The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

This paper describes the principles of a novel 3D PIV system based on the illumination, recording and reconstruction of tracer particles within a 3D measurement volume The technique makes use of several simultaneous views of the illuminated particles and their 3D reconstruction as a light intensity distribution by means of optical tomography The technique is therefore referred to as tomographic particle image velocimetry (tomographic-PIV) The reconstruction is performed with the MART algorithm, yielding a 3D array of light intensity discretized over voxels The reconstructed tomogram pair is then analyzed by means of 3D cross-correlation with an iterative multigrid volume deformation technique, returning the three-component velocity vector distribution over the measurement volume The principles and details of the tomographic algorithm are discussed and a parametric study is carried out by means of a computer-simulated tomographic-PIV procedure The study focuses on the accuracy of the light intensity field reconstruction process The simulation also identifies the most important parameters governing the experimental method and the tomographic algorithm parameters, showing their effect on the reconstruction accuracy A computer simulated experiment of a 3D particle motion field describing a vortex ring demonstrates the capability and potential of the proposed system with four cameras The capability of the technique in real experimental conditions is assessed with the measurement of the turbulent flow in the near wake of a circular cylinder at Reynolds 2,700

Tomographic particle image velocimetry

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

A direct numerical simulation of incompressible channel flow at a friction Reynolds number ( ) of 5186 has been performed, and the flow exhibits a number of the characteristics of high-Reynolds-number wall-bounded turbulent flows. For example, a region where the mean velocity has a logarithmic variation is observed, with von Karman constant . There is also a logarithmic dependence of the variance of the spanwise velocity component, though not the streamwise component. A distinct separation of scales exists between the large outer-layer structures and small inner-layer structures. At intermediate distances from the wall, the one-dimensional spectrum of the streamwise velocity fluctuation in both the streamwise and spanwise directions exhibits dependence over a short range in wavenumber . Further, consistent with previous experimental observations, when these spectra are multiplied by (premultiplied spectra), they have a bimodal structure with local peaks located at wavenumbers on either side of the range.

Direct numerical simulation of turbulent channel flow up to

A survey is given of the major developments in three-dimensional velocity field measurements using the tomographic particle image velocimetry (PIV) technique. The appearance of tomo-PIV dates back seven years from the present review (Elsinga et al 2005a 6th Int. Symp. PIV (Pasadena, CA)) and this approach has rapidly spread as a versatile, robust and accurate technique to investigate three-dimensional flows (Arroyo and Hinsch 2008 Topics in Applied Physics vol 112 ed A Schroder and C E Willert (Berlin: Springer) pp 127–54) and turbulence physics in particular. A considerable number of applications have been achieved over a wide range of flow problems, which requires the current status and capabilities of tomographic PIV to be reviewed. The fundamental aspects of the technique are discussed beginning from hardware considerations for volume illumination, imaging systems, their configurations and system calibration. The data processing aspects are of uppermost importance: image pre-processing, 3D object reconstruction and particle motion analysis are presented with their fundamental aspects along with the most advanced approaches. Reconstruction and cross-correlation algorithms, attaining higher measurement precision, spatial resolution or higher computational efficiency, are also discussed. The exploitation of 3D and time-resolved (4D) tomographic PIV data includes the evaluation of flow field pressure on the basis of the flow governing equation. The discussion also covers a-posteriori error analysis techniques. The most relevant applications of tomo-PIV in fluid mechanics are surveyed, covering experiments in air and water flows. In measurements in flow regimes from low-speed to supersonic, most emphasis is given to the complex 3D organization of turbulent coherent structures.

Tomographic PIV: principles and practice

Particle image velocimetry (PIV) has evolved to be the dominant method for velocimetry in experimental fluid mechanics and has contributed to many advances in our understanding of turbulent and complex flows. In this article we review the achievements of PIV and its latest implementations: time-resolved PIV for the rapid capture of sequences of vector fields; tomographic PIV for the capture of fully resolved volumetric data; and statistical PIV, designed to optimize measurements of mean statistical quantities rather than instantaneous fields. In each implementation, the accuracy and spatial resolution are limited. To advance the method to the next level, we need a completely new approach. We consider the fundamental limitations of two-pulse PIV in terms of its dynamic ranges. We then discuss new paths and developments that hold the promise of achieving a fundamental reduction in uncertainty.

