ANDERSON, S. MILES, AND A. C. CHARTERS. A fluid dynamics

OLOGY
LIMN
May 1982
AND
Volume
OCEANOGRAPHY/
Ltmnol.
Oceclnogr.,
27(3), 1982, 399412
@ 1982, by the American
Society of Limnology
and Oceanography,
Marine
Science
Institute,
3
Inc.
A fluid dynamics study of seawater
through Gelidium nudifronsl
S. Miles Anderson
Number
27
flow
and A. C. Charters
University
of California,
Santa Barbara 93106
Abstruct
Gelidium
nudifrons,
growing in the s&tidal
region of semiexposed
coasts has a thallus
composed of rodlike branches which are closely spaced, but the structure is sufficiently
open
so that water flows freely through it. The turbulence
of the water before and after passing
through the plant was measured by hot-film anemometry in a water tunnel in the laboratory
and observed by a dye stream technique in the tunnel and in situ in the sea. The flow after
passing through the plant was smooth at velocities
up to a critical value, but turbulent
for
velocities above an abrupt transition. Turbulence
in the flow entering the plant did not change
this result.
Analysis of the results demonstrates that the plant strongly suppresses turbulence
in the
flow entering its thallus and at the same time generates microturbulence
of its own at velocities above critical values varying from 6 to 12 cm. SK’, depending on the diameters and spatial
density of the branches, due to the formation
of systematic vorticity
in the wakes of the
individual
branches. The transition in the flow induced by the branches of a marine plant is
probably a phenomenon
of considerable
adaptive significance,
because the turbulence
generated by the plant itself or by neighboring
plants may be the only turbulence in water motion
past the plant that is of the right scale to enhance nutrient uptakes and effect the exchange
of gases and solutes.
Water motion is important to the life
processes of aquatic macrophytes.
Currents and wave surge stress the plants,
distribute their reproductive
propagules,
affect the adhesion
of spores to substrates, and enhance the uptake of nutrients and the output of gases and other
substances (e.g. see Charters et al. 1969,
1973; Gerard and Mann 1979; Koehl and
Wainwright
1977; Westlake 1967; Wheeler 1980). Water motion affects an aquatic
plant through the transport of mass, momentum, and energy from the main body
of the fluid through the boundary layer to
the plant’s surface. The transport rates
’ This research was supported
ENG76-22720 and OCE79-09241.
by NSF
grants
depend strongly on whether the flow in
the boundary layer is laminar or turbulent, because the rates in turbulent flow
with eddy diffusion are orders of magnitude greater than those in laminar flow
with molecular diffusion
(Prandtl 1935;
Schlichting
1968). Thus, transition from
laminar to turbulent
flow would imply
dramatic changes in hydrodynamic
forces
(transport of momentum) and in diffusion
of nutrients (transport of mass).
Gelidiunz is a bushy plant that grows in
the subtidal
regions
of semiexposed
coasts washed continually
by wave surge.
The seawater
flows back and forth
through the branched thallus of the plant.
Whether the flow through its thallus is
laminar or turbulent could influence the
forces and the uptake of nutrients. If the
flow entering the thallus is smooth, will
399
Gelidium
jiuid
bert Anchorage on the south side of Santa
Cruz, the largest of the Channel Islands
(33”59’N,
119”42’W),
and studied
the
morphology of many representative
specimens. They seemed similar in branching
structure;
the average diameter
of 10
fronds selected from five plants was 0.046
cm (SD = 0.01 cm), supporting the view
that G. nudifrons
has a uniform
morphology (Abbott and Hollcnberg
1976).
Laboratory experiments
were done in
our low-velocity
water tunnel (Charters
and Anderson 1980; Fig. 2). In normal
operation, the flow in the working section
is steady and uniform. The velocity
is
constant across 80% of the diameter, and
the turbulence
level is low, 0.5% or less.
A dye stream injected in the flow control
section upstream of the working section
flows down through the tunnel along its
centerline as a fine, straight, unwavering
line. In one series of tests, we artificially
introduced
turbulence
into the flow by
removing
the honeycomb
and screens
from the flow control section and placing
a large cylindrical
object directly over the
exit port from the reservoir to the flow
control section. The turbulence thus gcnerated had a topology similar to that in
the ocean (based on dye stream observations).
