Adaptive Local Grid Refinement Research Articles

Numerical simulation of a silicon-based latent heat thermal energy storage system operating at ultra-high temperatures

This study investigates numerically a silicon-based latent heat storage system operating at ultra-high temperatures (∼1410–2000 °C). Owing to the silicon’s high latent heat (1230 kWh·m−3), storage densities of almost an order of magnitude higher than the state-of-the-art molten salt-based systems can be achieved. Prior to fabricating this system, there is a necessity to scrutinize the complex heat transfer mechanisms occurring during silicon phase change and indicate a vessel design that enables quick charge rates. A validated transient computational fluid dynamics model, combining the multiphase volume of fluid method with the enthalpy porosity approach and an adaptive local grid refinement technique is applied. This model simultaneously takes into account the phase-change material (PCM) volumetric change and the effect of dendritic formations and buoyancy-driven natural convection on the melting process. Numerical results reveal that the system charge rate is highly dependent on the PCM permeability, vessel design and operating conditions. Between five shapes investigated, i.e. cube, truncated cone, sphere, cut-off sphere and cylinder of volume V = 3.75E−03 m3, the truncated cone is the most preferable considering both design flexibility- its height can be easily altered, without increasing its volume- and melting rates. Actually, the PCM melting time achieved with the cone is 21% quicker compared to sphere and 6% slower compared to cube. Concerning the vessel size, the melting time increases by almost 28%, when the vessel volume increases to V′ = 2V. Finally, reduction in the silicon melting time, almost by 31%, is achieved by increasing the Stefan number by 35%.

A numerical study on droplet-particle collision dynamics

The impact of liquid droplets onto spherical stationary solid particles under isothermal conditions is simulated. The CFD model solves the Navier-Stokes equations in three dimensions and employs the Volume of Fluid Method (VOF) coupled with an adaptive local grid refinement technique able to track the liquid-gas interface. A fast-marching algorithm suitable for the quick computation of distance functions required during the grid refinement in large 3-D computational domains is proposed. The numerical model is validated against experimental data for the case of a water droplet impact onto a spherical particle at low We number and room temperature conditions. Following that, a parametric study is undertaken examining (a) the effect of Weber number (= ρu2Do/σ) in the range of 8 to 80 and (b) the droplet to particle size ratio ranging in-between 0.31 and 1.24, on the impact outcome. This has resulted to the identification of two distinct regimes that form during droplet-particle collisions: the partial/full rebound and the coating regimes; the latter results to the disintegration of secondary satellite droplets from elongated expanding liquid ligaments forming behind the particle. Additionally, the temporal evolution of variables of interest, such as the maximum dimensionless liquid film thickness and the average wetting coverage of the solid particle by the liquid, have been quantified. The present study assists the understanding of the physical processes governing the impact of liquids onto solid spherical surfaces occurring in industrial applications, including fluid catalytic cracking (FCC) reactors.

Predicting the evaporation rate of stationary droplets with the VOF methodology for a wide range of ambient temperature conditions

This paper presents CFD predictions for the evaporation of nearly spherical suspended droplets for ambient temperatures in the range 0.56 up to 1.62 of the critical fuel temperature, under atmospheric pressures. The model solves the Navier-Stokes equations along with the energy conservation equation and the species transport equations; the Volume of Fluid (VOF) methodology has been utilized to capture the liquid-gas interface using an adaptive local grid refinement technique aiming to minimize the computational cost and achieve high resolution at the liquid-gas interface region. A local evaporation rate model independent of the interface shape is further utilized by using the local vapor concentration gradient on the droplet-gas interface and assuming saturation thermodynamic conditions. The model results are compared against experimental data for suspended droplet evaporation at ambient air cross flow including single- and multi-component droplets as well as experiments for non-convective conditions. It is proved that the detailed evaporation process under atmospheric pressure conditions can be accurately predicted for the wide range of ambient temperature conditions investigated.

