Passive Scalar Mixing Research Articles

Estimating the Rate of Field Line Braiding in the Solar Corona by Photospheric Flows

Abstract In this paper, we seek to understand the timescale in which the photospheric motions on the Sun braid coronal magnetic field lines. This is a crucial ingredient for determining the viability of the braiding mechanism for explaining the high temperatures observed in the corona. We study the topological complexity induced in the coronal magnetic field, primarily using plasma motions extracted from magneto-convection simulations. This topological complexity is quantified using the field line winding, finite time topological entropy (FTTE), and passive scalar mixing. With these measures, we contrast mixing efficiencies of the magneto-convection simulation, a benchmark flow known as a “blinking vortex”, and finally photospheric flows inferred from sequences of observed magnetograms using local correlation tracking. While the highly resolved magneto-convection simulations induce a strong degree of field line winding and FTTE, the values obtained from the observations from the plage region are around an order of magnitude smaller. This behavior is carried over to the FTTE. Nevertheless, the results suggest that the photospheric motions induce complex tangling of the coronal field on a timescale of hours.

Mixing in an agitated tubular reactor

AbstractWe analyze, through numerical simulations, the single‐phase liquid flow and associated passive scalar mixing in a tubular reactor that is agitated by lateral shaking, which induces the motion of a solid mixing element inside the reactor. The Reynolds number associated with the shaking motion is in the range 1200–5600. Dependent on its specific value, we perform direct or large eddy simulations. A fixed‐grid lattice‐Boltzmann method is used for solving the fluid flow. The moving boundary condition at the surface of the mixing element is dealt with by means of an immersed boundary method. To quantify mixing, a transport equation for a passive scalar is solved in conjunction with the flow dynamics.

Joint probability density function models for multiscalar turbulent mixing

Modeling multicomponent turbulent mixing is essential for simulations of turbulent combustion, which is controlled by mixing of fuel, oxidizer, combustion products, and intermediate species. One challenge is to find functions that can reproduce the joint probability density function (PDF) of scalar mixing states using only a small number of parameters. Even for mixing with only two independent scalars, several statistical distributions, including the Dirichlet, Connor–Mosimann (CM), five-parameter bivariate beta (BVB5), and statistically-most-likely distributions, have previously been proposed for this purpose, with minimal physical justification. This work uses the concept of statistical neutrality to relate these distributions to each other, relate the distributions to physical mixing configurations, and develop a systematic approach to model selection. This approach is validated by comparing the ability of these distributions to reproduce the evolution of the scalar PDF from Direct Numerical Simulations of three-component passive scalar mixing in isotropic turbulence with 11 different initial conditions that are representative of a wide range of mixing conditions of interest. The approach correctly identifies whether the Dirichlet, CM, and BVB5 distributions, which are increasingly complex bivariate generalizations of the beta distribution, can accurately model the joint PDFs, but knowledge of the mixing configuration is required to select the appropriate distribution. The statistically-most-likely distribution is generally less accurate than the appropriate bivariate beta distribution but still gives reasonable predictions and does not require knowledge of the mixing configuration, so it is a suitable model when no single mixing configuration can be identified.

Relative dispersion of particle pairs in turbulent channel flow

Lagrangian tracking of particle pairs is of fundamental interest in a large number of environmental applications dealing with contaminant dispersion and passive scalar mixing. The aim of the present study is to extend the observations available in the literature on relative dispersion of fluid particle pairs to wall-bounded turbulent flows, by means of particle pair tracking in direct numerical simulations (DNS) of a turbulent channel flow. The mean-square change of separation between particle pairs follows a clear ballistic regime at short times for all wall distances. The Eulerian structure functions governing this short-time separation are characterised in the channel, and allow to define a characteristic time scale for the ballistic regime, as well as a suitable normalisation of the mean-square separation leading to an overall collapse for different wall distances. Through fluid particle pair tracking backwards and forwards in time, the temporal asymmetry of relative dispersion is illustrated. At short times, this asymmetry is linked to the irreversibility of turbulence, as in previous studies on homogeneous isotropic flows. The influence of the initial separation (distance and orientation) as well as the influence of mean shear are addressed. By decomposing the mean-square separation into the dispersion by the fluctuating velocity field and by the average velocity, it is shown that the influence of mean shear becomes important at early stages of dispersion close to the wall but also near the channel centre. The relative dispersion tensor Δij is also presented and particularly the sign and time evolution of the cross-term Δxy are discussed. Finally, a ballistic cascade model previously proposed for homogeneous isotropic turbulence is adapted here to turbulent channel flows. Preliminary results are given and compared to the DNS. Future developments and assumptions in two particle stochastic models can be gauged against the issues and results discussed in the present study.

