Oil-jet lubrication and cooling of high-speed gears is frequently employed in aeronautical systems, such as novel high-bypass civil aero engines based on the geared turbofan technology. Using such oil-jet system, practitioners aim to achieve high cooling rates on the flanks of the highly thermally loaded gears with minimum oil usage. Thus, for an optimal design, detailed knowledge about the flow processes is desired. These involve the oil exiting the nozzle, the oil impacting on the gear teeth, the oil spreading on the flanks, the subsequent oil fling-off, as well as the effect of the design parameters on the oil flow. Better understanding of these processes will improve the nozzle design phase, e.g. regarding the nozzle positioning and orientation, as well as the nozzle sizing and operation.

Most related studies focus on the impingement depth to characterize the two-phase flow. However, the level of information of this scalar value is rather low for a complete description of the highly dynamic three-dimensional flow. Motivated by the advancements in numerical methods and the computational resources available nowadays, the investigation of the oil-jet gear interaction by means of computational fluid dynamics (CFD) has come into focus lately.

In this work, a numerical setup based on the volume-of-fluid method is presented and employed to investigate the two-phase flow phenomena occurring in the vicinity of the gear teeth. The setup consists of a single oil-jet impinging on a single rotating spur gear. By introducing new metrics for characterizing the flow phenomena, extensive use of the possibilities of modern CFD is made, allowing a detailed transient and spatially resolved flow analysis. Thus, not only the impingement depth, but also the temporal and spatial evolution of wetted areas on the gear flanks, as well as the evolution of the oil volume in contact with the gear flanks are extracted from the simulation data and compared in a CFD study.

The study consists of 21 different simulation cases, whereby the effect of varying the jet velocity, the jet inclination angle, the jet diameter, and the gear speed are examined. Consistent results compared to a simplified analytical approach for the impinging depth are obtained and the results for the newly introduced metrics are presented.

This paper presents topology optimization of capillary, the typical two-phase flow with immiscible fluids, where the level set method and diffuse-interface model are combined to implement the proposed method. The two-phase flow is described by the diffuse-interface model with essential no slip condition imposed on the wall, where the singularity at the contact line is regularized by the molecular diffusion at the interface between two immiscible fluids. The level set method is utilized to express the fluid and solid phases in the flows and the wall energy at the implicit fluid-solid interface. Based on the variational procedure for the total free energy of two-phase flow, the Cahn-Hilliard equations for the diffuse-interface model are modified for the two-phase flow with implicit boundary expressed by the level set method. Then the topology optimization problem for the two-phase flow is constructed for the cost functional with general formulation. The sensitivity analysis is implemented by using the continuous adjoint method. The level set function is evolved by solving the Hamilton-Jacobian equation, and numerical test is carried out for capillary to demonstrate the robustness of the proposed topology optimization method. It is straightforward to extend this proposed method into the other two-phase flows with two immiscible fluids.

In this article a comparison study of the numerical methods for compressible two-phase flows is presented. Although many numerical methods have been developed in recent years to deal with the jump conditions at the fluid-fluid interfaces in compressible multiphase flows, there is a lack of a detailed comparison of these methods. With this regard, the transport five equation model, the modified ghost fluid method and the cut-cell method are investigated here as the typical methods in this field. A variety of numerical experiments are conducted to examine their performance in simulating inviscid compressible two-phase flows. Numerical experiments include Richtmyer-Meshkov instability, interaction between a shock and a rectangle SF6 bubble, Rayleigh collapse of a cylindrical gas bubble in water and shock-induced bubble collapse, involving fluids with small or large density difference. Based on the numerical results, the performance of the method is assessed by the convergence order of the method with respect to interface position, mass conservation, interface resolution and computational efficiency.

A two-phase flow parametric study on an air induction nozzle with water and air as the working fluids is presented. Liquid was supplied at the pre-orifice with various inlet pressures ranging from 3 to 6 bar. The interaction between air and water at the molecular level at the orifice exit leads to formation of a strong shear layer that is intensified with the increase in inlet pressure. Thus, it is vital to regulate the ratio of the intake air to the supplied liquid so that the generated micro bubbles fit the design criteria. CFD analysis was conducted using the commercial software STAR CCM+ from Siemens and validated against experimental data to investigate the design parameters and their effect on the ALR. A volume of fluid (VOF) method of the RANS models was used to undertake the air-water interaction. Parameters such as the throat, air orifice, and air inlet diameter, along with the diffuser angle, were investigated. It was found that certain parameters such as the throat diameter have a more significant effect on the air/liquid entrainment ratio than other parameters.

