Experiments in Fluids

, Volume 49, Issue 3, pp 637–656

A tomographic PIV resolution study based on homogeneous isotropic turbulence DNS data

Authors

    • Cambridge University Engineering Department
  • T. B. Nickels
    • Cambridge University Engineering Department
  • N. Swaminathan
    • Cambridge University Engineering Department
Research Article

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

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 (Reλ = 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, RI, 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/cm3 and a mean size of dp  =  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.

In the DPIV experiment, a single 1, 024 × 1, 024 pixel CMOS Photron APX was positioned perpendicular to the light sheet (through alignment with the relevant tank face), at a distance of 1,100 mm from the measurement point. Details of the lenses, aperture settings, FOV, and resolution can be found in Table 1 for the different resolution cases.
Table 1

PIV camera configuration parameters

Case

lens(es)

f/#

FOV (mm2)

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 mm2 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.

Finite differencing must be applied to evenly spaced PIV data in order to calculate the velocity gradient tensor. A second-order accurate central differencing scheme was adopted (Eq. 1), with a truncation error of approximatly 0.7Enoise/x (Raffel et al. 1998), where Enoise is the measurement uncertainty due to noise. This scheme cannot be applied to the edge points, and therefore, either forward or backwards differencing must instead be used (Eqs. 2, 3). Due to the higher error associated with these alternative differencing methods (Raffel et al. 1998), these points are excluded from the statistical analysis and only used during visualisation (Sect. 4.1) where a large domain size is important.
$$ \frac{\rm {d}u}{\rm {d}x_i} = \frac{u_{i+1} - u_{i-1}}{2x} $$
(1)
$$ \frac{\rm {d}u}{\rm {d}x_i} = \frac{u_{i+1} - u_{i}}{x} $$
(2)
$$ \frac{\rm {d}u}{\rm {d}x_i} = \frac{u_{i} - u_{i-1}}{x} $$
(3)

3 Computational set-up

The DNS simulation solves the Navier-Stokes equations for a 4003 grid point domain using a spectral approach, achieving a Taylor microscale Reynolds number of Reλ = 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.

Flow spatial similarity is achieved by matching the DNS and experimental non-dimensional ratio of domain size to Kolmogorov microscale, L/η. Different sized regions are sampled to represent a range of spatial resolutions, with these sampled fields stretched to fit the DPIV and TPIV volumes of interest (detailed in Table 2). It should be noted that particle velocity averaging will take place over the entire interrogation window size and it is therefore appropriate to match this critical dimension to the DNS grid point spacing, which is 1.89 η. Therefore, the highest resolution case uses an interrogation window size, h/η, equal to the DNS resolution, which is the equivalent resolution. The 50% overlap gives a vector spacing, x/η, of half this value, which is also included in Table 2, in addition to the spectral cut-off frequency due to the windowing effect (Foucaut et al. 2004).
Table 2

Simulation case set-up details

Case ref.

DNS slice size

Fields

x

h

κcη

 D1

33 × 33 × 1

100

0.93

1.86

1.505

S − D3

103 × 103 × 1

100

2.97

5.94

0.471

 D5

171 × 171 × 1

100

4.95

9.90

0.283

 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

− T4

171 × 171 × 35

44

4.20

8.40

0.333

 T8

339 × 339 × 69

5

8.36

16.7

0.168

S simulation, D DPIV, T TPIV

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 Nppp = 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 Nppp = 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.

