Symmetries of pipe flow - Revision history

Apwillis at 00:13, 29 January 2020

2020-01-29T00:13:46Z

← Older revision		Revision as of 17:13, 28 January 2020
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	\begin{eqnarray}		\begin{eqnarray}
	\LieEl(\gSpace,\shift) \, [u,v,w,p](r,\theta,z)		\LieEl(\gSpace,\shift) \, [u,v,w,p](r,\theta,z)
	& = & [u,v,w,p](r,\theta-\gSpace,z-\shift)		& = & [u,v,w,p](r,\theta-\gSpace,z-\shift)
	\\		\\
	\sigma \, [u,v,w,p](r,\theta,z) & = & [u,-v,w,p](r,-\theta,z)		\sigma \, [u,v,w,p](r,\theta,z) & = & [u,-v,w,p](r,-\theta,z)
	\,.		\,.
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	is abelian and of order 8,		is abelian and of order 8,
	\begin{equation}		\begin{equation}
	G = \{e,\LieEl_\theta,\LieEl_z,\LieEl_\theta\LieEl_z,		G = \{e,\LieEl_\theta,\LieEl_z,\LieEl_\theta\LieEl_z,
	\sigma,\sigma\LieEl_\theta,\sigma\LieEl_z,\sigma\LieEl_\theta\LieEl_z\} .		\sigma,\sigma\LieEl_\theta,\sigma\LieEl_z,\sigma\LieEl_\theta\LieEl_z\} .
	\end{equation}		\end{equation}
	Focus lies on the following subgroups:		Focus lies on the following subgroups:
	\begin{equation}		\begin{equation}
	\label{ShiftRefl}		\label{ShiftRefl}
	Z = \{e,\sigma\}, \qquad		Z = \{e,\sigma\}, \qquad
	S = \{e,\sigma\LieEl_z\}, \qquad		S = \{e,\sigma\LieEl_z\}, \qquad
	\Omega_m \ = \{e,\LieEl_\theta\LieEl_z\}\, .		\Omega_m \ = \{e,\LieEl_\theta\LieEl_z\}\, .
	\end{equation}		\end{equation}
	The first is the `reflectional', or `mirror' symmetry,		The first is the `reflectional', or `mirror' symmetry,

Apwillis: /* Discrete symmetries */

2015-01-08T14:40:05Z

Discrete symmetries

← Older revision		Revision as of 07:40, 8 January 2015
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	the third is the `shift-and-rotate' symmetry.		the third is the `shift-and-rotate' symmetry.
	States invariant under $\LieEl_\theta$ or $\LieEl_z$ are		States invariant under $\LieEl_\theta$ or $\LieEl_z$ are
	invariant under $\Zn{2m,\theta}$ or $\Zn{2,z}$ and hence become redundant		invariant under $\Zn{2m,\theta}$ or $\Zn{2,z}$
	upon redefinition $m := 2m$ or $\alpha := 2\alpha$ ~~(i.e. they reduce to~~		(i.e. they reduce to half-cells) and hence become redundant
	~~half-cells)~~.		upon redefinition $m := 2m$ or $\alpha := 2\alpha$.
	It can also be shown that		It can also be shown that
	$\sigma\LieEl_\theta=\LieEl_\theta^{-1/2}\sigma\LieEl_\theta^{1/2}$,		$\sigma\LieEl_\theta=\LieEl_\theta^{-1/2}\sigma\LieEl_\theta^{1/2}$,

Apwillis: /* Discrete symmetries */

2015-01-08T13:05:11Z

Discrete symmetries

← Older revision		Revision as of 06:05, 8 January 2015
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	Z_m=\{e,\sigma\LieEl_\theta\}.		Z_m=\{e,\sigma\LieEl_\theta\}.
	\end{equation}		\end{equation}
	~~which has~~ mirror reflection planes located at		States invariant under $Z_m$ have mirror reflection planes located at
	$\theta=\pm\pi/(2m)$.		$\theta=\pm\pi/(2m)$.
	States invariant under $(S,Z)$		States invariant under $(S,Z)$

Apwillis at 12:35, 15 August 2014

2014-08-15T12:35:28Z

← Older revision		Revision as of 05:35, 15 August 2014
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	$		$

	For a discussion in the context of pipe-flow relative equilibria,		For a discussion in the context of pipe-flow relative equilibria and symmetry reduction,
	see Willis, Cvitanovic ~~and~~ Avila (2013)~~, Journal of Fluid Mechanics,~~		see Willis, Cvitanovic & Avila (2013) section 2.2 and Appendix [http://dx.doi.org/10.1017/jfm.2013.75] [http://arxiv.org/abs/1203.3701]
	~~721, 514-540,~~ section 2.2 and Appendix.

