database

# Differences

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 database [2009/06/05 09:08]gibson database [2010/02/02 07:55] Line 1: Line 1: - - ====== 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​(u) is the time-t map of the Navier-Stokes equations with plane Couette boundary conditions, ​ - - * equilibria satisfy ​ u - f<​sup>​t​(u) =  0  for all t - * traveling waves satisfy ​ u - τ(c<​sub>​x​ t, c<​sub>​z​ t) f<​sup>​t​(u) =  0 for all t and the translation symmetry τ, and - * (relative) periodic orbits satisfy ​ u - σ f<​sup>​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 ====== - - 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​ - 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 - - <​code>​ - 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<​sup>​T​(u) - u = 0, where f<​sup>​T​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) - ​ - - 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.