Wave Forces on a Body

Wave Forces on a Body

[math]\displaystyle{ U = \omega A \, }[/math]

[math]\displaystyle{ R_e = \frac{U\ell}{\nu} = \frac{\omega A \ell}{\nu} \, }[/math]

[math]\displaystyle{ K_C = \frac{UT}{\ell} = \frac{A\omega T}{\ell} = 2 \pi \frac{A}{\ell} \, }[/math]

[math]\displaystyle{ D_F = \frac{F}{\rho g A \ell^2} = f \left( \frac{}{} \right. }[/math]	[math]\displaystyle{ \underbrace{\frac{A}{\lambda}} \, }[/math],	[math]\displaystyle{ \underbrace{\frac{\ell}{\lambda}} \, }[/math],	[math]\displaystyle{ R_e \, }[/math],	[math]\displaystyle{ \frac{h}{\lambda} \, }[/math],	roughness,	[math]\displaystyle{ \ldots \left. \frac{}{} \right) \, }[/math]
	Wave	Diffraction
	steepness	parameter

Type of Forces

1. Viscous forces Form drag, viscous drag [math]\displaystyle{ = f ( R_e, K_c, \, }[/math] roughness, [math]\displaystyle{ \ldots ) }[/math].

• Form drag [math]\displaystyle{ ( C_D ) \, }[/math]

Associated primarily with flow separation -normal stresses.

• Friction drag [math]\displaystyle{ ( C_F ) \, }[/math]

Associated with skin friction [math]\displaystyle{ \tau_w, \ i.e., \ \vec{F} \, }[/math]	[math]\displaystyle{ dS \, }[/math].
body
(wetted surface)

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

Forces arising from potential flow wave theory,

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

For linear theory, the velocity potential [math]\displaystyle{ \phi \, }[/math] and the pressure [math]\displaystyle{ p \, }[/math] can be decomposed to

[math]\displaystyle{ \phi = \, }[/math]	[math]\displaystyle{ \underbrace{\phi_I} \, }[/math]	[math]\displaystyle{ + \, }[/math]	[math]\displaystyle{ \underbrace{\phi_D} \, }[/math]	[math]\displaystyle{ + \, }[/math]	[math]\displaystyle{ \underbrace{\phi_R} \, }[/math]
	Incident wave		Diffracted wave		Radiated wave
	potential [math]\displaystyle{ (a) \, }[/math]		potential [math]\displaystyle{ (b.1) \, }[/math]		potential [math]\displaystyle{ (b.2) \, }[/math]
[math]\displaystyle{ - \frac{p}{\rho} = \, }[/math]	[math]\displaystyle{ \frac{\partial\phi_I}{\partial t} \, }[/math]	[math]\displaystyle{ + \, }[/math]	[math]\displaystyle{ \frac{\partial\phi_D}{\partial t} \, }[/math]	[math]\displaystyle{ + \, }[/math]	[math]\displaystyle{ \frac{\partial\phi_R}{\partial t} \, }[/math]	[math]\displaystyle{ + \, }[/math]	[math]\displaystyle{ g y \, }[/math]

(a) Incident wave potential

• Froude-Krylov Force approximation When [math]\displaystyle{ \ell \ll \lambda \, }[/math], the incident wave field is not significantly modified by the presence of the body, therefore ignore [math]\displaystyle{ \phi_D \, }[/math] and [math]\displaystyle{ \phi_R \, }[/math]. Froude-Krylov approximation:

[math]\displaystyle{ \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]\displaystyle{ \Rightarrow \vec{F}_{FK} = \, }[/math]	[math]\displaystyle{ \iint \, }[/math]	[math]\displaystyle{ \underbrace{- \rho \left( \frac{\partial\phi_I}{\partial t} + g y \right)} }[/math]	[math]\displaystyle{ \hat{n} dS \leftarrow \, }[/math]	can calculate knowing (incident) wave kinematics (and body geometry)
		body surface	[math]\displaystyle{ \equiv p_I \, }[/math]

• Mathematical approximation After applying the divergence theorem, the [math]\displaystyle{ \vec{F}_{FK} \, }[/math] can be rewritten as [math]\displaystyle{ \vec{F}_{FK} = - \iint_{\mbox{body surface}} p_I \hat{n} dS = - \iiint_{\mbox{body volume}} \nabla p_I d\forall }[/math].

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

[math]\displaystyle{ \vec{F}_{FK} \cong \left( - \nabla_{p_I} \right) \left| {}_{\mbox{at body center}} \iiint_{\mbox{body volume}} d\forall = \underbrace{\forall}_{\mbox{body volume}} \left( - \nabla_{p_I} \right) \right| {}_{\mbox{at body center}} }[/math]

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]\displaystyle{ \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.

[math]\displaystyle{ \vec{F}_D \ = \ }[/math]	[math]\displaystyle{ \iint - \rho \left( \frac{\partial\phi_D}{\partial t} \right) \hat{n} dS }[/math]
	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.

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

Important parameters

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

• If [math]\displaystyle{ 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]\displaystyle{ \frac{\ell}{\lambda} \, }[/math],

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

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

[math]\displaystyle{ F = \frac{1}{2} \rho \ell^2 \, }[/math]	[math]\displaystyle{ \underbrace{U(t)} \, }[/math]	[math]\displaystyle{ \left| U(t) \right| C_D \left( R_e \right) \, }[/math]
	relative velocity

• Intermediate [math]\displaystyle{ K_c - \, }[/math] both viscous and inertial effects important, use Morrison's formula.

[math]\displaystyle{ 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]

• Summary

I. Use: [math]\displaystyle{ C D \, }[/math] and [math]\displaystyle{ F - K \, }[/math] approximation.

II. Use: [math]\displaystyle{ C F \, }[/math] and [math]\displaystyle{ F - K \, }[/math] approximation.

III. [math]\displaystyle{ C D \, }[/math] is not important and [math]\displaystyle{ F - K \, }[/math] approximation is not valid.