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This paper presents a finite-volume method for hexahedral multiblock grids to calculate multiphase flow in geologically complex reservoirs. Accommodating complex geologic and geometric features in a reservoir model (e.g., faults) entails non-orthogonal and/or unstructured grids in place of conventional (globally structured) Cartesian grids. To obtain flexibility in gridding as well as efficient flow computation, we use hexahedral multiblock grids. These grids are locally structured, but globally unstructured. One major advantage of these grids over fully unstructured tetrahedral grids is that most numerical methods developed for structured grids can be directly used for dealing with the local problems. We present several challenging examples, generated via a commercially available tool, that demonstrate the capabilities of hexahedral multiblock gridding. Grid quality is discussed in terms of uniformity and orthogonality. The presence of non-orthogonal grid and full permeability tensors requires the use of multi-point discretization methods. A flux-continuous finite-difference (FCFD) scheme, previously developed for stratigraphic hexahedral grid with full-tensor permeability, is employed for numerical flow computation. We extend the FCFD scheme to handle exceptional configurations (i.e. three- or five-cell connections as opposed to the regular four), which result from employing multiblock gridding of certain complex objects. In order to perform flow simulation efficiently, we employ a two-level preconditioner for solving the linear equations that results from the wide stencil of the FCFD scheme. The individual block, composed of cells that form a structured grid, serves as the local level; the higher level operates on the global block configuration (i.e. unstructured component). The implementation uses an efficient data structure where each block is wrapped with a layer of neighboring cells. We also examine splitting techniques [14] for the linear systems associated with the wide stencils of our FCFD operator. We present three numerical examples that demonstrate the method: (1) a pinchout, (2) a faulted reservoir model with internal surfaces and (3) a real reservoir model with multiple faults and internal surfaces.

The objective of this work is to develop an efficient predictive method to investigate the aerodynamic properties of a Micro Aerial Vehicle which is powered by a coaxial rotor. A simulation framework was developed which models the flow field around the rotor blades with the use of vortex methods. Aerodynamic properties of the helicopter rotor can be computed to investigate the performance of the rotor blades. Prescribed wake models, which adapt iteratively to the induced velocity field, are implemented to describe the geometry of the trailing vortex sheet behind the rotor blades. To allow flexible blade design, the framework is coupled to a 2D panel code which computes the performance data of the blade elements acting as boundary conditions for the implemented Lifting Line Method. The computational results are compared to experimental data of rotors in hover state. The numerical results show a good agreement with the measurements obtained from experiments.

Three different PDF algorithms have been applied to investigate a constant-density bluff-body stabilized flow using the same turbulence models and the same boundary conditions. The objectives of this paper are to compare the three algorithms in terms of numerical accuracy and efficiency and to demonstrate the ability of PDF methods to calculate this type of flow accurately. While one of the three algorithms is a stand-alone particle-mesh method, the other two are consistent hybrid algorithms, i.e., both are particle methods coupled with finite-volume schemes. The motivation for hybrid algorithms is to reduce the statistical and bias errors. Since the coupling between the finite-volume scheme and the particle method is a major numerical issue, different approaches have been investigated. It is shown that the results obtained from the three numerical algorithms are in good agreement with each other and with the experimental data.

In this paper a new particle-finite-volume hybrid algorithm for the joint velocityfrequency-composition PDF method for turbulent reactive flows is presented. This method is a combination of a finite-volume scheme and a particle method. The finitevolume scheme is used to solve the Reynolds averaged Navier‐Stokes equations and the particle method to solve the joint PDF transport equation. The motivation is to reduce the bias and the statistical error and to have an algorithm which is more efficient than stand-alone particle-mesh methods. Therefore, in the particle method we use the smoother mean densityh‰i and Favre averaged velocity ˜ U fields computed

The paper describes a new hybrid finite-volume (FV)/particle method developed for the solution of the PDF equations for statistically stationary turbulent reactive flows. In this approach, the conservation equations for mean mass, momentum, and energy conservation are solved by a FV method while a particle algorithm is employed to solve the fluctuating velocity-turbulence frequency-compositions joint PDF transport equation. The mean velocity and pressure are supplied to the particle code by the FV code which in turn obtains all the Reynolds stresses, the scalar fluxes, and the reaction terms needed in the FV code. An important feature of the method is the complete consistency between the set of equations solved by the FV and particle methods. The algorithmic and numerical issues arising in the development of the hybrid method are studied in the simple setting of the stochastic ideal flow equations. Numerical results are obtained for 1D reactive stochastic ideal flow to demonstrate numerical properties of the method. The total numerical error is identified as statistical error, bias, spatial truncation error, and temporal truncation error. In contrast to the self-contained particle method, the bias is found to be negligibly small. It is shown that all the numerical errors converge at the expected rates. Finally, the global convergence of the hybrid method is demonstrated and the optimal strategy for time-averaging that gives the best global convergence rate is investigated.

