[Nek5000-users] turbulent 3d box

Sun Jan 12 14:43:15 CST 2014

Barak,

Most of this is covered in Chap. 3 of Deville, Fischer, & Mund, High Order Methods for Incompressible
Flow.   As Tubois indicates, Canuto, Hussaini, Quarteroni & Zang - either first or 2nd edition also cover
these topics in varied levels of detail.

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: Saturday, January 11, 2014 4:59 AM
To: nek5000-users at lists.mcs.anl.gov
Subject: Re: [Nek5000-users] turbulent 3d box

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
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>>>>>>>>
>>>>>>> _______________________________________________
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>>>>>>> <UrlBlockedError.aspx>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>> _______________________________________________
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>>>>>
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>>>>
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