Journal of Plankton Research Vol.21 no.8 pp.1391–1421, 1999
Numerical study of the feeding current around a copepod
Houshuo Jiang1, Charles Meneveau2,3 and Thomas R.Osborn1,3
1Department of Earth and Planetary Sciences, 2Department of Mechanical
Engineering and 3Center for Environmental and Applied Fluid Mechanics,
The Johns Hopkins University, Baltimore, MD 21218, USA
Abstract. Three-dimensional, numerical simulations of the feeding current around a tethered
copepod were performed using a finite-volume code. The copepod’s body shape was modeled to
resemble Euchaeta norvegica, and was represented by a curvilinear body-fitted coordinate system. In
the simulations, the appendages that generate the feeding current were replaced by a distribution of
forces acting on the water adjacent and ventrally to the body. First, the accuracy of the code was
verified by simulating two viscous, zero-Reynolds-number flows for which analytical solutions are
available. Then, simulations with realistic body shape and Reynolds numbers were carried out. The
main features of the computed feeding current were compared with observations from Yen and
Strickler (Invert. Biol., 115, 191–205, 1996), and good agreement was obtained. The entrainment
region, as visualized by tracking particles in the feeding current and by plotting the resulting streamtube, is quite large. The result can be used to quantify how the copepod takes advantage of the feeding
current to trap the algal particles in its capture area. The configuration of the feeding current near to
the body surface of the copepod is controlled by how the copepod forces the feeding current and by
the copepod’s morphology. These parameters were varied and their effects studied in a systematic
manner. Specifically, by comparing various spatial distributions of the same amount of total force, it
was shown that a distributed force dissipates less energy (and increases the entrainment rate) than a
concentrated force, it is thus energetically more desirable. Variations of the copepod’s body shape
and of the distribution of forces showed little effect on the far field of the feeding current, and
therefore do not appear to affect the detectability by other mechano-receptional organisms. The
length scale of the influence field of the feeding current was shown to be anisotropic in three directions, extending 5–7 mm above or ventrally to the copepod, <1 mm dorsally to the copepod and >1 cm
down from the abdomen. The results also suggest that the net reaction force on the copepod from the
feeding current is of the same order of magnitude as the excess weight of the copepod, but is not
sufficient to balance the excess weight completely.
Introduction
Calanoid copepods are tiny planktonic crustaceans that play a key role in the food
chains of the oceans and lakes. Herbivorous calanoid copepods feed on algae and
some generate a feeding current (Strickler, 1982). Since understanding the
feeding current is crucial to answering the question ‘How do calanoid copepods
survive in a nutritionally dilute environment?’ (Conover, 1968), numerous efforts
have been undertaken to shed light on this important topic in trophic dynamics.
Early direct observations using high-speed microcinematography produced
important quantitative and qualitative information about the feeding current. For
example, Koehl and Strickler (1981) observed that the feeding current is a
laminar flow with low Reynolds number. The calanoid copepods generate the
feeding currents by employing the ‘fling and clap’ mechanism. The beating
frequency of the appendages is fairly high, ranging from 20 to 80 Hz, which
ensures the existence of a quasi-steady flow around the animal. Calanoid copepods are ‘suspension feeders’ (Alcaraz et al., 1980; Paffenhöfer et al., 1982).
Strickler (1982) reported that some copepods remain stationary or move at a
small translational speed in the water by generating a feeding current around the
© Oxford University Press
1391
H.Jiang, C.Meneveau and T.R.Osborn
body counteracting gravity. Physically, this result implies that the net force and
net torque on the body are zero or nearly zero. As to the biological function of
the feeding current, Strickler (1985) addressed two hypotheses. First, the feeding
current enables the animal, using either mechano- or chemoreception, or both,
to scan a large amount of water and hence benefits the animal’s feeding success.
Second, the feeding current also creates a hydrodynamic disturbance that may
alert the animal’s prey or arouse its potential predators.
In order to quantify the characteristics of the feeding current, one needs to
know the velocity field of the feeding current around the copepod. The laserilluminated video-imaging technique can determine the three-dimensional
coordinates of particles following the flow and then reconstruct the velocity field
from the coordinates. In the past 10 years, much work has been done with this
technique, providing quantitative information about the feeding current
(Strickler, 1982; Emlet, 1990; Yen et al., 1991; Yen and Fields, 1992; Fields and
Yen, 1993, 1997; Bundy and Paffenhöfer, 1996; Yen and Strickler, 1996). Yen
et al. (1991) reconstructed the feeding current fields around the copepods from
the algal particle paths. They also calculated the vorticity field, the shearing rate
field and the squared rates of strain. Bundy and Paffenhöfer (1996) analyzed the
flow fields associated with freely swimming calanoid copepods and concluded that
the observed differences in the flow fields between an omnivorous and a predominantly herbivorous calanoid are due to behavioral differences, i.e. changes in
swimming velocity, body orientation and the frequency of mouthpart movement,
and to differences in mouthpart morphology. Yen and Strickler (1996) qualitatively analyzed the small-scale fluid deformations created by copepods to document the complexity of the signal. Different species have different types of
motions and, therefore, different hydrodynamic patterns; different activities of
the same animal also lead to different hydrodynamic signals. Sometimes the
animal tries to conceal its hydrodynamic presence, sometimes it would like to
advertise. The Reynolds number, which reflects those differences in the hydrodynamic signals, spans from a low to an intermediate-to-high value corresponding to different species in different activities.
New techniques are available to observe and/or to analyze the feeding currents
around copepods. Very recently, van Duren et al. (1998) reported their analysis
of the feeding current around a fixed copepod by using laser sheet particle image
velocimetry (PIV). Their results show the difference between two velocity fields
generated by a foraging copepod and a hopping one, and support the argument
that the copepod activity determines the disturbance and hence the signal it
generates.
Theoretical and numerical methods are also employed to attack the problem.
Andrews (1983) coupled a feeding current into a chemical diffusion model and
found that the active space around an algal particle is deformed greatly by the
sheared feeding current and this effect benefits the copepod’s detection of algae.
Childress et al. (1987) studied the generation of scanning currents in Stokes flow
in a number of simple models. Osborn (1995) developed a model in which the food
is viewed as diffusing towards a copepod, which then uses the feeding current to
entrain the food particles, even though they are well beyond its perception radius.
1392
Numerical study of copepod feeding current
Such a model reveals the significant increase in food flux to the predator due to
the range of the feeding current.
The objective of the present work is to use computational fluid dynamics (CFD)
to gain additional insights into quantitative features of typical feeding currents
near some copepods. In particular, we wish to study the shape and extent of the
feeding current in three-dimensional space, and its relationship with how the
feeding current is generated through forcing of the water by the copepod. Effects
of various forcing distributions and copepod shapes will be studied, with particular emphasis on the feeding current, energetic efficiency, as well as far field
detectability by other predators or prey. The numerical methodology is based on
a state-of-the-art, finite-volume code, described in the next section. In the simulations, the appendages that generate the feeding current are represented by a
distribution of forces acting on the water adjacent and ventrally to the body. Thus,
a detailed force analysis is performed and presented, also in the next section. In
Results, the main features of the computed feeding current are compared to observations. Also, the relationship between the power expended by the animal and the
distribution of forces exerted on the water is examined. Moreover, the effects of
the distribution of forces and the copepod’s body shape on the drag and the viscous
dissipation are investigated. Conclusions are presented in the final section.
Dynamics, force analysis and numerical method
Forces on a copepod
Consider a freely swimming copepod in water with the flow around it, and assume
that the flow is generated by the animal. The forces acting on the body include
the buoyancy force, the body force due to gravity, and the forces exerted by the
flow on the copepod due to the distributions of the flow pressure and the shear
stress around the surface of the copepod. Figure 1 shows schematically the free
body diagram for the copepod. A ‘free body diagram’ represents the body under
consideration with the physical constraints removed and replaced by the forces
and torques they exert. Detailed descriptions of the forces and torques are given
in Table I.
