Boundarytrapped, inhalant siphon and drain flows - Wiley Online Library

ORIGINAL ARTICLE

Boundary-trapped, inhalant siphon and drain flows: Pipe entry revisited numerically Peter A. Jumars

Abstract Flows produced in suspension feeding and tube and burrow ventilation cause and influence many exchange processes and chemical reactions. I investigated two geometries, a capillary drawing water far from any other boundary and a constant-diameter drain flush with the bottom. Flow descriptions just inside these inlets have relied on solutions to pipe entry that posit uniform entrance velocity. I substituted the more realistic boundary condition of constant volumetric outflow rate in finite-element models for 0.01 # Re # 2000. Re is the pipe Reynolds number, the product of mean capillary flow speed and capillary inside diameter (D) divided by kinematic viscosity of the fluid. For the smallest Re in both geometries, axial velocity reached 99% of its maximal value at a distance 0.2725D into the entry—not the 0.619 D found with uniform entrance velocity—and flow entering the capillary originated from a small, cylindrical region centered on the pipe axis. Axial flow velocities approaching the inlets therefore decreased away from the opening more slowly than predicted by simple convergence. For Re . 330, flow converging on the capillary found traction on its outer wall (an apparent Coanda˘ effect), and flow separation had major effects inside. Flow entering the siphon at these high Re values originated below the capillary entrance, and subsequent decrease in Re failed to dislodge the flow from its boundary-trapped state. Such hysteresis and flow bifurcation is unusual at low Reynolds numbers and could affect suspension feeders that deploy siphons and water samplers that use suction. Keywords: hydraulic entrance length, siphon flows, pipe entry, inhalant siphon, bifurcation, Coanda˘ effect

Darling Marine Center, University of Maine, Walpole, Maine 04573, USA Correspondence to Peter A. Jumars [email protected]

Introduction [1] A nearby seabed suppresses vertical fluid motion. Mean flow parallels the interface and slows toward it; vertical turbulent fluctuations also diminish (Monin and Yaglom 1971). In this context, vertical flows produced by organisms, extending in many cases into the seabed, take on added importance for exchange. Benthic invertebrates from diverse phyla on and in both hard and soft substrata—both in the ocean and in fresh water—produce inhalant and exhalant flows, largely in suspension feeding and respi-

ration (Wildish and Kristmanson 1997). They greatly alter redox chemistries of impermeable coastal sediments, which would be anoxic almost immediately below a sediment–water interface lacking animal-generated conduits that import oxidants (e.g., Revsbech et al. 1980). In addition, benthic suspension feeders pose risks to larvae of benthos and plankton (Andre´ et al. 1993; Green et al. 2003) that depend on both escape capabilities of the larvae and geometries, velocities, and accelerations of flow fields produced by the feeders.

Limnology and Oceanography: Fluids and Environments † 3 (2013): 21–39 † DOI 10.1215/21573689-2082871 q2013 by the Association for the Sciences of Limnology and Oceanography, Inc.

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[2] Siphon flows are also used to evaluate responses of plankton and inanimate objects to fluid velocities, accelerations, shear stresses, and other stimuli. Assayed responses include passive reorientation of elongated prey (Visser and Jonsson 2000), escape reactions of motile plankton (Singarajah 1969), and bioluminescence in dinoflagellates (Christianson and Sweeney 1972; Anderson et al. 1988). The many applications of siphon flows to test escape reactions as “prey” are drawn toward a capillary tip have been briefly reviewed by Green et al. (2003). Flow within an analogous chamber having rectangular cross section has also been used to assess mechanical properties of fibers (Samuelsson 1963). While participating in a related study of mechanical properties of diatom chains (Young et al. 2012), I became aware of surprisingly fundamental gaps in understanding of both natural and laboratory entry flows. Siphonal Flows [3] Cylindrical coordinates are simplest for quantification. Because siphon tubes of both benthos and researchers are often oriented parallel with the gravity vector, I use z as the pipe axis and take it to be vertical. Assuming axial symmetry, I use r as radial distance from the z-axis and do not resolve any angular dependence. Because flow on each side of the siphon opening is of interest but is likely to behave differently, I set the origin (r ¼ 0, z ¼ 0) at the pipe entrance. Flow down the pipe (-z direction) and toward the axis (-r direction) is in a negative direction. Vertical and radial components of the velocity vector are denoted ~ zÞ and u~ ðr; zÞ; respectively. by wðr; [4] Well inside a capillary of constant diameter, steady flows at low to moderate flow speeds lack axial gradients in velocity, lack radial gradients in pressure, and follow the well-known Hagen– Poiseuille law with a volumetric flow rate, QV [L3 T-1], and parabolic velocity profile that both depend on the constant axial pressure ~ gradient, G[M L-2 T-2], maintained along the capillary (Sutera and Skalak 1993):

~ 4 pGa ; 8m

ð1Þ

~ 2 G ða - r 2 Þ: 4m

ð2Þ

Qv ¼ ~ ¼wðrÞ

Here m is dynamic viscosity, a is capillary inner radius, ~ and wðrÞ is vertical velocity as a function of r, radial distance from the capillary axis. Negative signs in Eqs. 1 and 2 indicate that flow opposes the pressure gradient. Maximal velocity is axial (r!¼ 0) and equals twice the mean downward velocity, w , over the whole cross section: !

~ max ¼ wð0Þ ~ w ¼ 2w :

ð3Þ

[5] Flow descriptions approaching a siphon opening are less standardized. Troost et al. (2009) reviewed prior descriptions of inhalant flow and published the most thorough analysis to date of acceleration leading toward the intakes of suspension feeders, analyzing flow fields produced by oysters, mussels, and cockles. They built on a flow model of hemispherical convergence suggested by Andre´ et al. (1993) for a siphon flush with the seabed but also included spherical convergence for projecting siphons. Spherical convergence is also the model generally proposed for analysis of responses to engineered siphon flows (Kiørboe et al. 1999): UðRÞ ¼ -

Qv ; 4pR 2

ð4Þ

where R is distance in any direction from that origin (uppercase R being used to distinguish this spherical geometry from the cylindrical geometry and r in Eq. 2), and U is flow velocity irrespective of the direction from which it approaches the siphon. The negative sign by convention indicates flow inward along all radii toward the point sink. For a hemisphere, the factor of 4 in the denominator is replaced by 2. Eq. 4 clearly oversimplifies: flow within the capillary goes away from the point sink, and none of the flow from the lower hemisphere can follow a straight line into the capillary. As noted by Kiørboe et al. (1999), Eq. 4 notably lacks dependence on viscosity or location of no-slip boundaries of the capillary. [6] Considerable effort has gone into estimating the distance that a flow must travel within a tube before its velocity profile fits the Hagen– Poiseuille parabola. This distance is often quantified as a hydrodynamic entrance length, Le, the distance from the entrance at which flow velocity along the axis reaches a given

q2013 by the Association for the Sciences of Limnology and Oceanography, Inc. / e-ISSN 2157-3689

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fraction (usually 0.99) of its equilibrium value as described by Eq. 3. Nearly universal assumptions in theoretical treatments have been that velocity at z ¼ 0 is invariant with r, lacks radial components, and is simply equal to the mean axial flow velocity corresponding to the volumetric flow rate (Fargie and Martin 1971): wðr; 0Þ ¼

