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The Vlasov-Fokker-Planck equation (together with Maxwell's equations) provides the basis for plasma flow calculations. While the terms accounting for long range forces are established, different drift and diffusion terms are used to describe Coulomb collisions. Here, linear drift and a constant diffusion coefficient are considered and the electromagnetic fields are imposed, i.e., plasma frequency is not addressed. The solution algorithm is based on evolving computational particles of a large ensemble according to a Langevin equation, whereas the time step size is typically limited by plasma frequency, Coulomb collision frequency and cyclotron frequency. To overcome the latter two time step size constraints, a novel time integration scheme for the particle evolution is presented. It only requires that gradients of mean velocity, bath temperature, magnetic field and electric field have to be resolved along the trajectories. In fact, if these gradients are zero, then the new integration scheme is statistically exact; no matter how large the time step is chosen. Obviously, this is a computational advantage compared to classical integration schemes, which is demonstrated with numerical experiments of isolated charged particle trajectories under the influence of constant magnetic- and electric fields. Besides single ion trajectories, also plasma flow in spatially varying electromagnetic fields was investigated, that is, the influence of time step size and grid resolution on the final solution was studied. • The research highlight presented in this paper is an analytical solution of the coupled Langevin equations as they result from the Vlasov-Fokker-Planck equation for plasma flow with constant electromagnetic fields and constant friction coefficient. • This analytical solution involves a transformation into complex space and allows to take extremely large time steps when used in a solution algorithm. • This is demonstrated in the paper with numerical studies. [ABSTRACT FROM AUTHOR]

Buoyant incompressible multiphase flow and transport in porous media typically is simulated by coupling an elliptic flow equation with a hyperbolic transport description. The tight coupling between flow and transport either requires a fully implicit solution algorithm or very small time steps, if solved sequentially. In any case, however, a large linear system has to be solved each time step. Here a new solution approach is devised, which relies on solving a coupled hyperbolic system of conservation laws with an explicit finite volume method. Consequently, only local operations have to be performed, which is a great advantage in case of massive parallel simulations and if GPUs are employed. The devised method is based on the isothermal Euler equations with momentum source terms accounting for resistance due to the porous medium and buoyancy. In this system the pressure is proportional to the density and in case of very small Mach numbers and relative density variation, the computed velocity field is divergence free and converges towards the mean total volumetric flow in the porous domain. To account for saturation transport, the system was augmented by an additional hyperbolic equation. In order to obtain the numerical fluxes, a characteristic based approximative Riemann solver was developed and 2nd order accuracy in space and time is achieved by piecewise linear reconstruction. Two-phase flow test cases with buoyant plumes and a lock exchange problem demonstrate that the devised hyperbolic approach recovers the correct incompressible solutions. Numerical experiments also demonstrate that the method is accurate and efficient. Thus the devised method is very promising, and conceptually it also can be applied for flow and transport in heterogeneous media; but it remains to be investigated how well it performs for such cases. • Flow and transport are described by a coupled hyperbolic system of PDEs. • Fully explicit solution scheme for strongly coupled two-phase flow in porous media. • An efficient approximate Riemann solver for the numerical fluxes is devised. • Second order in space and time is achieved by a TVD method. • The proposed method is an ideal candidate for GPUs. [ABSTRACT FROM AUTHOR]

Recently, a parallel Fokker–Planck-DSMC algorithm for rarefied gas flow simulation in complex domains at all Knudsen numbers was developed by the authors. Fokker–Planck-DSMC (FP-DSMC) is an augmentation of the classical DSMC algorithm, which mitigates the near-continuum deficiencies in terms of computational cost of pure DSMC. At each time step, based on a local Knudsen number criterion, the discrete DSMC collision operator is dynamically switched to the Fokker–Planck operator, which is based on the integration of continuous stochastic processes in time, and has fixed computational cost per particle, rather than per collision. In this contribution, we present an extension of the previous implementation with automatic local mesh refinement and parallel load-balancing. In particular, we show how the properties of discrete approximations to space-filling curves enable an efficient implementation. Exemplary numerical studies highlight the capabilities of the new code. [ABSTRACT FROM AUTHOR]

Pore-network models of porous media are useful in the study of pore-scale flow in porous media. In order to extract macroscopic properties from flow simulations in pore-networks, it is crucial the networks are large enough to be considered representative elementary volumes. However, existing two-phase network flow solvers are limited to relatively small domains. For this purpose, a multi-scale pore-network (MSPN) method, which takes into account flow-rate effects and can simulate larger domains compared to existing methods, was developed. In our solution algorithm, a large pore network is partitioned into several smaller sub-networks. The algorithm to advance the fluid interfaces within each subnetwork consists of three steps. First, a global pressure problem on the network is solved approximately using the multiscale finite volume (MSFV) method. Next, the fluxes across the subnetworks are computed. Lastly, using fluxes as boundary conditions, a dynamic two-phase flow solver is used to advance the solution in time. Simulation results of drainage scenarios at different capillary numbers and unfavourable viscosity ratios are presented and used to validate the MSPN method against solutions obtained by an existing dynamic network flow solver. [ABSTRACT FROM AUTHOR]

