WikiWaves - User contributions [en]

Linear and Second-Order Wave Theory

2019-07-17T09:03:11Z

Xjwiki: /* Dynamic condition */

{{Ocean Wave Interaction with Ships and Offshore Structures
| chapter title = [[Linear and Second-Order Wave Theory]]
| next chapter = [[Linear Plane Progressive Regular Waves]]
| previous chapter = [[Conservation Laws and Boundary Conditions]]
}}

{{complete pages}}

We saw in [[Conservation Laws and Boundary Conditions]] that the potential flow model for wave propagation is given Laplace's equation plus the free-surface conditions. In this section we present the linear and second order theory for these equations. The linear theory is valid for small wave heights and the second order theory is an improvement on this. However, neither of these theories work for very steep waves and of course the potential theory breaks down once the wave begins to break and completely different methods are required in this situation.

== Linearization of Free-surface Conditions ==

We use [http://en.wikipedia.org/wiki/Perturbation_theory perturbation theory] to expand the solution as follows
<center><math>
\begin{align}
\zeta &= \zeta_1 + \zeta_2 + \zeta_3 + \cdots \\
\Phi &= \Phi_1 + \Phi_2 + \Phi_3 + \cdots
\end{align}
</math></center>
where we are assuming that there exists a small parameter (the wave slope) and that with respect to this the
<math>\Phi_i</math> is proportional to <math>\epsilon^i</math> or <math>O(\epsilon^i)</math>. We then derive the boundary value problem for <math> \zeta_i,\Phi_i </math>. Rarely we need to go beyond <math> i = 3 </math> (in fact it is unlikely that the terms beyond
this will improve the accuracy.

In this section we will only derive the free-surface conditions up to second order. Remember that <math>\nabla^2 \Phi_i =0</math> for all <math>i</math>
We expand the kinematic and dynamic free surface conditions about the <math>z=0</math> plane and derive statements for the unknown pairs <math> (\Phi_1,\zeta_1 </math> and <math> (\Phi_2, \zeta_2) </math> at <math> z=0 </math>. The same technique can be used to linearize the body boundary condition at <math> U=0 </math> (zero speed) and <math> U>0 </math> (forward speed).

== Kinematic condition ==

The fully non-linear kinematic condition was derived in [[Conservation Laws and Boundary Conditions]] and we begin with this equation
<center><math> \left ( \frac{\partial \zeta}{\partial t} + \nabla \Phi \cdot \nabla \zeta \right )_{z=\zeta} = \left ( \frac{\partial \Phi}{\partial z} \right )_{z=\zeta} </math></center>
We expand this equation about <math>\zeta = 0</math>, which we can do because we have assumed that the slope is small. In fact the slope is our parameter <math>\epsilon</math>. It is obvious at this point that the theory does not apply to very steep waves. This gives us the following equation
<center><math>
\left( \frac{\partial\zeta}{\partial t} + \nabla \Phi \cdot \nabla \zeta \right)_{z=0} + \zeta \frac{\partial}{\partial z} \left( \frac{\partial\zeta}{\partial t} + \nabla \Phi \cdot \nabla \zeta \right)_{z=0} + \;\cdots = \left( \frac{\partial\Phi}{\partial z} \right)_{z=0} + \zeta \left( \frac{\partial^2 \Phi}{\partial z^2} \right)_{z=0} + \;\cdots
</math></center>
where we have only taken the first order expansion. We then substitute our expressions
<center> <math>
\begin{align}
\zeta &= \zeta_1 + \zeta_2 + \cdots \\
\Phi &= \Phi_1 + \Phi_2 + \cdots
\end{align}
</math></center>
and keep terms of <math>\ O(\varepsilon), \ O(\varepsilon^2)</math>, remembering that <math>\zeta_1\Phi_1</math> is <math>O(\varepsilon^2)</math> etc.

== Dynamic condition ==

The fully non-linear Dynamic condition was derived in [[Conservation Laws and Boundary Conditions]] and is given by
<center><math>
\zeta (x,y,t) = -\frac{1}{g} \left( \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi \right)_{z=\zeta}
</math></center>

<center><math>
\left . \begin{matrix}
\zeta =-\dfrac{1}{g} \left( \dfrac{\partial\Phi}{\partial t} + \dfrac{1}{2} \nabla\Phi \cdot \nabla\Phi \right)_{z=0}\\
-\dfrac{1}{g} \zeta \dfrac{\partial}{\partial z} \left( \dfrac{\partial\Phi}{\partial t} + \dfrac{1}{2} \nabla\Phi \cdot \nabla\Phi \right)_{z=0} + \cdots
\end{matrix} \right \}
\begin{matrix}
\zeta = \zeta_1 +\zeta_2 + \cdots \\
\Phi = \Phi_1 + \Phi_2 + \cdots
\end{matrix}
</math></center>

== Linear problem ==

The linear problem is the <math>O(\varepsilon)</math> problem derived by equating the terms which are proportional to <math>\epsilon</math>.

This can be done straight forwardly and gives the following expressions

<center><math> \frac{\partial\zeta_1}{\partial t} = \frac{\partial\Phi_1}{\partial z} , \ z=0; </math></center>
which follows from the Kinematic equation and
<center><math> \zeta_1 = -\frac{1}{g} \frac{\partial\Phi_1}{\partial t}, \ z=0; </math></center>
which follows from the Dynamic equation. These are the linear free surface conditions.

=== Derivation using Bernoulli's equation ===

The pressure from Bernoulli, <math> \omega </math> constant terms set equal to zero, at a fixed point in the fluid domain at <math> \mathbf{x}=(x,y,z) </math> is given by:
<center><math> P = - \rho \left( \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla\Phi \cdots \nabla\Phi + gz \right); </math></center>
When then make the perturbation expansion for the potential and the pressure
<center><math> \Phi = \Phi_1 + \Phi_2 + \cdots </math></center>
and
<center><math> P = P_0 + P_1 + P_2 + \cdots </math></center>.

This allows us to derive
<center><math> P_0 = -\rho g z \,, </math></center>
which is called the Hydrostatic pressure and
<center><math> P_1 = - \rho \frac{\partial\Phi_1}{\partial t} </math></center>
which is the linear pressure.

=== Classical linear free surface condition ===

If we eliminate <math> \zeta_1 </math> from the kinematic and dynamic free surface conditions, we obtain the classical linear free surface condition:
<center><math> \begin{cases}
\dfrac{\partial^2\Phi_1}{\partial t^2} + g \dfrac{\partial\Phi_1}{\partial z} = 0, \qquad z=0\\
\zeta_1 = - \dfrac{1}{g} \dfrac{\partial\Phi_1}{\partial t}, \qquad z=0
\end{cases} </math></center>

With:

<center><math> P_1 = - \rho \frac{\partial\Phi_1}{\partial t}, \qquad \mbox{at some fixed point} \ \mathbf{x} </math></center>

Note that on <math> z=0, \ P_1 \ne 0 </math> in fact it can obtained from the expressions above in the form

<center><math> P_1 = -\rho g \zeta_1, \qquad z=0 </math></center>

So linear theory states that the linear perturbation pressure on the <math> z=0 \, </math> plane due to a surface wave disturbance is equal to the positive (negative) "hydrostatic" pressure induced by the positive (negative) wave elevation <math> \zeta_1 \, </math>.

== Second-order problem ==

The second order equations can also be derived straight forwardly. The kinematic condition is
<center><math> \frac{\partial\zeta_2}{\partial t} + \nabla\Phi_1 \cdot \nabla\zeta_1 = \frac{\partial\Phi_2}{\partial z} + \zeta_1 \frac{\partial^2 \Phi_1}{\partial z^2}, \quad z=0 </math></center>
and the dynamic condition
<center><math> \zeta_2 = - \frac{1}{g} \left( \frac{\partial\Phi_2}{\partial t} + \frac{1}{2} \nabla\Phi_1 \cdot \nabla\Phi_1 \right)_{z=0} - \frac{1}{g} \zeta_1 \frac{\partial^2\Phi_1}{\partial z \partial t}, \quad z=0 </math></center>

Alternatively, the known linear terms may be moved in the right-hand side as forcing functions, leading to:

=== Kinematic second-order condition ===

<center>
<math> \frac{\partial\zeta_2}{\partial t} - \frac{\partial\Phi_2}{\partial z} = \zeta_1 \frac{\partial^2 \Phi_1}{\partial z^2} - \nabla\Phi_1 \cdot \nabla\zeta_1; \quad z=0 </math>
</center>

=== Dynamic second-order condition ===

<center>
<math> \zeta_2 + \frac{1}{g} \frac{\partial\Phi_2}{\partial t} = - \frac{1}{g} \left( \frac{1}{2} \nabla\Phi_1 \cdot \nabla\Phi_1 + \zeta_1 \frac{\partial^2\Phi_1}{\partial z \partial t} \right)_{z=0} </math>
</center>
where the second order pressure is given by
<center>
<math> P_2 = -\rho \left( \frac{\partial\Phi_2}{\partial t} + \frac{1}{2} \nabla\Phi_1 \cdot \nabla\Phi_1 \right); \quad \mbox{at} \ \mathbf{x} </math>
</center>

The very attractive feature of second order surface wave theory is that it allows the prior solution of the linear problem which is often possible analytically and numerically.
The linear solution is then used as a forcing function for the solution of the second order problem. This is often possible analytically and in most cases numerically in the absence or presence of bodies.
Linear and second-order theories are also very appropriate to use for the modeling of surface waves as stochastic processes.
Both theories are very useful in practice, particularly in connection with wave-body interactions.

-----

This article is based on the MIT open course notes and the original article can be found
[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/0B7683D3-9B31-453E-B98F-9F71A3C36C58/0/lecture2.pdf here]

[[Category:Linear Water-Wave Theory]]
[[Category:Nonlinear Water-Wave Theory]]

Second-Order Wave Theory

2019-07-17T08:41:58Z

Xjwiki: /* Second-Order Bi-Chromatic Wave Theory */

{{Ocean Wave Interaction with Ships and Offshore Structures
| chapter title = Second-Order Wave Theory
| next chapter = [[Second-Order Wave Effect On Offshore Platforms]]
| previous chapter = [[Wave Drift Forces]]
}}
{{incomplete pages}}

==Introduction==
Linear theory is a very powerful and useful means of modeling the responses of floating bodies in regular and random waves. Yet, for certain types of floating structures and certain types of effects nonlinear extensions are necessary. Mean drift forces in regular waves do not require an account of nonlinear effects. However, the definition of the quadratically nonlinear slow-drift excitation force in a seastate requires a second-order theory in principle. Tension-leg platforms that operate in large water depths for the exploration and extraction of hydrocarbons offshore often undergo resonant heave oscillations with periods of a few seconds which can only be excited by nonlinear wave effects. Their treatment requires a second-order theory and perhaps a more exact treatment.

==Second-Order Free Surface Condition==

The second-order free surface condition of free waves was derived earlier in the form:

<center><math> \phi = \phi_1 + \phi_2 + \phi_3 + \cdots \, </math></center>
<center><math> \zeta = \zeta_1 + \zeta_2 + \zeta_3 + \cdots \, </math></center>

Kinematic condition:

<center><math> \frac{\partial\zeta_2}{\partial t} - \frac{\partial\phi_2}{\partial t} = \zeta_1 \frac{\partial^2\phi_1}{\partial z^2} - \nabla\phi_1 \cdot \nabla\zeta_1 \quad \text{at }z=0 \, </math></center>

Dynamic condition:

<center><math> \zeta_2 + \frac{1}{g} \frac{\partial\phi_2}{\partial t} = - \frac{1}{g} \left( \frac{1}{2} \nabla\phi_1 \cdot \nabla\phi_1 + \zeta_1 \frac{\partial^2\phi_1}{\partial z \partial t} \right)_{z=0} </math></center>

Hydrodynamic pressure:

<center><math> P_2 = - \rho \left( \frac{\partial\phi_2}{\partial t} + \frac{1}{2} \nabla \phi_1 \cdot \nabla \phi_1 \right) </math></center>

at some fixed point in the fluid domain.

Assuming that the linear solution is known and eliminating <math>\zeta_2\,</math> from the kinematic condition, we obtain the more familiar form:

<center><math> \nabla^2\phi_2 = 0 \,</math></center>
<center><math> \frac{\partial^2\phi_2}{\partial t^2} + g \frac{\partial\phi_2}{\partial z} = - \frac{\partial}{\partial t} \left( \nabla\phi_1 \cdot \nabla\phi_1 \right) + \frac{1}{g} \frac{\partial\phi_1}{\partial t} \frac{\partial}{\partial z} \left( \frac{\partial^2\phi_1}{\partial t^2} + g \frac{\partial \phi_1}{\partial z} \right)_{z=0} </math></center>
<center><math> \zeta_2 = - \frac{1}{g} \left( \frac{\partial\phi_2}{\partial t} + \frac{1}{2} \nabla\phi_1\cdot\nabla\phi_1 + \zeta_1 \frac{\partial^2\phi_1}{\partial z \partial t} \right)_{z=0} </math></center>

The solution of the above boundary-value problem is available in closed from when the linear potential <math>\phi_1\,</math> is the linear superposition of regular plane progressive waves.

Very useful insights follow from this solution even in the absence of floating bodies.

The statement of the second-order problem when a body is present requires the derivation of a second-order body boundary condition which is involved. The solution of the second-order wave body interaction problem is only possible numerically and is in general a complex time consuming task.

==Second-Order Bi-Chromatic Wave Theory==

The forcing in the right-hand side of the second order free surface condition is a quadratic function of the linear solution. Hence, if the linear problem is written as the sum of monochromatic wave components, the treatment of the most general second order problem can be accomplished by treating the second-order free-surface condition by a bi-chromatic disturbance.

The above conclusion is analogous to the following identity:

<center><math> \mathrm{Re} \left\{ A_1 e^{i\omega t} \right\} \mathrm{Re} \left\{ A_2 e^{i\omega t} \right\} = \frac{1}{2} \mathrm{Re} \left\{ A_1 A_2 e^{i\left(\omega_1+\omega_2\right)t} + A_1 A_2^* e^{i\left(\omega_1-\omega_2\right)t}\right\} </math></center>

It follows that the forcing and hence the solution of the second-order problem is the linear super position of a sum and a difference-frequency components.

The velocity potential of a directional seastate represented by the linear superposition of plane progressive waves of various frequencies and wave headings takes the familiar form:

<center><math> \Phi_1 = \mathrm{Re} \sum_k \sum_m \frac{i g A_{km}}{\omega_k} e^{\nu_k z - i\nu_k x \cos\theta_m - i\nu_k y \sin\theta_m} e^{i\omega_k t+i\phi_{km}} </math></center>

where the summations are over a sufficiently large number of wave frequencies <math>\omega_k\,</math> and wave headings <math>\theta_m\,</math> necessary to characterize the ambient seastate.

The phase angles <math>\phi_{km}\,</math> are sampled from the uniform distribution over <math> (-\pi,\pi] \, </math> and the wave amplitudes <math>A_{km}\,</math> are chosen to conform to the ambient wave spectrum.

<math>A_{km}\,</math> may be selected to be deterministic or may be drawn from a Rayleigh distribution independently from the phase angle <math>\phi_{km}\,</math>. The latter choice has been found to be superior in practice when long records are needed with very long periodicity.

Upon substitution of <math> \Phi_1\,</math> into the right-hand side of the second-order free surface condition, sum- and difference- frequency terms arise as illustrated above.

We therefore define the yet unknown second-order potential as follows:

<center><math> \Phi_2 = \mathrm{Re} \left\{ \varphi_2^+ e^{i\left(\omega_k+\omega_\rho\right)t} + \varphi_2^- e^{i\left(\omega_k-\omega_\rho\right)t} \right\} </math></center>

So the principal task of second-order theory in the absence of floating bodies is to determine the complex second-order potentials <math>\varphi_2^+\,</math> and <math> \varphi_2^-\,</math>.

They satisfy second-order free-surface conditions stated below. Use is made of the following identity satisfied by regular plane progressive wave components:

<center><math> \frac{\mathrm{d}}{\mathrm{d}z} \left( \frac{\partial^2 \phi_1}{\partial t^2} + g \frac{\partial\phi_1}{\partial z} \right) = 0 , \quad \mbox{any} \ z \leq 0 \, </math></center>

Making use of the above identity it is easy to show that:

<center><math> - {\Omega^+}^2 \varphi_2^+ + g \frac{\partial \varphi_2^+}{\partial z} = - i \Omega^+ \nabla \varphi_{1km} \cdot \nabla \varphi_{2ln} \quad z=0 </math></center>

and

<center><math> - {\Omega^-}^2 \varphi_2^- + g \frac{\partial \varphi_2^-}{\partial z} = - i \Omega^- \nabla \varphi_{1km} \cdot \nabla \varphi_{2ln}^* \quad z=0 </math></center>

where <math> \varphi_{1km}\,</math> and <math> \varphi_{2ln}\,</math> are the complex velocity potentials of the regular plane progressive waves the ambient seastate consists of.

It follows that <math> \varphi_2^\pm \, </math> each depend on 4 indices, two for the frequencies <math> \omega_k \And \omega_l \, </math> and two for the respective headings <math> \theta_m \, </math> and <math> \theta_l \, </math>.

The solution of the above forced free surface problems together with <math> \nabla^2\varphi_2^\pm = 0 \, </math> in the fluid domain, is a straightforward exercise in Fourier Theory. The solutions for <math>\varphi_2^\pm \, </math> are:

<center><math> \varphi_2^\pm = \frac{\pm i}{2} \Omega^\pm A_k A_l \frac{\omega_k\omega_l}{g} \left[ 1 \mp \cos \left( \theta_m - \theta_l \right) \right] \cdot \frac{e^{z\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)}}{\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)^{1/2} - \frac{{\Omega^\pm}^2}{g}} e^{ - i \alpha^\pm x - i \beta^\pm y + i \left( \phi_k \pm \phi_l \right)} </math></center>

where:

<center><math> \alpha^\pm = \nu_k \cos \theta_m \pm \nu_l \cos\theta_n \, </math></center>
<center><math> \beta^\pm = \nu_k \sin\theta_m \pm \nu_l \sin\theta_n \, </math></center>

The real second-order potential follows by quadruple summation over all pairs of frequencies and wave headings and is not stated here.

==Special Cases==

A number of special cases are noteworthy to discuss since they convey the new physics introduced by the second-order potentials. All effects discussed below are additive to the linear solution by virtue of perturbation theory and <math> O \left(A_k A_l\right) = O \left(A^2\right) \,</math>. An in depth discussion of perturbation theory, its extensions and limitations in deep and shallow waters is presented in [[Mei 1983|Mei 1983]].

<center><math>\theta_m = \theta_n = \theta : \quad \mbox{unidirectional waves}\,</math></center>
<center><math> \varphi_2^+ \equiv 0 \, </math></center>
<center><math> \varphi_2^- = - i \Omega^- A_k A_l \frac{\omega_k\omega_l}{g} \frac{e^{Z\left|N^-\right|}}{\left|N^-\right|-\frac{{\Omega^-}^2}{g}} e^{-i N^- (x\cos\theta+y\sin\theta)=i\left(\phi_k-\phi_l\right)} </math></center>
where:
<center><math> N^- = \nu_k - \nu_l, \quad \Omega^- = \omega_k - \omega_l \, </math></center>

So the second-order potential is identically zero in unidirectional waves of the same frequency.

