A simple model for the effect of peak-locking on the accuracy of boundary layer turbulence statistics in digital PIV

A simple model was constructed to study the effect of peak-locking on the accuracy of particle image velocimetry (PIV) turbulence statistics. A crucial parameter is the ratio between the root-mean-square (rms) velocity and the discretization velocity, which reflects the number of peaks distributed over the velocity probability density functions. When the ratio of the discretization velocity, which is set by the PIV setup parameters, to the rms, given by the flow, is larger than two, the maximum errors introduced in the mean and rms values become significant (larger than 1%). The errors introduced also depend on the amplitude, or severity, of the peak-locking, and whether the mean displacement corresponds to an integer or a fractional number of pixels. The peak-locking affects the statistical moments of different order in such a way that the errors are phase shifted. The proposed model can be used to predict errors in the turbulence statistics in a laboratory PIV experiment. According to our model predictions, the most significant influence of peak-locking in a boundary layer type of flow is an overall underestimation of the wall-normal rms. Our predictions are in good agreement with our experimental results from turbulent boundary layers and the recent experimental results from a turbulent channel flow by Christensen (Exp Fluids 36:484–497, 2004) for a case of moderate peak-locking.

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An efficient anti-aliasing spectral continuous window shifting technique for PIV

A new sub-pixel correlation peak locating algorithm for PIV analysis is introduced. The method is theoretically consistent with the method of continuously shifting interrogation sub-windows by fractional displacements, which has proven to be an effective way to reduce the bias error associated with integer pixel aliasing, or peak-locking. However the proposed algorithm performs continuous window shifting in the spatial frequency domain using the shift property of the Fourier transform, thus it is equivalent to interpolating the original digital image with the Fourier transform reconstruction. Synthetic and real PIV images are used to test the new algorithms performance relative to that of traditional (non-iterative) peak-finding methods and other peak-locking reduction algorithms, such as the continuous window shifting technique. The resultant bias error of the proposed algorithm is smaller (by an order of magnitude in some cases), and importantly, the periodic nature of the bias error, the characteristic signature of peak-locking, is eliminated as long as the discrete particle images have been sampled at a rate greater than the Nyquist sampling frequency. Moreover, this new algorithm is shown to be computationally efficient and it converges faster than the competing algorithms.     

Simultaneous assessment of peak-locking and CCD readout errors through a multiple Δt strategy

The multiple ??t strategy is an inexpensive procedure that can be implemented with any usual particle image velocimetry (PIV) set up. The only requirement is acquiring different series of PIV image pairs, setting a different time between laser pulses for each series. With this additional information, robust procedures for error assessment are possible. Within this strategy, this paper offers new discoveries that correct and complement previous works by the authors. Nogueira et al. (Meas Sci Technol 20?C7:074001, 2009) addressed the tasks of assessing CCD readout and peak-locking bias errors separately. In this paper, a new approach, of general application to PIV, is proposed for assessing both errors simultaneously. Additionally, it unveils the effect of the flow variability on the local amplitude of the peak-locking bias. In a different work, Nogueira et al. (Exp Fluids. doi: 10.1007/s00348-011-1094-2, 2011) have focused on assessing peak-locking rms error. That paper achieved only order of magnitude estimations of the rms error because one of its components was overlooked. Here, the new term is included and the assessment obtained is good enough to correct rms systematic errors, improving the measurement of the turbulent kinetic energy or similar flow magnitudes. After describing the strategies, the application to selected synthetic and real cases is presented, validating the modeling of the errors behavior when dealing with turbulent flows. The results indicate the possibility to assess bias errors in the range from 0.01 to 0.2 pixels for both sources (peak locking and CCD readout) simultaneously. Furthermore, the previous works simply assessed the magnitude of the errors, but the results of the new procedures proposed here, in some cases, are good enough to correct the measurement itself. In addition, the determination of zones where these errors are not the dominant ones is presented.     