In this work, the zonal Multiscale Finite Volume (zMSFV) framework is presented. The framework generalizes previous extensions of the Multiscale Finite Volume (MSFV) method and allows to integrate additional models in a unified way. The new approach is based on splitting the domain of interest into a classical MSFV domain and additional zones. Within each zone, the solution space is spanned by so-called zonal functions. Coupling of the zonal solutions to the surrounding domain is achieved via extended zonal functions. The construction of basis, zonal and extended zonal functions can be performed such that the functions form a partition of unity. The zMSFV framework allows to include the influence of the zonal functions directly in the coarse-scale problem. Similar to previous MSFV method formulations, the zMSFV framework can ensure conservative coarse-scale solutions and allows for the construction of conservative fine-scale solutions. The main features of the zMSFV framework are used to derive robust preconditioners for iterative solutions of high-contrast problems as encountered in subsurface flow and transport simulations. It is shown that the obtained convergence rates are independent of the contrast. [ABSTRACT FROM AUTHOR]

A major challenge for the conventional Direct Simulation Monte Carlo ( DSMC ) technique lies in the fact that its computational cost becomes prohibitive in the near continuum regime, where the Knudsen number ( Kn )—characterizing the degree of rarefaction—becomes small. In contrast, the Fokker–Planck ( FP ) based particle Monte Carlo scheme allows for computationally efficient simulations of rarefied gas flows in the low and intermediate Kn regime. The Fokker–Planck collision operator—instead of performing binary collisions employed by the DSMC method—integrates continuous stochastic processes for the phase space evolution in time. This allows for time step and grid cell sizes larger than the respective collisional scales required by DSMC . Dynamically switching between the FP and the DSMC collision operators in each computational cell is the basis of the combined FP-DSMC method, which has been proven successful in simulating flows covering the whole Kn range. Until recently, this algorithm had only been applied to two-dimensional test cases. In this contribution, we present the first general purpose implementation of the combined FP-DSMC method. Utilizing both shared- and distributed-memory parallelization, this implementation provides the capability for simulations involving many particles and complex geometries by exploiting state of the art computer cluster technologies. [ABSTRACT FROM AUTHOR]

A Fokker–Planck based particle Monte Carlo algorithm was devised recently for simulations of rarefied gas flows by the authors [1–3] . The main motivation behind the Fokker–Planck (FP) model is computational efficiency, which could be gained due to the fact that the resulting stochastic processes are continuous in velocity space. This property of the model leads to simulations where the computational cost becomes independent of the Knudsen number (Kn) [3] . However, the Fokker–Planck model which can be seen as a diffusion approximation of the Boltzmann equation, becomes less accurate as Kn increases. In this study we propose a hybrid Fokker–Planck–Direct Simulation Monte Carlo (FP–DSMC) solution method, which is applicable for the whole range of Kn. The objective of this algorithm is to retain the efficiency of the FP scheme at low Kn ( Kn ≪ 1 ) and to employ conventional DSMC at high Kn ( Kn ≫ 1 ). Since the computational particles employed by the FP model represent the same data as in DSMC, the coupling between the two methods is straightforward. The new ingredient is a switching criterion which would ideally result in a hybrid scheme with the efficiency of the FP method and the accuracy of DSMC for the whole Kn-range. Here, we adopt the number of collisions in a given computational cell and for a given time step size as a decision criterion in order to switch between the FP model and DSMC. For assessment of the hybrid algorithm, different test cases including flow impingement and flow expansion through a slit were studied. Both accuracy and efficiency of the model are shown to be excellent for the presented test cases. [ABSTRACT FROM AUTHOR]

The multiscale finite volume (MSFV) method was designed to efficiently compute approximate numerical solutions of elliptic and parabolic problems with highly heterogeneous coefficients. For a wide range of test cases, it has been demonstrated that MSFV solutions are in excellent agreement with reference data obtained with a standard fine-scale finite-volume method. However, for some problems involving large coherent structures with strong contrasts (e.g., channels and shale layers), the MSFV method produces unsatisfactory results due to the dwindling validity of the localization assumption. Recently, the iterative MSFV (i-MSFV) method was introduced to iteratively improve the localization assumption, and it was shown that the method converges to the fine-scale reference solution. In this work, it is explained how the convergence rate of the i-MSFV method can be enhanced by consistent enrichment of the initial coarse space spanned by nodal basis functions. A new set of enrichment basis functions associated with additional coarse-scale degrees of freedom (DoF) is introduced. By construction, the sum of all nodal and enrichment basis functions is one in the entire computational domain. Further, a hybrid finite-volume/Galerkin formulation for the coarse-scale problem and a generic placement strategy for the additional DoF are devised. The resulting iterative Galerkin-enriched MSFV (i-Ge-MSFV) method has all features of the i-MSFV method, but is more robust and has improved convergence properties. [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to the algorithmic improvement and careful analysis of the Fokker–Planck kinetic model derived by Jenny et al. [1] and Gorji et al. [2]. The motivation behind the Fokker–Planck based particle methods is to gain efficiency in low Knudsen rarefied gas flow simulations, where conventional direct simulation Monte Carlo (DSMC) becomes expensive. This can be achieved due to the fact that the resulting model equations are continuous stochastic differential equations in velocity space. Accordingly, the computational particles evolve along independent stochastic paths and thus no collision needs to be calculated. Therefore the computational cost of the solution algorithm becomes independent of the Knudsen number. In the present study, different computational improvements were persuaded in order to augment the method, including an accurate time integration scheme, local time stepping and noise reduction. For assessment of the performance, gas flow around a cylinder and lid driven cavity flow were studied. Convergence rates, accuracy and computational costs were compared with respect to DSMC for a range of Knudsen numbers (from hydrodynamic regime up to above one). In all the considered cases, the model together with the proposed scheme give rise to very efficient yet accurate solution algorithms. [Copyright &y& Elsevier]