Equations and simplifications
The governing equations of the feeding current around the copepod are the
Navier–Stokes equation and the continuity equation:
∂u
r —— + ru ·=u = –=p + µ=2u
∂t
(1)
and
=·u = 0
(2)
where r is the density of the fluid (~1.030 3 103 kg m–3 for sea water) and µ is the
dynamic viscosity (~1.390 3 10–3 kg m–1 s–1 for sea water). The density of the fluid
1393
H.Jiang, C.Meneveau and T.R.Osborn
Fig. 1. The free body diagram for a freely swimming copepod in water. G stands for the body force
due to gravity which acts at the center of mass; B stands for the buoyancy force which acts at the
center of volume; Fpd stands for the force due to the integral of the flow pressure (the flow-induced
pressure deviation from the hydrostatic pressure) over the whole body surface; Fvd stands for the force
due to the integral of viscous stress over the whole body surface. The distributed force due to the flow
pressure or viscous stress can be represented by an integrated force and a torque. The point of application for the force is arbitrary (we chose the center of mass), but the torque must be calculated about
the point of application for the force. The curved arrow labeled Tpd indicates the torque on the
copepod by the flow pressure distribution on the surface of the body. The curved arrow labeled Tvd
indicates the torque on the copepod by the viscous stress on the surface of the body.
Table I. Forces on the copepod
Force
Value
Torque exerted by the force on
copepod
Force due to gravity (G)
rcopepodgVcopepod
Buoyancy force (B)
–eVPhydrostaticndV
= –rwatergVcopepod
eVn · TdV
None if the center of mass is
the application point
None if the center of volume is
the application point
Calculable in principle, it
depends on the application
point for the force, e.g. the
center of mass
Calculable in principle, it
depends on the application
point for the force, e.g. the
center of mass
Force due to the integral of
viscous stress over the
surface of the body (Fvd)
Force due to the integral of
flow pressure over the surface
of the body (Fpd)
–eVpndV
rcopepod is the density of the copepod; rwater is the density of water.
Vcopepod is the volume of the copepod. g (a vector) is the acceleration due to gravity.
Phydrostatic is the hydrostatic pressure due to the effect of gravity; p is the flow pressure.
V stands for the surface of the copepod’s body.
n is the outward unit vector normal to the element dV.
1 ∂ui
∂uj
T is the shear stress tensor: T = 2µS, where Sij = — —— + —— , and µ is the dynamic viscosity of
∂xi
water.
2 ∂xj
1
1394
2
Numerical study of copepod feeding current
is assumed to be uniform, and hence the gravity is not considered since it only
contributes to a hydrostatic pressure and has no effect on the flow field.
Boundary conditions are needed in addition to equations (1) and (2) in order
to determine the flow. The boundary condition is no relative motion at the surface
of the body and no flow at infinity. For a freely swimming copepod, which may
rotate its body while moving in a translation velocity in the water, this boundary
condition on the body–fluid interface is quite complicated. The situation is greatly
simplified if the copepod is stationary, e.g. by tethering the copepod (or by
considering a copepod that creates a feeding current while hovering). There are
many species of copepods, each with more or less unique behaviors and each with
more or less different fluid mechanical characteristics. Tiselius and Jonsson
(1990) discussed the various copepod behaviors in detail, but, as a first step, we
limit the situation to the case of a tethered copepod in order to simplify the
computation and to make the explanation of the computational method more
easy. However, our method has the ability to simulate a swimming copepod.
Results from modeling a swimming copepod will be reported in a forthcoming
paper. Variations also exist in copepod morphology. In order to compare the
numerical results to the feeding current generated by a tethered Euchaeta
norvegica observed by Yen and Strickler (1996), the simulated copepod shape is
modeled after that of an E.norvegica. Hereafter, ‘the copepod’ refers to the tethered E.norvegica.
The tether supplies an extra force and torque to make the net force and torque
on the copepod equal to zero. In this case, the no-slip boundary condition on the
copepod’s body becomes zero fluid velocity on the fixed surface. The feeding
current is generated by the movement of the feeding current-generating
appendages (Paffenhöfer et al., 1982; Cowles and Strickler, 1983; Price and Paffenhöfer, 1986; Fields and Yen, 1993). Since the no-slip boundary condition applies on
that moving boundary, the fluid velocity at the surface bounding the appendages
is equal to the (time-dependent) appendage velocity. The configuration of the
feeding current is determined not only by the movement and morphology of the
feeding current-generating appendages, but also by the body shape itself.
Summarizing, on the surface of the main body (not including the feeding
current-generating appendages), say Vmb, a no-slip condition is satisfied:
u = 0, at Vmb
(3)
On the surface of the moving appendages, say Va, which generate the feeding
current, the no-slip boundary condition is:
u = U(t), at Va(t)
(4)
where U(t) is the time-dependent velocity of the appendages and Va(t) stands
for the surface of the moving appendages. In addition, at infinity, the fluid is at
rest, i.e.
u → 0, at infinity
(5)
1395
H.Jiang, C.Meneveau and T.R.Osborn
It is complicated to deal with the unsteady Navier–Stokes equation with timedependent moving boundary conditions, but, according to observation, already
at a short distance from the moving appendages, the flow field around a feeding
copepod is quasi-steady. The reason is that the frequency of the beating movement is fairly high (20–80 Hz; Koehl and Strickler, 1981) compared to the characteristic inverse time scale of the feeding current. Actually, according to the scale
analysis and the solution for an incompressible fluid bounded by an infinite plane
surface which executes a simple harmonic oscillation in its own plane (Landau
and Lifshitz, 1959; see also Panton, 1996), the appropriate length scale is d = √2n/v
(where n = µ/r is the dynamic viscosity and v is the oscillatory frequency), beyond
which the fluctuating velocity profile dies away. For the present problem with
v = 20 Hz, one obtains d = 0.03 mm. Thus, the fluctuating motion is only seen at
a distance below ~0.1 mm into the fluid. Thus, we can simplify the problem by
removing the unsteady terms [essentially time-averaging equations (1), (2) and
(4)], and by replacing the effects of the time-dependent boundary condition with
a forcing term added to the right-hand side of the Navier–Stokes equation.
An alternative explanation of this approach is as follows. We choose to separate the appendages that generate the feeding current from the main body of the
copepod, and hence the surface force exerted by the water on these appendages
can be separated from the forces acting on the surface of the main body. The
mean effect of the beating movement of the appendages may be taken as a set of
distributed forces exerted by the appendages on the water adjacent and ventrally
to the body. On the other hand, by Newton’s third law, the adjacent water exerts
equal and opposite forces on these appendages. The separation enables us to
avoid the moving boundary condition that is difficult to incorporate into numerical simulations. However, the copepod’s main body remains in the model and a
no-slip boundary condition is satisfied on its surface.
At this point, the governing equations for the feeding current are written as:
ru · =u = –=p + µ=2u + fa
(6)
=·u=0
(7)
and
with the no-slip boundary condition on the surface of the main body, Vmb:
u = 0, at Vmb
(8)
and the boundary condition at infinity:
u → 0, at infinity
(9)
In equation (6), fa (N m–3) represents the force field that models the effect of
the distributed appendages. Since the feeding current-generating appendages are
spatially distributed ventrally to the copepod, fa(x) is interpreted as a distributed
force field.
1396
Numerical study of copepod feeding current
Lighthill (1975) solved the singularly forced Stokes equations that govern the
inertialess flow induced by a flagellum. However, we have to keep the inertia term
in equation (6) because the Reynolds number of the feeding current is typically
not negligible.
Numerical method
For realistic body shapes, Reynolds numbers and force distributions, equations
(6) and (7) with boundary conditions (8) and (9) may only be solved using numerical methods. In this work, a commercially available, state-of-the-art, finitevolume code, FLUENT™, has been used.
