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A New Explicit Algebraic Wall Model for LES of Turbulent Flows Under Adverse Pressure Gradient

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Abstract

A new explicit algebraic wall law for the Large Eddy Simulation of flows with adverse pressure gradient is proposed. This new wall law, referred as adverse pressure gradient power law (APGPL), is developed starting from the power-law of Werner and Wengle (Turbulent Shear Flows, vol 8, Springer, New York, pp 155–168, 1993) in order to mimic an implicit non-equilibrium log-law based on Afzal’s law (Afzal, IUTAM Symposium on Asymptotic Methods for Turbulent Shear Flows at High Reynolds Numbers, Kluwer Academic Publishers, Bochum, pp 95–118, 1996). No iterative method is needed for the evaluation of the wall shear stress from the APGPL contrary to the majority of models available in the literature. The APGPL model relies on the definition of three modes: the equilibrium power-law is used in regions of no or favourable pressure gradient, the APGPL is used in regions of adverse pressure gradient, and no wall model is used in separated flow regions. This model is assessed via Large Eddy Simulations of flows involving adverse pressure gradient and boundary layer separation using the Lattice Boltzmann Method on uniform nested grids. The flow around a clean and iced NACA23012 airfoil at Reynolds number \(Re = 1.88 \times 10^6\) and the flow over the LAGOON landing gear at \(Re = 1.59 \times 10^6\) are considered. Results are found in good agreement with those obtained by the non-equilibrium log-law and experimental and numerical data available in the literature.

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References

  • Afzal, N.: Wake layer in a turbulent boundary layer with pressure gradient: a new approach. In: Gersten, K. (eds.) IUTAM Symposium on Asymptotic Methods for Turbulent Shear Flows at High Reynolds Numbers, pp. 95–118. Kluwer Academic Publishers (1996)

  • Afzal, N.: Power law and log law velocity profiles in turbulent boundary-layer flow: equivalent relations at large Reynolds numbers. Acta Mech. 151(3–4), 195–216 (2001)

    MATH  Google Scholar 

  • Allmaras, S.R., Johnson, F.T., Spalart, P.R.: Modifications and clarifications for the implementation of the Spalart–Allmaras turbulence model. In: Seventh International Conference on Computational Fluid Dynamics (ICCFD7), pp. 1–11 (2012)

  • Barenblatt, G., Chorin, A., Prostokishin, V.: Scaling laws for fully developed turbulent flow in pipes. Appl. Mech. Rev. 50(7), 413–429 (1997)

    MATH  Google Scholar 

  • Berger, M., Aftosmis, M.: Progress towards a Cartesian cut-cell method for viscous compressible flow. Presented at 50th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Jan 9–12, 2012, Nashville, Tennessee, USA, AIAA Paper 2012-1301 (2012)  

  • Bernardini, M., Modesti, D., Pirozzoli, S.: On the suitability of the immersed boundary method for the simulation of high-Reynolds-number separated turbulent flows. Comput. Fluids 130, 84–93 (2016)

    MathSciNet  MATH  Google Scholar 

  • Bodart, J., Larsson, J., Moin, P.: Large eddy simulation of high-lift devices. Presented at 21st AIAA Computational Fluid Dynamics Conference, June 24–27, 2013, San Diego, CA, AIAA Paper 2013-2724 (2013)

  • Bose, S.T., Park, G.I.: Wall-modeled large-eddy simulation for complex turbulent flows. Annu. Rev. Fluid Mech. 50, 535–561 (2018)

    MathSciNet  MATH  Google Scholar 

  • Breuer, M., Kniazev, B., Abel, M.: Development of wall models for LES of separated flows. In: Lamballais, E., Friedrich, R., Geurts, B.J., Métais, O. (eds.) Direct and Large-Eddy Simulation VI, pp. 373–380. Springer, Dordrecht (2006)

    Google Scholar 

  • 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)

    MATH  Google Scholar 

  • Broeren, A.P., Addy, H.E., Lee, S., Monastero, M.C.: Validation of 3-D ice accretion measurement methodology for experimental aerodynamic simulation. Presented at 6th AIAA Atmospheric and Space Environments Conference, June 16–20, 2014, Atlanta, Georgia, AIAA paper 2014-2614 (2014)

  • Buschmann, M.H., Gad-el Hak, M.: Recent developments in scaling of wall-bounded flows. Prog. Aerosp. Sci. 42(5), 419–467 (2006)

    Google Scholar 

  • Capizzano, F.: Turbulent wall model for immersed boundary methods. AIAA J. 49(11), 2367–2381 (2011)

    Google Scholar 

  • Castro-Orgaz, O., Dey, S.: Power-law velocity profile in turbulent boundary layers: an integral Reynolds-number dependent solution. Acta Geophys. 59(5), 993–1012 (2011)

