Spatial resolution effects on the measurement of scalar variance and scalar gradient in turbulent nonpremixed jet flames

The effect of spatial averaging is important for scalar gradient measurements in turbulent nonpremixed flames, especially when the local dissipation length scale is small. Line imaging of Raman, Rayleigh and CO-LIF is used to investigate the effects of experimental resolution on the scalar variance and radial gradient in the near field of turbulent nonpremixed CH₄/H₂/N₂ jet flames at Reynolds numbers of 15,200 and 22,800 (DLR-A and B) and in piloted CH₄/air jet flames at Reynolds numbers of 13,400, 22,400 and 33,600 (Sandia flames C/D/E). The finite spatial resolution effects are studied by applying the Box filter with varying filter widths. The resulting resolution curves for both scalar variance and mean squared-gradient follow nearly the same trends as theoretical curves calculated from the model turbulence kinetic energy spectrum of Pope. The observed collapse of resolution curves of mean squared-gradient for nearly all studied cases implies the shape of the dissipation spectrum is approximately universal. Fluid transport properties are shown to have no effect on the dissipation resolution curve, which implies that the dissipation length scale inferred from the square gradient is equivalent to the length scale for the scalar dissipation rate, which includes the diffusion coefficient. With the Box filter, the required spatial resolution to resolve 98% of the mean dissipation rate is about one−two times of the dissipation cutoff length scale (analogous to the Batchelor scale in turbulent isothermal flows). The effects of resolution on the variances of mixture fraction, temperature, and the inverted Rayleigh signal are also compared. The ratio of the filtered variance to the true variance is shown to depend nearly linearly on the probe resolution. The inverted Rayleigh scattering signal can be used to study the resolution effect on temperature variance even when the Rayleigh scattering cross section is not constant. The experimental results also indicate that these laboratory scale turbulent jet flames have small effective Reynolds numbers, such that there is some direct interaction of the large (energy containing) and small (dissipative) scalar length scales, especially for the near field case at x/d = 7.5 of the piloted Sandia flames C/D/E.     

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.     

The influence of probe resolution on the measurement of a passive scalar and its derivatives

The influence of probe resolution on the statistical measurement of a passive scalar is reported. A spectral method is employed to simulate degradation of the spatial resolution of a probe on the measured variances of a fluctuating scalar and its streamwise derivative by low-pass filtering a time-series of data at different cutoff frequencies. Direct measurements are also employed by varying probe sensor separation. The far field of a circular jet and the near wake of a circular cylinder are both investigated using air as the working fluid. The use of this low-Schmidt number working fluid and relatively low turbulence Reynolds numbers allows for good resolution of small scales of scalar fluctuations. By comparison, the same level of resolution is much more difficult to achieve when utilising a high-Schmidt number working fluid. A small temperature differential above ambient is used to mark the passive scalar, which is measured using a cold-wire anemometer. Taylor's hypothesis is employed to determine length scales. The present results are in good agreement with previous direct measurements using both optical techniques and cold-wire probes. It is found that the spatial resolution required for accurate measurement of the scalar dissipation rate is well described by the characteristic smallest scale of the scalar fluctuation, i.e. 'the Batchelor scale'. However, an order of magnitude less resolution is required for the scalar variance. The effect of degrading resolution on the variance measurements is more significant in the near wake than the far-field jet, suggesting that these requirements may be flow-dependent.     

Skewness factor of turbulent velocity derivative

Using nonequilibrium statistical mechanics closure method, it is shown that the skewness factor of the velocity derivative of isotropic turbulence approaches a constant −0.515 when the Reynolds number is very high, which is in agreement with the DNS (direct numerical simulation) result of Vincent and Meneguzzi (1991). The project supported by the National Basic Research Program “Non-linear Science”     

Velocity derivative skewness in isotropic turbulence and its measurement with hot wires

We investigate the effect of the hot wire resolution on the measurement of the velocity derivative skewness in homogeneous isotropic turbulence. Single- and cross-wire configurations (with different lengths and separations of the wires, and temporal sampling resolution) are considered. Predictions of the attenuation on the basis of a model for the energy spectrum are compared to experimental and numerical data in grid and box turbulence, respectively. It is shown that the model-based correction is accurate for the single wire but not for the cross-wire. In the latter case, the effect of the separation between the wires is opposite to that found in the experiments and simulations. Moreover, the attenuation predicted by the numerical data is in good agreement with that observed in the experiment. For both probe configurations, the sampling resolution has a sizeable attenuation effect, but, for the X-probe, the impact of the separation between the wires is more important. In both cases, the length of the wires has only a minor effect, in the non-dimensional range of wire length investigated. Finally, the present experimental data support the conclusion that the skewness is constant with the Reynolds number, in agreement with Kolmogorov’s 41 theory.

