Aliasing errors arise in the multiplication of partial sums, such as those
encountered when numerically solving the Navier–Stokes equations, and can be
detrimental to the accuracy of a numerical solution. In this work, a
performance and cost analysis is proposed for widely used dealiasing schemes in
large-eddy simulation, focusing on a neutrally stratified, pressure-driven
atmospheric boundary-layer flow. Specifically, the exact

Tests are performed within a newly developed mixed pseudo-spectral finite differences large-eddy simulation code, parallelized using a two-dimensional pencil decomposition. A series of simulations are performed at varying resolution, and key flow statistics are intercompared among the considered runs and dealiasing schemes.

The three dealiasing methods compare well in terms of first- and second-order
statistics for the considered cases, with modest local departures that decrease
as the grid stencil is reduced. Computed velocity spectra using the

The past decades have seen significant progress in computer hardware in
remarkable agreement with Moore's law, which states that the number of nodes
in the discretization grids doubles every 18 months

At the same time, with increasing computer power, the range of scales and
applications in computational fluid dynamics (CFD) has significantly
broadened, allowing us to describe – at an unprecedented level of detail – complex
flow systems such as fluid–structure interaction

The Fourier-based pseudo-spectral collocation method

This has motivated efforts towards the development of approximate yet
computationally efficient dealiasing schemes, such as the Fourier truncation
(FT) method

In this work, we provide a cost–benefit analysis and a comparison of
turbulent flow statistics for the FT and FS dealiasing schemes in comparison
to the exact

Aliasing errors result from representing the product of two or more variables
by

Note that the corresponding expression for the Fourier transform of the
product

As a result, the energy contained within the remaining

The

Weighting functions used in the FT method (dashed line)
and the FS method (continuous line). The FT method filters scales
with

The LES approach consists of solving the filtered NS equations, where the
time and space evolution of the turbulent eddies larger than the grid size
are fully resolved, and the effect of the smaller ones is parameterized.
Mathematically, this is described through the use of a numerical filter that
separates the larger, energy-containing eddies from the smaller ones. Often,
the numerical grid of size

In these equations,

Note that the molecular viscous term has been neglected in the governing equations, including the wall-layer parameterization, which is equivalent to assuming that the surface drag is mostly caused by pressure (i.e., there are negligible viscous contributions). This is a typical situation in flow over natural surfaces where the surface is often in a fully rough aerodynamic regime.

The drag from the underlying surface is entirely modeled in this application
through the equilibrium logarithmic law for rough surfaces

In Eq. (

In addition, the corresponding vertical derivatives of the horizontal mean
velocity field are imposed at the first grid point of the vertically
staggered grid

This setup has now been extensively used to study neutrally stratified ABL
flows

The equations are solved using a pseudo-spectral approach, where the
horizontal derivatives are computed using discrete Fourier transforms and the
vertical derivatives are computed using second-order accurate centered finite
differences on a staggered grid. A projection fractional-step method is used
for time integration following Chorin's method

Next, an intermediary step is computed using an Adams–Bashforth scheme,
following

The new flow field for the complete time step is obtained by

Embedded within this approach, periodic boundary conditions are imposed on
the horizontal

Two-dimensional pencil decomposition of the computational domain with
the domain transposed into the three direction of space:

The code is parallelized following a 2-D pencil decomposition paradigm similar
to the one presented in

The LES algorithm can be separated into four distinct modules: (1) computation of the velocity gradients,
(2) evaluation of the SGS stresses and
(3) of the convective term, and (4) computation of the pressure field by
solving the Poisson equation. These four modules represent the bulk of the
computational cost of the code, in addition to MPI communication. Figure

Simplified flowchart of the main algorithm presenting the four modules that represent the bulk of the computational cost.

