Second-order accurate particle image velocimetry

 An adaptive, second-order accurate particle image velocimetry (PIV) technique is presented. The technique uses two singly exposed images that are interrogated using a modified cross-correlation algorithm. Consequently, any of the equipment commonly available for conventional PIV (such as dual head Nd: YAG lasers, interline transfer CCD cameras, etc.) can be used with this more accurate algorithm. At the heart of the algorithm is a central difference approximation to the flow velocity (accurate to order Δt ²) versus the forward difference approximation (accurate to order Δt) common in PIV. An adaptive interrogation region-shifting algorithm is used to implement the central difference approximation. Adaptive shifting algorithms have been gaining popularity in recent years because they allow the spatial resolution of the PIV technique to be maximized. Adaptive shifting algorithms also have the virtue of helping to eliminate velocity bias errors. The second- order accuracy resulting from the central difference approximation can be obtained with relatively little additional computational effort compared to that required for a standard first-order accurate forward difference approximation. The adaptive central difference interrogation (CDI) algorithm has two main advantages over adaptive forward difference interrogation (FDI) algorithms: it is more accurate, especially at large time delays between camera exposures; and it provides a temporally symmetric view of the flow. By comparing measurements of flow around a single red blood cell made using both algorithms, the CDI technique is shown to perform better than conventional FDI-PIV interrogation algorithms near flow boundaries. Cylindrical Taylor–Couette flow images, both experimental and simulated, are used to demonstrate that the CDI algorithm is significantly more accurate than conventional PIV algorithms, especially as the time delay between exposures is increased. The results of the interrogations are shown to agree quite well with analytical predictions and confirm that the CDI algorithm is indeed second-order accurate while the conventional FDI algorithm is only first-order accurate. Received: 15 June 2000/Accepted: 2 February 2001     

A correction method for measuring turbulence kinetic energy dissipation rate by PIV

The accuracy of turbulent kinetic energy (TKE) dissipation rate measured by PIV is studied. The critical issue for PIV-based dissipation measurements is the strong dependency on the spatial resolution, Δx, as reported by Saarenrinne and Piirto (Exp Fluids Suppl:S300–S307, 2000). When the PIV spacing is larger than the Kolmogorov scale, η, the dissipation is underestimated because the small scale fluctuations are filtered. For the case of Δx smaller than the Kolmogorov scale, the error rapidly increases due to noise. We introduce a correction method to eliminate the dominant error for the small Δx case. The correction method is validated by using a novel PIV benchmark, random Oseen vortices synthetic image test (ROST), in which quasi-turbulence is generated by randomly superposing multiple Oseen vortices. The error of the measured dissipation can be more than 1,000% of the analytical dissipation for the small Δx case, while the dissipation rate is underestimated for the large Δx case. Though the correction method does not correct the underestimate due to the low resolution, the dissipation was accurately obtained within a few percent of the true value by using the correction method for the optimal resolution of η/10 < Δx < η/2.     

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.     

Investigation of liquid–liquid drop coalescence using tomographic PIV

High-speed tomographic PIV was used to investigate the coalescence of drops placed on a liquid/liquid interface; the coalescence of a single drop and of a drop in the presence of an adjacent drop (side-by-side drops) was investigated. The viscosity ratio between the drop and surrounding fluids was 0.14, the Ohnesorge number (Oh = μ_d/(ρ_dσD)^1/2) was 0.011, and Bond numbers (Bo = (ρ _d  − ρ _s )gD ²/σ) were 3.1–7.5. Evolving volumetric velocity fields of the full coalescence process allowed for quantification of the velocity scales occurring over different time scales. For both single and side-by-side drops, the coalescence initiates with an off-axis film rupture and film retraction speeds an order of magnitude larger than the collapse speed of the drop fluid. This is followed by the formation and propagation of an outward surface wave along the coalescing interface with wavelength of approximately 2D. For side-by-side drops, the collapse of the first drop is asymmetric due to the presence of the second drop and associated interface deformation. Overall, tomographic PIV provides insight into the flow physics and inherent three-dimensionalities in the coalescence process that would not be achievable with flow visualization or planar PIV only.     

