Measurement of high-Re turbulent pipe flow using single-pixel PIV

The conventional PIV method determines the local instantaneous particle image displacement by computing the spatial correlation in typically 32\(\times\)32-pixel interrogation windows and identifying the location of the displacement correlation peak; the velocity is obtained by dividing the particle image displacement by the exposure time-delay between the two laser light pulses that illuminate the flow (Adrian and Westerweel 2011).

Alternatively, the particle image displacement can be found by correlating a single pixel in the first image frame with pixels in a small domain in the second frame, and summing the correlations over many frame pairs; see Fig. 4. When one uses 1024 frames, in principle the same amount of information is processed as for a single 32\(\times\)32-pixel (=1024 pixels) interrogation window. The gain is that the spatial resolution is now determined by the dimension of a single pixel, at the cost of losing the ability of recording time-resolved flows. This is attractive for stationary laminar and turbulent flows where a high spatial resolution is required.

In the present work, we make use of the fact that a fully developed turbulent pipe flow is homogeneous in the axial direction, and the single pixel approach is implemented by calculating the cross-correlation between a single image row of 922 pixels length in the first frame of a frame pair and a domain of a specified width and also a length of 922 pixels in the second frame. To accommodate the variations in particles-image displacement a domain width of 4 pixels is used for the mean flow quantities and a width of 8 pixels for the fluctuating quantities and spatial correlations. The computation of the correlation by means of Fourier transforms is implemented with zero padding (Adrian and Westerweel 2011). The particle image self-correlation is computed in a similar fashion, by means of auto-correlation of the second frame only. Both the cross-correlation and auto-correlation are summed over 20,000 recorded frame pairs for all cases. It is noted that in the vicinity of the reflections in the images (see Fig. 3), the resulting correlations appear to include a self-correlation peak at zero displacement that originates from the reflections. We therefore excluded in these cases a 3\(\times\)3-pixel domain centered around the zero displacement location in the spatial correlation to ignore the self-correlation peak.

Figure 5 shows a measured displacement correlation peak in a 5\(\times\)5 domain for Re\(_\tau\) = 10,254 at a radial distance of r = 85.2 mm from the pipe centerline, i.e. r/D = 0.414. A two-dimensional elliptical Gaussian is fitted to the correlation data, using the MATLAB routine lsqcurvefit and based on the routine D2GaussFit³ by Gero Nootz. The result shows that the correlation peak is elliptical, oriented with its long axis (red) over an angle of 0.269 rad (15.4°) with a half-width of 1.037 px and a short-axis (green) half-width of 0.763 px. The profiles to the right and on top compare the fitted curves to the measured correlation data along the two principal axes. The fit requires a priori estimates for the location, widths, and orientation of the Gaussian peak, for which we use the results of the 3-point Gaussian fits (Adrian and Westerweel 2011) and zero-angle for the orientation. In the remainder of this paper, we use 3\(\times\)3-data fits for estimating the mean displacement, and 5\(\times\)5-data fits for the normal and Reynolds stresses.

To correct for the effect of the finite particle image diameter on the measured probability density function, the measured displacement correlation peak needs to be deconvoluted with the (measured) particle image self-correlation peak. Given a large enough ensemble of image pairs, the displacement correlation peak can be expressed as the convolution of the particle image self-correlation \(F_\tau ({\textbf{s}})\) and the (in-plane) displacement distribution \(F_\varDelta ({\textbf{s}})\) (Westerweel 2008; Adrian and Westerweel 2011):where \({\textbf{s}}\) is a vector in the correlation plane. The exact derivation of this expression and a convenient generalized approximation are given by Westerweel (2008). To extract the displacement distribution, it is assumed that both \(F_\tau\) and \(F_\varDelta\) are two-dimensional Gaussian functions (Scharnowski et al. 2012); in the case of diffraction-limited imaging, the particle image self-correlation \(F_\tau ({\textbf{s}})\) is approximately a two-dimensional circular Gaussian function with a width \(d_\tau \sqrt{2}\), where \(d_\tau\) is the particle image diameter (Adrian and Westerweel 2011), while the displacement distribution \(F_\varDelta ({\textbf{s}})\) is approximately a two-dimensional elliptical Gaussian function where the widths in axial and radial directions are directly proportional to the local normal stresses and the orientation of the ellipse depends on the Reynolds stress.