Qv : pa 2

ð5Þ

[7] Both Eq. 2 and Eq. 4 have maxima on the capillary axis, however, so it is inconceivable that uniform entry flow could result from their viscous interaction. Not surprisingly, experimental fit to predictions assuming uniform entry flow has been notoriously poor. Durst et al. (2005) settled nagging questions about the theory by solving numerically for entry lengths under the assumption of uniform entry flow and fitting equations to predict Le as a function of pipe or capillary Reynolds number, Re, defined as Re ¼

w2a rw2a ; ¼ v m

ð6Þ

where r is fluid density, w is mean vertical capillary flow speed, and n combines fluid density and dynamic viscosity into the kinematic viscosity, effectively the diffusion coefficient for fluid momentum. In hydraulics, inside diameter (D) is used more frequently than inside radius (a) in defining pipe or capillary Reynolds numbers, so I insert the 2 in Eq. 6. [8] Durst et al. (2005) provided a simple expression that both fits their numerical results and scales consistently with known roles of advection and diffusion of momentum: 1=1:6 Le ¼ 0:6191:6 + ð0:0567 ReÞ1:6 : D

ð7Þ

The first term on the right dominates at low Re, with flow geometry determined by diffusion of momentum, and 0.619 is the low-Re asymptote for Le/D. The second term is needed at higher Re to describe a flow geometry produced by nonlinear interaction of diffusion of momentum with advection, and 0.0567 is the asymptote approached by ([Le/D]/Re) at high Re. Durst et al. (2005, p. 1159) concluded from historical scatter and their own failed attempts to verify Eq. 7 at Re , 200

that “there are insurmountable difficulties in determining the development length Le/D of laminar pipe and channel flows experimentally and/or analytically.” [9] Young et al.’s (2012) models did not assume uniform entry flow and displayed consistently shorter entrance lengths than Eq. 7 predicts. They made it obvious that the cause of the chronically poor fits of experiments to theory is the unrealistic assumption of uniform flow velocity at the pipe entrance. My purpose here is to extend results beyond the limited cases explored by Young et al. (2012) in order to examine the effects of relaxing the unrealistic assumption of uniform entry flow on siphonal flows. Methods [10] I chose two base cases (Fig. 1). The first was a capillary tube drawing water from the middle of a large volume, far from any boundaries other than the capillary walls. The second was a drain flush with the bottom in the center of a large tank. I built twodimensional, axially symmetric models of each through the graphical user interface in COMSOL Multiphysics (COMSOL, Inc., USA), version 4.3.0.184, on a Macintosh Pro with two 2.66-GHz 6-Core Intel Xeon processors and 12-GB 1333-MHz DDR3 memory. I assumed incompressible, laminar flow of constant kinematic viscosity that otherwise follows the Navier– Stokes equations. My capillary geometry resembles one that we recently used experimentally (Young et al. 2012) but is idealized in order to restrict effects of tank shape and wall proximity, that is, to make the results more generally useful in other geometries. It is also rotated by 908—in Young et al.’s (2012) application, the capillary was horizontal. Gravity and orientation do not affect the analysis; the orientation here conforms to the most frequent use. [11] The spherical (capillary) or hemispherical (drain) walls are further idealized as permeable inlets at constant reference pressure, p0. If pressure varied along the wall, resulting imbalance across the far field would drive its own circulation that would interact with the siphon flow. Rather than setting a constant velocity across the inlet, I set a volumetric outflow rate, v0 (¼Qv by continuity). To facilitate comparisons, my primary pressure results are given as p - p0 or, in the discussion


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behavior is observed (Avila et al. 2011). Specifically, I filled Re gaps of more than a factor of five in simulations run by Young et al. (2012) and added parallel simulations for the drain. [12] The size of the fluid sphere (capillary geometry) or hemisphere (drain geometry) was incremented until further increases left axial velocity results, in particular, entrance and decay lengths, unchanged within the first three significant figures. As expected, the lowest Re settings were most sensitive to domain size, but even they became insensitive by the time a sphere or hemisphere raA B dius of 300 mm (¼ 319 capillary z z diameters) was reached, so a HR SR 319D radius was used throughout. A background mesh for both the capillary and drain of Fig. 1 was implemented at COMr r SOL’s menu settings of “extremely fine” and “calibrated for fluid Length (mm) Length (mm) dynamics” for the entire fluid doSR = 300 HR = 300 OR = 0.60 IR = 0.47 main. To get reliable convergence IR = 0.47 CL = 300 on the Hagen–Poiseuille solCL = 300 ution, mesh resolution had to Drain OR be increased even more within 0 0 Capillary the capillary, and farther still near the capillary entrance to reIR IR solve the strong gradients there. For three regions in the capillary setting, maximal element size was set at 10 mm and minimal element size at 1 mm: a cylindrical region in r-z coordinates from r ¼ 0 to 1 mm and from z ¼ 0 to 1 mm over the top of the capv0 v0 illary opening, a cylindrical collar –z –z of fluid surrounding the capillary Fig. 1 Schematics of the capillary and drain geometries in r-z coordinates; r is radial distance from the axis of tip from r ¼ 0.6 to 1 mm and symmetry, and z is distance from the center of the capillary tip or drain. Suction, that is, a pressure gradient set up from z ¼ 0 to -1 mm, and the by the specified volumetric outflow rate, v0, moves water in the -z direction. The origin is indicated in each panel fluid in the uppermost 11 cm of as a small, white circle labeled with a zero. The two related, axisymmetric geometries are a spherical tank with a the capillary. In the rest of the capillary drawing fluid from its center and a hemispherical tank with a drain of similar geometry exiting the center of its flat, circular bottom. Solid lines indicate no-slip boundaries. Dotted lines indicate boundaries acting as diffuse capillary the same minimum inlets of fluid, set at constant pressure. Hemisphere (HR) and sphere (SR) radii and capillary lengths (CL) were was used, but the maximum was incremented until further increases yielded no further change in the precision of results reported here (i.e., HR ¼ raised to 0.1 mm. For most SR ¼ CL ¼ 300 mm ø 319D). The capillary and drain have the same inner radius (IR); the capillary outer radius no-slip boundaries, a maximal (OR) includes the capillary wall thickness. For clarity of labeling, radii and lengths are not drawn to scale. CL

Axis of symmetry

CL

Axis of symmetry

section, as p minus a different reference pressure that facilitates comparison with past solutions. I chose capillary inner (0.94 mm) and outer (1.2 mm) diameters and volumetric flow rates to equal those of Young et al. (2012). I also used the same fluid density, r (¼1021.9 kg m-3), and dynamic viscosity, m (¼ 1.065 · 10-3 pa s), although scaling against D and Re generalizes those arbitrary choices to other combinations of capillary diameters, viscosities, and fluid densities. I then extended the range to 0.01 #Re #2000, throughout which laminar


