Dissipation rate estimation from PIV in zero-mean isotropic turbulence

Measuring the turbulent kinetic energy dissipation rate in an enclosed turbulence chamber that produces zero-mean flow is an experimental challenge. Traditional single-point dissipation rate measurement techniques are not applicable to flows with zero-mean velocity. Particle image velocimetry (PIV) affords calculation of the spatial derivative as well as the use of multi-point statistics to determine the dissipation rate. However, there is no consensus in the literature as to the best method to obtain dissipation rates from PIV measurements in such flows. We apply PIV in an enclosed zero-mean turbulent flow chamber and investigate five methods for dissipation rate estimation. We examine the influence of the PIV interrogation cell size on the performance of different dissipation rate estimation methods and evaluate correction factors that account for errors related to measurement uncertainty, finite spatial resolution, and low Reynolds number effects. We find the Re _λ corrected, second-order, longitudinal velocity structure function method to be the most robust method to estimate the dissipation rate in our zero-mean, gaseous flow system.     

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.     

A turbulent boundary layer over a two-dimensional rough wall

Particle image velocimetry (PIV) measurements and planar laser induced fluorescence (PLIF) visualizations have been made in a turbulent boundary layer over a rough wall. The wall roughness consisted of square bars placed transversely to the flow at a pitch to height ratio of λ/k = 11 for the PLIF experiments and λ/k = 8 and 16 for the PIV measurements. The ratio between the boundary layer thickness and the roughness height k/δ was about 20 for the PLIF and 38 for the PIV. Both the PLIF and PIV data showed that the near-wall region of the flow was populated by unstable quasi-coherent structures which could be associated to shear layers originating at the trailing edge of the roughness elements. The streamwise mean velocity profile presented a downward shift which varied marginally between the two cases of λ/k, in agreement with previous measurements and DNS results. The data indicated that the Reynolds stresses normalized by the wall units are higher for the case λ/k = 16 than those for λ/k = 8 in the outer region of the flow, suggesting that the roughness density effects could be felt well beyond the near-wall region of the flow. As expected the roughness disturbed dramatically the sublayer which in turn altered the turbulence production mechanism. The turbulence production is maximum at a distance of about 0.5k above the roughness elements. When normalized by the wall units, the turbulence production is found to be smaller than that of a smooth wall. It is argued that the production of turbulence is correlated with the form drag.     

Extended self-similar scaling law of multi-scale eddy structure in wall turbulence

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Turbulence Modeling Effects on the Prediction of Equilibrium States of Buoyant Shear Flows

The effects of turbulence modeling on the prediction of equilibrium states of turbulent buoyant shear flows were investigated. The velocity field models used include a two-equation closure, a Reynolds-stress closure assuming two different pressure-strain models and three different dissipation rate tensor models. As for the thermal field closure models, two different pressure-scrambling models and nine different temperature variance dissipation rate ɛ_τ) equations were considered. The emphasis of this paper is focused on the effects of the ɛ_τ-equation, of the dissipation rate models, of the pressure-strain models and of the pressure-scrambling models on the prediction of the approach to equilibrium turbulence. Equilibrium turbulence is defined by the time rate of change of the scaled Reynolds stress anisotropic tensor and heat flux vector becoming zero. These conditions lead to the equilibrium state parameters, given by /ɛ, _τ/ɛ_τ, , Sk/ɛ and G/ɛ, becoming constant. Here, and _τ are the production of turbulent kinetic energy k and temperature variance , respectively, ɛ and ɛ_τ are their respective dissipation rates, R is the mixed time scale ratio, G is the buoyant production of k and S is the mean shear gradient. Calculations show that the ɛ_τ-equation has a significant effect on the prediction of the approach to equilibrium turbulence. For a particular ɛ_τ-equation, all velocity closure models considered give an equilibrium state if anisotropic dissipation is accounted for in one form or another in the dissipation rate tensor or in the ɛ-equation. It is further found that the models considered for the pressure-strain tensor and the pressure-scrambling vector have little or no effect on the prediction of the approach to equilibrium turbulence. Received 21 April 2000 and accepted 21 February 2001     

