Phase Flow Model Research Articles

Two-Phase Incompressible Flow with Dynamic Capillary Pressure in a Heterogeneous Porous Media

We prove the existence of weak solutions of a two-incompressible immiscible wetting and non-wetting fluids phase flow model in porous media with dynamic capillary pressure. This model is a coupled system which includes a nonlinear parabolic saturation equation and an elliptic pressure–velocity equation. In the regularized case, the existence and uniqueness of the weak solution are obtained. We let the regularization parameter be η→0 to prove the existence of weak solutions.

Optimizing thermal performance of nanofluid impinging upon a nonlinearly stretching sheet using Taguchi technique

The main purpose of this article is to optimize the thermal performance of system involving stagnation point flow of A l 2 O 3 + water based nanofluid over a nonlinearly stretching surface. The mathematical model for the problem under consideration utilizes the Tiwari and Das’s single phase flow model for the nanofluid. The aim is to find a similar solution of the problem using some suitable similarity variables. The partial differential equations dictating the flow situation are converted into dimensionless ordinary differential equations. Numerical solution of the system of dimensionless ordinary differential equations has been obtained with the help of MATLAB’s inbuilt bvp4c package. Further, Taguchi analysis along with ANOVA and regression analysis have been performed against the variable like Pr , γ , n and ϕ corresponding to response variable to predict the optimal heat transfer of the system. Outcomes of ANOVA indicate that Pr has maximum and ϕ has minimum influence over the Nusselt number.

Analysis of a two phase flow model of biofilm spread

We consider a quasi-stationary problem describing the status of velocities, pressures and chemicals affecting cell behavior within a biofilm. The model couples stationary transport equations and compressible Stokes systems with convection–reaction–diffusion equations. We establish existence, uniqueness and stability of solutions of the different submodels involved and then obtain well posedness results for the quasi-stationary system. Our analysis relies on the construction of weak solutions for the steady transport equations under sign assumptions and the reformulation of the compressible Stokes problem as an elliptic system with enhanced regularity properties on the pressure. We need to consider velocity fields whose divergence and normal boundary components satisfy sign conditions. Applications include the study of cells, biofilms and tissues, where one phase is a liquid solution, whereas the other one is assorted biomass.

Numerical Investigation of Oil–Air Flow Inside Tapered Roller Bearings with Oil Bath Lubrication

Oil–air flow within an oil bath lubrication tapered roller bearing is essential for the lubrication and cooling of the bearing. In this paper, we develop a simulation model to investigate the flow field of tapered roller bearings with oil bath lubrication. The multiple reference frame (MRF) approach is used to describe the physical motion of the bearing, and the volume of fluid (VOF) two–phase flow model is used to track the oil–air interface in the flow field. The effects of mesh scale, geometric gap, and oil reservoir size on calculation time and convergence accuracy are examined in detail, and the effects of inner ring rotational speed and lubricant viscosity on frictional torque are systematically studied. The results of the numerical simulation indicate that as the gap distance between the raceway and the rolling elements decreases, the frictional torque is mainly generated by churning losses at the inner raceway and the rolling elements. The frictional torque increases with increasing inner ring speed and lubricating oil viscosity, with the rolling element contributing the largest portion at approximately 50% of the total. We demonstrate the effectiveness of a method to reduce frictional torque by optimizing the internal structure of the bearing to control oil flow. By optimizing the cage structure and reducing the roller half-cone angle, frictional torque can be reduced by 29.1% and 26.2%, respectively.

