Influence of the spatial resolution on fine-scale features in DNS of turbulence generated by a single square grid

We focus in this paper on the effect of the resolution of direct numerical simulations (DNS) on the spatio-temporal development of the turbulence downstream of a single square grid. The aims of this study are to validate our numerical approach by comparing experimental and numerical one-point statistics downstream of a single square grid and then investigate how the resolution is impacting the dynamics of the flow. In particular, using the Q–R diagram, we focus on the interaction between the strain-rate and rotation tensors, the symmetric and skew-symmetric parts of the velocity gradient tensor, respectively. We first show good agreement between our simulations and hot-wire experiment for one-point statistics on the centreline of the single square grid. Then, by analysing the shape of the Q–R diagram for various streamwise locations, we evaluate the ability of under-resolved DNS to capture the main features of the turbulence downstream of the single square grid.     

Scaling of Conditional Lagrangian Time Correlation Functions of Velocity and Pressure Gradient Magnitudes in Isotropic Turbulence

We study Lagrangian statistics of the magnitudes of velocity and pressure gradients in isotropic turbulence by quantifying their correlation functions and their characteristic time scales. In a recent work (Yu and Meneveau, Phys Rev Lett 104:084502, 2010), it has been found that the Lagrangian time-correlations of the velocity and pressure gradient tensor and vector elements scale with the locally-defined Kolmogorov time scale, evaluated from the locally-averaged dissipation-rate (? _r ) and viscosity (ν) according to $\tau_{K,r}=\sqrt{\nu/\epsilon_r}$ . In this work, we study the Lagrangian time-correlations of the absolute values of velocity and pressure gradients. It has long been known that such correlations display longer memories into the inertial-range as well as possible intermittency effects. We explore the appropriate temporal scales with the aim to achieve collapse of the correlation functions. The data used in this study are sampled from the web-services accessible public turbulence database (http://turbulence.pha.jhu.edu). The database archives a 1024⁴ (space+time) pseudo-spectral direct numerical simulation of forced isotropic turbulence with Taylor-scale Reynolds number Re _λ ?=?433, and supports spatial differentiation and spatial/temporal interpolation inside the database. The analysis shows that the temporal auto-correlations of the absolute values extend deep into the inertial range where they are determined not by the local Kolmogorov time-scale but by the local eddy-turnover time scale defined as $\tau_{e,r}= r^{2/3}\epsilon_r^{-1/3}$ . However, considerable scatter remains and appears to be reduced only after a further (intermittency) correction factor of the form of (r/L) ^χ is introduced, where L is the turbulence integral scale. The exponent χ varies for different variables. The collapse of the correlation functions for absolute values is, however, less satisfactory than the collapse observed for the more rapidly decaying strain-rate tensor element correlation functions in the viscous range.     

Response of mean turbulent energy dissipation rate and spectra to concentrated wall suction

The response of mean turbulent energy dissipation rate and spectra to concentrated suction applied through a porous wall strip has been quantified. Both suction and no suction data of the spectra collapsed reasonably well for Kolmogorov normalised wavenumber k ₁^* > 0.2. Similar results were also observed for second-order structure functions (not shown) for Kolmogorov normalised radius r* < 10. Although, the quality of collapsed is poorer for transverse component, the result highlights that Kolmogorov similarity hypothesis is reasonably well satisfied. However, the suction results shows a significant departure from the no suction case of the Kolmogorov normalised spectra and second-order structure functions for k ₁^* < 0.2 and r* > 20, respectively. The departure at the larger scales with collapse at the small scales suggests that suction induce a change in the small-scale motion. This is also reflected in the alteration of mean turbulent energy dissipation rate and Taylor microscale Reynolds number. This change is a result of the weakening of the large-scale structures. The effect is increased as the suction rate is increased.     