The streamlines
of the flow in the
working section were recorded by dye
stream photographs taken with a 35-mm
camera and flash units backlighting
a diffusion screen. The dye, a 0.05% solution
of methyl blue in seawater, was injected
into the flow upstream of the working
section through an L-shaped hypodermic
tube (see Fig. 2). The tube was so thin
that its wake was smooth and laminar.
The position of the tube was adjustable
so that the dye stream could be positioned anywhere in the cross section of
the tunnel. In tests with artificial turbulence, we injected the dye just upstream
of the plant to prevent its dispersal by the
turbulent flow. The exact location of the
dye stream did not affect the topology of
the flow; i.e. whether the emergent flow
was laminar or turbulent did not depend
on the location of the dye stream.
Velocity
and turbulence
were mea-
dz.pnmics
401
c
Fig. 2. Sketch of low-velocity
water tunnel. Axis
of tunnel is vertical. Water flows from reservoir (H)
at top, through a honeycomb
(H), four fine Nytex
screens (SC), and a convcrg,rc’ncc section into the
working section, from thcrc through a second convergencc section to an orifice locntctl in the orifice
plate valve (OPV), emerges from the orifice as a jet
at ambient prossurc, and pours into the slunp (S).
Water from the sump is returned to the reservoir 1)~
a pump (I?). Water also flows from the reservoir over
a weir and returns directly to the sump, as shown.
A scparatc hydrnlllic
circuit circulates water from
the sump through a chiller (C) to maintain wntcr
temperatllre
at a desired vall~:. Main circuit of the
tunnel has a dye injector (DI) and an ancmometcr
probe (AI?), which can 1~ placed in a calibrate position (CAL) above the GeZidizm plant (G) or in a
test position (TEST) l~low the plant. Working section used is a plastic tube 5 cm in diamctcr and 100
cm long. Total height of tunnel from bottom of sump
to top of reservoir is 3 m.
402
Anderson
and Charters
sured with
a hot-film
anemometer
(McQuivey
1973; Hinze 1975). The anemometer probe was a cylinder (0.015 cm
in diameter and 0.200 cm long) made of
a quartz rod coated with a thin film of
platinum and covered with a thin layer of
quartz, which could be placed at several
stations along the working section and
moved across its diameter. The axis of the
cylindrical
sensing element was normal
to the flow direction. The probe was heated with an electric current and cooled by
the flow of water so that its temperature
was determined by the balance between
the two. The electric current was controlled by the anemometer circuit so that
the probe’s temperature
could be held
constant at a preset value.
The anemometer
was connected
to
three output devices simultaneously:
a
strip-chart
recorder for calibrating
and
monitoring,
a true RMS meter for measuring the intensity of turbulence,
and a
frequency meter for measuring the frequency of velocity changes when they
were sinusoidal. The output of the anemometer is a voltage that depends on the
probe temperature,
the water temperaturc, and the water velocity.. The velocity
of the flow being measured is determinecl
from the output voltage of the anemometer by calibrating
the anemometer in a
flow whose temperature and velocity are
known. In practice, the temperatures
of
the probe and the water were held constant during the calibration
and the run,
and only the velocity of the water was
varied during the calibration. The output
voltage was measured by a digital voltmeter and recorded on a strip-chart recorder.
The intensity of turbulence
is defined
as the root-mean-square
value of the velocity fluctuations,
namely
where u’ is the intensity
of turbulence
and U is the velocity. Usually, the value
of the turbulence
intensity is given as a
percent of the mean velocity, lOO(u’lu).
The intensity
of turbulence
was determined from the fluctuations in the output
voltage of the anemometer whose mean-
square value was measured directly by a
true RMS meter.
The basic anemometer circuit (model
1053B), the averaging
circuit
(model
1047), and the hot-film
probe (model
1210) were purchased from TSI, Inc. The
digital voltmeter (model SOOOA) and the
frequency
meter (model 1900A) were
purchased from the John Fluke Mfg. Co.
and the strip-chart
recorder
(model
Speed Servo II) from the Esterline Angus
Corp. The true RMS circuit was designed
and built
in-house
(UCSB,
Physics
Dept.).
The frequency response of the complete anemometer
circuit (probe, basic
anemometer
circuit,
averaging
circuit,
and true RMS meter) was flat from 1.0 to
200 IIz; the lower limit was set by averaging circuit, the upper limit determined
from the manufacturer’s
specifications.
