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Towards a parameter-free method for high Reynolds number turbulent flow simulation based on adaptive finite element approximation. (English) Zbl 1423.76240
Summary: This article is a review of our work towards a parameter-free method for simulation of turbulent flow at high Reynolds numbers. In a series of papers we have developed a model for turbulent flow in the form of weak solutions of the Navier-Stokes equations, approximated by an adaptive finite element method, where: (i) viscous dissipation is assumed to be dominated by turbulent dissipation proportional to the residual of the equations, and (ii) skin friction at solid walls is assumed to be negligible compared to inertial effects. The result is a computational model without empirical data, where the only model parameter is the local size of the finite element mesh. Under adaptive refinement of the mesh based on a posteriori error estimation, output quantities of interest in the form of functionals of the finite element solution converge to become independent of the mesh resolution, and thus the resulting method has no adjustable parameters. No ad hoc design of the mesh is needed, instead the mesh is optimized based on solution features, in particular no boundary layer mesh is needed. We connect the computational method to the mathematical concept of a dissipative weak solution of the Euler equations, as a model of high Reynolds number turbulent flow, and we highlight a number of benchmark problems for which the method is validated. The purpose of the article is to present the computational framework in a concise form, to report on recent progress, and to discuss open problems that are subject to ongoing research.

76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
76F65 Direct numerical and large eddy simulation of turbulence
FEniCS; PETSc; Unicorn
Full Text: DOI
[1] Abreu, R. V.; Jansson, N.; Hoffman, J., Adaptive computation of aeroacoustic sources for a rudimentary landing gear, Internat. J. Numer. Methods Fluids, 74, 6, 406-421, (2014)
[2] R.V. de Abreu, N. Jansson, J. Hoffman, Adaptive computation of aeroacoustic sources for a rudimentary landing gear using lighthills analogy, AIAA Paper 2942 (2011) 2011.
[3] R.V. de Abreu, N. Jansson, J. Hoffman, Computation of aeroacoustic sources for a complex nose landing gear geometry using adaptivity, in: Proceedings of the Second Workshop on Benchmark problems for Airframe Noise Computations, BANC-II, Colorado Springs, 2012. · Zbl 1390.76831
[4] J. Hoffman, J. Jansson, R.V. de Abreu, Computation of slat noise sources using adaptive fem and lighthills analogy, in: 19th AIAA/CEAS Aeroacoustics Conference, 2013.
[5] Hoffman, J.; Jansson, J.; Jansson, N.; de Abreu, R. V., Time-resolved adaptive FEM simulation of the dlr-f11 aircraft model at high Reynolds number, (AIAA 2014-0917, Proc. 52nd Aerospace Sciences Meeting, (2014), AIAA SciTech)
[6] Eriksson, K.; Estep, D.; Hansbo, P.; Johnson, C., Introduction to adaptive methods for differential equations, Acta Numer., 4, 1, 105-158, (1995) · Zbl 0829.65122
[7] Oden, J. T.; Prudhomme, S., Goal-oriented error estimation and adaptivity for the finite element method, Comput. Math. Appl., 41, 5-6, 735-756, (2001) · Zbl 0987.65110
[8] Becker, R.; Rannacher, R., An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 2001, 10, 1-102, (2001) · Zbl 1105.65349
[9] Hoffman, J.; Johnson, C., Adaptive finite element methods for incompressible fluid flow, (Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics, (2003), Springer), 97-157 · Zbl 1141.76420
