#include "advection.h"
#include "poisson.h"

Include dependency graph for stream.h:

Functions
void	event_defaults (void)
	Here we set the default boundary conditions for the streamfunction.

void	event_velocity (void)
	At every timestep we update the streamfunction field \(\psi\) by solving a Poisson equation with the updated vorticity field \(\omega\) (which has just been advected/diffused).

Variables
scalar	omega []
	We allocate the vorticity field \(\omega\), the streamfunction field \(\psi\) and a structure to store the statistics on the convergence of the Poisson solver.

scalar	psi []

mgstats	mgpsi

scalar *	tracers = {omega}
	Here we set the gradient functions for each tracer (as defined in the user-provided `tracers` list).

Function Documentation

◆ event_defaults()

void event_defaults ( void )

Here we set the default boundary conditions for the streamfunction.

Event: defaults (i = 0)

The default convention in Basilisk is no-flow through the boundaries of the domain, i.e. they are a streamline i.e. \(\psi=\)constant on the boundary. We set the default value for the CFL (the default in utils.h is 0.5). This is done once at the beginning of the simulation.

Event: defaults (i = 0)

The default display.

Definition at line 59 of file stream.h.

References CFL, and display().

Here is the call graph for this function:

◆ event_velocity()

void event_velocity ( void )

At every timestep we update the streamfunction field \(\psi\) by solving a Poisson equation with the updated vorticity field \(\omega\) (which has just been advected/diffused).

Event: velocity (i++)

Using the new streamfunction, we can then update the components of the velocity field. Since they are defined on faces we need to average the gradients, which gives the discrete expression below (for the horizontal velocity component). The expression for the vertical velocity component is obtained by automatic permutation of the indices but requires a change of sign: this is done through the pseudo-vector f.

Definition at line 74 of file stream.h.

References _i, f, mgpsi, omega, poisson, psi, u, and x.

Variable Documentation

◆ mgpsi

mgstats mgpsi

Definition at line 40 of file stream.h.

Referenced by event_velocity().

◆ omega

scalar omega[]

We allocate the vorticity field \(\omega\), the streamfunction field \(\psi\) and a structure to store the statistics on the convergence of the Poisson solver.

A streamfunction–vorticity solver for the Navier–Stokes equations

In two dimensions the incompressible, constant-density Navier–Stokes equations can be written

\[ \partial_t\omega + \mathbf{u}\cdot\nabla\omega = \nu\nabla^2\omega \]

\[ \nabla^2\psi = \omega \]

with \(\nu\) the viscosity coefficient. The vorticity \(\omega\) and streamfunction \(\psi\) are defined as

\[ \omega = \partial_x u_y - \partial_y u_x \]

\[ u_x = - \partial_y\psi \]

\[ u_y = \partial_x\psi \]

The equation for the vorticity is an advection–diffusion equation which can be solved using the flux–based advection scheme in advection.h. The equation for the streamfunction is a Poisson equation. The fields advected by the advection solver are listed in tracers.

Definition at line 39 of file stream.h.

Referenced by axistream(), event_tracer_advection(), event_velocity(), gotm_step(), and vorticity().