This database contains invariant solutions to plane Couette flow (equilibria, traveling waves, and periodic orbits) for a number of different parameters. Briefly, if f^t(u) is the time-t map of the Navier-Stokes equations with plane Couette boundary conditions,

• equilibria satisfy u - f^t(u) = 0 for all t
• traveling waves satisfy u - τ(c_x t, c_z t) f^t(u) = 0 for all t and the translation symmetry τ, and
• (relative) periodic orbits satisfy u - σ f^T(u) = 0 for a specific T and symmetry σ.

The solutions are given as differences from the laminar flow. The spatial periodicity of the solutions is defined in terms of either fundamental wavenumbers (α,γ) or periodic domain size (Lx,Lz), with the relations Lx = 2π/α, Lz = 2π/γ. Some authors prefer (α,γ), some (Lx,Lz). The author and year of each solution is listed. Multiple attributions indicate independent derivations; in such cases the author who provided the solution is listed in bold. For more information on the solutions, please see Data formats and References.

The W03 cell, defined as α=1.14, γ=2.5, was first studied by Waleffe (2003).¹⁾

These equilibria include the Nagata (1990) lower and upper branch (labelled EQ1 and EQ2 here), which were studied in detail by Clever and Busse (1997) and derived independently and extended to other boundary conditions by Waleffe (2003). Nagata's original derivation was for different α,γ values, but we have confirmed that Waleffe's solution is the same as Nagata by continuing Waleffe's solution to Nagata's parameters. Re=400 unless otherwise marked.

All traveling waves are for Re=400.

All periodic orbits are for Re=400.

HKW stands for Hamilton, Kim, Waleffe (1995), who first studied dynamics in this small periodic cell. In the literature the HKW cell is defined as either Lx=1.75π, Lz=1.2π or α=1.14, γ=1.67. Note that these definitions differ in the third digit.

These equilibria are for α=1.14, γ=1.67, to match Waleffe (2003). Re=400 for all.

These periodic orbits are for Lx=1.75π, Lz=1.2π, to match Viswanath (2007). All orbits are for Re=400.

The gzipped ASCII velocity files store gridpoint values of velocity fields in x,y,z,i order using the following C++ code

os << setprecision(16);
for (int nx=0; nx<Nx ++nx)
  for (int ny=0; ny<Ny ++ny)
    for (int nz=0; nz<Nz ++nz)
      for (int i=0; i<3 ++i)
        os << setw(23) << u(nx,ny,nz,i) << '\n';

The value u(nx,ny,nz,i) is the ith component of velocity at the gridpoint (nx,ny,nz). The (nx,ny,nz)th gridpoint has spatial coordinates (nx*Lx/Nx, cos(ny*pi/(Ny-1)), nz*Lz/Nz). The (u,v,w) components of velocity are i=0,1,2.

Geometrical and discretization parameters are stored as ASCII in *.geom files as follows

  32                      %Nx
  35                      %Ny
  32                      %Nz
  3                       %Nd
  5.511566058929462       %Lx
  2.513274122871834       %Lz
  0.8771929824561405      %lx=Lx/(2pi)
  0.4                     %lz=Lz/(2pi)
  1.14                    %alpha=2pi/Lx
  2.5                     %gamma=2pi/Lz

In channelflow, Lx and Lz are the canonical geometry specifications. The .geom files provide lx,lz and alpha,gamma for human convenience.

Periodic orbits require additional specification of symmetry parameters. These are stored as ASCII in *.symm files as follows

  35.862173675293143     %T
  1                      %s
  1                      %sx
  1                      %sy
  1                      %sz
  0.5                    %ax
  0                      %az

The interpretation is as follows. If u = [u,v,w](x,y,z) is an initial condition for a periodic orbit with symmetry parameters (T,s,sx,sy,sz,ax,az), then the orbit satisfies σ f^T(u) - u = 0, where f^Tis the time-T forward integration of the Navier-Stokes equations, and σ is a symmetry operation on velocity fields with action

The s,sx,sy,sz parameters take on values +/-1; ax and az are in [-0.5, 0.5)

The .ff files are in Channelflow FlowFields in binary format. The specification of the binary format is somewhat complicated. Suffice it to say that the channelflow binary format contains all geometrical and discretization information, but not orbit symmetries works transparently with channelflow codes and utilities is platform independent.