[10] Hoffman, J., On duality-based a posteriori error estimation in various norms and linear functionals for large eddy simulation, SIAM J. Sci. Comput., 26, 1, 178-195, (2004) · Zbl 1077.76041
[11] Hoffman, J.; Johnson, C., A new approach to computational turbulence modeling, Comput. Methods Appl. Mech. Engrg., 195, 23, 2865-2880, (2006) · Zbl 1176.76065
[12] Hoffman, J., Computation of mean drag for bluff body problems using adaptive dns/LES, SIAM J. Sci. Comput., 27, 1, 184-207, (2005) · Zbl 1149.65318
[13] Hoffman, J., Adaptive simulation of the subcritical flow past a sphere, J. Fluid Mech., 568, 77-88, (2006) · Zbl 1177.76157
[14] Hoffman, J., Efficient computation of mean drag for the subcritical flow past a circular cylinder using general Galerkin g2, Internat. J. Numer. Methods Fluids, 59, 11, 1241-1258, (2009) · Zbl 1409.76062
[15] Hoffman, J.; Jansson, J.; Vilela De Abreu, R., Adaptive modeling of turbulent flow with residual based turbulent kinetic energy dissipation, Comput. Methods Appl. Mech. Engrg., 200, 37, 2758-2767, (2011) · Zbl 1230.76025
[16] Sagaut, P., Large eddy simulation for incompressible flows, (2005), Springer
[17] Hoffman, J., Simulation of turbulent flow past bluff bodies on coarse meshes using general Galerkin methods: drag crisis and turbulent Euler solutions, Comput. Mech., 38, 4-5, 390-402, (2006) · Zbl 1188.76225
[18] Hoffman, J.; Jansson, N., A computational study of turbulent flow separation for a circular cylinder using skin friction boundary conditions, (Quality and Reliability of Large-Eddy Simulations II, (2011), Springer), 57-68 · Zbl 1303.76087
[19] Hoffman, J.; Johnson, C., Resolution of dalemberts paradox, J. Math. Fluid Mech., 12, 3, 321-334, (2010) · Zbl 1261.76005
[20] Piomelli, U.; Balaras, E., Wall-layer models for large-eddy simulations, Annu. Rev. Fluid Mech., 34, 1, 349-374, (2002) · Zbl 1006.76041
[21] Smagorinsky, J., General circulation experiments with the primitive equations: I. the basic experiment, Mon. Weather Rev., 91, 3, 99-164, (1963)
[22] VonNeumann, J.; Richtmyer, R. D., A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys., 21, 3, 232-237, (1950), URL http://scitation.aip.org/content/aip/journal/jap/21/3/10.1063/1.1699639 · Zbl 0037.12002
[23] Fureby, C.; Grinstein, F., Monotonically integrated large eddy simulation of free shear flows, AIAA J., 37, 5, 544-556, (1999)
[24] Bazilevs, Y.; Calo, V.; Cottrell, J.; Hughes, T.; Reali, A.; Scovazzi, G., Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows, Comput. Methods Appl. Mech. Engrg., 197, 1, 173-201, (2007) · Zbl 1169.76352
[25] Principe, J.; Codina, R.; Henke, F., The dissipative structure of variational multiscale methods for incompressible flows, Comput. Methods Appl. Mech. Engrg., 199, 13, 791-801, (2010) · Zbl 1406.76034
[26] Meyers, J.; Geurts, B.; Sagaut, P., A computational error-assessment of central finite-volume discretizations in large-eddy simulation using a smagorinsky model, J. Comput. Phys., 227, 1, 156-173, (2007) · Zbl 1280.76012
[27] Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 63, 1, 193-248, (1934) · JFM 60.0726.05
[28] Guermond, J.; Oden, J.; Prudhomme, S., An interpretation of the Navier-Stokes-alpha model as a frame-indifferent Leray regularization, Physica D, 177, 1, 23-30, (2003) · Zbl 1082.35120
[29] C. Fefferman, Official clay prize problem description: existence and smoothness of the Navier-Stokes equation (2000).
