Symmetry-preserving discretization of turbulent flow.

*(English)*Zbl 1062.76542Summary: We propose to perform turbulent flow simulations in such manner that the difference operators do have the same symmetry properties as the underlying differential operators, i.e., the convective operator is represented by a skew-symmetric coefficient matrix and the diffusive operator is approximated by a symmetric, positive-definite matrix. Mimicking crucial properties of differential operators forms in itself a motivation for discretizing them in a certain manner. We give it a concrete form by noting that a symmetry-preserving discretization of the Navier-Stokes equations is stable on any grid, and conserves the total mass, momentum and kinetic energy (for the latter the physical dissipation is to be turned off, of coarse). Being stable on any grid, the choice of the grid may be based on the required accuracy solely, and the main question becomes: how accurate is a symmetry-preserving discretization? Its accuracy is tested for a turbulent flow in a channel by comparing the results to those of physical experiments and previous numerical studies. The comparison is carried out for a Reynolds number of 5600, which is based on the channel width and the mean bulk velocity (based on the channel half-width and wall shear velocity the Reynolds number becomes 180). The comparison shows that with a fourth-order, symmetry-preserving method a \(64\times 64\times 32\) grid suffices to perform an accurate numerical simulation.

##### Keywords:

Higher-order discretization; Nonuniform grid; Conservation; Stability; Turbulent channel flow; Direct numerical simulation
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\textit{R. W. C. P. Verstappen} and \textit{A. E. P. Veldman}, J. Comput. Phys. 187, No. 1, 343--368 (2003; Zbl 1062.76542)

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##### References:

[1] | Teman, R., Navier – stokes equations and nonlinear functional analysis, (), 13 |

[2] | Manteufel, T.A.; White, A.B., The numerical solution of second-order boundary value problems on nonuniform meshes, Math. comput., 47, 511, (1986) · Zbl 0635.65092 |

[3] | Veldman, A.E.P.; Rinzema, K., Playing with nonuniform grids, J. eng. math., 26, 119, (1991) · Zbl 0747.76073 |

[4] | Morinishi, Y.; Lund, T.S.; Vasilyev, O.V.; Moin, P., Fully conservative higher order finite difference schemes for incompressible flow, J. comp. phys., 143, 90, (1998) · Zbl 0932.76054 |

[5] | Vasilyev, O.V., High order finite difference schemes on non-uniform meshes with good conservation properties, J. comp. phys., 157, 746, (2000) · Zbl 0959.76063 |

[6] | Nicoud, F., Conservative high-order finite-difference schemes for low-Mach number flows, J. comp. phys., 158, 71, (2000) · Zbl 0973.76068 |

[7] | Ducros, F.; Laporte, F.; Soulères, T.; Guinot, V.; Moinat, P.; Caruelle, B., High-order fluxes for conservative skew-symmetric-like schemes in structured meshes: application to compressible flows, J. comp. phys., 161, 114, (2000) · Zbl 0972.76066 |

[8] | A. Twerda, A.E.P. Veldman, S.W. de Leeuw, High order schemes for colocated grids: preliminary results. In: Proceedings of the 5th Annual Conference of the Advanced School for Computing and Imaging, 1999, p. 286 |

[9] | A. Twerda, Advanced computational methods for complex flow simulation, Ph.D. Thesis, Delft University of Technology, 2000 |

[10] | A. Jameson, W. Schmidt, E. Turkel, Numerical Solutions of the Euler Equations by Finite Volume Methods using Runge-Kutta Time Stepping, AIAA Paper 81-1259, 1981 |

[11] | Furihata, D., Finite difference schemes for \( ∂u ∂t=( ∂ ∂x)\^{}\{α\}δGδu\) that inherit energy conservation or dissipation property, J. comp. phys., 156, 181, (1999) · Zbl 0945.65103 |

[12] | Hyman, J.M.; Knapp, R.J.; Scovel, J.C., High order finite volume approximations of differential operators on nonuniform grids, Phys. D, 60, 112, (1992) · Zbl 0790.65089 |

[13] | Castillo, J.E.; Hyman, J.M.; Shaskov, M.J.; Steinberg, S., The sensitivity and accuracy of fourth order finite-difference schems on nonuniform grids in one dimension, Comput. math. appl., 30, 41, (1995) · Zbl 0836.65025 |

[14] | Hyman, J.M.; Shashkov, M., Natural discretizations for the divergence, gradient and curl on logically rectangular grids, Comput. math. appl., 33, 81, (1997) · Zbl 0868.65006 |

[15] | R.W.C.P. Verstappen, R.M. van der Velde, Symmetry-preserving discretization of heat transfer in a complex turbulent flow, submitted · Zbl 1095.76035 |

[16] | Verstappen, R.W.C.P.; Veldman, A.E.P., Direct numerical simulation of turbulence at lower costs, J. eng. math., 32, 143, (1997) · Zbl 0911.76072 |

[17] | Verstappen, R.W.C.P.; Veldman, A.E.P., Spectro-consistent discretization of navier – stokes: a challenge to RANS and LES, J. eng. math., 34, 163, (1998) · Zbl 0917.76059 |

[18] | Harlow, F.H.; Welsh, J.E., Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. fluids, 8, 2182, (1965) · Zbl 1180.76043 |

[19] | Antonopoulos-Domis, M., Large-eddy simulation of a passive scalar in isotropic turbulence, J. fluid mech., 104, 55-79, (1981) · Zbl 0456.76036 |

[20] | Kim, J.; Moin, P.; Moser, R., Turbulence statistics in fully developed channel flow at low Reynolds number, J. fluid mech., 177, 133, (1987) · Zbl 0616.76071 |

[21] | N. Gilbert, L. Kleiser, Turbulence model testing with the aid of direct numerical simulation results, in: Proceedings of the Turbulence Shear Flows 8, Paper 26-1, Munich, 1991 |

[22] | Kuroda, A.; Kasagi, N.; Hirata, M., Direct numerical simulation of turbulent plane couette – poisseuille flows: effect of Mean shear rate on the near-wall turbulence structures, (), 241-257 · Zbl 0832.76034 |

[23] | Kreplin, H.P.; Eckelmann, H., Behavior of the three fluctuating velocity components in the wall region of a turbulent channel flow, Phys. fluids, 22, 1233-1239, (1979) |

[24] | Eckelmann, H., The structure of the viscous sublayer and the adjacent wall region in a turbulent channel flow, J. fluid mech., 65, 439-459, (1974) |

[25] | Dean, R.B., Reynolds number dependence of skin friction and other bulk flow variables in two-dimensional rectangular duct flow, J. fluids eng., 100, 215-223, (1978) |

[26] | Hanratty, T.J.; Chorn, L.G.; Hatziavramidis, D.T., Turbulent fluctuations in the viscous wall region for Newtonian and drag reducing fluids, Phys. fluids, 20, S112, (1977) |

[27] | Finnicum, D.S.; Hanratty, T.J., Turbulent normal velocity fluctuations close to a wall, Phys. fluids, 28, 1654-1658, (1985) |

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