Energy Estimate of the Critical Reynolds Numbers in a Compressible Couette Flow. Effect of Bulk Viscosity

A variational problem of determining the critical Reynolds number of the laminar-turbulent transition is numerically solved within the framework of the nonlinear energy theory of stability of compressible flows. Stability of various modes in the Couette flow of a compressible gas is estimated by the method of collocations. It is demonstrated that the minimum critical Reynolds numbers in the range of the ratio of the bulk viscosity ηb to the shear viscosity η, which is realistic for diatomic gases, are reached for modes of streamwise disturbances. The critical Reynolds numbers increase as the bulk viscosity is increased in the interval η_b = 0-2η, with the maximum increase in the limit being approximately 30%.     

The Structure Functions of the Velocity and Temperature Fields from the Perspective of Dimensional Scaling

In this paper, dimensional scaling is used to describe the turbulence structure of the velocity and temperature fields in the inertial range and the far dissipation range as well as the intermediate transition range under locally isotropic conditions at sufficiently large Reynolds numbers. This kind of scaling is expressed in a strictly mathematical manner employing dimensional π -invariants analysis. It is shown that in the case of the asymptotic solutions for either the inertial range or the far dissipation range only one π number occurs that has to be considered as a non-dimensional universal constant. This π number may be determined theoretically or/and empirically. In the case of the transition range two π numbers occur. Consequently, a universal function is established that has to be derived theoretically or/and empirically, too. Here, Batchelor's [7] classical interpolation formula for the turbulence structure of the velocity field and the empirical one of Stolovitzky et al. [59], both may serve as universal functions, are compared with the results provided by numerical solutions of Kolmogorov′s [32] structure equation for the velocity field. It is shown that these interpolation formulae match not only the asymptotic solutions of the inertial range and the far dissipation range, respectively, but also these numerical results in an excellent manner. The former may be considered as necessary condition and the latter as sufficient condition. In the case of the temperature field results of the corresponding universal function are predicted using Yaglom's [63] structure equation. These results also match the corresponding asymptotic solutions of both the inertial range and the far dissipation range. However, in contrast to the case of the velocity field, the predicted universal function for the temperature field may notably overshoot its asymptotic solution for the inertial range. This overshooting occurs in the transition range and may be considered as an analogue to the so-called Hill ‘bump’ that usually occurs in the high-wave number portion of the temperature spectrum.     

Attributes of the large-scale coherent motions in a shear layer

PIV observations in a shear layer have been used to identify and characterize the discrete large-scale coherent motions (LSCMs) in the nominally self-preserving region: x/θ_o ≈ 450–610, of a shear layer. The LSCMs are given an objective definition wherein their centers are the (swirling flow pattern) nodes of the velocity-vector field as seen by an observer in the Galilean reference frame translating at an appropriately defined reference velocity. The statistical attributes of size, lateral location, and separation between these coherent motions (that exist in a single image) as well as their characteristic vorticity magnitude 〈ω_max〉 are reported.     

Measuring turbulence energy with PIV in a backward-facing step flow

Turbulence energy is estimated in a backward-facing step flow with three-component (3C, stereo) particle image velocimetry (PIV). Estimates of turbulence energy transport equation for convection, turbulence transport, turbulence production, viscous diffusion, and viscous dissipation in addition to Reynolds stresses are computed directly from PIV data. Almost all the turbulence energy terms in the backward-facing step case can be measured with 3C PIV, except the pressure-transport term, which is obtained by difference of the other turbulence energy terms. The effect of the velocity spatial sampling resolution in derivative estimations is investigated with four two-dimensional PIV measurement sets. This sampling resolution information is used to calibrate the turbulence energies estimated by 3C PIV measurements. The focus of this study is on the separated shear layer of the backward-facing step. The measurements with 3C PIV are carried out in a turbulent water flow at Reynolds number of about 15,000, based on the step height h and the inlet streamwise maximum mean velocity U₀. The expansion ratio (ER) is 1.5. Turbulence energy budget profiles in locations x/h=4, x/h=6, and x/h=10 are compared with DNS data of a turbulent flow. The shapes of profiles agree well with each other. Different ERs between the PIV case (1.5) and the DNS case (1.2) cause higher values for the turbulence energies measured by PIV than the energies by DNS when x/h=10 is approached. PIV results also show that the turbulence energy level in these experiments is generally higher than that of the DNS data.     

