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Turbulent reactive flows laden with droplets appear in various energy systems but are difficult to understand and parametrize. Such flows involve interactions of turbulent fluctuations, phase changes, and chemical reactions that give rise to complex phenomena. To improve our knowledge, we performed direct numerical simulations of a canonical shear flow. It is composed of a hot, quiescent outer layer and a cold, turbulent inner layer that is laden with droplets. Due to the turbulent fluctuations, the droplets form clusters. Due to the high temperatures, the droplets evaporate quickly and flames emerge spontaneously at the interface of the two layers. We observed premixed flames that enclose droplet clusters and diffusion flames that enclose vapor pockets or single droplets. To examine these flame structures in more detail, we varied the droplet size, droplet loading, and shear rate. We found that the droplet size and droplet loading have significant effects, whereas the shear rate has only subtle effects. [ABSTRACT FROM AUTHOR]

In this work, a data-based approach to gas-surface interaction modeling, which employs the recently introduced distribution element tree (DET) method, is proposed. The DET method allows efficient data-driven probability density function (PDF) estimations with the possibility of conditional and unconditional random number resampling from the constructed distributions. As part of our ongoing research on gas-surface interaction, a comprehensive molecular dynamics (MD) study was performed, where the scattering of a nitrogen molecule from a graphite surface was investigated. Our aim here is to demonstrate how the DET method can be used in combination with the obtained MD database for constructing a generalized kernel of gas-surface interaction and for generating postscattered samples directly from the MD data itself. The major benefit of this approach is that it preserves all the relevant physics contained within numerical or experimental data, without the need for new kernel developments or accommodation coefficient calibrations. A direct comparison between the proposed approach and a classical scattering kernel used in rarefied gas flow simulations was carried out in the case of molecular beam scattering of rotationally hot and cold nitrogen from a solid surface. A further comparison between the proposed method and the available experimental data was also performed. Additionally, the ability of the DET-based kernel to satisfy the reciprocity condition, which ensures energy conservation in the case of thermal equilibrium, is demonstrated. [ABSTRACT FROM AUTHOR]

We study the capability of the One-Dimensional-Turbulence (ODT) model to simulate the turbulent transport and mixing of multiple passive scalars in homogeneous isotropic stationary turbulence. To this end, forcing schemes that have been widely used for stationary three-dimensional direct numerical simulations are adapted to the ODT framework and the effects of the model and input parameters on the steady state properties are discussed. We observe the model's ability to convincingly represent the physical features of single-scalar mixing and variance decay, but also its limited qualitative accuracy in the multiple scalar case, especially concerning the scalar joint probability density function, which is of importance for reactive flow computations. [ABSTRACT FROM AUTHOR]

Highlights • Development of temporal velocity-based random walk model. • Applicable for mildly to highly heterogeneous formations. • Model has a clearly motivated and light-weight formulation. Abstract Recently, we have analytically derived a temporal velocity random-walk model for macrodispersion or transport uncertainty quantification based on velocity statistics from classical first-order perturbation expansions. The applicability of these expansions is limited to mildly heterogeneous formations with log-conductivity variances σ Y 2 < 1. In this work, we reformulate our model in order to account for velocity skewness and inter-component correlation that emerge as key drivers of non-Fickian dispersion at elevated heterogeneity levels σ Y. Eventually, we arrive at a light-weight parametrization of macrodispersion that is consistent with our earlier formulation, but at the same time is applicable for formations with multi-Gaussian log-conductivity characterization of variable heterogeneity, i.e. σ Y from 0 to 1 and beyond. [ABSTRACT FROM AUTHOR]

Recently, distribution element trees (DETs) were introduced as an accurate and computationally efficient method for density estimation. In this work, we demonstrate that the DET formulation promotes an easy and inexpensive way to generate random samples similar to a smooth bootstrap. These samples can be generated unconditionally, but also, without further complications, conditionally using available information about certain probability-space components. This article is accompanied by the R codes that were used to produce all simulation results. Supplementary material for this article is available online. [ABSTRACT FROM AUTHOR]