For the deconvolution of the displacement correlation peak we follow the method proposed by Strobl (2017): (i) first a two-dimensional elliptical Gaussian is fitted to the displacement correlation peak, which gives the orientation angle \(\alpha\) of the long axis of the ellipse; (ii) the elliptical Gaussian peak is rotated over \(-\alpha\), so that the main axes of the ellipse align with the horizontal (X) and vertical (Y) axes, and the widths of the Gaussian are given by \(d_{\textrm{X}}\) and \(d_\textrm{Y}\) respectively; (iii) given that \(F_\tau\) is well approximated by a two-dimensional circular Gaussian function with an \(e^{-2}\) width \(d_\tau \sqrt{2}\) (Adrian and Westerweel 2011), the widths after deconvolution are given by:respectively; (iv) the new two-dimensional ellipse is rotated back over an angle \(\alpha\) to its original orientation; see Fig. 6.

After deconvolution, the resulting two-dimensional Gaussian is normalized, so that it represents the local two-dimensional probability density function \(f(u^*,v^*)\), with: \(\iint f(u^*,v^*)du^* dv^* \equiv 1\), where \(f(u^*,v^*)du^* dv^*\) represents the probability that \(u^*< u < u^*+du^*\) and \(v^*< v < v^*+dv^*\) (Nieuwstadt et al. 2016). To obtain the mean velocities, i.e. first-order moments, and normal and Reynolds stresses, i.e. second-order moments, at each location the integration is carried out numerically as:In Fig. 7, the measured correlation function and the corresponding two-dimensional Gaussian representations, before and after deconvolution, are shown for case Re\(_D\) = 4.98\(\times\)10\(^5\) at a radial distance of r = 85.2 mm from the centerline. This approach appears to work well for the axial normal stress and the Reynolds stress, but appears to yield elevated values for the radial normal stress. It is conjectured that the variation in axial displacement is comparable to or larger than the particle image diameter, but this is not the case for the variation in the radial displacement, which appears to be smaller than the particle image diameter. As a result, the deconvolution procedure as described above may not yield accurate results for the radial normal turbulent stress. Instead, we use an effective particle image diameter \({\hat{d}}_\tau\) for the deconvolution in (2) that is found by matching the single-pixel result for the radial normal stress to the data that was obtained by the conventional PIV in the overlap region where both methods were applied.

In the work of Scharnowski et al. (2012), it is mentioned that bias effects may occur in estimates of the normal and Reynolds stresses due to large gradients in the particle image displacement in combination with a large particle image diameter. In the present measurements, the largest variation in displacements due to velocity gradients occurs near the pipe wall, where it is estimated to be at most 0.25 [px/px] (see Figs. 9 and 10 below), and the particle image diameter is measured at 1.7 px (see Fig. 5); hence, given these values, no significant bias is expected for the present measurement data.

In this paper, we introduce an extension to the single-pixel approach, and that is the evaluation of spatial correlations of the velocity fluctuations, defined as (Hinze 1975; Adrian and Westerweel 2011):where \(\rho _{uu}(\varDelta x)\) is the correlation coefficient for velocity data separated by a distance \(\varDelta x\) = \(x_2-x_1\) along a homogeneous direction in the flow (here the axial velocity fluctuations \(u'\) along the axial direction x of the pipe flow). Conventionally, \(Q_{uu}(\varDelta x)\) is estimated from PIV data by multiplying the velocity fluctuations at locations separated by a distance \(\varDelta x\), and then averaging over data along the homogeneous direction (if any) and over all recorded frames. This yields a biased estimate of \(Q_{uu}\) with an expectation value:(Adrian and Westerweel 2011), with \(|\varDelta x|<L\) where L is the length along the homogeneous direction of the region where PIV data is taken. The second factor on the righthand side is known as a Bartlett window.