The four modules have been individually timed to evaluate their corresponding
computational cost at a resolution of

Individual timing of the four modules of the time loop averaged
over 10 k steps:

The goal of this study is to develop a cost–benefit analysis for the
different, already established, dealiasing methods from a computational cost
standpoint as well as in terms of accuracy in reproducing turbulent flow
characteristics. For this reason, three different cases have been considered, corresponding to the three dealiasing methods considered: (a) the

For each dealiasing method, the simulations at

Simulations summary, each simulation was run with the three different dealiasing methods.

The computational cost of evaluating the convective term, dealiased via
the

Computational cost of the convective module as a
function of the horizontal resolution. The timing of the module
is presented on the left vertical axis and represented by
left-pointing arrows. The number of operations is shown on the
right vertical axis and represented by right-pointing arrows. The
three different dealiasing methods are plotted: the

Instantaneous streamwise velocity perturbation

In the following subsections, we compare the impact of the different
dealiasing schemes on flow statistics. Profiles from runs using the

Traditional flow metrics are investigated next and compared between the
different dealiasing schemes. Results for the

Top panels represent the plots of the non-dimensional mean
streamwise velocity profile for

The horizontally and temporally averaged velocity profiles are characterized
by an approximately logarithmic behavior within the surface layer (

Mean velocity gradient profiles (

Profiles of the non-dimensional variances

Figure

Mean error in the variance profiles between the

To complement the analysis of the effect of the different dealiasing methods
on the physical structure of the flow, the corresponding power spectra are investigated. According to Kolmogorov's energy cascade theory, the inertial
subrange of the power spectrum should be characterized by a power law of

Although the effect of the FT and FS methods on the small scale can be
clearly observed in Fig.

When using the FT method, energy at the low wavenumbers is underpredicted,
whereas energy at the large wavenumbers is overpredicted. Departures are in
general larger with decreasing resolution, with an excess of up to

Normalized streamwise spectra of the streamwise velocity
as a function of

Effect of the FT

In the development of this paper, focus has been directed to the study
of the advantages and disadvantages of different dealiasing methods. In this
regard, throughout the analysis, we have tried to keep the structure of the
LES configuration as simple and canonical as possible to remove the effect
of other add-on complexities. Additional complications might arise when
considering additional physics; here we discuss the potential effect that
these different dealiasing methods could have on them. One such element of added complexity is, for example, the use of more sophisticated subgrid-scale models based on dynamic approaches to determine the values of the
Smagorinsky constant

In general, we believe that it is not fair to advocate for one or other
dealiasing method based on the results of this analysis. Note that the goal
of this work is to provide an objective analysis of the advantages and
limitations that the different methods provide, leaving the readers with the
ultimate responsibility of choosing the option that will adjust better to their
application. For example, while having exact dealiasing (

The Fourier-based pseudo-spectral collocation method

The presented results show compelling evidence of the benefits of these
methods as well as some of their drawbacks. The advantage of using the FT or
the FS approximate dealiasing methods is their reduced computational cost
(cutback on the total simulation time of

Specifically, results illustrate that both the FT and FS methods
over-dissipate the turbulent motions in the near-wall region, yielding an
overall higher mass flux when compared to the reference one (

The sources of the LES code developed at the University of
Utah are accessible in prerelease at

Due to the large amount of data generated during this study, no lasting structure can be permanently supported where to freely access the data. However, access can be provided using the Temporary Guest Transfer Service of the Center of High Performance Computing of the University of Utah. To get access to the data, Marc Calaf (marc.calaf@utah.edu) will provide temporary login information for the sftp server.

The authors declare no competing interests

Fabien Margairaz and Marc Calaf acknowledge the Mechanical Engineering Department at the University of Utah for start-up funds. Marco G. Giometto acknowledges the Civil Engineering and Engineering Mechanics Department at Columbia University for start-up funds. Marc B. Parlange is grateful to NSERC and Monash University for their support. The authors would also like to recognize the computational support provided by the Center for High Performance Computing (CHPC) at the University of Utah as well as the Extreme Science and Engineering Discovery Environment (XSEDE) platform (project TG-ATM170018). Edited by: Simone Marras Reviewed by: Elie Bou-Zeid and one anonymous referee