280
9. Turbulent Flows
even further. One successful recipe is to borrow the van Driest damping that
has long been used to reduce the near-wall eddy viscosity in RANS models:
where n + is the distance from the wall in viscous wall units (n+ = nu,/v,
where u, is the shear velocity, u, = m, and T, is the shear stress a t the
wall) and A+ is a constant usually taken to be approximately 25. Although
this modification produces the desired results, it is difficult to justify in the
context of LES. An SGS model should depend solely on the local properties
of the flow and it is difficult to see how the distance from the wall qualifies
in this regard.
The purpose of the van Driest damping is to reduce the subgrid-scale eddy
viscosity near the wall; pt n3 in this region and models should respect this
property. An alternative is a subgrid-scale model which reduces the eddy
viscosity when the subgrid-scale Reynolds number, ISlA2/v, becomes small.
Models of this kind were suggested by McMillan and Ferziger (1980) and by
Yakhot and Orszag (1986); the latter used renormalization group theory to
derive their model.
A further problem is that, near a wall, the flow structure is very anisotropic.
Regions of low and high speed fluid (streaks) are created; they are approximately 1000 viscous units long and 30-50 viscous units wide in both
the spanwise and normal directions. Resolving the streaks requires a highly
anisotropic grid and the choice of length scale, A, to use in the SGS model is
not obvious. The usual choice is ( ~ l & A ~ ) l / ~
but (A? + A: +
is possible and others are easily constructed; here Ai is the width associated with
the filter in the ith coordinate direction. It is possible that, with a proper
choice of length scale, the damping (9.12) would become unnecessary. A fuller
discussion of this issue can be found in Piomelli et al. (1989).
In a stably-stratified fluid, it is necessary to reduce the Smagorinsky parameter. Stratification is common in geophysical flows; the usual practice
is to make the parameter a function of a Richardson or Froude number.
These are related non-dimensional parameters that represent the relative importance of stratification and shear. Similar effects occur in flows in which
rotation and/or curvature play significant roles. In the past, the Richardson
number was based on the properties of the mean flow field. Recent work indicates that it is better to base the parameter on properties of the turbulence
than on the applied forces; Ivey and Imberger (1991) suggested using the
turbulent Froude number.
Thus there are many difficulties with the Smagorinsky model. If we wish
to simulate more complex and/or higher Reynolds number flows, it may be
important to have a more accurate model. Indeed, detailed tests based on
results derived from DNS data, show that the Smagorinsky model is quite
poor in representing the details of the subgrid-scale stresses.
9. Turbulent Flows
even further. One successful recipe is to borrow the van Driest damping that
has long been used to reduce the near-wall eddy viscosity in RANS models:
where n + is the distance from the wall in viscous wall units (n+ = nu,/v,
where u, is the shear velocity, u, = m, and T, is the shear stress a t the
wall) and A+ is a constant usually taken to be approximately 25. Although
this modification produces the desired results, it is difficult to justify in the
context of LES. An SGS model should depend solely on the local properties
of the flow and it is difficult to see how the distance from the wall qualifies
in this regard.
The purpose of the van Driest damping is to reduce the subgrid-scale eddy
viscosity near the wall; pt n3 in this region and models should respect this
property. An alternative is a subgrid-scale model which reduces the eddy
viscosity when the subgrid-scale Reynolds number, ISlA2/v, becomes small.
Models of this kind were suggested by McMillan and Ferziger (1980) and by
Yakhot and Orszag (1986); the latter used renormalization group theory to
derive their model.
A further problem is that, near a wall, the flow structure is very anisotropic.
Regions of low and high speed fluid (streaks) are created; they are approximately 1000 viscous units long and 30-50 viscous units wide in both
the spanwise and normal directions. Resolving the streaks requires a highly
anisotropic grid and the choice of length scale, A, to use in the SGS model is
not obvious. The usual choice is ( ~ l & A ~ ) l / ~
but (A? + A: +
is possible and others are easily constructed; here Ai is the width associated with
the filter in the ith coordinate direction. It is possible that, with a proper
choice of length scale, the damping (9.12) would become unnecessary. A fuller
discussion of this issue can be found in Piomelli et al. (1989).
In a stably-stratified fluid, it is necessary to reduce the Smagorinsky parameter. Stratification is common in geophysical flows; the usual practice
is to make the parameter a function of a Richardson or Froude number.
These are related non-dimensional parameters that represent the relative importance of stratification and shear. Similar effects occur in flows in which
rotation and/or curvature play significant roles. In the past, the Richardson
number was based on the properties of the mean flow field. Recent work indicates that it is better to base the parameter on properties of the turbulence
than on the applied forces; Ivey and Imberger (1991) suggested using the
turbulent Froude number.
Thus there are many difficulties with the Smagorinsky model. If we wish
to simulate more complex and/or higher Reynolds number flows, it may be
important to have a more accurate model. Indeed, detailed tests based on
results derived from DNS data, show that the Smagorinsky model is quite
poor in representing the details of the subgrid-scale stresses.