The present paper investigates droplet laden turbulent gas flows with point droplet direct numerical simulations. A novel flow configuration is presented, which allows us to simulate statistically stationary homogeneous isotropic regions of turbulent sprays. This configuration enables a physical analysis with small statistical errors under controllable conditions. The length scales of clusters are analyzed with spectra of the droplet number density and vapor mass fraction fluctuations. The local conditions in clusters and voids are analyzed with probability density functions of the local droplet number density, local vapor mass fraction, and evaporation rate. Effects of the mass loading and Stokes number on the clustering of droplets and the mixing of vapor are characterized, and the compositions of clusters and voids are examined. As the mass loading increases from 2.5% to 12.5%, the fluctuations in the vapor mass fraction and droplet number density increase almost equally at all length scales. Clusters contain more droplets that together release more vapor, but their characteristic size remains unchanged. As the mean Stokes number increases from 1 to 4, the fluctuations in the vapor mass fraction and droplet number density decrease predominantly at small length scales. Clusters contain fewer droplets that together release less vapor, and their characteristic size increases. At a mass loading of 7.5% and a mean Stokes number of 2, clusters contain considerably more droplets and vapor than voids. Droplets located in clusters therefore evaporate more slowly than droplets located in voids. [ABSTRACT FROM AUTHOR]

Recently developed stochastic macro-dispersion models enable computationally inexpensive flow and transport predictions for highly heterogeneous formations with statistically non-stationary conductivity and flow statistics. So far, the random processes at the heart of these models have been calibrated numerically based on velocity statistics from Monte Carlo simulation studies. In this work, we provide a more rigorous foundation to some of these models by analytically relating the process definition to given velocity statistics from classical first-order perturbation expansions. [ABSTRACT FROM AUTHOR]

In most groundwater aquifers, dispersion of tracers is dominated by flow-field inhomogeneities resulting from the underlying heterogeneous conductivity or transmissivity field. This effect is referred to as macrodispersion. Since in practice, besides a few point measurements the complete conductivity field is virtually never available, a probabilistic treatment is needed. To quantify the uncertainty in tracer concentrations from a given geostatistical model for the conductivity, Monte Carlo (MC) simulation is typically used. To avoid the excessive computational costs of MC, the polar Markovian velocity process (PMVP) model was recently introduced delivering predictions at about three orders of magnitude smaller computing times. In artificial test cases, the PMVP model has provided good results in comparison with MC. In this study, we further validate the model in a more challenging and realistic setup. The setup considered is derived from the well-known benchmark macrodispersion experiment (MADE), which is highly heterogeneous and non-stationary with a large number of unevenly scattered conductivity measurements. Validations were done against reference MC and good overall agreement was found. Moreover, simulations of a simplified setup with a single measurement were conducted in order to reassess the model’s most fundamental assumptions and to provide guidance for model improvements. [ABSTRACT FROM AUTHOR]

Summary The multiscale nature of geological formations is reflected in the flow and transport behaviors of the pore fluids. For example, multiple pathways between different locations in the porous medium are usually present. The topology, length, and strength of these flow paths can vary significantly, and the total flow at a given location can be the result of contributions from a wide range of pathways between the points of interest. We use a high-resolution pore network of a natural porous formation as an example of the multiscale connectivity of the pore space. A single continuum model can capture the contributions from all the flow paths properly only if the control volume (computational cell) is much larger than the longest pathway. However, depending on the densities and lengths of these long pathways, choosing the appropriate size of the control volume that allows for a single continuum description of the properties, such as conductivity and transmissibility, may conflict with the desire to resolve the flow field properly. To capture the effects of the multiscale pathways on the flow, a non-local continuum model is described. The model can represent non-local effects, for which Darcy’s law is not valid. In the limit where the longest connections are much smaller than the size of the control volume, the model is consistent with Darcy’s law. The non-local model is used to describe the flow in complex pore networks. The pressure distributions obtained from the non-local model are compared with pore-network flow simulations, and the results are in excellent agreement. Importantly, such multiscale flow behaviors cannot be represented using the local Darcy law. [ABSTRACT FROM AUTHOR]

Highlights: [•] Particle-laden isotropic turbulence is simulated with kinematic simulation (KS). [•] A rigorous validation of KS with direct numerical simulation is presented. [•] KS provides collision frequencies in good agreement with DNS for St around 0 and >2. [Copyright &y& Elsevier]

