Passive Scalar Mixing Research Articles

Mixing in two-dimensional shear flow with smooth fluctuations.

Chaotic variations in flow speed up mixing of scalar fields via intensified stirring. This paper addresses the statistical properties of a passive scalar field mixing in a regular shear flow with random fluctuations against its background. We consider two-dimensional flow with shear component dominating over smooth fluctuations. Such flow is supposed to model passive scalar mixing, e.g., inside a large-scale coherent vortex forming in two-dimensional turbulence or in elastic turbulence in a microchannel. We examine both the decaying case and the case of the continuous forcing of the scalar variances. In both cases dynamics possesses strong intermittency, which can be characterized via the single-point moments and correlation functions calculated in our work. We present general qualitative properties of pair correlation function as well as certain quantitative results obtained in the framework of the model with fluctuations that are short correlated in time.

A priori direct numerical simulation assessment of MILD combustion modelling in the context of large eddy simulation

A priori Direct Numerical Simulation (DNS) assessment of the closures for the filtered reaction rate and the Favre-filtered scalar dissipation rate (SDR) for large eddy simulations (LES) of homogenous and stratified mixture MILD combustion has been performed. It has been found that the probability density function of the reaction progress variable can be adequately predicted using the beta function for both homogenous and stratified mixture MILD combustion. It has also been found that a scalar dissipation rate closure characteristic of passive scalar mixing may not be suitable for homogenous and stratified mixture MILD combustion. The predictions of different filtered reaction rate closures have been assessed with respect to DNS data in this study. The Eddy Dissipation Concept (EDC), Eddy-Break Up (EBU) model, and SDR-based reaction rate closures for Reynolds Averaged Navier-Stokes simulations have been extended for MILD combustion in the context of LES. The SDR-based reaction rate closure captures the qualitative behaviour but overpredicts the filtered reaction rate for both homogenous and stratified mixture MILD combustion. The EBU-based reaction rate and flame surface density (FSD) closures did not capture the qualitative behaviour of the filtered reaction rate for both homogenous and stratified mixture MILD combustion. By contrast, the EDC model with bridging functions, which ensures asymptotic behaviour, shows the potential to provide reasonable predictions of the filtered reaction rate for a wide range of filter widths in the cases investigated here.

Simulations of scalar transport in oscillatory flow over a wavy wall

A three-dimensional model that couples Reynolds–averaged Navier–Stokes equations with an advection–diffusion solver for flow and concentration fields over a fixed, non-permeable boundary is presented. The model simulates oscillatory flow over a wavy wall with a constant scalar source to assess the effects of a boundary-induced pressure gradient on passive scalar mixing. Results were compared to experimental data collected in an oscillating tray facility. Hydrodynamic quantities are in favourable agreement with observations. The model captured vortex dynamics correctly, and turbulence statistics were modelled reasonably well. Comparisons of time-resolved and bulk scalar quantities for crest and trough sources were in favourable agreement with observations. Results indicate that the model can be utilized to study processes with highly intermittent phenomena, such as the transport of wastes and dissolved nutrients in the coastal bottom.

The influence of resolved convective motions on scalar dispersion in hectometric‐scale numerical weather prediction models

AbstractThe UK Met Office has a 300‐m grid length numerical weather prediction (NWP) model running routinely over London and, in research mode, city‐scale hectometric grid length NWP has become commonplace. It is important to understand how moving from kilometre‐ to hectometre‐scale grid length NWP influences boundary‐layer vertical mixing. For a clear‐sky convective boundary layer (CBL) case study, using 55‐ and 100‐m grid length NWP, we demonstrate that CBL vertical mixing of passive scalar is almost fully resolved. Passive scalar converges near the surface after emission from an idealised pollution ground source representing city‐scale emissions, and is transported in updrafts preferentially into the upper boundary layer. Approximately 8 km downstream of the source edge, this causes lower near‐surface concentrations compared with 1.5‐km grid length NWP, where vertical mixing is fully parameterised. This demonstrates that resolving ballistic‐type dispersion, which is not typically represented in NWP vertical mixing parameterisations, can have a leading‐order influence on city‐scale near‐surface pollution concentration. We present a simple analytical model that is able to capture diffusive and ballistic dispersion behaviour in terms of effective timescales. The timescale controlling how long it takes passive scalar to become well mixed in the CBL is times longer for 1.5‐km compared with 100‐ and 55‐m grid length NWP.

