Table of Contents
DISCLAIMER 1: John von Neumann referred to potential flow theory as the study of
'dry water'.
DISCLAIMER 2: Richard Feynman stated that potential flow theory had nothing to
do with 'real fluids' and that it was more to do with solving beautiful
mathematical problems…
Despite the simplications, potential flow theory is incredibly important to WEC numerical modelling - enabling the modelling of a wide range of operating conditions in a fraction of the time of higher-fidelity approaches such as CFD and SPH.
Frequency-domain linear potential flow solvers such as WAMIT and NEMOH (so-called 'boundary element method (BEM) codes') are well-established in the hydrodynamics field, and the foundation of most time-domain solvers. However, they are often treated as a black box. This post aims to elucidate some of the underlying theory in a clear, succint manner.
1 Fundamentals and main assumptions
In order to model fluid motion, we essentially need to be able to determine the velocity vector \(\vec{v}(x, y, z, t)\) of a fluid particle at any point within the flow:
\begin{equation} \vec{v} = \frac{d\vec{x}}{dt} = \begin{bmatrix} u \\ v \\ w \end{bmatrix} \end{equation}- However, there are other fluid properties which may also vary throughout the flow, over time. To simplify the model, we can assume these remain constant.
If we assume the fluid to be incompressible, then its density must remain constant:
\begin{equation} \rho = const. \end{equation}This assumption means the divergence of the flow's velocity must be zero, hence the continuity (conservation of mass) equation is:
\begin{equation} \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \end{equation} \begin{equation} \nabla \cdot \vec{v} = 0 \end{equation}Potential flow theory also assumes that the the flow is inviscid (i.e. the fluid cannot experience any shear stresses) from \(t=0s\), and therefore the vorticity must be zero everywhere - hence the fluid is irrotational:
\begin{equation} \label{eq:irrotational} \nabla \times \vec{v} = 0 \end{equation}Under these simplifictions, the velocity vector, \(\vec{v}(x, y, z, t)\) may be replaced with a mathematical abstraction: the gradient of a scalar potential, \(\phi(x, y, z, t)\):
\begin{equation} \label{eq:scalar_potential} \vec{v} = \nabla\phi = \begin{bmatrix} \frac{\partial \phi}{\partial x} \\ \frac{\partial \phi}{\partial y} \\ \frac{\partial \phi}{\partial z} \end{bmatrix} \end{equation}- In this approach, instead of computing 3 unknown velocity vector components, only 1 unknown scalar quantity needs to be determined, from which all three velocity components can be derived.
Inserting Equation \ref{eq:scalar_potential} into Equation \ref{eq:irrotational} gives the Laplace equation:
\begin{equation} \label{eq:laplace} \nabla^2 \phi = 0 \end{equation}
2 Summary of boundary conditions
The free surface cannot withstand pressure differences, this is applied via Bernoulli's equation:
\begin{equation} \label{eq:Bernoulli} \frac{\partial \phi}{\partial t} + \frac{1}{2}(\nabla \phi)^2 + \frac{p_0}{\rho} + g\eta = C \end{equation}- Where the constant, \(C = \frac{p_0}{\rho}\). By assuming that the wavelength is much larger than the wave amplitude, the second order term in Equation \ref{eq:Bernoulli} can be assumed to be small, hence the boundary condition becomes:
The free surface is also impermeable, hence the fluid velocity normal to the surface must be equal to the surface velocity:
\begin{equation} \label{eq:fsImpermeable} \frac{\partial \eta}{\partial t} + \frac{\partial \phi}{\partial x}\frac{\partial \eta}{\partial x} + \frac{\partial \phi}{\partial y}\frac{\partial \eta}{\partial y} + \frac{\partial \phi}{\partial z} = 0 \end{equation}- Again, by assuming that the wavelength is much larger than the wave amplitude, the second order terms in Equation \ref{eq:fsImpermeable} can be assumed to be small, hence the boundary condition becomes:
