Summary
The Large-Eddy simulation (LES) method can be used in order to break the multi-scale complexity within turbulent flow simulations, since not all turbulent length scales have to be resolved, but will be given by an appropriate subgrid model. Beside this filtering in space, a filtering in time allows for larger time steps as well and gives rise to implicit methods, where an algebraic system of equations has to be solved. Multigrid as a numerical multi-scale approach matches LES quite well with that respect and will be applied. It is essential to control the numerical error introduced by the discretisation and numerical solver in order to minimize the influence on the turbulent solution and hence, being able to identify the model error of the subgrid model. Two different stabilization methods, that are used within the collocated Finite Volume dicretisation for unstructured grids, are investigated with respect to their mass conservation error. The obtained solutions will be compared with benchmark solutions found in literature. The used subgrid model takes advantage of mesh dependent parameters. A practical solution within the multigrid procedure is to derive the model parameter on the finest grid level and inject it successively to the coarser grid levels. By this strategy good convergence rates result.
This work has been supported by the German Research Foundation (DFG) through SFB 359 (Project A5) at the University of Heidelberg.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Alexander. Diagonally implicit Runge-Kutta methods for stiff o.d.e.’s. SIAM Journal on Numerical Analysis, 14:1006–1021, 1977.
B.J. Boersma, M.N. Kooper, F.T.M. Nieuwstadt, and P. Wesseling. Local grid refinement in large-eddy simulations. Journal of Engineering Mathematics, 32:161–175, 1997.
F. Brezzi. On the existence uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO Anal Numer, 8:129–151, 1974.
M. Germano. Turbulence: the filtering approach. Journal of Fluid Mechanics, 238:325–336, 1992.
U. Ghia, K.N. Ghia, and C.T. Shin. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. Journal of Compu-tational Physics, 48:387–411, 1982.
J.O. Hinze. Turbulence. McGraw Hill, 1975.
Volker John. Large eddy simulation of turbulent incompressible flows, analytical and numerical results for a class of LES models. Habilitationsschrift, 2002.
S.M.H. Karimian and G. Schneider. Pressure-based control-volume finite element method for flow at all speeds. AIAA Journal, 33:1611–1618, 1995.
Marcel Lesieur, Chantal Staquet, Pascal Le Roy, and Pierre Comte. The mixing layer and its coherence examined from the point of view of two-dimensional turbulence. Journal of Fluid Mechanics, 192:511–534, 1988.
D.K. Lilly. A proposed modification of the Germano subgrid-scale closure method. Physics of Fluids A, 4:633–634, 1992.
Robert D. Moser and Michael M. Rogers. The three-dimensional evolution of a plane mixing layer: pairing and transition to turbulence. Journal of Fluid Mechanics, 247:275–320, 1993.
Sandra Nägele. Mehrgitterverfahren für die inkompressiblen Navier-Stokes Gleichungen im laminaren und turbulenten Regime unter Berücucksichtigung verschiedener Stabilisierungsmethoden. PhD thesis, Universität Heidelberg, 2003.
Sandra Nägele and Gabriel Wittum. On the influence of different stabilisation methods for the incompressible Navier-Stokes equations, to be submitted.
Sandra Nägele and Gabriel Wittum. Large-eddy simulation and multigrid methods. Electronic Transactions on Numerical Analysis, 15:152–164, 2003.
G.D. Raithby. Skew upstream differencing schemes for problems involving fluid flow. Computer Methods in Applied Mechanics and Engineering, 9:153–164, 1976.
G.E. Schneider and M.J. Raw. Control volume finite-element method for heat transfer and fluid flow using colocated variables-1. computational procedure. Numerical Heat Transfer, 11:363–390, 1987.
J. Smagorinsky. General circulation experiments with the primitive equations i. the basic experiment. Monthly Weather Review, 9:99–164, 1963.
Bert Vreman, Bernard Geurts, and Hans Kuerten. Large-eddy simulation of the turbulent mixing layer. Journal of Fluid Mechanics, 339:357–390, 1997.
D.C. Wilcox. Turbulence Modeling for Computational Fluid Dynamics. DCW Industries, 1993.
Y. Zang, R.L. Street, and J.R. Koseff. A dynamic mixed subgrid-scale model and its application to turbulent recirculating flows. Physics of Fluids A, 12:3186–3195, 1993.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gordner, A., Nägele, S., Wittum, G. (2007). Multigrid Methods for Large-Eddy Simulation. In: Jäger, W., Rannacher, R., Warnatz, J. (eds) Reactive Flows, Diffusion and Transport. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-28396-6_14
Download citation
DOI: https://doi.org/10.1007/978-3-540-28396-6_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28379-9
Online ISBN: 978-3-540-28396-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)