A correlation-based continuous window-shift technique to reduce the peak-locking effect in digital PIV image evaluation

In this paper the peak-locking phenomenon is investigated in the evaluation of digital PIV recordings by using a correlation-based interrogation algorithm with a discrete window shift and a correlation-based tracking algorithm. Statistical analyses indicate that nonuniformly distributed bias errors are the main cause of the peak-locking effect, and the amplitude variation of the random error is also an important source of the peak locking. Simulations and experimental examples demonstrate that very strong peak-locking effects exist for the correlation-based interrogation algorithm with discrete window shift in the cases of large particle images, small interrogation windows, and very small particle images. Very strong peak-locking effects are also observed for the correlation-based tracking algorithm when the particle images are overexposed, binarized, or very small. These strong peak-locking effects can be avoided without loss of evaluation accuracy by using a continuous window-shift technique in combination with the correlation-based interrogation algorithm. Received: 2 July 2001 / Accepted: 28 November 2001     

Streamwise Vortices and Velocity Streaks in a Locally Drag-Reduced Turbulent Boundary Layer

This work aims to understand the changes associated with the near-wall streaky structures in a turbulent boundary layer (TBL) where the local skin-friction drag is substantially reduced. The Reynolds number is R e _?? = 1000 based on the momentum thickness or R e _τ = 440 based on the friction velocity of the uncontrolled flow. The TBL is perturbed via a local surface oscillation produced by an array of spanwise-aligned piezo-ceramic (PZT) actuators and measurements are made in two orthogonal planes using particle image velocimetry (PIV). Data analyses are conducted using the vortex detection, streaky structure identification, spatial correlation and proper orthogonal decomposition (POD) techniques. It is found that the streaky structures are greatly modified in the near-wall region. Firstly, the near-wall streamwise vortices are increased in number and swirling strength but decreased in size, and are associated with greatly altered velocity correlations. Secondly, the velocity streaks grow in number and strength but contract in width and spacing, exhibiting a regular spatial arrangement. Other aspects of the streaky structures are also characterized; they include the spanwise gradient of the longitudinal fluctuating velocity and both streamwise and spanwise integral length scales. The POD analysis indicates that the turbulent kinetic energy of the streaky structures is reduced. When possible, our results are compared with those obtained by other control techniques such as a spanwise-wall oscillation, a spanwise oscillatory Lorentz force and a transverse traveling wave.     

Spanwise vorticity measurements in a perturbed boundary layer

Hot-wire measurements of the spanwise vorticity fluctuation _z have been carried out in a turbulent boundary layer subjected to concentrated suction, applied through a porous wall strip. The results indicate that, relative to the no-suction case, the rms value of _z is significantly reduced in the near-wall region, with this reduction increasing with the suction rate . The reduction near the wall suggests an alteration in the dynamics of the layer. As the Reynolds number increases, the dynamics of the layer in the near-wall region become more intense and suction becomes less effective.     

Feasibility study of three-dimensional PIV by correlating images of particles within parallel light sheet planes

One approach to obtain information about the out-of-plane velocity component from PIV recordings is to analyze the height of the peak in the correlation plane. This value depends on the portion of paired particle images, which itself depends on the out-of-plane velocity component and on other parameters. To circumvent problems with other influences (e.g. background light, amount and size of images), images from another light sheet plane parallel to the first one were also captured for peak height normalization. Our experimental results show the feasibility of an out-of-plane velocity estimation by analyzing images of particles within parallel light sheets by spatial cross-correlation.List of Symbols C particle density in the flow - d particle image diameter - f ₀, f ₁ frames containing images of particles within the first light sheet at t=t ₀ (frame f ₀) and at t=t ₀ + t (frame f ₁) - f ₂ frame containing images of particles within the second light sheet parallel to the first one at t=t ₀ + 2t - F ₁ estimator of the loss of image pairs due to in-plane motion - F ₀ estimator of the loss of image pairs due to out-plane motion - F convolution of the particle image intensity distributions - K factor containing constant parameters in the correlation plane - M imaging magnification (image size/object size) - n ₀ number of particles in the measurement volume at t=t ₀ - n _0,1 number of particle image pairs in interrogation windows of f ₀ andf ₁ - n _1,2 number of particle image pairs in interrogation windows off ₁ and f ₂ - O _z overlap of the light sheets - R _C (s) convolution of the mean intensity distributions - R _D (s) correlation which gives the image displacement - R _F (s) fluctuating noise component of the cross correlation estimator - R _0,1(s _D ) cross-correlation peak height of interrogation windows off ₀ and f ₁ - R _1,2(s _iuD) cross-correlation peak height of interrogation windows of f ₁ and f ₂ - s two-dimensional separation vector in the correlation plane - s _D mean particle image displacement in the interrogation cell - t _e light pulse duration - t _f frame-transfer time of the video camera - u three-dimensional local flow velocity vector (u,v,w) - X _i position of the center of an interrogation window in the image plane (2d) - x _i position of the center of an interrogation volume in the flow (3d) - (z ₂ — Z ₁) displacement of the light sheets in z-direction - t separation time of the light pulses - x ₀ x-extension of an interrogation volume - y ₀ y-extension of an interrogation volume - z ₀ light sheet thickness The authors would like to thank DLR for supporting Markus Raffel's and Olaf Ronneberger's visit to Caltech (Center for Quantitative Visualisation), and the Office of Naval Research through the URI grant ONR-URI-N00014-92-J-1610. Dr. Alexander Weigand's generous offer of his experimental set-up and stimulating discussions with Dr. Jerry Westerweel and Dr. Thomas Roesgen are greatly appreciated. Special thanks also to Dr. Christian Willert for his advice regarding the modifications to the DPIV software.     

