Transport Equation For Volume Fraction Research Articles

An adaptive-correction algorithm for suppressing interface smearing in incompressible multiphase flows with complex interfacial behavior

In this paper, a numerical framework for modelling multiphase flows with interface correction is proposed. In original Volume-of-Fluid method, the interface smearing may appear since the interface profile deviates from its equilibrium state, especially in the simulations of complex interface deformation. To solve this problem, a novel correction step is introduced into the procedure after solving the volume fraction transport equation, and this step just works on the region of phase interface. The difference between the current method and the previous work is that the correction coefficient can be adjusted adaptively with the gradient of normal velocity at the interface, which is strongly correlated to the intensity of interface smearing and changes with the computational time and interface position. The validation computations are performed for bubble deformation (initial shapes of two-dimensional four-lobed-star and three-dimensional (3D) box with holes), bubble rising in a channel and Rayleigh–Taylor instability problems (Reynolds number of 100 and 1000). The obtained results show that the unphysical phenomenon of interface smearing is suppressed effectively, and the interface sharpness is improved greatly by the present method. In addition, the mass of bubble deformation by the original method will decrease by 13.3%. While these results obtained by the present method are in good agreement with the analytical solutions or published data.

Numerical model of an ultrasonically induced cavitation reactor and application to heavy oil processing

This study describes a numerical approach to model ultrasonically induced cavitation (UIC) reactors. UIC forms vapour-filled cavities in a liquid medium due to an applied acoustic field and their eventual collapse. UIC reactors are characterized by the presence of a vibrating probe that generates pressure waves by high-frequency oscillations (>20 kHz), which control the formation, dynamics, and eventual collapse of the vapour cavities. Those vapour cavities eventually enhance mixing and favour the occurrence of gas-liquid reactions. The zones of high mixing and reactivity coincide with the presence of the bubble cloud, which depends on the shape of the vessel and sonotrode. The development of advanced computational fluid dynamics (CFD) models is crucial to optimizing UIC processes’ geometry and operation parameters. A new algorithm for modelling UIC has been implemented within the OpenFOAM framework in the present study. The volume-of-fluid (VoF) method employs a diffuse interface approach for the volume fraction transport equation. The bubble dynamics are solved with sub-grid models, and the coupling between the main flow field and the sub-grid scales is performed through source terms in the transport equations. The source terms are de-coupled from convective and diffusive components of the volume fraction equation. The history of the bubbles is considered to consist of nucleation, oscillations, and collapse. The oscillations are resolved via the Rayleigh–Plesset equation. The concluding part of the work demonstrates the application of the algorithm to simulate the operation of an UIC reactor, which was designed to desulfurize fuels using the oxidative (ODS) process.

Heat transfer study of a new hybrid photovoltaic/thermal direct absorption parabolic solar collector by two-phase Buongiorno model.

Solar energy is one of the cleanest sources of renewable energy that is easily accessible in vast geographical areas. As the efficiency of the common solar collectors is very low and is limited by the absorption properties of working fluid, enhancing the thermal performance of these collectors is one of the major challenges of developing parabolic solar thermal power plants. In recent decades, researches revealed that utilizing nanofluid as a novel working fluid has a dramatic effect on the thermophysical and optical properties of the fluid. In this study, the flow and temperature fields of water/magnetite and water/aluminum nanofluids are evaluated by solving the steady form of governing equations including conservation laws of mass, volume fraction transport equation, momentum equation, energy equation, and radiation transfer equation. Moreover, the two-phase Buongiorno model is utilized and Brownian motion, thermophoresis effects, and magnetophoresis movement are taken into account in the nanofluid simulation. The numerical results demonstrate that increasing nanofluid volume fraction and flow rate can increase the thermal performance of the collector tube. It is found that the thermal efficiencies reach maximum values of 151.03% and 158.58% for water/aluminum and water/magnetite nanofluids, respectively. Furthermore, increasing the volume fraction from 0 to 0.3% leads to rise of 24.41% and 21.36% in the maximum temperature of the collector. The effects of different parameters such as nanoparticle volume fraction, flow rate, and nanoparticle kind on the collector thermal and electrical efficiencies, thermal distribution, and entropy production have been studied.

Numerical simulation of unsteady cavitation in liquid hydrogen flows

An unsteady cavitation model in liquid hydrogen flow is studied in the context of compressible, two-phase, one-fluid inviscid solver. This is accomplished by applying three conservation laws for mixture mass, mixture momentum and total energy along with gas volume fraction transport equation, with thermodynamic effects. Various mass transfers between phases are utilised to study the process under consideration. A numerical procedure is presented for the simulation of cavitation due to rarefaction and shock waves. Attention is focused on cavitation in which the simulated fluid is liquid hydrogen in cryogenic conditions. Numerical results are in close agreement with theoretical solutions for several test cases. The current numerical results show that liquid hydrogen flow can be accurately modelled using an accurate inviscid approach to describe the features of thermodynamic effects on cavitation.

