A tomographic PIV resolution study based on homogeneous isotropic turbulence DNS data
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DOI: 10.1007/s00348-010-0840-1
- Cite this article as:
- Worth, N.A., Nickels, T.B. & Swaminathan, N. Exp Fluids (2010) 49: 637. doi:10.1007/s00348-010-0840-1
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Abstract
In order to accurately assess measurement resolution and measurement uncertainty in DPIV and TPIV measurements, a series of simulations were conducted based on the flow field from a homogeneous isotropic turbulence data set (Re_{λ} = 141). The effect of noise and spatial resolution was quantified by examining the local and global errors in the velocity, vorticity and dissipation fields in addition to other properties of interest such as the flow divergence, topological invariants and energy spectra. In order to accurately capture the instantaneous gradient fields and calculate sensitive quantities such as the dissipation rate, a minimum resolution of x/η = 3 is required, with smoothing recommended for the TPIV results to control the inherently higher noise levels. Comparing these results with experimental data showed that while the attenuation of velocity and gradient quantities was predicted well, higher noise levels in the experimental data led increased divergence.
1 Introduction
Turbulent flow can be characterised by its three-dimensional nature. Recent advances in digital particle image velocimetry (DPIV) have extended the standard technique and enabled the capture of full three-dimensional velocity fields (defocussing PIV, Willert and Gharib (1992); particle tracking velocimetry (PTV), Maas et al. (2004); holographic and digital holographic PIV (HPIV and DHPIV), Hinsch (2002); scanning-PIV, Hori and Sakakibara (2004); and tomographic PIV (TPIV), Elsinga et al. (2006)).
In the most recent of these methods, TPIV, particles within a volume of interest are illuminated and then imaged from a number of discrete angles (typically employing between 3 and 5 cameras). A computational tomographic reconstruction algorithm is then used to reconstruct the light intensity through the volume. Full three-dimensional velocity vector fields can be obtained by directly cross-correlating light intensity volumes separated by a small temporal displacement, δt.
The method offers a number of potential advantages over other 3D techniques, which include higher temporal resolution and more simplistic set-up in comparison with HPIV and scanning-PIV; instantaneous flow field capture as opposed to quasi-instantaneous in scanning-PIV; and higher possible seeding densities in comparison with PTV and defocusing-PIV. However, the seeding density is still more limited than HPIV and scanning PIV (Elsinga et al. 2006); tomographic reconstruction is extremely computationally intensive and requires significant resources (Schröder et al. 2006; Worth and Nickels 2008; Atkinson and Soria 2009), and also technique accuracy is limited by imperfect reconstructions caused by optical defects (Wieneke and Taylor 2006) and relies heavily on high-precision calibration (Elsinga et al. 2006).
Furthermore, there have been relatively few studies to quantify the accuracy of TPIV measurements. Although several numerical parametric studies have been performed (Elsinga et al. 2006; Worth and Nickels 2007), these express accuracy in terms of the volume reconstruction success; a measure that is difficult to re-interpret in terms of velocity field error. The effect on the velocity field error was quantified by Worth and Nickels (2008) in a numerical parametric study of an angled line vortex flow field. However, experimental accuracy is highly dependent on the flow conditions and experimental set-up (Raffel et al. 1998), and therefore the simple ideal flow field used in these simulations may not be representative of a real turbulent flow field. Wieneke and Taylor (2006) also assessed the accuracy of TPIV in comparison to stereo-PIV both numerically and experimentally. While useful, the study was based on a ‘fat’ light sheet and as such their conclusions may not be applicable to full volume reconstructions, and again the measurement uncertainty associated with a relatively simplistic vortex ring flow field may not be representative of high Reynolds number turbulent mixing.
The effect of spatial resolution on both hot-wire anemometry (HWA) and DPIV measurements has been studied extensively. Wyngaard (1968), Antonia and Mi (1993), Zhu and Antonia (1996) simulate resolution effects through the application of a filter in the spectral domain corresponding to the dimensions of various hot-wire probes. These results can then be used to derive correction ratios for important quantities based on the resolution of the probes with respect to the smallest flow scale, η. This technique was extended to DPIV measurements by Lavoie et al. (2007), with the proposed corrections resulting in good agreement with HWA measurements. However, in comparison with HWA, the volume averaging present in PIV resulted in significantly higher attenuation of velocity and derivative statistics. Therefore, while these effects are well understood, it would be useful to extend these to the full 3D TPIV method.
