velocity field in the pressure-streamline coordinates the fast channels are magnified and the variation of the velocity field over the coarse scale is smaller. The flow is more or less uniform in the coarse cells of a coordinate system that follows the flow. We obtain the homogenization of the equation, using **two**-scale convergence that had been obtained in the Cartesian scheme in the nineties. The **two** results are different because they are valid under assumptions of **two** different multiscale structures. Yet we compare them and conclude that the approach in the pressure- srteamline **frame** leads to simpler equations that are easier to solve. We show weak convergence and rigorously estimate the **upscaling** error of the **two**-scale limit without using the assumption of periodicity and scale separation. Then we average out the fast dependence of the **two**-scale limit across the streamlines using Tartar’s method to obtain a fully homogenized equation. It is not clear how to design an efficient method using that equation, so we use an averaging approach to remove the fast dependence of the **two**-scale limit across the fast streamlines following Efendiev and Durlofsky [20, 21]. The advantages of the resulting method are demonstrated on realistic permeability fields in terms of accuracy and time of computation.

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The field of reservoir modeling is quite large and this thesis has addressed only a particular aspect of it. In particular, we assumed a specific model that ignored the effects of, for instance, cappillary pressure among other variables, and have focused our attention on the 2-D case. In modeling more realistic cases it is entirely possible that some of the problems faced here, such as the developement of sharp shocks in the solution for the saturation, would disappear. Nevertheless, within the current framework and numerical scheme there exist many potential avenues for improvement and further work. The most obvious of these is the development of a faster numerical scheme for the streamline projection. As mentioned in the section on numerical results, this was a major burden in the computations. Any improvements here could lead to a drastic cut in the run time for simulations. Also, coupling the scheme to an **adaptive** mesh refinement algorithm could also be useful. In this case, one can view the scheme as **upscaling** away from the oil-water front but also resolving features in this critical region.

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In Section 2 we present the general formulation of the governing equations for the modelling of mass, momentum and energy conservation of a compressible flow. In Section 3 the discretisation of these equations in the DG framework is described. In Section 4 we introduce the forest structure for the representation of an **adaptive** unstructured mesh. In Section 5 we describe the splitting and merging techniques occurring during the adaptation of the computational grid to the flow field solution. In Section 6 we describe the algorithm introduced for the e ffi cient assignment of the connectivity problem during the adaptation of the grid. In Section 7 we present a series of test cases for subsonic and supersonic configurations where we assess the accuracy of the algorithm, the e ffi ciency of the proposed method and the computational performance of the code. In Section 8 we demonstrate the capabilities of the code to capture moving structures for the problem of an inviscid supersonic flow around a cylinder in a duct. In Section 9 we demonstrate the resolution of the flow discontinuities and reaction regions arising from the interaction of the oblique shocks in the case of a hypersonic flow around a double cone configuration. In Section 10 we use the solver introduced in this paper for the solution of a dispersed multiphase flow arising during spray injection of gasoline fuel.

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Further development of radiometric methods is also associated with the application of the measurements of the absorption of gamma and X-rays in the flowing mixtures [2-6]. The application of mathematical analyses of signals derived from nuclear probes enables the determination of many parameters of multi-**phase** mixtures transported in open channels [7] and pipelines, the research and control of hydrometallurgical processes and the extraction of crude oil. What is also important is that the algorithms of the analysis of signals which are used may also be applied in the analysis of data, e.g. in optical and electrical methods [8-14].