The size of eddy (72)that can be measured
is approximately
related to the flow velocity (U) and frequency (f) of the velocity fluctuations
as
h = ulf.
With a representative
velocity
of 10
cm. s-l, effective eddy diameters rccorded by the anemometer ranged from 0.05
to 10 cm.
The hot-film anemometer is difficult to
USC because the calibration
is hard to
maintain. The heat balance of the probe
is sensitive to the condition of its surface,
and the formation of a bubble or the accretion of debris on the surface changes
the calibration.
The manufacturer
recommends that the probe be operated at
40°C above ambient water temperature,
but we found that bubbles tend to form
at this temperature and reduced the operating temperature
to 20°C above ambient (a loss in sensitivity offset by a gain
in stability). We also found it necessary
to filter the water to a mean particle size
of 0.45 p and an absolute size of 8 p on
filling the tunnel and to continue filtering
during operation (Browand pers. comm.;
Reischman
pers. comm.). If a bubble
forms or debris collects on the surface of
the probe, the output voltage decreases
slowly from its undisturbed
value. Re-
Gelidium
5
4
1
fluid
dynamics
.
7-
0
6-
t
5s
-+
0
0
403
-
"
I
4
I
8
I
I
12
I
16
20
U, cm/set
I
I
l
24
28
32
I
36
Fig. 3. Variation of intensity of turbulence, u’/u,
in flow of water downstream
of a Gelidium
nuclifrom with water velocity, U, for smooth flow in the
water
tunnel
(intensitv
of turbulence
< 0.5%).
Lines indicate ;mlv mahnitudes
of turbulence
intensitv in fiow and &rugt transition from smooth to
turbulent flow rather than exact variation of intensity with velocity.
I
cording the output voltage on a stripchart recorder (SCR) was invaluable
in
detecting
incipient
malfunction
of the .
anemometer.
For tests in the water tunnel, individual plants were supported in the center
of the working section by fine nylon line
tied to the walls of the tunnel and to the
base end of the stipe, which was severed
just above the holdfast
(see Fig. 2).
Groups of five plants were composed by
attaching individual
plants to a fine nylon
linc, one plant below another, and fastening the line to a yoke at the entrance to
the working section; the group was distributed over the length of the working
section .
Two rods were tested separately in the
water tunnel, spanning the tunnel’s diameter near the entrance of the working
section, one of 0.19-cm diameter
and
another of 0. lo-cm diameter.
All of the in situ flow observations
were made at Albert Anchorage, a typical
habitat for G. nudifrons.
On the day WC
made our observations,
a gentle wave
surge generated a slow flow of water back
and forth, around and through the plants
on the sea floor. Velocities were cstimatof 10
ed to range up to a maximum
cm. s-l. Dye from a large hypodermic
syringe was injected into the water near a
plant and observed as the wave surge car-
43-
14.3 cm/s
2-
12.8
cm/s
l-
0-
9.2 cm/s
-1.0
0
+1.0
y/r
Fig. 4. Variations
of intensity
of turbulence,
u’/U, in flow of water downstream
of a Gelidium
nuclijhas
across working section at three velocities
for smooth flow in the water tunnel (intensity
of
tl&ulcnce
< 0.5%). Abscissa is ratio of distance
from ccntcr, y, to radius of working section, r.
ried the clye stream through the water adjaccnt to it. Dye was also injected directly
into the “bushy”
thallus of a plant and
observed as the water flowed through the
plant and downstream.
The experiment
was rcpeatcd with three separate plants.
Results
Lnhorntory:
Plant in smooth flowThe flow emerging downstream from the
thallus of the plant was smooth at velocities < 12 cm. s- 1 and turbulent at velocitics > 14 cm. SK’. Figure 3 shows the
anemometer
measurements
of turbulence intensity on the tunnel’s centerline
10 cm downstream of the plant. Figure 4
shows similar measurements across a diameter of the tunnel at velocities of 9.2,
12.8, and 14.3 cm. s-‘. The rapid transition from smooth to turbulent flow is evident in both figures. The lines drawn in
Fig. 3 indicate only general levels of turbulence, because the actual values at velocitics
above the transition
will vary
with lateral position (see Fig. 4); our intent is to illustrate
the ordcr-of-magnitude increase in intensity of turbulence
through transition
from laminar to turbulent flow rather than to plot an exact
variation of intensity with velocity.
