Passive Scalar Spectrum Research Articles

Effect of Schmidt Number on Forced Isotropic Turbulence with Passive Scalars

Traditionally, Fourier spectra have been employed to gain a deeper understanding of turbulence flow structures. The investigation of isotropic forced turbulence with passive scalars offers a straightforward means to examine the disparities between velocity and passive scalar spectra. This flow configuration has been extensively studied in the past, encompassing a range of Reynolds and Schmidt numbers. In this present study, direct numerical simulations (DNS) of this flow are conducted at sufficiently high Reynolds numbers, enabling the formation of a wide inertial range. The primary focus of this investigation is to quantitatively assess the variations in scalar spectra with the Schmidt number (Sc). The spectra exhibit a transition from a k−5/3 scaling for low Sc to a k−4/3 scaling for high Sc. The emergence of the latter power law becomes evident at Sc = 2, with its width expanding as Sc increases. To gain further insights into the underlying flow structures, a statistical analysis is performed by evaluating quantities aligned with the principal axes of the strain field. The study reveals that enstrophy is primarily influenced by the vorticity aligned with the intermediate principal strain axis, while the scalar gradient variance is predominantly controlled by the compressive strain. To provide a clearer understanding of the differences between enstrophy and scalar gradient variance, joint probability density functions (PDFs) and visualizations of the budget terms for both quantities are presented. These visualizations serve to elucidate the distinctions between the two and offer insights into their respective behaviors.

A differential approximation model for passive scalar turbulence

Two-dimensional passive scalar turbulence is studied by means of a k-space diffusion model based on a third order differential approximation. This simple description of local nonlinear interactions in Fourier space is shown to provide a general expression, in line with previous seminal works, and appears to be suitable for various 2D turbulence problems. Steady state solutions for the spectral energy density of the flow are shown to recover the Kraichnan–Kolmogorov phenomenology of the dual cascade, while various passive scalar spectra, such as Batchelor or Obukhov–Corssin spectra are recovered as steady state solutions of the spectral energy density of the passive scalar. These analytical results are then corroborated by numerical solutions of the time evolving problem with energy and passive scalar injection and dissipation on a logarithmic wavenumber space grid over a large range of scales. The particular power law spectra that are obtained are found to depend mainly on the location of the kinetic and passive scalar energy injections. In particular, it is shown that by injecting the energy simultaneously at large and small scales, wave number spectra consistent with those of Nastrom–Gage can be obtained.

Scalar fluctuation and its dissipation in turbulent reacting flows

The dissipation rate of a scalar variance is related to the mean heat release rate in turbulent combustion. Mixture fraction is the scalar of interest for non-premixed combustion, and a reaction progress variable is relevant for premixed combustion. A great deal of work is conducted in past studies to understand the spectra of passive scalar transport in turbulent flows. A very brief summary of these studies to bring out the salient characteristics of the passive scalar spectrum is given first. Then, the classical analysis of the reactive scalar spectrum is revisited in the lights of recent understanding gained through analyzing the scalar spectrum deduced from direct numerical simulation data of both non-premixed and premixed combustion. The analysis shows that the reactive scalar spectral density in premixed combustion has a dependence on Karlovitz and Damköhler numbers, which comes through the mean scalar dissipation rate appearing in the spectral expression. In premixed combustion, the relevant scale for the scalar dissipation rate is shown to be of the order of the chemical length scale, and the dissipation rate is not influenced by the scales in the inertial-convective range unlike for the passive scalar dissipation rate. The scalar fluctuations produced near the chemical scales cascade exponentially to larger scales. These observations imply that the passive scalar models cannot be extended to premixed combustion.

Solenoidal Scaling Laws for Compressible Mixing.

Mixing of passive scalars in compressible turbulence does not obey the same classical Reynolds number scaling as its incompressible counterpart. We first show from a large database of direct numerical simulations that even the solenoidal part of the velocity field fails to follow the classical incompressible scaling when the forcing includes a substantial dilatational component. Though the dilatational effects on the flow remain significant, our main results are that both the solenoidal energy spectrum and the passive scalar spectrum assume incompressible forms, and that the scalar gradient essentially aligns with the most compressive eigenvalue of the solenoidal part, provided that only the solenoidal components are consistently used for scaling. A slight refinement of this statement is also pointed out.