Investigation of the vortex induced unsteadiness of a separation bubble via time-resolved and scanning PIV measurements

A transitional separation bubble on the suction side of an SD7003 airfoil is considered. The transition process that forces the separated shear layer to reattach seems to be governed by Kelvin–Helmholtz instabilities. Large scale vortices are formed due to this mechanism at the downstream end of the bubble. These vortices possess a three-dimensional structure and detach from the recirculation region, while other vortices are formed within the bubble. This separation of the vortex is a highly unsteady process, which leads to a bubble flapping. The structure of these vortices and the flapping of the separation bubble due to these vortices are temporally and spatially analyzed at angles of attack from 4° to 8° and chord-length based Reynolds numbers Re _c = 20,000–60,000 using time-resolved PIV measurements in a 2D and a 3D set-up, i.e., stereo-scanning PIV measurements are done in the latter case. These measurements complete former studies at a Reynolds number of Re _c = 20,000. The results of the time-resolved PIV measurements in a single light-sheet show the influence of the angle of attack and the Reynolds number. The characteristic parameters of the separation bubble are analyzed focusing on the unsteadiness of the separation bubble, e.g., the varying size of the main recirculation region, which characterizes the bubble flapping, and the corresponding Strouhal number are investigated. Furthermore, the impact of the freestream turbulence is investigated by juxtaposing the current and former results. The stereo-scanning PIV measurements at Reynolds numbers up to 60,000 elucidate the three-dimensional character of the vortical structures, which evolve at the downstream end of the separation bubble. It is shown that the same typical structures are formed, e.g., the c-shape vortex and the screwdriver vortex at each Reynolds number and angle of attack investigated and the occurrence of these patterns in relation to Λ-structures is discussed. To evidence the impact of the freestream turbulence, these results are compared with findings of former measurements.     

PIV measurements of a turbulent jet issuing from round sharp-edged plate

An experimental investigation is presented of a turbulent jet issuing from a round sharp-edged orifice plate (OP) into effectively unbounded surroundings. Planar measurements of velocity were conducted using Particle Image Velocimetry (PIV) in the near and transition regions. The Reynolds number, based on the jet initial diameter and velocity, is approximately 72,000. The instantaneous and mean velocities, Reynolds normal and shear stresses were obtained. The centerline velocity decay and the half-velocity radius were derived from the mean velocity. It is revealed that primary coherent structures occur in the near field of the OP jet and that they are typically distributed asymmetrically with respect to the nozzle axis. Comparison of the present PIV and previous hot-wire measurements for the OP jet suggests that high initial turbulence intensity leads to reduced rates of decay and spread of the mean flow field and moreover a lower rate of variation of the turbulence intensity. Results also show that self-similarity of the mean flow is well established from the transition region while the turbulent statistics are far from self-similar within the measured range to 16 diameters.     

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.     

Near-wall flow in a three-dimensional boundary layer on the endwall of a 30° bend

 Turbulence measurements are reported on the three-dimensional turbulent boundary layer along the centerline of the flat endwall in a 30° bend. Profiles of mean velocities and Reynolds stresses were obtained down to y ⁺≈2 for the mean flow and y ⁺≈8 for the turbulent stresses. Mean velocity data collapsed well on a simple law-of-the-wall based on the magnitude of the resultant velocity. The turbulence intensity and turbulent shear stress magnitude both increased with increased three-dimensionality. The ratio of these two quantities, the a ₁ structure parameter, decreased in the central regions of the boundary layer and showed profile similarity for y ⁺<50. The shear stress vector angle lagged behind the velocity gradient vector angle in the outer region of the boundary layer, however there was an indication that the shear stress vector tends to lead the velocity gradient vector close to the wall. Received: 16 July 1996/Accepted: 14 July 1997     