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Small-scale measurements in a partially stirred reactor

This paper describes an experimental study of a partially stirred reactor (PaSR). The reactor is a cubic box in which air (either pure or mixed with a tracer) is continuously injected through 12 jets situated in two opposite planes and impinging through the center. The flow in the reactor interior is well approximated as stationary, globally homogeneous and isotropic. Global properties of turbulent flow and passive scalar mixing are studied, in terms of length scales, characteristic times, spectra, etc. Particular attention has been paid to a proper determination of the mean value of the passive scalar variance dissipation rate 〈ε _Z 〉, in the central quasi-homogeneous zone of the reactor.     

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.     

A new method for probing the turbulence scalar spectrum by ultrasonic sonar scanning

 A new method is proposed to obtain a turbulent scalar spectrum and the energy dissipation rate in turbulent flow from ultrasonic frequency scanning data. A scanning sonar with frequency varying from 0.5 MHz to 5 MHz has been used to directly probe the energy dissipation rate ɛ and the three-dimensional scalar spectrum E _θ(k). Experiments were conducted in a laboratory open-channel flow in clear water with Reynolds numbers varying from 1.2×10⁵ to 6.9×10⁵. Good agreement is found between measured spectra and those predicted by the Batchelor theory. The energy dissipation rates compare favourably with those obtained from acoustic Doppler velocimeter measurements. Received: 20 March 1997/Accepted: 27 September 1997     

Testing of Model Equations for the Mean Dissipation using Kolmogorov Flows

Direct Numerical Simulations (DNS) of Kolmogorov flows are performed at three different Reynolds numbers Re _λ between 110 and 190 by imposing a mean velocity profile in y-direction of the form U(y) = F sin(y) in a periodic box of volume (2π)³. After a few integral times the turbulent flow turns out to be statistically steady. Profiles of mean quantities are then obtained by averaging over planes at constant y. Based on these profiles two different model equations for the mean dissipation ε in the context of two-equation RANS (Reynolds Averaged Navier–Stokes) modelling of turbulence are compared to each other. The high Reynolds number version of the k-ε-model (Jones and Launder, Int J Heat Mass Transfer 15:301–314, 1972), to be called the standard model and a new model by Menter et al. (2006), to be called the Menter–Egorov model, are tested against the DNS results. Both models are solved numerically and it is found that the standard model does not provide a steady solution for the present case, while the Menter–Egorov model does. In addition a fairly good quantitative agreement of the model solution and the DNS data is found for the averaged profiles of the kinetic energy k and the dissipation ε. Furthermore, an analysis based on flow-inherent geometries, called dissipation elements (Wang and Peters, J Fluid Mech 608:113–138, 2008), is used to examine the Menter–Egorov ε model equation. An expression for the evolution of ε is derived by taking appropriate moments of the equation for the evolution of the probability density function (pdf) of the length of dissipation elements. A term-by-term comparison with the model equation allows a prediction of the constants, which with increasing Reynolds number approach the empirical values.     

Performance of a transverse vorticity probe in a turbulent channel flow

 The performance of a four hot-wire transverse vorticity probe is tested by comparing measurements in a fully developed turbulent channel flow with corresponding data obtained from direct numerical simulations (DNS) of the same flow. In the inner region, the probe performs poorly, the rms vorticities being consistently smaller than the DNS values. In the outer region of the flow, there is reasonable agreement between measured and DNS vorticity statistics, especially after correcting the measurements for the effect of spatial resolution. In this region, the imbalance indicated by the vorticity form of the streamwise momentum equation is approximately constant. The magnitude of the imbalance can be reduced to an acceptable level of accuracy by considering sources of error which affect the velocity–vorticity correlations. Received: 17 March 1997/Accepted: 17 November 1997     

The effects of resolution and noise on kinematic features of fine-scale turbulence