Predicting droplet deformation and breakup for moderate Weber numbers

The present work examines numerically the deformation and breakup of free falling droplets subjected to a continuous cross flow. The model is based on the solution of the Navier–Stokes equations coupled with the Volume of Fluid (VOF) methodology utilized for tracking the droplet-air interface; an adaptive local grid refinement is implemented in order to decrease the required computational cost. Neglecting initially the effect of the vertical droplet motion, a 2D axisymmetric approximation is adopted to shed light on influential numerical parameters. Following that, 3D simulations are performed which include inertial, surface and gravitational forces. The model performance is assessed by comparing the results against published experimental data for the bag breakup and the sheet thinning breakup regimes. Furthermore, a parametric study reveals the model capabilities for a wider range of Weber numbers. It is proved that the model is capable of capturing qualitatively the breakup process, while the numerical parameters that best predict the experimental data are identified.

Numerical investigation of aerodynamic droplet breakup in a high temperature gas environment

The Navier–Stokes equations, energy and vapor transport equations coupled with the VOF methodology and a vaporization rate model are numerically solved to predict aerodynamic droplet breakup in a high temperature gas environment. The numerical model accounts for variable properties and uses an adaptive local grid refinement technique on the gas–liquid interface to enhance the accuracy of the computations. The parameters examined include Weber (We) numbers in the range 15–90 and gas phase temperatures in the range 400–1000K for a volatile n-heptane droplet. Initially isothermal flow conditions are examined in order to assess the effect of Weber (We) and Reynolds (Re) number. The latter was altered by varying the gas phase properties in the aforementioned temperature range. It is verified that the We number is the controlling parameter, while the Re number affects the droplet breakup at low We number conditions. The inclusion of droplet heating and evaporation mechanisms has revealed that heating effects have generally a small impact on the phenomenon due to its short duration except for low We number cases. Droplet deformation enhances heat transfer and droplet evaporation. An improved 0-D model is proposed, able to predict the droplet heating and vaporization of highly deformed droplets.

Coupling a local adaptive grid refinement technique with an interface sharpening scheme for the simulation of two-phase flow and free-surface flows using VOF methodology

This study presents the implementation of an interface sharpening scheme on the basis of the Volume of Fluid (VOF) method, as well as its application in a number of theoretical and real cases usually modelled in literature. More specifically, the solution of an additional sharpening equation along with the standard VOF model equations is proposed, offering the advantage of “restraining” interface numerical diffusion, while also keeping a quite smooth induced velocity field around the interface. This sharpening equation is solved right after volume fraction advection; however a novel method for its coupling with the momentum equation has been applied in order to save computational time. The advantages of the proposed sharpening scheme lie on the facts that a) it is mass conservative thus its application does not have a negative impact on one of the most important benefits of VOF method and b) it can be used in coarser grids as now the suppression of the numerical diffusion is grid independent. The coupling of the solved equation with an adaptive local grid refinement technique is used for further decrease of computational time, while keeping high levels of accuracy at the area of maximum interest (interface). The numerical algorithm is initially tested against two theoretical benchmark cases for interface tracking methodologies followed by its validation for the case of a free-falling water droplet accelerated by gravity, as well as the normal liquid droplet impingement onto a flat substrate. Results indicate that the coupling of the interface sharpening equation with the HRIC discretization scheme used for volume fraction flux term, not only decreases the interface numerical diffusion, but also allows the induced velocity field to be less perturbed owed to spurious velocities across the liquid–gas interface. With the use of the proposed algorithmic flow path, coarser grids can replace finer ones at the slight expense of accuracy.

An adaptive local grid refinement method for 2D diffusion equation with variable coefficients based on block-centered finite differences

In this paper, an adaptive local grid refinement method based on block-centered finite differences is proposed for 2D diffusion equation with Neumann boundary condition. The method first identifies the regions with large local error, then these regions are divided until a prescribed tolerance is satisfied. The numerical solutions of unknown variable along with its first derivatives are obtained simultaneously. Theoretical analysis shows that the proposed method is second order accurate both on uniform and non-uniform meshes. Some high gradient problems are carried out to verify the efficiency and reliability of the adaptive local grid refinement method.

