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- | ====== Database of invariant solutions ====== | ||
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- | 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. | ||
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- | |||
- | 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. | ||
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- | ^ 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 ===== | ||
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- | All periodic orbits are for Re=400. | ||
- | |||
- | ^ 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. | ||
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- | ^ 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. | ||
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- | ^ 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 | | ||
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- | ====== 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. |