TRANSPORT MODEL LIMITAIIONS I Y TURBULENCE
41
(50)
r(z, 0 ) = 3 C X ~
- z * , . 2 ~ ; }
instead of a Dirac function. To give smooth curves for r(z, t ) , the ensemble
of lattices is given a uniform distribution of phases, i.e. the lattice positioning
is dinerent (relative to the origin) in each realization, such that over the
ensemble of realizations the lattice points are uniformly distributed over one
step length (= one lattice spacing). Figure 4 shows the outcome (which
0.8
-
r
04
-
-
I
0.0 -20 1 -I5
A STEP LENGTH &t
FIG. 4. Mean concentration profiles resulting from dispersion of an initial Gaussian profile
by simple binary random walks whose step lengths I have dimerent ratios to the standard
drriiltion no or the initiel profile.
could, of course, be written out analytically) for three ratios l/ao of step
Icngth to standard deviation of initial field, compared with the diffusion
equation solution l/uo -+ 0. To maintain the same degree of dispersal for all
cases, the number of steps is chosen as
(51)
n = l6(0~/l)~,
thus ranging from 4 for the longest steps to 64 for the shortest.
The resemblance between the l/ao = 2 case (n = 4) and one of the telegraph equation dispersion solutions in Fig. 3 is evident.
The large-step-length random walk shows far less dramatic results f rhe
prohahilily deruity,function of velocities is not binary but normal (Gaussian),
because in that case it can be shown that an initially normal (or Diruc)
concentration ,field remains always normal, no matter how large a value we
choose for / / a o . A similar situation occurs during dispersion in isotropic
turbulence. I t is found empirically that the fluid velocity fluctuations are
41
(50)
r(z, 0 ) = 3 C X ~
- z * , . 2 ~ ; }
instead of a Dirac function. To give smooth curves for r(z, t ) , the ensemble
of lattices is given a uniform distribution of phases, i.e. the lattice positioning
is dinerent (relative to the origin) in each realization, such that over the
ensemble of realizations the lattice points are uniformly distributed over one
step length (= one lattice spacing). Figure 4 shows the outcome (which
0.8
-
r
04
-
-
I
0.0 -20 1 -I5
A STEP LENGTH &t
FIG. 4. Mean concentration profiles resulting from dispersion of an initial Gaussian profile
by simple binary random walks whose step lengths I have dimerent ratios to the standard
drriiltion no or the initiel profile.
could, of course, be written out analytically) for three ratios l/ao of step
Icngth to standard deviation of initial field, compared with the diffusion
equation solution l/uo -+ 0. To maintain the same degree of dispersal for all
cases, the number of steps is chosen as
(51)
n = l6(0~/l)~,
thus ranging from 4 for the longest steps to 64 for the shortest.
The resemblance between the l/ao = 2 case (n = 4) and one of the telegraph equation dispersion solutions in Fig. 3 is evident.
The large-step-length random walk shows far less dramatic results f rhe
prohahilily deruity,function of velocities is not binary but normal (Gaussian),
because in that case it can be shown that an initially normal (or Diruc)
concentration ,field remains always normal, no matter how large a value we
choose for / / a o . A similar situation occurs during dispersion in isotropic
turbulence. I t is found empirically that the fluid velocity fluctuations are