The effect of Weber number on the central binary collision outcome between unequal-sized droplets

The central binary collision of two unequal sized droplets is numerically investigated using the volume of fluid (V.O.F.) methodology. The numerical method based on the solution of the continuity and momentum equations in axi-symmetric formulation is coupled with a recently developed adaptive local grid refinement technique, thus allowing an accurate representation of the interface between the liquid and gas phase. Mass transfer mechanisms are reproduced by solving a transport equation for a colour function representing the mass of one of the colliding droplets before and after collision and mixing. The investigation is performed assuming either constant relative velocity of the colliding droplets or constant total energy of the system, thus creating a combination of the standard non-dimensional parameters affecting the collision process, i.e. Weber (We) and Ohnesorge (Oh) numbers as also droplet diameter ratio (Δ). The reliability of the procedure is first established by comparing predictions with available experimental data. The effect of the above mentioned parameters on ligament’s formation, maximum deformation of the two droplets, the penetration of one droplet into the other and satellite droplet formation is quantified.

Numerical investigation of the evaporation of two-component droplets

A numerical model for the complete thermo-fluid-dynamic and phase-change transport processes of two-component hydrocarbon liquid droplets consisting of n-heptane, n-decane and mixture of the two in various compositions is presented and validated against experimental data. The Navier–Stokes equations are solved numerically together with the VOF methodology for tracking the droplet interface, using an adaptive local grid refinement technique. The energy and concentration equations inside the liquid and the gaseous phases for both liquid species and their vapor components are additionally solved, coupled together with a model predicting the local vaporization rate at the cells forming the interface between the liquid and the surrounding gas. The model is validated against experimental data available for droplets suspended on a small diameter pipe in a hot air environment under convective flow conditions; these refer to droplet’s surface temperature and size regression with time. An extended investigation of the flow field is presented along with the temperature and concentration fields. The equilibrium position of droplets is estimated together with the deformation process of the droplet. Finally, extensive parametric studies are presented revealing the nature of multi-component droplet evaporation on the details of the flow, the temperature and concentration fields.

The effect of gas and liquid properties and droplet size ratio on the central collision between two unequal-size droplets in the reflexive regime

A numerical investigation of various parameters affecting the collision process of two unequal sized droplets in a gaseous environment in the reflexive regime is presented. The investigation is based on the finite volume numerical solution of the Navier–Stokes equations, in their axial-symmetric formulation, expressing the flow field of the two phases, liquid and gas, coupled with the volume of fluid method (VOF) for tracking the liquid–gas interfaces; a recently developed adaptive local grid refinement technique is used in order to track more accurately the liquid–gas interface with reduced computational cost. A colour function is used in order to follow the mixing process of the two droplets after collision. The reliability of the procedure is first established by comparing predictions with available experimental data and important details as regards the time evolution of the deformed shapes, the ligament formation, the maximum deformation of the droplets and the penetration of one droplet into the other are presented. The collision process and its outcome are investigated having the liquid properties (Ohnesorge number based on the liquid properties), the gas phase properties (Reynolds number based on gas properties) and droplet size ratio as the controlling parameters.

A PARAMETRIC NUMERICAL STUDYOFTHE HEAD-ON COLLISION BEHAVIOR OF DROPLETS

Head-on collision of droplets consisting of different liquids for a wide range of Weber numbers between 25 and 430 is studied numerically using the volume of fluid (VOF) methodology and by adopting an axisymmetric approximation. An adaptive local grid refinement technique is used to capture with precision the liquid–air interface at significantly reduced computational cost relative to a uniformly dense grid. The conditions setting the boundary between permanent coalescence and reflexive separation are defined; in addition, useful relationships determining the ligament’s elongation, the corresponding time of its breakup, the number of satellite droplets formed, the maximum droplet deformation, and the viscous dissipation coefficient are presented. Furthermore, the present study reveals the dependence of collision characteristics not only on Weber number, but also either on Reynolds, capillary, or Ohnesorge numbers.