An adaptive conservative time integration scheme (ACTI) for compressible flow simulations is devised. A finite volume method is used, in which every cell employs a time step size which is some power of two times smaller than a global time step. By first advancing those cells with smaller time steps, this allows to use the already computed accumulated fluxes when the adjacent cells with larger time steps catch up. Conservation and periodic synchronization are thus guaranteed by construction; both important advantages of the new scheme (especially in the presence of shock waves) compared to previously proposed adaptive time stepping methods for compressible flow. Accuracy and computational speedup are demonstrated for challenging 1D and 2D test cases. Comparisons of ACTI results with an analytical solution and with results obtained without ACTI show excellent agreement. Although maximum wave speeds and/or cell sizes vary by several orders of magnitude throughout the computational domains, the CFL numbers employed by ACTI in each cell vary not more than by approximately a factor of two. Without ACTI, on the other hand, the CFL numbers are tiny in most of the cells. Since this results in a dramatic reduction of required flux computations, high speedup gains are achieved due to ACTI; in particular, if only few cells experience a very small time step, while the time step size limitations are less severe for the majority of cells. • Explicit adaptive time integration for compressible flow. • Strictly conservative - is of great importance especially in the presence of shocks. • Piecewise linear reconstruction in space and time - is important for accuracy. • Total number of flux computations and cell value updates can be drastically reduced. • Proposed scheme is robust and can be integrated immediately into existing codes. [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]

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]

Abstract: In this work, we propose a consistent turbulence-modeling framework for hybrid LES/RANS modeling. In this framework, the filtered and Reynolds averaged Navier–Stokes (RANS) equations are solved simultaneously in the whole domain on their respective meshes. Consistency between the two solutions is achieved in terms of velocity, pressure, and turbulent quantities through additional drift terms in the corresponding equations. This approach leads to clean conditions at the LES/RANS interfaces. Note that this general framework does not depend on the specific choice of LES and RANS models. A hybrid LES/RANS solver is developed within this framework and used to simulate the flow in a plane channel and that in a channel with periodic hills. The results demonstrate that the hybrid solver leads to significantly improved results with moderate computational overhead compared to traditional LES, making it a promising candidate for industrial flow simulations. [Copyright &y& Elsevier]

Abstract: The multiscale finite volume (MSFV) method is a computationally efficient numerical method for the solution of elliptic and parabolic problems with heterogeneous coefficients. It has been shown for a wide range of test cases that the MSFV results are in close agreement with those obtained with a classical (computationally expensive) technique. The method, however, fails to give accurate results for highly anisotropic heterogeneous problems due to weak localization assumptions. Recently, a convergent iterative MSFV (i-MSFV) method was developed to enhance the quality of the multiscale results by improving the localization conditions. Although the i-MSFV method proved to be efficient for most practical problems, it is still favorable to improve the localization condition adaptively, i.e. only for a sub-domain where the original MSFV localization conditions are not acceptable, e.g. near shale layers and long coherent structures with high permeability contrasts. In this paper, a space–time adaptive i-MSFV (ai-MSFV) method is introduced. It is shown how to improve the MSFV results adaptively in space and simulation time. The fine-scale smoother, which is necessary for convergence of the i-MSFV method, is also applied locally. Finally, for multiphase flow problems, two criteria are investigated for adaptively updating the MSFV interpolation functions: (1) a criterion based on the total mobility change for the transient coefficients and (2) a criterion based on the pressure equation residual for the accuracy of the results. For various challenging test cases it is demonstrated that iterations in order to obtain accurate results even for highly anisotropic heterogeneous problems are required only in small sub-domains and not everywhere. The findings show that the error introduced in the MSFV framework can be controlled and improved very efficiently with very little additional computational cost compared to the original, non-iterative MSFV method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a stochastic model is presented to simulate the flow of gases, which are not in thermodynamic equilibrium, like in rarefied or micro situations. For the interaction of a particle with others, statistical moments of the local ensemble have to be evaluated, but unlike in molecular dynamics simulations or DSMC, no collisions between computational particles are considered. In addition, a novel integration technique allows for time steps independent of the stochastic time scale. The stochastic model represents a Fokker–Planck equation in the kinetic description, which can be viewed as an approximation to the Boltzmann equation. This allows for a rigorous investigation of the relation between the new model and classical fluid and kinetic equations. The fluid dynamic equations of Navier–Stokes and Fourier are fully recovered for small relaxation times, while for larger values the new model extents into the kinetic regime. Numerical studies demonstrate that the stochastic model is consistent with Navier–Stokes in that limit, but also that the results become significantly different, if the conditions for equilibrium are invalid. The application to the Knudsen paradox demonstrates the correctness and relevance of this development, and comparisons with existing kinetic equations and standard solution algorithms reveal its advantages. Moreover, results of a test case with geometrically complex boundaries are presented. [Copyright &y& Elsevier]

Abstract: This paper addresses the convergence properties of implicit numerical solution algorithms for nonlinear hyperbolic transport problems. It is shown that the Newton–Raphson (NR) method converges for any time step size, if the flux function is convex, concave, or linear, which is, in general, the case for CFD problems. In some problems, e.g., multiphase flow in porous media, the nonlinear flux function is S-shaped (not uniformly convex or concave); as a result, a standard NR iteration can diverge for large time steps, even if an implicit discretization scheme is used to solve the nonlinear system of equations. In practice, when such convergence difficulties are encountered, the current time step is cut, previous iterations are discarded, a smaller time step size is tried, and the NR process is repeated. The criteria for time step cutting and selection are usually based on heuristics that limit the allowable change in the solution over a time step and/or NR iteration. Here, we propose a simple modification to the NR iteration scheme for conservation laws with S-shaped flux functions that converges for any time step size. The new scheme allows one to choose the time step size based on accuracy consideration only without worrying about the convergence behavior of the nonlinear solver. The proposed method can be implemented in an existing simulator, e.g., for CO2 sequestration or reservoir flow modeling, quite easily. The numerical analysis is confirmed with simulation studies using various test cases of nonlinear multiphase transport in porous media. The analysis and numerical experiments demonstrate that the modified scheme allows for the use of arbitrarily large time steps for this class of problems. [Copyright &y& Elsevier]

Abstract: The multiscale finite-volume (MSFV) method was originally developed for the solution of heterogeneous elliptic problems with reduced computational cost. Recently, some extensions of this method for parabolic problems have been proposed. These extensions proved effective for many cases, however, they are neither general nor completely satisfactory. For instance, they are not suitable for correctly capturing the transient behavior described by the parabolic pressure equation. In this paper, we present a general multiscale finite-volume method for parabolic problems arising, for example, from compressible multiphase flow in porous media. Opposed to previous methods, here, the basis and correction functions are solutions of full parabolic governing equations in localized domains. At the same time, to enhance the computational efficiency of the scheme, the basis functions are kept pressure independent and do not have to be recalculated as pressure evolves. This general approach requires no additional assumptions and its good efficiency and high accuracy is demonstrated for various challenging test cases. Finally, to improve the quality of the results and also to extend the scheme for highly anisotropic heterogeneous problems, it is combined with the iterative MSFV (i-MSFV) method for parabolic problems. As one iterates, the i-MSFV solutions of compressible multiphase problems (parabolic problems) converge to the corresponding fine-scale reference solutions in the same way as demonstrated recently for incompressible cases (elliptic problems). Therefore, the proposed MSFV method can also be regarded as an efficient linear solver for parabolic problems and studies of its efficiency are presented for many test cases. [Copyright &y& Elsevier]

Abstract: The iterative-multi-scale-finite-volume (IMSFV) procedure is applied as an efficient solver for the pressure Poisson equation arising in numerical methods for the simulation of incompressible flows with the immersed-interface method (IIM). Motivated by the requirements of the specific IIM implementation, a modified version of the IMSFV algorithm is presented to allow the solution of problems, in which the varying coefficient of the elliptic equation (e.g. the permeability of the medium in the context of the simulation of flows in porous media) varies over several orders of magnitude or even becomes zero within the integration domain. Furthermore, a strategy is proposed to incorporate the iterative procedure needed by the IIM to converge out constraints at immersed boundaries into the iterative IMSFV cycle. No significant deterioration of performance of the IMSFV method is observed with respect to cases, in which no iterative improvement of the boundary conditions is considered. [Copyright &y& Elsevier]

Abstract: In the present study, a framework for modeling two-phase evaporating flow is presented, which employs an Eulerian–Lagrangian–Lagrangian approach. For the continuous phase, a joint velocity-composition probability density function (PDF) method is used. Opposed to other approaches, such PDF methods require no modeling for turbulent convection and chemical source terms. For the dispersed phase, the PDF of velocity, diameter, temperature, seen gas velocity and seen gas composition is calculated. This provides a unified formulation, which allows to consistently address the different modeling issues associated with such a system. Because of the high dimensionality, particle methods are employed to solve the PDF transport equations. To further enhance computational efficiency, a local particle time-stepping algorithm is implemented and a particle time-averaging technique is employed to reduce statistical and bias errors. In comparison to previous studies, a significantly smaller number of droplet particles per grid cell can be employed for the computations, which rely on two-way coupling between the droplet and gas phases. The framework was validated using established experimental data and a good overall agreement can be observed. [Copyright &y& Elsevier]

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: In this paper, an extension of the multi-scale finite-volume (MSFV) method is devised, which allows to simulate flow and transport in reservoirs with complex well configurations. The new framework fits nicely into the data structure of the original MSFV method and has the important property that large patches covering the whole well are not required. For each well, an additional degree of freedom is introduced. While the treatment of pressure-constraint wells is trivial (the well-bore reference pressure is explicitly specified), additional equations have to be solved to obtain the unknown well-bore pressure of rate-constraint wells. Numerical simulations of test cases with multiple complex wells demonstrate the ability of the new algorithm to capture the interference between the various wells and the reservoir accurately. [Copyright &y& Elsevier]

Abstract: Many of the complex physical processes relevant for compositional multi-phase flow in porous media are well understood at the pore-scale level. In order to study CO2 storage in sub-surface formations, however, it is not feasible to perform simulations at these small scales directly and effective models for multi-phase flow description at Darcy scale are needed. Unfortunately, in many cases it is not clear how the micro-scale knowledge can rigorously be translated into consistent macroscopic equations. Here, we present a new methodology, which provides a link between Lagrangian statistics of phase particle evolution and Darcy scale dynamics. Unlike in finite-volume methods, the evolution of Lagrangian particles representing small fluid phase volumes is modeled. Each particle has a state vector consisting of its position, velocity, fluid phase information and possibly other properties like phase composition. While the particles are transported through the computational domain according to their individual velocities, the properties are modeled via stochastic processes honoring specified Lagrangian statistics. Note that the conditional expectations of the particle velocities are different for different fluid phases. The goal of this paper is to present the general framework for this alternative modeling approach. Various one and two-dimensional numerical experiments demonstrate that with appropriate stochastic rules the particle solutions are consistent with a standard two-phase Darcy flow formulation. In the end, we demonstrate how to model non-equilibrium phenomena within the stochastic particle framework, which will be the main focus of the future work. [Copyright &y& Elsevier]

Abstract: The Multiscale Finite-Volume (MSFV) method has been recently developed and tested for multiphase-flow problems with simplified physics (i.e. incompressible flow without gravity and capillary effects) and proved robust, accurate and efficient. However, applications to practical problems necessitate extensions that enable the method to deal with more complex processes. In this paper we present a modified version of the MSFV algorithm that provides a suitable and natural framework to include additional physics. The algorithm consists of four main steps: computation of the local basis functions, which are used to extract the coarse-scale effective transmissibilities; solution of the coarse-scale pressure equation; reconstruction of conservative fine-scale fluxes; and solution of the transport equations. Within this framework, we develop a MSFV method for compressible multiphase flow. The basic idea is to compute the basis functions as in the case of incompressible flow such that they remain independent of the pressure. The effects of compressibility are taken into account in the solution of the coarse-scale pressure equation and, if necessary, in the reconstruction of the fine-scale fluxes. We consider three models with an increasing level of complexity in the flux reconstruction and test them for highly compressible flows (tracer transport in gas flow, imbibition and drainage of partially saturated reservoirs, depletion of gas–water reservoirs, and flooding of oil–gas reservoirs). We demonstrate that the MSFV method provides accurate solutions for compressible multiphase flow problems. Whereas slightly compressible flows can be treated with a very simple model, a more sophisticate flux reconstruction is needed to obtain accurate fine-scale saturation fields in highly compressible flows. [Copyright &y& Elsevier]

• A particle-particle hybrid approach by combining the unified stochastic particle BGK (USPBGK) method and DSMC is developed. • The newly proposed hybrid method has second-order accuracy in the continuum regime due to the advantage of the USPBGK method. • Simplification of the original USPBGK method is introduced to conform with the computational framework of DSMC. • An equilibrium-breakdown criterion, which refers to the Knudsen number and is easy to be calculated in particle methods, is proposed. For the simulation of multiscale gas flows, the numerical scheme should be valid and efficient in both rarefied and continuum regimes. For example, the Direct Simulation Monte Carlo (DSMC) method is not appropriate because of its huge computational cost in the continuum regime. Various kinds of hybrid methods with DSMC have been proposed to deal with this difficulty. One of them is the particle-particle hybrid method, which combines DSMC with another stochastic particle method that is based on an approximate kinetic model, such as the BGK-DSMC hybrid method. Since the same kind of computational particles is applied in the stochastic particle BGK (SPBGK) and DSMC methods, they can be implemented in a unified particle framework. Therefore, compared to the CFD-DSMC hybrid method, the BGK-DSMC hybrid method avoids difficulties caused by the amalgamation of two fundamentally different types of solvers. However, the traditional SPBGK method decouples the molecular motions and collisions in analogy to DSMC, and hence its transport properties deviate from physical values as the time step size increases. This defect significantly affects its computational accuracy and efficiency in the continuum regime. In the present paper, instead of the traditional SPBGK method, a unified stochastic particle BGK (USPBGK) method, which has second-order accuracy in time and space, is combined with DSMC. Comparing the computational performance of the USPBGK-DSMC and BGK-DSMC hybrid methods for numerical test cases of hypersonic flows past a cylinder and plume flows out of a planar micronozzle, one note that the proposed USPBGK-DSMC hybrid method can achieve higher efficiency for such multiscale gas flow simulations. In addition, an equilibrium-breakdown criterion based on the ratio of the time scales of the macroscopic flow field and molecule collision is proposed as a switching criterion for the new hybrid method. This criterion depends on the local Knudsen number and is easy to implement in a unified particle framework. [ABSTRACT FROM AUTHOR]

Unsteady flow computations typically rely on time integration schemes which employ the same step size everywhere. However, in cases with strong variations of wave speed and/or spatial resolution, local stability criteria and truncation errors would allow for much larger time steps in parts of the domain. To exploit this potential of improving the computational efficiency, adaptive time-stepping methods have been developed. Recently, an adaptive conservative time integration (ACTI) scheme for explicit finite volume methods was devised. Opposed to previous methods, ACTI guarantees exact conservation and periodic synchronization. Both are achieved by splitting a global time step into 2 L (L ≥ 0 is a cell dependent level) sub-steps, whereas the order of cell updates is critical. It has been demonstrated for flows in fractured porous media and for the Euler equations that ACTI can reduce the computational cost dramatically while preserving second order accuracy in space and time. In this paper, the ACTI scheme is extended to the compressible Navier-Stokes-Fourier system, and special attention is required for the spatio-temporal discretization of the convective and viscous fluxes. Numerical studies of unsteady flows involving shock boundary layer interaction demonstrate that the same accuracy is achieved with the new compressible ACTI flow solver as with classical time integration, but at much lower cost. Moreover, since the method is explicit, it has the potential for efficient computations on multiple CPUs and GPUs. • Adaptive conservative time integration for compressible Navier-Stokes-Fourier system. • Characteristic based Riemann solver with a simple multidimensional extension for convective flux discretization. • Second order time corrected stencil for viscous flux discretization. • Same accuracy but more efficient compared to a conventional finite volume method with spatially uniform time steps. [ABSTRACT FROM AUTHOR]

• An adaptive and still conservative time integration scheme for finite volume solvers. • In combination with a new flux solver the new sub-time stepping scheme is of almost second order (temporally and spatially). • The new adaptive conservative time integration scheme is robust and much faster (in some of the cases more than 50 times) than conventional time stepping (at almost the same accuracy). This is demonstrated in the paper with numerical studies. Finite volume methods for time dependent problems typically employ time integration schemes with constant time step sizes. The latter, in order to ensure stability and time accurate solutions, have to be chosen small enough, such that the Courant-Friedrichs-Lewy (CFL) number stays below a critical value everywhere. In many cases, however, this time step size limitation leads to huge numbers of very small time steps, which renders simulations very expensive. Motivated by this drawback of conventional time stepping schemes, various sub-time stepping algorithms have been proposed. However, most of them are inherently asynchronous, require small local CFL numbers or are not strictly conservative. In this paper a new adaptive time integration scheme for finite volume methods, which is conservative, of high spatial and temporal order, robust and easy to implement is presented. It relies on local sub-time steps which are fractions of a global time step by powers of two, i.e., the grid cells proceed asynchronously in the order of their termination time, but since they all synchronize at the end of each global time step, it is possible to guarantee continuity of the mean fluxes and thus strict conservation at the global time step resolution. Numerical experiments with 1D and 2D test cases demonstrate that the adaptive conservative time integration (ACTI) scheme can achieve extreme speed-up factors over conventional time integration, while still maintaining high spatial and temporal accuracy. [ABSTRACT FROM AUTHOR]

Data assimilation can reduce the model-form errors of RANS simulations. A spatially distributed corrective parameter field can be introduced into the closure model, whose optimal values can be efficiently found by an adjoint method and gradient-based optimization. When assimilating experimental data, which in most cases are sparsely distributed or based on a low-resolution grid, the inverse problem is highly underdetermined and thus ambiguous. Therefore, to reduce the ambiguity, regularization is required. Established regularization approaches such as total variation and Sobolev gradient methods can produce smooth and physically meaningful velocity fields in many cases. However, if the measurements are located close to walls, spiky and unphysical wall shear stress profiles can occur. A new regularization strategy based on a piecewise linear approximation of the corrective field is proposed. This method is shown to lead to a very accurate free stream velocity field and smooth wall shear stress profiles. The resulting skin friction drag error for the case of flow over periodic hills was around 1.38% which is seven times lower than the error obtained with the Sobolev gradient and two orders of magnitude lower than that obtained with the other two methods. [ABSTRACT FROM AUTHOR]

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. • We review the architecture and functionality of the cortical microvasculature. • We summarize topological characteristics of pial, penetrating and micro-vessels. • We compare vascular density over the cortical depth for different species. • We summarize the vascular and oxygenation response to neuronal activation. • We discuss the relevance of these factors for laminar fMRI. [ABSTRACT FROM AUTHOR]

Mathematical modelling allows to predict the state of a physical system based on a set of parameters. The corresponding inverse problem, i.e., when parameter values are inferred based on constraints imposed on the system state, is commonly ill-posed. Here, we consider consistent underdetermined problems, where infinitely many combinations of possible parameter values achieve a state that accurately matches all constraints. For such problems a unique regularization strategy can be used, where the ambiguity of the solution space is reduced by sequentially incorporating additional constraints. As this regularization approach always yields a consistent problem by construction, we now show that the Gauss-Newton method is the prime choice to achieve fast convergence. Moreover, using the adjoint method allows to efficiently compute the required Jacobian matrix, which makes the overall solution approach ideally suited for large test cases with many constraints and unknown parameters. We present results for several illustrative examples related to network flow, for which we visualize the solution manifolds of the regularized inverse problem. This provides an intuitive explanation of the hierarchical approach to reduce the ambiguity of the solution. Furthermore, we confirm that combining our regularization strategy with a Gauss-Newton method results in an order of magnitude lower computational cost compared to a gradient descent algorithm. This highlights the potential of our regularization strategy in combination with the Gauss-Newton method, which likely is beneficial for many comparable inverse problems, especially with large parameter spaces. • We present a fast convergence strategy for inverse problems with ambiguous solutions. • A hierarchical regularization approach is used to reduce the solution ambiguity. • We show that the Gauss-Newton method is optimally suited for such inverse problems. • We intuitively explain the solution strategy by discussing illustrative examples. • Fast convergence rates are achieved for inverse problems with large parameter spaces. [ABSTRACT FROM AUTHOR]

Turbulent flow simulations based on the Reynolds-averaged Navier–Stokes (RANS) equations continue to be the workhorse approach for industrial flow problems. However, due to the inherent averaging and the closure models required for the Reynolds stresses, their accuracy is limited. Experimental data, on the other hand, represents the true flow features but is potentially only available in limited locations or at low resolution. Data assimilation (DA) can be used to combine closure models with such experimental data to obtain accurate simulation results. In this scenario, the result of the DA process can be interpreted as a physics-based interpolation and serve as a basis for further studies of the flow. Moreover, such tuned models may be used in machine learning approaches aiming at a priori closure model enhancement. The main objective of this work is to recover a spatially varying eddy viscosity correction factor from sparsely distributed reference data. We use an efficient in-house data assimilation implementation based on the discrete adjoint method for RANS simulations in OpenFOAM to recover an optimal eddy viscosity field. A gradient optimization procedure is used to minimize a velocity-based cost function with a regularization term. All the partial derivatives in the gradient computation are evaluated with an efficient semi-analytical approach at the cost of approximately one forward solution step. The effectiveness of the proposed approach is demonstrated for a periodic hill test case at Re = 10595 with different spatial distributions of the reference data. The results show improvements in the agreement between simulation and reference for a range of reference data configurations and highlight the performance of linear eddy viscosity models. • We propose an efficient discrete adjoint gradient method for stationary problems. • We perform data assimilation of velocity reference data to correct the eddy viscosity in stationary RANS simulations. • We demonstrate the importance of regularization for sparsely distributed reference data. [ABSTRACT FROM AUTHOR]

• A particle-particle hybrid approach by combining the USP-BGK and DSMC methods is developed for polyatomic gases. • The USP-BGK method employs an ES-BGK model with discrete vibrational energy and can achieve second-order accuracy in the fluid limit. • The efficiency of the USPBGK-DSMC hybrid method is excellent for the simulation of multi-scale hypersonic flows. Simulating hypersonic flow around a space vehicle is challenging because of the multiscale and nonequilibrium nature inherent in these flows. To effectively deal with such flows, a hybrid scheme combining the stochastic particle Bhatnagar-Gross-Krook (BGK) method with direct simulation Monte Carlo (DSMC) was developed recently, but only for monatomic gases (Fei et al. (2021) [29]). In this paper, the particle-particle hybrid method is extended to polyatomic gas flows. In the near continuum regime, employing the Ellipsoidal–Statistical BGK model proposed by Dauvois et al. (2021) [22] with discrete levels of vibrational energy, the stochastic particle BGK method for polyatomic gases is first established following the idea of the unified stochastic particle BGK (USP-BGK) scheme. It is proven to be of second-order accuracy in the fluid limit. After that, the USP-BGK scheme with rotational and vibrational energies is combined with DSMC to construct a hybrid scheme. The present hybrid scheme for polyatomic gases is validated with numerical tests of homogeneous relaxation, 1D shock structure and 2D hypersonic flows past a wedge and a cylinder. Compared to the other stochastic particle methods, the proposed hybrid scheme can achieve higher accuracy at a much lower computational cost. Therefore, it is a more efficient tool to study multiscale hypersonic flows. [ABSTRACT FROM AUTHOR]

Recently, Fokker–Planck based particle Monte Carlo schemes have been proposed and evaluated for simulations of rarefied gas flows [1–3] . In this paper, the variance reduction for particle Monte Carlo simulations based on the Fokker–Planck model is considered. First, deviational based schemes were derived and reviewed, and it is shown that these deviational methods are not appropriate for practical Fokker–Planck based rarefied gas flow simulations. This is due to the fact that the deviational schemes considered in this study lead either to instabilities in the case of two-weight methods or to large statistical errors if the direct sampling method is applied. Motivated by this conclusion, we developed a novel scheme based on correlated stochastic processes. The main idea here is to synthesize an additional stochastic process with a known solution, which is simultaneously solved together with the main one. By correlating the two processes, the statistical errors can dramatically be reduced; especially for low Mach numbers. To assess the methods, homogeneous relaxation, planar Couette and lid-driven cavity flows were considered. For these test cases, it could be demonstrated that variance reduction based on parallel processes is very robust and effective. [ABSTRACT FROM AUTHOR]

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: An iterative multiscale finite volume (i-MSFV) method is devised for the simulation of multiphase flow in fractured porous media in the context of a hierarchical fracture modeling framework. Motivated by the small pressure change inside highly conductive fractures, the fully coupled system is split into smaller systems, which are then sequentially solved. This splitting technique results in only one additional degree of freedom for each connected fracture network appearing in the matrix system. It can be interpreted as an agglomeration of highly connected cells; similar as in algebraic multigrid methods. For the solution of the resulting algebraic system, an i-MSFV method is introduced. In addition to the local basis and correction functions, which were previously developed in this framework, local fracture functions are introduced to accurately capture the fractures at the coarse scale. In this multiscale approach there exists one fracture function per network and local domain, and in the coarse scale problem there appears only one additional degree of freedom per connected fracture network. Numerical results are presented for validation and verification of this new iterative multiscale approach for fractured porous media, and to investigate its computational efficiency. Finally, it is demonstrated that the new method is an effective multiscale approach for simulations of realistic multiphase flows in fractured heterogeneous porous media. [Copyright &y& Elsevier]

• Formal decoupling of the pulse generation, dispersion and scattering in the model. • New method to handle scatterers of relative refractive index lower than one. • Monte Carlo simulations are performed with a cloud of air bubbles in water. • Refracted photons exit the slab earlier than ballistic photon. • Bubbles cloud signature is visible in the time signal. The multiple scattering of an ultrashort laser pulse by a turbid dispersive medium (namely a cloud of bubbles in water) is investigated by means of Monte Carlo simulations. The theory of Gouesbet and Gréhan (2000) [14] is used to derive an energetic model of the scattering transient. It is shown that the spreading and extinction of the pulse can be decoupled from the transient of scattering, which allows to describe each phenomenon individually. The transient of scattering is modeled with the Lorenz-Mie Theory and thus is also valid for a relative refractive index lower than one, contrary to the Debye series expansion which does not converge close to the critical angle. This is made possible after the introduction of a new physical object, the Scattering Impulse Response Function (SIRF) which allows to detect the different modes of scattering transient, in time and direction. The present approach is more generic, as it enables to simulate clouds of air bubbles in water, which was not possible previously. Two different approaches are proposed within the Monte Carlo framework. The first is a pure Monte Carlo approach where the delay due to the scattering is randomly drawn at each event, while the second is based on the transport of the whole scattering signal. They are both embedded in the Monte Carlo code Scatter3D (Jenny et al., 2007) [21]. Both models produce equivalent trends and are validated against published numerical results. They are then applied to the multiple scattering of ultra short pulse by a cloud of bubble in water in the forward direction. The pulse spread due to the propagation in water is computed for a wide range of traveled distances and pulse durations, and the optimal pulse duration is given to minimize the pulse spread at a given distance. The main result is that the scattered photons exit the turbid medium earlier than the ballistic photons and produce a double peak related to the refraction in the bubble. This demonstrates the possibility to develop new diagnostics to characterize dynamic bubbly flows. [ABSTRACT FROM AUTHOR]

