On the Successive Impingement of Droplets Onto a Substrate

Numerical simulations of successive impingement of water droplets onto a substrate have been performed. The objective is to understand the hydrodynamics of the impingement process, particularly the interaction between successive droplets. The Navier-Stokes equations are solved using a finite-volume formulation. A two-step projection method is used along with a fixed, nonuniform, staggered rectangular grid. The free surface of the liquid is tracked by the volume-of-fluid method with a second-order-accurate piecewise-linear scheme. A continuum surface force model is used to calculate the surface tension force. The numerical results are in general agreement with experimental data.     

A sharp-interface level-set method for analysis of Marangoni effect on microdroplet evaporation

A level-set method is presented for computation of microdroplet evaporation including not only the effects of heat and mass transfer, phase change and contact line dynamics but also the Marangoni effect, which is a key parameter affecting the internal flow of the droplet and the particle deposition pattern. A sharp-interface formulation of the Marangoni force is derived and tested for two-phase Marangoni convection in a cavity. The computed results show good convergence in both the liquid and gas regions and are in excellent agreement with the analytical solutions. The level-set formulation is applied to microdroplet evaporation on a solid surface to investigate the Marangoni effect.     

FINITE ELEMENT ANALYSIS OF COUPLED HEAT AND MOISTURE TRANSFER IN CONCRETE SUBJECTED TO FIRE

A system of coupled transient differential equations governing heat, mass transfer, and pore pressure built up in porous media (concrete), subjected to intensive heating, is derived. Water vapor and liquid water are considered separately in the mass transfer formulation. The primary unknowns are temperature, water vapor content, and pore pressure of the gaseous mixture. A finite element formulation and corresponding flowchart of computations of all required data are presented. The numerical example solved represents a cross section of a concrete column exposed to fire. The domain and time distributions of temperature, pore pressure, water vapor, and liquid water content are presented. Computed pore pressure is higher than those usually reported by other analytical studies. The influence of some initial parameters (permeability, initial water content, and porosity) on maximum pore pressure is investigated.     

Computations of film boiling. Part I: numerical method

A numerical method for direct simulations of boiling flows is presented. The method is similar to the front tracking/finite difference technique of Juric and Tryggvason [Int. J. Multiphase Flow 24 (1998) 387], where one set of conservation equations is used to represent the mass transfer, heat transfer, and fluid flow in the liquid and the vapor, but improves on their numerical technique by elimination of their iterative algorithm. The justification of the mathematical formulation is presented and the numerical method and the code is validated by comparison of the results with the exact solutions of a few analytical problems. A grid refinement test for film boiling on a horizontal surface shows the convergence of results.     

Numerical simulation of a fountain flow on nonstaggered Cartesian grid system

A liquid–air fountain flow due to the downward motion of a rectangular sleeve over a stationary piston is studied in the paper. Two-dimensional incompressible laminar flows are assumed to prevail in both air and liquid regions. A single set of governing equations over the entire physical domain including the liquid, the air, and the liquid–air interface (free surface) is solved with the extended weighting function scheme and the NAPPLE (nonstaggered APPLE) algorithm on a fixed nonstaggered Cartesian grid system. To ensure the required dynamic contact angle, the liquid meniscus near the sleeve wall is corrected by solving the force balance equation with the geometry method. This is equivalent to introducing a slip condition at the contact line, and thus successfully removes the stress singularity. Steady state solution of the velocity and the pressure as well as the shape of the free surface is obtained. The numerical result evidences the existence of a toroidal-like motion on the free surface postulated by Dussan [E.B. Dussan V., Immiscible liquid displacement in a capillary tube: the moving contact line, AIChE J. 23 (1977) 131–133], although it is quite weak and thin. The resulting free surface profile agrees with the existing experimental observation excellently. Influence of the piston on the flow is discussed.     

A Simple Finite-Volume Formulation of the Lattice Boltzmann Method for Laminar and Turbulent Flows

A finite-volume formulation commonly employed in the well-known SIMPLE family algorithms is used to discretize the lattice Boltzmann equations on a cell-centered, non-uniform grid. The convection terms are treated by a higher-order bounded scheme to ensure accuracy and stability of solutions, especially in the simulation of turbulent flows. The source terms are linearized by a conventional method, and the resulting algebraic equations are solved by a strongly implicit procedure. A method is also presented to link the lattice Boltzmann equations and the macroscopic turbulence modeling equations in the frame of the finite-volume formulation. The method is applied to two different laminar flows and a turbulent flow. The predicted solutions are compared with the experimental data, benchmark solutions, and solutions by the conventional finite-volume method. The results of these numerical experiments for laminar flows show that the present formulation of the lattice Boltzmann method is slightly more diffusive than the finite-volume method when the same numerical grid and convection scheme are used. For a turbulent flow, the finite-volume lattice Boltzmann method slightly underpredicts the reattachment length in a separated flow. In general, the finite-volume lattice Boltzmann method is as accurate as the conventional finite-volume method in predicting the mean velocity and the pressure at the wall. These observations show that the present method is stable and accurate enough to be used in practical simulations of laminar and turbulent flows.     

