282
9. Turbulent Flows
large eddy simulation itself. In this way, a kind of self-consistent subgrid-scale
model is produced.
Thus the essential ingredient in this model is the assumption that the
same model with the same value of the parameter can be applied to both
the actual LES and LES done on a coarser scale. A secondary assumption
that is made more for convenience than necessity is that the parameter is
independent of location. Finally, we note that the dynamic procedure gives
the model parameter as the ratio of two quantities.
The actual procedure of Germano et al. is a bit more formal than what
we have just described but the result is the same; the model parameter is
computed, at every spatial grid point and every time step, directly from
results of the LES itself. We shall not present the formal procedure here. The
interested reader is referred to the original paper of Germano et al. (1990) or
the review by Ferziger (1995).
This process should be called a procedure rather than a model as any
subgrid-scale model can be used as a basis for it. A number of variations
are possible. One particularly significant improvement to the original model
proposed by Germano et al. (1990) is the least squares procedure suggested
by Lilly (1991). The dynamic procedure with the Smagorinsky model as its
basis removes many of the difficulties described earlier:
In shear flows, the Smagorinsky model parameter needs to be much smaller
than in isotropic turbulence. The dynamic model produces this change
automatically.
The model parameter has to be reduced even further near walls. The dynamic model automatically decreases the parameter in the correct manner
near the wall.
The definition of the length scale for anisotropic grids or filters is unclear.
This issue becomes moot with the dynamic model because the model compensates for any error in the length scale by changing the value of the
parameter.
Although it is a considerable improvement on the Smagorinsky model,
there are problems with the dynamic procedure. The model parameter it
produces is a rapidly varying function of the spatial coordinates and time so
the eddy viscosity takes large values of both signs. Although a negative eddy
viscosity has been suggested as a way of representing energy transfer from
the small scales to the large ones (this process is called backscatter), if the
eddy viscosity is negative over too large a spatial region or for too long a
time, numerical instability can and does occur. Ong cure is to set any eddy
viscosity pt < -p, the molecular viscosity, equal to -p; this is called clipping.
Another useful alternative is to employ averaging in space or time. For details,
the reader is referred to the papers cited above. These techniques produce
further improvements but are still not completely satisfactory; finding a more
robust model for the subgrid scale is the subject of current research.
9. Turbulent Flows
large eddy simulation itself. In this way, a kind of self-consistent subgrid-scale
model is produced.
Thus the essential ingredient in this model is the assumption that the
same model with the same value of the parameter can be applied to both
the actual LES and LES done on a coarser scale. A secondary assumption
that is made more for convenience than necessity is that the parameter is
independent of location. Finally, we note that the dynamic procedure gives
the model parameter as the ratio of two quantities.
The actual procedure of Germano et al. is a bit more formal than what
we have just described but the result is the same; the model parameter is
computed, at every spatial grid point and every time step, directly from
results of the LES itself. We shall not present the formal procedure here. The
interested reader is referred to the original paper of Germano et al. (1990) or
the review by Ferziger (1995).
This process should be called a procedure rather than a model as any
subgrid-scale model can be used as a basis for it. A number of variations
are possible. One particularly significant improvement to the original model
proposed by Germano et al. (1990) is the least squares procedure suggested
by Lilly (1991). The dynamic procedure with the Smagorinsky model as its
basis removes many of the difficulties described earlier:
In shear flows, the Smagorinsky model parameter needs to be much smaller
than in isotropic turbulence. The dynamic model produces this change
automatically.
The model parameter has to be reduced even further near walls. The dynamic model automatically decreases the parameter in the correct manner
near the wall.
The definition of the length scale for anisotropic grids or filters is unclear.
This issue becomes moot with the dynamic model because the model compensates for any error in the length scale by changing the value of the
parameter.
Although it is a considerable improvement on the Smagorinsky model,
there are problems with the dynamic procedure. The model parameter it
produces is a rapidly varying function of the spatial coordinates and time so
the eddy viscosity takes large values of both signs. Although a negative eddy
viscosity has been suggested as a way of representing energy transfer from
the small scales to the large ones (this process is called backscatter), if the
eddy viscosity is negative over too large a spatial region or for too long a
time, numerical instability can and does occur. Ong cure is to set any eddy
viscosity pt < -p, the molecular viscosity, equal to -p; this is called clipping.
Another useful alternative is to employ averaging in space or time. For details,
the reader is referred to the papers cited above. These techniques produce
further improvements but are still not completely satisfactory; finding a more
robust model for the subgrid scale is the subject of current research.