Lattice Boltzmann simulation of viscoplastic fluids on natural convection in inclined enclosure with inner cold circular/elliptical cylinders (Part II: Two cylinders)
Introduction
Natural convection of viscoplastic fluids in an enclosure due to its wide applications and interest in various chemical, metal, and food industries has been considered recently by researchers. Vola et al. [1] studied the natural convection in a cavity filled with a viscoplastic fluid using the Bingham model without any regularisation of the constitutive law. They applied a numerical method based on the combination of the characteristic/Galerkin method to cope with convection and of the Fortin-Glowinski decomposition/coordination method to deal with the non-differentiable and nonlinear terms that derive from the constitutive law. However, the streamlines and isotherms for various yield stress values were limited to one value of the Rayleigh number (Ra = ). Turan et al. [2] conducted a study into the simulations of natural convection in square enclosures filled with an incompressible Bingham fluid. The considered flow was laminar and steady. The commercial package FLUENT was utilized to solve the problem. In this study, a second-order central differencing scheme was used for the diffusive terms and a second order up-wind scheme for the convective terms. Coupling of the pressure and velocity fields was achieved using the SIMPLE algorithm. It should be noted that the default Bingham model in FLUENT is a bi viscosity model. The heat transfer and the flow velocities were investigated over a wide range of Rayleigh and Prandtl numbers. They found that the average Nusselt number augments with the rise of the Rayleigh number for both Newtonian and Bingham fluids, whereas the Nusselt numbers of Bingham fluids were smaller than those in Newtonian fluids for a fixed nominal Rayleigh number. They also mentioned that the mean Nusselt number of Bingham fluids decreased with an increase in the Bingham number. Moreover, it was observed that the conduction dominated regime occurs at large values of Bingham numbers. Finally, they reported that for low Bingham numbers, the mean Nusselt number increases with the enhancement of the Prandtl number; by contrast, the opposite behavior was observed for large values of Bingham numbers. Turan et al. [3] continued their studies with analysing the effect of different aspect ratios (the ratio of the height to the length) of the cavity, adding to their previous results that the average Nusselt number follows a non-monotonic pattern with the aspect ratio for specific values of the Rayleigh and Prandtl numbers for both Newtonian and Bingham fluids. At small aspect ratios, the conduction is dominant whereas convection remains predominantly responsible for the heat transfer for large values of aspect ratios. In addition, it was found that the conduction dominated regime occurred at higher values of the Bingham numbers for increasing values of the aspect ratio for a given value of the Rayleigh number. Turan et al. [4] scrutinised the laminar Rayleigh-Bnard convection of yield stress fluids in a square enclosure. The applied method and the achieved results were similar to the two previous studies. Huilgol and Kefayati [5] studied natural convection in a square cavity with differentially heated vertical sides and filled with a Bingham fluid without any regularisation. The finite element method (FEM) based on the operator splitting method was utilized to solve the problem. It was observed that for specific Rayleigh and Prandtl numbers, the increase in the Bingham number decreases the heat transfer. Furthermore, it was found that the growth of the Bingham number expands the unyielded sections in the cavity. Finally, they mentioned that for fixed Rayleigh and Bingham numbers, the unyielded regions grow with the augmentation of the Prandtl number. Karimfazli et al. [6] explored the feasibility of a novel method for the regulation of heat transfer across a cavity. They used computational simulations to resolve the Navier-Stokes and energy equations for different yield stresses.
Many studies have conducted the effect of the presence of the body inside the enclosure on the natural convection of Newtonian fluids and focused on the diverse body shapes such as circular, square and triangular cylinders [7], [8], [9], [10], [11], [12], [13], [14]. Just recently, natural convection of viscoplastic fluids is investigated, using regularized models inside cavities with hot and cold bodies. Baranwal and Chhabra [15] studied laminar natural convection heat transfer to Bingham plastic fluids from two differentially heated isothermal cylinders confined in a square enclosure. They utilized regularization approaches of biviscosity and the Bercovier and Engelman models. They used the finite element method-based solver, COMSOL Multiphysics (version 4.3a) to solve the governing equations. Dutta et al. [16] investigated the effects of tilt angle and fluid yield stress on the laminar natural convection from an isothermal square bar cylinder in a Bingham plastic fluid confined in a square duct. They also applied the same regularization approaches of biviscosity and the Bercovier and Engelman models. They also applied the finite element method-based solver, COMSOL Multiphysics (version 4.3a) to solve the governing equations.