Numerical simulations on mixing of passive scalars in river confluences

The study deals with the mixing of passive scalars (such as pollutants) in open-channel confluences when the two inflows exhibit different concentrations. The dispersion of such passive scalars is investigated through the analysis of the processes enhancing the mixing in the confluence in order to estimate the length for complete mixing Lm in the downstream branch. The aim of this study is then to establish a correlation between this length for complete mixing Lm and characteristics of the confluence, namely its angle and two hydrodynamic parameters that are the momentum ratio M* of the inflows and the width-to-depth ratio of the downstream branch b/h. In this work, the flow in the confluence is numerically simulated by RANS calculation coupled with the advection-diffusion equation at field scale, i.e. with Reynolds and Froude numbers corresponding to real rivers (Re = 107 and Fr=0.1). The role of other parameters such as bed discordance or bed forms... is not addressed here. The numerical results highlight that the mixing in the confluence and its downstream branch, is enhanced by the presence of secondary currents that are themselves strongly affected by the characteristics of the confluence. The results aim at getting an operative empirical law linking the geometric and hydrodynamic parameters of the confluence with Lm.

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.

High-order structure functions for passive scalar fed by a mean gradient

Transport equations for even-order structure functions are written for a passive scalar mixing fed by a mean scalar gradient, with a Schmidt number Sc=1. Direct numerical simulations (DNS), in a range of Reynolds numbers Rλ ∈ [88, 529] are used to assess the validity of these equations, for the particular cases of second-and fourth-order moments. The involved terms pertain to molecular diffusion, transport, production, and dissipative-fluxes. The latter term, present at all scales, is equal to: i) the mean scalar variance dissipation rate, ⟨χ⟩, for the second-order moments transport equation; ii) non-linear correlations between χ and second-order moments of the scalar increment, for the fourth-order moments transport equation.The equations are further analysed to show that the similarity scales (i.e., variables which allow for perfect collapse of the normalised terms in the equations) are, for second-order moments, fully consistent with Kolmogorov–Obukhov–Corrsin (KOC) theory. However, for higher-order moments, adequate similarity scales are built from ⟨χn⟩. The similarity is tenable for the dissipative range, and the beginning of the scaling range.

Quantitative Evaluation of Passive Scalar Flow Mixing – A Review of Recent Developments

AbstractFlow mixing has always been of particular interest for chemical engineering, combustion, and energy industries as well as other related engineering applications. Mixing always plays a crucial role because of its ability to uniformly distribute the passive scalar in the flow field without dynamic influences to the flow field itself. This paper reviews studies which quantitatively evaluated passive scalar mixing in various engineering applications. On the one hand, passive scalar mixing in turbulence has been reviewed from the aspect of energy spectrum cascades and small temporal‐spatial scale analysis, mixing time, unmixedness, mixing time‐scale ratio, and relative mixing intensity ratio using DNS. Secondly, several different concepts (percentage mixing, mixing intensity and degree of mixing) to quantify the flow mixing in microfluidic systems were introduced and compared thoroughly, and it is believed that these methods can be extended to other macro fluid engineering systems. Furthermore, a CFD‐based approach using the statistics of temporal and spatial distribution of fluid particles was reviewed, but cannot be extended to engineering applications with special requirements of mixing time.

Prandtl number effects in decaying homogeneous isotropic turbulence with a mean scalar gradient

ABSTRACTDecaying homogeneous isotropic turbulence with an imposed mean scalar gradient is investigated numerically, thanks to a specific eddy-damped quasi-normal Markovian closure developed recently for passive scalar mixing in homogeneous anisotropic turbulence (BGC). The present modelling is compared successfully with recent direct numerical simulations and other models, for both very large and small Prandtl numbers. First, scalings for the cospectrum and scalar variance spectrum in the inertial range are recovered analytically and numerically. Then, at large Reynolds numbers, the decay and growth laws for the scalar variance and mixed velocity–scalar correlations, respectively, derived in BGC, are shown numerically to remain valid when the Prandtl number strongly departs from unity. Afterwards, the normalised correlation ρwθ is found to decrease in magnitude at a fixed Reynolds number when Pr either increases or decreases, in agreement with earlier predictions. Finally, the small scales return to isotropy of the scalar second-order moments is found to depend not only on the Reynolds number, but also on the Prandtl number.

