Chapter 8 Fluid Dynamics in Seagrass Ecology
195
Bernoulli’s principle, which states that the sum of the
hydrostatic pressure and dynamic pressure along a
streamline are constant (Vogel, 1994), helps to explain flow-induced pressure changes (i.e. lift) that
occur within, around, and under seagrass canopies
(e.g. Nepf and Koch, 1999). Drag is another important force that acts downstream of obstacles. It has
two additive components: (i) the friction or viscous
drag that exists due to the interaction of the obstacle’s surface with the water, which can be defined
algebraically (i.e. F d = 1 / 2 C d ρAu
2 , where C d is the
drag coefficient, a shape and flow dependent constant); and (ii) the dynamic, form or pressure drag
(F p ) that exists under high flows when flows separate
from boundaries, which cannot be expressed algebraically and must, therefore, be determined empirically. As u increases, the dynamic drag contributes
a disproportionate fraction of the total drag. It is
important to note that drag is a force that operates opposite to the flow direction in that it “sucks”
a moving object upstream or a stationary object
downstream.
Water flow can exhibit a number of different
properties that depend on the temporal and spatial
scales under investigation. Water flow could either
be smooth and regular as if the fluid flows in layers
(i.e. laminar flow) or rough and irregular as if the
flow is “chaotic” (i.e. turbulent flow). This depends
on the velocity and the length scale (i.e. temporal and
spatial scale, respectively) under investigation as defined by the Reynolds number (Re = luρ/µ or more
simply Re = lu/ν; where l is the length scale appropriate for the hypothesis being tested). Re, which
is the non-dimensional ratio of inertial to viscous
forces in a fluid, defines four regimes that grade into
one another: (i) creeping flow (Re 1), which occurs at very low flows and spatial scales such as those
experienced by individual bacteria cells; (ii) laminar
flow (1 < Re < 10
3 ) as defined above; (iii) transitional flow (Re ∼ O(10
3 ); i.e. of the order of 10
3 ),
which involves the production of eddies and disturbances in the flow and is characterized by a critical
Re (Re crit ) defined for a particular geometry and flow;
and (iv) fully turbulent flow (Re 10
3 ). Associated
with these flow patterns are important differences
related to the fluid dynamic forces (e.g. friction vs.
pressure drag) and mass transfer processes (diffusion vs. advection) that operate under the different
regimes (see below; White, 1999; Kundu and Cohen,
2002). Moreover, because Re is scale dependent, it is
possible to experience multiple flow regimes simultaneously in the flow field depending on the spatial
scale under investigation. Consequently, flow is almost always turbulent at large spatial scales such
as seagrass beds, but it can also be laminar on the
scale of seagrass leaves and flowers (e.g. Ackerman
and Okubo, 1993; Koch, 1994). This not-so-subtle
distinction can influence the application and interpretation of physiological and ecological processes
in seagrass canopies (see Section III).
As indicated above, the flow conditions become
more complicated when water approaches a boundary (e.g. seagrass canopy, leaves, or seafloor, depending on the scale) or any obstacle for that matter. The
water cannot normally penetrate boundaries, except
for the most porous ones (see reviews in Boudreau
and Jørgensen, 2001; Okubo et al., 2002), and more
importantly, the water molecules directly next to a
boundary stick to the boundary rather than slip by
it. This no-slip condition leads to the development
of a velocity gradient perpendicular to the boundary (Fig. 1), as the velocity at the boundary will be
zero relative to the free stream velocity (U 0 ). As the
water flows downstream, the velocity gradient will
grow in size and a slower moving layer of fluid will
develop next to the boundary, which is referred to as
the boundary layer under turbulent conditions, otherwise technically it is a deformation layer (Prandtl
and Tietjens, 1934). This boundary layer, which is
defined by velocities <0.99 U 0 , has a thickness of
δ that is relatively small and can be expressed as a
function of Re and x. Initially it appears laminar in
nature, but the boundary layer will become turbulent when the local Re (Re x = ux/ν) approaches a
critical value of 3 to 5 × 10
5 , in the case of a flat
plate oriented parallel to the flow. In nature, this transition is accelerated by the presence of roughness
or obstacles on the boundary (Schlichting, 1979;
Nikora et al., 2002; Fig. 1) including undulations
on macroalgal blades (Hurd and Stevens, 1997). In
addition to the streamwise structure in a fully developed boundary layer, there is important vertical
structure as well. The first layer directly adjacent to
the boundary is the viscous sublayer (δ v ≈ 10ν/u ∗
where u ∗ is the friction velocity, which is a velocity
scale that provides an indication of the mass transfer
within the boundary layer) in which the forces (or
stresses if surface forces are considered) are largely
viscous, and consequently the mass transfer in this
layer is slow and dominated by diffusion, especially
within the thin diffusional sublayer (also called the
diffusive boundary layer; DBL) at the bottom of this
195
Bernoulli’s principle, which states that the sum of the
hydrostatic pressure and dynamic pressure along a
streamline are constant (Vogel, 1994), helps to explain flow-induced pressure changes (i.e. lift) that
occur within, around, and under seagrass canopies
(e.g. Nepf and Koch, 1999). Drag is another important force that acts downstream of obstacles. It has
two additive components: (i) the friction or viscous
drag that exists due to the interaction of the obstacle’s surface with the water, which can be defined
algebraically (i.e. F d = 1 / 2 C d ρAu
2 , where C d is the
drag coefficient, a shape and flow dependent constant); and (ii) the dynamic, form or pressure drag
(F p ) that exists under high flows when flows separate
from boundaries, which cannot be expressed algebraically and must, therefore, be determined empirically. As u increases, the dynamic drag contributes
a disproportionate fraction of the total drag. It is
important to note that drag is a force that operates opposite to the flow direction in that it “sucks”
a moving object upstream or a stationary object
downstream.
