Supporting Information for Thomson et al. Scale-dependent relationships between
suspension-feeding stream insects and velocity in spatially heterogeneous flow environments.
Appendix S1. Details of computational fluid dynamics (CFD) model used to estimate nearbed velocities above semi-cylinders.
Appendix S2. Additional result.
Appendix S3. A simple optimal foraging model to predict relative rates of emigration along
a velocity gradient for filter feeding simuliid larvae.
Appendix S1. Details of computational fluid dynamics (CFD) model used to estimate nearbed velocities above semi-cylinders.
List of symbols
k = turbulent kinetic energy
p = pressure
P = shear production of the turbulent kinetic energy
Re = Reynolds number of flow (uin_D/)
u = axial (x) component of velocity
v = normal (y) component of velocity
V= velocity vector
Subscripts
t = turbulent
in = inlet
Greek Symbols
 = dissipation rate of turbulent kinetic energy
 = dynamic viscosity
 = density
Modeling considerations
Turbulent flow of water is considered over a cylinder placed on a flat surface. The water
depth is low (as found in small streams) and the semi-cylinder is considered to be fully
immersed. For long semi-cylinders, a two dimensional formulation of the problem is
considered, thus neglecting end-effects. A schematic of the computational domain is shown
in Fig. S1. The free surface layer is assumed to be flat and is shown along the length (ef).
The computational domain includes sufficiently long upstream (ab = 40 cm) and downstream
(cd = 60 cm) distances so that the effects of idealized flow boundary conditions on the flow
behavior around the semi-cylinder are minimized. The diameter (bc) of the semi-cylinder is
10.9 cm. The lengths ab and cd are 40 and 60 cm respectively. The water depth over the top
of the semi-cylinder varies from 1 cm to 20 cm for the calculations. A parallel inlet flow is
considered along the inlet plane ad. The inlet velocity uin was varied from 3 cm/s to 50 cm/s.
Mathematical model
The mathematical formulation used in the study is based on the two-dimensional
incompressible Navier-Stokes equations for viscous flows. A k- turbulence model was used
to include the effects of turbulence on the flows considered. The mass conservation equation,
momentum equations and equations for the turbulence parameters were solved numerically to
determine the flow structure along the flat wall, particularly around the hemicylindrical
protrusion.
The governing equations in (two-dimensional Cartesian coordinates) for the flows considered
are presented below (Anderson et al., 1984):
Mass Conservation Equation
The continuity equation is given as:
 ( u) +  ( v) = 0
x
y
(1)
where the over-bar indicates a Reynolds averaged quantity.
Momentum Equations
The x- momentum equation can be written as
 ( u u) +  ( u v) = - p +  ( +  ) ( u )
t x
x
y
y
x
 ( +  ) ( u )
t y
y
(2)
and the y-momentum equation is given as:
p



v
( v u) +
( v v) = +
( + t) (
)
x
y
y
x
x

v
( + t) (
)
y
y
(3)
where t is the turbulent (eddy) viscosity, as computed by the k- model.
Equations for the Turbulence Parameters
The turbulence closure is obtained via a two-equation (k-) turbulence model. The transport
equation for the turbulent kinetic energy k is given by:
 ( u k) +  ( v k) =  ( +  ) ( k )
t x
x
y
x
 ( +  ) ( k )
t y
y
+ P-
(4)
The transport equation for the dissipation rate of the turbulent kinetic energy () is given by
 ( u ) +  ( v ) =  ( +  ) (  )
t x
x
y
x
 ( +  ) (  )
t y
y
+  ( C P - C  )
k 1
2
(5)
where P is the shear production of the turbulent kinetic energy and is given by
P = 2 (
u 2
u
v 2
) + (
+
) +
t x
y
t x
2 (
v 2
)
t y
(6)
The turbulent viscosity was calculated as
2
t = C k
(7)
In the model described above, standard values of the turbulence model constants have been
used. The k- model constants used are C1 = 1.44 , C2 = 1.92, and C_=_0.09.
Boundary Conditions
A flat velocity profile is given for the entrance flow condition. Non-slip boundary condition
is applied for all solid surfaces (abc, bc, cd) . The free surface (eg) is assumed flat and the
normal velocity gradient is zero across the boundary. A zero velocity gradient boundary is
employed at the exit of the computational domain (de).