    Google Scholar 

  • Catchirayer, M., Boussuge, J.F., Sagaut, P., Montagnac, M., Papadogiannis, D., Garnaud, X.: Extended integral wall-model for large-eddy simulations of compressible wall-bounded turbulent flows. Phys. Fluids 30(6), 065106 (2018)

    Google Scholar 

  • Chang, P.H., Liao, C.C., Hsu, H.W., Liu, S.H., Lin, C.A.: Simulations of laminar and turbulent flows over periodic hills with immersed boundary method. Comput. Fluids 92, 233–243 (2014)

    MathSciNet  MATH  Google Scholar 

  • Chen, S., Doolen, G.D.: Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30(1), 329–364 (1998)

    MathSciNet  MATH  Google Scholar 

  • Cheng, W., Samtaney, R.: Power-law versus log-law in wall-bounded turbulence: a large-eddy simulation perspective. Phys. Fluids 26(1), 011703 (2014)

    Google Scholar 

  • Coleman, G., Rumsey, C., Spalart, P.: Numerical study of turbulent separation bubbles with varying pressure gradient and Reynolds number. J. Fluid Mech. 847, 28–70 (2018)

    MathSciNet  MATH  Google Scholar 

  • Drela, M., Youngren, H.: XFOIL 6.94 User Guide (2001)

  • Duprat, C., Balarac, G., Métais, O., Congedo, P.M., Brugière, O.: A wall-layer model for large-eddy simulations of turbulent flows with/out pressure gradient. Phys. Fluids 23(1), 015101 (2011)

    Google Scholar 

  • Gungor, A., Maciel, Y., Simens, M., Soria, J.: Scaling and statistics of large-defect adverse pressure gradient turbulent boundary layers. Int. J. Heat Fluid Flow 59, 109–124 (2016)

    Google Scholar 

  • Hou, Y., Angland, D., Zhang, X.: A comparison of wall functions for bluff body aeroacoustic simulations. Presented at 22nd AIAA/CEAS Aeroacoustics Conference, 30 May–1 June, 2016, Lyon, France, AIAA Paper 2016-2771 (2016)

  • http://elsa.onera.fr/. Accessed 02 Oct 2019

  • Iaccarino, G., Verzicco, R.: Immersed boundary technique for turbulent flow simulations. Appl. Mech. Rev. 56(3), 331–347 (2003)

    Google Scholar 

  • Jacob, J., Malaspinas, O., Sagaut, P.: A new hybrid recursive regularised Bhatnagar–Gross–Krook collision model for Lattice Boltzmann method-based large eddy simulation. J. Turbul., 1–26 (2018)

  • Kalitzin, G., Iaccarino, G.: Turbulence modeling in an immersed-boundary RANS method. CTR Annual Research Briefs, pp. 415–426 (2002)

  • König, B., Fares, E., Broeren, A.P.: Lattice-Boltzmann analysis of three-dimensional ice shapes on a NACA 23012 airfoil. In: SAE Technical Paper 2015-01-2084 (2015). https://doi.org/10.4271/2015-01-2084.

  • König, B., Singh, D., Fares, E.: Lattice Boltzmann high-lift simulations-a step beyond classical CFD. Presented at 31st Congress of the International Council of the Aeronautical Sciences, September 09–14, 2018, Belo Horizonte, Brazil (2018)

  • Krüger, T., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G., Viggen, E.M.: The Lattice Boltzmann Method. Springer, New York (2017)

    MATH  Google Scholar 

  • Larsson, J., Kawai, S., Bodart, J., Bermejo-Moreno, I.: Large eddy simulation with modeled wall-stress: recent progress and future directions. Mech. Eng. Rev. 3(1), 15–00418 (2016)

    Google Scholar 

  • Lehmkuhl, O., Park, G., Moin, P.: LES of flow over the NASA Common Research Model with near-wall modeling. In: Center for Turbulence Research-Proceedings of the Summer Program, pp. 335–341 (2016)

  • Leveque, E., Touil, H., Malik, S., Ricot, D., Sengissen, A.: Wall-modeled large-eddy simulation of the flow past a rod-airfoil tandem by the Lattice Boltzmann method. Int. J. Numer. Methods Heat Fluid Flow 28(5), 1096–1116 (2018)

    Google Scholar 

  • Lucas, J.M., Cadot, O., Herbert, V., Parpais, S., Délery, J.: A numerical investigation of the asymmetric wake mode of a squareback Ahmed body-effect of a base cavity. J. Fluid Mech. 831, 675–697 (2017)

    MathSciNet  MATH  Google Scholar 

  • Maciel, Y., Wei, T., Gungor, A.G., Simens, M.P.: Outer scales and parameters of adverse-pressure-gradient turbulent boundary layers. J. Fluid Mech. 844, 5–35 (2018)