Multiphase flow gas transport in a deep geological repository

Gas transport in porous media is of interest in many industrial applications, such as the oil and gas industry, geological storage, and deep geological repositories for radioactive waste. In a deep geological repository, gas will be generated due to the corrosion of metallic components and the degradation of organic materials. This leads to a build-up of gas pressure, which may activate gas transport through the host rock as well as the excavation-damaged zone around backfilled galleries.
 In order to understand different transport mechanisms involved, numerical simulations were performed, and the results were compared with laboratory data. In the framework of the EURAD/GAS project, a gas pressure-dependent permeability model was implemented into the finite element code OpenGeoSys-6 (OGS-6) [1]. The permeability alteration in this model is a function of gas pressure. The laboratory experiments showed that the rate of permeability change are different at low and high gas pressures. Therefore, the permeability model employed a threshold pressure ( to categorize this behaviour.
 and are empirical parameters. Moreover, two other permeability models were employed to study the hydro-mechanical behaviour of the host rock and permeability changes. In the strain-dependent permeability model, the permeability change was related to the elasto-plastic behaviour of the host rock, and in the embedded fracture model, it was related to the opening and closure of fractures [3] [4]. Thus, volumetric elastic strain and equivalent plastic strain are employed to be the controlling variables.
 The initial permeability of the intact rock samples were determined by applying a constant pressure at the upstream and downstream of the samples (i.e. constant pressure gradient). The imperical parameters were determined by matching experimental and numerical results.
 Two types of gas injection tests carried out by the Institute for Rock Mechanics (IFG GmbH, See Figure 1) were used to investigate the gas transport through Opalinus Clay and to examine the permeability models [2]. The first experiment demonstrates an advective-diffusiion gas transport through the sample and an elastic deformation. The second experiment highlights the formation of a tensile fracture (plastic deformation and preferred flow path). In both experiments the advective transport is the dominant transport mechanism. The strain-dependent permeability model was successfully applied to reproduce the hydro-mechanical behaviour of the host rock in both elastic deformation test and tensile fracture test (see Figures 2 and 3). The hydro-mechanical response of a saturated single phase flow model was compared with the behaviour of a saturated two-phase flow model. Both single phase and two-phase flow models were able to describe the hydraulic as well as mechanical behaviour of the experiments performed. Therefore, one can conclude that in these experiments the water phase (wet-phase) was immobile.
 A gas injection test under triaxial conditions was performed by École Polytechnique Fédérale de Lausanne (EPFL) in saturated Opalinus Clay (Fig. 4). The numerical simulations reproduced the hydro-mechanical behaviour of the sample during the gas injection test. A two-phase flow model was applied to simulate the experiment. The relative permeabilities and capillary pressure functions followed Mualem approach and van Genuchten formulation, respectively. The outflow volume and mechanical response of the sample were measured. The experimental results were in good agreement with the numerical ones. The results of the modelling illustrated the penetration of gas into the sample and hence, the displacement of water (see Figures 5 and 6).

Shale gas transport in nanopores with mobile water films and water bridge

Gas flow properties in nanopores are significantly determined by the flow patterns. Slug flow pattern is a potential water–gas two phase flow pattern, in which gas molecules flow in form of gas slugs and water molecules separate gas slugs. Considering water slippage, a portion of water molecules accumulates at the wall with lower mobility, while the remaining water molecules take the shape of a water bridge. Adopting foam apparent viscosity model to represent slug rheological behavior, how water bridge disturbs on gas flow capacity is estimated. The results are compared with the water–gas two phase flow model that assumes annular flow pattern as well as the single gas flow model without the consideration of water. The comparison illustrates that gas molecular movement is significantly hindered by flow space reduction and loss of gas slippage. The impact from water phase of slug flow pattern is more significant than that of annular flow pattern on gas flow capacity. It is discovered that larger nanopores improves gas flow capacity while maintaining bulk water layer thickness and increasing water bridge thickness tend to reduce gas transport ability. A better understanding of the structure and transport of water and gas molecules are conducive to figure out the specific gas–water flow behavior and predict shale gas production.

A Study on the Thermal Properties of Oil-Film Viscosity in Squeeze Film Dampers

Considering the variation in temperature of the oil film in squeeze film dampers (SFDs) caused by squeezing, a more comprehensive analysis of SFD can be obtained. Aiming to investigate the viscosity thermal effect of the oil film in SFDs, this study established a 3D CFD solution model. Based on the total energy model, the viscosity thermal effect was performed. With the mixture multi–phase flow model, the Zwart–Gerber–Belamri (Z–G–B) cavitation model, and the use of the additional mass coefficient, the two–phase flow phenomenon in SFDs was discussed. The oil film at various temperatures and the temperature distribution of different oil types under different working conditions were analyzed. Furthermore, the influence of the SFD thermal effect on the two–phase flow phenomenon were particularly carried out. Meanwhile, the simulation results of the SFD thermal effect were experimentally verified. The results revealed that the maximum temperature of the SFD oil film was enhanced with the increase in the eccentricity ratio and precession frequency. The temperature of the oil film was closely related to the oil dynamic viscosity. The dynamic viscosity of the oil was reduced, which was caused by the SFD thermal effect, thus leading to decreased oil–film damping, enlarging the vaporization level and the range of vapor cavitation, as well as the ingested air amount. CFD simulation results of the thermal effect were in good agreement with the experimental data, which verified the accuracy of the proposed model.

