Large-eddy simulation of turbulent channel flow using relaxation filtering: Resolution requirement and Reynolds number effects
Introduction
Over the last two decades, computational fluid dynamics has become an efficient tool for the study of wall turbulence. In particular, wall-bounded flows at ever-higher Reynolds numbers have been simulated, which enabled the effects of the Reynolds number on flow statistics and coherent structures to be discussed [1]. It remains, however, difficult to reproduce the features of wall-bounded flows numerically, because wall turbulence is strongly influenced by the dynamics of the small scales developing close to the wall, which exhibit strong anisotropy and complex interactions with larger scales. These small scales must therefore be accurately calculated in simulations. This has been done in most cases using Direct Numerical Simulation (DNS) for channel flows [2], [3], [4], [5], [6], [7], [8] and boundary layers [9], [10], [11], [12], [13]. Unfortunately, as the Reynolds number increases, the computational cost of a DNS is rapidly prohibitive. As an illustration, note for instance that twenty years have elapsed between the DNS by Kim et al. [2] and by Hoyas and Jiménez [8] for channel flows at Reynolds numbers differing by one decade only.
In order to reduce the numerical cost, Large Eddy Simulations (LES), in which only the largest eddies are resolved, can be used. The effects of the under-resolved eddies are then taken into account by a so-called subgrid-scale model, which classically relies on the assumptions that the large scales carry energy, and that the small scales have mainly dissipative effects [14]. Depending on the possible near-wall resolution, wall-modelled or wall-resolved LES can be performed. In the first approach, only the outer part of wall-bounded flows is resolved, whereas the inner part is modelled [15]. In this way, very high Reynolds numbers can be reached [16], but the near-wall structures are not captured. In the second approach, both the outer and inner parts of the flows are computed at the expense of the computational cost. Accordingly, the range of Reynolds numbers affordable with wall-resolved LES is much smaller, and falls within the range of Reynolds numbers considered in DNS [17], [18], [19], [20]. The cost is however significantly lower using LES. For example, the number of grid points is about 10 times smaller in the LES of a boundary layer performed by Schlatter et al. [20] than in a DNS.
In wall-resolved LES, various numerical parameters such as the inflow and boundary conditions, the grid resolution, the subgrid-scale model and the discretization schemes can affect the calculation of the near-wall turbulent structures. It is consequently necessary to validate the simulation methods carefully. Regarding the impact of the inflow conditions, for example, Schlatter and Örlü [13] have reviewed data from several DNS of boundary layers, and pointed out some differences in basic integral quantities and in flow statistics. They showed in particular that flow features are significantly influenced by the inflow parameters and the boundary-layer tripping [21]. Such difficulties do not exist for fully-developed channel flows, where periodic conditions are imposed in the streamwise direction where turbulence is homogeneous. It appears therefore particularly interesting to study the quality of the LES of wall-bounded flows by simulating channel flows. This is the case for instance in the papers by Rasam et al. [22] and by Vuorinen et al. [23], who examined the effects of subgrid-scale model and grid resolution, and of a space discretization method, respectively.
In the present work, turbulent channel flows are simulated by LES using relaxation filtering as a subgrid-scale model. This LES approach was proposed by Visbal and Rizzetta [24], Mathew et al. [25] and Bogey and Bailly [26], among others. It consists in filtering the flow variables every n-th time step using a high-order low-pass filter at a strength between 0 and 1, in order to relax turbulent energy from the smallest discretized scales, characterized by wave numbers close to the grid cut-off wave number, while leaving larger scales mostly unaffected. In practice, the filtering is usually applied every time step at a fixed strength in order to ensure numerical stability, which is not guaranteed when low-dissipation and/or centered discretization schemes are used. Note, however, that dynamic procedures can be built to adjust the parameters of the filtering to the flow characteristics, e.g. in Tantikul and Domaradzki [27]. In previous studies, the validity of the LES approach was explored for a Taylor–Green vortex flow [28], free shear layers [29] and jets [26], [30], [31], [32]. The approach has also been successfully employed for a flow around an airfoil [33] or for a turbulent boundary layer [19]. Here, the performance of the LES method is investigated for wall-bounded flows by simulating fully developed channel flows on grids at different spatial resolutions and for different Reynolds numbers. The first objective is to determine for which mesh spacings accurate results, converged with respect to the grid, can be obtained. The second one is to check that Reynolds number effects [30], [34] on wall turbulence are reproduced. For this, velocity profiles and spectra obtained near the wall, and in particular in the buffer-layer region, where small scales play an important role, will be presented, and comparisons with DNS data of the literature will be provided. Transfer functions associated with molecular viscosity and relaxation filtering will also be shown in the wavenumber space.
The paper is organized as follows. The LES performed for a channel flow at different spatial resolutions are presented in Section 2. The LES of channel flows at different Reynolds numbers are reported in Section 3. Finally, concluding remarks are given in Section 4.
