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====== 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<sup>t</sup>(u) is the time-t map of the Navier-Stokes equations with plane Couette boundary conditions, 

    * equilibria satisfy  u - f<sup>t</sup>(u) =  0  for all t
    * traveling waves satisfy  u - τ(c<sub>x</sub> t, c<sub>z</sub> t) f<sup>t</sup>(u) =  0 for all t and the translation symmetry τ, and
    * (relative) periodic orbits satisfy  u - σ f<sup>T</sup>(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 [[:references|Waleffe (2003)]].((he would prefer 
we just used the α,γ values, but we find it convenient to have a short name :-))) 
===== Equilibria =====

These equilibria include the [[:references|Nagata (1990)]] lower and upper branch (labelled EQ1 and EQ2 here), 
which were studied in detail by  [[:references|Clever and Busse (1997)]] and derived independently and
extended to other boundary conditions by [[:references|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 | | {{database:w03:eq1.ff}} | {{database:w03:eq1.tgz}} | [[database:w03#equilibria|eq1]] | Nagata (1990)((independent derivation by Waleffe, who provided this numerical data)) | 2007-11-01 |
| EQ2 | | {{database:w03:eq2.ff}} | {{database:w03:eq2.tgz}} | [[database:w03#equilibria|eq2]] | Nagata (1990)((ibid)) | 2007-11-01 |
| EQ3 | | {{database:w03:eq3.ff}} | {{database:w03:eq3.tgz}} | [[database:w03#equilibria|eq3]] | Halcrow et al. (2008) | 2007-11-01 |
| EQ4 | | {{database:w03:eq4.ff}} | {{database:w03:eq4.tgz}} | [[database:w03#equilibria|eq4]] | Gibson  et al. (2008) | 2007-11-01 |
| EQ5 | | {{database:w03:eq5.ff}} | {{database:w03:eq5.tgz}} | [[database:w03#equilibria|eq5]] | Halcrow et al. (2008) | 2007-11-01 |
| EQ6 | 330 | {{database:w03:eq6.ff}} | {{database:w03:eq6.tgz}} | [[database:w03#equilibria|eq6]] |  "  | 2007-11-01 |
| EQ7 | | {{database:w03:eq7.ff}} | {{database:w03:eq7.tgz}} | [[database:w03#equilibria|eq7]] |  "  | 2008-05-09 |
| EQ8 | 270 | {{database:w03:eq8.ff}} | {{database:w03:eq8.tgz}} | [[database:w03#equilibria|eq8]] |  "  | 2008-05-09 |
| EQ9 | | {{database:w03:eq9.ff}} | {{database:w03:eq9.tgz}} | [[database:w03#equilibria|eq9]] |  "  | 2008-05-09 |
| EQ10 | | {{database:w03:eq10.ff}} | {{database:w03:eq10.tgz}} | [[database:w03#equilibria|eq10]] |  "  | 2008-05-11 |
| EQ11 | | {{database:w03:eq11.ff}} | {{database:w03:eq11.tgz}} | [[database:w03#equilibria|eq11]] |  "  | 2008-05-12 |

===== Traveling waves =====

All traveling waves are for Re=400.

^ name ^ binary data ^ ascii data ^ image ^ author ^ posted ^
| TW1 | {{database:w03:tw1.ff}} | {{database:w03:tw1.tgz}} | [[database:w03#traveling_waves|tw1]] | Halcrow et al. (2008) | 2008-01-29 |
| TW2 | {{database:w03:tw2.ff}} | {{database:w03:tw2.tgz}} | [[database:w03#traveling_waves|tw2]] | Viswanath (2007) | 2008-07-24 |
| TW3 | {{database:w03:tw3.ff}} | {{database:w03:tw3.tgz}} | [[database:w03#traveling_waves|tw3]] | Halcrow et al. (2008) | 2008-01-29 |


====== The HKW cell ======


HKW stands for [[#references|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. 

===== Equilibria =====

These equilibria are for α=1.14, γ=2.5, to match [[:references|Waleffe (2003)]]. Re=400 for all.

^ name ^ binary data ^ ascii data  ^ image ^ author ^ posted ^
|2 x EQ1| {{database:hkw:2xEQ1.ff}} | {{database:hkw:2xEQ1.tgz}} | [[database:hkw#2xeq1]] | Halcrow et al (2008) | 2008-01-29 |
|2 x EQ2| {{database:hkw:2xEQ2.ff}} | {{database:hkw:2xEQ2.tgz}} | [[database:hkw#2xeq2]] |   "   |  "  |
| EQ4 | {{database:hkw:EQ4.ff}} | {{database:hkw:EQ4.tgz}} | [[database:hkw#eq4]] |  "  |  "  |
| EQ7 | {{database:hkw:EQ7.ff}} | {{database:hkw:EQ7.tgz}} | [[database:hkw#eq7]] |  "  | 2008-05-13 |
| EQ9 | {{database:hkw:EQ9.ff}} | {{database:hkw:EQ9.tgz}} | [[database:hkw#eq9]] |  "  |  "  |