Therefore, in order to accurately quantify the effect of resolution and measurement uncertainty on high Reynolds number DPIV and TPIV turbulence measurements a more representative set of simulations has been conducted using velocity data from the homogeneous isotropic turbulence (HIT) DNS simulation (Tanahashi et al. 1999) to represent the ideal flow field. This approach was used previously by Lecordier et al. (2001) to assess the accuracy of DPIV measurements using different cross-correlation algorithms. By matching the experimental set-up and expected flow as closely as possible, a more accurate and applicable assessment can be conducted, which is an important step in the interpretation of any experimental results. This investigation extends the findings of previous studies, DPIV studies (Lecordier et al. 2001; Lavoie et al. 2007), demonstrating the effect of resolution on measurement uncertainty for the TPIV technique. The results of the simulation are also compared to a series of experimental measurements made in the Cambridge large mixing tank facility, in order to test the accuracy of the predictions made by these.
This paper begins by introducing the experimental set-up and data processing techniques. Next the ideal flow is detailed, followed by the computational set-up and flow matching strategy. After which, simulation results are assessed using a wide variety of measures and compared to experimental results before some brief conclusions are drawn.
2 Experimental set-up
A number of experimental studies investigating the small-scale structure of turbulence have measured the flow between between two counter-rotating disks or impellers (Douady et al. 1991; Cadot et al. 1995; Zocchi et al. 1994). This flow set-up is useful for generating high Reynolds numbers with constant energy injection at the largest scales, with the closed geometry permitting the spatial evolution of flow structures to be studied, without these being transported away too quickly by the mean flow.
The current measurements were made in a very large 2-m-diameter dodecahedral perspex water mixing tank. Two eight vane 0.8-m-diameter impellers, at a separation distance of 1.25 m, are counter-rotated causing the fluid nearest the impellers to spin in opposite directions, establishing a strong shear plane in the centre of the tank, which creates the majority of the turbulence. Radial baffles were fitted to remove any net rotation and further increase turbulence production. Two micro-stepping motors control impeller rotation, permitting accurate speed matching. While anisotropy may persist at the large scale, the flow is known to approximate homogeneous isotropic turbulence at least locally at the geometric centre of the tank, which is the measurement point. The flow can be characterised through the tank geometry, giving a Reynolds number based on impeller radius, R_{I}, and rate of rotation, \(\Upomega_I,\) as \(Re = \Upomega_I R_I^2/\nu.\)
The mixing tank was seeded with Dantec Dynamics spherical hollow borosilicate glass spheres, which have a density of 1.4 g/cm^{3} and a mean size of d_{p} = 10 μm. During preliminary testing, the current set-up geometry and PIV hardware were found to impose strict limits on the attainable volume and resolution of measurements. High seeding densities resulted in a rapid drop in SNR due to multiple scattering generated diffusion, limiting the maximum attainable resolution. A double-pulsed New Wave Gemini PIV 120-15 Nd:YAG laser was used to illuminate a thin volume at the centre of the tank. At the maximum repetition rate of 15 Hz, the energy per pulse is 120 mJ at a wavelength of 532 nm, with a pulse width of 5 ns.
A 1,000 frames were captured over ∼80 impeller rotations in order to achieve reasonable statistical convergence. A total of four DPIV experiments were conducted at different resolutions, which from lowest to highest resolution are denoted as cases D1, D2, D3 and D4. A single TPIV case was conducted, denoted as case T1.
PIV camera configuration parameters
Case | lens(es) | f/# | FOV (mm^{2}) | Resolution (pix/mm) |
---|---|---|---|---|
D1 | 60 mm Nikon | 2.8 | 212 × 212 | 4.8 |
D2 | 105 mm Sigma | 2.8 | 117 × 117 | 8.8 |
D3 | 180 mm Sigma | 3.5 | 77.4 × 77.4 | 13.2 |
D4 | 180 mm Sigma & 2 × teleconverter | 3.5 | 32.0 × 32.0 | 32.0 |
For the TPIV measurements four 1, 024 × 1, 024 pixel CMOS Photron APX cameras, fitted with Sigma 180 mm lenses, were positioned around the volume of interest. The use of Scheimpflug adapters permits aperture settings of f = 1/8. These were angled in the yaw and pitch planes by \(\theta=[-30,0,0,30]^{\circ}\) and \(\phi=[0,15,-15,0]^{\circ}\), respectively, and positioned 1,100 mm from the measurement volume, giving a field of view of 70 × 70 mm^{2} at a resolution of ≈15 pixels/mm. Water-filled perspex prisms are positioned between the central cameras and the tank, to correct for the refractive index effects through the air/perspex/water interface.
The cameras are calibrated using a clear perspex calibration object, 10 mm thick, with two regular 23 × 23 calibration mark arrays printed on each side, off-set to facilitate point detection. Real and image space co-ordinates are related through a third order polynomial calibration function (Soloff et al. 1997), which is defined for each camera and in each depth plane. Self-calibration on particle fields is used to refine the initial calibration (Wieneke 2007), reducing the mean calibration error to around 0.15 pixels.
