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The purpose of this study is to simulate the thermo-solutal convection resulting from a circular cylinder hanging in a rod inside a ∧-shaped cavity.

The two dimensional ∧-shaped cavity is filled by Al₂O₃-water nanofluid and saturated by three different levels of heterogeneous porous media. An incompressible smoothed particle hydrodynamics (ISPH) method is adopted to solve the governing equations of the present problem. The present simulations have been performed for the alteration of buoyancy ratio (−2≤N≤2), radius of a circular cylinder (0.05≤Rc≤0.3), a height of a rod (0.1≤Lh≤0.4), Darcy parameter (10−3≤Da≤10−5), Lewis number (1≤Le≤40), solid volume fraction (0≤ϕ≤0.06), porous levels (0≤η1=η2≤1.5)and various boundary-wall conditions.

The performed numerical simulations indicated the importance of embedded shapes on the distributions of temperature, concentration and velocity fields inside ∧-shaped cavity. Increasing buoyancy ratio parameter enhances thermo-solutal convection and nanofluid velocity. Adiabatic conditions of the vertical-walls of ∧-shaped cavity augment the distributions of the temperature and concentration. Regardless the Darcy parameter, a homogeneous porous medium gives the lowest values of a nanofluid velocity.

ISPH method is used to simulate thermo-solutal convection of a nanofluid inside a novel ∧-shaped cavity containing a novel embedded shape and heterogeneous porous media.

The purpose of the paper is to numerically simulate steady‐state thermo‐solutal convection in rectangular cavities with different aspect ratios, subject to horizontal temperature and concentration gradients, and validate the results against numerical and experimental data available from literature.

The fully explicit Artificial Compressibility (AC) version of the Characteristic Based Split (CBS) scheme is adopted to solve double diffusion (DD) problems. A stabilization analysis is carried out to efficiently solve the problems considered in the present work. The thermal and solutal buoyancy forces acting on the fluid have been taken into account in case of aiding and opposing flow conditions.

The stability limits derived by the authors for the thermo‐solutal convection assume a fundamental role to efficiently solve the DD problems considered. In the cases characterized by higher Rayleigh number the convergent solution is obtained only by employing the new stability conditions. The efficient matrix free procedure employed is a powerful tool to study complex DD problems.

In this paper, the authors extend the stabilization analysis for the AC‐CBS scheme to the solution of DD, fundamental to efficiently solve the present problems, and apply the present fully explicit matrix free scheme, based on finite elements, to the solution of DD natural convection in cavities.

This paper aims to tackle the problem of thermo‐solutal convection and macrosegregation during ingot solidification of metal alloys. Complex flow structures associated with the development of channels segregate and sharp gradients in the solutal field call for the implementation of accurate methods for numerical modeling of alloy solidification. In particular, the solute transport equation is convection dominated and requires special non‐oscillarity type high‐order schemes to handle the regions of channels segregates.

In the present study, a time‐splitting approach has been adopted to separately handle solute advection and diffusion. This splitting technique allows the application of accurate total variation dimensioning (TVD) schemes for solution of solute advection. Applications of second‐order Lax‐Wendroff TVD SUPERBEE and fifth‐order weighted essentially non‐oscillatory (WENO) schemes are described in the present article. Classical numerical solution of solute transport using hybrid and central‐difference schemes are also employed for the purpose of comparisons. Numerical simulations for solidification of Pb‐18%Sn in a two‐dimensional rectangular cavity have been carried out using different numerical schemes.

Numerical results show the difficulty of obtaining grid‐independent solutions with respect to local details in the region of channels. Grid convergence patterns and numerical uncertainty are found to be dependent on the applied scheme. In general, the first‐order hybrid scheme is diffusive and under predicts the formation of channels. The second‐order central‐difference scheme brings about oscillations with possible non‐physical extremes of solute composition in the region of channel segregates due to sharp gradients in the solutal field. The results obtained using TVD and WENO schemes contain no oscillations and show an excellent capture of channels formation and resolution of the interface between solute‐rich and depleted bands. Different stages of channels formation are followed by analyzing thermo‐solutal convection and macrosegregation at different times during solidification.

Accurate prediction of local variation in the solutal and flow fields in the channels regions requires grid refinement up to scales in the order of microscopic dendrite arm spacing. This imposes limitations in terms of large computational time and applicability of available macroscopic models based on classical volume‐averaging techniques.

