viscosity.h File Reference

#include "poisson.h"

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Data Structures
struct	Viscosity

Macros
#define	lambda   ((coord){1.,1.,1.})

Functions
static void	relax_viscosity (scalar *a, scalar *b, int l, void *data)
static double	residual_viscosity (scalar *a, scalar *b, scalar *resl, void *data)
trace mgstats	viscosity (vector u, vector mu, scalar rho, double dt, int nrelax=4, scalar *res=NULL)
trace mgstats	viscosity_explicit (vector u, vector mu, scalar rho, double dt)

Macro Definition Documentation

lambda

#define lambda   ((coord){1.,1.,1.})

Axisymmetry

In Basilisk, axisymmetric simulations are treated in a 2D Cartesian formulation for code generalisation purposes. The differential element \(dV\) is then accounted for in the metric (axi.h) incorporated in both \(\rho\) and \(\mu\). The strain rate tensor \(\boldsymbol{E}\) is of 3D nature:

\[ 2\boldsymbol{E} = \boldsymbol{\nabla u} + \boldsymbol{\nabla u}^T = \begin{bmatrix} 2\frac{\partial u_r}{\partial r} & 0 & \frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r}\\[.1cm] 0 & 2\frac{u_r}{r} & 0\\[.1cm] \frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r} & 0 & 2\frac{\partial u_z}{\partial z} \end{bmatrix}, \]

so that in cylindrical coordinates, the viscous diffusion equation reads,

\[ \rho\frac{\partial u_r}{\partial t} = \frac{\partial}{\partial r}\left(2\mu\frac{\partial u_r}{\partial r}\right) + \frac{\partial}{\partial z}\left[\mu\left(\frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r}\right)\right] + \frac{2\mu}{r}\left(\frac{\partial u_r}{\partial r} - \frac{u_r}{r}\right), \]

\[ \rho\frac{\partial u_z}{\partial t} = \frac{\partial}{\partial r}\left[\mu\left(\frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r}\right)\right] + \frac{\partial}{\partial z}\left(2\mu\frac{\partial u_z}{\partial z}\right) + \frac{\mu}{r}\left(\frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r}\right). \]

Conventionally, in basilisk, \(\boldsymbol{e}_y = \boldsymbol{e}_r\) and \(\boldsymbol{e}_x = \boldsymbol{e}_z\). For consistency reasons with 2D Cartesian simulations, the axisymmetric viscous diffusion equation in basilisk reads,

\[ \rho'\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}\left(2\mu'\frac{\partial u}{\partial x}\right) + \frac{\partial}{\partial y}\left[\mu'\left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\right)\right], \]

\[ \rho'\frac{\partial v}{\partial t} = \frac{\partial}{\partial x}\left[\mu'\left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\right)\right] + \frac{\partial}{\partial y}\left(2\mu'\frac{\partial v}{\partial y}\right) - \lambda^*, \]

where \(\rho' = \rho y\) and \(\mu' = \mu y\). If one performs the straightforward derivation, one finds that the equation along \(x\) is identical to the equation along \(z\) in cylindrical coordinates. This is not the case for the equations along \(y\) and \(r\). That is why the variable \(\lambda^*\) is added to the equation along \(y\) in the 2D Cartesian formulation. Performing the derivations and equating both equations yields

\[ \lambda^* = \frac{2\mu'}{y^2}v. \]

The expression under "lambda.y" below stems from \(\lambda^*\) after discretisation. In non AXI simulations, \(\rho' = \rho\), \(\mu' = \mu\) and \(\lambda^* = 0\).

Definition at line 115 of file viscosity.h.

Function Documentation

relax_viscosity()

static void relax_viscosity ( scalar *  a, scalar *  b, int  l, void *  data  )

static

Relaxation function

This function solves for the correction \(d\boldsymbol{a}\) in step 3 of the previously described algorithm. It is analogous to the relaxation function written for the Poisson-Helmholtz equation, with the only difference that the viscous diffusion equation is vectorial and yields a system of coupled scalar equations, the number of which is the dimension of the numerical simulation. This function is passed as an argument to the multigrid cycle.

We have the option of using red/black Gauss-Seidel relaxation or "re-use as soon as computed" relaxation. On GPUs (and probably also with OpenMP) red/black Gauss-Seidel converges much better (but requires two for (int _i = 0; _i < _N; _i++) /* foreach

Definition at line 131 of file viscosity.h.

References _GPU, _i, a, b, boundary_level, data, dimension, dt, Point::i, l, lambda, level, mu, p, point, rho, sq(), u, vector, w, vector::x, and x.

Referenced by viscosity().

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residual_viscosity()

static double residual_viscosity ( scalar *  a, scalar *  b, scalar *  resl, void *  data  )

static

Residual computation

This function computes the residual \(\boldsymbol{res}\) in step 1 of the previously described algorithm. It is passed as an argument to the multigrid solver.

We manually apply boundary conditions, so that all components are treated simultaneously. Otherwise (automatic) BCs would be applied component by component before each for (int _i = 0; _i < _N; _i++) /* foreach_face

Definition at line 222 of file viscosity.h.

References _i, a, b, boundary, d, data, dimension, dt, lambda, loop, max, mu, p, rho, sq(), u, vector, vector::x, and x.

Referenced by viscosity(), and viscosity_explicit().

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viscosity()

trace mgstats viscosity	(	vector	u,
		vector	mu,
		scalar	rho,
		double	dt,
		int	nrelax = 4,
		scalar *	res = NULL
	)

User interface

A user interface is provided for the solution of the viscous diffusion equation.

Implicit treatment

The velocity field \(\boldsymbol{u}_n\) is provided as an initial guess \(\tilde{\boldsymbol{a}}\).

We need \(\mu\) and \(\rho\) on all levels of the grid.

Definition at line 307 of file viscosity.h.

References _i, dimension, dt, mg_solve(), mu, p, relax_viscosity(), residual_viscosity(), restriction, rho, u, vector::x, and x.

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viscosity_explicit()

trace mgstats viscosity_explicit	(	vector	u,
		vector	mu,
		scalar	rho,
		double	dt
	)

Explicit treatment

This function does not make use of the multigrid solver. Instead, it explicitly advances the velocity field in time, in only one iteration.

Definition at line 337 of file viscosity.h.

References _i, dimension, dt, mu, p, residual_viscosity(), rho, u, vector::x, and x.

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