Abstract
Wall models reduce the computational cost of large eddy simulations (LES) by modeling the near-wall energetic scales and enable the application of LES to complex flow configurations of engineering interest. However, most wall models assume that the boundary layer is fully turbulent, at equilibrium, and attached. Such models have also been successfully applied to turbulent boundary layers under moderated adverse pressure gradients. When the adverse pressure gradient becomes too strong, and the boundary layer separates, equilibrium wall models are no longer applicable. In this work, the relations between the instantaneous wall shear stress, velocity field, and pressure gradients are evaluated using space-time correlations for the purpose of analyzing the near-wall physics in different flow configurations. These correlations are extracted from two wall-resolved LES: a channel flow at a friction Reynolds number \({\rm {Re}_\tau} \) of 950 and the two-dimensional periodic hill at a bulk Reynolds number Re\(_b\) of 10595. This analysis highlights that no instantaneous and local correlation is observed in the vicinity of the separation. The domain of high correlation appears to be shifted downstream. This study of the near-wall physics is a step for developing a data-driven wall model applied to separated flows and, in particular, selecting suitable input parameters for the training of neural networks.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availability of data and materials
No but can be made available if needed.
Code availability
No but can be made available if needed
References
Abel, M., Stojkovic, D., Breuer, M.: Nonlinear stochastic estimation of wall models for LES. Heat Fluid Flow 27(2), 267–278 (2006). https://doi.org/10.1016/j.ijheatfluidflow.2005.10.011
Balaras, E., Benocci, C., Piomelli, U.: Two-layer approximate boundary conditions for large-eddy simulations. AIAA J. 34(6), 1111–1119 (1996). https://doi.org/10.2514/3.13200
Benocci, C., Pinelli, A.: The role of the forcing term in the large-eddy simulation of equilibrium channel flow. In: Rodi, W. (ed.) Engineering Turbulence Modeling and Experiments, pp. 287–296. Elsevier, Amsterdam (1990)
Bose, S.T., Park, G.I.: Wall-modeled large-eddy simulation for complex turbulent flows. Annu. Rev. Fluid Mecha. 50, 535–561 (2018). https://doi.org/10.1146/annurev-fluid-122316-045241
Breuer, M., Peller, N., Rapp, Ch., Manhart, M.: Flow over periodic hills - numerical and experimental study in a wide range of Reynolds numbers. Comput. Fluids 38(2), 433–457 (2009). https://doi.org/10.1016/j.compfluid.2008.05.002
Breuer, M., Kniazev, B., Abel, M.: Development of wall models for LES of separated flows using statistical evaluations. Comput. Fluids 36(5), 817–837 (2007). https://doi.org/10.1016/j.compfluid.2006.09.001
Brunton, S.L., Noack, B.R., Koumoutsakos, P.: Machine learning for fluid mechanics. Annu. Rev. Fluid Mech. 52, 477–508 (2020). https://doi.org/10.1146/annurev-fluid-010719-060214
Cadieux, F., Sadique, J., Yang, X. I., Meneveau, C., Mittal, R.: Wall-modeled large eddy simulation of laminar and turbulent separation bubble flows. In: 46th AIAA Fluid Dynamics Conference. AIAA 2016-3189. (2016). https://doi.org/10.2514/6.2016-3189
Carton de Wiart, C., Hillewaert, K., Bricteux, L., Winckelmans, G.: LES using a discontinuous Galerkin method: isotropic turbulence, channel flow and periodic hill flow. In: Fröhlich, J., Kuerten, H., Geurts, B., Armenio, V. (eds.) Direct and Large-Eddy Simulation IX. ERCOFTAC Series, vol. 20. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-14448-1_13
Carton de Wiart, C., Hillewaert, K., Bricteux, L., Winckelmans, G.: Implicit LES of free and wall bounded turbulent flows based on the discontinuous Galerkin/symmetric interior penalty method. Int. J. Numer. Methods Fluids 78(6), 335–354 (2014)
Chaudhuri, A., Hu, W.: A fast algorithm for computing distance correlation. Comput. Stat. Data Anal. 135, 15–24 (2019). https://doi.org/10.1016/j.csda.2019.01.016