Global existence and large time behaviour for the pressureless Euler–Naver–Stokes system in ℝ3

We investigate the global Cauchy problem for a two–phase flow model consisting of the pressureless Euler equations coupled with the isentropic compressible Navier–Stokes equations through a drag forcing term. This model was first derived by Choi–Kwon [J. Differential Equations, 261(1) (2016), pp. 654–711] by taking the hydrodynamic limit of the Vlasov/compressible Navier–Stokes equations. Under the assumption that the initial perturbation is sufficiently small, Choi–Kwon [J. Differential Equations, 261(1) (2016), pp. 654–711] established the global well–posedness and large time behaviour for the three dimensional periodic domain $\mathbb {T}^3$. However, up to now, the global well–posedness and large time behaviour for the three dimensional Cauchy problem still remain unsolved. In this paper, we resolve this problem by proving the global existence and optimal decay rates of classic solutions for the three dimensional Cauchy problem when the initial data is near its equilibrium. One of key observations here is that to overcome the difficulties arising from the absence of pressure in the Euler equations, we make full use of the drag forcing term and the dissipative structure of the Navier–Stokes equations to closure the energy estimates of the variables for the pressureless Euler equations.

Study on Gas–Solid Two–Phase Flow Characteristics of One–Furnace with Two–Tower Semi–Dry Desulfurization in Circulating Fluidized Bed Boiler

A single–phase flow model was used to analyze the uniformity of the flow field in the desulfurization tower under different baffle combinations, and a multiphase flow model was used to explain the gas and solid two–phase flow characteristics and chemical reaction characteristics in the tower. The stability of the flow behavior of gas and solids in one furnace and two towers was discussed. The results show that the installation of shielding plates at appropriate positions in the tower for sulfur removal is beneficial to enhance the uniform distribution of flow in space, reduce the pulsation interference of bed pressure in the tower, keep the state of gas and solid flow unchanged, and maintain the efficiency of desulfurization at a high level. Reducing the instability of gas–solid two–phase flow is the basis of ensuring the stable switching between single and double towers.

Exact and Numerical Solutions of the Riemann Problem for a Conservative Model of Compressible Two-Phase Flows

In this work we study the solution of the Riemann problem for the barotropic version of the conservative symmetric hyperbolic and thermodynamically compatible (SHTC) two-phase flow model introduced in Romenski et al. (J Sci Comput 42(1):68, 2009, Quart Appl Math 65(2):259–279, 2007). All characteristic fields are carefully studied and explicit expressions are derived for the Riemann invariants and the Rankine–Hugoniot conditions. Due to the presence of multiple characteristics in the system under consideration, non-standard wave phenomena can occur. Therefore we briefly review admissibility conditions for discontinuities and then discuss possible wave interactions. In particular we will show that overlapping rarefaction waves are possible and moreover we may have shocks that lie inside a rarefaction wave. In contrast to nonconservative two phase flow models, such as the Baer–Nunziato system, we can use the advantage of the conservative form of the model under consideration. Furthermore, we show the relation between the considered conservative SHTC system and the corresponding barotropic version of the nonconservative Baer–Nunziato model. Additionally, we derive the reduced four equation Kapila system for the case of instantaneous relaxation, which is the common limit system of both, the conservative SHTC model and the non-conservative Baer–Nunziato model. Finally, we compare exact solutions of the Riemann problem with numerical results obtained for the conservative two-phase flow model under consideration, for the non-conservative Baer–Nunziato system and for the Kapila limit. The examples underline the previous analysis of the different wave phenomena, as well as differences and similarities of the three systems.

Differentiation time scales of small rocky bodies

The petrologic and geochemical diversity of meteorites is a function of the bulk composition of their parent bodies, but also the result of how and when internal differentiation took place. Here we focus on this second aspect considering the two principal parameters involved: size and accretion time of the body. We discuss the interplay of the various time scales related to heating, cooling and drainage of silicate liquids. Based on two phase flow modeling in 1-D spherical geometry, we show that drainage time is proportional to two independent parameters: μm/R2, the ratio of the matrix viscosity to the square of the body radius and μf/a2, the ratio of the liquid viscosity to the square of the matrix grain size. We review the dependence of these properties on temperature, thermal history and degree of melting, demonstrating that they vary by several orders of magnitude during thermal evolution. These variations call into question the results of two phase flow modeling of small body differentiation that assume constant properties. For example, the idea that liquid migration was efficient enough to remove 26Al heat sources from the interior of bodies and dampen their melting (e.g. Moskovitz and Gaidos, 2011; Neumann et al., 2012) relies on percolation rates of silicate liquids overestimated by six to eight orders of magnitude. In bodies accreted during the first few million years of solar-system history, we conclude that drainage cannot prevent the occurrence of a global magma ocean. These conditions seem ideal to explain the generation of the parent-bodies of iron meteorites. A map of the different evolutionary scenarios of small bodies as a function of size and accretion time is proposed.

