Uniform Mean Gradient 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.

Péclet-number dependence of small-scale anisotropy of passive scalar fluctuations under a uniform mean gradient in isotropic turbulence

We study passive scalar fluctuations convected by statistically stationary homogeneous isotropic turbulence under a uniform mean scalar gradient. In order to elucidate the parameter dependence of small-scale statistics of scalar fluctuations, we conduct direct numerical simulations of passive scalar turbulence with 59 different combinations of Reynolds number and Schmidt number. For all the cases, we compute time-average statistics of various quantities, which include the scalar derivative skewness and flatness, the ratio of parallel-to-perpendicular scalar-gradient variances, and the anisotropy parameter recently proposed (Hill, Phys. Rev. Fluids, vol. 2, 2017, 094601). Notably, the degree of small-scale anisotropy of passive scalar fluctuation is characterised by a universal function of the Peclet number , where is the root mean square velocity, the Taylor microscale of scalar fluctuation, the mass diffusivity. In the definition of the Peclet number, the use of , rather than the Taylor microscale of velocity fluctuation, is key to collapsing the data of different Reynolds and Schmidt numbers. When the Peclet number is low, large-scale anisotropic scalar structures emerge irrespective of the Reynolds number. These structures are elongated along the direction of the uniform mean scalar gradient, and their size is significantly larger than the integral length scale of velocity fluctuation.

Anisotropic fluctuations in passive scalar turbulence with a uniform mean gradient

Anisotropic fluctuations in passive scalar turbulence with a uniform mean gradient

A Lagrangian study of turbulent mixing: forward and backward dispersion of molecular trajectories in isotropic turbulence

Statistics of the trajectories of molecules diffusing via Brownian motion in a turbulent flow are extracted from simulations of stationary isotropic turbulence, using a postprocessing approach applicable in both forward and backward reference frames. Detailed results are obtained for Schmidt numbers ($Sc$) from 0.001 to 1000 at Taylor-scale Reynolds numbers up to 1000. The statistics of displacements of single molecules compare well with the earlier theoretical work of Saffman (J. Fluid Mech. vol. 8, 1960, pp. 273–283) except for the scaling of the integral time scale of the fluid velocity following the molecular trajectories. For molecular pairs we extend Saffman’s theory to include pairs of small but finite initial separation, which is in excellent agreement with numerical results provided that data are collected at sufficiently small times. At intermediate times the separation statistics of molecular pairs exhibit a more robust Richardson scaling behaviour than for the fluid particles. The forward scaling constant is very close to 0.55, whereas the backward constant is approximately 1.53–1.57, with a weak Schmidt number dependence, although no scaling exists if $Sc\ll 1$ at the Reynolds numbers presently accessible. An important innovation in this work is to demonstrate explicitly the practical utility of a Lagrangian description of turbulent mixing, where molecular displacements and separations in the limit of small backward initial separation can be used to calculate the evolution of scalar fluctuations resulting from a known source function in space. Lagrangian calculations of the production and dissipation rates of the scalar fluctuations are shown to agree very well with Eulerian results for the case of passive scalars driven by a uniform mean gradient. Although the Eulerian–Lagrangian comparisons are made only for $Sc\sim O(1)$, the Lagrangian approach is more easily extended to both very low and very high Schmidt numbers. The well-known scalar dissipation anomaly is accordingly also addressed in a Lagrangian context.

Structure functions and applicability of Yaglom's relation in passive-scalar turbulent mixing at low Schmidt numbers with uniform mean gradient

An extensive direct numerical simulation database over a wide range of Reynolds and Schmidt numbers is used to examine the Schmidt number dependence of the structure function of passive scalars and the applicability of the so-called Yaglom's relation in isotropic turbulence with a uniform mean scalar gradient. For the moderate Reynolds numbers available, the limited range of scales in scalar fields of very low Schmidt numbers (as low as 1/2048) is seen to lead to weaker intermittency, and weaker alignment between velocity gradients and principal strain rates. Strong departures from both Obukhov-Corrsin scaling for second-order structure functions and Yaglom's relation for the mixed velocity-scalar third-order structure function are observed. Evaluation of different terms in the scalar structure function budget equation assuming statistical stationarity in time shows that, if the Schmidt number is very low, at intermediate scales production and diffusion terms (instead of advection) are major contributors in the balance against dissipation.

