*} \pd{\phi}{t} + \frac{1}{2}|\nabla\phi|^2 + \frac{p_{\text{atm}}}{\rho} + g\eta = F(t), \qquad \text{on $y = \eta$}. \end{equation*}

It is convenient to set the arbirary function $F(t) = \frac{p_{\text{atm}}}{\rho}\text{,}$ yielding the final result:

\begin{equation*} \pd{\phi}{t} + \frac{1}{2}|\nabla\phi|^2 + g\eta = 0, \qquad \text{on $y = \eta$}, \end{equation*}

which is a re-phrasing of the necessary dynamic boundary condition on the free surface.

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Subsubsection 5.1.1.3 Kinematic boundary conditions

In the case of fluids near solid surfaces, we require a no-penetration condition, which enforces that the fluid velocity is zero in a direction normal to the surface, $\bu \cdot \bn = 0\text{.}$ For the case of a free boundary, such as the one at $y = \eta(x, t)\text{,}$ there is a similar constraint, and it may be shown that this is equivalent to imposing that the material fluid fluid elements on the free surface must remain on the free surface.

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Thus, if $y = \eta$ for some fluid particle at time $t\text{,}$ then $y = \eta$ for the same particle for all time. Thus we need

\begin{equation*} \DD{(y - \eta)}{t} = 0, \qquad \text{on $y = \eta$}. \end{equation*}

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Expansion and use of the material derivative then implies the kinematic boundary condition

\begin{equation} v = \pd{\eta}{t} + u \pd{\eta}{x}, \qquad \text{at $y = \eta$}.\tag{5.1.5} \end{equation}

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Subsection 5.1.2 Summary of the governing equations

Let us summarise the governing water-wave equations in one section.

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Theorem 5.1.2. Water wave equations in potential flow.

Consider two-dimensional free-surface flow of a potential (inviscid, incompressible, irrotational) fluid. The fluid satisfies Laplace’s equation,

\begin{equation*} \nabla^2 \phi = 0, \qquad \text{in the fluid domain}. \end{equation*}

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On the free surface, we have the kinematic boundary condition and dynamic boundary conditions:

\begin{align} \pd{\eta}{t} + u \pd{\eta}{x} = v, \qquad & y = \eta(x, t)\tag{5.1.6}\\ \pd{\phi}{t} + \frac{1}{2}|\nabla \phi|^2 + g\eta = 0, \qquad & y = \eta(x, t).\tag{5.1.7} \end{align}

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Finally there is typically a condition in the far field, either at infinity in the case of infinite depth, or a condition on a channel bottom in the case of finite depth. For example, in the case of infinite depth and an otherwise motionless fluid at infinity, we would require

\begin{equation*} \nabla \phi \to 0, \qquad \text{as $y \to -\infty$}. \end{equation*}

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While for the case of a finite channel bottom at $y = -h\text{,}$ we would require

\begin{equation*} \pd{\phi}{y} = 0 \qquad \text{on $y = -h$}. \end{equation*}

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