A novel technique is introduced to accelerate the convergence of compressible low Mach number flow computations to the steady state. The stiffness due to the large disparity of flow velocity and acoustic wave speeds is bypassed by artificially reducing the speed of sound and thereby increasing the Mach number. This Mach number transformation is achieved by subtracting a constant value from the pressure in the entire flow field. Only the inviscid terms of the energy equation are influenced by that pressure decrease. The steady-state error is corrected by solving a scalar equation after each time step such that the steady-state solutions of the modified and non-modified schemes coincide. Thus, for each low Mach number simulation, one can obtain a convergence performance comparable to the corresponding simulation with a Mach number of about 0.4. This convergence acceleration is demonstrated for premixed laminar flames. If the present technique is implemented without time splitting, it corresponds to a novel low Mach number preconditioning.

Journal of Fluid Mechanics, 749, ISSN:0022-1120, ISSN:1469-7645

Conservative Euler solvers for gas mixtures produce numerical errors near contact discontinuities, if the temperature and the ratio of specific heats are not constant there. For mixtures of perfect gases, a simple correction of the total energy per unit volume is proposed to avoid these errors. This is done in a physical way and only the total energy looses some of its conservativity. Numerical simulations of contact discontinuity convection, a shock tube problem, and shock-interface interactions in 1D and 2D yield much more accurate solutions, if the correction is applied. The straightforward extension to 3D is outlined.

In this work, we present a hybrid algorithm based on the Fokker-Planck (FP) kinetic model and direct simulation Monte Carlo (DSMC) for studies of rarefied gas flows. A particle based FP solution algorithm for rarefied gas flow simulations has recently been devised by the authors. The motivation behind the FP approximation is purely computational, i.e. due to the fact that the resulting random processes are continuous in time the computational cost of the corresponding time integration becomes independent of the Knudsen number. However, the method faces limitations for flows with very high Knudsen numbers (larger than approximately 5). In the method presented here, the FP approach is coupled with DSMC in order to gain from the efficiency of the FP model and from the accuracy of DSMC at small and large cell based Knudsen numbers, respectively.Copyright © 2013 by ASME

In this work a phenomenon arising from gas-wall interaction in rarefied regimes is presented and discussed. It is hypothesized that the effect of energy transfer between different modes of molecules scattered from walls can be used for demixing of different gas species. In order to study this effect, first a kinetic wall boundary model for diatomic gas molecules is devised. The model is derived by generalizing the Cercignani–Lampis gas-surface interaction kernel in order to embody the gas internal degrees of freedom (DoF). Here, opposed to the extensions of e.g. [5], energy exchange between different molecular modes are honored and thus different physical phenomena arising from inelastic gas–surface collisions can be observed.

Journal of Intelligent & Robotic Systems, 69 (1-4), ISSN:0921-0296, ISSN:1573-0409

Potential of morphing was investigated for a UAV ying in a spatially and temporally changing wind eld. Trajectory optimization was performed for a 5 meter span UAV with morphing capability as well as without it. Results show that energy extraction from inhomogeneity in spanwise direction is of little practical relevance, but morphing can be e ectively used for drag reduction while performing gust soaring. Additional optimization was done for a 15 meter span UAV ying in an arti cial static wind eld to demonstrate that actual energy extraction is theoretically possible. Results for the latter case show, that the spacial distribution of the wind eld can have profound e ect on the amount of extracted energy.

M Mass of the aircraft L, d Annular wing perimeter and pro le depth T Duration of apping period n, m Parameters of mesh for aerodynamic modeling N Number of control points on half of the wing Attached circulation around annular wing cvd Viscous drag coe cient (wing) cl Inclination of the lifting polar c vd Viscous drag coe cient (fuselage) J Target functional Vx Horizontal speed of ight F Total force , Air density and kinematic viscosity v, r UAV speed and position aero Total aerodynamic angle of attack C(s; t)Wing contour parametrization u On ow velocity

In this work, we introduce a recently proposed framework for hybrid LES/RANS modeling and its preliminary applications on simple flows. In this framework, the filtered and Reynolds averaged Navier-Stokes (RANS) equations are solved simultaneously in the whole domain. The novelty of the framework is the dual-solution approach and the consistency between the two solutions achieved via additional drift terms. A hybrid LES/RANS solver is developed within this framework and used to simulate flows in a plane channel and flows in a channel with periodic hills. The results demonstrate that the hybrid solver leads to significantly improved results compared to traditional LES on the same grid.