A direct comparison can be made between the ideal and PIV velocity, vorticity and dissipation fields using Eq. 4. This represents the local error, which is the modulus of the difference between ideal and PIV fields at each point in the domain, summed and normalised by the mean ideal values. The necessary modulus use in this error estimate precludes the determination of the error direction (under or over prediction). Except for the highest resolution cases, the ideal and PIV data points are at different spatial locations. Therefore, in order to compare the local error variation with resolution, cubic spline interpolation is used to evaluate the ideal velocity field at PIV node locations.
$$ E_{\rm {local}} = \frac{\sum \left(|u_{\rm {ideal}} - u_{\rm {PIV}}| \right) }{\sum |u_{\rm {ideal}}|} \times 100 $$
(4)
The global error is the mean ideal value minus the mean PIV value, again normalised by the mean ideal value (Eq. 5). Positive or negative values for this error indicate PIV under-or over-prediction respectively. While the local error term includes the influence of both random and bias errors and is representative of the instantaneous error, the averaging used in the global error calculation removes the latter, providing an accuracy measurement applicable to statistical analysis.
$$ E_{\rm {global}} = \frac{\sum u_{\rm {ideal}}/N_{\rm {ideal}} - \sum u_{\rm {PIV}}/N_{\rm {PIV}} }{\sum u_{\rm {ideal}}/N_{\rm {ideal}}} \times 100 $$
(5)

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, Sij2, 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

Figures 1, and 2 show a comparison of the u velocity component and squared strain rate at two spatial resolutions for the DPIV simulation. Flow details are well predicted in the high-resolution S − D1 case. However, with reducing spatial resolution, the fields show the effect of spatial averaging, resulting in smoothing and a loss of fine-scale information. This can be further illustrated for example by direct spatially filtering of the ideal field with a kernel width equal to the interrogation region (taking the mean value within an interrogation window sized region at each interrogation window location on the ideal field) and comparing it to the PIV results, as shown in Fig. 3. The effect of this direct spatial filtering is extremely similar to the DPIV simulation, demonstrating the isolated effect of resolution reduction.
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Fig. 1

DPIV u velocity comparison; displacement in pixels

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Fig. 2

DPIV squared strain rate rate comparison; squared strain rate (pixels/pixel)2

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Fig. 3

Example field created by spatially filtering the ideal S − D10 field with a kernel width equal to the interrogation window size; displacement in pixels

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).

Figure 4a shows the velocity error distribution for the S − D1 case. The largest errors appear to be concentrated in regions of highest gradient [demonstrated through comparison with Fig. 4b], as a result of correlation peak broadening in these areas (Raffel et al. 1998). However, as the gradient magnitude is higher in these regions, the relative gradient error magnitude reduces with increasing gradient magnitude (Ganapathisubramani et al. 2007). This trend is demonstrated later in Sect. 4.5. Reducing the measurement resolution is expected to increase error magnitude significantly, which is discussed further in Sect. 4.4.
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Fig. 4

Spatial location of errors, absolute error magnitude; a displacement in pixels; b pixels per pixel

Figures 5 and 6 show u velocity and squared strain rate slices for the S − T1 TPIV simulation, with and without Gaussian smoothing. The ideal and TPIV velocity vector locations are not coincident, and therefore, slices are taken at the central depth plane of the ideal fields, with cubic interpolation used to obtain values for the TPIV fields at these locations. As the volume edges are trimmed where all camera lines of sight fail to converge, these plots have a smaller domain size in comparison with the DPIV simulation.
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Fig. 5

TPIV u velocity comparison; displacement in voxels; GS 2-pass Gaussian smoothing

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Fig. 6

TPIV dissipation rate comparison; squared strain rate (voxels/voxel)2; GS 2-pass Gaussian smoothing

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).

Figure 7 shows iso-surfaces of vorticity magnitude above a threshold value of twice the mean vorticity for the S − T1 and S − T3 cases. Reducing resolution reduces the peak vorticity magnitude, resulting in the identification of fewer high vorticity magnitude structures. However, if the known reduction in vorticity magntiude is taken into account and the threshold based instead on the mean TPIV vorticity (Fig. 7c–d), the same high vorticity structures can be identified at high resolution. As the spatial resolution is reduced, spatial averaging effects again remove the finer flow features.
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Fig. 7

TPIV iso-surfaces of vorticity magnitude (voxels/voxel); (ab) |ωideal| > 2 〈|ωideal|〉; (cd) |ωTPIV| > 2〈|ωTPIV|〉