Patterns of the flow downstream of the
Gelidium
fluid
10 to 12 cm. s-l; with groups of plants it
was lower, varying from 6 to 8 cm. s-l.
The decrease in transition
velocity
on
going from single plants to groups of
plants suggests that the transition value
depends on the density of branches in
the flow path, not only on a single dimension characteristic of the plant’s morphology.
Laboratory:
Plant in turbulent flowArtificial turbulence was generated in the
flow through the control and working sections, and the previous experiments were
repeated. The intensity
of artificial turbulence in the flow upstream of the plant
is shown as a function of velocity by the
dashed line in Fig. 6. With turbulent flow
in the tunnel, the condition
of the flow
emerging downstream of the thallus was
the same as with smooth flow in the tunnel; that is, the emerging
flow was
smooth at velocities < 12 cm * s-’ and turbulent
at velocities
> 14 cm. s-l with
either smooth or turbulent flow entering
the plant. Figure 6 shows the anemometer measurements of the intensity of turbulence on the tunnel’s centerline 10 cm
downstream of the plant. Figures 3 and
6 show the same rapid transition
from
smooth to turbulent flow downstream regardless of whether the flow entering the
plant is smooth or turbulent.
Again, the
solid lines drawn in Fig. 6 indicate only
general levels of turbulence.
Patterns of flow downstream
of the
plant were marked with dye streams as
in the previous experiment, and the flow
patterns observed resembled
those in
Fig. 5: both dye stream patterns and anemometer records show that the emergent
flow from the thallus of this particular
plant is smooth at velocities < 11 cm * s-l
and turbulent
at velocities
> 14 cm. SC’,
regardless of turbulence
in the entering
flow.
Laboratory:
Rod in smooth flow-The
dynamics of the flow around a slender,
circular cylinder
oriented with its axis
normal to the direction of a uniform flow
depends on the Reynolds number:
Re = dUlv
where
Re is Reynolds
number,
d is rod
405
dynamics
2l0
0
I
4
u
I
6
Y
I
12
I
16
I
20
I
24
I
28
I
32
I
36
U. cm/set
Fig. 6. Variation of intensity of turbulence, t&‘/U,
in flow downstream
of a Gdidium
nudifrons
(0)
with water velocity,
U, for turbulent
flow in the
water tunnel. Intensity
of turl~ulcncc
in flow in
working section upstream of plant is shown 1,~ triangles and dashed line. Solid lines indicntc only
magnitudes
of turbulence
intensity
in flow and
abrupt transition
from smooth to turMent
flow,
rather than exact variation of intensity with velocity.
diameter, U is flow velocity, and v is kinematic viscosity (the dynamic viscosity
divided by the density). The Reynolds
number is a measure of the ratio of inertial to viscous forces in the flow field: at
low Reynolds numbers, viscous forces
predominate;
at high Reynolds numbers,
inertial
forces are controlling
(Prandtl
1935).
In our experiments,
a dye stream impinging on the rod showed that the flow
passed smoothly
around the rod, converged a short distance behind, and left
a smooth , laminar wake free from discrete vortices at low Reynolds numbers
up to a value of about 40. When we increased the Reynolds number to a value
a little >40, the dye stream showed a dramatic change in flow pattern. A series of
vortices formed at the rear of the rod, first
a vortex on one side and then a vortex on
the other. The vortices separated from
the rod and traveled with the flow, forming an evenly spaced series alternating
from side to side of the wake. The series
of periodic vortices is shown in Fig. 7; it
is similar to a phenomenon known as the
Karman vortex street (Karman
1911,
1912).
The velocity fluctuations
generated in
the wake by the vortices were sinusoidal
Gelidium
fluid
incremental change with the onset of vorticity in the wakes of the fronds. The energy in the turbulence also varies directly
with the square of the velocity. IIcnce,
the intensity
of turbulence,
defined as
the ratio of the root-mean-square
of the
velocity fluctuations
to the mean vclocity, is suppressed uniformly
at all velocities, and this process is not velocity-dependent.
The two processes seem to act independently,
and the net cffcct is the sum
of the two. If the turbulence
of the entering flow is suppressed strongly, as it
was in our experiments,
the only turbulence in the flow leaving the thallus is
the microturbulence
generated by the plant itself at velocities above the critical
value. Turbulence
in the entering flow
therefore has little effect on whether the
flow leaving the thallus is laminar or turbulent.
