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A consistent dual-mesh hybrid LES/RANS framework for turbulence modeling has been proposed recently (H. Xiao, P. Jenny, A consistent dual-mesh framework for hybrid LES/RANS modeling, J. Comput. Phys. 231 (4) (2012)). To better enforce componentwise Reynolds stress consistency between the LES and the RANS simulations, in the present work the original hybrid framework is modified to better exploit the advantage of more advanced RANS turbulence models. In the new formulation, the turbulent stresses in the filtered equations in the under-resolved regions are directly corrected based on the Reynolds stresses provided by the RANS simulation. More precisely, the new strategy leads to implicit LES/RANS consistency, where the velocity consistency is achieved indirectly via imposing consistency on the Reynolds stresses. This is in contrast to the explicit consistency enforcement in the original formulation, where forcing terms are added to the filtered momentum equations to achieve directly the desired average velocity and velocity fluctuations. The new formulation keeps the averaging procedure for the filtered quantities and at the same time preserves the ability of the original formulation to conform with the physical differences between LES and RANS quantities. The modified formulation is presented, analyzed, and then evaluated for plane channel flow and flow over periodic hills. Improved predictions are obtained compared with the results obtained using the original formulation.

Mechanisms of mass transfer intensification in the process of membrane cleaning using supercritical fluids were investigated. Transport properties, hydrodynamics of the solvent flow, mutual solubility of the solvent and the oil contaminant, and capillary effects occurring inside the porous membrane as factors affecting the overall process performance were studied. The analysis was performed using empirical correlations for the transport property coefficients and a model of the process implemented to CFD code developed using the OpenFOAM environment. Supercritical carbon dioxide exhibits favourable transport properties, which are highly tunable with process parameters and contribute to low mass transfer resistance. The investigated process is controlled by diffusive mass transfer inside the membrane pores, so increasing solvent flow rate has limited impact on the overall process rate. Mutual solubility of the oil and solvent phase leads to the effect of swelling of the oil phase, which promotes faster completion of cleaning. Capillary effects inside the pores may be another factor of process acceleration which requires further investigation.

Summary We consider two-phase flow and transport in heterogeneous porous media, where the permeability is modeled as a random field. To assess the resulting uncertainty of flow and transport, multilevel Monte Carlo (MLMC) is used. MLMC involves two types of errors: a sampling error and a discretization error, which, under reasonable assumptions, can be conveniently estimated during execution. In this work, we address the question of how to balance the two errors. Furthermore, a simple MLMC parallelization strategy is presented. A numerical comparison of MLMC with standard Monte Carlo (MC) reveals significant gains in computational efficiency at equivalent accuracy.

In this paper, the influence of the gas-surface interaction on the time-dependent rarefied gas flow through a short tube into vacuum is investigated. Due to a significant scale separation, the flow is simulated using a hybrid scheme proposed by Vargas et al. [1, 2], in which the pressure change in the upstream chamber is coupled to the flow rate obtained by the Direct Simulation Monte Carlo (DSMC) computations. First, the influence of gas-surface interaction is demonstrated through comparison of DSMC simulation results obtained for different values of accommodation coefficients. The obtained dataset is then used to show how the boundary conditions can affect the time-dependent gas flow, even when applied on the small surface area of the device. It is argued that the domain of possible solutions is bounded by the cases of specular and diffuse scattering. Furthermore, it is investigated how this domain can be affected by measurement uncertainties present in the experimental setup. Next, the analysis of the gas-surface interaction is extended to binary mixtures. In the end, it is discussed how the obtained results can be used in order to improve the efficiency of gas separation mechanisms.

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.

Journal of Computational Physics: X, 2, ISSN:2590-0552

In this work, we propose an approach for Temporal Large-Eddy Simulation (TLES) with direct deconvolution. In contrast to previous approaches such as the Temporal Approximate Deconvolution Model (TADM) by Pruett et al. [“A temporal approximate deconvolution model for large-eddy simulation,” Phys. Fluids 18, 028104 (2006)], the non-filtered velocity field is recovered using the differential form of the filter operation rather than from a truncated series expansion of the inverse filter operator. This direct deconvolution is used to obtain formal closure of an analytic evolution equation of the temporal residual-stress tensor. Thus, the Temporal Direct Deconvolution Model (TDDM) has advantages relative to the TADM in being both more accurate and requiring less computational effort. As for the TADM, a secondary regularization term based on selective frequency damping is employed. The TDDM was implemented in the spectral element code Nek5000 (Argonne National Laboratory, NEK5000 Version 17.0, 2019, https://nek5000.mcs.anl.gov) to simulate two canonical incompressible flows as three test cases: an a priori test case of Homogeneous Isotropic Turbulence (HIT) at Reλ = 50, a greatly coarsened a posteriori HIT case at Reλ = 190, and an a posteriori highly anisotropic turbulent channel flow at Reτ = 180. Analyses of the energy spectrum, the mean flow, the root-mean-square of the velocity fluctuations, and the Reynolds stresses are presented. The results demonstrate a significant improvement compared to no-model solutions regarding the mean flow in the turbulent channel and the energy spectrum in the HIT case, while the computational cost is reduced dramatically compared to the direct numerical simulation.