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condition in each subsequent computation. For each Re, I iterated, starting with the previous solution, to achieve convergence #10-10 as reported by COMSOL. Convergence is reached when relative error in the subsequent calculation falls to this specified relative tolerance. For my computations, this criterion was estimated in COMSOL as relative change in subsequent LU-factorized (with partial pivoting) matrices of equation outputs (e.g., Quintana-Ortı´ and van de Geijn 2008). I discovered that for Re in the mid-hundreds, flow solutions for the capillary could suddenly flip to a different geometry, and that if I decremented rather than incremented Re, I could maintain that flipped geometry all the way back down to Re ¼ 0.01. For Re #330, I was thereby able to identify two alternate, stable solutions for the capillary case: one that drew fluid primarily from above the capillary opening and was “free” of no-slip boundaries, and one that drew fluid primarily from below, along the capillary outer wall, referred to as “trapped.” [15] Entrance length, Le, calculated as the distance from the pipe entrance where axial velocity reached 99% of its maximum, was plotted as both Le/D and (Le/D)/Re against Re. Expectation at low Re is that Le/D will remain constant because diffusion of momentum over length scales of the A B capillary radius is substantially fas300 300 Re = 0.01 Re = 2000 ter than advection over the same distance. At the higher Re used here, both advection and diffusion are important in determining Le, z and expectation is that (Le/D)/Re 150 D 150 will reach a constant value (Durst et al. 2005). Outside the capillary, I used the same model runs to calculate decay length, Ld, a parameter analogous to entrance length but 150 300 150 300 instead applied to falloff in velocity. r/D r/D I chose it as the upstream distance Fig. 2 Streamlines showing acceptance regions in the drain geometry, that is, the volume from which fluid from the capillary tip (r, z ¼ 0) at enters at Re ¼ 0.01 (A) and 2000 (B). Radial (r) and axial (z) coordinates have been nondimensionalized which axial flow speed fell to 1% of through division by the capillary diameter (D). Red streamlines spanning the acceptance region were produced mean flow speed in the capillary. by specifying 20 equally spaced points across the mouth (z ¼ 0) of the drain radius from r ¼ 0 to r ¼ a - 0.0011 mm and using COMSOL to draw flow trajectories through them. Blue streamlines indicate a recirAnalogous off-axis decay distances culation region and were generated by choosing points by eye to plumb its extent; COMSOL will draw a would in general differ. streamline through any specified point. At higher Re, the recirculation region is small but continues to circulate [16] Two checks proved usecounterclockwise. At all Re, a hemispherical region of omnidirectional convergence whose radius is much larger ful in quality control. The first was than Ld contains streamlines that have little curvature (dotted lines).

element size of 0.1 mm and a minimal size of 1 mm sufficed, but for the axis-orthogonal surface of the capillary tip, spanning its inner and outer radii, the maximum needed to be lowered to 5 mm to get convergence. [13] Meshing for the drain geometry (Fig. 1) differed slightly. For a cylindrical region in r-z coordinates from r ¼ 0 to 1 mm and from z ¼ 0 to 1 mm over the top of the drain opening and for the fluid in the uppermost 11 cm of the capillary, maximal element size was set at 10 mm and minimal element size at 1 mm. The noslip boundaries adjacent to these fluid regions had minimal element size set at 0.1 mm and maximal element size set at 0.1 mm. Remaining no-slip boundaries had maxima and minima each set an order of magnitude larger. Mesh sizes so fine for each geometry were needed to get accurate parameter estimates at only the highest few Re. For most Re, larger meshes would have sufficed. Decreasing mesh sizes further failed to change the first three significant figures in estimated variables. [14] At Re ¼ 0.01, I used zero flow and uniform pressure (¼ p0) of the entire fluid volume as starting conditions. For computational efficiency, however, I used the solution for the next lowest Re as the starting


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numerical coefficients in Eq. 7 precisely, confirming numerical precision at both ends of the Re range studied.

30 Ld/D trapped Ld/D free Ld/D drain

10 Ld

5 3.54 2.50

D

1 0.5 0.01

0.1

1

10

100

Results Flow Approaching a Drain [17] Flow patterns approaching a drain showed remarkably little diversity (Fig. 2). To distances of Ld/D #75, they roughly approximated hemispherical convergence, that is,

1000

Re

Fig. 3 Nondimensional decay lengths (Ld/D) versus Re for the drain and capillary geometries. The bottom, no-slip boundary in the drain geometry severely damps changes in flow geometry and Ld/D (black circles). Starting from an initial condition of zero velocity and zero volumetric outflow rate (v0), and then incrementing outflow rates, produces streamlines that enter the capillary opening from above (the free state, blue triangles with one vertex pointing upward) and that transition from a low-Re configuration to a high-Re configuration over the range 1 # Re # 100 (thick blue arrow). Free flows become unstable at Re . 330, flipping to a trapped state (moving upward, trapped on the outside of the capillary; red triangles with one vertex pointing downward and short, thick pink arrow pointing to the right). If flow rate is decremented in any trapped configuration, flow remains trapped on the outside capillary wall and follows the trajectory shown in red and paralleled by the thick, pink arrow pointing left and down, with marked divergence in Ld/D at low Re between trapped and free states. At high Re where only the trapped state is stable, flow in the drain geometry asymptotes toward hemispherical convergence (black dashed line, Ld/D ¼ 3.54) and in the capillary geometry toward spherical convergence (dashed red line, Ld/D ¼ 2.50).

for convergence on the Hagen –Poiseuille maximum of twice the mean capillary flow speed. That test flagged data entry errors (e.g., incorrect keyboard entry of the volumetric outflow rate) and indicated mesh adequacy. Meshes much too large produced noisy or periodic variation in flow speed along the axis, whereas meshes only slightly too large in the capillary or drain produced asymptotes above the theoretical maximum of twice mean flow speed. A second check was a rerun with boundary conditions altered to have uniform inflow flow at the capillary or drain entrance (but the same Re) for comparison of numerical accuracy against Eq. 7. Although Durst et al. (2005) noted that the equation fitted their numerical results only within 3.8%, deviations that large occurred only in the transition region near Re ¼ 10 where Le/D shifts from approximate constancy near 0.619 to dependence on Re. My geometries and meshings reproduced the two

UðRÞ ¼ -

wD Qv 2 ¼ : 2pR 2 8R 2

ð8Þ

If I then set the rightmost fraction equal to 0.01 times w; Ld ¼ 3.54D. This equation, however, overestimates flow speed at shallow angles to the tank bottom, where it is affected by the no-slip condition. Resulting shear stress is evident in the form of more strongly diverging streamlines away from the drain at shallow angles to the r ¼ 0 axis at all Re (Fig. 2). By continuity, however, average flow speed toward the origin over the entire hemisphere must equal the quantity on the right side of Eq. 8. Consequently, Eq. 8 also underestimates flow speed for at least some smaller angles to the drain axis. Directional bias in favor of greater speeds along the axis is greater at lower Re, accounting for the somewhat higher decay lengths at lower Re (Fig. 3). Not all the fluid accelerated downward in this way can enter the drain, especially at low Re, so at some distance from the drain an attachment circle forms beyond which near-bottom flow is abaxial (in the +r direction) and forms part of a toroidal vortex surrounding the drain. In cross section, at low Re, streamlines in this convergence region resemble viscous corner flows (Fig. 2). [18] Bottom friction restricts flow change in the drain geometry. Ld/D varied from 5.73 at Re ¼ 0.01 to 3.42 at Re ¼ 2000, only a 40% change in nondimensional decay length (Ld/D) over five orders of magnitude in Re (Fig. 3). For convenience, I refer to the volume from which fluid is drawn into a drain or capillary as the “acceptance region.” Another consequence of bottom friction is that a fluid parcel entering the drain may have originated from anywhere in a large fraction of the entire hemispherical domain, albeit a smaller fraction at lower Re (Fig. 2). Well beyond nondimensional distances of R ¼ 10Ld/D, streamlines lack significant


