8.3 Concentration of Matter for Molecular and Turbulent Diffusion
263
After substituting Eq. (8.23) into Eq. (8.24) and integrating we obtain:
M = A.;;Jj.
(8.25)
Thus, Eq. (8.23) becomes:
M
(z2 )
c(z, t) = - - exp - - .
VIr Dt
4Dt
(8.26)
Equation (8.24) indicates that the amount of diffusing substance remains constant and equal to the amount originally deposited at the surface, z = O.
Concentration, c, tends to zero, as the distance from the surface increases. For
t = 0 it vanishes everywhere except at z = 0, where it becomes infinite.
In Fig. 8.4, vertical distributions of concentration, c, are shown for some time
instants after to. It is assumed that the diffusion coefficient D = 2 X 10- 9
m 2 /s. This is a typical value for the diffusion coefficient for oxygen in water of
temperature 20°C (Denny, 1993). Therefore, the values of Dt equal to 0.1, 1,
5 and 10 correspond to times of 5 x 10 7 S ("-' 579 days), 2.5 X 10 8 s (2893 days),
0.5x10 9 s (5787 days), and 5x10 4 s (57870 days). It takes almost 8 years for
particles to descend 3.2 m below the water surface. Thus, molecular diffusion
is an extremely inefficient mechanism of transport. However, as we showed in
the previous section, flows in the ocean are generally turbulent, and we need
to define a turbulent equivalent of the molecular diffusion coefficient, and find
methods for the calculation of substance concentration in turbulent flows.
8.3.2 Concentration of Matter for Turbulent Diffusion
Governing Equations. Turbulent flow is a random motion characterized by
fluctuations of water velocity as well as fluctuations of concentration of matter
suspended in the fluid. Therefore, we are not able to determine the instantaneous values of concentration at a given point and at a given time. However, in
defining the turbulence equivalent of a concentration equation we can exploit
the analogy between the transport of mass and the transport of momentum,
a process we examined in Sect. 2.4. Thus, we represent the instantaneous
concentration and instantaneous velocity components as a summation of their
mean values and fluctuation components, i.e.:
c
u
c + c'
ii + u ' , V = V + Vi, w = tV +w ' }.
After substituting the representation (8.27) into Eq. (8.17) we obtain:
(8.27)
(8.28)
263
After substituting Eq. (8.23) into Eq. (8.24) and integrating we obtain:
M = A.;;Jj.
(8.25)
Thus, Eq. (8.23) becomes:
M
(z2 )
c(z, t) = - - exp - - .
VIr Dt
4Dt
(8.26)
Equation (8.24) indicates that the amount of diffusing substance remains constant and equal to the amount originally deposited at the surface, z = O.
Concentration, c, tends to zero, as the distance from the surface increases. For
t = 0 it vanishes everywhere except at z = 0, where it becomes infinite.
In Fig. 8.4, vertical distributions of concentration, c, are shown for some time
instants after to. It is assumed that the diffusion coefficient D = 2 X 10- 9
m 2 /s. This is a typical value for the diffusion coefficient for oxygen in water of
temperature 20°C (Denny, 1993). Therefore, the values of Dt equal to 0.1, 1,
5 and 10 correspond to times of 5 x 10 7 S ("-' 579 days), 2.5 X 10 8 s (2893 days),
0.5x10 9 s (5787 days), and 5x10 4 s (57870 days). It takes almost 8 years for
particles to descend 3.2 m below the water surface. Thus, molecular diffusion
is an extremely inefficient mechanism of transport. However, as we showed in
the previous section, flows in the ocean are generally turbulent, and we need
to define a turbulent equivalent of the molecular diffusion coefficient, and find
methods for the calculation of substance concentration in turbulent flows.
8.3.2 Concentration of Matter for Turbulent Diffusion
Governing Equations. Turbulent flow is a random motion characterized by
fluctuations of water velocity as well as fluctuations of concentration of matter
suspended in the fluid. Therefore, we are not able to determine the instantaneous values of concentration at a given point and at a given time. However, in
defining the turbulence equivalent of a concentration equation we can exploit
the analogy between the transport of mass and the transport of momentum,
a process we examined in Sect. 2.4. Thus, we represent the instantaneous
concentration and instantaneous velocity components as a summation of their
mean values and fluctuation components, i.e.:
c
u
c + c'
ii + u ' , V = V + Vi, w = tV +w ' }.
After substituting the representation (8.27) into Eq. (8.17) we obtain:
(8.27)
(8.28)