SIMULATION OF METAL DROPLET DEPOSITION WITH SOLIDIFICATION INCLUDING UNDERCOOLING AND CONTACT RESISTANCE EFFECTS

A numerical procedure for the deformation and solidification of a metal droplet impinging on a flat surface is developed and a sample calculation is presented. A previously derived second-order ordinary differential equation (ODE) that approximates the splat as a cylinder and describes the droplet size evolution based on the mechanical energy equation in conjunction with kinematic and geometrical compatibility is used. The thermal energy equations for the liquid and solid regions of the splat and the substrate are separately solved coupled by boundary conditions such as contact resistance and undercooling in a regularized calculation domain produced by algebraic grid generation. The solidified layer thickness is calculated by solving a hyperbolic partial differential equation (PDE) resulting from the interface energy equation at the phase change boundary. Physical processes such as convective heat loss, substrate heat loss, viscous dissipation, and surface tension are modeled through appropriate nondimensional parameters.     

Numerical simulation of drop deformation and breakup in shear flow

Three‐dimensional numerical simulation of the deformation and breakup of an isolated liquid drop suspended in immiscible viscous fluid under shear flow was performed with diffuse interface method. The governing equations of the model were described by Navier– Stokes– Cahn– Hilliard equations. The surface tension was treated as a modified stress. In this paper, a uniform staggered Cartesian grid was used. The transient Navier– Stokes equations were solved by an approximation projection method based on pressure increment formulation, while the Cahn– Hilliard equations were solved by a nonlinear full approximation multigrid method. The numerical results of the drop deformation and breakup were in good agreement with the experimental measurements. Therefore, the present model could be perfectly applied to study the mechanism of drop deformation and breakup. © 2007 Wiley Periodicals, Inc. Heat Trans Asian Res, 36(5): 286– 294, 2007; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20160     

Conjugate heat and mass transfer in a cross-flow hollow fiber membrane contactor for liquid desiccant air dehumidification

The fluid flow and conjugate heat and mass transfer in a cross-flow hollow fiber membrane contactor are investigated. The shell-and-tube like contactor is used for liquid desiccant air dehumidification, where numerous fibers are packed into the shell and air flows across the fiber bank. To overcome the difficulties in the direct modeling of the whole contactor, a representative cell, which comprises of a single fiber, a liquid solution inside the fiber, and an air stream across the fiber, is selected as the calculation domain. The air stream in the cell is surrounded by an assumed outer free surface. The equations governing the fluid flow and heat and mass transfer in the two cross-flow streams are solved together with the heat and mass diffusion equations in the membrane. The friction factor and the Nusselt and Sherwood numbers on the air and stream sides are then calculated and experimentally validated.     

Numerical Analysis of Film Boiling in Liquid Jet Impingement

A numerical approach is presented for computing film boiling in liquid jet impingement on a high-temperature plate. The conservation equations of mass, momentum and energy are numerically solved in the liquid, vapor, and air phases. The sharp-interface level-set formulation is employed to track the liquid-air interface, as well as the liquid-vapor interface with phase change. A simplified analytical model for a thin vapor film, whose thickness is several orders of magnitude smaller than the liquid layer, is incorporated into the level-set formulation. The multiscale approach is tested through the computations of film boiling in a circular water jet.     

A study on transient ice formation of laminar flow inside externally supercooled rectangular duct

The transient process of solidification of laminar liquid flow (water) submitted to super-cooling was investigated both theoretically and experimentally. In this study an alternative analytical formulation and numerical approach were adopted resulting in the unsteady model with temperature dependent thermophysical properties in the solid region. The proposed model is based upon the fundamental equations of energy balance in the solid and liquid regions as well as across the solidification front. The basic equations and the associated boundary and initial conditions were made dimensionless by using the Landau transformation to immobilize the moving front and render the problem to a fixed plane type problem. A laminar velocity profile is admitted in the liquid domain and the resulting equations were discretized using the finite difference approach. The numerical predictions obtained were compared with the available results based on other models and concepts such as Neumann analytical model, the apparent thermal capacity model due to Bonacina and the conventional fixed grid energy model due to Goodrich. To obtain further comparisons and more validation of the model and the numerical solution, an experimental rig was constructed and instrumented permitting very well controlled experimental measurements. The numerical predictions were compared with the experimental results and the agreement was found satisfactory.     