Dynamic behaviors of the turbulent cavitating flows based on the Eulerian and Lagrangian viewpoints

In order to investigate the structures of the cavitating flow, a volume fraction transport equation with a hybrid turbulence model has been used to simulate the dynamics of the cavitation phenomenon over a two-dimensional ClarkY hydrofoil (AoA=8°, σ=0.8, and Re=7·105). From the Eulerian viewpoint, the interactions of pressure, vortex structure, and volume fraction have been evaluated, and the results have been validated carefully with the experimental observations. Four different flow stages can be categorized accordingly based on the development of the attached cavity, trailing edge cavities, and re-entrant jet.Furthermore, the Finite-Time Lyapunov Exponent (FTLE) and the corresponding Lagrangian Coherent Structures (LCSs) have been used to separate dynamically distinct regions. Above the upper surface, the liquid flow captured by LCS A could travel along the cavity interface to the trailing edge. Similarly, the LCS C captures the liquid flow below the lower surface that can be attracted into the upper surface. From the corresponding particle tracking, these two flows meet near the trailing edge and mix together to form the re-entrant jet, which can be represented by the LCS B.The current study shows that the LCS approach together with the Eulerian method can help us to have better understandings of the cavitating flow. The Lagrangian analysis especially indicates the underlying flow physics about the mixing process and bubble growth and decline behaviors. Most of the previous related studies only focus on the flow above the upper surface. The LCSs shown in this study also emphasize the importance of the flow structure of the lower surface, which provides more insightful information for the flow control and is worth further investigation.

오픈 소스 라이브러리를 이용한 수치수조 구현 및 적용

In this paper, ship performance prediction solver was developed using open source computational fluid dynamics (CFD) libraries. The mass- and momentum-conservation equations and turbulent model with a wall function for the turbulent closer were considered. The volume fraction transport equation with a high-resolution interface capturing scheme were selected for free-surface simulation. The predicted wave pattern around KRISO container ship (KCS) using developed program showed good agreement against existing experimental data. For the revolution of a propeller in the propulsive test, general grid interface (GGI) library was used. The predicted propulsive performance showed 7 % difference against experimental data. Two-phase mixture model was developed to simulate a cavitation and applied to a propeller. The sheet cavitation on the propeller was predicted well. From results, the potential of the numerical tank developed by open source libraries was verified by applying it to KCS.

볼륨비 이송방정식의 소스항을 이용한 자유수면 유동 해석의 해 확산 감소

Accurate simulation of free-surface wave flows around a ship is very important for better hull-form design. In this paper, a computational fluid dynamics (CFD) code, termed SNUFOAM, which is based on the open source libraries, OpenFOAM, was developed to predict the wave patterns around a ship. Additional anti-diffusion source term for minimizing a numerical diffusion, which was caused by convection differencing scheme, was considered in the volume-fraction transport equation. The influence of the anti-diffusion source term was tested by applying it to free-surface wave flow around the Wigley model ship. In results, the band width of the volume fraction contours between 0.1 to 0.9 at the hull surface was narrowed by considering the anti-diffusion term.

Numerical Investigation of Binary Droplet Collisions in All Relevant Collision Regimes

A numerical investigation of binary droplet collisions in a gaseous phase has been conducted. A volume of fluid (VOF) based interface capturing method, which is characterized by introducing an extra artificial compression term into the volume-fraction transport equation, is employed to capture the liquid-gas interface. The full Navier-Stokes equations coupled with the volume-fraction transport equation are discretized on a fixed, uniform grid using the finite volume method. The solution of the volume-fraction transport equation is bounded by an explicit universal multidimensional limiter. The simulations cover five major regimes of binary droplet collisions: (I) coalescence with minor deformation, (II) bouncing, (III) coalescence with major deformation, (IV) reflexive separation and (V) stretching separation. A ghost cell method is introduced in order to simulate bouncing and retarded coalescence in head-on collisions. Especially, for retarded coalescence collisions, the volume-fraction boundary condition on the symmetry plane is switched to numerically control coalescence at a critical time which is determined by tracking the interface front. The simulated results are compared with the attainable experimental data and provide detailed pictures exhibiting the droplet collision processes. The inter-droplet pressure, defined by the static pressure on the symmetry plane in head-on collisions, is also evaluated and analyzed.

