# Introduction to multigrid methods

Ravichandran, KS (1993) Introduction to multigrid methods. In: CFD: Advances and Applications, 1993, India.

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## Abstract

The principal aim of computational fluid dynamics is to compute flows in multicomponent 3-D geometries using accurate and reliable mathematical models. A major or perhaps even the single most important parameter governing fluid flow is the Reynolds number Re = UL/nu where U is the typical flow speed,L is a length (of an aircraft,say), and nu is the kinematic viscosity.Typical Reynolds numbers encountered in airflow range from 10(exp 5) for flow past a flat plate to 10(exp 7)for flow past an aircraft. It is well known that for high Reynolds number flow becomes chaotic and turbulent and energy transfer processes take place through the large eddies and dissipation through the small eddies. The ratio of the sizes of the small and large eddies is given by eta/L = O(Re(exp -3/4)). If one intends to resolve the smallest eddies then the mesh size is of the order of eta and to cover the entire flow domain with such a mesh in three dimensions, we require (L /eta)(exp 3) grid points. In other words the number of mesh points required to resolve the flow fully is O(Re(exp 9/4) and for a Reynolds number of one million, the computing resource required is certainly beyond current projections. Consequently, even if the task is scaled down to the computation of averaged flow with turbulence modeling for the small eddies, one needs to approach steady states rather efficiently. Typically, as Brandt quantifies it, we need methods that would solve a problem of N unknowns with O(N) work and storage. Multigrid offers a way of doing this for different problems.We explain the multigrid principle and technique beginning with simple linear problems and two grid levels.

Item Type: Conference or Workshop Item (Paper) Computational fluid dynamics;Computational grids;Multigrid methods;Reynolds number;Turbulence;Three dimensional flow; Turbulent flow;Vortices AERONAUTICS > Aerodynamics M/S ICAST NAL 17 Mar 2008 24 May 2010 04:25 http://nal-ir.nal.res.in/id/eprint/4412 View Item