2.1 Experimental facility

Measurements were performed in a $40\times 40\times 40~\text{cm}^{3}$ turbulence chamber (Hwang & Eaton Reference Hwang and Eaton2004; Sabban & van Hout Reference Sabban and van Hout2011) having transparent acrylic windows to ensure optical access (see figure 1). Near homogeneous, isotropic air turbulence was generated by eight woofers mounted on the corners of the chamber. The woofers were forced at a randomly changing frequency (90–110 Hz) and phase in order to avoid standing waves. Input signals to the woofers were generated by an AtoD board (National Instruments) controlled by LabView software. Near-HIT in the centre of the chamber was validated by stereoscopic particle image velocimetry (SPIV) and the reader is referred to Sabban & van Hout (Reference Sabban and van Hout2011) for further details. In the following, fluctuating air velocities are denoted by $u_{i}$ ( $=U_{i}-\overline{U_{i}}$ ), and root-mean-square (r.m.s.) values by a prime ‘ $\,^{\prime }\,$ ’; an overbar, ‘ $\,\bar{\;}\,$ ’, denotes temporal and/or ensemble averaging and angle brackets, ‘ $\langle \cdots \rangle$ ’, indicate spatial averaging. For normalization purposes we also used, $u^{\prime }=(\langle u_{1}^{\prime }\rangle +\langle u_{2}^{\prime }\rangle )/2$ . The temporally and spatially averaged air velocities and r.m.s. values of fluctuating velocities from the SPIV pertinent to the current measurements are summarized in table 1. Statistical uncertainties in table 1 were evaluated assuming uncorrelated and normally distributed velocities (Benedict & Gould Reference Benedict and Gould1996). Under these constraints and for 95 % confidence intervals, the uncertainty of $\langle \overline{U_{i}}\rangle$ is $\pm 1.96\langle u_{i}^{\prime }\rangle /\sqrt{N}$ , where $N$ is the number of independent samples and the uncertainty of $\langle u_{i}^{\prime }\rangle$ is $\pm 1.96\langle u_{i}^{\prime }\rangle /\sqrt{2N}$ .

Table 1.

Mean and r.m.s. values of air velocities. SPIV data are taken from Sabban & van Hout (Reference Sabban and van Hout2011).

2.2 Time-resolved PIV set-up

The TR-PIV system (figure 1) comprised a CMOS camera (Photron Ultima APX, $512\times 1024$  pixels at 3 kHz), a high-speed laser (NewWave Pegasus, 527 nm, $10~\text{mJ}~\text{pulse}^{-1}$ ), laser sheet optics, acquisition and data processing software (LaVision DaVis 7.4). The laser sheet (less than 1 mm thick) crossed the centre of the turbulence chamber such that the field of view (FOV) fell within the region of near-HIT (SPIV, Sabban & van Hout Reference Sabban and van Hout2011). The CMOS camera was equipped with a long working distance, microscopic lens (Infinity video lens model $\text{K2}/\text{S}^{\text{TM}}$ with K2 close-focus CF-2 objective) providing a FOV of $6.3\times 12.6~\text{mm}^{2}$ . Flow tracers generated by a smoke machine (Colt 4, tracer size 0.2– $0.3~\unicode[STIX]{x03BC}\text{m}$ , Glycerine BP/purified water mixture, ‘Smoke Fluid A’, Concept Engineering Limited) were introduced into the chamber after which the woofers were activated. The tracer particle size was approximately $0.3~\unicode[STIX]{x03BC}\text{m}$ corresponding to a Stokes number less than $10^{-4}$ , and resulting in a near perfect response to the highest frequency, $1/\unicode[STIX]{x1D70F}_{k}$ , in the present turbulent flow (Melling Reference Melling1997). Two sets of nylon fibres (material density, $\unicode[STIX]{x1D70C}=1.15~\text{g}~\text{cm}^{-3}$ , courtesy of Dr Ir. R. Delfos, Delft University of Technology) having the same nominal length, $L=0.5$  mm, but different diameters were released from the top of the chamber using a custom-made particle dispenser that consisted of a small cylindrical container. The dispenser was ‘shaken’ by a small load, eccentrically placed on the axis of an electric motor attached to the dispenser. The bottom part of the dispenser consisted of replaceable sieves with laser-drilled holes (Sabban & van Hout Reference Sabban and van Hout2011). In the present case, in order to prevent clogging, two holes with diameters of 2 mm were used.