Implementation of phase change thermodynamic probability for unsteady simulation of cavitating flows

AbstractThe aim of this work is to investigate the non‐equilibrium effects of phase change in cavitating flows. For this purpose, the concept of phase change thermodynamic probability is used along with homogeneous model to simulate two‐phase cavitating flows. For simulation of unsteady behaviors of cavitation, which have practical applications, unsteady PISO algorithm based on the non‐conservative approach is utilized. For multi‐phase simulation, single‐fluid Navier–Stokes equations, along with the volume fraction transport equation, are employed.In this paper, phase change thermodynamics probabilities and cavitation model is briefly summarized. Thus, derivation of the cavitation model, starting from the basic thermodynamic equations to the mass and momentum conservation equations at a liquid–vapor two‐phase flow, is presented to explain the numerical model. Unsteady simulations of cavitation around a flat plate normal to flow direction are presented to clarify the accuracy of the model. The accuracy of the numerical results is good, and it is possible to apply this method to more complex geometries. Copyright © 2010 John Wiley & Sons, Ltd.

LARGE EDDY SIMULATION OF LIQUID JET ATOMIZATION

In the past decade, numerical methods for two-phase flows, which are now capable of fully describing both phases and the interface, have progressed to a state of maturity. At present, direct numerical simulations (DNS) are possible even for complex configurations such as atomization processes. However, the numerical resources necessary to completely compute the atomization up to the formation of the final spray are still out of reach. Therefore, the LES approach is very appealing for two-phase flows. Depending on the assumption concerning the relative size of the LES filter compared to the characteristic scale of interface wrinkles, two classes of LES methods have emerged: the first class of LES method is required close to the injector while the second class is needed far from the injector when the spray is formed. The present work describes a modeling proposal that solves this problem by focusing on the transition between these two classes of method for LES of two phase flows. A postulated transport equation of the surface density is used to describe the subgrid spray formation ranging from interface wrinkling, ligaments, and sheets up to the droplets. Issued from RANS formalisms, this transport equation extends the usual spherical geometry given to the liquid structures that compose the subgrid spray. This equation contains several source terms that act on the creation and destruction of the liquid/gas interface. Considering that major breakup processes are due to turbulence in the field near the injector, an appropriate source term is proposed. This source term, also derived from RANS models, introduces a local critical value of surface density related to a critical Weber number that provides information concerning the local equilibrium between turbulent kinetic energy and surface energy. Concerning the velocity field, a standard Smagorinsky approach is considered. The initial results, obtained through the simulation of the atomization of a liquid jet, confirm that the subgrid term of the liquid volume fraction transport equation determines the transition between the resolved interface and the subgrid spray. A LES model for continuous two-phase flows, termed CLES in this paper, is proposed. The CLES model allows for a better description of the liquid dispersion within the calculation domain from both qualitative and quantitative points of view when compared with the usual LES formalism that does not consider subgrid terms for the interfacial contribution. Mean liquid volume fraction profiles are well captured with CLES when compared to '{\it a posteriori}' DNS simulations, and only a weak grid dependency is observed. Grid dependency is generally a major drawback when carrying out LES of interfacial flows. The surface density and behavior are well estimated and described by using a dedicated transport equation. Based on this parameter and thanks to a good estimation of the critical Weber number value, relevant characteristic scales of the subgrid spray in the atomization regime can be obtained.

Two‐fluid model with interface sharpening

AbstractTwo‐fluid models are applicable for simulations of all types of two‐phase flows ranging from separated flows with large characteristic interfacial length scales to highly dispersed flows with very small characteristic interfacial length scales. The main drawback of the two‐fluid model, when used for simulations of stratified flows, is the numerical diffusion of the interface. Stratified flows can be easily and more accurately solved with interface tracking methods; however, these methods are limited to the flows, that do not develop into dispersed types of flows. The present paper describes a new approach, where the advantage of the two‐fluid model is combined with the conservative level set method for interface tracking. The advection step of the volume fraction transport equation is followed by the interface sharpening, which preserves the thickness of the interface during the simulation. The proposed two‐fluid model with interface sharpening was found to be more accurate than the existing two‐fluid models. The mixed flow with both: stratified and dispersed flow, is simulated with the coupled model in this paper. In the coupled model, the dispersed two‐fluid model and two‐fluid model with interface sharpening are used locally, depending on the parameter which recognizes the flow regime. Copyright © 2010 John Wiley & Sons, Ltd.