The configuration of the feeding current is controlled by two factors. One is
body shape, which determines the stress applied in the water around the surface
of the copepod’s body through the no-slip boundary condition. The other is the
distribution of forces exerted by the copepod on some small parts of water
adjacent to the feeding current-generating appendages. FLUENT™ allows the
representation of both these effects. (i) By using curvilinear body-fitted coordinates, the body shape of the copepod can be smoothly portrayed without jagged
edges that would arise with Cartesian coordinates. (ii) It allows us to apply
momentum to any desired finite-volume cell. Thus, the discretized force distribution is applied on the small computational cells that are located near where the
copepod’s appendages are known to exist.
In order to limit the computational resources used, the domain is a 10 3 10 3
10 cm cubic box (Figure 2). The outer boundary condition at the six surfaces of
the cubic box is chosen as a fixed static pressure condition. A fixed-pressure
boundary condition is more accurate than a zero-velocity boundary condition,
since at large distance the decay of pressure is faster than that of velocity [see
equations (A8) and (A9)]. The effectiveness of the code and of using a fixed static
pressure condition has been verified, as described in the Appendix where two
zero-Reynolds-number flows for which analytical solutions are available are
simulated accurately.
The computational domain is discretized on 40 3 40 3 40 nodes. The copepod
is located at the center of the box. The body shape of the copepod is composed
first of a sphere of radius 1 mm, and located at the center of the domain. The
sphere occupies 5 3 5 3 5 cells, with indices in the range (I, J, K) = (19–23, 19–23,
19–23). The copepod contains two antennae and an abdomen, which have been
added to the sphere. The length from the anterior to the posterior is ~4 mm. The
whole computational domain with the copepod located at the center of the box
and the magnified copepod with the grid distribution along the grid slice I = 21
are shown in Figure 2.
Results
Features of the computed feeding current
As a viscous shear flow, the feeding current mainly depends on two factors. One
is the body shape of the copepod, which is the internal boundary of the feeding
1397
H.Jiang, C.Meneveau and T.R.Osborn
Fig. 2. (a) The cubical box-shaped computational domain with a copepod located at the center of the
box; the grid distribution along the grid slice I = 21 is also shown. (b) The magnified grid distribution
along the grid slice I = 21 with a focused ventral view of the copepod located at the center.
current. The other is the distribution of forces, which is the dynamic mechanism
generating the flow. Specifically, the feeding current depends on the magnitude
of the total force applied, on its spatial distribution, and on the distances between
the applied forces and the body surface of the copepod.
Flow. With the grid system and copepod shape shown in Figure 2, we calculate
the feeding currents for several cases with different distributions and magnitudes
of forces. Our approach is to choose a total force magnitude and spatial distribution that reproduces the general appearance and peak velocity observed in the
measurements by Yen and Strickler (1996). This case is then considered the baseline case. After some numerical experimentation, the distribution of forces illustrated in Figure 3 is chosen as the baseline configuration, with a total force of
efad3x = 1.21 3 10–6 N. The velocity field of the computed feeding current is
shown in Figure 4. Two peak velocity regions are observed, which are located
1398
Numerical study of copepod feeding current
ventrally to the body with a side-to-side symmetry about the median plane. The
maximum velocity obtained from the simulation is 19.8 mm s–1, and the horizontal
distance in the y-direction between points where the velocity decays from 19.0 to
1.5 mm s–1 is ~2 mm. All of these features are comparable to the observational
results obtained by Yen and Strickler (1996). The computed flow field ventrally
to the copepod shows the so-called double-shear field, just like the observation
by Strickler (1982): ‘One extending laterally from the median plane and another
parallel to the median plane’. The velocity vector field clearly shows that the
feeding current is bounded internally by the body surface of the copepod. The
flow goes smoothly around the copepod from the top to the bottom of the body.
Since the forces are exerted ventrally to the body, the flow field ventrally to the
body is much stronger than that dorsally to the body.
Vorticity. Contours of vorticity are shown in Figure 5. Figure 5a shows the
∂w
∂v
x-direction vorticity component —– – —– along the plane at x = 1.0 mm,
∂y
∂z
∂u ∂w
Figure 5b shows the y-direction vorticity component —– – —– along the plane
∂z
∂x
∂v
∂u
at y = 0.0 and Figure 5c shows the z-direction vorticity component —– – —–
∂x
∂y
along the plane at z = 0.0. It can be seen that the vorticity field extends around
the body and that the horizontal components of vorticity are one order of magnitude larger than the vertical components. There are regions of strong vorticity at
the places where the copepod moves its feeding current-generating appendages
to create the flow. The fields of x-direction and z-direction vorticity components
Fig. 3. Illustration of the distribution of forces that is used to produce the feeding current whose main
features are comparable to the observations. Forces are applied on 56 cells each on four I-planes:
I = 24, I = 25, I = 26, I = 27. The magnitude of each force applied on K = 18, 19 is 1.7 3 10–9 N, on
K = 20, 21 is 3.7 3 10–9 N, on K = 22, 23 is 5.7 3 10–9 N, and on K = 24, 25 is 7.7 3 10–9 N. The total
force is 1.21 3 10–6 N. For simplicity, all forces are applied along the negative z-direction.
1399
H.Jiang, C.Meneveau and T.R.Osborn
1400
Numerical study of copepod feeding current
are anti-symmetrical about the median plane, which implies a symmetrical flow
field about this plane. Simple Stokes flow models may not be able to reproduce
the spatial complexity of the observed vorticity field, since such simple analytical
models cannot represent the complex morphology of the body and the finite
Reynolds number of the flow.
Entrainment. The entrainment region is determined by particle tracking. Particle
locations xp(t) are determined from integrating
dxp
—— = u(xp)
dt
(10)
based on the calculated u, v and w velocity component fields. The time integration
scheme for equation (10) is a fourth-order Runge–Kutta method. At each time
step, the velocity needs to be interpolated onto the current particle location. Since
numerical calculation of the feeding current around the copepod is based on a
body-fitted coordinate system, non-rectangular cells cause some difficulties for
interpolation. Thus, a method for particle location and field interpolation on
complex, three-dimensional computational meshes (Oliveira et al., 1997) has
been used. It is based on an iterative procedure that uses transformed coordinates
defined by tri-linear isoparametric functions. The method has the ability to deal
with fairly arbitrary three-dimensional computational meshes. We have tested the
particle tracking and interpolation scheme for a particle in a Burgers vortex (see
Panton, 1996). The computational trajectory coincides almost exactly with the
analytical trajectory.
As initial condition, we chose a capture area consisting of a small ellipse
ventrally to the copepod. [The ellipse is centered at (1.8 mm, 0.0, 0.0) with a semimajor axis of 1.0 mm in the y-direction and a semi-minor axis of 0.5 mm in the xdirection.] Integrating equation (10) backward in time, we determined the particle
trajectories leading to the capture area, i.e. from where particles that end up in the
capture area have come. The union of such trajectories forms the streamtube
shown in Figure 6. The top, ventral and lateral views of the streamtube are also
Fig. 4. Velocity field for the simulated feeding current. (a) Ventral view, the velocity vectors are along
the plane of x = 1.6 mm ventrally to the copepod. (b) Lateral view, the velocity vectors are along the
median plane y = 0.0.
Fig. 5. Vorticity contours for the simulated feeding current. (a) x-direction, i.e. the dorsal–ventral
direction vorticity component along the plane x = 1.0 mm. (b) y-direction, i.e. right–left lateral direction vorticity component along the plane y = 0.0. (c) z-direction, i.e. the bottom–top direction vorticity component along the plane z = 0.0.
Fig. 6. A streamtube ventrally to the copepod, which is calculated from the simulated feeding current.
The time integration for particle tracking is from t = 0.0 s to t = –25.0 s. The red-colored ellipse labeled
t = 0.0 s is the capture area of the copepod. The yellow-colored closed curve labeled t = –1.0 s is the
positions of particles at 1 s before they reach the capture area, and so on for t = –4.0 s, t = –12.0 s and
t = –25.0 s. In order to express the three-dimensional structure of the streamtube, the center of the
capture area (the red star) is tracked from t = 0.0 s to t = –25.0 s, the positions are connected by the
red-colored line. The yellow star is the position at t = –1.0 s, the blue star is the position at t = –4.0 s,
the violet star is the position at t = –12.0 s and the green star is the position at t = –25.0 s.