Fibres can be characterized by their linear mass density in units of dtex ( $=\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}D^{2}/4\times 10^{-7}$  g per 10 000 m). According to the manufacturer’s data sheet, the two used fibre sets were characterized by 1.7 and 3.3 dtex, leading to nominal diameters of $D=13.7~\unicode[STIX]{x03BC}\text{m}$ and $19.1~\unicode[STIX]{x03BC}\text{m}$ , respectively. In the following, we refer to the two fibre sets as dtex1.7 and dtex3.3 fibres. Detailed information on actual fibre length and diameter distributions (see figure 2) was obtained by analysing microscope images taken at 200 $\times$ magnification ( $0.022~\unicode[STIX]{x03BC}\text{m}~\text{pix}^{-1}$ , DeltaPix Invenio 2D II) using Matlab based, image processing software. Results indicate that the manufacturer’s nominal values are within 3 % of the measured mean diameters and lengths with maximum standard deviations of $0.07\bar{D}$ and $0.18\bar{L}$ , respectively. Note that the fibre diameter was measured at its centre of mass and was not affected by its edges that were slightly lopsided due to cutting (figure 2 c). In the following, nominal values are used for normalization purposes.

Figure 2.

Measured histograms of fibre (a) length and (b) diameter; (c) fibre and magnified fibre edge imaged under a microscope.

The response time associated with the translation of randomly oriented fibres in Stokes flow depends on the aspect ratio and is given by (Bernstein & Shapiro Reference Bernstein and Shapiro1994; Zhang et al. Reference Zhang, Ahmadi, Fan and Mclaughlin2001):

where $\unicode[STIX]{x1D707}$ is the air dynamic viscosity. In the present case, $\unicode[STIX]{x1D70F}=2.82$  ms and 5.08 ms for dtex1.7 and 3.3, respectively, and corresponding Stokes numbers are $St=1.35$ and 2.44.

In all cases, measurements were started at least one minute after activation of the woofers (in order to reach a statistically steady state) and just after releasing the fibres. Volume fractions were dilute, $\unicode[STIX]{x1D6F7}\approx 2\times 10^{-6}$ (estimated from the holography measurements described in § 3), just on the border between one- and two-way coupling (Elghobashi Reference Elghobashi1994; Poelma, Westerweel & Ooms Reference Poelma, Westerweel and Ooms2007). In total 22 data sets (10 for each dtex value and 2 unladen sets, 1.37 s per set) were acquired at 3 kHz resulting in a time separation of 0.33 ms, i.e. ${\sim}\unicode[STIX]{x1D70F}_{k}/6$ as will be discussed in the following.

2.3 TR-PIV data processing

Data processing consisted of distinguishing and separating fibres from flow tracers using image processing techniques (e.g. Parsheh Reference Parsheh, Brown and Aidun2005; Carlsson et al. Reference Carlsson, Håkansson, Kvick, Lundell and Söderberg2011) after which images containing fibres and flow tracers were processed separately. Examples of cropped PIV images containing a single fibre as well as flow tracers depicted in figure 3(a) clearly show that the intensity signature of the fibre changes significantly due to rotation and translation in and out of the light sheet. PIV images were first converted to binary images by a user-defined threshold (figure 3 b) and fibres were subsequently detected by applying the ‘Hough’ transform that identifies line elements in binary images (Duda & Hart Reference Duda and Hart1972). Note that detected line elements could have widths up to 15 pixels due to light reflections (see figure 3 a). Gaps (5–7 pixels) between two adjacent line elements were bridged. Subsequently, the fibre centroid position and in-plane angle, $\unicode[STIX]{x1D719}_{1{-}2}$ (figure 1), were calculated based on a line connecting the two fibre extremities (indicated by ‘ $\times$ ’ in figure 3 c). In total, 705 and 974 fibres were detected for $\mathit{dtex}1.7$ and 3.3, respectively.

Figure 3.

Procedure of fibre detection and masking of the original PIV data ( $\unicode[STIX]{x0394}t=2$  ms between rows). (a) Cropped raw PIV image, (b) binarized image, (c) binarized image plus superimposed detected line element by Hough transform, (d) applied mask, (e) (d) multiplied by (a): dashed white rectangle denotes mask boundary, (f) resulting vector maps plus superimposed fibres. Red lines are original detected fibres while blue lines in (f) denote interpolated fibre positions.

