4.3.1 Spatial resolution of velocity measurements
Since the velocity is averaged across the interrogation window, DCIV resolution is an important issue.
One of the major difficulties in flow imaging is achieving adequate spatial and temporal resolution. This is particularly the case when flows are turbulent because the resolution requirements are typically very severe if it is desired to resolve the smallest scales at which fluctuations occur. Laminar flows, however, pose substantially less stringent requirements on resolution, compared to turbulent flows. The primary issue when considering the resolution requirements is the gradient of the flow property that is being measured because the gradient determines the amount of averaging that occurs across the resolution volume.
In turbulent flows, the spatial fluctuations in flow properties, such as velocity, range in scale from the largest physical dimension of the flow (e.g., the local width of the boundary layer or jet) to the scale at
which diffusion acts to remove all gradients. The largest scales are often called the macro scales or outer scales, whereas the smallest scales are the micro or inner or dissipation scales because these are the scales at which the energy of fluctuations, whether kinetic or scalar, is dissipated.
The resolution that one can achieve in practice using these image-based techniques depends on a number of factors, including but not limited to camera resolution, lens optical quality, and marker size and quality.
In the present study velocity fields were calculated based on the FFT-correlation algorithms. Similar to the work by Raffel et al. [122], each pair of images was processed from an interrogation window size of 32×32 pixels using 75 % overlap. The interrogation window was moved vertically and horizontally in steps of 8 pixels producing vectors which describe the velocity flow field for the entire image. The resulting velocity vector field has a vector arrow density of 1 vector per 8×8 pixels. Using calibration factors of 19.5 and 16.0 pixels/cm obtained for the xand z directions, respectively, this produced for each image pair, a 91×56 array of velocity vector maps with a spatial resolution of 4.1 mm×5.0 mm for the horizontal and vertical directions, respectively. This spatial resolution does not resolve velocities at the scale of dissipation. However, dissipation can be estimated from the velocity measurements by other means such using the slope of the Fourier spectra in the frequency range above the micro scales (e.g Govender et al.[73],Huang et al.[115]). The choice of a 32×32 interrogation window was based on our sampling time of 1.84 ms which resulted in maximum particle displacements of 16 pixels. The sensitivity of interrogation window depends on having a significant number of particles within the window averaged over a number of waves. Previous analysis of window size sensitivity has been done byGovender [94] and since we are using similar settings, the optimum window size they determined is employed here. Many previous researchers such asTing [77], Govender et al. ([42], [73], [79], [78]) and Huang et al. [44] have used 32×32 pixel size interrogation windows.
4.3.2 Experimental accuracy
The experimental nature of this DCIV study may affect the results with several error sources. Considering the complexity of quantifying a priori all the error sources, only the most relevant ones will be briefly analyzed. A more detailed analysis can be found in the work of Silva [144]. A primary error source is related to inertial effects between the particles displacement and the fluid motion. At sub-micron scales Brownian motion can assume a significant influence in the physical description of the tracer particles motion (Devasenathipathy et al.[145]). For the present case, where tracer particles mean diameter is 0.5 mm, and their displacement is computed through an ensemble time-average algorithm, Brownian motion effects have a negligible influence. For a cross-correlation algorithm the in-plane velocity is computed by locating the highest peak in the correlation plane. The uncertainty in locating this correlation peak is closely related to the particles image size, de. As demonstrated by Prasad et al. [146], as long as de is resolved by 3 to 4 pixels, the uncertainty, ∆u, in locating the correlation peak and thus the error estimate in computing the in-plane velocity can be estimated as :
∆u= de
10M (4.10)
where M is the magnification of the camera lens, which is 19.5 pixels/cm in this case. This gives ∆u
= 0.2 mm which is about one order of magnitude smaller than the characteristic flow length-scale of 5.0 mm, chosen herein as the distance between adjacent velocity vectors. Thus this uncertainty does not influence significantly the in-plane velocity results. The presence of out-of-plane particles contributing to the in-plane velocity can bias the computation of the latter if the measurement volume is too thick and strong velocity gradients in the out-of plane direction exist simultaneously (Silva et al.[147].
4.3.3 Assessment of the DCIV technique
Since its introduction, DPIV has seen a series of improvements in both the experimental and the post- processing stage (Westerweel et al. [130]; Jones [148]; Scarano & Riethmuller [149]), increasing the accuracy and reducing noise in the PIV representation of the actual flow field. Improvements like these are particularly very useful for comparisons of experimental results with theoretical models or Navier- Stokes simulations. Experimental accuracy in DCIV tends to be good, provided that the experimental parameters, such as : seed particle diameter, interrogation window size, etc. are suitably chosen. While no straight-forward formula for predicting accuracy exists, some estimate are available from Raffel et al.[122]. Their Monte Carlo simulations suggest that, for a 32×32 interrogation window and particle image diameters of 13 pixels, displacement uncertainty is around 0.02 - 0.04 pixels. For the present experiments, computed mean particle displacements were consistently around 3-4 pixels, indicating that the uncertainty was of the order of a few percent. Hyun et al. [150] provide a discussion of the factors that determine the accuracy of PIV data.
In PIV measurements, seed particle size must be small enough to faithfully follow the flow without disrupting the flow field, and producing unnecessarily large images. At the same time, particles must be large enough to be adequately imaged and to dampen the effects of Brownian motion (Santiago et al.[151]). Figure 4.10 shows two marked subsections in one of the images chosen to closely examine the DCIV results. Two boxes marked A and B in the image show 32×32 pixel sub-images (not drawn to scale) in the trough region and the crest region, respectively. The crest region consists mainly of bubbles while the trough region consists of beads. The intention is to determine the correlation peaks and the resultant displacement for these two different sections of the wave.
