Wave Forces on a Body

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Wave Forces on a Body

$U = \omega A \,$

$R_e = \frac{U\ell}{\nu} = \frac{\omega A \ell}{\nu} \,$

$K_C = \frac{UT}{\ell} = \frac{A\omega T}{\ell} = 2 \pi \frac{A}{\ell} \,$

$C_F = \frac{F}{\rho g A \ell^2} = f \left( \frac{}{} \right.$	$\underbrace{\frac{A}{\lambda}} \,$ ,	$\underbrace{\frac{\ell}{\lambda}} \,$ ,	$R_e \,$ ,	$\frac{h}{\lambda} \,$ ,	roughness,	$\ldots \left. \frac{}{} \right) \,$
	Wave	Diffraction
	steepness	parameter

Type of Forces

1. Viscous forces Form drag, viscous drag $= f ( R_e, K_c, \,$ roughness, $\ldots )$ .

Form drag $( C_D ) \,$

Associated primarily with flow separation -normal stresses.

Friction drag $( C_F ) \,$

Associated with skin friction $\tau_w, \ i.e., \ \,$	$\vec{F} \sim \iint \tau_w \,$	$dS \,$ .
	body
	(wetted surface)

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

Forces arising from potential flow wave theory,

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

For linear theory, the velocity potential $\phi \,$ and the pressure $p \,$ can be decomposed to

$\phi = \,$	$\underbrace{\phi_I} \,$	$+ \,$	$\underbrace{\phi_D} \,$	$+ \,$	$\underbrace{\phi_R} \,$
	Incident wave		Scattered wave		Radiated wave
	potential $(a) \,$		potential $(b.1) \,$		potential $(b.2) \,$
$- \frac{p}{\rho} = \,$	$\frac{\partial\phi_I}{\partial t} \,$	$+ \,$	$\frac{\partial\phi_D}{\partial t} \,$	$+ \,$	$\frac{\partial\phi_R}{\partial t} \,$	$+ \,$	$g y \,$

(a) Incident wave potential

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

$\left. \begin{matrix} & \phi \approx \phi_I \\ & p \approx - \rho \left( \frac{\partial\phi_I}{\partial t} + g y \right) \end{matrix} \right\}$	$\Rightarrow \vec{F}_{FK} = \,$	$\iint \underbrace{- \rho \left( \frac{\partial\phi_I}{\partial t} + g y \right)}$	$\hat{n} dS \leftarrow \,$	can calculate knowing (incident) wave kinematics (and body geometry)
		body $. \qquad \equiv p_I \,$
		surface

Mathematical approximation After applying the divergence theorem, the $\vec{F}_{FK} \,$ can be rewritten as

$\vec{F}_{FK} \,$	$= - \iint p_I \hat{n} \,$	$dS = - \iiint \nabla p_I d\forall \,$
	body surface	body volume

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

$\vec{F}_{FK} \cong \left( - \nabla_{p_I} \right) \left. \frac{}{} \right\| \,$	at body	$\iiint d\forall = \,$	$\underbrace{\forall} \,$	$\left( - \nabla_{p_I} \right) \left. \frac{}{} \right\| \,$	at body
	center	body volume	body volume		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 $\ell \not\ll \lambda \,$ , the wave field near the body will be affected even if the body is stationary, so that no-flux B.C. is satisfied.

$\vec{F}_D \ = \$	$\iint - \rho \left( \frac{\partial\phi_D}{\partial t} \right) \hat{n} dS$
	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.

$\vec{F}_R = \,$	$\iint - \rho \left( \frac{\partial\phi_R}{\partial t} \right) \hat{n} dS = -$	$\underbrace{m_{ij}} \,$	$\dot{U}_j \ - \,$	$\underbrace{d_{ij}} \,$	$U_j \,$
	body surface	added mass		wave radiation damping

Important parameters

$(1) K_C = \frac{UT}{\ell} = 2 \pi \frac{A}{\ell} \,$	$\left. \begin{matrix} \\ \\ \\ \\ \\ \\ \end{matrix} \right\} \,$	Interrelated through maximum wave steepness
		$\frac{A}{\lambda} \leq 0.07 \,$
(2)diffraction parameter $\frac{\ell}{\lambda} \,$		$\left( \frac{A}{\ell} \right) \left( \frac{\ell}{\lambda} \right) \leq 0.07 \,$

If $K_c \leq 1 \,$ : 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 $\frac{\ell}{\lambda} \,$ ,

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

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