Kinematics and dynamics of small-scale vorticity and strain-rate structures in the transition from isotropic to shear turbulence

We applied a technique that defines and extracts “structures” from a DNS dataset of a turbulence variable in a way that allows concurrent quantitative and visual analysis. Local topological and statistical measures of enstrophy and strain-rate structures were compared with global statistics to determine the role of mean shear in the dynamical interactions between fluctuating vorticity and strain-rate during transition from isotropic to shear-dominated turbulence. We find that mean shear adjusts the alignment of fluctuating vorticity, fluctuating strain-rate in principal axes, and mean strain-rate in a way that (1) enhances both global and local alignments between vorticity and the second eigenvector of fluctuating strain-rate, (2) two-dimensionalizes fluctuating strain-rate, and (3) aligns the compressional components of fluctuating and mean strain-rate. Shear causes amalgamation of enstrophy and strain-rate structures, and suppresses the existence of strain-rate structures in low-vorticity regions between enstrophy structures. A primary effect of shear is to enhance “passive” strain-rate fluctuations, strain-rate kinematically induced by local vorticity concentrations with negligible enstrophy production, relative to “active,” or vorticity-generating strain-rate fluctuations. Enstrophy structures separate into “active” and “passive” based on the level of the second eigenvalue of fluctuating strain-rate. We embedded the structure-extraction algorithm into an interactive visualization-based analysis system from which the time evolution of a shear-induced hairpin enstrophy structure was visually and quantitatively analyzed. The structure originated in the initial isotropic state as a vortex sheet, evolved into a vortex tube during a transitional period, and developed into a well-defined horseshoe vortex in the shear-dominated asymptotic state.     

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.     

Spectral Vorticity and Lagrangian Velocity Measurements in Turbulent Jets

In this paper we report an experimental investigation of various statistical properties of the spatial Fourier modes of the vorticity field in turbulent jets for a large range of Reynolds numbers (530 ≤R _λ≤ 6100). The continuous time evolution of a spatial Fourier mode of the vorticity distribution, characterized by a well-defined wavevector, is obtained from acoustic scattering measurements. The spatial enstrophy spectrum, as a function of the spatial wave-vector, is determined by scanning the incoming sound frequencies. Time-frequency analysis of the turbulent vorticity fluctuations is also performed for different length scales of the flows. Vorticity time-correlations show that the characteristic time of a Fourier mode behaves as the sweeping time. Finally, we report preliminary Lagrangian velocity measurements obtained using acoustic scattering by soap bubbles inflated with helium. Gathering a large number of passages of isolated bubbles in the scattering volume, one is able to compute the Lagrangian velocity PDF and velocity spectrum. Despite the spatial filtering due to the finite size of the bubble, the latter exhibits a power law, with the -2 exponent predicted by the Kolmogorov theory, over one decade of frequencies.     

Reynolds number dependence of the inertial range scaling of energy dissipation rate and enstrophy in a cylinder wake

The full energy dissipation rate and enstrophy are measured simultaneously using a probe consisting of four X-wires in the intermediate region of a cylinder wake for Taylor microscale Reynolds number in the range of 120–320. Longitudinal and transverse velocity increments are also obtained temporally using Taylor’s hypothesis. The inertial range scaling exponents indicate that the full enstrophy field has a stronger intermittency than does the full dissipation field for all the Reynolds numbers considered. The approximations of the energy dissipation rate and enstrophy based on isotropy are more intermittent than their corresponding true values. While the scaling exponents of the full energy dissipation rate remain approximately constant for different Reynolds numbers, those of the enstrophy decrease slightly and consistently with the increase of Reynolds number. It is conjectured that the scaling of the energy dissipation rate and the enstrophy may be the same when Reynolds number is extremely high, a trend that is consistent with that suggested by Nelkin (Phys Fluids 11:2202–2204, 1999; Am J Phys 68:310–318, 2000).     

Vorticity dynamics in turbulence growth

The mechanisms of vorticity amplification in the formation of turbulence are investigated by means of direct numerical simulations of the Navier–Stokes equations with different initial conditions and Reynolds numbers. The simulations show good universality of the enstrophy evolution, that occurs in two stages. The first stage is dominated by the effect of vortex stretching, and it finishes with a k ^?3 power-law energy spectrum. The second stage is dominated by the action of viscosity on the small scales, and it finishes with a Kolmogorov k ^?5/3 energy spectrum.     

