Off topic - Some advice on solving Navier-Stokes with FiniteDifference
Matthew Knepley
knepley at gmail.com
Thu Aug 13 06:30:42 CDT 2009
On Thu, Aug 13, 2009 at 2:02 AM, Stephen Wornom <
stephen.wornom at sophia.inria.fr> wrote:
> Matthew Knepley wrote:
>
>> On Wed, Aug 12, 2009 at 10:39 AM, Stephen Wornom <
>> stephen.wornom at sophia.inria.fr <mailto:stephen.wornom at sophia.inria.fr>>
>> wrote:
>>
>> Matthew Knepley wrote:
>>
>> 1) You should really handle this by creating the constant
>> vector on the pressure
>> space and using MatNullSpaceCreate()
>>
>> 2) You can also easily handle this by fixing the pressure at
>> one point
>>
>> At what indices or location does one fix the pressure? What value
>> is it set. Usually pressure is part of the solution. It would seem
>> to introduce an inconsistency.
>> I would like to understand how to do it.
>>
>>
>> It does not matter where you fix it or to what value. The absolute value
>> of the pressure is not physically
>> relevant, only pressure differences (which is why only grad p appears).
>>
> Thanks for clarification on this point which is confusing to many of us. So
> why is it necessary to fix a value at a point since as you point out that
> only the grad p appears? If the pressure is not set in the numerical eqns,
> will you agree that the initial pressure sets the value for the numerical
> pressure at least in time accurate solutions.
>
No! This is the fundamental mathematical difference between compressible and
incompressible flow (see article by Weinan E). In compressible flow,
pressure is a true dynamical
variable, and this is why we need an equation of state to close the system.
In incompressible flow, pressure is merely the Lagrange multiplier that
enforces the divergence-free
constraint. It is not a dynamical variable, and does not have initial
conditions.
> Fixing the pressure at any point
>> to any value just sets the scale. There is no inconsistency. This is in
>> many many elementary fluid mechanics
>> books.
>>
> Pass along a reference for myself and students.
There is a nice review of incompressible flow by Stefan Turek which goes
over every solution method I have seen.
>
> Thanks again,
> Stephen
>
>>
>> To do this mechanically. Change one row in the operator div (from the div
>> u = 0 equation) to the identity.
>>
>> Matt
>>
>>
>> Stephen
>>
>>
>> Matt
>>
>>
>> On Wed, Aug 12, 2009 at 10:19 AM, William A. Perkins
>> <william.perkins at pnl.gov <mailto:william.perkins at pnl.gov>
>> <mailto:william.perkins at pnl.gov
>> <mailto:william.perkins at pnl.gov>>> wrote:
>>
>>
>> Stephen,
>>
>> There are two ways that I know of to deal with pressure checker
>> boarding: staggered grids or some form of Rhie-Chow
>> interpolation.
>> IMO, these are simple only for uniform, Cartesian grids.
>> For grids
>> that are curvilinear, unstructured, non-uniform, and/or
>> non-orthogonal, things get real complicated. There may be other
>> methods, but something is required.
>>
>> Regarding boundary conditions, I would suggest this text book:
>>
>> H. K. Versteeg and W. Malalasekera. An Introduction to
>> Computational Fluid Dynamics, the Finite Volume Method. 2nd
>> edition. Prentice-Hall. 2007
>>
>> While this book uses the finite volume method, the
>> explanation of
>> boundary conditions and staggered grids is very good and
>> relatively
>> easy to interpret for finite difference. I would also
>> recommend
>>
>> Joel H. Ferziger and Milovan Peric. Computational
>> Methods for
>> Fluid Dynamics. Springer-Verlag, 3rd edition, 2002.
>>
>> This is a little more general with regard to method
>> discussing finite
>> difference and finite volume, but still settling on finite
>> volume.