4.2 Local error variation with respect to noise

Figure 8 shows mean error variation in the local velocity, velocity gradient, vorticity and squared strain rate estimates for the S − D1 and S − T1 cases with increasing noise. Figure 8a shows the absolute velocity error. The accuracy of the DPIV simulations is similar to previous studies (Raffel et al. 1998; Nobach and Bodenschatz 2009), taking a value of around 0.1 pixels. Even after smoothing the TPIV accuracy is found to be lower, with a mean error of around 0.2 voxels in the u component and 0.3 voxels in the w component, which is again in line with previous experiments (Wieneke and Taylor 2006; Worth and Nickels 2008). Figure 8b shows the velocity in percentage terms, with a DPIV measurement uncertainty of around 3–4%, and after smoothing around 6 and 11% for the TPIV in and out-of-plane components, respectively. The effect of noise is slight below a value of 20%, becoming increasingly significant at higher levels, showing a similar trend to the previous study (Worth and Nickels 2008). The effect of Gaussian smoothing on the TPIV velocity field is shown to increase accuracy by several per cent and also reduce the measurement uncertainty noise dependence.
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Fig. 8

Noise variation in local errors for S − D1 and S − T1 cases; GS 2-pass Gaussian smoothing

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 9 shows the global RMS velocity, velocity gradients and squared strain rate variation with reducing resolution for the DPIV and TPIV techniques in terms of percentage error magnitude.
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Fig. 9

Global percentage errors; GS 2-pass Gaussian smoothing

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

Figure 10 shows mean error variation in the local velocity, velocity gradient and squared strain rate estimates with increasing resolution. Figure 10a shows that as the resolution is reduced, the measurement error increases significantly in both DPIV and TPIV simulations, quantifying the behaviour observed previously in Sect. 4.1. The effect of smoothing on the TPIV data only permits a small error reduction. The inclusion of random errors in this metric results in higher error magnitudes in comparison with the global errors shown in Sect. 4.3.
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-010-0840-1/MediaObjects/348_2010_840_Fig10_HTML.gif
Fig. 10

Resolution variation in local errors; GS 2-pass Gaussian smoothing

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

Another stringent accuracy test can be conducted by assessing velocity gradient variation from the incompressible flow zero divergence condition (∇·u = 0). Figure 11a shows the variation in mean normalised divergence ratio, ξ (Zhang et al. 1997; defined in Eq. 6), with measurement resolution for the TPIV simulation. In comparison with the ideal data (zero divergence), the simulation divergence is high. However, as this accuracy criterion is based on squared velocity gradients, it is very stringent, with the unsmoothed divergence values similar to the full resolution value of ξ = 0.74 obtained by Zhang et al. (1997). Smoothing is shown to significantly reduce the divergence.
$$ \xi = \frac{\left(\partial u / \partial x + \partial v / \partial y + \partial w / \partial z \right)^2}{\left(\partial u / \partial x \right)^2 + \left(\partial v / \partial y\right)^2 + \left(\partial w / \partial z \right)^2} $$
(6)
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-010-0840-1/MediaObjects/348_2010_840_Fig11_HTML.gif
Fig. 11

Divergence analysis

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.

As PIV error is a function of gradient magnitude, JPDFs were constructed to correlate the normalised divergence values with the velocity gradient magnitude at each point, as shown in Fig. 12. The ideal data have been filtered using a kernel width equal to the interrogation window size to demonstrate the increase in divergence due to the effects of spatial averaging. The JPDFs show that regions of high gradient magnitude, which may be associated with high vorticity magnitude coherent structures, have lower divergence errors and therefore lower relative uncertainty. The distribution is similar to that presented by Ganapathisubramani et al. (2007) for stereo-PIV measurements of a turbulent jet. As measurement resolution is reduced, the characteristic triangular distribution loses definition, becoming increasingly circular as a result of high gradient magnitude region smoothing; a trend which can be independently observed in the ideal data. However, comparison with the ideal data suggests that the divergence error is dominated not by spatial averaging effects but rather through noise associated with TPIV.
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-010-0840-1/MediaObjects/348_2010_840_Fig12_HTML.gif
Fig. 12