Lattice Boltzmann method (LBM) has been demonstrated to be a very effective mesoscopic numerical method to model a broad variety of complex fluid flow phenomena [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]. Lattice Boltzmann method (LBM) combined with Finite Difference Method (FDM) has been applied for this problem [29]. It was demonstrated to be a successful mesoscopic method for simulation of Non-Newtonian fluids. Independency of the method to the relaxation time in contrast with common LBM provokes the method to solve different non-Newtonian fluid energy equations successfully as the method protects the positive points of LBM simultaneously. Huilgol and Kefayati [30] explained and derived the two and three dimensional equations of continuum mechanics for this method and demonstrated that the theoretical development can be applied to all fluids, whether they be Newtonian, or power law fluids, or viscoelastic and viscoplastic fluids. Following the previous study, Huilgol and Kefayati [31] derived the two and three dimensional equations of this method for the cartesian, cylindrical and spherical coordinates. Double-diffusive natural convection of non-Newtonian power-law fluid in a square cavity was studied by the cited method while entropy generations through fluid friction, heat transfer, and mass transfer were analyzed [32]. In the following study, heat and mass transfer as well as entropy generations in natural convection of non-Newtonian power-law fluids in an inclined porous cavity were studied, applying the method [33], [34]. Kefayati and Huilgol [35] applied this method to simulate the steady flow in a pipe of square cross-section when the pipe is filled with a Bingham fluid. The problem was solved employing the Bingham model without any regularisation. In the next step, Kefayati and Huilgol [36] utilized the mesoscopic method to conduct a two-dimensional simulation of steady mixed convection in a square enclosure with differentially heated sidewalls when the enclosure is filled with a Bingham fluid. The problem was solved by the Bingham model without any regularisations and also by applying the regularised Papanatasiou model. Double-diffusive natural convection and entropy generations of Bingham fluid in a square cavity and open cavities were simulated by this method [37], [38]. Double-diffusive natural convection and entropy generations, studying Soret and Dufour effects and viscous dissipation in a heated enclosure with an inner cold cylinder filled with non-Newtonian Carreau fluid were simulated by the method [39], [40].
The main aim of this study is to simulate natural convection of Bingham fluid in a heated cavity with two inner cold cylinders as the yielded/unyielded sections have been displayed. In this study, the Bingham model without any regularization has been studied and moreover viscous dissipation effect also has been analyzed. LBM has been employed to study the problem numerically. Moreover, it is endeavored to express the effects of different parameters on heat transfer as well as yielded/unyielded zones. The obtained results were validated with previous numerical investigations and the effects of the main parameters (Rayleigh number, Bingham number, Eckert number, the size and position of the cold cylinder) on unyielded parts and heat transfer are researched.
Section snippets
Theoretical formulation
The geometry of the present problem is shown in Fig. 1. The temperatures of the enclosure walls have been considered to be maintained at high temperature of as the inner cylinders are kept at low temperature of . The cylinders have been set in a vertical positions with the same area. The lengths of the enclosure sidewalls are L where the first and second cylinders centers are defined by (, ) and (, ); respectively. The first cylinder is located on the top side of the cavity
The numerical method
The LBM equations and their relationships with continuum equations have been explained in details in Huilgol and Kefayati [30]. Here, just a brief description about the main equations would be cited. In addition, the applied algorithm has been described and the studied problem equations in the LBM are mentioned.
Applied parameters, code validation and grid independence
Lattice Boltzmann Method (LBM) scheme is utilized to simulate laminar natural convection in a heated enclosure with two inner cold cylinders that is filled with a viscoplastic fluid in the presence of the viscous dissipation in the energy equation. The Prandtl number is fixed at Pr = 0.1. This problem is investigated at different parameters of Rayleigh number (Ra = , and ), Eckert number, the size of the inner cylinder, various inclined angles of the cavity (, 40°, 80°, 120°), the
Results and discussion
Fig. 4 shows the isotherms, streamlines and yielded/unyielded parts in different Rayleigh numbers at = 0.4, Bn = 1, Ec = 0, , and a = b = 0.1 L. At Ra = , the temperature contours are circular around the cylinder which demonstrate the conduction is dominated in the enclosure. As the Rayleigh number increases, the movements of the isotherms between the cold cylinder and hot walls ameliorate significantly and they become progressively curved. Moreover, the gradient of temperature on the
Concluding remarks
Natural convection of viscoplastic fluids in an inclined heated enclosure with two inner cold cylinders in the presence of viscous dissipation has been analyzed by Lattice Boltzmann method (LBM). In this study, the Bingham model without any regularization has been studied for the simulation of viscoplastic fluids. Fluid flow, heat transfer, and yielded/unyielded have been conducted for certain pertinent parameters of Rayleigh number (Ra= and ), Bingham number (Bn = ), Eckert
Acknowledgements
The first author’s (GH. R. K) gratefully acknowledges funding support in the form of a Research Fellowship awarded him by the Hong Kong Polytechnic University.
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