170
J. L. LLJMI.EY A N l ) El. KHAJEH-NOLIRI
The next most successful approach is direct siniulation with so-called subgrid scale modeling (e.g., Deardorff. 1974). This recognircs that it is economically impossible to carry in the computation the smallest scales at high
Rcynolds numbers; the grid scnle is made as small as is economically feasiblc, and the motion on scales below the grid scale is modeled, making iisc of
the wcll-supported property of turbulence dynamics (Tennekes and Luniley,
1972) that the precise nature of the dissipative mechanism does not influence
the large scales of the motion, if the Reynolds number is high enough.
The success of this method depends not only on having a sufficiently high
Reynolds number to have an incrtial subrange (so that energy-containing
and dissipative scales are. to a first approximation, dynamically related only
by the value of the spectral energy flux) but also on being able to use a grid
scale that is small enough to lie in the inertial subrange. It suffers from the
disadvantage that one calculation is not sufficient: each calculation is a
realization in An ensemble, and a sufficient number of independent runs
must be made to obtain stable statistics; for example, of the order of 00
runs .to obtain lo"$ accuracy in second order quantities. To date, this
technique has been used primarily for flows having a homogeneous direction; in such flows, the number of runs necessary to obtain stable statistics
can be substantially reduced by spacial averaging in the homogeneous
direction.
If one is attempting to model the flow in a fully three-dimensional region
such as an urban environment, two facts rapidly become clear: the number
of points'required to make even a crude model of the region precludes the
use of a grid scale lying in the inertial subrange, and a single calculation is so
expensive as to preclude the possibility of doing statistics on an ensemble of
them (sinm there is no homogeneous direction for averaging). We must then
w e u grid scale lying in the energy-containing range and compute only the
stittistical properties of the turbulence. The so-called subgrid scale motions
now contain virtually all the turbulence, and the dynamical modeling becomes much more critical. In fact, since the entire influence of the turbulence
is being computed through the moments, it is no longer correct to think of
turbulence quantities as being subgrid scale; for example, the scale of the
turbulence is not now related to the grid scale, but must be obtained from
dynamical considerations.
No good direct model of second order turbulence quantities exists. The
only practical model is the so-called eddy diffusivity, or " K-theory " model,
which has been used with some success in simple situations (Tennekes and
Lumley, 1972) to predict first order quantities. This model is known to fail,
however, in situations which are rapidly changing in space or time. A
number of authors, realizing that a good model is desirable but that good
second order models are not available, have decided to carry the equations
J. L. LLJMI.EY A N l ) El. KHAJEH-NOLIRI
The next most successful approach is direct siniulation with so-called subgrid scale modeling (e.g., Deardorff. 1974). This recognircs that it is economically impossible to carry in the computation the smallest scales at high
Rcynolds numbers; the grid scnle is made as small as is economically feasiblc, and the motion on scales below the grid scale is modeled, making iisc of
the wcll-supported property of turbulence dynamics (Tennekes and Luniley,
1972) that the precise nature of the dissipative mechanism does not influence
the large scales of the motion, if the Reynolds number is high enough.
The success of this method depends not only on having a sufficiently high
Reynolds number to have an incrtial subrange (so that energy-containing
and dissipative scales are. to a first approximation, dynamically related only
by the value of the spectral energy flux) but also on being able to use a grid
scale that is small enough to lie in the inertial subrange. It suffers from the
disadvantage that one calculation is not sufficient: each calculation is a
realization in An ensemble, and a sufficient number of independent runs
must be made to obtain stable statistics; for example, of the order of 00
runs .to obtain lo"$ accuracy in second order quantities. To date, this
technique has been used primarily for flows having a homogeneous direction; in such flows, the number of runs necessary to obtain stable statistics
can be substantially reduced by spacial averaging in the homogeneous
direction.
If one is attempting to model the flow in a fully three-dimensional region
such as an urban environment, two facts rapidly become clear: the number
of points'required to make even a crude model of the region precludes the
use of a grid scale lying in the inertial subrange, and a single calculation is so
expensive as to preclude the possibility of doing statistics on an ensemble of
them (sinm there is no homogeneous direction for averaging). We must then
w e u grid scale lying in the energy-containing range and compute only the
stittistical properties of the turbulence. The so-called subgrid scale motions
now contain virtually all the turbulence, and the dynamical modeling becomes much more critical. In fact, since the entire influence of the turbulence
is being computed through the moments, it is no longer correct to think of
turbulence quantities as being subgrid scale; for example, the scale of the
turbulence is not now related to the grid scale, but must be obtained from
dynamical considerations.
No good direct model of second order turbulence quantities exists. The
only practical model is the so-called eddy diffusivity, or " K-theory " model,
which has been used with some success in simple situations (Tennekes and
Lumley, 1972) to predict first order quantities. This model is known to fail,
however, in situations which are rapidly changing in space or time. A
number of authors, realizing that a good model is desirable but that good
second order models are not available, have decided to carry the equations