Wave Forces on a Body

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Wave Forces on a Body

[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]
[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]
Wave Diffraction
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]
Associated with skin friction [math] \tau_w, \ i.e., \ \, [/math] [math] \vec{F} \sim \iint \tau_w \, [/math] [math] dS \, [/math].
(wetted surface)

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

Forces arising from potential flow wave theory,

[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]
body [math] =0 \, [/math] , for linear theory,
(wetted surface) small amplitude waves

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

[math] \phi = \, [/math] [math] \underbrace{\phi_I} \, [/math] [math] + \, [/math] [math] \underbrace{\phi_D} \, [/math] [math] + \, [/math] [math] \underbrace{\phi_R} \, [/math]
Incident wave Scattered wave Radiated wave
potential [math] (a) \, [/math] potential [math] (b.1) \, [/math] potential [math] (b.2) \, [/math]
[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:
[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] can calculate knowing (incident)
wave kinematics (and body geometry)
body [math] . \qquad \equiv p_I \, [/math]
  • Mathematical approximation After applying the divergence theorem, the [math] \vec{F}_{FK} \, [/math] can be rewritten as
[math] \vec{F}_{FK} \, [/math] [math] = - \iint p_I \hat{n} \, [/math] [math] dS = - \iiint \nabla p_I d\forall \, [/math]

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

[math] \vec{F}_{FK} \cong \left( - \nabla_{p_I} \right) \left. \frac{}{} \right| \, [/math] at body [math] \iiint d\forall = \, [/math] [math] \underbrace{\forall} \, [/math] [math] \left( - \nabla_{p_I} \right) \left. \frac{}{} \right| \, [/math] at body
center body

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.

[math] \vec{F}_D \ = \ [/math] [math] \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] \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]
body surface added mass wave radiation damping

Important parameters

[math] (1) K_C = \frac{UT}{\ell} = 2 \pi \frac{A}{\ell} \, [/math] [math] \left. \begin{matrix} \\ \\ \\ \\ \\ \\ \end{matrix} \right\} \, [/math] Interrelated through maximum wave steepness
[math] \frac{A}{\lambda} \leq 0.07 \, [/math]
(2)diffraction parameter [math] \frac{\ell}{\lambda} \, [/math] [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.
[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]
relative velocity
  • Intermediate [math] K_c - \, [/math] both viscous and inertial effects important, use Morrison's formula.
[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]
  • 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 here.

Marine Hydrodynamics