1401
H.Jiang, C.Meneveau and T.R.Osborn
Fig. 7. Plane views of the streamtube in Figure 6. (a) Top view. (b) Ventral view. (c) Lateral view.
1402
Numerical study of copepod feeding current
shown in Figure 7 in order to visualize its three-dimensional structure more clearly.
The results show, among others, that particles at a distance of 2.5 mm above the
copepod require ~1 s to arrive at the capture area of the copepod (defined as the
small red-colored ellipse labeled t = 0.0 s in Figures 6 and 7). At a distance of 1 cm
above the copepod, the width of the streamtube is of the order of 1 cm, and it takes
a particle ~25 s to reach the capture area. The time labels corresponding to the
several different distances away from the copepod shown in Figures 6 and 7 also
show that the time that it takes a particle to be entrained into the capture area
increases rapidly with increasing distance. The figures quantitatively show that the
copepod takes advantage of the feeding current to feed on particles. Thus, the
numerical simulations provide us with a detailed flow field of a feeding current
that has features comparable to the observations.
In the following sections, the distribution of forces and the body shape of the
copepod in the numerical model are varied, and their effects on the configuration
of the feeding current, energetic efficiency and straining fields are studied. The
main advantage of the numerical simulation is that it allows us to vary parameters
and to study their impact in a systematic manner.
Power and distributions of forces
In this section, based on the shape of the copepod portrayed in Figure 2, we vary
the distribution of forces to examine the relationship between the power that the
copepod invests in generating the feeding current and the distribution of forces.
Six distributions of forces are illustrated in Figure 8. Each distribution has the
same total force applied to the water. For distribution N = 1, the entire force is
applied on the cell indicated in Figure 8 (top left) with a downward arrow. The
cell is located 0.82 mm (or three cells) ventrally to the body surface. For distribution N = 36, the same amount of force is divided equally among the 36 cells in
the same plane as before shown in Figure 8 (bottom right). Intermediate cases
for N = 4, N = 9, N = 16 and N = 25 are also shown in Figure 8. For simplicity, all
forces are applied along the negative z-direction. Then, a feeding current is
·
computed numerically for each distribution. The power input into the flow W (i.e.
the power that the copepod uses to generate the feeding current) is calculated
using the formula:
·
W=
N
o Fziwi
i=1
(11)
where the sum is taken over all the cells on which a force is applied, Fzi is the
force applied at the ith cell and wi is the z-direction velocity at the ith cell center,
as obtained from the numerical simulations. Figure 9 shows the normalized power
versus the index of distribution of forces, i.e. N = 1, 4, 9, 16, 25 and 36. As can be
seen, the power decreases rapidly with increasing N. If the copepod spreads out
a fixed amount of total force over a larger volume of water, it expends less energy
in generating the feeding current. The next question is whether the feeding
current for large N is as effective in bringing in food particles towards the capture
1403
H.Jiang, C.Meneveau and T.R.Osborn
Fig. 8. Illustration of six types of distributions of forces. The force is applied on the grid I = 26 for
each distribution. For N = 1, the total force is applied on the cell indicated by the downward arrow.
For N = 36, the total force is divided equally and applied on 36 cells indicated by downward arrows.
The same procedure is followed for N = 4, N = 9, N = 16 and N = 25.
area as it is for the low N case. To answer this question, we calculate the volume
flux through an area ventrally to the copepod using the data obtained from the
numerical simulations. The volume flux Q is defined as:
1404
Numerical study of copepod feeding current
Q = ee wdxdy
(12)
A
where A is near the capture area ventrally to the copepod. For simplicity in
numerically integrating over A, a rectangular region is chosen. The area is 1 mm
in the x-direction and 3 mm in the y-direction, and centered at the point (1.8 mm,
0.0, 0.0). The normalized volumetric flux as a function of the index of the distribution of forces is shown in Figure 10. Interestingly, the most spread-out distribution of forces, i.e. N = 36, results in the largest volume flux. Thus, a distributed
force dissipates less energy and results in a larger entrainment rate than a concentrated one does, and is thus energetically much more desirable.
This result may help us to understand the evolution of appendages better. The
copepod has several appendages to generate the feeding current, and on these
appendages there are many setae. The appendages, together with the setae on
Fig. 9. Normalized power input as a function of the index of the distribution of forces. The power is
normalized by the power for the distribution index 1, which is the largest power required.
Fig. 10. Normalized volumetric flux as a function of the index of the distribution of forces. The flux
is normalized by the flux for the distribution index 36, which is the largest resulting flux.
1405
H.Jiang, C.Meneveau and T.R.Osborn
them, serve to spread out the force and to exert the force on a fairly large volume
of water adjacent to the body in order to save energy and at the same time to
increase the flux.
Owing to the viscous nature of the feeding current, the energy that the copepod
spends to generate the feeding current must be dissipated by viscosity. Consequently, the more concentrated force generates a flow with a higher viscous dissipation rate. We will discuss this issue in more detail in the section on viscous
dissipation.
Effects of the distribution of forces and the body shape on drag
As the internal boundary of a shear flow, the body surface of the copepod is
subjected to a surface force (the drag) from the viscous feeding current. There
are two components contributing to the total drag: the friction drag and the pressure drag. As summarized in Table I, the friction drag can be calculated as the
surface integral of the shear force tangential to the body–fluid interface. The
surface integral of the flow pressure over the interface yields the pressure drag.
The vector sum of the friction drag and the pressure drag is the total drag on the
copepod.
The spatial distribution of the forces and the distance between the forces and
the surface of the main body control the magnitude of the velocity and the gradient of the velocity field. In addition, the shape of the main body affects the flow
field through the no-slip boundary condition. Thus, the distribution of the forces
and the shape of the main body determine the shear and normal stress field
around the copepod, and control the drag acting on the copepod.
In order to examine the effects of the applied force distribution and the body
shape on the drag, two types of distributions and three kinds of body shapes are
considered, and hence six cases are implemented and simulated. These six cases
are numbered A1, A2, B1, B2, C1 and C2, respectively (see Figure 11). (Actually, case A1 is the case already illustrated in Figure 3.) For distribution 1, forces
are applied on 56 cells each on four I-planes, say I = 24, I = 25, I = 26, I = 27. The
magnitude of each force applied on K = 18, 19 is 1.7 3 10–9 N, on K = 20, 21 is 3.7
3 10–9 N, on K = 22, 23 is 5.7 3 10–9 N, and on K = 24, 25 is 7.7 3 10–9 N. The
total force is 1.21 3 10–6 N. For distribution 2, four forces are applied on cells
numbered (24, 21, 21), (25, 21, 21), (26, 21, 21) and (27, 21, 21), respectively. The
magnitude of each force is 3.03 3 10–7 N and the total force is still 1.21 3 10–6 N.
Shape A corresponds to a full copepod with antennae and abdomen, shape B is
a copepod without the first antennae at the top but with the abdomen, and shape
C is a copepod without the first antennae and abdomen, i.e. a purely spherical
‘copepod’. For each case, a feeding current is obtained from the direct numerical
simulation and the drag forces on the surface of the main body are calculated
from the simulation results. The values are listed in Table II.
A direct conclusion drawn from Table II is that the y-direction drag, i.e. the
horizontally side-to-side drag, is negligible compared to the forces in the other
two directions. This result is expected from symmetry. Comparing the z-direction
drag forces of case A1 with A2, case B1 with B2, and case C1 with C2, we
1406
Numerical study of copepod feeding current
Fig. 11. Illustration of the six cases used to examine the effects of the distribution of forces and the
copepod’s body shape on the drag and viscous dissipation.
conclude that for the same shape of the main body and a given amount of total
force, different distributions of applied forces yield almost the same vertical drag.
