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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]

A recently introduced subphase framework for modeling the nonwetting phase relative permeability is extended to the wetting phase. Within this framework, the wetting phase is divided into four subphases, which are distinguished by their connectivity; backbone, dendritic, isolated and corner-film subphases. The subphase saturations evolve according to inter-subphase volume transfer terms, which require modeling. An advantage of distinguishing the subphases is that wetting phase relative permeability relations as functions of these constituent subphases can be developed. In order to develop models for the inter-subphase volume transfer and the wetting phase relative permeability in a strongly wetted system, quasi-static flow simulations in pore networks were conducted to analyze the evolution of the wetting subphases during drainage and imbibition. The simulation results suggest that hysteresis trends apparent in experimentally obtained wetting phase relative permeability curves for Berea sandstone may be explained by accounting for corner-film flow. [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 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]

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]

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. [ABSTRACT FROM AUTHOR]

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

Abstract: For the 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. [Copyright &y& Elsevier]

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

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]