In this paper, we present some efficient numerical schemes to solve a two-phase hydrodynamics coupled phase field model with moving contact line boundary conditions. The model is a nonlinear coupling system, which consists the Navier-Stokes equations with the general Navier Boundary conditions or degenerated Navier Boundary conditions, and the Allen-Cahn type phase field equations with dynamical contact line boundary condition or static contact line boundary condition. The proposed schemes are linear and unconditionally energy stable, where the energy stabilities are proved rigorously. Various numerical tests are performed to show the accuracy and efficiency thereafter.

Semi-Lagrangian (S-L) methods have no CFL stability constraint, and are more stable than the Eulerian methods. In the literature, the S-L method for the level-set re-initialization equation was complicated, which may be unnecessary. Since the re-initialization procedure is auxiliary, we propose to use the first-order S-L scheme coupled with a projection technique to improve the accuracy at the grid points just adjacent to the interface. Standard second-order S-L method is used for evolving the level-set convection equation. The implementation is simple, including on the block-structured adaptive mesh. The efficiency of the S-L method is demonstrated by extensive numerical examples including passive convection of interfaces with corners/kinks/large deformation under given velocity fields, a geometrical flow with topological changes, simulations of bubble/ droplet dynamics in incompressible two-phase flows. In terms of accuracy it is comparable to the other existing methods.

A method based on the Eulerian two-phase flow theory to numerically simulate three-dimensional rime ice accretions on an aircraft wing is presented in this paper. The governing equations for supercooled droplet motion under Eulerian framework are established using the droplet pseudo-fluid model. A permeable wall boundary condition is proposed to simulate the phenomenon of droplets impinging on the wing in solving the governing equations for droplets. The local droplet collection efficiency is readily obtained from the droplet flowfield solution in the control volume adjacent to the wing surface. The rime ice accretion can be simulated under the assumption that the droplets freeze immediately as they impinge on the wing surface since the environment temperature is low enough (typically below –15°C). A method to build the ice shape is proposed based on the assumption that ice grows in the direction normal to the wing surface. The rime ice accretion on a GLC-305 swept wing model under some specific conditions has been simulated to validate the present method. Furthermore, different flight conditions, namely, different angles of attack and different angles of sideslip, have been dealt with to investigate their effects on rime ice accretion as well as the corresponding aerodynamic effects.

In the first part, we study the convergence of discrete solutions to splitting schemes for two-phase flow with different mass densities suggested in [Guillen-Gonzalez, Tierra, J.Comput.Math. (6)2014]. They have been formulated for the diffuse interface model in [Abels, Garcke, Grün, M3AS, 2012, DOI:10.1142/S0218202511500138] which is consistent with thermodynamics. Our technique covers various discretization methods for phase-field energies, ranging from convex-concave splitting to difference quotient approaches for the double-well potential. In the second part of the paper, numerical experiments are presented in two space dimensions to identify discretizations of Cahn-Hilliard energies which are ϕ-stable and which do not reduce the acceleration of falling droplets. Finally, 3d simulations in axial symmetric geometries are shown to underline even more the full practicality of the approach.

We develop a numerical method to simulate a two-phase compressible flow with sharp phase interface on Eulerian grids. The scheme makes use of a levelset to depict the phase interface numerically. The overall scheme is basically a finite volume scheme. By approximately solving a two-phase Riemann problem on the phase interface, the normal phase interface velocity and the pressure are obtained, which is used to update the phase interface and calculate the numerical flux between the flows of two different phases. We adopt an aggregation algorithm to build cell patches around the phase interface to remove the numerical instability due to the breakdown of the CFL constraint by the cell fragments given by the phase interface depicted using the levelset function. The proposed scheme can handle problems with tangential sliping on the phase interface, topological change of the phase interface and extreme contrast in material parameters in a natural way. Though the perfect conservation of the mass, momentum and energy in global is not achieved, it can be quantitatively identified in what extent the global conservation is spoiled. Some numerical examples are presented to validate the numerical method developed.