Global Existence and Large–Time Behavior to a Two–Phase Flow Model with Magnetic Field

Global Existence and Large–Time Behavior to a Two–Phase Flow Model with Magnetic Field

Analysis and control of annular pressure caused by leakage effect of completion string for HP/HT/HHS gas wells in Sichuan Basin

Completion string leakage will result in sustained annular pressure, H2S entering the annulus and a drop in annular liquid level. Based on choke flow model, gas-liquid two phase flow model, and gas state equation, prediction model of sustained annular pressure caused by completion string leakage is firstly established. Then, buildup law, typical buildup patterns and judgment chart, and influence law of control parameters on sustained annular pressure are acquired through case study and influential factors analysis. Finally, three control measures against completion string leakage are put forward and applied in actual gas wells in Sichuan Basin. Analysis results indicate that the smaller the depth of leakage point, the greater the wellbore safety risk. Reducing the equivalent diameter of leakage point is beneficial to control sustained annular pressure. The annulus needs not to be full filled with protective fluid. Field application verifies that judgment chart and control measures have successfully realized quick judgment of wellbore safety risk and control of sustained annular pressure, ensuring production safety of actual gas wells in Sichuan Basin. The management and control methods of sustained annular pressure caused by completion string leakage validated in this work can be extended to similar marine carbonate gas fields.

A SINGLE PHASE MODEL FOR ANALYZING GAS PIPELINE NETWORKS

A single phase flow model has been developed for gas distribution pipeline networks. The model is developed based on looped-system approach with some modifications. In this model, a equation of state model is implemented for predicting the gas properties required for the governing equations of the network system. By utilizing the Linear Theory Method, the Panhandle’s single phase gas flow model is implemented in this model to predict the hydrodynamic variables in each leg of the network using the iterative technique which is developed in this study. A generalization of the single phase network model is provided thereby making it possible for the single flow model used to be replaced by another one that may be more applicable for a particular situation. Using the iterative procedure developed, pressure at all nodes, gas flow rate at each leg can be predicted. The test results demonstrate that the model can serve as a predictive and design tool for solving a single phase gas flow problem in pipeline network.

Advanced two phase flow model for transient molten salt receiver system simulation

In order to realistically predict and optimize the actual performance of a concentrating solar power (CSP) plant sophisticated simulation models and methods are required. This paper presents a detailed dynamic simulation model for a Molten Salt Solar Tower (MST) system, which is capable of simulating transient operation including detailed startup and shutdown procedures including drainage and refill. For appropriate representation of the transient behavior of the receiver as well as replication of local bulk and surface temperatures a discretized receiver model based on a novel homogeneous two-phase (2P) flow modelling approach is implemented in Modelica Dymola®. This allows for reasonable representation of the very different hydraulic and thermal properties of molten salt versus air as well as the transition between both. This dynamic 2P receiver model is embedded in a comprehensive one-dimensional model of a commercial scale MST system and coupled with a transient receiver flux density distribution from raytracing based heliostat field simulation. This enables for detailed process prediction with reasonable computational effort, while providing data such as local salt film and wall temperatures, realistic control behavior as well as net performance of the overall system. Besides a model description, this paper presents some results of a validation as well as the simulation of a complete startup procedure. Finally, a study on numerical simulation performance and grid dependencies is presented and discussed.

Universality of Riemann solutions in porous media

Universality, a desirable feature in any system. For decades, elusive measurements of three-phase flows have yielded countless permeability models that describe them. However, the equations governing the solution of water and gas co-injection has a robust structure. This universal structure stands for Riemann problems in green oil reservoirs. In the past we established a large class of three phase flow models including convex Corey permeability, Stone I and Brooks–Corey models. These models share the property that characteristic speeds become equal at a state somewhere in the interior of the saturation triangle. Here we construct a three-phase flow model with unequal characteristic speeds in the interior of the saturation triangle, equality occurring only at a point of the boundary of the saturation triangle. Yet the solution for this model still displays the same universal structure, which favors the two possible embedded two-phase flows of water-oil or gas-oil. We focus on showing this structure under the minimum conditions that a permeability model must meet. This finding is a guide to seeking a purely three-phase flow solution maximizing oil recovery.