Assessment of tomographic PIV in wall-bounded turbulence using direct numerical simulation data

Simulations of tomographic particle image velocimetry (Tomo-PIV) are performed using direct numerical simulation data of a channel flow at Reynolds number of Re _τ = 934, to investigate the influence of experimental parameters such as camera position, seeding density, interrogation volume size and spatial resolution. The simulations employ camera modelling, a Mie scattering illumination model, lens distortion effects and calibration to realistically model a tomographic experiment. Results are presented for camera position and orientation in three-dimensional space, to obtain an optimal reconstruction quality. Furthermore, a quantitative analysis is performed on the accuracy of first and second order flow statistics, at various voxel sizes normalised using the viscous inner length scale. This enables the result to be used as a general reference for wall-bounded turbulent experiments. In addition, a ratio relating seeding density and the interrogation volume size is proposed to obtain an optimal reference value that remains constant. This can be used to determine the required seeding density concentration for a certain interrogation volume size.     

A tomographic PIV resolution study based on homogeneous isotropic turbulence DNS data

In order to accurately assess measurement resolution and measurement uncertainty in DPIV and TPIV measurements, a series of simulations were conducted based on the flow field from a homogeneous isotropic turbulence data set (Re _λ = 141). The effect of noise and spatial resolution was quantified by examining the local and global errors in the velocity, vorticity and dissipation fields in addition to other properties of interest such as the flow divergence, topological invariants and energy spectra. In order to accurately capture the instantaneous gradient fields and calculate sensitive quantities such as the dissipation rate, a minimum resolution of x/η = 3 is required, with smoothing recommended for the TPIV results to control the inherently higher noise levels. Comparing these results with experimental data showed that while the attenuation of velocity and gradient quantities was predicted well, higher noise levels in the experimental data led increased divergence.     

Effects of the Local Flow Topologies Upon the Structure of a Premixed Methane-air Turbulent Jet Flame

Local flow topologies have been identified and their interactions with the iso-scalar surfaces geometries have been investigated using the results of a three-dimensional direct numerical simulation (DNS) of a turbulent premixed methane-air flame in a piloted Bunsen burner configuration with tabulated chemistry. The universal teardrop shape of the joint probability density function (jpdf) of the second and third invariants of the velocity-gradient tensor disappears in the different flame regions under study. A ‘canonical’ vortex, which affects the fine-scale structure of the turbulent premixed flame, has been identified and analyzed at three times, differing by increments of the order of the Kolmogorov time micro-scale.     

Stereoscopic PIV measurements of a turbulent boundary layer with a large spatial dynamic range

The flow in a streamwise/wall-normal plane of a turbulent boundary layer at moderate Reynolds number (Re _θ = 2,200) is characterized using two stereo PIV systems just overlapping in the streamwise direction. The aim is to generate SPIV data for near-wall turbulence with enough spatial dynamic range to resolve most of the coherent structures present in the flow and to facilitate future comparisons with direct numerical simulations. This is made possibly through the use of four cameras with large CCD arrays (4,008 px × 2,672 px) and through a rigorous experimental procedure designed to minimize the impact of measurement noise on the resolution of the small scales. For the first time, both a large field of view [S _x ; S _y ] = [2.6δ; 0.75δ] and a high spatial resolution (with an interrogation window size of 13.6⁺) have been achieved. The quality of the data is assessed through an analysis of some of the statistical results such as the mean velocity profile, the rms and the PDF of the fluctuations, and the power spectra.     

An automated vortex detection scheme using the wavelet transform of the<Emphasis Type="Italic"> d</Emphasis><Subscript>2</Subscript> field

A vortex detection scheme using the wavelet transform of the discriminant of the eigenvalues of the velocity gradient matrix (d ₂) is presented in this paper. The use of d ₂ field results in better eduction capability over the previously used test fields such as enstrophy since it automatically distinguishes shear layers from vortices. Level sets are used to refine the shape of the vortex without causing a huge computational penalty. Further, the algorithm is easily automated to aid batch processing. The detection scheme was applied to swirl flow fields, successfully estimating the vortex location, the core radius and the vortex shape.     

