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9. Turbulent Flows
The presence of the Reynolds stresses and turbulent scalar flux in the
conservation equations means that the latter are not closed, that is t o say,
they contain more variables than there are equations. Closure requires use of
some approximations, which usually take the form of prescribing the Reynolds
stress tensor and turbulent scalar fluxes in terms of the mean quantities.
It is possible to derive equations for the higher order correlations e g , for
the Reynolds stress tensor, but these contain still more (and higher-order)
unknown correlations that require modeling approximations. These equations
will be introduced later but the important point is that it is impossible to
derive a closed set of exact equations. The approximations introduced are
called turbulence models in engineering or parametrizations in the geosciences.
9.4.2 Simple Turbulence Models and their Application
To close the equations we must introduce a turbulence model. To see what
a reasonable model might be, we note, as we did in the preceding section,
that in laminar flows, energy dissipation and transport of mass, momentum,
and energy normal to the streamlines are mediated by the viscosity, so it
is natural to assume that the effect of turbulence can be represented as an
increased viscosity. This leads to the eddy-viscosity model for the Reynolds
stress:
and the eddy-diffusion model for a scalar:
In Eq. (9.34), k is the turbulent kinetic energy:
The last term in Eq. (9.34) is required to guarantee that, when both sides of
the equation are contracted (the two indices are set equal and summed over),
the equation remains correct. Although the eddy-viscosity hypothesis is not
correct in detail, it is easy to implement and, with careful application, can
provide reasonably good results for many flows.
In the simplest description, turbulence can be characterized by two parameters: its kinetic energy, k, or a velocity, q = *,
and a length scale, L.
Dimensional analysis shows that:
where C, is a dimensionless constant whose value will be given later.
9. Turbulent Flows
The presence of the Reynolds stresses and turbulent scalar flux in the
conservation equations means that the latter are not closed, that is t o say,
they contain more variables than there are equations. Closure requires use of
some approximations, which usually take the form of prescribing the Reynolds
stress tensor and turbulent scalar fluxes in terms of the mean quantities.
It is possible to derive equations for the higher order correlations e g , for
the Reynolds stress tensor, but these contain still more (and higher-order)
unknown correlations that require modeling approximations. These equations
will be introduced later but the important point is that it is impossible to
derive a closed set of exact equations. The approximations introduced are
called turbulence models in engineering or parametrizations in the geosciences.
9.4.2 Simple Turbulence Models and their Application
To close the equations we must introduce a turbulence model. To see what
a reasonable model might be, we note, as we did in the preceding section,
that in laminar flows, energy dissipation and transport of mass, momentum,
and energy normal to the streamlines are mediated by the viscosity, so it
is natural to assume that the effect of turbulence can be represented as an
increased viscosity. This leads to the eddy-viscosity model for the Reynolds
stress:
and the eddy-diffusion model for a scalar:
In Eq. (9.34), k is the turbulent kinetic energy:
The last term in Eq. (9.34) is required to guarantee that, when both sides of
the equation are contracted (the two indices are set equal and summed over),
the equation remains correct. Although the eddy-viscosity hypothesis is not
correct in detail, it is easy to implement and, with careful application, can
provide reasonably good results for many flows.
In the simplest description, turbulence can be characterized by two parameters: its kinetic energy, k, or a velocity, q = *,
and a length scale, L.
Dimensional analysis shows that:
where C, is a dimensionless constant whose value will be given later.