Turbulent Scalar Fields Research Articles

Intermittency of turbulent velocity and scalar fields using three-dimensional local averaging

An efficient approach for extracting 3D local averages in spherical subdomains is proposed and applied to study the intermittency of small-scale velocity and scalar fields in direct numerical simulations of isotropic turbulence. We focus on the inertial-range scaling exponents of locally averaged energy dissipation rate, enstrophy and scalar dissipation rate corresponding to the mixing of a passive scalar $\theta$ in the presence of a uniform mean gradient. The Taylor-scale Reynolds number $R_\lambda$ goes up to $1300$, and the Schmidt number $Sc$ up to $512$ (albeit at smaller $R_\lambda$). The intermittency exponent of the energy dissipation rate is $\mu \approx 0.23$, whereas that of enstrophy is slightly larger; trends with $R_\lambda$ suggest that this will be the case even at extremely large $R_\lambda$. The intermittency exponent of the scalar dissipation rate is $\mu_\theta \approx 0.35$ for $Sc=1$. These findings are in essential agreement with previously reported results in the literature. We further show that $\mu_\theta$ decreases monotonically with increasing $Sc$, either as $1/\log Sc$ or a weak power law, suggesting that $\mu_\theta \to 0$ as $Sc \to \infty$, reaffirming recent results on the breakdown of scalar dissipation anomaly in this limit.

Triple Hill’s Vortex Synthetic Eddy Method

The generation of initial or inflow synthetic turbulent velocity or scalar fields reproducing statistical characteristics of realistic turbulence is still a challenge. The synthetic eddy method, previously introduced in the context of inflow conditions for large eddy simulations, is based on the assumption that turbulence can be regarded as a superposition of coherent structures. In this paper, a new type of synthetic eddy method is proposed, where the fundamental eddy is constructed by superposing three Hill's vortices, with their axes orthogonal to each other. A distribution of Hill's vortices is used to synthesize an anisotropic turbulent velocity field that satisfies the incompressibility condition and match a given Reynolds stress tensor. The amplitudes of the three vortices that form the fundamental eddy are calculated from known Reynolds stress profiles through a transformation from the physical reference frame to the principal-axis reference frame. In this way, divergence-free anisotropic turbulent velocity fields are obtained that can reproduce a given Reynolds stress tensor. The model was tested on both isotropic and anisotropic turbulent velocity fields, in the framework of grid turbulence decay and turbulent channel flow, respectively. The transition from artificial to realistic turbulence in the proximity to the inflow boundary was found to be small in all test cases that were considered.

Data assimilation for turbulent mean flow and scalar fields with anisotropic formulation

This work presents an anisotropic formulation for data assimilation (DA) of turbulent flows using the continuous adjoint system. This DA scheme serves as a tool to complement flow measurement data (the observations), which are usually limited in spatial range and measurable quantities, with the help of a predictive model in the framework of the Reynolds-averaged Navier–Stokes approach (the estimator). Herein, the measured data profiles at several locations are used as observations. In the estimator, a forcing term $${\varvec{F}}$$ is added to the momentum equation to compensate for the contribution of the anisotropic eddy viscosity, whilst the isotropic part of the eddy viscosity is determined by conventional turbulence models. $${\varvec{F}}$$ is optimised using the continuous adjoint equations, driving the predicted flow quantities towards the observations. Similar treatments are conducted for the scalar prediction. Three test cases are used for assessment and validation of the present DA scheme. The results of a circular jet demonstrate that the mean flow and scalar fields can be perfectly reproduced from the observations due to the $${\varvec{F}}$$ optimisation. The flow over a blunt plate demonstrates the ability of the present DA scheme to reconstruct global flow fields from several measured velocity profiles. The flow and heat transfer on a ribbed wall reveal that once the global flow field is well recovered, the wall heat transfer characteristics can be correctly determined with a limited number of observations. The present anisotropic DA approach is quite practical for complex flow applications, serving as a complement to our previous isotropic adjoint-based data assimilation for further DA work.