The effect of spatial resolution and experimental noise on the kinematic fine-scale features in shear flow turbulence is investigated by means of comparing numerical and experimental data. A direct numerical simulation (DNS) of a nominally two-dimensional planar mixing layer is mean filtered onto a uniform Cartesian grid at four different, progressively coarser, spatial resolutions. Spatial gradients are then calculated using a simple second-order scheme that is commonly used in experimental studies in order to make direct comparisons between the numerical and previously obtained experimental data. As expected, consistent with other studies, it is found that reduction of spatial resolution greatly reduces the frequency of high magnitude velocity gradients and thereby reduces the intermittency of the scalar analogues to strain (dissipation) and rotation (enstrophy). There is also an increase in the distances over which dissipation and enstrophy are spatially coherent in physical space as the resolution is coarsened, although these distances remain a constant number of grid points, suggesting that the data follow the applied filter. This reduction of intermittency is also observed in the eigenvalues of the strain-rate tensor as spatial resolution is reduced. The quantity with which these eigenvalues is normalised is shown to be extremely important as fine-scale quantities, such as the Kolmogorov length scale, are showed to change with different spatial resolution. This leads to a slight change in the modal values for these eigenvalues when normalised by the local Kolmogorov scale, which is not observed when they are normalised by large-scale, resolution-independent quantities. The interaction between strain and rotation is examined by means of the joint probability density function (pdf) between the second and third invariants of the characteristic equation of the velocity gradient tensor, Q and R respectively and by the alignments between the eigenvectors of the strain-rate tensor and the vorticity vector. Gaussian noise is shown to increase the divergence error of a dataset and subsequently affect both the Q–R joint pdf and the magnitude of the alignment cosines. The experimental datasets are showed to behave qualitatively similarly to the numerical datasets to which Gaussian noise has been added, confirming the importance of understanding the limitations of coarsely resolved, noisy experimental data.     

Noise Correction and Length Scale Estimation for Scalar Dissipation Rate Measurements in Turbulent Partially Premixed Flames

A recently developed conditional sampling-based method for correcting noise effects in scalar dissipation rate measurements and for estimating the extent of resolution of the dissipation rate is employed to analyze the data obtained in turbulent partially premixed (Sandia) flames. The method uses conditional sampling to select instantaneous fully resolved local scalar fields, which are analyzed to determine the measurement noise and to correct the Favre mean, conditional, and conditionally filtered dissipation rates. The potentially under-resolved local scalar fields, also selected using conditional sampling, are corrected for noise and are analyzed to examine the extent of resolution. The error function is used as a model for the potentially under-resolved local scalar to evaluate the scalar dissipation length scales and the percentage of the dissipation resolved. The results show that the Favre mean dissipation rate, the mean dissipation rate conditional on the mixture fraction, and dissipation rate filtered conditionally on the mixture fraction generally are well resolved in the flames. Analyses of the dissipation rates filtered conditionally on the mixture fraction and temperature show that the length scale increases with temperature, due to lower dissipation rate and higher diffusivity. The dissipation rate is well resolved for temperatures above 1,300 K but is less resolved at lower temperatures, although the probability of very low temperature events is low. To fully resolve these rare events the sample spacing needs to be reduced by approximately one half. The present study further demonstrates the effectiveness of the new noise correction and length scale estimation method.     

Effects of Lewis Number on Scalar Variance Transport in Premixed Flames

The influences of differential diffusion rates of heat and mass on the transport of the variances of Favre fluctuations of reaction progress variable and non-dimensional temperature have been studied using three-dimensional simplified chemistry based Direct Numerical Simulation (DNS) data of statistically planar turbulent premixed flames with global Lewis number ranging from Le?= 0.34 to 1.2. The Lewis number effects on the statistical behaviours of the various terms of the transport equations of variances of Favre fluctuations of reaction progress variable and non-dimensional temperature have been analysed in the context of Reynolds Averaged Navier Stokes (RANS) simulations. It has been found that the turbulent fluxes of the progress variable and temperature variances exhibit counter-gradient transport for the flames with Lewis number significantly smaller than unity whereas the extent of this counter-gradient transport is found to decrease with increasing Lewis number. The Lewis number is also shown to have significant influences on the magnitudes of the chemical reaction and scalar dissipation rate contributions to the scalar variance transport. The modelling of the unclosed terms in the scalar variance equations for the non-unity Lewis number flames have been discussed in detail. The performances of the existing models for the unclosed terms are assessed based on a-priori analysis of DNS data. Based on the present analysis, new models for the unclosed terms of the active scalar variance transport equations are proposed, whenever necessary, which are shown to satisfactorily capture the behaviours of unclosed terms for all the flames considered in this study.     

