Case Of Two-phase Flow Research Articles

Analytical solutions to the Euler equations for Chaplygin gas

In this paper, we study the analytical solutions to the Euler equations for Chaplygin gas. First, we construct two exact solutions for the one-dimensional system by using a self-similar ansatz. Second, we present some analytical solutions for the N-dimensional radially symmetric system. Third, we extend the above results to the two-phase flow case. The concentration and cavitation phenomena are observed from the constructed solutions.

A coupled point particle two-phase heat and mass transfer model for dispersed flows based on Boundary Element Methods

In dispersed multiphase flow processes we encounter coupled heat, mass and momentum transfer between the disoersed and the continuous phase. In the context of the subdomain Boundary Domain Integral Method (BDIM) solution of the Navier-Stokes equations a two-way coupling model is presented based on the use of the elliptic fundamental solution and the Dirac delta function properties which leads to accurate evaluation of the heat and mass point particle source impacts on the continuous (air) phase. In addition, the two-phase flow case under consideration is extended to the case of porous spherical particle drying with internal moving drying front, which is solved by the Boundary Element Method (BEM).

Drag coefficient for micron-sized particle in high-speed flows

The drag force on the small particle in high-speed flows is influenced by the combined effects of fluid viscosity, compressibility, and rarefaction. The existing drag coefficient models are still insufficient in accuracy and efficiency for gas-particle flow simulation. This study comprehensively considers these effects and conducts high-fidelity numerical simulations. A new drag coefficient is generated using a symbolic regression method reasonably based on the particle Mach number, Reynolds number, and Knudsen number, which are related to particle diameter, gas-particle relative velocity, and other parameters. The new drag coefficient possesses clear physical significance, high predictive accuracy, low computational cost, and consistency with theory in limiting conditions. The application of the new drag coefficient to three typical gas-solid two-phase flow cases demonstrated its excellent performance.

Study of the Effect of Static Eccentricity on Vibration Damping Properties of Squeeze Film Dampers Considering the Two-Phase Flow Case

To analyze the effect of static eccentricity on the air ingestion distribution and vibration damping properties of the SFD, a numerical simulation study of SFDs considering two-phase flow was carried out based on CFD using a transient solution method and dynamic mesh technique. The results show that the angle between the static eccentricity direction and the circumferential direction of the oil supply hole increases and the air ingestion area in the oil film expands. In contrast, the oil film damping decreases, and the larger the static eccentricity distance, the greater its effect on the air ingestion area in the oil film. When the circumferential angle is small, the oil film damping increases with the increase of static eccentricity distance, and when the circumferential angle is large, the oil film damping decreases with the increase of static eccentricity distance and then increases. With the increase of static eccentricity distance, the air ingestion area at both ends of the oil film increases. At the same time, studying the effect of dynamic eccentricity shows that as the dynamic eccentricity increases, the oil film damping first decreases and then increases, and the air ingestion area increases. Comparing the 1 hole, the 2 hole, and the 3 hole oil supplies, the air ingestion area is significantly larger in the 1 hole oil supply than in the 2 hole or the 3 hole oil supplies, and the oil film damping of the 1 hole oil supply is smaller than the oil film damping of the 2 hole or the 3 hole oil supplies. It can be seen from the present study that in the actual installation of the SFD, when the circumferential angle is less than 60°, the static eccentricity can be increased appropriately. When the circumferential angle is greater than 60°, the static eccentricity can be appropriately reduced.

Regularity for a special case of two-phase Hele-Shaw flow via parabolic integro-differential equations

We establish that the C1,γ regularity theory for translation invariant fractional order parabolic integro-differential equations (via Krylov-Safonov estimates) gives an improvement of regularity mechanism for solutions to a special case of a two-phase free boundary flow related to Hele-Shaw. The special case is due to both a graph assumption on the free boundary of the flow and an assumption that the free boundary is C1,Dini in space. The free boundary then must immediately become C1,γ for a universal γ depending upon the Dini modulus of the gradient of the graph. These results also apply to one-phase problems of the same type.

