MODELING OF TURBULENT TRANSPORT
I85
this assignment of the value at the second time step. This manifests itself by a
gradual separation of the values of the solution at alternate time steps. This
is a well understood phenomenon, and we used the classical cure: the solutions at K and K + I were averaged and the process restarted every 10 time
steps.
This differencing scheme is only conditionally stable, so that a limitation
must be placed on the size of the time step. The classical limitation is
AT 4 f Ay2, in dimensionless variables in which the diffusion coefficients are
unity. In our variables, normalized by the velocity defect on the centerline
and the standard deviation of the defect profile, the diffusion coefficient is
12.5 (roughly) so that At I 3.1 Ay2 is the appropriate restriction. We used
At = 3 Ay', and Ay = 0.1.
Because the wake width grows continually, and the centerline defect
shrinks, development is slower and slower, and larger and larger time and
space steps can be used as time progresses, both from the viewpoint of
truncation error and from the viewpoint of stability. In addition, the size of
the computational mesh re uired constantly increases. Consequently, as the
point; when it had grown to a value about 1 of the peak, a new, empty
mesh point was added. When the number of meshpoints had doubled, the
computation was stopped, every other mesh point was discarded, the values
renormalized by the new velocity and length scales, and the computation
restarted, effectively increasing Ay by 2 and At by 4. Each of these doublings
corresponded to a dimensionless time lapse of about 5 (defined by d t W ,
where F = q2/& on the centerline). About three doublings are required to
get within 10 Ox;, of the final values.
If the initial wake width is taken as 8, the momentum thickness, then three
doublings corresponds to .x/d = 500 (cf. Tennekes and Lumley, 1970). Each
successive doubling corresponds to a fourfold increase in streamwise distance. To get within 1 % of the final values, roughly two more doublings
are required, that is, an .x/d of roughly 8OOO (see Fig. 1).
Coefficients were approximated by the following techniques: in the wake,
outboard of the zero of the advection - (cf. Tennekes and Lumley, 1972) there
is a region in which -uu - q2 - 2, production - dissipation,
advection - transport, and hence 2.9- - const. In this region, the equations may be solved exactly; comparability among the equations requires
that the coefficient in the net q2 transport be unity, that in the2 transport be
f and that in the - U P transport be zJ2/q2. Using this, plus the concepts
described in Eqs. (41 )-(a), and in addition neglecting pressure transport
[it., requirinl: threefold symmetry of (33)] reduced the transport coefficients
to fivc, two of known magnitude.
We found generally that the values of the various transport coefficients are
computation progressed, 9 u was continually monitored at the last mesh
I85
this assignment of the value at the second time step. This manifests itself by a
gradual separation of the values of the solution at alternate time steps. This
is a well understood phenomenon, and we used the classical cure: the solutions at K and K + I were averaged and the process restarted every 10 time
steps.
This differencing scheme is only conditionally stable, so that a limitation
must be placed on the size of the time step. The classical limitation is
AT 4 f Ay2, in dimensionless variables in which the diffusion coefficients are
unity. In our variables, normalized by the velocity defect on the centerline
and the standard deviation of the defect profile, the diffusion coefficient is
12.5 (roughly) so that At I 3.1 Ay2 is the appropriate restriction. We used
At = 3 Ay', and Ay = 0.1.
Because the wake width grows continually, and the centerline defect
shrinks, development is slower and slower, and larger and larger time and
space steps can be used as time progresses, both from the viewpoint of
truncation error and from the viewpoint of stability. In addition, the size of
the computational mesh re uired constantly increases. Consequently, as the
point; when it had grown to a value about 1 of the peak, a new, empty
mesh point was added. When the number of meshpoints had doubled, the
computation was stopped, every other mesh point was discarded, the values
renormalized by the new velocity and length scales, and the computation
restarted, effectively increasing Ay by 2 and At by 4. Each of these doublings
corresponded to a dimensionless time lapse of about 5 (defined by d t W ,
where F = q2/& on the centerline). About three doublings are required to
get within 10 Ox;, of the final values.
If the initial wake width is taken as 8, the momentum thickness, then three
doublings corresponds to .x/d = 500 (cf. Tennekes and Lumley, 1970). Each
successive doubling corresponds to a fourfold increase in streamwise distance. To get within 1 % of the final values, roughly two more doublings
are required, that is, an .x/d of roughly 8OOO (see Fig. 1).
Coefficients were approximated by the following techniques: in the wake,
outboard of the zero of the advection - (cf. Tennekes and Lumley, 1972) there
is a region in which -uu - q2 - 2, production - dissipation,
advection - transport, and hence 2.9- - const. In this region, the equations may be solved exactly; comparability among the equations requires
that the coefficient in the net q2 transport be unity, that in the2 transport be
f and that in the - U P transport be zJ2/q2. Using this, plus the concepts
described in Eqs. (41 )-(a), and in addition neglecting pressure transport
[it., requirinl: threefold symmetry of (33)] reduced the transport coefficients
to fivc, two of known magnitude.
We found generally that the values of the various transport coefficients are
computation progressed, 9 u was continually monitored at the last mesh