Once detected, fibres were removed from the PIV images by placing at the centroid of each fibre, a rectangular mask of 52 by 16 pixels (approximately $0.6\times 0.2$  mm) rotated according to $\unicode[STIX]{x1D719}_{1{-}2}$ (see figure 3 d). The masked images (only containing flow tracers, figure 3 e) were pre-processed to remove any spatial non-uniformities in the light intensity and to improve the signal to noise ratio. Subsequently, velocity fields were obtained by PIV algorithms (LaVision DaVis 7.4, based on fast Fourier transforms) using multi-pass processing and decreasing interrogation window size ( $64\times 64$ to $32\times 32$  pixels with 50 % overlap). The final vector spacing was $197~\unicode[STIX]{x03BC}\text{m}$ , i.e. comparable to $\unicode[STIX]{x1D702}_{k}$ (see table 2). In between passes and at the post-processing stage, outliers were removed by applying a $3\times 3$ median filter (Nogueira, Lecuona & Rodríguez Reference Nogueira, Lecuona and Rodríguez1997) and in the final step, a $3\times 3$ smoothing filter was applied. Examples of the resulting vector field in the vicinity of the fibre and superposed fibre are depicted in figure 3(f). Since each vector field was determined from two consecutive PIV images taken at times $t_{1}$ and $t_{2}$ , cubic interpolation was performed to position the fibre at $(t_{1}+t_{2})/2$ (blue line segment in figure 3 f). Note that in order to ensure uncorrelated vector maps, TR-PIV data were temporally averaged over vector maps spaced $25\unicode[STIX]{x1D70F}_{k}$ apart. At this spacing the velocity autocorrelation coefficient dropped to zero (not shown). TR-PIV ensemble and spatially averaged air velocities and r.m.s. values are summarized in table 1. Further note that convergence of first- and second-order moments was validated.

Since the FOV of the TR-PIV is very small, the mean velocities and r.m.s. velocities are compared to those obtained in the SPIV measurements performed by Sabban & van Hout (Reference Sabban and van Hout2011) in the same facility and flow conditions albeit at a much larger FOV ( ${\sim}45\times 45~\text{mm}^{2}$ ). Comparing between $(\langle u_{1}^{\prime }\rangle +\langle u_{2}^{\prime }\rangle )/2$ for the SPIV and unladen TR-PIV (table 1) shows that values are almost the same while $\langle u_{1}^{\prime }\rangle$ and $\langle u_{2}^{\prime }\rangle$ differ at most by 4 %. Differences between the SPIV and unladen TR-PIV results for $\langle \overline{U_{i}}\rangle$ and $\langle u_{i}^{\prime }\rangle$ are the result of the difference in FOVs, i.e. for a larger FOV (SPIV) spatial inhomogeneities are better ‘averaged out’. PDFs of normalized, fluctuating air velocities, $u_{i}/\langle u_{i}^{\prime }\rangle$ , for SPIV and unladen TR-PIV nearly collapsed onto normal distributions (not shown). The spatial distributions of the ratios $u_{1}^{\prime }/u_{2}^{\prime }$ , $u_{1}^{\prime }/\langle u_{1}^{\prime }\rangle$ and $u_{2}^{\prime }/\langle u_{2}^{\prime }\rangle$ covered the same range for TR-PIV and SPIV with maximum deviations up to 20 % of the spatially averaged values, similar to reports by Hwang & Eaton (Reference Hwang and Eaton2004) and Gopalan, Malkiel & Katz (Reference Gopalan, Malkiel and Katz2008), ensuring that near HIT was reached in the TR-PIV measurements.

Dispensing the fibres did not affect $\langle \overline{U_{1}}\rangle$ but $\langle \overline{U_{2}}\rangle$ increasingly decreased upon dispensing heavier fibres and reached $\langle \overline{U_{2}}\rangle =-0.44~\text{m}~\text{s}^{-1}$ for dtex3.3, only 22 % smaller than $\langle u_{2}^{\prime }\rangle$ . The latter observation can be attributed to secondary flow induced by the settling fibres similar as discussed by Tanaka & Eaton (Reference Tanaka and Eaton2010) for large spheres at increasing mass fraction. Furthermore, while values of $\langle u_{i}^{\prime }\rangle$ for dtex1.7 are similar to those obtained in the unladen case, for dtex3.3 both $\langle u_{1}^{\prime }\rangle$ and $\langle u_{2}^{\prime }\rangle$ increased 10 % and 17 %, respectively.

