Elsevier

European Journal of Mechanics - B/Fluids

Volume 67, January–February 2018, Pages 185-197
European Journal of Mechanics - B/Fluids

Large-eddy simulation of a spatially-evolving supersonic turbulent boundary layer at M=2

https://doi.org/10.1016/j.euromechflu.2017.09.005Get rights and content

Abstract

The ability of large-eddy simulations (LES) to resolve the most energetic coherent structures of a spatially-evolving supersonic turbulent boundary layer over a flat plate at M=2 and Reθ6000 is analyzed using three different local subgrid scale models. Additionally, an Implicit LES (ILES), which relies on the intrinsic numerical dissipation to act as a subgrid model, is investigated to assess the consistency and the accuracy of the method. Direct comparison with data from high resolution DNS calculations (Pirozzoli and Bernardini, 2011) provides validation of the different modeling approaches. Turbulent statistics up to the fourth-order are reported, which help emphasizing some salient features related to near-wall asymptotic behavior, mesh resolution and models prediction. Detailed analysis of the near-wall asymptotic behavior of all relevant quantities shows that the models are able to correctly reproduce the near-wall tendencies. The thermodynamic fluctuations, Trms and ρrms, show a lack of independence from SGS modeling and grid refinement in contrast to the velocity fluctuations. The pressure fluctuations, which are associated with the acoustic mode, are not significantly affected by the modeling and the mesh resolution. Furthermore, the comparison of different contributions to the viscous dissipation reveals that the solenoidal dissipation plays the most dominant role regardless of the model. Finally, it is found that the ILES is more likely to produce consistent near-wall behavior even with a numerical scheme that has a small amount of numerical dissipation to emulate the effects of unresolved scales.

Introduction

The study of supersonic turbulent boundary layers (STBL) is crucial for understanding basic flow physics in turbulent wall-bounded flows. The study has also a great importance in many industrial applications, such as high speed external and internal aerodynamics [2], [3], [4], [5], combustion and detonation [6], [7]. For adiabatic STBL, and due to viscous heating, compressibility effects arise mainly from the large change in the fluid properties (variable-density flow). It is then commonly concluded that adiabatic supersonic turbulent boundary layers at moderate Mach numbers (typically M5) can be studied using the same models as low-speed flows, as long as the variations in the mean flow properties are accounted for (see for example Morkovin 1961 [8], Bradshaw, 1977 [9] and Smits & Dussauge, 2006 [10]). Adiabatic supersonic turbulent boundary layers were first investigated through experiments, in order to validate the Morkovin’s hypothesis (for a large data compilation, see Fernholz & Finley, 1977 [11]).

Three-dimensional numerical simulations of turbulent boundary layer are usually classified as direct Navier–Stokes simulations (DNS) and large-eddy simulations (LES). In DNS, all relevant scales of motions are numerically resolved and therefore a detailed representation of a turbulent flow field can be obtained. In LES, only large energy-containing eddies are numerically resolved. This is accomplished by filtering-out the high-frequency component of the flow field and using the low-pass-filtered form of the Navier–Stokes equations to solve for the large-scale component only. The effects of the filtered-out small-scale fields on the resolved fields are accounted for through the so-called subgrid-scale (SGS) model. Many different LES approaches have been developed for the construction of SGS models; some of these are described in this paper, whose objective is to address their applicability in the case of supersonic turbulent flow over a flat plate at a zero-pressure-gradient. Our focus here is on developing a methodology for assessing LES approaches on a representative flow, which is the obvious pre-requisite before applying these models to more complex geometries.

