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database [2009/06/05 09:08]
gibson
database [2010/02/02 07:55]
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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 |===== Periodic orbits ===== 
- 
-We currently have five periodic orbits in the W03 cell.  
- 
-^ name ^ binary data ^ ascii data ^ symmetry ^ movie ^ author ^ posted ^ 
-| P35.77 | {{database:​w03:​P35p77.ff}} | {{database:​w03:​P35p77.tgz}} | {{database:​w03:​P35p77symm.asc}} | [[database:​w03#​p35.77|P35.77 movie]] | Gibson et al.       | 2009-06-05 | 
-| P47.18 | {{database:​w03:​P47p18.ff}} | {{database:​w03:​P47p18.tgz}} | {{database:​w03:​P47p18symm.asc}} | [[database:​w03#​p47.18|P47.18 movie]] |        " ​           | 2009-06-05 | 
-| P50.16 | {{database:​w03:​P50p16.ff}} | {{database:​w03:​P50p16.tgz}} | {{database:​w03:​P50p16symm.asc}} | [[database:​w03#​p50.16|P50.16 movie]] |        " ​           | 2009-06-05 | 
-| P82.36 | {{database:​w03:​P82p36.ff}} | {{database:​w03:​P82p36.tgz}} | {{database:​w03:​P82p36symm.asc}} | [[database:​w03#​p82.36|P82.36 movie]] |                     | 2009-06-05 | 
-| P83.60 | {{database:​w03:​P83p60.ff}} | {{database:​w03:​P83p60.tgz}} | {{database:​w03:​P83p60symm.asc}} | [[database:​w03#​p83.60|P83.60 movie]] |                     | 2009-06-05 | 
- 
- 
-====== 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=1.67, 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#​equilibria|2xeq1]] | Halcrow et al (2008) | 2008-01-29 | 
-|2 x EQ2| {{database:​hkw:​2xeq2.ff}} | {{database:​hkw:​2xeq2.tgz}} | [[database:​hkw#​equilibria|2xeq2]] |   " ​  ​| ​ " ​ | 
-| EQ4 | {{database:​hkw:​eq4.ff}} | {{database:​hkw:​eq4.tgz}} | [[database:​hkw#​equilibria|eq4]] |  " ​ |  " ​ | 
-| EQ7 | {{database:​hkw:​eq7.ff}} | {{database:​hkw:​eq7.tgz}} | [[database:​hkw#​equilibria|eq7]] |  " ​ | 2008-05-13 | 
-| EQ9 | {{database:​hkw:​eq9.ff}} | {{database:​hkw:​eq8.tgz}} | [[database:​hkw#​equilibria|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}} | [[database:​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}} | [[database:​hkw#​p19.06|P19.06 movie]] |        " ​           | 2009-01-28 | 
-| P31.81 | {{database:​hkw:​P31p81.ff}} | {{database:​hkw:​P31p81.tgz}} | {{database:​hkw:​p31p81symm.asc}} | [[database:​hkw#​p31.81|P31.81 movie]] |        " ​           | 2009-01-28 | 
-| P41.36 | {{database:​hkw:​P41p36.ff}} | {{database:​hkw:​P41p36.tgz}} | {{database:​hkw:​p41p36symm.asc}} | [[database:​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}} | [[database:​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}} | [[database:​hkw#​p62.13|P62.13 movie]] |         " ​          | 2009-01-28 | 
-| P68.07 | {{database:​hkw:​p68p07.ff}} | {{database:​hkw:​p68p07.tgz}} | {{database:​hkw:​p68p07symm.asc}} | [[database:​hkw#​p68.07|P68.07 movie]] |         " ​          | 2008-03-20 | 
-| P75.35 | {{database:​hkw:​p75p35.ff}} | {{database:​hkw:​p75p35.tgz}} | {{database:​hkw:​p75p35symm.asc}} | [[database:​hkw#​p75.35|P75.35 movie]] |         " ​          | 2008-03-20 | 
-| P76.82 | {{database:​hkw:​p76p82.ff}} | {{database:​hkw:​p76p82.tgz}} | {{database:​hkw:​p76p82symm.asc}} | [[database:​hkw#​p76.82|P76.82 movie]] |         " ​          | 2008-03-20 | 
-| P76.85 | {{database:​hkw:​p76p85.ff}} | {{database:​hkw:​p76p85.tgz}} | {{database:​hkw:​p76p85symm.asc}} | [[database:​hkw#​p76.85|P76.85 movie]] |         " ​          | 2008-03-20 | 
-| P85.27 | {{database:​hkw:​p85p27.ff}} | {{database:​hkw:​p85p27.tgz}} | {{database:​hkw:​p85p27symm.asc}} | [[database:​hkw#​p85.27|P85.27 movie]] |         " ​          | 2009-01-28 | 
-| P87.89 | {{database:​hkw:​p87p89.ff}} | {{database:​hkw:​p87p89.tgz}} | {{database:​hkw:​p87p89symm.asc}} | [[database:​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}} | [[database:​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}} | [[database:​hkw#​p90.31|P90.31 movie]] |         " ​          | 2008-01-28 | 
-| P90.52 | {{database:​hkw:​p90p52.ff}} | {{database:​hkw:​p90p52.tgz}} | {{database:​hkw:​p90p52symm.asc}} | [[database:​hkw#​p90.52|P90.52 movie]] |         " ​          | 2009-01-28 | 
-| P99.70 | {{database:​hkw:​p99p70.ff}} | {{database:​hkw:​p99p70.tgz}} | {{database:​hkw:​p99p70symm.asc}} | [[database:​hkw#​p99.70|P99.70 movie]] |         " ​          | 2009-01-28 | 
-| P121.4 | {{database:​hkw:​p121p4.ff}} | {{database:​hkw:​p121p4.tgz}} | {{database:​hkw:​p121p4symm.asc}} | [[database:​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. 
database.txt · Last modified: 2010/02/02 07:55 (external edit)