Verify that this is not however the case for the second-order wave elevation and second-order pressure.

Derive as an exercise the second-order wave elevation when <math> \omega_k = \omega_l = \omega \, </math> and show that it contributes a correction to the linear sinusoidal solution which produces more steepness at the crests and less at at troughs.

Carrying the perturbation solution when <math>\omega_k = \omega_l =\omega \, </math> to arbitrarily high order (it has been applied numerically with over 100 terms!). The famous solution by Stokes is recovered:

The crests in the limit become cusped with an enclosed angle of <math>120^\circ\,</math> and the limiting value of the <math> \frac{H}{\lambda}\,</math> ratio is about <math>\frac{1}{7}\,</math>.

So to the extent that periodic waves of this form can be created or observed these limiting properties are useful to remember and know they can be obtained by perturbation theory as well.

When Stokes waves are reflected off a wall the limiting enclosed angle is <math>90^\circ\,</math>.

<center><math> \theta_m + \pi = \theta_n = \theta \, </math></center>

In this case we have two waves propagating in opposite directions.

It can be shown that now:

<center><math> \varphi_2^- \equiv 0 \, </math></center>

Yet the corresponding components of the wave elevation and pressure are again now zero.

The sum-frequency potential becomes:

<center><math> \varphi_2^+ = - i \Omega^+ A_k A_l \frac{\omega_k\omega_l}{g} \frac{e^{Z\left|N^-\right|}}{\left|N^-\right|-\frac{{\Omega^-}^2}{g}} e^{i N^- (x\cos\theta+y\sin\theta)+i\left(\phi_k-\phi_l\right)} </math></center>
where <math> N^- = \omega_k - \omega_l \, </math>.

A noteworthy property of <math>\varphi_2^+\,</math> is that when <math> \omega_k = \omega_l \, </math> or <math> N^- = 0 \, </math>, the velocity potential and the term in the second order hydrodynamic pressure proportional to <math>\frac{\partial\phi_2}{\partial t}\,</math>, does not attenuate with depth!

This is a nonlinear wave property when waves propagate in opposite directions and has been measured in ocean pressure fields at large depths. It has been considered responsible of microseisms on the ocean floor.

This lack of attenuation of the hydrodynamic second-order pressure also arises when diffraction occurs off large volume floating platforms, like TLP's. The possible result is a large enough high-frequency excitation in heave & pitch which could contribute appreciably to the resonant responses of TLP tethers.

In order to assess the full effect of second-order nonlinearities on floating platforms (and certain large ships that may undergo flexural vibrations) the complete solution of the sum-frequency second-order problem is necessary.

In random waves, it is also believed that the third-order solution is necessary. This being a formidable task, fully nonlinear solutions may be appropriate.

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]

Second-Order Wave Theory

2019-07-17T08:41:04Z

Xjwiki: /* Second-Order Bi-Chromatic Wave Theory */

{{Ocean Wave Interaction with Ships and Offshore Structures
| chapter title = Second-Order Wave Theory
| next chapter = [[Second-Order Wave Effect On Offshore Platforms]]
| previous chapter = [[Wave Drift Forces]]
}}
{{incomplete pages}}

==Introduction==
Linear theory is a very powerful and useful means of modeling the responses of floating bodies in regular and random waves. Yet, for certain types of floating structures and certain types of effects nonlinear extensions are necessary. Mean drift forces in regular waves do not require an account of nonlinear effects. However, the definition of the quadratically nonlinear slow-drift excitation force in a seastate requires a second-order theory in principle. Tension-leg platforms that operate in large water depths for the exploration and extraction of hydrocarbons offshore often undergo resonant heave oscillations with periods of a few seconds which can only be excited by nonlinear wave effects. Their treatment requires a second-order theory and perhaps a more exact treatment.

==Second-Order Free Surface Condition==

The second-order free surface condition of free waves was derived earlier in the form:

<center><math> \phi = \phi_1 + \phi_2 + \phi_3 + \cdots \, </math></center>
<center><math> \zeta = \zeta_1 + \zeta_2 + \zeta_3 + \cdots \, </math></center>

Kinematic condition:

<center><math> \frac{\partial\zeta_2}{\partial t} - \frac{\partial\phi_2}{\partial t} = \zeta_1 \frac{\partial^2\phi_1}{\partial z^2} - \nabla\phi_1 \cdot \nabla\zeta_1 \quad \text{at }z=0 \, </math></center>

Dynamic condition:

<center><math> \zeta_2 + \frac{1}{g} \frac{\partial\phi_2}{\partial t} = - \frac{1}{g} \left( \frac{1}{2} \nabla\phi_1 \cdot \nabla\phi_1 + \zeta_1 \frac{\partial^2\phi_1}{\partial z \partial t} \right)_{z=0} </math></center>

Hydrodynamic pressure:

<center><math> P_2 = - \rho \left( \frac{\partial\phi_2}{\partial t} + \frac{1}{2} \nabla \phi_1 \cdot \nabla \phi_1 \right) </math></center>

at some fixed point in the fluid domain.

Assuming that the linear solution is known and eliminating <math>\zeta_2\,</math> from the kinematic condition, we obtain the more familiar form:

<center><math> \nabla^2\phi_2 = 0 \,</math></center>
<center><math> \frac{\partial^2\phi_2}{\partial t^2} + g \frac{\partial\phi_2}{\partial z} = - \frac{\partial}{\partial t} \left( \nabla\phi_1 \cdot \nabla\phi_1 \right) + \frac{1}{g} \frac{\partial\phi_1}{\partial t} \frac{\partial}{\partial z} \left( \frac{\partial^2\phi_1}{\partial t^2} + g \frac{\partial \phi_1}{\partial z} \right)_{z=0} </math></center>
<center><math> \zeta_2 = - \frac{1}{g} \left( \frac{\partial\phi_2}{\partial t} + \frac{1}{2} \nabla\phi_1\cdot\nabla\phi_1 + \zeta_1 \frac{\partial^2\phi_1}{\partial z \partial t} \right)_{z=0} </math></center>

The solution of the above boundary-value problem is available in closed from when the linear potential <math>\phi_1\,</math> is the linear superposition of regular plane progressive waves.

Very useful insights follow from this solution even in the absence of floating bodies.

The statement of the second-order problem when a body is present requires the derivation of a second-order body boundary condition which is involved. The solution of the second-order wave body interaction problem is only possible numerically and is in general a complex time consuming task.

==Second-Order Bi-Chromatic Wave Theory==

The forcing in the right-hand side of the second order free surface condition is a quadratic function of the linear solution. Hence, if the linear problem is written as the sum of monochromatic wave components, the treatment of the most general second order problem can be accomplished by treating the second-order free-surface condition by a bi-chromatic disturbance.

The above conclusion is analogous to the following identity:

<center><math> \mathrm{Re} \left\{ A_1 e^{i\omega t} \right\} \mathrm{Re} \left\{ A_2 e^{i\omega t} \right\} = \frac{1}{2} \mathrm{Re} \left\{ A_1 A_2 e^{i\left(\omega_1+\omega_2\right)t} + A_1 A_2^* e^{i\left(\omega_1-\omega_2\right)t}\right\} </math></center>

It follows that the forcing and hence the solution of the second-order problem is the linear super position of a sum and a difference-frequency components.

The velocity potential of a directional seastate represented by the linear superposition of plane progressive waves of various frequencies and wave headings takes the familiar form:

<center><math> \Phi_1 = \mathrm{Re} \sum_k \sum_m \frac{i g A_{km}}{\omega_k} e^{\nu_k z - i\nu_k x \cos\theta_m - i\nu_k y \sin\theta_m} e^{i\omega_k t+i\phi_{km}} </math></center>

where the summations are over a sufficiently large number of wave frequencies <math>\omega_k\,</math> and wave headings <math>\theta_m\,</math> necessary to characterize the ambient seastate.

The phase angles <math>\phi_{km}\,</math> are sampled from the uniform distribution over <math> (-\pi,\pi] \, </math> and the wave amplitudes <math>A_{km}\,</math> are chosen to conform to the ambient wave spectrum.

<math>A_{km}\,</math> may be selected to be deterministic or may be drawn from a Rayleigh distribution independently from the phase angle <math>\phi_{km}\,</math>. The latter choice has been found to be superior in practice when long records are needed with very long periodicity.

Upon substitution of <math> \Phi_1\,</math> into the right-hand side of the second-order free surface condition, sum- and difference- frequency terms arise as illustrated above.

We therefore define the yet unknown second-order potential as follows:

<center><math> \Phi_2 = \mathrm{Re} \left\{ \varphi_2^+ e^{i\left(\omega_k+\omega_\rho\right)t} + \varphi_2^- e^{i\left(\omega_k-\omega_\rho\right)t} \right\} </math></center>

So the principal task of second-order theory in the absence of floating bodies is to determine the complex second-order potentials <math>\varphi_2^+\,</math> and <math> \varphi_2^-\,</math>.

They satisfy second-order free-surface conditions stated below. Use is made of the following identity satisfied by regular plane progressive wave components:

<center><math> \frac{\mathrm{d}}{\mathrm{d}z} \left( \frac{\partial^2 \phi_1}{\partial t^2} + g \frac{\partial\phi_1}{\partial z} \right) = 0 , \quad \mbox{any} \ z \leq 0 \, </math></center>

Making use of the above identity it is easy to show that:

<center><math> - {\Omega^+}^2 \varphi_2^+ + g \frac{\partial \varphi_2^+}{\partial z} = - i \Omega^+ \nabla \varphi_{1km} \cdot \nabla \varphi_{2ln} \quad z=0 </math></center>

and

<center><math> - {\Omega^-}^2 \varphi_2^- + g \frac{\partial \varphi_2^-}{\partial z} = - i \Omega^- \nabla \varphi_{1km} \cdot \nabla \varphi_{2ln}^* \quad z=0 </math></center>

where <math> \varphi_{1km}\,</math> and <math> \varphi_{2ln}\,</math> are the complex velocity potentials of the regular plane progressive waves the ambient seastate consists of.

It follows that <math> \varphi_2^\pm \, </math> each depend on 4 indices, two for the frequencies <math> \omega_k \And \omega_l \, </math> and two for the respective headings <math> \theta_m \, </math> and <math> \theta_l \, </math>.

The solution of the above forced free surface problems together with <math> \nabla^2\varphi_2^\pm = 0 \, </math> in the fluid domain, is a straightforward exercise in Fourier Theory. The solutions for <math>\varphi_2\pm \, </math> are:

<center><math> \varphi_2^\pm = \frac{\pm i}{2} \Omega^\pm A_k A_l \frac{\omega_k\omega_l}{g} \left[ 1 \mp \cos \left( \theta_m - \theta_l \right) \right] \cdot \frac{e^{z\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)}}{\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)^{1/2} - \frac{{\Omega^\pm}^2}{g}} e^{ - i \alpha^\pm x - i \beta^\pm y + i \left( \phi_k \pm \phi_l \right)} </math></center>

where:

<center><math> \alpha^\pm = \nu_k \cos \theta_m \pm \nu_l \cos\theta_n \, </math></center>
<center><math> \beta^\pm = \nu_k \sin\theta_m \pm \nu_l \sin\theta_n \, </math></center>

The real second-order potential follows by quadruple summation over all pairs of frequencies and wave headings and is not stated here.

==Special Cases==

A number of special cases are noteworthy to discuss since they convey the new physics introduced by the second-order potentials. All effects discussed below are additive to the linear solution by virtue of perturbation theory and <math> O \left(A_k A_l\right) = O \left(A^2\right) \,</math>. An in depth discussion of perturbation theory, its extensions and limitations in deep and shallow waters is presented in [[Mei 1983|Mei 1983]].

<center><math>\theta_m = \theta_n = \theta : \quad \mbox{unidirectional waves}\,</math></center>
<center><math> \varphi_2^+ \equiv 0 \, </math></center>
<center><math> \varphi_2^- = - i \Omega^- A_k A_l \frac{\omega_k\omega_l}{g} \frac{e^{Z\left|N^-\right|}}{\left|N^-\right|-\frac{{\Omega^-}^2}{g}} e^{-i N^- (x\cos\theta+y\sin\theta)=i\left(\phi_k-\phi_l\right)} </math></center>
where:
<center><math> N^- = \nu_k - \nu_l, \quad \Omega^- = \omega_k - \omega_l \, </math></center>

So the second-order potential is identically zero in unidirectional waves of the same frequency.

Verify that this is not however the case for the second-order wave elevation and second-order pressure.

Derive as an exercise the second-order wave elevation when <math> \omega_k = \omega_l = \omega \, </math> and show that it contributes a correction to the linear sinusoidal solution which produces more steepness at the crests and less at at troughs.

Carrying the perturbation solution when <math>\omega_k = \omega_l =\omega \, </math> to arbitrarily high order (it has been applied numerically with over 100 terms!). The famous solution by Stokes is recovered:

The crests in the limit become cusped with an enclosed angle of <math>120^\circ\,</math> and the limiting value of the <math> \frac{H}{\lambda}\,</math> ratio is about <math>\frac{1}{7}\,</math>.

So to the extent that periodic waves of this form can be created or observed these limiting properties are useful to remember and know they can be obtained by perturbation theory as well.

When Stokes waves are reflected off a wall the limiting enclosed angle is <math>90^\circ\,</math>.

<center><math> \theta_m + \pi = \theta_n = \theta \, </math></center>

In this case we have two waves propagating in opposite directions.

It can be shown that now:

<center><math> \varphi_2^- \equiv 0 \, </math></center>

Yet the corresponding components of the wave elevation and pressure are again now zero.

The sum-frequency potential becomes:

<center><math> \varphi_2^+ = - i \Omega^+ A_k A_l \frac{\omega_k\omega_l}{g} \frac{e^{Z\left|N^-\right|}}{\left|N^-\right|-\frac{{\Omega^-}^2}{g}} e^{i N^- (x\cos\theta+y\sin\theta)+i\left(\phi_k-\phi_l\right)} </math></center>
where <math> N^- = \omega_k - \omega_l \, </math>.

A noteworthy property of <math>\varphi_2^+\,</math> is that when <math> \omega_k = \omega_l \, </math> or <math> N^- = 0 \, </math>, the velocity potential and the term in the second order hydrodynamic pressure proportional to <math>\frac{\partial\phi_2}{\partial t}\,</math>, does not attenuate with depth!

This is a nonlinear wave property when waves propagate in opposite directions and has been measured in ocean pressure fields at large depths. It has been considered responsible of microseisms on the ocean floor.

This lack of attenuation of the hydrodynamic second-order pressure also arises when diffraction occurs off large volume floating platforms, like TLP's. The possible result is a large enough high-frequency excitation in heave & pitch which could contribute appreciably to the resonant responses of TLP tethers.

In order to assess the full effect of second-order nonlinearities on floating platforms (and certain large ships that may undergo flexural vibrations) the complete solution of the sum-frequency second-order problem is necessary.

In random waves, it is also believed that the third-order solution is necessary. This being a formidable task, fully nonlinear solutions may be appropriate.

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]

Second-Order Wave Theory

2019-07-17T08:38:32Z

Xjwiki: /* Second-Order Bi-Chromatic Wave Theory */

{{Ocean Wave Interaction with Ships and Offshore Structures
| chapter title = Second-Order Wave Theory
| next chapter = [[Second-Order Wave Effect On Offshore Platforms]]
| previous chapter = [[Wave Drift Forces]]
}}
{{incomplete pages}}

==Introduction==
Linear theory is a very powerful and useful means of modeling the responses of floating bodies in regular and random waves. Yet, for certain types of floating structures and certain types of effects nonlinear extensions are necessary. Mean drift forces in regular waves do not require an account of nonlinear effects. However, the definition of the quadratically nonlinear slow-drift excitation force in a seastate requires a second-order theory in principle. Tension-leg platforms that operate in large water depths for the exploration and extraction of hydrocarbons offshore often undergo resonant heave oscillations with periods of a few seconds which can only be excited by nonlinear wave effects. Their treatment requires a second-order theory and perhaps a more exact treatment.

==Second-Order Free Surface Condition==

The second-order free surface condition of free waves was derived earlier in the form:

<center><math> \phi = \phi_1 + \phi_2 + \phi_3 + \cdots \, </math></center>
<center><math> \zeta = \zeta_1 + \zeta_2 + \zeta_3 + \cdots \, </math></center>

Kinematic condition:

<center><math> \frac{\partial\zeta_2}{\partial t} - \frac{\partial\phi_2}{\partial t} = \zeta_1 \frac{\partial^2\phi_1}{\partial z^2} - \nabla\phi_1 \cdot \nabla\zeta_1 \quad \text{at }z=0 \, </math></center>

Dynamic condition:

<center><math> \zeta_2 + \frac{1}{g} \frac{\partial\phi_2}{\partial t} = - \frac{1}{g} \left( \frac{1}{2} \nabla\phi_1 \cdot \nabla\phi_1 + \zeta_1 \frac{\partial^2\phi_1}{\partial z \partial t} \right)_{z=0} </math></center>

Hydrodynamic pressure:

<center><math> P_2 = - \rho \left( \frac{\partial\phi_2}{\partial t} + \frac{1}{2} \nabla \phi_1 \cdot \nabla \phi_1 \right) </math></center>

at some fixed point in the fluid domain.

Assuming that the linear solution is known and eliminating <math>\zeta_2\,</math> from the kinematic condition, we obtain the more familiar form:

<center><math> \nabla^2\phi_2 = 0 \,</math></center>
<center><math> \frac{\partial^2\phi_2}{\partial t^2} + g \frac{\partial\phi_2}{\partial z} = - \frac{\partial}{\partial t} \left( \nabla\phi_1 \cdot \nabla\phi_1 \right) + \frac{1}{g} \frac{\partial\phi_1}{\partial t} \frac{\partial}{\partial z} \left( \frac{\partial^2\phi_1}{\partial t^2} + g \frac{\partial \phi_1}{\partial z} \right)_{z=0} </math></center>
<center><math> \zeta_2 = - \frac{1}{g} \left( \frac{\partial\phi_2}{\partial t} + \frac{1}{2} \nabla\phi_1\cdot\nabla\phi_1 + \zeta_1 \frac{\partial^2\phi_1}{\partial z \partial t} \right)_{z=0} </math></center>

The solution of the above boundary-value problem is available in closed from when the linear potential <math>\phi_1\,</math> is the linear superposition of regular plane progressive waves.

Very useful insights follow from this solution even in the absence of floating bodies.

The statement of the second-order problem when a body is present requires the derivation of a second-order body boundary condition which is involved. The solution of the second-order wave body interaction problem is only possible numerically and is in general a complex time consuming task.

==Second-Order Bi-Chromatic Wave Theory==

The forcing in the right-hand side of the second order free surface condition is a quadratic function of the linear solution. Hence, if the linear problem is written as the sum of monochromatic wave components, the treatment of the most general second order problem can be accomplished by treating the second-order free-surface condition by a bi-chromatic disturbance.

