## 1. Introduction

Natural convection heat transfer in an annular space between two concentric cylinders (also known as concentric annuli) is one of the most studied problems in the field of heat transfer. This fundamental problem has attracted many researchers because of its significance in many engineering applications such as the design of heat exchanger devices, solar energy collectors, cooling of electric power cables, nuclear and chemical reactors, food processing devices, aircraft cabin insulation, etc. [

1,

2]. For early research works of theoretical and/or experimental investigations on natural convection in an annular space between cold outer and hot inner cylinders, one can find the literature [

1,

2,

3,

4,

5,

6,

7,

8]. Studying the behavior of natural convection flow in an annulus with irregular geometries (other than the square or rectangular such as circular, elliptical, triangular, and hexagonal) by using the numerical simulations is highly challenging due to complex irregular boundaries. The irregular boundary problems are generally treated with unstructured (body-fitted) grid methods, also known as conforming-mesh methods, which are very complicated and are computationally intensive. In the past two decades, many researchers have paid attention to develop non-conforming-mesh methods, which use a fixed Cartesian grid to simulate the fluid flow in complex geometries. Some of those numerical methods are immersed boundary method (IBM) [

9], distributed Lagrange multiplier method or fictitious domain method [

10], and smoothed profile method (SPM) [

11,

12,

13].

In the past three decades, the lattice Boltzmann method (LBM) has evolved as a powerful computational technique for solving fluid flow and heat transfer problems. In LBM, one gets the solution for the particle density distribution functions (PDF) (by solving Boltzmann kinetic equation on a discrete lattice mesh) instead of directly solving the pressure and velocity fields. The macroscopic variables (such as pressure, velocity, and temperature) are obtained by calculating the hydrodynamic moments of PDF [

14]. Because of its many advantages [

15] compared to the classical Navier–Stokes equations solvers, LBM has been successfully used to simulate various Multiphysics problems such as multiphase flows [

15,

16,

17], magnetohydrodynamic (MHD) flows [

18,

19], micro- and nano-flows [

20,

21,

22] and fluid-solid interactions [

13,

23,

24,

25,

26]. LBM has also been successfully implemented to predict the behavior of the fluid flow due to natural convection in complex geometries [

27,

28,

29,

30,

31,

32,

33,

34,

35,

36,

37]. In the above-mentioned research works on natural convection, the following techniques have been used to handle the no-slip and constant temperature BC on complex irregular surfaces: the bounce back (BB) scheme [

27,

28,

29,

30,

31,

32], IBM [

33,

34,

35], and SPM [

36,

37].

BB rule was first proposed by the Ladd [

38,

39] to impose the no-slip BC at curved surfaces of solid particles. In this scheme, the irregular surface of a solid body is imagined as a flat edge that lies in-between two neighboring solid and fluid grid points. The no-slip BC can then be achieved with the help of the standard mid-plane BB scheme which bounces back the missing distribution functions coming from the solid nodes to the fluid nodes. Later, Bouzidi et al. [

40] and Yu et al. [

41] developed an improved version of the Ladd scheme, known as the interpolated bounce back (IBB) scheme to achieve the second-order accuracy for the fluid velocity and temperature. Sheikholeslami et al. [

27,

28] investigated MHD flow and heat transfer in an annular space between a heated inner circular and a cold outer square cylinder and they reported the fluid flow and heat transfer results at various Ra and

AR values. Lin et al. [

29] performed simulation of natural convection flow in an annulus between a heated inner circular cylinder, which is located eccentrically, and a cold square enclosure. Bararnia et al. [

30] simulated the natural convection between a heated inner elliptical cylinder and a square outer cylinder. They reported the fluid flow and heat transfer characteristics for various combinations of the vertical positions of the inner cylinder and Ra. Sheikholeslami et al. [

31] studied the effect of a magnetic field on the fluid flow and heat transfer characteristics of a nanofluid inside a circular cylinder with an inner triangular cylinder. Moutaouakil et al. [

32] conducted lattice Boltzmann simulations of natural convection in an annulus between an inner hexagonal cylinder and an outer square cavity. In all the above-mentioned articles [

27,

28,

29,

30,

31,

32], IBB scheme was used to treat the complex boundaries of circular, triangular, elliptical, and hexagonal geometries. Even though IBB scheme can effectively be used for treating the complex curved boundary problems, the main drawback is that there may be fluctuations in the velocity and temperature fields at fluid-solid interfaces especially when the solid boundary is moving with a certain velocity.

In IBM, the complex irregular boundaries of solid bodies are represented with a set of Lagrangian nodes while the evaluation of the fluid flow is considered on a fixed Eulerian grid. To enforce the no-slip and constant temperature BC on the solid nodes, artificial body force and heat source terms are added to fluid momentum and energy equations, respectively. On can refer to the review article by Mittal and Iaccarino [

42] for a clear discussion on different approaches for calculating the body force terms. Hu et al. [

33] simulated natural convection in a concentric annulus of circular cylinders using LBM and they used IBM to treat the no-slip and isothermal BC on the curved boundaries. Hu et al. [

34] developed an immersed boundary lattice Boltzmann method (IBLBM) for simulating fluid flow due to natural convection in a cold square cavity with a heated inner circular cylinder covered by a porous layer. They investigated the effects of thermal conductivity ratio, Ra, and Darcy number on the behavior of fluid flow and heat transfer. Khazaeli et al. [

35] used IBLBM to simulate the natural convection due to a hot circular cylinder inside a square and circular enclosures (cold) for different Ra values. IBM could resolve the problem of fluctuations in the velocity field of IBB scheme. However, the main disadvantage of IBM is that it requires complex interpolation functions, and needs a lot of data exchange between the fluid (Eulerian) and solid (Lagrangian) nodes. Therefore, the parallel computational performance of the scheme based on the LBM and IBM becomes lower as the global data communication between the neighboring grid points increases [

13].