A study of the three-dimensional spectral energy distribution in a zero pressure gradient turbulent boundary layer

Time-resolved particle image velocimetry (PIV) measurements performed in wall parallel planes at three wall normal locations, y ⁺ = 34, 108, and 278, in a zero pressure gradient turbulent boundary layer at Re _τ = 470 are used to illuminate the distribution of streamwise velocity fluctuations in a three-dimensional energy spectrum (2D in space and 1D in time) over streamwise, spanwise, and temporal wavelengths. Two high-speed cameras placed side by side in the streamwise direction give a 10δ × 5δ streamwise by spanwise field of view with a vector spacing of _boxclose = z^+ 37\Updelta x^+ = \Updelta z^+ \approx 37 and a time step of \Updelta t⁺=0.5\Updelta t^+=0.5. Although 3D wavenumber--frequency spectra have been calculated in acoustics studies, to the authors’ knowledge this is the first time they has been calculated and presented for a turbulent boundary layer. The calculation and normalization of this spectrum, its relation to 2D and 1D spectra, and the effects of the PIV algorithm on its shape are carefully analyzed and outlined.     

Estimating the viscoelastic moduli of complex fluids using the generalized Stokes–Einstein equation

We obtain the linear viscoelastic shear moduli of complex fluids from the time-dependent mean square displacement, <Δr ²(t)>, of thermally-driven colloidal spheres suspended in the fluid using a generalized Stokes–Einstein (GSE) equation. Different representations of the GSE equation can be used to obtain the viscoelastic spectrum, G˜(s), in the Laplace frequency domain, the complex shear modulus, G ^*(ω), in the Fourier frequency domain, and the stress relaxation modulus, G _r (t), in the time domain. Because trapezoid integration (s domain) or the Fast Fourier Transform (ω domain) of <Δr ²(t)> known only over a finite temporal interval can lead to errors which result in unphysical behavior of the moduli near the frequency extremes, we estimate the transforms algebraically by describing <Δr ²(t)> as a local power law. If the logarithmic slope of <Δr ²(t)> can be accurately determined, these estimates generally perform well at the frequency extremes. Received: 8 September 2000/Accepted: 9 March 2000     

Computer simulated analyses on deformation and fracture of non-cracked and cracked specimens

A unified damage and fracture model, the combinatory work density model, which is suitable for either non-cracked body or cracked body has been suggested^[t−7]. In the present paper, the deformation and fracture of the two kinds of tensile spceimen and TPB specimen made of 40Cr steel have been simulated by using the new model together with the large elastic-plastic deformation finite element method. The results give a good picture of the whole deformation and fracture processes of the specimens in experiments; especially, the results on the TPB specimen can be used to obtain the relationship between load and displacement at the loading pointP-Δ, and between crack extension and displacement at the loading point Δa-Δ, the resistance curveJ _R -Δa and the fracture toughnessJ _1C . All the results are in remarkable agreement with those obtained by experiments. Therefore the model suggested here can be used to simulate crack initiation and propagation in non-cracked body and fracture initiation and crack stable propagation in cracked body. The project supported by National Natural Science Foundation of China     

On asymptotic properties of solutions of linear differential functional equations with linearly transformed argument

We establish new properties of C ¹[−1, +∞)-solutions of the linear functional differential equation ẋ(t) = ax(t) + bx(qt) + hx(t−1) + cẋ(qt) + rẋ(t−1) in the neighborhood of the singular point t = +∞. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 2, pp. 170–177, April–June, 2006.     

Characterization of the flow properties of sodium carboxymethylcellulose via mechanical and optical techniques

Sodium carboxymethylcellulose (NaCMC) in solution represents a complex rheological system, since it forms aggregates and associations and hence higher-level structures and, depending on the synthesis, is only found in a molecularly dispersed form in exceptional cases. Rheo-mechanical investigations of the viscoelasticity showed that the Cox-Merz rule is not fulfilled. The aim was therefore to examine whether rheo-optics could be employed to provide more detailed conclusions about the parameters that influence the flow behavior of NaCMC than has hitherto been available with mechanical methods. The flow birefringence, Δn ^′, rises as the degree of polymerization increases, and exhibits the same dependence on molar mass as does the viscosity: Δn ^′∝M _w ^3.4. As the degree of polymerization increases while the shear rate remains constant, the polymer segments become more distinctly aligned in the direction of shear. Hence increasing the degree of polymerization also affects the solution structure, i.e. the interaction of the molecules with one another. The stress-optical rule only applies to a limited extent for this system. The stress-optical coefficient, C, is almost independent of the shear rate, but is strongly influenced by the concentration and attains a limiting value of 3 × 10⁻⁸ Pa⁻¹. C was determined for a polymer in dilute solution and the curve obtained also enabled transitions in the solution structure to be recognized. Received: 1 May 1998 Accepted: 5 October 1998     

BV Solutions of the Semidiscrete Upwind Scheme

  We consider the semidiscrete upwind scheme

An Example of Finite-time Singularities in the 3d Euler Equations

Let be the exterior of the closed unit ball. Consider the self-similar Euler system