Dissipation Element Analysis of Turbulent Premixed Jet Flames

ABSTRACTDissipation element (DE) analysis is a method for analyzing scalar fields in turbulent flows. DEs are defined as a coherent region in which all gradient trajectories of a scalar field reach the same extremal points. Therefore, the scalar field can be compartmentalized in monotonous space-filling regions.The DE analysis is applied to a set of spatially evolving premixed jet flames at different Reynolds numbers. The simulations feature finite rate chemistry with 16 species and 73 reactions. The jet consists of a methane/air mixture with an equivalence ratio φ = 0.7.Statistics of DE parameters are shown and compared to those of a DNS of a non-reacting spatial jet, a non-reacting temporally evolving jet and isotropic homogeneous turbulence.The invariance of the normalized length distribution of the DEs toward changes in Reynolds number observed in non-reacting flows holds for the reacting cases and the characteristic scaling with Kolmogorov micro-scale is reproduced. Furthermore, the DE statistics reflect the influence of the premixed flame structure on the turbulent scalar fields.

Persistence of local anisotropy of passive scalars in wall-bounded flows

A persistent small-scale anisotropy of turbulent passive scalar fields, which is independent of the scalar field initial conditions, is observed in direct numerical simulations of fully developed channel flow, even after the flow has undergone substantial mixing, and is attributed to the mean velocity gradient.

Subgrid-scale stresses and scalar fluxes constructed by the multi-scale turnover Lagrangian map

The multi-scale turnover Lagrangian map (MTLM) [C. Rosales and C. Meneveau, “Anomalous scaling and intermittency in three-dimensional synthetic turbulence,” Phys. Rev. E 78, 016313 (2008)] uses nested multi-scale Lagrangian advection of fluid particles to distort a Gaussian velocity field and, as a result, generate non-Gaussian synthetic velocity fields. Passive scalar fields can be generated with the procedure when the fluid particles carry a scalar property [C. Rosales, “Synthetic three-dimensional turbulent passive scalar fields via the minimal Lagrangian map,” Phys. Fluids 23, 075106 (2011)]. The synthetic fields have been shown to possess highly realistic statistics characterizing small scale intermittency, geometrical structures, and vortex dynamics. In this paper, we present a study of the synthetic fields using the filtering approach. This approach, which has not been pursued so far, provides insights on the potential applications of the synthetic fields in large eddy simulations and subgrid-scale (SGS) modelling. The MTLM method is first generalized to model scalar fields produced by an imposed linear mean profile. We then calculate the subgrid-scale stress, SGS scalar flux, SGS scalar variance, as well as related quantities from the synthetic fields. Comparison with direct numerical simulations (DNSs) shows that the synthetic fields reproduce the probability distributions of the SGS energy and scalar dissipation rather well. Related geometrical statistics also display close agreement with DNS results. The synthetic fields slightly under-estimate the mean SGS energy dissipation and slightly over-predict the mean SGS scalar variance dissipation. In general, the synthetic fields tend to slightly under-estimate the probability of large fluctuations for most quantities we have examined. Small scale anisotropy in the scalar field originated from the imposed mean gradient is captured. The sensitivity of the synthetic fields on the input spectra is assessed by using truncated spectra or model spectra as the input. Analyses show that most of the SGS statistics agree well with those from MTLM fields with DNS spectra as the input. For the mean SGS energy dissipation, some significant deviation is observed. However, it is shown that the deviation can be parametrized by the input energy spectrum, which demonstrates the robustness of the MTLM procedure.

