6.6: Wave Loading of Stationary and Moving Bodies

The elegance of the linear wave theory permits explicit estimation of wave loads on structures, usually providing reasonable first approximations. We break the forces on the body into three classes:

1. The dynamic pressure load integrated over the body surface, with the assumption that the presence of the body does not affect the flow - it is a "ghost" body. We call this the incident wave force.
2. The flow is deflected from its course because of the presence of the body, assuming here that the body is stationary. This is the diffraction wave force.
3. Forces are created on the body by its moving relative to still water. This is wave-making, due to the body pushing fluid out of the way. We call this the radiation wave force.

This separation of effects clearly depends on linearizing assumptions. Namely, the moving flow interacts with a stationary body in the incident wave and diffraction forces, whereas the stationary flow interacts with a moving body in the radiation force. Further, among the first two forces, we decompose them into a part that is unaffected by the "ghost" body and a part that exists only because of the body’s presence.

Without proof, we will state simple formulas for the diffraction and radiation loads, and then go into more detail on the incident wave (pressure) force.

As a prerequisite, we need the concept of added mass: it can be thought of as the fluid mass that goes along with a body when it is accelerated or decelerated. Forces due to added mass will be seen most clearly in experiments under conditions when the body has a low instantaneous speed, and separation drag forces are minimal. The added mass of various two-dimensional sections and three-dimensional shapes can be looked up in tables. As one simple example, the added mass of a long cylinder exposed to crossflow is precisely the mass of the displaced water: \(A_m = \pi r^2 \rho\) (per unit length).

A very interesting and important aspect of added mass is its connection with the Archimedes force. We observe that the added mass force on a body accelerating in a still fluid is only onehalf that which is seen on a stationary body in an accelerating flow. Why is this? In the case of the accelerating fluid, and regardless of the body shape or size, there must be a pressure gradient in the direction of the acceleration - otherwise the fluid would not accelerate. This non-uniform pressure field integrated over the body will lead to a force. This is entirely equivalent to the Archimedes explanation of why, in a gravitational field, objects float in a fluid. This effect is not at all present if the body is accelerating in a fluid having no pressure gradient. The ”mass” that explains this Archimedes force as an inertial effect is in fact the same as the added mass, and hence the factor of two.

For the development of a simple model, we will focus on a body moving in the vertical direction; we term the vertical motion \(\xi(t)\), and it is centered at \(x = 0\). The vertical wave elevation is \(\eta(t, \, x)\), and the vertical wave velocity is \(w(t, \, x, \, z)\). The body has beam \(2b\) and draft \(T\); its added mass in the vertical direction is taken as \(A_m\). The objective is to write an equation of the form

\[ m \xi_t t + C \xi \, = \, F_I + F_D + F_R, \]

where \(m\) is the material (sectional) mass of the vessel, and \(C\) is the hydrostatic stiffness, the product of \(\rho g\) and the waterplane area: \(C = 2b \rho g\).

The diffraction force is

\[ F_D (t) \, = \, A_m w_t (t, \, x = 0, \, z = -T/2). \]

In words, this force pushes the body upward when the wave is accelerating upward. Note the wave velocity is referenced at the center of the body. This is the effect of the accelerating flow encountering a fixed body - but does not include the Archimedes force. The Archimedes force is derived from the dynamic pressure in the fluid independent of the body, and captured in the incident wave force below. The radiation force is

\[ F_R (t) \, = \, - A_m \xi_{tt}. \]

This force pulls the body downward when it is accelerating upward; it is the effect of the body accelerating through still fluid. Clearly there is no net force when the acceleration of the wave is matched by the acceleration of the body: \(F_D + F_R = 0\).

Now we describe the incident wave force using the available descriptions from the linear wave theory:

\begin{align} \eta (t, \, x) \, &= \, a \cos (\omega t - kx + \psi) \textrm{ and} \\[4pt] p (t, \, x, \, z) \, &= \, \rho ga e^{kz} \cos (\omega t - kx + \psi) - \rho gz. \end{align}

We will neglect the random angle \(\psi\) and the hydrostatic pressure \(−\rho gz\) in our discussion. The task is to integrate the pressure force on the bottom of the structure:

\begin{align} F_I \, &= \, \int\limits_{-b}^{b} p(t, \, x, \, z = -T) \, dx \\[4pt] &= \, \rho ag e^{-kT} \int\limits_{-b}^{b} \cos (\omega t - kx) \, dx \\[4pt] &= \, \dfrac{2 \rho ag}{k} e^{-kT} \cos (\omega t) \sin(kb). \end{align}

As expected, the force varies as \(\cos (\omega t)\). The effect of spatial variation in the \(x\)-direction is captured in the \(\sin (kb)\) term.