Water flow can exhibit a number of different
properties that depend on the temporal and spatial
scales under investigation. Water flow could either
be smooth and regular as if the fluid flows in layers
(i.e. laminar flow) or rough and irregular as if the
flow is “chaotic” (i.e. turbulent flow). This depends
on the velocity and the length scale (i.e. temporal and
spatial scale, respectively) under investigation as defined by the Reynolds number (Re = luρ/µ or more
simply Re = lu/ν; where l is the length scale appropriate for the hypothesis being tested). Re, which
is the non-dimensional ratio of inertial to viscous
forces in a fluid, defines four regimes that grade into
one another: (i) creeping flow (Re 1), which occurs at very low flows and spatial scales such as those
experienced by individual bacteria cells; (ii) laminar
flow (1 < Re < 10
3 ) as defined above; (iii) transitional flow (Re ∼ O(10
3 ); i.e. of the order of 10
3 ),
which involves the production of eddies and disturbances in the flow and is characterized by a critical
Re (Re crit ) defined for a particular geometry and flow;
and (iv) fully turbulent flow (Re 10
3 ). Associated
with these flow patterns are important differences
related to the fluid dynamic forces (e.g. friction vs.
pressure drag) and mass transfer processes (diffusion vs. advection) that operate under the different
regimes (see below; White, 1999; Kundu and Cohen,
2002). Moreover, because Re is scale dependent, it is
possible to experience multiple flow regimes simultaneously in the flow field depending on the spatial
scale under investigation. Consequently, flow is almost always turbulent at large spatial scales such
as seagrass beds, but it can also be laminar on the
scale of seagrass leaves and flowers (e.g. Ackerman
and Okubo, 1993; Koch, 1994). This not-so-subtle
distinction can influence the application and interpretation of physiological and ecological processes
in seagrass canopies (see Section III).
As indicated above, the flow conditions become
more complicated when water approaches a boundary (e.g. seagrass canopy, leaves, or seafloor, depending on the scale) or any obstacle for that matter. The
water cannot normally penetrate boundaries, except
for the most porous ones (see reviews in Boudreau
and Jørgensen, 2001; Okubo et al., 2002), and more
importantly, the water molecules directly next to a
boundary stick to the boundary rather than slip by
it. This no-slip condition leads to the development
of a velocity gradient perpendicular to the boundary (Fig. 1), as the velocity at the boundary will be
zero relative to the free stream velocity (U 0 ). As the
water flows downstream, the velocity gradient will
grow in size and a slower moving layer of fluid will
develop next to the boundary, which is referred to as
the boundary layer under turbulent conditions, otherwise technically it is a deformation layer (Prandtl
and Tietjens, 1934). This boundary layer, which is
defined by velocities <0.99 U 0 , has a thickness of
δ that is relatively small and can be expressed as a
function of Re and x. Initially it appears laminar in
nature, but the boundary layer will become turbulent when the local Re (Re x = ux/ν) approaches a
critical value of 3 to 5 × 10
5 , in the case of a flat
plate oriented parallel to the flow. In nature, this transition is accelerated by the presence of roughness
or obstacles on the boundary (Schlichting, 1979;
Nikora et al., 2002; Fig. 1) including undulations
on macroalgal blades (Hurd and Stevens, 1997). In
addition to the streamwise structure in a fully developed boundary layer, there is important vertical
structure as well. The first layer directly adjacent to
the boundary is the viscous sublayer (δ v ≈ 10ν/u ∗
where u ∗ is the friction velocity, which is a velocity
scale that provides an indication of the mass transfer
within the boundary layer) in which the forces (or
stresses if surface forces are considered) are largely
viscous, and consequently the mass transfer in this
layer is slow and dominated by diffusion, especially
within the thin diffusional sublayer (also called the
diffusive boundary layer; DBL) at the bottom of this