At the inlet, the k and in values are based on average flow characteristics. Uniform profiles
for k and  are approximated as:
kin = 0.002 uin2
in = kin1.5/ 0.3H
where H is the water depth-layer at inlet af (see Fig. S1). The logarithmic boundary-layer
profile is used to derive the boundary conditions for k and _along the solid walls (Launder
and Spalding, 1974).
Solution technique
Two-dimensional incompressible Navier-Stokes equations along with the prescribed
boundary conditions were solved numerically. For the numerical solution, the conservation
equations are discretized using a finite volume technique. To accommodate the semicylinder within a rectangular computational domain, a multi-block body-fitted grid system
was employed. In the multi-block approach employed in this study, the computational
domain is divided into a five blocks (Fig. S1). Body fitted grids are then generated within
each block such that the grids are not restricted to be topologically rectangular. In multiblock grids, data is transferred from one block to another using a generalization of the
periodic boundary condition. The blocks are arranged to overlap such that a boundary
surface of one block is situated in the interior of another. For the present study, the
computational domain was sub-divided into five adjoining block structures. Three blocks
were used in the vicinity of the semi-cylinder (B, C, D in Fig. S1) whereas the other two
were used to cover the upstream and the downstream regions (A and E in Fig. S1). The grid
size is for each block was  40 x 30 cells. For the range of parameters considered (entrance
velocity and water depth) the above grid was found to be adequate to provide gridindependent solutions..
A non-staggered grid is used for locating the pressure and velocity components. The RhieChow interpolation procedure is used to prevent checkerboard oscillations of pressure on the
co-located grids. The hybrid differencing scheme (Patankar, 1980) was used to model the
convective terms of all transport equations. All computations were performed on an IBMRISC 6000 workstation computer. The typical CPU time used for one complete case is about
200 seconds.
Figure S1. Schematic of the problem domain for CFD model
a
B
b
40.0 cm
C
40
90
115
A
65
Fixed
Velocity
Inlet, uin
e
Depth, d
f
14
0
10.9 cm
D
E
d
c
60.0 cm
Fixed
Pressure
Outlet
References:
Anderson, D.A., Tannehill J. C. & Pletcher R.H.. (1984) Computational fluid mechanics and
heat transfer. Hemisphere Publishing, Washington, D.C., USA.
Launder, B.E. & Spalding D.B. (1974) The numerical computation of turbulent flows.
Computational Methods in Applied Mechanical Engineering, 3, 269-289.
Patankar S.V. (1980) Numerical heat transfer and fluid flow. Hemisphere Publishing,
Washington, D.C., USA.
Appendix S2. Additional results.
Table S1. Estimated velocity coefficients from four different model structures, comprising all
combinations of a) separate or combined models for riffles and pools, and b) multivariate
normal (MVN) or conditionally independent (and identically distributed, IID) residual errors.
Type
Scale
Pools
Microhabitat
(position)
Microhabitat
(position)
Substrate
(cylinder)
Substrate
(cylinder)
Riffle
Pools
Riffle
Combined models
IID errors
MVN errors
4.08
3.71
(1.28, 6.90)
(1.03, 6.27)
3.84
3.79
(1.34, 5.97)
(0.87, 6.12)
2.76
3.54
(0.03, 4.81)
(1.93, 5.22)
-1.56
-1.44
(-2.91, 0.25)
(-2.90, 0.22)
Separate models
IID errors
MVN errors
5.03
3.94
(3.26, 8.11)
(3.06, 4.82)
2.68
2.66
(-0.02, 4.51)
(0.36, 4.47)
7.52
7.51
(5.26, 9.77)
(4.93, 10.10)
-3.94
-3.94
(-5.69, -2.18) (-5.71, -2.17)
Table S2. Correlation among raw densities and among residual errors (residual correlation)
when densities were regressed against water velocity in hierarchical models. Values are
correlation coefficients derived from estimated variance-covariance matrices for multivariate
normal (MVN) error terms in separate models for riffles and pools (combined MVN models
produced similar results but had poor mixing for covariance and related parameters).