Targeting of permissive entry sites is crucial for bacterial infection. The targeting mechanisms are incompletely understood. We have analyzed target-site selection by S. Typhimurium. This enteropathogenic bacterium employs adhesins (e.g. fim) and the type III secretion system 1 (TTSS-1) for host cell binding, the triggering of ruffles and invasion. Typically, S. Typhimurium invasion is focused on a subset of cells and multiple bacteria invade via the same ruffle. It has remained unclear how this is achieved. We have studied target-site selection in tissue culture by time lapse microscopy, movement pattern analysis and modeling. Flagellar motility (but not chemotaxis) was required for reaching the host cell surface in vitro. Subsequently, physical forces trapped the pathogen for ∼1.5–3 s in “near surface swimming”. This increased the local pathogen density and facilitated “scanning” of the host surface topology. We observed transient TTSS-1 and fim-independent “stopping” and irreversible TTSS-1-mediated docking, in particular at sites of prominent topology, i.e. the base of rounded-up cells and membrane ruffles. Our data indicate that target site selection and the cooperative infection of membrane ruffles are attributable to near surface swimming. This mechanism might be of general importance for understanding infection by flagellated bacteria., PLoS Pathogens, 8 (7), ISSN:1553-7374, ISSN:1553-7366

Journal of Fluid Mechanics, 688, ISSN:0022-1120, ISSN:1469-7645

For the simulation of fluid flows, probability and mass density function (PDF/MDF) methods have advantageous properties compared to moment-based approaches or purely deterministic methods and are applicable in different fields. For example, PDF and MDF methods are used for the quantification of uncertainty in turbulent or subsurface flows, and the simulation of multi-phase flows or rarefied fluids. In this chapter, differences of these methods compared to other solution techniques are discussed and illustrated by application examples. Moreover, the theory behind PDF and MDF methods is outlined. Finally, a PDF method for uncertainty quantification in subsurface flows and an MDF method for the simulation of rarefied fluid flows are discussed in more details.

Journal of Fluid Mechanics, 680, ISSN:0022-1120, ISSN:1469-7645

The hierarchical fracture modeling approach is extended to a loosely coupled strategy in order to enhance its efficiency for highly conductive fractures and to easily integrate it into the iterative Multiscale Finite Volume (i-MSFV) framework. The fracture and matrix equations are decoupled and a sequential implicit approach is used to solve them. The i-MSFV method is then introduced for the efficient solution of the matrix equation which also involves fracture terms. Very important it that only one degree of freedom per connected fracture network appears in the coarse multiscale operator. This results in very efficient simulations for realistic problems, and also ensures total mass balance for each fracture network during the iterations. Local fracture functions are introduced in order to capture the fractures at the coarse scale accurately; similar to the complex well functions developed by Jenny and Lunati (2009). Numerical results show that the i-MSFV is an efficient conservative multiscale solver for simulations of flow in fractured porous media.

The multiscale finite volume (MSFV) method is extended to account for non-matching coarse grids arising from faulted porous media. To obtain accurate quantities at the coarse scale, basis and correction functions are computed in extended local domains near the fault regions. With the new framework it is shown that the MSFV results are very close to the fine-scale reference solutions. Finally, to improve the quality and to extend the applicability of the framework, an iterative MSFV method for faulted reservoirs is devised; similar to the recently proposed i-MSFV method for conforming coarse grids by Hajibeygi et al. (2008). Convergence histories of the new i-MSFV method are shown for different homogeneous and heterogeneous test cases. It is concluded that this extension of the MSFV method is very flexible and allows to deal with complex geometries such as faulted reservoirs.

[1] The probability density function (PDF) of the local concentration of a contaminant, or tracer, is an important component of risk assessment in applications that involve flow in heterogeneous subsurface formations. In this paper, a novel joint velocity-concentration PDF method for tracer flow in highly heterogeneous porous media is introduced. The PDF formalism accounts for advective transport, pore-scale dispersion (PSD), and molecular diffusion. Low-order approximations (LOAs), which are usually obtained using a perturbation expansion, typically lead to Gaussian one-point velocity PDFs. Moreover, LOAs provide reasonable approximations for small log conductivity variances (i.e., σY2 < 1). For large σY2, however, the one-point velocity PDFs deviate significantly from the Gaussian distribution as demonstrated convincingly by several Monte Carlo (MC) simulation studies. Furthermore, the Lagrangian velocity statistics exhibit complex correlations that span a wide range of scales, including long-range correlations due to the formation of preferential flow paths. Both non-Gaussian PDFs and complex long-range correlations are accurately represented using Markovian velocity processes (MVPs) in the proposed joint PDF method. LOA methods can be generalized to some extent by presuming a certain shape for the concentration PDF (e.g., a β PDF fully characterized by the concentration mean and variance). The joint velocity-concentration PDF method proposed here does not require any closure assumptions on the shape of the marginal concentration PDF. The Eulerian joint PDF transport equation is solved numerically using a computationally efficient particle-based approach. The PDF method is validated with high-resolution MC reference data from Caroni and Fiorotto (2005) for saturated transport in velocity fields, which are stationary in space and time, for domains with σY2 = 0.05, 1, and 2 and Peclet numbers ranging from 100 to 10,000. PSD is modeled using constant anisotropic dispersion coefficients in both the reference MC computations and our PDF method.