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Non-equilibrium **phase** behavior may result from interaction of fluids with porous media. The possible causes are adsorption, capillary condensation, wettability-controlled **phase** transitions, layered crystallization, critical adsorption, etc. [8]. The most significant factor not necessarily associated with the influence of porous media is the limited area of interphase contact which hinders components' mass transfer between phases. In this case, scale factor is of great importance in **phase** behavior and compositional flow simulations. The processes which can be successfully simulated with an equilibrium model on small scale require the use of a non-equilibrium model on larger scale. In numerical simulations, this effect is associated with model grid cell sizes. In addition to **upscaling** methods based on tuning parameters, such as the well-known method of alpha-factors [4, 5, 29], **two** thermodynamically-consistent procedures were independently presented in [19, 33]. Further progress was reported in [16, 17]. Both procedures are thermodynamically-consistent and based on similar modifications to the flash problem formulation as far as **upscaling** is considered. Here we demonstrate that the method of [16, 19] provides clear physical interpretation of both sides of the flash equations and can be readily incorporated in a unified thermodynamically-consistent framework for non-equilibrium simulations at different scales.

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p4est is one recent example of a cell-based AMR implementation that uses linear storage given by a space-filling curve. The primary usage of space filling curves in numerical simulation, is to provide a simple and efficient way of partitioning data for load balancing in distributed computing, but they can also be used for organizing data memory layout as in p4est. Indeed, many space filling curves have a nice property called compactness, which can be stated as: contiguity along the space-filing curve index implies, to a certain extend, contiguity in the N-dimensional space of the Cartesian mesh. As a consequence of the compactness property, one can expect also improved AMR performance due to a better cache memory usage resulting from a certain degree of preserved locality between the computational mesh and the data memory layout. p4est also implements specialized refining, coarsening and iterating algorithms for its specific choice of linear storage that have proven scalability [9]. These points are real advantages in the **frame** of HPC where time and space complexity have to be handled with extreme care. On the top of that, the concept of forest of trees, allows to use multiple deformed but conforming and adjacent meshes (each tree), enabling, to a certain point, to represent complex geometries, although this method does not offer the same flexibility as unstructured meshes.

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The mathematical equations are derived for three phases to **two**-**phase** transformation using pseudo reference **frame** is a slightly complex method. In this paper, the suggested mathematical model of the three-**phase** asynchronous electrical machine is depending on **two**-**phase** electrical machine mathematical model at stator reference **frame** coordinate system axis 𝛼𝛽, which are associated with stator **phase** axis ABC for the conversion of three phases to **two** phases asynchronous mathematical machine model [3]. A pseudo-reference **frame** in which maximum control algorithms are designed, and is able to express analytically and derive torque and MMF equations. As compared to other models, the analytical mathematical model is most suitable for design considerations like flux observer and machine controllers. For loss minimization control algorithms, the optimal control variables can be solved by using mathematical analysis, numerical or iteratively based on the analytical connection between the machine loss and control variables [1]. The **two**- **phase** reference **frame** is done with the provision of a sinusoidally scattered MMF wave in the machine air gap, it also done mathematical analysis of **two** windings on stator and **two** windings on rotor of induction machine which contains a transformation to **two**-axis components of stator and the **two**- axis space vector components transformation of rotor [2]. The existence reference model phenomenon that is normally neglected in the modelling of the induction machine is more difficult to deal with, meanwhile these characteristically require a suitable modification of the basic induction machine model. The available models of induction machine (steady state and dynamic), is account for various stray losses components under the study of conditions of inverter and grid supply [5]. **Two**-**phase** reference **frame** mathematical models are used for the study of various solutions for steady-state and transient problems require machine parameters that are usually considered for mathematical modelling parameters. These machine parameters are including various resistance and inductance equations in matrix form which representing stator, rotor and magnetizing branches [6]. For mathematical modelling of the asynchronous machine, dynamic machine modelling attracts much attention due to their substantial impacts on the correctness of reference **frame** modelling [8], ________________________________