Stochastic modeling of turbulent mixing based on a hierarchical swapping of fluid parcels

AbstractTurbulent mixing is an omnipresent phenomenon that constantly affects our everyday life and plays an important role in a variety of industrial applications. The simulation of turbulent mixing poses great challenges, since the full resolution of all relevant length and time scales is associated with an immense computational effort. This limitation can be overcome by only resolving the large‐scale effects and completely model the sub‐grid scales. The development of an accurate sub‐grid mixing model is therefore a key challenge to capture all interactions in the sub‐grid scales. At this place, the hierarchical parcel‐swapping (HiPS) model formulated by A.R. Kerstein [J. Stat. Phys. 153, 142–161 (2013)] represents a computationally efficient and scale‐resolving turbulent mixing model. HiPS mimics the effects of turbulence on time‐evolving, diffusive scalar fields. In HiPS, the diffusive scalar fields or a state space is interpreted as a binary tree structure, which is an alternative approach compared to the most common mixing models. Every level of the tree represents a specific length and time scale, which is based on turbulence inertial range scaling. The state variables are only located at the base of the tree and are treated as fluid parcels. The effects of turbulent advection are represented by stochastic swaps of sub‐trees at rates determined by turbulent time scales associated with the sub‐trees. The mixing only takes places between adjacent fluid parcels and at rates consistent with the prevailing diffusion time scales. In this work, the HiPS model formulation for the simulation of passive scalar mixing is detailed first. Preliminary results for the mean square displacement, passive scalar probability density function (PDF) and scalar dissipation rate are given and reveal the strengths of the HiPS model considering the reduced order and computational efficiency. These model investigations are an important step of further HiPS advancements. The integrated auxiliary binary tree structure allows HiPS to satisfy a large number of criteria for a good mixing model. From this point of view, HiPS is an attractive candidate for modeling the mixing in transported PDF methods.

On the non-self-adjoint and multiscale character of passive scalar mixing under laminar advection

Except in the trivial case of spatially uniform flow, the advection–diffusion operator of a passive scalar tracer is linear and non-self-adjoint. In this study, we exploit the linearity of the governing equation and present an analytical eigenfunction approach for computing solutions to the advection–diffusion equation in two dimensions given arbitrary initial conditions, and when the advecting flow field at any given time is a plane parallel shear flow. Our analysis illuminates the specific role that the non-self-adjointness of the linear operator plays in the solution behaviour, and highlights the multiscale nature of the scalar mixing problem given the explicit dependence of the eigenvalue–eigenfunction pairs on a multiscale parameter $q=2{\rm i}k\,{\textit {Pe}}$ , where $k$ is the non-dimensional wavenumber of the tracer in the streamwise direction, and ${\textit {Pe}}$ is the Péclet number. We complement our theoretical discussion on the spectra of the operator by computing solutions and analysing the effect of shear flow width on the scale-dependent scalar decay of tracer variance, and characterize the distinct self-similar dispersive processes that arise from the shear flow dispersion of an arbitrarily compact tracer concentration. Finally, we discuss limitations of the present approach and future directions.

Passive scalar small-scale anisotropy and mixing characteristics in magnetohydrodynamic turbulent channel flow

In wall-bounded turbulent flows, both velocity and scalar fluctuations exhibit inhomogeneity and anisotropy. This study investigates the statistical properties of the small-scale scalar fluctuations in a turbulent channel flow at Reτ≈585 using direct numerical simulations with and without a magnetic field. The influence of the Hartmann, Ha, and Prandtl, Pr, numbers on turbulent velocity and passive scalar fields is examined at Ha=0, 20, and 40 and Pr=0.7 and 1.4. Small-scale dynamics of the passive scalar and velocity fields are studied, analyzing their probability density functions and higher-order moments, as well as their gradients. We observed that the magnetic field substantially changes flow dynamics such as the typical cliff-and-ramp type structures. The presence of the magnetic field led to statistical anisotropy, even at small-scale gradient fields. The findings reveal that the skewness of the normal derivative of scalar fluctuations remains at the order of 2. We investigated mixing characteristics by analyzing scalar dissipation rates. Scalar dissipation rates near the wall remain close to unity and decrease sharply toward the channel center, reaching a minimum value. Moreover, an increase in scalar dissipation rates leads to a decrease in the corresponding mixing timescale of the flow. This could suggest a connection between an increase in the Lorentz force and potential adjustments in the mixing timescale, potentially contributing to enhance overall mixing. Additionally, we argue that combined effects of strong intermittency and persistency of anisotropy at small scales can influence the mixing characteristics of magnetohydrodynamic turbulent flow.

