[Nek5000-users] turbulent 3d box

Sun Dec 22 15:22:44 CST 2013

Hi Barak,

Derivatives are best computed using gradm1().   See earlier posts on this topic.

Interpolation is spectrally accurate, but you can also just integrate your fields against
cos(kx), sin(kx), etc. and compute energy at a particular wave number in the old-fashioned
way.   The advantage of this approach is that it is trivially parallel -- but it's not optimal for
performance if you want the full spectrum.   At present, we don't have a full set of tools for
3D Fourier analysis of SEM flows because such flows are not standard;  normally this
code is applied in unstructured geometries.   I think if you review the users lists you might
find some discussion of transformations in a single direction, and we certainly do have tools
that you could adapt for such purposes.   If I were doing it, that's the route I would follow --- i.e.,
I would transform in one direction to get planes of differing wave numbers on each processor,
then locally transform in the other directions.  I would also first interpolate onto a uniform 
mesh in each element --- (grep -i zuni *.f in the /nek directory will lead you to calls to implement
such a transformation).   Note that you can interpolate from GLL to uniform --- that direction
is stable.  You cannot, however, transform from uniform to GLL.

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 2:59 PM
To: nek5000-users at lists.mcs.anl.gov
Subject: Re: [Nek5000-users] turbulent 3d box

Dear Paul,

You're right.I check your calculations and they are correct!!!.

A huge mistake of mine. Sorry for bothering you. In fact what I meant
was the energy in the velocity field which is
sqrt(3)/2=\int (ux^2+uy^2+uz^2)dxdydz
  So now I am going to recheck your code for nu<1/12 and see if indeed I
obtain those periodic solutions. Did you see the papers?

Another question, again with the interpolation: is it spectrally accurate?
How I compute velocity field derivatives? such as vorticity, pressure,
or strain field in a spectral accuracy?
Furthermore: how in general one computes the energy spectrum of the
velocity field, and the energy in a particular wavenumber ? (in the
spectral methods it is quite straightforward).

Many thanks,
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
>>>>
>>>>
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>>
>>
>>
>>
>>
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