SINGLE DROPLET IMPACTS ONTO DEPOSITED DROPS. NUMERICAL ANALYSIS AND COMPARISON

The impact of a spherical water droplet onto a stationary sessile droplet lying on a solid wall is studied numerically using the volume-of-fluid methodology. The governing Navier-Stokes equations are solved both for the gas and liquid phase coupled with an additional equation for the transport of the liquid interface. An unstructured numerical grid is used along with an adaptive local grid refinement technique, which enhances the accuracy of the numerical results along the liquid-gas interface and decreases the computational cost. The stationary sessile droplet has been created from the prior impact of one or two water droplets falling onto the solid wall, while two solid walls have been studied−an aluminum substrate and a glass substrate. The material of the wall plays an important role because it has an impact on the droplet's wetting behavior. The numerical model is validated against corresponding experimental data presented in the first part of the present work (Nikolopoulos et al., 2010), showing good agreement. Furthermore, the numerical investigation sheds light on the governing physics of the phenomenon.

Off-centre binary collision of droplets: A numerical investigation

The paper presents results from a numerical investigation of the non-central binary collision of two equal size droplets in a gaseous phase. The flow field is two phase and three dimensional; the investigation is based on the finite volume numerical solution of the Navier–Stokes equations, coupled with the Volume of Fluid Method (VOF), expressing the unified flow field of the two phases, liquid and gas. A recently developed adaptive local grid refinement technique is used, in order to increase the accuracy of the solution particularly in the region of the liquid–gas interface. The reliability of the solution procedure is tested by comparing predictions with available experimental data. The numerical results predict the collision process of the two colliding droplets (permanent coalescence or separation) and in the case of separation the formation and the size of the satellite droplets. The time evolution of the geometrical characteristics of the ligament, for various Weber numbers and impact parameters, is calculated and details are shown of the velocity and pressure fields particularly at the ligament pinch off location not hitherto available. Gas bubbles due to collision are trapped within the liquid phase as it has also been observed in experiments and their volume is calculated.

A numerical investigation of central binary collision of droplets

The paper presents a numerical investigation of the central collision of two equal-sized droplets in a gaseous phase. The investigation is based on the numerical solution of the Navier–Stokes equations in their axi-symmetric form using the finite volume technique. The Volume of Fluid Method (V.O.F) is employed for tracking the liquid–gas interface. An adaptive local grid refinement technique developed recently is used in order to increase the resolution around the interface. By using two V.O.F indicator functions the identity of each droplet is preserved and can be detected after droplet contact until coalescence. The results are compared with available experimental data and provide a very detailed picture of the collision process, the ligament formation and dimensions, the pinch off mechanism and the creation of the satellite droplet. The conversion of the droplet’s kinetic energy to the surface energy and vise versa, the energy viscous dissipation as well as the maximum deformation of the droplets are also evaluated.

Modelling of high-pressure dense diesel sprays with adaptive local grid refinement

An Eulerian–Lagrangian fluid dynamics model simulating the development of dense liquid plumes formed during injection of fuels against compressed air is described and assessed against experimental data. The numerical model employs an adaptive local grid refinement methodology combined with a calculation procedure distributing the mass, momentum and energy exchanged between the liquid and gaseous phases in the numerical cells found in the vicinity of the moving droplets. The use of appropriate weighting functions resolves numerical as well as physical problems realised when the interaction volume available between the two phases is limited to the cell-containing parcel, whose volume may become comparable to that of the dispersed phase. Calculation of ‘virtual’ cell properties provide better estimates for the flow variables realised by the droplets crossing cells in the wake of those upstream and allows for larger time steps to be employed in the solution of the carrier phase conservation equations. The results suggest that the proposed methodology offers significant improvements compared to the standard Lagrangian one frequently adopted in simulation of combustion systems, without the need to use Eulerian flow models in dense spray regions.

An adaptive local grid refinement and peak/valley capture algorithm to solve nonlinear transport problems with moving sharp-fronts

Highly nonlinear advection–dispersion-reaction equations govern numerous transport phenomena. Robust, accurate, and efficient algorithms to solve these equations hold the key to the success of applying numerical models to field problems. This paper presents the development and verification of a computational algorithm to approximate the highly nonlinear transport equations of reactive chemical transport and multiphase flow. The algorithm was developed based on the Lagrangian-Eulerian decoupling method with an adaptive ZOOMing and Peak/valley Capture (LEZOOMPC) scheme. It consists of both backward and forward node tracking, rough element determination, peak/valley capturing, and adaptive local grid refinement. A second-order tracking was implemented to accurately and efficiently track all fictitious particles. Shanks’ method was introduced to deal with slowly converging case. The accuracy and efficiency of the algorithm were verified with the Burger equation for a variety of cases. The robustness of the algorithm to achieve convergent solutions was demonstrated by highly nonlinear reactive contaminant transport and multiphase flow problems.