However, the distribution of forces has some effect on the magnitude of the xdirection drag, i.e. the horizontally dorsal–ventral drag. The positive x-direction
1407
H.Jiang, C.Meneveau and T.R.Osborn
Table II. Drag forces on the surface of the main body caused by the feeding current
Pressure drag (N)
Friction drag (N)
Total drag (N)
Case A1
x-direction
y-direction
z-direction
2.100 3 10–8
–3.977 3 10–10
–1.288 3 10–7
1.578 3 10–8
4.688 3 10–11
–3.640 3 10–7
3.68 3 10–8
–3.51 3 10–10
–4.93 3 10–7
Case A2
x-direction
y-direction
z-direction
1.426 3 10–7
–5.821 3 10–10
–7.958 3 10–8
1.847 3 10–9
–5.022 3 10–10
–4.141 3 10–7
1.44 3 10–7
–1.08 3 10–9
–4.94 3 10–7
Case B1
x-direction
y-direction
z-direction
–1.461 3 10–8
–3.516 3 10–10
–1.124 3 10–7
2.911 3 10–8
–3.224 3 10–11
–3.284 3 10–7
1.45 3 10–8
–3.84 3 10–10
–4.41 3 10–7
Case B2
x-direction
y-direction
z-direction
1.339 3 10–7
–5.547 3 10–10
–7.335 3 10–8
7.615 3 10–9
–5.527 3 10–10
–4.084 3 10–7
1.42 3 10–7
–1.11 3 10–9
–4.82 3 10–7
Case C1
x-direction
y-direction
z-direction
1.508 3 10–8
–8.843 3 10–10
–1.095 3 10–7
3.158 3 10–8
–9.854 3 10–12
–2.984 3 10–7
4.67 3 10–8
–8.94 3 10–10
–4.08 3 10–7
Case C2
x-direction
y-direction
z-direction
1.860 3 10–7
–8.562 3 10–10
–6.526 3 10–8
9.638 3 10–9
–7.320 3 10–10
–3.457 3 10–7
1.96 3 10–7
–1.59 3 10–9
–4.11 3 10–7
drag, i.e. the drag in the direction from the dorsal side to the ventral side, is due
to the dorsal–ventral asymmetry of the feeding current. If the copepod concentrates the forces, the pressure drag by the feeding current in the dorsal–ventral
direction increases, while the friction drag decreases. However, the net drag by
the feeding current in the dorsal–ventral direction is increased.
On the other hand, for a fixed distribution of forces, the change in body shape
also has some effects on the drag, which can be seen directly by comparing cases
A1, B1 and C1, or comparing cases A2, B2 and C2. For example, the x-component
of pressure drag for case B1 reverses direction compared to cases A1 and C1. The
reason is quite complicated. Without the first antennae, the spread-out distribution of forces in case B1 is able to increase the flow going from the dorsal side
to the ventral side around the top of the body, and hence the pressure at the dorsal
side decreases. At the same time, the existence of the abdomen makes the effect
of the positive pressure ventrally to the lower body become dominant, and therefore results in the reverse of the direction of the pressure drag.
As to the vertical force balance, the copepod exerts a distributed downward
force on the adjacent water using its appendages to generate the feeding current
and, at the same time, the copepod receives an upward force of equal magnitude.
In addition, there is the downward drag acting on the copepod by the water due
to the flow of the feeding current, listed in Table II. Another force is the downward excess weight. With an excess density relative to sea water of 30 kg m–3
1408
Numerical study of copepod feeding current
(Tiselius and Jonsson, 1990), the excess weight is 1.57 3 10–6 N (the volume of
the copepod is 5.325 3 10–9 m3 for case A1). The vector sum of the downward
drag due to the feeding current and the upward reaction force is a net upward
force of 7.20 3 10–7 N. Therefore, the net vertical reaction force on the copepod
from the feeding current is a significant fraction of the excess weight of the
copepod, balancing about half of the excess weight. However, Strickler (1982)
observed that a copepod generating a feeding current can remain stationary or
move horizontally at a small translational speed in water, which indicates that the
net force and net torque on the body are zero, and that the force by the feeding
current balances the excess weight. The discrepancy with present results comes
from the fact that the calculated feeding current is generated by moving some of
the appendages ventrally to the copepod. In the real case, the copepod has many
other appendages (e.g. moving abdomen; Strickler, 1982) and may take advantage of the complex arrangement and movement of all the appendages to force
and adjust the flow around itself, and thus control the net force on the body in
order to balance the excess weight fully and to make the total force and torque
zero.
Viscous dissipation
Any shearing motion in the fluid is inevitably accompanied by a one-way transfer of energy from the mechanical agencies causing the motion to internal energy
of the fluid (Batchelor, 1967). The rate of this transfer of energy per unit mass of
fluid is called the viscous dissipation rate, defined for incompressible fluid as:
µ
f = 2 — SijSij
r
(13)
1 ∂ui ∂uj
where Sij = — —– + —– stands for the components of the strain rate tensor
2 ∂xj ∂xi
S, and the summation convention applies. SijSij is the trace of the tensor SS, and
is a useful measure of the (squared) magnitude of the flow deformation irrespective of directional information.
The viscous dissipation rate is calculated for case A1, in which a full body shape
of the copepod and a fairly realistic distribution of forces are considered. (Recall
that case A1 yields results comparable with observations on a real copepod; see
the previous section ‘Features of the computed feeding current’). As shown in
Figure 12, which shows filled contours of dissipation, the highest viscous dissipation rate (9.82 3 10–4 m2 s–3) occurs ventrally to the copepod, corresponding to
the region where the force is applied. This rate of energy dissipation is significantly higher than coastal oceanic energy dissipation rates in turbulent regimes,
which are typically in the range 10–7–10–5 m2 s–3 (Gargett et al., 1984).
Figure 13a shows the filled contours of the viscous dissipation rate for case A2,
which has the same body shape, but a more concentrated distribution of forces.
The latter leads to a higher valued and more concentrated distribution of the
viscous dissipation rate. (We have shown in the previous section, ‘Power and
1
2
1409
H.Jiang, C.Meneveau and T.R.Osborn
Fig. 12. Filled contours of viscous dissipation rate for the simulated feeding current of case A1. (a)
Contours along the plane x = 0.0. (b) Contours along the plane y = 0.0. (c) Contours along the plane
z = 0.0.
1410
Numerical study of copepod feeding current
distributions of forces’, that the more concentrated distribution of forces expends
more energy. The reason is that it must supply more energy to the flow to offset
the increased losses due to the higher viscous dissipation.) The highest viscous
dissipation rate is now 1.10 3 10–2 m2 s–3, due to the much higher shear around
the copepod. As to the effect of body shape, Figure 13b shows contours of the
viscous dissipation rate for case B2, which has the same distribution of forces as
case A2, but a body shape without the first antennae. Figure 13c shows the dissipation contours for case C2, which has the same distribution of forces as cases A2
and B2, but a body shape without the first antennae and the abdomen. The
highest viscous dissipation rates for cases B2 and C2 are quite the same as that
for case A1. The minor modifications of body shape slightly change the spatial
distribution of shear and dissipation. However, all changes are confined to a
region fairly close to the body. Neither the force distribution nor the body shape
appear to affect the far field of the feeding current very much.