Images were written to disk as loss-less 16 bit TIFF files. Blank images are defined as the mean image intensity from all particle fields. Subtracting this from each image reduces camera noise due to light sheet diffusion and hot-pixels. As discussed by Nobach and Bodenschatz (2009), variations in particle size and intensity between exposures limit PIV accuracy. Therefore, to reduce this effect, image pre-processing is applied to improve particle image size and intensity homogeneity. The DPIV images are pre-processed by sliding background subtraction and particle intensity normalisation operations. A custom image pre-processing routine is applied to the TPIV images: first, bandpass filtering is performed using a 3 × 3 Gaussian kernel and background noise subtracted; next, local intensity normalisation is performed, before repeating the bandpass filtering and background noise removal operations.
Vector fields were computed from the images using Davis 7.2 for the DPIV and and an in-house algorithm for the TPIV simulations, with processing options chosen to match the experimental cases. The DPIV simulations are evaluated using an FFT-based 3-pass recursive window shifting cross-correlation algorithm employing window deformation, Gaussian sub-pixel peak estimation and Whittaker reconstruction, with a final interrogation size of 32 × 32 pixels at 50% overlap, giving 64 × 64 vectors.
The TPIV code is an implementation of the MART algorithm (Herman and Lent 1976), using a multiplicative first guess (MFG) of the intensity field (Worth and Nickels 2008) followed by three MART iterations. Use of the MFG has been shown to accelerate convergence allowing fewer iterations to be used (Worth and Nickels 2008). The volume discretisation of 15 voxels/mm was set to maintain a pixel/voxel ratio of unity. The code employs an FFT-based 3-pass recursive window shifting cross-correlation algorithm with Gaussian sub-pixel peak estimation with a final interrogation window size of 32 × 32 × 32 voxels at 50% overlap, giving 51 × 51 × 8 vectors. Correlation plane smoothing is performed before the inverse transformation to reduce the effect of correlation plane noise.
Additional noise introduced through the TPIV technique will produce significant errors during gradient calculation, and therefore smoothing with a 3 × 3 × 3 Gaussian kernel is recommended for 3D data (Ganapathisubramani et al. 2007). Schröder et al. (2006) use a similar smoothing kernel for the presentation of TPIV data. It should be noted that the smoothing kernel width is the same size as the interrogation window size, and therefore, the associated cut-off frequency remains unaffected.
As shown by Lecordier et al. (2001), when employing relatively large interrogation windows, the improvement due to window deformation can be slight. After a comparison of window deformation effects in the DPIV results, the presence of moderate gradients resulted in only slight improvements, which did not justify implementing these in the TPIV code. Furthermore, as noted by Lecordier et al. (2001), increased uncertainty in regions of high gradient magnitude is caused by high frequency noise. Therefore, the use of correlation plane smoothing during cross-correlation and velocity field smoothing during post-processing applied in the current investigation are likely to reduce the effect of this noise and the benefit of deforming windows. Therefore, while window deformation may increase accuracy, the TPIV cross-correlation was performed without this. Vector validation is based on a median criterion (Westerweel 1994) applied over a 3 × 3 × 3 mask.
3 Computational set-up
The DNS simulation solves the Navier-Stokes equations for a 400^{3} grid point domain using a spectral approach, achieving a Taylor microscale Reynolds number of Re_{λ} = 141 (Tanahashi et al. 1999). The domain size was 2π, and the ratio of this to the integral length scale was L/l = 3.8. This Reynolds number is similar to the lowest Reynolds number experimental results (see Table 3), allowing a comparison to be made at this point through a careful choice of set-up parameters.
Simulation case set-up details
Case ref. | DNS slice size | Fields | x/η | h/η | κ_{c}η |
---|---|---|---|---|---|
S − D1 | 33 × 33 × 1 | 100 | 0.93 | 1.86 | 1.505 |
S − D3 | 103 × 103 × 1 | 100 | 2.97 | 5.94 | 0.471 |
S − D5 | 171 × 171 × 1 | 100 | 4.95 | 9.90 | 0.283 |
S − D10 | 339 × 339 × 1 | 100 | 9.84 | 19.7 | 0.142 |
S − T1 | 39 × 39 × 7 | 50 | 0.94 | 1.88 | 1.489 |
S − T3 | 103 × 103 × 21 | 50 | 2.52 | 5.04 | 0.556 |
S − T4 | 171 × 171 × 35 | 44 | 4.20 | 8.40 | 0.333 |
S − T8 | 339 × 339 × 69 | 5 | 8.36 | 16.7 | 0.168 |
The DNS non-dimensional RMS velocity was scaled to match the experimental RMS displacement of ∼5 pixels (or voxels). Matching the velocity in this manner will produce similar particle displacements, permitting a reasonable comparison with the experimental results. It should be noted that this image plane displacement was maintained irrespective of the effective resolution.