2.4 Dissipation rate estimate

For the present planar TR-PIV measurements and assuming isotropy, the average TKE is defined as, $k\equiv 3\langle \overline{(u_{1}^{2}+u_{2}^{2})}\rangle /4$ . The dissipation rate, $\unicode[STIX]{x1D700}\equiv 2\unicode[STIX]{x1D708}\langle \overline{s_{ij}s_{ij}}\rangle$ where $s_{ij}$ denotes the fluctuating rate of strain, can be estimated by (Tanaka & Eaton Reference Tanaka and Eaton2010):

However, direct calculation of $\unicode[STIX]{x1D700}_{2D}$ using (2.2) is affected by a random error when the spatial measurement resolution, $\unicode[STIX]{x0394}x$ , is smaller than $\unicode[STIX]{x1D702}_{k}$ (Tanaka & Eaton Reference Tanaka and Eaton2007), and a bias error when $\unicode[STIX]{x0394}x>\unicode[STIX]{x1D702}_{k}$ (Lavoie et al. Reference Lavoie, Avallone, De Gregorio, Romano and Antonia2007). In order to reduce the random error, Tanaka & Eaton (Reference Tanaka and Eaton2007) recommended calculating the dissipation rate by:

where the superscripts ‘ $\unicode[STIX]{x0394}x$ ’ and ‘ $2\unicode[STIX]{x0394}x$ ’ denote evaluating (2.2) at once and twice the spatial resolution. De Jong et al. (Reference de Jong, Cao, Woodward, Salazar, Collins and Meng2009) further suggested correcting both errors irrespective of spatial resolution constraints by first reducing the random error as proposed by Tanaka & Eaton (Reference Tanaka and Eaton2007) and subsequently applying the Lavoie et al. (Reference Lavoie, Avallone, De Gregorio, Romano and Antonia2007) correction factors that depend on $\unicode[STIX]{x0394}x/\unicode[STIX]{x1D702}_{k}$ . For the present measurements ( $\unicode[STIX]{x0394}x/\unicode[STIX]{x1D702}_{k}\approx 2$ ), $\unicode[STIX]{x1D700}_{T}$ (2.3) was divided by correction factors taken from Lavoie et al. (Reference Lavoie, Avallone, De Gregorio, Romano and Antonia2007) of 0.81 and 0.92–0.93 for the SPIV and TR-PIV measurements, respectively, in order to obtain $\unicode[STIX]{x1D700}_{TL}$ . Results are summarized in table 2, including Kolmogorov scales based on $\unicode[STIX]{x1D700}_{TL}$ ; the Taylor microscale, $\unicode[STIX]{x1D706}$ ; the Taylor microscale Reynolds number, $Re_{\unicode[STIX]{x1D706}}=\unicode[STIX]{x1D706}u^{\prime }/\unicode[STIX]{x1D708}$ ; and the integral length scale $L_{T}=u^{\prime 3}/\unicode[STIX]{x1D700}_{TL}$ .

Table 2.

Summary of turbulence characteristics at the centre of the turbulence chamber.

3.1 Digital holographic cinematography set-up

The 3-D fibre orientational and translational dynamics were measured using two orthogonal view, digital inline holographic cinematography (van Hout, Sabban & Cohen Reference van Hout, Sabban and Cohen2013), schematically depicted in figure 4. The system comprised a high-speed laser (Crystalaser, $10~\unicode[STIX]{x03BC}\text{J}~\text{pulse}^{-1}$ at 10 kHz), two high-speed, lensless CMOS cameras (Photron UltimaAPX, $1024\times 1024$  pixels at 2 kHz, $17~\unicode[STIX]{x03BC}\text{m}~\text{pixel}^{-1}$ ), a neutral density filter and a spatial filter (aspherical focusing lens and $10~\unicode[STIX]{x03BC}\text{m}$ pinhole), a collimating lens having a focal length of 200 mm, a beam splitter (50–50 %, 54 mm cube, Newport) and two mirrors to direct the laser beams to the cameras. The intersection between the two collimated beams comprised the volume of interest (VOI) having an approximate size of $17\times 17\times 17~\text{mm}^{3}$ (see figure 4 b). Note that the TR-PIV’s FOV (see figure 1) was positioned inside the holographic VOI.