A point particle source model for conjugate heat and mass transfer in dispersed two-phase flows by BEM based methods

The paper reports on development of a Boundary Element Method (BEM) based two phase dispersed flow model, that allows an accurate and efficient heat and mass transfer coupling by using a moving point source model to account for particle–fluid interaction. The two-way coupling model highlights the efficient use of the elliptic fundamental solution and the Dirac delta distribution properties to accurately evaluate the heat and mass point particle source impact on the continuous phase, solved by the Boundary Domain Integral Method (BDIM). In addition to the BDIM model of the particle–fluid heat and mass transfer interaction, the two-phase flow case under consideration is extended to the case of porous spherical particle drying with internal moving drying front, which is solved by the Boundary Element Method. As the two-phase flow is considered to be dilute, the particle–fluid momentum exchange is covered by a one-way coupling algorithm, with Lagrangian particle tracking used for determination of particle positions and velocities in the flow. Two computational cases are presented, where 1000 and 10000 particles are dried in a stream of hot air. Comparison between the obtained drying times for the cases of the one-way and the two-way heat and mass transfer coupling results shows, that the developed two-way coupling model accurately captures the effect of moisture accumulation and temperature decrease in the fluid phase, leading to realistic computations of drying rates of particles in the flow.

Modelling reinjection of two-phase non-condensable gases and water in geothermal wells

Steam production from high enthalpy geothermal systems is frequently accompanied by the emission of non-condensable gases (NCGs), initially dissolved in the liquid phase or mixed in the vapour phase at depth in the reservoir. Capturing and reinjecting geothermal gases (CO2, CH4, NH3, H2S, H2, …) together with condensed steam leads to a significant improvement of the environmental profile of geothermal power systems and helps with reservoir recharge and pressure support. Nowadays, there are several ongoing projects targeting the minimalization of the environmental impact of geothermal exploitation through reinjection of the NCGs. For low NCG content, the gases can be fully dissolved into the condensed water and reinjected. However, for a high concentration of NCGs, the dissolution is only partial, and a two-phase mixture needs to be reinjected to achieve this goal. We present here the numerical results for the reinjection process in two different geothermal sites and for two different injection scenarios, focusing on the case of two-phase (liquid–gas) flows. Three commercial software (UniSim®, PIPESIM, and OLGA) and one in-house developed code (GWellFM) were used, and their results were compared. The injection should always take place inside a liquid column in the well to avoid degassing and accumulation of the NCGs. When targeting high pressure reservoirs or when injecting high ratios of water to NCGs flow rates, mixing of the two phases should take place on the surface. Whereas for low pressure reservoirs or for high gas contents mixing must occur deep in the well through one or several mixing points to ensure either complete dissolution of the gas or the downward flow of a two-phase mixture with minimum compression on the surface. The uncertainties of two-phase downward flows and water/NCGs mixtures characterisations introduce additional complexities in the modelling of the reinjection process.

Numerical Prediction of Internal Flows in He/LOx Seals for Liquid Rocket Engine Cryogenic Turbopumps

Cryogenic turbopumps are used in high-performance, lightweight liquid rocket engines for space applications. The development of bearings and shaft seals for cryogenic turbopumps requires detailed characterization of the internal flow, taking into account the effects of boiling and multi-component two-phase flow. Here, a flow network solver was developed to analyse the secondary flow circuit of a cryogenic turbopump where the propellant is mixed with high-temperature helium after bearing cooling. The network solver is based on an extension of a classic 1D homogeneous model, originally developed for a pure substance, to the case of two-phase multi-component flow. The solver is capable of predicting pressures, temperatures, flow rates, and species concentrations in a complex two-phase flow in the presence of non-condensable gases. The unsteady mass, momentum, and energy conservation equations are implemented in conjunction with the thermodynamic equations of state using a general-purpose finite volume formulation, where the pressure drop and the heat transfer are calculated using correlations. The numerical tool was validated by comparing its predictions with experimental data obtained during tests on the secondary circuit of an oxygen turbopump developed at Avio S.p.A. A number of engine operating conditions were considered (inlet helium temperature in the range of 250–280 K, helium/liquid oxygen drain in the range of 165–230 K). The predicted temperature values showed good agreement with the experimental data in most conditions.

Development of drift-flux correlations for vertical forward bubble column-type gas-liquid lead-bismuth two-phase flow

Bubble columns represent an extreme case of gas-liquid two-phase flow, where net liquid velocity is zero and the gas simply bubbles up through the liquid. The bubble column-type gas-liquid metal two-phase flow always appears in an accident scenario for pool-type lead-bismuth eutectic cooled fast breeder reactors. To accurately predict the void fraction for the accident evaluation and design of a reactor system, nine existing drift-flux type constitutive correlations are evaluated with a nitrogen-liquid heavy metal two-phase flow test. Few correlations give a relatively good prediction and the basic assumption in the distribution parameter calculation is not applicable for a bubble column. To solve this problem, analysis based on Clark’s theoretical model is carried out. The results show that the distribution parameter assumes very high values at a low Re number. As the Froude number increases, the distribution parameter tends to decrease. At lower void factions, the distribution parameter is also assumed to be at high values. This indicates that the pipe size, flow rate, and void fraction can all influence the distribution parameter. Considering the quantitative laws of these influence factors obtained by theoretical analysis and fitting the data of Ariyoshi’s test, a new correlation for a bubble column-type gas-LBE two-phase flow is developed and evaluated. The results of the statistical analysis show that the new correlation gives the best prediction for gas-LBE two-phase flow in the void fraction range of 0.018–0.313.