Cheng, C., Li, W., Lozano-Duran, A., Liu, H.: On the structure of streamwise wall-shear stress fluctuations in turbulent channel flows. J. Fluid Mech. 903, A29 (2020). https://doi.org/10.1017/jfm.2020.639
Colella, K.J., Keith, W.L.: Measurements and scaling of wall shear stress fluctuations. Exp. Fluids 34, 253–260 (2003). https://doi.org/10.1007/s00348-002-0552-2
Dawson, S.T.M., McKeon, B.J.: On the shape of resolvent modes in wall-bounded turbulence. J. Fluid Mech. 877, 682–716 (2019). https://doi.org/10.1017/jfm.2019.594
Deardoff, J.W.: A numerical study of three-dimensional turbulence channel flow at large Reynolds numbers. J. Fluid Mech. 41(2), 453–480 (1970). https://doi.org/10.1017/S0022112070000691
Duraisamy, K., Iaccarino, G., Xiao, H.: Turbulence modeling in the age of data. Annu. Rev. Fluid Mech. 51, 357–377 (2019)
Edelmann, D., Mori, T.F., Székely, G.J.: On relationships between the Pearson and the distance correlation coefficients. Stat. Probab. Lett. 169, 108960 (2021). https://doi.org/10.1016/j.spl.2020.108960
Frère, A.: Towards wall-modeled Large-Eddy Simulations of high Reynolds number airfoils using a Discontinuous Galerkin method (Unpublished doctoral dissertation). Université catholique de Louvain (2018)
Frère, A., Carton de Wiart, C., Hillewaert, K., Chatelain, P., Winckelmans, G.: Application of wall-models to discontinuous Galerkin LES channel flow. Phys. Fluids 9(8), 085111 (2017). https://doi.org/10.1063/1.4998977
Frère, A., Hillewaert, K., Chatelain, P., Winckelmans, G.: High Reynolds number airfoil: from wall-resolved to wall-modeled LES. Flow Turbul. Combust. 101, 457–476 (2018)
Fröhlich, J., Mellen, C.P., Rodi, W., Temmerman, L., Leschziner, M.A.: Highly resolved large-eddy simulation of separated flow in a channel with streamwise periodic constrictions. J. Fluid Mech. 526, 19–66 (2005). https://doi.org/10.1017/S0022112004002812
Gloerfelt X., Cinnella P.: Investigation of the flow dynamics in a channel constricted by periodic hills. In: 45th AIAA Fluid Dynamics Conference. Dallas. United States. 2015-2480 (2015). https://doi.org/10.2514/6.2015-2480
Grötzbach, G.: Encyclopedia of Fluid Mechanics, edited by N. P. Cheremisinoff (Gulf, West Orange, NJ), Vol. 6, (1987)
Hoffmann, G., Benocci, C.: Approximate wall boundary conditions for large eddy simulations. In: Benzi, R. (ed.) Advances in Turbulence V. Fluid Mechanics and Its Applications, vol. 24. Springer, Dordrecht (1995). https://doi.org/10.1007/978-94-011-0457-9_40
Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Netw. 4(2), 251–257 (1991). https://doi.org/10.1016/0893-6080(91)90009-T
Hoyas, S., Jimenez, J.: Reynolds number effects on the Reynolds-stress budgets in turbulent channels. Phys. Fluids 20, 101511 (2008). https://doi.org/10.1063/1.3005862
Hillewaert, K.: Development of the Discontinuous Galerkin method for high-resolution, large scale CFD and acoustics in industrial geometries. PhD thesis. Ecole polytechnique de Louvain/iMMC (2013)
Krank, B., Kronbichler, M., Wall, W.A.: A multiscale approach to hybrid RANS/LES wall modeling within a high-order discontinuous Galerkin scheme using function enrichment. Numer. Methods Fluids 90(2), 81–113 (2019). https://doi.org/10.1002/fld.4712
Lozano-Durán, A., Bae, H. J.: Self-critical machine-learning wall-modeled LES for external aerodynamics. arXiv:2012.10005 [physics] (2021)
Marusic, I., Monty, J.P.: Attached Eddy model of wall turbulence. Annu. Rev. Fluid Mech. 51, 49–74 (2019). https://doi.org/10.1146/annurev-fluid-010518-040427
Mason, P.J., Callen, N.S.: On the magnitude of the subgrid-scale eddy coefficient in large eddy simulation of turbulent channel flow. J. Fluid Mech. 162, 439–462 (1986). https://doi.org/10.1017/S0022112086002112
McKeon, B.J.: The engine behind (wall) turbulence: perspectives on scale interactions. J. Fluid Mech. (2017). https://doi.org/10.1017/jfm.2017.115
Nicoud, F., Baggett, J.S., Moin, P., Cabot, W.: Large eddy simulation wall-modeling based on suboptimal control theory and linear stochastic estimation. Phys. Fluids 13, 2968 (2001). https://doi.org/10.1063/1.1389286