Large eddy simulation of two isothermal and reacting turbulent separated oxy-fuel jets

In this work, a Large eddy simulation (LES) method and a tabulated chemistry approach according to the Flamelet Generated Manifold (FGM) strategy are coupled to numerically study the interactions of turbulent isothermal and reacting flows stemming from two aligned jets providing alternately fuel (natural gas) and oxidant (pure oxygen gas).The jets feature different geometries and deliver unequal momentums at the boundaries. The effect of oxygen in comparison to air environment on the FGM tabulation and results is pointed out. In addition, the impact of combustion on the flow and mixing field evolvement is analyzed. The LES relies on a dynamic Smagorinsky subgrid scale (SGS) model and a linear eddy diffusivity ansatz to close the SGS stresses and the SGS scalar fluxes for describing the turbulent flow field and the turbulent scalar field, respectively. For model assessment, available laser-based experimental data are used for model validation. In particular the numerical results are compared with available experimental data for the flow field. The latter are gained experimentally by the Particle Image Velocimetry (PIV) and laser tomography, respectively. In the first part of this paper, the jets interaction process is studied for the isothermal case while the oxy-fuel combustion in the reacting case is analyzed in the second part. The analysis is achieved in terms of statistical quantities for the flow velocity, mixture fraction, chemical species and temperature. An overall satisfactory agreement is reported.

Line segments in homogeneous scalar turbulence

The local structure of a turbulent scalar field in homogeneous isotropic turbulence is analyzed by direct numerical simulations (DNS) with different Taylor micro-scale based Reynolds numbers between 119 and 529. A novel signal decomposition approach is introduced where the signal of the scalar along a straight line is partitioned into segments based on the local extremal points of the scalar field. These segments are then parameterized by the distance ℓ between adjacent extremal points and the scalar difference Δϕ at the extrema. Both variables are statistical quantities and a joint distribution function of these quantities contains most information to statistically describe the scalar field. It is highlighted that the marginal distribution function of the length becomes independent of Reynolds number when normalized by the mean length ℓm. From a statistical approach, it is further shown that the mean length scales with the Kolmogorov length, which is also confirmed by DNS. For turbulent mixing, the scalar gradient plays a paramount role. Turbulent scalar fields are characterized by cliff-ramp-like structures manifesting the occurrence of localized large scalar gradients. To study turbulent mixing, a segment-based gradient is defined as Δϕ/ℓ. Joint statistics of the length and the segment-based gradient provide novel understanding of cliff-ramp-like structures. Ramp-like structures are unveiled by the asymmetry of the joint distribution function of the segment-based gradient and the length. Cliff-like structures are further analyzed by conditional statistics and it is shown from DNS that the width of cliffs scales with the Kolmogorov length scale.

Research on the Separation Performance of Deoiling Hydrocyclones

This paper uses FLUENT as the researching tools of 3D numerical simulation of deoiling hydrocyclones, analysis on the geometric structure of deoiling Hydrocyclone, summed up a set of grid partition method based on the basic theory and turbulence simulation of CFD theory, to determine a reasonable mathematical model, boundary conditions, convection diffusion the discrete format and pressure velocity coupling algorithm. This paper establishes the mathematical model, calculation method and the optimization principle for the separation mechanism; it also provides basic theory and experience for studying on deoiling hydrocyclone such as separation mechanism, flow and turbulent scalar field characteristics and structure optimization design.

Direct numerical simulation of turbulent mixing at very low Schmidt number with a uniform mean gradient

In a recent direct numerical simulation (DNS) study [P. K. Yeung and K. R. Sreenivasan, “Spectrum of passive scalars of high molecular diffusivity in turbulent mixing,” J. Fluid Mech. 716, R14 (2013)] with Schmidt number as low as 1/2048, we verified the essential physical content of the theory of Batchelor, Howells, and Townsend [“Small-scale variation of convected quantities like temperature in turbulent fluid. 2. The case of large conductivity,” J. Fluid Mech. 5, 134 (1959)] for turbulent passive scalar fields with very strong diffusivity, decaying in the absence of any production mechanism. In particular, we confirmed the existence of the −17/3 power of the scalar spectral density in the so-called inertial-diffusive range. In the present paper, we consider the DNS of the same problem, but in the presence of a uniform mean gradient, which leads to the production of scalar fluctuations at (primarily) the large scales. For the parameters of the simulations, the presence of the mean gradient alters the physics of mixing fundamentally at low Peclet numbers. While the spectrum still follows a −17/3 power law in the inertial-diffusive range, the pre-factor is non-universal and depends on the magnitude of the mean scalar gradient. Spectral transfer is greatly reduced in comparison with those for moderately and weakly diffusive scalars, leading to several distinctive features such as the absence of dissipative anomaly and a new balance of terms in the spectral transfer equation for the scalar variance, differing from the case of zero gradient. We use the DNS results to present an alternative explanation for the observed scaling behavior, and discuss a few spectral characteristics in detail.