The issue on the muzzle blast flow with gas-particle mixtures was numerically investigated in this paper. The propellant gas in the cannon was assumed to be gas-particle mixtures consisting of a variety of gaseous species and particles. The model made use of the Eulerian-Eulerican approach, where the particle were modeled as a second fluid with parameters like bulk density, velocity and temperature, interacting with the gas flow. A high-resolution upwind scheme(AUSMPW+) and detailed reaction kinetics model were employed to solve the chemical non-equilibrium Euler equations for gas phase. The Euler equations for particle phase were solved by MacCormack scheme. The particle diameter and the mass fraction of particle were tested to show their effects on the development process of muzzle blast flow with gas-particle mixtures. The distribution of the main flow parameters of both gas and particle were obtained at different time intervals. The results show the evolution of the muzzle blast flow with gas-particle mixtures and demonstrate the effects of key parameter on the flow field of the gas-particle flow. This paper is a significant investigation for understanding the muzzle blast flow with gas-particle mixtures, which can provide valuable reference for the research on the muzzle blast flow.

In this paper, we compute a phase field (diffuse interface) model of Cahn-Hilliard type for moving contact line problems governing the motion of isothermal multiphase incompressible fluids. The generalized Navier boundary condition proposed by Qian et al. [1] is adopted here. We discretize model equations using a continuous finite element method in space and a modified midpoint scheme in time. We apply a penalty formulation to the continuity equation which may increase the stability in the pressure variable. Two kinds of immiscible fluids in a pipe and droplet displacement with a moving contact line under the effect of pressure driven shear flow are studied using a relatively coarse grid. We also derive the discrete energy law for the droplet displacement case, which is slightly different due to the boundary conditions. The accuracy and stability of the scheme are validated by examples, results and estimate order.

Various conceptual models exist for numerical simulation of fluid flow in fractured porous media, such as dual-porosity model and equivalent continuum model. As a promising model, the discrete-fracture model has been received more attention in the past decade. It can be used both as a stand-alone tool as well as for the evaluation of effective parameters for the continuum models. Various numerical methods have been applied to the discrete-fracture model, including control volume finite difference, Galerkin and mixed finite element methods. All these methods have inherent limitations in accuracy and applicabilities. In this work, we developed a new numerical scheme for the discrete-fracture model by using mimetic finite difference method. The proposed numerical model is applicable in arbitrary unstructured gridcells with full-tensor permeabilities. The matrix-fracture and fracture-fracture fluxes are calculated based on powerful features of the mimetic finite difference method, while the upstream finite volume scheme is used for the approximation of the saturation equation. Several numerical tests in 2D and 3D are carried out to demonstrate the efficiency and robustness of the proposed numerical model.

Numerical simulation of two-phase flow in fractured karst reservoirs is still a challenging issue. The triple-porosity model is the major approach up to now. However, the triple-continuum assumption in this model is unacceptable for many cases. In the present work, an efficient numerical model has been developed for immiscible two-phase flow in fractured karst reservoirs based on the idea of equivalent continuum representation. First, based on the discrete fracture-vug model and homogenization theory, the effective absolute permeability tensors for each grid blocks are calculated. And then an analytical procedure to obtain a pseudo relative permeability curves for a grid block containing fractures and cavities has been successfully implemented. Next, a full-tensor simulator has been designed based on a hybrid numerical method (combining mixed finite element method and finite volume method). A simple fracture system has been used to demonstrate the validity of our method. At last, we have used the fracture and cavity statistics data from TAHE outcrops in west China, effective permeability values and other parameters from our code, and an equivalent continuum simulator to calculate the water flooding profiles for more realistic systems.

Two-phase flow and heat transfer, such as boiling and condensing flows, are complicated physical phenomena that generally prohibit an exact solution and even pose severe challenges for numerical approaches. If numerical solution time is also an issue the challenge increases even further. We present here a numerical implementation and novel study of a fully distributed dynamic one-dimensional model of two-phase flow in a tube, including pressure drop, heat transfer, and variations in tube cross-section. The model is based on a homogeneous formulation of the governing equations, discretized by a high resolution finite difference scheme due to Kurganov and Tadmore.