The above conclusion is analogous to the following identity:

<center><math> \mathrm{Re} \left\{ A_1 e^{i\omega t} \right\} \mathrm{Re} \left\{ A_2 e^{i\omega t} \right\} = \frac{1}{2} \mathrm{Re} \left\{ A_1 A_2 e^{i\left(\omega_1+\omega_2\right)t} + A_1 A_2^* e^{i\left(\omega_1-\omega_2\right)t}\right\} </math></center>

It follows that the forcing and hence the solution of the second-order problem is the linear super position of a sum and a difference-frequency components.

The velocity potential of a directional seastate represented by the linear superposition of plane progressive waves of various frequencies and wave headings takes the familiar form:

<center><math> \Phi_1 = \mathrm{Re} \sum_k \sum_m \frac{i g A_{km}}{\omega_k} e^{\nu_k z - i\nu_k x \cos\theta_m - i\nu_k y \sin\theta_m} e^{i\omega_k t+i\phi_{km}} </math></center>

where the summations are over a sufficiently large number of wave frequencies <math>\omega_k\,</math> and wave headings <math>\theta_m\,</math> necessary to characterize the ambient seastate.

The phase angles <math>\phi_{km}\,</math> are sampled from the uniform distribution over <math> (-\pi,\pi] \, </math> and the wave amplitudes <math>A_{km}\,</math> are chosen to conform to the ambient wave spectrum.

<math>A_{km}\,</math> may be selected to be deterministic or may be drawn from a Rayleigh distribution independently from the phase angle <math>\phi_{km}\,</math>. The latter choice has been found to be superior in practice when long records are needed with very long periodicity.

Upon substitution of <math> \Phi_1\,</math> into the right-hand side of the second-order free surface condition, sum- and difference- frequency terms arise as illustrated above.

We therefore define the yet unknown second-order potential as follows:

<center><math> \Phi_2 = \mathrm{Re} \left\{ \varphi_2^+ e^{i\left(\omega_k+\omega_\rho\right)t} + \varphi_2^- e^{i\left(\omega_k-\omega_\rho\right)t} \right\} </math></center>

So the principal task of second-order theory in the absence of floating bodies is to determine the complex second-order potentials <math>\varphi_2^+\,</math> and <math> \varphi_2^-\,</math>.

They satisfy second-order free-surface conditions stated below. Use is made of the following identity satisfied by regular plane progressive wave components:

<center><math> \frac{\mathrm{d}}{\mathrm{d}z} \left( \frac{\partial^2 \phi_1}{\partial t^2} + g \frac{\partial\phi_1}{\partial z} \right) = 0 , \quad \mbox{any} \ z \leq 0 \, </math></center>

Making use of the above identity it is easy to show that:

<center><math> - {\Omega^+}^2 \varphi_2^+ + g \frac{\partial \varphi_2^+}{\partial z} = - i \Omega^+ \nabla \varphi_{1km} \cdot \nabla \varphi_{2ln} \quad z=0 </math></center>

and

<center><math> - {\Omega^-}^2 \varphi_2^- + g \frac{\partial \varphi_2^-}{\partial z} = - i \Omega^- \nabla \varphi_{1km} \cdot \nabla \varphi_{2ln}^* \quad z=0 </math></center>

where <math> \varphi_{1km}\,</math> and <math> \varphi_{2ln}\,</math> are the complex velocity potentials of the regular plane progressive waves the ambient seastate consists of.

It follows that <math> \varphi_2^\pm \, </math> each depend on 4 indices, two for the frequencies <math> \omega_k \And \omega_l \, </math> and two for the respective headings <math> \theta_m \, </math> and <math> \theta_l \, </math>.

The solution of the above forced free surface problems together with <math> \nabla_2\varphi_2^\pm = 0 \, </math> in the fluid domain, is a straightforward exercise in Fourier Theory. The solutions for <math>\varphi_2\pm \, </math> are:

<center><math> \varphi_2^\pm = \frac{\pm i}{2} \Omega^\pm A_k A_l \frac{\omega_k\omega_l}{g} \left[ 1 \mp \cos \left( \theta_m - \theta_l \right) \right] \cdot \frac{e^{z\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)}}{\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)^{1/2} - \frac{{\Omega^\pm}^2}{g}} e^{ - i \alpha^\pm x - i \beta^\pm y + i \left( \phi_k \pm \phi_l \right)} </math></center>

where:

<center><math> \alpha^\pm = \nu_k \cos \theta_m \pm \nu_l \cos\theta_n \, </math></center>
<center><math> \beta^\pm = \nu_k \sin\theta_m \pm \nu_l \sin\theta_n \, </math></center>

The real second-order potential follows by quadruple summation over all pairs of frequencies and wave headings and is not stated here.

==Special Cases==

A number of special cases are noteworthy to discuss since they convey the new physics introduced by the second-order potentials. All effects discussed below are additive to the linear solution by virtue of perturbation theory and <math> O \left(A_k A_l\right) = O \left(A^2\right) \,</math>. An in depth discussion of perturbation theory, its extensions and limitations in deep and shallow waters is presented in [[Mei 1983|Mei 1983]].

<center><math>\theta_m = \theta_n = \theta : \quad \mbox{unidirectional waves}\,</math></center>
<center><math> \varphi_2^+ \equiv 0 \, </math></center>
<center><math> \varphi_2^- = - i \Omega^- A_k A_l \frac{\omega_k\omega_l}{g} \frac{e^{Z\left|N^-\right|}}{\left|N^-\right|-\frac{{\Omega^-}^2}{g}} e^{-i N^- (x\cos\theta+y\sin\theta)=i\left(\phi_k-\phi_l\right)} </math></center>
where:
<center><math> N^- = \nu_k - \nu_l, \quad \Omega^- = \omega_k - \omega_l \, </math></center>

So the second-order potential is identically zero in unidirectional waves of the same frequency.

Verify that this is not however the case for the second-order wave elevation and second-order pressure.

Derive as an exercise the second-order wave elevation when <math> \omega_k = \omega_l = \omega \, </math> and show that it contributes a correction to the linear sinusoidal solution which produces more steepness at the crests and less at at troughs.

Carrying the perturbation solution when <math>\omega_k = \omega_l =\omega \, </math> to arbitrarily high order (it has been applied numerically with over 100 terms!). The famous solution by Stokes is recovered:

The crests in the limit become cusped with an enclosed angle of <math>120^\circ\,</math> and the limiting value of the <math> \frac{H}{\lambda}\,</math> ratio is about <math>\frac{1}{7}\,</math>.

So to the extent that periodic waves of this form can be created or observed these limiting properties are useful to remember and know they can be obtained by perturbation theory as well.

When Stokes waves are reflected off a wall the limiting enclosed angle is <math>90^\circ\,</math>.

<center><math> \theta_m + \pi = \theta_n = \theta \, </math></center>

In this case we have two waves propagating in opposite directions.

It can be shown that now:

<center><math> \varphi_2^- \equiv 0 \, </math></center>

Yet the corresponding components of the wave elevation and pressure are again now zero.

The sum-frequency potential becomes:

<center><math> \varphi_2^+ = - i \Omega^+ A_k A_l \frac{\omega_k\omega_l}{g} \frac{e^{Z\left|N^-\right|}}{\left|N^-\right|-\frac{{\Omega^-}^2}{g}} e^{i N^- (x\cos\theta+y\sin\theta)+i\left(\phi_k-\phi_l\right)} </math></center>
where <math> N^- = \omega_k - \omega_l \, </math>.

A noteworthy property of <math>\varphi_2^+\,</math> is that when <math> \omega_k = \omega_l \, </math> or <math> N^- = 0 \, </math>, the velocity potential and the term in the second order hydrodynamic pressure proportional to <math>\frac{\partial\phi_2}{\partial t}\,</math>, does not attenuate with depth!

This is a nonlinear wave property when waves propagate in opposite directions and has been measured in ocean pressure fields at large depths. It has been considered responsible of microseisms on the ocean floor.

This lack of attenuation of the hydrodynamic second-order pressure also arises when diffraction occurs off large volume floating platforms, like TLP's. The possible result is a large enough high-frequency excitation in heave & pitch which could contribute appreciably to the resonant responses of TLP tethers.

In order to assess the full effect of second-order nonlinearities on floating platforms (and certain large ships that may undergo flexural vibrations) the complete solution of the sum-frequency second-order problem is necessary.

In random waves, it is also believed that the third-order solution is necessary. This being a formidable task, fully nonlinear solutions may be appropriate.

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]

Second-Order Wave Theory

2019-07-17T08:35:37Z

Xjwiki: /* Second-Order Bi-Chromatic Wave Theory */

{{Ocean Wave Interaction with Ships and Offshore Structures
| chapter title = Second-Order Wave Theory
| next chapter = [[Second-Order Wave Effect On Offshore Platforms]]
| previous chapter = [[Wave Drift Forces]]
}}
{{incomplete pages}}

==Introduction==
Linear theory is a very powerful and useful means of modeling the responses of floating bodies in regular and random waves. Yet, for certain types of floating structures and certain types of effects nonlinear extensions are necessary. Mean drift forces in regular waves do not require an account of nonlinear effects. However, the definition of the quadratically nonlinear slow-drift excitation force in a seastate requires a second-order theory in principle. Tension-leg platforms that operate in large water depths for the exploration and extraction of hydrocarbons offshore often undergo resonant heave oscillations with periods of a few seconds which can only be excited by nonlinear wave effects. Their treatment requires a second-order theory and perhaps a more exact treatment.

==Second-Order Free Surface Condition==

The second-order free surface condition of free waves was derived earlier in the form:

<center><math> \phi = \phi_1 + \phi_2 + \phi_3 + \cdots \, </math></center>
<center><math> \zeta = \zeta_1 + \zeta_2 + \zeta_3 + \cdots \, </math></center>

Kinematic condition:

<center><math> \frac{\partial\zeta_2}{\partial t} - \frac{\partial\phi_2}{\partial t} = \zeta_1 \frac{\partial^2\phi_1}{\partial z^2} - \nabla\phi_1 \cdot \nabla\zeta_1 \quad \text{at }z=0 \, </math></center>

Dynamic condition:

<center><math> \zeta_2 + \frac{1}{g} \frac{\partial\phi_2}{\partial t} = - \frac{1}{g} \left( \frac{1}{2} \nabla\phi_1 \cdot \nabla\phi_1 + \zeta_1 \frac{\partial^2\phi_1}{\partial z \partial t} \right)_{z=0} </math></center>

Hydrodynamic pressure:

<center><math> P_2 = - \rho \left( \frac{\partial\phi_2}{\partial t} + \frac{1}{2} \nabla \phi_1 \cdot \nabla \phi_1 \right) </math></center>

at some fixed point in the fluid domain.

Assuming that the linear solution is known and eliminating <math>\zeta_2\,</math> from the kinematic condition, we obtain the more familiar form:

<center><math> \nabla^2\phi_2 = 0 \,</math></center>
<center><math> \frac{\partial^2\phi_2}{\partial t^2} + g \frac{\partial\phi_2}{\partial z} = - \frac{\partial}{\partial t} \left( \nabla\phi_1 \cdot \nabla\phi_1 \right) + \frac{1}{g} \frac{\partial\phi_1}{\partial t} \frac{\partial}{\partial z} \left( \frac{\partial^2\phi_1}{\partial t^2} + g \frac{\partial \phi_1}{\partial z} \right)_{z=0} </math></center>
<center><math> \zeta_2 = - \frac{1}{g} \left( \frac{\partial\phi_2}{\partial t} + \frac{1}{2} \nabla\phi_1\cdot\nabla\phi_1 + \zeta_1 \frac{\partial^2\phi_1}{\partial z \partial t} \right)_{z=0} </math></center>

The solution of the above boundary-value problem is available in closed from when the linear potential <math>\phi_1\,</math> is the linear superposition of regular plane progressive waves.

Very useful insights follow from this solution even in the absence of floating bodies.

The statement of the second-order problem when a body is present requires the derivation of a second-order body boundary condition which is involved. The solution of the second-order wave body interaction problem is only possible numerically and is in general a complex time consuming task.

==Second-Order Bi-Chromatic Wave Theory==

The forcing in the right-hand side of the second order free surface condition is a quadratic function of the linear solution. Hence, if the linear problem is written as the sum of monochromatic wave components, the treatment of the most general second order problem can be accomplished by treating the second-order free-surface condition by a bi-chromatic disturbance.

The above conclusion is analogous to the following identity:

<center><math> \mathrm{Re} \left\{ A_1 e^{i\omega t} \right\} \mathrm{Re} \left\{ A_2 e^{i\omega t} \right\} = \frac{1}{2} \mathrm{Re} \left\{ A_1 A_2 e^{i\left(\omega_1+\omega_2\right)t} + A_1 A_2^* e^{i\left(\omega_1-\omega_2\right)t}\right\} </math></center>

It follows that the forcing and hence the solution of the second-order problem is the linear super position of a sum and a difference-frequency components.

The velocity potential of a directional seastate represented by the linear superposition of plane progressive waves of various frequencies and wave headings takes the familiar form:

<center><math> \Phi_1 = \mathrm{Re} \sum_k \sum_m \frac{i g A_{km}}{\omega_k} e^{\nu_k z - i\nu_k x \cos\theta_m - i\nu_k y \sin\theta_m} e^{i\omega_k t+i\phi_{km}} </math></center>

where the summations are over a sufficiently large number of wave frequencies <math>\omega_k\,</math> and wave headings <math>\theta_m\,</math> necessary to characterize the ambient seastate.

The phase angles <math>\phi_{km}\,</math> are sampled from the uniform distribution over <math> (-\pi,\pi] \, </math> and the wave amplitudes <math>A_{km}\,</math> are chosen to conform to the ambient wave spectrum.

<math>A_{km}\,</math> may be selected to be deterministic or may be drawn from a Rayleigh distribution independently from the phase angle <math>\phi_{km}\,</math>. The latter choice has been found to be superior in practice when long records are needed with very long periodicity.

Upon substitution of <math> \Phi_1\,</math> into the right-hand side of the second-order free surface condition, sum- and difference- frequency terms arise as illustrated above.

We therefore define the yet unknown second-order potential as follows:

<center><math> \Phi_2 = \mathrm{Re} \left\{ \varphi_2^+ e^{i\left(\omega_k+\omega_\rho\right)t} + \varphi_2^- e^{i\left(\omega_k-\omega_\rho\right)t} \right\} </math></center>

So the principal task of second-order theory in the absence of floating bodies is to determine the complex second-order potentials <math>\varphi_2^+\,</math> and <math> \varphi_2^-\,</math>.

They satisfy second-order free-surface conditions stated below. Use is made of the following identity satisfied by regular plane progressive wave components:

<center><math> \frac{\mathrm{d}}{\mathrm{d}z} \left( \frac{\partial \phi_1}{\partial t^2} + g \frac{\partial\phi_1}{\partial z} \right) = 0 , \quad \mbox{any} \ z \leq 0 \, </math></center>

Making use of the above identity it is easy to show that:

<center><math> - {\Omega^+}^2 \varphi_2^+ + g \frac{\partial \varphi_2^+}{\partial z} = - i \Omega^+ \nabla \varphi_{1km} \cdot \nabla \varphi_{2ln} \quad z=0 </math></center>

and

<center><math> - {\Omega^-}^2 \varphi_2^- + g \frac{\partial \varphi_2^-}{\partial z} = - i \Omega^- \nabla \varphi_{1km} \cdot \nabla \varphi_{2ln}^* \quad z=0 </math></center>

where <math> \varphi_{1km}\,</math> and <math> \varphi_{2ln}\,</math> are the complex velocity potentials of the regular plane progressive waves the ambient seastate consists of.

It follows that <math> \varphi_2^\pm \, </math> each depend on 4 indices, two for the frequencies <math> \omega_k \And \omega_l \, </math> and two for the respective headings <math> \theta_m \, </math> and <math> \theta_l \, </math>.

The solution of the above forced free surface problems together with <math> \nabla_2\varphi_2^\pm = 0 \, </math> in the fluid domain, is a straightforward exercise in Fourier Theory. The solutions for <math>\varphi_2\pm \, </math> are:

<center><math> \varphi_2^\pm = \frac{\pm i}{2} \Omega^\pm A_k A_l \frac{\omega_k\omega_l}{g} \left[ 1 \mp \cos \left( \theta_m - \theta_l \right) \right] \cdot \frac{e^{z\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)}}{\left( {\alpha^\pm}^2 + {\beta^\pm}^2 \right)^{1/2} - \frac{{\Omega^\pm}^2}{g}} e^{ - i \alpha^\pm x - i \beta^\pm y + i \left( \phi_k \pm \phi_l \right)} </math></center>

where:

<center><math> \alpha^\pm = \nu_k \cos \theta_m \pm \nu_l \cos\theta_n \, </math></center>
<center><math> \beta^\pm = \nu_k \sin\theta_m \pm \nu_l \sin\theta_n \, </math></center>

The real second-order potential follows by quadruple summation over all pairs of frequencies and wave headings and is not stated here.

==Special Cases==

A number of special cases are noteworthy to discuss since they convey the new physics introduced by the second-order potentials. All effects discussed below are additive to the linear solution by virtue of perturbation theory and <math> O \left(A_k A_l\right) = O \left(A^2\right) \,</math>. An in depth discussion of perturbation theory, its extensions and limitations in deep and shallow waters is presented in [[Mei 1983|Mei 1983]].

<center><math>\theta_m = \theta_n = \theta : \quad \mbox{unidirectional waves}\,</math></center>
<center><math> \varphi_2^+ \equiv 0 \, </math></center>
<center><math> \varphi_2^- = - i \Omega^- A_k A_l \frac{\omega_k\omega_l}{g} \frac{e^{Z\left|N^-\right|}}{\left|N^-\right|-\frac{{\Omega^-}^2}{g}} e^{-i N^- (x\cos\theta+y\sin\theta)=i\left(\phi_k-\phi_l\right)} </math></center>
where:
<center><math> N^- = \nu_k - \nu_l, \quad \Omega^- = \omega_k - \omega_l \, </math></center>

So the second-order potential is identically zero in unidirectional waves of the same frequency.

Verify that this is not however the case for the second-order wave elevation and second-order pressure.

Derive as an exercise the second-order wave elevation when <math> \omega_k = \omega_l = \omega \, </math> and show that it contributes a correction to the linear sinusoidal solution which produces more steepness at the crests and less at at troughs.

Carrying the perturbation solution when <math>\omega_k = \omega_l =\omega \, </math> to arbitrarily high order (it has been applied numerically with over 100 terms!). The famous solution by Stokes is recovered:

The crests in the limit become cusped with an enclosed angle of <math>120^\circ\,</math> and the limiting value of the <math> \frac{H}{\lambda}\,</math> ratio is about <math>\frac{1}{7}\,</math>.

So to the extent that periodic waves of this form can be created or observed these limiting properties are useful to remember and know they can be obtained by perturbation theory as well.

When Stokes waves are reflected off a wall the limiting enclosed angle is <math>90^\circ\,</math>.

<center><math> \theta_m + \pi = \theta_n = \theta \, </math></center>

In this case we have two waves propagating in opposite directions.

It can be shown that now:

<center><math> \varphi_2^- \equiv 0 \, </math></center>

Yet the corresponding components of the wave elevation and pressure are again now zero.