This paper deals with an efficient and accurate simulation algorithm to solve the vector Boltzmann equation for polarized light transport in scattering media. The approach is based on a stencil method, which was previously developed for unpolarized light scattering and proved to be much more efficient (speedup factors of up to 10 were reported) than the classical Monte Carlo while being equally accurate. To validate what we believe to be the new stencil method, a substrate composed of spherical non-absorbing particles embedded in a non-absorbing medium was considered. The corresponding single scattering Mueller matrix, which is required to model scattering of polarized light, was determined based on the Lorenz–Mie theory. From simulations of a reflected polarized laser beam, the Mueller matrix of the substrate was computed and compared with an established reference. The agreement is excellent, and it could be demonstrated that a significant speedup of the simulations is achieved due to the stencil approach compared with the classical Monte Carlo.

A second-order accurate numerical procedure is presented for the solution of the incompressible Navier-Stokes equations with immersed boundaries in the formulation proposed by Peller et al. [5]. In particular we derive exact constraints for the pressure at immersed solid walls and show that the resulting Poisson equation is formally identical to the elliptic problems governing flows in porous media. An efficient iterative procedure for the computation of the pressure is then obtained by adapting the iterative-multi-scale-finite-volume procedure by Hajibeygi et al. [2].

At its most fundamental level, cerebral blood flow (CBF) may be modeled as fluid flow driven through a network of resistors by pressure gradients. The composition of the blood as well as the cross-sectional area and length of a vessel are the major determinants of its resistance to flow. Here, we introduce a vascular graph modeling framework based on these principles that can compute blood pressure, flow and scalar transport in realistic vascular networks. By embedding the network in a computational grid representative of brain tissue, the interaction between the two compartments can be captured in a truly three-dimensional manner and may be applied, among others, to simulate oxygen extraction from the vessels. Moreover, we have devised an upscaling algorithm that significantly reduces the computational expense and eliminates the need for detailed knowledge on the topology of the capillary bed. The vascular graph framework has been applied to investigate the effect of local vascular dilation and occlusion on the flow in the surrounding network.

This work deals with the efficient and accurate modeling of fluorescence in the context of stochastic Monte Carlo methods for which we propose a novel multiscale method. As in other approaches of this category, the transport theory is employed to describe the physics. The new framework was successfully applied for a quantitative assessment of halftone reflectance measurements with three different devices. It could be demonstrated that the described method is faster than classical Monte Carlo by multiple orders of magnitude, and that it is capable of correctly handling the geometrical device differences. It is also shown that optical dot gain is accurately predicted for the whole ink coverage range.

Spray combustion is a complex phenomena finding its application in many combustion devices like gas turbines, diesel engines, liquid-fueled rocket engines etc. This phenomena is characterised by the break up of the liquid fuel sheet, followed by atomisation, droplet dispersion and its evaporation, mixing of the fuel vapour with the oxidiser and eventually its combustion. However, the present study is focused on the turbulence modification of the carrier phase (i.e. gas) and the fuel vapour mixing. Flows seeded with droplet volume fraction less than 10-3 are considered so that the inter-droplet collision effects are negligible.

The Multiscale Finite-Volume (MSFV) method was originally proposed to solve elliptic problems arising from incompressible multiphase flow in highly heterogeneous porous media at reduced computational cost. However, when phases are compressible, especially when the reservoir contains gas, mathematical formulations lead to a parabolic equation to be solved for pressure. In this paper we introduce a general MSFV method to deal with such parabolic problems. In this scheme, the basis and correction functions are numerical solutions of the full parabolic problems in localized domains. Hence compressibility effects are represented by a stencil in the coarse-system matrix, i.e. not only by a diagonal entry. Furthermore, to enhance the computational efficiency of the scheme, the basis functions are kept independent of the pressure field. As a result, only the correction functions (1 per dual-coarse cell) have to be updated during the iterative procedure. It is an important property of this approach that it requires no additional simplifications. Finally, its good efficiency is demonstrated for a number of challenging test cases.