	A flow $\dot{\ssp}= \vel(\ssp)$ is said to be $\Group$-equivariant		A flow $\dot{\ssp}= \vel(\ssp)$ is said to be $\Group$-equivariant

Apwillis at 10:54, 15 August 2014

2014-08-15T10:54:33Z

← Older revision		Revision as of 03:54, 15 August 2014
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	subscripts $z,\theta$ indicate stream-wise translation and azimuthal		subscripts $z,\theta$ indicate stream-wise translation and azimuthal
	rotation respectively.		rotation respectively.
			Only the laminar Hagen–Poiseuille equilibrium is invariant under all of $\Group$,
			a turbulent state has only the trivial symmetry group $\{e\}$.


	=== Discrete symmetries ===		=== Discrete symmetries ===
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	half-shift in azimuth, $\sigma$ and $\LieEl_\theta$ commute so that		half-shift in azimuth, $\sigma$ and $\LieEl_\theta$ commute so that
	\begin{equation}		\begin{equation}
	G = D_1 \times \Zn{m,\theta} \times \Zn{1,z} ~~\subset \Gamma~~		G = D_1 \times \Zn{m,\theta} \times \Zn{1,z}
	\end{equation}		\end{equation}
	is abelian and of order 8,		is abelian and of order 8,

Apwillis: Created page with "{{latexPreamble}} $ \newcommand{\LieEl}{g} \newcommand{\bCell}{\Omega} \def\Zn#1{C_{#1}} \newcommand{\Group}{G} \newcommand{\vel}{v} \newcommand{\ssp}{a} \newcommand{\gSpace}{..."

2014-08-15T10:18:46Z

Created page with "{{latexPreamble}} $ \newcommand{\LieEl}{g} \newcommand{\bCell}{\Omega} \def\Zn#1{C_{#1}} \newcommand{\Group}{G} \newcommand{\vel}{v} \newcommand{\ssp}{a} \newcommand{\gSpace}{..."

New page

{{latexPreamble}}
$
\newcommand{\LieEl}{g}
\newcommand{\bCell}{\Omega}
\def\Zn#1{C_{#1}}
\newcommand{\Group}{G}
\newcommand{\vel}{v}
\newcommand{\ssp}{a}
\newcommand{\gSpace}{\phi}
\newcommand{\shift}{l}
$

For a discussion in the context of pipe-flow relative equilibria,
see Willis, Cvitanovic and Avila (2013), Journal of Fluid Mechanics,
721, 514-540, section 2.2 and Appendix.

A flow $\dot{\ssp}= \vel(\ssp)$ is said to be $\Group$-equivariant
if the form of evolution equations is left invariant by the set of
transformations $\LieEl$ that form the group of symmetries of the
dynamics $\Group$,
\begin{equation}
\vel(\ssp)=\LieEl^{-1} \, \vel(\LieEl \, \ssp)
\,,\qquad \mbox{for all } \LieEl \in {\Group}
\,.
\end{equation}
On an infinite domain and in the absence of boundary conditions, the
Navier-Stokes equations are equivariant under translations, rotations,
and inversion through the origin.
Time reversal symmetry is broken for non-zero Reynolds number.

=== Continuous symmetries ===

In pipe flow
the cylindrical wall restricts the rotation symmetry to rotation about
the $z$-axis, and translations along it. Let $\LieEl(\gSpace,\shift)$ be
the shift operator such that $\LieEl(\gSpace,0)$ denotes an azimuthal
rotation by $\gSpace$ about the pipe axis, and $\LieEl(0,\shift)$ denotes
the stream-wise translation by $\shift$; let $\sigma$ denote reflection
about the $\theta=0$ azimuthal angle:
\begin{eqnarray}
\LieEl(\gSpace,\shift) \, [u,v,w,p](r,\theta,z)
& = & [u,v,w,p](r,\theta-\gSpace,z-\shift)
\\
\sigma \, [u,v,w,p](r,\theta,z) & = & [u,-v,w,p](r,-\theta,z)
\,.
\end{eqnarray}
The Navier--Stokes equations for pipe flow are equivariant under these
transformations. The symmetry group of stream-wise periodic pipe flow is
thus $\Group = O(2)_\theta \times SO(2)_z = D_1 \ltimes
SO(2)_{\theta} \times SO(2)_z$, where $D_1 = \{ e,\, \sigma \}$
denotes azimuthal reflection, $\ltimes$ stands for a semi-direct product
(in general, reflections and rotations do not commute), and the
subscripts $z,\theta$ indicate stream-wise translation and azimuthal
rotation respectively.