Investigation of turbulent mixing using a stochastic hierarchical parcel swapping mixing model

AbstractTurbulent mixing plays an important role in a variety of applications ranging from astrophysics to combustion and even pollutant dispersion. The Direct Numerical Simulation (DNS) that resolves all scales is not feasible for most engineering applications since the flow has a wide range of length and time scales, which yields extremely high resolution requirements. Large Eddy Simulations (LES) overcome this limitation by modeling the sub‐grid scale effects. In transported Probability Density Function (PDF) methods, the key challenge is to develop an accurate mixing model. At this point, the Hierarchical Parcel‐Swapping (HiPS) model, introduced by A.R. Kerstein [J. Stat. Phys. 153, 142‐161 (2013)], is an attractive candidate. It is characterized by a computationally efficient representation of the effects of turbulence on a time‐evolving structure of diffusive scalar fields. The interpretation of the diffusive scalar fields or a state space as a binary tree structure is an alternative approach compared to the most common mixing models. The characteristic feature of HiPS is that every level of the tree corresponds to a specific length and time scale, which is based on turbulence inertial range scaling. The state variables only reside at the base of the tree and are understood as fluid parcels. The effects of turbulent advection are represented by stochastic swaps of sub‐trees at rates determined by prevailing turbulent time scales associated with the sub‐trees. The mixing of adjacent fluid parcels is done either instantaneously or at rates consistent with the corresponding diffusion time scales. In this work, HiPS is detailed for the simulation of passive scalar mixing first. Preliminary results for the scalar power spectra, mean square displacement and scalar dissipation rate are shown and reveal a reasonable agreement with experimental findings. Furthermore, the integrated binary tree structure allows to satisfy a large number of criteria for a good mixing model. Considering the reduced order and computational efficiency, HiPS is an attractive candidate for modeling the mixing in transported PDF methods.

Mixing mechanisms in the view of mixing indicators: from passive-scalar mixing to variable-density mixing

<abstract><p>Mixing is the basis of stable and efficient combustion in air-breathing power systems, and it is also an important problem in fluid mechanics, which has been extensively studied from various perspectives. The purpose of this review is to investigate mixing mechanisms based on two commonly-used mixing indicators, namely $ c^i $ ($ c $ refers to concentration, and $ i $ is either 1 or 2, indicating first- or second-order statistics), with a focus on passive-scalar (PS) and variable-density (VD) mixing. For PS mixing, the flow is not influenced by the mixing process. By using first-order statistics with concentration as the core, the PS mixing mechanisms on lamella structures can be described as stretching enhancing diffusion and promoting mixing. On the other hand, second-order statistics represented by the scalar dissipation rate can investigate mixing mechanisms on specific type of flow structures described by the invariants of velocity gradient tensors and the rotation of principal strain axis. As such, it has been found that strain-dominated flow structures can promote mixing, while rotation-dominated flow structures hinder it. For VD mixing, it has two distinct characteristics: flow changes due to baroclinic vorticity, and the inherent velocity divergence alters the mixing indicators. Studies using first-order statistics center on the mixing time in different types of VD flows, leading to the discovery of new phenomena. For instance, the second baroclinic vorticity can promote stretching in shock bubble interactions. Studies on second-order statistics for VD mixing have defined several mixing indicators from the component-transport equation, which have been utilized in phenomenological studies on VD mixing. This review aims to provide an overview of mixing phenomena, mixing indicators, and mixing mechanisms, and proposes research directions for understanding the mixing characteristics, flow structures, and their relationship with specific combustion phenomena particularly by second-order statistics.</p></abstract>

Bounds on the Rate of Enhanced Dissipation

We are concerned with flow enhanced mixing of passive scalars in the presence of diffusion. Under the assumption that the velocity gradient is suitably integrable, we provide upper bounds on the exponential rates of enhanced dissipation. Recent constructions indicate the optimality of our results.

