[Nek5000-users] Resolution problems at computation of the energy spectrum

[nek5000-users at lists.mcs.anl.gov]

David,

You're at the marginally-resolved limit of the Nyquist sampling theorem. 

The only numerical method that will work in this regime is a Fourier one, and even
then the quadratic nonlinearity will lead to aliasing or truncation of the higher
wave numbers (depending on whether you dealias, or not;  in Nek, we typically
dealias, which implies truncation of the higher modes rather than aliasing of
the same).

Next point - all high-order polynomial based methods have a further under sampling that
and be alternately viewed as a limit in the accuracy of the under-resolved modes ---
The heuristic is that the Gauss points (all flavors) have a separation of pi/N in the middle
of the reference domain [-1 1] as opposed to 2/N that you get for uniform points.
The advantage gained is numerical stability and exponentially convergent dispersion
for the resolved modes, which amounts to k_max < N/pi, instead of the usual Nyquist N/2.

There have been many studies comparing Nek and spectral codes  (e.g., ZENG, L., BALACHANDAR, S., & FISCHER, P. (2005). Wall-induced forces on a rigid sphere at finite Reynolds number. Journal of Fluid Mechanics, 536, 1-25. ,
for channel flows)   -- There is no particular difficulty here provided you understand the resolution issues.

hth,

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, May 19, 2017 7:33 AM
To: nek5000-users at lists.mcs.anl.gov
Subject: [Nek5000-users] Resolution problems at computation of the energy       spectrum

Hi Neks,

I am currently performing DNS of a 3d periodic box. It seems I might have
encountered a resolution problem while working on the spectral analysis of
the velocity field.
In short: The high wavenumber modes appear to be insufficiently resolved
by the GLL points / polynomials causing effects similar to aliasing and
therefore a significant distortion of the energy spectrum, which can also
be observed away from the high wavenumber end.

The comparison of my energy spectra with literature values revealed that
the energy in the high wavenumber end is significantly underestimated. In
order to make sure that this deviation is not caused by a bug in my post
processing, I bypassed the routine by analyzing the NEK output of a simple
test case in MATLAB. In this simple 3d 2pi box the velocity is initialized
as vx=sin(30*x), vy=vz=0.0 on a grid with N=65 (nelx=8,lx1=9). The regular
grid output (ifreguo=.true.) of this initial time step (zeroth time step -
no equations were solved) was compared to sin(30*x). Here is a plot of
this comparison http://imgur.com/a/2c1q5. The deviations lead to a
redistribution of the energy among several modes.

Currently I am wondering if there is any solution which allows for a
computation of a less distorted energy spectrum without increasing the
number of elements or polynomial order, and to subsequently cutting off
the high modes. I already increased the number of equidistant grid points
by increasing lxo to 1.5*lx1, however the quality of the energy spectrum
is not improved as the interpolation deviates nonetheless.

I am looking forward to helpful suggestions or an interesting discussion,

Daniel
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