Cascades of temperature and entropy fluctuations in compressible turbulence

Cascades of temperature and entropy fluctuations are studied by numerical simulations of stationary three-dimensional compressible turbulence with a heat source. The fluctuation spectra of velocity, compressible velocity component, density and pressure exhibit the $-5/3$ scaling in an inertial range. The strong acoustic equilibrium relation between spectra of the compressible velocity component and pressure is observed. The $-5/3$ scaling behaviour is also identified for the fluctuation spectra of temperature and entropy, with the Obukhov–Corrsin constants close to that of a passive scalar spectrum. It is shown by Kovasznay decomposition that the dynamics of the temperature field is dominated by the entropic mode. The average subgrid-scale (SGS) fluxes of temperature and entropy normalized by the total dissipation rates are close to 1 in the inertial range. The cascade of temperature is dominated by the compressible mode of the velocity field, indicating that the theory of a passive scalar in incompressible turbulence is not suitable to describe the inter-scale transfer of temperature in compressible turbulence. In contrast, the cascade of entropy is dominated by the solenoidal mode of the velocity field. The different behaviours of cascades of temperature and entropy are partly explained by the geometrical properties of SGS fluxes. Moreover, the different effects of local compressibility on the SGS fluxes of temperature and entropy are investigated by conditional averaging with respect to the filtered dilatation, demonstrating that the effect of compressibility on the cascade of temperature is much stronger than on the cascade of entropy.

Mixing, scalar boundedness, and numerical dissipation in large-eddy simulations

Numerical schemes for scalar transport and mixing in turbulent flows must be high-order accurate, and observe conservation and boundedness constraints. Discretization accuracy can be evaluated from the truncation error, and assessed by its dispersion and dissipation properties. Dispersion errors can cause violation of physical scalar bounds, whereas numerical dissipation is key to mitigating those violations. Numerical dissipation primarily alters the energy at small scales that are critical to turbulent mixing. Influence of additional dissipation on scalar mixing in large-eddy simulations (LES) of incompressible temporally evolving shear flow is examined in terms of the resolved passive-scalar field, Z¯. Scalar fields in flows with different mixing behavior, exhibiting both uniform and non-uniform mixed-fluid composition across a shear layer, are compared for different grid resolutions, subgrid-scale models, and scalar-convection schemes. Scalar mixing is assessed based on resolved passive scalar probability density function (PDF), variance, and spectra. The numerical-dissipation influence on mixing is found to depend on the nature of the flow. Mixing metrics sensitive to numerical dissipation are applied to examine the performance of limiting methods employed to mitigate unphysical scalar excursions. Two approaches, using a linear-scaling limiter for finite-volume schemes and a monotonicity-preserving limiter for finite-difference schemes, are studied. Their performance with respect to accuracy, conservation, and boundedness is discussed.

Passive scalar mixing and decay at finite correlation times in the Batchelor regime

An elegant model for passive scalar mixing and decay was given by Kraichnan (Phys. Fluids, vol. 11, 1968, pp. 945–953) assuming the velocity to be delta correlated in time. For realistic random flows this assumption becomes invalid. We generalize the Kraichnan model to include the effects of a finite correlation time, $\unicode[STIX]{x1D70F}$, using renewing flows. The generalized evolution equation for the three-dimensional (3-D) passive scalar spectrum $\hat{M}(k,t)$ or its correlation function $M(r,t)$, gives the Kraichnan equation when $\unicode[STIX]{x1D70F}\rightarrow 0$, and extends it to the next order in $\unicode[STIX]{x1D70F}$. It involves third- and fourth-order derivatives of $M$ or $\hat{M}$ (in the high $k$ limit). For small-$\unicode[STIX]{x1D70F}$ (or small Kubo number), it can be recast using the Landau–Lifshitz approach to one with at most second derivatives of $\hat{M}$. We present both a scaling solution to this equation neglecting diffusion and a more exact solution including diffusive effects. To leading order in $\unicode[STIX]{x1D70F}$, we first show that the steady state 1-D passive scalar spectrum, preserves the Batchelor (J. Fluid Mech., vol. 5, 1959, pp. 113–133) form, $E_{\unicode[STIX]{x1D703}}(k)\propto k^{-1}$, in the viscous–convective limit, independent of $\unicode[STIX]{x1D70F}$. This result can also be obtained in a general manner using Lagrangian methods. Interestingly, in the absence of sources, when passive scalar fluctuations decay, we show that the spectrum in the Batchelor regime at late times is of the form $E_{\unicode[STIX]{x1D703}}(k)\propto k^{1/2}$ and also independent of $\unicode[STIX]{x1D70F}$. More generally, finite $\unicode[STIX]{x1D70F}$ does not qualitatively change the shape of the spectrum during decay. The decay rate is however reduced for finite $\unicode[STIX]{x1D70F}$. We also present results from high resolution ($1024^{3}$) direct numerical simulations of passive scalar mixing and decay. We find reasonable agreement with predictions of the Batchelor spectrum during steady state. The scalar spectrum during decay is however dependent on initial conditions. It agrees qualitatively with analytic predictions when power is dominantly in wavenumbers corresponding to the Batchelor regime, but is shallower when box-scale fluctuations dominate during decay.

