Marine Outfalls 32.3 Predicting Near-Field Dilutions 717
Part C | 32.3
In the far-field, the wastewater is passively transported by the ambient currents and the rates of dilution
are much lower than in the near-field. Far-field mixing
is dominated by natural processes, over which the design engineer has little control.
While the use of both near-field and far-field models may be necessary for the detailed design of a marine
outfall, we argue that near-field modeling alone may be
adequate for the initial design and emphasis in the following sections is placed on near-field modeling.
This section provides a broad introduction to nearfield modeling. Wood et al. [32.4] provide considerable
detail on near-field modeling and many of the problems
that may be encountered in the design of a marine outfall.
32.3.1 Physical Models
While the focus of this section is on near-field numerical modeling, it is recognized that physical modeling
can also play an important role in the design of a marine outfall. Scaled physical models of prototype marine
outfalls are sometimes constructed in the laboratory
and used to examine the behavior of jets and plumes
in the near-field. They provide good visualization of
the plumes, particularly interactions between multiple
plumes and include effects that are not found in most
numerical models.
The fluids used in the model are typically fresh
and saline water. The scales for such models are expressed as a ratio of a prototype quantity to a model
quantity. Model design requires the selection of (i) the
fluids which yield the reduced gravity ratio (g
0 defined
below), (ii) the length scale to ensure that the model
Reynolds numbers are sufficiently high to guarantee
turbulent model flows and (iii) a scaling criterion which,
in this case, is the densimetric Froude number (FR ),
i. e., there is a point-to-point correspondence of FR in
prototype and model. This scaling criterion, together
with the length scale, yields the velocity scale. Other
scales for time, pressure and buoyancy force can then
be determined. Although the inclusion of ambient currents in physical models is possible, it places greater
demands on laboratory facilities and data acquisition
systems.
Results from physical models may include information on dilutions, trajectories, the velocity field, and
interactions between neighboring plumes.
32.3.2 Positively Buoyant Jets and Plumes
Two basic approaches to near-field numerical modeling
are available: Eulerian and Lagrangian. A Lagrangian
approach is followed in both Lee and Cheung [32.5] and
Tate and Middleton [32.1, 6]. Central to either approach
are the conservation equations for mass, momentum,
and buoyancy. In a Lagrangian framework, they are:
Mass conservation
@ ..V/
@t
D a f ent U ent A
Momentum conservation
@ ..Vu i /
@t
D U i
@ ..V/
@t
C g
0 V
Buoyancy conservation
@ .g
0 V/
@t
D DN
2 u i V ;
where is the density of the jet/plume, a is the density
of the ambient fluid, f ent is an entrainment function, U ent
is the entrainment velocity, V is the volume of a buoyant fluid element, A is the cross-sectional area through
which ambient water is entrained, u i is the velocity of
the buoyant fluid, U i is the velocity of the ambient fluid,
g
0 is the buoyancy modified gravity (D .g/ == ref ,
where ref is a reference density), and N is the Brunt–
Väisälä frequency
N D
s
 g
ref
à @@ a
@z
:
These governing equations are also applicable to negatively buoyant jets and plumes.
If the buoyant jet/plume (a) lies well away from
its source (i. e., beyond the influence of the initial momentum), (b) is moving with the ambient fluid, and
(c) the Boussinesq approximation is applied, then the
above equations can be solved analytically to give what
is known as the asymptotic results. These equations
(Table 32.2) are equivalent to the advected thermal
equations in Wood et al. [32.4] and the corresponding
flow classifications are detailed in Jirka and Akar [32.7]
and Jirka and Doneker [32.8].
Solutions to the asymptotic governing equations for
positively buoyant plumes emerging from round (i. e.,
axisymmetric) outlet ports and from a slot (i. e., line
source), in a flowing ambient fluid, with both linearly
stratified or nonstratified marine waters, are presented
in Table 32.2. It should be noted that these asymptotic
solutions below should only be used at the conceptual
stage of outfall design. For preliminary and detailed
design, the full set of conservations equations above
should be used and solved numerically.