Determination of complete velocity gradient tensor by using cinematographic stereoscopic PIV in a turbulent jet

Cinematographic stereoscopic PIV measurements were performed in the far field of an axisymmetric co-flowing turbulent round jet (Re _T ≈ 150, where Re _T is the Reynolds number based on Taylor micro scale) to resolve small and intermediate scales of turbulence. The time-resolved three-component PIV measurements were performed in a plane normal to the axis of the jet and the data were converted to quasi-instantaneous three-dimensional (volumetric) data by using Taylor’s hypothesis. The availability of the quasi-three-dimensional data enabled the computation of all nine components of the velocity gradient tensor over a volume. The use of Taylor’s hypothesis was validated by performing a separate set of time-resolved two component “side-view” PIV measurements in a plane along the jet axis. Probability density distributions of the velocity gradients computed using Taylor’s hypothesis show good agreement with those computed directly with the spatially resolved data. The overall spatial structure of the gradients computed directly exhibits excellent similarity with that computed using Taylor’s hypothesis. The accuracy of the velocity gradients computed from the pseudo-volume was assessed by computing the divergence error in the flow field. The root mean square (rms) of the divergence error relative to the magnitude of the velocity gradient tensor was found to be 0.25, which is consistent with results based on other gradient measurement techniques. The velocity gradients, vorticity components and mean dissipation in the self-similar far field of the jet were found to satisfy the axisymmetric isotropy conditions. The divergence error present in the data is attributed to the intrinsic uncertainty associated with performing stereoscopic PIV measurements and not to the use of Taylor’s hypothesis. The divergence error in the data is found to affect areas of low gradient values and manifests as nonphysical values for quantities like the normalized eigenvalues of the strain-rate tensor. However, the high gradients are less affected by the divergence error and so it can be inferred that structural features of regions of intense vorticity and dissipation will be faithfully rendered.     

On pre-dissipative ‘bumps' and a Reynolds-number-dependent spectral parameterization of turbulence

Considerable experimental, numerical and theoretical evidence has accumulated during the last two decades for the presence of a maximum above the right end of the inertial plateau in compensated high-Reynolds-number turbulence spectra k^+5/3E(k). This energy pileup, due to the reduced nonlocal triadic interactions near the viscous cut-off, complies with Kolmogorov's 1941 theory but hampers experimental interpretation about its intermittency corrections. It has been included in a semi-empirical Reynolds-number-dependent complete (i.e. from the largest to the smallest scales) spectral model of isotropic turbulence. This simple parameterization is shown to represent satisfactorily well experimental data over a large variety of situations.     

Effect of particle number density in in-line digital holographic particle velocimetry

A digital in-line holographic particle tracking velocimetry (HPTV) system was developed to measure 3D (three-dimensional) velocity fields of turbulent flows. The digital HPTV (DHPTV) procedure consists of four steps: recording, numerical reconstruction, particle extraction and velocity extraction. In the recording step, a digital CCD camera was used as a recording device. Holograms contained many unwanted images or noise. To get clean holograms, digital image processing techniques were adopted. In the velocity extraction routine, we improved the HPTV algorithm to extract 3D displacement information of tracer particles. In general, the results obtained using HPTV were not fully acceptable due to technical limitations such as low spatial resolution, small volume size, and low numerical aperture (NA). The problems of spatial resolution and NA are closely related with a recording device. As one experimental parameter that can be optimized, we focused on the particle number density. Variation of the reconstruction efficiency and recovery ratio were compared quantitatively with varying particle number density to check performance of the developed in-line DHPTV system. The reconstruction efficiency represented the particle number distribution acquired through the numerical reconstruction procedure. In addition the recovery ratio showed the performance of 3D PTV algorithm employed for DHPTV measurements. The particle number density in the range of C _o = 13–17 particles/mm³ was found to be optimum for the DHPTV system tested in this study.     