Global implicit solver for multiphase multicomponent flow in porous media with multiple gas components and general reactions

In order to study the efficiency of the various forms of trapping including mineral trapping scenarios for CO2 storage behavior in deep layers of porous media, highly nonlinear coupled diffusion-advection-reaction partial differential equations (PDEs) including kinetic and equilibrium reactions modeling the miscible multiphase multicomponent flow have to be solved. We apply the globally fully implicit PDE reduction method (PRM) developed 2007 by Kräutle and Knabner for one-phase flow, which was extended 2019 to the case of two-phase flow with a pure gas in the study of Brunner and Knabner. We extend the method to the case of an arbitrary number of gases in gaseous phase, because CO2 is not the only gas that threats the climate, and usually is accompanied by other climate killing gases. The application of the PRM leads to an equation system consisting of PDEs, ordinary differential equations, and algebraic equations. The Finite Element discretized / Finite Volume stabilized equations are separated into a local and a global system but nevertheless coupled by the resolution function and evaluated with the aid of a nested Newton solver, so our solver is fully global implicit. For the phase disappearance, we use persistent variables which lead to a semismooth formulation that is solved with a semismooth Newton method. We present scenarios of the injection of a mixture of various gases into deep layers, we investigate phase change effects in the context of various gases, and study the mineral trapping effects of the storage technique. The technical framework also applies to other fields such as nuclear waste storage or oil recovery.

Discontinuous and continuous Galerkin methods for compressible single-phase and two-phase flow in fractured porous media

Accurate simulation of flow behaviors in fractured porous media is challenging. We present a discontinuous Galerkin (DG) approximation and continuous Galerkin (CG) approximation for compressible single- and two-phase flow in porous media with conducting (high permeable) and blocking (low permeable) fractures using a mixed-dimensional approach in which the fracture is described as a reduced-dimensional interface coupled with linear transmission conditions. The proposed DG/CG method was first verified with single-phase fractured flow benchmark cases and then applied to time-dependent single-phase flow cases. The simulated results demonstrate that the DG/CG method is capable of capturing the continuity as well as the jump in pressure between the two sides of the matrix-fracture interface. For the two-phase flow cases, we verify the DG/CG method with a reference case involving a complex fracture configuration. Subsequently, we analyze several cases to study two-phase flow through a single fracture and a discrete fracture network in two dimensions. Overall, the simulation results show that the developed DG/CG approach can reliably predict the flow behaviors for single- and two-phase flow in fractured porous media.

Development and Verification of an Enhanced Equation of State in TOUGH2

Abstract A reservoir-geomechanics coupled simulation tool is required in the interpretation and prediction of stimulation and production performance of unconventional reservoirs in a physically rigorous manner. One such a coupled simulation tool developed at Lawrence Berkeley National Laboratory (LBNL), TOUGH2-FLAC3D, has been extensively adopted in many scientific research and practical application projects in last two decades. In TOUGH2's most often used equation of state (EOS), EOS4, the vapor pressure is used as a primary variable, however, the vapor pressure lowering effect is not the primary interest in a typical reservoir flow simulation. Instead, the temperature should be a natural and physically correct primary variable, considering the convenience of model input, and the understanding and interpretation of simulation results. In this work, the vapor pressure is replaced by temperature in the EOS implementation. To be consistent with the change of primary variable, the Jacobian matrix is modified correspondingly and illustrated using a case of two-phase flow in one grid block. The development work was verified in three well-defined problems that are related to fluid diffusion, thermal conduction, thermal fluid conduction and convection, respectively, through comparing the modeling results with the corresponding analytical solutions.