The present study is very useful for numerical simulation of macrosegregation during ingot casting of metal alloys.

The paper provides the methodology and application of TVD schemes to predict channel segregates during columnar solidification of metal alloys. It also demonstrates the limitations of classical schemes for simulation of alloy solidification.

In the present work, a numerical method, based on the well established enthalpy technique, is developed to simulate the growth of binary alloy equiaxed dendrites in presence of melt convection. The paper aims to discuss these issues.

The principle of volume-averaging is used to formulate the governing equations (mass, momentum, energy and species conservation) which are solved using a coupled explicit-implicit method. The velocity and pressure fields are obtained using a fully implicit finite volume approach whereas the energy and species conservation equations are solved explicitly to obtain the enthalpy and solute concentration fields. As a model problem, simulation of the growth of a single crystal in a two-dimensional cavity filled with an undercooled melt is performed.

Comparison of the simulation results with available solutions obtained using level set method and the phase field method shows good agreement. The effects of melt flow on dendrite growth rate and solute distribution along the solid-liquid interface are studied. A faster growth rate of the upstream dendrite arm in case of binary alloys is observed, which can be attributed to the enhanced heat transfer due to convection as well as lower solute pile-up at the solid-liquid interface. Subsequently, the influence of thermal and solutal Peclet number and undercooling on the dendrite tip velocity is investigated.

As the present enthalpy based microscopic solidification model with melt convection is based on a framework similar to popularly used enthalpy models at the macroscopic scale, it lays the foundation to develop effective multiscale solidification.

Double-diffusive convection within a tri-dimensional in a horizontal annulus partially filled with a fluid-saturated porous medium is numerically investigated. The aim of this work is to understand the effects of a source of heat and solute on the fluid flow and heat and mass transfer rates.

In the formulation of the problem, the Darcy–Brinkman–Forchheimer model is adopted to the fluid flow in the porous annulus. The laminar flow regime is considered under steady state conditions. Moreover, the transport equation for continuity, momentum, energy and mass transfer are solved using the Patankar–Spalding technique.

Through this investigation, the predicted results for both average Nusselt and Sherwood numbers were correlated in terms of Lewis number, thermal Grashof number and buoyancy ration. A comparison was made with the published results and a good agreement was found.

The paper’s results are validated by favorable comparisons with previously published results. The results of the problem are presented in graphical forms and discussed. This paper aims to study the behavior of the flow structure and heat transfer and mass for different parameters.

The entropy and thermal behavior analyses of non-Newtonian nanofluid double-diffusive natural convection inside complex domains may captivate a bunch of scholars’ attention because of the potential utilizations that they possess in modern industries, for example, heat exchangers, solar energy collectors and cooling of electronic apparatuses. This study aims to investigate the second law and thermal behavior of non-Newtonian double-diffusive natural convection (DDNC) of Al₂O₃-H₂O nanofluid within a C-shaped cavity emplacing two hot baffles and impacted by a magnetic field.

For the governing equations of the complicated and practical system with all considered parameters to be solved via a formidable numerical approach, the finite element method acts as an approach to achieving the desired solution. This method allows us to gain a detailed solution to the studied geometry.

This investigation has been executed for the considered parameters of range, such as power-law index, baffle length, Lewis number, buoyancy ratio, Hartmann number and Rayleigh number. The main results reveal that isothermal and concentration lines are significantly more distorted, indicating intensified concentration and temperature distributions because of the growth of baffle length (L). Nu_ave decreases by 8.4% and 0.8% while it enhances by 49.86% and 33.87%, respectively, because of growth in the L from 0.1 to 0.2 and 0.2 to 0.3.

Such a comprehensive study on the second law and thermal behavior of DDNC of Al₂O₃-H₂O nanofluid within a C-shaped cavity emplacing two hot baffles and impacted by magnetic field has not yet been carried out.

The purpose of this paper is to present a new numerical approach for modeling the multi‐phase flow during an alloy solidification process. In many solidification processes, advection of solid may have a dramatic effect on bulk convection field as well as on the solid front growth and hence on the macro‐segregation pattern. In the present work, a numerical model is developed to simulate directional solidification in presence of melt convection as well as solid advection in the form of sedimentation. A 2D cavity filled with hyper‐eutectic aqueous ammonium chloride solution (25 wt.% of ammonium chloride) being chilled from one of the side walls has been chosen as the model problem for the numerical simulation.

