Equations and parameters

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Governing equations

Non-dimensionalisation / scales

The scales used are

• $R$, the radius of the pipe.
• $U_{cl}$, the centre-line velocity for laminar flow.
• $R/U_{cl}$ for time.

In computational units,

• the non-dimensional radius is 1 and
• the non-dimensional laminar flow is $W(r)=1-r^2$.

Note that $R=D/2$, where $D$ is the diameter. When the bulk flow rate is fixed, so that the mean axial flow speed $U_b$ is constant, we have $U_{cl}=2U_b$.

For 'lab-units', based on $D$ and $U_b$,

• 1 advection time unit $D/U_b$ is equivalent to 4 code time units $R/U_{cl}$,
• the bulk velocity corresponds to $\frac{1}{2}$ in code units.

Dimensionless parameters

Reynolds number, fixed flux, $Re_m = 2 U_b R / \nu = DU_b\rho/ \mu$.

Reynolds number, fixed pressure, $Re = U_{cl} R / \nu$.

For fixed flux (constant flow rate), $U_{cl}=2\,U_b$ at all times. The Reynolds number $Re_m$ is more commonly defined in terms of the constant mean speed $U_b$.

For fixed pressure gradient, $U_b$ is a time-dependent quantity that depends on the flow pattern. We define the Reynolds number $Re$ in terms of the unique $U_{cl}$ for the given pressure gradient.

$1+\beta = Re / Re_m$ is an observed quantity. For fixed flux, $1+\beta=\langle\partial p/\partial z\rangle \,/\, (dP/dz)$, where $(dP/dz)$ is the laminar pressure gradient and $\langle\partial p/\partial z\rangle$ is the average pressure gradient observed. For fixed pressure, $1+\beta=U_{cl}/(2U_b)$, where $U_b$ is the observed bulk speed.

The 'wall-Reynolds number' $Re_\tau=u_\tau R/\nu$, where $u_\tau$ is the 'wall-velocity' [1], is given by $Re_\tau = (2\,Re_m\,(1+\beta))^\frac{1}{2}=(2\,Re)^\frac{1}{2}$.

Evolution equations

Fixed flux,

$(\partial_{t} + \vec{u}\cdot\bnabla) \vec{u} = -\bnabla \hat{p} + \frac{4}{\Rey_m}\,(1+\beta)\vechat{z} + \frac{1}{\Rey_m}\bnabla^2 \vec{u}$ and $\bnabla\cdot\vec{u}=0$.

Fixed pressure

$(\partial_{t} + \vec{u}\cdot\bnabla) \vec{u} = -\bnabla \hat{p} + \frac{4}{\Rey}\vechat{z} + \frac{1}{\Rey}\bnabla^2 \vec{u}$ and $\bnabla\cdot\vec{u}=0$.

Let $\vec{u}=W(r)\vechat{z}+\vec{u}'$. Using the scaling above, the laminar flow is $W(r) = 1-r^2$. The equation, in rotational form, for the evolution of the perturbation $\vec{u}'$ is then

$(\partial_{t} - \frac{1}{\Rey_m}\bnabla^2)\,\vec{u}' = \vec{u}' \wedge (\bnabla \wedge\vec{u}') - \frac{\mathrm{d}W}{\mathrm{d}r}\,u'_r \vechat{z} - W\,\partial_{z}\vec{u}' + \frac{4\,\beta}{\Rey_m}\vechat{z} - \bnabla\hat{p}' \, .$

Boundary conditions

The no-slip boundary conditions are $\vec{u}=\vec{0}$ at the wall, $r=1$. There is no boundary condition explicitly on the pressure. Indirectly, the pressure must ensure that $\bnabla\cdot\vec{u}=0$ is satisfied everywhere, i.e. also on the boundary.

At the axis $r=0$, symmetry implies that functions are odd or even across the axis. For a Fourier mode with azimuthal index $m$, each mode is odd/even if $m$ is odd/even for the variables $u_z$ and $p$ (and other scalars). For $u_r$ and $u_\theta$, each mode is even/odd if $m$ is odd/even.

Decoupling the equations

The equations for $u_r$ and $u_\theta$ are coupled in the Laplacian. They can be separated in a Fourier decompositon by considering

$u_\pm = u_r \pm \mathrm{i} \, u_\theta,$

for which the $\pm$ are considered respectively. Original variables are easily recovered

$u_r = \frac{1}{2} ( u_+ + u_-), \qquad u_\theta = -\,\frac{\mathrm{i}}{2}(u_+ - u_- ) .$

Governing equations are then decoupled in the linear part and take the form

$\begin{eqnarray*} (\partial_{t} - \nabla^2_\pm)\, u_\pm & = & N_\pm - (\bnabla p)_\pm , \\ (\partial_{t} - \nabla^2 )\, u_z & = & N_z - (\bnabla p)_z ,\end{eqnarray*}$

where

$\nabla^2_\pm = \nabla^2 - \frac{1}{r^2} \pm \frac{2\,\mathrm{i}}{r^2}\partial_{\theta}$