Among others, Spyropoulos & Braisdell (1998) [12] reported LES of spatially-evolving supersonic turbulent boundary layer at Mach number M=2.25. Because of the low considered Mach number, the modeling of the isotropic part of the shear stresses was not found to have a considerable effect on the skin-friction coefficient, Cf. The insufficient amount of turbulent transport was attributed to the use of the dynamic Smagorinsky model (DSM), in which the eddy viscosity is computed using the smallest resolved scales. Hadjadj et al. (2015) [13] recently studied spatially-evolving STBL with cooled walls via well-resolved LES’s at low Reynolds numbers. Also, supersonic flat-plate boundary layers have been investigated by Yan et al. (2002) [14] using monotonically integrated large-eddy simulation (MILES) approach. In this simulation, the numerical dissipation induced by the scheme substitutes the SGS eddy viscosity, mimicking from an energetic view-point the action of SGS terms on the flow dynamics. Their results indicate that the subgrid-scale effects can be adequately modeled using MILES without the need for the Smagorinsky model.

In LES, the accuracy of the resolved scales highly relies on the mesh size. Locally refined grids usually lead to more resolved turbulent energy but with costly CPU time and memory requirements. The strategy in LES is then to make the best compromise between accuracy and computational costs. Dissipation of a given SGS model may originate, in different proportions, either from the resolved velocity fluctuations or from the mean-averaged velocity gradients. In the recent work of Ben-Nasr et al. (2016) [15], we presented a detailed study of a spatially-evolving STBL over a flat plate at M=2 and Reθ2600. Different SGS models (namely the DSM, the wall-adapting local eddy-viscosity (WALE) model and the Coherent Structures model (CSM)) as well as grid resolutions were used in order to compare the contribution of the SGS modeling on turbulence. The advantage and superiority of CSM and WALE in resolving high-speed compressible turbulent boundary layer over flat plate is clearly established in that study over computationally costly DSM. It is also interesting to mention the performance of Implicit LES (ILES) with respect to other models. In the present study, we extend the previous work for higher Reθ, while using wider spanwise domain for various LES models. Much coarser grid resolutions have been employed to assess the effectiveness of the LES modeling compared to well-resolved LESs presented in Ben-Nasr et al., (2016) [15]. We further highlight the interesting features of these LESs in light of the near wall interactions of the fluctuating quantities as a natural sequel of the previous work. After a brief description of the numerical methodology and the problem setup in Section 2, we present the flow analysis in Section 3. Turbulence statistics up to fourth-order moments are reported in order to assess more specifically the near-wall behavior of the SGS models. Finally, the conclusions are drawn in Section 4.

Section snippets

Numerical methodology and SGS modeling

For sake of brevity, we restrict the description of the governing equations used for the present study. The details of the numerical methodology as well as the modeling aspect can be found in Ben-Nasr et al., (2016) [15]. The convective fluxes are discretized using a sixth-order locally-conservative skew-symmetric split-centered formulation [16]. The viscous fluxes are discretized using a fourth-order compact central differences scheme. Time advancement is assessed by a standard explicit

Basic flow organization

It is known that the inner part of the boundary layer is occupied by alternating streaks of high- and low-speed fluids. These streaks are presumed to derive from elongated, counter-rotating streamwise vortices near the wall. At y+<100, those streaks are shown to significantly contribute to the turbulence production, which occurs during the bursting process: low-speed streaks would gradually lift up from the wall, oscillate, and then break up violently, ejecting fluid away from the wall and into

Conclusion

In this paper, large-eddy simulations of a spatially-evolving supersonic turbulent boundary layer over a flat plate are performed using three different SGS models. An Implicit LES (a subset of under-resolved DNS) is also investigated to assess its applicability and to see whether small truncation terms of a sixth-order numerical scheme would themselves serve as SGS models. The results are compared to both DNS and theoretical considerations and showed an overall acceptable agreement. The study

Acknowledgments

This work was performed using HPC resources from GENCI [CCRT/CINES/IDRIS] (Grant 2016-0211640) and from CRIANN (Centre Régional Informatique et d’Applications Numériques de Normandie, Rouen. The authors acknowledge Prof. Sergio Pirozzoli, University of Rome La Sapienza, for providing his DNS data for validation.

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