268
9. Turbulent Flows
times the integral scale. A valid simulation must also capture all of the kinetic energy dissipation. This occurs on the smallest scales, the ones on which
viscosity is active, so the size of the grid must be no larger than a viscously
determined scale, called the Kolmogoroff scale, 7.
For homogeneous isotropic turbulence, the simplest type of turbulence,
there is no reason t o use anything other than a uniform grid. In this case,
the argument just given shows that number of grid points in each direction
must be at least Llq; it can be shown (Tennekes and Lumley, 1976) that this
ratio is proportional to ~ e 3 L / ~ .
Here ReL is a Reynolds number based on the
magnitude of the velocity fluctuations and the integral scale; this parameter
is typically about 0.01 times the macroscopic Reynolds number engineers use
to describe a flow. Since this number of points must be employed in each of
the three coordinate directions, and the time step is related to the grid size,
the cost of a simulation scales as ~ e i .
In terms of the Reynolds number that
an engineer would use to describe the flow, the scaling of the cost may be
somewhat different.
Since the number of grid points that can be used in a computation is
limited by the processing speed and memory of the machine on which it
is carried out, direct numerical simulation is possible only for flows at relatively low Reynolds numbers and in geometrically simple domains. On present
machines, it is possible to make direct numerical simulations of homogeneous flows at turbulent Reynolds numbers up to a few hundred. As noted in
the preceding paragraph, this corresponds to overall flow Reynolds numbers
about two orders of magnitude larger and allows DNS to reach the low end
of the range of Reynolds numbers of engineering interest, making it a useful
method in some cases. In other cases, it may be possible to extrapolate from
the Reynolds number of the simulation to the Reynolds number of actual
interest by using some kind of extrapolation. For further details about DNS,
see the recent review by Leonard (1995).
The results of a DNS contain very detailed information about the flow.
This can be very useful but, on the one hand, it is far more information than
any engineer needs and, on the other, DNS is too expensive to be employed
very often and cannot be used as a design tool. One must then ask what
DNS can be used for. With it, we can obtain detailed information about the
velocity, pressure, and any other variable of interest at a large number of grid
points. These results may be regarded as the equivalent of experimental data
and can be used to produce statistical information or to create a 'numerical
flow visualization.' From the latter, one can learn a great deal about the
coherent structures that exist in the flow. This wealth of information can
then be used to develop a qualitative understanding of the physics of the
flow or to construct a quantitative model, perhaps of the RANS type, which
will allow other, similar, flows to be computed.
We thus conclude that the major role that DNS can fill is as a research
tool. Some examples of kinds of uses to which DNS has been put are:
9. Turbulent Flows
times the integral scale. A valid simulation must also capture all of the kinetic energy dissipation. This occurs on the smallest scales, the ones on which
viscosity is active, so the size of the grid must be no larger than a viscously
determined scale, called the Kolmogoroff scale, 7.
For homogeneous isotropic turbulence, the simplest type of turbulence,
there is no reason t o use anything other than a uniform grid. In this case,
the argument just given shows that number of grid points in each direction
must be at least Llq; it can be shown (Tennekes and Lumley, 1976) that this
ratio is proportional to ~ e 3 L / ~ .
Here ReL is a Reynolds number based on the
magnitude of the velocity fluctuations and the integral scale; this parameter
is typically about 0.01 times the macroscopic Reynolds number engineers use
to describe a flow. Since this number of points must be employed in each of
the three coordinate directions, and the time step is related to the grid size,
the cost of a simulation scales as ~ e i .
In terms of the Reynolds number that
an engineer would use to describe the flow, the scaling of the cost may be
somewhat different.
Since the number of grid points that can be used in a computation is
limited by the processing speed and memory of the machine on which it
is carried out, direct numerical simulation is possible only for flows at relatively low Reynolds numbers and in geometrically simple domains. On present
machines, it is possible to make direct numerical simulations of homogeneous flows at turbulent Reynolds numbers up to a few hundred. As noted in
the preceding paragraph, this corresponds to overall flow Reynolds numbers
about two orders of magnitude larger and allows DNS to reach the low end
of the range of Reynolds numbers of engineering interest, making it a useful
method in some cases. In other cases, it may be possible to extrapolate from
the Reynolds number of the simulation to the Reynolds number of actual
interest by using some kind of extrapolation. For further details about DNS,
see the recent review by Leonard (1995).
The results of a DNS contain very detailed information about the flow.
This can be very useful but, on the one hand, it is far more information than
any engineer needs and, on the other, DNS is too expensive to be employed
very often and cannot be used as a design tool. One must then ask what
DNS can be used for. With it, we can obtain detailed information about the
velocity, pressure, and any other variable of interest at a large number of grid
points. These results may be regarded as the equivalent of experimental data
and can be used to produce statistical information or to create a 'numerical
flow visualization.' From the latter, one can learn a great deal about the
coherent structures that exist in the flow. This wealth of information can
then be used to develop a qualitative understanding of the physics of the
flow or to construct a quantitative model, perhaps of the RANS type, which
will allow other, similar, flows to be computed.
We thus conclude that the major role that DNS can fill is as a research
tool. Some examples of kinds of uses to which DNS has been put are: