Numerical simulation of thermo‐solutal‐capillary migration of a dissolving drop in a cavity

In the present paper the thermo‐solutal‐capillary migration of a dissolving liquid drop, composed by a binary mixture having a miscibility gap, injected in a closed cavity with differentially heated end walls, is studied. The main goal of the analysis is to clarify if and how the drop migration is affected by the dissolution process. The numerical code is based on a finite volume formulation. A level‐set technique is used for describing the dynamics of the interface separating the different phases. A thermodynamic constraint fixes the concentration jump between the interface sides. This jump, together with that of the concentration normal derivatives, in turn defines the entity of the dissolution cross‐flow through the interface and the interface velocity relative to the fluid. Since the jump singularity of normal derivatives cannot be easily mollified, while retaining the necessary accuracy, a scheme for the species equation is elaborated that allows sharp jumps and has subcell resolution. Steady migration speeds are determined after the start‐up phase for different radii and temperature differences. The results will be used for the preparation of a sounding rocket space experiment. Copyright © 2003 John Wiley & Sons, Ltd.     

Level‐set based flamelet approach for simulating turbulent lifted jet flame

The present study focuses on numerically investigating the flame structure, flame liftoff, and stabilization in a lifted turbulent H₂/N₂ jet flame with a vitiated coflow. To realistically represent the turbulent partially premixed nature in the flow region between nozzle exit and flame base, the level‐set approach coupled with the conserved scalar flamelet model has been applied. The unstructured‐grid level‐set approach has been developed to allow the geometric flexibility and computational efficiency for the solution of the physically and geometrically complex reacting flows. The pressure–velocity coupling is handled by the multiple pressure‐correction method. The predicted flame pattern is in good conformity with the measured one. In terms of the liftoff height, the agreement between prediction and experiment is quite good. Even if there are noticeable deviations in a certain region, the predicted profiles for the overall flame structure agree reasonably well with the experimental data. These numerical results indicate that the present level‐set‐based flamelet approach in conjunction with the unstructured‐grid finite‐volume method is capable of realistically predicting the essential features and precise structure of the turbulent‐lifted jet flame with computational efficiency. Copyright © 2008 John Wiley & Sons, Ltd.     

The effect of solutal undercooling on double‐diffusive convection and macrosegregation during binary alloy solidification: a numerical investigation

In this paper, we present a macroscopic numerical model that is capable of capturing the interaction between the double‐diffusive convective field and a localized fluid flow on account of solutal undercooling during non‐equilibrium solidification of binary alloys. The model is essentially based on a fixed‐grid enthalpy based control volume approach. In the present model, microscopic features pertaining to non‐equilibrium effects on account of solutal undercooling are incorporated through the formulation of a modified partition‐coefficient. The effective partition‐coefficient is numerically modelled by means of a number of macroscopically observable parameters related to the solidifying domain. This feature has made the present treatment different from micro‐macro modelling of alloy solidification, which involves certain parameters that may not be macroscopically resolvable. Numerical simulations are performed for the case of two‐dimensional transient solidification of Pb–Sn alloys (both hypoeutectic and hypereutectic) in a rectangular cavity, employing the present model. The simulation results are also compared with the corresponding experimental results quoted in the literature, and the agreement is excellent. From the results, it can be concluded that non‐equilibrium effects on account of solutal undercooling result in a more enhanced macrosegregation. Copyright © 2002 John Wiley & Sons, Ltd.     

A hybrid Lagrangian–Eulerian particle‐level set method for numerical simulations of two‐fluid turbulent flows

A coupled Lagrangian interface‐tracking and Eulerian level set (LS) method is developed and implemented for numerical simulations of two‐fluid flows. In this method, the interface is identified based on the locations of notional particles and the geometrical information concerning the interface and fluid properties, such as density and viscosity, are obtained from the LS function. The LS function maintains a signed distance function without an auxiliary equation via the particle‐based Lagrangian re‐initialization technique. To assess the new hybrid method, numerical simulations of several ‘standard interface‐moving’ problems and two‐fluid laminar and turbulent flows are conducted. The numerical results are evaluated by monitoring the mass conservation, the turbulence energy spectral density function and the consistency between Eulerian and Lagrangian components. The results of our analysis indicate that the hybrid particle‐level set method can handle interfaces with complex shape change, and can accurately predict the interface values without any significant (unphysical) mass loss or gain, even in a turbulent flow. The results obtained for isotropic turbulence by the new particle‐level set method are validated by comparison with those obtained by the ‘zero Mach number’, variable‐density method. For the cases with small thermal/mass diffusivity, both methods are found to generate similar results. Analysis of the vorticity and energy equations indicates that the destabilization effect of turbulence and the stability effect of surface tension on the interface motion are strongly dependent on the density and viscosity ratios of the fluids. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical simulation of free‐surface flow using the level‐set method with global mass correction

A new numerical method that couples the incompressible Navier–Stokes equations with the global mass correction level‐set method for simulating fluid problems with free surfaces and interfaces is presented in this paper. The finite volume method is used to discretize Navier–Stokes equations with the two‐step projection method on a staggered Cartesian grid. The free‐surface flow problem is solved on a fixed grid in which the free surface is captured by the zero level set. Mass conservation is improved significantly by applying a global mass correction scheme, in a novel combination with third‐order essentially non‐oscillatory schemes and a five stage Runge–Kutta method, to accomplish advection and re‐distancing of the level‐set function. The coupled solver is applied to simulate interface change and flow field in four benchmark test cases: (1) shear flow; (2) dam break; (3) travelling and reflection of solitary wave and (4) solitary wave over a submerged object. The computational results are in excellent agreement with theoretical predictions, experimental data and previous numerical simulations using a RANS‐VOF method. The simulations reveal some interesting free‐surface phenomena such as the free‐surface vortices, air entrapment and wave deformation over a submerged object. Copyright © 2009 John Wiley & Sons, Ltd.     

Front‐tracking by the level‐set and the volume penalization methods in a two‐phase microfluidic network

Two‐phase immiscible fluids in a two‐dimensional micro‐channels network are considered. The incompressible Stokes equations are used to describe the Newtonian fluid flow, while the Oldroyd‐B rheological model is used to capture the viscoelastic behavior. In order to perform numerical simulations in a complex geometry like a micro‐channels network, the volume penalization method is implemented. To follow the interface between the two fluids, the level‐set method is used, and the dynamics of the contact line is modeled by Cox law. Numerical results show the ability of the method to simulate two‐phase flows and to follow properly the contact line between the two immiscible fluids. Finally, simulations with realistic parameters are performed to show the difference when a Newtonian fluid is pushed by a viscoelastic fluid instead of a Newtonian one. Copyright © 2015 John Wiley & Sons, Ltd.     

Effect of thermo‐solutal stratification on recirculation flow patterns in a backward‐facing step channel flow

This paper presents results on the combined effect of thermo‐solutal buoyancy forces on the recirculatory flow behavior in a horizontal channel with backward‐facing step and the ensuing impact on heat and mass transfer phenomena. The governing equations for double diffusive mixed convection are represented in velocity–vorticity form of momentum equations, velocity Poisson equations, energy and concentration equations. Galerkin's finite‐element method has been employed to solve the governing equations. Recirculatory flow fields with heat and mass transfer are simulated for opposing and aiding thermo‐solutal buoyancy forces by assuming suitable boundary conditions for energy and concentration equations. The effect of Richardson number (0.1?Ri?10) and buoyancy ratio (?10?N?10) on the recirculation bubble and Nusselt and Sherwood numbers are studied in detail. For Richardson number greater than unity, distinct variations in the gradients of Nusselt number and Sherwood number with buoyancy ratio are observed for flow regimes with opposing and aiding buoyancy forces. Copyright © 2009 John Wiley & Sons, Ltd.     

Accelerated convergence of the numerical simulation of incompressible flow in general curvilinear co‐ordinates by discretizations on the double‐staggered grids

The convergence rate of a methodology for solving incompressible flow in general curvilinear co‐ordinates is analyzed. Double‐staggered grids (DSGs), each defined by the same boundaries as the physical domain, are used for discretization. Both grids are MAC quadrilateral meshes with scalar variables (pressure, temperature, etc.) arranged at the center and the Cartesian velocity components at the middle of the sides of the mesh cells. The problem was checked against benchmark solutions of natural convection in a squeezed cavity, heat transfer in concentric horizontal cylindrical annuli, and a hot cylinder in a duct. Poisson's pressure‐correction equations that arise from the SIMPLE‐like procedure are solved by several methods: successive overrelaxation, symmetric overrelaxation, modified incomplete factorization preconditioner, conjugate gradient (CG), and CG with preconditioner. A genetic algorithm was developed to solve problems of numerical optimization of SIMPLE‐like calculation time in a space of iteration numbers and relaxation parameters. The application provides a means of making an unbiased comparison between the DSGs method and the widely used interpolation method. Furthermore, the convergence rate was demonstrated by application to the calculation of natural convection heat transfer in concentric horizontal cylindrical annuli. Calculation times when DSGs were used were 2–10 times shorter than those achieved by interpolation. With the DSGs method, calculation time increases slightly with increasing non‐orthogonality of the grids, whereas an interpolation method calls for very small iteration parameters that lead to unacceptable calculation times. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical study of flow and thermal behaviour of lid‐driven flows in cavities of small aspect ratios