Identification of a new source of peak locking, analysis and its removal in conventional and super-resolution PIV techniques

One of the key factors that limit accuracy of particle image velocimetry (PIV) is the peak-locking effect. In this paper, a previously uncharacterised source of peak locking is presented. This source is neither related to the sensor geometry nor the subpixel resolution peak-fitting algorithms. It is present even when the particles are well described in terms of sensor spatial resolution (i.e. for particle diameters larger than 2 pixels). If no specific actions to avoid it are taken, its effect is especially important in those super-resolution systems that are based on iteratively reducing the size of the interrogation window. In this work, the mentioned source and its effects are studied and modelled. Based on this study, the actions required to avoid this type of peak locking are described. This includes the most usual correlation-based PIV systems, as well as super-resolution ones. Once this source of inaccuracy is avoided, it is possible to discriminate the performance of different types of correlation algorithms. As a consequence, specific proposals for the algorithms in the last steps of multigrid super-resolution PIV systems are given. The performances of the proposed solutions are verified using both synthetic and real PIV images. Received: 31 January 2000/Accepted: 2 May 2000     

Quantitative evaluation of PIV peak locking through a multiple Δ<Emphasis Type="Italic">t</Emphasis> strategy: relevance of the rms component

The possibility of using different times between laser pulses (Δt) in a PIV (Particle Image Velocimetry) measurement of the same real flow field for error assessment has already been proposed by the authors in a recent paper Nogueira et al. (Meas Sci Technol 20, 2009). It is a simple procedure that is available with the usual PIV setup. In that work, peak locking was considered basically as a bias error. Later measurements indicated that, using appropriate processing algorithms, this error is not the main peak-locking effect. Scenarios with the rms (root mean square) error due to peak locking as the most relevant contribution are more common than initially expected and require a differentiated approach. This issue is relevant due to the impact of the rms error in the evaluation of flow quantities like turbulent kinetic energy. The first part of this work is centred on showing that peak-locking error in PIV is not always a measurement bias towards the closest pixel integer displacement. Insight in the subject indicates that this is the case only for algorithm-induced peak locking. The peak locking coming out of image acquisition limitations (i.e. resolution) is not ‘a priory’ biased. It is a random error with a peculiar probability density function. Discussion on the subject is offered, and a particular approach to use a simple multiple Δt strategy to asses this error is proposed. The results reveal that in real images where amplitude of the peak-locking bias error is assessed to be as small as 0.02 pixels, rms errors can be in the order of 0.1 pixels. As PIV approaches maturity, providing a quantitative confidence interval by estimating measurement error seems essential. The method developed is robust enough to quantify these values in the presence of turbulence with rms up to ~0.6 pixels. This proposal constitutes a relevant step forward from the traditional histogram-based considerations that only reveal whether strong peak-locking error is present or not, without any information on its magnitude or whether its origin is bias or rms.     