• Novel lattice Boltzmann method (LBM) for temporal large eddy simulation (TLES). • Based on multiple-relaxation-time (MRT) collision, consistent filtering and temporal direct deconvolution. • Evaluated for decaying homogeneous isotropic turbulence with Taylor–Green vortex for Reynolds numbers 800 and 3000. • Numerically approved consistency and qualitatively good agreement to reference results. • Subgrid activity and error measurements for dissipation rate and energy spectrum as proof of concept. We provide a first investigation of using lattice Boltzmann methods (LBM) for temporal large eddy simulation (TLES). The temporal direct deconvolution model (TDDM) is injected as a closure for the filtered discrete velocity Bhatnagar–Gross–Krook (BGK) Boltzmann equation with orthogonal multiple-relaxation-time (MRT) collision. The novel combination of methods is calibrated for decaying homogeneous isotropic turbulence. The Taylor–Green vortex flow is used as a benchmark for the Reynolds numbers 800 and 3000. Various turbulence quantities are numerically evaluated. The numerical results obtained with the MRT LBM are validated against a well-established spectral element method both, with and without the proposed turbulence model. A qualitatively good agreement to reference results of direct numerical simulation is observed in terms of dissipation rate, energy spectrum and dissipation spectrum. Whereas the latter two show marginal differences compared to underresolved simulations without TLES, the dissipation rate exhibits substantial improvement through the model in capturing the peak region for low and intermediate resolutions. The consistency of the TDDM with first and second order discretization is numerically demonstrated for single-relaxation-time and MRT collision via computing the total dissipation rate error. Measuring the interaction between subgrid activity and energy spectrum error serves as a proof of concept for the proposed MRT LBM TLES. Conclusively, the model recovers and enhances the expected numerical features of LBM with respect to the target equation. [ABSTRACT FROM AUTHOR]

Abstract: The multiscale finite-volume (MSFV) method for the solution of elliptic problems is extended to an efficient iterative algorithm that converges to the fine-scale numerical solution. The localization errors in the MSFV method are systematically reduced by updating the local boundary conditions with global information. This iterative multiscale finite-volume (i-MSFV) method allows the conservative reconstruction of the velocity field after any iteration, and the MSFV method is recovered, if the velocity field is reconstructed after the first iteration. Both the i-MSFV and the MSFV methods lead to substantial computational savings, where an approximate but locally conservative solution of an elliptic problem is required. In contrast to the MSFV method, the i-MSFV method allows a systematic reduction of the error in the multiscale approximation. Line relaxation in each direction is used as an efficient smoother at each iteration. This smoother is essential to obtain convergence in complex, highly anisotropic, heterogeneous domains. Numerical convergence of the method is verified for different test cases ranging from a standard Poisson equation to highly heterogeneous, anisotropic elliptic problems. Finally, to demonstrate the efficiency of the method for multiphase transport in porous media, it is shown that it is sufficient to apply the iterative smoothing procedure for the improvement of the localization assumptions only infrequently, i.e. not every time step. This result is crucial, since it shows that the overall efficiency of the i-MSFV algorithm is comparable with the original MSFV method. At the same time, the solutions are significantly improved, especially for very challenging cases. [Copyright &y& Elsevier]

We present a new method for modeling tissue perfusion on the capillary scale. The microvasculature is represented by a network of one-dimensional vessel segments embedded in the extra-vascular space. Vascular and extra-vascular space exchange fluid over the vessel walls. This exchange is modeled by distributed sources using smooth kernel functions for the extra-vascular domain. It is shown that the proposed method may significantly improve the approximation of the exchange flux, in comparison with existing methods for mixed-dimension embedded problems. Furthermore, the method exhibits better convergence rates of the relevant quantities due to the increased regularity of the extra-vascular pressure solution. Numerical experiments with a vascular network from the rat cortex show that the error in the approximation of the exchange flux for coarse grid resolutions may be decreased by a factor of 3. This may open the way for computing on larger network domains, where a fine grid resolution cannot be achieved in practical simulations due to constraints in computational resources, for example in the context of uncertainty quantification. • A new computational method for 1d-3d mixed-dimension problems with accurate approximation of the coupling term. • Tissue perfusion model with distributed source term regularizes pressure solution and improves convergence rates. • We show cases where the reconstruction of the source term from the regularized solution is exact. • Numerical experiment with realistic capillary network shows reduced error w.r.t. other methods by factor 3 for coarse grids. • More accurate description of the source term for coarse grids may allow simulation with larger vessel networks in the future. [ABSTRACT FROM AUTHOR]