Unsteady Mixed Convection from Two Isothermal Semicircular Cylinders in Tandem Arrangement

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 1⁄4 uDD=υ 1⁄4 200, Prandtl number of Pr 1⁄4 7, blockage ratio of BR 1⁄4 D=H 1⁄4 0:2 and nondimensional pitch ratio of σ 1⁄4 L=D 1⁄4 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.


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][2][3][4][5][6], elliptic [7][8][9][10], rectangular [11][12][13][14][15] and square cross-sections [16][17][18][19][20], there are fewer studies on the semicircular cylinder geometry [21][22][23][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 0 o ≤ α ≤ 180 o . 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 increase in the value of the Reynolds, Prandtl and Richardson numbers.
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 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).

Governing equations and boundary conditions
Consider a 2D steady, Newtonian, incompressible Poiseuille flow fluid with a mean mainstream velocity u 0 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 (L tot ¼ 30D and H,r e s p e ctively). 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 1 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 T 0 , and the semicylinders have a wall temperature of T w . Flow rectifiers are placed at the channel exit producing a parallel flow at x ¼ S 1 .T h e 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 (1) where V ¼ðU, VÞ is the dimensionless velocity vector and θ is the dimensionless temperature. In Eqs.
(1)-(3), U and V are the X and Y components of V, respectively. All velocity components are scaled with the oncoming mean bulk velocity u 0 ; the longitudinal and transverse coordinates are scaled with the semicylinder diameter D; the time is scaled with the residence time D=u 0 , τ ¼ tu 0 =D;thetemperature is normalized as θ ¼ðT−T 0 Þ=ðT w −T 0 Þ.Intheaboveequations,thenondimensionalparameters are the Reynolds number, Re ¼ u 0 D=ν, the Prandtl number Pr ¼ ν=α and the Richardson number, Ri ¼ gβðT w −T 0 ÞD=u 2 0 , respectively (frequently, instead of using the Richardson number, the Grashof number is employed, Gr ¼ RiRe 2 ¼ gβðT w −T 0 ÞD 3 =ν 2 Þ.Here,g is the acceleration due to gravity, α is 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: The inflow boundary is specified by a developed velocity profile at the channel inlet For the channel walls, Á , respectively. Vorticity at the walls is evaluated using Thom's first-order formula [31], Heat Exchangers-Design, Experiment and Simulation 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.
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 q j is obtained in nondimensional form with the local Nusselt number Nu j ,w i t hj ¼ 1, 2 for the upstream and downstream semicylinder, respectively. The local Nusselt numbers are evaluated from the following equation 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

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 −7 is used, with an optimal time step of Δτ ¼ 5×10 −4 . 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 In Eq. (10), u is the vertical component of the velocity field specified on the upstream boundary and u D 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].

Results and discussion
The numerical results presented in this work correspond in all cases to 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.

Response characteristics for assisting flow ðRi ¼ −1Þ
In this section, the response characteristics for assisting flow are presented. Figure 2 shows the resulting nondimensional mean flow and thermal profiles at      Figure 5 shows the nondimensional mean flow values for an isothermal flow ðRi ¼ 0Þ.I n the absence of buoyancy, the mean flow solution is symmetric. Although the recirculation zone of the upstream semicylinder still occupies the total space within the gap, its width is now larger than the semicylinder diameter. In addition, the length of the near wake of the downstream semicylinder extends to X≈4:5 and a slight decrease in vorticity strength takes place.       Heat Exchangers-Design, Experiment and Simulation

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), St ¼ fD=u 0 for Ri ¼ −1, 0 and 1, respectively.
These images show how for Ri ¼ −1, 0 and 1 and for selected locations within the gap and downstream of the lower semicylinder, there is a sharp peak at St ¼ 0:32111, 0:29448, and 0:22295, respectively, indicating that the wake vortex shedding of both semicylinders is time-periodic and is dominated by a single fundamental frequency. These images exemplify how for the three values of the buoyancy parameter studied, the recirculation zone of the upstream semicylinder locks on to the shedding frequency of the downstream one.
In addition, these images show how the Strouhal number decreases for increasing values of the buoyancy parameter. The right images in Figure 11 show the corresponding phasespace relation between the longitudinal and transverse velocity signals after the vortex shedding reaches an established periodicity. The inset of these figures describe the fluctuations at a location of ðX, YÞ¼ð1:5, 0Þ. For all cases, the single orbit with a double loop Heat Exchangers-Design, Experiment and Simulation 232 illustrates how the periodic alternate shedding of vortices takes place at the space within the gap and downstream of the lower semicylinder.

Heat transfer
In this section, the heat transfer characteristics of the semicylinder array are presented for buoyancy assisting and opposing flow.  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 τ ¼ 180 to 200. It is worth to mention that the discernible periodic oscillations of the mean Nusselt number of the lower semicylinder are closely related to flow oscillation due to vortex shedding for both cases.

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-todiameter are fixed at BR ¼ 0:2, Pr ¼ 7a n dσ ¼ 3, respectively. Numerical simulations are performed using the control-volume method on a nonuniform orthogonal Cartesian grid. The immersed-boundary method is employed to identify the semicylinders confined inside the channel. The influence of buoyancy has been assessed on the resulting mean and instantaneous flow, vortex shedding properties, nondimensional oscillation frequencies (Strouhal numbers), phase-space portraits of flow oscillation, thermal fields and local and overall nondimensional heat transfer rates (Nusselt numbers) from each semicylinder. Results show that in this parameter space, the flow patterns reach a time-periodic oscillatory state, the recirculation zone of the upper semicylinder completely fills the space within the gap and vortex shedding from the lower semicylinder occurs. For values of the Richardson number of for Ri ¼ −1a n dRi ¼ 1, steady-state and time periodic oscillations of the mean Nusselt number are observed for the upstream and downstream semicylinder, respectively.
Nomenclature BR blockage ratio, D=H D semicylinder diameter (characteristic length) f vortex shedding frequency (Hz) g gravity acceleration Gr Grashof number based on semicylinder diameter, Gr ¼ gβðT w −T 0 ÞD 3 =ν 2 h local heat transfer coefficient H width of computational domain k thermal conductivity of fluid L pitch (centre-to-centre distance between two semicylinders) L tot length of computational domain n normal direction Nu local Nusselt number (see Eq. (8)