CONVECTIVE FLOW AND HEAT TRANSFER ANALYSIS BY USING THERMAL LATTICE BOLTZMANN METHOD

Abstract

Over the years, great extent of research efforts are devoted towards the investigation of hydrodynamic characteristics of thermally driven flows and transport processes, due to their fundamental and pragmatic significance. Thermally (or buoyancy) driven flows are widely encountered in diverse fields of nuclear reactor systems, meteorology, geophysics, energy storage and conservation, fire control, and chemical, food, and metallurgical industries, as well as in the conventional fields of the fluid and heat transfer processes (Roy and Basak, 2005). Among others, investigation of natural convection heat transfer in closed, as well as open, ended cavities is considered as an important research field due to the wide ranges of the industrially important applications namely chemical vapor deposition (Spall, 1996), cooling devices in electronic equipment (Bilgen and Muftuoglu, 2008; Hsu and Wang, 2000; Du et al., 1998), polymer and material processing (Hsiao, 2007; Habib et al., 2005), solar collectors (Hobbi and Siddiqui, 2009), electronic card arrays (Manca and Nardini, 2010) and domestic refrigerators, oven (Skok et al., 1990). It is not to be mentioned that the flow and heat transfer in cavity is also considered as one of the bench marking problem in the development and testing of numerical and computational fluid dynamics solver. An ideal representation of natural convection heat transfer is generally based on the thermal conditions on the cavity walls, i.e., one wall maintained isothermally at higher temperature while other walls are either kept isothermally at lower temperature or maintained adiabatically or open to the ambient. In chemical and process industries, however, such ideal conditions deviate due to the practical and measurement limitations and they, in turn, leads to the non-linear heating/cooling of the cavity walls. It, therefore, necessitates the investigation of the partially/non-linearly heated cavities and their influences on the natural convection characteristics. In spite of their wide occurrence in the ranges of practical/industrial applications, very limited results are available for cavity having varied iii iv combinations of partial heating arrangements (Varol et al., 2008; Aghajani Delavar et al., 2011; Sankar et al., 2011; Nikbakhti and B., 2012; Jmai et al., 2013). Thus, in this work, an attempt have been made to fulfill the gap available in literature for convective heat transfer in enclosures. The natural convection heat transfer in enclosure (closed as well as open ended) is, therefore, studied herein for laminar range of Rayleigh number (heat intensity parameter for buoyancy driven flows), Prandtl numbers, heating size and locations. Further attempts are also made to investigate the magneto-hydrodynamic (MHD) effects on natural convection heat transfer in partially heated square cavity. Similarly, the flow and heat transfer across bluff bodies in particular cylinders and spheres, is considered as one of the fundamental as well as classical problem in the area of fluid mechanics. Owing to its fundamental and practical significance, the flow across bluff bodies (cylinder of circular and non-circular cross-sections and spheres) have been explored well over the centuries, for instance, see (Zdravkovich, 1997a,b; Chhabra, 1996, 1999; Dhiman et al., 2006a,b, 2007; Bharti et al., 2007). The review of the available literature suggests that the flow across circular cylinders have been explored in greater details in comparison of rectangular cylinders (?Sharma and Eswaran, 2005; Dhiman, 2006; Bharti et al., 2007; Dhiman et al., 2007; Sahu, 2010), etc. It is, however, greatly acknowledged that the flow characteristics of square cylinders, i.e., gross engineering parameters such as drag coefficient, Nusselt number, wake size, etc., are often used in the designing of cooling towers, antennas, chimneys, support structures, high rise buildings, etc (Chatterjee et al., 2009; Sharma et al., 2012). Though reasonable amount of information is available for flow past bluff bodies other than circular cylinder, it is neither extensive and comprehensible. The available literature encompasses the influence of wall blockage on flow and thermal characteristics have been explored, but for the limited range of blockage (β ≤ 1/8) and/or aspect ratio (AR ≤ 6). The present work aims to extend the literature knowledge for the wide ranges of both blockage and aspect ratios of a rectangular cylinder. In particular, the influences of wall blockage and aspect ratio on forced convection flow and heat transfer from rectangular cylinders have been investigated numerically for the wide ranges of the flow governing parameters. Over the past decades, the lattice Boltzmann method (LBM) has been established as a promising numerical tool of computational fluid dynamics (CFD) for solving various problems of complex fluid flows and heat transfer. The lattice Boltzmann method has derived from Boolean variables based lattice gas automata (LGA). It is, therefore, considered as an alternative numerical tool to conventional CFD numerical tools, which are based on the macroscopic continuum equations (Mishra et al., 2005; Mishra and Roy, 2007; Mishra v et al., 2008; Mondal and Mishra, 2009). The lattice Boltzmann method basically solves a kinetic and discrete velocity based Boltzmann equation (in statistical physics) (Succi et al., 1989; Chen and Doolen, 1998). The LBM has been successfully applied in the varieties of complex fluid flows involving