OrcaFlex Manual - Orcina

w
Repenetration Offset After Uplift
149
Theory, Environment Theory
This is the parameter λrep that controls the penetration at which the repenetration resistance limit Pmax(z) in
equation (3b) in Repenetration mode merges with the bounding curve for initial penetration resistance, PIP(z). A
smaller value results in less penetration past zP=0 before the repenetration resistance after uplift merges with the
bounding curve of initial penetration resistance. A higher value leads to greater penetration before the bounding
curve is reached.
5.10.5 Morison's Equation
OrcaFlex calculates hydrodynamic loads on lines, 3D buoys and 6D buoys using an extended form of Morison's
Equation. See Morison, O'Brien, Johnson and Schaaf.
Morison's equation was originally formulated for calculating the wave loads on fixed vertical cylinders. There are
two force components, one related to water particle acceleration (the 'inertia' force) and one related to water
particle velocity (the 'drag' force). For moving objects, the same principle is applied, but the force equation is
modified to take account of the movement of the body.
The extended form of Morison's equation used in OrcaFlex is:
where
Fw = (Δ.aw + Ca.Δ.ar) + ½.ρ.Cd.A.Vr|Vr|
Fw is the fluid force
Δ is the mass of fluid displaced by the body
aw is the fluid acceleration relative to earth
Ca is the added mass coefficient for the body
ar is the fluid acceleration relative to the body
ρ is the density of water
Vr is the fluid velocity relative to the body
Cd is the drag coefficient for the body
A is the drag area
The term in parentheses is the inertia force, the other term is the drag force. The drag force is familiar to most
engineers, but the inertia force can cause confusion.
The inertia force consists of two parts, one proportional to fluid acceleration relative to earth (the Froude-Krylov
component), and one proportional to fluid acceleration relative to the body (the added mass component).
To understand the Froude-Krylov component, imagine the body being removed and replaced with an equivalent
volume of water. This water would have mass Δ and be undergoing an acceleration aw. It must therefore be
experiencing a force Δ.aw.
Now remove the water and put the body back: the same force must now act on the body. This is equivalent to saying
that the Froude-Krylov force is the integral over the surface of the body of the pressure in the incident wave,
undisturbed by the presence of the body. (Note the parallel with Archimedes' Principle: in still water, the integral of
the fluid pressure over the wetted surface must exactly balance the weight of the water displaced by the body.)
The added mass component is due to the distortion of the fluid flow by the presence of the body. A simple way to
understand it is to consider a body accelerating through a stationary fluid. The force required to sustain the
acceleration may be shown to be proportional to the body acceleration and can be written:
where
F = (m + Ca.Δ).a
F is the total force on the body
m is the mass of the body
(Ca.Δ) is a constant related to the shape of the body and its displacement
a is the acceleration of the body.