## Abstract

In this chapter, two-dimensional mixed convection heat transfer in a laminar cross-flow from two heated isothermal semicircular cylinders in tandem arrangement with their curved surfaces facing the oncoming flow and confined in a channel is studied numerically. The governing equations are solved using the control-volume method on a nonuniform orthogonal Cartesian grid. Using the immersed-boundary method for fixed Reynolds number of ReD=uDD/υ=200, Prandtl number of Pr=7, blockage ratio of BR=D/H = 0.2 and nondimensional pitch ratio of σ=L/D=3, the influence of buoyancy and the confinement effect are studied for Richardson numbers in the range −1≤Ri≤1. Here, uD is the average longitudinal velocity based on the diameter of the semicylinder. The variation of the mean and instantaneous nondimensional velocity, vorticity and temperature distributions with Richardson number is presented along with the nondimensional oscillation frequencies (Strouhal numbers) and phase-space portraits of flow oscillation from each semicylinder. In addition, local and averaged Nusselt numbers over the surface of the semicylinders are also obtained. The results presented herein demonstrate how the buoyancy and wall confinement affect the wake structure, vortex dynamics and heat transfer characteristics.

### Keywords

- bluff bodies
- tandem arrangement
- blockage ratio
- interference effects
- wall effects

## 1. Introduction

The flow and heat transfer past bluff bodies of various cross-sectional geometries is important because of advances in heat exchanger technology, cooling of electronic components and chips 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 *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

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 *n* and that the drag coefficients and the averaged Nusselt number increased with increasing blockage ratio for any value of *n*.

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 tandem arrangement confined inside a vertical channel of finite length using fixed Reynolds and Prandtl numbers, fixed blockage ratio and gap width and several values of the buoyancy parameter (Richardson number).

## 2. Formulation of the problem

### 2.1. Governing equations and boundary conditions

Consider a 2D steady, Newtonian, incompressible Poiseuille flow fluid with a mean mainstream velocity

where

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

For the channel walls,

where

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.

At the surface of the semicylinders,

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

where

### 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 *et al*. [35] defined a Reynolds number,

In Eq. (10),

## 3. Results and discussion

The numerical results presented in this work correspond in all cases to

### 3.1. Response characteristics for assisting flow ( R i = − 1 )

In this section, the response characteristics for assisting flow are presented. Figure 2 shows the resulting nondimensional mean flow and thermal profiles at

Figure 3 shows typical instantaneous flow and thermal patterns for

Figure 4 shows the time variations of the nondimensional longitudinal and transverse velocity components at the symmetry plane and selected positions inside the channel. Clearly, the velocity fluctuations depict a harmonic behavior after a short induction time of

### 3.2. Response characteristics for isothermal flow ( R i = 0 )

Figure 5 shows the nondimensional mean flow values for an isothermal flow

Figure 6 shows typical instantaneous patterns of velocity and vorticity illustrating how vortex shedding takes place at the rear of the downstream semicylinder. The third strip illustrates how in the absence of buoyancy, the interaction between the shear layers generated by the upstream semicylinder and the confining walls reduces.

Figure 7 shows the time variations of the nondimensional longitudinal and transverse velocity components at the symmetry plane and selected positions inside the channel. This image shows how after an induction time of

### 3.3. Response characteristics for opposing flow ( R i = 1 )

In this section, the response characteristics for opposing flow are presented. Figure 8 shows the nondimensional mean flow values at

Figure 9 shows a typical instantaneous flow and thermal pattern at

Figure 10 shows the time variations of the nondimensional longitudinal and transverse velocity components at the symmetry plane and selected longitudinal positions inside the channel. Clearly, time-periodic flow oscillation sets in after an induction time. The inset of the lower left image shows how the recirculation zone within the gap depicts periodic flow oscillation of relatively small amplitude.

### 3.4. Strouhal number and phase space plots

The left images in Figure 11 show (from top to bottom) the normalized spectrum of the transverse velocity component as a function of the nondimensional frequency (Strouhal number),

These images show how for

## 4. Heat transfer

In this section, the heat transfer characteristics of the semicylinder array are presented for buoyancy assisting and opposing flow.

### 4.1. Local Nusselt numbers

Figures 12a and **b** show representative distributions of the local Nusselt number defined in Eq. (8) over the curve length **A-B-C-D** (body contour of each semicylinder) for **C**. Beyond point **C**, as the warm/cold downward-flowing fluid travels through the front half of the semicylinder along the surface **B-C-D**, it yields/picks up thermal energy and the local Nusselt number gradually decreases toward points **B** and **D**. The cold/warm upward flow present between both semicylinders impinges the rear of the upstream one, a local maximum is reached at point **A** and a progressive increase in the local Nusselt number is observed over the curve length **B-D**. Depending on whether buoyancy assists/opposes the flow and because of the presence of the recirculation zone within the gap that yields/picks up thermal energy from the wake of the upstream semicylinder, a local minimum of the local Nusselt number is reached at the front stagnation point of the downstream semicylinder. Thus, the local Nusselt number beyond point **C** gradually increases toward points **B** and **D**. As the flow detaches from the tip of the downstream semicylinder (points **B** and **D**), the local Nusselt number reaches a local/global maximum for assisting/opposing buoyancy, respectively.

### 4.2. Overall Nusselt number

Figure 13 shows the time variation of the surface-averaged Nusselt number of both semicylinders with Richardson number. In these figures, the broken and continuous lines correspond to the upstream and downstream semicylinder, respectively. Figure 13 shows how the presence of the upstream semicylinder has a significant effect on the heat transfer characteristics of the downstream semicylinder and lower heat transfer rates are achieved by the latter. For clarity, in the inset of Figure 13, the value of the mean Nusselt number of both semicylinders is plotted in a limited range of the nondimensional time, from

## 5. Conclusions

In this work, numerical simulations have been carried out to study the unsteady flow and heat transfer characteristics around two identical isothermal semicylinders arranged in tandem and confined in a channel. The blockage ratio, Prandtl number and pitch-to-diameter are fixed at

## Acknowledgments

This research was supported by the Consejo Nacional de Ciencia y Tecnología (CONACYT), Grant No. 167474.