In both DPIV and TPIV cases, the simulations were set-up by creating sets of artificial images. In the DPIV simulation, the DNS velocity field slice was mapped into image space using a third order polynomial Taylor’s series expansion calibration function obtained from the experimental case. Particle centres were randomly distributed in a volume matching the experimental domain size in image space, assuming a light sheet thickness of 1.5 mm, with a seeding density of N_{ppp} = 0.01 chosen to match the experimental conditions. The velocity at each particle position was calculated using bi-cubic interpolation and used to linearly displace the particles, removing those displaced beyond the volume bounds. The original and displaced particle positions in the x and y directions were then used directly as particle image co-ordinates to create two images, with particles modelled as blobs of Gaussian intensity approximately 3 pixels in diameter. Modelling the particles in this manner neglects particle sizes and intensity effects (Nobach and Bodenschatz 2009). However, the image pre-processing applied to the experimental images reduce these effects considerably, with the remaining error expected to be small in comparison with other more dominant effects.
The linear displacement approximation is justified through the combination of reasonably low magnitude gradients, coupled with a moderate particle displacement. Based on mean displacement and gradient magnitudes, a particle displacement error of approximately 1.5% may be expected. The effect of this error on the results was assessed by running the S − T8 case (a case containing the highest gradient magnitudes) with both linear and quadratic particle displacement models. The quadratic model was found to change all local and global accuracy metrics (including sensitive measures such as the dissipation and vorticity) by less than 1%, and therefore the effects of this approximation can be considered negligible. Accurate particle motion integration [as performed by Lecordier et al. (2001)] is unlikely to significantly affect the results of the current study, which tend to be dominated by other sources of measurement uncertainty.
In the TPIV simulation, particles centres were randomly distributed in a volume matching the real space experimental domain size of 60 × 60 × 12 mm. The velocity at the particle positions was found using tri-cubic interpolation, which as before, was used to linearly displace the particles whilst removing particles displaced beyond the volume bounds. The original and displaced particle positions were then transformed into image space for four cameras using a set of polynomial calibration functions obtained from an actual experimental case (4 1, 024 × 1, 024 pixel Photron APX cameras, with Sigma 180 mm lenses, at a distance 1,000 mm from the measurement point, and at pitch and yaw angles of \({\theta}=[-30,0,0,30]^{\circ}\) and \({\phi}=[0,15,-15,0]^{\circ}\)). Projecting the particles into image space in this manner will eliminate the calibration error, which has a significant effect on the measurement uncertainty (Elsinga et al. 2006; Worth and Nickels 2008). Therefore, disparity maps obtained from the camera self-calibration [see Wieneke (2005) for disparity map examples] were used to displace the particle image co-ordinates, modelling the camera misalignment error based on the known experimental accuracy (∼0.1–0.2 pixels) applied according to the disparity map. Artificial images were created from these particle image co-ordinates using the same modelling assumptions as the DPIV simulation, with a seeding density of N_{ppp} = 0.02.
Vector and gradient fields are evaluated from the artificial images for both DPIV and TPIV simulations using the procedures detailed previously in Sect. 2.
Where available 50 and 100 velocity fields were taken at regularly spaced intervals throughout the DNS data cube for the DPIV and TPIV simulations, respectively, to produce reasonably well-converged statistics and permit analysis of quantities such as the energy spectrum. Due to the size of the largest field slices (see Table 2) fewer samples could be taken and most significantly only five fields were used for the TPIV S − T8 case. The resolutions quoted in Table 2 refer to the spacing between vectors.
To simulate the effect of camera noise on the PIV accuracy, a 10% level of random noise was added to each set of images. The effect of noise is investigated more fully for the S − D1 and S − T1 cases, where varying levels of noise ranging from 0 to 50% are added to the images.
Although representative as far as possible, Reynolds number differences between the ideal and real flow and the different forcing methods (the non-physical energy input in DNS compared to the experimental turbulence generation) may affect the results. Additionally, these simulations neglect the following effects that may be present during experimentation: particle loss and uneven illumination due to laser sheet non-uniformity, sheet misalignment between pulses, and irregular particle shape and size; any inhomogeneity in the particle distribution; persistent non-random noise due to camera hot-pixels and light diffusion due to multiple scattering effects; use of interpolation to displacement particles, particularly using the coarse ideal data in the high-resolution case. Furthermore, the velocity matching employed in the current simulation matches the mean value calculated experimentally, which may itself be under-predicted due to spatial averaging effects, affecting accuracy predictions accordingly.