Summary There is strong interest to design Sequential Fully Implicit (SFI) methods for compositional flow simulations with convergence properties that are comparable to Fully Implicit (FI) methods. SFI methods decompose the fully coupled system into a pressure equation and a transport system of the components. During the pressure update, the compositions are frozen, and during the transport calculations, both the pressure and total-velocity are kept constant. The two systems are solved sequentially, and the solution, which is a fully implicit one, is obtained by controlling the splitting errors due to the decoupling. Having an SFI scheme that enjoys a convergence rate similar to FI makes it possible to design specialized numerical methods optimized for the different parabolic and the hyperbolic operators, as well as the use of high-order spatial and temporal discretization schemes. Here, we show that phase-potential upwinding is incompatible with the total-velocity formulation of the fluxes, which is common in SFI schemes. We observe that in cases with strong gravity or capillary pressure, it is possible to have flow reversals. These reversals can strongly affect the convergence rate of SFI methods. In this work, we employ implicit hybrid upwinding (IHU) with a SFI method. IHU determines the upwinding direction differently for the viscous, buoyancy, and capillary pressure terms in the phase velocity expressions. The use of IHU leads to a consistent SFI scheme in terms of both pressure and compositions, and it improves the SFI convergence significantly in settings with strong buoyancy or capillarity. We demonstrate the robustness of the IHU-based SFI algorithm across a wide parameter range. Realistic compositional models with gas and water injection are presented and discussed.

In this work, the energy transfer in gas-surface collisions is investigated using the molecular dynamics method. The numerical setup consists of a nitrogen molecule scattering from a graphite surface. The main focus is put on the energy redistribution between different molecular kinetic modes and the surface for the case of strong thermal non-equilibrium. The thermal non-equilibrium is defined as the state when either translational or rotational temperature of impinging molecules differs significantly from that of the surface. Accordingly, two different scenarios have been examined, including rotational and translational excitation of the initial molecular state. In contrast to the molecular beam method, the initial molecular velocities are sampled from the equilibrium Maxwellian distribution, ensuring isotropic incidence angles and energies. The obtained results are expressed in the form of energy transfer coefficients, which are used to quantify the normalized energy loss or gain in a specific mode. Furthermore, the velocity distributions of reflected molecules are analyzed and compared with some of the available wall kernels, providing further insight into the nature of the energy transfer dynamics and scattering process. Additionally, the numerical setup is tested against the available molecular beam experimental data and the obtained results were used to select a suitable numerical force field.

The Sequential Fully Implicit (SFI) method was proposed to simulate coupled immiscible multiphase fluid flow in porous media. Later, it was extended to the black-oil model, whereby the gas component is allowed to dissolve in the oil phase. Most recently, the SFI approach was extended to fully compositional isothermal displacements. SFI schemes solve the fully coupled system in two steps: (1) Construct and solve the pressure equation (flow problem). (2) Solve the coupled species transport equations for the phase saturations and phase compositions. Experience indicates that complex interphase mass transfer behaviors often lead to large numbers of SFI outer iterations compared with the Fully Implicit (FI) method. Here, we demonstrate that the convergence difficulties are directly related to the treatment of the coupling between the flow and transport problems, and we propose a new SFI variant based on a nonlinear overall-volume balance equation. The first step consists of forming and solving a nonlinear pressure equation, which is a weighted sum of all the component mass conservation equations. The second step of the new SFI scheme entails introducing the overall-mass density as a degree-of-freedom, and solving the full set of component conservation equations cast in the natural-variables form. During the second step, the pressure and the total-velocity fields are fixed. We analyze the `splitting errors' associated with the compositional SFI scheme, and we show how to control these errors in order to converge to the same solution as the Fully Implicit (FI) method. This robust sequential-implicit solution scheme allows for designing numerical methods and linear solvers that are optimized for the sub-problems of flow and transport., Comment: 52 pages, 36 figures, preprint

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.

Several existing models for immiscible two-phase flow identify trapping as the major cause of nonwetting relative permeability hysteresis. These models usually assume that relative permeability is a nonhysteretic function of the connected saturation. However, existing experimental results indicate that this assumption is not necessarily true. It is observed that while relative permeability models based on trapping, e.g., Land’s model, may capture the correct hysteresis behavior in consolidated porous media, they may be qualitatively inaccurate for the case of unconsolidated porous media, which a have lower pore-body to pore-throat aspect ratio. In order to bridge this gap, we present a novel framework for immiscible two-phase flow in which one can model relative permeability hysteresis patterns for both consolidated and unconsolidated porous media. An important aspect of this framework is the subdivision of the nonwetting phase into backbone, dendritic and trapped subphases. The closure problem now consists in modeling the volume transfer between the subphases and the relationship between the relative permeability and these subphases. For the purpose of developing, calibrating and validating our models in this framework, pore-network simulations of drainage and imbibition cycles are conducted for artificial networks as well as for a network representing Berea sandstone. Confirming results from previous works, we find a nearly nonhysteretic relationship between the nonwetting phase relative permeability and its backbone subphase. Additionally, we also observe a nonhysteretic relationship between the nonwetting backbone and trapped saturations. These observations are used to motivate constitutive relations for the proposed framework.

Recent studies suggest that pericytes around capillaries are contractile and able to alter the diameter of capillaries. To investigate the effects of capillary dilation on network dynamics, we performed simulations in artificial capillary networks of different sizes and complexities. The unequal partition of hematocrit at diverging bifurcations was modeled by assuming that each red blood cell (RBC) enters the branch with the faster instantaneous flow. Network simulations with and without RBCs were performed to investigate the effect of local dilations. The results showed that the increase in flow rate due to capillary dilation was less when the effects of RBCs are included. For bifurcations with sufficient RBCs in the parent vessel and nearly equal flows in the branches, the flow rate in the dilated branch did not increase. Instead, a self-regulation of flow was observed due to accumulation of RBCs in the dilated capillary. A parametric study was performed to examine the dependence on initial capillary diameter, dilation factor, and tube hematocrit. Furthermore, the conditions needed for an efficient self-regulation mechanism are discussed. The results support the hypothesis that RBCs play a significant role for the fluid dynamics in capillary networks and that it is crucial to consider the blood flow rate and the distribution of RBCs to understand the supply of oxygen in the vasculature. Furthermore, our results suggest that capillary dilation/constriction offers the potential of being an efficient mechanism to alter the distribution of RBCs locally and hence could be important for the local regulation of oxygen delivery.

American Journal of Physiology. Heart and Circulatory Physiology, 308 (3), ISSN:0363-6135, ISSN:1522-1539

The Rankine-Hugoniot-Riemann (RHR) solver has been designed to solve steady multidimensional conservation laws with source terms. The solver uses a novel way of incorporating cross fluxes as source terms. The combined source term from the cross fluxes and normal source terms is imposed in the middle of a cell, causing a jump in the solution according to the Rankine-Hugoniot condition. The resulting Riemann problems at the cell faces are then solved by a conventional Riemann solver. We prove that the solver is of second order accuracy for rectangular grids and confirm this by its application to the 2D scalar advection equation, the 2D isothermal Euler equations and the 2D shallow water equations. For these cases, the error of the RHR solver is comparable to or smaller than that of a standard Riemann solver with a MUSCL scheme. The RHR solver is also applied to the 2D full Euler equations for a channel flow with injection, and shown to be comparable to a MUSCL solver. Copyright © 2014 Published by Elsevier Ltd. Copyright © 2014 Elsevier Ltd. All rights reserved. This is the authors' accepted and refereed manuscript to the article.

In this work, we investigate a resolution evaluation criterion based on the ratio between turbulent length-scales and grid spacing within the context of dynamic resolution evaluation in hybrid LES/RANS simulations. A modified version of the commonly used length-scale criterion is adopted. The modified length-scale criterion is evaluated for a plane channel flow and compared to the criterion based on two-point correlations. Simulation results show qualitative agreement between the two criteria and physical predictions from both resolution indicators. These observations are confirmed by simulations of flows over periodic hills. It is further demonstrated that the length-scale based criterion is relatively less sensitive on variation of model parameters compared to criteria based on resolved percentage of turbulent quantities. The improved resolution criterion is applied in a dual-mesh hybrid LES/RANS solver. Numerical simulations with the hybrid solver suggest that the interactions between the length-scale resolution indicator and the solution are moderate, and that favorable comparisons with benchmark results are obtained. In summary, we demonstrate that the modified length-scale based resolution indicator performs satisfactorily in both pure LES and hybrid simulations. Therefore, it is selected as a promising candidate to provide reliable predictions of resolution adequacy for individual cells in hybrid LES/RANS simulations., Flow, Turbulence and Combustion, 93 (1), ISSN:1386-6184, ISSN:1573-1987

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 ana...

Direct simulation Monte-Carlo (DSMC) is the most established method for rarefied gas flow simulations. It is valid from continuum to near vacuum, but in cases involving small Knudsen numbers (Kn), it suffers from high computational cost. The Fokker-Planck (FP) method, on the other hand, is almost as accurate as DSMC for small to moderate Kn, but it does not have the computational drawback of DSMC, if Kn is small [P. Jenny, M. Torrilhon, and S. Heinz, “A solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion,” J. Comput. Phys. 229, 1077–1098 (2010) and H. Gorji, M. Torrilhon, and P. Jenny, “Fokker–Planck model for computational studies of monatomic rarefied gas flows,” J. Fluid Mech. 680, 574–601 (2011)]. Especially attractive is the combination of the two approaches leading to the FP-DSMC method. Opposed to other hybrid methods, e.g., coupled DSMC/Navier-Stokes solvers, it is relatively straightforward to couple DSMC with the FP method since both are based on particle solution algorithms sharing the same data structure and having similar components. Regarding the numerical accuracy of such particle methods, one has to distinguish between spatial truncation errors, time stepping errors, statistical errors and bias errors. In this paper, the bias error of the FP method is analyzed in detail, and it is shown how it can be reduced without increasing the particle number to an exorbitant level. The effectiveness of the discussed bias error reduction scheme is demonstrated for uniform shear flow, for which an analytical reference solution was derived.Direct simulation Monte-Carlo (DSMC) is the most established method for rarefied gas flow simulations. It is valid from continuum to near vacuum, but in cases involving small Knudsen numbers (Kn), it suffers from high computational cost. The Fokker-Planck (FP) method, on the other hand, is almost as accurate as DSMC for small to moderate Kn, but it does not have the computational drawback of DSMC, if Kn is small [P. Jenny, M. Torrilhon, and S. Heinz, “A solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion,” J. Comput. Phys. 229, 1077–1098 (2010) and H. Gorji, M. Torrilhon, and P. Jenny, “Fokker–Planck model for computational studies of monatomic rarefied gas flows,” J. Fluid Mech. 680, 574–601 (2011)]. Especially attractive is the combination of the two approaches leading to the FP-DSMC method. Opposed to other hybrid methods, e.g., coupled DSMC/Navier-Stokes solvers, it is relatively straightforward to couple DSMC with the FP method since both are based on p...