New instability and mixing simulations using SPH and a novel mixing measure

This paper assesses the ability of smoothed particle hydrodynamics (SPH) to simulate mixing of two-phase flows and their transition to instabilities under different flow regimes. A new measure for quantification of the degree of mixing between phases in a Lagrangian framework is also developed. The method is validated using the lid-driven cavity and two-phase Poiseuille flow cases. The velocity along the centre of the cavity is compared with results from the literature, whilst commercial volume-of-fluid code STAR-CCM+ provides a benchmark for the mixing and different mixing measures are considered. The velocity of two-phase Poiseuille flow along the channel is compared to the analytical solution, and the appearance of interfacial instabilities with perturbation theory. This is the first time SPH has been used to investigate the onset and development of these instabilities. In particular, it is able to model the deforming shape of the interface, which is not given by analytical studies, while also offering improved predictions over conventional mesh-based computational fluid dynamics simulations.

Computationally efficient CFD prediction of bubbly flow using physics-guided deep learning

To realize efficient computational fluid dynamics (CFD) prediction of two-phase flow, a multi-scale framework was proposed in this paper by applying a physics-guided data-driven approach. Instrumental to this framework, Feature Similarity Measurement (FSM) technique was developed for error estimation in two-phase flow simulation using coarse-mesh CFD, to achieve a comparable accuracy as fine-mesh simulations with fast-running feature. By defining physics-guided parameters and variable gradients as physical features, FSM has the capability to capture the underlying local patterns in the coarse-mesh CFD simulation. Massive low-fidelity data and respective high-fidelity data are used to explore the underlying information relevant to the main simulation errors and the effects of phenomenological scaling. By learning from previous simulation data, a surrogate model using deep feedforward neural network (DFNN) can be developed and trained to estimate the simulation error of coarse-mesh CFD. In a demonstration case of two-phase bubbly flow, the DFNN model well captured and corrected the unphysical “peaks” in the velocity and void fraction profiles near the wall in the coarse-mesh configuration, even for extrapolative predictions. The research documented supports the feasibility of the physics-guided deep learning methods for coarse mesh CFD simulations which has a potential for the efficient industrial design.

Stability of a vertical Couette flow in the presence of settling particles

We consider the linear stability of a particle-laden vertical Couette flow between two flat plates. The flow is described in the framework of a two-fluid approach with a negligibly small volume fraction of the dispersed phase (the “dusty-gas model”). The carrier-fluid velocity profile in the main flow is modified by the presence of settling particles distributed non-uniformly between the plates. Two types of particle number density distributions are considered, namely, a single layer with a maximum at the flow center line and two symmetric layers located between the center line and the walls. Linearized governing equations are reduced to a modified Orr–Sommerfeld equation for the amplitude of a disturbance in the form of a normal mode. On the basis of a parametric study of the eigenvalues, it is shown that, in contrast to the case of two-phase Couette flow with zero particle velocity slip, the vertical disperse flow is unstable over a wide range of governing parameters, including small Reynolds numbers. In flows with both kinds of particle concentration profiles considered, there are two types of growing modes. A gravitational mode is triggered at low Reynolds numbers and typically small wavenumbers, while a shear mode with the wavelength of the order of the channel width and larger time-amplification rates is triggered at large Reynolds numbers. It is found that both modes are amplified with an increase in the particle mass loading or the ratio of the Reynolds number to squared Froude number, determining the carrier-fluid velocity profile. The results of the study can be used to control the stability of various technological processes involving particle-laden flows.

Air entrainment and pressure drop in low-cost ejectors.

This study investigated experimentally the air entrainment and pressure drop in low-cost ejectors composed of two pieces shaped from PVC bars inserted in a 25 mm T-junction of the same material. The hydraulic behavior was very similar for the different ejector designs, and linear relationships between the water and air flow rates were fitted. However, when a rotameter was installed at the air line, the head losses resulted in a pronounced decrease (3-fold) in the air entrainment rate. The maximum air-water entrainment ratios reached by the low-cost ejectors was 1.7, while the pressure drop was about 80% of the upstream pressure. The results suggest that these ejectors have a better benefit-cost ratio than conventional ones for applications such as aeration and mixing in reactors, tanks and water bodies. Comparing our results with those obtained previously by using water both as primary and suction fluids, it was shown that under gas-liquid flow conditions the entrainment ratio was about 2.5 times larger than that for the single-phase case, while the pressure drop was about 15% higher. This was attributed to the lower density of the air and the higher dissipation of turbulent kinetic energy due to bubble-liquid interactions in the two-phase flow case.

