Nstrophy is a tool to solve the two-dimensional Navier-Stokes equation as well as Gallavotti's reversible Navier-Stokes equation and compare them.

Nstrophy is under active development

Building

Compile Nstrophy with

make

which will place a binary at build/nstrophy.

The syntax for the execution of Nstrophy is

./build/nstrophy [-p parameters] [-s savefile] <command>

Nstrophy is written in C. The Makefile uses the GNU C Compiler.

Nstrophy depends on fftw: [https://fftw.org]

Commands

The available commands are

• enstrophy: to compute the enstrophy, the parameter alpha and alpha times the enstrophy. This command prints step_index time average_alpha average_enstrophy average_alphaxenstrophy alpha enstrophy alphaxenstrophy where the averages are running averages over print_freq steps.

• uk: to compute the Fourier transform of the solution.

• quiet: does not print anything, useful for debugging.

Parameters

The parameters can be specified using the -p flag. The parameter string should be a ; sperated list of key=value pairs. The possible keys are

• equation: either irreverisible (default) or reversible.

• K (int, default 16): cutoff in momentum space: -K<=k_i<=K

• K1 (int, default K): cutoff in momentum space for the x component: -K<=k_x<=K

• K2 (int, default K): cutoff in momentum space for the y component: -K<=k_y<=K

• N (int, default smallest power of 2 that is larger than 3K): size of fft vectors: must be at least 3 times K to avoid aliasing.

• N1 (int, default N): same as N but only for x component.

• N2 (int, default N): same as N but only for y component.

• nsteps (long int, default 10000000): number of steps in the computation. Set to 0 to keep on going forever.

• nu (double, default 0.00048828125): viscosity.

• delta (double, default 0.0001220703125): step size.

• L (double, default 2pi): size of box.

• print_freq (long int, default 1000): only print every print_freq steps.

• starting_time (long int, default 0): start the computation at this step.

• driving: either zero for no driving, test (default) for a testing driving force or file:<filename> or file_txt:<filename> to read the driving force from a file. When using file:<filename> the file should be binary, whereas with file_txt:<filename> it should be plaintext. The binary file format is (double)(double) for each entry of the driving force, excluding kx<0 and kx=0&&ky<=0. The plaintext file format is kx ky real_part imag_part.

• init: either random for a random initialization, gaussian (default) for a Gaussian initial condition or file:<filename> or file_txt:<filename> to read the driving force from a file. When using file:<filename> the file should be binary, whereas with file_txt:<filename> it should be plaintext. The binary file format is (double)(double) for each entry of the driving force, excluding kx<0 and kx=0&&ky<=0. The plaintext file format is kx ky real_part imag_part.

• init_en (double, default 1.54511597324389e+02): initial value of the energy if equation=irreversible or of the enstrophy if equation=reversible.

• random_seed (int, default ): seed for random initialization.

• algorithm: either RK4 for Runge-Kutta 4, or RK2 for Runge-Kutta 2.

Interrupting/resuming the computation

The computation can be interrupted by sending Nstrophy the SIGINT signal (e.g. by pressing Ctrl-C.) When Nstrophy receives the SIGINT signal, it finishes its current step and writes the value of uk, either to savefile if such a file was specified on the command line (using the -s flag), or to stderr. In addition, when a savefile is specified it writes the command that needs to be used to resume the computation (which essentially just sets the appropriate starting_time and init:file:<savefile> parameters. The data written to the savefile is binary.