4 Simulation results
DPIV and TPIV accuracy is assessed using a number of measures, visually comparing local fields of velocity, vorticity and squared strain rate, before examining trends in local and global mean quantities. The squared strain rate, S_{ij}^{2}, is representative of the dissipation rate and was chosen over the latter quantity to give a more generally applicable representation of the mean local errors. Given that the velocity is expressed as a displacement in pixels, the squared strain rate is therefore non-dimensional. The one-dimensional energy spectrum is also calculated and compared, as are topological invariants and divergence for the 3D TPIV fields.
4.1 Visual comparison of fields
The effect of resolution reduction on the velocity magnitude is moderate, with a reasonably small reduction in the peak velocities. However, the effect on the squared strain rate is more pronounced, and despite similar spatial distributions, the peak magnitude is increasingly under-predicted as the spatial resolution is reduced. Therefore, despite the ability of high-resolution PIV to identify the spatial distribution of important quantities such as dissipation, at low-resolution spatial averaging significantly reduces the magnitude of these predictions. These findings are confirm previous trends identified during HWA and PIV studies (Zhu and Antonia 1996; Lavoie et al. 2007).
Although the main flow features are predicted well, the fields are considerably less smooth than the DPIV simulation, as a result of the noise introduced through the TPIV technique (Elsinga et al. 2006). However, after smoothing, the fields show significantly better agreement with the ideal flow and improved gradient predictions (Fig. 6).
4.2 Local error variation with respect to noise
Figure 8c shows the measurement uncertainty associated with each gradient component. The TPIV gradient fields are subject to extremely large errors, which can be controlled to a certain degree through smoothing. The DPIV gradient uncertainty is around 20% for reasonable noise levels, which is half that of the smoothed in-plane TPIV gradients. As expected, the highest uncertainty is associated with w component velocity gradients, particularly those out-of-plane.
Figure 8d and e demonstrate gradient error effects on local vorticity and squared strain rate prediction, producing errors of around 17 and 27% in the former and around 22 and 40% in the latter for the DPIV and smoothed TPIV simulations, respectively, at reasonable noise levels. As with the velocity gradients, smoothing dramatically reduces the measurement uncertainty and noise only produces a significant accuracy reduction at very high levels. The known velocity gradient sensitivity to noise and error amplification through use of squared gradient terms produces the high measurement uncertainty, which although substantial, is again in line with previous experiments (Wieneke and Taylor 2006; Ganapathisubramani et al. 2007).
4.3 Mean global error variation with respect to resolution
Figure 9a shows that at high resolution, both the DPIV and TPIV results give good RMS velocity predictions. As resolution is reduced, the velocity is increasingly under-estimated, resulting in increasing errors. The DPIV accuracy decrease is slight, showing ∼7% error for the S − D10 case. The increase in TPIV measurement uncertainty is more significant, however, with an error magnitude of ∼20% for the smoothed S − T8 case. The erroneous over-prediction of velocity in the unsmoothed TPIV data counters the under-prediction due to spatial averaging, resulting in lower measurement uncertainty for the unsmoothed results.
This noise-related over-prediction error is much more prominent in the unsmoothed field velocity gradients in Fig. 9b, resulting in poor accuracy at high measurement resolution. As measurement resolution is reduced the gradient under-prediction due to spatial averaging increases, with the two effects balancing each other in the unsmoothed TPIV results at a resolution of x/η∼3. Beyond this value the spatial averaging error dominates, resulting in increasing gradient under-prediction. Despite the apparent low error magnitude resulting from this balance, it is not clear that this relationship will hold and as such the smoothed results are preferable. However, it is unlikely that all noise-related gradient over-prediction can be eliminated through smoothing, and therefore, care should also be exercised in the interpretation of smoothed squared strain rate estimates which may also be subject to a balance of error sources.
At the lowest resolution, the DPIV and smoothed TPIV results produce gradient errors of around 80% for the lowest resolution cases. The effect of these on the squared strain rate estimate is shown in Fig. 9c, producing errors of up to 90%. This analysis confirms the previous observations (see Sect. 4.1) that at low resolution, the effects of spatial filtering remove the smallest scale high gradient regions responsible for the majority of the high magnitude squared strain rate, resulting in severe under-prediction. Similarly, Lavoie et al. (2007) demonstrated increased attenuation of velocity and gradient quantities due to volume averaging over the interrogation region comparing PIV and HWA measurements. The TPIV averaging over a large volume in the current simulation results in a further increase in attenuation. However, it should also be noted that the use of a smoothing kernel will also reduce the magnitude of these gradients, contributing towards this under-prediction.