Planar measurements of the full three-dimensional scalar dissipation rate in gas-phase turbulent flows

A simultaneous planar Rayleigh scattering and planar laser-induced fluorescence (PLIF) technique is described which allows planar measurement of the full three-dimensional scalar gradient, ∇C (x, t), and scalar energy dissipation rate, χ≡D ∇C·∇C, in gas-phase turbulent flows. The conserved scalar used is the jet fluid concentration, where the jet consists of propane and seeded acetone. The propane serves as the primary Rayleigh scattering medium, while the acetone is used for fluorescence. For a given amount of available laser energy, this planar Rayleigh scattering/PLIF technique yields much higher signals levels than would, for example, a two-plane Rayleigh scattering technique. By applying the current technique to a single spatial plane, the errors incurred in measuring a spatial derivative across distinct planes are quantified. The errors are found to be well described by a random distribution, and the magnitude of these errors is found to be smaller than the magnitude of significant events in the true scalar gradient fields. Sample results for the fields of the three-dimensional scalar gradient and scalar energy dissipation in a planar turbulent jet, with outer scale Reynolds numbers between 3200 and 8400, are shown, demonstrating the applicability of these measurements to analyses of the fine scale mixing in turbulent flows. The application of these measurements to determination of the scaling properties of the dissipation rate is also discussed. Received: 3 June 1998/Accepted: 12 February 1999     

Simultaneous measurements of temperature and its dissipation using pairs of parallel cold wires

 The link between fluctuations of a passive scalar and its dissipation is an important problem for various aspects of turbulent flow modelling both with and without chemical reaction. This paper first reports experimental methods for simultaneous measurements of the three derivatives involved in temperature dissipation using cold wire probes in a boundary layer over a weakly heated flat plate. Particular attention is paid to the determination of the optimal length and spacing between sensors, including a four-wire probe of fixed dimensions, for such gradient measurements. Corrections to be applied to account for the effect of the fixed geometry are analyzed. Results about joint statistics, including conditional temperature pdfs, between the scalar and its dissipation are then reported and compared to previous results which mainly concerned conditional dissipation rates. Received: 15 October 1996 / Accepted: 13 January 1997     

Effects of imaging system blur on measurements of flow scalars and scalar gradients

Planar imaging of flow scalars is widely used in fluid mechanics, but the effects of imaging system blur on the measured scalar and its gradients are often inadequately quantified. Here, we present a 1-D analytical study that uses simplified models of the scalar profiles and imaging system blur to estimate the measurement errors caused by finite resolution. One objective of this paper is to give the experimentalist a methodology for quantitatively assessing the impact of imaging system blur on the accuracy of scalar measurements. The scalar profiles are modeled as either error or Gaussian functions, and the imaging system resolution is cast in terms of the line-spread function (LSF), which is modeled as Gaussian. The analysis gives the errors induced in the scalar structure thickness, gradient, and dissipation, for varying degrees of blur, the latter of which is quantified by , the standard deviation of the Gaussian LSF. The results show that, to keep errors in the peak scalar gradients and dissipation to less than 10%, the 20%-width of the scalar structures should be at least 7.5. Typical flow imaging experiments require fast (i.e., low f/#) optics that may suffer from significant blur and, therefore, this requirement may be difficult to meet in many applications. It is also shown that the resolution requirements for measuring the dissipation are more restrictive than for structure thicknesses. Further simulations were made to assess the effects of having clustered, or closely spaced, dissipation structures. Compared to the single structure results, there is a less severe resolution requirement to obtain scalar structure length scales, but a more severe requirement on the scalar gradient and dissipation.     