Scalar gradient trajectory measurements using high-frequency cinematographic planar Rayleigh scattering

In this work, we perform an experimental investigation into statistics based on scalar gradient trajectories in a turbulent jet flow, which have been suggested as an alternative means to analyze turbulent flow fields by Wang and Peters (J Fluid Mech 554:457–475, 2006, 608:113–138, 2008). Although there are several numerical simulations and theoretical works that investigate the statistics along gradient trajectories, only few experiments in this area have been reported. To this end, high-frequency cinematographic planar Rayleigh scattering imaging is performed at different axial locations of a turbulent propane jet issuing into a CO2 coflow at nozzle-based Reynolds numbers Re 0 = 3,000–8,600. Taylor’s hypothesis is invoked to obtain a three-dimensional reconstruction of the scalar field in which then the corresponding scalar gradient trajectories can be computed. These are then used to examine the local structure of the mixture fraction with a focus on the scalar turbulent/non-turbulent interface. The latter is a layer that is located between the fully turbulent part of the jet and the outer flow. Using scalar gradient trajectories, we partition the turbulent scalar field into these three regions according to an approach developed by Mellado et al. (J Fluid Mech 626:333–365, 2009). Based on the latter, we investigate the probability to find the respective regions as a function of the radial distance to the centerline, which turns out to reveal the meandering nature of the scalar T/NT interface layer as well as its impact on the local structure of the turbulent scalar field.

A novel forcing technique to simulate turbulent mixing in a decaying scalar field

To realize the full potential of Direct Numerical Simulation in turbulent mixing studies, it is necessary to develop numerical schemes capable of sustaining the flow physics of turbulent scalar quantities. In this work, a new scalar field forcing technique, termed “linear scalar forcing,” is presented and evaluated for passive scalars. It is compared to both the well-known mean scalar gradient forcing technique and a low waveshell spectral forcing technique. The proposed forcing is designed to capture the physics of one-time scalar variance injection and the subsequent self-similar turbulent scalar field decay, whereas the mean scalar gradient forcing and low waveshell forcing techniques are representative of continuous scalar variance injection. The linear scalar forcing technique is examined over a range of Schmidt numbers, and the behavior of the proposed scalar forcing is analyzed using single and two-point statistics. The proposed scalar forcing technique is found to be perfectly isotropic, preserving accepted scalar field statistics (fluxes) and distributions (scalar quantity, dissipation rate). Additionally, it is found that the spectra resulting from the three scalar forcing techniques are comparable for unity Schmidt number conditions, but differences manifest at high Schmidt numbers. These disparities are reminiscent of those reported between scaling arguments suggested by theoretical predictions and experimental results for the viscous-convective subrange.

Hierarchical Parcel-Swapping Representation of Turbulent Mixing. Part 1. Formulation and Scaling Properties

An economical representation of effects of turbulence on the time-evolving structure of diffusive scalar fields is obtained by introducing a hierarchical (tree) network connecting fluid parcels, with effects of turbulent advection represented by swapping pairs of sub-trees at rates determined by turbulence time scales associated with the sub-trees. The fluid parcels reside at the base of the tree. The tree structure partitions the fluid parcels into adjacent pairs (or more generally, p-tuples). Adjacent parcels intermix at rates governed by diffusion time scales based on molecular diffusivities and parcel sizes. This simple procedure efficiently accomplishes long-standing objectives of turbulent mixing model development, such as generating physically based time histories of fluid-parcel nearest-neighbor encounters and the associated spatial structure of turbulent scalar fields. Correspondences between features of the hierarchical formulation and turbulent mixing phenomenology, both generic and case-specific, are noted.