Scaling laws of passive-scalar diffusion in the interstellar medium

Passive scalar mixing (metals, molecules, etc.) in the turbulent interstellar medium (ISM) is critical for abundance patterns of stars and clusters, galaxy and star formation, and cooling from the circumgalactic medium. However, the fundamental scaling laws remain poorly understood in the highly supersonic, magnetized, shearing regime relevant for the ISM. We therefore study the full scaling laws governing passive-scalar transport in idealized simulations of supersonic turbulence. Using simple phenomenological arguments for the variation of diffusivity with scale based on Richardson diffusion, we propose a simple fractional diffusion equation to describe the turbulent advection of an initial passive scalar distribution. These predictions agree well with the measurements from simulations, and vary with turbulent Mach number in the expected manner, remaining valid even in the presence of a large-scale shear flow (e.g. rotation in a galactic disk). The evolution of the scalar distribution is not the same as obtained using simple, constant "effective diffusivity" as in Smagorinsky models, because the scale-dependence of turbulent transport means an initially Gaussian distribution quickly develops highly non-Gaussian tails. We also emphasize that these are mean scalings that only apply to ensemble behaviors (assuming many different, random scalar injection sites): individual Lagrangian "patches" remain coherent (poorly-mixed) and simply advect for a large number of turbulent flow-crossing times.

Direct numerical simulation of a low momentum round jet in channel crossflow

Results of a direct numerical simulation of a jet in crossflow with passive scalar mixing are presented. The laminar jet issues from a circular exit into the channel crossflow with a low jet-to-crossflow velocity ratio of 1/6. The governing equations are solved by Incompact3d, an open-source code combining the high-order compact scheme and Poisson spectral solver. An internal recycling approach is used to generate the fully turbulent channel flow profile. Four main flow structures are identified: 1) a large recirculation seen immediately downstream of the jet-exit; 2) a contour-rotating vortex pair formed from the stretching and reorientation of the injection-flow vorticity; 3) a horseshoe vortex generated as a result of the stretching of the vorticity at the jet-exit windward side; 4) shear layer vortices coming from the lifted and shed crossflow boundary layer vorticity. Passive scalar profiles show the mixing are strong in the shear layer where the crossflow fluid encounters the jet fluid. The database is made available online for public access.

Influence of flow topology and dilatation on scalar mixing in compressible turbulence

The kinematics of passive scalar mixing and its relation to local flow topology and dilatation in compressible turbulence are examined using direct numerical simulations (DNS) of decaying isotropic turbulence. The main objective of this work is to characterize the dependence of various evolution mechanisms of scalar dissipation on local streamline topology and normalized dilatation. The DNS results indicate that the topology has a stronger influence on the nonlinear amplification mechanism than the dilatation level of a fluid element. In its appropriately normalized form, the amplification mechanism is found to be fairly independent of the Mach and Reynolds numbers. Non-focal topologies (the so called unstable-node/saddle/saddle and stable-node/saddle/saddle) are found to be associated with more intense mixing than the focal topologies at almost all dilatation levels. Alignment tendencies (jointly conditioned upon topology and dilatation) between the scalar-gradient vector and the strain-rate eigenvectors are shown to play a key role in shaping the observed behaviour in compressible turbulence. Finally, some modelling implications of these findings are discussed.

Passive scalar mixing in Mc

We investigate passive scalar mixing in spatially developing supersonic shear layer flows formed by two planar streams by means of large eddy simulation. After validation of the numerical procedures employing a high-order hybrid WENO/compact scheme by comparing the results of simulation with the results obtained from a well characterized experimental case, the effects of the convective Mach number, Mc, flow velocity ratio, r = U1/U2, and fluid density ratio, s = ρ1/ρ2, between Stream 1 and Stream 2 on passive scalar mixing are considered by examining the variation of mixing layer thickness and the mixing efficiency represented by the transport of the passive scalar. Mc is specified from 0.2 to 0.8. The evolution of large-scale coherent structures is well reproduced, with vortices undergoing rolling up, pairing, merging, and breaking up. The mixing layer thickness and mixing efficiency both decrease as Mc increases. As s increases, the mixing layer thickness increases, and the mixing efficiency of an entrained fluid decreases. As r increases, the mixing layer thickness decreases while the mixing efficiency increases. While the present results are not applicable to the examination of micro-mixing properties, which require assessment by experimental methods or high computing cost direct numerical simulation, they are useful for evaluating the effects of different flow parameters on macro-performance, which is equally important for scramjet combustor design and evaluation in engineering.