Limits on the resolution of correlation PIV iterative methods. Fundamentals

The spatial resolution of correlation particle image velocimetry (PIV) is a frequently addressed issue that still raises scientific interest. In conventional non-iterative PIV, the spatial resolution limits are of common knowledge (Willert and Gharib (1991) Exp Fluids 10:181–193; Raffel et al. (1998) ISBN 3-540-63683-8, Springer, Berlin Heidelberg New York, among others). On the contrary, those advanced iterative multipass methods that use image distortion techniques or multigrid techniques present a more complex scenario. One of the concepts that raises more debate is the limiting effect of the interrogation window size. This paper focuses on the subject, trying to clarify key points. The results indicate that iterative algorithms using an appropriate weighting function eliminate the window size from the ensemble of spatial resolution limits.     

Performances of feature tracking in turbulent boundary layer investigation

In this paper, we describe the application of a feature tracking (FT) algorithm for the measurement of velocity statistics in a turbulent boundary layer over a flat plate at Re _θ ≃ 3,700. The feature tracking algorithm is based on an optical flow approach. Displacements are obtained by searching the parameters of the mapping between interrogation windows in the first and second image which minimize a correlation distance between them. The correlation distance is here defined as the minimum of the sum of squared differences of interrogation windows intensities. The linearized equation which governs the minimization problem is solved with an iterative procedure only where the solution is guaranteed to exist, thus maximizing the signal-to-noise ratio. In this process, the interrogation window first undergoes a pure translation, and then a complete affine deformation. Final mapping parameters provide the velocity and velocity gradients values in a lagrangian framework. The interpolation inherent to window-deforming algorithms represents a critical factor for the overall accuracy and particular attention must be devoted to this step. In this paper different schemes are tested, and their effects on algorithm accuracy are first discussed by looking at the distribution of systematic and random errors computed from synthetic images. The same analysis is then performed on the turbulent boundary layer data, where the effects associated with the use of a near-wall logical mask are also investigated. The comparison with single-point data gathered from the literature demonstrate the overall ability of the FT technique to correctly extract all relevant statistical quantities, including the spanwise vorticity distribution. Concerning the mean velocity profile, no evident influence of the interpolation scheme appears, while the near-wall accuracy is improved by the application of the logical mask. On the contrary, for the fluctuating components of the velocity, the interpolation based on B-Spline basis functions is found to perform better compared to the classical Bicubic scheme, particularly in the highly sheared region close to the wall.

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Landslide generated impulse waves.

