272
9. Turbulent Flows
smaller than that at the peak in the energy spectrum, it is probably safe to
assume that the flow has been well resolved.
The accuracy requirement makes use of spectral methods common in DNS
and LES. These methods were described briefly earlier, in Sect. 3.10. In
essence, they use Fourier series as a means of computing derivatives. The
use of Fourier transforms is feasible only because the fast Fourier transform
algorithm (Cooley and Tukey, 1965) reduces the cost of computing a Fourier
transform to n log, n operations. Unfortunately, this algorithm is applicable
only for equi-spaced grids and a few other special cases. A number of specialized methods of this kind have been developed for solving the Navier-Stokes
equations; the reader interested in more details of spectral methods is referred
to the book by Canuto et al. (1987).
We briefly mention one special type of method. Rather than directly approximating the Navier-Stokes equations, one could multiply them by a sequence of 'test functions' and integrate over the entire domain and then find
a solution that satisfies the resulting equations. This procedure is similar
to one used in deriving finite element methods. Functions which satisfy this
form of the equations are known as 'weak solutions'. One can represent the
solution of the Navier-Stokes equations as a series of vector functions, each of
which has zero divergence. This choice removes the pressure from the integral
form of the equations, thereby reducing the number of dependent variables
that need to be computed and stored. The set of dependent variables can be
further reduced by noting that, if a function has zero divergence, its third
component can be computed from the other two. The result is that only
two sets of dependent variables need to be computed, reducing the memory
requirements by half. As these methods are quite specialized and their development requires considerable space, they are not given in detail here; the
interested reader is referred to the paper by Moser et al. (1983).
Another difficulty in DNS is that of generating initial and boundary conditions. The former must contain all the details of the initial three-dimensional
velocity field. Since coherent structures are an important component of the
flow, it is difficult to construct such a field. Furthermore, the effects of initial
conditions are typically 'remembered' by the flow for a considerable time,
usually a few 'eddy-turnover times.' An eddy-turnover time is essentially the
integral time scale of the flow or the integral length scale divided by the rootmean-square velocity (9). Thus the initial conditions have a significant effect
on the results. Frequently, the first part of a simulation that is started with
artificially constructed initial conditions must be discarded because it is not
faithful to the physics. The question of how to select initial conditions is as
much art as science and no unique prescriptions applicable to all flows can
be given but we shall give some examples.
For homogeneous isotropic turbulence, the simplest case, periodic boundary conditions are used and it is easiest to construct the initial conditions
in Fourier space i.e., we need to create &(k). This is done by giving the
9. Turbulent Flows
smaller than that at the peak in the energy spectrum, it is probably safe to
assume that the flow has been well resolved.
The accuracy requirement makes use of spectral methods common in DNS
and LES. These methods were described briefly earlier, in Sect. 3.10. In
essence, they use Fourier series as a means of computing derivatives. The
use of Fourier transforms is feasible only because the fast Fourier transform
algorithm (Cooley and Tukey, 1965) reduces the cost of computing a Fourier
transform to n log, n operations. Unfortunately, this algorithm is applicable
only for equi-spaced grids and a few other special cases. A number of specialized methods of this kind have been developed for solving the Navier-Stokes
equations; the reader interested in more details of spectral methods is referred
to the book by Canuto et al. (1987).
We briefly mention one special type of method. Rather than directly approximating the Navier-Stokes equations, one could multiply them by a sequence of 'test functions' and integrate over the entire domain and then find
a solution that satisfies the resulting equations. This procedure is similar
to one used in deriving finite element methods. Functions which satisfy this
form of the equations are known as 'weak solutions'. One can represent the
solution of the Navier-Stokes equations as a series of vector functions, each of
which has zero divergence. This choice removes the pressure from the integral
form of the equations, thereby reducing the number of dependent variables
that need to be computed and stored. The set of dependent variables can be
further reduced by noting that, if a function has zero divergence, its third
component can be computed from the other two. The result is that only
two sets of dependent variables need to be computed, reducing the memory
requirements by half. As these methods are quite specialized and their development requires considerable space, they are not given in detail here; the
interested reader is referred to the paper by Moser et al. (1983).
Another difficulty in DNS is that of generating initial and boundary conditions. The former must contain all the details of the initial three-dimensional
velocity field. Since coherent structures are an important component of the
flow, it is difficult to construct such a field. Furthermore, the effects of initial
conditions are typically 'remembered' by the flow for a considerable time,
usually a few 'eddy-turnover times.' An eddy-turnover time is essentially the
integral time scale of the flow or the integral length scale divided by the rootmean-square velocity (9). Thus the initial conditions have a significant effect
on the results. Frequently, the first part of a simulation that is started with
artificially constructed initial conditions must be discarded because it is not
faithful to the physics. The question of how to select initial conditions is as
much art as science and no unique prescriptions applicable to all flows can
be given but we shall give some examples.
For homogeneous isotropic turbulence, the simplest case, periodic boundary conditions are used and it is easiest to construct the initial conditions
in Fourier space i.e., we need to create &(k). This is done by giving the