The homogeneous formulation requires a set of thermodynamic relations to cover the entire range from liquid to gas state. This leads a number of numerical challenges since these relations introduce discontinuities in the derivative of the variables and are usually very slow to evaluate. To overcome these challenges, we use an interpolation scheme with local refinement.

The simulations show that the method handles crossing of the saturation lines for both liquid to two-phase and two-phase to gas regions. Furthermore, a novel result obtained in this work, the method is stable towards dynamic transitions of the inlet/outlet boundaries across the saturation lines. Results for these cases are presented along with a numerical demonstration of conservation of mass under dynamically varying boundary conditions. Finally we present results for the stability of the code in a case of a tube with a narrow section.

We investigate the motion of two immiscible fluids in a porous medium described by a two-phase flow system. In the capillary pressure relation, we include static and dynamic hysteresis. The model is well established in the context of the Richards equation, which is obtained by assuming a constant pressure for one of the two phases. We derive an existence result for this hysteresis two-phase model for non-degenerate permeability and capillary pressure curves. A discretization scheme is introduced and numerical results for fingering experiments are obtained. The main analytical tool is a compactness result for two variables that are coupled by a hysteresis relation.

The lattice Boltzmann method (LBM) with an elastic model is applied to the simulation of two-phase flows containing a deformable body with a viscoelastic membrane. The numerical method is based on the LBM for incompressible two-phase fluid flows with the same density. The body has an internal fluid covered by a viscoelastic membrane of a finite thickness. An elastic model is introduced to the LBM in order to determine the elastic forces acting on the viscoelastic membrane of the body. In the present method, we take account of changes in surface area of the membrane and in total volume of the body as well as shear deformation of the membrane. By using this method, we calculate two problems, the behavior of an initially spherical body under shear flow and the motion of a body with initially spherical or biconcave discoidal shape in square pipe flow. Calculated deformations of the body (the Taylor shape parameter) for various shear rates are in good agreement with other numerical results. Moreover, tank-treading motion, which is a characteristic motion of viscoelastic bodies in shear flows, is simulated by the present method.

Free energy lattice Boltzmann methods are well suited for the simulation of two phase flow problems. The model for the interface is based on well understood physical grounds. In most cases a numerical interface is used instead of the physical one because of lattice resolution limitations. In this paper we present a framework where we can both follow the droplet behavior in a coarse scale and solve the interface in a fine scale simultaneously. We apply the method for the simulation of a droplet using an interface to diameter size ratio of 1 to 280. In a second simulation, a small droplet coalesces with a 42 times larger droplet producing on it only a small capillary wave that propagates and dissipates.

L’alimentation en air de l’habitacle d’une automobile se fait par l’intermédiairede la boîte à eau (BAO), réceptacle également destinaé recevoirl’eau de pluie ruisselant du pare-brise. Une meilleure connaissance des écoulements d’airet d’eau et des instabilités générées à l’intérieur de cette BAO est nécessaire pourassurer à la fois une bonne évacuation de l’eau vers l’extérieur et une alimentation enair sec, dépourvue de gouttelettes, vers l’intérieur. Les résultats tirés de cetteapproche expérimentale doivent permettre de faire évoluer les modèles numériques utilisésdans l’entreprise.

We consider a dense granular shear flow in a two-dimensional system. Granular systems(composed of a large number of macroscopic particles) are far from equilibrium due toinelastic collisions between particles: an external driving is needed to maintain themotion of particles. Theoretical description of driven granular media is especiallychallenging for dense granular flows. This paper focuses on a gravity-driven densegranular Poiseuille flow in a channel. A special focus here is on the intriguingphenomenon of fluid-solid coexistence: a solid plug in the center of the system,surrounded by fluid layers. To find and analyze various flow regimes, a multi-scaleapproach is taken. On macro scale, granular hydrodynamics is employed. On micro scale,event-driven molecular dynamics simulations are performed. The entire phase diagram ofparameters is explored, in order to determine which flow regime occurs in various regionsin the parameter space.