Effective Asymptotic Model of Two-Phase Flow through Fractured-Porous Media

The solution of the Buckley-Leverett problem classical for the theory of flow through a porous medium, generalized to the case of two-phase flows in fractured-porous media, is considered. In this case, immiscible displacement of fluid in the porous medium is complicated by the absence of local capillary equilibrium between pore spaces of different scales and in the generic case the solution of problem is not self-similar. The flow through a porous medium is considered in the limiting case of large time scales when capillary equilibrium is established and the flow parameter distributions, as shown in the study, tend to self-similar asymptotics. For the effective ordinary porous medium the average equations of equilibrium flow through the porous medium which describe these asymptotics are obtained

1-D two-phase flow analysis for interlocking double layer counter flow mini-channel heat sink

Mini and micro-channel heat sinks with two-phase flow boiling are considered as one of the promising cooling methods due to their ability to efficiently manage high heat flux heat dissipation. However, it is characterized by a significant decrease in the heat transfer coefficient of the liquid deficient portion where the quality of flow is increased by the boiling in the channel. The heat transfer coefficient at which the wall dry-out occurs sharply decreases, which can cause a large temperature gradient in the direction of flow. Such a dry-out condition may correspond to the critical heat flux state, and a drastic decrease in cooling performance is inevitable. To alleviate the temperature gradient increase and to improve the cooling performance, the two-phase counter flow mini-channel heat sink with interlocking double layer structure has been proposed. The proposed mini-channel heat sink was designed based on the commercial IGBT module size and numerically analyzed by applying 1-D two-phase flow in each counter flow direction. Two-phase boiling heat transfer correlation and momentum equation were solved to calculate the pressure drop and analyze the cooling performance benefits of proposed counter flow heat sink. The applied analysis method was extended to the single-phase counter flow situation and the results were compared with two-phase flow case. From the results of the analysis, it was confirmed that the counter flow heat sink can obtain a more uniform temperature distribution than the conventional unidirectional heat sink. The non-uniformity of the temperature distribution due to the uneven flow rate of the parallel channels can be improved by applying the counter flow configuration.

An improved Multiphase Moving Particle Semi-implicit method in bubble rising simulations with large density ratios

Abstract In this study, an improved MMPS-CA method is developed against the inaccuracy of MMPS-CA (Continuous Acceleration) method in simulating bubble rising cases with large density ratios. A higher order Laplacian model is adopted to discretize pressure Poisson equation and a ECS source term with compressible form is adopted. A new multi-viscosity model is developed based on the Shepherd smoothing function, and the performances of different multi-viscosity models in bubble rising cases are investigated in detail. A higher order accuracy density smoothing scheme base on the Gaussian function is employed to avoid the unphysical invasions through the interface, and a dynamic displacement modification technique is developed to make the particle distribution more uniform in the calculation. Then several multiphase simulation cases including the stratified pool case, deformation of 2D square droplet case, single bubble rising case, two-phase Poiseuille flow case and the bubble rising benchmark case are employed to verify the robustness of the improved MMPS-CA method. Bubble pairs rising and coalescence cases are carried out to further verify the ability of the proposed method in simulating bubbles’ behaviors. Results of the improved MMPS method fit well enough to other numerical and experimental results. At the end of the paper, a 2D closed air-water separator case is carried out to investigate the robustness of the improved MMPS-CA method in general bubble rising simulations with large density ratios.

A discrete fracture model for two-phase flow in fractured porous media

Abstract A discrete fracture model on the basis of a cell-centered finite volume scheme with multi-point flux approximation (MPFA) is presented. The fractures are included in a d-dimensional computational domain as ( d − 1 )-dimensional entities living on the element facets, which requires the grid to have the element facets aligned with the fracture geometries. However, the approach overcomes the problem of small cells inside the fractures when compared to equi-dimensional models. The system of equations considered is solved on both the matrix and the fracture domain, where on the prior the fractures are treated as interior boundaries and on the latter the exchange term between fracture and matrix appears as an additional source/sink. This exchange term is represented by the matrix-fracture fluxes, computed as functions of the unknowns in both domains by applying adequate modifications to the MPFA scheme. The method is applicable to both low-permeable as well as highly conductive fractures. The quality of the results obtained by the discrete fracture model is studied by comparison to an equi-dimensional discretization on a simple geometry for both single- and two-phase flow. For the case of two-phase flow in a highly conductive fracture, good agreement in the solution and in the matrix-fracture transfer fluxes could be observed, while for a low-permeable fracture the discrepancies were more pronounced. The method is then applied two-phase flow through a realistic fracture network in two and three dimensions.