2. Formulation of the problem

of different shapes and sizes. Although the majority of these studies have focused on studying the cross-flow past bluff bodies such as cylinders of circular [1–6], elliptic [7–10], rectangular [11–15] and square cross-sections [16–20], there are fewer studies on the semicircular cylinder geometry [21–24]. Gode et al. [25] studied numerically the momentum and heat transfer characteristics of a two-dimensional (2D), incompressible and steady flow over a semicircular cylinder and pointed out that the wake ceases to be steady somewhere in the range of 120 ≤Re ≤ 130. Boisaubert and Texier [26] performed solid tracer visualizations to assess the effect of a splitter plate on the near-wake development of a semicircular cylinder for Reynolds numbers of Re ¼ 200 and 400 and three splitter plate configurations. Their results show that for Re ¼ 400, the splitter plate causes an increase in near-wake length, a decrease in near-wake maximum width, a secondary vortex formation and a decrease of the maximum velocity in the recirculating zone, while for Re ¼ 200, the near-wake keeps its symmetry and vortex shedding is inhibited. Nalluri et al. [27] solved numerically the coupled momentum and energy equations for buoyancy-assisted mixed convection from an isothermal hemisphere in Bingham plastic fluids and reported results for streamline and isotherm contours, local and mean Nusselt number as a function of the Reynolds, Prandtl, Richardson and Bingham numbers. Bhinder et al. [28] studied numerically the wake dynamics and forced convective heat transfer past an unconfined semicircular cylinder at incidence using air as the working fluid for Reynolds numbers in the range of 80 ≤ Re ≤ 180 and angles of incidence in the range of

. Based on the flow pattern and the angle of incidence, they identified three flow

distinct zones and proposed a correlation for the Strouhal and averaged Nusselt number as a function of Re and α. Chandra and Chhabra [29] performed a numerical study to assess the flow and thermal characteristics from a heated semicircular cylinder immersed in power-law fluids under laminar free and mixed convection for the case of buoyancy-assisted flow. Their results show that as the value of the Richardson and Reynolds numbers increase, the drag coefficient shows a monotonic increase and that the average Nusselt number increases with an

The foregoing survey of literature reveals that although the great majority of research for the flow and heat transfer past a heated hemisphere in cross-flow has been made for an unbounded domain, there are relatively few studies that deal with the investigation of the blockage constraints present in the confined hemisphere problem. Kumar et al. [30] performed a numerical analysis to investigate the forced convection of power-law fluids (power-law index varying from 0.2 to 1.8) around a confined heated semicircular cylinder for Reynolds numbers between 1 and 40 and Prandtl number of 50. They assessed the effects of blockage ratios ranging from 0.16 to 0.50 and found that for a fixed value of Re, the length of the recirculation zone decreased with an increase in the value of n and that the drag coefficients and the averaged Nusselt number

From the foregoing discussion, it is clear that no prior results are available on the flow and heat transfer characteristics past a confined tandem hemisphere array under buoyancy-assisted and buoyancy-opposing conditions. This study aims to fill this void in the existing literature. In this work, we numerically investigate the transient fluid flow and thermal characteristics in the mixed convection regime around two isothermal semicylinders of the same diameter in

increase in the value of the Reynolds, Prandtl and Richardson numbers.

increased with increasing blockage ratio for any value of n.

0<sup>o</sup> ≤ α ≤ 180<sup>o</sup>

222 Heat Exchangers– Design, Experiment and Simulation

#### 2.1. Governing equations and boundary conditions

Consider a 2D steady, Newtonian, incompressible Poiseuille flow fluid with a mean mainstream velocity u<sup>0</sup> at the entrance of a vertical duct over infinitely long semicylinders of diameter D placed symmetrically between two parallel plane walls as shown schematically in Figure 1. A system of Cartesian coordinates ðx, yÞ is used with its origin located at the centre point of the upstream hemisphere. The length and height of the computational domain are defined in terms of the axial and lateral dimensions (Ltot ¼ 30D and H, respectively). The pitch-to-diameter ratio is σ ¼ L=D ¼ 3 and the blockage ratio BR ¼ D=H ¼ 0:2, where L is the longitudinal spacing between semicylinders. The upstream hemisphere is placed at a distance of 5:5D from the inlet to its centre and at a distance S<sup>1</sup> ¼ 24:5D from its centre to the outflow boundary. These values were chosen as they were estimated to be sufficiently large to allow the wake behind the downstream semicylinder to develop properly and to exit the domain without producing observable reflections. The forced flow enters the channel at ambient temperature T0, and the semicylinders have a wall temperature of Tw. Flow rectifiers are placed at the channel exit producing a parallel flow at x ¼ S1. The thermophysical properties of the fluid are assumed to be constant except for the variation of density in the buoyancy term of the axial momentum equation (Boussinesq approximation) and the effect of viscous dissipation is neglected. Using the vorticity (Ω ¼ ∂V=∂X− ∂U=∂Y) and stream function formulation (U ¼ ∂ψ=∂Y, V ¼ −∂ψ=∂X), the flow is described by the nondimensional equations

