284
9. Turbulent Flows
procedure the advantage of not requiring any externally provided information
such as model constants.
We shall describe a simple version of the model in order to give a flavor of
the approach. The unfiltered velocity on the right side of Eq. (9.5) is expanded
in a Taylor series about the point x. Truncating the series (usually keeping
only terms up to second order) gives a differential equation for the unfiltered
velocity in terms of the filtered velocity. We give the result for the simplest
case in which the filter kernel is symmetric about the point x. We have:
which is the desired differential equation. Shah (1998) used an approximate
inversion of this equation that consisted of using approximate factorization (see Chap. 5) and performing just one iteration. He computed several
flows with the resulting model, obtaining very good results. Katapodes et
al. (2000) used a simpler approximate inversion which consists of simply iterating Eq. (9.14) to get:
They also preesented more complex versions of the model.
A still more complex, but more accurate, approach to the deconvolution
modeling concept has been presented by Domaradzki and coworkers (Domaradzki and Saiki, 1997).
This ends the presentation of subgrid-scale models. At present, reasonable subgrid-scale models exist and they generally produce good simulations.
However, the models are not sufficiently precise to be trusted to simulate a
flow that has never been treated before. There is need for further improvements and there is a considerable amount of ongoing research.
9.3.4 Example: Flow Over a Wall-Mounted Cube
As an example of the method, we shall use the flow over a cube mounted on
one wall of a channel. The geometry is shown in Fig. 9.4. For the simulation
shown, which was made by Shah and Ferziger (1997), the Reynolds number
based on the maximum velocity a t the inflow and the cube height is 3200.
The inflow is fully developed channel flow and was taken from a separate
simulation of that flow, the outlet condition was the convective condition
given above. Periodic boundary conditions were used in the spanwise direction
and no-slip conditions a t all wall surfaces.
The LES used a grid of 240 x 128 x 128 control volumes with second order
accuracy. The time advancement method was of the fractional step type.
The convective terms were treated explicitly by a third order Runge-Kutta
method in time while the viscous terms were treated implicitly. In particular,
9. Turbulent Flows
procedure the advantage of not requiring any externally provided information
such as model constants.
We shall describe a simple version of the model in order to give a flavor of
the approach. The unfiltered velocity on the right side of Eq. (9.5) is expanded
in a Taylor series about the point x. Truncating the series (usually keeping
only terms up to second order) gives a differential equation for the unfiltered
velocity in terms of the filtered velocity. We give the result for the simplest
case in which the filter kernel is symmetric about the point x. We have:
which is the desired differential equation. Shah (1998) used an approximate
inversion of this equation that consisted of using approximate factorization (see Chap. 5) and performing just one iteration. He computed several
flows with the resulting model, obtaining very good results. Katapodes et
al. (2000) used a simpler approximate inversion which consists of simply iterating Eq. (9.14) to get:
They also preesented more complex versions of the model.
A still more complex, but more accurate, approach to the deconvolution
modeling concept has been presented by Domaradzki and coworkers (Domaradzki and Saiki, 1997).
This ends the presentation of subgrid-scale models. At present, reasonable subgrid-scale models exist and they generally produce good simulations.
However, the models are not sufficiently precise to be trusted to simulate a
flow that has never been treated before. There is need for further improvements and there is a considerable amount of ongoing research.
9.3.4 Example: Flow Over a Wall-Mounted Cube
As an example of the method, we shall use the flow over a cube mounted on
one wall of a channel. The geometry is shown in Fig. 9.4. For the simulation
shown, which was made by Shah and Ferziger (1997), the Reynolds number
based on the maximum velocity a t the inflow and the cube height is 3200.
The inflow is fully developed channel flow and was taken from a separate
simulation of that flow, the outlet condition was the convective condition
given above. Periodic boundary conditions were used in the spanwise direction
and no-slip conditions a t all wall surfaces.
The LES used a grid of 240 x 128 x 128 control volumes with second order
accuracy. The time advancement method was of the fractional step type.
The convective terms were treated explicitly by a third order Runge-Kutta
method in time while the viscous terms were treated implicitly. In particular,