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in order to determine validity range of **two**-fluid the model. Complex roots of characteristic equation leads to an ill-posed initial value problem and failure to achieve converged and consistent results. Numerical approachs are divided into **two** categories in terms of pressure term: The first category of algorithms is based on pressure. Among them, Interphase Slip Algorithm [5] and Simple Algorithm can be referred [1, 6]. The second category of algorithms are algorithms based Riemann solver. Among these algorithms, Conservative Shock Capturing Method [7] and Flux Splitting Methods [8] can be referred. In this study, we focussed on this category. This category of approaches were developed form generalizing of solving methods of Shock flow to **two**-**phase** flow problems. Due to their capturing shock, these approaches are used in order to predict discontinuities in the interface of **two** **phase** **flows**. The second category of algorithms are considered as explicit numerical schemes. In order to calculate time step, they are dependent on maximum value of wave velocity. The maximum value of wave velocity for **two**- fluid model is equal to maximum value of the roots of the characteristic equation of governing equations in solution field. Derivative by varying the terms of the **two**-fluid model, which generally are related to assumptions of pressure and pressure correction terms, the **two**-fluid model requires to new hyperbolic analysis and determining real roots of the characteristic equation. In the **two**-fluid model, there is a difference between pressure of each **phase** and pressure of the same **phase** at the interface. The pressure difference between pressure of each **phase** and pressure of the same **phase** at the interface is called “term correction pressure”.

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Within the context of quaternary onium salts, which are the catalysts of choice in many reaction systems, the exact choice of the catalyst depends on the system under consider- ation. In general, it is usually recommended that a variety of both ammonium and phosphonium salts be screened for a given reaction system. However, some generalized observa- tions can be made from various studies. Usually, a highly lipophilic cation is used to ensure high compatibility with the organic **phase** and efficient anion transfer. In gen- eral, asymmetric quats like cetyltrimethylammonium halide are found to be more active than symmetric quats like traalkylammonium salts due to the lipophilicity imparted by the long chain alkyl group. However, in the case of solid-liquid systems, where the quat has to approach the solid surface to pick up the reactive anion, symmetric quats per- form better than asymmetric ones. In either case, the catalyst must be with sufficient organic structure (large alkyl groups) so that the cation-anion pair is substantially parti- tioned into the organic **phase**. Thus, tetramethyl ammonium chloride which is soluble more in water than in the organic **phase** is not a good PT agent, whereas tetrabutyl ammonium bromide has a high partition coefficient in the organic **phase** due to its higher lipophilic nature and is an effective and widely used PT catalyst. Besides the structure of the quat cation, the choice of the anion associated with the ternary cation is also crucial since decreasing catalyst activity is observed with increasing tendency of to associate with the quaternary cation. Phosphonium salts, which are slightly more expensive than their ammonium analogs, are stable only under very mild conditions 15% and high concentrations of (Landini et al., 1986). Some other aspects like the choice of anion, cation structure, solvent, and so on which influence the choice of the catalyst are discussed in the Section on Mechanisms of PTC.

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ratios of the interfacial tensions, seven possible equilibrium morphologies are success- fully reproduced, which are consistent with the theoretical stability diagram by Gu- zowski et al.[15]. For various geometry configurations of general and perfect Janus droplets, good agreemento between simulated results and analytical solutions shows the present model is accurate when three interfacial tensions yield a Neumann’s trian- gle. In addition, we also simulate the near-critical and critical states of double droplets, which is challenging since the outcomes are very sensitive to the model accuracy. It is found that the simulated results from the present model agree well with the ana- lytical solutions, while the simulated results from the existing color-gradient models significantly deviate from the analytical solutions, especially in critical states. In sum- mary, the present work provides the first LB multiphase model that allows for accurate simulation of ternary fluid **flows** with a full range of interfacial tensions.

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The investigation of **flows** in porous media is motivated by various engineering applications, such as migration of moisture in fibrous insulation, grain storage, transport in contaminated soils, underground disposal of nuclear wastes, transport in drying processes, etc. Nield and Bejan [1] summarized the state-of-art on this topic. Another quickly developing research field related to porous media is concerned with biomedical applications. In a clotted artery, the lesions or "plaques" within the artery wall consist of localized deposits of fat compounds (lipids) surrounded by cells recruited from the blood stream and scar tissue; this acts as a porous medium that may diminish or completely eliminate the blood flow. The coronary arteries surrounding the heart are curved and at least segments of them can be modeled as helical.