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.

Bounded stochastic shell mixing model for turbulent mixing of multiple scalars with arbitrary diffusivities

In the paper by Xia et al. [Flow, Turbul. Combust. 85, 689 (2010)], we derived a stochastic shell mixing model (SSMM) for turbulent mixing of multiple scalars with arbitrary diffusivities. The framework uses a Monte Carlo scheme to advance notional particles that move with the fluid and has scalar concentrations that are subdivided into “shells” that loosely represent a spectral decomposition of the scalar fluctuations. The variance within each shell is equivalent to the scalar spectrum evaluated at the corresponding wavenumber. Small-scale phenomena such as differential diffusion and the dependence of mixing on the scalar length scale are captured by the inherently spectral nature of the model. Furthermore, the sum of the scalar over shells for each notional particle simultaneously provides a fine-grained representation of the joint scalar probability density function (PDF). This aspect of the model was introduced, but not fully explored in the earlier paper, which focused on scalar fluctuations arising from a uniform mean gradient. At the level of this model, the scalar PDF reduces to a joint Gaussian. In this study, we explore more complex mixing scenarios, including the initial double delta function, which arises when two streams are brought into sudden contact. This introduces two important complexities that are the focus of this paper. The first is the shape changes to the PDF that result as the scalar PDF relaxes into a Gaussian form. The second is due to the bounds that the scalar must satisfy. Monte Carlo schemes inherently violate bounds; we have modified the model so as to ensure the bounds are satisfied by nearly all of the particles (bounded SSMM or BSSMM). Extensive comparisons between the BSSMM predictions and direct numerical simulations (DNS) are made for two decaying scalars (with different diffusivities) in forced, isotropic turbulence. Overall the agreement is very good. Additionally, we test a conjecture made in deriving the BSSMM model, namely the rate of mixing of a scalar is assumed to be a function of its spectrum, but not its PDF. Using DNS, we contrast the mixing rate of two scalars with identical PDFs, but different spectra against two scalars with identical spectra, but different PDFs. We conclude from this comparison that the mixing rate is much more sensitive to the scalar spectrum than to its PDF, in support of the assumption used to derive the new mixing model.

Universality and anisotropy in passive scalar fluctuations in turbulence with uniform mean gradient

Passive scalar and velocity fluctuations in homogeneous isotropic turbulence with or without mean scalar gradient are studied by using high-resolution direct numerical simulation (DNS). The local scaling exponents of the velocity structure functions are newly computed at larger Reynolds number, and the values at their plateau in the inertial range are found to be consistent with the previous DNS and experimental data. The 4/3 law for the velocity–scalar mixed correlation and the high-order structure functions of the passive scalar increment are analyzed in terms of the Legendre polynomial expansion. It is shown theoretically that the contributions from the second order in the expansion can be removed by taking the average over three directions parallel and perpendicular to the mean gradient, and it is found numerically that the contributions higher than the fourth order are negligible. The scaling exponents of the isotropic part of the scalar structure functions under the mean gradient are found to have the well-developed scaling range and are smaller than those in the isotropic case. The different behavior in the local scaling exponents with or without the mean gradient is examined and the universality of the scaling exponents is discussed.