===== Periodic orbits =====


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

^ name ^ binary data ^ ascii data ^ symmetry ^ movie ^ author ^ posted ^
| P19.02 | {{database:hkw:P19p02.ff}} | {{database:hkw:P19p02.tgz}} | {{database:hkw:p19p02symm.asc}} | [[movies:hkw#p19.02|P19.02 movie]] | Gibson et al.       | 2008-04-04 |
| P19.06 | {{database:hkw:P19p06.ff}} | {{database:hkw:P19p06.tgz}} | {{database:hkw:p19p02symm.asc}} | [[movies:hkw#p19.06|P19.06 movie]] |        "            | 2009-01-28 |
| P31.81 | {{database:hkw:P31p81.ff}} | {{database:hkw:P31p81.tgz}} | {{database:hkw:p31p81symm.asc}} | [[movies:hkw#p31.81|P31.81 movie]] |        "            | 2009-01-28 |
| P41.36 | {{database:hkw:P41p36.ff}} | {{database:hkw:P41p36.tgz}} | {{database:hkw:p41p36symm.asc}} | [[movies:hkw#p41.36|P41.36 movie]] | Kawahara, Kida 2001 ((found independently by Viswanath, who provided this data)) | 2008-04-04 |
| P46.23 | {{database:hkw:p46p23.ff}} | {{database:hkw:p46p23.tgz}} | {{database:hkw:p46p23symm.asc}} | [[movies:hkw#p46.23|P46.23 movie]] | Gibson et al.       | 2008-03-20 |
| P62.13 | {{database:hkw:p62p13.ff}} | {{database:hkw:p62p13.tgz}} | {{database:hkw:p62p13symm.asc}} | [[movies:hkw#p62.13|P62.13 movie]] |         "           | 2009-01-28 |
| P68.07 | {{database:hkw:p68p07.ff}} | {{database:hkw:p68p07.tgz}} | {{database:hkw:p68p07symm.asc}} | [[movies:hkw#p68.07|P68.07 movie]] |         "           | 2008-03-20 |
| P75.35 | {{database:hkw:p75p35.ff}} | {{database:hkw:p75p35.tgz}} | {{database:hkw:p75p35symm.asc}} | [[movies:hkw#p75.35|P75.35 movie]] |         "           | 2008-03-20 |
| P76.82 | {{database:hkw:p76p82.ff}} | {{database:hkw:p76p82.tgz}} | {{database:hkw:p76p82symm.asc}} | [[movies:hkw#p76.82|P76.82 movie]] |         "           | 2008-03-20 |
| P76.85 | {{database:hkw:p76p85.ff}} | {{database:hkw:p76p85.tgz}} | {{database:hkw:p76p85symm.asc}} | [[movies:hkw#p76.85|P76.85 movie]] |         "           | 2008-03-20 |
| P85.27 | {{database:hkw:p85p27.ff}} | {{database:hkw:p85p27.tgz}} | {{database:hkw:p85p27symm.asc}} | [[movies:hkw#p85.27|P85.27 movie]] |         "           | 2009-01-28 |
| P87.89 | {{database:hkw:p87p89.ff}} | {{database:hkw:p87p89.tgz}} | {{database:hkw:p87p89symm.asc}} | [[movies:hkw#p87.89|P87.89 movie]] | Viswanath 2007      | 2008-03-20 |
| P88.90 | {{database:hkw:p88p90.ff}} | {{database:hkw:p88p90.tgz}} | {{database:hkw:p88p90symm.asc}} | [[movies:hkw#p88.90|P88.90 movie]] | Gibson et al.       | 2008-03-20 |
| P90.31 | {{database:hkw:p90p31.ff}} | {{database:hkw:p90p31.tgz}} | {{database:hkw:p90p31symm.asc}} | [[movies:hkw#p90.31|P90.31 movie]] |         "           | 2008-01-28 |
| P90.52 | {{database:hkw:p90p52.ff}} | {{database:hkw:p90p52.tgz}} | {{database:hkw:p90p52symm.asc}} | [[movies:hkw#p90.52|P90.52 movie]] |         "           | 2009-01-28 |
| P99.70 | {{database:hkw:p99p70.ff}} | {{database:hkw:p99p70.tgz}} | {{database:hkw:p99p70symm.asc}} | [[movies:hkw#p99.70|P99.70 movie]] |         "           | 2009-01-28 |
| P121.4 | {{database:hkw:p121p4.ff}} | {{database:hkw:p121p4.tgz}} | {{database:hkw:p121p4symm.asc}} | [[movies:hkw#p121.4|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
<code>
  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
</code>

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

<code>
  35.862173675293143     %T
  1                      %s
  1                      %sx
  1                      %sy
  1                      %sz
  0.5                    %ax
  0                      %az
</code>

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<sup>T</sup>(u) - u = 0, where f<sup>T</sup>is the time-T forward integration of the 
Navier-Stokes equations, and σ is a symmetry operation on velocity fields with action

<latex> 
\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)
</latex>

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.
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