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First published online August 17, 2007
Journal of Experimental Biology 210, 2969-2978 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.005942
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Analytical and numerical investigation of the flow past the lateral antennular flagellum of the crayfish Procambarus clarkii

Joseph A. C. Humphrey1,2,* and DeForest Mellon, Jr1

1 Department of Biology, University of Virginia, Charlottesville, VA 22904, USA
2 Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904, USA


Figure 1
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  		Fig. 1. Schematic of a downward flicking flagellum rotating at angular velocity {Omega} relative to the pivot point at its base in a fixed X–Y coordinate system. Relative to an x–y coordinate system fixed at point z along the flagellum's arc length, the approaching local fluid velocity normal to the secant R(z) is Uyf. This generates components Unf=Uyfcos{gamma} and Utf=Uyfsin{gamma} along the normal (nf) and tangent (tf) to the flagellum. Aesthetascs and hydrodynamic sensilla on the flagellum are not shown.

 

Figure 2
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  		Fig. 2. Schematic of an aesthetasc or mechanoreceptor sensillum on a downward flicking flagellum rotating at angular velocity {Omega} relative to its base (see Fig. 1). The fluid velocity approaching the flagellum, Uyf, generates components Uns=Uyfcos{varphi} and Uts=Uyfsin{varphi} along the normal (ns) and tangent (ts) to the sensillum, where {varphi}={gamma}–{psi}.

 

Figure 3
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  		Fig. 3. Variation of the Reynolds number Ref along the dimensionless length of a flicking flagellum for downward ({Omega}=5.24 rad s–1) and upward ({Omega}=3.14 rad s–1) flicks; flagellum dimensions are given in the text. The decreases in flagellum diameter and of the normal component of velocity with increasing distance toward the flagellum tip account for the reduction in Ref with increasing z/Lf.

 

Figure 4
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  		Fig. 4. S-shaped form of the dimensionless velocity (Uin/Uo, continuous line) approximating the rate of acceleration of the far-field flow approaching a flicking flagellum as a function of dimensionless time, t* ({equiv}tUo/df). (In the calculations the flagellum is fixed and the flow accelerates past it.) Also shown is the dimensionless distance (xf/df, broken line) traveled by the approaching flow in units of the flagellum diameter. For the conditions of interest in this work, Uo=8.63x10–2 m s–1 and df=5x10–4 m.

 

Figure 5
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  		Fig. 5. Near-field flow streamlines with dimensionless velocity magnitude superimposed for the 2D flow accelerating (from left to right) past a flagellum (approximated as a long cylinder) according to the far field approaching velocity S-curve plotted in Fig. 4. Results are shown at times t*=1 (A; Ref=2.6), t*=2 (B; Ref=25), t*=3 (C; Ref=47.1) and t*=4 (D; Ref=50). Between t*=3 and t*=4 the flow separates at the top and bottom of the flagellum to form a recirculating flow region containing two vortices downstream of the flagellum.

 

Figure 6
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  		Fig. 6. Profiles of the dimensionless Ux velocity component for the flow approaching (left profiles) and moving away from (right profiles) a flagellum (circle) plotted along the x/df axis at y/df=10 (passing through the front and back stagnation points of the flagellum). Results are given for four dimensionless times (t*=1 to t*=4) for a subregion of the entire 20dfx20df calculation domain, and at t*=4 the flagellum Reynolds number is Ref=50. At each time, the Ux velocity component approaching the flagellum drops from the free stream value Uo(t) for that time to the stagnation point value of 0 within less than two flagellum diameters. The region of reversed (negative) flow in the wake of the flagellum grows asymptotically with time (see Fig. 5 also). The black bars denote the lengths, to scale, of 0.25xdf mechanoreceptor (MR) sensilla oriented normal to the flagellum surface.