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.

In a recently proposed dual-mesh hybrid framework (Xiao and Jenny, J. Comput. Phys. 231 (4) (2012)), LES and RANS simulations are conducted simultaneously on the same domain, but on different meshes. In the current work, this framework is further extended to allow for non-conforming computational domains for the LES and the RANS simulations. With this extension we developed a hybrid solver coupling a high-order LES code based on Cartesian meshes with a general-purpose RANS solver based on body-fitting meshes. A relaxation approach is used to enforce the solid boundary conditions in the LES. Plane channel flow at Re τ = 590 and flows over periodic hills at two Reynolds numbers ( Re = 2800 and 10,595) are investigated with the new solver. The adequacy of the boundary representation and forcing strategy is shown. The numerical studies also demonstrate the flexibility of the extended solver and the predictive capability of the new hybrid framework, which consists of two solvers operating on the same physical domain, but with non-conforming computational domains (i.e., a Cartesian mesh based LES solver combined with a body-fitting mesh based RANS solver). The extensions explored in this study are of practical importance for industrial CFD applications as they successfully demonstrate how academic, very accurate, massively parallel LES solvers can be coupled with flexible RANS solvers. Since the coupling strategy is minimally intrusive, it is attractive for industrial purposes. With the current framework, the potential of many existing academic codes for practical flow simulations, where complex geometries and wall resolution requirements represent major hurdles, can be explored.

[1] In subsurface aquifers, dispersion of contaminant, or tracer, is mainly driven by spatial fluctuations in the flow field caused by heterogeneity of the hydraulic conductivity. Measurements of conductivity, however, are usually sparse. To assess the resulting uncertainty in the transport of tracers, Monte Carlo (MC) methods are usually applied, where the transport statistics are sampled over a large number of probable hydraulic conductivity realizations. In this paper, an alternative method is described that provides accurate transport statistics at a computational expense that is 3 orders of magnitude lower than conventional MC. The new method is applicable for conductivity fields with multivariate Gaussian characterization involving conductivity measurements for both small and high log-conductivity variances. The new method is validated against MC for different dispersion scenarios, where the region of interest spans tens of log-conductivity correlation lengths.

An amazingly wide range of complex behavior emerges from the cerebral cortex. Much of the information processing that leads to these behaviors is performed in neocortical circuits that span throughout the six layers of the cortex. Maintaining this circuit activity requires substantial quantities of oxygen and energy substrates, which are delivered by the complex yet well-organized and tightly-regulated vascular system. In this review, we provide a detailed characterization of the most relevant anatomical and functional features of the cortical vasculature. This includes a compilation of the available data on laminar variation of vascular density and the topological aspects of the microvascular system. We also review the spatio-temporal dynamics of cortical blood flow regulation and oxygenation, many aspects of which remain poorly understood. Finally, we discuss some of the important implications of vascular density, distribution, oxygenation and blood flow regulation for (laminar) fMRI.

In this paper we propose a novel model for turbulent premixed flames in the corrugated flamelet regime. The model combines the premixed flamelet approach with a BML ansatz in a PDF framework. For this purpose we introduce a progress variable, which is zero for unburnt and equal to one for reacting and burnt state. The probability that the progress variable switches from zero to one depends on the flame surface density (FSD). The FSD itself is modeled based on a modified flame residence time starting when the progress variable switches to one. To account for flame curvature, collapse and cusp formation, a stretch factor consistent with existing FSD transport equation source terms is employed to modify the flame residence time. To evolve the sensible enthalpy, conditional mixing is applied, which prevents mixing across the flame front. If the progress variable is one, a premixed flamelet is employed. With this approach, some model difficulties related to solving transport equations for the mean FSD and progress variable are overcome. For example, counter-gradient diffusion and FSD transport due to flame propagation appear in closed form. The combination of the BML approach with a flamelet model leads to more accurate temperature predictions, which is demonstrated for the Aachen flames F1–F3.