On the role of vortical structures for turbulent mixing using direct numerical simulation and wavelet-based coherent vorticity extraction

The influence of vortical structures on the transport and mixing of passive scalars is investigated. Initial conditions are taken from a direct numerical simulation database of forced homogeneous isotropic turbulence, with passive scalar fluctuations, driven by a uniform mean gradient, are performed for Taylor microscale Reynolds numbers (R λ) of 140 and 240, and Schmidt numbers 1/8 and 1. For each R λ, after reaching a fully developed turbulent regime, which is statistically steady, the Coherent Vorticity Extraction is applied to the flow. It is shown that the coherent part is able to preserve the vortical structures with only less than 4% of wavelet coefficients while retaining 99.9% of energy. In contrast, the incoherent part is structureless and contains negligible energy. By taking the total, coherent and incoherent velocity fields in turn as initial conditions, new simulations were performed without forcing while the uniform mean scalar gradient is maintained. It is found that the results from simulations with total and coherent velocity fields as initial conditions are very similar. In contrast, the time integration of the incoherent flow exhibits its primarily dissipative nature. The evolutions of passive scalars at Schmidt numbers 1/8 and 1 advected by the total, coherent or incoherent velocity suggest that the vortical structures retained in the coherent part play a dominant role in turbulent transport and mixing. Indeed, the total and coherent flows give almost the same time evolution of the scalar variance, scalar flux and mean scalar dissipation, while the incoherent flow only gives rise to weak scalar diffusion.

Scalar flux spectrum in isotropic steady turbulence with a uniform mean gradient

The scaling law of a scalar flux spectrum (velocity-scalar cospectrum) in the inertial convective range of passive scalar turbulence under a uniform mean scalar gradient is examined using direct numerical simulation with a resolution of up to 20483 grid points. When the Reynolds number Reλ is increased up to Reλ=585, the scalar flux spectrum tends to obey the power law k−7∕3, as predicted by Lumley [J. Atmos. Sci. 21, 99 (1964); Phys. Fluids 10, 855 (1967)], with a nondimensional constant of Cuθ=1.50±0.08 at Reλ=585. The Reλ effect on the scaling of the scalar flux spectrum is well compensated using the mean molecular destruction of the scalar flux ϵ¯uθ. The Reλ dependence of Cuθ is also compared with the results of previous studies, and its asymptotic state at an infinite Reynolds number is discussed.

Schmidt number effects on turbulent transport with uniform mean scalar gradient

We study by direct numerical simulations the effects of Schmidt number (Sc) on passive scalars mixed by forced isotropic and homogeneous turbulence. The scalar field is maintained statistically stationary by a uniform mean gradient. We consider the scaling of spectra, structure functions, local isotropy and intermittency. For moderately diffusive scalars with Sc=1/8 and 1, the Taylor-scale Reynolds number of the flow is either 140 or 240. A modest inertial-convective range is obtained in the spectrum, with a one-dimensional Obukhov–Corrsin constant of about 0.4, consistent with experimental data. However, the presence of a spectral bump makes a firm assessment somewhat difficult. The viscous-diffusive range is universal when scaled by Obukhov–Corrsin variables. In a second set of simulations we keep the Taylor-microscale Reynolds number fixed at 38 but vary Sc from 1/4 to 64 (a range of over two decades), roughly by factors of 2. We observe a gradual evolution of a −1 roll-off in the viscous-convective region as Sc increases, consistent with Batchelor’s predictions. In the viscous-diffusive range the spectra follow Kraichnan’s form well, with a coefficient that depends weakly on Sc. The breakdown of local isotropy manifests itself through differences between structure functions with separation distances in directions parallel and perpendicular to the mean scalar gradient, as well as via finite values of odd-order moments of scalar gradient fluctuations and of mixed velocity-scalar gradient correlations. However, all these indicators show, to varying degrees, an increasing tendency to isotropy with increasing Sc. The moments of scalar gradients and the scalar dissipation rate peak at Sc≈4. The intermittency exponent for the scale-range between the Kolmogorov and Batchelor scales is found to decrease with Sc, suggesting qualitative consistency with previous dye experiments in water [Sc=O(1000)].