Figure 4.

Schematic layout of (a) the experimental facility and digital holography set-up (top view, not to scale) and (b) VOI and employed coordinate system.

Holograms were recorded at the cameras’ sensor planes as a result of the interference between diffracted object beams (due to the fibres) and the undiffracted reference beam (Vikram Reference Vikram1992; Rabencov, Arca & van Hout Reference Rabencov, Arca and van Hout2014). Subsequently, the holograms were digitally reconstructed onto planes parallel but away from the sensor plane in order to determine the fibre’s orientation and position as will be discussed in § 3.2. Note that it would be difficult using the present inline set-up to measure both fibre dynamics as well as the flow field since a hologram ‘integrates’ all particle contributions (i.e. diffracted object beams) along the laser beam path. Therefore, also (tracer) particles outside the VOI will contribute to the recorded interference pattern (i.e. the hologram) leading to noisy speckle patterns that cannot be reconstructed clearly (Vikram Reference Vikram1992).

Figure 5.

Schematic of fibre and associated coordinate systems (a) projected onto the fields of views of cameras I and II (not to scale); (b) definition of fibre polar angle and unit orientation vector in a co-moving coordinate system.

The employed coordinate systems and definitions of the fibre orientation angles are schematically depicted in figure 5 together with the FOVs of cameras I and II. Subscripts ‘I’ and ‘II’ denote camera I and II, respectively, and $\unicode[STIX]{x0394}x_{i}$ denote the projected, measured fibre lengths onto the corresponding $x_{i}$ -axes. The spatial measurement resolution was equal to the square pixel size of $17~\unicode[STIX]{x03BC}\text{m}$ , insufficient to resolve $D$ but clearly sufficient to measure $L$ as will be discussed in the next section. As in the TR-PIV measurements, experiments were started at least one minute after activation of the woofers (in order to reach a statistically steady state) and just after releasing the fibres. In total 8 data sets were acquired for each dtex value at an acquisition frequency, $f=2$  kHz, resulting in 4096 holograms (2048 for each camera) per data set. Note that $f\unicode[STIX]{x1D70F}_{k}^{o}=4.16$ indicating that the measurements were time resolved. The superscript ‘ $o$ ’ denotes ‘determined for the unladen TR-PIV’ (see table 2). In the following, instantaneous fibre velocities are denoted by $V_{i}$ , fibre fluctuating velocities by $v_{i}$ ( $=V_{i}-\overline{V_{i}}$ ) and $\unicode[STIX]{x1D719}_{1{-}2}$ and $\unicode[STIX]{x1D719}_{2{-}3}$ denote the fibre angles in the $x_{1}$ – $x_{2}$ and $x_{2}$ – $x_{3}$ planes, respectively (figure 5). Note that the two in-plane fibre angles were measured directly from the reconstructed holograms. The fibre polar angle, $\unicode[STIX]{x1D703}$ , defined in a spherical coordinate system is also depicted in figure 5(b).

3.2 Data processing

Acquired holograms were digitally reconstructed by numerically solving the Fresnel diffraction formula through convolution of the hologram intensity distribution with an impulse response function (e.g. Milgram & Li Reference Milgram and Li2002; Schnars & Jueptner Reference Schnars and Jueptner2005; van Houtet al. Reference van Hout, Sabban and Cohen2013). We used hologram reconstruction software based on fast Fourier transform developed by Sheng, Malkiel & Katz (Reference Sheng, Malkiel and Katz2006). Due to spatial and temporal changes in the laser light intensity, reconstructed holograms suffered from poor signal-to-noise ratio that was improved by subtracting a unique background image prior to reconstruction (Rabencov et al. Reference Rabencov, Arca and van Hout2014). Data processing was initiated by a ‘coarse’ reconstruction of holograms acquired by one of the cameras (in this case camera I, see figures 4 and 5). ‘Coarse’ reconstruction (figure 6) means that all fibres within the VOI were detected without determining their ‘accurate’ in-focus position (in case of camera I, the position of the fibre along $x_{3}$ , figure 5 a). Note that determining the in-focus position is computationally highly demanding and suffers from a relatively large uncertainty inherent to single-view holography (Vikram Reference Vikram1992). Instead, the present two-view set-up relies on determining the accurate in-depth fibre position from the second camera’s FOV (figure 5). In order to detect all fibres in the VOI, the distance between ‘coarse’ reconstructions, $\unicode[STIX]{x1D6FF}x_{3}=1.75$  mm, was chosen carefully based on the spatial imprint of reconstructed fibres along $x_{3}$ (see figure 6 a). At this ‘coarse’ in-depth spatial resolution only 16 planes were reconstructed at each time instant that were subsequently binarized using a local ( $15\times 15$  pixels), greyscale threshold value (figure 6 b) (http://www.mathworks.com/matlabcentral/fileexchange/41787-meanthresh-local-image-thresholding). The total reconstruction depth (28 mm) exceeded the depth of the intersecting VOIs of cameras I and II (see figure 4) in order to better match tracks as will be discussed in the following.