If \(kb < 0.6\) or so, then \(\sin (kb)\approx kb\). This is the case when \(b\) is about one-tenth of the wavelength or less, and quite common for smaller vessels in beam seas. Furthermore, \(e^{-kT} \approx 1 - kT\) if \(kT < 0.3\) or so. This is true if the draft is less than about one-twentieth of the wavelength, which is also quite common. Under these conditions, we can rewrite \(F_I\):

\begin{align} F_I \, &\approx \, 2 \rho ga(1 - kT)b \cos (\omega t) \\[4pt] &= \, 2b \rho ga \cos (\omega t) - 2bT \rho \omega^2 a \cos (\omega t) \\[4pt] &= \, C \eta(t, \, x=0) + \nabla \rho w_t(t, \, x=0, \, z=0). \end{align}

Here \(\nabla\) is the (sectional) volume of the vessel. Note that to obtain the second line we used the deepwater dispersion relation \(\omega^2 = kg\).

We can now assemble the complete equation of motion:

\begin{align} m \xi_{tt} + C \xi \, &= \, F_I + F_D + F_R \\[4pt] &= \, C \eta(t, \, x=0) + \nabla \rho w_t(t, \, x=0, \, z=0) + A_m w_t(t, \, x=0, \, z=-T/2) - A_m \xi_{tt}, \textrm{ so that} \\[4pt] (m + A_m) \xi_{tt} + C \xi \, &\approx \, C \eta(t, \, x=0) + (\nabla \rho + A_m) w_t (t, \, x=0, \, z=-T/2). \end{align}

Note that in the last line we have equated the \(z\)-locations at which the fluid acceleration \(w_t\) is taken to \(z = −T/2\). It may seem arbitrary at first, but if we chose the alternative of \(w_t(t, \, x=0, \, z=0)\), we would obtain

\begin{align*} (m + A_m) \xi_{tt} + C \xi \, &= \, C \eta(t, \, x=0) + (\nabla \rho + A_m) w_t (t, \, x=0, \, z=0) \\[4pt] (-(m + A_m) \omega^2 + C) \xi \, &= \, (C - (\nabla \rho + A_m) \omega^2) \eta(t, \, x=0) \longrightarrow \\[4pt] \dfrac{\xi (j \omega)}{\eta (j \omega)} \, &= \, 1, \end{align*}

since \(m\) is equal to \(\nabla \rho\) for a neutrally buoyant body. Clearly the transfer function relating vehicle heave motion to wave elevation cannot be unity - the vessel does not follow all waves equally! If we say that \(w_t(t, \, x=0, \, z=-T/2) = \gamma w_t(t, \, x=0, \, z=0)\), where \(\gamma < 1\)is a function of the wavelength and \(T\), the above equation becomes more suitable:

\begin{align*} (-(m + A_m) \omega^2 + C)\xi \, &= \, (C - (\gamma \nabla \rho + A_m) \omega^2) \eta(t, \, x=0) \longrightarrow \\[4pt] \dfrac{\xi (j \omega)}{\eta (j \omega)} \, &= \, \dfrac{C - (\gamma \nabla \rho + A_m) \omega^2}{C - (m + A_m) \omega^2} \end{align*}

This transfer function has unity gain at low frequencies and gain \( (\gamma \nabla \rho + A_m) / (m + A_m) \) at high frequencies. It has zero magnitude at \(\omega = \sqrt{C / (\gamma \nabla \rho + A_m)}\), but very high magnitude (resonance) at \(\omega = \sqrt{C / (m + A_m)}\). The zero occurs at a higher frequency than the resonance because \(\gamma < 1\).

In practice, the approximation that \(w_t\) should be taken at \(z = −T/2\) is reasonable. However, one significant factor missing from our analysis is damping, which depends strongly on the specific shape of the hull. Bilge keels and sharp corners cause damping, as does the creation of radiated waves.