TPIV gradient and divergence magnitude JPDF, contours of base 10 exponentials, SF spatially filtered, GS 2-pass Gaussian Smoothing

Another way of expressing the divergence error is through JPDFs of ∂u/∂x against − (∂v/∂y + ∂w/∂z), in which the degree of divergence is represented by the scatter of data away from the zero divergence line where the components are equal, shown in Fig. 13. The divergence is quantified through the use of a correlation coefficient, Q, between the components. Reducing resolution manifests in the spatially filtered ideal data as an increase in scattering from the zero divergence line, reducing the almost perfect correlation value of Q = 0.99 for the S − T1 ideal data to Q = 0.95 for the S − T4 ideal case. However, TPIV noise is shown to dominate, with a much lower correlation value of Q = 0.66 for the smoothed S − T1 case. Although this is lower than the value of 0.82 calculated by Ganapathisubramani et al. (2007) and the value of 0.7 calculated from the multi-probe hot-wire results of Tsinober et al. (1992), a slight reduction in accuracy may be expected moving from single point and planar measurements using hot-wires and stereo-PIV respectively to fully volumetric TPIV. Noise is controllable using increased smoothing, with the twice smoothed TPIV simulation predicting a similar correlation coefficient to Tsinober et al. (1992).
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Fig. 13

TPIV divergence correlation JPDF, contours of base 10 exponentials, SF spatially filtered, GS 2-pass Gaussian smoothing

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.

Figure 14a shows a comparison between ideal and PIV spectra for the S − T8 case with and without Gaussian smoothing. The ideal spectra closely follow the Pao spectra (Pao 1965) over a wide range of wavenumbers, only diverging at high wavenumbers. This peel off can be mainly attributed to the process of re-sampling used. Although the original ideal data set was periodic, 1D lines are taken from the data slices (see Sect. 3), which are less than the original width of the periodic cube. Therefore, truncation errors are introduced, which manifest as the observed roll-off.
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-010-0840-1/MediaObjects/348_2010_840_Fig14_HTML.gif
Fig. 14

Effect of low resolution on the energy spectra, S − T8 case

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.

Figure 15 shows the effect of image noise for the S − D1 and S − T1 cases. The ideal spectrum for both DPIV and TPIV cases is extremely similar, and therefore, only the former is plotted. In comparison with the Pao spectrum, the ideal spectrum contains more energy and also rolls-off at the high wavenumber end. The latter effect can be attributed to severe spatial truncation and non-periodic effects, as shown by Foucaut et al. (2004).
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-010-0840-1/MediaObjects/348_2010_840_Fig15_HTML.gif
Fig. 15

Effect of noise on the energy spectra, S − T1 case; N noise level

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

As described by Chong et al. (1990), the flow can be characterised topologically using invariants of the symmetric parts of the velocity gradient tensor. Figure 16 shows JPDFs of two invariant quantities \(Q_A^{\ast}\) and \(R_A^{\ast},\) normalised by the mean second invariant of vorticity, \(\langle \overline{Q_W} \rangle,\) as \(Q_A^{\ast} = Q_A/\langle \overline{Q_W} \rangle\) and \(R_A^{\ast}=R_A/\langle \overline{Q_W} \rangle^{3/2};\) a plot which has been shown to take a characteristic teardrop shape for a number of DNS studies (Ooi et al. 1999; O’Neill and Soria 2005). This distinctive distribution is clearly shown in the ideal data. As larger volumes are sampled in cases corresponding to lower resolutions the larger number of points makes this distribution clearer. The simulation ability, to reproduce this distribution is again highly noise sensitive, requiring smoothing to reduce this sensitivity and improve gradient accuracy (shown in Fig. 16). Severe noise effects in the unsmoothed results remove much of the distribution definition, leaving it almost circular. The effect spatial averaging due to reducing measurement resolution manifests as gradient magnitude reductions, resulting in smaller lower magnitude distributions.
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-010-0840-1/MediaObjects/348_2010_840_Fig16_HTML.gif
Fig. 16

