Channelflow: database of invariant solutions

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 boundry 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.

If you have a new solution you would like added to the database, please send it to gibson@cns.physics.gatech.edu.

Tabel of contents

Invariant solutions

W03 box α = 1.14, γ=2.50, Re = 400

HKW box α = 1.14, γ=1.67, Re = 400

HKW box Lx = 1.75 π, Lz = 1.2 π, Re = 400

Data formats
References

Invariant solutions

W03 box: α=1.14, γ=2.5. Re=400 unless otherwise noted

Equilibria
eqb 1, "lower branch"		LB.ff	LB.asc.gz	LB.geom	image	Nagata (1990); Waleffe (2003)	2007-11-01
eqb 2, "upper branch"		UB.ff	UB.asc.gz	UB.geom	image	Nagata (1990); Waleffe (2003)	2007-11-01
eqb 3, "newbie 2"		NB2.ff	NB2.asc.gz	NB2.geom	image	Halcrow et al. (2008)	2007-11-01
eqb 4, "newbie"		NB.ff	NB.asc.gz	NB.geom	image	Gibson et al. (2008)	2007-11-01
eqb 5		EQ5.ff	EQ5.asc.gz	EQ5.geom	image	Halcrow et al. (2008)	2007-11-01
eqb 6	Re=330	EQ6.ff	EQ6.asc.gz	EQ6.geom	image	Halcrow et al. (2008)	2007-11-01
eqb 7		EQ7.ff	EQ7.asc.gz	EQ7.geom	image	Halcrow et al. (2008)	2008-05-09
eqb 8	Re=270	EQ8.ff	EQ8.asc.gz	EQ8.geom	image	Halcrow et al. (2008)	2008-05-09
eqb 9		EQ9.ff	EQ9.asc.gz	EQ9.geom	image	Halcrow et al. (2008)	2008-05-09
eqb 10		EQ10.ff	EQ10.asc.gz	EQ10.geom	image	Halcrow et al. (2008)	2008-05-11
eqb 11		EQ11.ff	EQ11.asc.gz	EQ11.geom	image	Halcrow et al. (2008)	2008-05-12
Traveling waves
tw 1		TW1.ff	TW1.asc.gz	TW1.geom	image	Halcrow et al. (2008)	2008-1-29
tw 2		TW2.ff	TW2.asc.gz	TW2.geom	image	Halcrow et al. (2008)	2008-1-29
Periodic orbits ...coming soon...

HKW box, &alpha=1.14, &gamma=1.67, Re=400

Equilibria
doubled lower branch	2LB.ff	2LB.asc.gz	2LB.geom	image	Halcrow et al. (2008)	2008-1-29
doubled upper branch	2UB.ff	2UB.asc.gz	2UB.geom	image	Halcrow et al. (2008)	2008-1-29
newbie	NB.ff	NB.asc.gz	NB.geom	image	Halcrow et al. (2008)	2008-1-29
eqb 7	EQ7.ff	EQ7.asc.gz	EQ7.geom	image	Halcrow et al. (2008)	2008-05-13
eqb 9	EQ9.ff	EQ9.asc.gz	EQ9.geom	image	Halcrow et al. (2008)	2008-05-13

HKW box, Lx=1.75π, Lz=1.2π, Re=400

This definition of the HKW box differs from the &alpha=1.14, &gamma=1.67 HKW box in the third digits of Lx and Lz (compare the .geom files). We calculated periodic orbits with this definition to match Viswanath.

Periodic Orbits
P19.02	P19p02.ff	P19p02.asc.gz	P19p02.geom	P19p02.symm	animation	Gibson et al. (2008)	2008-04-04
P41.36	P41p36.ff	P41p36.asc.gz	P41p36.geom	P41p36.symm	animation	Kawahara,Kida (2001); Viswanath (2007)	2008-03-20
P46.23	P46p23.ff	P46p23.asc.gz	P46p23.geom	P46p23.symm	animation	Gibson et al. (2008)	2008-03-20
P68.07	P68p07.ff	P68p07.asc.gz	P68p07.geom	P68p07.symm	animation	Gibson et al. (2008)	2008-03-20
P75.35	P75p35.ff	P75p35.asc.gz	P75p35.geom	P75p35.symm	animation	Gibson et al. (2008)	2008-03-20
P76.82	P76p82.ff	P76p82.asc.gz	P76p82.geom	P76p82.symm	animation	Gibson et al. (2008)	2008-03-20
P76.85	P76p85.ff	P76p85.asc.gz	P76p85.geom	P76p85.symm	animation	Gibson et al. (2008)	2008-03-20
P85.27	P85p27.ff	P85p27.asc.gz	P85p27.geom	P85p27.symm	animation	Gibson et al. (2008)	2008-03-20
P87.89	P87p89.ff	P87p89.asc.gz	P87p89.geom	P87p89.symm	animation	Viswanath (2007), Gibson et al. (2008)	2008-03-20
P88.90	P88p90.ff	P88p90.asc.gz	P88p90.geom	P88p90.symm	animation	Gibson et al. (2008)	2008-03-20
P90.31	P90p31.ff	P90p31.asc.gz	P90p31.geom	P90p31.symm	animation	Gibson et al. (2008)	2008-03-20
P121.4	P121p4.ff	P121p4.asc.gz	P121p4.geom	P121p4.symm	animation	Gibson et al. (2008)	2008-03-20

Data formats

ASCII velocity fields

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.

ASCII geometry and discretization

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.

ASCII orbit symmetries

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^T is the time-T forward integration of the Navier-Stokes equations, and σ is a symmetry operation on velocity fields: σ [u,v,w](x,y,z) = (s)[sx u, sy v, sz w](sx+ax*x/Lx, sy y, sz+az*z/Lz).

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

Binary FlowField format

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

References

M. Nagata, "Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity," J. Fluid Mech. 217 (1990).
G. Kawahara, S. Kida, "Periodic motion embedded in Plane Couette turbulence: regeneration cycle and burst", J. Fluid Mech. 449 (2001).
F. Waleffe, "Homotopy of exact coherent structures in plane shear flows," Phys. Fluids 15 (2003).
D. Viswanath, "Recurrent motions within plane Couette turbulence," J. Fluid Mech. 580 (2007) [arXiv:/physics/0604062].
J. F. Gibson, J. Halcrow, P. Cvitanović, "Visualizing the geometry of state space in plane Couette flow," accepted by J. Fluid Mech. pending revisions [PDF]. [arXiv:0705.3957 (low-res figs)].