A Comparison between Statistical Behaviours of Scalar Dissipation Rate between Homogeneous MILD Combustion and Premixed Turbulent Flames

Three-dimensional Direct Numerical Simulations (DNS) data has been utilised to analyse statistical behaviours of the scalar dissipation rate (SDR) and its transport for homogeneous methane-air mixture turbulent Moderate or Intense Low oxygen Dilution (MILD) combustion for different O2 dilution levels and turbulence intensities for different reaction progress variable definitions. Additional DNS has been conducted for turbulent premixed flames and passive scalar mixing for the purpose of comparison with the SDR statistics of the homogeneous mixture MILD combustion with that in conventional premixed combustion and passive scalar mixing. It has been found that the peak mean value of the scalar dissipation rate decreases with decreasing O2 concentration for MILD combustion cases. Moreover, SDR magnitudes increase with increasing turbulence intensity for both MILD and conventional premixed combustion cases. The profiles and mean values of the scalar dissipation rate conditioned upon the reaction progress variable are found to be sensitive to the choice of the reaction progress variable definition. This behaviour arises due to the differences in the distributions of the species mass fractions within the flame. The strain rate contribution and the molecular dissipation term are found to be the leading order contributors in the scalar dissipation rate transport for MILD combustion; whereas, in conventional premixed flames, the terms rising from density variation and reaction rate gradient also play leading roles in addition to the strain rate and molecular dissipation contributions. By contrast, the terms due to density gradient and reaction rate gradient remain negligible in comparison to the leading order contributors in MILD combustion cases due to small density variation because of moderate temperature rise and small reaction rate gradient magnitudes. Furthermore, the qualitative behaviour of the strain rate contribution to the SDR transport in premixed flames is significantly different to that in the case of MILD combustion and passive scalar mixing. The findings of the current analysis indicate that the scalar dissipation rate statistics in MILD combustion show several qualitative similarities to the passive scalar mixing despite major differences with the SDR transport in conventional turbulent premixed flames. This further suggests that the scalar dissipation rate models, which were originally proposed in the context of passive scalar mixing, have the potential to be applicable for MILD combustion but the models for the premixed turbulent combustion may not be applicable for MILD combustion of homogeneous mixtures.

Structure of iso-scalar sets

An analytical framework is proposed to explore the structure and kinematics of iso-scalar fields. It is based on a two-point statistical analysis of the phase indicator field which is used to track a given iso-scalar volume. The displacement speed of the iso-surface, i.e. the interface velocity relative to the fluid velocity, is explicitly accounted for, thereby generalizing previous two-point equations dedicated to the phase indicator in two-phase flows. Although this framework applies to many transported quantities, we here focus on passive scalar mixing. Particular attention is paid to the effect of Reynolds (the Taylor based Reynolds number is varied from 88 to 530) and Schmidt numbers (in the range 0.1 to 1), together with the influence of flow and scalar forcing. It is first found that diffusion in the iso-surface tangential direction is predominant, emphasizing the primordial influence of curvature on the displacement speed. Second, the appropriate normalizing scales for the two-point statistics at either large, intermediate and small scales are revealed and appear to be related to the radius of gyration, the surface density and the standard deviation of mean curvature, respectively. Third, the onset of an intermediate ‘scaling range’ for the two-point statistics of the phase indicator at sufficiently large Reynolds numbers is observed. The scaling exponent complies with a fractal dimension of 8/3. A scaling range is also observed for the transfer of iso-scalar fields in scale space whose exponent can be estimated by simple scaling arguments and a recent closure of the Corrsin equation. Fourth, the effects of Reynolds and Schmidt numbers together with flow or scalar forcing on the different terms of the two-point budget are highlighted.

Robust and efficient identification of optimal mixing perturbations using proxy multiscale measures.

Understanding and optimizing passive scalar mixing in a diffusive fluid flow at finite Péclet number (where and are characteristic velocity and length scales, and is the molecular diffusivisity of the scalar) is a fundamental problem of interest in many environmental and industrial flows. Particularly when , identifying initial perturbations of given energy that optimally and thoroughly mix fluids of initially different properties can be computationally challenging. To address this challenge, we consider the identification of initial perturbations in an idealized two-dimensional flow on a torus that extremize various measures over finite time horizons. We identify such ‘optimal’ initial perturbations using the ‘direct-adjoint looping’ method, thus requiring the evolving flow to satisfy the governing equations and boundary conditions at all points in space and time. We demonstrate that minimizing multiscale measures commonly known as ‘mix-norms’ over short time horizons is a computationally efficient and robust way to identify initial perturbations that thoroughly mix layered scalar distributions over relatively long time horizons, provided the magnitude of the mix-norm’s index is not too large. Minimization of such mix-norms triggers the development of coherent vortical flow structures which effectively mix, with the particular properties of these flow structures depending on and also the time horizon of interest.This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.

