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In the framework of the spectral element method, a comparison is carried out on turbulent first-and second-order statistics generated by large eddy simulation (LES), under-resolved (UDNS) and fully resolved direct numerical simulation (DNS). The LES is based on classical models like the dynamic Smagorinsky approach or the approximate deconvolution method. Two test problems are solved: the lid-driven cubical cavity and the differentially heated cavity. With the DNS data as benchmark solutions, it is shown that the numerical results produced by the UDNS calculation are of the same accuracy, even in some cases of better quality, as the LES computations. The conclusion advocates the use of UDNS and calls for improvement of the available algorithms.

The numerical simulation of turbulent flows still remains a major challenge, especially at high values of the Reynolds number. While direct numerical simulation (DNS) is feasible at the expense of large computational resources for moderate Reynolds numbers of the order

In a LES, the dynamics of the gross structures of the flow is computed by integrating the filtered Navier- Stokes (NS) equations, while the fine structures of the flow that cannot be resolved by the computational grid are modeled. To obtain the LES equations, a low-pass filter built through a convolution operator is applied to the NS equations. In the context of high-order methods like the spectral element method (SEM) [

A fundamental issue of LES consists in checking the convergence of the model used with respect to some reference benchmark like experimental results or DNS data. Here the two DNS test cases are the lid-driven cavity (LDC) problem [

In this paper we want to examine another viewpoint of comparison between UDNS and LES. Namely, we will examine the question: Can a UDNS with a coarse grid yield comparable results with the LES calculations? Phrased another way, do we need an LES model if we can achieve through a UDNS the same results?

The paper is organized as follows. Section 2 presents the two test cases: the LDC and DHC problems. In Section 3 the filtered equations are given with the various LES models used in this study. The spectral element method is briefly summarized in Section 4 with the space and time discretizations and the associated filters. Section 5 treats the LDC results, while the DHC is the subject of Section 6. Conclusions are drawn in Section 7.

We will describe the geometrical features of the test problems and the associated mathematical models.

The first numerical test treats the lid-driven cubical cavity as shown in

where U is the maximum wall velocity.

The mathematical model is given by the Navier-Stokes equations for a viscous Newtonian incompressible fluid

where

expressed in terms of the characteristic length

In this case the geometry is a cube where all walls are fixed. The flow is generated by a temperature difference between the hot left side wall and the cold right side wall. All other walls are insulated.

With the Boussinesq approximation the fluid is considered as incompressible. The mathematical model includes the advection-diffusion temperature equation. Therefore the buoyancy term in the momentum equations incorporates the influence of the temperature field. The non-dimensional Boussinesq equations read

where T is the temperature, the air Prandtl number

with

For high Rayleigh numbers of

In this section we present the LES Boussinesq equations to get acquainted with the filtering procedure and the LES modeling. For the LDC case, the LES NS equations are an isothermal subset of the Boussinesq equations as the model does no longer contain the buoyancy term in the momentum equations and the temperature equation.

Filtering the Navier-Stokes and temperature equations, we obtain the filtered Boussinesq equations

with the subgrid scale (SGS) tensor

and the subgrid scale heat flux:

The filtered variables denoted by an overbar are computed as a convolution with a filter kernel G. If we filter the variable

The filters will be presented in section 4.2.

By the filtering operation (14), any variable is decomposed into a resolved part

The decomposition of a statistically stationary flow field

In a LES, in general only the filtered flow solution

In the following, we assume that the turbulent flow has reached a statistically steady state, and the Reynolds average is computed as a time average

The additional variables

The subgrid scale tensor uses the dynamic Smagorinsky model [

where the SGS viscosity is computed by the relation

Here the quantity

The subgrid heat flux is modelled by a subgrid diffusivity

where the SGS diffusivity is evaluated as

and is based on a Reynolds analogy assumption.

The Smagorinsky constant

culated with the help of the dynamic procedure that uses a coarser test filter denoted by

process allows for the adaption of these parameters to the characteristics of the local flow. The assumption behind this approach rests on a scale-similarity hypothesis which considers that the behavior of the smallest resolved scales is similar to the modeled subgrid scales. The application of the test filter to the filtered Boussinesq equations produces twice filtered equations. The Germano identities allow the evaluation of the difference between the residual stress tensor and the residual heat flux resulting from the double-filtered quantities, namely

We note that

The approximate deconvolution method (ADM) defilters the filtered fields. Following the lead of Stolz et al. [

where

where I is the identity operator.

