9.4 RANS Models
297
other turbulence model) are much stiffer than the laminar equations. Thus,
there is little difficulty in the discretization of these equations other than
one to be discussed below but the solution method has to take the increased
stiffness into account.
For this reason, in the numerical solution procedure, one first performs an
outer iteration of the momentum and pressure correction equations in which
the value of the eddy viscosity is based on the values of k and E at the end of
the preceding iteration. After this has been completed, an outer iteration of
the turbulent kinetic energy and dissipation equations is made. Since these
equations are highly nonlinear, they have to be linearized prior to iteration.
After completing an iteration of the turbulence model equations, we are ready
to recalculate the eddy viscosity and start a new outer iteration.
The stiffness is the reason why the mean flow and turbulence equations
are treated separately in the method just described; coupling the equations
would make convergence very difficult to obtain. Too large a time step (or its
equivalent in an iterative method) can lead to negative values of either k of E
and numerical instability. It is therefore necessary to use under-relaxation in
the iterative method for these quantities; the values of the under-relaxation
parameters are similar to the ones used in the momentum equations (typically
0.6-0.8).
The profiles of the turbulent kinetic energy and its dissipation are typically much more peaked near the wall than the mean velocity profile. These
peaks are difficult to capture; one should probably use a finer grid for the
turbulence quantities than for the mean flow but this is rarely done. If the
same grid is used for all quantities, the resolution may be insufficient for the
turbulence quantities and there is a chance that the solution will contain wiggles which can lead t o negative values of these quantities in this region. This
possibility can be avoided by locally blending the central difference scheme
with a low order upwind discretization for the convective terms in the k and
E equations. This, of course, decreases the accuracy to which these quantities
are calculated but is necessary if the same grid is used for all quantities.
Boundary conditions are needed for the model equations. These are generally similar to the conditions applied to any scalar equation. However, a t
solid walls there may be significant differences. One possibility is to solve the
equations accurately right up to the wall. Then the conditions to be applied
are the standard no-slip ones for the velocity. In the k-E model, it is appropriate to set k = 0 at the wall but the dissipation is not zero there; instead
one can use the condition:
wall
wall
When this is done, it is generally necessary to modify the model itself near
the wall. It is argued that the effects that need to be modeled are due to the
low Reynolds number of the turbulence near the wall and a number of low
297
other turbulence model) are much stiffer than the laminar equations. Thus,
there is little difficulty in the discretization of these equations other than
one to be discussed below but the solution method has to take the increased
stiffness into account.
For this reason, in the numerical solution procedure, one first performs an
outer iteration of the momentum and pressure correction equations in which
the value of the eddy viscosity is based on the values of k and E at the end of
the preceding iteration. After this has been completed, an outer iteration of
the turbulent kinetic energy and dissipation equations is made. Since these
equations are highly nonlinear, they have to be linearized prior to iteration.
After completing an iteration of the turbulence model equations, we are ready
to recalculate the eddy viscosity and start a new outer iteration.
The stiffness is the reason why the mean flow and turbulence equations
are treated separately in the method just described; coupling the equations
would make convergence very difficult to obtain. Too large a time step (or its
equivalent in an iterative method) can lead to negative values of either k of E
and numerical instability. It is therefore necessary to use under-relaxation in
the iterative method for these quantities; the values of the under-relaxation
parameters are similar to the ones used in the momentum equations (typically
0.6-0.8).
The profiles of the turbulent kinetic energy and its dissipation are typically much more peaked near the wall than the mean velocity profile. These
peaks are difficult to capture; one should probably use a finer grid for the
turbulence quantities than for the mean flow but this is rarely done. If the
same grid is used for all quantities, the resolution may be insufficient for the
turbulence quantities and there is a chance that the solution will contain wiggles which can lead t o negative values of these quantities in this region. This
possibility can be avoided by locally blending the central difference scheme
with a low order upwind discretization for the convective terms in the k and
E equations. This, of course, decreases the accuracy to which these quantities
are calculated but is necessary if the same grid is used for all quantities.
Boundary conditions are needed for the model equations. These are generally similar to the conditions applied to any scalar equation. However, a t
solid walls there may be significant differences. One possibility is to solve the
equations accurately right up to the wall. Then the conditions to be applied
are the standard no-slip ones for the velocity. In the k-E model, it is appropriate to set k = 0 at the wall but the dissipation is not zero there; instead
one can use the condition:
wall
wall
When this is done, it is generally necessary to modify the model itself near
the wall. It is argued that the effects that need to be modeled are due to the
low Reynolds number of the turbulence near the wall and a number of low