Lower bounds on mixing norms for the advection diffusion equation in mathbb {R}^d

An algebraic lower bound on the energy decay for solutions of the advection-diffusion equation in mathbb {R}^d with d=2,3 is derived using the Fourier-splitting method. Motivated by a conjecture on mixing of passive scalars in fluids, a lower bound on the L^2- norm of the inverse gradient of the solution is obtained via gradient estimates and interpolation.

Scalar Forcing Methodology for Direct Numerical Simulations of Turbulent Stratified Mixture Combustion

Scalar forcing in the context of turbulent stratified flame simulations aims to maintain the fuel-air inhomogeneity in the unburned gas. With scalar forcing, stratified flame simulations have the potential to reach a statistically stationary state with a prescribed mixture fraction distribution and root-mean-square value in the unburned gas, irrespective of the turbulence intensity. The applicability of scalar forcing for Direct Numerical Simulations of stratified mixture combustion is assessed by considering a recently developed scalar forcing scheme, known as the reaction analogy method, applied to both passive scalar mixing and the imperfectly mixed unburned reactants of statistically planar stratified flames under low Mach number conditions. The newly developed method enables statistically symmetric scalar distributions between bell-shaped and bimodal to be maintained without any significant departure from the specified bounds of the scalar. Moreover, the performance of the newly proposed scalar forcing methodology has been assessed for a range of different velocity forcing schemes (Lundgren forcing and modified bandwidth forcing) and also without any velocity forcing. It has been found that the scalar forcing scheme has no adverse impact on flame-turbulence interaction and it only maintains the prescribed root-mean-square value of the scalar fluctuation, and its distribution. The scalar integral length scale evolution is shown to be unaffected by the scalar forcing scheme studied in this paper. Thus, the scalar forcing scheme has a high potential to provide a valuable computational tool to enable analysis of the effects of unburned mixture stratification on turbulent flame dynamics.

Almost-sure exponential mixing of passive scalars by the stochastic Navier–Stokes equations

We deduce almost-sure exponentially fast mixing of passive scalars advected by solutions of the stochastically-forced 2D Navier–Stokes equations and 3D hyper-viscous Navier–Stokes equations in Td subjected to nondenegenerate Hσ-regular noise for any σ sufficiently large. That is, for all s>0 there is a deterministic exponential decay rate such that all mean-zero Hs passive scalars decay in H−s at this same rate with probability one. This is equivalent to what is known as quenched correlation decay for the Lagrangian flow in the dynamical systems literature. This is a follow-up to our previous work, which establishes a positive Lyapunov exponent for the Lagrangian flow—in general, almost-sure exponential mixing is much stronger than this. Our methods also apply to velocity fields evolving according to finite-dimensional models, for example, Galerkin truncations of Navier–Stokes or the Stokes equations with very degenerate forcing. For all 0≤k<∞, this exhibits many examples of CtkCx∞ random velocity fields that are almost-sure exponentially fast mixers.

On mixing enhancement by secondary baroclinic vorticity in a shock–bubble interaction

To investigate the intrinsic mechanism for mixing enhancement by variable-density (VD) behaviour, a canonical VD mixing extracted from a supersonic streamwise vortex protocol, a shock–bubble interaction (SBI), is numerically studied and compared with a counterpart of passive-scalar (PS) mixing. It is meaningful to observe that the maximum concentration decays much faster in a VD SBI than in a PS SBI regardless of the shock Mach number ( $Ma=1.22 - 4$ ). The quasi-Lamb–Oseen-type velocity distribution in the PS SBI is found by analysing the azimuthal velocity that stretches the bubble. Meanwhile, for the VD SBI, an additional stretching enhanced by the secondary baroclinic vorticity (SBV) production contributes to the faster-mixing decay. The underlying mechanism of the SBV-enhanced stretching is further revealed through the density and velocity difference between the light shocked bubble and the heavy ambient air. By combining the SBV-accelerated stretching model and the initial shock compression, a novel mixing time estimation for VD SBI is theoretically proposed by solving the advection–diffusion equation under a deformation field of an axisymmetric vortex with the additional SBV-induced azimuthal velocity. Based on the mixing time model, a mixing enhancement number, defined by the ratio of VD and PS mixing time further, reveals the contribution from the VD effect, which implies a better control of the density distribution for mixing enhancement in a supersonic streamwise vortex.