 

Figure 7
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  		Fig. 7. Profiles of the dimensionless Ux velocity component at locations –90° (left profiles) and + 90° (right profiles) relative to the stagnation point (bottom of circle) for the flow past a flagellum (circle) plotted along the y/df axis at x/df=5 (passing through the ±90° locations). Results are given for four dimensionless times (t*=1 to t*=4) for a subregion of the entire 20dfx20df calculation domain, and at t*=4 the flagellum Reynolds number is Ref=50. The symmetrical profiles show the Ux velocity component increasing with time. At each time, the Ux velocity component maximizes near the flagellum surface (a characteristic of this class of flows) and decreases to the free stream value Uo(t) for that time within less than two flagellum diameters. The black bars denote the lengths, to scale, of 0.25xdf mechanoreceptor (MR) sensilla oriented normal to the flagellum surface.

 

Figure 8
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  		Fig. 8. Schematic (not to scale) of a simplified model of the flagellum-in-tube experiment and definition of the (x, y, z) coordinate system used for plotting the velocity components Ux, Uy and Uz. The case shown corresponds to proximal-to-distal (P->D) flow, entry port A being to the left and exit port B to the right in the figure. Values for the geometrical dimensions and volumetric flow rate are given in the text. N, north; S, south; E, east; W, west.

 

Figure 9
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  		Fig. 9. Dimensionless Ux, Uy and Uz velocity component profiles for the flow in the annular space between a rod-like flagellum and a tube around it along the S–N (A) and W–E (B) compass directions (see Fig. 8). Continuous lines denote velocity profiles corresponding to the P->D flow and broken lines to the D->P flow. Calculation conditions, given in the text, correspond closely to the experimental, and the results shown are typical of the flow in the annular space in the region 0.20<=z/Lt<=0.60 or, equivalently, 0.33<=z/Lf<=1 for times t>=0.025 s. In this region the only significant velocity component is the axial, Uz, which presents the skewed parabolic profile shape characteristic of the developed flow through an annular passage. The black and green bars denote the lengths, to scale, of 0.25xdf mechanoreceptor (MR) sensilla oriented normal to the flagellum surface.

 

Figure 10
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  		Fig. 10. Dimensionless Ux, Uy and Uz velocity component profiles for the flow in the annular space between a rod-like flagellum and a tube around it along the S->N (A) and W->E (B) compass directions (see Fig. 8). Continuous lines denote velocity profiles corresponding to the P->D flow and broken lines to the D->P flow. Calculation conditions, provided in the text, correspond closely to the experimental, and the results shown are typical of the flow in the annular space in the region 0<=z/Lt<=0.20 or, equivalently, 0<=z/Lf<=0.33 for times t>=0.025 s. In this region the three velocity components are roughly comparable in magnitude, with the P->D flow profiles in the vicinity of the mechanoreceptors corresponding closely to the flow around a downward flicking flagellum and the D->P flow profiles to an upward-flicking flagellum. The black and green bars denote the lengths, to scale, of 0.25xdf mechanoreceptor (MR) sensilla oriented normal to the flagellum surface.

 

Figure 11
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  		Fig. 11. Schematic, not to scale, for evaluating the drag force and torque acting on a medial or lateral mechanoreceptor sensillum contained in a plane passing through the flagellum axis and oriented at an angle {psi} with respect to the local tangent to the flagellum surface. The velocity component, Ux(y), inducing the drag and torque is normal to the plane of the figure and, at any location {eta} along the sensillum, Uns({eta})=Ux(y) where y={eta}sin{psi}.

 

Figure 12
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  		Fig. 12. Schematic, not to scale, for evaluating the drag force and torque acting on a ventral or dorsal mechanoreceptor sensillum contained in a plane passing through the flagellum axis and oriented at an angle {psi} with respect to the local tangent to the flagellum surface. The velocity components, Ux(x) and Uz(x), inducing the drag and torque are respectively normal and parallel to the local tangent to the flagellum surface and, at any location {eta} along the sensillum, Uns({eta})=Ux(x)cos{psi}+Uz(x)sin{psi} where x={eta}sin{psi}.

 

Figure 13
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  		Fig. 13. Maximum values of the drag forces (FMR, N) and torques (TMR, Nm) acting on the medial and lateral (M/L) and ventral (V) mechanoreceptor sensilla of a downward flicking flagellum calculated as a function of time. Calculation conditions and methodology are given in the text.

 




© The Company of Biologists Ltd 2007