Figure 4.11 shows an expanded view of the subimages from the two positions and the resulting surface and intensity plot of the normalized cross-correlation results. On average there are more than 40 beads in each 32×32 pixel interrogation subimage and each bead in the image occupies about 3-4 pixels. In Figure 4.11(a) it is possible to identify corresponding particles in both subimages labeled by 1 and 1′; and 2 and 2′. The displacement of the peak from the center is equivalent to the displacement of beads and bubble structures during a certain time interval. Figure 4.11 was included to show the effects of air bubbles to the overall flow. During a small inter image time of 1.84 ms, the bubbles buoyancy effects are negligible as can be seen from Figure 4.11(b) and so the bubbles are considered to be locked with the flow. Govender et al.[42], [73], [79] used similar air bubbles together with seed particles to track the flow patterns and obtained results that compare well with PIV measurements.
Figure 4.10: Typical images for a 0.4 Hz plunging breaking wave captured by a monochrome digital camera. Zones markedA andBare 32×32 pixel sub images consisting of mainly beads for areaA, and bubbles forB.
Figure 4.11: Expanded view showing 32×32 pixel subimages shown in Figure 4.10 and surface plots of the resulting cross- correlation functions,crf g(m, n) and its intensity plot for : (a)-beads only area (b)-bubbles and beads area.
The spatial cross-correlations between the subimages A1 and A2 and between B1 and B2 are shown near the bottom. A perfect match in the two subimages gives rise to a sharp peak in the cross correlation result as seen for Figure 4.11(a). A plot of the intensity profile for subimages A, show a well defined point. The actual subpixel displacements in the xand z direction are 6.03 pixels and 1.16 pixels, re- spectively. In Figure 4.11(b) a few matching particles such as 2 and 2′ and bubble structures a and a′ are observed. Bubble structures are spread out, that is why the resultant cross correlation is not as peaked as in (a). The resulting plot of intensity profile is thus smeared out. The horizontal displace- ment is 8.03 pixels while for the vertical it is -1.42 pixels. A ”signal-to-noise” ratio, (SNR), defined here
as the ratio of the cross-correlation peak to the second highest (noise) peak was used to detect spurious vectors. In the analysis, velocity vectors with aSNRratio under 1.5 were removed as explained previously.
4.3.4 Error analysis in the velocity measurements
Because PIV is based on the statistical correlation of imaged subregions to determine local flow velocities, it is subject to inherent errors that arise from finite tracer particle numbers, sample volume size, and image resolution. These errors, in extreme cases, are relatively easy to detect as they tend to vary substantially from neighboring vectors in both magnitude and direction. One fundamental source of these errors arises from the implementation of cross correlation. Other major sources of these errors arise from the peak-finding scheme, which locates the correlation peak with a sub-pixel accuracy, and noise within the particle images (Huang et al.[132],Fairweather &Hargrave [152].
There are two major sources of error that may be of concern in our DCIV measurements of fluid velocity in the breaking wave crests. The first is ther.m.serror associated with the accuracy of estimation of the location of the displacement correlation peak. The second is downward biasing of the velocity measure- ment due to gradients in the velocity field. Appropriate references areHuang et al.[132],Westerweel et al.[130] andKeane&Adrian[153]. Generally one would not expect the total velocity measurement error to exceed 2% - 3% in well-prepared PIV experiments (Stansell [154]). From Figure 1. of Westerweel et al. [130] it is possible to estimate the r.m.s error. Typical pixel displacements in our PIV images are about 5 pixels. Seeding particle image diameters are about 3-4 pixels. From Figure 1. ofWesterweel et al.[130] it is observed that the expected value of therms displacement error will be no more than about 0.2 pixels. This gives an upper bound on the rms error in the velocity measurements of about 2%. A typical value for the minimum measurement error is 0.05 to 0.1 pixel units for a 32×32 pixel inter- rogation region. This implies a relative measurement error of about 1% for a displacement that is one quarter of the interrogation window size (8 pixels) (Westerweel et al.[130]). When setting up a DCIV measurements, the side length of the interrogation area,dI, and the image magnification,M are balanced against the size of the flow structures to be resolved. One way of expressing this is to require the velocity gradient to be small within the interrogation area. According toKeane&Adrian [153] the dimensionless parameter,pvthat represents the performance of DCIV with regard to velocity biasing is given by:
pv=M|∆u|∆t dI
(4.11) whereM is the magnification factor in units ofpix/m,|∆u|is the maximum variation of velocity across the interrogation region of sizedI pixels and ∆tis the time interval in seconds between light pulses. The condition given byKeane&Adrian[153] for optimum performance of PIV with regard to velocity biasing is :
M|∆u|∆t dI
<0.05 (4.12)
The highest measurable velocity is constrained by particles traveling further than the size of the interro- gation area within the time, ∆t. The result is lost correlation between the two image frames and thus loss of velocity information. The upper bound on|∆u|is given by :
|∆u|< 0.05dI
M∆t (4.13)
For the present experiments, dI =32 pixels, M = 19.5 pixels/cm, and ∆t = 0.00184 s. This sets the upper bound of|∆u|<0.45 m/s. The maximum observed|∆u| is about 0.25 m/s, which is well within the optimum limit.
The uncertainty in instantaneous velocity measurements, ∆uand ∆w, estimated from ther.m.sdeviation is about 2.0 cm/s. The mean velocities< u > and < w >have an uncertainty : ∆u/√
N = 0.2 cm/s, whereN is the sampling number used in the ensemble average.