    MathSciNet  MATH  Google Scholar 

  • Malaspinas, O., Sagaut, P.: Advanced large-eddy simulation for lattice boltzmann methods: the approximate deconvolution model. Phys. Fluids 23(10), 105103 (2011)

    MATH  Google Scholar 

  • Malaspinas, O., Sagaut, P.: Consistent subgrid scale modelling for lattice boltzmann methods. J. Fluid Mech. 700, 514–542 (2012)

    MathSciNet  MATH  Google Scholar 

  • Malaspinas, O., Sagaut, P.: Wall model for large-eddy simulation based on the lattice Boltzmann method. J. Comput. Phys. 275, 25–40 (2014)

    MathSciNet  MATH  Google Scholar 

  • Manhart, M., Peller, N., Brun, C.: Near-wall scaling for turbulent boundary layers with adverse pressure gradient. Theor. Comput. Fluid Dyn. 22(3), 243–260 (2008)

    MATH  Google Scholar 

  • Manoha, E., Bulté, J., Caruelle, B.: LAGOON: an experimental database for the validation of CFD/CAA methods for landing gear noise prediction. Presented at 14th AIAA/CEAS Aeroacoustics Conference (29th AIAA Aeroacoustics Conference), May 05–07, 2008, Vancouver, British Columbia, Canada, AIAA Paper 2008-2816 (2008)

  • Manoha, E., Bulté, J., Ciobaca, V., Caruelle, B.: LAGOON: further analysis of aerodynamic experiments and early aeroacoustics results. Presented at 15th AIAA/CEAS Aeroacoustics Conference (30th AIAA Aeroacoustics Conference), May 11–13, 2009, Miami, Florida, AIAA Paper 2009-3277 (2009)

  • Marié, S., Ricot, D., Sagaut, P.: Comparison between lattice Boltzmann method and Navier–Stokes high order schemes for computational aeroacoustics. J. Comput. Phys. 228(4), 1056–1070 (2009)

    MathSciNet  MATH  Google Scholar 

  • Marsden, O., Bogey, C., Bailly, C.: Direct noise computation of the turbulent flow around a zero-incidence airfoil. AIAA J. 46(4), 874–883 (2008)

    Google Scholar 

  • Marusic, I., McKeon, B., Monkewitz, P.A., Nagib, H., Smits, A., Sreenivasan, K.: Wall-bounded turbulent flows at high Reynolds numbers: recent advances and key issues. Phys. Fluids 22(6), 065103 (2010)

    MATH  Google Scholar 

  • Marusic, I., Monty, J.P., Hultmark, M., Smits, A.J.: On the logarithmic region in wall turbulence. J. Fluid Mech. 716 (2013)

  • Mellor, G.L.: The effects of pressure gradients on turbulent flow near a smooth wall. J. Fluid Mech. 24(2), 255–274 (1966)

    Google Scholar 

  • Monfort, D., Benhamadouche, S., Sagaut, P.: Meshless approach for wall treatment in large-eddy simulation. Comput. Methods Appl. Mech. Eng. 199(13–16), 881–889 (2010)

    MathSciNet  MATH  Google Scholar 

  • Murakami, S., Mochida, A., Hibi, K.: Three-dimensional numerical simulation of air flow around a cubic model by means of large eddy simulation. J. Wind Eng. Ind. Aerodyn. 25(3), 291–305 (1987)

    Google Scholar 

  • Park, G.I.: Wall-modeled large-eddy simulation of a high reynolds number separating and reattaching flow. AIAA J. 55(11), 3709–3721 (2017)

    Google Scholar 

  • Patel, V., Sotiropoulos, F.: Longitudinal curvature effects in turbulent boundary layers. Prog. Aerosp. Sci. 33(1–2), 1–70 (1997)

    Google Scholar 

  • Piomelli, U.: Wall-layer models for large-eddy simulations. Prog. Aerosp. Sci. 44(6), 437–446 (2008)

    Google Scholar 

  • Piomelli, U., Balaras, E.: Wall-layer models for large-eddy simulations. Annu. Rev. Fluid Mech. 34(1), 349–374 (2002)

    MathSciNet  MATH  Google Scholar 

  • Roman, F., Armenio, V., Fröhlich, J.: A simple wall-layer model for large eddy simulation with immersed boundary method. Phys. Fluids 21(10), 101701 (2009)

    MATH  Google Scholar 

  • Sagaut, P.: Large Eddy simulation for incompressible flows: an introduction. Springer, New York (2006)

    MATH  Google Scholar 

  • Schlichting, H., Gersten, K., Krause, E., Oertel, H.: Boundary-layer theory, 7th edn. Springer, New York (1955)

    Google Scholar 

  • Sengissen, A., Giret, J.C., Coreixas, C., Boussuge, J.F.: Simulations of LAGOON landing-gear noise using Lattice Boltzmann solver. Presented at 21st AIAA/CEAS Aeroacoustics Conference, June 22–26, 2015, Dallas, TX, AIAA Paper 2015-2993 (2015)