4.4 Local error variation with respect to resolution
The variation of local velocity gradient errors are shown in Fig. 10b. With the exception of the unsmoothed S − T1 case, the distribution is similar to those shown in previously in Fig. 9b, although the magnitude is considerably larger. As shown in Sect. 4.3, the unsmoothed S − T1 case gradients are significantly over-predicted. However, the necessary use of the modulus in the local error definition results in a positive error for this case in Fig. 10b. As shown previously, when resolution is reduced, the balance of different error sources changes, with the over-prediction of the unsmoothed gradients balancing the increasing under-prediction due to spatial filtering. Therefore, the apparent drop in error of these unsmoothed TPIV results as the resolution is reduced must again be treated with caution.
The effect of resolution on the local squared strain rate error is shown in Fig. 10c. Again, despite higher error magnitudes, a similar trend to the global error variation is observed in the DPIV and smoothed TPIV cases. The variation in the unsmoothed TPIV cases is again due to the balancing of error sources and use of error modulus in the local error definition.
4.5 Divergence
The divergence was also quantified by Mullin and Dahm (2006) for dual plane stereo-PIV measurements using the local divergence value normalised by the local velocity gradient tensor norm ∇· u/(∇u:∇u)^{1/2}. The normalised divergence was found to be normally distributed with a mean of zero, and an RMS variation of 0.35, which is similar to that found in the present investigation after smoothing (see Fig. 11b). The variation in the S − T8 case is caused by an insufficient quantity of data resulting in poor statistical convergence, making it difficult to draw conclusions from this case.
4.6 Effect on energy spectrum
The 1D energy spectrum is calculated directly from the PIV data using a similar approach to Foucaut et al. (2004). 1D lines of the instantaneous velocity are transformed using the FFT and multiplied by their conjugate values, with the energy averaged over each field according to a homogeneity hypothesis and ensemble averaged over all fields. PIV windowing spectral cut-off frequencies are detailed in Table 2.
The unsmoothed TPIV spectra take a lower energy level in comparison with the ideal spectra at low wavenumbers; a relationship which is then reversed as the simulation spectra rapidly rolls off at wavenumbers significantly lower than the cut-off wavenumber. The effect of Gaussian smoothing is dramatic, with the reduction in noise generated during the tomographic technique significantly altering the shape of the spectrum and shifting roll-off towards the cut-off frequency at the high wavenumber end. The severe under-prediction of velocity shown previously for the low resolution cases (see Sect. 4.3) result in an under-prediction of the energy content, similar to that noted by Lecordier et al. (2001), placing the PIV spectra below the ideal. The limited resolution also truncates the spectra to much lower wavenumbers, causing the redistribution of energy through aliasing.
The effect of aliasing on Kolmogorov power law scaling was examined by studying the pre-multiplied spectra shown in Fig. 14b. Despite a reasonably high degree of fluctuation in the ideal data, which may be due to insufficient low wavenumber modes [similar fluctuations can be observed in Ishihara et al. (2009)], the brief plateau indicates Kolmogorov inertial range scaling. However, in the smoothed TPIV spectra, energy losses through low resolution and energy redistribution through aliasing result in a small persistent negative gradient. The unsmoothed spectra is dominated by noise for all but the lowest wavenumber components.
Although it contains fractionally less energy, the S − D1 spectrum agrees relatively well with the ideal curve. In agreement with the the findings in Sect. 4.2, image plane noise appears to have little effect, with both zero and 50% noise level cases agreeing closely. The unsmoothed TPIV spectrum begins to roll-off at a reasonably low wavenumber. However, after smoothing, this follows the DPIV spectrum relatively well, albeit with a slight loss of energy that can be attributed to the smoothing operation.