JPDF of normalised topological invariants \(Q_A^{\ast}\) and \(R_A^{\ast},\) contours of base 10 exponentials

5 Comparison with experimental data

The predictions made by the numerical analysis can now compared with the experimental measurements. A summary of these measurements can be found in Table 3, which also details the experimental resolution in terms of the interrogation window size and vector spacing, and equivalent spectral cut-off frequencies. The dissipation rates in Table 3 are calculated using a scaling argument estimate, \(\epsilon = A u^3/\Uplambda,\) where the value of the constant, A, has been shown to be ≈0.5 from high Reynolds number homogeneous isotropic turbulence DNS (Rλ  >  200) (Sreenivasan 1998; Davidson 2004).
Table 3

Experimental properties

Case

\({\Omega_I}R_I\) (mm/s)

Re

Rλ

urms (mm/s)

η (mm)

ε (m2/s3)

τ (ms)

ν × 106 (m2/s)

x

h

κcη

D1− 1

56

3.77 × 104

240

17.4

0.536

2.01 × 10−5

243

1.18

6.71

13.4

0.209

D1− 2

112

7.55 × 104

334

33.6

0.327

1.45 × 10−4

90.5

1.18

11.0

22.0

0.127

D1− 3

223

1.51 × 105

479

69.1

0.191

1.26 × 10−3

30.7

1.18

18.9

37.8

0.0741

D1− 4

335

2.26 × 105

581

102

0.143

4.01 × 10−3

17.2

1.18

25.2

50.4

0.0556

D1 − 5

447

2.94 × 105

651

131

0.120

8.67 × 10−3

11.8

1.22

30.0

60.0

0.0467

D2 − 1

56

3.67 × 104

236

17.3

0.550

1.97 × 10−5

248

1.22

3.60

7.20

0.389

D2 − 2

112

7.35 × 104

332

34.2

0.330

1.52 × 10−4

89.3

1.22

6.01

12.0

0.233

D2 − 3

223

1.47 × 105

432

57.7

0.222

7.36 × 10−4

40.7

1.22

8.90

17.8

0.157

D2 − 4

335

2.20 × 105

584

105

0.142

4.48 × 10−3

16.5

1.22

14.0

28.0

0.100

D2 − 5

447

2.94 × 105

633

124

0.125

7.30 × 10−3

12.9

1.22

15.8

31.6

0.0886

D3 − 1

28

1.91 × 104

174

9.04

0.867

2.82 × 10−6

643

1.17

1.51

3.02

0.9272

D3 − 2

56

3.82 × 104

248

18.3

0.511

2.34 × 10−5

223

1.17

2.57

5.14

0.545

D3 − 3

112

7.65 × 104

356

37.8

0.296

2.06 × 10−4

75.2

1.17

4.43

8.86

0.316

D3 − 4

223

1.53 × 105

478

68.1

0.191

1.21 × 10−3

31.1

1.17

6.88

13.8

0.203

D3 − 5

335

2.29 × 105

628

117

0.127

6.18 × 10−3

13.7

1.17

10.4

20.8

0.135

D4 − 1

112

8.06 × 104

335

31.7

0.326

1.21 × 10−4

95.6

1.11

1.67

3.34

0.838

D4 − 2

223

1.61 × 105

501

70.8

0.178

1.36 × 10−3

28.6

1.11

3.05

6.10

0.459

D4 − 3

335

2.42 × 105

548

84.7

0.156

2.32 × 10−3

21.9

1.11

3.49

6.98

0.401

D4− 4

447

3.22 × 105

700

138

0.108

1.01 × 10−2

10.5

1.11

5.04

10.1

0.277

T1 − 1

28

1.86 × 104

162

7.98

0.972

1.94 × 10−6

787

1.2

1.09

2.18

1.28

T1 − 2

56

3.72 × 104

224

15.3

0.595

1.38 × 10−5

295

1.2

1.79

3.58

0.782

T1 − 3

112

7.45 × 104

323

31.8

0.344

1.23 × 10−4

98.7

1.2

3.10

6.20

0.452

T1 − 4

223

1.49 × 105

458

64.1

0.204

1.00 × 10−3

34.6

1.2

5.24

10.5

0.267

T1 − 5

335

2.23 × 105

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

The effect of resolution on experimental dissipation rate estimates is assessed using two approaches: first by dropping an increasing numbers of vectors from the highest resolution D4 case to artificially reduce the resolution (shown in Fig. 17a) and secondly, by comparing all experimental results, which include a reasonable range of measurement resolutions (shown in Fig. 17b). In both of these comparisons, the dissipation rate is calculated directly and normalised by the scaling argument estimates defined in Table 3.