A fixed grid volume averaging technique has been used for solving mass, momentum, energy, and species equation while taking into account the solid phase advection and local re‐melting. Two different criteria have been identified for the solid particles in the mushy zone to be mobile. These two criteria are represented by a critical solid fraction, and a critical velocity. Based on these two criteria, the mushy zone has been subdivided into two different regions namely, an immobile coherent zone consisting of packed equiaxed crystals and a mobile non‐coherent zone where the solid crystals are able to move.

The numerical results are compared with corresponding experimental observations.

The solid advection velocity and source terms dealing with solid velocity have been calculated using an explicit scheme, whereas the main conservation equations are solved using an implicit scheme.

This paper aims to consider numerical analysis of laminar double-diffusive natural convection inside a non-homogeneous closed medium composed of a saturated porous matrix and a clear binary fluid under spatial sinusoidal heating/cooling on one side wall and uniform salting.

The domain of interest is a partially square porous enclosure with sinusoidal wall heating and cooling. The fluid flow, heat and mass transfer dimensionless governing equations associated with the corresponding boundary conditions are discretized using the finite volume method. The resulting algebraic equations are solved by an in-house FORTRAN code and the SIMPLE algorithm to handle the non-linear character of conservation equations. The validity of the in-house FORTRAN code is checked by comparing the current results with previously published experimental and numerical works. The effect of the porous layer thickness, the spatial frequency of heating and cooling, the Darcy number, the Rayleigh number and the porous to fluid thermal conductivity ratio is analyzed.

The results demonstrate that for high values of the spatial frequency of heating and cooling (f = 7), temperature contours show periodic variations with positive and negative values providing higher temperature gradient near the thermally active wall. In this case, the temperature variation is mainly in the porous layer, while the temperature of the clear fluid region is practically the same as that imposed on the left vertical wall. This aspect can have a beneficial impact on thermal insulation. Besides, the porous to fluid thermal conductivity ratio, Rk, has practically no effect on Shhot wall, contrary to Nuinterface where a strong increase is observed as Rk is increased from 0.1 to 100, and much heat transfer from the hot wall to the clear fluid via the porous media is obtained.

The findings are useful for devices working on double-diffusive natural convection inside non-homogenous cavities.

The authors believe that the presented results are original and have not been published elsewhere.

The purpose of this study is to study the impacts of exponential space-dependent heat source (ESHS) and thermal radiation in Marangoni convective flow of Cross fluid. The passively controlled model is developed to exhibit the nanoparticles’ concentration on the surface.

The resulting problem under consideration is tackled by using the shooting approach.

Temperature field augments with enhancement of the thermophoretic, exponential-based space heat source (ESHS) and radiation parameters, whereas it decays with the increase of the Marangoni ratio parameter.

To the best of the authors’ knowledge, no such analysis has yet been reported.

This paper aims to tackle the problem of some ambiguity of the momentum equation formulation in the commonly used macroscopic models of two‐phase solid/liquid region, developing during alloy solidification. These different appearances of the momentum equation are compared and the issue is addressed of how the choice of the particular form affects velocity and temperature fields.

Attention is focused on the ensemble averaging method, which, owing to its stochastic nature, is a new promising tool for setting up the macroscopic transport equations in highly inhomogeneous multiphase micro‐ and macro‐structures, with morphology continuously changing in time when the solidification proceeds. The basic assumptions of the two other continuum models, i.e. based on the classical mixture theory and on the volume‐averaging technique, are also unveiled. These three different forms of the momentum equation are then compared analytically and their impact on calculated velocity and temperature distribution in the mushy zone is studied for the selected test problem of binary alloy solidification driven by diffusion and thermal natural convection in a square mould.

It is found that a chosen appearance of the momentum equation mildly affects temporal velocity/temperature, and shapes of the phase interface at longer times of the solidification.

This mainly results from small variations of the liquid fraction across the mushy zone and from a low solidification rate, and it may change drastically when anisotropic properties of the mushy zone, solutal convection, different phase densities and cooling conditions are considered. Therefore, further comprehensive study is needed.

The paper addresses how the different focus of the momentum equation for liquid flow is compared.