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Experimental techniques such as particle image velocimetry (PIV) and hot wire anemometry (HWA) have been used to evaluate the liquid energy spectrum in pseudo-turbulent bubbly **flows**. Lance and Bataille [50] found the classical -5/3 power law gradually replaced by a -8/3 slope in the high frequency range as void fraction was increased in bubbly turbulence and these results were reproduced by Wang et al. [1]. Rensen et al. [71] were not able to reproduce these results and instead reported a slope slightly less than -5/3 and attributed their differences to the bubblance parameter. Shawkat et al. [51] experimentally investigated the energy spectrum of air-water flow in a vertical pipe and found slopes ranging from -8/3 to -10/3 dependent on the void fraction. They proposed that the traditional inertial subrange might not be valid for **two**-**phase** bubbly **flows** since there is energy production due to liquid-bubble interactions at the traditional inertial subrange length scales. Bolotnov et al. [10] made similar arguments that bubble contributions to the liquid turbulence occur at length scales on the order of the bubble diameter in their spectral cascade- transport model development. Mercado et al. [73] found a slope of -3.18 for the energy spectrum with no dependency on the void fraction. Mendez-Diaz et al. [52] experimentally analyzed pseudo- turbulent flow for **two** different bubble sizes and three different viscosity liquids and found a slope close to -3 for all cases. A slope of -5/3 transitioning to -3 at higher wave numbers was reported by Sathe et al. [72].

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The formulation presented in (2.3.1) utilises a free energy approach, first applied to the kinetics of surfactant adsorption in Diamant and Andelman [1996]; Diamant, Ariel, and Andelman [2001], to model instantaneous adsorption kinetics. At adsorption/desorption equilibrium, the chemical potentials γ 0 (c Γ ) and G 0 (c) must be equal [Zhdanov, 2001; Liu and Zhang, 2010; van der Sman and van der Graaf, 2006] and thus this approach allows us to cover the adsorption isotherms often used in the literature by selecting suitable functional forms for γ and G. Hence, α (i) > 0 can be seen as a kinetic factor which relates the speed of adsorption to the inter- face or desorption from the interface to the deviation from local thermodynamical equilibrium. Let us summarise the governing equations of the general model for **two**-**phase** flow with soluble surfactant: For i = 1, 2,

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When applying engineering processes to porous media involving **two**-**phase** **flows**, e.g., enhanced oil recovery, geological carbon sequestration, water infiltration, and groundwater remediation, a fractured pore structure is commonly observed. To understand the fluid interactions in a fractured porous material, a uniform pore-pattern embedded with a single macropore channel is commonly used to represent the pore geometry of the fractured porous medium [1–4]. A thorough conceptual and quantitative understanding of the flow behavior of fluids in pore spaces forms the basis of multiphase flow in porous materials. These subsurface multiphase **flows** are of interest to hydrologists, agricultural/environmental engineers, and petroleum engineers. Nevertheless, since most soils and geomaterials are not transparent, visualizing their flow behavior in porous materials is a long-term persistent challenge.

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There are many examples of **adaptive** sampling in environmental science. Here we review some of the designs that can be useful for sampling populations that are rare and clustered. A text on **adaptive** sampling was published in 1996 by Thompson and Seber [39], which presents much of the theoretical background and development of estimators. The emphasis is on rare and clustered populations is simply because these are often the most challenging to survey and are also very common in environmental science [17]. There is no one definition for rare and clustered populations that all statisticians and ecologists agree on, but generally these are populations that are difficult to detect (so in fact may not be rare, but sightings are rare), or the population that are sparse in some sense. Sparseness can be because there are few individuals in the population or because the population covers a very large area. McDonald [17] has an excellent discussion of the variety of rare and clustered populations ranging from snakes, and freshwater mussels to polar bears and walruses.