An alternative approach is to estimate \(Q_{uu}(\varDelta x)\) from the joint probability density function of the velocity fluctuations \((u'_1,u'_2)\) at two distinct points \((x_1,x_2)\) in the flow:The two-point joint pdf \(f(u_1^*,u_2^*|x_1,x_2)\) is constructed as illustrated in Fig. 8: We multiply the instantaneous intensity in a single pixel at \(x_1\) in frame A with the intensities in a given number of pixels in frame B. When a particle image is present at \(x_1\) in frame A and also in the range of pixels in frame B, then this gives a peak at an offset of \(u_1\) equal to the instantaneous particle image displacement; see the red pixels and signal peak in Fig. 8. At the same time, we multiply the instantaneous intensity of a single pixel at \(x_2\) in frame A with the intensities of a given number of pixels in frame B. If there happens to be a particle image present at \(x_2\), this gives a signal peak at an offset \(u_2\) equal to the instantaneous particle image displacement; see the blue pixels and correlation peak in Fig. 8. We subsequently perform a dyadic multiplication of the two signals, i.e. \(f_1(u_1|x_1) \otimes f_1(u_2|x_2)\) (where each signal is represented as a vector), to construct a matrix containing a peak in the \((u_1,u_2)\) plane; green dot in Fig. 8. This is then repeated for all pixels along a row in frames A and B, which is the homogeneous direction for the current flow, and over all frame pairs (A,B). This then builds up an estimate for the joint probability density function \(f(u_1^*,u_2^*|x_1,x_2)\):where the sum over i means averaging data in the homogeneous direction and over all frame pairs. Note that only when there is a particle image present at \(x_1\) and at the same moment another particle image at \(x_2\) that there is a contribution to the estimation of the joint pdf; without a particle image present in either \(x_1\) or \(x_2\) the net contribution to the estimate is effectively nihil. Hence, a large volume of data is required to obtain a reliable unbiased estimate \(Q^*_{uu}(\varDelta x)\). Given that the image density for a 32\(\times\)32-pixel area is 10, the probability to find a particle image in a single pixel is 0.010; to find two particle images simultaneously in two pixels is then only 10\(^{-4}\). In the present data we have around 900 pixels per line, and 20,000 frame pairs, yielding a total of 18\(\times\)10\(^6\) sample pairs, so that one can expect 1800 simultaneously matching particle image pairs (or more when the image density is above 10). This is why a large number of image frame pairs is required for the estimation of the spatial correlation. Furthermore, there is a finite probability that erroneous matches occur, but these are expected to be randomly distributed over the \((u_1,u_2)\) domain and do not seriously affect the estimation of spatial correlation. The inset in Fig. 8 shows the convergence of the relative statistical error for the two-point spatial correlation as a function of the number of frames (each containing 900 pixels per row) that was processed. The data in this graph were determined via so-called bootstrapping. The results indicate that the error converges proportional to \(1/\sqrt{N}\), where N is the number of pixels; this is indicative of a linear process for correlation averaging (Meinhart et al. 2000; Westerweel et al. 2004; Scharnowski et al. 2012).

A typical result for the joint PDF is shown in the right inset in Fig. 8. The joint PDF appears as an elongated 2D Gaussian distribution that is always oriented along the diagonal. The correlation estimate \(Q^*_{uu}\) follows from the mixed moment, defined in (10).

As for the point-wise PDF’s, the joint PDF’s are convoluted by the particle image self-correlation. To correct for this, we normalize the measured correlations \(Q^*_{uu}\) with those of the conventional PIV analysis, similar to the normalization of the wall-normal stresses. However, the conventional PIV data yields a biased estimate \({\hat{Q}}_{uu}\) of the spatial correlation (Priestley 1992; Adrian and Westerweel 2011), so in order to compare the two results, the estimate \(Q^*_{uu}\) is multiplied with a Bartlett window to match the biased estimate \({\hat{Q}}_{uu}\).

In the current work, the spatial correlation is determined for an axial spatial range of 800 pixels. To properly estimate the integral scale from the spatial correlation, its tail is extended by fitting of an exponential tail from the data between offsets of 400 to 800 pixels. In a region of approximately 7 pixels near \(\varDelta x = 0,\) the spatial correlation shows erratic results; see “Appendix D”. Hence, the data for the spatial correlation over a separation of less than 7 pixels is ignored and replaced by the results of a parabolic fit over the remaining data. Finally, the Fourier transform is taken of the single-pixel spatial correlation data to yield the power spectrum of axial velocity fluctuations. To attenuate the noise in the tails of the spatial correlation, a moving mean filter with a variable filter size ranging from 4 to 132 pixels is applied. Furthermore, the spectral data is averaged in equally spaced spectral windows in logarithmic wavenumber-space (Priestley 1992, p. 580).