Our hypothesis is based on the unique
morphology
of this plant. The slender,
rodlikc segments of the fronds are its basic units, and the structure of the plant
can bc viewed as a three-dimensional,
irregular latticework of short, slender rods.
Our tests of a slender rod in the water
tunnel, under flow conditions
similar to
those of the rodlike
segments of the
plant, gave us an immediate familiarity
with the flow around a rod. The fact that
our results were in excellent agreement
with those of Kovasznay (1949) and Roshko (1954) supported our belief that flow
in the tunnel’s working section is suitable for fluid dynamics tests-a
belief
previously
based only on measurements
of velocity and turbulence
(Charters and
Anderson 1980).
A rod segment, more favorably oriented than its neighbors, probably gcneratcd
the vortex street shown in Fig. 5 (center).
The Reynolds number, based on the average branch diameter of 0.046 cm and
the flow velocity of 12.8 cm-s-l, was 59,
a value above the critical value of 40. A
dye stream impinging
on a leading
branch showed systematic vorticity in the
wake at a velocity of 12.8 cm. s-r. The
shedding frequency was estimated to be
dynamics
407
34 Hz; the shedding frequency
of the
Karman vortex street of a 0.046-cm-diameter circular cylinder normal to a flow
of water at 12.8 cm-s-’ would have been
38 Hz (Roshko 1954). The vortex pattern
differs from the Karman vortex street (cf.
Figs. 5 and 7), but the difference is probably due to the finite length of the branch
segment. The circular rod tested in the
tunnel spans the tunnel and is effectively
infinite in length, because the vortices in
its wake can terminate
on the tunnel
walls. The branch segment does not span
the tunnel and is finite in length; hence,
the vortices in its wake cannot terminate
on the tunnel’s walls and must form a
continuous, interlocking
pattern of loops
with one vortex joined to the next, as
seen in Fig. 5, because vortex lines cannot begin or end in the fluid but only on
its boundaries (Goldstein 1965).
Increasing the flow velocity from 12.8
to 14.3 cm. s-l brought
the Reynolds
numbers of many branch segments above
their critical values, and vortex streets
were generated in many regions of the
thallus. These vortex streets merged and
interacted
to produce
turbulent
flow
(Roshko 1976; Liepmann
1979). The
branches were closely spaced and turbulent flow probably began well within
the thallus.
Multiscreen
analog
of plccnt-Our
view of the plant as a three-dimensional
latticework
of short, slender rods suggests that a mechanical
analog of the
plant would be a structure consisting of
several screens spaced one after another
in the direction
of flow. Thus, we can
compare our experimental
results for
plants with theoretical results for a set of
screens.
Each screen is a substructure
composed of two sets of evenly spaced, parallel, circular rods; the sets arc oriented
perpendicular
to one another. The series
of screens gives the structure its threedimensional
form. The plant differs from
its analog in the arrangement of the basic
units -the
rods: a disorderly
arrangement in the plant, an orderly arrangement in the screen analog.
408
Anderson
and Charters
Screens have been used for many years
to produce or suppress turbulence
in
wind tunnels. The flow through a screen
exhibits the same two fluid dynamics processcs that we postulate for flow through
a G. nudifrons.
There is a transition from
laminar to turbulent
flow downstream of
a screen at a critical Reynolds number
(Dryden and Schubaucr 1947; Schubauer
et al. 1950). At low Reynolds numbers,
the flow passes smoothly
through the
screen and the rods of the screens leave
smooth wakes free from discrete vortices.
At a critical Reynolds number, the rods
of the screens generate vortex streets,
which merge to generate turbulent
flow
a short distance downstream.
The mcasurements of Schubauer et al. (1950) of
turbulence
intensity
in the downstream
flow show a transition that is almost the
same as that shown in Fig. 3 (their fig. 8).
The value of the critical Reynolds number depends not only on the diameter of
the rods (d) but also on the spacing between rods (M, mesh spacing); the critical value decreases from 65 at M/d = 9.5
to 30 at M/d = 2.8.