Study of the interference of plumes released from two near-ground point sources in an open channel

The dispersion and mixing of passive scalars released from two near-ground point sources into an open-channel flow are studied using direct numerical simulation. A comparative study based on eight test cases has been conducted to investigate the effects of Reynolds number and source separation distance on the dispersion and interference of the two plumes. In order to determine the nonlinear relationship between the variance of concentration fluctuations of the total plume and those produced by each of the two plumes, the covariance of the two concentration fields is studied in both physical and spectral spaces. The results show that at the source height, the streamwise evolution of the cross correlation between the fluctuating components of the two concentration fields can be classified into four stages, which feature zero, destructive and constructive interferences and a complete mixing state. The characteristics of these four stages of plume mixing are further confirmed through an analysis of the pre-multiplied co-spectra and coherency spectra. From the coherency spectrum, it is observed that there exists a range of ‘leading scales’, which are several times larger than the Kolmogorov scale but are smaller than or comparable to the scale of the most energetic eddies of turbulence. At the leading scales, the mixing between the two interfering plumes is the fastest and the coherency spectrum associated with these scales can quickly approach its asymptotic value of unity.

A computational study of syngas auto-ignition characteristics at high-pressure and low-temperature conditions with thermal inhomogeneities

A computational study was conducted to investigate the characteristics of auto-ignition in a syngas mixture at high-pressure and low-temperature conditions in the presence of thermal inhomogeneities. Highly resolved one-dimensional numerical simulations incorporating detailed chemistry and transport were performed. The temperature inhomogeneities were represented by a global sinusoidal temperature profile and a local Gaussian temperature spike (hot spot). Reaction front speed and front Damköhler number analyses were employed to characterise the propagating ignition front. In the presence of a global temperature gradient, the ignition behaviour shifted from spontaneous propagation (strong) to deflagrative (weak), as the initial mean temperature of the reactant mixture was lowered. A predictive Zel'dovich–Sankaran criterion to determine the transition from strong to weak ignition was validated for different parametric sets. At sufficiently low temperatures, the strong ignition regime was recovered due to faster passive scalar dissipation of the imposed thermal fluctuations relative to the reaction timescale, which was quantified by the mixing Damköhler number. In the presence of local hot spots, only deflagrative fronts were observed. However, the fraction of the reactant mixture consumed by the propagating front was found to increase as the initial mean temperature was lowered, thereby leading to more enhanced compression-heating of the end-gas. Passive scalar mixing was not found to be important for the hot spot cases considered. The parametric study confirmed that the relative magnitude of the Sankaran number translates accurately to the quantitative strength of the deflagration front in the overall ignition advancement.

A recycling method for the large-eddy simulation of plumes in the atmospheric boundary layer

A method for the large-eddy simulation (LES) of dispersion and mixing of passive scalars is developed and evaluated. The new method addresses the requirements of tracking the evolution of plumes for large distances from their sources while attaining a low computational cost. To reduce computational cost, the velocity and thermodynamic fields are solved on a doubly periodic domain in the horizontal directions. In contrast, when the plume reaches the downstream end of the computational domain, it is reintroduced at the upstream plane but as a different scalar field. The same procedure is repeated when the new scalar field reaches the downstream boundary. By using several scalar fields to describe the evolution of a single plume, the simulation is computationally cheaper since the same velocity and thermodynamic fields are reused, or recycled, when computing the plume evolution. The recycling method is verified by showing that low-order plume statistics are identical to a single-domain LES. Three cases of dispersion and mixing from a point ground source in diverse boundary layer conditions (stable, convectively unstable, and shallow cumulus convection) are considered. Moreover, the LES results are compared with the predictions a Gaussian plume model, which is found to perform satisfactorily in all cases when accurate information about the state of the boundary layer is provided.