Abstract: We consider two phase flow and transport in heterogeneous porous media with uncertain permeability distribution. The resulting transport uncertainty is assessed by means of multilevel Monte Carlo (MLMC). In contrast to the Monte Carlo (MC) method, which operates on one specific numerical grid with one numerical solver, MLMC samples from a hierarchy of grids or numerical solvers. In this work, the MLMC performance resulting from a hierarchy consisting of a finite volume transport solver and a streamline-based solver is compared to a purely grid-based hierarchy. Unlike the established grid-based MLMC method, our solver-based MLMC method operates on the same numerical grid and therefore avoids difficulties related to the upscaling of permeability fields or boundary conditions on coarser grids. For a two dimensional test case with log-normal permeability distribution, both MLMC approaches are compared to a MC reference run. At equivalent accuracy, significant speedups of MLMC with respect to MC are achieved. [Copyright &y& Elsevier]

Abstract: Different simulation methods are applicable to study turbulent mixing. When applying probability density function (PDF) methods, turbulent transport, and chemical reactions appear in closed form, which is not the case in second moment closure methods (RANS). Moreover, PDF methods provide the entire joint velocity-scalar PDF instead of a limited set of moments. In PDF methods, however, a mixing model is required to account for molecular diffusion. In joint velocity-scalar PDF methods, mixing models should also account for the joint velocity-scalar statistics, which is often under appreciated in applications. The interaction by exchange with the conditional mean (IECM) model accounts for these joint statistics, but requires velocity-conditional scalar means that are expensive to compute in spatially three dimensional settings. In this work, two alternative mixing models are presented that provide more accurate PDF predictions at reduced computational cost compared to the IECM model, since no conditional moments have to be computed. All models are tested for different mixing benchmark cases and their computational efficiencies are inspected thoroughly. The benchmark cases involve statistically homogeneous and inhomogeneous settings dealing with three streams that are characterized by two passive scalars. The inhomogeneous case clearly illustrates the importance of accounting for joint velocity-scalar statistics in the mixing model. Failure to do so leads to significant errors in the resulting scalar means, variances and other statistics. [Copyright &y& Elsevier]

Simulations of spatially inhomogeneous turbulent mixing in decaying grid turbulence with a joint velocity-concentration probability density function (PDF) method were conducted. The inert mixing scenario involves three streams with different compositions. The mixing model of Meyer ['A new particle interaction mixing model for turbulent dispersion and turbulent reactive flows,' Phys. Fluids 22(3), 035103 (2010)], the interaction by exchange with the mean (IEM) model and its velocity-conditional variant, i.e., the IECM model, were applied. For reference, the direct numerical simulation data provided by Sawford and de Bruyn Kops ['Direct numerical simulation and lagrangian modeling of joint scalar statistics in ternary mixing,' Phys. Fluids 20(9), 095106 (2008)] was used. It was found that velocity conditioning is essential to obtain accurate concentration PDF predictions. Moreover, the model of Meyer provides significantly better results compared to the IECM model at comparable computational expense. [ABSTRACT FROM AUTHOR]

Probability density function (PDF) methods are an established tool applied for the simulation of turbulent mixing and turbulent reactive flows. Mixing models are required to close the molecular diffusion term in the PDF transport equation. From the nature of molecular diffusion, several requirements or design criteria can be derived for mixing models. All current models have certain shortcomings with respect to these requirements. A new mixing model is presented which fully satisfies almost all requirements. It conserves the mean of an inert scalar, reduces its scalar variance, and relaxes closely to a Gaussian scalar PDF. Multiple inert scalars without differential diffusion effects evolve independently and are kept bounded within their allowable region. Mixing is conditional on the velocity and particle scalar trajectories are continuous in time leading to a model that is local in a weak sense. Validation tests show that the model can reproduce differential diffusion effects and mixing rate dependencies due to variable initial scalar length scales or Reynolds and Schmidt number variations. [ABSTRACT FROM AUTHOR]