The sum-frequency potential becomes:

<center><math> \varphi_2^+ = - i \Omega^+ A_k A_l \frac{\omega_k\omega_l}{g} \frac{e^{Z\left|N^-\right|}}{\left|N^-\right|-\frac{{\Omega^-}^2}{g}} e^{i N^- (x\cos\theta+y\sin\theta)+i\left(\phi_k-\phi_l\right)} </math></center>
where <math> N^- = \omega_k - \omega_l \, </math>.

A noteworthy property of <math>\varphi_2^+\,</math> is that when <math> \omega_k = \omega_l \, </math> or <math> N^- = 0 \, </math>, the velocity potential and the term in the second order hydrodynamic pressure proportional to <math>\frac{\partial\phi_2}{\partial t}\,</math>, does not attenuate with depth!

This is a nonlinear wave property when waves propagate in opposite directions and has been measured in ocean pressure fields at large depths. It has been considered responsible of microseisms on the ocean floor.

This lack of attenuation of the hydrodynamic second-order pressure also arises when diffraction occurs off large volume floating platforms, like TLP's. The possible result is a large enough high-frequency excitation in heave & pitch which could contribute appreciably to the resonant responses of TLP tethers.

In order to assess the full effect of second-order nonlinearities on floating platforms (and certain large ships that may undergo flexural vibrations) the complete solution of the sum-frequency second-order problem is necessary.

In random waves, it is also believed that the third-order solution is necessary. This being a formidable task, fully nonlinear solutions may be appropriate.

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]

Wave Momentum Flux

2019-01-02T08:26:40Z

Xjwiki: /* Simplification */

{{Ocean Wave Interaction with Ships and Offshore Structures
| chapter title = Wave Momentum Flux
| next chapter = [[Wavemaker Theory]]
| previous chapter = [[Wave Energy Density and Flux]]
}}

{{incomplete pages}}

== Introduction ==

The momentum is important to determine the forces.

== Momentum flux in potential flow ==

The momentum flux (the time derivative of the [http://en.wikipedia.org/wiki/Momentum momentum] <math>\mathcal{M}</math>) is given by
<center><math> \frac{\mathrm{d}\mathcal{M}(t)}{\mathrm{d}t} = \rho \frac{\mathrm{d}}{\mathrm{d}t} \iiint_{\Omega(t)} \mathbf{v} \mathrm{d}V
= \rho \iiint_{\Omega(t)} \frac{\partial\mathbf{v}}{\partial t} \mathrm{d}V + \rho \iint_{\partial\Omega(t)} \mathbf{v} U_n \mathrm{d}S,
</math></center>
where <math> U_n \, </math> is the outward normal velocity of the surface <math> \partial\Omega(t)\, </math>.
This equation follows from the [http://en.wikipedia.org/wiki/Reynolds_transport_theorem transport theorem].

We begin with [http://en.wikipedia.org/wiki/Euler_equations Euler's equation] in the absence of viscosity
<center><math>
\frac{\partial\mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla ) \mathbf{v} =
- \nabla \left(\frac{P}{\rho} + g z\right)
</math></center>
where we have defined the direction of gravity to be in the negative <math>z</math>
direction.
We may recast the momentum flux in the form
<center><math>
\frac{\mathrm{d}\mathcal{M}(t)}{\mathrm{d}t} = - \rho \iiint_{\Omega(t)} \left( \nabla ( \frac{P}{\rho} + g z ) + ( \mathbf{v} \cdot \nabla ) \mathbf{v} \right)
\mathrm{d}V + \rho \iint_{\partial\Omega(t)} \mathbf{v} U_n \mathrm{d}S
</math></center>
So far <math> \Omega(t)\, </math> is an arbitrary closed time dependent volume bounded by the time dependent surface <math> \partial\Omega(t)\, </math>.
We have however defined the gravitational acceleration to be in the negative <math>z</math> direction.

=== Simplification ===

We use the following vector theorem
<center><math> \nabla \cdot( \mathbf{v} \mathbf{v} )=(\mathbf{v} \cdot \nabla ) \mathbf{v}+ \mathbf{v} (\nabla \cdot \mathbf{v} ) </math></center>
If we have potential flow then <math> \mathbf{v} = \nabla \Phi \,</math> and for solenoidal flow (<math>\nabla \cdot \mathbf{v}=0 </math>), we can use Gauss's vector theorem:
<center><math> \iiint_{\Omega(t)} \nabla \cdot( \mathbf{v} \mathbf{v} ) \mathrm{d}V = \iint_{\partial\Omega(t)}
\left(\mathbf{v} \cdot \mathbf{n} \right) \mathbf{v} \mathrm{d}S </math></center>
In potential flow it follows that
<center><math> \iint_{\partial\Omega(t)} ( \mathbf{v} \cdot \mathbf{n} ) \mathbf{v} \mathrm{d}S = \iint_{\partial\Omega(t)} \frac{\partial\Phi}{\partial n} \nabla \Phi \mathrm{d}S
= \iint_{\partial\Omega(t)} V_n \mathbf{v} \mathrm{d}S </math></center>
In the above <math> \mathbf{n} \, </math> is the unit vector pointing out of the volume <math> \Omega(t) </math> and <math> V_n = \mathbf{n} \cdot \nabla \Phi </math>

Upon substitution in the momentum flux formula, we obtain:
<center><math> \frac{\mathrm{d}\mathcal{M}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega(t)} \left( ( \frac{P}{\rho} + gz ) \mathbf{n} + \mathbf{v} (V_n - U_n )\right) \mathrm{d}S </math></center>

This formula is of central importance in potential flow marine hydrodynamics because the rate of change of the linear momentum defined above is just <math> \pm \,</math> the force acting on the fluid volume. When its mean value can be shown to vanish, important force expressions on solid boundaries follow and will be derived in what follows.

== Hydrostatic Term ==

Consider separately the term in the momentum flux expression involving the hydrostatic pressure:
<center><math> \frac{\mathrm{d}\mathcal{M}_H}{\mathrm{d}t} = - \rho \iint_{\partial\Omega} gz \mathbf{n},
</math></center>
We break the boundary up into the free surface, ends, body surface, and sea floor at infinite depth, i,.e.
<math>\partial\Omega=\partial\Omega_F + \partial\Omega^{\pm} + \partial\Omega_B + \partial\Omega_{\infty} </math>
The integral over the body surface, assuming a fully submerged body is:
<center><math> \frac{\mathrm{d}\mathcal{M}_{H,B}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega_B} gz \mathbf{n} = \rho g V \mathbf{k} </math></center>
where <math>V</math> is the volume. This follows from the vector theorem of Gauss and is the [http://en.wikipedia.org/wiki/Displacement_%28fluid%29 principle of Archimedes]. That is,
the momentum flux is equal to the buoyancy force.

We may therefore consider the second part of the integral involving wave effects independently and in the absence of the body, assumed fully submerged. In the case of a surface piercing body and in the fully nonlinear case matters are more complex. Consider the application of the momentum conservation theorem in the case of a submerged or floating body in steep waves.

[[Image:Momentum.png|600px|right|thumb|Momentum boundaries]]

Here we consider the two-dimensional case in order to present the concepts. Extensions to three dimensions are then trivial. Note that unlike the [[Wave Energy Density and Flux|energy conservation principle]], the momentum conservation theorem derived above is a vector identity with a horizontal and a vertical component. The integral of the hydrostatic term over the remaining surfaces leads to:
<center><math> \frac{\mathrm{d} \mathcal{M}_{H,S}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega_F+\partial\Omega^+ + \partial\Omega^- +\partial\Omega_{\infty}} gz \mathbf{n} \mathrm{d}S = - \rho g V_{\text{Fluid}} \mathbf{k} </math></center>
where <math>V_{\mbox{Fluid}} </math> is the fluid volume.
This is simply the static weight of the volume of fluid bounded by <math> \partial\Omega_F, \partial\Omega^+, \partial\Omega^- \,</math> and <math> \partial\Omega_{\infty}.</math> With no waves present, this is simply the weight of the ocean water "column" bounded by <math> \partial\Omega\, </math> which does not concern us here. This weight does not change in principle when waves are present at least when <math> \partial\Omega^+, \partial\Omega^- \,</math> are placed sufficiently far away that the wave amplitude has decreased to zero. So "in principle" this term being of hydrostatic origin may be ignored. However, it is in principle more "rational" to apply the momentum conservation theorem over the "linearized" volume <math> V_L(t) \, </math> which is perfectly possible within the framework derived above. In this case <math> \frac{\mathrm{d}\mathcal{M}_H}{\mathrm{d}t} \,</math> is exactly equal to the static weight of the water column and can be ignored in the wave-body interaction problem.

On <math> S_F; P=P_a=0 \, </math> and hence all terms within the free-surface integral and over <math> S_{\infty} \, </math> (seafloor) can be neglected. It follows that:
<center><math> \frac{\mathrm{d} \mathcal{M}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega^\pm +\partial\Omega_B} \left[ \frac{P}{\rho} \mathbf{n} + \mathbf{v} (V_n -U_n) \right] \mathrm{d}S </math></center>

Note that the free surface integrals also vanish for the horizontal component since the hydrostatic force is always vertical. This momentum flux formula is of central importance in wave-body interactions and has many important applications, some of which are discussed bellow. Note that the mathematical derivations involved in its proof apply equally when the volume <math> \Omega </math> and its enclosed surface are selected to be at their linearized positions. In such a case it is essential to set <math> U_n=0\,</math> and <math> V_n \ne 0 </math>. Let the math take over and suggest the proper expression for the force. In the fully nonlinear case, <math>U_n \ne 0 \, </math> on <math> \partial\Omega_F\, </math> and <math> P=0 \,</math> on <math> \partial\Omega_F \, </math>!

On a solid boundary:
<center><math> U_n =V_n \,</math></center>
and
<center><math> \overrightarrow{F}_B = \iint_{S_B} P \vec{n} dS </math></center>
With <math> \vec{n} \,</math> pointing inside the body. We may therefore recast the momentum conservation theorem in the form:
<center><math> \overrightarrow{F}_B (t) = - \frac{\mathrm{d} \overrightarrow{M}}{\mathrm{d}t} - \rho \iint_{S^\pm} \left[ \frac{P}{\rho} \vec{n} + \overrightarrow{V} (V_n - U_n) \right] \mathrm{d}S </math></center>
Where <math> S^\pm \, </math> are fluid boundaries at some distance from the body. If the volume of fluid surrounded by the body, free surface and the furfaces <math> S^\pm \, </math> does not grow in time, then the momentum of the enclosed fluid cannot grow either, so <math> \overrightarrow{M}(t) \, </math> is a stationary physical quantity. It is a well known result that the mean value in time of the time derivative of a stationary quantity is zero. So:
<center><math> {\overline{\frac{\mathrm{d}\overrightarrow{M}}{\mathrm{d}t}}}^t = 0 </math></center>

 Proof :

<center><math> {\overline{\frac{\mathrm{d}F}{\mathrm{d}t}}}^t = \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^{T} \frac{\mathrm{d}F{\tau}}{\mathrm{d}\tau} \mathrm{d}\tau = \lim_{T\to\infty} \frac{1}{2T} [F(T) - F(-T) ] </math></center>

Since <math> F(\pm \tau) \, </math> must be bounded for a stationary signal <math> F(t) \, </math>, it follows that <math> {\overline{\frac{\mathrm{d}F}{\mathrm{d}t}}}^t = 0 \, </math>.

Taking mean values, it follows that:

<center><math> {\overline{\overrightarrow{F}_B (t)}}^t = - \rho \ {\overline{\iint_{S^\pm} \left[ \frac{P}{\rho} \vec{n} + \overrightarrow{V} (V_n - U_n ) \right] \mathrm{d}S}}^t </math></center>

This is the fundamental formula underlying the definition of the mean wave drift forces acting on floating bodies. Such forces are very imprtant for stationary floating structures and can be expressed in terms of integrals of wave effects over control surfaces <math> S^\pm \, </math> which may be located at infinity.

The extension of the above formula for <math> {\overline{\overrightarrow{F}_B}}^t \, </math> in three dimensions is trivial. Simply replace <math> S^\pm \, </math> by <math> S_{\infty} \, </math>, a control surface at infinity. Common choices are a vertical cylindrical boundary or two vertical planes paraller to the axis of forward motion of a ship.

== Applications ==

=== Mean horizontal momentum flux due to a [[Plane Progressive Regular Waves| Plane Progressive Regular Wave]] ===

[[Image:Plane_wave_momentum.png|thumb|right|600px|Plane progressive wave momentum]]

We can determine the mean horizontal momentum flux due to a [[Plane Progressive Regular Waves| Plane Progressive Regular Wave]] with surface displacement
<center><math> \zeta = A \cos (\omega t - k x) \,</math></center>
where <math>\omega</math> and <math>k</math> are related by the [[Dispersion Relation for a Free Surface]]
<math> \omega^2 = gk \tanh kh \, </math>

The momentum flux across <math> \Omega^+ \, </math> is given by

<center><math> \frac{\mathrm{d}M_x}{\mathrm{d}t} = - \int_{-h}^{\zeta} ( P + \rho u^2 ) \mathrm{d}z </math></center>
The pressure is given by [[Conservation Laws and Boundary Conditions|Bernoulli's equation]]
<center><math> P = - \rho \frac{\partial\Phi}{\partial t} - \frac{1}{2} \rho ( u^2 + v^2 ) - \rho g z </math></center>
so that the momentum flux is
<center><math>
\begin{align}
\frac{dM_x}{dt} &= \rho \int_{-h}^{\zeta} \left[ \frac{\partial\Phi}{\partial t} + \frac{1}{2} ( v^2 - u^2 ) + g z \right] \mathrm{d}z \\
&= \rho \left( \int_{-h}^{0} + \int_{0}^{\zeta} \right) \left[ \frac{\partial\Phi}{\partial t} + \frac{1}{2} (v^2 - u^2) + gz \right] \mathrm{d}z
\end{align}
</math></center>

Taking mean values in time and keeping terms of <math> O(A^2) </math> we obtain:

<center><math> {\overline{\frac{\mathrm{d}M_x}{\mathrm{d}t}}}^t = {\overline{\rho \zeta(t) \left. \frac{\partial\Phi}{\partial t} \right |_{h=0}}}^t + \frac{1}{2} \rho \ {\overline{\int_{-h}^{0} (v^2 -u^2 ) \mathrm{d}z}}^t + {\overline{\frac{1}{2} \rho g \zeta^2}}^t </math></center>

Invoking the [[Linear and Second-Order Wave Theory|linearized dynamic free surface condition]] we obtain

<center><math> \left. \frac{\partial\Phi}{\partial t} \right |_{z=0} = - g \zeta </math></center>

It follows that:

<center><math> {\overline{\frac{\mathrm{d}M_x}{\mathrm{d}t}}}^t = {-\frac{1}{2} \rho g \zeta^2 (t)}^t + \frac{1}{2} \rho {\overline{\int_{-h}^0 (v^2-u^2)\mathrm{d}z}}^t </math></center>

In deep water <math> \overline{v^2} = \overline{u^2} \, </math> and the second term is identically zero, so

<center><math> {\overline{\frac{\mathrm{d}M_x}{\mathrm{d}t}}}^t = - \frac{1}{2} \rho g {\overline{\zeta^2 (t)}}^t = - \frac{1}{4} \rho g A^2 </math></center>

In water of finite depth the wave particle trajectories are elliptical with the mean horizontal velocities larger than the mean vertical velocities. So:

<center><math> \overline{V_H^2} < \overline{U_H^2} </math></center>

Therefore, in finite depth the modulus of the mean momentum flux is higher than in deep water for the same A.
So the mean horizontal momentum flux due to a plane progressive wave against its direction of propagation and equal to <math> - \frac{1}{4} \rho g A^2 </math>.

=== [[Wavemaker Theory]] ===

[[Image:Wave_maker_momentum.png|600px|right|thumb|Wavemaker momentum]]
Consider a wave maker shown in the figure generating a wave of amplitude <math>A</math> at infinity.

What is the mean horizontal force on the wavemaker? From the momentum conservation theorem the mean horizontal flux of momentum to the left must flow into the wavemaker. This mean flux translates into a mean horizontal force in the same direction, as shown in the figure. Not an easy conclusion without using some basic fluid mechanics!

-----

This article is based on the MIT open course notes and the original article can be found
[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/A939D3A9-1F49-46F5-BEB5-0F6646CE340E/0/lecture5.pdf here]

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]

Wave Momentum Flux

2019-01-02T08:23:40Z

Xjwiki: /* Momentum flux in potential flow */ \iiint_{\partial\Omega} \frac{1}{2} ( \nabla\Phi \cdot \nabla\Phi) \mathbf{n} \mathrm{d}S \nequiv \iint_{\partial\Omega} \frac{\partial\Phi}{\partial n} \na

{{Ocean Wave Interaction with Ships and Offshore Structures
| chapter title = Wave Momentum Flux
| next chapter = [[Wavemaker Theory]]
| previous chapter = [[Wave Energy Density and Flux]]
}}

{{incomplete pages}}

== Introduction ==

The momentum is important to determine the forces.

== Momentum flux in potential flow ==

The momentum flux (the time derivative of the [http://en.wikipedia.org/wiki/Momentum momentum] <math>\mathcal{M}</math>) is given by
<center><math> \frac{\mathrm{d}\mathcal{M}(t)}{\mathrm{d}t} = \rho \frac{\mathrm{d}}{\mathrm{d}t} \iiint_{\Omega(t)} \mathbf{v} \mathrm{d}V
= \rho \iiint_{\Omega(t)} \frac{\partial\mathbf{v}}{\partial t} \mathrm{d}V + \rho \iint_{\partial\Omega(t)} \mathbf{v} U_n \mathrm{d}S,
</math></center>
where <math> U_n \, </math> is the outward normal velocity of the surface <math> \partial\Omega(t)\, </math>.
This equation follows from the [http://en.wikipedia.org/wiki/Reynolds_transport_theorem transport theorem].

We begin with [http://en.wikipedia.org/wiki/Euler_equations Euler's equation] in the absence of viscosity
<center><math>
\frac{\partial\mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla ) \mathbf{v} =
- \nabla \left(\frac{P}{\rho} + g z\right)
</math></center>
where we have defined the direction of gravity to be in the negative <math>z</math>
direction.
We may recast the momentum flux in the form
<center><math>
\frac{\mathrm{d}\mathcal{M}(t)}{\mathrm{d}t} = - \rho \iiint_{\Omega(t)} \left( \nabla ( \frac{P}{\rho} + g z ) + ( \mathbf{v} \cdot \nabla ) \mathbf{v} \right)
\mathrm{d}V + \rho \iint_{\partial\Omega(t)} \mathbf{v} U_n \mathrm{d}S
</math></center>
So far <math> \Omega(t)\, </math> is an arbitrary closed time dependent volume bounded by the time dependent surface <math> \partial\Omega(t)\, </math>.
We have however defined the gravitational acceleration to be in the negative <math>z</math> direction.