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curvature. The near field of quasi-linear streamlines extends farther with increasing Re. Flow Approaching a Capillary [19] Flows converging on a capillary opening displayed much greater diversity. Starting either from zero flow velocity or from a solution reached by incrementing velocities from any lower-Re solution, solutions for Re # 330 revealed flow entering the capillary from only a relatively small cylinder of fluid extending above the capillary (with the fluid flowing downward). The acceptance region beyond R . 35D is not attached to any no-slip boundary, so I refer to this geometry as “free capillary flow.” For Re # 5, that cylinder was roughly 20D in diameter. The siphon flow drives downward fluid motion well outside this acceptance region and beyond the modeled region of Fig. 1, but that flow has no opportunity to enter the capillary. With further

C

330

100

100 130

B 0.01 5.0 20 30 44 57 70

A

increase in Re, the pressure gradient near the mouth of the capillary steepens, extends farther outside the capillary, and also begins to draw more water upward past the outer edge of the capillary and around the solid tip. The acceptance region expands gradually with increasing Re, reaching about 140D in diameter for Re ¼ 330 (Fig. 4A). For Re ¼ 330, the acceptance region also extends about 30D below the capillary tip, where flow attaches to the outside wall of the capillary and streamlines sharply change direction to flow upward (Fig. 4A). With both the drain geometry and free flow into the capillary, scaled decay distances (Ld/D) are roughly constant for Re # 1, where the flow pattern is dominated by diffusion of momentum. For Re . 100, they asymptote toward lower plateaus in Ld/D that approach expectation for hemispherical convergence on the drain and spherical convergence on the capillary tip (Fig. 3). For Re increasing from 3 to 100, decay

20 130 200

0.4

80

0

5

150

20

60

–20

2000

57 0.01

100 z D

40 z 1000 D 50 800

20

z –40 D

1300

–60

300

0 0

130 –80

–50 –20 –35

–100 0

20

40 r D

60

0

50

100

150 r D

200

250

300

–100

0

20 r D

40

Fig. 4 Bounding streamlines of acceptance regions for the capillary geometry as Re (the number closest to each curve) changes. Red lines represent the capillary; they are drawn to scale 1D wide and centered on the z-axis. As Re increases in the free state (A), bounding streamlines reach farther downward below z ¼ 0 and experience more shear stress near the outside of the capillary until they become unstable in that configuration and flip to flow from below, upward, trapped along the outer capillary wall (B). Decrementing Re in this downward configuration retains a trapped configuration (B, C), narrower in r than in the corresponding upward configuration (A). Bulges in the streamlines at small, negative values of z (C) accommodate an attached toroidal vortex on the outer wall of the capillary. For flows from above (A), illustrated streamlines shown pass through r ¼ a - 0.0001 mm, z ¼ 0. For flows from below (B, C), streamlines pass through r ¼ 0.0001 mm, z ¼ 0.


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lengths show increasing direct dependence on Re as ad20 20 vection takes on added importance. Re = 44 Re = 14.3 [20] Above Re ¼ 330, downward flows into the capillary became unstable as boundary-layer flow over 10 10 the exterior of the capillary extended farther down the z z outer capillary wall, gaining more traction and flipping 0 D D the flow field to a downward-directed, “trapped” accep0 tance region (with upward flow; Fig. 4B). Attempts to –10 achieve COMSOL model convergence of 10-10 failed for all upward-directed acceptance regions at Re .330 and –10 –20 0 10 20 0 10 20 ended after sufficient iterations in the trapped state, r /D r /D explaining termination of the blue curve at Re ¼ 330 in Fig. 3. Acceptance regions and flow patterns for Fig. 5 Hairpin flow structure in cross sections of the acceptance region in the trapped state for Re ¼ 44 (A) and 14.3 (B). Red streamlines were produced by trapped capillary flows varied considerably (Fig. 4B,C), specifying 20 equally spaced points across the mouth of the drain radius from as the unusual pattern in Ld suggests (Fig. 3). Boundaryr ¼ 0.0001 mm to r ¼ a - 0.0001 mm at z ¼ 0 and using COMSOL to draw flow trapped flow solutions for Re # 330 could be reached trajectories through them. They represent the acceptance region. When decreonly by using another trapped flow solution as the menting Re for a flow trapped on the outside surface of the capillary, no recirculation region is evident for Re ¼ 57; a recirculation region (blue streamlines) first appears initial condition. For the broad range 44 $ Re $ 0.065, near Re ¼ 44 (A) and last appears near Re ¼ 0.065. Flow in these sections of the trapped flows produced an attached, toroidal recirculation region is counterclockwise, constantly accelerated by the adjacent recirculation region surrounding the capillary near its flow entering the capillary and decelerated by friction with the outer capillary wall. tip (Fig. 5). Over most of this range, a cross section of For 20 $ Re $ 0.1, turning of the flow around the attached vortex produces a shape that in cross section (B) resembles a human ear. At Re ¼ 14.3 where the size the acceptance region in an r-z plane had characteristic of the recirculation region nears its maximum, flow detaches from the outer wall of hairpin structure, with its limiting streamlines looping the capillary at about z/D ¼ - 13, marking the lower limit of the lower “lobe” of around this attached vortex (Fig. 5). For 20 $ Re $ 0.1, the flow. the recirculating vortex in cross section resembled a human ear, its upper lobe extending above z ¼ 0 (Fig. 5B). Streamline patterns for boundary-trapped flows on A B a capillary at Re where the Re = 0.0 0.0 2000 externally attached vortex 44 –0.2 –0.2 was absent resembled 14.3 Best-fit parabola 0.01 –0.4 –0.4 a Rankine half-body, that w –0.6 is, inviscid flow past a 2 w –0.6 Re = 0.1 point source (e.g., Batche–0.8 –0.8 Hagen–Poiseuille lor 1967, p. 461), but with equilibrium parabola –1.0 –1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 flow directions reversed. r r Flow not entering the sia a phon is driven upward by Fig. 6 Radial profiles of vertical velocity (w) relative to the Hagen – Poiseuille speed maximum, j2w j; at the entrance flow within the acceptance (z ¼ 0) for various Re. The classic solution assumes that entry flow velocity everywhere equals mean flow velocity for the region. Most unusual is the capillary (dashed lines). Where viscosity plays a large role (low Re), this assumption is a poor one, and flow at entry is very region of direct dependence nearly parabolic (A). All solutions shown are for the capillary geometry with flow from below (upward along the outer capillary wall), but results differ little in the other geometry or flow configuration (except for the capillary at Re ¼ 2000, of Ld/D on Re for Re #1.3 ! A

B

where only the shown flow orientation is stable). Uniform vertical inflow is most accurate as an approximation for w ðr; 0Þ near Re ¼ 2000 (B), just short of Re ¼ 2037, above which laminar behavior breaks down and the Hagen– Poiseuille solution no longer applies, but note that only the vertical component is shown and that the radial component exceeds mean vertical pipe velocity at some locations across the entrance.