In SPM, a smoothed profile function is used to recognize the complex surfaces of a solid body, and the same grid system is used for fluid and solid. The no-slip and isothermal BC at the complex surfaces are implemented by adding a hydrodynamic force and heat source terms to the fluid momentum and energy equations, respectively. The main advantage of SPM over the other non-conforming-mesh methods is that all operations are completely local to a grid point as both fluid and solid are represented with the same grid system; so, the implementation of this scheme to parallel computing applications is easier [

13]. Also, SPM does not need any complex interpolation function as needed by IBM. Although SPM is computationally more efficient and easier to apply than IBB and IBM schemes, till now, only a few researchers have used the method based on LBM and SPM to study the fluid flow and heat transfer behavior in complex boundaries. Hu et al. [

36] used the LBM combined with SPM for simulating natural convection in complex irregular geometries. They reported the simulation results for the velocity and temperature inside a square enclosure with a hot circular cylinder for different values of Ra and

AR. All the above-mentioned research works [

27,

28,

29,

30,

31,

32,

33,

34,

35,

36] considered the double populations model (DPM) (where two sets of PDF are used: one set for solving the velocity field and another one for the temperature field). Recently, Alapati et al. [

37] developed a numerical technique based on the combination of LBM and SPM to simulate particulate flows with heat transfer. They used a hybrid method (HM), which solves the fluid flow by LBM with a set of PDF and the temperature field by FDM and concluded that LBM-SPM method based on HM is computationally more efficient than the method based on DPM.

Fluid flow and heat transfer thorough or over hexagonal-shaped geometries is a ubiquitous problem in many engineering applications such as solar energy collectors [

43,

44], nuclear power plants [

45], microfluidic heat sinks [

46], lamella type compact heat exchangers [

47], air-conditioning applications [

48], etc. In solar energy collectors, to minimize the radiation and convection losses to the surrounding atmosphere, an array of transparent tubes, arranged in a hexagonal honeycomb pattern, is used in-between the absorbing surface and cover plate. Marshall et al. [

43] and Buchberg et al. [

44] found that the thermal efficiencies of honeycomb solar collectors were higher compared to the collectors without the honeycomb layer. The fuel rods of a nuclear reactor core are stacked in the form of a hexagonal lattice and are located inside a circular or hexagonal channel to pass a coolant longitudinally over them [

45]. In a ministered heat sink used for electronic systems cooling, an array of pin-fins of various cross-sectional shapes are attached to a microchannel wall. Aliabadia et al. [

44] found that hydrothermal (hydraulic and thermal) performance was best for the pin-fins with the circular and hexagonal cross-sections. Hexagonal duct shape is a commonly used shape in lamella type compact heat exchangers, which is used in many industries such as pulp and paper, alcohol, petrochemical, and other chemical industries [

47]. A desiccant disk used in air-conditioning applications consists of an array of several ducts packed in the form of honeycomb pattern. Zhang [

48] found that the heat and mass transfer efficiencies of ducts with hexagonal cross-section were higher compared to circular and rectangular ducts as the hexagonal duct walls are more uniformly placed in the desiccant wheel.

Even though it has great significance and applications only a few researchers have investigated the behavior of natural convection flow in an annulus with a heated hexagonal cylinder. Boyd [

3] experimentally investigated natural convection in an annulus with a heated hexagonal inner cylinder and cold outer circular cylinder. Raithby et al. [

6] simulated the natural convection in an annulus bounded by a circular cylinder outside and horizontal hexagonal cylinder inside by using an orthogonal curvilinear coordinates system (a body-fitted grid system). Galkape and Asfaw [

7] employed a non-orthogonal coordinate system to study the same problem of Raithby et al. [

6]. More recently, Moutaouakil et al. [

32] studied the natural convection due to a hot hexagonal cylinder inside a square enclosure (cold) with LBM. They presented the fluid flow and heat transfer characteristics for different combinations of

AR and Ra by considering two orientations for the hexagonal cylinder. As mentioned earlier, they used DPM with IBB scheme to treat BC on hexagonal surfaces.

The literature review showed that the problem of natural convection in an annulus space between two hexagonal cylinders (concentric with each other) has not been studied. The main objective of this work is to solve the problem of two-dimensional natural convection flow caused by a hot hexagonal cylinder placed concentrically inside a cold hexagonal cylinder. The numerical technique that combines LBM, SPM, and FDM methods is employed because it offers many advantages over the other methods. In the present work, an equation for smoothed profile function that identifies the hexagonal boundaries is proposed. Assessing the accuracy and robustness of the present simulation technique by comparing results given by the present method with ANSYS-Fluent

^{®} results is also another purpose of this work. The remainder of this paper is arranged as follows. A brief description of the simulation technique is given in

Section 2. A discussion on numerical results is presented in

Section 3. At first, the validation results, by applying the present simulation scheme to a standard benchmarking test, are provided. Later, the simulation results of streamlines, isotherms, and temperature and velocity distributions inside the concentric hexagonal annulus are presented. The concluding remarks of the present study are provided in

Section 4.