Symmetries of pipe flow

$ \renewcommand{\vec}[1]{ {\bf #1} } \newcommand{\bnabla}{ \vec{\nabla} } \newcommand{\Rey}{Re} \def\vechat#1{ \hat{ \vec{#1} } } \def\mat#1{#1} $ $ \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. Only the laminar Hagen–Poiseuille equilibrium is invariant under all of $\Group$, a turbulent state has only the trivial symmetry group $\{e\}$.

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} \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.