Residual trapping is one of the CO2 retention mechanism during CO2 storage in saline aquifers. In a region where the CO2 saturation falls below the residual saturation, isolated ganglias of CO2 are formed which can become immobile (trapped). In this paper, we propose a stochastic model for trapping and release of non-wetting phase ganglias. Opposed to traditional trapping models, which only take into account of the mean quantities, the proposed stochastic model contains additional statistical informations, e.g., correlation time. By selecting appropriate model parameters, both drainage and imbibition can be described without a priori knowledge of the displacement direction.

A new approach for uncertainty assessment in porous media is devised. The goal is to study tracer or phase transport while assuming that the multi-point velocity statistics is known. The method depends on solving a transport equation for the joint probability density function (PDF) of velocity and concentration (or phase saturation). Similar PDF methods have been developed for turbulent flow simulations and proved extremely successful in providing joint statistics of species concentration and velocity required for turbulent combustion modeling. As in the case of turbulent flows, the joint PDF equation for uncertainty assessment of transport in porous media is defined over a high dimensional space (physical, velocity, concentration or phase, and time). This results in severe computational limitations, if a finite-volume, finite-difference or finite-element method is employed, which is the motivation for the use of a particle method. The crucial advantage of the PDF modeling approach is its flexibility and high level of closure. For example, no modeling is required for macro-dispersion (ensemble mean of the product of velocity-concentration fluctuations). Moreover, no assumptions are made regarding correlation length scales and unlike perturbation-based Statistical Moment Equation (SME) methods, the PDF method is not restricted to small input variance. Here, the PDF methodology is presented for incompressible single-phase tracer transport in heterogeneous porous media, but more general applications involving multi-phase transport are discussed. A detailed description of the PDF method is given and its accuracy is demonstrated by comparison with published Monte Carlo results.

We developed an adaptive reservoir simulator for accurate modeling of multiphase flow and transport in large scale heterogeneous reservoirs. The simulator is based on a multiscale finite volume (MSFV) method. We describe both IMPES and sequential implicit formulations. The algorithms are sensitive to the specific characteristics of flow (i.e., pressure and total velocity) and transport (i.e., saturation). To obtain the fine scale (i.e., fine grid) 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 transmissibilities. 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 simulator 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 and triggers selective updates 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.

A simulator with hexahedral multiblock grids has been developed to simulate flow in reservoirs with geometrically complex features (i.e., faults and wells). A reservoir model can always be gridded with multiblock grids with matching grid lines between blocks. However, this matching requirement may introduce many small cells that have little impact on the simulation results as well as unnecessarily increase the number of blocks. We thus relax the matching requirement by allowing for nonmatching grid lines at block boundaries. This generalization simplifies the construction of multiblock grids greatly as each block can now be gridded independently and patched together afterwards. For instance, a new grid around a well can be inserted into an existing stratigraphic grid. The result is much simpler grids with fewer blocks and/or cells as compared to original matched multiblock grids. The complexity shifts from the grid construction to the calculation of transmissibilities. To this end, we need an accurate, consistent discretization scheme for nonmatching grids. We first derive a flux continuous discretization scheme for convex polyhedra grids. For general geometries a prismatic interface grid is constructed that establishes cell connectivities across the unmatched interface. Applying a piecewise constant approximation, we compute transmissibilities for this interface grid and map them back to the original hexahedral cells. We analyze the accuracy of this method via comparisons to matched fine grids. The effectiveness and applicability of this new method is demonstrated with reservoir models with wells and faults.

In multiblock reservoir simulation techniques, the reservoir model is divided into a number of subdomains or blocks. In our current implementation, these blocks are locally structured but globally unstructured. This representation enables the modeling of geometrically complex reservoir features such as fault surfaces and nonconventional wells while avoiding many of the complications of fully unstructured formulations. In this paper, we present several important developments within this frame-work. These include the extension of a previous two phase flow finite volume formulation to the general black oil case, the implementation of techniques for treating systems in which grid lines do not match between adjacent subdomains, and the application of a near-well radial upscaling technique to the multi-block paradigm. Simulation results illustrating the accuracy and efficiency of these new capabilities are presented. These include a black oil simulation for a local well study, flow through a realistic reservoir with several wells and faults, flow through a fault surface represented by nonmatching grid lines, and a two phase flow simulation demonstrating the applicability of the near-well upscaling procedure to multiblock models. With the new developments presented in this paper, the finite volume based multiblock simulator can be applied to a variety of problems of practical interest.

A black oil reservoir simulator was developed for hexahedral multi-block grids. An object-oriented approach for the design and implementation of the simulator was adopted. We discuss the general features and object-oriented (O/O) design of the simulator.

This work presents a kinetic wall boundary model for diatomic gas molecules. The model is derived by generalizing the Cercignani-Lampis-Lord gas-surface interaction kernel in order to account for the gas internal degrees of freedom. Here, opposed to the extensions by Lord [“Some extensions to the Cercignani-Lampis gas-surface scattering kernel,” Phys. Fluids 3, 706–710 (1991)], energy exchange between different molecular modes is honored and thus, different physical phenomena arising from inelastic gas–surface collisions can be described. For practical implementations of the model, a Monte–Carlo algorithm was devised, which significantly reduces the computational cost associated with sampling. Comparisons of model predictions with experimental and molecular dynamics data exhibit good agreement. Moreover, simulation studies are performed to demonstrate how energy transfers between different modes due to wall collisions can be exploited for gas separation.

A hexahedral multi-block grid formulation for the modeling of two phase reservoir flows is developed and applied. Multi-block approaches are well suited to the modeling of geometrically complex features while avoiding many of the complications of fully unstructured techniques. Important implementation issues, including the accurate treatment of grid non-orthogonality 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 which allows for the accurate resolution of near-well effects, are discussed. 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 multi-block approach.

A new approach for a flux solver is presented which takes source terms into account. The source terms are distributed from the volumes to the volume interfaces, where they define the flux jumps of the Rankine-Hugoniot conditions. Thereby within the cells, homogeneous hyperbolic conditions are obtained. These are used by a linearized Riemann solver to yield the additional relations to determine the left and right states at the cell interfaces. For viscous and multidimensional flows, the viscous flux balance and the flux difference in the other coordinate directions, respectively, are considered as parts of the source terms. Compared with conventional Riemann solvers not taking into account source terms, the new Rankine-Hugoniot-Riemann solver significantly improves the accuracy for a 2D Euler test case without source terms and for 1D and 2D combustion simulations with stiff source terms and viscous terms. The present consideration of source terms by the flux solver can be generalized to hyperbolic systems other than the Euler equations.

Turbulent combustion is commonly categorized into premixed, non-premixed and partially premixed combustion. For nonpremixed combustion simulations the laminar flamelet concept proved to be very valuable while for the more complex case of partially premixed combustion this model shows considerable deficiencies. Here, the classical laminar flamelet approach is extended to the partially premixed combustion regime. For that, the joint statistics of mixture fraction, scalar dissipation rate and a progress variable, calculated with a joint probability density function (PDF) method, is used to get the statistics of the compositions and of the chemical energy source term from pre-processed flame tables. This approach can be compared with the unsteady flamelet concept; the main differences consists of the way the progress variable evolution is computed and in the pre-computed flame tables. The progress variable describes the point of time a fluid parcel is consumed by a flame front. The fluid parcels are represented by computational particles, which are used for PDF methods. The pre-computed flame tables are computed from steady solutions 2D stabilized flames propagating into an unburnt mixture with varying mixture fraction. The corresponding position of a fluid particle in such a 2D laminar flame is determined by its mixture fraction and a burning time; both to be modeled for each computational particle in the PDF simulation. Numerical experiments of turbulent diffusion jet flames demonstrate that this approach can be employed for challenging test cases. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A new model for turbulent premixed combustion is presented which is based on a joint velocity composition probability density function (JPDF) method. The key idea is a scale separation approach. The method combines the model by Bray, Moss and Libby [1] (BML) for premixed combustion with the flamelet approach for nonpremixed combustion. Here, a Lagrangian formulation of the BML model is considered. The progress variable used by the BML model becomes a computational particle property and its value is triggered by the arrival of the flame front at the particle’s position. Similar as in the flamelet approach we assume that the smallest eddies are not small enough to disturb the reactive diffusive flame structure. To resolve the (embedded) quasi laminar flame structure, a flame residence time is introduced. With that residence time, the evolution of the particle composition, including enthalpy, can be determined from precomputed laminar 1D flames. The main challenge with this approach is to model the probability that an embedded flamefront arrives at the particle location, which is necessary to close the chemical source term. Numerical experiments of a turbulent premixed flame show good agreement with experimental data. 1 Combustion Model In many combustion applications the flow field is highly turbulent. Turbulence, chemical reactions and their interactions have to be modeled accurately and reliably. A common model for turbulent premixed combustion is the one by Bray, Moss and Libby (BML), where a progress variable c distinguishes between burnt (c =1 ) and unburnt (c =0 ) gas. The transport equation for the Favre-averaged progress variable is not in closed form, and therefore, the turbulent convection and the source term have to be modeled. Here we present a JPDF method for turbulent premixed combustion which is based on the BML model and the flamelet approach. In this Lagrangian framework, the progress variable c is a computational particle property. The resulting averaged transport equation for c is consistent with the BML model with the advantage that the turbulent convection term is closed. Moreover, the use of the burning time τ , which starts to run when c switches from 0 to 1, allows to resolve the embedded quasi laminar flame structure, i.e. the composition of a particle evolves with τ according to the composition in a precomputed and tabulated quasi laminar 1D flame. This scale separation approach is valid under the assumption that the flame structure is not disrupted by the turbulence. Note that this can be done very naturally in a Lagrangian framework. The remaining task is to determine when a flame front arrives at the particle position. Here, this is modeled with the ignition probability

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 it has the important property that no large patches covering the whole well are required. For each well, an additional degree of freedom is introduced. While the treatment of pressure-constraint wells is trivial, since the well-bore reference pressure is explicitly specified, additional equations have to be solved to obtain the unknown pressure values of rate-constraint wells. Numerical simulations of a difficult test case with multiple complex wells demonstrate the ability of the new framework to capture the interference between the various wells and the reservoir accurately. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Joint composition probability density function (PDF) methods are used for the numerical simulation of turbulent reactive flows. Here, other than in classical Reynolds averaged Navier–Stokes (RANS) or large eddy simulation (LES) approaches, the highly non-linear chemical source term appears in closed form. On the other hand, mixing models are required for the closure of the molecular diffusion term. In the present work, the joint statistics of the scalar and the scalar dissipation rate provided by the parameterized scalar profile (PSP) mixing model are validated. The goal is to combine the PDF method with a flamelet approach, where the scalar dissipation rate plays a crucial role in determining the contribution of the chemical source term. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Non-equilibrium phenomena arising from the pore scale dynamics can have considerable effect on the large scale dynamics of multiphase flow in porous media. In such cases, the relative permeabilities and capillary pressure curves are not just simple functions of phase saturations, rather are dynamic functions of space and time. The present work proposes a stochastic approach in which particle mobility can be modelled based on the actual conductance of fluid volume inside a pore space. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Simulation of sub-surface flow in geologically complex formations is just one example in computational science, where efficient and accurate solutions of heterogeneous elliptic problems are of great interest. Often it is not feasible to resolve the whole range of relevant length scales associated with the spatial distribution of the highly varying coefficients. The MSFV method was originally developed for multi-phase flow in porous media. In the more general form presented here, it can be applied for solving large elliptic problems in various areas of computational science. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In turbulent combustion one distinguishes between premixed, non-premixed and partially premixed combustion. While laminar flamelet models proved to be extremely valuable for a wide range of non-premixed flame simulations, similar approaches are more problematic in the partially premixed regime. Here the laminar flamelet concept for non-premixed turbulent combustion simulations is generalized for the partially premixed regime. Similar as in the unsteady flamelet approach, the joint statistics of a progress variable, mixture fraction and scalar dissipation rate is used to obtain the joint statistics of the compositions from pre-computed flame tables. The required distribution is computed with a joint PDF method and the main differences between the new approach and previous ones, are the pre-computed tables and the way the evolution of the progress variable is calculated. Instead of evolving 1D flamelets, steady 2D solutions of burning flamelets propagating into unburned mixtures with varying mixture fraction are considered. The location of a fluid particle in this 2D laminar flame is defined by its mixture fraction and a burning time, which are modeled for each computational particle used in the PDF method. Numerical experiments of a turbulent lifted diffusion flame and a premixed Bunsen flame demonstrate that this approach can be employed for a wide range of applications. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The composition fields in turbulent reacting flows are affected by turbulent transport (macromixing), molecular diffusion (micromixing), and chemical reactions. In the joint velocity-composition probability density function transport equation the highly non-linear macromixing and chemical reaction terms appear in closed form. This is a considerable advantage over moment closure methods. Micromixing on the other hand requires modeling and especially for turbulent combustion accurate mixing models are crucial. Our approach to model the mixing of scalars, e.g. species mass fractions or temperature, is based on considering one-dimensional parameterized scalar profiles (PSP). Here, an extension of the PSP mixing model to inhomogeneous flows is presented.