A spectral study of differential diffusion of passive scalars in isotropic turbulence

In a companion paper, Ulitsky & Collins (2000) applied the eddy-damped quasi-normal Markovian (EDQNM) turbulence theory to the mixing of two inert passive scalars with different diffusivities in stationary isotropic turbulence. Their paper showed that a rigorous application of the EDQNM approximation leads to a scalar covariance spectrum that violates the Cauchy–Schwartz inequality over a range of wavenumbers. The violation results from the improper functionality of the inverse diffusive time scales that arise from the Markovianization of the time evolution of the triple correlations. The modified inverse time scale they proposed eliminates this problem and allows meaningful predictions of the scalar covariance spectrum both with and without a uniform mean gradient.This study uses the modified EDQNM model to investigate the spectral dynamics of differential diffusion. Consistent with recent DNS results by Yeung (1996), we observe that whereas spectral transfer is predominantly from low to high wavenumbers, spectral incoherence, being of molecular origin, originates at high wavenumbers and is transferred in the opposite direction by the advective terms. Quantitative comparisons between the EDQNM model and the DNS show good agreement. In addition, the model is shown to give excellent estimates for the dissipation coefficient defined by Yeung (1998).We show that the EDQNM scalar covariance spectrum for two scalars with different molecular diffusivities can be approximated by the EDQNM autocorrelation spectrum for a scalar with molecular diffusivity equal to the arithmetic mean of the first two scalars. The result is exact for the case of an isotropic scalar and is shown to be a very good approximation for the scalar with a uniform mean gradient. We then exploit this relationship to derive a simple formula for the correlation coefficient of two differentially diffusing scalars as a function of their two Schmidt numbers and the turbulent Reynolds number. A comparison of the formula with the EDQNM model shows the model predicts the correct Reynolds number scaling and qualitatively predicts the dependence on the Schmidt numbers.To investigate the degree of local versus non-local transfer of the scalar covariance spectrum, we divided the energy spectrum into three ranges corresponding to the energy-containing eddies, the inertial range, and the dissipation range. Then, by conditioning the scalar transfer on the energy contained within each of the three ranges, we have determined that the transfer process is dominated first by local interactions (local transfer) followed by non-local interactions leading to local transfer. Non-local interactions leading to non-local transfer are found to be significant at the higher wavenumbers. This result has important implications for defining simpler spectral models that potentially can be applied to more complex engineering flows.

Dependence of turbulent scalar flux on molecular Prandtl number

The dependence of turbulent scalar flux on molecular Prandtl number is studied by direct numerical simulation of statistically stationary isotropic turbulence with uniform mean gradient of temperature. Both Reynolds averaged scalar flux and subgrid scalar flux are investigated at molecular Prandtl numbers ranging from 0.1 to 3.0. In order to consider the Reynolds number effect, two cases of grid resolution are computed, i.e., 1283 and 2563, with the Taylor-scale Reynolds numbers equaling 30 and 50, respectively. The turbulent Prandtl number is used to characterize the turbulent scalar flux. It is found that both Reynolds averaged turbulent Prandtl number (simplified as turbulent Prandtl number hereafter) and subgrid Prandtl number change with molecular Prandtl number significantly. The turbulent Prandtl number has been found to be a linearly reciprocal function of molecular Prandtl number, whereas the subgrid Prandtl number takes a minimum around Pr=1. The appearance of minimum subgrid Prandtl number around Pr=1 can be well understood based on the analysis of transfer spectrum of scalar flux.

Dynamics of scalar dissipation in isotropic turbulence: a numerical and modelling study