In Figure 14, the viscous dissipation rate is plotted along lines in three orthogonal directions for cases A1, A2 and C2, to see how it decays with increasing
distance away from the copepod and how it depends on the distribution of forces
and shape. These results allow us to identify the distance below which the shear
or dissipation induced by the feeding current exceeds that of background turbulence. One can agree that at smaller distances the induced fluid deformation could
be sensed by other mechano-receptional organisms, while at larger distances the
signal may get swamped by the turbulence. Figure 14a shows the viscous dissipation rate along the line at y = 0.0 and z = 0.0. For x > 0, i.e. ventrally to the
copepod, ~7 mm away from the copepod, the viscous dissipation rate of the
feeding current is below the typical range of coastal oceanic energy dissipation
rate. For x < 0, i.e. dorsally to the copepod, the entire range is below the typical
range of coastal oceanic energy dissipation rate. Figure 14b shows the viscous
dissipation rate along the line at x = 1.76 mm and z = 0.0. In both ranges of y > 0
and y < 0, ~5 mm away from the copepod, the viscous dissipation rate of the
feeding current is below the typical range of coastal oceanic energy dissipation
rate. Figure 14c shows the viscous dissipation rate along the line at x = 1.76 mm
and y = 0.0. Above the copepod, i.e. z > 0, up to ~7 mm from the copepod, the
viscous dissipation rate of the feeding current exceeds the typical oceanic energy
dissipation rate. However, at z < 0, due to the persistence of the wake below the
abdomen of the copepod, even at 1 cm away from the copepod, the dissipation
of the feeding current is still not below the typical oceanic dissipation rates. The
influence field of the feeding current is quite anisotropic: above and ventrally to
the copepod, it extends perhaps to a distance of 5–7 mm. Dorsally to the copepod,
it is smaller than 1 mm. On the other hand, below the abdomen of the copepod,
the influence field extends over 1 cm. Notice that the asymmetry is due to the
finite Reynolds number of the simulations (in the case of a spherical copepod—
in the other cases, body shape can also introduce some asymmetry). Comparison
of cases A1 and A2 shows that the different spatial distributions of the same total
force do not change the length scale of the influence field very much. However,
case A2 results in a larger and more concentrated viscous dissipation rate near
the copepod. Thus, variation of the copepod’s body shape and the behavior
1411
H.Jiang, C.Meneveau and T.R.Osborn
Fig. 13. Filled contours of viscous dissipation rate for the simulated feeding currents with different
body shapes. (a) Contours for case A2 along the plane x = 0.0. (b) Contours for case B2 along the
plane x = 0.0. (c) Contours for case C2 along the plane x = 0.0.
1412
Numerical study of copepod feeding current
Fig. 14. Plots of viscous dissipation rate. (a) Along the line at y = 0.0 and z = 0.0. (b) Along the line
at x = 1.76 mm and z = 0.0. (c) Along the line at x = 1.76 mm and y = 0.0. The solid line is case A1,
i.e. the case with a full body shape of the copepod and a fairly realistic distribution of forces. The
dashed line is case A2, i.e. the case with a full body shape of the copepod, but a more concentrated
distribution of forces. The diamonds are case C2, i.e. the case with the same concentrated distribution
of forces as case A2, but a body shape without the first antennae and abdomen. The shaded regions
indicate the range of oceanic energy dissipation rate, i.e. 10–7–10–5 m2 s–3.
difference in generating the feeding current may not have a strong effect on the
detectability by other mechano-receptional organisms several centimeters away.
Summary and conclusions
1. The feeding current around a tethered copepod was simulated numerically
using a finite-volume code. The accuracy of the code was verified by simulating two viscous, zero-Reynolds-number flows for which the analytical solutions
are known. In the simulations, the body shape of the copepod was represented
by a curvilinear body-fitted coordinate system. The appendages that generate
the feeding current were replaced by a distribution of forces exerted on the
water adjacent and ventrally to the copepod.
2. Based on a realistic body shape and a distribution of forces that simulates the
spatial distribution of feeding appendages, we calculated a feeding current
with main features comparable to the observations by Yen and Strickler
(1996). The entrainment region of the computed feeding current was visualized by tracking particles in the feeding current and by plotting the resulting
1413
H.Jiang, C.Meneveau and T.R.Osborn
streamtube. The results quantitatively display the time and length scales of the
feeding current that entrains the algal particles into the copepod’s capture
area. The net reaction force on the copepod from the computed feeding
current was shown to be of the same order of magnitude as the excess weight
of the copepod, but was not sufficient to balance the excess weight completely.
3. The feeding current is a viscous shear flow that is controlled by two important
factors. The first is the body shape of the copepod. Since the surface of the
body is the internal boundary of the flow, a no-slip condition must be satisfied
on the surface. The second factor is the distribution of forces, which represents
the copepod’s activities that generate the feeding current. The magnitude of
the velocity and the normal and shear stress fields around the body surface of
the copepod are strongly dependent on the spatial distribution of forces
together with their distances from the surface of the body. In the numerical
simulations, we varied the distribution of forces and the copepod’s morphology, and studied their effects on the configuration of the feeding current in
a systematic manner. The results are stated as follows.
i(i) By comparing various distributions of forces, it was shown that a distributed force dissipates less energy and results in a higher entrainment rate
than a concentrated force. This situation is energetically more desirable
and probably led to the evolution of the complicated and delicate feeding
appendages.
.(ii) The variations in the distribution of forces and in the body shape lead to
changes in the flow field very close to the copepod, and thus affect the drag
on the surface of the copepod. The viscous dissipation rate of the water
volume very close to the copepod is also dependent on the distribution of
forces and the body shape of the copepod. However, the feeding currents
for three shapes of the main body or several configurations of the forcing
showed little difference in the far field. Consequently, changes in the shape
of the main body and in the activities for generating the feeding current
may not affect the detectability by other mechano-receptional organisms
several centimeters away. By calculating the viscous dissipation rates
around the copepod, the influence field of the computed feeding current
was shown to be anisotropic, extending 5–7 mm above and ventrally to the
copepod, less than 1 mm dorsally to the copepod, and more than 1 cm
down from the abdomen, in the wake.
4. The present approach of using detailed CFD can be used to study the effects
of many other parameters. Forthcoming work will focus on simulations of flow
around a moving copepod with a morphology much more realistic than the
present one, the interactions of the feeding currents of multiple copepods close
to each other, as well as the effects of turbulence on the feeding current.
Acknowledgements
The authors would like to thank Professors G.-A.Paffenhöfer and J.Yen for their
most helpful suggestions and discussions. Professors J.R.Strickler and T.Kiørboe
reviewed the manuscript and provided valuable suggestions and comments that
1414
Numerical study of copepod feeding current
improved the manuscript. We also thank Dr L.A.van Duren for her valuable
suggestions and comments that improved the manuscript. The financial support
of the Office of Naval Research (contract number N000149710429) is gratefully
acknowledged.
References
Alcaraz,M., Paffenhöfer,G.-A. and Strickler,J.R. (1980) Catching the algae: A first account of
visual observations on filter-feeding calanoids. Am. Soc. Limnol. Oceanogr. Spec. Symp., 3,
241–248.
Andrews,J.C. (1983) Deformation of the active space in the low Reynolds number feeding current of
calanoid copepod. Can. J. Fish. Aquat. Sci., 40, 1293–1302.
Batchelor,G.K. (1967) An Introduction of Fluid Dynamics. Cambridge University Press, Cambridge.
Bundy,M.H. and Paffenhöfer,G.-A. (1996) Analysis of flow fields associated with freely swimming
calanoid copepods. Mar. Ecol. Prog. Ser., 133, 99–113.
Childress,S., Koehl,M.A.R. and Miksis,M. (1987) Scanning currents in Stokes flow and the efficient
feeding of small organisms. J. Fluid Mech., 177, 407–436.
Conover,R.J. (1968) Zooplankton—Life in a nutritionally dilute environment. Am. Zool., 8, 107–118.
Cowles,T.J. and Strickler,J.R. (1983) Characterization of feeding activity patterns in the planktonic
copepod Centropages typicus Kroyer under various food conditions. Limnol. Oceanogr., 28, 105–115.
Emlet,R.B. (1990) Flow fields around ciliated larvae: effects of natural and artificial tethers. Mar. Ecol.
Prog. Ser., 63, 211–225.
Fields,D.M. and Yen,J. (1993) Outer limits and inner structure: the 3-dimensional flow fields of
Pleuromamma xiphias. Bull. Mar. Res., 53, 84–95.