Three-dimensional numerical investigation of a droplet impinging normally onto a wall film

The paper presents a three-dimensional numerical investigation of a droplet impinging normally onto a wall film. The numerical method is based on the finite volume solution of the Navier–Stokes equations coupled with the volume of fluid method (VOF) and utilizing an adaptive local grid refinement technique for tracking more accurately the liquid–gas interface. The results are compared with available experimental data for integral quantities such as the lamella temporal development. Two mechanisms are identified leading to secondary droplet formation; in the initial and intermediate stages of splashing secondary droplet formation is according to Rayleigh instability while at later times surface tension effects contribute further to secondary atomization. Moreover, the influence of Weber number on the impingement process is investigated and correlations for the diameter and number of secondary droplets are proposed.

Towards Hybrid LES-RANS-Coupling for Complex Flows with Separation

The paper is concerned with a new hybrid LES–RANS approach, which splits up the simulation into a near–wall RANS part and an outer LES part, both modes relying on a one–equation model for the turbulent kinetic energy. This model has been tested on the standard plane channel flow test case as well as on the flow over a periodic arrangement of hills. Encouraging results were achieved. In an additional study based on the detached–eddy simulation (DES) concept attempts have been made to improve the accuracy of the simulation by using adaptive local grid refinement. For this purpose, a criterion based on the residual of the budget of the mean momentum equations has been studied.

An adaptive multigrid approach for the simulation of contaminant transport in the 3D subsurface

This paper presents an adaptive multigrid approach, combining adaptive local grid refinement and multigrid methods, in conjunction with the Lagrangian–Eulerian finite element method to simulate contaminant transport in the 3D subsurface. Adaptive local grid refinement can improve solution accuracy without sacrificing computational efficiency because computer efforts are focused on the rough regions (i.e., requiring high spatial resolution) of the problem domain. To implement adaptive grids, a backward/forward particle tracking technique is applied in the Lagrangian step, and the interpolation errors of the Lagrangian concentrations are compared with prescribed error tolerances to determine rough regions. A modular setting of the grid generation is then used to generate locally zooming grids and to prepare information for applying multigrid methods. The Lagrangian concentrations of the newly generated nodes at the finest grid level are also evaluated by performing a backward tracking. Multigrid strategies which can effectively eliminate the smooth component error through coarse grid correction are finally applied in the Eulerian step to solve the matrix equations for further saving of computer time. Example problems are used to demonstrate the success of this integrated approach.

Incomplete Multilevel Cholesky Factorizations

Adaptive in-time local grid refinement techniques use multilevel local discretizations designed to achieve local accuracy. The changing nature of the matrix structure of the linear systems arising from the multilevel local discretizations requires flexible approximate factorizations that focus on local components and coordinate their interaction. The solution of these composite grid systems with Krylov solvers is considered. The selection of an adequate preconditioner is crucial. The incomplete Cholesky (IC) factorization of the composite matrix and the inexact BEPS preconditioner are two such potential preconditioners. The inexact BEPS preconditioner can be constructed and applied with significantly more flexibility than the IC factorizations of the composite multilevel grid matrix. An extension of the IC factorizations for matrices arising from discretizations of self-adjoint PDEs on multilevel composite grids is proposed. The resulting factorization is referred to as the incomplete multilevel Cholesky (IMC) factorization. The IMC factorization is spectrally equivalent to the IC factorization of the matrix of the composite grid system. IMC factorization can be constructed with the same flexibility as the inexact BEPS preconditioner. The application of IMC factorization is achieved via a multilevel LLT-cycle consisting of a forward elimination pass proceeding downward on the grids from fine to coarse followed by an reverse-order upward back substitution pass. The application of multilevel factorization as a preconditioner requires roughly one-half as many operations as the inexact BEPS preconditioner. The numerical results illustrate the potential of the method.