=== Simplification ===

We use the following vector theorem
<center><math> \nabla ( \mathbf{v} \mathbf{v} )=(\mathbf{v} \cdot \nabla ) \mathbf{v}+ \mathbf{v} (\nabla \cdot \mathbf{v} ) </math></center>
If we have potential flow then <math> \mathbf{v} = \nabla \Phi \,</math> and for solenoidal flow (<math>\nabla \cdot \mathbf{v}=0 </math>), we can use Gauss's vector theorem:
<center><math> \iiint_{\Omega(t)} \nabla \cdot( \mathbf{v} \mathbf{v} ) \mathrm{d}V = \iint_{\partial\Omega(t)}
\left(\mathbf{v} \cdot \mathbf{n} \right) \mathbf{v} \mathrm{d}S </math></center>
In potential flow it follows that
<center><math> \iint_{\partial\Omega(t)} ( \mathbf{v} \cdot \mathbf{n} ) \mathbf{v} \mathrm{d}S = \iint_{\partial\Omega(t)} \frac{\partial\Phi}{\partial n} \nabla \Phi \mathrm{d}S
= \iint_{\partial\Omega(t)} V_n \mathbf{v} \mathrm{d}S </math></center>
In the above <math> \mathbf{n} \, </math> is the unit vector pointing out of the volume <math> \Omega(t) </math> and <math> V_n = \mathbf{n} \cdot \nabla \Phi </math>

Upon substitution in the momentum flux formula, we obtain:
<center><math> \frac{\mathrm{d}\mathcal{M}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega(t)} \left( ( \frac{P}{\rho} + gz ) \mathbf{n} + \mathbf{v} (V_n - U_n )\right) \mathrm{d}S </math></center>

This formula is of central importance in potential flow marine hydrodynamics because the rate of change of the linear momentum defined above is just <math> \pm \,</math> the force acting on the fluid volume. When its mean value can be shown to vanish, important force expressions on solid boundaries follow and will be derived in what follows.

== Hydrostatic Term ==

Consider separately the term in the momentum flux expression involving the hydrostatic pressure:
<center><math> \frac{\mathrm{d}\mathcal{M}_H}{\mathrm{d}t} = - \rho \iint_{\partial\Omega} gz \mathbf{n},
</math></center>
We break the boundary up into the free surface, ends, body surface, and sea floor at infinite depth, i,.e.
<math>\partial\Omega=\partial\Omega_F + \partial\Omega^{\pm} + \partial\Omega_B + \partial\Omega_{\infty} </math>
The integral over the body surface, assuming a fully submerged body is:
<center><math> \frac{\mathrm{d}\mathcal{M}_{H,B}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega_B} gz \mathbf{n} = \rho g V \mathbf{k} </math></center>
where <math>V</math> is the volume. This follows from the vector theorem of Gauss and is the [http://en.wikipedia.org/wiki/Displacement_%28fluid%29 principle of Archimedes]. That is,
the momentum flux is equal to the buoyancy force.

We may therefore consider the second part of the integral involving wave effects independently and in the absence of the body, assumed fully submerged. In the case of a surface piercing body and in the fully nonlinear case matters are more complex. Consider the application of the momentum conservation theorem in the case of a submerged or floating body in steep waves.

[[Image:Momentum.png|600px|right|thumb|Momentum boundaries]]

Here we consider the two-dimensional case in order to present the concepts. Extensions to three dimensions are then trivial. Note that unlike the [[Wave Energy Density and Flux|energy conservation principle]], the momentum conservation theorem derived above is a vector identity with a horizontal and a vertical component. The integral of the hydrostatic term over the remaining surfaces leads to:
<center><math> \frac{\mathrm{d} \mathcal{M}_{H,S}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega_F+\partial\Omega^+ + \partial\Omega^- +\partial\Omega_{\infty}} gz \mathbf{n} \mathrm{d}S = - \rho g V_{\text{Fluid}} \mathbf{k} </math></center>
where <math>V_{\mbox{Fluid}} </math> is the fluid volume.
This is simply the static weight of the volume of fluid bounded by <math> \partial\Omega_F, \partial\Omega^+, \partial\Omega^- \,</math> and <math> \partial\Omega_{\infty}.</math> With no waves present, this is simply the weight of the ocean water "column" bounded by <math> \partial\Omega\, </math> which does not concern us here. This weight does not change in principle when waves are present at least when <math> \partial\Omega^+, \partial\Omega^- \,</math> are placed sufficiently far away that the wave amplitude has decreased to zero. So "in principle" this term being of hydrostatic origin may be ignored. However, it is in principle more "rational" to apply the momentum conservation theorem over the "linearized" volume <math> V_L(t) \, </math> which is perfectly possible within the framework derived above. In this case <math> \frac{\mathrm{d}\mathcal{M}_H}{\mathrm{d}t} \,</math> is exactly equal to the static weight of the water column and can be ignored in the wave-body interaction problem.

On <math> S_F; P=P_a=0 \, </math> and hence all terms within the free-surface integral and over <math> S_{\infty} \, </math> (seafloor) can be neglected. It follows that:
<center><math> \frac{\mathrm{d} \mathcal{M}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega^\pm +\partial\Omega_B} \left[ \frac{P}{\rho} \mathbf{n} + \mathbf{v} (V_n -U_n) \right] \mathrm{d}S </math></center>

Note that the free surface integrals also vanish for the horizontal component since the hydrostatic force is always vertical. This momentum flux formula is of central importance in wave-body interactions and has many important applications, some of which are discussed bellow. Note that the mathematical derivations involved in its proof apply equally when the volume <math> \Omega </math> and its enclosed surface are selected to be at their linearized positions. In such a case it is essential to set <math> U_n=0\,</math> and <math> V_n \ne 0 </math>. Let the math take over and suggest the proper expression for the force. In the fully nonlinear case, <math>U_n \ne 0 \, </math> on <math> \partial\Omega_F\, </math> and <math> P=0 \,</math> on <math> \partial\Omega_F \, </math>!

On a solid boundary:
<center><math> U_n =V_n \,</math></center>
and
<center><math> \overrightarrow{F}_B = \iint_{S_B} P \vec{n} dS </math></center>
With <math> \vec{n} \,</math> pointing inside the body. We may therefore recast the momentum conservation theorem in the form:
<center><math> \overrightarrow{F}_B (t) = - \frac{\mathrm{d} \overrightarrow{M}}{\mathrm{d}t} - \rho \iint_{S^\pm} \left[ \frac{P}{\rho} \vec{n} + \overrightarrow{V} (V_n - U_n) \right] \mathrm{d}S </math></center>
Where <math> S^\pm \, </math> are fluid boundaries at some distance from the body. If the volume of fluid surrounded by the body, free surface and the furfaces <math> S^\pm \, </math> does not grow in time, then the momentum of the enclosed fluid cannot grow either, so <math> \overrightarrow{M}(t) \, </math> is a stationary physical quantity. It is a well known result that the mean value in time of the time derivative of a stationary quantity is zero. So:
<center><math> {\overline{\frac{\mathrm{d}\overrightarrow{M}}{\mathrm{d}t}}}^t = 0 </math></center>

 Proof :

<center><math> {\overline{\frac{\mathrm{d}F}{\mathrm{d}t}}}^t = \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^{T} \frac{\mathrm{d}F{\tau}}{\mathrm{d}\tau} \mathrm{d}\tau = \lim_{T\to\infty} \frac{1}{2T} [F(T) - F(-T) ] </math></center>

Since <math> F(\pm \tau) \, </math> must be bounded for a stationary signal <math> F(t) \, </math>, it follows that <math> {\overline{\frac{\mathrm{d}F}{\mathrm{d}t}}}^t = 0 \, </math>.

Taking mean values, it follows that:

<center><math> {\overline{\overrightarrow{F}_B (t)}}^t = - \rho \ {\overline{\iint_{S^\pm} \left[ \frac{P}{\rho} \vec{n} + \overrightarrow{V} (V_n - U_n ) \right] \mathrm{d}S}}^t </math></center>

This is the fundamental formula underlying the definition of the mean wave drift forces acting on floating bodies. Such forces are very imprtant for stationary floating structures and can be expressed in terms of integrals of wave effects over control surfaces <math> S^\pm \, </math> which may be located at infinity.

The extension of the above formula for <math> {\overline{\overrightarrow{F}_B}}^t \, </math> in three dimensions is trivial. Simply replace <math> S^\pm \, </math> by <math> S_{\infty} \, </math>, a control surface at infinity. Common choices are a vertical cylindrical boundary or two vertical planes paraller to the axis of forward motion of a ship.

== Applications ==

=== Mean horizontal momentum flux due to a [[Plane Progressive Regular Waves| Plane Progressive Regular Wave]] ===

[[Image:Plane_wave_momentum.png|thumb|right|600px|Plane progressive wave momentum]]

We can determine the mean horizontal momentum flux due to a [[Plane Progressive Regular Waves| Plane Progressive Regular Wave]] with surface displacement
<center><math> \zeta = A \cos (\omega t - k x) \,</math></center>
where <math>\omega</math> and <math>k</math> are related by the [[Dispersion Relation for a Free Surface]]
<math> \omega^2 = gk \tanh kh \, </math>

The momentum flux across <math> \Omega^+ \, </math> is given by

<center><math> \frac{\mathrm{d}M_x}{\mathrm{d}t} = - \int_{-h}^{\zeta} ( P + \rho u^2 ) \mathrm{d}z </math></center>
The pressure is given by [[Conservation Laws and Boundary Conditions|Bernoulli's equation]]
<center><math> P = - \rho \frac{\partial\Phi}{\partial t} - \frac{1}{2} \rho ( u^2 + v^2 ) - \rho g z </math></center>
so that the momentum flux is
<center><math>
\begin{align}
\frac{dM_x}{dt} &= \rho \int_{-h}^{\zeta} \left[ \frac{\partial\Phi}{\partial t} + \frac{1}{2} ( v^2 - u^2 ) + g z \right] \mathrm{d}z \\
&= \rho \left( \int_{-h}^{0} + \int_{0}^{\zeta} \right) \left[ \frac{\partial\Phi}{\partial t} + \frac{1}{2} (v^2 - u^2) + gz \right] \mathrm{d}z
\end{align}
</math></center>

Taking mean values in time and keeping terms of <math> O(A^2) </math> we obtain:

<center><math> {\overline{\frac{\mathrm{d}M_x}{\mathrm{d}t}}}^t = {\overline{\rho \zeta(t) \left. \frac{\partial\Phi}{\partial t} \right |_{h=0}}}^t + \frac{1}{2} \rho \ {\overline{\int_{-h}^{0} (v^2 -u^2 ) \mathrm{d}z}}^t + {\overline{\frac{1}{2} \rho g \zeta^2}}^t </math></center>

Invoking the [[Linear and Second-Order Wave Theory|linearized dynamic free surface condition]] we obtain

<center><math> \left. \frac{\partial\Phi}{\partial t} \right |_{z=0} = - g \zeta </math></center>

It follows that:

<center><math> {\overline{\frac{\mathrm{d}M_x}{\mathrm{d}t}}}^t = {-\frac{1}{2} \rho g \zeta^2 (t)}^t + \frac{1}{2} \rho {\overline{\int_{-h}^0 (v^2-u^2)\mathrm{d}z}}^t </math></center>

In deep water <math> \overline{v^2} = \overline{u^2} \, </math> and the second term is identically zero, so

<center><math> {\overline{\frac{\mathrm{d}M_x}{\mathrm{d}t}}}^t = - \frac{1}{2} \rho g {\overline{\zeta^2 (t)}}^t = - \frac{1}{4} \rho g A^2 </math></center>

In water of finite depth the wave particle trajectories are elliptical with the mean horizontal velocities larger than the mean vertical velocities. So:

<center><math> \overline{V_H^2} < \overline{U_H^2} </math></center>

Therefore, in finite depth the modulus of the mean momentum flux is higher than in deep water for the same A.
So the mean horizontal momentum flux due to a plane progressive wave against its direction of propagation and equal to <math> - \frac{1}{4} \rho g A^2 </math>.

=== [[Wavemaker Theory]] ===

[[Image:Wave_maker_momentum.png|600px|right|thumb|Wavemaker momentum]]
Consider a wave maker shown in the figure generating a wave of amplitude <math>A</math> at infinity.

What is the mean horizontal force on the wavemaker? From the momentum conservation theorem the mean horizontal flux of momentum to the left must flow into the wavemaker. This mean flux translates into a mean horizontal force in the same direction, as shown in the figure. Not an easy conclusion without using some basic fluid mechanics!

-----

This article is based on the MIT open course notes and the original article can be found
[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/A939D3A9-1F49-46F5-BEB5-0F6646CE340E/0/lecture5.pdf here]

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]

Wave Momentum Flux

2018-12-31T12:26:08Z

Xjwiki: /* Simplification */

{{Ocean Wave Interaction with Ships and Offshore Structures
| chapter title = Wave Momentum Flux
| next chapter = [[Wavemaker Theory]]
| previous chapter = [[Wave Energy Density and Flux]]
}}

{{incomplete pages}}

== Introduction ==

The momentum is important to determine the forces.

== Momentum flux in potential flow ==

The momentum flux (the time derivative of the [http://en.wikipedia.org/wiki/Momentum momentum] <math>\mathcal{M}</math>) is given by
<center><math> \frac{\mathrm{d}\mathcal{M}(t)}{\mathrm{d}t} = \rho \frac{\mathrm{d}}{\mathrm{d}t} \iiint_{\Omega(t)} \mathbf{v} \mathrm{d}V
= \rho \iiint_{\Omega(t)} \frac{\partial\mathbf{v}}{\partial t} \mathrm{d}V + \rho \iint_{\partial\Omega(t)} \mathbf{v} U_n \mathrm{d}S,
</math></center>
where <math> U_n \, </math> is the outward normal velocity of the surface <math> \partial\Omega(t)\, </math>.
This equation follows from the [http://en.wikipedia.org/wiki/Reynolds_transport_theorem transport theorem].

We begin with [http://en.wikipedia.org/wiki/Euler_equations Euler's equation] in the absence of viscosity
<center><math>
\frac{\partial\mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla ) \mathbf{v} =
- \nabla \left(\frac{P}{\rho} + g z\right)
</math></center>
where we have defined the direction of gravity to be in the negative <math>z</math>
direction.
We may recast the momentum flux in the form
<center><math>
\frac{\mathrm{d}\mathcal{M}(t)}{\mathrm{d}t} = - \rho \iiint_{\Omega(t)} \left( \nabla ( \frac{P}{\rho} + g z ) + ( \mathbf{v} \cdot \nabla ) \mathbf{v} \right)
\mathrm{d}V + \rho \iint_{\partial\Omega(t)} \mathbf{v} U_n \mathrm{d}S
</math></center>
So far <math> \Omega(t)\, </math> is an arbitrary closed time dependent volume bounded by the time dependent surface <math> \partial\Omega(t)\, </math>.
We have however defined the gravitational acceleration to be in the negative <math>z</math> direction.

=== Simplification ===

We use the following vector theorem
<center><math> \nabla ( \mathbf{v} \mathbf{v} )=(\mathbf{v} \cdot \nabla ) \mathbf{v}+ \mathbf{v} (\nabla \cdot \mathbf{v} ) </math></center>
If we have potential flow then <math> \mathbf{v} = \nabla \Phi \,</math> and for solenoidal flow (<math>\nabla \cdot \mathbf{v}=0 </math>), we can use Gauss's vector theorem:
<center><math> \iiint_{\Omega(t)} \nabla \cdot( \mathbf{v} \mathbf{v} ) \mathrm{d}V = \iint_{\partial\Omega(t)}
\left(\mathbf{v} \cdot \mathbf{n} \right) \mathbf{v} \mathrm{d}S </math></center>
In potential flow it follows that
<center><math> \iint_{\partial\Omega(t)} ( \mathbf{v} \cdot \mathbf{n} ) \mathbf{v} \mathrm{d}S = \iint_{\partial\Omega(t)} \frac{\partial\Phi}{\partial n} \nabla \Phi \mathrm{d}S
= \iint_{\partial\Omega(t)} V_n \mathbf{v} \mathrm{d}S </math></center>
since for <math> \nabla^2 \Phi = 0 \, </math>;
<center><math> \iiint_{\partial\Omega} \frac{1}{2} ( \nabla\Phi \cdot \nabla\Phi) \mathbf{n} \mathrm{d}S \equiv \iint_{\partial\Omega} \frac{\partial\Phi}{\partial n} \nabla\Phi \mathrm{d}S. </math></center>
In the above <math> \mathbf{n} \, </math> is the unit vector pointing out of the volume <math> \Omega(t) </math> and <math> V_n = \mathbf{n} \cdot \nabla \Phi </math>

Upon substitution in the momentum flux formula, we obtain:
<center><math> \frac{\mathrm{d}\mathcal{M}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega(t)} \left( ( \frac{P}{\rho} + gz ) \mathbf{n} + \mathbf{v} (V_n - U_n )\right) \mathrm{d}S </math></center>

This formula is of central importance in potential flow marine hydrodynamics because the rate of change of the linear momentum defined above is just <math> \pm \,</math> the force acting on the fluid volume. When its mean value can be shown to vanish, important force expressions on solid boundaries follow and will be derived in what follows.

== Hydrostatic Term ==

Consider separately the term in the momentum flux expression involving the hydrostatic pressure:
<center><math> \frac{\mathrm{d}\mathcal{M}_H}{\mathrm{d}t} = - \rho \iint_{\partial\Omega} gz \mathbf{n},
</math></center>
We break the boundary up into the free surface, ends, body surface, and sea floor at infinite depth, i,.e.
<math>\partial\Omega=\partial\Omega_F + \partial\Omega^{\pm} + \partial\Omega_B + \partial\Omega_{\infty} </math>
The integral over the body surface, assuming a fully submerged body is:
<center><math> \frac{\mathrm{d}\mathcal{M}_{H,B}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega_B} gz \mathbf{n} = \rho g V \mathbf{k} </math></center>
where <math>V</math> is the volume. This follows from the vector theorem of Gauss and is the [http://en.wikipedia.org/wiki/Displacement_%28fluid%29 principle of Archimedes]. That is,
the momentum flux is equal to the buoyancy force.

We may therefore consider the second part of the integral involving wave effects independently and in the absence of the body, assumed fully submerged. In the case of a surface piercing body and in the fully nonlinear case matters are more complex. Consider the application of the momentum conservation theorem in the case of a submerged or floating body in steep waves.

[[Image:Momentum.png|600px|right|thumb|Momentum boundaries]]

Here we consider the two-dimensional case in order to present the concepts. Extensions to three dimensions are then trivial. Note that unlike the [[Wave Energy Density and Flux|energy conservation principle]], the momentum conservation theorem derived above is a vector identity with a horizontal and a vertical component. The integral of the hydrostatic term over the remaining surfaces leads to:
<center><math> \frac{\mathrm{d} \mathcal{M}_{H,S}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega_F+\partial\Omega^+ + \partial\Omega^- +\partial\Omega_{\infty}} gz \mathbf{n} \mathrm{d}S = - \rho g V_{\text{Fluid}} \mathbf{k} </math></center>
where <math>V_{\mbox{Fluid}} </math> is the fluid volume.
This is simply the static weight of the volume of fluid bounded by <math> \partial\Omega_F, \partial\Omega^+, \partial\Omega^- \,</math> and <math> \partial\Omega_{\infty}.</math> With no waves present, this is simply the weight of the ocean water "column" bounded by <math> \partial\Omega\, </math> which does not concern us here. This weight does not change in principle when waves are present at least when <math> \partial\Omega^+, \partial\Omega^- \,</math> are placed sufficiently far away that the wave amplitude has decreased to zero. So "in principle" this term being of hydrostatic origin may be ignored. However, it is in principle more "rational" to apply the momentum conservation theorem over the "linearized" volume <math> V_L(t) \, </math> which is perfectly possible within the framework derived above. In this case <math> \frac{\mathrm{d}\mathcal{M}_H}{\mathrm{d}t} \,</math> is exactly equal to the static weight of the water column and can be ignored in the wave-body interaction problem.