Abstract: In transported probability density function and filtered density function methods, micromixing models are required to close the molecular mixing term. The accuracy and computational efficiency of improved versions of the parameterized scalar profile (PSP) model are assessed and compared with commonly used mixing models such as Curl, modified Curl, interaction by exchange with the mean and Euclidean minimum spanning tree. Different generalizations of the PSP mixing model for spatially inhomogeneous flow configurations are presented. The selected test cases focus on molecular mixing and avoid interference with other models. Simulation results for a three-stream problem, involving two inert scalars, and a multi-scalar test case with mean-scalar-gradients are presented. [Copyright &y& Elsevier]

Abstract: For the calculation of nonpremixed turbulent flames with thin reaction zones the joint probability density function (PDF) of the mixture fraction and its dissipation rate plays an important role. The corresponding PDF transport equation involves a mixing model for the closure of the molecular mixing term. Here, the parameterized scalar profile (PSP) mixing model is extended to provide the required joint statistics. Model predictions are validated using direct numerical simulation (DNS) data of a passive scalar mixing in a statistically homogeneous turbulent flow. Comparisons between the DNS and the model predictions are provided, which involve different initial scalar-field lengthscales. [Copyright &y& Elsevier]

• Produce large random flow or pore networks from a given template or base network. • Spatial heterogeneity of nodes or pores are preserved from the base network. • Connectivity and throat-/pore-statistics are preserved as well. • Bounded and unbounded, that is periodic networks can be generated. • Tests involving homogeneous and highly heterogeneous networks are documented. Over the past years, tomographic scanning techniques like micro-CT have become popular for the acquisition of high-fidelity void-space geometries of natural porous media (e.g., Bultreys et al., 2016; Raeini et al., 2017). Limitations both in computing time and memory prohibit, however, direct numerical simulations of flow and transport in large resp. detailed sample geometries. Flow or pore networks derived from scans alleviate this limitation, but still necessitate a methodology to extrapolate to larger samples. In this work, we present a network generation algorithm that is particularly suited for heterogeneous irregular networks. While emulating from an existing base network new networks of equal or larger sizes, the outlined algorithm scales approximately linearly with the network node or pore count and maintains (1) node connectivity resp. pore coordination-number statistics, (2) geometrical pore/throat properties, as well as (3) the potentially inhomogeneous spatial clustering of pores. While existing methods address the first two properties, the third point is crucial especially in heterogeneous media to match flow/transport properties like the permeability that have a strong dependence on the spatial distance between connected pores. Moreover, the cubical networks generated by our algorithm are periodic in all spatial directions, thus eliminating topological boundary effects, which are not present in natural media. Bounded networks of arbitrary sizes can then be recovered by cutting the generated networks and thus flow/transport processes at larger scales can be studied while incorporating physically-based descriptions of pore-scale processes. [ABSTRACT FROM AUTHOR]

• Non-local Darcy theory with conductivity distribution is extended for finite domains. • A robust method for the extraction of the conductivity distribution is presented. • Validation of extended theory against data from networks with different connectivity. This work is concerned with the intricate interplay between node or pore pressures and connection or throat conductivities in flow or pore networks. A setting similar to pore networks is given by fracture networks. Recently, a non-local generalization of Darcy's law for flow and transport in porous media was presented in the context of unbounded or periodic pore networks. In this work, we first outline a robust method for the extraction of the hydraulic conductivity distribution, which is at the heart of the non-local Darcy formulation. Second, a theory for mean pressure and flow in bounded networks is outlined. Predictions of that theory are validated against numerical network results and it is demonstrated that the theory works well for networks with high connectivity involving pores with high coordination numbers. For other networks, improvements to the outlined theory are proposed and their accuracy is assessed. [ABSTRACT FROM AUTHOR]

This paper examines droplets that cluster and evaporate in reactive turbulence with direct numerical simulations. The flows are statistically homogeneous and isotropic with mass loadings of about 0.1, Stokes numbers of about 1, and Taylor-scale Reynolds numbers of about 40. Our simulation results reveal diffusion and premixed flames. When the mass loading is small or the Stokes number is large, clusters contain few droplets such that diffusion flames surround single droplets. However, when the mass loading is large or the Stokes number is small, clusters contain many droplets such that premixed flames propagate through clusters and diffusion flames surround clusters. [ABSTRACT FROM AUTHOR]