Pools
Riffles
65°
90°
115°
140°
65°
90°
115°
140°
Correlation of Raw densities
40°
65°
90°
115°
0.81
0.65
0.73
-0.21
-0.34
-0.25
-0.22
-0.33
-0.24
0.30
0.08
0.84
0.59
0.53
-0.51
0.15
0.70
-0.54
0.21
0.78
40°
0.04
-0.10
-0.05
-0.01
0.02
0.00
0.02
0.01
Residual correlation
65°
90°
115°
-0.05
-0.07
-0.02
0.03
0.01
0.06
0.01
0.05
-0.01
0.03
0.01
0.01
0
0.04
0.08
0.12
0.16
0.2
location
−15%
geo.unit
43%
geo.type
78%
cylinder
44%
position
92%
pos. x geo.type
69%
pos. x geo.unit
20%
pos. x cylinder
44%
0
1
2
3
4
5
Figure S2. Sources of variation in near-bed velocity (white boxes, top axis), larval density (dark grey,
bottom axis) and residual larval density after accounting for velocity effects (i.e. variation not
explained by velocity in the model; light grey, bottom axis). Boxplots summarizes posterior
distributions of finite population standard deviations for each source of variation (i.e. variance
components on the scale of the variable): solid line =posterior median, box = interquartile range,
whiskers = 95% intervals. Values below velocity model boxes show the % variance explained
(posterior median) by velocity at each scale. The velocity model explained 65% of that total variance
in larval density.
Appendix S3. A simple optimal foraging model to predict relative rates of emigration along
a velocity gradient for filter feeding simuliid larvae.
If larvae seek to maximize ingestion rates, their tendency to leave substrata and enter
the drift should depend on the expected increase in performance (ingestion rate) associated
with drifting to a new substrate (Charnov 1976, Beachly et al. 1995). The expected net gain
in total food ingested over time period T for a larva dispersing via the drift from a substrate
with local velocity 𝑣𝑐 may be expressed as
𝐸[∆] = 𝑔(𝑣𝑐 ) = E[𝑓(𝑣). (𝑇 − 𝑡𝑣 )] − 𝑓(𝑣𝑐 )𝑇
= (E[𝑓(𝑣)] − 𝑓(𝑣𝑐 )). 𝑇 − E[𝑓(𝑣). 𝑡𝑣 ]
𝑏
𝑏
∞
= 𝑇 (∫ 𝑓(𝑣) 𝑝(𝑣) . 𝑑𝑣 − 𝑓(𝑣𝑐 )) − ∫ ∫ 𝑓(𝑣). 𝑡𝑣 . 𝑝(𝑡𝑣 |𝑣). 𝑑𝑡 . 𝑑𝑣
𝑎
𝑎
0
where 𝐸[∆] is the expected change in total ingestion, which is a function, 𝑔(𝑣𝑐 ), of the
current feeding velocity 𝑣𝑐 . f(v) is the functional relationship between ingestion rate and
velocity, tv is the time taken to disperse and recommence feeding, p(v) is the probability of
settlement on a site with velocity v, p(tv|v) is the conditional probability distribution of drift
times for larvae settling on substrates with velocity v (i.e. probability it takes tv seconds to
disperse to a new location with velocity v), and a and b are the minimum and maximum
velocities in a habitat, respectively. We assume here that p(v) and p(tv|v) are independent of
departure velocity vc: while this may not be strictly true, we believe an assumption of
effective independence is reasonable, at least within geomorphic units, because 1) passively
drifting larvae should be well mixed in turbulent flow environments, so that the eventual
settlement point may be independent of the point of entry, and 2) the ability of larvae to settle
is affected by flow conditions (shear stresses, turbulence etc) at the point of attempted
attachment – hence probabilities and time to attachment should depend more on the
destination velocity than on the departure velocity. Given the assumption that p(v) and p(tv|v)
are effectively independent of vc, the integrals in (1) are constant, and give the expected (i.e.
𝑏
mean) new ingestion rate for a dispersing larva, 𝐸[𝑓(𝑣)] = ∫𝑎 𝑓(𝑣) 𝑝(𝑣) . 𝑑𝑣 , and the
𝑏
∞
expected loss of ingestion while dispersing, 𝐸[𝑓(𝑣)𝑡𝑣 ] = ∫𝑎 ∫0 𝑓(𝑣). 𝑡𝑣 . 𝑝(𝑡𝑣 |𝑣). 𝑑𝑡 . 𝑑𝑣.
