Line theory: Contents flow effects

Introduction

OrcaFlex allows you to specify contents flow for pipes. In many cases the contents flow effects are not significant, and may be neglected simply by setting the flow rate to zero. For pipes carrying high-density contents at rapid flow rates, however, the flow effects can be significant and should not be ignored.

The published literature shows that there are three forces due specifically to contents flow – centrifugal, Coriolis, and flow friction forces. If contents flow is non-zero then OrcaFlex calculates the resulting centrifugal and Coriolis effects.

Theory

We here describe in detail the theory behind the modelling in OrcaFlex of the centrifugal and Coriolis forces due to contents flow effects. First, we will introduce some notation \begin{align*} \rho&=\text{contents density}\\ a&=\text{internal cross sectional area}\\ s&=\text{contents flow speed}\\ r&=\text{mass flow rate = }\rho a s\\ l&=\text{segment length}\\ \vec{p}&=\text{position of node relative to global axes}\\ \vec{v}&=\text{velocity of node relative to global axes}\\ \vec{u}&=\text{unit vector in the contents flow direction}\\ \vec{\omega}&=\text{angular velocity of local (moving) frame relative to $\href{CoordinateSystems.htm}{\text{ global frame }}$}\GXYZ\\ \ODt{x}&=\text{rate of change of any quantity $x$ relative to global axes}\\ x'&=\text{rate of change of any quantity $x$ relative to local axes} \end{align*}

Centrifugal force on a node due to flow through a node

Consider a node with flow arriving from one direction, $\ui$ say, and leaving in another direction, $\uo$. For a mid-node $\ui$ and $\uo$ are simply the unit vectors in the directions of the segments before (into) and after (out of) the node. For the first node $\ui$ is the end direction – if the end is free this is taken to be the same as $\uo$, otherwise it is in the no-moment direction. At the last node, we define $\uo$ analogously.

Similarly let $a_\mathrm{i}$ and $a_\mathrm{o}$ denote the into and out-of internal cross sectional areas respectively. We define $a_\mathrm{i}$ for the first node, and $a_\mathrm{o}$ for the last node, to be the same as the internal cross section area of the corresponding end segment – i.e. we assume no change in internal cross sectional area at the line ends.

The contents flow into the node with velocity $s_\mathrm{i} \ui$, so the rate of input of momentum is $\rho a_\mathrm{i} s_\mathrm{i}^2 \ui$. The corresponding rate of output of momentum is $\rho a_\mathrm{o} s_\mathrm{o}^2 \uo$. The force on the contents that is required to achieve the change in flow direction, from $\ui$ to $\uo$, must therefore be the net change in rate of momentum \begin{equation} \rho a_\mathrm{o} s_\mathrm{o}^2 \uo - \rho a_\mathrm{i} s_\mathrm{i}^2 \ui \end{equation} The resulting centrifugal force $\vec{f}_\mathrm{ce}$ on the node must be equal and opposite to this, so \begin{equation} \label{fce} \vec{f}_\mathrm{ce} = \rho (a_\mathrm{i} s_\mathrm{i}^2 \ui - a_\mathrm{o} s_\mathrm{o}^2 \uo) \end{equation} This theory 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 equation (\ref{fce}) simplifies to \begin{equation} \vec{f}_\mathrm{ce} = \rho\ a\ s^2 (\ui - \uo) \end{equation} 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 $n_i$ and $n_{i+1}$ and introduce, in addition to the fixed global frame of reference $\GXYZ$, a moving local frame $\Lxyz$. This moving frame has an origin which moves with node $n_i$ and its $z$-axis always points in direction $\vec{u}=$ unit vector from $n_i$ towards $n_{i+1}$.

Consider the contents of a segment. Its velocity relative to the local moving axes is \begin{equation} \vec{p}' = \frac{r}{a\rho}\vec{u} \end{equation} so its velocity relative to the global axes is \begin{equation} \vec{v}_i + \ODt{\vec{p}} = \vec{v}_i + \vec{p}' + \vec{\omega}{\times}\vec{p} \end{equation} Therefore its acceleration relative to global axes is \begin{align} \ODt{}(\vec{v}_i + \vec{p}' + \vec{\omega}{\times}\vec{p}) &= (\vec{v}_i + \vec{p}' + \vec{\omega}{\times}\vec{p})' + \vec{\omega} {\times} (\vec{v}_i + \vec{p}' + \vec{\omega}{\times}\vec{p}) \\ &= 0 + 0 + \vec{\omega}'{\times}\vec{p} + \vec{\omega}{\times}\vec{p}' + \vec{\omega}{\times}\vec{v}_i + \vec{\omega}{\times}\vec{p}' + \vec{\omega}{\times}(\vec{\omega}{\times}\vec{p}) \\ &= \vec{\omega}'{\times}\vec{p} + 2\vec{\omega}{\times}\vec{p}' + \vec{\omega}{\times}\vec{v}_i + \vec{\omega}{\times}(\vec{\omega}{\times}\vec{p}) \end{align} and of these terms the only new one, i.e. that is dependent on the rate of flow $\vec{p}'$ rather than $\vec{p}$, is the term $2\vec{\omega}{\times}\vec{p}' = 2\vec{\omega}{\times}\frac{r}{a\rho}\vec{u}$. When multiplied by the mass of contents in the segment, $l a \rho$, this gives the Coriolis force $\vec{f}_\mathrm{co}$ on the segment \begin{equation} \vec{f}_\mathrm{co} = 2lr(\vec{\omega}{\times}\vec{u}) \end{equation} But $\vec{\omega}$ is given by \begin{equation} \vec{\omega} = \frac1l\ \vec{u}\times(\vec{v}_{i+1}-\vec{v}_i) \end{equation} so the Coriolis force is \begin{align} \vec{f}_\mathrm{co} &= 2r\ [\vec{u}\times(\vec{v}_{i+1}-\vec{v}_i)]\times\vec{u} \\ &= 2r\ [(\vec{u}{.}\vec{u})\ (\vec{v}_{i+1}-\vec{v}_i) - (\vec{u}{.}(\vec{v}_{i+1}-\vec{v}_i))\ \vec{u}] \\ &= 2r\ [(\vec{v}_{i+1}-\vec{v}_i) - (\vec{u}\text{-direction component of }(\vec{v}_{i+1}{-}\vec{v}_i))] \\ &= 2r\ (\text{component of }(\vec{v}_{i+1}{-}\vec{v}_i) \text{ normal to }\vec{u}) \end{align} The resulting Coriolis force $\vec{f}_\mathrm{co}$ on the node must be equal and opposite to this, so \begin{equation} \vec{f}_\mathrm{co} = 2r\ (\text{component of }(\vec{v}_i{-}\vec{v}_{i+1}) \text{ normal to }\vec{u}) \end{equation} We then apportion this total Coriolis force on the segment into two equal parts, a force of \begin{equation} r\ (\text{component of }(\vec{v}_i{-}\vec{v}_{i+1})\text{ normal to }\vec{u}) \end{equation} on each of the two nodes at the ends of the segment.

This result agrees with the Coriolis term included in equation 10 of Gregory & Paidoussis, 1996.

Other relevant references are Paidoussis M P, 1970, Paidoussis M P & Deksnis E B, 1970, and Paidoussis M P & Lathier B E, 1976.