270
8 Transport in the Oceans and Coastal Zone
The solution of Eq. (8.47), under the condition ac' /az = 0 at z = -h/2 and
z = h/2, is:
1 ac jZ jZ2
c'(z) = - -
u'(zl)dz1dz2 + c'( -h/2).
D ax -h/2 -h/2
(8.48)
Now, the rate of mass transport in the streamwise direction becomes:
j
h/2
1 ac jh/2
jZ jZ2
Q =
u'c'dz = - -
u'(z)
u'(zl)dz1dz2dz.
-h/2
D ax -h/2
-h/2 -h/2
(8.49)
Thus, the total mass transport along the stream is proportional to the concentration gradient in the streamwise direction, which is exactly the same as
for molecular diffusion (see Eq. 8.12). However, now the result is related to
the flow integrated over the whole cross-section. Using this similarity, we can
rewrite Eq. (8.49) in the form:
ac
Q = -hKx-·
ax
(8.50)
The coefficient Kx, although similar to the diffusion coefficient, expresses the
diffusive property of the velocity distribution; it is known as the longitudinal
dispersion coefficient. So, from Eqs. (8.49) and (8.50) we have:
-1 jh/2
jZ jZ2
Kx = -
u'(z)
u'(zl)dz1dz2dz.
hD -h/2
-h/2 -h/2
(8.51 )
The longitudinal dispersion coefficient, Kx, is a measure of the diffusive property of the velocity on the macroscopic scale of all sections, while the diffusion
coefficient, D, is a measure of the molecular diffusivity of fluid on a microscopic
scale. Using Eq. (8.50), we can stipulate that the one-dimensional dispersion
equation has a form similar to Eq. (8.19):
(8.52)
It is straightforward to extend similar analysis to turbulent shear flow. The
only differences are a somewhat different velocity profile and the possibility that
the turbulent dispersion coefficient, K, may be a function of cross-sectional position, y. For uniform flow and in the absence of density gradients, the dispersion coefficient is constant and equal to the turbulent diffusion coefficient. In
non-uniform flow, when the transported constituent induces density gradients,
the dispersion coefficient depends on the distance.
Extension of Taylor's method for unsteady shear flow and dispersion in two
dimensions are treated in the books by Csanady (1973) and Fisher et al. (1979).
8 Transport in the Oceans and Coastal Zone
The solution of Eq. (8.47), under the condition ac' /az = 0 at z = -h/2 and
z = h/2, is:
1 ac jZ jZ2
c'(z) = - -
u'(zl)dz1dz2 + c'( -h/2).
D ax -h/2 -h/2
(8.48)
Now, the rate of mass transport in the streamwise direction becomes:
j
h/2
1 ac jh/2
jZ jZ2
Q =
u'c'dz = - -
u'(z)
u'(zl)dz1dz2dz.
-h/2
D ax -h/2
-h/2 -h/2
(8.49)
Thus, the total mass transport along the stream is proportional to the concentration gradient in the streamwise direction, which is exactly the same as
for molecular diffusion (see Eq. 8.12). However, now the result is related to
the flow integrated over the whole cross-section. Using this similarity, we can
rewrite Eq. (8.49) in the form:
ac
Q = -hKx-·
ax
(8.50)
The coefficient Kx, although similar to the diffusion coefficient, expresses the
diffusive property of the velocity distribution; it is known as the longitudinal
dispersion coefficient. So, from Eqs. (8.49) and (8.50) we have:
-1 jh/2
jZ jZ2
Kx = -
u'(z)
u'(zl)dz1dz2dz.
hD -h/2
-h/2 -h/2
(8.51 )
The longitudinal dispersion coefficient, Kx, is a measure of the diffusive property of the velocity on the macroscopic scale of all sections, while the diffusion
coefficient, D, is a measure of the molecular diffusivity of fluid on a microscopic
scale. Using Eq. (8.50), we can stipulate that the one-dimensional dispersion
equation has a form similar to Eq. (8.19):
(8.52)
It is straightforward to extend similar analysis to turbulent shear flow. The
only differences are a somewhat different velocity profile and the possibility that
the turbulent dispersion coefficient, K, may be a function of cross-sectional position, y. For uniform flow and in the absence of density gradients, the dispersion coefficient is constant and equal to the turbulent diffusion coefficient. In
non-uniform flow, when the transported constituent induces density gradients,
the dispersion coefficient depends on the distance.
Extension of Taylor's method for unsteady shear flow and dispersion in two
dimensions are treated in the books by Csanady (1973) and Fisher et al. (1979).