Numerical solution of Laplace's equation in cylindrical coordinates
Karmox
My purpose is to modify this code for fluid flow problems in cylindrical ducts with moving boundaries, which is why I can't use analytic solutions.
Thanks!
Greg Heath
You may want to investigate the conformal transformation of
coordinates
x = log(r/r0)
y = theta-theta0
Hope this helps.
Greg
Karmox
Any more ideas or links? If you know of a web site or reference which just shows the discretized equations corresponding to this PDE in cylindrical coordinates, this would also be helpful.
Thanks again.
Greg Heath <he...@alumni.brown.edu> wrote in message <6f6f8f26-684d-4021...@34g2000pru.googlegroups.com>...
Bruno Luong
>
> Any more ideas or links? If you know of a web site or reference which just shows the discretized equations corresponding to this PDE in cylindrical coordinates, this would also be helpful.
>
http://mathworld.wolfram.com/CylindricalCoordinates.html
The laplacian operator of antisymmetric function f in cylindrical coordinates is
delta f = d^2f/dr^2 + 1/r * df/dr + d^2f/dz^2
The asymmetric condition to impose is
df/dr = 0 at r = 0
That should be enough for you to start with.
Bruno
Greg Heath
On Mar 25, 2:28 am, "Karmox " <karmoweld...@gmail.com> wrote:
> Greg Heath <he...@alumni.brown.edu> wrote in message <6f6f8f26-684d-4021-a2ea-57f4e18ee...@34g2000pru.googlegroups.com>...
> > On Mar 24, 8:12 am, "Karmox " <karmoweld...@gmail.com> wrote:
> > > Please point me to a free Matlab code which numerically
(by e.g. the finite difference method) solves the Laplace
equation in cylindrical coordinates. Further, I'd appreciate an
academic textbook reference.
>
> > > My purpose is to modify this code for fluid flow problems in cylindrical ducts with moving boundaries, which is why I can't use analytic solutions.
>
> > > Thanks!
>
> > You may want to investigate the conformal transformation of
> > coordinates
>
> > x = log(r/r0)
> > y = theta-theta0
> Thanks for the suggestion, Greg. However, I don't think this is
exactly what I'm looking for. Maybe I just didn't understand your
point, but I'll be attempting to get rid of the angular dependence
>altogether (axisymmetric solution) in the end.
http://groups.google.com/group/comp.soft-sys.matlab/...
msg/9e2eb370164c5085
My point is the log(r) transformation will get rid
of the nasty (1/r) coefficients in the Laplacian
and you will just get the cartesian coordinate
Laplacian in the transformed coordinate.
For incompressible flow the PDE for pressure is
Poisson, not Laplace
http://en.wikipedia.org/wiki/Discrete_Poisson_equation
I did a lot of 2-D plasma and electron beam
quasistatic (i.e., electrostatic Poisson instead of
electrodynamic Maxwell) simulations in cylindrical
coordinates ~ 45 years ago. The log(r)
transformation made the simulations much easier.
> Any more ideas or links?
If you know of a web site or
reference which just shows the discretized equations
corresponding to this PDE in cylindrical coordinates,
this would also be helpful.
>
> Thanks again.
>
Check out the references in the wikipedia article.
Hope this helps.
Greg
Bruno Luong
>
> I did a lot of 2-D plasma and electron beam
> quasistatic (i.e., electrostatic Poisson instead of
> electrodynamic Maxwell) simulations in cylindrical
> coordinates ~ 45 years ago. The log(r)
> transformation made the simulations much easier.
Transforming with log(r) made simulation *Much* easier is debatable. The problem of singularity at r=0 is shifted rather than solved: The Finite domain suddenly becomes infinite. The Mesh would deform greatly and the boundary condition at r=0 becomes some kind of infinity radiation...
A full of multitude traps happens with log(r) transformation.
But it's surely something to look at.
Bruno
Greg Heath
wrote:
> Greg Heath <he...@alumni.brown.edu> wrote in message <be8c9403-791e-4736-82d7-3fc14adba...@n10g2000yqf.googlegroups.com>...
You are right. All of my simulations excluded the origin.
with a charged cylindrical conducting boundary at
r = r1 > 0 and dealt with the instability of hollow beams
containing charged particles with perturbed helical orbits.
If I had included the case of solid beams with particle
motion allowed at r = 0, I would have had to use an equation
of motion patch to deal with particles in the region 0<=r<=r1.
I don't remember details, but I don't think the problem at
the origin goes away for particles or fluids in the original
cylindrical coordinate system.
Hope this helps.
Greg