Symmetries of pipe flow

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For a discussion in the context of pipe-flow relative equilibria and symmetry reduction, see Willis, Cvitanovic & Avila (2013) section 2.2 and Appendix [1] [2]

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 $\Group$, $$\vel(\ssp)=\LieEl^{-1} \, \vel(\LieEl \, \ssp) \,,\qquad \mbox{for all } \LieEl \in {\Group} \,.$$ 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 $$G = D_1 \times \Zn{m,\theta} \times \Zn{1,z}$$ is abelian and of order 8, $$G = \{e,\LieEl_\theta,\LieEl_z,\LieEl_\theta\LieEl_z, \sigma,\sigma\LieEl_\theta,\sigma\LieEl_z,\sigma\LieEl_\theta\LieEl_z\} .$$ Focus lies on the following subgroups: $$\label{ShiftRefl} Z = \{e,\sigma\}, \qquad S = \{e,\sigma\LieEl_z\}, \qquad \Omega_m \ = \{e,\LieEl_\theta\LieEl_z\}\, .$$ 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 $\Zn{2,z}$ (i.e. they reduce to half-cells) and hence become redundant upon redefinition $m := 2m$ or $\alpha := 2\alpha$. 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 $$Z_m=\{e,\sigma\LieEl_\theta\}.$$ States invariant under $Z_m$ have 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.