By introducing Equation \ref{eq:fsImpermeableLinear} into the time-derivative of Equation \ref{eq:BernoulliLinear}, the generalized form of the free surface boundary condition is:
\begin{equation} \label{eq:fsBoundaryCondition} \frac{\partial^2 \phi}{\partial t^2} - g \frac{\partial \phi}{\partial z} = 0 \end{equation}Assuming the seabed is flat and impermeable, there must be no vertical velocity component:
\begin{equation} \frac{\partial \phi}{\partial z} = 0 \end{equation}Similarly, the wetted surface of the body in the fluid is impermeable, hence the fluid velocity normal to the body surface, \(u_n\) must be equal to the body velocity in the direction normal to its surface:
\begin{equation} \frac{\partial \phi}{\partial n} = u_n \end{equation}The final condition is that radiated waves must decay with increasing distance from the body:
\begin{equation} \phi \propto (kr)^{\frac{1}{2}}e^{ikr} \end{equation}As \(r \rightarrow \infty\), where \(r\) is the radial distance and \(k\) is the wave number:
\begin{equation} \frac{\omega^2}{g} = ktanh(kh) \end{equation}
3 Solving for hydrodynamic forces
To obtain the hydrodynamic forces experienced by a body in the flow, we essentially need to integrate the dynamic pressure exerted on the (mean) wetted body surface, \(S_b\):
\begin{equation} \label{eq:potentialHydrodynamics} \vec{f}_{hd} = \rho \int_{S_b}\frac{\partial \phi}{\partial t} n dS_b \end{equation}Where \(n\) is the unit vector normal to the body surface.
Under the linearity assumption, the velocity potential can be decomposed into 2 components describing the diffracted and radiated wave fields:
\begin{equation} \phi = \phi_D + \phi_{rad} \end{equation}Where the diffracted wave potential is:
\begin{equation} \phi_D = \phi_0 + \phi_s \end{equation}\(\phi_0\) represents the incident wave potential and \(\phi_s\) represents the scattered wave potential.
Hence, Equation \ref{eq:potentialHydrodynamics} can be expressed in terms of the complex vectors of the excitation and radiation components:
\begin{equation} \hat{f}_{hd} = \hat{f}_{ex} + \hat{f}_{rad} \end{equation}\(f_{ex}\) has two components: the Froude-Krylov force, \(f_{FK}\) - associated with the undisturbed incident wave. And the scatter force, \(f_s\), which results from waves interacting with the body and diffracting/scattering across the free surface1. In the frequency-domain:
\begin{equation} \hat{f}_{FK}(\omega) = i\omega\rho \int_{S_b} \hat{\phi}_0 n dS_b \end{equation} \begin{equation} \hat{f}_{s}(\omega) = i\omega\rho \int_{S_b} \hat{\phi}_s n dS_b \end{equation}As a body oscillates in a fluid, it radiates waves - which results in a damping force experienced by the body, \(f_{rad}\). There is also an added mass effect due the body forcing the motion of some fluid in its vicinity. The analogy of electrical impedance (resistance and reactance) is commonly used (denoted by \(Z(\omega)\)):
\begin{equation} \hat{f}_{rad}(\omega) = -i \omega Z(\omega) \hat{\vec{s}}(\omega) \end{equation} \begin{equation} Z(\omega) = -i\omega\rho \int_{S_b} \phi_{rad} n dS_b = \mathbf{B}(\omega) + i\omega\mathbf{A}(\omega) \end{equation}(Where \(\mathbf{B}(\omega)\) is radiation damping and \(\mathbf{A}(\omega)\) is added mass).
The radiation and diffraction potentials can be represented by the potential formulation of Green's theorem:
\begin{equation} \label{eq:Green} \begin{pmatrix} 2 \pi \\ 4 \pi \end{pmatrix} \phi + \int \int_{S_b} \phi G_{n\xi} dS_{\xi} = \int \int_{S_b} \phi_n G dS_{\xi} \end{equation}- The Green function, \(G(\vec{\xi}, \vec{x})\), corresponds to the potential of an oscillatory source, located at the point \(\vec{\xi}=\vec{x}\) (where \(\vec{\xi}\) is the position vector of a point on the body's surface and \(\vec{x}\) is the position vector of a point within the fluid domain).
Footnotes:
The scatter term, \(f_{s}\) can be seen as a correction to the Froude-Krylov force, to account for body's presence.