The physical mechanisms underlying the dynamics of the dissipation of passive scalar fluctuations with a uniform mean gradient in stationary isotropic turbulence are studied using data from direct numerical simulations (DNS), at grid resolutions up to 5123. The ensemble-averaged Taylor-scale Reynolds number is up to about 240 and the Schmidt number is from ⅛ to 1. Special attention is given to statistics conditioned upon the energy dissipation rate because of their important role in the Lagrangian spectral relaxation (LSR) model of turbulent mixing. In general, the dominant physical processes are those of nonlinear amplification by strain rate fluctuations, and destruction by molecular diffusivity. Scalar dissipation tends to form elongated structures in space, with only a limited overlap with zones of intense energy dissipation. Scalar gradient fluctuations are preferentially aligned with the direction of most compressive strain rate, especially in regions of high energy dissipation. Both the nature of this alignment and the timescale of the resulting scalar gradient amplification appear to be nearly universal in regard to Reynolds and Schmidt numbers. Most of the terms appearing in the budget equation for conditional scalar dissipation show neutral behaviour at low energy dissipation but increased magnitudes at high energy dissipation. Although homogeneity requires that transport terms have a zero unconditional average, conditional molecular transport is found to be significant, especially at lower Reynolds or Schmidt numbers within the simulation data range. The physical insights obtained from DNS are used for a priori testing and development of the LSR model. In particular, based on the DNS data, improved functional forms are introduced for several model coefficients which were previously taken as constants. Similar improvements including new closure schemes for specific terms are also achieved for the modelling of conditional scalar variance.

Direct numerical simulation of a statistically stationary, turbulent reacting flow

An inhomogeneous, non-premixed, stationary, turbulent, reacting model flow that is accessible to direct numerical simulation (DNS) is described for investigating the effects of mixing on reaction and for testing mixing models. The mixture-fraction-progress-variable approach of Bilger is used, with a model, finite-rate, reversible, single-step thermochemistry, yielding non-trivial stationary solutions corresponding to stable reaction and also allowing local extinction to occur. There is a uniform mean gradient in the mixture fraction, which gives rise to stationarity as well as a flame brush. A range of reaction zone thicknesses and Damkohler numbers are examined, yielding a broad spectrum of behaviour, including thick and thin flames, local extinction and near equilibrium. Based on direct numerical simulations, results from the conditional moment closure (CMC) and the quasi-equilibrium distributed reaction (QEDR) model are evaluated. Large intermittency in the scalar dissipation leads to local extinction in the DNS. In regions of the flow where local extinction is not present, CMC and QEDR based on the local scalar dissipation give good agreement with the DNS.M This article features multimedia enhancements available from the supplemental page in the online journal.

Application of the eddy damped quasi-normal Markovian spectral transport theory to premixed turbulent flame propagation

The eddy damped quasi-normal Markovian (EDQNM) turbulence theory was applied to a modified Kuramoto–Sivashinsky field equation to develop a spectral model for investigating the single and two-point scalar statistics associated with a flame front (treated as a passive scalar interface) propagating through isotropic turbulence. As a result of the presence of a uniform mean gradient in the scalar field, all correlations involving the scalar were found to be functions of both the wave number, k, and μ, the cosine of the angle between the k vector and the mean gradient vector. An infinite Legendre expansion separated out the wave number and angle dependencies, where the first term in each series accounted for the isotropic contribution to the correlations and the higher order terms accounted for the anisotropy introduced as a result of the mean gradient. It was found that while strong anisotropy existed in the scalar field at short times, at steady state the scalar field became nearly isotropic. A parameter study was then conducted to ascertain the effect of independently varying u′/sL and Reλ (where u′ is the rms velocity, sL is the laminar burning velocity, and Reλ is the Reynolds number based on the Taylor microscale). The turbulent burning velocity increased with increases in either u′/sL or Reλ; however, the model predicted a finite turbulent burning velocity as u′/sL→∞, even though flame quenching was not accounted for. This finite asymptote for the burning velocity was traced to the constitutive relationship used for the flame thickness and the ratio of the Markstein length to the flame thickness. It was also shown that the dominant wrinkling of the flame surface and subsequent contribution to the turbulent burning velocity occurred at smaller and smaller length scales as the inertial range of the scalar spectrum increased. Single point models will therefore have great difficulty reproducing this significant result. Scalar spectra exhibited changes over all wave numbers as either u′/sL or Reλ was modified. Transfer spectra, which arose in the form of convolution integrals as a result of the advection and propagation processes, were also analyzed and separated into their pairwise spectral interactions to determine which nonlinear terms in each integral were dominant.

