36
STANLEY CORRSIN
where V and 7 are nonzero and noninfinite constants, essentially wave
s p e d and relaxation time. respectively. Then
(39)
The exact solutions to Eqs. (35), (37), and (39) need not be repeated here,
but their contrasting traits can be inferred from the qualitative sketches in
Fig.2. For “small times” the telegraph solution is wavelike; for large
times” it is diffusionlike. It is this combination of asymptotic attributes
which resembles turbulent diffusion (Taylor, 1921). The initial conditions
here are a Dirac function for P(z, 0) and, for tbe equations which require an
addition condition, a finite P,,(r, 0). Figure 3 shows the solution of the telegraph equation for a normal (Gaussian) initial distribution,
P,, + T ’ P , = V2P,, .
(40)
P(e, 0) = expI +},
and for two different dimensionless times. The solid c u m is for t/T = 2, the
dashed one for t/T = 5. Both ordinates are scaled so that P,, = 1. Here the
wavelike propagation of the initially localized P(z, 0) function has nearly
disappeared by the time r = 5T. The abcissa widths ate nearly the same
because t is the same for the two curves; T was varied.
The solution to Eq. (39), even with a Dirac function initial condition,
implied for Fig. 2. is a bit complicated (Goldstein, 1951; Monin and Yaglom,
1965, 197 I), but ordinary differential equations for the second moments are
easily deduced from the three transport equations, Eqs. ( 3 9 (37), and (39).
They arc, writing D as V’T,
d d / d t - 2V2T = 0,
d2a2/dt2 - 2V2 = 0,
and
(43)
d2a2Jdt2 + T - ’ dUa/dt - 2V2 = 0.
With initial conditions a(0) = 0 [for all] and u(0) + V [like turbulence
(Taylor, 1921). for Eqs. (42) and (43)]. the solutions are’
(4)
t
?
= 2V2Tt,
(45 )
a’ = v5’.
’ Of course. a great variety of n(r) behaviors can bc achieved with the diffusion equation by
making the diffuaivity a function of time. Suppose Eq. (35) is generalized to P, = D(r)P,,.
Taylor (IYM ) pointed out that, at least for the unshcclrd caw. turbulent dispersion from a
point source can be empirically described this way, with D(r) = 4 d Z2/dr. where Z is the fluid
point diapliicement. Solutions to the dificrential equation are obtained simply by observing that
thr D ( i ) can be iibaorbed into il rescaled time variable, (11, f D(r)dt. which gives the classical
diffusion eqiiation. Only the initial conditions pose a rlipht chllcnge.
STANLEY CORRSIN
where V and 7 are nonzero and noninfinite constants, essentially wave
s p e d and relaxation time. respectively. Then
(39)
The exact solutions to Eqs. (35), (37), and (39) need not be repeated here,
but their contrasting traits can be inferred from the qualitative sketches in
Fig.2. For “small times” the telegraph solution is wavelike; for large
times” it is diffusionlike. It is this combination of asymptotic attributes
which resembles turbulent diffusion (Taylor, 1921). The initial conditions
here are a Dirac function for P(z, 0) and, for tbe equations which require an
addition condition, a finite P,,(r, 0). Figure 3 shows the solution of the telegraph equation for a normal (Gaussian) initial distribution,
P,, + T ’ P , = V2P,, .
(40)
P(e, 0) = expI +},
and for two different dimensionless times. The solid c u m is for t/T = 2, the
dashed one for t/T = 5. Both ordinates are scaled so that P,, = 1. Here the
wavelike propagation of the initially localized P(z, 0) function has nearly
disappeared by the time r = 5T. The abcissa widths ate nearly the same
because t is the same for the two curves; T was varied.
The solution to Eq. (39), even with a Dirac function initial condition,
implied for Fig. 2. is a bit complicated (Goldstein, 1951; Monin and Yaglom,
1965, 197 I), but ordinary differential equations for the second moments are
easily deduced from the three transport equations, Eqs. ( 3 9 (37), and (39).
They arc, writing D as V’T,
d d / d t - 2V2T = 0,
d2a2/dt2 - 2V2 = 0,
and
(43)
d2a2Jdt2 + T - ’ dUa/dt - 2V2 = 0.
With initial conditions a(0) = 0 [for all] and u(0) + V [like turbulence
(Taylor, 1921). for Eqs. (42) and (43)]. the solutions are’
(4)
t
?
= 2V2Tt,
(45 )
a’ = v5’.
’ Of course. a great variety of n(r) behaviors can bc achieved with the diffusion equation by
making the diffuaivity a function of time. Suppose Eq. (35) is generalized to P, = D(r)P,,.
Taylor (IYM ) pointed out that, at least for the unshcclrd caw. turbulent dispersion from a
point source can be empirically described this way, with D(r) = 4 d Z2/dr. where Z is the fluid
point diapliicement. Solutions to the dificrential equation are obtained simply by observing that
thr D ( i ) can be iibaorbed into il rescaled time variable, (11, f D(r)dt. which gives the classical
diffusion eqiiation. Only the initial conditions pose a rlipht chllcnge.