Implementation of PISO algorithm for simulating unsteady cavitating flows

The present study focuses on the implementation of PISO algorithm to simulate cavitating flows. For simulation of unsteady behaviors of cavitation which have practical applications, the development of unsteady PISO algorithm based on the non-conservative approach is investigated. The effects of mixture compressibility are considered to improve accuracy of simulations. For multi-phase simulation, single-fluid Navier–Stokes equations, along with the volume fraction transport equation, are employed. The bubble dynamics model is utilized to simulate phase change. To prove capabilities of the developed PISO algorithm to simulate cavitating flows, unsteady simulation of cavitation around NACA0015 hydrofoil, a two dimensional flat plat, and a three dimensional circular disk are performed. The frequency of flow, pressure distributions, cavitating vortex shedding, and cavity characteristics are analyzed to discern results accuracy. To investigate accuracy of results, comparisons with available published experimental data are made and good agreement is achieved.

Drag force on a flat plate in cavitating flows

The present study focuses on the simulation of two dimensional unsteady cavitating flows. For simulation of unsteady behaviors of cavitation which have practical applications, the development of unsteady PISO algorithm based on the non-conservative approach is utilized. For multi-phase simulation, single-fluid Navier–Stokes equations, along with the volume fraction transport equation, are employed. The bubble dynamics model is utilized to simulate phase change between vapor and liquid phases of the water. Unsteady simulation of cavitation around NACA66(MOD) and supercavitation around a flat plate normal to flow direction are performed to clarify accuracy of presented model. Numerical results and comparisons with experimental data are provided. The accuracy is good, and it is possible to apply this method to more complex shapes.

Numerical simulation of three‐dimensional interfacial flows

PurposeThis study aims to present compatible computational fluid dynamics procedure for calculation of incompressible three‐dimensional time‐dependent flow with complicated free surface deformation. A computer software is developed and validated using a variety of academic test cases.Design/methodology/approachTwo fluids are modeled as a single continuum with a fluid property jump at the interface by solving a scalar transport equation for volume fraction. In conjunction, the conservation equations for mass and momentum are solved using fractional step method. Here, a finite volume discretisation and colocated arrangement are used.FindingsThe developed code results in accurate simulation of interfacial flows, e.g. Rayleigh‐Taylor instability, sloshing and dambreaking problems. All results are in good concordance with experimental data especially when there are two phases with high density ratio.Research limitations/implicationsTurbulence, which has great importance in a wide variety of real world phenomena, is not considered in the present formulation and left for future researches.Originality/valueHere, an integrated numerical simulation for transient interfacial flows is presented. In this way, the pressure integral term in Navier‐Stokes equation is discretised based on a newly developed interpolation which results in non‐oscillative velocity field especially in free surface.

Development of a VoF-fractional step solver for floating body motion simulation

Numerical simulation of floating or submerged body motions is presented based on a Volume of Fluid (VoF)-fractional step coupling. Solving a scalar transport equation for volume fraction of two phases results in a single continuum with a fluid property jump at the interface. In addition, velocity and pressure fields are coupled using the fractional step method. Based on integration of stresses over a body, acting forces and moments are calculated. Using the strategy of non-orthogonal body-attached mesh and calculation of motions in each time step result in time history of hydrodynamic motions. To verify the accuracy of the numerical procedure in simulation of two-phase flow, sloshing and dam breaking with obstacle problems are investigated. Besides, motions simulation strategy is evaluated by using a cylinder water entry test case. To demonstrate the capability of the simulation, barge resistance is calculated in two cases of fixed and free motion (2-DoF). All of the results are in good concordance with experimental data. The present method can be extended for full nonlinear motion of ships in waves.

A Pressure-Based Method for Turbulent Cavitating Flow Computations

A pressure-based algorithm is presented for turbulent cavitating flow computations. Single-fluid Navier–Stokes equations cast in their conservative form, along with a volume fraction transport equation, are employed. The flow field is computed in both phases with the vapor pressure recovered inside the cavity via a mass transfer model. A pressure–velocity–density coupling scheme is developed to handle the large density ratio associated with cavitation. While no temperature, and hence Mach number, effect is considered in the cavitation model, the resulting pressure–correction equation shares common features with that of high-speed flows, exhibiting a convective–diffusive type, instead of only a diffusive type. Furthermore, similar to high-speed cases, upwinded density interpolation in mass flux computations also aids convergence of the cavitating flow computations. The nonequilibrium effect in the context of the k-ε turbulence model, the grid distribution, and the choice of convection schemes have been computationally examined in projectile flows. While satisfactory predictions in wall pressure distribution can be made with variations in grid resolution and parameters in the cavitation model, other aspects, such as the density distribution and detailed streamline characteristics, are found to exhibit higher sensitivity to them.