To improve the performance of particle image velocimetry in measuring instantaneous velocity fields, direct cross-correlation of image fields can be used in place of auto-correlation methods of interrogation of double- or multiple-exposure recordings. With improved speed of photographic recording and increased resolution of video array detectors, cross-correlation methods of interrogation of successive single-exposure frames can be used to measure the separation of pairs of particle images between successive frames. By knowing the extent of image shifting used in a multiple-exposure and by a priori knowledge of the mean flow-field, the cross-correlation of different sized interrogation spots with known separation can be optimized in terms of spatial resolution, detection rate, accuracy and reliability.

Theory of cross-correlation analysis of PIV images : Image analysis as measuring technique in flows

We review wall-bounded turbulent flows, particularly high–Reynolds number, zero–pressure gradient boundary layers, and fully developed pipe and channel flows. It is apparent that the approach to an asymptotically high–Reynolds number state is slow, but at a sufficiently high Reynolds number the log law remains a fundamental part of the mean flow description. With regard to the coherent motions, very-large-scale motions or superstructures exist at all Reynolds numbers, but they become increasingly important with Reynolds number in terms of their energy content and their interaction with the smaller scales near the wall. There is accumulating evidence that certain features are flow specific, such as the constants in the log law and the behavior of the very large scales and their interaction with the large scales (consisting of vortex packets). Moreover, the refined attached-eddy hypothesis continues to provide an important theoretical framework for the structure of wall-bounded turbulent flows.

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High–Reynolds Number Wall Turbulence

Wall-bounded turbulent flows at high Reynolds numbers have become an increasingly active area of research in recent years. Many challenges remain in theory, scaling, physical understanding, experimental techniques, and numerical simulations. In this paper we distill the salient advances of recent origin, particularly those that challenge textbook orthodoxy. Some of the outstanding questions, such as the extent of the logarithmic overlap layer, the universality or otherwise of the principal model parameters such as the von Karman “constant,” the parametrization of roughness effects, and the scaling of mean flow and Reynolds stresses, are highlighted. Research avenues that may provide answers to these questions, notably the improvement of measuring techniques and the construction of new facilities, are identified. We also highlight aspects where differences of opinion persist, with the expectation that this discussion might mark the beginning of their resolution.

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Wall-bounded turbulent flows at high Reynolds numbers: Recent advances and key issues

This thesis deals with the problem of high Reynolds number zero pressuregradient turbulent boundary layers in an incompressible flow without any effects of heat-transfer. The zero-pressure gradient turbulent boundary layer is one of the canonical shear flows important in many applications and of large theoretical interest. The investigation was carried out through an experimental study in the MTL wind-tunnel at KTH, where the fluctuating velocity components and the fluctuating wall-shear stress in a turbulent boundary layer were measured using hot-wire and hot-film anemometry. Attempts were made to answer some basic and “classical” questions concerning turbulent boundary boundary layers. The classical two layer theory was confirmed and constant values of the slope of the logarithmic overlap region (i.e. the von Karman constant) and the additive constants were found and estimated to κ = 0.38, B = 4.1 and B1 = 3.6 (δ = δ95). The inner limit of overlap region was found to scale on the viscous length scale (ν/uτ) and was estimated to be y = 200, i.e. considerably further out compared to previous knowledge. The outer limit of the overlap region was found to scale on the outer length scale and was estimated to be y/δ = 0.15. This also means that a universal overlap region only can exist for Reynolds numbers of at least Reθ ≈ 6000. The values of the newly determined limits explain the Reynolds number variation found in some earlier experiments. Measurements of the fluctuating wall-shear stress using the hot-wire-onthe-wall technique and a MEMS hot-film sensor show that the turbulence intensity τr.m.s./τw is close to 0.41 at Reθ ≈ 9800. A numerical and experimental investigation of the behavior of double wire probes were carried out and showed that the Peclet number based on wire separation should be larger than about 50 to ensure an acceptably low level of thermal interaction. Results are presented for two-point correlations between the wall-shear stress and the streamwise velocity component for separations in both the wallnormal-streamwise plane and the wall-normal-spanwise plane. Turbulence producing events are further investigated using conditional averaging of isolated shear-layer events. Comparisons are made with results from other experiments and numerical simulations. Descriptors: Fluid mechanics, turbulence, boundary layers, high Reynolds number, zero-pressure gradient, hot-wire, hot-film anemometry, oil-film interferometry, structures, streak spacing, micro-electro-mechanical-systems.