Toward low‐noise synthetic turbulent inflow conditions for aeroacoustic calculations

SUMMARYThe accuracy of boundary conditions for computational aeroacoustics is a well‐known challenge, due in part to the necessity of truncating the flow domain and replacing the analytical boundary conditions at infinity with numerical boundary conditions. In particular, the inflow boundary condition involving turbulent velocity or scalar fields is likely to introduce spurious waves into the domain, therefore degrading the flow behavior and deteriorating the physical acoustic waves. In this work, a method to generate low‐noise, divergence‐free, synthetic turbulence for inflow boundary conditions is proposed. It relies on the classical view of turbulence as a superposition of random eddies convected with the mean flow. Within the proposed model, the vector potential and the requirement that the individual eddies must satisfy the linearized momentum equations about the mean flow are used. The model is tested using isolated eddies convected through the inflow boundary and an experimental benchmark data for spatially decaying isotropic turbulence. Copyright © 2013 John Wiley & Sons, Ltd.

Self-similar decay and mixing of a high-Schmidt-number passive scalar in an oscillating boundary layer in the intermittently turbulent regime

AbstractWe performed numerical simulations of dissolved oxygen (DO) transfer from a turbulent flow, driven by periodic boundary-layer turbulence in the intermittent regime, to underlying DO-absorbing organic sediment layers. A uniform initial distribution of oxygen is left to decay (with no re-aeration) as the turbulent transport supplies the sediment with oxygen from the outer layers to be absorbed. A very thin diffusive sublayer at the sediment–water interface (SWI), caused by the high Schmidt number of DO in water, limits the overall decay rate. A decomposition of the instantaneous decaying turbulent scalar field is proposed, which results in the development of similarity solutions that collapse the data in time. The decomposition is then tested against the governing equations, leading to a rigorous procedure for the extraction of an ergodic turbulent scalar field. The latter is composed of a statistically periodic and a steady non-decaying field. Temporal averaging is used in lieu of ensemble averaging to evaluate flow statistics, allowing the investigation of turbulent mixing dynamics from a single flow realization. In spite of the highly unsteady state of turbulence, the monotonically decaying component is surprisingly consistent with experimental and numerical correlations valid for steady high-Schmidt-number turbulent mass transfer. Linearly superimposed onto it is the statistically periodic component, which incorporates all the features of the non-equilibrium state of turbulence. It is modulated by the evolution of the turbulent coherent structures driven by the oscillating boundary layer in the intermittent regime, which are responsible for the violent turbulent production mechanisms. These cause, in turn, a rapid increase of the turbulent mass flux at the edge of the diffusive sublayer. This outer-layer forcing mechanism drives a periodic accumulation of high scalar concentration levels in the near-wall region. The resulting modulated scalar flux across the SWI is delayed by a quarter of a cycle with respect to the wall-shear stress, consistently with the non-equilibrium state of the turbulent mixing.