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1000

100

100

Le D

(Le /D)/Re trapped

Le /D trapped

(Le /D)/Re free

Le /D free

(Le /D)/Re drain

Le /D drain

10

1

10

Le/D Re

1

0.1

Durst et al. (2005) Uniform inflow, Le /D

0.01

0.1

(Le /D)/Re uniform

1

10

100

1000

0.1

0.01

Re

Fig. 7 Relative entry length (Le/D) and Re-scaled relative entry length ((Le/D)/Re), versus Re for the capillary and drain flow geometries and compared with model results for the classic pipe entrance problem (uniform flow at entry). The brown dashed curve plots Eq. 7 (Durst et al. 2005), and dividing by Re gives the corresponding solid brown line having asymptote at (Le/D)/Re ¼ 0.0567 at high Re. Results for both flow geometries used here and both trapped and free capillary flows are very similar except at low Re, where D/Re for the trapped flow picks up a direct dependence on Re. With constant volumetric outflow rate as the boundary condition, transition from dominance by diffusion of momentum is delayed to higher Re than in the classic solution described by Durst et al. (2005), creating a distinct minimum in the curve of (Ld/D)/Re near Re ¼ 15.

in the trapped state, where Ld ¼ 4:434 Re 0:409 D

ð9Þ

fits numerical results within 2.5%. [21] Radii of acceptance regions are smaller for the trapped state than for the untrapped at a given Re (Fig. 4). In the untrapped state, the downward-flowing fluid above the capillary tip can support little shear, and the highest-velocity fluid lies closest to the axis (r ¼ 0). In contrast, in the trapped state, the no-slip condition enforces a strong velocity gradient with faster flow at greater r. The volume of fluid contained increases as the square of the radius, so the combination of rapidly increasing velocity and volume with distance from the capillary outer wall meets the need set by v0 within a smaller value of r than for free capillary flow. Turn of the fluid over the capillary lip dynamically resembles half a toroidal vortex. Entrance Lengths [22] Flow at and just inside the capillary entrance at Re #5 varied little in pattern and differed little between capillary and drain geometries. Velocity at the capillary

entrance was much closer to the equilibrium Hagen– Poiseuille profile than to uniform flow (Fig. 6). For Re #0.1, axial velocity at the entrance (origin) was 87.5% of the Hagen– Poiseuille maximum rather than the 50% that characterizes uniform flow. Not until Re rose above 7 did that figure drop below 80%. Departure from a best-fit parabola in the model results shows a more uniform profile near the capillary axis (dw/dr smaller than for the parabola) and a steeper profile near the wall (Fig. 6A). However, because the profiles at low Re are so nearly parabolic at the entrance, nondimensional entrance lengths at low Re are very short, plateauing at Le/D ¼ -0.2725, rising to -0.274 at Re ¼ 0.2 and to -0.335 at Re ¼ 5 (Fig. 7). These scaled entrance lengths at the lowest Re are roughly 44% as long as for hypothetical, uniform flow entering a pipe (Fig. 7). [23] At higher Re, vertical velocities at the capillary entrance more closely approach uniform flow (Fig. 6B) and nearly achieve it. Over the entire range of laminar pipe Re, Le is shorter in the flow geometries explored here than in the hypothetical case of uniform flow at the pipe entrance. An equation of the form of Eq. 7 fits high and low Re entry lengths but seriously overestimates entry lengths for 1 , Re , 100. A slightly modified equation fits entry lengths for Re #15: 1=1:6 Le ¼ 0:27251:6 + ð0:052 · 0:6 ReÞ1:6 : D

ð10Þ

In effect, the ability of the flow to adjust upstream of the pipe entrance and the added no-slip boundaries cause transition to high Re behavior at Re 40% larger than in the classic case lacking upstream behavior and boundaries. The first constant on the right is stated with excess precision because it is a compromise between the value (0.272) that is more accurate for the two trapped states and the value (0.273) that is more accurate for free capillary flow. Reduction of Re by a constant rather than proportional amount serves to taper apparent reduction in Re for Re $ 15: 1=1:6 Le ¼ 0:27251:6 + ½0:052 · ðRe - 6Þ1:6 : D

ð11Þ

[24] Eqs. 10 and 11 match at Re ¼ 15. They fit the numerical results for Le/D within 3% for flows coming


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from above into a capillary and into a drain. However, for boundary-trapped capillary flows (with flow coming from below) at Re so small that the attached vortex has disappeared and streamlines effectively must turn two right angles, there is a systematic offset (Fig. 7). For Re ¼ 0.01 with the flow in this configuration, Eq. 9 overestimates the numerically calculated Le/D by 10.7%, but this weak dependence of Le/D on Re does not extend to Re .0.1, and the estimation error can be largely eliminated by including an additional term of - 0.00026/Re on the right side of Eq. 10. Unlike the classic case analyzed by Durst et al. (2005), (Le/D)/Re shows a distinct minimum at intermediate Re (ø 15; Fig. 7). Discussion Reduced Entrance Lengths at Low Re [25] Entrance lengths for both the capillary and drain geometries show distinct behaviors at high and low , Re . , 100. Substantial Re, with a transitional region, 8 . departure of entry lengths in Fig. 7 from numerical results of Durst et al. (2005) comes as no surprise at low Re, where viscosity dominates and makes uniform flow profiles at entry all but impossible. Effects of wall friction propagate upstream of the pipe entrance through diffusion of momentum so that some of the transition to a parabolic flow profile occurs before flow reaches the pipe entrance—a mechanism unavailable in the classic model, where uniform flow at entry is the starting condition. Added friction with the tip and outside wall of the capillary and with the bottom in the drain geometry does not exist in the hypothetical pipe entry problem or its classic solution. The results presented here thus help to clarify why so much deviation from prediction—based on

assumed uniform entry velocity—has been observed (Durst et al. 2005). [26] Characteristic of the classic entry-flow solution at low Re is a local maximum in pressure at a small distance ( ø 0.15D) from the entrance along the axis (e.g., Dombrowski et al. 1993). Pressure first spikes in the incoming flow at the entry wall (r ¼ a, z ¼ 0) because the no-slip condition stalls the flow there (Fig. 8). The resulting pressure gradient drives radial convergence from this pressure maximum toward the pipe axis, as well as a maximum in axis-parallel flow speed near but not next to the pipe wall, sometimes referred to as a velocity overshoot (Durst et al. 2005). Just inside the entry, the inward radial component of velocity reaches a substantial fraction of the mean axial velocity in magnitude (Fig. 8). That convergence results in the local pressure maximum along the axis (Figs. 8, 9). Friction with the wall and the fact that by continuity the same net volume of fluid must move per unit of time through every cross section of pipe combine to cause acceleration of fluid closer to the axis as the wall boundary layer grows to occupy more, and eventually all, of the pipe radius, but for low Re there is no “inviscid” central core even at the entry. The convergence that is forced to be located inside the entry in the classic solution is pushed outside, upstream of the opening, through diffusion of momentum (Fig. 8). As a consequence, the global maximum in pressure at r ¼ a, z ¼ 0 that is characteristic of the classic solution is much reduced or eliminated entirely in a contracting flow geometry (fluid entering a pipe from a larger volume). Large Radial Components of Inflow at High Re [27] Apparent convergence of the present flow model on dynamics of the classic solution at high Re, suggested in

Fig. 8 Flow at Re ¼ 0.01 for boundary conditions of uniform pipe entry velocity (A - D) and constant volumetric outflow rate from capillary (E - H). Contrasts a projecting shown are pressure ( p - p0, Pa) (A, E), absolute value of the ratio of downward vertical speed to mean flow speed in the capillary ( w=w ) (B, F), absolute value of the ratio of radial speed to mean vertical flow speed in the capillary ( u=w ) (C, G), and streamline patterns (c) (D, H). Isopleths of pressure are linearly spaced. White arrows show flow directions. Uniform entrance velocity yields a global pressure maximum just inside the lip of the pipe (A), whereas constant outflow rate produces a global pressure maximum just above the inside corner of the lip (E). Black streamlines originate at 20 equally spaced points that span half the capillary opening. The dotted red line is a vertical to show that the outermost streamline has some curvature. Magenta streamlines show flow driven (downward) by the fluid converging on the opening but deflected by the outer wall of the capillary and are drawn through points spaced 0.04 mm apart at z ¼ 0, starting at r ¼ 0.6001 mm. The divergence of a black from a magenta streamline at the pipe edge marks the presence of a stagnation circle and a secondary pressure minimum on the tip near its outer edge. Note substantial adjustment by the fluid in advance of the capillary opening (H).



