#include "poisson.h"

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Functions
static void	relax_psi (scalar al, scalar bl, int l, void *data)

static double	residual_psi (scalar al, scalar bl, scalar resl, void data)

mgstats	axistream (vector u, scalar psi)
	The function axistream() takes as input the \(u\) velocity field (with \(x=z\) and \(y=r\)) and returns the corresponding axisymmetric stream function.

Function Documentation

◆ axistream()

mgstats axistream	(	vector	u,
		scalar	psi
	)

The function axistream() takes as input the \(u\) velocity field (with \(x=z\) and \(y=r\)) and returns the corresponding axisymmetric stream function.

Definition at line 84 of file axistream.h.

References _i, mg_solve(), omega, psi, relax_psi(), residual_psi(), u, and y.

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◆ relax_psi()

static void relax_psi	(	scalar *	al,
		scalar *	bl,
		int	l,
		void *	data
	)

static

Axisymmetric stream function

The axisymmetric or Stokes stream function \(\psi\) verifies

\[ u_r = -\frac{1}{r}\partial_z\psi \]

\[ u_z = +\frac{1}{r}\partial_r\psi \]

with \(u_r\) and \(u_z\) the radial and longitudinal velocity components of an incompressible axisymmetric flow.

Given the definition of the vorticity \(\omega\)

\[ \omega = \partial_zu_r - \partial_ru_z \]

\(\psi\) verifies the variable-coefficient Poisson equation

\[ \partial_z\left(\frac{1}{r}\partial_z\psi\right) + \partial_r\left(\frac{1}{r}\partial_r\psi\right) = - \omega \]

This is not well-conditioned due to the divergence of the coefficients at \(r = 0\) and can be rewritten instead as

\[ \frac{\partial^2\psi}{\partial z^2} + \frac{\partial^2\psi}{\partial r^2} - \frac{1}{r}\partial_r\psi = - \omega r \]

which is better behaved.

This equation can be inverted using the multigrid solver combined with the relaxation and residual functions defined below.

Definition at line 41 of file axistream.h.

References _i, a, b, l, sq(), and y.

Referenced by axistream().

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◆ residual_psi()

static double residual_psi	(	scalar *	al,
		scalar *	bl,
		scalar *	resl,
		void *	data
	)