  • Shih, T.H., Povinelli, L.A., Liu, N.S., Potapczuk, M.G., Lumley, J.: A generalized wall function. Tech. rep, NASA TM-1999-209398 (1999)

  • Simpson, R.L.: A model for the backflow mean velocity profile. AIAA J. 21(1), 142–143 (1983)

    Google Scholar 

  • Skote, M., Henningson, D.S.: Direct numerical simulation of a separated turbulent boundary layer. J. Fluid Mech. 471, 107–136 (2002)

    MATH  Google Scholar 

  • Spalding, D.: A single formula for the ”law of the wall”. J. Appl. Mech. 28(3), 455–458 (1961)

    MATH  Google Scholar 

  • Tamaki, Y., Harada, M., Imamura, T.: Near-wall modification of Spalart-Allmaras turbulence model for immersed boundary method. AIAA J. 55(9), 3027–3039 (2017)

    Google Scholar 

  • Temmerman, L., Leschziner, M.A., Mellen, C.P., Fröhlich, J.: Investigation of wall-function approximations and subgrid-scale models in large eddy simulation of separated flow in a channel with streamwise periodic constrictions. Int. J. Heat Fluid Flow 24(2), 157–180 (2003)

    Google Scholar 

  • Tessicini, F., Iaccarino, G., Fatica, M., Wang, M., Verzicco, R.: Wall modeling for large-eddy simulation using an immersed boundary method. Annual Research Briefs, pp. 181–187. Stanford University Center for Turbulence Research, Stanford (2002)

    Google Scholar 

  • Vreman, A.: An eddy-viscosity subgrid-scale model for turbulent shear flow: algebraic theory and applications. Phys. Fluids 16(10), 3670–3681 (2004)

    MATH  Google Scholar 

  • Wang, M., Moin, P.: Dynamic wall modeling for large-eddy simulation of complex turbulent flows. Phys. Fluids 14(7), 2043–2051 (2002)

    MathSciNet  MATH  Google Scholar 

  • Werner, H., Wengle, H.: Large-eddy simulation of turbulent flow over and around a cube in a plate channel. In: Durst, F., Friedrich, R., Launder, B.E., Schmidt, F.W., Schumann, U., Whitelaw, J.H. (eds.) Turbulent Shear Flows 8, pp 155–168. Springer, Berlin, Heidelberg (1993)

    MATH  Google Scholar 

  • Wilhelm, S., Jacob, J., Sagaut, P.: An explicit power-law-based wall model for lattice Boltzmann method-Reynolds-averaged numerical simulations of the flow around airfoils. Phys. Fluids 30(6), 065111 (2018)

    Google Scholar 

  • Yang, X., Sadique, J., Mittal, R., Meneveau, C.: Integral wall model for large eddy simulations of wall-bounded turbulent flows. Phys. Fluids 27(2), 025112 (2015)

    Google Scholar 

  • Zagarola, M., Perry, A., Smits, A.: Log laws or power laws: the scaling in the overlap region. Phys. Fluids 9(7), 2094–2100 (1997)

    MathSciNet  MATH  Google Scholar 

  • Zhang, C., Sanjose, M., Moreau, S.: Improvement of the near wall treatment in large eddy simulation for aeroacoustic applications. Presented at 2018 AIAA/CEAS Aeroacoustics Conference, June 25–29, 2018, Atlanta, Georgia, AIAA Paper 2018-3795 (2018)

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Acknowledgements

The authors acknowledge C.Gacherieu from Airbus for the eslA reference data. The authors would like to thank J.Basirico for his help on this study.

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  1. Aix Marseille Univ, CNRS, Centrale Marseille, M2P2, Marseille, France

    Sylvia Wilhelm, Jerome Jacob & Pierre Sagaut

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  1. Sylvia Wilhelm
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Correspondence to Sylvia Wilhelm.

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Conflicts of interest

P. Sagaut has received research grants from Airbus, Renault, Safran and CS. Other authors declare no conflict of interest.

Funding

This work was supported by the project OMEGA3 ”Outil de ModElisation de nouvelle Génération pour l’Aérodynamique Appliquée à l’Aéronautique“ (No. 2018-16), with the financial support of DGAC. This work was performed using HPC resources from GENCI-TGCC/CINES (Grant 2018-A0052A07679).

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All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by S. Wilhelm, J. Jacob and P. Sagaut. The first draft of the manuscript was written by S. Wilhelm and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Wilhelm, S., Jacob, J. & Sagaut, P. A New Explicit Algebraic Wall Model for LES of Turbulent Flows Under Adverse Pressure Gradient. Flow Turbulence Combust 106, 1–35 (2021). https://doi.org/10.1007/s10494-020-00181-7

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