Piomelli, U., Moin, P., Ferziger, J.H., Kim, J.: New approximate boundary conditions for large-eddy simulations of wall-bounded flows. Phys. Fluids A Fluid Dyn. 1, 1061 (1989). https://doi.org/10.1063/1.857397
Piomelli, U., Balaras, E.: Wall-layer models for large Eddy simulations. Annu. Rev. Fluid Mech. 34, 349–374 (2002). https://doi.org/10.1146/annurev.fluid.34.082901.144919
Piomelli, U.: Wall-layer models for large-eddy simulations. Prog. Aerosp. Sci. 44(6), 437–446 (2008). https://doi.org/10.1016/j.paerosci.2008.06.001
Rajagopalan, S., Antonia, R.A.: Some properties of the large structure in a fully developed turbulent duct flow. Phys. Fluids 22, 614 (1979). https://doi.org/10.1063/1.862643
Song, S., Eaton, J.: Flow structures of a separating, reattaching, and recovering boundary layer for a large range of Reynolds number. Exp. Fluids 36, 642–653 (2004). https://doi.org/10.1007/s00348-003-0762-2
Spalart, P. R., Jou W. H., Strelets M., Allmaras S. R.: Comments on the feasibility of LES for wings and on a hybrid RANS/LES approach. Advances in DNS/LES: Direct numerical simulation and large eddy simulation, 137-148 (1997)
Schumann, U.: Subgrid-scale model for finite-difference simulations of turbulent flows in plane channels and annuli. J. Comput. Phy. 18(4), 376–404 (1975). https://doi.org/10.1016/0021-9991(75)90093-5
Székely, G.J., Rizzo, M.L., Bakirov, N.K.: Measuring and testing dependence by correlation of distances. Ann. Statistics. 35(6), 2769–2794 (2007). https://doi.org/10.1214/009053607000000505
Temmerman, L., Leschziner, M., Mellen, C., Fröhlich, J.: Investigation of subgrid-scale models and wall-function approximations in large eddy simulation of separated flow in a channel with streamwise periodic constrictions. Int. J. Heat Fluid Flow 24(2), 157–180 (2003). https://doi.org/10.1016/S0142-727X(02)00222-9
Townsend, A.A.: The structure of turbulent boundary layer. Math. Proc. Camb. Philos. Soc. 47, 375–95 (1951)
Townsend, A.A.: The Structure of Turbulent Shear Flows. Cambridge University Press, , CambridgeCambridge (1961)
Townsend, A.A.: The structure of turbulent shear flows, 2nd edn. Cambridge University Press, Cambridge (1976)
Wang, M.: LES with wall models for trailing-edge aeroacoustics. In: Annual Research Briefs. Center Turbulence Research. Stanford University California 355-364 (1999)
Werner, H., Wengle, H.: Large-eddy simulation of turbulent flow around a cube in a plane channel. In: Durst, F., Friedrich, R., Launder, B.E., Schumann, U., Whitelaw, J.H. (eds.) Selected Papers from the 8th Symposium on Turbulent Shear Flows, pp. 155–168. Springer, New York (1993)
Yang, X.I.A., Zafar, S., Wang, J.-X., Xiao, H.: Predictive large-eddy-simulation wall modeling via physics-informed neural networks. Phys. Rev. Fluids 4, 034602 (2019). https://doi.org/10.1103/PhysRevFluids.4.034602
Zhou, Z., He, G., Yang, X.: Wall model based on neural networks for LES of turbulent flows over periodic hills. Phys. Rev. Fluids 6, 054610 (2021). https://doi.org/10.1103/PhysRevFluids.6.054610
Acknowledgements
The present research benefited from computational resources made available on the Tier-1 supercomputer of the Fédération Wallonie-Bruxelles, infrastructure funded by the Walloon Region under the Grant Agreement No. 1117545. The funding of M. Boxho by Safran Tech is gratefully acknowledged.
Funding
Safran Tech and computational resources made available on the Tier-1 supercomputer of the Fédération Wallonie-Bruxelles, infrastructure funded by the Walloon Region under the Grant Agreement No. 1117545.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
Not applicable.
Ethics approval
Not applicable.
Consent to participate
Not applicable.
Consent for publication
Not applicable.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Boxho, M., Rasquin, M., Toulorge, T. et al. Analysis of Space-Time Correlations to Support the Development of Wall-Modeled LES. Flow Turbulence Combust 109, 1081–1109 (2022). https://doi.org/10.1007/s10494-022-00365-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10494-022-00365-3