A

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3 (2013)

Re = 0.01 –0.0002

–0.0004

–0.0006

0.05 p*

–2.0

1 rw 2= 0.00566 Pa 2 –1.5

–1.0

–0.5

0

0.5

–0.0008

0

B

0

0.0300

–0.0010 1.0

Re = 2000 –2000 Uniform entry flow Numerical solution

–4000 –6000 –8000

1 p* 1 rw 2= 1133 Pa 2 –60

–50

–40

z /D

–30

–20

–10

–10,000

0

Pressure deviation from p0 (Pa)

†

Pressure deviation from p0 (Pa)

32

–12,000

z /D

Fig. 9 Contrasting axial pressure gradients (Pa, for r ¼ 0) at the pipe entrance under the classic boundary condition (uniform entrance velocity; dashed curves) and under constant volumetric outflow rate (solid curves) for Re ¼ 0.01 (A) and 2000 (B). For ease of comparison curves have been shifted to show deviation from pressure at the pipe entrance (classic model) or from far-field pressure (constant volumetric outflow rate). At the downstream (-z) end in both plots and with both boundary conditions, the fluid is close enough to its Hagen– Poiseuille equilibrium to have the same pressure gradient (slope). The solid line in A is for free capillary flow, whereas the solid line in B is for trapped capillary flow. For the new boundary conditions, pressure loss includes costs of driving the flow outside of the entrance. Pressure can be nondimensionalized by dividing by the quantity shown in the equation; scale bars at the left express pressure in these nondimensional units ( p*).

particular by Fig. 7, proved illusory. My initial expectation was that Eq. 10 would hold over the whole range of Re, reflecting the ability of momentum to diffuse upstream of the capillary or drain entrance, but otherwise would follow the functional form well explained by Durst et al. (2005). Nothing in the dynamics analyzed by Durst et al. (2005), however, could explain the global minimum in (Le/D)/Re near Re ¼ 15, that is, the divergence of results between Eqs 10 and 11 for Re .15. Although w(r, 0) approached mean pipe flow speed, radial velocities increased in both absolute and relative magnitudes with Re, even exceeding mean axial velocity (Fig. 10). Nothing comparable to the radial velocities of Fig. 10C at the entrance occurs in the classic model. Not only did they increase, but significant radial velocities were also drawn further into the capillary or drain. Reflecting the stronger inertial forces at these Re and the pressure gradient associated with acceleration of fluid around the tip and into the capillary, this inrush created a local minimum in pressure just below the lip, and this adverse pressure gradient (opposing the flow direction along the pipe wall) led to flow separation (Fig. 11). At Re ¼ 2000, an axially stretched torus at its widest occupied 0.16 of the pipe radius (at z ¼ -0.406D; Fig. 11). Because there is no net axial velocity in the region occupied by the recirculation vortex, it effectively constricts the pipe, raising mean velocity in

the narrower, central remainder by a full 30%. Acceleration caused by this flow constriction resulted in a local maximum in axial velocity and corresponding local minimum in pressure (Fig. 9). Subsequent flow deceleration produced a local axial minimum in axial flow speed and axial maximum in pressure at z ¼ - 2.66D. It would not be surprising if this deceleration led to turbulent bursting at Re , 2037. The meticulous experiments of Avila et al. (2011) used long tubes and took care to work far from the entrance and any such source of instability. A further caution is that the solutions given here exclude swirl (flow components with circulation about the z-axis). At the higher Re, instabilities may well result in swirl. [28] An adverse pressure gradient and flow detachment at entry apparently underlie the need for Eq. 11 when Re .15. Approximate convergence of entrylength solutions between the classic case and the capillary and drain (Fig. 7) at high Re are thus caused by the balance of competing forces absent from the classic problem. No-slip boundaries outside the inner diameter of the pipe and adjustment of the upstream flow through diffusion of momentum stabilize the flow. Radial inrush of fluid and flow separation destabilize it. That the classic solution (Durst et al. 2005) gives an approximately correct solution for Le /D when Re .100 simply reflects the fact that with increasing Re it takes a


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Fig. 10 Isopleths of vertical and horizontal velocities scaled against mean vertical flow speed in the capillary at Re ¼ 2000: downward components (A), upward components (B), and radial components (C ). Solid white and black arrows indicate the sole direction of the components shown in each panel. All isopleths are in increments of 5% of mean vertical flow speed except for the smaller abaxial velocity within the capillary (C). Note the local maximum and minimum in panel A, where values . 1.55 times the mean flow speed have been circled in white to draw attention to the local flow acceleration. Upward speeds (B) exceed 0.20 times the mean within the capillary and 0.45 times outside. Upward flow inside the capillary is the wall-proximate part of an attached vortex; the dotted vertical line helps read its radial extent on the scale below. Nonzero upward components exist throughout the gray and red regions. The radial inrush at its peak exceeds 1.25 times mean vertical speed in the capillary (C ), and a weak abaxial component is present where the axial flow of panel A is decelerating past the recirculation vortex. The color bar applies only to panel C.

longer stretch of pipe to converge through radial diffusion of momentum on Hagen –Poiseuille flow from any entrance condition that departs from it. [29] A recirculation region near the pipe entrance is particularly problematic for sampling of solutes because the volume contained in the recirculation region under steady flow will exchange solutes through molecular diffusion. New water will be contaminated with material diffusing out from this “dead space.” More subtly, acceptance regions vary with Re. It is not yet clear to what extent these issues can be alleviated or exacerbated by altering design of the entrance geometry or to what

extent they are made moot by the presence of ambient flows. [30] Contrast with the classic flow solution at high Re is stark (Fig. 12). Whereas capillary and drain models show distinct local pressure minima just inside the pipe entrance at the wall, the classic solution shows a global maximum. The relative importance of radial flow is much reduced at high Re in the classic case (Fig. 12), and the high axial flow speed prevents convergence near the entrance from producing any local pressure maximum along the axis (Fig. 9). Nothing in the boundary conditions or flow evolution for the classic


