Equations and parameters
$ \renewcommand{\vec}[1]{ {\bf #1} } \newcommand{\bnabla}{ \vec{\nabla} } \newcommand{\Rey}{Re} \def\vechat#1{ \hat{ \vec{#1} } } \def\mat#1{#1} $ (As implemented in openpipeflow.org. For a reminder of the code parameter names see Getting_started#parameters.)
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, $Re_m = 2 U_b R / \nu = DU_b / \nu$, where the kinematic viscosity $\nu = \mu / \rho$.
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
$\nabla^2_\pm = \nabla^2 - \frac{1}{r^2} \pm \frac{2\,\mathrm{i}}{r^2}\partial_{\theta}$