The effect of cold-work on the electrical resistivity of alloys and the law of recovery

Summary The change in resistivity due to cold-work has been determined for a number of binary alloy series having copper, silver or gold as basic metal. The effects observed are very different in the different alloy systems. A change as large as 42% of the total resistivity is found in one case. Alloys from the systems gold-chromium and gold-iron show a decrease of resistivity when cold-worked. In addition to the binary series, some alloys containing three or four components have also been investigated, and on the basis of the results obtained the question of the influence of the different solutes upon the change of resistivity is discussed. Results are given regarding the recovery of the resistivity changes as a function of time for different annealing temperatures. The resistivity change Δϱ as a function of temperatureT and time τ is found to obey the law Δϱ=cτ ⁿ exp (−nE/RT), wheren is a number of the order 0.1–0.4 andE is the activation energy of the process;R is the gas constant, andc a quantity of proportionality. Alloys belonging to the following binary systems have been investigated: Cu-Al, Cu-Si, Ag-Al, Ag-Mn, Ag-Sn, Au-Al, Au-Cr, Au-Mn, Au-Fe and Au-Sn, and the following multi-component systems: Au-Mn-Cr, Au-Mn-Cr-Fe and Au-Al-Cr.     

Experimental investigation on turbulence modification in a horizontal channel flow at relatively low mass loading

Particle-laden flows in a horizontal channel were investigated by means of a two-phase particle image velocimetry (PIV) technique. Experiments were performed at a Reynolds number of 6 826 and the flow is seeded with polythene beads of two sizes, 60 μm and 110 μm. One was slightly smaller than and the other was larger than the Kolmogorov length scale. The particle loadings were relatively low, with mass loading ratio ranging from 5×10⁻⁴ to 4×10⁻² and volume fractions from 6×10⁻⁷ to 4.8×10⁻⁵, respectively. The results show that the presence of particles can dramatically modify the turbulence even under the lowest mass loading ratio of 5×10⁻⁴. The mean flow is attenuated and decreased with increasing particle size and mass loading. The turbulence intensities are enhanced in all the cases concerned. With the increase of the mass loading, the intensities vary in a complicated manner in the case of small particles, indicating complicated particle-turbulence interactions; whereas they increase monotonously in the case of large particles. The particle velocities and concentrations are also given. The particles lag behind the fluid in the center region but lead in the wall region, and this trend is more prominent for the large particles. The streamwise particle fluctuations are larger than the gas fluctuations for both sizes of particles, however their varying trend with the mass loadings is not so clear. The wall-normal fluctuations increase with increasing mass loadings. They are smaller in the 60 μm particle case but larger in the 110 μm particle case than those of the gas phase. It seems that the small particles follow the fluid motion to certain extent while the larger particles are more likely dominated by their own inertia. Finally, remarkable non-uniform distributions of particle concentration are observed, especially for the large particles. The inertia of particles is proved to be very important for the turbulence modification and particles behaviors and thus should be considered in horizontal channels. The project supported by the National Natural Science Foundation of China (50276021), and Program for New Century Excellent Talents in University, Ministry of Education (NCET-04-0708) The English text was polished by Yunming Chen.     

A class of two-level explicit difference schemes for solving three dimensional heat conduction equation

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｉｓｐａｐｅｒｄｅａｌｓｗｉｔｈｔｈｅｉｎｉｔｉａｌ＿ｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｏｆｔｈｒｅｅ＿ｄｉｍｅｎｓｉｏｎａｌｈｅａｔｃｏｎｄｕｃｔｉｏｎｅｑｕａｔｉｏｎｉｎｔｈｅｒｅｇｉｏｎＤ :0≤ｘ,ｙ ,ｚ≤Ｌ ,0 ≤ｔ≤Ｔ ｕ ｔ= 2 ｕ ｘ2 2 ｕ ｙ2 2 ｕ ｚ2 ,ｕ｜ｘ=0 =ｆ1(ｙ,ｚ,ｔ) ,　ｕ｜ｘ=Ｌ =ｆ2 (ｙ ,ｚ,ｔ) ,ｕ｜ｙ=0 =ｇ1(ｚ,ｘ,ｔ) ,　ｕ｜ｙ=Ｌ =ｇ2 (ｚ,ｘ,ｔ) ,ｕ｜ｚ=0 =ｈ1(ｘ ,ｙ ,ｔ) ,　ｕ｜ｚ=Ｌ =ｈ2 (ｘ ,ｙ ,ｔ) ,ｕ｜ｔ=0 =φ(ｘ ,ｙ,ｚ) .(1 )(2 )…     

