# Wave Forces on a Body

## 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.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.
 $F = \frac{1}{2} \rho \ell^2 \,$ $\underbrace{U(t)} \,$ $\left| U(t) \right| C_D \left( R_e \right) \,$ relative velocity
• Intermediate $K_c - \,$ both viscous and inertial effects important, use Morrison's formula.
$F= \frac{1}{2} \rho \ell^2 U(t) \left| U(t) \right| C_D \left( R_e \right) + \rho \ell^3 \dot{U} C_m \left( R_e, K_C \right)$
• Summary

I. Use: $C D \,$ and $F - K \,$ approximation.

II. Use: $C F \,$ and $F - K \,$ approximation.

III. $C D \,$ is not important and $F - K \,$ approximation is not valid.