A finite volume based numerical modeling framework using a hierarchical fracture representation has been developed to compute flow induced shear failure. To accurately capture the mechanics near fracture manifolds, discontinuous basis functions are employed which ensure continuity of the displacement gradient across fractures. With these special basis functions, traction and compressive forces on the fracture segment can be calculated without any additional constraints, which is extremely useful for estimating the irreversible slip based on a constitutive friction law. Unlike other models, here asymptotic dilation of fracture aperture due to shear failure is considered. To solve the resulting linear system, a sequential approach is used, that is, first the flow- and then the mechanics problems are solved. The new modeling framework is very useful to predict seismicity, permeability- and flow evolution in geological reservoirs. This is demonstrated with numerical simulations of enhancing a geothermal system. Novelties of this approach are (i) that due to the special basis functions only one additional degree of freedom per fracture segment is introduced (opposed to four in the XFEM method) and (ii) achieving consistent coupling with the flow solver using asymptotic aperture relaxation.

The Fully Implicit (FI) and the Adaptive Implicit (AI) methods are widely used for general-purpose reservoir simulation. There has been growing interest, however, in sequential-implicit schemes. The common sequential solution strategies are: IMplicit Pressure Explicit Saturations (IMPES) and Sequential Fully Implicit (SFI) methods. In highly heterogeneous domains with tight coupling between the multiphase flow and the multi-component transport, IMPES suffers from severe restrictions on the size of the stable timestep, and SFI suffers from slow convergence of the sequential updating between the flow and transport problems. Here, we describe a modified SFI (m-SFI) scheme that improves the convergence behavior substantially. The modification entails additional coupling terms to the pressure equation that are limited in both space and time. Specifically, the pressure equation is complemented with a local approximation of the pressure-saturation/composition coupling terms that are brought about by the appearance of the gas-phase during iterations. This modification is also applied to the modified Sequential Adaptive Implicit (m-SAI) scheme. We consider several very challenging compositional processes whereby the SFI method suffers from severe nonlinear convergence behaviors, and we demonstrate using numerical experiments and analysis that the modified algorithms have convergence properties that are quite close to those of the FI and AI methods.

In the context of flow and transport in porous and fractured media, Darcy-based continuum models, while computationally inexpensive, are of limited use when the scale of interest is of similar size or smaller than the characteristic network connection length. Recently, we have outlined a non-local Darcy model that bridges the gap between network and Darcy-based descriptions. This formulation is able to account for non-local pressure effects that are not accounted for in a classical Darcy description. At the heart of this non-local flow formulation is a conductivity distribution or kernel that is related to the scalar permeability in the classical Darcy law. In this paper, ensembles of flow networks are considered, of which the necessary statistical information is assumed to be known. In order to relate the conductivity distribution with the flow statistics, a stochastic transport model for fluid particles, termed generalized continuous time random walk (g-CTRW), which is a generalization of correlated continuous time random walk, is introduced. Note that similar assumptions as for correlated CTRW are made, i.e., that lengths and velocities of connections between successive nodes along the trajectories can be described by Markov processes. In order to proceed with a theoretical analysis, a Boltzmann equation is presented, which is consistent with the particle time marching algorithm based on g-CTRW. An important outcome of the analysis is an expression relating the joint probability density function of velocity and connection length in the networks with the conductivity kernel. A numerical, stationary flow example demonstrates how the kernel can be extracted. Further, an algorithm is proposed to compute consistent velocity statistics, mean pressure distribution, and spatially varying conductivity kernel in the case of non-stationary flow. This coupled iterative approach is an attempt to consistently compute stochastic flow and transport in large network ensembles.

A recently introduced subphase framework for modeling immiscible two phase flow in porous media has been extended. In this framework the nonwetting and wetting phases are divided into subphases distinguished by their connectivity. The nonwetting phase is divided into three subphases: backbone, dendritic, and trapped subphases. Similarly, the wetting phase is divided into four subphases; backbone, dendritic, film, and isolated subphases. The subphase saturations evolve according to volume transfer terms, which require modeling. Within this framework, relative permeability models can be developed, which take into account the contributions of the different subphases appropriately. For example, only the nonwetting backbone subphase contributes to nonwetting relative permeability. Quasi-static flow network simulations of several drainage-imbibition cycles are conducted to analyze the evolution of the subphases in three different pore-networks. Furthermore, a relative permeability model for the wetting phase as a function of the subphase saturations is proposed. The resulting model can capture complex hysteretic behavior present in relative permeability-saturation curves.

Objective Oxygen transport to parenchymal cells occurs mainly at the microvascular level, and depends on convective red blood cell (RBC) flux, which is proportional in an individual capillary to the product of capillary hematocrit and red blood cell velocity. This study investigates the relative influence of these two factors on tissue oxygen partial pressure (PO2). Methods A simple analytical model is used to quantify the respective influences of hematocrit, RBC velocity and flow on tissue oxygenation around capillaries. Predicted tissue PO2 levels are compared with a detailed computational model. Results Hematocrit is shown to have a larger influence on tissue PO2 than RBC velocity. The effect of RBC velocity increases with distance from the arterioles. Good agreement between analytical and numerical results is obtained and the discrepancies are explained. Significant dependence of mass transfer coefficients on RBC velocity at low hematocrit is demonstrated. Conclusions For a given RBC flux in a capillary, the PO2 in the surrounding tissue increases with increasing hematocrit, as a consequence of decreasing intravascular resistance to diffusive oxygen transport from RBCs to tissue. These results contribute to understanding the effects of blood flow changes on oxygen transport, such as occur in functional hyperemia in the brain. This article is protected by copyright. All rights reserved.

