Symmetry of flows in channel geometries

The symmetry group of 3D fields in channel geometries is generated by

$\begin{eqnarray*} [ u,v,w ](x,y,z) & \rightarrow [-u, v, w](-x,y,z) \\ [ u,v,w ](x,y,z) & \rightarrow [ u, -v, w](x,-y,z) \\ [ u,v,w ](x,y,z) & \rightarrow [ u, v, -w](x, y,-z) \\ [ u,v,w ](x,y,z) & \rightarrow [-u,-v,-w](x,y,z) \\ [ u,v,w ](x,y,z) & \rightarrow [ u, v, w](x+\ell_x, y, z+\ell_z) \end{eqnarray*}$

“Channel geometry” means a domain that is periodic or infinite in x and z and bounded in y, with $-L_y/2 \leq y \leq L_y/2$ and Dirichlet or Neumann boundary conditions at the bounds in y. The symmetry groups of velocity fields for specific flows, with constraints such as incompressibility and specific boundary conditions, are subgroups of the group generated by the above symmetries.

Symmetry of plane Couette flow

For the full description of 67 isotropy subgroups of plane Couette, see J. Halcrow, J. F. Gibson, and P. Cvitanović, Equilibrium and traveling-wave solutions of plane Couette flow, arXiv:0808.3375, J. Fluid Mech. (to appear, 2009), and J. Halcrow, "Charting the state space of plane Couette flow: Equilibria, relative equilibria, and heteroclinic connections" (Georgia Tech Ph.D. thesis, Aug 2008). Here are some highlights.

Invariance

Plane Couette flow is invariant under the following symmetries

$\begin{eqnarray*} \sigma_x \, [u,v,w](x,y,z) &= [-u,-v,w](-x,-y,z) \\ \sigma_z \, [u,v,w](x,y,z) &= [u, v,-w](x,y,-z) \\ \tau(\ell_x, \ell_z) [u,v,w](x,y,z) &= [u, v,-w](x+\ell_x, y, z+\ell_z) \\ \end{eqnarray*}$

That is, if f^t(u) is the time-t map of plane Couette flow, then

$\begin{eqnarray*} f^t(s u) = s f^t(u) \end{eqnarray*}$

for any s in group G generated by $\{\sigma_x, \sigma_z, \tau(\ell_x, \ell_z)\}.$

Let u(t) be a solution of Navier Stokes with initial condition u(0),

$\begin{eqnarray*} u(t) = f^t(u(0)) \end{eqnarray*}$

then

$\begin{eqnarray*} s u(t) = s f^t(u(0)) = f^t(s u(0)) \end{eqnarray*}$

is a solution of Navier-Stokes with initial condition s u(0).

Isotropy

Suppose u(0) is invariant under a symmetry s in G, i.e.

$\begin{eqnarray*} s u(0) = u(0) \end{eqnarray*}$

Then u(t) satisfies that symmetry for all t, since

$\begin{eqnarray*} s u(t) = s f^t(u(0)) = f^t(s u(0)) = f^t(u(0)) = u(t) \end{eqnarray*}$

The set of all symmetries s in G satisfied by u forms a subgroup H ⊂ G, called the isotropy group group of u. Isotropy groups are useful because they form invariant subspaces of the flow.

Isotropy groups of known solutions

The isotropy group most known equilibria and periodic orbits of plane Couette flow is

$\begin{eqnarray*} S = \{1, s_1, s_2, s_3 \} \end{eqnarray*}$

where

$\begin{eqnarray*} s_1 \, [u, v, w](x,y,z) &= [u, v, -w](x+L_x/2, y, -z) \\ s_2 \, [u, v, w](x,y,z) &= [-u, -v, w](-x+L_x/2,-y,z+L_z/2) \\ s_3 \, [u, v, w](x,y,z) &= [-u,-v,-w](-x, -y, -z+L_z/2) \\ \end{eqnarray*}$

It is helpful to express these symmetries in terms of σ_x, σ_z, and translations. Let

$\begin{eqnarray*} \tau_x &= \tau(L_x/2, 0) \\ \tau_z &= \tau(0, L_z/2) \\ \tau_{xz} &= \tau_x \tau_z \end{eqnarray*}$

then

$\begin{eqnarray*} S = \{1, \, \tau_x \sigma_z, \, \tau_{xz} \sigma_x, \, \tau_z \sigma_{xz} \} \end{eqnarray*}$

Fun facts

1. If u has isotropy group S, then

$\begin{eqnarray*} \tau_x u, \, \tau_z u, \, \text{ and } \, \tau_{xz} u \end{eqnarray*}$

also have isotropy group S. Thus for each equilibrium or periodic orbit with isotropy group S, there are four half-box shifted partners.

2. Since s^2 = 1 for s ∈ S, the eigenfunctions v of the linearized dynamics about any solution u with isotropy group S are either symmetric or antisymmetric with respect each symmetry s in S. (I.e. sv = ±v)

3. σ_x defines a center of symmetry in x, σ_z in z, and σ_xz in both. Therefore the presence of σ_x in an isotropy group rules out traveling waves in x (similarly, z, and xz).

4. The S isotropy group admits of no traveling wave solutions and relative periodic orbits only of the form

$\begin{eqnarray*} \tau f^t(u) - u = 0 \text{ for } \tau \in T = \{1, \, \tau_x, \, \tau_z, \, \tau_{xz} \} \end{eqnarray*}$

Isotropy groups and invariant solutions

So far we have restricted most of our attention to the solutions with S isotropy. We have a few solutions with other isotropies.

One of the main simplifications of the restriction to S is that reduces the number of free parameters in the search for good initial guesses for invariant solutions. E.g. we don't have to provide a guess for the wave speed of traveling waves, and for periodic orbits, there are only four choices for the symmetry σ in

$\begin{eqnarray*} \sigma f^t u - u = 0 \end{eqnarray*}$

namely, $\sigma = 1, \tau_x, \, \tau_z, \, \text{ or } \tau_{xz}$ , rather than the continuum $\tau(\ell_x, \ell_z)$ .

To search for initial guesses for periodic orbits, we define a measure of close recurrence within a trajectory u(t) by

$\begin{eqnarray*} r(t,T) &= min_{\tau} \| \sigma f^T u(t) - u(t) \| \\ &= min_{\tau} \| \sigma u(t+T) - u(t) \| \end{eqnarray*}$

for $\tau \in \{1, \, \tau_x, \, \tau_z, \tau_{xz}\}$ . We can compute r(t,T) from a time series of u(t) and look for places where r(t,T) « 1 for stretches of t and constant T. Those will be good guesses for periodic orbits.

channelflow.org

Table of Contents

Symmetry of flows in channel geometries

Symmetry of plane Couette flow

Invariance

Isotropy

Isotropy groups of known solutions

Fun facts

Isotropy groups and invariant solutions

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Table of Contents

Symmetry of flows in channel geometries

Symmetry of plane Couette flow

Invariance

Isotropy

Isotropy groups of known solutions

Fun facts

Isotropy groups and invariant solutions

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