Turbulence
is also suppressed in flow
passing through a screen (Dryden and
Schubauer 1947; Schubaucr et al. 1950;
Taylor and Batchclor
1949; Townsend
1951). The energy of the turbulence
is
dissipated by the resistance of the screen
to the flow passing through it and by the
action of the screen in straightening
the
flow. The energy of the main flow is dissipated also and this loss of energy generates a pressure drop across the screen,
if the screen spans the flow passage (as
it does in the control section ofthe water
tunnel: see Fig. 2). A series of screens
both damps the turbulence and generates
a pressure drop in the flow through the
series, as does a single screen, but the
series is more effective
than a single
screen in damping
turbulence,
even
though the series of screens is designed
so that the cumulative
pressure drop
through the series is the same as the pressure drop through the single screen. Dryden and Schubauer (1947) gave an example in which a single screen reduced
the turbulence intensity by a factor of 0.3,
and 10 screens, having the same pressure
drop as the single screen, reduced the
turbulence
intensity by a factor of 0.03,
ten times greater than the single screen.
Theoretical
fluid dynamics model of
the plant-The
multiple
screen analog
can be carried a step further by designing
the screens on the basis of dimensions of
the plant. The analog then becomes a
fluid dynamics model of the plant amenable to quantitative
analysis. The model
may be useful in studies of the flow of
seawater through plant communities
on
the sea floor. The properties of the flow
through the model can be determined
theoretically
from published
studies:
damping of turbulence
from the work of
Schubauer et al. (1950) and Drydcn and
Schubauer (1947), turbulence
generation
from that of Townsend
(1951), Stewart
and Townsend (1951), and Batchelor and
Townsend (1948). The properties of the
flow through the model are determined
from theory; the properties
of the flow
through the plant were measured in the
water tunnel. Hence, comparison of the
flow properties computed for the model
with those measured for the plant will
validate the model.
The first step is to determine the diameter of the rod and the total length of
rod available
for manufacture
of the
screens. The rod diameter is assumed to
be the average diameter of the plant’s
branches. The total length of rod is assumed to be equal to the combined
length of all of the branches; its value can
be computed
from the weight
of the
plant, assuming that all branches have
the same diameter and that the material
of the plant has the density of water.
The second step is to determine
the
shape and size of each of the screens. It
is reasonable to assume that the external
contour of the model approximates
the
envelope of the thallus. A representative
plant filled the cross section of the working section of the tunnel (see Fig. 1,
right);
accordingly,
the area of each
screen is assumed equal to the cross-sectional area of the working section; however, the shape of each screen is assumed
to be a square (in order to simplify the
Celidium
fluid
calculation).
This step determines
the
length of each rod and the total number
of rods.
The third step is to determine the spacing between rods. The value of the ratio
of rod spacing (M) to rod diameter (cl) is
assumed equal to 16:3. This value is arbitrary and is not directly related to the
plant’s morphology,
in contrast to our
previous assumptions, but is a value used
in many experiments
reported in the literature. Thus, the assumption of 16:3 for
the spacing ratio greatly facilitates computation of the fluid dynamics properties
of the model from published
results.
Comparison
of computed
(model) and
measured (plant) values will assess its
validity.
This step dctcrmines
the numbcr of rods in each screen and the number of screens.
The final step is to determine the spacing between screens in the streamwise
direction. The streamwise spacing is not
critical
to the fluid dynamics
performance of the series of screens as long as
it is greater than a few mesh spaces. The
streamwise distance from the first to the
last screen is assumed to be twice the
screen width (see Fig. 1, right). Computing the spacing between screens checks
that the spacing is several mesh spaces or
greater.
Model design exercise-The
following
exercise illustrates the design of a multiple-screen
model of one of the plants
tested in the water tunnel (Figs. 1, 3, 4,
5, 6). The measured value of the average
branch diameter
was 0.046 cm (using
0.04663 cm will simplify
the calculations); the plant weighed 1.80 g; hence,
the total length of rod should be 1,054
cm. The cross-sectional arca of the working section of the tunnel was 19.6 cm;
hence, the width of each screen is 4.43
cm, the length of each rod is also 4.43 cm,
and 238 rods should be used to build the
screens. If we assume that 17 rods are
spaced across one dimension
of each
screen (34 rods per screen), the spacing
between rods is 0.246 cm, because the
walls of the tunnel bound the flow and
w = M(n + l), where w is screen width,
M is mesh spacing, and n is half the num-
409
dynamics
ber of rods per screen. Hence, the spacing ratio is 5.28 (sufficiently
close to the
assumed value of 16:3 = 5.33 for design
purposes), and the number of screens is
7. If the streamwise distance from screen
1 to screen 7 is twice the screen width of
4.43 cm, then the strcamwisc spacing between screens is 1.5 cm, a value equal to
six mesh spaces.