Landslide generated impulse waves were investigated in a two-dimensional physical laboratory model based on the generalized Froude similarity. Digital particle image velocimetry (PIV) was applied to the landslide impact and wave generation. Areas of interest up to 0.8 m by 0.8 m were investigated. The challenges posed to the measurement system in an extremely unsteady three-phase flow consisting of granular matter, air, and water were considered. The complex flow phenomena in the first stage of impulse wave initiation are: high-speed granular slide impact, slide deformation and penetration into the fluid, flow separation, hydrodynamic impact crater formation, and wave generation. During this first stage the three phases are separated along sharp interfaces changing significantly within time and space. Digital masking techniques are applied to distinguish between phases thereafter allowing phase separated image processing. PIV provided instantaneous velocity vector fields in a large area of interest and gave insight into the kinematics of the wave generation process. Differential estimates such as vorticity, divergence, elongational, and shear strain were extracted from the velocity vector fields. The fundamental assumption of irrotational flow in the Laplace equation was confirmed experimentally for these non-linear waves. Applicability of PIV at large scale as well as to flows with large velocity gradients is highlighted.List of symbols a wave amplitude (L) - c wave celerity (LT^–1) - d_diff diffraction limited minimum particle image diameter (L) - d_e diffracted particle image diameter (L) - d_g granulate grain diameter (L) - d_p seeding particle diameter (L) - d recorded particle image diameter (L) - f focal length (L) - f_# f number (-) - F slide Froude number (-) - g gravitational acceleration (LT^–2) - h still-water depth (L) - H wave height (L) - l_s slide length (L) - L wavelength (L) - M magnification (-) - m_s slide mass (M) - n refractive index (-) - n_por slide porosity (-) - N_iw number of seeding particles in an interrogation window (-) - N_pair number of detected particle image pairs in window (-) - p interrogation window size p×p pixels; 1 pixel=9 m (L) - P probability (-) - P_il probability of in-plane loss of particle (-) - P_ol probability of out-of-plane loss of particle (-) - s slide thickness (L) - S relative slide thickness (-) - t time after impact (T) - T wave period (T) - v velocity (LT^–1) - v_p particle velocity (LT^–1) - v_px streamwise horizontal component of particle velocity (LT^–1) - v_py crosswise horizontal component of particle velocity (LT^–1) - v_pz vertical component of particle velocity (LT^–1) - v_s slide centroid velocity at impact (LT^–1) - V dimensionless slide volume (-) - V_iw interrogation volume (L³) - V_s slide volume (L³) - x streamwise coordinate (L) - x_ip area of view x dimension in image plane (L) - z vertical coordinate (L) - slide impact angle (°) - bed friction angle (°) - y depth of field (L) - t laser pulse separation (T) - x mean particle image x displacement in interrogation window (L) - _x random displacement x error (L) - _v random velocity v error (LT^–1) - _tot total random velocity v error (LT^–1) - _bias velocity v error due to biased correlation analysis (LT^–1) - _optics velocity v error due to optical imaging errors (LT^–1) - _track velocity v error due to particle flow tracking error (LT^–1) - _xx streamwise horizontal elongational strain component (1/T) - _xz shear strain component (1/T) - _zx shear strain component (1/T) - _zz vertical elongational strain component (1/T) - water surface displacement (L) - wavelength (L) - dynamic viscosity (ML^–1T^–1) - density (ML^–3) - _g granulate density (ML^–3) - _p particle density (ML^–3) - _s mean slide density (ML^–3) - _w water density (ML^–3) - granulate internal friction angle (°) - _y vorticity vector component (out-of-plane) (1/T)     

Optimization procedure for pulse separation in cross-correlation PIV

In PIV, the optimal time separation (t) between successive laser pulses is influenced by a number of parameters. In the present paper, only two kinds of error affecting the choice of t are studied: (i) random error arising from noise during recording of the flow seeded with tracer particles and subsequent interrogation of the particle images, and (ii) acceleration error arising from approximation of the local Eulerian velocity based on small (but non-zero) particle displacements. These two kinds of error place conflicting requirements on t. A model to optimize t with respect to these errors is described, and the model is confirmed by the results of a Monte Carlo simulation. This model for optimal t is extended to various acceleration distributions. An estimate for the spatial resolution of the velocity field resulting from cross-correlation PIV is proposed.We wish to thank the University of Delaware and the Department of Mechanical Engineering for providing a graduate fellowship to support this work.     

Re-understanding the law-of-the-wall for wall-bounded turbulence based on in-depth investigation of DNS data

Based on direct numerical simulation (DNS) data of the straight ducts, namely square and rectangular annular ducts, detailed analyses were conducted for the mean streamwise velocity, relevant velocity scales, and turbulence statistics. It is concluded that turbulent boundary layers (TBL) should be broadly classified into three types (Type-A, -B, and -C) in terms of their distribution patterns of the time-averaged local wall-shear stress (\(\tau _\mathrm{w} )\) or the mean local frictional velocity (\(u_\tau )\). With reference to the Type-A TBL analysis by von Karman in developing the law-of-the-wall using the time-averaged local frictional velocity (\(u_\tau )\) as scale, the current study extended the approach to the Type-B TBL and obtained the analytical expressions for streamwise velocity in the inner-layer using ensemble-averaged frictional velocity (\(\bar{{u}}_\tau )\) as scale. These analytical formulae were formed by introducing the general damping and enhancing functions. Further, the research applied a near-wall DNS-guided integration to the governing equations of Type-B TBL and quantitatively proved the correctness and accuracy of the inner-layer analytical expressions for this type.     