Passive scalars: Mixing, diffusion, and intermittency in helical and nonhelical rotating turbulence.

We use direct numerical simulations to compute structure functions, scaling exponents, probability density functions, and effective transport coefficients of passive scalars in turbulent rotating helical and nonhelical flows. We show that helicity affects the inertial range scaling of the velocity and of the passive scalar when rotation is present, with a spectral law consistent with ∼k_{⊥}^{-1.4} for the passive scalar variance spectrum. This scaling law is consistent with a phenomenological argument [P. Rodriguez Imazio and P. D. Mininni, Phys. Rev. E 83, 066309 (2011)PLEEE81539-375510.1103/PhysRevE.83.066309] for rotating nonhelical flows, which follows directly from Kolmogorov-Obukhov scaling and states that if energy follows a E(k)∼k^{-n} law, then the passive scalar variance follows a law V(k)∼k^{-n_{θ}} with n_{θ}=(5-n)/2. With the second-order scaling exponent obtained from this law, and using the Kraichnan model, we obtain anomalous scaling exponents for the passive scalar that are in good agreement with the numerical results. Multifractal intermittency models are also considered. Intermittency of the passive scalar is stronger than in the nonhelical rotating case, a result that is also confirmed by stronger non-Gaussian tails in the probability density functions of field increments. Finally, Fick's law is used to compute the effective diffusion coefficients in the directions parallel and perpendicular to rotation. Calculations indicate that horizontal diffusion decreases in the presence of helicity in rotating flows, while vertical diffusion increases. A simple mean field argument explains this behavior in terms of the amplitude of velocity fluctuations.

Spectral modelling for passive scalar dynamics in homogeneous anisotropic turbulence

The present work aims at developing a spectral model for a passive scalar field and its associated scalar flux in homogeneous anisotropic turbulence. This is achieved using the paradigm of eddy-damped quasi-normal Markovian (EDQNM) closure extended to anisotropic flows. In order to assess the validity of this approach, the model is compared to several detailed direct numerical simulations (DNS) and experiments of shear-driven flows and isotropic turbulence with a mean scalar gradient at moderate Reynolds numbers. This anisotropic modelling is then used to investigate the passive scalar dynamics at very high Reynolds numbers. In the framework of homogeneous isotropic turbulence submitted to a mean scalar gradient, decay and growth exponents for the cospectrum and scalar energies are obtained analytically and assessed numerically thanks to EDQNM closure. With the additional presence of a mean shear, the scaling of the scalar flux and passive scalar spectra in the inertial range are investigated and confirm recent theoretical predictions. Finally, it is found that, in shear-driven flows, the small scales of the scalar second-order moments progressively return to isotropy when the Reynolds number increases.

Passive scalar convective-diffusive subrange for low Prandtl numbers in isotropic turbulence.

In this Rapid Communication, we study the behavior of a strongly diffusive passive scalar field T submitted to a freely decaying, homogeneous and isotropic turbulence with eddy-damped quasinormal Markovian simulations. We present a new subrange located between the k(-17/3) inertial-diffusive subrange and the Kolmogorov wave number k(η). This subrange is generated by small-scale convection linked to k(η) that balances diffusion effects. Thus, we build a typical length scale k(CD)(-1) based on convection and diffusion and give an expression for the shape of the passive scalar spectrum in this subrange E(T)∼√[Pr]k(-11/3) using physical arguments. This result unifies two different theories coming from Batchelor [G. K. Batchelor, J. Fluid. Mech. 5, 113 (1959)] and Chasnov [J. Chasnov et al., Phys. Fluids A 1, 1698 (1989)] and explains results previously obtained experimentally.