On <math> S_F; P=P_a=0 \, </math> and hence all terms within the free-surface integral and over <math> S_{\infty} \, </math> (seafloor) can be neglected. It follows that:
<center><math> \frac{\mathrm{d} \mathcal{M}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega^\pm +\partial\Omega_B} \left[ \frac{P}{\rho} \mathbf{n} + \mathbf{v} (V_n -U_n) \right] \mathrm{d}S </math></center>

Note that the free surface integrals also vanish for the horizontal component since the hydrostatic force is always vertical. This momentum flux formula is of central importance in wave-body interactions and has many important applications, some of which are discussed bellow. Note that the mathematical derivations involved in its proof apply equally when the volume <math> \Omega </math> and its enclosed surface are selected to be at their linearized positions. In such a case it is essential to set <math> U_n=0\,</math> and <math> V_n \ne 0 </math>. Let the math take over and suggest the proper expression for the force. In the fully nonlinear case, <math>U_n \ne 0 \, </math> on <math> \partial\Omega_F\, </math> and <math> P=0 \,</math> on <math> \partial\Omega_F \, </math>!

On a solid boundary:
<center><math> U_n =V_n \,</math></center>
and
<center><math> \overrightarrow{F}_B = \iint_{S_B} P \vec{n} dS </math></center>
With <math> \vec{n} \,</math> pointing inside the body. We may therefore recast the momentum conservation theorem in the form:
<center><math> \overrightarrow{F}_B (t) = - \frac{\mathrm{d} \overrightarrow{M}}{\mathrm{d}t} - \rho \iint_{S^\pm} \left[ \frac{P}{\rho} \vec{n} + \overrightarrow{V} (V_n - U_n) \right] \mathrm{d}S </math></center>
Where <math> S^\pm \, </math> are fluid boundaries at some distance from the body. If the volume of fluid surrounded by the body, free surface and the furfaces <math> S^\pm \, </math> does not grow in time, then the momentum of the enclosed fluid cannot grow either, so <math> \overrightarrow{M}(t) \, </math> is a stationary physical quantity. It is a well known result that the mean value in time of the time derivative of a stationary quantity is zero. So:
<center><math> {\overline{\frac{\mathrm{d}\overrightarrow{M}}{\mathrm{d}t}}}^t = 0 </math></center>

 Proof :

<center><math> {\overline{\frac{\mathrm{d}F}{\mathrm{d}t}}}^t = \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^{T} \frac{\mathrm{d}F{\tau}}{\mathrm{d}\tau} \mathrm{d}\tau = \lim_{T\to\infty} \frac{1}{2T} [F(T) - F(-T) ] </math></center>

Since <math> F(\pm \tau) \, </math> must be bounded for a stationary signal <math> F(t) \, </math>, it follows that <math> {\overline{\frac{\mathrm{d}F}{\mathrm{d}t}}}^t = 0 \, </math>.

Taking mean values, it follows that:

<center><math> {\overline{\overrightarrow{F}_B (t)}}^t = - \rho \ {\overline{\iint_{S^\pm} \left[ \frac{P}{\rho} \vec{n} + \overrightarrow{V} (V_n - U_n ) \right] \mathrm{d}S}}^t </math></center>

This is the fundamental formula underlying the definition of the mean wave drift forces acting on floating bodies. Such forces are very imprtant for stationary floating structures and can be expressed in terms of integrals of wave effects over control surfaces <math> S^\pm \, </math> which may be located at infinity.

The extension of the above formula for <math> {\overline{\overrightarrow{F}_B}}^t \, </math> in three dimensions is trivial. Simply replace <math> S^\pm \, </math> by <math> S_{\infty} \, </math>, a control surface at infinity. Common choices are a vertical cylindrical boundary or two vertical planes paraller to the axis of forward motion of a ship.

== Applications ==

=== Mean horizontal momentum flux due to a [[Plane Progressive Regular Waves| Plane Progressive Regular Wave]] ===

[[Image:Plane_wave_momentum.png|thumb|right|600px|Plane progressive wave momentum]]

We can determine the mean horizontal momentum flux due to a [[Plane Progressive Regular Waves| Plane Progressive Regular Wave]] with surface displacement
<center><math> \zeta = A \cos (\omega t - k x) \,</math></center>
where <math>\omega</math> and <math>k</math> are related by the [[Dispersion Relation for a Free Surface]]
<math> \omega^2 = gk \tanh kh \, </math>

The momentum flux across <math> \Omega^+ \, </math> is given by

<center><math> \frac{\mathrm{d}M_x}{\mathrm{d}t} = - \int_{-h}^{\zeta} ( P + \rho u^2 ) \mathrm{d}z </math></center>
The pressure is given by [[Conservation Laws and Boundary Conditions|Bernoulli's equation]]
<center><math> P = - \rho \frac{\partial\Phi}{\partial t} - \frac{1}{2} \rho ( u^2 + v^2 ) - \rho g z </math></center>
so that the momentum flux is
<center><math>
\begin{align}
\frac{dM_x}{dt} &= \rho \int_{-h}^{\zeta} \left[ \frac{\partial\Phi}{\partial t} + \frac{1}{2} ( v^2 - u^2 ) + g z \right] \mathrm{d}z \\
&= \rho \left( \int_{-h}^{0} + \int_{0}^{\zeta} \right) \left[ \frac{\partial\Phi}{\partial t} + \frac{1}{2} (v^2 - u^2) + gz \right] \mathrm{d}z
\end{align}
</math></center>

Taking mean values in time and keeping terms of <math> O(A^2) </math> we obtain:

<center><math> {\overline{\frac{\mathrm{d}M_x}{\mathrm{d}t}}}^t = {\overline{\rho \zeta(t) \left. \frac{\partial\Phi}{\partial t} \right |_{h=0}}}^t + \frac{1}{2} \rho \ {\overline{\int_{-h}^{0} (v^2 -u^2 ) \mathrm{d}z}}^t + {\overline{\frac{1}{2} \rho g \zeta^2}}^t </math></center>

Invoking the [[Linear and Second-Order Wave Theory|linearized dynamic free surface condition]] we obtain

<center><math> \left. \frac{\partial\Phi}{\partial t} \right |_{z=0} = - g \zeta </math></center>

It follows that:

<center><math> {\overline{\frac{\mathrm{d}M_x}{\mathrm{d}t}}}^t = {-\frac{1}{2} \rho g \zeta^2 (t)}^t + \frac{1}{2} \rho {\overline{\int_{-h}^0 (v^2-u^2)\mathrm{d}z}}^t </math></center>

In deep water <math> \overline{v^2} = \overline{u^2} \, </math> and the second term is identically zero, so

<center><math> {\overline{\frac{\mathrm{d}M_x}{\mathrm{d}t}}}^t = - \frac{1}{2} \rho g {\overline{\zeta^2 (t)}}^t = - \frac{1}{4} \rho g A^2 </math></center>

In water of finite depth the wave particle trajectories are elliptical with the mean horizontal velocities larger than the mean vertical velocities. So:

<center><math> \overline{V_H^2} < \overline{U_H^2} </math></center>

Therefore, in finite depth the modulus of the mean momentum flux is higher than in deep water for the same A.
So the mean horizontal momentum flux due to a plane progressive wave against its direction of propagation and equal to <math> - \frac{1}{4} \rho g A^2 </math>.

=== [[Wavemaker Theory]] ===

[[Image:Wave_maker_momentum.png|600px|right|thumb|Wavemaker momentum]]
Consider a wave maker shown in the figure generating a wave of amplitude <math>A</math> at infinity.

What is the mean horizontal force on the wavemaker? From the momentum conservation theorem the mean horizontal flux of momentum to the left must flow into the wavemaker. This mean flux translates into a mean horizontal force in the same direction, as shown in the figure. Not an easy conclusion without using some basic fluid mechanics!

-----

This article is based on the MIT open course notes and the original article can be found
[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/A939D3A9-1F49-46F5-BEB5-0F6646CE340E/0/lecture5.pdf here]

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]

Wave Momentum Flux

2018-12-31T12:25:42Z

Xjwiki: /* Simplification */

{{Ocean Wave Interaction with Ships and Offshore Structures
| chapter title = Wave Momentum Flux
| next chapter = [[Wavemaker Theory]]
| previous chapter = [[Wave Energy Density and Flux]]
}}

{{incomplete pages}}

== Introduction ==

The momentum is important to determine the forces.

== Momentum flux in potential flow ==

The momentum flux (the time derivative of the [http://en.wikipedia.org/wiki/Momentum momentum] <math>\mathcal{M}</math>) is given by
<center><math> \frac{\mathrm{d}\mathcal{M}(t)}{\mathrm{d}t} = \rho \frac{\mathrm{d}}{\mathrm{d}t} \iiint_{\Omega(t)} \mathbf{v} \mathrm{d}V
= \rho \iiint_{\Omega(t)} \frac{\partial\mathbf{v}}{\partial t} \mathrm{d}V + \rho \iint_{\partial\Omega(t)} \mathbf{v} U_n \mathrm{d}S,
</math></center>
where <math> U_n \, </math> is the outward normal velocity of the surface <math> \partial\Omega(t)\, </math>.
This equation follows from the [http://en.wikipedia.org/wiki/Reynolds_transport_theorem transport theorem].

We begin with [http://en.wikipedia.org/wiki/Euler_equations Euler's equation] in the absence of viscosity
<center><math>
\frac{\partial\mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla ) \mathbf{v} =
- \nabla \left(\frac{P}{\rho} + g z\right)
</math></center>
where we have defined the direction of gravity to be in the negative <math>z</math>
direction.
We may recast the momentum flux in the form
<center><math>
\frac{\mathrm{d}\mathcal{M}(t)}{\mathrm{d}t} = - \rho \iiint_{\Omega(t)} \left( \nabla ( \frac{P}{\rho} + g z ) + ( \mathbf{v} \cdot \nabla ) \mathbf{v} \right)
\mathrm{d}V + \rho \iint_{\partial\Omega(t)} \mathbf{v} U_n \mathrm{d}S
</math></center>
So far <math> \Omega(t)\, </math> is an arbitrary closed time dependent volume bounded by the time dependent surface <math> \partial\Omega(t)\, </math>.
We have however defined the gravitational acceleration to be in the negative <math>z</math> direction.

=== Simplification ===

We use the following vector theorem
<center><math> \nabla ( \mathbf{v} \mathbf{v} )=(\mathbf{v} \cdot \nabla ) \mathbf{v}+ \mathbf{v} (\nabla \cdot \mathbf{v} ) </math></center>
If we have potential flow then <math> \mathbf{v} = \nabla \Phi \,</math> and for solenoidal flow (<math>\nabla \cdot \mathbf{v} </math>), we can use Gauss's vector theorem:
<center><math> \iiint_{\Omega(t)} \nabla \cdot( \mathbf{v} \mathbf{v} ) \mathrm{d}V = \iint_{\partial\Omega(t)}
\left(\mathbf{v} \cdot \mathbf{n} \right) \mathbf{v} \mathrm{d}S </math></center>
In potential flow it follows that
<center><math> \iint_{\partial\Omega(t)} ( \mathbf{v} \cdot \mathbf{n} ) \mathbf{v} \mathrm{d}S = \iint_{\partial\Omega(t)} \frac{\partial\Phi}{\partial n} \nabla \Phi \mathrm{d}S
= \iint_{\partial\Omega(t)} V_n \mathbf{v} \mathrm{d}S </math></center>
since for <math> \nabla^2 \Phi = 0 \, </math>;
<center><math> \iiint_{\partial\Omega} \frac{1}{2} ( \nabla\Phi \cdot \nabla\Phi) \mathbf{n} \mathrm{d}S \equiv \iint_{\partial\Omega} \frac{\partial\Phi}{\partial n} \nabla\Phi \mathrm{d}S. </math></center>
In the above <math> \mathbf{n} \, </math> is the unit vector pointing out of the volume <math> \Omega(t) </math> and <math> V_n = \mathbf{n} \cdot \nabla \Phi </math>

Upon substitution in the momentum flux formula, we obtain:
<center><math> \frac{\mathrm{d}\mathcal{M}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega(t)} \left( ( \frac{P}{\rho} + gz ) \mathbf{n} + \mathbf{v} (V_n - U_n )\right) \mathrm{d}S </math></center>

This formula is of central importance in potential flow marine hydrodynamics because the rate of change of the linear momentum defined above is just <math> \pm \,</math> the force acting on the fluid volume. When its mean value can be shown to vanish, important force expressions on solid boundaries follow and will be derived in what follows.

== Hydrostatic Term ==

Consider separately the term in the momentum flux expression involving the hydrostatic pressure:
<center><math> \frac{\mathrm{d}\mathcal{M}_H}{\mathrm{d}t} = - \rho \iint_{\partial\Omega} gz \mathbf{n},
</math></center>
We break the boundary up into the free surface, ends, body surface, and sea floor at infinite depth, i,.e.
<math>\partial\Omega=\partial\Omega_F + \partial\Omega^{\pm} + \partial\Omega_B + \partial\Omega_{\infty} </math>
The integral over the body surface, assuming a fully submerged body is:
<center><math> \frac{\mathrm{d}\mathcal{M}_{H,B}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega_B} gz \mathbf{n} = \rho g V \mathbf{k} </math></center>
where <math>V</math> is the volume. This follows from the vector theorem of Gauss and is the [http://en.wikipedia.org/wiki/Displacement_%28fluid%29 principle of Archimedes]. That is,
the momentum flux is equal to the buoyancy force.

We may therefore consider the second part of the integral involving wave effects independently and in the absence of the body, assumed fully submerged. In the case of a surface piercing body and in the fully nonlinear case matters are more complex. Consider the application of the momentum conservation theorem in the case of a submerged or floating body in steep waves.

[[Image:Momentum.png|600px|right|thumb|Momentum boundaries]]

Here we consider the two-dimensional case in order to present the concepts. Extensions to three dimensions are then trivial. Note that unlike the [[Wave Energy Density and Flux|energy conservation principle]], the momentum conservation theorem derived above is a vector identity with a horizontal and a vertical component. The integral of the hydrostatic term over the remaining surfaces leads to:
<center><math> \frac{\mathrm{d} \mathcal{M}_{H,S}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega_F+\partial\Omega^+ + \partial\Omega^- +\partial\Omega_{\infty}} gz \mathbf{n} \mathrm{d}S = - \rho g V_{\text{Fluid}} \mathbf{k} </math></center>
where <math>V_{\mbox{Fluid}} </math> is the fluid volume.
This is simply the static weight of the volume of fluid bounded by <math> \partial\Omega_F, \partial\Omega^+, \partial\Omega^- \,</math> and <math> \partial\Omega_{\infty}.</math> With no waves present, this is simply the weight of the ocean water "column" bounded by <math> \partial\Omega\, </math> which does not concern us here. This weight does not change in principle when waves are present at least when <math> \partial\Omega^+, \partial\Omega^- \,</math> are placed sufficiently far away that the wave amplitude has decreased to zero. So "in principle" this term being of hydrostatic origin may be ignored. However, it is in principle more "rational" to apply the momentum conservation theorem over the "linearized" volume <math> V_L(t) \, </math> which is perfectly possible within the framework derived above. In this case <math> \frac{\mathrm{d}\mathcal{M}_H}{\mathrm{d}t} \,</math> is exactly equal to the static weight of the water column and can be ignored in the wave-body interaction problem.

On <math> S_F; P=P_a=0 \, </math> and hence all terms within the free-surface integral and over <math> S_{\infty} \, </math> (seafloor) can be neglected. It follows that:
<center><math> \frac{\mathrm{d} \mathcal{M}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega^\pm +\partial\Omega_B} \left[ \frac{P}{\rho} \mathbf{n} + \mathbf{v} (V_n -U_n) \right] \mathrm{d}S </math></center>

Note that the free surface integrals also vanish for the horizontal component since the hydrostatic force is always vertical. This momentum flux formula is of central importance in wave-body interactions and has many important applications, some of which are discussed bellow. Note that the mathematical derivations involved in its proof apply equally when the volume <math> \Omega </math> and its enclosed surface are selected to be at their linearized positions. In such a case it is essential to set <math> U_n=0\,</math> and <math> V_n \ne 0 </math>. Let the math take over and suggest the proper expression for the force. In the fully nonlinear case, <math>U_n \ne 0 \, </math> on <math> \partial\Omega_F\, </math> and <math> P=0 \,</math> on <math> \partial\Omega_F \, </math>!

On a solid boundary:
<center><math> U_n =V_n \,</math></center>
and
<center><math> \overrightarrow{F}_B = \iint_{S_B} P \vec{n} dS </math></center>
With <math> \vec{n} \,</math> pointing inside the body. We may therefore recast the momentum conservation theorem in the form:
<center><math> \overrightarrow{F}_B (t) = - \frac{\mathrm{d} \overrightarrow{M}}{\mathrm{d}t} - \rho \iint_{S^\pm} \left[ \frac{P}{\rho} \vec{n} + \overrightarrow{V} (V_n - U_n) \right] \mathrm{d}S </math></center>
Where <math> S^\pm \, </math> are fluid boundaries at some distance from the body. If the volume of fluid surrounded by the body, free surface and the furfaces <math> S^\pm \, </math> does not grow in time, then the momentum of the enclosed fluid cannot grow either, so <math> \overrightarrow{M}(t) \, </math> is a stationary physical quantity. It is a well known result that the mean value in time of the time derivative of a stationary quantity is zero. So:
<center><math> {\overline{\frac{\mathrm{d}\overrightarrow{M}}{\mathrm{d}t}}}^t = 0 </math></center>

 Proof :

<center><math> {\overline{\frac{\mathrm{d}F}{\mathrm{d}t}}}^t = \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^{T} \frac{\mathrm{d}F{\tau}}{\mathrm{d}\tau} \mathrm{d}\tau = \lim_{T\to\infty} \frac{1}{2T} [F(T) - F(-T) ] </math></center>

Since <math> F(\pm \tau) \, </math> must be bounded for a stationary signal <math> F(t) \, </math>, it follows that <math> {\overline{\frac{\mathrm{d}F}{\mathrm{d}t}}}^t = 0 \, </math>.

Taking mean values, it follows that:

<center><math> {\overline{\overrightarrow{F}_B (t)}}^t = - \rho \ {\overline{\iint_{S^\pm} \left[ \frac{P}{\rho} \vec{n} + \overrightarrow{V} (V_n - U_n ) \right] \mathrm{d}S}}^t </math></center>

This is the fundamental formula underlying the definition of the mean wave drift forces acting on floating bodies. Such forces are very imprtant for stationary floating structures and can be expressed in terms of integrals of wave effects over control surfaces <math> S^\pm \, </math> which may be located at infinity.

