Penalized Wall Function Method for Turbulent Flow Modeling

Natalia S. Zhdanova; Oleg V. Vasilyev

doi:10.14529/jsfi220406

Authors

Natalia S. Zhdanova Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russian Federation https://orcid.org/0000-0001-8712-2817
Oleg V. Vasilyev Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russian Federation https://orcid.org/0000-0003-0294-6097

DOI:

https://doi.org/10.14529/jsfi220406

Keywords:

turbulence modeling, wall function, wall-bounded compressible turbulent flow, volume penalization

Abstract

A novel penalized wall function method for simulations of wall-bounded compressible turbulent flows is proposed. The new approach is based on the Reynolds-averaged Navier–Stokes (RANS) equations to model the outer region of the turbulent boundary layer, while the inner part is approximated by the equilibrium wall function model. The differential formulation to match the external and the wall function solutions is reformulated in a form of the generalized characteristic-based volume penalization method to model the transfer of the shear stress from the outer region of the boundary layer to the wall and to impose the wall-stress boundary conditions on the RANS solution. The exchange location is specified implicitly through a localized source term in the boundary layer equation, written as a function of the normalized distance from the wall. The wall-stress condition is determined by solving an auxiliary equation for the wall-stress, ensuring the correct matching of the RANS and the wall function solutions at the exchange layer. The proposed method noticeably reduces the near-wall mesh resolution requirements without significant modification of the RANS solver and removes the ill-defined explicit matching procedure, commonly used by traditional wall function-based methods. The penalized wall function approach is implemented using the vertex-centered control volume method on unstructured computational grids. The effectiveness of the developed penalized wall function method is demonstrated for twodimensional bump-in-channel flow for the Spalart–Allmaras turbulence model.

References

NASA Langley research center turbulence modeling resource, https://turbmodels.larc.nasa.gov, accessed: 2022-11-07

Abalakin, I.V., Vasilyev, O.V., Zhdanova, N.S., Kozubskaya, T.K.: Characteristic based volume penalization method for numerical simulation of compressible flows on unstructured meshes. Comput. Math. and Math. Phys. 61(8), 1315–1329 (2021). https://doi.org/10.1134/S0965542521080029

Bakhvalov, P., Abalakin, I., Kozubskaya, T.: Edge-based reconstruction schemes for unstructured tetrahedral meshes. Int. J. Numer. Meth. Fluids 81(6), 331–356 (2016). https://doi.org/10.1002/fld.4187

Bardina, J., Huang, P., Coakley, T., et al.: Turbulence modeling validation. In: 28th Fluid dynamics conference. p. 2121 (1997)

Beaugendre, H., Morency, F.: Penalization of the Spalart–Allmaras turbulence model without and with a wall function: Methodology for a vortex in cell scheme. Computers & Fluids 170, 313–323 (Jul 2018), https://hal.inria.fr/hal-01963687

Beaugendre, H., Morency, F.: Penalization of the Spalart–Allmaras turbulence model without and with a wall function: Methodology for a vortex in cell scheme. Computers & Fluids 170, 313–323 (2018). https://doi.org/10.1016/j.compfluid.2018.05.012

Bodart, J., Larsson, J.: Wall-modeled large eddy simulation in complex geometries with application to high-lift devices. Annual Research Briefs, Center for Turbulence Research, Stanford University pp. 37–48 (2011)

Brown-Dymkoski, E., Kasimov, N., Vasilyev, O.V.: A characteristic based volume penalization method for general evolution problems applied to compressible viscous flows. J. Comp. Phys. 262, 344–357 (2014). https://doi.org/10.1063/1.4825260

Brown-Dymkoski, E., Kasimov, N., Vasilyev, O.V.: A characteristic-based volume penalization method for arbitrary mach flows around solid obstacles. In: Fröhlich, J., Kuerten, H., Geurts, B.J., Armenio, V. (eds.) Direct and Large-Eddy Simulation IX. pp. 109–115. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-14448-1_15

Cai, S.G., Degrigny, J., Boussuge, J.F., Sagaut, P.: Coupling of turbulence wall models and immersed boundaries on cartesian grids. J. Comp. Phys. 429, 109995 (2021). https://doi.org/10.1016/j.jcp.2020.109995

Craft, T., Gant, S., Gerasimov, A., et al.: Development and application of wall-function treatments for turbulent forced and mixed convection flows. Fluid Dyn. Res. 38(2), 127–144 (2006). https://doi.org/10.1016/j.fluiddyn.2004.11.002

Dhamankar, N., Blaisdell, G., Lyrintzis, A.: Implementation of a wall-modeled sharp immersed boundary method in a high-order large eddy simulation tool for jet aeroacoustics. In: 54th AIAA Aerospace Sciences Meeting (01 2016). https://doi.org/10.2514/6.2016-0257

Duben, A.P., Abalakin, I.V., Tsvetkova, V.O.: On boundary conditions on solid walls in viscous flow problems. Math. Models and Comput. Simul. 13(4), 591–603 (2021). https://doi.org/10.1134/S2070048221040128

Durbin, P.A., Reif, B.A.P.: Statistical Theory and Modeling for Turbulent Flows. Wiley (2001)

