[Nek5000-users] turbulent 3d box

nek5000-users at lists.mcs.anl.gov nek5000-users at lists.mcs.anl.gov
Fri Jan 10 09:07:55 CST 2014 

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
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>>>>>>> https://lists.mcs.anl.gov/mailman/listinfo/nek5000-users
>>>>>>>
>>>>>> _______________________________________________
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>>>>>> <mailto:Nek5000-users at lists.mcs.anl>.govhttps://lists.mcs.anl.gov/mailman/listinfo/nek5000-users
>>>>>>
>>>>>> <UrlBlockedError.aspx>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
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>>>>
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