A numerical model for transport in flat heat pipes considering wick microstructure effects

A transient, three-dimensional model for thermal transport in heat pipes and vapor chambers is developed. The Navier–Stokes equations along with the energy equation are solved numerically for the liquid and vapor flows. A porous medium formulation is used for the wick region. Evaporation and condensation at the liquid–vapor interface are modeled using kinetic theory. The influence of the wick microstructure on evaporation and condensation mass fluxes at the liquid–vapor interface is accounted for by integrating a microstructure-level evaporation model (micromodel) with the device-level model (macromodel). Meniscus curvature at every location along the wick is calculated as a result of this coupling. The model accounts for the change in interfacial area in the wick pore, thin-film evaporation, and Marangoni convection effects during phase change at the liquid–vapor interface. The coupled model is used to predict the performance of a heat pipe with a screen-mesh wick, and the implications of the coupling employed are discussed.     

A Unified Generalized Thermoelasticity Formulation; Application to Thick Functionally Graded Cylinders

In this article a new unified formulation for the generalized coupled thermoelasticity theories is presented. The generalized coupled thermoelasticity theories proposed by Lord–Shulman, Green–Lindsay, and Green–Naghdi are combined into a unified formulation introducing the unified parameters. The formulation is given for the general anisotropic heterogeneous linear thermoelastic materials and then is reduced to the system of equations for the isotropic heterogeneous materials. As an example, a functionally graded cylinder is considered and the thermoelastic waves based on the new unified formulation, using the generalized theories, are obtained and discussed. Utilizing the transfinite element method, the equations for a long thick circular cylinder are solved in the Laplace domain and the results are inverted to the real time domain using a numerical inversion technique of the Laplace transform. The results for the propagation of thermoelastic waves based on the Lord–Shulman, Green–Lindsay, and Green–Naghdi models are derived and compared.     

蒸发液滴内部非稳态流动的数值计算

在气液两相流VOF(volume of fluid,VOF)模型的基础上耦合CSF(continuum surface force,CSF)表面张力模型,建立了高温平板上的铺展液滴与高温空气中悬浮液滴蒸发过程中内部非稳态流动模型,对液滴蒸发过程中内部非稳态流动进行了研究。基于相变理论,采用用户自定义函数将流体相变模型加入非稳态流动模型中进行耦合计算,获得了高温平板上的铺展液滴与高温空气中悬浮液滴蒸发过程中的内部流动及变化过程。液滴蒸发过程中非稳态内部流动由液滴表面的温度梯度引发,Marangoni流动在液滴内部形成的时间非常短,流体从液滴表面高温区域流向低温区域。计算结果表明:高温平板上随着液滴蒸发的进行,液滴内部一直保持两个对称的涡流,Marangoni流动比较稳定;高温空气环境中随着液滴蒸发的进行,液滴内部四个涡流逐渐转变成两个对称的涡流;液滴内部温度分布因Marangoni流动加强传热而变得均匀,同时由于温度分布变得均匀,Marangoni流动被削弱。     

A Fast Numerical Simulation for Modeling Simultaneous Metal Flow and Solidification in Thin Cavities Using the Lubrication Approximation

A numerical algorithm for modelling steady flow of liquid metal accompanied by solidification in a thin cavity is presented. The problem is closely related to a die cast process and in particular to the metal flow phenomenon observed in thin ventilation channels. Using the fact that the rate of metal flow in the channel is much higher than the rate of solidification, a numerical algorithm is developed by treating the metal flow as steady in a given time-step while treating the heat transfer in the thickness direction as transient. The flow in the thin cavity is treated as two dimensional after integrating the momentum and continuity equations over the thickness of the channel, while the heat transfer is modelled as a one-dimensional phenomenon in the thickness direction. The presence of a moving solid-liquid interface introduces non-linearity in the resulting set of equations, and which are solved iteratively. The location and shape of the solid-liquid interface are found as a part of the solution. The staggered grid arrangement is used to discretize the flow governing equations and the resulting set of partial differential equations is solved using the SIMPLE algorithm. The thickness direction heat-transfer problem accompanied by phase change is solved using a control volume formulation. The results are compared with the predictions of the commercial software FLOW3D® which solves the full three-dimensional set of flow and heat transfer equations accompanied with solidification. The Reynolds's lubrication equations accompanied by the through-the-thickness heat loss and solidification model can be successfully implemented to analyze flow and solidification of liquid metals in thin channel during the die cast process. The results were obtained with significant savings in CPU time.     