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Experimental studies of zero pressure-gradient turbulent boundary layer flow

The friction factor relationship for high-Reynolds-number fully developed turbulent pipe flow is investigated using two sets of data from the Princeton Superpipe in the range 31×10^3 ≤ ReD ≤ 35×10^6. The constants of Prandtl’s ‘universal’ friction factor relationship are shown to be accurate over only a limited Reynolds-number range and unsuitable for extrapolation to high Reynolds numbers. New constants, based on a logarithmic overlap in the mean velocity, are found to represent the high-Reynolds-number data to within 0.5%, and yield a value for the von Karman constant that is consistent with the mean velocity profiles themselves. The use of a generalized logarithmic law in the mean velocity is also examined. A general friction factor relationship is proposed that predicts all the data to within 1.4% and agrees with the Blasius relationship for low Reynolds numbers to within 2.0%.

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A new friction factor relationship for fully developed pipe flow

New scaling laws for turbulent boundary layers recently derived (see Oberlack 2000) using Lie group symmetry methods have been tested against experimental data from the KTH database for zero-pressure-gradient turbulent boundary layers. The most significant new law predicts an exponential variation of the mean velocity defect in the outer (wake) region. It was shown to fit the experimental data very well over a large part of the boundary layer, from the outer part of the overlap region to about half the boundary layer thickness ( $\delta_{99}$ ). In the outermost part of the boundary layer the velocity defect falls more rapidly than predicted by the exponential law. This can partly be attributed to intermittency in that region but the main cause stems from non-parallel effects that are not accounted for in the derivation of the exponential law. The two-point correlation function behaviour in the outer region, where an exponential velocity defect law is observed, was found to be very different from that derived under the assumption of parallel flow. It is found to be plausible that this indeed can be attributed to non-parallel effects. A small modification of the innermost part of the log-layer in the form of an additive constant within the log-function is predicted by the Lie group symmetry method. A qualitative agreement with such a behaviour just below the overlap region was found. The derived scaling law behaviour in the overlap region for the two-point correlation functions was also verified by the experimental data.

Evaluation of scaling laws derived from Lie group symmetry methods in zero-pressure-gradient turbulent boundary layers

Numerical and experimental study of separated flow in a plane asymmetric diffuser

The probability density functions (PDFs) of the streamwise mean velocity in high Reynolds number turbulent boundary layers have been studied. The hypothesis of self-similar, normalized with the root mean square velocity, PDFs has been tested using the KTH database of high Reynolds number zero pressure-gradient turbulent boundary layer flow. The self-similarity was tested using the Kullback–Leibler divergence measure and it was found that the region of self-similar PDFs extends from about 160 viscous wall units to about 0.3 boundary layer thicknesses (in δ95). This region is somewhat larger than the universal overlap region. The PDF shape in the universal overlap region is close to Gaussian allowing for a Gram–Charlier expansion approximation of the measured PDFs. A remarkable collapse was found for 57 normalized PDF distributions for different positions within the universal overlap region and Reynolds numbers based on the momentum-loss thickness between 4300 and 12 600, strongly indicating a high degree o...

Universality of probability density distributions in the overlap region in high Reynolds number turbulent boundary layers

A new low-speed, closed-circuit, closed-test-section wind tunnel, called BLT, has been designed and built at KTH. The turbulence intensity in the test section is <0.04%, the total pressure varia ...

Evaluation of a new wind tunnel with expanding corners

The near-wall region of zero-pressure gradient turbulent boundary layers was studied through correlation- and other two-point measurements over a wide range of Reynolds numbers. The requirements of high spatial resolution were met by use of a MEMS-type of hot-film sensor array together with a small, in-house built hot-wire probe. Streak-spacing and characteristics of buffer region shear-layer events were studied. At high Reynolds numbers the motions that are of substantially larger scale than the streaks have a significant influence on the near-wall dynamics. By removing such scales through high-pass filtering a streak spacing was recovered that is close to that found in low Reynolds number flows. The frequency of occurrence of shear-layer events was found to scale with a mixed time scale, in analogy with earlier findings in channel flow, again indicating the increasing relative influence of large scales with increasing Reynolds number.

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Björn Lindgren

Papers

Evaluation of scaling laws derived from Lie group symmetry methods in zero-pressure-gradient turbulent boundary layers

Numerical and experimental study of separated flow in a plane asymmetric diffuser

Universality of probability density distributions in the overlap region in high Reynolds number turbulent boundary layers

Evaluation of a new wind tunnel with expanding corners

Flow structures in zero pressure-gradient turbulent boundary layers at high Reynolds numbers