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} $ For a reminder of the code's parameters see Getting_started#parameters

Governing equations

Non-dimensionalisation / scales

• Length $R$ (radius=$D/2$).
• Velocity $2\, U_b$, equivalent to $U_{cl}$ of laminar Hagen-Poiseuille flow (centreline, bulk=mean).
• The non-dimensional radius is 1, laminar centreline speed is 1, and bulk speed is 1/2.
• Note that 1 'advection-time' unit $D/U_b$ is equivalent to 4 code time units $R/2U_b$.

Dimensionless parameters

Reynolds number, fixed flux, $Re_m = 2 U_b R / \nu$
Reynolds number, fixed pressure, $Re = U_{cl} R / \nu$
$1+\beta = Re / Re_m$
$Re_\tau=u_\tau R/\nu = (2\,Re_m\,(1+\beta))^\frac{1}{2}=(2\,Re)^\frac{1}{2}$

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} $

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} $

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'_z \vechat{z}
  - W\,\partial_{z}\vec{u}' + \frac{4\,\beta}{\Rey_m}\vechat{z} - \bnabla\hat{p}' \, . $

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{\mathrm{i}}{r^2}\partial_{\theta}$