Differentiating 𝑔(𝑣𝑐 ) with respect to the current (i.e. departure) velocity shows that the rate
of change in the expected gain in ingestion rate, 𝑑𝐸[∆]/𝑑𝑣 = 𝑔′(𝑣𝑐 ), is determined by the
rate of change in ingestion rate with increasing velocity: 𝑔′(𝑣𝑐 ) = −𝑇𝑓′(𝑣𝑐 ). Note that the
constant integrals drop out of the derivative with respect to 𝑣𝑐 .
If the functional relationship f(·) is monotonic (𝑓′(𝑣𝑐 ) is positive for all 𝑣𝑐 ), which is a
reasonable assumption here because measured velocities were below optimal feeding
velocities, then 𝑔(𝑣𝑐 ) must decline with increasing vc (𝑔′(𝑣𝑐 ) is negative) and therefore
emigration should decline with increasing velocity, as observed empirically (Fonseca and
Hart 1996, Fonseca and Hart 2001). Importantly, if the marginal increase in ingestion rate
declines with increasing velocity, as found for S. tribulatum by Finelli et al (2002), then the
expected gain in ingestion rate decreases most rapidly at low velocities (𝑔′(𝑣𝑐 ) = −𝑇𝑓′(𝑣𝑐 ) is
most negative at low 𝑣𝑐 ). Therefore if emigration rates are proportional to the expected gain
in ingestion rate, we would expect more rapid decline in emigration rates over low velocity
ranges. Furthermore, once the expected gain in ingestion rate, 𝑔(𝑣𝑐 ), becomes negative larvae
should never enter the drift as a foraging strategy. Based on empirical evidence that
settlement rates decline with velocity (Fonseca 1999, Fingerut et al. 2011), we expect p(v) to
decline with velocity, and / or p(tv|v) to increase with velocity: the former lowers the
expected new ingestion rate (𝐸[𝑓(𝑣)]), the later increases the expected cost of dispersal
(𝐸[𝑓(𝑣)𝑡𝑣 ]). Low 𝐸[𝑓(𝑣)] or high [𝑓(𝑣)𝑡𝑣 ]) reduce the expected gain for all departure
velocities, and therefore lowers the threshold at which the expected gain becomes negative.
Larvae should never leave substrates (as a foraging strategy) if the expected gain is negative
𝑔(𝑣𝑐 ), and so emigration rates should be independent of velocity (and near zero) for all
velocities above the threshold at which 𝑔(𝑣𝑐 ) becomes negative. Positive density velocity
relationships cannot occur over velocity ranges where immigration rate declines and
emigration rates are constant.
The simple theoretical model presented here ignores the direct costs and mortality risks
associated with drifting, but the general conclusion would change only if those costs 1)
declined with departure velocity, and 2) were large relative to potential energy gains. Neither
condition seems plausible.
References
Beachly, W.M., Stephens D.W. & Toyer K.B. (1995) On the economics of sit-and-wait
foraging: site selection and assessment. Behavioral Ecology, 6, 258-268.
Charnov, E. L. (1976) Optimal foraging: the marginal value theorem. Theoretical
Population Biology, 9, 129-136.
Finelli, C. M., Hart, D. D. & Merz, R. (2002) Stream insects as passive suspension feeders:
effects of velocity and food concentration on feeding performance. Oecologia, 131,
145-153.
Fingerut, J.T., Hart, D.D & Thomson J.R. (2011) Larval settlement in benthic environments:
the effects of velocity and bed element geometry. Freshwater Biology, 56, 904–915.
Fonseca, D. M. (1999) Fluid-mediated dispersal in streams: models of settlement from the
drift. Oecologia, 121, 212-223.
Fonseca, D.M. & Hart D.D. (2001) Colonization history masks habitat preferences in local
distributions of stream insects. Ecology, 82, 2897–2910.
Fonseca, D.M. & Hart D.D. (1996) Density-dependent dispersal of black fly neonates is
mediated by flow. Oikos, 75, 49-56.