Synchronized analysis of an unsteady laminar flow downstream of a circular cylinder centred between two parallel walls using PIV and mass transfer probes

Experiments are carried out in the wake of a cylinder of d _c  = 10 mm diameter placed symmetrically between two parallel walls with a blockage ratio r = 1/3 and a Reynolds number varying between 75 ≤ Re ≤ 277. Particle image velocimetry is exerted to obtain the instantaneous velocity components in the cylinder wake. A snapshot proper orthogonal decomposition (POD) is also applied to these PIV results in order to extract the dominant modes through the implementation of an inhomogeneous filtering of these different snapshots, apart from an interpolation to estimate the wall shear rate at the lower wall downstream the cylinder. Mass transfer circular probes are placed at the lower wall downstream this obstacle so as to further determine the time evolution of the wall shear rate, by bringing the inverse method to bear on the convective-diffusion equation. Comparisons between the two synchronized techniques demonstrate that electrochemical method can give more accurate information about the coherent structures present in the flow and about the interaction of the von Kármán vortices with the walls of the tunnel as well. The comparison between the two measurement techniques in the flow regions concerns the spatiotemporal evolutions of the wall shear rate obtained from PIV measurements and the wall shear rate using mass transfer probes. Discrepancy between the PIV measurements and the electrochemical ones near the wall, where the secondary vortices P ₁′ are generated at wall, are caused by a PIV bias and a limitations of the singular mass transfer probes.     

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).     

On the Shear Number Effect in Stratified Shear Flow

The influence of the shear number on the turbulence evolution in a stably stratified fluid is investigated using direct numerical simulations on grids with up to 512 × 256 × 256 points. The shear number SK/ε is the ratio of a turbulence time scale K/ε to the shear time scale 1/S. Simulations are performed at two initial values of the Reynolds number Re _Λ= 44.72 and Re _Λ= 89.44. When the shear number is increased from small to moderate values, the nondimensional growth rate γ= (1/SK)dK/dt of the turbulent kinetic energy K increases since the shear forcing and its associated turbulence production is larger. However, a further increase of the shear number from moderate to large values results in a reduction of the growth rate γ and the turbulent kinetic energy K shows long-time decay for sufficiently large values of the shear number. The inhibition of turbulence growth at large shear numbers occurs for both initial values of the Reynolds number and can be explained with the predominance of linear effects over nonlinear effects when the shear number is sufficiently high. It is found that, at the higher initial value of the Reynolds number, the reduction of the growth rate occurs at a higher value of the shear number. The shear number is found to affect spectral space dynamics. Turbulent transport coefficients decrease with increasing shear number. Received 23 June 1998 and accepted 25 February 1999     

Orthogonal wavelet analysis of turbulent wakes behind various bluff bodies

The multi-scale structures of turbulent wakes generated by three kinds of bluff body, i.e. circular cylinder, square cylinder and compound of cylinder and square (CS) cylinders, have been experimentally investigated in this paper. Firstly, the instantaneous velocity fields and vorticity were measured by the high-speed PIV technique in a circulating water channel. The instantaneous streamlines and corresponding normalized vorticity contours are obtained at a Reynolds number of 5600. Then one- and two-dimensional wavelet multi-resolution technique was used to analyze the instantaneous velocities and vorticity measured by the high-speed PIV. The turbulence structures were separated into a number of subsets based on their central frequencies, which are linked with the turbulence scales. The instantaneous vorticity and Reynolds shear stresses of various scales were examined and compared between the three generators. It is found that the large-scale turbulent structure makes the largest contribution to the vorticity and Reynolds shear stresses for the three wake generators and exhibits a strong dependence upon the initial conditions or the wake generators. The large-scale vorticity and the sizes of vortex in the circular and square cylinders are larger than those in the CS cylinder wake. The contributions to the Reynolds shear stresses from the large-scale turbulent structures account for 90-96% to the measured maximum Reynolds shear stresses for the three wakes. However, the small-scale structures make less contribution to the vorticity and Reynolds shear stresses.     