Measurements are performed at bulk Reynolds numbers Re\(_D\) (\(= 2U_bR/\nu\)) between 3.4\(\times\)10\(^5\) and 6.9\(\times\)10\(^5\) (where \(U_b\) is the bulk velocity, \(R = \frac{1}{2}D\) is the pipe radius, and \(\nu\) the kinematic viscosity of the fluid). The main flow conditions are shown in Table 2. The wall shear stress is calculated from the measured pressure drop and used to obtain the friction velocity as \(u_\tau =\sqrt{\tau _w/\rho }\). The friction factor is determined via \(c_f = 8 (u_\tau / U_b)^2\). The water density and kinematic viscosity are taken at the measured mean temperature.

For each Reynolds number a total of 20,000 double-frame images are recorded. We mainly present results obtained at Re\(_D\) = 4.98\(\times\)10\(^5\), with a corresponding shear Reynolds number Re\(_\tau\) (\(\equiv u_\tau R/\nu\)) = 10.3\(\times\)10\(^3\).

For the conventional PIV processing, the images are pre-processed using a 5\(\times\)5-px min-max filter (Adrian and Westerweel 2011). A multi-grid approach is used with initially 64\(\times\)64-px interrogation areas and 50% overlap. After determining the median vectors in a 5\(\times\)5 neighborhood, the result is interpolated to a 32\(\times\)32-px grid with 50% overlap for the second interrogation pass. After the second pass, any outliers are detected using universal outlier detection (Westerweel and Scarano 2005) and replaced by means of linear interpolation. The 32\(\times\)32-pixel interrogation result for the case Re\(_D\) = 4.98\(\times\)10\(^5\) contains 1.8% spurious vectors (67 outliers per image pair yielding 3763 vectors each, with 45 and 96 as the 5- and 95-percentiles, respectively); this indicates that there is sufficient seeding and negligible loss-of-correlation due to out-of-plane motion. These numbers are also representative for the measurements taken at the other Reynolds numbers presented in Table 2, with the exception of the image pairs for the highest flow rate (i.e., 97 lps), where the working fluid became slightly polluted.

The final grid consists of 71\(\times\)53 interrogation areas leading to a vector spacing of 2.94 mm. Profiles of the mean velocity, normal stresses and Reynolds stress are determined by pointwise averaging of the interrogation results over all image pairs followed by averaging the data along rows (in the homogeneous direction of the turbulent pipe flow). For the conventional PIV results only the first 5000 images are used for all cases. All PIV processing is performed using in-house MATLAB software based on the methods documented by Adrian and Westerweel (2011).

The spatial resolution of the conventional PIV is given by the dimensions of the 32\(\times\)32-pixel interrogation window. For the measurement presented here this corresponds to a resolution of \(\varDelta y^+ \cong\) 600. Evidently, this is too crude to make measurements in the near-wall region. The single-pixel approach gives a measurement at each (radial) pixel location, and thus gives a spatial resolution of \(\varDelta y^+\) = 19. This would in principle be sufficient to determine the flow profiles in the entire log-region.

Figure 9 shows the measured velocity profiles with both the conventional and the single-pixel PIV approaches. The results cover a displacement range between 4 and 11 pixels and appear to be free from pixel locking errors. The conventional PIV was performed over the entire flow, while the single-pixel results were only done for the near-wall flow region, with a sufficient overlap with the conventional PIV data.

Figure 10 shows the velocity profiles for all Reynolds numbers, but now in a semi-log plot, which illustrates the high level of accuracy of the single-pixel PIV results and the extension of the range in the log-region by a decade in wall units. Note that the results are converged up to the size of the symbols and also note that the data point nearest to the wall for case \(Re_\tau\) = 7162 is the only near-wall data point that is increasing in value during the convergence process. To assess the quality of the mean flow, the data of McKeon et al. (2004) for \(Re_{\tau } = 10,914\) is compared with the present case \(Re_{\tau } = 10,254\). The agreement up to symbol size indicates that the quality of the mean flow is comparable. The profiles coincide with the log profile with a von-Kármán constant of k = 0.405. This value for k is in agreement with the value found by Nagib and Chauhan (2008) for pipe flow.