Institute of Fluid Dynamics, ETH Zurich, Sonneggstr. 3, CH-8092 ZurichA model for premixed turbulent combustion based on a joint velocity probability density function (PDF) method and aprogress variable is presented. Compared with other methods employing progress variables, the advantage here is that turbu-lentmixing ofthe progress variable requires no modeling. Moreover, by applying scale separation, theLagrangian frameworkallows to account for the embedded, quasi laminar flame structure in a very natural way. The numerical results presented hereare based on a simple closure of the progress variable source term and it is demonstrated that the new modeling approach isrobust and shows the correct qualitative behavior

The composition fields in turbulent reacting flows are affected by turbulent transport (macromixing), molecular diffusion (micromixing), and chemical reactions. In the joint velocity-composition probability density function transport equation the highly nonlinear macromixing and chemical reaction terms appear in closed form. This is a considerable advantage over second moment closure methods. Micromixing on the other hand requires modelling and especially for turbulent combustion accurate mixing models are crucial. In this paper we present an approach to model the mixing of scalars, e.g. species mass fractions or temperature, based on considering one-dimensional parameterized scalar profiles (PSP). In probability density function (PDF) methods the velocity, U(x ,t ), and scalar fields, φ(x ,t ), are viewed as random quantities [4, 1]. Here, x and t are points in physical space and time, respectively. From the conservation equations of mass, momentum and scalars, a transport equation for the joint one-point, one-time PDF of U and φ can be derived. This equation is solved using realization based techniques. Here, the evolution of notional particles is calculated based on stochastic differential equations (SDEs). Since the joint PDF lacks information about spatial correlations of the scalars, the molecular diffusion term, Γ∇ 2 φ, is not calculable from the one-point PDF (note that Γ is the diffusion coefficient). Therefore, a mixing model is required to close the molecular diffusion term. An early attempt was the one suggested by Villermaux and Devillon [6] who developed the interaction by exchange with the mean (IEM) model. This model is simple but not general. More sophisticated models are the Euclidean minimum spanning tree model by Subramaniam and Pope [5] and the Fokker–Planck model by Fox [2]. However, no general model exists which fulfills all desirable requirements and the closure of molecular mixing remains an active research area. A summary of these requirements (derived from the nature of the molecular diffusion term) is given by Fox [1]. In this paper we present a new mixing model which is based on a parameterization of one-dimensional scalar profiles. In section 2, a physical interpretation of the widely used interaction by exchange with the mean (IEM) mixing model is given. A generalization of the IEM model is outlined in section 3, leading to the PSP mixing model. The accuracy of the PSP mixing model is demonstrated by comparison with DNS data in section 4.

In hybrid joint probability density function (joint PDF) algorithms for turbulent reactive flows the equations for the mean flow discretized with a classical grid based method (e.g. finite volume methods (FVM)) are solved together with a Monte Carlo (particle) method for the joint velocity composition PDF. When applied for complex geometries, the solution strategy for such methods which aims at obtaining a converged solution of the coupled problem on a sufficiently fine grid becomes very important. This paper describes one important aspect of this solution strategy, i.e. multigrid computing, which is well known to be very efficient for computing numerical solutions on fine grids. Two sets of grid based variables are involved: cell-centered variables from the FVM and node-centered variables, which denote the moments of the PDF extracted from the particle fields. Starting from a given multiblock grid environment first a new (refined or coarsened) grid is defined retaining the grid quality. The projection and prolongation operators are defined for the two sets of variables. In this new grid environment the particles are redistributed. The effectiveness of the multigrid algorithm is demonstrated. Compared to solely solving on the finest grid, convergence can be reached about one order of magnitude faster when using the multigrid algorithm in three stages. Computation time used for projection or prolongation is negligible. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A Fokker–Planck based kinetic model is presented here, which also accounts for internal energy modes characteristic for diatomic gas molecules. The model is based on a Fokker–Planck approximation of the Boltzmann equation for monatomic molecules, whereas phenomenological principles were employed for the derivation. It is shown that the model honors the equipartition theorem in equilibrium and fulfills the Landau–Teller relaxation equations for internal degrees of freedom. The objective behind this approximate kinetic model is accuracy at reasonably low computational cost. This can be achieved due to the fact that the resulting stochastic differential equations are continuous in time; therefore, no collisions between the simulated particles have to be calculated. Besides, because of the devised energy conserving time integration scheme, it is not required to resolve the collisional scales, i.e., the mean collision time and the mean free path of molecules. This, of course, gives rise to much more efficient simulations with respect to other particle methods, especially the conventional direct simulation Monte Carlo (DSMC), for small and moderate Knudsen numbers. To examine the new approach, first the computational cost of the model was compared with respect to DSMC, where significant speed up could be obtained for small Knudsen numbers. Second, the structure of a high Mach shock (in nitrogen) was studied, and the good performance of the model for such out of equilibrium conditions could be demonstrated. At last, a hypersonic flow of nitrogen over a wedge was studied, where good agreement with respect to DSMC (with level to level transition model) for vibrational and translational temperatures is shown.