[Nek5000-users] turbulent 3d box

Sun Jan 12 14:14:53 CST 2014

Hi,

This is a very nice book about SEM. It could be a good start, if you
already know something about finite element method and spectral method.

http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780198528692.001.0001/acprof-9780198528692

The second book is by CHQZ, a classic of spectral method, but also talks
about SEM.
http://www.springer.com/mathematics/computational+science+%26+engineering/book/978-3-540-30727-3

Good lecture!

Tubois

On Sat, Jan 11, 2014 at 11:59 AM, <nek5000-users at lists.mcs.anl.gov> wrote:

> Dear Paul,
> Thank you for the information. I am a bit unfamiliar with the basic
> knowledge in SEM. Could you recommend me a nice short introduction for the
> theory? especially that deals with the issues below.
>
> Many thanks,
> Barak
>
>
> On 01/10/2014 05:07 PM, nek5000-users at lists.mcs.anl.gov wrote:
>
>>
>> Hi Barak,
>>
>> If the elements are of uniform size in a given direction then, to leading
>> order, the
>> resolution is
>>
>>      dx ~  Lx / (N*Ex)
>>
>> where Lx = length in x, Ex=# of elements in x-direction, and N is
>> polynomial order.
>>
>> Technically, the resolution is a bit less -- like (2/pi).
>>
>> There are two errors of primary concern --- approximation error and
>> dispersion.
>>
>> If n=number of points in a given direction (n=N*ex the SEM case), then
>> you have dx=L/n,
>> and Nyquist says you can't resolve wavelengths shorter than  2*dx.   This
>> restriction applies
>> to all methods, including Fourier spectral and SEM.
>>
>> For Fourier, all wavenumbers are transported with zero phase error
>> (because of the periodic
>> BCs + fact that they are eigenfunctions of the operators, etc.).
>>
>> For SEM, only 2/pi have diminishing error as N increases, and it is in
>> fact not as small
>> as machine epsilon (as is the case with Fourier spectral, only).
>>  However, the SEM is
>> far superior to low-order approaches, and that's where it's strength
>> lies.   The primary
>> reason to use SEM over Fourier is when you don't have homogeneous
>> geometries or
>> constant coefficient operators.
>>
>> Paul
>>
>> ________________________________________
>> From: nek5000-users-bounces at lists.mcs.anl.gov [
>> nek5000-users-bounces at lists.mcs.anl.gov] on behalf of
>> nek5000-users at lists.mcs.anl.gov [nek5000-users at lists.mcs.anl.gov]
>> Sent: Friday, January 10, 2014 8:30 AM
>> To: nek5000-users at lists.mcs.anl.gov
>> Subject: Re: [Nek5000-users] turbulent 3d box
>>
>> Hi Paul,
>> Thanks for your help till now.
>> Regarding the proper discretization of the 3d turbulent box setup. When
>> simulating using a standard spectral method the smallest scale defines
>> the highest wavenumber kmax. What is its equivalent in SEM?
>> Happy new year 2014,
>> Barak
>>
>> On 01/04/2014 03:40 AM, nek5000-users at lists.mcs.anl.gov wrote:
>>
>>>
>>> Barak,
>>>
>>> I suggest comparing the SIZE files for various examples.  From there you
>>> should
>>> be able to see how to properly modify SIZE to change the resolution.
>>>
>>> Of course, you'll need to reduce dt to maintain CFL ~ 0.5
>>>
>>> Paul
>>>
>>> ________________________________________