porous structures (Succi et al., 1989; Kao et al., 2007), magneto-hydrodynamic (Chen et al., 1991a; Sheikholeslami et al., 2012), non-Newtonian rheology Delouei et al. (2014); Nazari and Ramzani (2014), reaction-diffusion (Dawson et al., 1993), diffusion-dispersion (Mohamad et al., 2009), suspension flows (Sankaranarayanan et al., 2002), compressible flows (Yu and Zhao, 2000), multiphase flows (Chen et al., 1991b), nanotube effect (Jafari et al., 2014) etc. The advantages of the LBM are simplicity of coding and algorithm, ease in application of boundary conditions (thus, suitable for complex fluid flow problems), ease of parallel computing, an adroitness estimation of pressure field as compared to conventional CFD tools, etc (Chen and Doolen, 1998). Keeping in mind the simplicity and efficiency, aforementioned investigations have been carried out by using lattice Boltzmann method (LBM) based computational flow solver, developed in C++ programming language in the present work. The flow and thermal field in LBM can be simulated by using three approaches, viz., multispeed, double distribution function (DDF) approach and passive scalar (or simplified DDF). In the present work, passive scalar- thermal lattice Boltzmann method (PS-TLBM) based on simplified double distribution function model (He et al., 1998; Peng et al., 2003b) is used to solve field equations. The basic validation of the present LBM code is ascertained through the standard benchmark problems of 2D lid driven cavity (Ghia et al., 1982b) and flow between parallel walls. For validation of flow through channel, the comparison of analytical solution of the fully developed velocity profile along vertical axis of channel is carried out. The comparison of present results with available for both cases shows close agreement, thus lending the credibility in the reliability and accuracy of the numerical results developed by in-house LBM code. Further, optimum grid size is chosen by carrying out grid independence for all considered problems herein. For the problems of flow past a rectangular cylinders, the proper choice of domain parameters (upstream length, downstream length, etc.) is very important as it has influence on the accuracy of the solution. Thus, systematic study is carried out for the selection of these parameters. In this work, the extensive results elucidating the influence of flow governing parameters on the local and global flow and thermal characteristics of flow problems (briefed in vi Table 1: Flow problems considered in this work with their ranges of parameters. Sr. No. Problem Physical parameters 1. Differentially heated cavity∗ 0.71 ≤ Pr ≤ 100 104 ≤ Ra ≤ 106 2. Partially-differentially heated cavity∗ 104 ≤ Ra ≤ 106 Pr = 0.71; Lh = Lc = 1/2 3. Magneto-hydrodynamic partially heated cavity∗ Pr = 1; 103 ≤ Ra ≤ 105 0 ≤ Ha ≤ 120; θ = 0o, 45o, 90o 1/6 ≤ Lc ≤ 1; Lh = 1/2 4. Partially heated open ended cavity∗ 0.71 ≤ Pr ≤ 7; 103 ≤ Ra ≤ 106 Ll=Middle, top, bottom Lh = 1/4, 1/2, 3/4 5. Square cavity with built-in heated square block∗ 0.71 ≤ Pr ≤ 10 104 ≤ Ra ≤ 106 Hs = 0.15H 6. Flow past rectangular cylinder# 5 ≤ Re ≤ 40; 1/8 ≤ β ≤ 1/20 Pr = 1; AR = 1, 2, 4, 6 *: Natural convection, #: Forced convection, θ: Angle of magnetic field, Ll : Heating location, Lh : Heater size, AR: aspect ratio of rectangular cylinder (width/height), β(b/H): Blockage ratio, b: side of square, H: Height of channel, Hs: height of square cylinder next paragraphs) are obtained by using the in-house developed PS-TLBM solver. In particular, dependence of local characteristics (streamlines, vorticity, pressure, isotherm profiles) and gross engineering parameters (individual and total drag coefficients, local and average Nusselt numbers, etc.) on the flow and geometrical parameters (Reynolds number, Rayleigh number, Prandtl number, heater and cooler size, heating location, Hartmann number etc.) are presented. The ranges of conditions used in various problem is detailed in Table 5.1. A brief description of the problems considered herein is presented below. 1. Natural convection in differentially heated square cavity: Effect of Prandtl and Rayleigh numbers The influence of wide range of Prandtl numbers on natural convective heat transfer in differentially heated closed cavity have been elucidated by using thermal lattice Boltzmann method (TLBM) for laminar range of Rayleigh number. Natural convection effect increases with the increase in Prandtl number (Pr) for all values of the Rayleigh number (Ra) due to the increasing dominance of viscous forces over the inertial forces. As thermal diffusion is inversely proportional to Prandtl number, velocity is more diffused than thermal energy. The average Nusselt number (dimensionless heat transfer coefficient) of vii isothermal wall is seen to increase with increasing value of both the Prandtl and Rayleigh numbers. 2. Natural convection in partially-differentially-simultaneously heated/cooled square cavity The influence of one wall of cavity exposed to contrast (i.e., both hot and cold) thermal conditions on natural convective heat transfer have been explored. The one wall of cavity is equally exposed to hot and ambient conditions and other wall exposed to ambient. The flow governing parameters used for numerical experimentation are Rayleigh number in laminar range with heater size, Lh = 1 2 with air (Pr = 0.71) as a working fluid. The results indicated the formation of convection cell near lower part of mixed heated wall of cavity is observed for Ra ≥ 104, as low temperature fluid retained in that region. The size of convection cell increases with the increase in Rayleigh number (Ra). The average Nusselt number (Nu) and overall Nusselt number (dN u) value show linear increase with Rayleigh number. 