Numerical study has been performed to investigate the effects of cavity shape on flow and heat transfer characteristics of the lid‐driven cavity flows. Dependence of flow and thermal behaviour on the aspect ratio of the cavities is also evaluated. Three types of the cross‐sectional shape, namely, circular, triangular, and rectangular, and four aspect ratios, 0.133, 0.207, 0.288, and 0.5, are taken into account to construct twelve possible combinations; however, attention is focused on the small‐aspect‐ratio situations. Value of the Reynolds number considered in this study is varied between 100 and 1800. For the cases considered in this study a major clockwise vortex driven by the moving lid prevailing in the cavity is always observed. When the Reynolds number is fixed, the rectangular cavity produces strongest lid‐driven flow, and the triangular cavity weakest. For the cases at small aspect ratio and low Reynolds number, the streamlines appear symmetric fore‐and‐aft with respect to the central line at x/L = 0.5. Data for the local and average Nusselt numbers are also provided. For rectangular cavities, it is observed that case 1/5R produces the highest average Nusselt number at any Reynolds number. Among the twelve possible geometric cases considered herein, the highest and lowest average Nusselt numbers are found with cases 1/6T and 1/2C, respectively. Copyright © 2006 John Wiley & Sons, Ltd.     

A new VOF‐based numerical scheme for the simulation of fluid flow with free surface. Part II: application to the cavity filling and sloshing problems

Finite element analysis of fluid flow with moving free surface has been performed in 2‐D and 3‐D. The new VOF‐based numerical algorithm that has been proposed by the present authors (Int. J. Numer. Meth. Fluids, submitted) was applied to several 2‐D and 3‐D free surface flow problems. The proposed free surface tracking scheme is based on two numerical tools; the orientation vector to represent the free surface orientation in each cell and the baby‐cell to determine the fluid volume flux at each cell boundary. The proposed numerical algorithm has been applied to 2‐D and 3‐D cavity filling and sloshing problems in order to demonstrate the versatility and effectiveness of the scheme. The proposed numerical algorithm resolved successfully the free surfaces interacting with each other. The simulated results demonstrated applicability of the proposed numerical algorithm to the practical problems of large free surface motion. It has been also demonstrated that the proposed free surface tracking scheme can be easily implemented in any irregular non‐uniform grid systems and can be extended to 3‐D free surface flow problems without additional efforts. Copyright © 2003 John Wiley & Sons, Ltd.     

High‐order upwind compact scheme and simulation of turbulent premixed V‐flame

A high‐order accurate upwind compact difference scheme with an optimal control coefficient is developed to track the flame front of a premixed V‐flame. In multi‐dimensional problems, dispersion effect appears in the form of anisotropy. By means of Fourier analysis of the operators, anisotropic effects of the upwind compact difference schemes are analysed. Based on a level set algorithm with the effect of exothermicity and baroclinicity, the flame front is tracked. The high‐order accurate upwind compact scheme is employed to approximate the level set equation. In order to suppress numerical oscillations, the group velocity control technique is used and the upwind compact difference scheme is combined with the random vortex method to simulate the turbulent premixed V‐flame. Distributions of velocities and flame brush thickness are obtained by this technique and found to be comparable with experimental measurement. Copyright © 2005 John Wiley & Sons, Ltd.     

A simulation of free surface waves for incompressible two‐phase flows using a curvilinear level set formulation

A level set formulation in a generalized curvilinear coordinate is developed to simulate the free surface waves generated by moving bodies or the sloshing of fluid in a container. The Reynolds‐averaged Navier–Stokes (RANS) equations are modified to account for variable density and viscosity in two‐phase (i.e. water–air) fluid flow systems. A local level set method is used to update the level set function and a least square technique adopted to re‐initialize it at each time step. To assess the developed algorithm and its versatility, a selection of different fluid–structure interaction problems are examined, i.e. an oscillating flow in a two‐dimensional square tank, a breaking dam involving different density fluids, sloshing in a two‐dimensional rectangular tank and a Wigley ship hull travelling in calm water. Copyright © 2005 John Wiley & Sons, Ltd.     

