libMesh: A C++ Finite Element Library

On Sun, Jun 26, 2011 at 1:37 PM, Saumil Patel <saumil.patel134@...> wrote:
> Hello,
>
> I am getting familiar with the LIBMESH code and I was wondering if the FEM
> formulation for the Unsteady NSE in example 13 are in a non-dimensional
> form?  If so, I would just need to multiply the diffusion terms in the Kuu
> and Kvv submatrices by the inverse Reynolds number factor, correct?

Well the *dimensional* incompressible NS equations typically contain
only a single parameter: "nu", the kinematic viscosity, anyway.

So if you multiply the diffusion terms in ex13 by some parameter, you
can think of it as a dimensional viscosity or an inverse Reynold's
number.

> Furthermore, I wouldn't need to change the B.C. condition for the top lid,
> correct?

The velocity on the top lid, the size of the domain, and the timestep
will all take on different physical meaning depending on whether the
value multiplying the diffusion terms is thought of as the Reynolds
number or the kinematic viscosity.  Consider:

The default values for ex13 are domain size = "1", lid velocity="1".

If the number multiplying the diffusion terms is thought of as the
kinematic viscosity (in SI units of m^2/s, say) then the domain size
is 1 meter, the lid velocity 1 m/s.

On the other hand, if the number multiplying the diffusion terms is
thought of as 1/Re, then setting the lid velocity="1" just means the
lid velocity is equal to the "characteristic velocity used in defining
the Reynold's number" and does not imply any particular units.  In
this case, one can always assign a physical value to, for example, the
lid velocity "U" by specifying particular values for the Re, the
kinematic viscosity of the fluid, and the domain size "L":

Ex1, air @ 300K (nu=15.68e-6 m^2/s) in a L=1cm enclosure, Re=1: U = Re
* (nu/L) ~ 0.15 cm/s

Ex2, water @ 50C (nu=0.553e-6 m^2/s) in a L=1m enclosure, Re=1: U ~ .55 micron/s

That is, the same Re=1 solution could represent either of the two
physical situations (and infinitely many others!) above.

-- 
John

Thank You, John.  This makes sense. I was able to get results for Re 100.

I have been trying to get results for Re = 1000.  I've tried two approaches:
1) to multiply the diffusion term by 1/Re and 2) Change the boundary
condition to U=1000.  BC's are still based on the Penalty formulation.

In both cases I'm using a refined mesh of 500 by 500.

I'm having trouble getting accurate results so I was wondering if the
K-matrix is becoming ill-conditioned when I multiply the diffusion term by
.001 (1/Re)?  Do you have any suggestions on getting around it this issue?

Thanks for your help,

Saumil

On Mon, Jun 27, 2011 at 10:43 AM, John Peterson <jwpeterson@...:

> On Sun, Jun 26, 2011 at 1:37 PM, Saumil Patel <saumil.patel134@...>
> wrote:
> > Hello,
> >
> > I am getting familiar with the LIBMESH code and I was wondering if the
> FEM
> > formulation for the Unsteady NSE in example 13 are in a non-dimensional
> > form?  If so, I would just need to multiply the diffusion terms in the
> Kuu
> > and Kvv submatrices by the inverse Reynolds number factor, correct?
>
> Well the *dimensional* incompressible NS equations typically contain
> only a single parameter: "nu", the kinematic viscosity, anyway.
>
> So if you multiply the diffusion terms in ex13 by some parameter, you
> can think of it as a dimensional viscosity or an inverse Reynold's
> number.
>
> > Furthermore, I wouldn't need to change the B.C. condition for the top
> lid,
> > correct?
>
> The velocity on the top lid, the size of the domain, and the timestep
> will all take on different physical meaning depending on whether the
> value multiplying the diffusion terms is thought of as the Reynolds
> number or the kinematic viscosity.  Consider:
>
> The default values for ex13 are domain size = "1", lid velocity="1".
>
> If the number multiplying the diffusion terms is thought of as the
> kinematic viscosity (in SI units of m^2/s, say) then the domain size
> is 1 meter, the lid velocity 1 m/s.
>
> On the other hand, if the number multiplying the diffusion terms is
> thought of as 1/Re, then setting the lid velocity="1" just means the
> lid velocity is equal to the "characteristic velocity used in defining
> the Reynold's number" and does not imply any particular units.  In
> this case, one can always assign a physical value to, for example, the
> lid velocity "U" by specifying particular values for the Re, the
> kinematic viscosity of the fluid, and the domain size "L":
>
> Ex1, air @ 300K (nu=15.68e-6 m^2/s) in a L=1cm enclosure, Re=1: U = Re
> * (nu/L) ~ 0.15 cm/s
>
> Ex2, water @ 50C (nu=0.553e-6 m^2/s) in a L=1m enclosure, Re=1: U ~ .55
> micron/s
>
> That is, the same Re=1 solution could represent either of the two
> physical situations (and infinitely many others!) above.
>
> --
> John
>

On Tue, Jun 28, 2011 at 8:03 AM, Saumil Patel <saumil.patel134@...> wrote:
> Thank You, John.  This makes sense. I was able to get results for Re 100.
>
> I have been trying to get results for Re = 1000.  I've tried two approaches:
> 1) to multiply the diffusion term by 1/Re and 2) Change the boundary
> condition to U=1000.  BC's are still based on the Penalty formulation.
>
> In both cases I'm using a refined mesh of 500 by 500.
>
> I'm having trouble getting accurate results so I was wondering if the
> K-matrix is becoming ill-conditioned when I multiply the diffusion term by
> .001 (1/Re)?  Do you have any suggestions on getting around it this issue?