Mixing of a passive scalar across a thin shearless layer: concentration of intermittency on the sides of the turbulent interface

The advection of a passive scalar through an initial flat interface separating two different isotropic decaying turbulent fields is investigated in two and three dimensions. Simulations have been performed for a range of Taylor’s microscale Reynolds numbers from 45 to 250 and for a Schmidt number equal to 1. Different to the case where the transport involves the momentum and kinetic energy only and one intermittency layer is formed in the low-turbulent energy side of the system, in the passive scalar concentration field two intermittent layers are observed to develop at the sides of the interface. The layers move normally to the interface in opposite directions. The dimensionality produces different time scaling of the passive scalar diffusion, which is much faster in the two-dimensional case. In two dimensions, the propagation of the intermittent layers exhibits a significant asymmetry with respect to the initial position of the interface and is deeper for the layer which moves towards the high kinetic energy side of the system. In three dimensions, the two intermittent layers propagate nearly symmetrically with respect the centre of the mixing region. During the temporal decay, inside the mixing, which is both inhomogeneous and anisotropic but devoid of a mean velocity shear, the passive scalar spectra are computed. In three dimensions, the exponent in the scaling range gets in time a value close to that of the kinetic energy spectrum of isotropic turbulence (−5/3). In two dimensions, instead the exponent settles down to a value that is about one-half of the corresponding isotropic case. By means of an analysis based on simple wavy perturbations of the interface we show that the formation of the double layer of intermittency is a dynamic general feature not specific to the turbulent transport. These results of our numerical study are discussed in the context of experimental results and numerical simulations.

Fluctuations of a passive scalar in a turbulent mixing layer

The turbulent flow originating downstream of the Kelvin-Helmholtz instability in a mixing layer has great relevance in many applications, ranging from atmospheric physics to combustion in technical devices. The mixing of a substance by the turbulent velocity field is usually involved. In this paper, a detailed statistical analysis of fluctuations of a passive scalar in the fully developed region of a turbulent mixing layer from a direct numerical simulation is presented. Passive scalar spectra show inertial ranges characterized by scaling exponents -4/3 and -3/2 in the streamwise and spanwise directions, in agreement with a recent theoretical analysis of passive scalar scaling in shear flows [Celani et al., J. Fluid Mech. 523, 99 (2005)]. Scaling exponents of high-order structure functions in the streamwise direction show saturation of intermittency with an asymptotic exponent ζ_{∞}=0.4 at large orders. Saturation of intermittency is confirmed by the self-similarity of the tails of the probability density functions of the scalar increments at different scales r with the scaling factor r^{-ζ_{∞}} and by the analysis of the cumulative probability of large fluctuations. Conversely, intermittency saturation is not observed for the spanwise increments and the relative scaling exponents agree with recent results for homogeneous isotropic turbulence with mean scalar gradient. Probability density functions of the scalar increments in the three directions are compared to assess anisotropy.

Passive scalar cascades in rotating helical and non-helical flows

We study how helicity affects the spectrum of a passive scalar in rotating turbulent flows, using numerical simulations of turbulent flows with or without rotation and with or without injection of helicity. Scaling laws for energy and passive scalar spectra in the direction perpendicular to the rotation axis differ in rotating helical flows from those found in the non-helical case, with the spectrum of passive scalar variance in the former case being shallower than in the latter. A simple phenomenological model that links the effects of helicity on the energy spectrum with the passive scalar spectrum is presented.

Spectrum of passive scalars of high molecular diffusivity in turbulent mixing

AbstractWe consider the mixing of passive scalars transported in turbulent flow, with a molecular diffusivity that is large compared to the kinematic viscosity of the fluid. This particular case of mixing has not received much attention in experiment or simulation even though the first putative theory, due to Batchelor, Howells & Townsend (J. Fluid Mech., vol. 5, 1959, pp. 134–139), is now more than 50 years old. We study the problem using direct numerical simulation of decaying scalar fields in steadily sustained homogeneous turbulence as the Schmidt number (the ratio of the kinematic viscosity of the fluid to the molecular diffusivity of the scalar) is allowed to vary from $1/ 8$ to $1/ 2048$ for two values of the microscale Reynolds number, ${R}_{\lambda } \approx 140$ and $\approx $240. The simulations show that the passive scalar spectrum assumes a slope of $- 17/ 3$ in a range of scales, as predicted by the theory, when the Schmidt number is small and the Reynolds number is simultaneously large. The observed agreement between theory and simulation in the prefactor in the spectrum is not perfect. We assess the reasons for this discrepancy by a careful examination of the scalar evolution equation in the light of the assumptions of the theory, and conclude that the finite range of scales resolved in simulations is the main reason. Numerical issues specific to the regime of very low Schmidt numbers are also addressed briefly.