In this paper, a novel model for turbulent premixed combustion in the corrugated flamelet regime is presented, which is based on transporting a joint probability density function (PDF) of velocity, turbulence frequency and a scalar vector. Due to the high dimensionality of the corresponding sample space, the PDF equation is solved with a Monte-Carlo method, where individual fluid elements are represented by computational particles. Unlike in most other PDF methods, the source term not only describes reaction rates, but accounts for “ignition” of reactive unburnt fluid elements due to propagating embedded quasi laminar flames within a turbulent flame brush. Unperturbed embedded flame structures and a constant laminar flame speed (as expected in the corrugated flamelet regime) are assumed. The probability for an individual particle to “ignite” during a time step is calculated based on an estimate of the mean flame surface density (FSD), latter gets transported by the PDF method. Whereas this model concept has recently been published [21], here, a new model to account for local production and dissipation of the FSD is proposed. The following particle properties are introduced: a flag indicating whether a particle represents the unburnt mixture; a flame residence time, which allows to resolve the embedded quasi laminar flame structure; and a flag indicating whether the flame residence time lies within a specified range. Latter is used to transport the FSD, but to account for flame stretching, curvature effects, collapse and cusp formation, a mixing model for the residence time is employed. The same mixing model also accounts for molecular mixing of the products with a co-flow. To validate the proposed PDF model, simulation results of three piloted methane-air Bunsen flames are compared with experimental data and very good agreement is observed.

In a real turbulent spray flame, dispersion, continuous phase turbulence modification, dispersed phase inter-particle collisions, evaporation, mixing and combustion occur simultaneously. Dealing with all these complexities and their interactions poses a tremendous modeling task. Therefore, in order to advance current modeling capabilities, it seems reasonable to aim for progress in individual sub-areas like breakup, dispersion, mixing and combustion, which however cannot be viewed in complete isolation. Further, one has to consider advantages and disadvantages of the general modeling approaches, which are direct numerical simulation (DNS), large eddy simulation (LES), simulations based on Reynolds averaged equations and probability density function (PDF) methods. Not least one also has to distinguish between Eulerian and Lagrangian dispersed phase descriptions. The goal of this paper is to provide a review of computational model developments relevant for turbulent dilute spray combustion, i.e. the dense regime, including collisions as well as primary and secondary atomization, is not covered. Also not considered is breakup in dilute sprays, which can occur in the presence of sufficiently high local turbulence. It is intended to guide readers interested in theory, in the development and validation of predictive models, and in planning new experiments. In terms of physical phenomena, the current understanding regarding turbulence modification due to droplets, preferential droplet concentration, impact on evaporation and micro-mixing, and different spray combustion regimes is summarized. In terms of modeling, different sets of equations are discussed, i.e. the governing conservation laws without and with point droplet approximation as employed by DNS, the filtered equations considered in LES, the Reynolds averaged equations, and Lagrangian evolution equations. Further, small scale models required in the context of point droplet approximations are covered. In terms of computational studies and method developments, progress is categorized by the employed approaches, i.e. DNS, LES, simulations based on Reynolds averaged equations, and PDF methods. In terms of experiments, various canonical spray flame configurations are discussed. Moreover, some of the most important experiments in this field are presented in a structured way with the intention to provide a database for model validation and a guideline for future investigations.

Probability density function (PDF) methods are especially suited for turbulent combustion calculations, since the averaging of the reaction source term in the governing equation poses no closure problem. Molecular mixing and computing thereof in the presence of chemical kinetics, however, remains a major modeling challenge; not only for PDF methods. In this work we present a new model for partially premixed combustion based on the joint statistics of mixture fraction, scalar dissipation rate and a burning indicator, which is used to evolve notional fluid particles in mixture fraction-reactive scalar space (e.g., enthalpy). Therefore, the parameterized scalar profile (PSP) mixing model was extended for reactive scalars. As in the PSP mixing model for inert scalars, each fluid particle is associated with a representative profile in physical space. Different than in the standard PSP model, the reactive profiles get modified by chemical reactions. The decision whether a fluid particle is reacting or not is based on a burning indicator, the local scalar dissipation rate and the profile boundaries. The burning indicator accounts for the flame propagation in the partially premixed environment and controls extinction and re-ignition; together with the scalar dissipation rate. The solutions of the reaction–diffusion equation for different scalar dissipation rates is used to determine the state of the embedded flame associated with a reacting particle, which allows to construct reactive scalar profiles and on the other hand pure diffusion is assumed for extinct particles. The new combustion model was validated with the Sandia flame F data and comparisons demonstrate its ability to account for the relevant phenomena in partially premixed flames.

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.