The damping of turbulence by the seven-screen model can be computed from
data on the pressure drop across a screen
given by Schubauer et al. (1950) and from
values of the turbulence reduction factors
given by Dryden and Schubauer (1947).
Our calculations give a turbulence rcduction factor for the model of 0.11; the measured turbulence reduction factor for the
plant was 0.13 (see Fig. 6). Computed and
measured values for turbulence damping
are in good agreement.
At flow velocities corresponding
to supercritical
Reynolds numbers, the intcnsity of turbulence
generated
by the
downstream screen, its principal source,
can be computed from the formula developed
by Batchelor
and Townsend
(1948) for a screen with a mesh spacing
to rod diameter ratio of 5.58. The formula
gives the variation of the_reciprocal of the
intensity of turbulence, U/u’, with the ratio of the distance downstream
of the
screen, X, to the mesh spacing, M, namely
(l%~‘)~ = 147[(x/M)
- (x,/M)]
where the quantity, x0/M, is an arbitrary
constant which dcpcnds on the Reynolds
number
of the mesh spacing,
Re,+l =
MU/v (U, velocity;
v, kinematic viscosity). The smallest value of Re, in the experiments
of Batchelor
and Townsend
was 2,800, for which they list x,/M = 20.
Using x,/M = 20 and x/M = 37, corresponding to the distance of the hot-film
anemometer probe downstream from the
bottom of the plant, we compute the intensity of turbulence
to bc 2.1%. The
measured value of turbulence
intensity
at the highest value of Re, = 836, corresponding
to U = 34 cm. s-l, was 3.2%
(see Fig. 6). The value computed from
Batchelor and Townsend’s formula agrees
with the measured value if x,/M = 29, an
410
Anderson
acceptable number considering
the difference in Re, and the fact that their
measurements show an increase in x,/M
with decrease in Re, (see table 1: Batchelor and Townsend
1948). Hence, computed (model) and measured (plant) values for the intensity
of turbulence
generated by the screen are in satisfactory agreement,
as arc computed
and
measured values of the damping of turbulence, thereby validating our model.
Possible effects of transition
on the hiology of Gelidium nudifrons and similar
macroalgae-The
effects of transition
from smooth to turbulent flow in the flow
of seawater through G. nzrdifrons or similar macroalgae on the uptake of nutrients
and other life processes arc a matter of
conjecture. A careful survey of the litcrature of both marine and terrestrial plants
has left us with few guidelines. The most
evident fluid dynamics effect is the ordcr-of-magni tude increase in transport
rates with transition from smooth to turbulent flow, but fluid dynamics is only
one aspect of the complex of interactions
of the mechanical, chemical, and biological systems of living organisms. Estimating the change in transport
rates
caused by transition is difficult because
suitable examples are lacking. Most studies of mass transfer are concerned with
the flow over streamlined
bodies or
through pipes at high Reynolds numbers
(Eckert and Drake 1959; Schlichting
1968). The flow through the thallus of a
macroalga, on the other hand, involves
flow over a bluff body at low Reynolds
numbers.
The change in mass transport
with
transition from laminar to turbulent flow
in a smooth, round tube may serve to illustrate the effect of transition,
even
though the flow configuration
is entirely
different from the case in hand. Flow in
a tube offers the opportunity
to study the
effect of transition alone, apart from the
effects of changes in the dimensional
parameters of velocity
and diameter, because the flow may be either laminar or
turbulent
in the same tube at the same
velocity under circumstances
depending
and Charters
on the condition of the flow entering the
tube (Schlichting
1968).
A formula for the ratio of mass transport
in turbulent
flow to that in laminar flow
can be derived from the theory presented
by Eckert and Drake (1959):
JT/JI, = (0.0064)Re7/8 SP
where JL is mass transfer rate for laminar
flow and JT is mass transfer rate for turbulent flow; Re is Reynolds number, defined by Re = Udlv, where U is velocity,
d is diameter of the tube, and v is kinematic viscosity; SC is the Schmidt number, defined by SC = v/D, where D is the
diffusion
coefficient
for the solute (nutrient) in water and v is kinematic viscosity, as before. For nutrient solutes in
water, D has the order of magnitude of
lo-” cm* * SC’ and v of 10Pz cm”. s-l; hence,
SC has the order of magnitude of 1,000.