The matrix condition number goes like h^-2 for the Laplacian, so the
fine mesh will certainly hurt you there.

The cell Reynolds number, Re*(h/L) = 1000/500 = 2 for your case, which
is right on the borderline of Galerkin instability.

The timestep, linear, and nonlinear tolerances will all need to be
different from what they were in the Re=1 case.  You'll need to play
with those and see which ones work.

-- 
John

Saumil,
           If you are using PETSc, you can tack on the command line
option -ksp_monitor_singular_value and check whether the linear solver
is running into trouble.

If that is indeed the case, you can try increasing the number of GMRES
restart steps or use modified Gram Schmidt.

Another approach is to use continuation in parameters, where you start
with a smaller Reynolds number and then increment the Reynold number
iteratively to the desired value. The solution at Re(i) is used as the
starting guess for the solver at Re(i+1).

Thanks.

On Tue, Jun 28, 2011 at 9:03 AM, Saumil Patel <saumil.patel134@...> wrote:
>
> Thank You, John.  This makes sense. I was able to get results for Re 100.
>
> I have been trying to get results for Re = 1000.  I've tried two approaches:
> 1) to multiply the diffusion term by 1/Re and 2) Change the boundary
> condition to U=1000.  BC's are still based on the Penalty formulation.
>
> In both cases I'm using a refined mesh of 500 by 500.
>
> I'm having trouble getting accurate results so I was wondering if the
> K-matrix is becoming ill-conditioned when I multiply the diffusion term by
> .001 (1/Re)?  Do you have any suggestions on getting around it this issue?
>
> Thanks for your help,
>
> Saumil
>
> On Mon, Jun 27, 2011 at 10:43 AM, John Peterson <jwpeterson@...:
>
> > On Sun, Jun 26, 2011 at 1:37 PM, Saumil Patel <saumil.patel134@...>
> > wrote:
> > > Hello,
> > >
> > > I am getting familiar with the LIBMESH code and I was wondering if the
> > FEM
> > > formulation for the Unsteady NSE in example 13 are in a non-dimensional
> > > form?  If so, I would just need to multiply the diffusion terms in the
> > Kuu
> > > and Kvv submatrices by the inverse Reynolds number factor, correct?
> >
> > Well the *dimensional* incompressible NS equations typically contain
> > only a single parameter: "nu", the kinematic viscosity, anyway.
> >
> > So if you multiply the diffusion terms in ex13 by some parameter, you
> > can think of it as a dimensional viscosity or an inverse Reynold's
> > number.
> >
> > > Furthermore, I wouldn't need to change the B.C. condition for the top
> > lid,
> > > correct?
> >
> > The velocity on the top lid, the size of the domain, and the timestep
> > will all take on different physical meaning depending on whether the
> > value multiplying the diffusion terms is thought of as the Reynolds
> > number or the kinematic viscosity.  Consider:
> >
> > The default values for ex13 are domain size = "1", lid velocity="1".
> >
> > If the number multiplying the diffusion terms is thought of as the
> > kinematic viscosity (in SI units of m^2/s, say) then the domain size
> > is 1 meter, the lid velocity 1 m/s.
> >
> > On the other hand, if the number multiplying the diffusion terms is
> > thought of as 1/Re, then setting the lid velocity="1" just means the
> > lid velocity is equal to the "characteristic velocity used in defining
> > the Reynold's number" and does not imply any particular units.  In
> > this case, one can always assign a physical value to, for example, the
> > lid velocity "U" by specifying particular values for the Re, the
> > kinematic viscosity of the fluid, and the domain size "L":
> >
> > Ex1, air @ 300K (nu=15.68e-6 m^2/s) in a L=1cm enclosure, Re=1: U = Re
> > * (nu/L) ~ 0.15 cm/s
> >
> > Ex2, water @ 50C (nu=0.553e-6 m^2/s) in a L=1m enclosure, Re=1: U ~ .55
> > micron/s
> >
> > That is, the same Re=1 solution could represent either of the two
> > physical situations (and infinitely many others!) above.
> >
> > --
> > John
> >
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--
Vikram Garg
PhD Candidate
Institute for Computational and Engineering Sciences
The University of Texas at Austin
http://users.ices.utexas.edu/~vikram/
http://www.runforindia.org/runners/vikramg