A consistent method to derive the parameterized NO source term from a given flamelet library is presented. The approach was tested by conducting PDF simulations and comparing computed solutions of the Sandia D flame with measurements. The parameterization is based on laminar flamelets with additional parameters for NO formation and radiation. To obtain feedback of the NO concentration on the NO source term, evolution to equilibrium is calculated with a complex chemical mechanism (including nitrogen chemistry), whereas the concentration of all non-nitrogen species are parameters obtained from flamelet profiles. With this method, which requires an appropriate initialization of the nitrogen containing species, unrealistically large NO source terms are avoided, which is a shortcoming of previously devised methods. In addition, to properly account for radiative heat losses, an additional progress variable was introduced. Simulation results obtained with our PDF method are in excellent agreement with experimental data. While the NO concentration values appear too high (by approximately 30% at the exit of the burner), if radiation effects are ignored, their match is very good, if radiation effects are taken into account.

The design of an airfoil structure involves the disciplines of aerodynamics and structural mechanics, both of which are considered in the design methodology presented in this article. The approach described in this article starts from a requirement formulation based on a time-series of spanwise lift distributions on a morphing wing, representing the mission profile of the aircraft as a whole. This allows to specify goals based directly on aerodynamic performances instead of prescribing fixed geometrical shapes. Using the aero-structural analysis tool presented here, together with a parametrization representing the airfoil outer shape as well as its mechanical properties, allows the formulation of a combined aero-structural optimization problem. Promising aerodynamic and structural morphing performances have been obtained by applying the method to a morphing concept using Dielectric Elastomers (DEs) as actuators. Although the coupled physics are considered and a detailed material model has been used, results can be obtained within reasonable computational time by parallel evaluation of the candidate solutions. Improved aerodynamic performances have been obtained using this concurrent coupled method, in comparison to a sequential aerodynamic and structural optimization.

Transport in Porous Media, 87 (2), ISSN:0169-3913, ISSN:1573-1634

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.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.

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.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 ...

A detailed model of red blood cell (RBC) transport in a capillary network is an indispensable element of a comprehensive model for the supply of the human organism with oxygen and nutrients. In this paper, we introduce a two-phase model for the perfusion of a capillary network. This model accounts for the special role of RBCs, which have a strong influence on network dynamics. Analytical results and numerical simulations with a discrete model and a generic network topology indicate that there exists a local self-regulation mechanism for the flow rates and a global de-mixing process that leads to an inhomogeneous haematocrit distribution. Based on the results from the discrete model, we formulate an efficient algorithm suitable for computing the pressure and flow field as well as a continuous haematocrit distribution in large capillary networks at steady state.

The authors use a probability density function method in which a transport equation for thejoint probability density function of velocity, turbulence frequency, progress variable, mixture fraction, and enthalpy was solved. Based on a new combustion model, this information was used to obtain the joint statistics of the compositions. The extinction and reignition of fluid particles is controlled by a particle progress variable and its scalar dissipation rate. Precomputed flamelets are only employed for particles in the flammable mixture fraction range. Outside of this range, the particle enthalpy is evolved by a mixing model (similar to the mixture fraction). Numerical test cases of lifted turbulent diffusion flames with a considerable amount of local extinction show the ability of the model to account for the major physical effects and comparisons with experimental data show good agreement.

For the simulation of turbulent nonpremixed flames the joint statistics of mixture fraction and scalar dissipation rate are of major importance. If thin reaction zones are present, these quantities determine essentially the composition and thermal state in the flow domain. It has been argued that in the transported probability density function (PDF) context the commonly used mixing models do not provide such statistics even though PDF methods are applied with some success for the simulation of flames involving local extinction. In this work, we propose an extension of the parameterized scalar profile (PSP) mixing model, which is able to provide the joint scalar–scalar dissipation rate PDF. The model predictions are compared with direct numerical simulation (DNS) data, where the influence of different initial scalar lengthscales on the mixing dynamics was investigated. In a small study involving a partially stirred reactor (PaSR), the suggested mixing model is combined with a laminar flamelet approach forming a closure for the reaction/diffusion dynamics.

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.

A previously presented hybrid finite volume/particle method for the solution of the joint-velocity-frequency-composition probability density function (JPDF) transport equation in complex 3D geometries is extended for parallel computing. The parallelization strategy is based on domain decomposition. The finite volume method (FVM) and the particle method (PM) are parallelized separately and the algorithm is fully synchronous. For the FVM a standard method based on transferring data in ghost cells is used. Moreover, a subdomain interior decomposition algorithm to efficiently solve the implicit time integration for hyperbolic systems is described. The parallelization of the PM is more complicated due to the use of a sub-time stepping algorithm for the particle trajectory integration. Hereby, each particle obeys its local CFL criterion, and the covered distances per global time step can vary significantly. Therefore, an efficient algorithm which deals with this issue and has minimum communication effort was devised and implemented. Numerical tests to validate the parallel vs. the serial algorithm are presented, where also the effectiveness of the subdomain interior decomposition for the implicit time integration was investigated. A 3D dump-combustor configuration test case with about 2.5 × 10 5 cells was used to demonstrate the good performance of the parallel algorithm. The hybrid algorithm scales well and the maximum speedup on 60 processors for this configuration was 50 (≈80% parallel efficiency).