[30] Scheffer, V., Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math., 66, 2, 535-552, (1976) · Zbl 0325.35064
[31] Caffarelli, L.; Kohn, R.; Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure. Appl. Math., 35, 6, 771-831, (1982) · Zbl 0509.35067
[32] Guermond, J.-L., On the use of the notion of suitable weak solutions in cfd, Internat. J. Numer. Methods Fluids, 57, 9, 1153-1170, (2008) · Zbl 1140.76365
[33] Duchon, J.; Robert, R., Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations, Nonlinearity, 13, 1, 249, (2000) · Zbl 1009.35062
[34] Onsager, L., Statistical hydrodynamics, Il Nuovo Cimento, 1943-1954, 6, 279-287, (1949)
[35] Eyink, G. L.; Sreenivasan, K. R., Onsager and the theory of hydrodynamic turbulence, Rev. Modern Phys., 78, 1, 87, (2006) · Zbl 1205.01032
[36] Hoffman, J.; Johnson, C., Computational turbulent incompressible flow: applied mathematics: body and soul 4, vol. 4, (2007), Springer · Zbl 1114.76002
[37] Hoffman, J.; Johnson, C., Blow up of incompressible Euler solutions, BIT, 48, 2, 285-307, (2008) · Zbl 1143.76012
[38] Spalart, P. R., Detached-eddy simulation, Annu. Rev. Fluid Mech., 41, 181-202, (2009) · Zbl 1159.76036
[39] Schlichting, H.; Gersten, K.; Gersten, K., Boundary-layer theory, (2000), Springer · Zbl 0940.76003
[40] Moin, P.; Kim, J., Tackling turbulence with supercomputers, Sci. Am., 276, 1, 46-52, (1997)
[41] Prandtl, L., On fluid motions with very small friction, (Verhldg. 3 Int. Math. Kongress, (1904), Wiley), 484-491
[42] Bangerth, W.; Rannacher, R., Adaptive finite element methods for differential equations, (2003), Springer · Zbl 1020.65058
[43] Hoffman, J.; Jansson, J.; Vilela de Abreu, R.; Degirmenci, N. C.; Jansson, N.; Müller, K.; Nazarov, M.; Spühler, J. H., Unicorn: parallel adaptive finite element simulation of turbulent flow and fluid-structure interaction for deforming domains and complex geometry, Comput. & Fluids, 80, 310-319, (2013) · Zbl 1284.76223
[44] The fenics project.
[45] Jansson, N.; Hoffman, J.; Jansson, J., Framework for massively parallel adaptive finite element computational fluid dynamics on tetrahedral meshes, SIAM J. Sci. Comput., 34, 1, C24-C41, (2012) · Zbl 1237.68244
[46] Jansson, N.; Hoffman, J.; Nazarov, M., Adaptive simulation of turbulent flow past a full car model, (2011 International Conference for High Performance Computing, Networking, Storage and Analysis, SC, (2011), IEEE), 1-8
[47] Parmetis — parallel graph partitioning and fill-reducing matrix ordering.
[48] Petsc — portable, extensible toolkit for scientific computation.
[49] Zdravkovich, M., Flow around circular cylinders, vol. 1. fundamentals, J. Fluid Mech., 350, 377-378, (1997) · Zbl 0882.76004
[50] Korotkin, A., The three-dimensional character of a cross flow around a circular cylinder, TsAGI, Uchenye Zap., 4, 5, 26-33, (1973)
[51] Humphreys, J. S., On a circular cylinder in a steady wind at transition Reynolds numbers, J. Fluid Mech., 9, 04, 603-612, (1960) · Zbl 0091.19305
[52] Gölling, B., Experimentelle untersuchungen des laminar-turbulenten überganges der zylindergrenzschichtströmung, (2001), DLR, (Ph.D. thesis) · Zbl 1021.76001
[53] Schewe, G., Reynolds-number effects in flow around more-or-less bluff bodies, J. Wind Eng. Ind. Aerodyn., 89, 14, 1267-1289, (2001)
[54] J. Jansson, J. Hoffman, N. Jansson, Simulation of 3d unsteady incompressible flow past a naca 0012 wing section, CTL Technical Report kth-ctl-4023.
[55] Fidkowski, K. J.; Darmofal, D. L., Review of output-based error estimation and mesh adaptation in computational fluid dynamics, AIAA J., 49, 4, 673-694, (2011)
[56] Wang, Q.; Gao, J.-H., The drag-adjoint field of a circular cylinder wake at Reynolds numbers 20, 100 and 500, J. Fluid Mech., 730, 145-161, (2013) · Zbl 1291.76117
[57] Nazarov, M.; Hoffman, J., On the stability of the dual problem for high Reynolds number flow past a circular cylinder in two dimensions, SIAM J. Sci. Comput., 34, 4, A1905-A1924, (2012) · Zbl 1250.76132
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