9.3 Large Eddy Simulation (LES)
289
mean density (assumed constant) and p is the deviation from that mean and
includes both the mean stratification and the fluctuations.
We would like to impose periodic boundary conditions on the flow because
this will permit use of spectral methods. To do this, we must be sure that the
domain is sufficiently large that the velocity is not correlated across it. If this
is not the case, the simulation would simply represent the behavior of a single
eddy, not real turbulence. We also note that periodic conditions imply that
the parts of the fluid on opposite ends of the domain are in contact. That
makes sense only if they move at the same speed, which is not the case when
a mean velocity field is imposed. To allow periodic conditions, it is necessary
to do the simulation in a coordinate system moving with the mean velocity;
the best method for achieving this was presented by Rogallo (1981) and is
briefly described below.
The equations are first transformed to a coordinate system that moves
with the mean flow. The terms that represent advection by the mean flow
are thereby eliminated. Then the production term can be formally integrated
and also eliminated. Finally, since a spectral method is used to solve the
equations and the viscous term takes a simple form in Fourier space, it too
can be integrated. The remaining equations are then advanced in time using
a Runge-Kutta method.
For a stratified flow t o be homogeneous, the mean density must also have
a linear profile but, because gravity acts in only one direction, the density
profile must be:
where the constant mean density has been removed. It can be treated in
exactly the same way that the mean velocity profile was and the resulting
equations can be integrated in the same way.
The presence of both stable stratification and shear means that there are
two competing forces. The shear increases the intensity of the turbulence
while stratification reduces it by converting some of its kinetic energy into
potential energy. The interplay of the two forces is what makes this flow
interesting. The parameter traditionally used to characterize their relative
importance is the gradient Richardson number:
which represents the relative strengths of the two forces. It can be shown that
Rig determines whether a laminar flow is stable or not (Drazin and Reid,
1981). This parameter can also be interpreted as the ratio of the squares of
the time scales associated with the two forces. The time scale associated with
the stratification is the inverse of the Brunt-Vaisala frequency,
289
mean density (assumed constant) and p is the deviation from that mean and
includes both the mean stratification and the fluctuations.
We would like to impose periodic boundary conditions on the flow because
this will permit use of spectral methods. To do this, we must be sure that the
domain is sufficiently large that the velocity is not correlated across it. If this
is not the case, the simulation would simply represent the behavior of a single
eddy, not real turbulence. We also note that periodic conditions imply that
the parts of the fluid on opposite ends of the domain are in contact. That
makes sense only if they move at the same speed, which is not the case when
a mean velocity field is imposed. To allow periodic conditions, it is necessary
to do the simulation in a coordinate system moving with the mean velocity;
the best method for achieving this was presented by Rogallo (1981) and is
briefly described below.
The equations are first transformed to a coordinate system that moves
with the mean flow. The terms that represent advection by the mean flow
are thereby eliminated. Then the production term can be formally integrated
and also eliminated. Finally, since a spectral method is used to solve the
equations and the viscous term takes a simple form in Fourier space, it too
can be integrated. The remaining equations are then advanced in time using
a Runge-Kutta method.
For a stratified flow t o be homogeneous, the mean density must also have
a linear profile but, because gravity acts in only one direction, the density
profile must be:
where the constant mean density has been removed. It can be treated in
exactly the same way that the mean velocity profile was and the resulting
equations can be integrated in the same way.
The presence of both stable stratification and shear means that there are
two competing forces. The shear increases the intensity of the turbulence
while stratification reduces it by converting some of its kinetic energy into
potential energy. The interplay of the two forces is what makes this flow
interesting. The parameter traditionally used to characterize their relative
importance is the gradient Richardson number:
which represents the relative strengths of the two forces. It can be shown that
Rig determines whether a laminar flow is stable or not (Drazin and Reid,
1981). This parameter can also be interpreted as the ratio of the squares of
the time scales associated with the two forces. The time scale associated with
the stratification is the inverse of the Brunt-Vaisala frequency,