The extension of the above formula for <math> {\overline{\overrightarrow{F}_B}}^t \, </math> in three dimensions is trivial. Simply replace <math> S^\pm \, </math> by <math> S_{\infty} \, </math>, a control surface at infinity. Common choices are a vertical cylindrical boundary or two vertical planes paraller to the axis of forward motion of a ship.

== Applications ==

=== Mean horizontal momentum flux due to a [[Plane Progressive Regular Waves| Plane Progressive Regular Wave]] ===

[[Image:Plane_wave_momentum.png|thumb|right|600px|Plane progressive wave momentum]]

We can determine the mean horizontal momentum flux due to a [[Plane Progressive Regular Waves| Plane Progressive Regular Wave]] with surface displacement
<center><math> \zeta = A \cos (\omega t - k x) \,</math></center>
where <math>\omega</math> and <math>k</math> are related by the [[Dispersion Relation for a Free Surface]]
<math> \omega^2 = gk \tanh kh \, </math>

The momentum flux across <math> \Omega^+ \, </math> is given by

<center><math> \frac{\mathrm{d}M_x}{\mathrm{d}t} = - \int_{-h}^{\zeta} ( P + \rho u^2 ) \mathrm{d}z </math></center>
The pressure is given by [[Conservation Laws and Boundary Conditions|Bernoulli's equation]]
<center><math> P = - \rho \frac{\partial\Phi}{\partial t} - \frac{1}{2} \rho ( u^2 + v^2 ) - \rho g z </math></center>
so that the momentum flux is
<center><math>
\begin{align}
\frac{dM_x}{dt} &= \rho \int_{-h}^{\zeta} \left[ \frac{\partial\Phi}{\partial t} + \frac{1}{2} ( v^2 - u^2 ) + g z \right] \mathrm{d}z \\
&= \rho \left( \int_{-h}^{0} + \int_{0}^{\zeta} \right) \left[ \frac{\partial\Phi}{\partial t} + \frac{1}{2} (v^2 - u^2) + gz \right] \mathrm{d}z
\end{align}
</math></center>

Taking mean values in time and keeping terms of <math> O(A^2) </math> we obtain:

<center><math> {\overline{\frac{\mathrm{d}M_x}{\mathrm{d}t}}}^t = {\overline{\rho \zeta(t) \left. \frac{\partial\Phi}{\partial t} \right |_{h=0}}}^t + \frac{1}{2} \rho \ {\overline{\int_{-h}^{0} (v^2 -u^2 ) \mathrm{d}z}}^t + {\overline{\frac{1}{2} \rho g \zeta^2}}^t </math></center>

Invoking the [[Linear and Second-Order Wave Theory|linearized dynamic free surface condition]] we obtain

<center><math> \left. \frac{\partial\Phi}{\partial t} \right |_{z=0} = - g \zeta </math></center>

It follows that:

<center><math> {\overline{\frac{\mathrm{d}M_x}{\mathrm{d}t}}}^t = {-\frac{1}{2} \rho g \zeta^2 (t)}^t + \frac{1}{2} \rho {\overline{\int_{-h}^0 (v^2-u^2)\mathrm{d}z}}^t </math></center>

In deep water <math> \overline{v^2} = \overline{u^2} \, </math> and the second term is identically zero, so

<center><math> {\overline{\frac{\mathrm{d}M_x}{\mathrm{d}t}}}^t = - \frac{1}{2} \rho g {\overline{\zeta^2 (t)}}^t = - \frac{1}{4} \rho g A^2 </math></center>

In water of finite depth the wave particle trajectories are elliptical with the mean horizontal velocities larger than the mean vertical velocities. So:

<center><math> \overline{V_H^2} < \overline{U_H^2} </math></center>

Therefore, in finite depth the modulus of the mean momentum flux is higher than in deep water for the same A.
So the mean horizontal momentum flux due to a plane progressive wave against its direction of propagation and equal to <math> - \frac{1}{4} \rho g A^2 </math>.

=== [[Wavemaker Theory]] ===

[[Image:Wave_maker_momentum.png|600px|right|thumb|Wavemaker momentum]]
Consider a wave maker shown in the figure generating a wave of amplitude <math>A</math> at infinity.

What is the mean horizontal force on the wavemaker? From the momentum conservation theorem the mean horizontal flux of momentum to the left must flow into the wavemaker. This mean flux translates into a mean horizontal force in the same direction, as shown in the figure. Not an easy conclusion without using some basic fluid mechanics!

-----

This article is based on the MIT open course notes and the original article can be found
[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/A939D3A9-1F49-46F5-BEB5-0F6646CE340E/0/lecture5.pdf here]

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]

Wave Momentum Flux

2018-12-30T21:18:09Z

Xjwiki: /* Simplification */ Correcting the formulas

{{Ocean Wave Interaction with Ships and Offshore Structures
| chapter title = Wave Momentum Flux
| next chapter = [[Wavemaker Theory]]
| previous chapter = [[Wave Energy Density and Flux]]
}}

{{incomplete pages}}

== Introduction ==

The momentum is important to determine the forces.

== Momentum flux in potential flow ==

The momentum flux (the time derivative of the [http://en.wikipedia.org/wiki/Momentum momentum] <math>\mathcal{M}</math>) is given by
<center><math> \frac{\mathrm{d}\mathcal{M}(t)}{\mathrm{d}t} = \rho \frac{\mathrm{d}}{\mathrm{d}t} \iiint_{\Omega(t)} \mathbf{v} \mathrm{d}V
= \rho \iiint_{\Omega(t)} \frac{\partial\mathbf{v}}{\partial t} \mathrm{d}V + \rho \iint_{\partial\Omega(t)} \mathbf{v} U_n \mathrm{d}S,
</math></center>
where <math> U_n \, </math> is the outward normal velocity of the surface <math> \partial\Omega(t)\, </math>.
This equation follows from the [http://en.wikipedia.org/wiki/Reynolds_transport_theorem transport theorem].

We begin with [http://en.wikipedia.org/wiki/Euler_equations Euler's equation] in the absence of viscosity
<center><math>
\frac{\partial\mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla ) \mathbf{v} =
- \nabla \left(\frac{P}{\rho} + g z\right)
</math></center>
where we have defined the direction of gravity to be in the negative <math>z</math>
direction.
We may recast the momentum flux in the form
<center><math>
\frac{\mathrm{d}\mathcal{M}(t)}{\mathrm{d}t} = - \rho \iiint_{\Omega(t)} \left( \nabla ( \frac{P}{\rho} + g z ) + ( \mathbf{v} \cdot \nabla ) \mathbf{v} \right)
\mathrm{d}V + \rho \iint_{\partial\Omega(t)} \mathbf{v} U_n \mathrm{d}S
</math></center>
So far <math> \Omega(t)\, </math> is an arbitrary closed time dependent volume bounded by the time dependent surface <math> \partial\Omega(t)\, </math>.
We have however defined the gravitational acceleration to be in the negative <math>z</math> direction.

=== Simplification ===

We use the following vector theorem
<center><math> \nabla ( \mathbf{v} \mathbf{v} )=(\mathbf{v} \cdot \nabla ) \mathbf{v}+ \mathbf{v} (\nabla \cdot \mathbf{v} ) </math></center>
If we have potential and solenoidal flow then <math> \mathbf{v} = \nabla \Phi \,</math> and we can use Gauss's vector theorem:
<center><math> \iiint_{\Omega(t)} \nabla \cdot( \mathbf{v} \mathbf{v} ) \mathrm{d}V = \iint_{\partial\Omega(t)}
\left(\mathbf{v} \cdot \mathbf{n} \right) \mathbf{v} \mathrm{d}S </math></center>
In potential flow it follows that
<center><math> \iint_{\partial\Omega(t)} ( \mathbf{v} \cdot \mathbf{n} ) \mathbf{v} \mathrm{d}S = \iint_{\partial\Omega(t)} \frac{\partial\Phi}{\partial n} \nabla \Phi \mathrm{d}S
= \iint_{\partial\Omega(t)} V_n \mathbf{v} \mathrm{d}S </math></center>
since for <math> \nabla^2 \Phi = 0 \, </math>;
<center><math> \iiint_{\partial\Omega} \frac{1}{2} ( \nabla\Phi \cdot \nabla\Phi) \mathbf{n} \mathrm{d}S \equiv \iint_{\partial\Omega} \frac{\partial\Phi}{\partial n} \nabla\Phi \mathrm{d}S. </math></center>
In the above <math> \mathbf{n} \, </math> is the unit vector pointing out of the volume <math> \Omega(t) </math> and <math> V_n = \mathbf{n} \cdot \nabla \Phi </math>

Upon substitution in the momentum flux formula, we obtain:
<center><math> \frac{\mathrm{d}\mathcal{M}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega(t)} \left( ( \frac{P}{\rho} + gz ) \mathbf{n} + \mathbf{v} (V_n - U_n )\right) \mathrm{d}S </math></center>

This formula is of central importance in potential flow marine hydrodynamics because the rate of change of the linear momentum defined above is just <math> \pm \,</math> the force acting on the fluid volume. When its mean value can be shown to vanish, important force expressions on solid boundaries follow and will be derived in what follows.

== Hydrostatic Term ==

Consider separately the term in the momentum flux expression involving the hydrostatic pressure:
<center><math> \frac{\mathrm{d}\mathcal{M}_H}{\mathrm{d}t} = - \rho \iint_{\partial\Omega} gz \mathbf{n},
</math></center>
We break the boundary up into the free surface, ends, body surface, and sea floor at infinite depth, i,.e.
<math>\partial\Omega=\partial\Omega_F + \partial\Omega^{\pm} + \partial\Omega_B + \partial\Omega_{\infty} </math>
The integral over the body surface, assuming a fully submerged body is:
<center><math> \frac{\mathrm{d}\mathcal{M}_{H,B}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega_B} gz \mathbf{n} = \rho g V \mathbf{k} </math></center>
where <math>V</math> is the volume. This follows from the vector theorem of Gauss and is the [http://en.wikipedia.org/wiki/Displacement_%28fluid%29 principle of Archimedes]. That is,
the momentum flux is equal to the buoyancy force.

We may therefore consider the second part of the integral involving wave effects independently and in the absence of the body, assumed fully submerged. In the case of a surface piercing body and in the fully nonlinear case matters are more complex. Consider the application of the momentum conservation theorem in the case of a submerged or floating body in steep waves.

[[Image:Momentum.png|600px|right|thumb|Momentum boundaries]]

Here we consider the two-dimensional case in order to present the concepts. Extensions to three dimensions are then trivial. Note that unlike the [[Wave Energy Density and Flux|energy conservation principle]], the momentum conservation theorem derived above is a vector identity with a horizontal and a vertical component. The integral of the hydrostatic term over the remaining surfaces leads to:
<center><math> \frac{\mathrm{d} \mathcal{M}_{H,S}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega_F+\partial\Omega^+ + \partial\Omega^- +\partial\Omega_{\infty}} gz \mathbf{n} \mathrm{d}S = - \rho g V_{\text{Fluid}} \mathbf{k} </math></center>
where <math>V_{\mbox{Fluid}} </math> is the fluid volume.
This is simply the static weight of the volume of fluid bounded by <math> \partial\Omega_F, \partial\Omega^+, \partial\Omega^- \,</math> and <math> \partial\Omega_{\infty}.</math> With no waves present, this is simply the weight of the ocean water "column" bounded by <math> \partial\Omega\, </math> which does not concern us here. This weight does not change in principle when waves are present at least when <math> \partial\Omega^+, \partial\Omega^- \,</math> are placed sufficiently far away that the wave amplitude has decreased to zero. So "in principle" this term being of hydrostatic origin may be ignored. However, it is in principle more "rational" to apply the momentum conservation theorem over the "linearized" volume <math> V_L(t) \, </math> which is perfectly possible within the framework derived above. In this case <math> \frac{\mathrm{d}\mathcal{M}_H}{\mathrm{d}t} \,</math> is exactly equal to the static weight of the water column and can be ignored in the wave-body interaction problem.

On <math> S_F; P=P_a=0 \, </math> and hence all terms within the free-surface integral and over <math> S_{\infty} \, </math> (seafloor) can be neglected. It follows that:
<center><math> \frac{\mathrm{d} \mathcal{M}}{\mathrm{d}t} = - \rho \iint_{\partial\Omega^\pm +\partial\Omega_B} \left[ \frac{P}{\rho} \mathbf{n} + \mathbf{v} (V_n -U_n) \right] \mathrm{d}S </math></center>

Note that the free surface integrals also vanish for the horizontal component since the hydrostatic force is always vertical. This momentum flux formula is of central importance in wave-body interactions and has many important applications, some of which are discussed bellow. Note that the mathematical derivations involved in its proof apply equally when the volume <math> \Omega </math> and its enclosed surface are selected to be at their linearized positions. In such a case it is essential to set <math> U_n=0\,</math> and <math> V_n \ne 0 </math>. Let the math take over and suggest the proper expression for the force. In the fully nonlinear case, <math>U_n \ne 0 \, </math> on <math> \partial\Omega_F\, </math> and <math> P=0 \,</math> on <math> \partial\Omega_F \, </math>!

On a solid boundary:
<center><math> U_n =V_n \,</math></center>
and
<center><math> \overrightarrow{F}_B = \iint_{S_B} P \vec{n} dS </math></center>
With <math> \vec{n} \,</math> pointing inside the body. We may therefore recast the momentum conservation theorem in the form:
<center><math> \overrightarrow{F}_B (t) = - \frac{\mathrm{d} \overrightarrow{M}}{\mathrm{d}t} - \rho \iint_{S^\pm} \left[ \frac{P}{\rho} \vec{n} + \overrightarrow{V} (V_n - U_n) \right] \mathrm{d}S </math></center>
Where <math> S^\pm \, </math> are fluid boundaries at some distance from the body. If the volume of fluid surrounded by the body, free surface and the furfaces <math> S^\pm \, </math> does not grow in time, then the momentum of the enclosed fluid cannot grow either, so <math> \overrightarrow{M}(t) \, </math> is a stationary physical quantity. It is a well known result that the mean value in time of the time derivative of a stationary quantity is zero. So:
<center><math> {\overline{\frac{\mathrm{d}\overrightarrow{M}}{\mathrm{d}t}}}^t = 0 </math></center>

 Proof :

<center><math> {\overline{\frac{\mathrm{d}F}{\mathrm{d}t}}}^t = \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^{T} \frac{\mathrm{d}F{\tau}}{\mathrm{d}\tau} \mathrm{d}\tau = \lim_{T\to\infty} \frac{1}{2T} [F(T) - F(-T) ] </math></center>

Since <math> F(\pm \tau) \, </math> must be bounded for a stationary signal <math> F(t) \, </math>, it follows that <math> {\overline{\frac{\mathrm{d}F}{\mathrm{d}t}}}^t = 0 \, </math>.

Taking mean values, it follows that:

<center><math> {\overline{\overrightarrow{F}_B (t)}}^t = - \rho \ {\overline{\iint_{S^\pm} \left[ \frac{P}{\rho} \vec{n} + \overrightarrow{V} (V_n - U_n ) \right] \mathrm{d}S}}^t </math></center>

This is the fundamental formula underlying the definition of the mean wave drift forces acting on floating bodies. Such forces are very imprtant for stationary floating structures and can be expressed in terms of integrals of wave effects over control surfaces <math> S^\pm \, </math> which may be located at infinity.

The extension of the above formula for <math> {\overline{\overrightarrow{F}_B}}^t \, </math> in three dimensions is trivial. Simply replace <math> S^\pm \, </math> by <math> S_{\infty} \, </math>, a control surface at infinity. Common choices are a vertical cylindrical boundary or two vertical planes paraller to the axis of forward motion of a ship.

== Applications ==

=== Mean horizontal momentum flux due to a [[Plane Progressive Regular Waves| Plane Progressive Regular Wave]] ===

[[Image:Plane_wave_momentum.png|thumb|right|600px|Plane progressive wave momentum]]

We can determine the mean horizontal momentum flux due to a [[Plane Progressive Regular Waves| Plane Progressive Regular Wave]] with surface displacement
<center><math> \zeta = A \cos (\omega t - k x) \,</math></center>
where <math>\omega</math> and <math>k</math> are related by the [[Dispersion Relation for a Free Surface]]
<math> \omega^2 = gk \tanh kh \, </math>

The momentum flux across <math> \Omega^+ \, </math> is given by

<center><math> \frac{\mathrm{d}M_x}{\mathrm{d}t} = - \int_{-h}^{\zeta} ( P + \rho u^2 ) \mathrm{d}z </math></center>
The pressure is given by [[Conservation Laws and Boundary Conditions|Bernoulli's equation]]
<center><math> P = - \rho \frac{\partial\Phi}{\partial t} - \frac{1}{2} \rho ( u^2 + v^2 ) - \rho g z </math></center>
so that the momentum flux is
<center><math>
\begin{align}
\frac{dM_x}{dt} &= \rho \int_{-h}^{\zeta} \left[ \frac{\partial\Phi}{\partial t} + \frac{1}{2} ( v^2 - u^2 ) + g z \right] \mathrm{d}z \\
&= \rho \left( \int_{-h}^{0} + \int_{0}^{\zeta} \right) \left[ \frac{\partial\Phi}{\partial t} + \frac{1}{2} (v^2 - u^2) + gz \right] \mathrm{d}z
\end{align}
</math></center>

Taking mean values in time and keeping terms of <math> O(A^2) </math> we obtain:

<center><math> {\overline{\frac{\mathrm{d}M_x}{\mathrm{d}t}}}^t = {\overline{\rho \zeta(t) \left. \frac{\partial\Phi}{\partial t} \right |_{h=0}}}^t + \frac{1}{2} \rho \ {\overline{\int_{-h}^{0} (v^2 -u^2 ) \mathrm{d}z}}^t + {\overline{\frac{1}{2} \rho g \zeta^2}}^t </math></center>

Invoking the [[Linear and Second-Order Wave Theory|linearized dynamic free surface condition]] we obtain

<center><math> \left. \frac{\partial\Phi}{\partial t} \right |_{z=0} = - g \zeta </math></center>

It follows that:

<center><math> {\overline{\frac{\mathrm{d}M_x}{\mathrm{d}t}}}^t = {-\frac{1}{2} \rho g \zeta^2 (t)}^t + \frac{1}{2} \rho {\overline{\int_{-h}^0 (v^2-u^2)\mathrm{d}z}}^t </math></center>

In deep water <math> \overline{v^2} = \overline{u^2} \, </math> and the second term is identically zero, so

<center><math> {\overline{\frac{\mathrm{d}M_x}{\mathrm{d}t}}}^t = - \frac{1}{2} \rho g {\overline{\zeta^2 (t)}}^t = - \frac{1}{4} \rho g A^2 </math></center>

In water of finite depth the wave particle trajectories are elliptical with the mean horizontal velocities larger than the mean vertical velocities. So:

<center><math> \overline{V_H^2} < \overline{U_H^2} </math></center>

Therefore, in finite depth the modulus of the mean momentum flux is higher than in deep water for the same A.
So the mean horizontal momentum flux due to a plane progressive wave against its direction of propagation and equal to <math> - \frac{1}{4} \rho g A^2 </math>.

=== [[Wavemaker Theory]] ===

[[Image:Wave_maker_momentum.png|600px|right|thumb|Wavemaker momentum]]
Consider a wave maker shown in the figure generating a wave of amplitude <math>A</math> at infinity.

