From: bds@rzg.mpg.de (Bruce Scott TOK)
Subject: Re: Navier-Stokes equation formulations
Date: 4 Jan 2000 16:53:24 +0100
Newsgroups: sci.math.num-analysis

In article <84r784$r8v$1@news.uky.edu>,
Jun Zhang <jzhang@cs.engr.uky.edu> wrote:
>Hi, Everyone:
>
>The streamfunction and vorticity formulation of the incompressible
>Navier-Stokes equation in 2D is well known. But I was told that
>such formulation is not extendible to 3D. However, I did see at
>least one paper by L. Fuchs and H.-S. Zhao, "Solution of Three-
>Dimensional Viscous Incompressible Flows by A Multigrid Methods",
>Int. J. for Numer. Methods in Fluids, Vol. 4, 539-555 (1984), using
>the streamfunction vorticity formulation.
>
>Could someone explain to me what are the problems of using
>streamfunction vorticity formulation in 3D Navier-Stokes
>equation?

I don't have access to that journal...

The usual problem is that the stream function as a scalar only describes
a vector field in one plane, obviously the way to go for 2D
incompressible.  Some 3D incompressible treatments calculate a potential
flow component and then subtract it out:

    v = u - grad phi

where the new velocity, v, is found as the incompressible projection of
the predicted velocity, u, ``onto the space of divergence free vector
fields'' just a bit of fancy terminology.

Note Bien: this is equivalent to the stream function form in 2D but not
in 3D.

The 3D flow my MG method applies easily to the projection method since
the equation you get for phi is just Poisson's.  In fact, this is how
the Colella et al crowd do it (see papers in J Comput Phys 1989, 1992).

In more generality, you can always pick a direction, z, and then do the
Hodge decompotision (or its generalisation... I am not sure of the
terminology),

    u = z cross grad W + z cross z cross grad phi + V z

Here, W is the stream function of the 2D incompressible part of the
flow, phi is the 2D compressible part, and V is the ``parallel'' part.
This is used in plasma physics where the background magnetic field is
the obvious choice for z, but it does not make sense for a neutral fluid
unless the background rotates strongly enough for that to be important.
In the latter case you can take the rotation axis for z.

For a 3D incompressible vector field, you need in general a vector
potential, like for the magnetic field,

    B = curl A

Note that A need not in general be itself divergence free, although it
is so for many applications in plasma physics.


J. B. Bell, P. Colella, and H. M. Glaz,
A Second Order Projection Method for Two-dimensional Incompressible Flow
J. Comput. Phys. 85 (1989) 257-283.

J. B. Bell and D. L. Marcus
A Second Order Projection Method for Variable Density Flows
J. Comput. Phys. 101 (1992) 334-348.

For plasma physics (``Hodge'' decomposition) see these:

incompressible, W. Park, D. A. Monticello, and R. B. White,
              Phys. Fluids 27 (1984) 137-149.

compressible, W. Park, D. A. Monticello, and T. K. Chu,
              Phys. Fluids 30 (1987) 285.

-- 
cu,
Bruce

drift wave turbulence:  http://www.rzg.mpg.de/~bds/