3. Magneto-hydrodynamic natural convection in partially heated square cavity In this problem, the influence of cooler size, Hartmann number, Rayleigh number and angle of magnetic field direction on natural convection heat transfer in differentially as well as partially heated cavity is elucidated. The cavity considered is partially heated at middle location (1/4 ≤ Lh ≤ 3/4) at one wall while other wall is partially cooled for different cooling length (Lc). The other part of vertical walls except heated/cooled are kept at adiabatic thermal condition. The top and bottom walls are also maintained adiabatically. It is observed that temperature contours move towards partially heated wall, which increases the temperature gradients, hence, enhancing the rate of heat transfer (average Nusselt number values). Also the rate of heat transfer increases with both Hartmann and Rayleigh number, while the angle of magnetic field has marginal influence on heat transfer rate. 4. Natural convection in partially heated open ended square cavity The natural convection heat transfer analysis in a partially heated open ended square cavity have been carried out to elucidate the influence of heater size and heating location. First, effect of three heating locations (middle, top, bottom) and heater size (Lc = 1/4, 1/2, 3/4) for Pr=0.71 and, secondly, effect of Prandtl number (0.71 ≤ Pr ≤ 7) on partially heated open ended cavity (heated at the middle location of vertical wall), on heat transfer characteristics are analyzed herein. Linear dependence of the average Nusselt number (Nu) on the Rayleigh number is observed, irrespective of the heating locations and heater size. However, average Nusselt number (Nu) shows a proportional viii dependence for the bottom and middle locations and inversely proportional dependence for the top heating location on the heater size, i.e., an increasing value of Lh enhanced Nu for the bottom and middle locations and deteriorated Nu for the top heating location. Over the range of Rayleigh number, middle partial heating location shows higher heat transfer rate followed by bottom and top heating locations. The results also indicated the strong influence of Prandtl numbers on rate of heat transfer. As expected, the average Nusselt number values increased with both Prandtl and Rayleigh number. Finally, a closure relationship between average Nusselt number with Prandtl and Rayleigh numbers have developed in standard form. 5. Natural convection in square cavity with built-in square block In this problem, the vertical walls of square cavity is exposed to the ambient (Tc) with horizontal walls maintained at adiabatic condition. A heated square block (Th) is placed at the center of cavity. The natural convection characteristics have been explored for range of fluids (0.71 ≤ Pr ≤ 10). It is observed that the heated block has significant effect on the nature of flow inside cavity. The circulation of fluid between active wall causes formation of plume over the top wall of square block. The Prandtl number variation causes significant change in structure of the plume. With the increase in Prandtl number the length of plume decreases. Moreover, the increase in Prandtl number causes isotherms patterns to be more confined towards the heated walls. The circulation of fluid between cold cavity walls with heated square block is decipited in the form of streamlines. The Prandtl number has remarkable influence on the size of this quasi-motionless region, i.e., increasing in Prandtl number decreases the size of this region. 6. Wall effects on forced convection flow and heat transfer from channel builtin rectangular cylinder The effect of wall confinement on the momentum and heat transfer characteristics of a channel built-in rectangular cylinder (1 ≤ AR ≤ 6) for blockage ratios (1/8 ≤ β ≤ 1/20), Reynolds numbers (5 ≤ Re ≤ 40) and Prandtl number (Pr = 1) have been explored. The results indicated that the increase in blockage ratio causes marginal increase in recirculation length for considered range of Reynolds number. The drag coefficient values are found to be in inverse proportion with blockage ratio and Reynolds number. Furthermore, for a fixed Reynolds number, increase in blockage ratio causes crowding of isotherms in the vicinity of cylinder. Higher surface pressure coefficient (CP ) values are obtained for front face of cylinder at low blockage ratio. Thus, increasing blockage ratio reduces CP values along the cylinder surface. Linear increase in average Nusselt number (Nu) is observed with Reynolds number and lower blockage ratio. Thus, increase in blockage ratio impedes ix rate of heat transfer. The Colburn heat transfer factor jH is strongly dependent on blockage ratio. Finally, an empirical correlations relating total drag coefficient (CD) and average Nusselt number (Nu) with blockage ratio (β) and Reynolds number (Re) have been developed for its possible use in engineering design purpose. It is observed that drag as well as average Nusselt number have linear dependence on aspect ratio of rectangular cylinder. In summary, the detailed insights of the natural and forced convection flow and heat transfer have been gained and presented for wide ranges of flow governing parameters and geometrical parameters. In addition, the present study also successfully developed and utilized the passive scalar thermal lattice Boltzmann method (PS-TLBM) with acceptable level of accuracy.