Summary A multiscale finite-volume (MSFV) framework for reservoir simulation is described. This adaptive MSFV formulation is locally conservative and yields accurate results of both flow and transport in large-scale highly heterogeneous reservoir models. IMPES and sequential implicit formulations are described. The algorithms are sensitive to the specific characteristics of flow (i.e., pressure and total velocity) and transport (i.e., saturation). To compute the fine-scale flow field, two sets of basis functions - dual and primal - are constructed. The dual basis functions, which are associated with the dual coarse grid, are used to calculate the coarse scale transmissi-bilities. The fine-scale pressure field is computed from the coarse grid pressure via superposition of the dual basis functions. Having a locally conservative fine scale velocity field is essential for accurate solution of the saturation equations (i.e., transport). The primal basis functions, which are associated with the primal coarse grid, are constructed for that purpose. The dual basis functions serve as boundary conditions to the primal basis functions. To resolve the fine scale flow field in and around wells, a special well basis function is devised. As with the other basis functions, we ensure that the support for the well basis is local. Our MSFV framework is designed for adaptive computation of both flow and transport in the course of a simulation run. Adaptive computation of the flow field is based on the time change of the total mobility field, which triggers the selective updating of basis functions. The key to achieving scalable (efficient for large problems) adaptive computation of flow and transport is the use of high fidelity basis functions with local support. We demonstrate the robustness and computational efficiency of the MSFV simulator using a variety of large heterogeneous reservoir models, including the SPE 10 comparative solution problem.

In this paper we present a multi-scale finite-volume (MSFV) method to solve elliptic problems with many spatial scales arising from flow in porous media. The method efficiently captures the effects of small scales on a coarse grid, is conservative, and treats tensor permeabilities correctly. The underlying idea is to construct transmissibilities that capture the local properties of the differential operator. This leads to a multi-point discretization scheme for the finite-volume solution algorithm. Transmissibilities for the MSFV have to be constructed only once as a preprocessing step and can be computed locally. Therefore this step is perfectly suited for massively parallel computers. Furthermore, a conservative fine-scale velocity field can be constructed from the coarse-scale pressure solution. Two sets of locally computed basis functions are employed. The first set of basis functions captures the small-scale heterogeneity of the underlying permeability field, and it is computed in order to construct the effective coarse-scale transmissibilities. A second set of basis functions is required to construct a conservative fine-scale velocity field. The accuracy and efficiency of our method is demonstrated by various numerical experiments.

In this chapter, mathematical and computational light scattering models relevant for halftone reproduction and other simulation studies in digital imaging are discussed. While in principle Maxwell's electromagnetic equations provide a very accurate and general description of light propagation, their numerical solution in highly heterogeneous domains is computationally too expensive for the kind of practical applications considered in this book. Therefore, the focus here will be on the computationally much more efficient transport theory. Besides discussing its limitations, the most general solution approach, that is, the Monte Carlo method, is explained and compared with more simple models, which involve further approximations. Keywords: probability density function method; point spread function; light scattering; transport theory; Monte Carlo; stencil method; multi–scale method; optical dot gain; Kubelka-Munk; random walk; diffusion model; polarization; fluorescence

Summary In this paper we present a 3D flux-continuous finite-difference formulation designed for flow simulation of models with nonorthogonal hexahedron grids with general tensor permeability. Our development follows that of Aavatsmark et al.,1 but we do not operate in transformed space. The new 27-point discretization formula has been implemented in a finite-difference reservoir simulator. This stencil has many desirable properties, including collapsing into a consistent form in two dimensions. We demonstrate that there are many practical situations in which neglecting the influence of nonorthogonality and general tensors results in firstorder errors in flow predictions. A rigorous implementation for this 27-point difference operator as a control-volume finite-difference method determines the upwinding of convection terms associated with multiphase computations. Results and issues associated with implementation of this operator in a conventional finite-difference reservoir simulator are discussed. As an alternative to directly solving the linear matrix associated with the 27-point stencil of the flux-continuous operator, we examine iterative methods that split the matrix into a 7-point stencil part and a remainder. The 7-point stencil part is solved by a direct or iterative method, with the remainder part updated from the previous timestep or iteration. This split operator may permit retention of the linear solver for the standard 7-point formulation while retaining nonorthogonal grid and tensor information.

Summary A hexahedral multiblock grid (MBG) formulation for the modeling of two-phase reservoir flows is developed and applied. Multiblock approaches are well suited to the modeling of geometrically complex features while avoiding many of the complications of fully unstructured techniques. Important implementation issues are discussed, including the accurate treatment of grid nonorthogonality and full tensor permeability (through use of a 27-point finite difference stencil), the treatment of exceptional cases arising when five blocks intersect, and a new well model that allows for the accurate resolution of near-well effects. Solution efficiency issues, including the use of tensor splitting and the linear solution technique, are also discussed. The actual grid generation step is accomplished using a commercial package. Results for a variety of cases involving flow through geologically complex systems and reservoirs with horizontal and deviated wells are presented. Comparison with analytical results demonstrates the high level of accuracy of the 27-point formulation and the new well model. The method is also applied to a realistic example involving flow through a heterogeneous faulted system. This example illustrates the types of geometrically complex systems that can be modeled using the hexahedral multiblock approach.