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**A new approach to computational turbulence modeling.**
*(English)*
Zbl 1176.76065

Summary: We present a new approach to computational fluid dynamics (CFD) using adaptive stabilized Galerkin finite element methods with duality based a posteriori error control for chosen output quantities of interest. We address the basic question of computability in CFD: For a given flow, what quantity is computable to what tolerance to what cost? We focus on incompressible Newtonian flow with medium to large Reynolds numbers involving both laminar and turbulent flow features. We estimate a posteriori the output of the computed solution with the output based on the exact solution to the Navier-Stokes equations, thus circumventing introducing and modeling Reynolds stresses in averaged Navier-Stokes equations. Our basic tool is a representation formula for the error in the quantity of interest in terms of a space-time integral of the residual of a computed solution multiplied by weights related to derivatives of the solution of an associated dual problem with data connected to the output. We use the error representation formula to derive an a posteriori error estimate combining residuals with computed dual weights, which is used for mesh adaptivity in space-time with the objective of satisfying a given error tolerance with minimal computational effort. We show in a concrete example that outputs such as a mean value in time of drag of a turbulent flow around a bluff body are computable on a PC with a tolerance of a few percent using a few hundred thousand mesh points in space. We refer to our methodology as adaptive DNS/LES, where automatically by adaptivity certain features of the flow are resolved in a direct numerical simulation (DNS), while certain other small scale turbulent features are left unresolved in a large eddy simulation (LES). The stabilization of the Galerkin method giving a weighted least square control of the residual acts as the subgrid model in the LES. The a posteriori error estimate takes into account both the error from discretization and the error from the subgrid model. We pay particular attention to the stability of the dual solution from (i) perturbations replacing the exact convection velocity by a computed velocity, and (ii) computational solution of the dual problem, which are the crucial aspects entering by avoiding using averaged Navier-Stokes equations including Reynolds stresses. A crucial observation is that the contribution from subgrid modeling in the a posteriori error estimation is small, making it possible to simulate aspects of turbulent flow without accurate modeling of Reynolds stresses.

### MSC:

76M10 | Finite element methods applied to problems in fluid mechanics |

76F65 | Direct numerical and large eddy simulation of turbulence |

### Keywords:

adaptive DNS/LES; adaptivity; computability; adaptive finite element method; posteriori error estimate; turbulence; incompressible flow; DNS; LES
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\textit{J. Hoffman} and \textit{C. Johnson}, Comput. Methods Appl. Mech. Eng. 195, No. 23--24, 2865--2880 (2006; Zbl 1176.76065)

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### References:

[1] | Cde-forum, Available from: <http://www.phi.chalmers.se/cdeforum/; Cde-forum, Available from: <http://www.phi.chalmers.se/cdeforum/ |

[2] | Becker, R.; Rannacher, R., A feed-back approach to error control in adaptive finite element methods: Basic analysis and examples, East-West J. Numer. Math., 4, 237-264 (1996) · Zbl 0868.65076 |

[3] | Becker, R.; Rannacher, R., A posteriori error estimation infinite element methods, Acta Numer., 10, 1-103 (2001) |

[4] | S. Buijssen, S. Turek, personal communication, 2004.; S. Buijssen, S. Turek, personal communication, 2004. |

[5] | Eriksson, K.; Estep, D.; Hansbo, P.; Johnson, C., Introduction to adaptive methods for differential equations, Acta Numer., 4, 105-158 (1995) · Zbl 0829.65122 |

[6] | Eriksson, K.; Estep, D.; Johnson, C., Applied Mathematics Body and Soul, vols. I-III (2003), Springer-Verlag Publishing |

[7] | M. Giles, M. Larson, M. Levenstam, E. Süli, Adaptive error control for finite element approximations of the lift and drag coefficients in viscous flow, Technical Report Na-76/06, Oxford University Computing Laboratory, 1997.; M. Giles, M. Larson, M. Levenstam, E. Süli, Adaptive error control for finite element approximations of the lift and drag coefficients in viscous flow, Technical Report Na-76/06, Oxford University Computing Laboratory, 1997. |