>>> From: nek5000-users-bounces at lists.mcs.anl.gov [
>>> nek5000-users-bounces at lists.mcs.anl.gov] on behalf of
>>> nek5000-users at lists.mcs.anl.gov [nek5000-users at lists.mcs.anl.gov]
>>> Sent: Sunday, December 22, 2013 4:25 PM
>>> To: nek5000-users at lists.mcs.anl.gov
>>> Subject: Re: [Nek5000-users] turbulent 3d box
>>>
>>> It didn't work (I tried both options). Do you have working example?
>>> (see my previous email and my attachment).
>>> Barak
>>>
>>> On 12/22/2013 11:20 PM, nek5000-users at lists.mcs.anl.gov wrote:
>>>
>>>>
>>>> Hi Paul,
>>>> On 12/22/2013 11:14 PM, nek5000-users at lists.mcs.anl.gov wrote:
>>>>
>>>>>
>>>>> Hi Barak,
>>>>>
>>>>> You can either increase lx1,ly1, lz1 in SIZE and re-make, or you
>>>>> can change nelx,nely,nelz in the .box file.
>>>>>
>>>>>  Thanks.
>>>>
>>>>> Did you resolve the issue with the initial condition?
>>>>>
>>>>>  See my previous email. I have inside further questions.
>>>> Best regards,
>>>> Barak
>>>>
>>>>
>>>>  Paul
>>>>> ________________________________________
>>>>> From: nek5000-users-bounces at lists.mcs.anl.gov
>>>>> [nek5000-users-bounces at lists.mcs.anl.gov] on behalf of
>>>>> nek5000-users at lists.mcs.anl.gov [nek5000-users at lists.mcs.anl.gov]
>>>>> Sent: Sunday, December 22, 2013 3:05 PM
>>>>> To: nek5000-users at lists.mcs.anl.gov
>>>>> Subject: Re: [Nek5000-users] turbulent 3d box
>>>>>
>>>>> Hi Paul,
>>>>> Before I run problems with smaller viscosity: how I increase the
>>>>> resolution? and remesh the box?
>>>>> Barak
>>>>>
>>>>> On 12/22/2013 10:24 PM, nek5000-users at lists.mcs.anl.gov wrote:
>>>>>
>>>>>>
>>>>>> Hi Barak,
>>>>>>
>>>>>> When I compute the L2 norm of the function, given as:
>>>>>>
>>>>>>            v2 = [ sum_i  vx_i b_i vx_i + vy_i b_i vy_i + vz_i b_i vz_i
>>>>>> / sum_i b_i ]^(1/2)
>>>>>>
>>>>>> it comes out to ~ 2.23
>>>>>>
>>>>>> Note that there are some places where ux==2, uy==0, uz==1,
>>>>>> which would give an L2 norm of
>>>>>>
>>>>>>          sqrt (2*2 + 0*0 + 1*1) = sqrt(5) > sqrt(3)
>>>>>>
>>>>>> ?
>>>>>>
>>>>>> Paul
>>>>>>
>>>>>> ________________________________________
>>>>>> From: nek5000-users-bounces at lists.mcs.anl.gov
>>>>>> [nek5000-users-bounces at lists.mcs.anl.gov] on behalf of
>>>>>> nek5000-users at lists.mcs.anl.gov [nek5000-users at lists.mcs.anl.gov]
>>>>>> Sent: Sunday, December 22, 2013 11:04 AM
>>>>>> To: nek5000-users at lists.mcs.anl.gov
>>>>>> Subject: Re: [Nek5000-users] turbulent 3d box
>>>>>>
>>>>>> Hi Paul,
>>>>>> My definition for velocity magnitude is:
>>>>>> |v|=sqrt(ux^2+uy^2+uz^2)
>>>>>> so if you see, there are two components in the form:
>>>>>> cos^2(x)+sin^2(x)
>>>>>> etc in each direction
>>>>>> so in fact
>>>>>> |v|=sqrt(3) for any x,y,z.
>>>>>> Isn't correct?
>>>>>> Barak
>>>>>>
>>>>>> On 12/22/2013 05:05 PM, nek5000-users at lists.mcs.anl.gov wrote:
>>>>>>
>>>>>>>
>>>>>>> Hi Barak,
>>>>>>>
>>>>>>> If your initial condition is:
>>>>>>>
>>>>>>>            ux=(cos(y)+sin(z))
>>>>>>>            uy=(cos(z)+sin(x))
>>>>>>>            uz=(cos(x)+sin(y))
>>>>>>>
>>>>>>> and your domain is [0,2pi]^3, how is it possible that your velocity
>>>>>>> magnitude
>>>>>>> is less than 2?
>>>>>>>
>>>>>>>