On the Development of a Mesoscale Friction Tester

MEMS and NEMS devices typically have a large surface area to volume ratio. As a result, a major concern in the development of such devices is friction. Contact radii in MEMS and NEMS devices are expected to range from 10⁻⁸<a<10⁻⁵ m. This regime, which generally lies between the limits of single asperity and macroscopic contact, has yet to be explored because the apparati used to characterize friction at these limits do not operate in the range of forces appropriate to these length scales. A Mesoscale Friction Tester (MFT) with smooth probe tip radii from 50 nm to 50 μm and capable of applying forces ranging from 10 nN to l mN over contact radii from 10 nm to 10 μm has been developed to address this need. With carefully planned experiments, this device has the potential to help answer unresolved questions regarding friction mechanisms in the mesoscale range.     

A unified approach to closed-form solutions of moving heat-source problems

A closed-form model for the computation of temperature distribution in an infinitely extended isotropic body with a time-dependent moving-heat sources is discussed. The temperature solutions are presented for the sources of the forms: (i) ₀₁(t)=₀ exp(−λt), (ii) ₀₂(t) =₀(t/t ^*)exp(−λt), and ₀₃(t)=₀[1+a cos(ωt)], where λ and ω are real parameters and t ^* characterizes the limiting time. The reduced (or dimensionless) temperature solutions are presented in terms of the generalized representation of an incomplete gamma function Γ(α,x;b) and its decomposition C _Γ and S _Γ. The solutions are presented for moving, -point, -line, and -plane heat sources. It is also demonstrated that the present analysis covers the classical temperature solutions of a constant strength source under quasi-steady state situations. Received on 13 June 1997     

Variational optical flow estimation for particle image velocimetry

We introduce a novel class of algorithms for evaluating PIV image pairs. The mathematical basis is a continuous variational formulation for globally estimating the optical flow vector fields over the whole image. This class of approaches has been known in the field of image processing and computer vision for more than two decades but apparently has not been applied to PIV image pairs so far. We pay particular attention to a multi-scale representation of the image data so as to cope with the quite specific signal structure of particle image pairs. The experimental evaluation shows that a prototypical variational approach competes in noisy real-world scenarios with three alternative approaches especially designed for PIV-sequence evaluation. We outline the potential of the variational method for further developments.The publications of the CVGPR Group are listed under .

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Dynamical deformation of a flat liquid–liquid interface

The passage of solid spheres through a liquid–liquid interface was experimentally investigated using a high-speed video and PIV (particle image velocimetry) system. Experiments were conducted in a square Plexiglas column of 0.1 m. The Newtonian Emkarox (HV45 50 and 65% wt) aqueous solutions were employed for the dense phase, while different silicone oils of different viscosity ranging from 10 to 100 mPa s were used as light phase. Experimental results quantitatively reveal the effect of the sphere’s size, interfacial tension and viscosity of both phases on the retaining time and the height of the liquid entrained behind the sphere. These data were combined with our previous results concerning the passage of a rising bubble through a liquid–liquid interface in order to propose a general relationship for the interface breakthrough for the wide range of Mo ₁/Mo ₂ ∈ [2 × 10⁻⁵–5 × 10⁴] and Re ₁/Re ₂ ∈ [2 × 10⁻³–5 × 10²].     

The Logarithmic Tail of Néel Walls

  We study the multiscale problem of a parametrized planar 180° rotation of magnetization states in a thin ferromagnetic film. In an appropriate scaling and when the film thickness is comparable to the Bloch line width, the underlying variational principle has the form

Spatial resolution of PIV for the measurement of turbulence

Recent technological advancements have made the use of particle image velocimetry (PIV) more widespread for studying turbulent flows over a wide range of scales. Although PIV does not threaten to make obsolete more mature techniques, such as hot-wire anemometry (HWA), it is justifiably becoming an increasingly important tool for turbulence research. This paper assesses the ability of PIV to resolve all relevant scales in a classical turbulent flow, namely grid turbulence, via a comparison with theoretical predictions as well as HWA measurements. Particular attention is given to the statistical convergence of mean turbulent quantities and the spatial resolution of PIV. An analytical method is developed to quantify and correct for the effect of the finite spatial resolution of PIV measurements. While the present uncorrected PIV results largely underestimate the mean turbulent kinetic energy and energy dissipation rate, the corrected measurements agree to a close approximation with the HWA data. The transport equation for the second-order structure function in grid turbulence is used to establish the range of scales affected by the limited resolution. The results show that PIV, due to the geometry of its sensing domain, must meet slightly more stringent requirements in terms of resolution, compared with HWA, in order to provide reliable measurements in turbulence.

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