Enhanced mixing by optimized streamwise and angular positioning of longitudinal vorticity

Streamwise vortices can be readily generated to enhance the mass and momentum transfer in continuous static mixers. The topology of these vortices is related to the shape and dimensions of the vortex generator (VG). It is essential to design the VG in such a way that the mass and momentum transfer are obtained for relatively small power dissipation. In the present paper, a streamwise and angular optimization study is performed for the passive scalar mixing enhancement downstream from rectangular wings inserted in a laminar circular pipe flow. The rectangular wing generates streamwise vortices which size is of the same order of the pipe radius. The optimal distance between two successive VGs is determined by analyzing the streamwise intensity of segregation and the pressure drop. The angular positioning of the vortices is also determined via local analysis of the scalar intensity of segregation. Such a study may be applied for other configurations of VGs where streamwise vorticity is used to enhance the mixing process in circular pipe flows.

Large Eddy Simulation of the Mixing of a Passive Scalar in a High-Schmidt Turbulent Jet

Accurate subgrid-scale (SGS) scalar flux models are essential when large eddy simulation (LES) is used to represent flow, mixing and transport of passive and active scalars in engineering, and environmental applications in turbulent regime. Many SGS scalar flux models have been developed for flows with low Schmidt numbers (Sc), but their application to high Sc flows has important limitations. In high Sc flows, the behavior of the scalar field becomes anisotropic because of intermittency effects, phenomenon that must be accounted for by SGS scalar flux models. The objective of this paper is to evaluate the ability of three SGS scalar flux models to predict the scalar behavior of a high Sc-number flow configuration, namely the anisotropy-resolved SGS scalar flux model: (1) appropriate for high Sc-number flow configurations, and two additional SGS models (linear eddy diffusivity based SGS models) with (2) constant, and (3) dynamically calculated turbulent Schmidt number. The LES simulation results accomplished by these models are compared to each other and to experimental data of a turbulent round jet discharging a diluted scalar into a low-velocity coflowing water stream. The comparison of simulation results and experimental observations shows that, in general, all SGS models reproduce the mean filtered concentration distribution in radial direction. The dynamic eddy diffusivity and anisotropy models reproduce the rms of the concentration and SGS scalar fluxes distribution. In particular, the anisotropy model improves the prediction reliability of LES. However, the three models evaluated in this study cannot accurately predict the scalar behavior at the superviscous layer. Finally, this work demonstrates that complex models can achieve reliable predictions on reasonable grids using less computational effort, while simple models require fine grids with increased computational costs.

Mixing, Transport and Coherent Structures

The subject of this workshop was coherent structures, which play a significant role in the transport and mixing of passive and active scalars in fluids on a wide range of spatial scales. Participants coming from diverse scientific communities such as experimental fluid flows, dynamical systems theory, computational mathematics and visualisation, reported on recent developments and open problems.

Algebraic Closure of Scalar Dissipation Rate for Large Eddy Simulations of Turbulent Premixed Combustion

An a-priori analysis of the filtered reaction rate closure based on Favre-filtered scalar dissipation rate (SDR) for large eddy simulations (LES) of turbulent premixed combustion has been conducted using a direct numerical simulation (DNS) database of freely propagating statistically planar flames for a range of different values of heat release parameter τ, global Lewis number Le, and turbulent Reynolds number Ret. It has been found that an existing SDR based reaction rate closure for Reynolds averaged Navier–Stokes (RANS) simulations is also valid for LES, when the filter width is larger than the flame thickness. This RANS-based reaction rate closure has been extended here for LES, and a satisfactory performance of this LES closure is observed for the values of filter widths, τ, Le, and Ret investigated here. A-priori DNS assessment of the SDR closures based on passive scalar mixing model and a power-law has been conducted. Moreover, an existing algebraic model of Favre-averaged SDR for RANS simulations has been extended here for LES. The performances of the algebraic closures of have been assessed with respect to Favre-filtered SDR extracted from the DNS data. It has been found that the newly proposed model for , which was extended here for LES from an existing RANS-based closure, predicts both local and volume-averaged behaviors of SDR satisfactorily for a range of for flames with different values of τ, Le, and Ret. The sensitivity of sub-grid turbulent velocity fluctuation modeling on the newly developed algebraic SDR closure has also been analyzed and it has been observed that the modeling does not significantly affect the performance of the algebraic SDR model proposed in this study.