KOLMOGOROV BEHAVIOR OF NEAR-WALL TURBULENCE AND ITS APPLICATION IN TURBULENCE MODELING

The near-wall behavior of turbulence is re-examined in a way different from that proposed by Hanjalic and Launder¹ and followers^2,3,4,5. It is shown that at a certain distance from the wall, all energetic large eddies will reduce to Kolmogorov eddies (the smallest eddies in turbulence). All the important wall parameters, such as friction velocity, viscous length scale, and mean strain rate at the wall, are characterised by Kolmogorov microscales. According t o this Kolmogorov behavior of near-wall turbulence, the turbulence quantities, such as turbulent kinetic energy, dissipation rate, etc. at the location where the large eddies become “Kolmogorov” eddies, can be estimated by using both direct numerical simulation (DNS) data and asymptotic analysis of near-wall turbulence. This information will provide useful boundary conditions for the turbulent transport equations. As a n example, the concept is incorporated in the standard κ - εmodel which is then applied t o channel and boundary layer flows. Using appropriate boundary conditions (based on Kolmogorov behaviour of near-wall turbulence), there is no need for any wall-modification to the κ - ε equations (including model constants). Results compare very well with the DNS and experimental data.     

Development and assessment of a single-image fringe projection method for dynamic applications

A single-image fringe projection profiling method suitable for dynamic applications was developed by combining an accurate camera calibration procedure and improved phase extraction procedures. The improved phase extraction process used a modified Hilbert transform with Laplacian pyramid algorithms to improve measurement accuracy. The camera calibration method used an accurate pinhole camera model and pixel-by-pixel calibration of the phase-height relationship. Numerical simulations and controlled baseline experiments were performed to quantify key error sources in the measurement process and verify the accuracy of the approach. Results from numerical simulations indicate that the resulting phase error can be reduced to less than 0.02 radians provided that parameters such as fringe spacing, random measured intensity noise, fringe contrast and frequency of spatial intensity noise are carefully controlled. Experimental results show that the effects of random temporal and spatial noise in typical CCD cameras for single fringe images limits the accuracy of the method to 0.04 radians in most applications. Quantitative results from application of the fringe projection method are in very good agreement with numerical predictions, demonstrating that it is possible to design both a fringe projection system and a measurement process to achieve a prespecified accuracy and resolution in the point-to-point measurement of the spatial (X, Y, Z) positions.     

Assessment of dual plane PIV measurements in wall turbulence using DNS data

Experimental dual plane particle image velocimetry (PIV) data are assessed using direct numerical simulation (DNS) data of a similar flow with the aim of studying the effect of averaging within the interrogation window. The primary reason for the use of dual plane PIV is that the entire velocity gradient tensor and hence the full vorticity vector can be obtained. One limitation of PIV is the limit on dynamic range, while DNS is typically limited by the Reynolds number of the flow. In this study, the DNS data are resolved more finely than the PIV data, and an averaging scheme is implemented on the DNS data of similar Reynolds number to compare the effects of averaging inherent to the present PIV technique. The effects of averaging on the RMS values of the velocity and vorticity are analyzed in order to estimate the percentage of turbulence intensity and enstrophy captured for a given PIV resolution in turbulent boundary layers. The focus is also to identify vortex core angle distributions, for which the two-dimensional and three-dimensional swirl strengths are used. The studies are performed in the logarithmic region of a turbulent boundary layer at z ⁺ = 110 from the wall. The dual plane PIV data are measured in a zero pressure gradient flow over a flat plate at Re _τ = 1,160, while the DNS data are extracted from a channel flow at Re _τ = 934. Representative plots at various wall-normal locations for the RMS values of velocity and vorticity indicate the attenuation of the variance with increasing filter size. Further, the effect of averaging on the vortex core angle statistics is negligible when compared with the raw DNS data. These results indicate that the present PIV technique is an accurate and reliable method for the purposes of statistical analysis and identification of vortex structures.