4.7 Topological analysis
5 Comparison with experimental data
Experimental properties
Case | \({\Omega_I}R_I\) (mm/s) | Re | R_{λ} | u_{rms} (mm/s) | η (mm) | ε (m^{2}/s^{3}) | τ (ms) | ν × 10^{6} (m^{2}/s) | x/η | h/η | κ_{c}η |
---|---|---|---|---|---|---|---|---|---|---|---|
D1− 1 | 56 | 3.77 × 10^{4} | 240 | 17.4 | 0.536 | 2.01 × 10^{−5} | 243 | 1.18 | 6.71 | 13.4 | 0.209 |
D1− 2 | 112 | 7.55 × 10^{4} | 334 | 33.6 | 0.327 | 1.45 × 10^{−4} | 90.5 | 1.18 | 11.0 | 22.0 | 0.127 |
D1− 3 | 223 | 1.51 × 10^{5} | 479 | 69.1 | 0.191 | 1.26 × 10^{−3} | 30.7 | 1.18 | 18.9 | 37.8 | 0.0741 |
D1− 4 | 335 | 2.26 × 10^{5} | 581 | 102 | 0.143 | 4.01 × 10^{−3} | 17.2 | 1.18 | 25.2 | 50.4 | 0.0556 |
D1 − 5 | 447 | 2.94 × 10^{5} | 651 | 131 | 0.120 | 8.67 × 10^{−3} | 11.8 | 1.22 | 30.0 | 60.0 | 0.0467 |
D2 − 1 | 56 | 3.67 × 10^{4} | 236 | 17.3 | 0.550 | 1.97 × 10^{−5} | 248 | 1.22 | 3.60 | 7.20 | 0.389 |
D2 − 2 | 112 | 7.35 × 10^{4} | 332 | 34.2 | 0.330 | 1.52 × 10^{−4} | 89.3 | 1.22 | 6.01 | 12.0 | 0.233 |
D2 − 3 | 223 | 1.47 × 10^{5} | 432 | 57.7 | 0.222 | 7.36 × 10^{−4} | 40.7 | 1.22 | 8.90 | 17.8 | 0.157 |
D2 − 4 | 335 | 2.20 × 10^{5} | 584 | 105 | 0.142 | 4.48 × 10^{−3} | 16.5 | 1.22 | 14.0 | 28.0 | 0.100 |
D2 − 5 | 447 | 2.94 × 10^{5} | 633 | 124 | 0.125 | 7.30 × 10^{−3} | 12.9 | 1.22 | 15.8 | 31.6 | 0.0886 |
D3 − 1 | 28 | 1.91 × 10^{4} | 174 | 9.04 | 0.867 | 2.82 × 10^{−6} | 643 | 1.17 | 1.51 | 3.02 | 0.9272 |
D3 − 2 | 56 | 3.82 × 10^{4} | 248 | 18.3 | 0.511 | 2.34 × 10^{−5} | 223 | 1.17 | 2.57 | 5.14 | 0.545 |
D3 − 3 | 112 | 7.65 × 10^{4} | 356 | 37.8 | 0.296 | 2.06 × 10^{−4} | 75.2 | 1.17 | 4.43 | 8.86 | 0.316 |
D3 − 4 | 223 | 1.53 × 10^{5} | 478 | 68.1 | 0.191 | 1.21 × 10^{−3} | 31.1 | 1.17 | 6.88 | 13.8 | 0.203 |
D3 − 5 | 335 | 2.29 × 10^{5} | 628 | 117 | 0.127 | 6.18 × 10^{−3} | 13.7 | 1.17 | 10.4 | 20.8 | 0.135 |
D4 − 1 | 112 | 8.06 × 10^{4} | 335 | 31.7 | 0.326 | 1.21 × 10^{−4} | 95.6 | 1.11 | 1.67 | 3.34 | 0.838 |
D4 − 2 | 223 | 1.61 × 10^{5} | 501 | 70.8 | 0.178 | 1.36 × 10^{−3} | 28.6 | 1.11 | 3.05 | 6.10 | 0.459 |
D4 − 3 | 335 | 2.42 × 10^{5} | 548 | 84.7 | 0.156 | 2.32 × 10^{−3} | 21.9 | 1.11 | 3.49 | 6.98 | 0.401 |
D4− 4 | 447 | 3.22 × 10^{5} | 700 | 138 | 0.108 | 1.01 × 10^{−2} | 10.5 | 1.11 | 5.04 | 10.1 | 0.277 |
T1 − 1 | 28 | 1.86 × 10^{4} | 162 | 7.98 | 0.972 | 1.94 × 10^{−6} | 787 | 1.2 | 1.09 | 2.18 | 1.28 |
T1 − 2 | 56 | 3.72 × 10^{4} | 224 | 15.3 | 0.595 | 1.38 × 10^{−5} | 295 | 1.2 | 1.79 | 3.58 | 0.782 |
T1 − 3 | 112 | 7.45 × 10^{4} | 323 | 31.8 | 0.344 | 1.23 × 10^{−4} | 98.7 | 1.2 | 3.10 | 6.20 | 0.452 |
T1 − 4 | 223 | 1.49 × 10^{5} | 458 | 64.1 | 0.204 | 1.00 × 10^{−3} | 34.6 | 1.2 | 5.24 | 10.5 | 0.267 |
T1 − 5 | 335 | 2.23 × 10^{5} | 555 | 94.1 | 0.153 | 3.18 × 10^{−3} | 19.4 | 1.2 | 6.99 | 14 | 0.200 |
In this section comparisons are made to experimental measurements of turbulent energy dissipation rate, divergence and 1D energy spectra.
5.1 Experimental dissipation rate estimates
Despite the considerable scatter shown in Fig. 17a, the estimates show increasing dissipation under-prediction as the measurement resolution is artificially reduced. Lavoie et al. (2007) demonstrate the effect of spatial filtering and finite difference gradient calculation on PIV dissipation rate prediction, by filtering a known spectrum using a function based on interrogation window dimensions. This spectral correction agrees closely with the correction ratio computed from the current simulation results (Sect. 4.3)), indicating that the under-prediction of dissipation is dominated by spatial resolution effects, as opposed to other sources of PIV measurement uncertainty. The reasonably close agreement between the experimental DPIV measurements and the correction ratios confirms the source of the experimental under-prediction, providing a means of correction.
