Numerical issues in DNS It is of course beyond the scope of this introduction to go into the details of
3.9.1 Numerical issues in DNS It is of course beyond the scope of this introduction to go into the details of
the methods used for DNS, but it is worth touching on the specific require- ments for this type of computation. The review by Moin and Mahesh (1998) highlights the following issues being tackled in the DNS research literature.
CHAPTER 3 TURBULENCE AND ITS MODELLING
Spatial discretisation The first DNS simulations were performed with spectral methods (Orszag
and Patterson, 1972). These are based on Fourier series decomposition in periodic directions and Chebyshev polynomial expansions in directions with solid walls. The methods are economical and have high convergence rates, but they are difficult to apply in complex geometry. Nevertheless, they are still widely used in research on transitional flows and turbulent flows with simple geometries: some recent applications are flow-induced vibrations (Evangelinos et al., 2000), strained two-dimensional wake flow (Rogers, 2002) and transition in rotor–stator cavity flow (Serre et al., 2002, 2004). Early recognition of the limitations of spectral methods led to the development of spectral element methods (Orszag and Patera, 1984; Patera, 1986). These combine the geometric flexibility of the finite element method with the good convergence properties of the spectral method. These methods have been developed for complex turbulent flows by Karniadakis and co-workers (e.g. Karniadakis, 1989, 1990).
Higher-order finite difference methods (Moin, 1991) are now widely used for problems with more complex geometry. Particular attention needs to be paid to the design of the spatial and temporal discretisation schemes to ensure that the method is stable and to make sure that numerical dissipation does not swamp turbulent eddy dissipation. A sample of recent work illus- trates the range of applications: turbulent flow in a pipe rotating about its axis (Orlandi and Fatica, 1997), flow around square cylinders (Tamura et al., 1998), plumes ( Jiang and Luo, 2000) and diffusion flames (Luo et al., 2005).
Spatial resolution Above we have noted that the spatial mesh for DNS is determined at one end
by the largest geometrical features that need to be resolved and at the other end by the finest turbulence scales that are generated. Research has shown that the grid point requirement N ∝ Re 9/4 can be somewhat relaxed, because
most of the dissipation actually takes place at scales that are substantially larger than of the order of the Kolmogorov length scale η, say 5η–15η (Moin and Mahesh, 1998). As long as the bulk of the dissipation process is adequately represented, the number of grid cells can be reduced. In typical finite difference calculations reduction by a factor of around 100 is possible without significant loss of accuracy.
Temporal discretisation There is a wide range of time scales in a turbulent flow, so the system of
equations is stiff. Implicit time advancement and large time steps are routinely used for stiff systems in general-purpose CFD, but these are unsuitable in DNS because complete time resolution is needed to describe the energy dissipation process accurately. Specially designed implicit and explicit methods have been developed to ensure time accuracy and stability (see e.g. Verstappen and Veldman, 1997).
Temporal resolution Reynolds (in Lumley, 1989) noted that it is essential to have accurate time
resolution of all the scales of turbulent motion. The time steps must be
3.10 SUMMARY 113
adjusted so that fluid particles do not move more than one mesh spacing. Moin and Mahesh (1998) demonstrated the strong influence of time step size on small-scale amplitude and phase error.
Initial and boundary conditions Issues relating to initial and boundary conditions are similar to those in LES.
The reader is referred to section 3.8.5 for a relevant discussion.