OrcaFlex Manual - Orcina

Theory, Line Theory
5.12.12 Contents Flow Effects
Introduction
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Contents flow effects are normally neglected when modelling pipes in OrcaFlex. However for pipes carrying high
contents density at rapid flow rates the flow effects can be significant. The published literature show that there are
three extra forces introduced by contents flow – a centrifugal force, a Coriolis force and a flow friction force.
OrcaFlex includes the facility to specify contents flow. If contents flow is included then the resulting centrifugal and
Coriolis effects are modelled. Note that the flow friction effects are not included in OrcaFlex.
Theory
This section documents the theory behind the modelling of the centrifugal and Coriolis forces in OrcaFlex. This
theory is technical and specific to the way pipes are modelled in OrcaFlex.
Notation
ρ = contents density
a = internal cross-sectional area
s = contents flow velocity
r = mass flow rate = ρas
l = segment length
p = position of node relative to fixed axes
v = velocity of node relative to fixed axes
u = unit vector in downstream direction of line
ω = angular velocity of moving frame relative to fixed frame
dx/dt = rate of change of any variable x relative to fixed axes
x' = rate of change of any variable x relative to moving axes
Centrifugal force on a node due to flow through a node
First consider a node with flow arriving from one direction, ui say, and leaving in another direction, uo. For a midnode
ui and uo are simply the unit vectors in the directions of the segments before and after the node. For the first
node ui is the end direction – this is taken to be the same as uo if the end is free and otherwise is taken to be the nomoment
direction. Similar treatment is applied to uo at the last node.
Similarly let ai and ao denote the internal cross sectional areas on the input side and output side, respectively. We
define ai for the first node, and ao for the last node, to be the same as the internal cross section area of the end
segment – i.e. we assume no change in internal cross sectional area at the line ends.
Contents flow into the node at velocity siui so the rate of input of momentum is ρaisi 2 ui. Likewise the rate of output of
momentum is ρaoso 2 uo. The force on the contents that is required to achieve this change in flow direction must
therefore be the rate of output of momentum minus the rate of input of momentum, i.e.
ρaoso 2 uo - ρaisi 2 ui
The resulting centrifugal force on the node must be equal and opposite to this, so
Centrifugal Force on Node = ρ(aisi 2 ui - aoso 2 uo)
The theory above caters for the fully general situation where the internal cross-section may vary along the line. For
the common case of a uniform internal cross-section the equation simplifies to
Centrifugal Force on Node = ρas 2 (ui - uo)
This result agrees with the centrifugal term included in equation 10 of Gregory & Paidoussis, 1996.
Coriolis force due to movement of a segment
Now consider a segment between two nodes n1 and n2 and consider the following two frames of reference: a fixed
global frame and a moving local frame whose origin moves with node n1 and whose z-axis always points in direction
u = unit vector from n1 towards n2.
Consider the contents of a segment. Its velocity relative to the moving axes is
p' = (r/aρ)u
So its velocity relative to the fixed axes is
v1 + dp/dt