Figure 6.

Illustration of the ‘coarse’ fibre detection scheme for camera I. (a) Cropped, background subtracted, reconstructed holograms of the same fibre at different reconstruction distances, (b) binarized images, (c) morphologically ‘cleaned’ binary images and (d) collapsed binary image.

In contrast to the TR-PIV images (see figure 3), the reconstructed holograms (figure 6 a) do not suffer from light reflections and binarized ‘blobs’ clearly show fibre outlines (figure 6 b). Therefore, most ‘blobs’ representing fibres were easily distinguished from noise based on geometrical properties such as ‘area’ (between 20 and 200 pixels) and ‘eccentricity’ exceeding 0.97 (Matlab image processing toolbox, figure 6 c). Note that as a result of applying ‘area’ and ‘eccentricity’ thresholds, tracks of fibres that came close to alignment with the $x_{3}$ -axis (camera I) and the $x_{1}$ -axis (camera II) were terminated. In practice this meant that conical volumes (in total approximately 7 % of the VOI) extending from both these axes at a polar angle of about $20^{\circ }$ were not sampled. Subsequently, at each time instant, binarized and morphologically cleaned, ‘coarse’ reconstructions were collapsed onto a single binary image containing the detected fibre (figure 6 d). Based on these collapsed images, fibre centroids were determined by the first moments of the ‘blob’ areas while in-plane orientation angles and fibre lengths were determined based on an ellipse having the same second moment as the ‘blob’. After determining the centroids, 2-D fibre tracks were constructed for each camera separately using a nearest neighbour algorithm (Ouellette, Xu & Bodenschatz Reference Ouellette, Xu and Bodenschatz2006; Rabencov et al. Reference Rabencov, Arca and van Hout2014) that performed well at the present dilute volume fraction. In order to discard spurious noise, only tracks longer than 3 instances were taken into account for further processing. Furthermore, in order to reduce random measurement noise, a smoothing spline fit (Matlab’s curve fitting toolbox, smoothing parameter between 0.4 and 0.5; Green & Silverman Reference Green and Silverman1993) was applied to the fibre centroid positions and in-plane angles along each fibre track.

Figure 7.

Comparison between fibre (a,b) centroid positions, (c) in-plane angles, (d,e) translational velocities and (f) in-plane rotation rates along a fibre track determined by camera I based on ( $\times$ ) in-focus reconstructions and (○) collapsed ‘coarse’ reconstructions. Symbols denote raw data while curves based on smoothed data in (d) to (f) are denoted by dashed (in-focus) and solid lines (collapsed).

In order to validate that the above described procedure provided accurate results, fibre centroid positions, translational velocities, in-plane angles and rotation rates based on in-focus and collapsed ‘coarse’ reconstructions were compared and an example for a single fibre track is shown in figure 7. Results show that besides a small bias in $x_{1}$ (figure 7 a, of the order of the measurement resolution, no correction was attempted), data points based on in-focus and ‘coarse’ reconstructions collapse (figure 7 a–c). Fibre translational velocities and in-plane rotation rates were calculated using a centre difference scheme while forward and backward difference schemes were applied at the start and end of the track, respectively. Higher-order difference schemes were tried but did not provide any substantial improvement. Note that in-plane and polar angle rotation rates were always assumed to change smoothly along a track (see figure 7 f). Figure 7(d–f) shows that irrespective of being based on in-focus or collapsed, ‘coarse’ reconstructions, curves based on smoothed centroid positions and fibre angles (dashed and solid lines) nearly collapse.