[8] | J. Hoffman, Adaptive finite element methods for LES: Computation of the mean drag coefficient in a turbulent flow around a surface mounted cube using adaptive mesh refinement, Courant Mathematics and Computing Laboratory DOE Technical Report 03-010, New York University, 2003.; J. Hoffman, Adaptive finite element methods for LES: Computation of the mean drag coefficient in a turbulent flow around a surface mounted cube using adaptive mesh refinement, Courant Mathematics and Computing Laboratory DOE Technical Report 03-010, New York University, 2003. |

[9] | J. Hoffman, Computation of functionals in 3d incompressible flow for stationary benchmark problems using adaptive finite element methods, Courant Mathematics and Computing Laboratory DOE Technical Report 03-003, New York University, 2003.; J. Hoffman, Computation of functionals in 3d incompressible flow for stationary benchmark problems using adaptive finite element methods, Courant Mathematics and Computing Laboratory DOE Technical Report 03-003, New York University, 2003. |

[10] | J. Hoffman, Computation of mean drag for bluff body problems using adaptive DNS/LES, SIAM J. Sci. Comput., accepted for publication.; J. Hoffman, Computation of mean drag for bluff body problems using adaptive DNS/LES, SIAM J. Sci. Comput., accepted for publication. · Zbl 1149.65318 |

[11] | J. Hoffman, C. Johnson, Stability of the dual Navier-Stokes equations and efficient computation of mean output in turbulent flow using adaptive DNS/LES, Comput. Methods Appl. Mech. Engrg., to appear.; J. Hoffman, C. Johnson, Stability of the dual Navier-Stokes equations and efficient computation of mean output in turbulent flow using adaptive DNS/LES, Comput. Methods Appl. Mech. Engrg., to appear. · Zbl 1115.76037 |

[12] | Hoffman, J., On duality based a posteriori error estimation in various norms and linear functionals for LES, SIAM J. Sci. Comput., 26, 1, 178-195 (2004) · Zbl 1077.76041 |

[13] | Hoffman, J.; Johnson, C., Adaptive finite element methods for incompressible fluid flow, (Earth, T. J.; Deconinck, H., Error Estimation and Solution Adaptive Discretization in Computational Fluid Dynamics, Lecture Notes in Computational Science and Engineering (2002), Springer-Verlag Publishing: Springer-Verlag Publishing Heidelberg) · Zbl 1141.76420 |

[14] | Hoffman, J.; Johnson, C., Applied Mathematics Body and Soul, Fluid Dynamics A: Incompressible Flow, vol. 5 (2005), Springer-Verlag Publishing |

[15] | Hoffman, J.; Johnson, C., Computability and Adaptivity in CFD, (Stein, E.; de Borst, R.; Hughes, T. J.R., Encyclopedia of Computational Mechanics (2004), John Wiley and Sons) · Zbl 1190.76001 |

[16] | J. Hoffman, C. Johnson, On the uniqueness of weak solutions of Navier-Stokes equations: Remarks on a clay mathematics institute prize problem, Chalmers Finite Element Center Preprint 2004-07, Chalmers University of Technology, 2004.; J. Hoffman, C. Johnson, On the uniqueness of weak solutions of Navier-Stokes equations: Remarks on a clay mathematics institute prize problem, Chalmers Finite Element Center Preprint 2004-07, Chalmers University of Technology, 2004. |

[17] | Johnson, C., Adaptive finite element methods for conservation laws, (Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Springer Lecture Notes in Mathematics (1998), Springer-Verlag), 269-323 · Zbl 0927.65121 |

[18] | Krajnović, S.; Davidson, L., Large-eddy simulation of the flow around a bluff body, AIAA J., 40, 927-936 (2002) |

[19] | R. Rannacher, Finite element methods for the incompressible Navier-Stokes equations, Preprint Institute of Applied Mathematics, University of Heidelberg, 1999.; R. Rannacher, Finite element methods for the incompressible Navier-Stokes equations, Preprint Institute of Applied Mathematics, University of Heidelberg, 1999. · Zbl 1107.76353 |

[20] | Sagaut, P., Large Eddy Simulation for Incompressible Flows (2001), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0964.76002 |

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