Gradient trajectory analysis in a Jet flow for turbulent combustion modelling

Based on planar high-speed Rayleigh scattering measurements of the mixture fraction Z of propane discharging from a turbulent round jet into co-flowing carbon dioxide at nozzle-based Reynolds numbers Re 0 = 3000–8600, we use scalar gradient trajectories to investigate the local structure of the turbulent scalar field with a focus on the scalar turbulent/non-turbulent interface. The latter is located between the fully turbulent part of the jet and the outer flow. Using scalar gradient trajectories, we partition the turbulent scalar field into these three regions according to an approach developed by Mellado et al. (J.P. Mellado, L. Wang, and N. Peters, Gradient trajectory analysis of a scalar field with external intermittency, J. Fluid Mech. 626 (2009), pp. 333–365.). Based on these different regions, we investigate in a next step zonal statistics of the scalar probability density function (pdf) P(Z) as well as the scalar difference along the trajectory ΔZ and its mean scalar value Zm , where the latter two quantities are used to parameterize the scalar profile along gradient trajectories. We show that the scalar pdf P(Z) can be reconstructed from zonal gradient trajectory statistics of the joint pdf P(Zm , ΔZ). Furthermore, on the one hand we relate our results for the scalar turbulent/non-turbulent interface to the findings made in other experimental and numerical studies of the turbulent/non-turbulent interface, and on the other hand discuss them in the context of the flamelet approach and the modelling of pdfs in turbulent non-premixed combustion. Finally, we compare the zonal statistics for P(Z) with the composite model of Effelsberg and Peters (E. Effelsberg and N. Peters, A composite model for the conserved scalar pdf, Combust. Flame 50 (1983), pp. 351–360) and observe a very good qualitative and quantitative agreement.

Dissipation element analysis in experimental and numerical shear flow

Tomographic particle-image velocimetry (TPIV) measurements and direct numerical simulations (DNS) are performed to analyze the small and large scale structure of channel flow data, i.e., the fields of the fluctuations of the three velocity components and the turbulent kinetic energy, respectively, are investigated by dissipation element analysis which is a new method to gain statistical information from turbulent scalar fields based on distinct, space-filling elements.In the inertial range, the conditional mean of the experimental turbulent kinetic energy difference reveals a scaling in the range of the expected characteristic 2/3-Kolmogorov scaling. This is in excellent agreement with the numerical results from the channel flow DNS and comparative data from the DNS of a homogeneous shear flow. Furthermore, the shape of the marginal pdfs of the normalized element length and the scalar difference, i.e., the linear rise and the exponential decay, is shown to agree with the findings from the associated DNS. On average, the dissipation elements possess an orientation over a certain bandwidth in the mean flow direction.

Structure Function Analysis of Two-Scale Scalar Ramps. Part II: Ramp Characteristics and Surface Renewal Flux Estimation

Ramp features in the turbulent scalar field are associated with turbulent coherent structures, which dominate energy and mass fluxes in the atmospheric surface layer. Although finer scale ramp-like shapes embedded within larger scale ramp-like shapes can readily be perceived in turbulent scalar traces, their presence has largely been overlooked in the literature. We demonstrate the signature of more than one ramp scale in structure functions of the turbulent scalar field measured from above bare ground and two types of short plant canopies, using structure-function time lags ranging in scale from isotropic to larger than the characteristic coherent structures. Spectral analysis of structure functions was used to characterize different scales of turbulent structures. By expanding structure function analysis to include two ramp scales, we characterized the intermittency, duration, and surface renewal flux contribution of the smallest (i.e., Scale One) and the dominant (i.e., Scale Two) coherent structure scales. The frequencies of the coherent structure scales increase with mean wind shear, implying that both Scale One and Scale Two are shear-driven. The embedded Scale One turbulent structure scale is ineffectual in the surface-layer energy and mass transport process. The new method reported here for obtaining surface renewal-based scalar exchange works well over bare ground and short canopies under unstable conditions, effectively eliminating the α calibration for these conditions and forming the foundation for analysis over taller and more complex surfaces.