Hydro-mechanical modelling of gas transport in clayey host rocks for nuclear waste repositories

Host rocks, as the final impediment to waste migration, play a significant role in the nuclear waste repositories. Modelling of gas migration in saturated host rocks as well as its coupled hydro-mechanical (HM) behaviors is of importance to the assessment of geological disposal facilities. A comprehensive literature review is carried out on the models for simulating gas migration in saturated rock materials. Several aspects have been provided and discussed, including the material properties and experimental interpretations, governing equations, constitutive models for hydraulic and mechanical processes, fracture propagation models. Specifically, these models are discussed in detail with respect to their performance in describing the recorded experimental behaviors. It is found that the visco-capillary two phase flow model with enriched intrinsic permeability is commonly used to describe the advective gas transport in saturated host rocks. The embedded fracture model (EFM) or enriched EFM seems to be the most favored model as it accounts for the fracturing mechanism which is more representative to implicitly simulate the preferential gas pathways. To describe the mechanical behavior of rocks during the gas migration process, linear/nonlinear elastic, elastoplastic and damage models have been included in the mechanical processes. The HM models incorporating plasticity or damage may not be applicable in most experimental studies, which are not competent to simulate the key experimental behavior associated with the development of gas preferential pathways. Current models that can explicitly describe the gas induced micro-fracturing are seldomly reported, the existed ones are not able to represent all the key experimental behaviors related to preferential gas flow. Advanced approaches, i.e., phase field (PF) method, extended finite element method (XFEM), discrete element method (DEM), hybrid finite discrete element method (FDEM) can be integrated with cohesive zone model (CZM) to provide some promising aspects in representing the development of preferential gas pathways. Lastly, the conclusions and recommendations for future modelling are given.

Eigenfunction expansion method for peristaltic flow of hybrid nanofluid flow having single-walled carbon nanotube and multi-walled carbon nanotube in a wavy rectangular duct.

In this study, "peristaltic transport of hybrid nanofluid" inside a rectangular duct is examined. Water (base fluid) is used with two types of nanoparticles, namely, single-walled carbon nanotube (SWCNT) and multi-walled carbon nanotube (MWCNT). The viscous dissipation effect comes out as the prime heat generation source as compared to the conduction of molecules. After using some suitable dimensionless quantities, we obtained the nonlinear partial differential equations in a coupled form which are then solved exactly by the Eigenfunction expansion method. Velocity distribution, pressure gradient, and pressure rise phenomena are also discussed graphically through effective physical parameters. The heat transfer rate is high for the phase flow (single-walled carbon nanotube/water) model as compared to the hybrid (single-walled carbon nanotube + multi-walled carbon nanotube/water) model due to the enhanced thermal conductivity of the hybrid model.

Review on the local weak form-based meshless method (MLPG): Developments and Applications in Ocean Engineering

This paper reviews the developments and applications of meshfree method based on MLPG (Meshless Local Petrov Galerkin) in ocean engineering, primarily from the work carried out at IIT Madras and City, University of London, UK. Apart from discussing the various stages in the model development, this paper will also reports its applications to small amplitude waves, wave overtopping and breaking, porous layers, long wave run-up, vegetation, floating bodies in waves, wave interaction with elastic structure and two phase flow modelling. Generally, the Navier – Stokes equation will lead to numerical dissipation for long distance propagations and increase in computational time. In order to avoid this, one needs to look for a physics based approach. One of the successfully implemented approaches in MLPG was by coupling with fully nonlinear potential flow theory (FNPT), either one-way or two-way. In this paper, we bring out the advantages and implementation issues of MLPG. The paper also discuss the relationship with ISPH/MPS methods and some concepts that are adopted in both formulations. The paper ends with the key challenges and future directions in the development of the numerical method.

Numerical and dimensional analysis for the jet buckling of highly viscous fluid

Jet buckling of highly viscous fluid is a common phenomenon both in nature and industry. In order to effectively understand this phenomenon, experiments, numerical simulation and dimensional analysis are employed with the jet buckling of rubber melt and in injection molding. To begin with the construction of two high-resolution discretization schemes for convection terms, a three dimensional algorithm for liquid jet buckling phenomenon is established based on two phase flow model and discretization via finite volume method. The developed algorithm shows its good accuracy and stability for jet buckling simulation. The numerical results are in good agreement with experimental results obtained by camera directly and short shot experiments. It is found shear rate and gravity have an effect on jet buckling. Compared to non-buckling, buckling of liquid occurs at high shear rate. Direction of gravity influences the length of jet and the pattern of buckling. Furthermore, dimensional analysis of buckling is carried out based on experimental and numerical results. Results reveal that the influence of gravity on buckling is greater when gravity is vertical than parallel to velocity. Dimensional analysis illustrates that shear rate has a greater effect on buckling frequency than hydraulic radius of inlet, length of cavity, gravitational force, viscous force and kinematic viscosity. The buckling frequency is approximately proportional to the square of the shear rate.