The flow is always laminar for Reynolds
numbers
~2,000
(Schlichting
1968);
hence, Re = 2,000 is the minimum value
for turbulent
flow. With SC = 1,000 and
Rc = 2,000 WC find that the value ofJT:J,
is 16, indicating
that the mass transport
rate in turbulent flow will bc an order of
magnitude
greater than in laminar flow
for the case of flow in a smooth, round
tube.
Oceanic waters are nearly always turbulent
(McLellan
1965). How might
suppression
of this turbulence
benefit
the plant? The answer to this question
depends on the scale as well as the intensity of the turbulence,
because fluid
turbulence
is like light in that its complete specification
involves not only the
intensity
but also the spectral distribution of energy. For turbulence to interact
with the flow around or through a body,
the scale of the turbulence must be comparable to that of the body. For example,
if the scale is much larger than the dimensions of the body, the turbulence
will not affect the fluid dynamics processes in the immediate neighborhood
of
the body; the magnitude and direction of
the main flow will vary in response to its
large-scale turbulence,
but the flow in
the boundary layer, where the transport
Gelidium
fluid
of fluid properties takes place, will retain
the same character, laminar or turbulent,
that it would have had if the main flow
were completely
smooth (Dryden
and
Schubauer 1947). Aeronautics provides a
cogent example. To quote from Drydcn
and Schubaucr (1947, p. 221): “In free
flight it has been found that there are no
disturbances of sufficiently
small scale to
produce
appreciable
aerodynamic
effects-i.e.,
the turbulence
of the atmosphere may be regarded as zero.”
The similarity of the fluid atmospheres
of air over the earth’s surface and of water
over the sea floor leaves us with a disturbing thought. For a plant like G. nudifrons,
whose representative
dimensions have the order of 0.1 cm, the flow
due to currents and wave surge may be
completely
smooth insofar as the effects
of turbulence
are concerned. Hence, the
only turbulence
with
a scale small
enough to modify the transport process
may be the turbulence
gcnerntcd by the
plant itself or by neighboring
plants.
Should this be the case, the transition
due to the generation of periodic vorticity
in the flow over the plant’s branches
could play a critical role in the life processes of the plant, because this turbulence may be the only source of eddy diffusivity
available
to it. Othcrwisc,
molecular
diffusivity
will prevail,
because the oceanic water motions have effectively zero turbulence and the flow in
the boundary layers of the plant will always be laminar at the small Reynolds
numbers involved.
Conclusions
The latticework
structure
of bare
branches comprising the thallus of G. nudifrons interacts with the water motions
of waves and currents by modifying
the
turbulence
of the flow passing through
the plant’s structure. If the external flow
is turbulent, this turbulence is damped as
the water flows through.
At the same
time, turbulence
is generated in the flow
through the plant if the velocity of the
flow is greater than a critical value. Because turbulence
in the entering flow is
dynamics
411
actually heavily damped, the net effect is
a smooth internal and exiting flow at velocities below the critical value and an
abrupt transition to turbulent flow at velocities
above. The critical
velocity
ranges from 6 to 12 cm* s-l depending on
the spacing density of the branches.
A series of screens placed one after
another both damps and generates turbulence in the flow through them, as the
plant does. The screen analog can be developed into a fluid dynamics model of
the plant by designing the screens from
the plant’s morphology.
The simulation
is quantitative,
and the model is amcnablc to numerical analysis.
How transition from smooth to turbulent flow will affect the life processes of
the plant is not known. Experience with
mechanical systems suggests that transition may enhance the uptake of nutrients
an d exchange of gases and other substances. In any case, it is possible that the
only effective turbulence experienced by
the plant may bc that generated by the
plant itself or by neighboring
plants. No
disturbances
are of sufficiently
small
scale to produce
appreciable
aerodynamic effects and the turbulence
of the
atmosphere can bc regarded as zero. The
correspondence
between the fluid atmospheres of air over the earth and of water
over the ocean floor suggests that the
flow due to currents and wave surge in
the ocean may effectively
be completely
smooth in fluid dynamics
interactions
with bushy macroalgae such as G. nudifrons.
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Submitted:
Accepted:
27 December
17 September
1979
1981