Uniform mean scalar gradient in grid turbulence: Asymptotic probability distribution of a passive scalar

An asymptotic self-similar solution is obtained for the one-point probability density function (pdf) equation, of a passive scalar with uniform mean gradient in incompressible homogeneous turbulence. It is argued that the same solution should be a valid approximation when turbulence is generated in a high-quality wind tunnel. The asymptotic pdf shape is a unique function of the conditional expectation of the normalized scalar dissipation rate. The mean scalar gradient modifies the scalar pdf shape only if the conditional expected velocity component in the direction of the mean gradient is a nonlinear function of scalar fluctuation value. Experimental data from wind tunnel studies are consistent with the sign and scale of the changes produced by these nonlinearities.

EDQNM model of a passive scalar with a uniform mean gradient

Dynamic equations for the scalar autocorrelation and scalar-velocity cross correlation spectra have been derived for a passive scalar with a uniform mean gradient using the Eddy Damped Quasi Normal Markovian (EDQNM) theory. The presence of a mean gradient in the scalar field makes all correlations involving the scalar axisymmetric with respect to the axis pointing in the direction of the mean gradient. Equivalently, all scalar spectra will be functions of the wave number k and the cosine of the azimuthal angle designated as μ. In spite of this complication, it is shown that the cross correlation vector can be completely characterized by a single scalar function Q(k). The scalar autocorrelation spectrum, in contrast, has an unknown dependence on μ. However, this dependency can be expressed as an infinite sum of Legendre polynomials of μ, as first suggested by Herring [Phys. Fluids 17, 859 (1974)]. Furthermore, since the scalar field is initially zero, terms beyond the second order of the Legendre expansion are shown to be exactly zero. The energy, scalar autocorrelation, and scalar-velocity cross correlation were solved numerically from the EDQNM equations and compared to results from direct numerical simulations. The results show that the EDQNM theory is effective in describing single-point and spectral statistics of a passive scalar in the presence of a mean gradient.

Similarity states of passive scalar transport in isotropic turbulence

By simple analytical and large-eddy simulations, the time evolution of the kinetic energy and scalar variance in decaying isotropic turbulence transporting passive scalars are determined. The evolution of a passive scalar field with and without a uniform mean gradient is considered. First, similarity states of the flow during the final period of decay are discussed. Exact analytical solutions may be obtained, and these depend only on the form of the energy and scalar-variance spectra at low wave numbers, and the molecular transport coefficients. The solutions for a passive scalar field with mean-scalar gradient are of special interest, and we find that the scalar variance may grow or decay asymptotically in the final period, depending on the initial velocity distribution. Second, similarity states of the flow at high Reynolds and Péclet numbers are considered. Here it is assumed that the solutions also depend on the low-wave-number spectral coefficients, but not on the molecular transport coefficients. This results in a nonlinear dependence of the kinetic energy and scalar variance on the spectral coefficients, in contrast to the final period results. The analytical results obtained may be exact when the similarity solutions depend only on spectral coefficients that are time invariant. The present analysis also leads directly to a similarity state for a passive scalar field with uniform mean scalar gradient. Last, large-eddy simulations of the flow field are performed to test the theoretical results. Asymptotic similarity states at large times in the simulations are obtained and found to be in good agreement with predictions of the analysis. Several dimensionless quantities are also determined, which compare favorably to earlier experimental results. An argument for the inertial subrange scaling of the scalar-flux spectrum is presented, which yields a spectrum proportional to the scalar gradient and decaying as k−7/3. This result is partially supported by the small-scale statistics of the large-eddy simulations.

Passive scalar fluctuations with and without a mean gradient: A numerical study.

We report a numerical study of a passive scalar advected by a random incompressible Gaussian velocity field. The calculations are carried out on a two-dimensional square lattice. Depending on the two dimensionless parameters in the problem, the scalar fluctuations at the center can be Gaussian, exponential, or stretched exponential. This is true not only when the passive scalar is subjected to a uniform mean gradient but also true when a new alternating boundary condition is used which results in a mean scalar profile that has a vanishing gradient in the central region.