Fields,D.M. and Yen,J. (1997) Implications of the feeding current structure of Euchaeta rimana, a
carnivorous pelagic copepod, on the spatial orientation of their prey. J. Plankton Res., 19, 79–95.
Gargett,A.E., Osborn,T.R. and Nasmyth,P.W. (1984) Local isotropy and the decay of turbulence in a
stratified fluid. J. Fluid Mech., 144, 231–280.
Gaskell,P.H. and Lau,A.K.C. (1988) Curvature-compensated convective transport: SMART, a new
boundedness preserving transport algorithm. Int. J. Numer. Methods Fluids, 8, 617–641.
Higdon,J.J.L. (1979) A hydrodynamic analysis of flagellar propulsion. J. Fluid Mech., 90, 685–711.
Koehl,M.A.R. and Strickler,J.R. (1981) Copepod feeding currents: Food capture at low Reynolds
number. Limnol. Oceanogr., 26, 1062–1073.
Landau,L.D. and Lifshitz,E.M. (1959) Fluid Mechanics. Pergamon Press, Oxford.
Leonard,B.P. (1979) A stable and accurate convective modelling procedure based on quadratic
upstream interpolation. Comput. Methods Appl. Mech. Eng., 19, 59–98.
Lighthill,J. (1975) Mathematical Biofluiddynamics. Society for Industrial and Applied Mathematics,
Philadelphia.
Oliveira,P.J., Gosman,A.D. and Issa,R.I. (1997) A method for particle location and field interpolation on complex, three-dimensional computational meshes. Adv. Eng. Software, 28, 607–614.
Osborn,T.R. (1995) Copepod behavior and turbulence: an evolutionary response? In Skjoldal,H.R.,
Hopkins,C., Erikstad,K.E. and Leinaas,H.P. (eds), Ecology of Fjords and Coastal Waters. Elsevier
Science B.V., pp. 27–32.
Paffenhöfer,G.-A., Strickler,J.R. and Alcaraz,M. (1982) Suspension-feeding by herbivorous calanoid
copepods: a cinematographic study. Mar. Biol., 67, 193–199.
Panton,R.L. (1996) Incompressible Flow. John Wiley & Sons, Inc.
Patankar,S.V. (1980) Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing, Washington,
DC.
Pozrikidis,C. (1992) Boundary Integral and Singularity Methods for Linearized Viscous Flow.
Cambridge University Press, Cambridge.
Pozrikidis,C. (1997) Introduction to Theoretical and Computational Fluid Dynamics. Oxford
University Press, New York.
Price,H.J. and Paffenhöfer,G.-A. (1986) Capture of small cells by the copepod Eucalanus elongatus.
Limnol. Oceanogr., 31, 189–194.
Strickler,J.R. (1982) Calanoid copepods, feeding currents, and the role of gravity. Science, 218,
158–160.
Strickler,J.R. (1985) Feeding currents in calanoid copepods: two new hypotheses. In Lavarack,M.S.
(ed.), Physiological Adaptations of Marine Animals. Symp. Soc. Exp. Biol., 23, 459–485.
1415
H.Jiang, C.Meneveau and T.R.Osborn
Tiselius,P. and Jonsson,P.R. (1990) Foraging behaviour of six calanoid copepods: observations and
hydrodynamic analysis. Mar. Ecol. Prog. Ser., 66, 23–33.
van Duren,L.A., Stamhuis,E.J. and Videler,J.J. (1998) Reading the copepod personal ads: increasing
encounter probability with hydromechanical signals. Phil. Trans. R. Soc. London Ser. B, 353,
691–700.
Yen,J. and Fields,D.M. (1992) Escape responses of Acartia hudsonica nauplii from the flow field of
Temora longicornis. Arch. Hydrobiol. Beih., 36, 123–134.
Yen,J. and Strickler,J.R. (1996) Advertisement and concealment in the plankton: what makes a
copepod hydrodynamically conspicuous? Invert. Biol., 115, 191–205.
Yen,J., Sanderson,B., Strickler,J.R. and Okubo,A. (1991) Feeding currents and energy dissipation by
Euchaeta rimana, a subtropical pelagic copepod. Limnol. Oceanogr., 36, 362–369.
Received on November 24, 1998; accepted on March 16, 1999
Appendix: Validation of finite-volume code on simple low-Reynoldsnumber three-dimensional flows
In order to test the performance of the finite-volume code FLUENT™ for viscous, low-Reynoldsnumber flows, several simple test cases for which analytical solutions are available are simulated. The
validation of the code is a crucial step to establish suitable schemes and parameters for an accurate
feeding current simulation.
Stokes flow due to a point force
The first flow considered is Stokes flow in an infinite domain, with a point force located at the origin.
Analytical solution. The velocity and modified pressure fields due to a point force singularity are
found by solving the singularly forced Stokes equation together with the continuity equation:
–=p + µ=2u + bd(x – x0) = 0, and =· u = 0
(A1)
In the equation, x0 is the location of the point force, the vector b represents the direction and magnitude of the point force (with units of force) and d is the three-dimensional delta function (with units
of inverse volume). The solution of equation (A1) must be found subject to appropriate boundary
conditions. By using Green’s function method (see Pozrikidis, 1997), the solution for an infinite
domain with no interior boundaries can be written as follows:
1
ui(x) = —— Gij(x,x0)bj
8pµ
(A2)
1
p(x) = —— Pj(x,x0)bj
8p
(A3)
x^i x^j
dij
Gij(x,x0) = —— + ———
r
r3
(A4)
x^i
Pj(x,x0) = 2 —–
r3
(A5)
where the Green’s functions are:
and
with x^i = xi – x0i, and r = |x^|. In terms of (x, y, z) coordinates, with the point force located at the origin
and directed along –z, i.e. b = (0, 0, –b), where b > 0, the solution is:
1416
Numerical study of copepod feeding current
b xz
u(x,y,z) = – —— ——
8pµ r3
(A6)
b yz
v(x,y,z) = – —— ——
8pµ r3
(A7)
b z2 + r2
w(x,y,z) = – —— ———
8pµ
r3
(A8)
b
z
p(x,y,z) = – —— ——
4p r3
(A9)
where r3 = (x2 + y2 + z2)3/2.
Computational domain, boundary conditions and parameters. The computational domain consists of
a 10 3 10 3 10 cm cubic box discretized with 40 3 40 3 40 nodes. The point force is located at the
center of the box. Strong grid stretching is employed so that the size of the center mesh element, where
the force is applied, is only 0.29 3 0.29 3 0.29 mm. The system, together with a cross-section of the
grid, is illustrated in Figure A1. A constant zero relative static pressure boundary condition is applied
on the six surfaces of the box.
For the problem of singularly forced Stokes flow, a Reynolds number based on the forcing can be
defined according to:
br
Re = ——–
4pµ2
(A10)
because based on equation (A8), one can show that the mean velocity over a sphere of radius r
b
– = —–—.
centered at the origin is given by w
In order to compare with Stokes flow, a Reynolds number
4pµr
much smaller than one must be chosen. However, FLUENT™ has numerical difficulties handling
small velocities (due to the digital precision), so that dimensional analysis is used to rescale the
problem. Choosing b = 10–2 N, µ = 10 kg m–1 s–1 and r = 103 kg m–3, we obtain Re ≈ 0.008 and a mean
–
velocity of 0.32 mm s–1 (taking r to be the smallest half mesh size multiplied by √3, here r = 0.25 mm).
These are the values chosen for the simulation. The simulated results are then rescaled to be consistent with the viscosity of water, µ = 10–3 kg m–1 s–1, which implies a value of b = 10–10 N.
Fig. A1. Point force at the center of the cubical domain together with a cross-section of the grid along
the grid slice J = 21. Note that each grid slice is the same.