The 3-D fibre tracks were constructed by matching the in-plane tracks of both cameras based on the common $x_{2}$ -coordinate (figure 5). Examples displayed in figure 8 illustrate (i) a small systematic error in the common $x_{2}$ coordinate, $x_{2,II}$ – $x_{2,I}$ (figure 5) due to camera misalignment and (ii) the possibility of multiple fibres being present in the VOI at the same time and similar height. The first bias was easily removed and when multiple fibres were present, the pair with minimal vertical velocity difference was picked. In total 641 and 538 different 3-D fibre tracks were matched, totalling 7496 and 6619 instantaneous instances for dtex1.7 and 3.3, respectively (i.e. a mean track length of ${\sim}12$ instances).

Figure 8.

Example of combining in-plane tracks from both cameras based on their common $x_{2}$ coordinate.

Once 3-D fibre tracks were constructed, the instantaneous fibre length and polar angle were calculated by $L=\sqrt{\unicode[STIX]{x0394}x_{1,I}^{2}+\unicode[STIX]{x0394}x_{2,I}^{2}+\unicode[STIX]{x0394}x_{3,II}^{2}}$ , and $\unicode[STIX]{x1D703}=\cos ^{-1}(\unicode[STIX]{x0394}x_{3,II}/L)$ , respectively, where $\unicode[STIX]{x0394}x_{2}$ was based on camera I (similar results were obtained for camera II). The fibre unit orientation vector, $\boldsymbol{p}$ , defined in terms of $\unicode[STIX]{x1D719}_{1{-}2}$ and $\unicode[STIX]{x1D703}$ (figure 5 b) is given by (Olson Reference Olson2001):

where $\hat{\boldsymbol{j}},\hat{\boldsymbol{k}},\hat{\boldsymbol{l}}$ are unit vectors in the $x_{i}^{\ast }$ ( $i=1,2,3$ ) directions, respectively. The fibre rotation (‘tumbling’) rate is then given by:

where $\hat{\unicode[STIX]{x1D719}}_{1{-}2}=[-\text{sin}\,\unicode[STIX]{x1D719}_{1{-}2};\cos \unicode[STIX]{x1D719}_{1{-}2};0]$ and $\hat{\unicode[STIX]{x1D703}}=[\cos \unicode[STIX]{x1D719}_{1{-}2}\cos \unicode[STIX]{x1D703};\sin \unicode[STIX]{x1D719}_{1{-}2}\cos \unicode[STIX]{x1D703};-\text{sin}\,\unicode[STIX]{x1D703}]$ are the corresponding unit vectors and $\dot{\unicode[STIX]{x1D719}}_{1{-}2}$ and $\dot{\unicode[STIX]{x1D703}}$ are the fibre in-plane and polar angular velocities.

Figure 9.

Example of a 3-D fibre track (dtex3.3), colour coded by $\dot{\unicode[STIX]{x1D703}}/\dot{\unicode[STIX]{x1D703}}^{\prime }$ . Black lines are the fibre projections onto planes subtended by the intersecting coordinate axes.

An example of a 3-D fibre track is shown in figure 9 while the associated time series of $\unicode[STIX]{x1D703}$ and $L$ are displayed in figure 10. The fibre clearly rotates in 3-D space as illustrated by its projections (solid black lines) onto the planes subtended by the intersecting coordinate axes. The fibre does not rotate ‘violently’ and $\dot{\unicode[STIX]{x1D703}}$ reaches a maximum equalling its r.m.s. value. Furthermore, while $\unicode[STIX]{x1D703}$ is expected to change along the fibre’s trajectory (figure 10 a), $L$ should remain constant and the scatter observed in figure 10(b) is due to noise characterized by an r.m.s. of 0.02 mm (for the smoothed data), i.e. comparable to the measurement resolution.

Figure 10.

Time sequence of fibre (a) polar angle and (b) length along the fibre track depicted in figure 9. Based on ○ raw data and $\times$ smoothed data.

Table 3.

Mean and r.m.s. values of fibre centroid velocities. Uncertainties include estimates by bootstrapping clustered data (Field & Welsh Reference Field and Welsh2007) and the ‘acceleration error’ (Boillot & Prasad Reference Boillot and Prasad1996).

Histograms of fibre lengths determined from the reconstructed holograms are depicted in figure 2(a) together with those from the microscope images. The former are shifted to slightly larger  $L$ , however, differences in mean values and standard deviations are small and do not exceed 6 % and 14 %, respectively.