What is the mean horizontal force on the wavemaker? From the momentum conservation theorem the mean horizontal flux of momentum to the left must flow into the wavemaker. This mean flux translates into a mean horizontal force in the same direction, as shown in the figure. Not an easy conclusion without using some basic fluid mechanics!

-----

This article is based on the MIT open course notes and the original article can be found
[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/A939D3A9-1F49-46F5-BEB5-0F6646CE340E/0/lecture5.pdf here]

[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]

Talk:Wave Drift Forces

2011-06-22T06:36:49Z

Xjwiki: Created page with 'I would like to use the term "diffraction" wave including scattering and incident waves, which is consistent with NEWMAN's paper and WAMIT, as well as FALTINSEN's.'

I would like to use the term "diffraction" wave including scattering and incident waves, which is consistent with NEWMAN's paper and WAMIT, as well as FALTINSEN's.

Memory Effect Function

2011-03-22T11:30:27Z

Xjwiki: /* Frequency-domain equation of motion */

{{incomplete pages}}

==Frequency-domain equation of motion==

The radiation force coefficients also play a significant role in determining the motion of a floating structure possibly subject to mooring and applied forces. To obtain the correct equation of motion in the frequency domain it is important to adopt the approach similar to that taken by [[McIver 2005]], i.e. keep the initial condition terms in the Fourier transform operations. Therefore, if the Fourier transform of the velocity is <math>v_{\mu}(\omega)</math> then the Fourier transform of the acceleration is given by

<center><math>

\int_{0}^{\infty}\ddot{X}_{\mu}(t)e^{i\omega t}\mathrm{d}t=-i\omega v_{\mu}(\omega) - V_{\mu}(0)

</math></center>

the time-derivative of the potential obeys a similar relation

<center><math>

\int_{0}^{\infty}\frac{\partial\Phi}{\partial t}e^{i\omega t}\mathrm{d}t=-i\omega \phi_{\mu}(\omega) - \Phi(\mathbf{x},0).

</math></center>

Therefore, the initial conditions of the structure <math>(X_{\mu}(0),V_{\mu}(0))</math> and of the potential <math>\Phi(\mathbf{x},0)</math> will be present in the Fourier transform of the equation motion which will be referred to as the frequency-domain equation of motion. If the frequency domain potential is decomposed in the same manner as before then the Fourier transform of the equation of motion for the structure is
<center><math>
M\left[-V_{\mu}(0)-i\omega v_{\mu}(\omega)\right]
=-\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}\mathrm{d}S+F^{S}_{\mu}(\omega)+
i\omega \sum_{\nu} (f_{\mu\nu}(\omega)+i\gamma_{\mu\nu}/\omega)v_{\nu}(\omega)-
\sum_{\nu} c_{\mu\nu}\frac{v(\omega)+X(0)}{-i\omega}+f^{A}_{\mu}(\omega)
</math></center>
for <math>\mu=1,\ldots,6</math>, where <math>f^{A}_{\mu}(\omega)</math> is the Fourier transform of the applied force <math>F_{\mu}(t)</math> in equation~(\ref{linearisedmotion}). Although it is assumed that <math>\Phi(\mathbf{x},t)=0</math> for <math>t<0</math>, for a non-zero initial velocity <math>\lim_{t^{+}\rightarrow 0}\Phi(\mathbf{x},t)\neq 0</math> because of the impulsive pressure on the free-surface resulting from the initial motion of the pressure. Therefore, the first term on the right-hand side of the above equation is not identically zero, rather is given by
<center><math>
\rho\iint_{\Gamma}(-\Phi(\mathbf{x},0))n_{\mu}\mathrm{d}S=a(\infty)V(0)
</math></center>
[[Mei 1983]]
where <math>a(\infty)=\lim_{\omega\rightarrow\infty}a(\omega)</math> is the infinite frequency added mass.

The frequency-domain equation is usually re-expressed in the following form
<center><math>
\sum_{\nu} \{c_{\mu\nu}-\omega^{2}\left[M_{\mu\mu}\delta_{\mu\nu}+a_{\mu\nu}(\omega)+i(b_{\mu\nu}+\gamma_{\mu\nu})/\omega\right]\}v_{\nu}(\omega)
= -i\omega\left[ F_{\mu}^{S}(\omega)+f_{\mu}(\omega)+(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)\right]-\sum_{\nu}c_{\mu\nu}X_{\nu}(0)
</math></center>
for <math>\mu=1,\ldots,6</math>. If it is assumed that the structure can only move in one mode of motion and that there is no incident wave or applied force then the complex velocity will be given by
<center><math>
v_{\mu}(\omega)=\frac{-i\omega(M_{\mu\mu}+a_{\mu\mu}(\infty))V_{\mu}(0)-c_{\mu\mu}X_{\mu}(0)}{c_{\mu\mu}-\omega^{2}\left[M_{\mu\mu}+a_{\mu\mu}(\omega)+i(b_{\mu\mu}+\gamma_{\mu\mu})/\omega\right]}.
</math></center>

As described by [[McIver 2006]], the zero of the denominator gives the location of the pole referred to as the motion resonance. The complex force coefficient <math>f_{\mu\nu}(\omega)</math> will also possess a pole, however this resonance is annulled by the frequency-dependence of the complex velocity (see [[McIver 2006]]) and so the motion, in the absence of incident waves, is dominated by the motion resonance term. Solving for <math>v(\omega)</math> then the time-domain velocity can be recovered using the inverse transform
<center><math>
V(t)=\frac{1}{\pi} Re \int^{\infty}_{0}v(\omega)e^{-i\omega t} \mathrm{d}t.
</math></center>

==Integro-differential equation method==

The widespread availability of numerical and analytical methods for determining hydrodynamic coefficients in the frequency domain facilitated the development of a solution method for the motion of a floating body in the time domain. The principles of this method are described by \citeasnoun{ccmei}~in~\S7.11 based on work by \citeasnoun{cummins} and \citeasnoun{wehausen1971} among others. The method requires the consideration of the fluid response to a displacement given impulsively to a structure denoted by <math>\delta(t-\tau)V_{\alpha}(\tau)</math> where <math>\dot{X_{\alpha}}=V_{\alpha}</math>. The resultant impulse response functions for the velocity potential and the hydrodynamic force are written as the sum of an infinite frequency component and a time-dependent component using the Cummins' decomposition. To obtain the fluid response to a general continuous velocity function <math>V_{\alpha}(t)</math> an integral of the impulse response over the range <math>-\infty<\tau<\infty</math> must be calculated. The hydrodynamic force on the structure due to the forced motion of the structure is then shown to be

<center><math>(11)

\mathcal{F}_{\beta}^{R}(t)=-m_{\beta\alpha}(\infty)\ddot{X_{\alpha}}-\int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,\mathrm{d}\tau

</math></center>

where <math>m_{\beta\alpha}(\infty)</math> is an infinite frequency added mass coefficient and <math>L_{\beta\alpha}(t-\tau)</math> is referred to as the impulse response function.

By considering the velocity of the body to be time-harmonic, i.e. <math>V_{\alpha}(t)=Re\{ v_{\alpha} e^{-i\omega t} \}</math>, the impulse response function <math>L_{\beta\alpha}</math> can be related to the added mass and damping coefficients by comparing the force expression~(11) to (5). The comparison yields the following relations

<center><math>
\begin{align}

a_{\alpha\beta}-m_{\alpha\beta}(\infty)&=\int^{\infty}_{0}L_{\alpha\beta}(\tau)\cos\omega\tau\, \mathrm{d}\tau \\

b_{\alpha\beta}&=\omega\int^{\infty}_{0}L_{\alpha\beta}(\tau)\sin\omega\tau\, \mathrm{d}\tau

\end{align}
</math></center>

and so if the added mass or damping coefficients are known for all frequencies <math>0\leq\omega\leq\infty</math>, the impulse response or memory function <math>L_{\alpha\beta}(t)</math> can be found by inverse cosine or sine transform. Therefore, it is be possible to determine the radiation force due to a non-harmonic forcing by using frequency-domain information. The exciting force can also be determined in a similar manner; to obtain it the incident wave-form and the radiation impulse response function are required.

By expressing the total hydrodynamic force in terms of the radiation force~(11) and the exciting force, the equation of motion~(\ref{linearisedmotion}) for transient motions of a floating structure becomes

<center><math>

\left[M_{\alpha\beta}+m_{\alpha\beta}(\infty)\right]\ddot{X}_{\beta} + \int^{t}_{-\infty}L_{\beta\alpha}(t-\tau)\ddot{X}_{\alpha}(\tau)\,\mathrm{d}\tau + c_{\alpha\beta}X_{\beta}=\mathcal{F}^{D}_{\alpha}+F_{\mu}(t),

</math></center>
for <math>\alpha=1,\ldots,6</math>, where repeated indices implies summation. This set of integro-differential equations is to be solved given the position and velocity of the body. It is assumed that the impulse response function and infinite frequency added mass are known from Fourier transforms of the frequency domain hydrodynamic coefficients.

Thus, it is possible, in principle, to calculate the transient response of a floating structure from the frequency-domain responses. Numerically, an accurate calculation of transient response requires that the hydrodynamic coefficients be computed at a large number of discrete values of the frequency <math>\omega</math> on the interval <math>[0,\infty)</math>. This time-domain solution method has been in use since the work by Cummins and Wehausen. As numerical methods and computing power have improved, more complex problems can be solved to a higher degree of accuracy. A recent application of the method can be found in the investigations into latching control by~\citeasnoun{babarit2006}. The principal drawback of the application of this method is that the motion of the fluid cannot be determined from the computational results. Therefore, if the fluid response is required, for example in the investigation of trapped modes, further computations are necessary.

Wave Forces on a Body

2009-07-01T09:46:05Z

Xjwiki: /* Type of Forces */

== Wave Forces on a Body ==

{| border="0" align="center"
| <math> U = \omega A \, </math>
|-
| <math> R_e = \frac{U\ell}{\nu} = \frac{\omega A \ell}{\nu} \, </math>
|-
| <math> K_C = \frac{UT}{\ell} = \frac{A\omega T}{\ell} = 2 \pi \frac{A}{\ell} \, </math>
|}

{| border="0" align="center"
|- align="center"
| <math> C_F = \frac{F}{\rho g A \ell^2} = f \left( \frac{}{} \right. </math>
| <math> \underbrace{\frac{A}{\lambda}} \,</math>,
| <math> \underbrace{\frac{\ell}{\lambda}} \, </math>,
| <math> R_e \, </math>,
| <math> \frac{h}{\lambda} \, </math>,
| roughness,
| <math> \ldots \left. \frac{}{} \right) \, </math>
|- align="center"
| || Wave || Diffraction
|- align="center"
| || steepness || parameter
|}

=== Type of Forces ===

1. '''Viscous forces''' Form drag, viscous drag <math> = f ( R_e, K_c, \, </math> roughness, <math> \ldots ) </math>.

* ''Form drag'' <math> ( C_D ) \, </math>

Associated primarily with flow separation -normal stresses.

* ''Friction drag'' <math> ( C_F ) \, </math>

{| border="0"
| Associated with skin friction <math> \tau_w, \ i.e., \ \, </math>
| <math> \vec{F} \sim \iint \tau_w \, </math>
| <math> dS \, </math>.
|- align="center"
|
| body
|- align="center"
|
| (wetted surface)
|}

2. '''Inertial forces''' Froude-Krylov forces, diffraction forces, radiation forces.

Forces arising from potential flow wave theory,

{| border="0" align="center"
| <math> \vec{F} = \iint p \hat{n} \, </math>
| <math> dS \, </math>,
| where <math> \ p = - \rho \left( \frac{\partial\phi}{\partial t} + g y \right. \, </math>
| <math> + \left. \underbrace{ \frac{1}{2} \left| \nabla \phi \right|^2} \right) \, </math>
|- align="center"
| body || || || <math> =0 \, </math> , for linear theory,
|- align="center"
| (wetted surface) || || || small amplitude waves
|}

For linear theory, the velocity potential <math> \phi \, </math> and the pressure <math> p \, </math> can be decomposed to

{| border="0" align="center"
|- align="center"
| <math> \phi = \, </math> || <math> \underbrace{\phi_I} \, </math> || <math> + \, </math> || <math> \underbrace{\phi_D} \, </math> || <math> + \, </math> || <math> \underbrace{\phi_R} \, </math>
|- align="center"
| || Incident wave || || Scattered wave || || Radiated wave
|- align="center"
| || potential <math> (a) \, </math> || || potential <math> (b.1) \, </math> || || potential <math> (b.2) \, </math>
|- align="center"
| <math> - \frac{p}{\rho} = \, </math> || <math> \frac{\partial\phi_I}{\partial t} \, </math> || <math> + \, </math> || <math> \frac{\partial\phi_D}{\partial t} \, </math> || <math> + \, </math> || <math> \frac{\partial\phi_R}{\partial t} \, </math> || <math> + \,
</math> || <math> g y \, </math>
|}

(a) Incident wave potential

* ''Froude-Krylov Force approximation'' When <math> \ell \ll \lambda \, </math>, the incident wave field is not significantly modified by the presence of the body, therefore ignore <math> \phi_D \,</math> and <math> \phi_R \, </math>. Froude-Krylov approximation:

{| border="0"
|- align="center"
| <math> \left. \begin{matrix} & \phi \approx \phi_I \\ & p \approx - \rho \left( \frac{\partial\phi_I}{\partial t} + g y \right) \end{matrix} \right\} </math>
| <math> \Rightarrow \vec{F}_{FK} = \, </math>
| <math> \iint \underbrace{- \rho \left( \frac{\partial\phi_I}{\partial t} + g y \right)} </math>
| <math> \hat{n} dS \leftarrow \, </math>
| width="225" | can calculate knowing (incident) wave kinematics (and body geometry)
|-
|
|
| body <math> . \qquad \equiv p_I \, </math>
|-
|
|
| width="25" | surface
|}

* Mathematical approximation After applying the divergence theorem, the <math> \vec{F}_{FK} \, </math> can be rewritten as
{| border'"0" align="center"
| <math> \vec{F}_{FK} \, </math>
| <math> = - \iint p_I \hat{n} \, </math>
| <math> dS = - \iiint \nabla p_I d\forall \, </math>
|- align="center"
|
| body surface
| body volume
|}

If the body dimensions are very small comparable to the wave length, we can assume that <math> \nabla_{p_I} \, </math> is approximately constant through the body volume <math> \forall \, </math> and 'pull' the <math> \nabla_{p_I} \, </math> out of the integral. Thus, the <math> \vec{F}_{FK} \, </math> can be approximated as

{| border="0" align="center"
| <math> \vec{F}_{FK} \cong \left( - \nabla_{p_I} \right) \left. \frac{}{} \right| \, </math>
| valign="bottom" align="center" | at body
| <math> \iiint d\forall = \, </math>
| <math> \underbrace{\forall} \, </math>
| <math> \left( - \nabla_{p_I} \right) \left. \frac{}{} \right| \, </math>
| valign="bottom" | at body
|-
|
| align="right" valign="top" | center
| body volume
| body volume
|
| align="right" valign="top" | center
|}

The last relation is particularly useful for small bodies of non-trivial geometry for 13.021, that is all bodies that do not have a rectangular cross section.

(b) Diffraction and Radiation Forces

(b.1) Diffraction or scattering force When <math> \ell \not\ll \lambda \, </math>, the wave field near the body will be affected even if the body is stationary, so that no-flux B.C. is satisfied.

{| border="0" align="center"
| <math> \vec{F}_D \ = \ </math>
| <math> \iint - \rho \left( \frac{\partial\phi_D}{\partial t} \right) \hat{n} dS </math>
|-
| || width="50" | body surface
|}

(b.2) '''Radiation Force -added mass and damping coefficient''' Even in the absence of an incident wave, a body in motion creates waves and hence inertial wave forces.

{| border="0" align="center"
|-
| <math> \vec{F}_R = \, </math> || <math> \iint - \rho \left( \frac{\partial\phi_R}{\partial t} \right) \hat{n} dS = - </math>
| <math> \underbrace{m_{ij}} \, </math> || <math> \dot{U}_j \ - \, </math>
| <math> \underbrace{d_{ij}} \, </math> || <math> U_j \, </math>
|- valign="top"
| || width="50" | body surface || width="50" | added mass || || width="50" | wave radiation damping
|}

=== Important parameters ===

{| border="0" align="center"
| <math> (1) K_C = \frac{UT}{\ell} = 2 \pi \frac{A}{\ell} \, </math>
| rowspan="3" valign="top" | <math> \left. \begin{matrix} \\ \\ \\ \\ \\ \\ \end{matrix} \right\} \, </math>
| Interrelated through maximum wave steepness
|-
|
| align="center" | <math> \frac{A}{\lambda} \leq 0.07 \, </math>
|-
| (2)diffraction parameter <math> \frac{\ell}{\lambda} \, </math>
| align="center" | <math> \left( \frac{A}{\ell} \right) \left( \frac{\ell}{\lambda} \right) \leq 0.07 \, </math>
|}

* If <math> K_c \leq 1 \, </math>: no appreciable flow separation, viscous effect confined to boundary layer (hence small), solve problem via potential theory. In addition, depending on the value of the ratio <math> \frac{\ell}{\lambda} \, </math>,

* If <math> \frac{\ell}{\lambda} \ll 1 \, </math>, ignore diffraction , wave effects in radiation problem (i.e., <math> d_{ij} \approx 0, \ m_{ij} \approx m_{ij} \, </math> infinite fluid added mass). F-K approximation might be used, calculate <math> \vec{F}_{FK} \, </math>.
* If <math> \frac{\ell}{\lambda} \gg 1/5 \, </math>, must consider wave diffraction, radiation <math> \left( \frac{A}{\ell} \leq \frac{0.07}{\ell / \lambda} \leq 0.035 \right) \, </math>.

* If <math> K_C \gg 1 \, </math>: separation important, viscous forces can not be neglected. Further on if <math> \frac{\ell}{\lambda} \leq \frac{0.07}{A/\ell} \, </math> so <math>
\frac{\ell}{\lambda} \ll 1 \, </math> ignore diffraction, i.e., the Froude-Krylov approximation is valid.

{| border="0" align="center"
| <math> F = \frac{1}{2} \rho \ell^2 \, </math>
| <math> \underbrace{U(t)} \, </math>
| <math> \left| U(t) \right| C_D \left( R_e \right) \, </math>
|-
| || width="50" | relative velocity
|}

* Intermediate <math> K_c - \, </math> both viscous and inertial effects important, use Morrison's formula.

<center><math> F= \frac{1}{2} \rho \ell^2 U(t) \left| U(t) \right| C_D \left( R_e \right) + \rho \ell^3 \dot{U} C_m \left( R_e, K_C \right) </math></center>

* Summary

I. Use: <math> C D \, </math> and <math> F - K \, </math> approximation.

II. Use: <math> C F \, </math> and <math> F - K \, </math> approximation.

III. <math> C D \, </math> is not important and <math> F - K \, </math> approximation is not valid.

-----

This article is based on the MIT open course notes and the original article can be found [http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-20Spring-2005/E5599747-6768-4532-9601-8E4297E8DAC2/0/lecture22.pdf here].

[[Marine Hydrodynamics]]