Suppression of Marangoni convection of silicon melt by a non-contaminating method

A set of numerical simulation of the effect of the gas shearing flow over a silicon melt free surface on Marangoni convection under microgravity condition was conducted by using finite element method. For given gas channel width, Marangoni number and aspect ratio a remarkable reduction of Marangoni convection in silicon liquid bridge can be achieved by choosing the optimal gas velocity in accordance with the correlation proposed in the paper. The effectiveness of the reduction of the gas flow under different conditions shows that, in some cases, Marangoni convection reduction of 99% can be realized by this non-contaminating method.     

On the impingement heat transfer of an oblique free surface plane jet

The objective of the present study is to understand the hydrodynamics and heat transfer of the impingement process, particularly the complexities attributable to the asymmetric geometry of an oblique liquid plane jet. The Navier-Stokes equations are solved using a finite-volume formulation with a two-step projection method on a fixed non-uniform rectangular grid. The free surface of the jet is tracked by the volume-of-fluid method with a second order accurate piecewise-linear scheme. The energy equation is modeled by using an enthalpy-based formulation. The method provides a state-of-the-art comprehensive model of the dynamic and thermal aspects of the impinging process. Nusselt number plots and pressure distributions on the substrate are obtained. The locations of the maximum Nusselt number as well as maximum pressure on the surface are identified and compared with the geometric jet impingement point. Results for normal impingement are also obtained and are used as reference. The effects of several parameters are examined. These include jet Reynolds number, jet impingement angle and jet inlet velocity profile. Experimental and analytical data from the literature are also included for comparison.     

Direct Numerical Simulation of Air Heated Cylinder Wake in Transitional State. Part II: Visualization and Mechanism of Unified 3-D Streaklines,Vortex Dislocation,and Turbulence Wreck

Abstract

A comparison of two frequently used computational techniques for solving phase-change problems is presented. The governing equations for the conservation of mass, momentum, and energy are solved using a control-volume-based discretization scheme. In Ike first approach, the physical space is mapped onto a simpler domain and the moving boundary is immobilized using Landau transformation. The computations are carried out on a uniform orthogonal grid in the transformed space using the stream function-vorticity formulation. The need to retain all the terms in the governing equations arising from the transformation, for an accurate simulation, is investigated. Simplifications in the governing equations have been used in the literature and are discussed. Both implicit and explicit methods are used to track the phase front. In the second approach, the computations are carried out on a uniform fixed grid in the physical space with primitive variables. The enthalpy-porosity formulation, with appropriate source terms to account for the phase change, is employed. Numerical results yield the temperature distribution and the buoyancy-induced velocity field. The test problems used are the melting of gallium and tin in a rectangular cavity with isothermal side walls and adiabatic top and bottom walls. Comparisons are made between the numerical predictions and experimental data on the morphology and position of the phase front for cavities of different aspect ratios, and the computational times are recorded. Heat transfer rates and velocity field results obtained are also presented. The study indicates the range of applicability and computational complexity of the two approaches.     

Study of wind flow and pollutant dispersion by newly developed precision-improving methods

The paper presents a three-dimensional transient numerical model for atmospheric wind flow and industry and/or traffic pollutant dispersion over terrains having a complex topography. The model is based on a finite-volume integration of the equations governing mass, momentum, heat and pollutant transport within the earth's atmospheric boundary layer, using a collocated grid arrangement. The instability provoked by such a formulation was avoided by using a special pressure-velocity coupling. Local refinement of the grid was achieved via a domain decomposition method. The technique of “porosity” used to approximate curved three-dimensional boundaries is incorporated in the procedure thus avoiding the less accurate and more common approximation by a broken surface with segments parallel to the coordinate lines. The method was validated by simulating the flow over the Attica peninsula for which measurements of wind speed and pollutant emissions are available.     

Conjugate heat and mass transfer in a hollow fiber membrane module for liquid desiccant air dehumidification: A free surface model approach

Conjugate heat and mass transfer in a hollow fiber membrane module used for liquid desiccant air dehumidification is investigated. The module is like a shell-and-tube heat exchanger where the liquid desiccant stream flows in the tube side, while the air stream flows in the shell side in a counter flow arrangement. Due to the numerous fibers in the shell, a direct modeling of the whole module is difficult. This research takes a new approach. A representative cell comprising of a single fiber, the liquid desiccant flowing inside the fiber and the air stream flowing outside the fiber, is considered. The air stream outside the fiber has an outer free surface (Happel’s free surface model). Further, the equations governing the fluid flow and heat and mass transfer in the two streams are combined together with the heat and mass diffusion equations in membranes. The conjugate problem is then solved to obtain the velocity, temperature and concentration distributions in the two fluids and in the membrane. The local and mean Nusselt and Sherwood numbers in the cell are then obtained and experimentally validated.