Froehlich, J., von Terzi, D.: Hybrid LES/RANS methods for the simulation of turbulent flows. Progress in Aerospace Sciences 44(5), 349–377 (2008). https://doi.org/10.1016/j.paerosci.2008.05.001

Gatski, T.B., Hussaini, M.Y., Lumley, J.L.: Simulation and Modeling of Turbulent Flows. Oxford (1996)

Gorobets, A., Bakhvalov, P.: Heterogeneous CPU+GPU parallelization for high-accuracy scale-resolving simulations of compressible turbulent flows on hybrid supercomputers. Comput. Phys. Commun. 271, 108231 (2022). https://doi.org/10.1016/j.cpc.2021.108231

Gorobets, A., Duben, A.: Technology for supercomputer simulation of turbulent flows in the good new days of exascale computing. Supercomput. Front. Innov. 8(4), 4–10 (Feb 2021). https://doi.org/10.14529/jsfi210401

Kasimov, N., Dymkoski, E., De Stefano, G., Vasilyev, O.V.: Galilean-invariant characteristic-based volume penalization method for supersonic flows with moving boundaries. Fluids 6(8) (2021). https://doi.org/10.3390/fluids6080293

Kawai, S., Larsson, J.: Wall-modeling in large eddy simulation: length scales, grid resolution, and accuracy. Phys. Fluids. 24(1), 015105 (2012). https://doi.org/10.1063/1.3678331

Kawai, S., Larsson, J.: Dynamic non-equilibrium wall-modeling for large eddy simulation at high Reynolds numbers. Phys. Fluids. 25(1), 015105 (2013). https://doi.org/10.1063/1.4775363

Liu, Q., Vasilyev, O.V.: Hybrid adaptive wavelet collocation – Brinkman penalization method for unsteady RANS simulations of compressible flow around bluff bodies. In: 36th AIAA Fluid Dynamics Conference and Exhibit, San Francisco, California, USA, June 5–8, 2006 (2006). https://doi.org/10.2514/6.2006-3206

Liu, Q., Vasilyev, O.V.: A Brinkman penalization method for compressible flows in complex geometries. J. Comp. Phys. 227(2), 946–966 (2007). https://doi.org/10.1016/j.jcp.2007.07.037

Moin, P., Mahesh, K.: Direct numerical simulation: A tool in turbulence research. Annual Rev. Fluid Mech. 30, 539–578 (1998). https://doi.org/10.1146/annurev.fluid.30.1.539

Nichols, R.H., Nelson, C.C.: Wall function boundary conditions including heat transfer and compressibility. AIAA Journal 42(6), 1107–1114 (2004). https://doi.org/10.2514/1.3539

Park, G.I., Moin, P.: An improved dynamic non-equilibrium wall-model for large eddy simulation. Phys. Fluids. 26(1), 37–48 (2014). https://doi.org/10.1063/1.4861069

Patankar, S.V., Spalding, D.B.: Heat and Mass Transfer in Boundary Layers. Morgan-Grampia (1968)

Reichardt, H.: Vollständige darstellung der turbulenten geschwindigkeitsverteilung in glatten leitungend. Zeitschrift für Angewandte Mathematik und Mechanik 31(7), 208–219 (1951)

Rumsey, C., Gatski, T., Sellers, W., et al.: Summary of the 2004 CFD validation workshop on synthetic jets and turbulent separation control. In: 2nd AIAA Flow Control Conference, Portland, Oregon, June 28 – July 1, 2004. p. 2217 (2004). https://doi.org/10.2514/6.2004-2217

Shur, M.L., Spalart, P.R., Strelets, M.K., Travin, A.K.: A hybrid RANS-LES approach with delayed-DES and wall-modelled LES capabilities. Int. J. Heat Fluid Flow 29(6), 1638–1649 (2008). https://doi.org/10.1016/j.ijheatfluidflow.2008.07.001

Spalart, P.R., Allmaras, S.R.: A one equation turbulence model for aerodinamic flows. AIAA journal 94 (1992). https://doi.org/10.2514/6.1992-439

Spalart, P.R., Deck, S., Shur, M.L., et al.: A new version of detached-eddy simulation, resistant to ambiguous grid densities. Theoretical and computational fluid dynamics 20(3), 181–195 (2006). https://doi.org/10.1007/s00162-006-0015-0

Spalart, P.R., Jou, W.H., Strelets, M., et al.: Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach. In: First AFOSR international conference on DNS/LES, Ruston, Louisiana. vol. 1, pp. 4–8. Greyden Press, Columbus, OH (1997)

van der Vorst, H.A.: BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13(2), 631–644 (1992). https://doi.org/10.1137/0913035

Wilcox, D.C.: Formulation of the k−w turbulence model revisited. AIAA Journal 46(11), 2823–2838 (2008). https://doi.org/10.2514/1.36541

Xiao, H., Jenny, P.: A consistent dual-mesh framework for hybrid LES/RANS modeling. J. Comp. Phys. 231(4), 1848–1865 (feb 20 2012). https://doi.org/10.1016/j.jcp.2011.11.009

Zhdanova, N.S., Abalakin, I.V., Vasilyev, O.V.: Generalized Brinkman volume penalization method for compressible flows around moving obstacles. Math. Models. and Comput. Simul. 14(5), 716–726 (2022). https://doi.org/10.1134/S2070048222050