Experimental study of near-wall turbulent characteristics in an open-channel with gravel bed using an acoustic Doppler velocimeter

This experimental study investigated the mean velocity profiles, skin friction and turbulent characteristics of a gravel bed over a wide range of roughness using an acoustic Doppler velocimeter (ADV). The median diameter of bed material ranged from 2 to 40 mm, and the normalized roughness heights ranged from 47 to 4,881 mm. The flow regime was fully developed turbulence with a Reynolds number in the range of 4.2 × 10⁴–9.86 × 10⁴. All velocity curves exhibited logarithmic distributions, and the log-law region was influenced greatly by both the roughness and the Reynolds number. Moreover, the roughness of the gravel bed exerted a strong effect on Reynolds stress, and the turbulence tended towards isotropic with increasing roughness. Using statistical analyses, the third-order turbulence moments, sweep, and ejection motions were also examined. The results of this experimental analysis present a contrast to the classical wall similarity hypothesis.     

Error quantification of 3D homogeneous and isotropic turbulence measurements using 2D PIV

Particle image velocimetry (PIV) has become a popular non-intrusive tool for measuring various types of flows. However, when measuring three dimensional flows with 2D PIV, there is inherent measurement error due to out-of-plane motion. Errors in the measured velocity field propagate to turbulence statistics. Since this can distort the overall flow characteristics, it is important to understand the effect of this out-of-plane error. In this study, the effect of out-of-plane motion on turbulence statistics is quantified. Using forced isotropic turbulence direct numerical simulation (DNS) flow field data provided by the Johns Hopkins turbulence database (JHTDB), synthetic image tests are performed. Turbulence statistics such as turbulence kinetic energy, dissipation rate, Taylor microscale, Kolmogorov scale, and velocity correlations are calculated. Various test cases were simulated while controlling three main parameters which affect the out-of-plane motion: PIV interrogation window size, camera inter-frame time, and laser sheet thickness. The amount of out-of-plane motion was first quantified, and then the error variation according to these parameters was examined. This information can be useful when examining fully three dimensional flows such as homogeneous and isotropic turbulence via 2D PIV.     

Dissipation rate correction methods

Dissipation rates of the turbulent kinetic energy and of the scalar variance are underestimated when the measurement resolution of the small scales of a turbulent flow field are insufficient. Results are presented of experiments conducted in a salt-stratified water tunnel (Schmidt number ∼700). Dissipation rates are determined to be underestimated, and thus correction techniques based on velocity structure functions and mixed-moment functions are proposed. Dissipation rates in laboratory experiments of shear-free, grid-generated turbulence are determined from balance calculations of the kinetic energy and scalar variance evolution equations. Comparisons between the structure function and balance estimates of dissipation show that the corrections are O(1) for the kinetic energy dissipation rate, and are O(100) for the scalar variance dissipation rate. This difference is due to the lack of resolution down to the Batchelor scales that is required for a high Schmidt number flow. Simple correction functions based on microscale Reynolds numbers are developed for both turbulent kinetic energy and scalar variance dissipation rates. Application of the technique to the results of laboratory experiments of density stratified turbulence, sheared turbulence, and sheared density stratified turbulence yields successful corrections. It is also demonstrated that the Karman–Howarth equality (and the analogous Yaglom equation) that relates second and third-order structure functions to dissipation rates is valid for both unstrained (decaying grid-generated turbulence) and density stratified and sheared turbulence at least up to the magnitudes of strains of the current experiments Nt∼10, St∼10, respectively. This is helpful for it allows the use of these equations in the analysis of turbulence even when the large scale background profiles of velocity and scalar are unknown.