Publisher's Note: “Scaling range of velocity and passive scalar spectra in grid turbulence” [Phys. Fluids 24, 075101 (2012)

Publisher's Note: “Scaling range of velocity and passive scalar spectra in grid turbulence” [Phys. Fluids 24, 075101 (2012)

Scaling range of velocity and passive scalar spectra in grid turbulence

Isotropic velocity and scalar fluctuations are closely approximated by slightly stretching a heated grid flow through a short (1.36:1) contraction. The heating is such that temperature serves as a passive scalar, and the velocity/scalar time scale ratio is about one. At small values of Taylor microscale Reynolds number (10 < Rλ < 102), the spectrum of the temperature fluctuations has a more discernible scaling range than the spectrum of the velocity fluctuations. The scaling-range exponent for the thermal spectrum, mθ, exhibits a power-law function of Rλ and tends to the Kolmogorov value of 5/3 more rapidly than that for the velocity spectrum, mu. Both mθ and mu are closer to the Kolmogorov value with the contraction than with no contraction. The trends for the present measurements supplemented with previously published data for larger Rλ (>102) indicate that, to obtain a 5/3 scaling range, Rλ must exceed 103. The ratio (5/3 + mu)/mθ is approximately 2, in close conformity with the proposal of Danaila and Antonia [“Spectrum of a passive scalar in moderate Reynolds number homogeneous isotropic turbulence,” Phys. Fluids 21, 111702 (2009)10.1063/1.3264881].

Spectrum of a passive scalar in moderate Reynolds number homogeneous isotropic turbulence

For moderate Reynolds numbers, the spectral scaling exponents of both velocity [E(k)∝k−mu] and the transported passive scalar [Eθ(k)∝k−mθ] fields exhibit departures from the asymptotic prediction mu=mθ=5/3. However, at the same Reynolds number, the passive scalar spectrum for homogeneous isotropic turbulence is closer to the universal asymptotic state than the dynamic velocity field that transports it. This paper provides a possible explanation for this behavior, in the case of a gaseous mixing with Prandtl (or Schmidt) number Pr≃1. A scenario of the scalar energy transfer toward higher wavenumbers is proposed and validated using experimental data, in which the velocity field itself is actively involved via its characteristic time. A direct relationship between velocity and scalar spectra and therefore between mu and mθ is thus established.

Passive scalars in stratified turbulence

Statistics of a passive scalar (or tracer) with a horizontal mean gradient in randomly forced and strongly stratified turbulence are investigated by numerical simulations. We observe that horizontal isotropy of the passive scalar spectrum is satisfied in the inertial range. The spectrum has the form Eθ(kh) = CθɛθɛK−1/3kh−5/3, where ɛθ, ɛK are the dissipation of scalar variance and kinetic energy respectively, and Cθ ≃ 0.5 is a constant. This spectrum is consistent with atmospheric measurements in the mesoscale range with wavelengths 1–500 km. We also calculate the fourth‐order passive scalar structure function and show that intermittency effects are present in stratified turbulence.

Passive scalar spectrum in high-Schmidt-number stationary and nonstationary turbulence

Chasnov [Phys. Fluids 10, 1191 (1998)] reviewed the results for passive scalar spectra in high-Schmidt-number stationary turbulence as derived by Kraichnan [J. Fluid Mech. 64, 737 (1974)] and generalized them to simple nonstationary flows. In two-dimensional turbulence, the Kraichnan spectra are usually fitted by numerically solving the spectral equation using the derived asymptotic behavior for small and large wave numbers. In this Brief Report, we show that the Kraichnan passive scalar spectrum over the entire range of k is essentially a modified Bessel function of the second kind. We also present analytical forms of the spectra in three-dimensional nonstationary turbulence, where as shown by Chasnov, the nonstationarity can be responsible for different asymptotic behavior than the usual Kraichnan's three-dimensional stationary form. Our results considerably simplify the "fitting" of passive scalar spectra from experimental and numerical data, with the simple analytical form valid for the whole range of k , instead of just the asymptotes, which are usually valid only for a small fraction of resolved wave numbers.

Local and nonlocal advection of a passive scalar

Passive and active scalar mixing is examined in a simple one-parameter family of two-dimensional flows based on quasi-geostrophic dynamics, in which the active scalar, the quasi-geostrophic potential vorticity, is confined to a single horizontal surface (so-called surface quasi-geostrophic dynamics) and in which a passive scalar field is also advected by the (horizontal, two-dimensional) velocity field at a finite distance from the surface. At large distances from the surface the flow is determined by the largest horizontal scales, the flow is spectrally nonlocal, and a chaotic advection-type regime dominates. At small distances, z, scaling arguments suggest a transition wavenumber kc∼1∕2z, where the slope of the passive scalar spectrum changes from k−5∕3, determined by local dynamics, to k−1, determined by nonlocal dynamics, analogous to the transition to a k−1 slope in the Batchelor regime in three-dimensional turbulence. Direct numerical simulations reproduce the qualitative aspects of this transition. Other characteristics of the simulated scalar fields, such as the relative dominance of coherent or filamentary structures, are also shown to depend strongly on the degree of locality.