This paper investigates droplets that evaporate and cluster in shear turbulence with direct numerical simulations. The flows are statistically stationary and homogeneous, which reduces the physical complexity and simplifies the statistical analysis. The mass loadings are about 0.1, the Stokes numbers are about 1, and the Taylor-scale Reynolds numbers are about 60. The simulations show that the clusters are anisotropic and inclined toward the flow direction on large scales, but isotropic on small scales. When the mass loading increases, the clusters contain more droplets, but their size remains unchanged, and the droplets in clusters experience higher vapor mass fractions. When the Stokes number increases, the clusters contain fewer droplets and become larger, and the droplets in clusters experience lower vapor mass fractions. When the Reynolds number increases, the clusters contain more, smaller droplets and become smaller, and the inclination angles of the clusters change. [ABSTRACT FROM AUTHOR]

This letter investigates the clustering of particles in isotropic turbulence sustained by linear and spectral forcing. The Stokes numbers range from 0.1 to 100. The time series of particles are analyzed with autocorrelation functions of the particle velocity and local fluid velocity. The spatial distributions of particles are analyzed with probability density functions of Voronoï volumes. These statistics show significant differences between the clustering of particles in linearly and spectrally forced turbulence. [ABSTRACT FROM AUTHOR]

We devise an efficient methodology to provide a universal statistical description of advection-dominated dispersion (Péclet→∞) in natural porous media including carbonates. First, we investigate the dispersion of tracer particles by direct numerical simulation (DNS). The transverse dispersion is found to be essentially determined by the tortuosity and it approaches a Fickian limit within a dozen characteristic scales. Longitudinal dispersion was found to be Fickian in the limit for bead packs and superdiffusive for all other natural media inspected. We demonstrate that the Lagrangian velocity correlation length is a quantity that characterizes the spatial variability for transport. Finally, a statistical transport model is presented that sheds light on the connection between pore-scale characteristics and the resulting macroscopic transport behavior. Our computationally efficient model accurately reproduces the transport behavior in longitudinal direction and approaches the Fickian limit in transverse direction. [ABSTRACT FROM AUTHOR]

Stochastic Markov processes are used very frequently to model, for example, processes in turbulence and subsurface flow and transport. Based on the weak Chapman-Kolmogorov equation and the strong Markov condition, we present methods to test the Markov hypothesis that is at the heart of these models. We demonstrate the capabilities of our methodology by testing the Markov hypothesis for fluid and inertial particles in turbulence, and fluid particles in the heterogeneous subsurface. In the context of subsurface macrodispersion, we find that depending on the heterogeneity level, Markov models work well above a certain scale of interest for media with different log-conductivity correlation structures. Moreover, we find surprising similarities in the velocity dynamics of the different media considered. [ABSTRACT FROM AUTHOR]

Advection dominated transport processes in sub-surface formations are characterized by discontinuities in the fields of transported quantities, and realistic predictions are challenging for Eulerian transport schemes because they suffer from numerical diffusion. Henceforth, we have focused on developing a Lagrangian particle-tracking scheme for modeling advective solute transport in fractured media. To this end, we adopt an Embedded Discrete Fracture Model (EDFM) for fractured media with a permeable matrix. The flexibility to use non-conformal fracture–matrix discretizations makes EDFMs a compelling choice in field-scale flow problems. Unaffected by the numerical diffusion, Lagrangian transport schemes complement the potential of EDFMs by allowing the use of sufficiently large grid cell sizes for flow field computations. In an EDFM framework, the inter-continuum fluid mass exchange cannot be quantified by the particle trajectories/pathlines due to the unresolved fracture–matrix interfaces and different dimensionalities of the matrix and fracture discretizations. These constraints motivate the use of a stochastic particle-tracking scheme, and thus, we formulated a pathline-specific probability of inter-continuum particle transfer based on mass conservation of an elementary solute/fluid mass. The particle's transfer probability is calculated for the maximum residence time period in its associated fracture/matrix control volume, thus making the scheme time-adaptive. In addition, a conditional residence time distribution was derived, which dictates the timestamp of the particle transfer. First, we showcase that the tracking scheme preserves the initially homogeneous solute concentration field, suggesting that the probabilistic rules are consistent with the inter-continuum fluxes. Additionally, we illustrate the estimation of an evolving solute plume and compare the results with those of an Eulerian counterpart. The presented stochastic approach enables straightforward formulations of Lagrangian models for dynamic and sub-grid processes, e.g., solute interactions with the solid phase, the mapping of their effects onto large scale transport and additionally, be included in random walk models for dispersion. • Particle-tracking scheme suited for EDFMs, involving probabilistic particle transfers. • Formulation of particle-transfer probability and residence time distribution. • The probabilistic scheme is consistent with the fluxes and thus, conservative. [ABSTRACT FROM AUTHOR]