$$
\frac{
\partial^2 \psi
}{
\partial X^2
} + \frac{
\partial^2 \psi
}{
\partial Y^2
} = -\Omega,
\tag{1}
$$

$$
\frac{\partial \mathfrak{Q}}{\partial \pi} + \frac{\partial \psi}{\partial Y} \frac{\partial \mathfrak{Q}}{\partial X} - \frac{\partial \psi}{\partial X} \frac{\partial \mathfrak{Q}}{\partial Y} = \frac{1}{Re} \left( \frac{\partial^2 \mathfrak{Q}}{\partial X^2} + \frac{\partial^2 \mathfrak{Q}}{\partial Y^2} \right) + Ri \frac{\partial \theta}{\partial Y}, \tag{2}
$$

$$
\frac{\partial \Theta}{\partial \pi} + \frac{\partial \psi}{\partial Y} \frac{\partial \Theta}{\partial X} - \frac{\partial \psi}{\partial X} \frac{\partial \Theta}{\partial Y} = \frac{1}{RePr} \left( \frac{\partial^2 \theta}{\partial X^2} + \frac{\partial^2 \theta}{\partial Y^2} \right). \tag{3}
$$

where V ¼ ðU, VÞ is the dimensionless velocity vector and θ is the dimensionless temperature. In Eqs. (1)–(3),U andV are theX andY components ofV, respectively. All velocity components are scaled with the oncoming mean bulk velocity u0; the longitudinal and transverse coordinates are scaled with the semicylinder diameter D; the time is scaled with the residence time D=u0, τ ¼ tu0=D; the temperature is normalized as θ ¼ ðT−T0Þ=ðTw−T0Þ. In the above equations, the nondimensional parameters are the Reynolds number, Re ¼ u0D=ν, the Prandtl number Pr ¼ ν=α and the Richardson number,

Ri <sup>¼</sup> <sup>g</sup>βðTw−T0ÞD=u<sup>2</sup> <sup>0</sup>, respectively (frequently, instead of using the Richardson number, the Grashof number is employed, Gr <sup>¼</sup> RiRe<sup>2</sup> <sup>¼</sup> <sup>g</sup>βðTw−T0ÞD<sup>3</sup> <sup>=</sup>ν<sup>2</sup>Þ. Here, <sup>g</sup> is the acceleration due to gravity, <sup>α</sup> is the thermal diffusivity, β is the thermal expansion coefficient of the fluid and ν is the kinematic viscosity.

Figure 1. Schematic diagram of the computational domain and the configuration of the semicylinders inside the channel at BR ¼ 0:2 and σ ¼ 3.

Eqs. (1)–(3) have to be solved with the following boundary conditions:

The inflow boundary is specified by a developed velocity profile at the channel inlet

$$\left[\psi - 1/2[1/BR + 3Y - 4(BR)^2Y^3] = \Omega - 12(BR)^2Y = \theta = 0,\tag{4}$$

For the channel walls, <sup>ψ</sup> <sup>¼</sup> <sup>0</sup>, <sup>þ</sup> <sup>1</sup>=BR at the left Y ¼ −1=ð2BRÞ and right walls Y ¼ þ1= ð2BRÞ , respectively. Vorticity at the walls is evaluated using Thom's first-order formula [31],

Unsteady Mixed Convection from Two Isothermal Semicircular Cylinders in Tandem Arrangement http://dx.doi.org/10.5772/66692 225

$$
\Omega\_w = \mathcal{Z}(\psi\_{w+1} - \psi\_w) / \Delta n^2,\tag{5}
$$

where Δn is the grid space normal to the wall. Adiabatic channel walls are assumed, ∂θ=∂Y ¼ 0. Homogeneous Neumann-type boundary conditions are adopted at the channel exit, provided that the outlet boundary is located sufficiently far downstream from the region of interest.

$$
\left. \partial \psi / \partial X \right|\_{x=s\_1} = \partial^2 \psi / \partial X \partial Y \vert\_{x=s\_1} = \partial \Theta / \partial X \vert\_{x=s\_1} = 0,\tag{6}
$$

At the surface of the semicylinders,

Ri <sup>¼</sup> <sup>g</sup>βðTw−T0ÞD=u<sup>2</sup>

number is employed, Gr <sup>¼</sup> RiRe<sup>2</sup> <sup>¼</sup> <sup>g</sup>βðTw−T0ÞD<sup>3</sup>

224 Heat Exchangers– Design, Experiment and Simulation

<sup>0</sup>, respectively (frequently, instead of using the Richardson number, the Grashof

the thermal diffusivity, β is the thermal expansion coefficient of the fluid and ν is the kinematic viscosity.

