The Reynolds-averaged approach to turbulence modeling requires that the
Reynolds stresses in Equation
12.2-4 be appropriately modeled. A common
method employs the Boussinesq hypothesis [
142] to relate the Reynolds
stresses to the mean velocity gradients:
The Boussinesq hypothesis is used in the Spalart-Allmaras model, the
models, and the
models. The advantage of this approach is the
relatively low computational cost associated with the computation of the
. In the case of the Spalart-Allmaras model, only
one additional transport equation (representing turbulent viscosity) is solved.
In the case of the
models, two additional transport equations
(for the turbulence kinetic energy,
, and either the turbulence dissipation
, or the specific dissipation rate,
) are solved, and
is computed as a function of
. The disadvantage of
the Boussinesq hypothesis as presented is that it assumes
isotropic scalar quantity, which is not strictly true.
The alternative approach, embodied in the RSM, is to solve transport equations
for each of the terms in the Reynolds stress tensor. An additional
scale-determining equation (normally for
) is also required. This
means that five additional transport equations are required in 2D flows and
seven additional transport equations must be solved in 3D.
In many cases, models based on the Boussinesq hypothesis perform very well, and
the additional computational expense of the Reynolds stress model is not
justified. However, the RSM is clearly superior for situations in which the
anisotropy of turbulence has a dominant effect on the mean flow. Such cases
include highly swirling flows and stress-driven secondary flows.