=== Discrete symmetries ===

In addition to azimuthal reflection, invariant solutions can exhibit
further discrete symmetries that derive from azimuthal and stream-wise
periodicities over the computational cell $\bCell$.

Periodicity in the azimuthal direction allows for solutions with discrete
cyclic symmetry $\LieEl(2\pi/m, 0)$, defined for integer $m$.
Velocity fields invariant under such rational azimuthal shifts
are said to be invariant under the discrete cyclic group $\Zn{m,\theta}$.
Note that all solutions are invariant under $\Zn{1,\theta}$, and
given the assumed stream-wise periodicity, under $\Zn{1,z}$ as well.
This permits the study of states in the reduced computational cells
$\bCell=[0,1/2]\times[0,2\pi/m]\times[0,\pi/\alpha]$, where $L=\pi/\alpha$.
Calculations in larger domains are required to determine subharmonic bifurcations.

Consider states invariant under $\Zn{m,\theta}$ and $\Zn{1,z}$, and
denote half-shifts within our reduced cell, in $\theta$ and $z$ respectively,
by $\LieEl_\theta=\LieEl(\pi/m,0)$ and $\LieEl_z=\LieEl(0,L/2)$.
For the special case of a
half-shift in azimuth, $\sigma$ and $\LieEl_\theta$ commute so that
\begin{equation}
G = D_1 \times \Zn{m,\theta} \times \Zn{1,z} \subset \Gamma
\end{equation}
is abelian and of order 8,
\begin{equation}
G = \{e,\LieEl_\theta,\LieEl_z,\LieEl_\theta\LieEl_z,
\sigma,\sigma\LieEl_\theta,\sigma\LieEl_z,\sigma\LieEl_\theta\LieEl_z\} .
\end{equation}
Focus lies on the following subgroups:
\begin{equation}
\label{ShiftRefl}
Z = \{e,\sigma\}, \qquad
S = \{e,\sigma\LieEl_z\}, \qquad
\Omega_m \ = \{e,\LieEl_\theta\LieEl_z\}\, .
\end{equation}
The first is the `reflectional', or `mirror' symmetry,
the second is the `shift-and-reflect' symmetry, and
the third is the `shift-and-rotate' symmetry.
States invariant under $\LieEl_\theta$ or $\LieEl_z$ are
invariant under $\Zn{2m,\theta}$ or $\Zn{2,z}$ and hence become redundant
upon redefinition $m := 2m$ or $\alpha := 2\alpha$ (i.e. they reduce to
half-cells).
It can also be shown that
$\sigma\LieEl_\theta=\LieEl_\theta^{-1/2}\sigma\LieEl_\theta^{1/2}$,
where $\LieEl_\theta^{1/2}$ is the half-half-shift,
and therefore that $\sigma\LieEl_\theta\LieEl_z=\LieEl_\theta^{-1/2}
\sigma\LieEl_z\,\LieEl_\theta^{1/2}$. Invariance under these combinations
is conjugate to $Z$ and $S$. We use, however, the `rotate-and-reflect'
subgroup, denoted by
\begin{equation}
Z_m=\{e,\sigma\LieEl_\theta\}.
\end{equation}
which has mirror reflection planes located at
$\theta=\pm\pi/(2m)$.
States invariant under $(S,Z)$
implies invariance under $\sigma\sigma\LieEl_z=\LieEl_z$, and
hence under $\Zn{2,z}$, reducing to the half-length pipe. Invariance
under $(S,\Omega_m)$ is permissible, however, and
using the combinations above it can be calculated that
$(S,\Omega_m)=(S,Z_m)=(Z_m,\Omega_m)$.
Such states have been termed `highly symmetric' by
Pringle et al (2009).
As reflection is arguably easier to visualise than shift-and-rotate,
we use the notation $(S,Z_m)$ for these states.