Particle Image Velocimetry for Complex and Turbulent Flows

An experimental study is carried out to investigate the three-dimensional instantaneous structure of an incident shock wave/turbulent boundary layer interaction at Mach 2.1 using tomographic particle image velocimetry. Large-scale coherent motions within the incoming boundary layer are observed, in the form of three-dimensional streamwise-elongated regions of relatively low- and high-speed fluid, similar to what has been reported in other supersonic boundary layers. Three-dimensional vortical structures are found to be associated with the low-speed regions, in a way that can be explained by the hairpin packet model. The instantaneous reflected shock wave pattern is observed to conform to the low- and high-speed regions as they enter the interaction, and its organization may be qualitatively decomposed into streamwise translation and spanwise rippling patterns, in agreement with what has been observed in direct numerical simulations. The results are used to construct a conceptual model of the three-dimensional unsteady flow organization of the interaction.

/pdf/three-dimensional-instantaneous-structure-of-a-shock-wave-n1fsupdwce.pdf

Three-dimensional instantaneous structure of a shock wave/turbulent boundary layer interaction

Tomographic particle image velocimetry was used to quantitatively visualize the three-dimensional coherent structures in a supersonic (Mach 2) turbulent boundary layer in the region between y/? = 0.15 and 0.89. The Reynolds number based on momentum thickness Re? = 34000. The instantaneous velocity fields give evidence of hairpin vortices aligned in the streamwise direction forming very long zones of low-speed fluid, consistent with Tomkins & Adrian (J. Fluid Mech., vol. 490, 2003, p. 37). The observed hairpin structure is also a statistically relevant structure as is shown by the conditional average flow field associated to spanwise swirling motion. Spatial low-pass filtering of the velocity field reveals streamwise vortices and signatures of large-scale hairpins (height > 0.5?), which are weaker than the smaller scale hairpins in the unfiltered velocity field. The large-scale hairpin structures in the instantaneous velocity fields are observed to be aligned in the streamwise direction and spanwise organized along diagonal lines. Additionally the autocorrelation function of the wall-normal swirling motion representing the large-scale hairpin structure returns positive correlation peaks in the streamwise direction (at 1.5? distance from the DC peak) and along the 45° diagonals, which also suggest a periodic arrangement in those directions. This is evidence for the existence of a spanwise–streamwise organization of the coherent structures in a fully turbulent boundary layer.

/pdf/three-dimensional-vortex-organization-in-a-high-reynolds-y8jb2yqin5.pdf

Three-dimensional vortex organization in a high-Reynolds-number supersonic turbulent boundary layer

In this feasibility study the tomographic PIV technique has been applied to time resolved PIV recordings for the study of the growth of a turbulent spot in a laminar flat plate boundary layer and to visualize the topology of coherent flow structures within a tripped turbulent flat plate boundary layer flow. The experiments are performed around (Re
 x
 )1/2 ≈ 450 in a low speed wind-tunnel using four high speed CMOS cameras operating up to 5 kHz. The volume illumination required a multiple-reflection system able to intensify light intensity within the measurement volume. This aspect is deemed essential when a high-speed tomographic PIV system is applied in air flows. The particle image recordings are used for a three dimensional tomographic reconstruction of the light intensity distribution within the illuminated volume. Each pair of reconstructed three-dimensional light distributions is analyzed by 3D spatial cross-correlation using iterative multi-grid schemes with volume-deformation, yielding a correlated time sequence of three-dimensional instantaneous velocity vector volumes. The coherent structures organization is analyzed by 3D-vorticity and -swirling-strength iso-surfaces visualization. In both flow types streaks and hairpin-like or arch vortical structures are most prominent. The data gives insight into the role of these structures for the spatio-temporal arrangement of the wall normal flow exchange mechanisms, especially of the instantaneous Reynolds stress events Q2 and Q4. A description of different self-sustainable flow organizations based on modifications of the hairpin-vortex- and streak-models is given. Two preliminary results are essential: Self-sustainability of a coherent vortical structure depends on the ability to entrain high momentum fluid, initially Q4. And, stream-wise swirl at the near-wall region of arch or hairpin-like vortices has been observed to be rare.

/pdf/investigation-of-a-turbulent-spot-and-a-tripped-turbulent-2gnn553jyp.pdf

Gerrit E. Elsinga

Papers

Tomographic particle image velocimetry

Particle Image Velocimetry for Complex and Turbulent Flows

Three-dimensional instantaneous structure of a shock wave/turbulent boundary layer interaction

Three-dimensional vortex organization in a high-Reynolds-number supersonic turbulent boundary layer

Investigation of a turbulent spot and a tripped turbulent boundary layer flow using time-resolved tomographic PIV