>>
>> My $0.02: I question the use of finite difference. For
>> Navier-Stokes,
>> the use of finite volume is much more prevalent in
>> commercial and
>> research codes. If your student follows Versteeg and
>> Malalasekera a
>> simple, working, staggered grid FV code could be built in a
>> very short
>> time. If something more complicated is needed, it's probably
>> explained in Ferziger and Peric.
>>
>> Also My $0.02: Unless the point of your student's work is to
>> experience building her own code, why not download
>> something like
>> OpenFOAM (http://www.opencfd.co.uk/openfoam/) and just use
>> it? I
>> expect the effort to learn something like OpenFOAM for a simple
>> application will be much less than writing a new code.
>>
>> Hope this helps.
>>
>> Bill
>>
>> >>>>> "Stephen" == Stephen Wornom
>> <stephen.wornom at sophia.inria.fr
>> <mailto:stephen.wornom at sophia.inria.fr>
>> <mailto:stephen.wornom at sophia.inria.fr
>> <mailto:stephen.wornom at sophia.inria.fr>>> writes:
>>
>> Stephen> Shengyong wrote:
>> >> Hi, Farshid
>> >>
>> >> Maybe she should use the staggered grid method which
>> is very
>> simple to
>> >> implement.
>> Stephen> Does it remain simple for curvilinear meshes?
>> Stephen> Stephen
>> >>
>> >> On Tue, Aug 11, 2009 at 5:16 AM, Farshid Mossaiby
>> <mossaiby at yahoo.com <mailto:mossaiby at yahoo.com>
>> <mailto:mossaiby at yahoo.com <mailto:mossaiby at yahoo.com>>
>> >> <mailto:mossaiby at yahoo.com
>> <mailto:mossaiby at yahoo.com> <mailto:mossaiby at yahoo.com
>> <mailto:mossaiby at yahoo.com>>>> wrote:
>> >>
>> >> Hi all,
>> >>
>> >> Sorry for this off-topic post.
>> >>
>> >> I am helping a master studnet which is working on solving
>> >> Navier-Stokes equation with Finite Difference method.
>> She is
>> >> trying to eliminate spourious pressure modes from the
>> solution.
>> >> She needs to know some details that are not usually
>> found in the
>> >> papers but important when programming, e.g. boundary
>> condition for
>> >> pressure. If someone has expertise on this or know a
>> *simple* FD
>> >> code, I would be thankful to let me know.
>> >>
>> >> Best regards,
>> >> Farshid Mossaiby
>> >>
>> >>
>> >>
>> >>
>> >>
>> >>
>> >> --
>> >> Pang Shengyong
>> >> Solidification Simulation Lab,
>> >> State Key Lab of Mould & Die Technology,
>> >> Huazhong Univ. of Sci. & Tech. China
>>
>>
>> --
>> Bill Perkins
>> Research Engineer
>> Hydrology Group
>>
>> Pacific Northwest National Laboratory
>> 902 Battelle Boulevard
>> P.O. Box 999, MSIN K9-36
>> Richland, WA 99352 USA
>> Tel: 509-372-6131
>> Fax: 509-372-6089
>> william.perkins at pnl.gov <mailto:william.perkins at pnl.gov>
>> <mailto:william.perkins at pnl.gov <mailto:william.perkins at pnl.gov>>
>> www.pnl.gov <http://www.pnl.gov> <http://www.pnl.gov>
>>
>>
>>
>>
>>
>> -- What most experimenters take for granted before they
>> begin
>> their experiments is infinitely more interesting than any
>> results to which their experiments lead.
>> -- Norbert Wiener
>>
>>
>>
>>
>>
>> --
>> What most experimenters take for granted before they begin their
>> experiments is infinitely more interesting than any results to which their
>> experiments lead.
>> -- Norbert Wiener
>>
>
>
--
What most experimenters take for granted before they begin their experiments
is infinitely more interesting than any results to which their experiments
lead.
-- Norbert Wiener
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