Equations and parameters: Difference between revisions

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{{latexPreamble}}
{{latexPreamble}}
For a reminder of the code's parameters see [[Getting_started#parameters]]
(As implemented in openpipeflow.org.  For a reminder of the code parameter names see [[Getting_started#parameters]].)


== Governing equations ==
== Governing equations ==
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=== Non-dimensionalisation / scales ===
=== Non-dimensionalisation / scales ===


* Length $R$ (radius=$D/2$).
The scales used are
* Velocity $2\, U_b$, equivalent to $U_{cl}$ of laminar Hagen-Poiseuille flow ('''c'''entre'''l'''ine, '''b'''ulk=mean).
* $R$, the radius of the pipe.  ($R=D/2$, where $D$ is the diameter.)
* The non-dimensional radius is 1, laminar centreline speed is 1, and bulk speed is 1/2.
* $U_{cl}$, the centre-line velocity for laminar flow.
* Note that 1 'advection-time' unit $D/U_b$ is equivalent to 4 code time units $R/2U_b$.
* $R/U_{cl}$ for time.
 
In computational units,  
* the non-dimensional radius is 1,
* the non-dimensional laminar flow is $W(r)=1-r^2$,
* the non-dimensional bulk speed, when fixed, is $\frac{1}{2}$.
 
Note that when the mean axial flow speed $U_b$ is constant, we have $U_{cl}=2U_b$.
 
The streamwise wavenumber is $\alpha=2\pi/L$, where $L$ is the periodic length in units $R$.  In papers where $D$ has been used as the scale, $\alpha$ may have been defined to give consistent values, i.e. $\alpha=\pi/L$.
 
For 'lab-units', based on $D$ and $U_b$, see [[Table of unit conversions]].
1 advection time unit $D/U_b$ is equivalent to 4 code time units $R/U_{cl}$.


=== Dimensionless parameters ===
=== Dimensionless parameters ===


Reynolds number, fixed flux, $Re_m = 2 U_b R / \nu$<br />
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, $Re = U_{cl} R / \nu$<br />
 
$1+\beta = Re / Re_m$<br />
Reynolds number, fixed pressure, $Re = U_{cl} R / \nu$.
$Re_\tau=u_\tau R/\nu = (2\,Re_m\,(1+\beta))^\frac{1}{2}=(2\,Re)^\frac{1}{2}$
 
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' [https://en.wikipedia.org/wiki/Shear_velocity],
is given by $Re_\tau = (2\,Re_m\,(1+\beta))^\frac{1}{2}=(2\,Re)^\frac{1}{2}$.


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


Fixed pressure
Fixed pressure


$ (\partial_{t} + \vec{u}\cdot\bnabla) \vec{u}
$ (\partial_{t} + \vec{u}\cdot\bnabla) \vec{u}
  = -\bnabla \hat{p} + \frac{4}{\Rey}\vechat{z}
= -\bnabla \hat{p} + \frac{4}{\Rey}\vechat{z}
    + \frac{1}{\Rey}\bnabla^2 \vec{u} $
+ \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, $W(r) = 1-r^2$. Then
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}'  
$ (\partial_{t} - \frac{1}{\Rey_m}\bnabla^2)\,\vec{u}'  
  = \vec{u}' \wedge (\bnabla \wedge\vec{u}')  
= \vec{u}' \wedge (\bnabla \wedge\vec{u}')  
  - \frac{\mathrm{d}W}{\mathrm{d}r}\,u'_z \vechat{z}
- \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}' \, . $
- 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 ==
== Decoupling the equations ==
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$u_r = \frac{1}{2} ( u_+ + u_-),
$u_r = \frac{1}{2} ( u_+ + u_-),
  \qquad
\qquad
  u_\theta = -\,\frac{\mathrm{i}}{2}(u_+ - u_- ) .$
u_\theta = -\,\frac{\mathrm{i}}{2}(u_+ - u_- ) .$


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


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


where
where


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

Latest revision as of 06:52, 5 November 2024

$ \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. ($R=D/2$, where $D$ is the diameter.)
  • $U_{cl}$, the centre-line velocity for laminar flow.
  • $R/U_{cl}$ for time.

In computational units,

  • the non-dimensional radius is 1,
  • the non-dimensional laminar flow is $W(r)=1-r^2$,
  • the non-dimensional bulk speed, when fixed, is $\frac{1}{2}$.

Note that when the mean axial flow speed $U_b$ is constant, we have $U_{cl}=2U_b$.

The streamwise wavenumber is $\alpha=2\pi/L$, where $L$ is the periodic length in units $R$. In papers where $D$ has been used as the scale, $\alpha$ may have been defined to give consistent values, i.e. $\alpha=\pi/L$.

For 'lab-units', based on $D$ and $U_b$, see Table of unit conversions. 1 advection time unit $D/U_b$ is equivalent to 4 code time units $R/U_{cl}$.

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

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