Geometry and dynamics of passive scalar structures in compressible turbulent mixing

We present a structure-based numerical analysis of passive scalar mixing in decaying homogeneous isotropic turbulence (DHIT) and shock-turbulence interaction canonical configurations. The analysis focuses on the temporal evolution of ensembles of passive scalar structures, initialized as spheres of different sizes relative to the Taylor microscale. An algorithm is introduced to track the evolution of each individual structure and the interactions with other structures in the ensemble, relating changes in the surface geometry and the underlying physical processes (turbulent transport, scalar dissipation, and shock compression). The tracking algorithm is applied to datasets from shock-capturing direct numerical simulations of DHIT, with Taylor microscale Reynolds number Reλ=40 and turbulence Mach number Mt=0.2, and STI cases in which the turbulence is processed by a shock wave at Mach numbers M = 1.5 and 3.0. Temporal surface convolution increases for initially larger structures, resulting in a higher probability of locally hyperbolic geometries where breakup into smaller structures occurs. Shock-induced deformation of the structures amplifies breakup processes, enhancing mixing, particularly for larger structures. Mixing enhancement by the shock is manifested as an amplification of the surface-averaged scalar gradient, which increases for initially larger structures. The alignment between the scalar gradient and the most extensional strain-rate eigendirection on the scalar isosurfaces also increases across the shock. Larger magnitudes of the scalar gradient and its alignment with the most compressive strain-rate eigendirection correlate with flatter surface regions. Shock-induced structure compression increases the area coverage of flat regions, where the amplification of scalar gradient is localized.

Bacteria hinder large-scale transport and enhance small-scale mixing in time-periodic flows

Understanding mixing and transport of passive scalars in active fluids is important to many natural (e.g., algal blooms) and industrial (e.g., biofuel, vaccine production) processes. Here, we study the mixing of a passive scalar (dye) in dilute suspensions of swimming Escherichia coli in experiments using a two-dimensional (2D) time-periodic flow and in a simple simulation. Results show that the presence of bacteria hinders large-scale transport and reduces overall mixing rate. Stretching fields, calculated from experimentally measured velocity fields, show that bacterial activity attenuates fluid stretching and lowers flow chaoticity. Simulations suggest that this attenuation may be attributed to a transient accumulation of bacteria along regions of high stretching. Spatial power spectra and correlation functions of dye-concentration fields show that the transport of scalar variance across scales is also hindered by bacterial activity, resulting in an increase in average size and lifetime of structures. On the other hand, at small scales, activity seems to enhance local mixing. One piece of evidence is that the probability distribution of the spatial concentration gradients is nearly symmetric with a vanishing skewness. Overall, our results show that the coupling between activity and flow can lead to nontrivial effects on mixing and transport.

The Batchelor Spectrum of Passive Scalar Turbulence in Stochastic Fluid Mechanics at Fixed Reynolds Number

In 1959 Batchelor predicted that the stationary statistics of passive scalars advected in fluids with small diffusivity k should display a power spectrum along an inertial range contained in the viscous‐convective range of the fluid model. This prediction has been extensively tested, both experimentally and numerically, and is a core prediction of passive scalar turbulence.In this article we provide a rigorous proof of a version of Batchelor's prediction in the limit when the scalar is subjected to a spatially smooth, white‐in‐time stochastic source and is advected by the 2D Navier‐Stokes equations or 3D hyperviscous Navier‐Stokes equations in forced by sufficiently regular, nondegenerate stochastic forcing. Although our results hold for fluids at arbitrary Reynolds number, this value is fixed throughout. Our results rely on the quantitative understanding of Lagrangian chaos and passive scalar mixing established in our recent works. Additionally, in the limit, we obtain statistically stationary, weak solutions in to the stochastically forced advection problem without diffusivity. These solutions are almost‐surely not locally integrable distributions with nonvanishing average anomalous flux and satisfy the Batchelor spectrum at all sufficiently small scales. We also prove an Onsager‐type criticality result that shows that no such dissipative, weak solutions with a little more regularity can exist. © 2021 Wiley Periodicals LLC.