>>>>>>> What is your measure of velocity magnitude?   When I compute the L2
>>>>>>> norm of this initial condition I'm seeing about 2.3+
>>>>>>>
>>>>>>> Paul
>>>>>>>
>>>>>>> ------------------------------------------------------------
>>>>>>> ------------
>>>>>>>
>>>>>>> *From:* nek5000-users-bounces at lists.mcs.anl.gov
>>>>>>> [nek5000-users-bounces at lists.mcs.anl.gov] on behalf of
>>>>>>> nek5000-users at lists.mcs.anl.gov [nek5000-users at lists.mcs.anl.gov]
>>>>>>> *Sent:* Sunday, December 22, 2013 7:53 AM
>>>>>>> *To:* nek5000-users at lists.mcs.anl.gov
>>>>>>> *Subject:* Re: [Nek5000-users] turbulent 3d box
>>>>>>>
>>>>>>> Hi Paul,
>>>>>>> It seems that something does not work, comparing my simulations
>>>>>>> using a
>>>>>>> standard spectral code (pseudo-spectral). For nu=0.1, I used 16
>>>>>>> points
>>>>>>> for each space direction. Now for this case, the nonlinear terms
>>>>>>> vanish
>>>>>>> (just cancel each other) and the solution is just the ABC flow if you
>>>>>>> set it as initial condition. Then since the forcing is f=nu*ABC the
>>>>>>> dissipation term (Laplacian) dissipation=-nu*ABC and therefore the
>>>>>>> initial value (ABC) remains as it is. If we decrease the viscosity
>>>>>>> above
>>>>>>> the critical value of 1/12 (if I remember correctly) there is a
>>>>>>> bifurcation (Hopf...) and we obtain a new periodic solution (in
>>>>>>> time),
>>>>>>> where the ABC modes are dominating combined with some high wave
>>>>>>> numbers
>>>>>>> that are oscillating.
>>>>>>>
>>>>>>>      >From my first tests using NEK, I find that the velocity field
>>>>>>> (for
>>>>>>> nu>1/12) does not agree with the above characteristics. For example,
>>>>>>> the
>>>>>>> max magnitude of the velocity field should be for the ABC flow
>>>>>>> sqrt(3),
>>>>>>> whereas here I get a value close to 2.5.
>>>>>>>
>>>>>>> Do you have any idea what's wrong?
>>>>>>> If you wish to read some literature about the ABC flow as a forcing
>>>>>>> for
>>>>>>> Navier Stokes or as a prescribed velocity for generation of magnetic
>>>>>>> field (i.e., kinematic generation of magnetic field MHD):
>>>>>>>
>>>>>>> 1. D. Galloway and U. Frisch. A numerical investigation of magnetic
>>>>>>> field generation in a flow with chaotic streamlines. Geophysical &
>>>>>>> Astrophysical Fluid Dynamics, 29(1):13–18, 1984.
>>>>>>>
>>>>>>> 2. D. Galloway and U. Frisch. Dynamo action in a family of flows with
>>>>>>> chaotic streamlines.
>>>>>>> Geophysical & Astrophysical Fluid Dynamics, 36(1):53–83, 1986.
>>>>>>>
>>>>>>> 3. Podvigina, Pouquet, On the non-linear stability of the 1:1:1 ABC
>>>>>>> flow,
>>>>>>> Phys. D, Vol. 75, 4, 1994.
>>>>>>>
>>>>>>>
>>>>>>> With my best wishes,
>>>>>>> Barak
>>>>>>>
>>>>>>>
>>>>>>> Quoting nek5000-users at lists.mcs.anl.gov
>>>>>>> <mailto:nek5000-users at lists.mcs.anl.gov>:
>>>>>>>
>>>>>>>  Hi Barak,
>>>>>>>>
>>>>>>>> Thanks for the note about the ABC flow -- that's very useful.
>>>>>>>>
>>>>>>>> What is your target viscosity?
>>>>>>>> If you want it to be 0.1, then you would set param 2 in the .rea
>>>>>>>> file
>>>>>>>> to be 0.1.
>>>>>>>> Setting param 2 = -0.1 would imply that Re=0.1 or equivalently
>>>>>>>> nu=10.
>>>>>>>> Does this resolve the issue for you?