A similar trend is shown in Fig. 17b, with under-prediction increasing as measurement resolution is reduced. The more severe under-prediction of the D3 DPIV case may result from RMS velocity over-prediction and corresponding under-prediction of the scaling factor in this case.
A reduction in TPIV measurement resolution was shown to cause more severe dissipation rate under-prediction in comparison with DPIV, as a result of spatial averaging over a larger volume (see Sect. 4.3). This is reflected in the experimental measurements, which show similar under-prediction to the TPIV correction curve. Slightly higher than expected dissipation rate estimates may be a result of increased experimental noise in comparison with the simulations, resulting in gradient over-prediction, which compensate slightly (albeit erroneously) for the effects of spatial averaging. The source of this additional uncertainty may stem from the additional error sources not considered in the current simulations (detailed in Sect. 3).
5.2 Experimental divergence measurements
5.3 Experimental energy spectra
The inertial range agreement with Kolmogorov scaling is assessed more rigourously through the pre-multiplied spectra shown in Fig. 20b, which for clarity shows only the highest Reynolds number results containing the lowest wavenumber components. The failure of these pre-multiplied spectra to plateau demonstrates similarity with the simulation results, and the absence of Kolmogorov scaling in the inertial range is attributed to aliasing effects produced by insufficient resolution at the highest Reynolds numbers (note that since these are spatial spectra then reducing the spatial resolution also reduces the spatial sampling “frequency” leading to aliasing). The roll-off at high wavenumbers is clearly identified on the pre-multiplied spectra, which in similarity with the simulations appears at around the cut-off frequency for all cases, including the TPIV results after smoothing.
6 Conclusions
A series of simulations have been conducted to investigate the accuracy of DPIV and TPIV measurements, at varying levels of measurement resolution and noise, using an ideal flow representative of the experimental flow in the Cambridge large mixing tank facility. Accuracy has been assessed through examination of instantaneous velocity, vorticity and dissipation fields and statistically by comparing mean local and global estimates of these values. Comparisons of the flow divergence, energy spectra and some topological analysis have also been conducted.
Despite the close matching of experimental and simulation details differences between the ideal and experimental Reynolds number and flow fields and the use of modelling assumptions may affect these accuracy predictions. Due to a number of unconsidered error sources and the possible under-prediction of velocity, the accuracy estimates presented in this paper may be slightly conservative, with these extra error sources further increasing measurement uncertainty.
At high resolution, the ability of both DPIV and TPIV to accurately predict the velocity and gradient fields is demonstrated. Although measurement uncertainty increases in TPIV due to additional noise introduced during tomographic reconstruction, which manifests particularly in sensitive quantities such as the divergence and topological invariants, this can be partially controlled with velocity field smoothing.
The effect of reducing measurement resolution is shown to act in a similar way to spatial averaging over interrogation window sized regions, significantly reducing peak gradient magnitude and resulting in increasing under-prediction of important quantities such as dissipation and vorticity. This under-prediction is particularly severe for measurement resolutions lower than x/η = 3, suggesting this represents a minimum requirement for estimating these quantities and studying their spatial distribution, with higher resolution desirable.
Direct energy spectrum calculation from PIV data is subject to aliasing and spatial truncation effects, reducing agreement with the expected trend. Furthermore, the use of windowing functions to reduce truncation effects appears to be non-trivial, with accuracy dependent on the degree of aliasing caused by the spatial resolution.
Comparison with experimental data demonstrated that while resolution effects are reasonably predicted, other sources of error increase noise considerably, resulting in significantly higher levels of divergence, which is a particularly sensitive quantity. Experimental spectra harboured similar trends to the simulation results, remaining largely unaffected by the higher noise levels, which may only manifest at wavenumbers larger than the cut-off frequency.
The quantification of measurement uncertainty in this manner, through the careful matching of experimental conditions and use of a representative turbulent velocity field, has enabled accurate DPIV and TPIV error estimates to be made; an important step in analysing and understanding experimental data. Although this paper goes towards addressing the sparsity of literature on TPIV accuracy, which should help users during both the selection of experimental technique and subsequent interpretation of results, further studies across a range of flows would be useful, including variation of other parameters of interest such as the seeding density and mean particle displacement.
Acknowledgments
Dr. Tanahashi of Tokyo Tech is acknowledged for the DNS data. The first author wishes to acknowledge funding from the Engineering and Physical Sciences Research Council, through a Cambridge University Doctoral Training Award.