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0.5

2.0 –1

A

B

1.5

0

C 200

–100 1.0

–500

z D

100

–0.5

0.5 –1000

0

0 –1.0

–7600

–0.5

–1.0 –5500 –5000 0

0.5

–100 1.0

1.5

2.0

r a

2.5

3.0

3.5

4.0

0

0.2 0.4 0.6 0.8 1.0

0

r a

200

400

600

r a

Fig. 11 Pressure and streamline fields associated with flow into a capillary at Re ¼ 2000. Isopleths of pressure deviation from the constant pressure imposed at the outer model boundaries (A) are linearly spaced at 500-Pa increments except for the outer two. Note the local pressure minimum where a global maximum is found under uniform entrance velocity. Blue shading highlights a local minimum that extends all the way to the axis and is created in part by fluid acceleration past the attached recirculation region (B), indicated in deep royal blue. The streamlines shown were created through 20 equally spaced points across the full span of the radius at z ¼ 0.001 mm. A largescale view of those same streamlines as they diverge outside the capillary (C) resembles a Rankine half-body, that is, inviscid flow past a point source (e.g., Batchelor 1967, p. 461), but with flow directions reversed. The blue streamlines were created through points spaced at 10D intervals from the axis along the dashed line shown. All their flow components are in the + z and -r directions, driven by the flow entering the siphon.

case leads to an adverse pressure gradient or flow separation for Re #2000. Flow Upstream of the Pipe Entrance [31] Because the classic case of pipe entry starts at the entrance and works downstream, no comparable body of literature and numerical models treats flow toward a pipe opening. Happel and Brenner (1983, p. 153), however, give the stream function and pressure drop for flow through a circular aperture at low Re, which have been used in estimating pumping costs (Riisga˚rd and Larsen 1995). Upstream streamlines near the pipe entrance (R ,D) for the drain geometry appear approximately hyperbolic, as they are for a circular aperture (Happel and Brenner 1983). Happel and Brenner (1983, p. 140) also treat a point source emanating from a wall as a conical diffuser. In creeping flow, direction is reversible, so the results also apply to a wall sink (Happel and

Brenner 1983, p. 141). For distances near Ld at low Re, conical spreading appears to be a reasonable approximation compatible with an approximately constant value near 5.73D for Ld (Figs. 2, 3). That result equates with a cone whose half-acceptance angle (u0 in Happel and Brenner 1983, Fig. 4-24.1) is 76.28 and draws water through 38.1% of the surface area of a sphere rather than the 50% expected of a hemisphere. Conical acceptance likewise is a reasonable first approximation for free capillary flows at low Re and R of about 10D (Fig. 4). The cone is much narrower, however, than that for the drain. The implied half-acceptance angle (by Ld ¼ 12.58D; cf. Fig. 3) at low Re is 32.68, drawing water through only 7.9% of the sphere area surrounding the capillary tip. [32] The positive slope for Ld versus Re (Fig. 3) at low Re in the trapped capillary state, by contrast, reflects a rapidly changing geometry of acceptance that is more


35

A

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0

1.0

B

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Peter A. Jumars

C 1.7 0

1.1

0

0.2

0.4

r a

0.6

1.0 79,800

–12.0 9000

–0.1 –1.0

0.8

8900

1.2 8800 –0.2

z D

–13.0 1.3

–0.3 8700

–2.0 –0.4 –14.0 –0.5

1.4

z D

8600

1.8

–3.0

D

E

0

0

0.15

–15.0

0.10

–4.0

0.05

1.5 –0.05

–0.5

–16.0

z D

z D

–5.0 –0.10

for Re , 4 (Fig. 4). Both trapped and free capillary flows at large R from the tip show slower than conical expansion ($100D for Re #1300 and .180D for Re ¼ 2000; cf. Fig. 4), asymptoting toward cylindrical shape. There apparently is no published analytic theory for trapped capillary flows, but they may be the products of a Coanda˘ effect (Tritton 1977, Section 22.7). For free capillary flow at low Re, however, w(r) at large values of z in the numerical models is distributed approximately as expected when the half-acceptance angle, u0, approaches zero (Happel and Brenner 1983, p. 140).

–17.0 –1.0

Caveats [33] Far-field inhalant 1.9 –6.0 flows are generally very 0 1.0 0 1.0 0.90 0.95 1.00 0 0.2 0.4 0.6 0.8 1.0 slow and thus should r r r a a a be interpreted cautiously. Fig. 12 Flow in a capillary under uniform entrance velocity at Re ¼ 2000. The vertical downward component of flow speed They are particularly senshows characteristic minima at the wall and maxima near the wall near the entrance (A) before adjusting through diffusion of sitive to numerical issues, momentum toward the Hagen – Poiseuille equilibrium (B, bottom). Numbers in A, B, and D are speeds divided by mean especially as the outer downward flow speed in the capillary. Radial components are relatively small (D); note the restricted range of radii drawn. boundaries of the model Isopleths of pressure deviation (Pa) from its value at the inlet (C) are linearly spaced so far as drawn, but the gradient is too steep near the wall at the inlet to continue the spacing shown. Pressure spikes at z ¼ 0, r ¼ a. Only very modest adjustment space are approached. In occurs in streamlines over the first diameter-equivalent of vertical distance (E) because the high flow speeds allow little analog experiments, they diffusion of momentum over that distance. would also be especially sensitive to ambient flows, complex than spherical spreading or spreading in a con- including thermal convection. A little more subtly, ical section of a sphere (Fig. 4) and includes reversal of although Re suffices for both geometric and dynamic flow direction. The z intercepts of the acceptance curves similitude in the classic solution and for drains, of Fig. 4B,C are just below the point where vertical flow matching Re alone is inadequate to assure geometric or speed drops to zero before reversing and flowing up- dynamic similitude for flow from a large tank into a ward. Reversal is more abrupt and spreading is slower capillary even if the tank walls are far away. An additional 1.6


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length scale—pipe or capillary wall thickness (or, equivalently, outer pipe radius or diameter)—enters this problem. My results strictly apply only to the case where wall thickness is 0.28 of the inner radius (or, equivalently, where outer diameter is 1.28 times inner diameter). Entry length (Le), however, appears remarkably insensitive to this variable (Fig. 7), as the drain geometry could be considered a capillary with wall thickness q D). Insensitivity does not extend to Ld, however (Fig. 3). [34] The dichotomy of capillary versus drain hides yet another quantitative variable: capillary height. It is not clear how many capillary diameters a capillary would need to extend above the bottom to behave more like the capillary geometry and less like the drain. Again, my cases are only for heights of zero (the drain) and heights q D (the capillary far from any wall). Fig. 4A, however, suggests that deviations from free capillary flow geometry will occur when the greatest downward extent of the acceptance region in that figure exceeds capillary height above the bottom, likely increasing the acceptance volume and perhaps effecting trapping at Re #330. My results are also shape dependent. They hold explicitly for a capillary wall that is rectangular in axis-parallel section (blunt tipped, not rounded). Location of the flow separation at high Re will be sensitive to shape of the entrance. Two further, substantial caveats are that the present results apply strictly only to the steady state and only in the absence of ambient flows, including those created by the exhalant siphon. Extant observations nevertheless suggest that coherent inhalant flows in the presence of ambient flows extend to R ,10D (Ertman and Jumars 1988; Troost et al. 2009). Despite these many and significant limitations, however, the results given here should have much broader applicability than the ubiquitously reproduced case of steady flow with uniform entrance velocity. They should be far easier to test than those based on an untenable entrance condition. Biological Implications [35] In support of biological relevance of my limited Re range, it is worth noting that ascidians cover the bulk of my modeled Re range (10-1 to 102 cf. Sherrard and LaBarbera 2005), that alpheid shrimp respiratory pumping occasionally exceeds upper limits of laminar