Measurements of Scalar Variance, Scalar Dissipation, and Length Scales in Turbulent Piloted Methane/Air Jet Flames

One-dimensional (line) measurements of mixture fraction, temperature, and scalar dissipation in piloted turbulent partially premixed methane/air jet flames (Sandia flames C, D, and E) are presented. The experimental facility combines line imaging of Raman scattering, Rayleigh scattering, and laser-induced CO fluorescence. Simultaneous single-shot measurements of temperature and the mass fractions of all the major species (N₂, O₂, CH₄, CO₂, H₂O, CO, and H₂) are obtained along 7 mm segments with a nominal spatial resolution of 0.2 mm. Mixture fraction, ξ, is then calculated from the measured mass fractions. Ensembles of instantaneous mixture fraction profiles at several streamwise locations are analyzed to quantify the effect of spatial averaging on the Favre average scalar variance, which is an important term in the modeling of turbulent nonpremixed flames. Results suggest that the fully resolved scalar variance may be estimated by simple extrapolation of spatially filtered measurements. Differentiation of the instantaneous mixture fraction profiles yields the radial contribution to the scalar dissipation, χ _r = 2D _ξ(?ξ/?r)², and radial profiles of the Favre mean and rms scalar dissipation are reported. Scalar length scales, based on autocorrelation of the spatial profiles of ξ and χ _r , are also reported. These new data on this already well-documented series of flames should be useful in the context of validating models for sub-grid scalar variance and for scalar dissipation in turbulent flames.     

High-resolution measurements of the spatial and temporal scalar structure of a turbulent plume

 Two techniques are described for measuring the scalar structure of turbulent flows. A planar laser-induced fluorescence technique is used to make highly resolved measurements of scalar spatial structure, and a single-point laser-induced fluorescence probe is used to make highly resolved measurements of scalar temporal structure. The techniques are used to measure the spatial and temporal structure of an odor plume released from a low-momentum, bed-level source in a turbulent boundary layer. For the experimental setup used in this study, a spatial resolution of 150 μm and a temporal resolution of 1,000 Hz are obtained. The results show a wide range of turbulent structures in rich detail; the nature of the structure varies significantly in different regions of the plume. Received: 8 May 2000/Accepted: 15 November 2000     

On the Relationship Between the Statistics of the Resolved and True Rate of Dissipation of Mixture Fraction

The relationship between the one-point probability-density-function (PDF) of the dissipation rate of mixture fraction fluctuations and the corresponding resolved quantity available in large eddy simulation (LES) is analyzed. The investigation pursues two fronts: an a priori study using direct numerical simulation (DNS), and an analytic development that, using common turbulence physics simplifications, relates the one-point statistics of the resolved and true scalar dissipations. Particularly, the analysis reveals the connection between the multi-point correlations of the mixture fraction gradient and the one-point PDF of the resolved scalar dissipation. A DNS of a temporally evolving shear layer with and without heat release is used to quantify the accuracy of the analytical result. It is verified, both by filtering the DNS and from the theory, that increasing the filter cutoff width reduces the magnitude of the resolved scalar dissipation fluctuations, as expected and observed experimentally. Comparison with DNS indicates that the analytical relationship predicts the behavior of the resolved scalar dissipation PDF well at the center planes of the shear layer, where turbulence is locally more isotropic and homogeneous. Large-scale anisotropy and inhomogeneities in the DNS degrade the accuracy of the approximate analytical result close to the edges of the shear layer. These results may be improved with future investigations to account fully for the missing statistics in LES, which have the potential to allow a more accurate quantification of finite-rate chemistry effects in reacting flows.     

Implementation Issues of the Conditional Moment Closure Model in Large Eddy Simulations

Different ways of transferring information regarding the mixture fraction, its sub-grid scale variance and the scalar dissipation rate are examined in terms of a Large Eddy Simulation (LES)/Conditional Moment Closure (CMC) calculation. In such a simulation, information must be transferred from a fine LES grid to a usually coarser CMC grid. Different options of calculating conditional and unconditional quantities in the CMC resolution are assessed by filtering experimental mixture fraction and scalar dissipation rate data at various resolutions. It was found that when a presumed shape for the Filtered Density Function at the CMC resolution is used, special care must be given to the mixture fraction variance. It was also found that the Amplitude Mapping Closure model can be used for the conditional scalar dissipation rate. LES/CMC with detailed chemistry of a bluff-body stabilised burner was performed using two different ways of calculating the turbulent diffusivity. The structure of the flame is realistic, with little difference noticed when using the two diffusivities.