Abstract: Monte Carlo (MC) is a well known method for quantifying uncertainty arising for example in subsurface flow problems. Although robust and easy to implement, MC suffers from slow convergence. Extending MC by means of multigrid techniques yields the multilevel Monte Carlo (MLMC) method. MLMC has proven to greatly accelerate MC for several applications including stochastic ordinary differential equations in finance, elliptic stochastic partial differential equations and also hyperbolic problems. In this study, MLMC is combined with a streamline-based solver to assess uncertain two phase flow and Buckley–Leverett transport in random heterogeneous porous media. The performance of MLMC is compared to MC for a two dimensional reservoir with a multi-point Gaussian logarithmic permeability field. The influence of the variance and the correlation length of the logarithmic permeability on the MLMC performance is studied. [Copyright &y& Elsevier]

Solute transport observed in subsurface formations shows complex behavior, particularly in the presence of fractures. In this work, we focus on formations with fractures that are smaller compared to the domain of interest and which are distributed in a heterogeneous matrix at densities below the percolation threshold. We developed a simplified Lagrangian approach to characterize and predict advective transport in 2-D domains containing differently oriented fractures. To this end, we performed Monte Carlo simulation (MCS) studies using ensembles of random domain realizations and gathered/analyzed tracer particle displacement statistics. The series of displacement steps were defined by the locations of particles entering the matrix after traversing through one or several connected fractures. Then, we identify key correlation structures in the evolution of and between displacement step coordinates, namely, the step length, its orientation and the traverse time. Subsequently, a correlated random walk model was derived which is able to accurately reproduce longitudinal and transverse macrodispersion as recorded in the MCS. Finally, we explored the predictive capabilities of our stochastic model by simulating macrodispersion in stratified media composed of heterogeneous zones with different fracture orientations. • Statistical analysis of particle motion in 2-D heterogeneous continua with fractures. • Guided by the statistical analysis, development of a stochastic model for macrodispersion. • Longitudinal and transverse plume spreading are captured well by the model. [ABSTRACT FROM AUTHOR]

Abstract: The Karhunen–Loeve (KL) decomposition and the polynomial chaos (PC) expansion are elegant and efficient tools for uncertainty propagation in porous media. Over recent years, KL/PC-based frameworks have successfully been applied in several contributions for the flow problem in the subsurface context. It was also shown, however, that the accurate solution of the transport problem with KL/PC techniques is more challenging. We propose a framework that utilizes KL/PC in combination with sparse Smolyak quadrature for the flow problem only. In a subsequent step, a Lagrangian sampling technique is used for transport. The flow field samples are calculated based on a PC expansion derived from the solutions at relatively few quadrature points. To increase the computational efficiency of the PC-based flow field sampling, a new reduction method is applied. For advection dominated transport scenarios, where a Lagrangian approach is applicable, the proposed PC/Monte Carlo method (PCMCM) is very efficient and avoids accuracy problems that arise when applying KL/PC techniques to both flow and transport. The applicability of PCMCM is demonstrated for transport simulations in multivariate Gaussian log-conductivity fields that are unconditional and conditional on conductivity measurements. [Copyright &y& Elsevier]

Particles or droplets dispersed in turbulent flows at sufficiently high volume loadings lead to modifications of turbulence characteristics. More specifically, in a detailed experimental investigation by Poelma and coworkers, where the particle phase moves with a non-zero mean velocity relative to the fluid phase, it was found that anisotropic Reynolds-stress dissipation is induced. Recently, we have proposed a model that can account for this effect in RANS-based and PDF method simulations. In our previous work, however, no simulation results of the RANS model formulation were presented. In the present work, a new compact tensorial RANS formulation is presented and the new formulation is validated against the experimental data of Poelma and coworkers. [ABSTRACT FROM AUTHOR]