Table of Contents
couette
couette integrates plane Couette or channel flow from a given initial condition and save velocity fields to disk at a regular interval. (It should probably be called integrate; I might change it to that in the next release.)
main options
-T0 --T0 <real> default == 0 start time -T1 --T1 <real> default == 100 end time -dt --dt <real> default == 0.03125 timestep -dT --dT <real> default == 1 save interval -ts --timestepping <string> default == sbdf3 timestepping algorithm -nl --nonlinearity <string> default == rot method of calculating nonlinearity, -o --outdir <string> default == data/ output directory -R --Reynolds <real> default == 400 Reynolds number -c --channel channelflow instead of plane Couette -b --bulkvelocity hold bulk velocity fixed, not pressure gradient -P --dPdx <real> default == 0 value for fixed pressure gradient -U --Ubulk <real> default == 0 value for fixed bulk velocity -symms --symmetries <string> default == "" filename for generators of flow-invariant symmetry subspace <flowfield> (trailing arg 1) initial condition
time intervals
The integration starts at t=T0 and saves data at times t = T0 + n dT, exactly. The dt
value is approximate. Channelflow actually rounds dt to the closest integral divisor of dT
so that the save interval is exactly what you asked for. Data is saved to disk in files named u0.ff, u1.ff, u2.ff, … if t takes on integer values or filenames with four-digit precision if t otherwise, e.g. u0.000.ff, u0.125.ff, u0.250.ff, …. The integration stops when t = T0 + n dT > T1.
timestepping
The -ts or –timestepping sets the finite-difference time stepping algorithm. It takes the following
values
| value | O(dt^n) | algorithm |
|---|---|---|
cnfe1 | 1 | Crank-Nicolson Forward Euler |
cnab2 | 2 | Crank-Nicolson Adams-Bashforth |
smrk2 | 2 | Spalart-Moser Runge-Kutta |
sbdf1 | 1 | Semi-implict Backwards Differentiation |
sbdf2 | 2 | Semi-implict Backwards Differentiation |
sbdf3 | 3 | Semi-implict Backwards Differentiation |
sbdf4 | 4 | Semi-implict Backwards Differentiation |
nonlinearity
The -nl or –nonlinearity specifies the method for calculating the
nonlinearity u⋅∇u in the Navier-Stokes equation.
| value | method | |
|---|---|---|
rot | (∇×u)×u | rotational |
conv | u⋅∇u | convection |
div | ∇⋅(uu) | divergence |
skew | (u⋅∇u + ∇⋅(uu))/2 | skew-symmetric |
alt | alternating between conv and div |
mean constraints
symmetries
The -symms filename.asc option will cause the numerically integrated solution to be periodically projected onto a flow-invariant symmetric subspace specified by the named file. The file should list the generators of the subspace's symmetry group. For example, suppose you want to restrict integration to the symmetry group S = {s1,s2,s3}
 &= [u, v, -w](x+L_x/2, y, -z) \\
s_2 \, [u, v, w](x,y,z) &= [-u, -v, w](-x+L_x/2,-y,z+L_z/2) \\
s_3 \, [u, v, w](x,y,z) &= [-u,-v,-w](-x, -y, -z+L_z/2) \\
\end{align*}
$](/dokuwiki/lib/exe/fetch.php?media=wiki:latex:/img125bd2c7da8a2a24d1f85f0b34f887c7.png "LaTeX")
s1 and s2 are generators of S. So construct a file s1s2.asc with these contents
% 2 1 1 1 -1 0.5 0.0 1 -1 -1 1 0.5 0.5
The first line specifies the number of generators, and the following lines specify the generators, according to
c sx sy sz ax az
where
(c sx sy sz ax az) [u, v, w](x, y, z) → c [sx u, sy v, sz w](sx x + ax Lx, sy y, sz z + az Lz)
This functionality is provided because numerical errors and instability of invariant symmetric subspaces will cause numerical solutions to drift out of the invariant subspaces. The integration routines do the project at intervals dT.
Usage examples