Section snippets
Parameters
Large-eddy simulations of a turbulent channel flow are performed by solving the three-dimensional compressible Navier–Stokes equations on Cartesian meshes. The channel flow is at a Reynolds number of and a Mach number of , where is the centerline velocity, c is the speed of sound, h is the channel half-width, is the friction velocity, is the wall shear stress, and and are the kinematic molecular viscosity and the density of the flow. The streamwise,
Parameters
Three large-eddy simulations of turbulent channel flows at a Mach number of and at Reynolds numbers of , 600 and 960, referred to as Re350, Re600 and Re960, are performed. The Mach number is slightly higher than the Mach number of considered in Section 2. However, both are low enough so that compressibility effects are very weak, and that the flow features do not appreciably depend on the Mach number [45]. In the three LES, the dimensions of the computational domain are
Conclusion
In this paper, LES of fully developed channel flows using relaxation filtering as subgrid model are reported. The simulations are performed using different grid resolutions and for various Reynolds numbers, in order to assess the validity of the LES approach for turbulent wall-bounded flows.
For the LES at a fixed Reynolds number carried out with different spatial resolutions, the mean and rms velocity profiles are found not to change significantly with the grid for mesh spacings
Acknowledgments
The first author is grateful to the Direction Générale de l’Armement (DGA) and the Centre National de la Recherche Scientifique (CNRS) for financial support for his doctoral studies. This work was granted access to the HPC resources of the Institut du Développement et des Ressources en Informatique Scientifique (IDRIS) under the allocation 2012-020204 made by GENCI (Grand Equipement National de Calcul Intensif). The authors would like to thank Olivier Marsden for his help in parallelizing the
References (45)
- et al.
A high-resolution code for turbulent boundary layers
J Comput Phys
(2009) - et al.
Spectral features of the wall-pressure fluctuations in turbulent wall flows with and without perturbations using LES
Int J Heat Fluid Flow
(2001) - et al.
Compressible large eddy simulations of wall-bounded turbulent flows using a semi-implicit numerical scheme for low Mach number aeroacoustics
J Comput Phys
(2006) - et al.
Simulations of spatially evolving turbulent boundary layers up to
Int J Heat Fluid Flow
(2010) - et al.
Filter shape dependence and effective scale separation in large-eddy simulations based on relaxation filtering
Comput Fluids
(2011) - et al.
Large eddy simulations of round free jets using explicit filtering with/without dynamic Smagorinsky model
Int J Heat Fluid Flow
(2006) - et al.
A family of low dispersive and low dissipative explicit schemes for flow and noise computations
J Comput Phys
(2004) - et al.
Low-dissipation and low-dispersion 4th-order Runge–Kutta algorithm
Comput Fluids
(2006) - et al.
A shock-capturing methodology based on adaptative spatial filtering for high-order non-linear computations
J Comput Phys
(2009) - et al.
Direct numerical simulation of compressible turbulent channel flows using the discontinuous Galerkin method
Comput Fluids
(2011)
Wall-bounded turbulent flows at high Reynolds numbers: recent advances and key issues
Phys Fluids
Turbulence statistics in fully developed channel flow at low Reynolds number
J Fluid Mech
Direct numerical simulation of turbulent channel flow up to = 590
Phys Fluids
Aeroacoustics of wall-bounded turbulent flows
AIAA J
Spectra of the very large anisotropic scales in turbulent channels
Phys Fluids
Scaling of the energy spectra of turbulent channels
J Fluid Mech
Wall pressure and shear stress spectra from direct simulations of channel flow
AIAA J
Scaling of the velocity fluctuations in turbulent channels up to = 2003
Phys Fluids
Direct simulation of a turbulent boundary layer up to = 1410
J Fluid Mech
Turbulent boundary layers up to = 2500 studied through simulation and experiment
Phys Fluids
Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer
J Fluid Mech
Assessment of direct numerical simulation data of turbulent boundary layers
J Fluid Mech
Cited by (52)
Assessment of implicit LES modelling for bypass transition of a boundary layer
2023, Computers and FluidsCitation Excerpt :In particular, it was demonstrated in [22] and [21] that such an implicit model seems preferable to explicit models when used in combination with high-order schemes, thanks to the control of the numerical dissipation at the smallest resolved scales, which is designed to mimic Spectral Vanishing Viscosity (SVV). It is also noticeable that the relation of this approach with the solution filtering approach (e.g. [24]) has recently been analysed, and an original way of introducing this numerical dissipation together with the molecular dissipation through filtering was recently proposed in [25], with substantial benefits in terms of stability. It should be noted that the class of transitional flows considered here is highly challenging for LES, first because of the diversity of flow regimes (FST, laminar and turbulent boundary layer), and also because the small scales in the transition region are a consequence of instabilities rather than a kinetic energy cascade.
Viscous and hyperviscous filtering for direct and large-eddy simulation
2021, Journal of Computational PhysicsCitation Excerpt :In this study, we are interested in the technique that consists in the application of spatial filtering every time step on the solution during the time advancement of the governing equations. This technique is sometimes referred as “solution filtering” [5] or as “relaxation filtering” [6,7]. It can also be designated as “explicit filtering”, but this term is potentially confusing with the SGS modelling approach where the non-linear convective terms, based on the implicitly filtered (i.e. large-scale) solution, are themselves filtered with a given operator to define an alternative SGS tensor in a modified closure problem [8–10].
Numerical assessments of a parametric implicit large eddy simulation model
2020, Journal of Computational and Applied Mathematics