flow (Gust and Harrison 1981), and that many clams of modest size reach Re of 102 (e.g., Mya arenaria; cf. Jørgensen and Riisga˚rd 1988). Although data are lacking to provide strong tests of my current models by covering both low and high Re ranges modeled here, limited testing of consistency for my higher range of Re is possible with the data of Kiørboe et al. (1999). I retained my 300-mm radius for the model space but ran the capillary model for the actual inner radii used by Kiørboe et al. (1999). In addition to the information in their paper (temperature ¼ 198C), they conveyed that the outer radius of the smaller tube was 1.4 mm and that salinity was near 33 Sp (T. Kiørboe, pers. comm.). For the larger tube, I assumed an outer diameter of 3.8 mm. Of five cases that they plotted (their figure 3), three fell in the laminar range for which my model is suited. For both the runs with the larger tube, their empirical equations match my predictions closely (Fig. 13). For the one run with the smaller tube, their empirical results are closer to expectation from spherical convergence but still are not far from the model predictions. A larger Re range would be more revealing, but the fits are encouraging. [36] The litany of caveats underscores the need for explicit experimental testing that will lead to confirmation or further modifications. The value of the model results is clear, however. First, through consistency for both capillary and drain geometries, they make it clear that for low Re, the conceptual construct of an “inviscid core” (e.g., Fargie and Martin 1971) that has not yet been affected by diffusion of momentum from the pipe walls is misleading. Entrance lengths for the present geometries should be substantially shorter at low Re than indicated by numerical models assuming uniform entrance velocity. The present results comprise both qualitative and quantitative predictions that were not obvious without the model. Two prominent examples that need empirical testing are the existence of two alternate stable states (free and trapped) for flows into a capillary and the existence of a recirculation region on the outer capillary wall for 0.065 #Re #44 in the trapped state (Fig. 5). [37] Matching flow models inside and outside the pipe, that is, using the principle of continuity to constrain them simultaneously, also suggests improvements


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3.0 2.0

B

A 2.5

U (cm s–1)

1.5

1.0

U (cm s–1)

Re = 551 U = 0.151R –1.95 2 U = D w2 16R Numerical solution

Re = 1542 U = 0.375R –2.09

2.0

2 U = D w2 16R Numerical solution

1.5

1.0 0.5 0.5

0.0

0.5

1.0

1.5

2.0

0.5

0.0

1.0

2.5

2.0

0.12

Spherical convergence

C 0.10

2.0

Re = 1406 U = 0.110R –1.97

D

Re = 551 Re = 1406 Re = 1542 Re = 3049 Re = 3970

0.08

2 U = D w2 16R Numerical solution

1.5

U/ w

U (cm s–1)

1.5

z or R (cm)

z or R (cm)

0.06

1.0 0.04 0.5

0.0

0.02

0.5

1.0

1.5

2.0

0

1

z or R (cm)

2

3

4

5

z /D or R /D

Fig. 13 Fits of spherical convergence (dashed black lines, for R in any direction) and my model results (filled blue circles, modeled along the z-axis) to data in Kiørboe et al. (1999, figure 3). A and B correspond to the same designators in Kiørboe et al. (1999), but my panel C corresponds with their panel E. Red lines in A – C summarize their experimental results for U versus R. For A and B, correspondence is good between the numerical and experimental results. In C, experimental results fall closer to spherical convergence. Most of Kiørboe et al.’s (1999) results collapse onto nearly the same curve when plotted nondimensionally (D), reflecting their uniformly high Re relative to the range treated in this article. No comparable published results for lower Re that include volumetric flow rates as well as flow velocities appear to exist.

to empirical quantification of siphon flows. Kiørboe et al. (1999) fitted measured flow speeds to curves of the form aR b, and Troost et al. (2009) fitted them to curves of the form ce-dR. In these equations, a, b, c, and d are obtained from best fits. For both spherical and hemispherical convergence, b should be exactly -2. For the high Re used by Kiørboe et al. (1999), results shown (their figure 3) do fall in a narrow range between 1.95 and 2.13. Moreover, if I write aR -2 ¼

ðw pD 2 =4Þ ; 4pR 2

ð12Þ

based on Eq. 4, then I can solve for a and find that it equals D 2 w=16: [38] Solutions for the base cases of capillary and drain geometries enable the framing of hypotheses and understanding and quantification of structural and functional deviations from those base cases. If flow separation were not observed inside a blunttipped capillary, then the functional role of rounded siphon edges could not be appreciated. A steady-state solution is now in hand to compare with future time series in unsteady flow. Acceptance regions for siphons and drains give a first approximation of the volume from which benthic suspension feeders draw their food and from which respiratory currents come.

Significance to Aquatic Environments [39] Uniform entrance flow is a poor approximation for flow into a tube drawing water from a large volume and of a drain flush with the bottom at high Reynolds numbers (Re .100) because much of the flow entering faster, less viscous, and larger-diameter pipe flows enters with appreciable radial velocity that affects flow well into the pipe. Numerical models of lower and also ecologically relevant laminar conditions (Re ,5) indicate that flow adjustment occurs upstream of the pipe entrance, an impossibility under the assumption of uniform entrance velocity. Flow at low Re entering the pipe comes from a relatively small, cylindrical region parallel with the pipe rather than entering isotropically. Moreover, in some cases the flow can be trapped on the


38

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external, no-slip boundary of the siphon. In addition to some biological consequences, presented below, the results demonstrate the necessity for most applications to replace textbook models based on uniform entry flow. [40] Chemosensing is a dominant sensory modality in the ocean. Whereas tentacles on exhalant siphons are often subterminal, tentacles on inhalant siphons are usually terminal and held so that they provide a grating over the inhalant opening (e.g., Sartori et al. 2008). Straining out large particles is an obvious function, but the present modeling suggests that chemical sensors that vary in radial position could also give information on different parts of the spatial field in the acceptance region. The present results also lead to other interesting questions, such as whether animals can achieve the trapped capillary state and thereby obtain information on, for example, the nearby presence of epibenthic predators. Clams do respond to chemical cues from predators (e.g., Smee and Weissburg 2006a, 2006b). Can an unsteady, accelerating siphon flow separate chemical information from the near field from that from the far field? [41] Even the present results mark an advance over using the classic model for entrance length, which has been virtually the sole model employed to estimate hydrodynamic costs of pumping in the entrance regions of suspension-feeder (e.g., Riisga˚rd and Larsen 2005) and liquid-feeder (Loudon and McCulloh 1999) siphons. Total axial pressure drop gives a first approximation of those costs (Fig. 9), but a better accounting awaits a fuller analysis of the costs of producing flows off the pipe axis and on both sides of the siphon entrance. Acknowledgments This work stemmed in large measure from penetrating questions from Ashley Young (cf. Young et al. 2012). It was supported by Division of Ocean Sciences grants 0724744 to P. Jumars and L. Karp-Boss and 0851172 to P. Jumars and S. Lindsay, both from the U.S. National Science Foundation. I thank T. Kiørboe and E. Saiz for generously providing unpublished details of their studies (Kiørboe et al. 1999) and thank K. Du Clos, J. Crimaldi, P. S. Larsen, and an anonymous reviewer for their many constructive suggestions for improve-

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Received: 11 July 2012 Amended: 24 September 2012 Accepted: 19 November 2012