A novel architecture for a super-resolution PIV algorithm developed for the improvement of the resolution of large velocity gradient measurements

Three different particle image processing algorithms have been developed for the improvement of PIV velocity measurements characterized by large velocity gradients. The objectives of this study are to point out the limitations of the standard processing methods and to propose a complete algorithm to enhance the measurement accuracy. The heart of the PIV image processing is a direct cross-correlation calculation in order to obtain complete flexibility in the choice of the size and the shape of the interrogation window (IW). An iterative procedure is then applied for the reduction of the size of IW at each measurement location. This procedure allows taking into account the local particle concentration in the image. The results of this first iterative processing, applied to synthetic images, show both a significant improvement of measurement accuracy and an increase of the spatial resolution. Finally, a super-resolution algorithm is developed to further increase the spatial resolution of the measurement by determining the displacement of each particle. The computer time for a complete image processing is optimized by the introduction of original data storage in Binary Space Partitions trees. It is shown that measurement errors for large velocity gradient flows are similar to those obtained in simpler cases with uniform translation displacements. This last result validates the ability of the developed super-resolution algorithm for the aerodynamic characterization of large velocity gradient flows.     

Near-wall turbulence models for 3D boundary layers

Calculations of the three-dimensional boundary layer in an S shaped duct are performed with various – models. Three different near-wall models are used for the – model, of which one is using a new set of near-wall damping functions deduced from direct numerical simulations of turbulent channel flow available in the literature. The results show that it is possible to obtain damping functions giving better agreement, especially for and , with direct simulation data and experiments than with damping functions deduced from trial and error.     

Experimental investigations on two-phase flow past a sphere using digital Particle-Image-Velocimetry

Digital Particle-Image-Velocimetry was applied to investigate particle trajectories in a gas flow past a sphere. The particle displacement was determined by autocorrelation analysis of image sections. To enhance the signal/noise ratio a synthetic image with idealized particle pictures was generated from the real image. The autocorrelation function (ACF) was calculated using the Fast Hartley Transformation (FHT). The desired secondary maximum of this function was detected by an algorithm with subpixel resolution. A data validation step testing the plausibility of the velocity vectors completes the image analysis. Particle trajectories are traced with help of the particles' velocity vectors. The particle deposition on a sphere can be deduced from the course of these trajectories.List of symbols Cu Cunningham correction - e double distance between the limiting particle trajectory and the stagnation point axis - f focal length - H Hartley-Transform - M enlargement factor - S interference band spacing - x coordinate - y coordinate - x _p particle diameter - x _T droplet or sphere diameter - V image as grey value function - _.rel face velocity of droplet or sphere - particle image displacement - x particle image displacement in x-direction - y particle image displacement in y-direction - collection efficiency - wavelength of the laser light - _L fluid viscosity - _L fluid density - _p particle density     

Particle tracking velocimetry in three-dimensional flows

The photogrammetric determination of three-dimensional particle coordinates from a 3-camera system is described in Part I. In Part II we describe a fully automated tracking scheme for the determination of a sequence of velocity vectors within a three-dimensional observation volume of a fluid flow. From this sequence long-time particle trajectories are reconstructed.The tracking scheme is tested on trajectories obtained using the Kinematic Simulation Inertial Model (KSIM). Estimates of the yield of links between adjacent data sets of particle positions and of the yield of long-time particle trajectories are obtained. The limits of efficient tracking as a function of the spacing-displacement ratio p = _o/ut are also obtained. The effect of noise, in the form of the apparent appearance and disappearance of particles between one image and the next, and of jitter, which is the error in the determination of particle coordinates, is examined. It is shown that noise reduces the number of links per frame, but does not increase the number of erroneous links which is always small. However, the yield of long trajectories drops sharply with increasing noise. A small level of jitter, on the other hand, does not significantly influence any of the results.The tracking scheme is used on two sets of particle coordinate data obtained from real flows: a non-turbulent flow in a small water tank and a turbulent open channel flow.     

Optimal subpixel interpolation in particle image velocimetry

It is shown that among the different techniques for particle image velocimetry subpixel interpolation, only the "sinc"-kernel creates an optimal result in that it completely suppresses spurious spectral sidelobes. An efficient method is introduced for the computation of the subpixel-accurate correlation peak position without any systematic errors. A connection is made with the kernel-dependent observation of the peak-locking phenomenon.