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{ C_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., \ \, }[/math] | [math]\displaystyle{ \vec{F} \sim \iint \tau_w \, }[/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} = \iint 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 | Scattered 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 \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 [math]\displaystyle{ . \qquad \equiv p_I \, }[/math] | ||||
| surface |
- Mathematical approximation After applying the divergence theorem, the [math]\displaystyle{ \vec{F}_{FK} \, }[/math] can be rewritten as
| [math]\displaystyle{ \vec{F}_{FK} \, }[/math] | [math]\displaystyle{ = - \iint p_I \hat{n} \, }[/math] | [math]\displaystyle{ dS = - \iiint \nabla p_I d\forall \, }[/math] |
| body surface |
body volume |
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. \frac{}{} \right| \, }[/math] | at body | [math]\displaystyle{ \iiint d\forall = \, }[/math] | [math]\displaystyle{ \underbrace{\forall} \, }[/math] | [math]\displaystyle{ \left( - \nabla_{p_I} \right) \left. \frac{}{} \right| \, }[/math] | 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 [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.
- 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.
This article is based on the MIT open course notes and the original article can be found here.
