Numerical investigation of convective heat transfer in fluid flow past a tandem of triangular and square cylinders in channel

A numerical study of a two-dimensional laminar flow and heat transfer characteristics in a horizontal channel containing two in-line obstacles of different shapes, namely an upstream triangular cylinder and a downstream square cylinder, was carried out. It is based on a coupling between the Lattice Boltzmann Method (LBM) and the Finite Difference Method (FDM). The airflow (Pr = 0.71) is assumed to be laminar and incompressible. All physical properties of the fluid, expect its density, are supposed to be constant. The two cylinders are kept at a constant hot temperature, while the incoming flow is at a cold temperature. The Reynolds number and the horizontal separation distance between the cylinders are varied to investigate their influences on fluid flow and heat transfer.


Introduction
Flow around cylinders has been a subject of research for many scientists for a long time due to its practical applications. The majority of numerical or experimental investigation has focused on the effects of the form, number, and position of cylinders, Reynolds number, and the boundary of the computational domain. Many applications can be found in heat exchangers, buildings, bridges, mechanical systems, and electronic devices. Looking at the literature, one can conclude that the majority of numerical papers are about the flow passes on bluff bodies with the same shape [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] such as circular cylinders [1][2][3][4] square cylinders [5][6][7][8] rectangular cylinders [9][10][11][12], and triangular cylinders [12][13][14][15][16]. All these works are done with a variety of numerical methods. From the above, it can be concluded that very limited attention has been focused on problems associated with the flow around two cylinders composed by the square cylinder and triangular cylinders or used triangular cylinders as disturber cylinders in front of square cylinders. For this reason, this study concentrates on the effects of Reynolds number and horizontal distance separation in a channel containing two in-line obstacles of different shapes, namely a triangular cylinder upstream and a square cylinder downstream. Fig. 1 illustrates the physical model of the plane channel with the two obstacles. The air flow (Pr = 0.71) is considered laminar and incompressible. With the exception of its density, all the physical properties of the fluid are considered invariable. The L/d ratio is fixed at 30.75 and the blockage ratio IOP Publishing doi:10.1088/1757-899X/1091/1/012058 2 is d/H=1/4. The triangular cylinder is located at a distance Xin/d = 8 downstream of the inlet section of the channel. Both cylinders are maintained at a constant hot temperature h = 0.5, whereas the incoming flow is at a cold temperature c = -0.5. The input velocity profile is parabolic. The boundary conditions implemented on the top and bottom walls of the channel are u = v = 0, for velocity, and adiabatic for temperature. For the boundary conditions, the rebound scheme and a spatial quadratic interpolation [17,18] are used.

Numerical simulation
To simulate the fluid flow, the numerical method used is the multi relaxation time (MRT) lattice Boltzmann method [19] which can be formulated as follows: Where M is the 9×9 transformation matrix such as m = M.f and S is the relaxation matrix where is a diagonal matrix, i.e., S = diag(0,s 1 ,s 2 ,0,s 4 ,0,s 6 ,s 7 ,s 8 ).
We consider a D2Q9 model and the particle speed 2 cos (2 9) / 4 ,sin (2 9) / 4 , 5 8 Where c= dx/dt is the lattice speed, and dx and dt are the lattice width and time step, respectively. Here, dt is chosen to be equal to dx, thus c=1.
The macroscopic fluid density and moment flux are calculated by: The finite difference method [19,20] is used to solve the macroscopic temperature. An explicit coupling between both schemes MRT and FDM is carried out. The energy equation is given by: The above equation is resolved explicitly by using first-order direct time difference scheme and the second-order central difference scheme for space discretization, with the same grid points as for the LBM scheme.

Results and discussion
The numerical code (HTLBM) has been validated in the previous work (see [21][22][23][24][25]). A preliminary work was performed to identify the optimum grid. Various simulations were carried out for w= 4d, Re = 100, and for different uniform grids (Nx×Ny) in order to examine the grid independence. To optimize grid refinement with simulations efficiency, the grid 1567×203 was selected for all the further calculations.
The local Nusselt number (Nu) based on square cylinder height is expressed by the following relationship:  (Re=20, 40, 80, and 100). In general, when a cylinder is positioned in the wake of another cylinder in the transverse flow, its unsteady charge becoming dependent not only on the flow characteristics in its wake but also on those in the wake of the upstream cylinder. For Re=20, It is apparent that behind each cylinder, two symmetrical vortex zones appear on either side of the wake which turns in opposite senses. The velocities vectors present a perfect symmetry in the channel.  For Re=80 and 100, the flow structure is changed to be asymmetric. The shape of the temperature fields reflects the fluid motion. An indicative temperature gradient is generated around the cylinders reflecting the presence of a significant heat transfer in these zones. In order to to understand the influence of varying gap spacing on fluid flow and heat transfer, Figure 4 and 5 illustrate the instantaneous velocity profiles and isotherms at Re= 100 and for various spacing ratios W=2d, 4d and 6d. Let us note that at low Reynolds number (Re=20), the flow present a constant symmetry and the gap spacing don't affect the behaviors of the fluid flow and the heat transfer. When Re=100, two modes are observed depending on the gap space between the two cylinders. When w=2d, the square cylinder is very close to the triangular; which prevents the appearance of vortex in the space between them. As w increases to w=4d and 6d, the flow the flow pattern is qualitatively changed. An alternate vortex are created between the cylinders and in the downstream of the second cylinder. At w=2d there is no flow separation and the isotherms have diffusion-type profiles while they become more non-uniform at w=4d and 6d because of the flow separation.    Re=100, a higher fluid velocity results in a stronger temperature gradient, which means an increased heat transfer from the cylinder surface. On the front face, Nu increases as d increases, but decreases on both the side faces comparing w=2d and w=4d.

Conclusion
The 2D HTLBM was implemented to laminar flow (air: Pr=0.71) and the heat transfer in a channel containing a tandem of triangular cylinder and square cylinder for different Reynolds number and gap spacing. Based on the results, one can conclude that the flow structure is strongly influenced by the Reynolds number and the gap space. The heat transfer is clearly affected by varying gap spaces between the two cylinders, particularly for large Reynolds numbers.