Typical deconvolutions to third or fifth order are given by

with the notation

The subgrid scale tensor is computed as

Because the approximate deconvolution method does not take account of interactions from the computational grid unresolved scales, it needs to be supplemented with a dissipative term. Stolz et al. used an empirical relaxation term to stabilize the computation. Another possibility is to combine the ADM model with a subgrid- viscosity model. The subgrid scale tensor is modeled by the relation

The ADM-DMS was introduced by Bouffanais in his thesis [

The ADM-DMS method with

The spectral element method (SEM) decomposes the computational domain into

where

In order to avoid spurious pressure modes, the pressure p is staggered and approximated on a Gauss-Legendre (GL) grid based on points that are the roots of the equation

From the functional point of view velocity and pressure are in

The discrete equations are designed using the weak formulation of the Galerkin method. The continuous integrals of the weak formulation are approximated by Gauss-Legendre numerical quadratures. With the notations borrowed from the monograph [

where

The time integration scheme rests upon an implicit treatment of the transient Stokes operator and the linear diffusive terms in order to avoid the stringent stability restrictions. This is performed by an Euler backward scheme of order two (Euler2). The non-linear terms are treated explicitly by extrapolation in time (EXT2). The global scheme Euler2/EXT2 has no splitting error and is globally of second order accuracy. A real advantage of SEM comes from the minimal numerical dissipation and dispersion. However the explicit treatment of the advection term imposes a

The spectral element method is implemented in the toolbox Speculoos [

Two types of filters may be used in the SEM methodology.

The nodal filter due to Fischer and Mullen [

Therefore, the matrix operator of order M

interpolates on the GLL grid of degree M a function defined on the GLL grid of degree N and transfers the data back to the original grid. This process eliminates the highest modes of the polynomial representation. A one- dimensional representation of the filter is given by the relation

with

Here the variable is filtered in the spectral Legendre space that is built on the hierarchical basis (cf. [

In the spectral space a low-pass filter is easily implemented and allows to prune the high-wave number spurious modes. A fully detailed description is yielded in [

For the ADM-DMS model, the modal filter was used on a polynomial space of degree 8.

The ADM-DMS model was first applied to the lid-driven cavity problem as reported in [

For a quantitative comparison between the UDNS and the DNS, we plot first-order statistics in the mid-plane

Elements | Polynomial degree | Time step | Ave. time | ||
---|---|---|---|---|---|

DNS | 12000 | 1 | 128 | 1000 | |

UDNS | 12000 | 8 | 319 |

Now we plot second-order statistics in the mid-plane

The comments made about the transfer function in section 4.2.2 raise the question, if it would be a good idea to use the exact filter inverse to model the subgrid-scale tensor (12). Such a model could be readily implemented. But since no information is lost, the filtering can be considered as a change of variables and according to Domaradzki et al. [

In order to better understand the efficiency and performance of the LES models, we carried out two under- resolved direct numerical simulations (UDNS) of the DHC. These simulations used the same computational parameters as the LES simulations published in [

We evaluate the number of grid points for the case UDNS1000 in one direction as the product of the number of elements in one direction times the polynomial degree (The number of grid points in a direction within one element is equal the polynomial degree plus one, but values at the element boundaries need to be counted only once). Therefore, the total number of grid points for the case UDNS1000 is 512,000. For the DNS, the total number of grid points is 4,826,809 or a factor 9.4 larger. The two cases, UDNS512 and UDNS1000, correspond to the two LES computations (cf. [

In

Elements | Polynomial degree | Time step | Sampl. freq. | Ave. time | ||
---|---|---|---|---|---|---|

DNS | 10^{9} | 1 | 169 | 20 | 470 | |

UDNS512 | 10^{9} | 8 | 10 | unstable | ||

UDNS1000 | 10^{9} | 8 | 10 | 200 |

the walls (cfr.

In

The comparison for the velocity component

In

The UDNS was done with a Legendre spectral element code presented in [

Although it is unclear how the accuracy of two different numerical methods can be compared, there is no doubt that the resolution of UDNS1000 is coarser than the one in the DNS. It is not possible to perform a

simulation with significantly less grid points using the Chebyshev spectral program as the computation blows up after a while due to the under-resolution. In our opinion, the high accuracy of the SEM UDNS is linked to the fact that a higher order method with domain decomposition is used. It seems that the Chebyshev spectral method is more sensitive to the under-resolution than the Legendre spectral element method, especially as the grid points scale like

The improvement of the available algorithms should involve more stable time schemes in order to avoid the stringent CFL condition of the explicit treatment of the non-linear terms. That could be achieved by explicit schemes with a larger stability region or by resorting to implicit schemes.

This research has been funded by the ARTIST Consortium Project headed by the Paul Scherrer Institute and its support is greatly acknowledged. The LES simulations were carried out on the Pleiades-2 cluster at EPFL-STI- IGM. For the largest simulations the supercomputers Blue Gene/P at EPFL and the Rosa-Cray XT 5 computer at CSCS, Manno, Switzerland were used. The financial support for CADMOS and the Blue Gene/P system is provided by the Canton of Geneva, Canton of Vaud, Hans Wilsdorf Foundation, Louis-Jeantet Foundation, University of Geneva,University of Lausanne, and Ecole Polytechnique Fédérale de Lausanne. This work was supported by a grant from the Swiss National Supercomputing Centre (CSCS) under project ID s328. E. Leriche would like to acknowledge the ERCOFTAC visiting program for several stays at EPFL.

ChristophBosshard,Michel O.Deville,AbdelouahabDehbi,EmmanuelLeriche, (2015) UDNS or LES, That Is the Question. Open Journal of Fluid Dynamics,05,339-352. doi: 10.4236/ojfd.2015.54034