High‐order detached eddy simulation,zonal LES and URANS of cavity and labyrinth seal flows

Hybrid numerical large eddy simulation (NLES), detached eddy simulation (DES) and URANS methods are assessed on a cavity and a labyrinth seal geometry. A high sixth‐order discretization scheme is used and is validated using the test case of a two‐dimensional vortex. The hybrid approach adopts a new blending function. For the URANS simulations, the flow within the cavity remains steady, and the results show significant variation between models. Surprisingly, low levels of resolved turbulence are observed in the cavity for the DES simulation, and the cavity shear layer remains two dimensional. The hybrid RANS–NLES approach does not suffer from this trait. For the labyrinth seal, both the URANS and DES approaches give low levels of resolved turbulence. The zonal Hamilton–Jacobi approach on the other had given significantly more resolved content. Both DES and hybrid RANS–NLES give good agreement with the experimentally measured velocity profiles. Again, there is significant variation between the URANS models, and swirl velocities are overpredicted. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical simulation of bubble and droplet deformation by a level set approach with surface tension in three dimensions

In this paper we present a three‐dimensional Navier–Stokes solver for incompressible two‐phase flow problems with surface tension and apply the proposed scheme to the simulation of bubble and droplet deformation. One of the main concerns of this study is the impact of surface tension and its discretization on the overall convergence behavior and conservation properties. Our approach employs a standard finite difference/finite volume discretization on uniform Cartesian staggered grids and uses Chorin's projection approach. The free surface between the two fluid phases is tracked with a level set (LS) technique. Here, the interface conditions are implicitly incorporated into the momentum equations by the continuum surface force method. Surface tension is evaluated using a smoothed delta function and a third‐order interpolation. The problem of mass conservation for the two phases is treated by a reinitialization of the LS function employing a regularized signum function and a global fixed point iteration. All convective terms are discretized by a WENO scheme of fifth order. Altogether, our approach exhibits a second‐order convergence away from the free surface. The discretization of surface tension requires a smoothing scheme near the free surface, which leads to a first‐order convergence in the smoothing region. We discuss the details of the proposed numerical scheme and present the results of several numerical experiments concerning mass conservation, convergence of curvature, and the application of our solver to the simulation of two rising bubble problems, one with small and one with large jumps in material parameters, and the simulation of a droplet deformation due to a shear flow in three space dimensions. Furthermore, we compare our three‐dimensional results with those of quasi‐two‐dimensional and two‐dimensional simulations. This comparison clearly shows the need for full three‐dimensional simulations of droplet and bubble deformation to capture the correct physical behavior. Copyright © 2009 John Wiley & Sons, Ltd.     

Wetting boundary conditions in numerical simulation of binary fluids by using phase‐field method: some comparative studies and new development

We studied several wetting boundary conditions (WBCs) in the numerical simulation of binary fluids by using phase‐field method. Five WBCs, three using the linear, cubic, and sine form surface energy (LinSE, CubSE, and SinSE), the other two using the geometric formulation (Geom) and the characteristic interpolation (CI), were compared through the study of several problems: (1) the static contact angle (CA) of a drop; (2) a Poiseuille flow‐driven liquid column; (3) a wettability gradient (WG)‐driven liquid column; and (4) drop dewetting. It was found that while all WBCs can predict the static CA fairly accurately, they may affect the simulation outcomes of dynamic problems differently, depending on the CA. For the flow‐driven problem with a CA near 90°, using different WBCs had almost no effect on the flow characteristics over a large scale. For the WG‐driven problem, to use different WBCs may lead to different steady drop velocities, and all WBCs except LinSE can give reasonably consistent prediction between the drop velocity and dynamic CAs. For drop dewetting, Geom led to the most violent drop motion, whereas CubSE caused the weakest motion. For several problems, CubSE and SinSE gave almost the same results, and those by Geom and CI were also close, possibly due to similar consideration in their design. Besides, a new implementation that may be used for all WBCs was proposed to mimic the wall energy relaxation and control the degree of slip. This new procedure made it possible to allow the simulations to match experimental measurements well. Copyright © 2014 John Wiley & Sons, Ltd.     

Lattice Boltzmann simulation of MHD natural convection in a cavity with porous media and sinusoidal temperature distribution

The lattice Boltzmann method (LBM) is used to simulate the effect of magnetic field on the natural convection in a porous cavity. The sidewalls of the cavity are heated sinusoidally with a phase derivation, whereas the top and bottom walls are thermally insulated. Numerical simulation is performed, and the effects of the pertinent parameters, e.g., the Hartmann number, the porosity, the Darcy number, and the phase deviation, on the fluid flow and heat transfer are investigated. The results show that the heat transfer is affected by the temperature distribution on the sidewalls clearly. When the Hartmann number is 0, the maximum average Nusselt number is obtained at the phase deviation 90°. Moreover, the heat transfer enhances when the Darcy number and porosity increase, while decreases when the Hartman number increases.