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The application of air in micro-bubbles DR is environmentally friendly and cheaper compared to polymers and surfactants [77–78]. Micro-bubbles have diameters less than ten-microns and exhibit behaviours different from those of larger size bubbles. These differences are seen in their chemical and physical characteristics such as the tendency to be suspended in the liquid **phase** over a longer time [78]. The first work published on the application of micro-bubbles as DRAs was by [79]. The mechanism of DR by micro-bubbles is not yet well understood. Similar to other DR techniques, the essence of micro-bubble injection is to change the boundary layer structure. It had been suggested that micro-bubbles reduce drag by altering both laminar and turbulent boundary-layer characteristics [79]. It has been reported that, injecting air bubbles increases kinematic viscosity and decreases turbulent Reynolds number in the buffer layer [80]. This results in enlarging the viscous sub-layer thereby decreasing the velocity gradient at the wall. [77] used particle image velocimetry (PIV) to investigate the effect of injecting low void fraction micro-bubbles into the boundary layer of a channel flow. Some of their results showed some similarities with drag reduction behaviour by polymers or surfactants as well as reports of some earlier investigations [80–81]. These similarities include thickening of the buffer layer as well as upward shift of the log-law region. They stated that the micro- bubble layer formed at the top of the channel was not responsible for the drag reduction recorded. This micro-bubble layer served to reduce the slip between the micro-bubbles and the liquid. The major contribution to drag reduction is the accumulation of micro-bubbles in a critical zone within the buffer layer. The interaction of micro-bubbles with turbulence in the buffer layer is responsible for the observed DR. in general, injection of micro bubbles reduces turbulent energies with the shear in the boundary layer remaining unchanged [82]. There appears to be some agreement on the mechanism of micro-bubble drag reduction especially as it relates to thickening of the viscous sub- layer and turbulence suppression.

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For modelling cryogenic **flows**, either **two**-fluid or one-fluid models have been employed in numerous past studies. The **two**-fluid model, which was originally introduced by Wallis in 1980 [15], considers slip velocity and non-equilibrium effects. However, such models strongly depend on closure models describing the interaction between the **two** phases, thus rendering the relevant methodologies problem dependent. Maksic and Mewes [16] simulated flashing in converging-diverging nozzles by employing a transport equation for the bubble number density (4-equation model) to the continuity, momentum and energy equations. Later on, a 6-equation model (continuity, momentum and enthalpy equations for each **phase** separately) has been used by Marsh, O’ Mahony [17] for flashing **flows** and by Mimouni [18] for boiling **flows**, where Janet et al. [19] utilised a 7-equation model and they considered nucleation during flashing flow in converging-diverging nozzles. In the latter, a bubble number transport equation was added in the system of 6 equations.

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The set of hydromechanical equations for the i-th **phase** (2), (5), (9) and the medium as a whole (3), (7), (10), (11) add a generalized system of equations for **two**-**phase** **flows** with external heat and mass exchange and are valid for Description of the environment with favorite physical properties. The analysis showed that these equations are general, since the proposed earlier different forms of equations of **two**-**phase** media [1-7] can be obtained from these as special cases by appropriate transformations (under certain assumptions).However, the system of equations for **two**-**phase** media with external heat and mass transfer is uncertain. For its closure it is necessary to attract thermodynamic and rheological equations of state, as well as expressions for the heat flux, interphase forces and heat transfer between phases. These additional (defining) relations are established in the construction of a mathematical model of a particular studied medium.

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χ for different compositions of liquid-water systems based on pure water, will be useful in predicting **two**-**phase** heat transfer coefficients using pure **phase** thermo-physical properties. The predicted values can be used for designing heat exchangers for a specific **two**-**phase** duty in the Reynolds numbers range investigated. Based on the summary in Table 1, it can be concluded that water is a better reference fluid compared to other organic liquids. The developed correlation predicts the MTPM for **two**-**phase** liquid system in tube side with a maximum error of ± 15 % for wide range of data points covering 7 different **two**-**phase** systems.

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Functions F and G are very complex by their defi- nition, especially for non integer values of . They in- volve the integral-defined function K . They e diffi- cult to handle algebraically. For the present work, we sketch the surfaces representing the **two** functions, above the region R , for some valu of

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