Figure 11 shows the results for the normal stresses and the Reynolds stress, as determined by the procedure explained in Sect. 3. The conventional PIV method using 32\(\times\)32-pixel interrogation windows shows erroneous results for the radial normal stress and the Reynolds stress close to the pipe wall (for \(|r|/D > 0.43\)), which corresponds to a distance \(y^+ < 10^3\) from the wall. The single-pixel PIV results appear to give reliable results when approaching the wall. As explained in Sect. 3, the results for the radial normal stress \(\overline{{v'}^2}\) are deconvoluted by matching the deconvoluted single-pixel correlation results with the conventional 32\(\times\)32-pixel results for \(\overline{{v'}^2}\). The single-pixel PIV results for the Reynolds stress appear to deviate occasionally from the surrounding data, which can be attributed to interference of the remaining reflections with the single-pixel estimate of the displacement correlation peak.

One of the features observed in high-Re turbulent pipe flow is the emergence of a so-called outer peak in the streamwise normal stress \(\overline{{u'}^2}\) (Hultmark et al. 2013; Willert et al. 2017). The outer peak begins to emerge at a Reynolds number Re\(_\tau\) larger than 2\(\times\)10\(^4\), which is just outside the range of our measurements. However, at a Reynolds number Re\(_\tau \sim 10^4\) a clear ‘plateau’ is visible in the profile for \(\overline{{u'}^2}\), which is evident when comparing with results for much lower Reynolds numbers (e.g., den Toonder and Nieuwstadt 1997).

In Fig. 12, the normalized normal stress \(\overline{{u'}^2}^+\) is plotted as a function of the distance from the wall \(y^+\). The data is compared with data obtained by Hultmark et al. (2013) at Re\(_\tau\) = 10.5\(\times\)10\(^3\) in the Princeton Superpipe and by Willert et al. (2017) at Re\(_\tau\) = 11.7\(\times\)10\(^3\) in the CICLoPE facility. The single- pixel results match the conventional 32\(\times\)32-px PIV data that is available for \(y^+>\) 665 only. The original single-pixel PIV data show occasional fluctuations in the range \(250< y^+ < 10^3\). This can be attributed to the reflections that occur close to the pipe wall; see Fig. 3. By expanding the number of modes in the Fourier filter to six in the images prior to the single-pixel correlation in the affected area, the effect of these reflections could be substantially suppressed only at the locations in between the reflections. A tenth order polynomial is fitted to the PIV and single-pixel data. The results show an acceptable agreement with the Superpipe and CICLoPE data to within an error margin that corresponds to variations in the particle displacement of 0.11 px. Note that the so-called inner peak, located at \(y^+ \approx\) 10–20 could not be resolved. The plateau in \(\overline{u'^2}^+\) that occurs at these high Reynolds numbers can be resolved down to \(y^+ \cong 150\) from the pipe wall. This is approximately 500 wall units closer to the wall than for conventional PIV.

In Fig. 13, we present the results for the spatial correlation coefficient \(\rho _{uu}(\varDelta x)\) of the axial velocity fluctuations at five wall distances, ranging from the pipe centerline to \(y^+ = 172\). Upon integrating the area below the spatial correlation we obtain the longitudinal integral turbulence length scale L/D. The inset in Fig. 13 shows the integral length scale as a function of the distance from the wall. The length scale decreases in size upon approaching the wall, as is expected. Towards the pipe centerline the length scale again decreases.

By taking the Fourier transform of \(\rho _{uu}(\varDelta x)\) we find the longitudinal spectrum \(F_{uu}(\kappa )\). In Fig. 14, the spectrum as derived from the single pixel data is shown, along with the binned results and an indication of a \(-5/3\) slope. The single-pixel PIV result extends the range in wavenumber by more than a decade (owing to the 16–32 times higher resolution compared to the conventional 32\(\times\)32-px PIV data), but also becomes noisy at the higher wave numbers. By using binning over equal sections along the logarithmic axis (red dots in Fig. 14), a consistent result is obtained for \(\kappa D < 10^3\), whereas the conventional PIV data appears to give consistent results for \(\kappa D < 100\). This demonstrates the potential of single-pixel PIV to provide information of the spatial structure of the turbulence. The results of Laufer (1954) for the measured turbulence spectrum at the centerline (\(y/R=1.0\)) and \(y/R=0.074\) at Re\(_D\) = 500\(\times\)10\(^3\) are included in Fig. 14.