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-010-0840-1/MediaObjects/348_2010_840_Fig17_HTML.gif
Fig. 17

Effect of equivalent measurement resolution on dissipation rate estimation

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

Figure 18a shows the variation in mean normalised divergence ratio, ξ, with measurement resolution for the experimental TPIV results. In comparison with the simulation divergence, the experimental divergence is significantly higher. Although smoothing is shown to reduce divergence, the smoothed results still contain significant errors. While the normalised divergence (shown in Fig. 18b) is again found to be normally distributed, the RMS variation is again significantly higher than predicted in the simulation. From the ideal simulation data, the effect of reducing resolution on divergence was shown to be slight. Therefore, while the simulation appears to accurately predict reduced resolution effects, it appears to under-predict the extremely noise-sensitive divergence effects. Furthermore, the dominance of noise appears to remove the resolution dependence of these divergence quantities.
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-010-0840-1/MediaObjects/348_2010_840_Fig18_HTML.gif
Fig. 18

Experimental divergence analysis

JPDFs of normalised divergence against velocity gradient magnitude, shown for three cases in Fig. 19, demonstrate the relative uncertainty of the measurements. Before smoothing, Fig. 19a–c show that the original distributions are relatively flat, with even high gradient regions containing reasonably large divergence. However, after smoothing, the characteristic triangular distribution again indicates flow regions containing high gradient magnitude are associated with lower divergence error, and therefore lower relative uncertainty, with these distributions qualitatively similar to those in Sect. 4.5. The distribution deformation with reducing resolution is not clear, which as predicted by the simulation results is due to the dominance of noise.
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-010-0840-1/MediaObjects/348_2010_840_Fig19_HTML.gif
Fig. 19

Experimental TPIV gradient and divergence magnitude JPDFs, contours of base 10 exponentials; GS 2-pass Gaussian smoothing

5.3 Experimental energy spectra

The 1D energy spectrum was calculated directly from the experimental DPIV and TPIV data using the method previously described in Sect. 4.6. Figure 20a shows a selection of these results, demonstrating a reasonable collapse of data between cases and reasonable agreement with the Pao result (Pao 1965) at low wavenumbers. Qualitatively similar behaviour is observed between these measurements, previous experimental investigations (Hwang and Eaton 2004; Liu et al. 1999), and the simulations (see Sect. 4.6), with noise resulting in spectrum roll-off at high wavenumber. The effect of smoothing on the TPIV data reduces the noise present in this data, improving the collapse of this curve, particularly at the noise-dominated high wavenumber end.
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-010-0840-1/MediaObjects/348_2010_840_Fig20_HTML.gif
Fig. 20

Energy spectra without windowing function; GS 2-pass Gaussian smoothing

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.

Copyright information

© Springer-Verlag 2010