>>>>>>>>
>>>>>>>> Once it's working we could add it as a documented example in the
>>>>>>>> example
>>>>>>>> suite.
>>>>>>>>
>>>>>>>> Thanks!
>>>>>>>>
>>>>>>>> Paul
>>>>>>>> ________________________________________
>>>>>>>> From: nek5000-users-bounces at lists.mcs.anl.gov
>>>>>>>> <mailto:nek5000-users-bounces at lists.mcs.anl.gov>
>>>>>>>> [nek5000-users-bounces at lists.mcs.anl.gov
>>>>>>>> <mailto:nek5000-users-bounces at lists.mcs.anl.gov>] on behalf of
>>>>>>>> nek5000-users at lists.mcs.anl.gov
>>>>>>>> <mailto:nek5000-users at lists.mcs.anl.gov>
>>>>>>>> [nek5000-users at lists.mcs.anl.gov
>>>>>>>> <mailto:nek5000-users at lists.mcs.anl.gov>]
>>>>>>>> Sent: Saturday, December 21, 2013 5:24 PM
>>>>>>>> To: nek5000-users at lists.mcs.anl.gov
>>>>>>>> <mailto:nek5000-users at lists.mcs.anl.gov>
>>>>>>>> Subject: Re: [Nek5000-users] turbulent 3d box
>>>>>>>>
>>>>>>>> Hi Paul,
>>>>>>>> Regarding the 3dbox:
>>>>>>>> The forcing which I propose is known as the ABC flow
>>>>>>>> (Arnold-Beltrami-Childress) and has steady solutions (in the form
>>>>>>>> of ABC
>>>>>>>> flow) with Re=1/nu (nu the viscosity) Re=<12 (in periodic box).
>>>>>>>> The forcing should be in the form:
>>>>>>>> f=Re*ABC in order to balance the dissipation term.
>>>>>>>> This settings can serve as a benchmark for pure hydro/mhd problems.
>>>>>>>>
>>>>>>>> I tried to fix the example which you have sent me, but I don't
>>>>>>>> succeed
>>>>>>>> retreiving the expected values.
>>>>>>>> I attach the relevant files( tarball file).
>>>>>>>> Best regards,
>>>>>>>> Barak
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>> On 12/18/2013 03:29 AM, nek5000-users at lists.mcs.anl.gov
>>>>>>>> <mailto:nek5000-users at lists.mcs.anl.gov> wrote:
>>>>>>>>
>>>>>>>>  Hi Barak,
>>>>>>>>>
>>>>>>>>> I added a 3d periodic box under "3dbox" in the examples
>>>>>>>>> directory.
>>>>>>>>>
>>>>>>>>> Paul
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> Dear nek's,
>>>>>>>>> I would like to run the following problem:
>>>>>>>>> a 3D periodic box, small viscosity and the following forcing:
>>>>>>>>>
>>>>>>>>> fx=cos(y)+sin(z)
>>>>>>>>> fy=cos(z)+sin(x)
>>>>>>>>> fz=cos(x)+sin(y)
>>>>>>>>>
>>>>>>>>> Do you have a similar example?
>>>>>>>>>
>>>>>>>>> Many thanks,
>>>>>>>>> Barak
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> _______________________________________________
>>>>>>>>> Nek5000-users mailing list
>>>>>>>>> Nek5000-users at lists.mcs.anl.gov
>>>>>>>>> <mailto:Nek5000-users at lists.mcs.anl.gov>
>>>>>>>>> https://lists.mcs.anl.gov/mailman/listinfo/nek5000-users
>>>>>>>>> _______________________________________________
>>>>>>>>> Nek5000-users mailing list
>>>>>>>>> Nek5000-users at lists.mcs.anl.gov
>>>>>>>>> <mailto:Nek5000-users at lists.mcs.anl.gov>
>>>>>>>>> https://lists.mcs.anl.gov/mailman/listinfo/nek5000-users
>>>>>>>>>
>>>>>>>>>  _______________________________________________
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>>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> _______________________________________________
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>>>>>>>
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