User Tools

Site Tools


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 ft(u) is the time-t map of the Navier-Stokes equations with plane Couette boundary conditions,

  • equilibria satisfy u - ft(u) = 0 for all t
  • traveling waves satisfy u - τ(cx t, cz t) ft(u) = 0 for all t and the translation symmetry τ, and
  • (relative) periodic orbits satisfy u - σ fT(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

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.

name Re binary data ascii data image author posted
EQ1 eq1.ff eq1.tgz eq1 Nagata (1990)2) 2007-11-01
EQ2 eq2.ff eq2.tgz eq2 Nagata (1990)3) 2007-11-01
EQ3 eq3.ff eq3.tgz eq3 Halcrow et al. (2008) 2007-11-01
EQ4 eq4.ff eq4.tgz eq4 Gibson et al. (2008) 2007-11-01
EQ5 eq5.ff eq5.tgz eq5 Halcrow et al. (2008) 2007-11-01
EQ6 330 eq6.ff eq6.tgz eq6 2007-11-01
EQ7 eq7.ff eq7.tgz eq7 2008-05-09
EQ8 270 eq8.ff eq8.tgz eq8 2008-05-09
EQ9 eq9.ff eq9.tgz eq9 2008-05-09
EQ10 eq10.ff eq10.tgz eq10 2008-05-11
EQ11 eq11.ff eq11.tgz eq11 2008-05-12

Traveling waves

All traveling waves are for Re=400.

name binary data ascii data image author posted
TW1 tw1.ff tw1.tgz tw1 Halcrow et al. (2008) 2008-01-29
TW2 tw2.ff tw2.tgz tw2 Viswanath (2007) 2008-07-24
TW3 tw3.ff tw3.tgz tw3 Halcrow et al. (2008) 2008-01-29

Periodic orbits

All periodic orbits are for Re=400.

name binary data ascii data symmetry movie author posted
P35.77 p35p77.ff p35p77.tgz p35p77symm.asc P35.77 movie Gibson et al. 2009-06-05
P47.18 p47p18.ff p47p18.tgz p47p18symm.asc P47.18 movie 2009-06-05
P50.16 p50p16.ff p50p16.tgz p50p16symm.asc P50.16 movie 2009-06-05
P82.36 p82p36.ff p82p36.tgz p82p36symm.asc P82.36 movie 2009-06-05
P83.60 p83p60.ff p83p60.tgz p83p60symm.asc P83.60 movie 2009-06-05

The HKW cell

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.

name binary data ascii data image author posted
2 x EQ1 2xeq1.ff 2xeq1.tgz 2xeq1 Halcrow et al (2008) 2008-01-29
2 x EQ2 2xeq2.ff 2xeq2.tgz 2xeq2
EQ4 eq4.ff eq4.tgz eq4
EQ7 eq7.ff eq7.tgz eq7 2008-05-13
EQ9 eq9.ff eq8.tgz eq9

Periodic orbits

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

name binary data ascii data symmetry movie author posted
P19.02 p19p02.ff p19p02.tgz p19p02symm.asc P19.02 movie Gibson et al. 2008-04-04
P19.06 p19p06.ff p19p06.tgz p19p02symm.asc P19.06 movie 2009-01-28
P31.81 p31p81.ff p31p81.tgz p31p81symm.asc P31.81 movie 2009-01-28
P41.36 p41p36.ff p41p36.tgz p41p36symm.asc P41.36 movie Kawahara, Kida 2001 4) 2008-04-04
P46.23 p46p23.ff p46p23.tgz p46p23symm.asc P46.23 movie Gibson et al. 2008-03-20
P62.13 p62p13.ff p62p13.tgz p62p13symm.asc P62.13 movie 2009-01-28
P68.07 p68p07.ff p68p07.tgz p68p07symm.asc P68.07 movie 2008-03-20
P75.35 p75p35.ff p75p35.tgz p75p35symm.asc P75.35 movie 2008-03-20
P76.82 p76p82.ff p76p82.tgz p76p82symm.asc P76.82 movie 2008-03-20
P76.85 p76p85.ff p76p85.tgz p76p85symm.asc P76.85 movie 2008-03-20
P85.27 p85p27.ff p85p27.tgz p85p27symm.asc P85.27 movie 2009-01-28
P87.89 p87p89.ff p87p89.tgz p87p89symm.asc P87.89 movie Viswanath 2007 2008-03-20
P88.90 p88p90.ff p88p90.tgz p88p90symm.asc P88.90 movie Gibson et al. 2008-03-20
P90.31 p90p31.ff p90p31.tgz p90p31symm.asc P90.31 movie 2008-01-28
P90.52 p90p52.ff p90p52.tgz p90p52symm.asc P90.52 movie 2009-01-28
P99.70 p99p70.ff p99p70.tgz p99p70symm.asc P99.70 movie 2009-01-28
P121.4 p121p4.ff p121p4.tgz p121p4symm.asc P121.4 movie 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 fT(u) - u = 0, where fTis the time-T forward integration of the Navier-Stokes equations, and σ is a symmetry operation on velocity fields with action

\sigma [u,v,w](x,y,z) = (s)[s_x u, s_y v, s_z w](s_x + a_x x/L_x, \,  s_y y, \, s_z + a_z z/L_z)

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.

he would prefer we just used the α,γ values, but we find it convenient to have a short name :-)
independent derivation by Waleffe, who provided this numerical data
found independently by Viswanath, who provided this data
database.txt · Last modified: 2010/02/02 07:55 (external edit)