The length distribution of streamline segments in homogeneous isotropic decaying turbulence

Based on the profile of the absolute value u of the velocity field ui along streamlines, the latter are partitioned into segments at their extreme points as proposed by Wang [J. Fluid Mech. 648, 183–203 (2010)]10.1017/S0022112009993041. It is found that the boundaries of all streamline segments, i.e., points where the gradient projected in streamline direction ∂u/∂s vanishes, define a surface in space. This surface also contains all local extreme points of the scalar u-field, i.e., points where the gradient in all directions of the field of the absolute value of the velocity and thereby those of the turbulent kinetic energy (k = u2/2, where k is the instantaneous turbulent kinetic energy) vanishes. Such points also include stagnation points of the flow field, which are absolute minimum points of the turbulent kinetic energy. As local extreme points are the ending points of dissipation elements, an approach for space-filling geometries in turbulent scalar fields, such elements in the turbulent kinetic energy field also end and begin on the surface and the temporal evolution of dissipation elements and streamline segments are intimately related. Streamline segments by construction evolve both morphologically and topologically. A morphological evolution of a streamline segment corresponds to a continuous deformation when it is subject to stretching or compression and thus also implies a continuous evolution of the arclength l with time. Such an evolution does not change the number of the overall streamline segments and hence does not involve counting. On the other hand a topological evolution corresponds to either a cutting of a large segment into smaller ones or a connection of two smaller ones to form a larger segment. Such a process changes the number of segments and thus involves counting. This change of integer variables (i.e., counting) yields discrete jumps in the length of the streamline segment which are discontinuous in time. Following the terminology by Schaefer et al. [“Fast and slow changes of the length of gradient trajectories in homogenous shear turbulence,” in Advances in Turbulence XII, edited by B. Eckhardt (Springer-Verlag, Berlin, 2009), pp. 565–572] we will refer to the morphological part of the evolution of streamline segments as slow changes while the topological part of the evolution is referred to as fast changes. This separation yields a transport equation for the probability density function (pdf) P(l) of the arclength l of streamline segments in which the slow changes translate into a convection and a diffusion term when terms up to second order are included and the fast changes yield integral terms. The overall temporal evolution (morphological and topological) of the arclength l of streamline segments is analyzed and associated with the motion of the above isosurface. This motion is diffusion controlled for small segments, while large segments are mainly subject to strain and pressure fluctuations. The convection velocity corresponds to the first order jump moment, while the diffusion term includes the second order jump moment. It is concluded, both theoretically and from direct numerical simulations (DNS) data of homogeneous isotropic decaying turbulence at two different Reynolds numbers, that the normalized first order jump moment is quasi-universal, while the second order one is proportional to the inverse of the square root of the Taylor based Reynolds number \documentclass[12pt]{minimal}\begin{document}$Re_{\lambda }^{-1/2}$\end{document}Reλ−1/2. Its inclusion thus represents a small correction in the limit of large Reynolds numbers. Numerical solutions of the pdf equation yield a good agreement with the pdf obtained from the DNS data. The interplay of viscous drift acting on small segments and linear strain acting on large segments yield, as it has already been concluded for dissipation elements, that the mean length of streamline segments should scale with Taylor microscale.

High-resolution turbulent scalar field measurements in an optically accessible internal combustion engine

High-resolution planar laser-induced fluorescence (PLIF) measurements were performed in an optically accessible internal combustion engine to investigate the evolution of the turbulent mixing process during the intake and compression strokes. The PLIF measurements were used to analyze the important turbulent length scales, scalar energy and dissipation spectra, and mean scalar gradients. The fluorescence images had sufficient spatial resolution and integrity to resolve all of the fine-scale features of the flow, allowing for direct determination of the Batchelor length scale. The integral and Taylor scales were also determined from two-point spatial correlations of the fluctuating scalar field using an appropriately defined mean scalar value. The general morphology of the scalar field and the measured integral, Taylor and Batchelor length scales were found to be largely independent of engine speed and intake pressure, but increased as the engine cycle progressed through the intake and compression strokes. The measured Batchelor scales ranged from 22 to 54 μm; the integral scales ranged from 1.8 to 3.5 mm; and the Taylor microscales ranged from 0.6 to 1.2 mm. The Taylor and integral scale values were comparable to values reported in the literature from in-cylinder velocity measurements. The mean scalar gradient, a measure of the fine-scale mixing rate, monotonically decreased as the engine cycle advanced. High-resolution measurements of this type are important in the development and validation of future engine combustion models used in computer simulations.