1417
H.Jiang, C.Meneveau and T.R.Osborn
Computational schemes. The present problem is relatively simple and convergence is mainly limited
by the pressure–velocity coupling. We chose SIMPLEC (Semi-Implicit Method for Pressure-Linked
Equations-Consistent) (Patankar, 1980) as the pressure–velocity coupling algorithm. As for the interpolation scheme, QUICK (Quadratic Upwind Interpolation) (Leonard, 1979; Gaskell and Lau, 1988)
was selected. In order to solve the resulting discretized equations, Line–Gauss–Seidel (LGS), a lineby-line iterative solution technique, is used.
Results. Figure A2 shows plots of pressure and velocities along a line across the inner part of the
domain, i.e. the 2 3 2 3 2 cm subdomain. Because of the rescaling, velocities computed by
FLUENT™ have been rescaled by u = uFLUENT 3 10–4 and pressure by p = pFLUENT 3 10–8. By comparing the u, v, w and p calculated by FLUENT™ with their analytical counterparts in the inner part
of the domain, we find generally good agreement. However, as seen from Figure A3, which shows the
plots in the entire domain, when approaching the outer boundary, the pressure no longer follows the
analytical solution. Consequently, the computed w component decreases faster than the analytical
solution at large distances. The reason is that the imposed boundary condition forces the computed
pressure to be zero at the boundary surfaces, while, for the analytical solution, the pressure at the
boundary surfaces is non-zero. Moreover, close to the origin, the fact that the small but finite cell on
which a force is applied is not exactly a point may introduce small errors near the origin. Nevertheless, in absolute terms, the errors are small. For instance, even in Figure A3d, the discrepancy is <1%
of the largest computed w component in Figure A2d.
Stokes flow due to a point force outside a solid sphere
The second flow considered is flow bounded internally by a solid sphere centered at the origin with
a point force located outside the sphere at x0.
Fig. A2. Plots of pressure and velocities along the line at y = 0.433 mm and z = 0.433 mm for the case
of a point force located at the origin. Solid lines, analytical predictions; diamonds, numerical results
at grid points. (a) Pressure; (b) u velocity; (c) v velocity; (d) w velocity.
1418
Numerical study of copepod feeding current
Fig. A3. Plots of pressure and velocities along the line at x = 9.36 mm and y = 9.36 mm for the case
of a point force located at the origin. Solid lines, analytical predictions; diamonds, numerical results
at grid points. (a) Pressure; (b) u velocity; (c) v velocity; (d) w velocity.
Analytical solution. The Green’s function for an infinite flow that is bounded internally by a solid
sphere was obtained by Oseen in 1927 (Higdon, 1979; see also Pozrikidis, 1992). The results are:
dij
x^i x^j
a dij
a3 x^i*x^*j
GijSPH(x,x0) = ( —– + ——
) – —— —–
– ——
—–—
3
*
r
r
)x0) r
)x0)3 (r*)3
)x0)2 – a2
– ————
)x0)
XX
a
2X X
X x
– ———— (X x + X x ) + –——— ———4
3 ———
a
(r )
ar
)x ) (r )
* *
i j
3 *
0
2
* 3
* ^*
i j
* ^*
j i
* *
i j
3
* ^*
k k
* 3
(A11)
()x)2 – a2)()x0)2 – a2)
∂fj
– —————————
———
2)x0)3
∂xi
with
3Xjx^i*
adij
3ax^i*x^*j
2XjXi*
6Xjx^i*x^k*Xk*
∂fj
—— = – ———
+ ——–
– ———
– ———
+ ———–—–
*
3
*
3
*
5
*
3
∂xi
a(r )
(r )
(r )
a(r )
a(r*)5
3a Xj*x^i(r*)2 + x^i*x^*j )X*)2 + 1r* – )X*)2(r*)2)X*)dij
+ ——– —–—————————————————
(r*)3)X*)1)X*)r* + xkXk* – )X*)22
)X*)
3a
1)X*)x^i* + r*Xi*23Xj*(r*)2 – x^*j )X*)2 + 1x^*j – Xj*2r*)X*)4
– ——– ————–——————————————————
(r*)2)X*)1)X*)r* + xkXk* – )X*)222
)X*)
(A12)
1419
H.Jiang, C.Meneveau and T.R.Osborn
3a
xiXj* + )x))X*)dij
3a
1)X*)xi + )x)Xi*21)X*)xj + )x)Xj*2
– ——– ——————–——— + —— ——–———————————
*
*
*
*
*
*
*
* 2
)x))X )1)x))X ) + xkXk 2
)x))X )1)x))X ) + xkXk 2
)X )
)X )
where
x^ = x – x0
a2
X* = ——
x0
)x0)2
r = )x^)
x^* = x – X* r* = )x^*)
(A13)
a is the radius of the sphere. The velocity field due to a point force outside the solid sphere can be
written as:
1
ui(x) = ——GijSPH(x,x0)bj
8pµ
(A14)
Computational domain, boundary conditions, schemes and parameters. As in the previous section on
Stokes flow due to a point force, we chose a core computational region of 2 3 2 3 2 cm, contained in
a 10 3 10 3 10 cm cubic region discretized with 40 3 40 3 40 nodes. The boundary condition at the
surface of this region is a constant relative static pressure of 0 Pa. The no-slip solid sphere is centered
at the midpoint of the cubic region. The radius of the sphere is 1.0 mm. A body-fitted coordinate grid
is generated in which the sphere is divided into six equal-area regions and mapped into a cube. Strong
grid stretching toward the solid sphere is employed. Figure A4 shows a magnified view of the grid
distribution around the sphere.
The point force is located at (1.8 mm, 0.0, 0.0) or at the center of the cell numbered as (27, 21, 21).
The problem is rescaled for the same reasons as in the section on Stokes flow due to a point force,
with b = 10–2 N, µ = 10 kg m–1 s–1 and r = 103 kg m–3. According to equation (A10), one has a Reynolds
number of 0.008 (<<1.0) and therefore the computational results can be compared to the analytical
results for Stokes flow (A14).
The numerical schemes are the same as those used in the section on Stokes flow due to a point
force.
Results. The simulated results are rescaled to be consistent with the viscosity of water, µ = 10–3 kg
m–1 s–1, which implies a value of b = 10–10 N, u = uFLUENT 3 10–4 and p = pFLUENT 3 10–8. Figure A5
shows plots of velocities along one of the grid lines across the inner part of the domain, namely the
2 3 2 3 2 cm subdomain. Those lines are straight in the mapped computational domain, but are curved
in physical space due to the body-fitted curvilinear grid. Nevertheless, if the analytical expressions are
evaluated at exactly the same points as the grid points, a meaningful comparison can be made. As can
Fig. A4. Magnified view of the grid distribution along the grid plane J = 21 of the three-dimensional
grid system for the simulation of Stokes flow due to a point force outside a solid sphere. The shaded
cell with a downward arrow shows where the force is applied.
1420
Numerical study of copepod feeding current
Fig. A5. Plots of velocities along the grid line J = 22 and K = 22 for the flow bounded internally by a
1-mm-radius solid sphere centered at the origin with a point force located at x = 1.8 mm, y = 0.0 mm,
z = 0.0 mm. Solid lines, analytical predictions; diamonds, numerical results at grid points. (a) u velocity; (b) v velocity; (c) w velocity.
be seen in Figure A5, comparing the u, v and w components calculated by FLUENT™ with their analytical counterparts in the inner part of the domain, we find generally good agreement, except again
for the w velocity approaching the outer boundary (not shown). The deviation is again due mainly to
the zero pressure boundary condition applied at a finite distance as opposed to at infinity.
We conclude that for low-Reynolds-number three-dimensional flows due to a point force in free
space and near a solid sphere, the computational results show good agreement with the analytical
results in the internal subdomain of main interest.
However, typical maximum velocities expected near microorganisms due to feeding currents are of
the order of 1 mm s–1. A typical Reynolds number based on a velocity of 1 mm s–1 and a size of 1 mm
in water is Re ≈ 1. Therefore, the flow around a microorganism may not be approximated as a Stokes
flow and direct numerical simulation based on the full Navier–Stokes equations is needed.
1421