The entrainment function has evolved from the constants used in Morton et al. [32.9] to a complex function
of the densimetric Froude number, plume geometry, the
velocity of the fluid inside the plume, and the velocity of the ambient current [32.5]. Wood et al. [32.4] use
Part C | 32.3
In the far-field, the wastewater is passively transported by the ambient currents and the rates of dilution
are much lower than in the near-field. Far-field mixing
is dominated by natural processes, over which the design engineer has little control.
While the use of both near-field and far-field models may be necessary for the detailed design of a marine
outfall, we argue that near-field modeling alone may be
adequate for the initial design and emphasis in the following sections is placed on near-field modeling.
This section provides a broad introduction to nearfield modeling. Wood et al. [32.4] provide considerable
detail on near-field modeling and many of the problems
that may be encountered in the design of a marine outfall.
32.3.1 Physical Models
While the focus of this section is on near-field numerical modeling, it is recognized that physical modeling
can also play an important role in the design of a marine outfall. Scaled physical models of prototype marine
outfalls are sometimes constructed in the laboratory
and used to examine the behavior of jets and plumes
in the near-field. They provide good visualization of
the plumes, particularly interactions between multiple
plumes and include effects that are not found in most
numerical models.
The fluids used in the model are typically fresh
and saline water. The scales for such models are expressed as a ratio of a prototype quantity to a model
quantity. Model design requires the selection of (i) the
fluids which yield the reduced gravity ratio (g
0 defined
below), (ii) the length scale to ensure that the model
Reynolds numbers are sufficiently high to guarantee
turbulent model flows and (iii) a scaling criterion which,
in this case, is the densimetric Froude number (FR ),
i. e., there is a point-to-point correspondence of FR in
prototype and model. This scaling criterion, together
with the length scale, yields the velocity scale. Other
scales for time, pressure and buoyancy force can then
be determined. Although the inclusion of ambient currents in physical models is possible, it places greater
demands on laboratory facilities and data acquisition
systems.
Results from physical models may include information on dilutions, trajectories, the velocity field, and
interactions between neighboring plumes.
32.3.2 Positively Buoyant Jets and Plumes
Two basic approaches to near-field numerical modeling
are available: Eulerian and Lagrangian. A Lagrangian
approach is followed in both Lee and Cheung [32.5] and
Tate and Middleton [32.1, 6]. Central to either approach
are the conservation equations for mass, momentum,
and buoyancy. In a Lagrangian framework, they are:
Mass conservation
@ ..V/
@t
D a f ent U ent A
Momentum conservation
@ ..Vu i /
@t
D U i
@ ..V/
@t
C g
0 V
Buoyancy conservation
@ .g
0 V/
@t
D DN
2 u i V ;
where is the density of the jet/plume, a is the density
of the ambient fluid, f ent is an entrainment function, U ent
is the entrainment velocity, V is the volume of a buoyant fluid element, A is the cross-sectional area through
which ambient water is entrained, u i is the velocity of
the buoyant fluid, U i is the velocity of the ambient fluid,
g
0 is the buoyancy modified gravity (D .g/ == ref ,
where ref is a reference density), and N is the Brunt–
Väisälä frequency
N D
s
 g
ref
à @@ a
@z
:
These governing equations are also applicable to negatively buoyant jets and plumes.
If the buoyant jet/plume (a) lies well away from
its source (i. e., beyond the influence of the initial momentum), (b) is moving with the ambient fluid, and
(c) the Boussinesq approximation is applied, then the
above equations can be solved analytically to give what
is known as the asymptotic results. These equations
(Table 32.2) are equivalent to the advected thermal
equations in Wood et al. [32.4] and the corresponding
flow classifications are detailed in Jirka and Akar [32.7]
and Jirka and Doneker [32.8].
Solutions to the asymptotic governing equations for
positively buoyant plumes emerging from round (i. e.,
axisymmetric) outlet ports and from a slot (i. e., line
source), in a flowing ambient fluid, with both linearly
stratified or nonstratified marine waters, are presented
in Table 32.2. It should be noted that these asymptotic
solutions below should only be used at the conceptual
stage of outfall design. For preliminary and detailed
design, the full set of conservations equations above
should be used and solved numerically.
The entrainment function has evolved from the constants used in Morton et al. [32.9] to a complex function
of the densimetric Froude number, plume geometry, the
velocity of the fluid inside the plume, and the velocity of the ambient current [32.5]. Wood et al. [32.4] use