Eqs. (1)–(3) have to be solved with the following boundary conditions:

ψ−1=2½1=BR þ 3Y−4ðBRÞ

For the channel walls, <sup>ψ</sup> <sup>¼</sup> <sup>0</sup>, <sup>þ</sup> <sup>1</sup>=BR at the left

ð2BRÞ 

at BR ¼ 0:2 and σ ¼ 3.

The inflow boundary is specified by a developed velocity profile at the channel inlet

2 Y3

Figure 1. Schematic diagram of the computational domain and the configuration of the semicylinders inside the channel

, respectively. Vorticity at the walls is evaluated using Thom's first-order formula [31],

� ¼ Ω−12ðBRÞ

2

Y ¼ −1=ð2BRÞ

Y ¼ θ ¼ 0, (4)

Y ¼ þ1=

and right walls

<sup>=</sup>ν<sup>2</sup>Þ. Here, <sup>g</sup> is the acceleration due to gravity, <sup>α</sup> is

$$
\Omega \text{--} \mathcal{Q} (\psi\_{w+1} - \psi\_w) / \Delta n^2 = \theta \text{--} 1 = 0. \tag{7}
$$

No-normal and no-slip boundary conditions are enforced at the surface of each semicylinder. Due to the fact that the value of the stream function is an unknown constant along the surface of each hemisphere, its value is determined at each time step as part of the solution process [32].

With the temperature field known, the rate of heat flux qj is obtained in nondimensional form with the local Nusselt number Nuj, with j ¼ 1, 2 for the upstream and downstream semicylinder, respectively. The local Nusselt numbers are evaluated from the following equation

$$N\mu\_{\dot{f}}(\mathcal{S},\tau) = \frac{|q\_{\dot{f}}(\mathcal{S},t)|D}{(T\_w - T\_0)k} = \left|\frac{\partial\theta}{\partial n}\right|\_{\mathcal{S}}\tag{8}$$

where k is the thermal conductivity of the fluid and S is the surface of the immersed semicylinders. The surface-averaged (mean) Nusselt number is obtained by integrating the local Nusselt number along the surface of each semicylinder

$$\sqrt{N\mu}\_{\circ}(\tau) = \frac{1}{S} \int\_{S} N u\_{\circ}(S, \tau) dS \tag{9}$$

#### 2.2. Numerical solution

The governing equations are discretized using the power-law scheme described by Patankar [33] using a nonuniform staggered Cartesian grid with local grid refinements near the immersed semicylinders and near the channel walls. Eqs. (1)–(3) along with their corresponding boundary conditions are solved using a finite volume-based numerical method developed in Fortran 90 using parallel programming (OpenMP). Internal flow boundaries in the flow field are specified using the immersed boundary method [34]. For all computations, water is used as the cooling agent ðPr ¼ 7Þ. A stringent convergence criteria of the dependent variables of 1×10<sup>−</sup><sup>7</sup> is used, with an optimal time step of <sup>Δ</sup><sup>τ</sup> <sup>¼</sup> <sup>5</sup>×10<sup>−</sup><sup>4</sup> . A fully developed base flow is assigned as the initial value to each grid point in the domain, which physically means that both semicylinders are introduced into an isothermal fully developed cross-flow. For a given value of the Richardson number, computation is started immediately after the sudden imposition of a uniform wall nondimensional temperature from 0 to 1 on both semicylinders at time τ ¼ 0. Transient calculations are performed up to 500 nondimensional time units. In order to make comparisons with experimental results obtained on what are effectively unbounded domains, Chen et al. [35] defined a Reynolds number, ReD ¼ uDD=ν, where

$$u\_D = \frac{1}{D} \int\_{-D/2}^{D/2} u(y) dy. \tag{10}$$

In Eq. (10), u is the vertical component of the velocity field specified on the upstream boundary and uD is the average longitudinal velocity based on the diameter of the semicylinder. The accuracy of the numerical algorithm was tested by comparing results of the mean Nusselt number against available analytical [2] and numerical results [35] for the standard case of a symmetrically confined isothermal circular cylinder in a plane channel. Details about the numerical solution, validation of the algorithm and the grid employed can be found elsewhere [36, 37].

### 3. Results and discussion

The numerical results presented in this work correspond in all cases to ReD ¼ 200, Pr ¼ 7, BR ¼ 0:2, and σ ¼ L=D ¼ 3. In this section, results are presented for the mean and instantaneous flow and thermal characteristics under varying thermal buoyancy. For clarity, only a portion of the computational domain is shown. The images display (from left to right) the nondimensional longitudinal and transverse velocity components with superimposed streamlines, the nondimensional vorticity field and the temperature field with superimposed velocity profiles. The color scales below each image map the velocity, vorticity and temperature contours, with red/yellow coloration representing positive vorticity or counterclockwise fluid rotation and the green regions reflecting a lack of rotational motion.
