c Pleiades Publishing, Ltd., 2013. ISSN 1810-2328, Journal of Engineering Thermophysics, 2013, Vol. 22, No. 2, pp. 122–133. 

Numerical Study of Flow and Heat Transfer from a Torus Placed in a Uniform Flow N. P. Moshkin1* , J. Sompong2, 3 , and P. Suwannasri4 1

Institute of Science, School of Mathematics, Suranaree University of Technology, Nakhon Ratchasima, Thailand 2 Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000 Thailand 3 Center of Excellence in Mathematics, CHE, Si Ayutthaya Rd., Bangkok, 10400 Thailand 4 Department of Mathematics, Statistics and Computer, Ubon Ratchathani University, Ubon Ratchathani, 34190 Thailand Received December 11, 2011

Abstract—Forced convection heat transfer characteristics of a torus (maintained at a constant temperature) immersed in a streaming fluid normal to the plane of the torus are studied numerically. The governing equations, namely, continuity, momentum and thermal energy in toroidal coordinate system, are solved using a finite difference method over ranges of parameters (aspect ratio of torus, 1.4 ≤ Ar ≤ 20; Reynolds number, 20 ≤ Re ≤ 40; Prandtl number, 0.7 ≤ Pr ≤ 10). Over the ranges of parameters considered herein, the nature of flow is assumed to be steady. In particular, numerical results elucidating the influence of Reynolds number, Prandtl number and aspect ratio on the isotherm patterns, local and average Nusselt numbers for the constant temperature (on the surface of the torus) boundary condition. As expected, at large aspect ratio the flow pattern and heat transfer are similar to the case of flow and heat transfer over a single circular cylinder. DOI: 10.1134/S1810232813020045

1. INTRODUCTION The forced convection heat transfer and flow of fluids across a heated bluff body have been the subject of considerable research interest because of relevance in many engineering applications. The flow past a circular cylinder (or a sphere) is considered to be an ideal bluff body to study the important phenomena of heat and mass transfer. The hydrodynamics and heat transfer of a torus is important in two aspects. Firstly, the ring described at aspect ratio limits a sphere and a circular cylinder. Secondly, a torus has the simplest geometry, which represents not simply connected domain in the three dimensions. Significant research works are available in the literature in regard to the fluid flow and heat transfer over a circular cylinder and a sphere. Fluid flow around and heat transfer from circular cylinders in cross flow to viscous fluid have been extensively studied theoretically, experimentally, and numerically by many researchers (see, for example, paper of Khan et al. [1] and references therein) The convective heat and mass transfer from isothermal spherical particles surrounded by a flowing fluid is involved in many engineering industries and has been the subject of several investigations in the past. Many researchers have investigated steady laminar forced convection heat transfer from an isothermal sphere into a substantial amount of air or water experimentally, analytically and numerically. Many of available studies pertain to rigid spheres in creeping flows. Studies dealing in detail with scalar transport are largely confined to a steady flow [2–4]. In a recent numerical study conducted by Feng and Michaelides [5], the Navier–Stokes equations has been solved in the range of Reynolds numbers from 0 to 4000 and the wide range of Peclet numbers. Recently, Moshkin and his coworkers [6–8] numerically investigated the characteristics of forced convection heat transfer and flow over two side-by-side rotating circular cylinders in the range of low *

E-mail: [email protected]

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Fig. 1. Sketch of the geometry of the torus.

Reynolds number (10 ≤ Re ≤ 40). The variation in the drag coefficient for low-Reynolds-number flow past rings orientated normal to the direction of flow has been investigated numerically by Sheard et al. [9]. The self-propelled motion of a torus rotating about its centerline in a viscous incompressible fluid is studied numerically by Moshkin and Suvnnasri [10]. However, the problem of fluid flow and heat transfer from a torus placed normal to the direction of flow stream has not been studied so far. A review of existing literature reveals that there are no experimental/numerical studies regarding the configuration of the present problem. The present study is primarily concerned with flow and heat transfer in the laminar flow regime around a torus. 2. MATHEMATICAL FORMULATION Assuming the flow remains axisymmetric for all time, makes the following toroidal coordinate system: x=

c sinh η cos ϕ , cosh η − cos ξ

y=

c sinh η sin ϕ , cosh η − cos ξ

z=

c sin ξ , cosh η − cos ξ

(1)

where ξ ∈ (0, 2π], η ∈ (−∞, ∞) and ϕ ∈ [0, 2π), c > 0 is the characteristic length, the natural choice. The surface η = η0 defines a torus, z 2 + (r − c coth η0 )2 = c2 csch2 η0 , and the surface ξ = ξ0 defines  c sinh η . Figure 1 a spherical bowl, (z − c cot η0 )2 + r 2 = c2 csc2 ξ0 , where r 2 = x2 + y 2 = cosh η − cos ξ shows the torus with radius b = c coth η0 and the circular cross section with radius a = c cschη0 . If a and b are given, one can find c and η0 as the following:

c=



b2 − a2 ,

⎛ ⎞    2 b b − η0 = ln ⎝ − 1⎠ . a a

The torus geometry is described by the aspect ratio parameter Ar = b/a, which is the ratio of the torus diameter 2b to the cross section diameter 2a. In terms of the toroidal coordinates and the assumption of axisymmetry, the governing Navier–Stokes equations in dimensionless form are given by

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 ∂vξ ∂vξ 1 2 + vη + vη sin ξ − vξ vη sinh η ∂ξ ∂η c   2   ∂vξ ∂ 2 vξ ∂vη ∂vη 1 2 1 ∂ vξ 1 ∂p − + sin ξ + 2 sinh η − 2 sin ξ + =− h ∂ξ Re h2 ∂ξ 2 ∂η 2 ch ∂ξ ∂ξ ∂η      ∂vξ sin ξ 1 coth η sinh η + (2 − 2 cosh η cos ξ) + sinh2 η vη − + 2 2 h ch ∂η c sinh η     2 2 1 1 2 cosh η − 2 sin ξ + sinh η + 2 (cosh η cos ξ − 1) vξ , (2) + ch c c

∂vξ 1 + ∂t h

vξ

  1 2 ∂vη ∂vη vξ + vη + vξ sinh η − vξ vη sin ξ ∂ξ ∂η c  2    ∂vξ ∂vξ 2 1 ∂ vη ∂vη ∂ 2 vη 1 1 ∂p + sin ξ − 2 sinh η + 2 sin ξ + − =− h ∂η Re h2 ∂ξ 2 ∂η 2 ch ∂ξ ∂ξ ∂η      cosh η 2 coth η sinh η ∂vη + − 2 sin2 ξ + sinh2 η vη − + 2 h ch ∂η ch c     1 (1 − cosh η cos ξ)2 sin ξ sinh η 2 vη − + vξ , (3) sin ξ + (cosh η cos ξ − 1) + c2 c2 sinh2 η

1 ∂vη + ∂t h



1 h 1 ∂T + ∂t h



∂vξ ∂vη + ∂ξ ∂η

 −

  2h sin ξ 2h sinh η vξ + coth η − vη = 0, c c

       2 1 ∂ ∂T ∂ ∂T ∂T ∂T vξ + vη = h sinh η + h sinh η , ∂ξ ∂η RePr h3 sinh η ∂ξ ∂ξ ∂η ∂η

(4)

(5)

where p is the pressure, vξ and vη are the velocity components in ξ and η directions, respectively, and h = c/(cosh η − cos ξ). The governing equations are made dimensionless by using the following scaling variables: radius of a cross section of torus a for length variables, U∞ for velocities, a/U∞ 2 for pressure, ρ is the fluid density. The temperature is non-dimensionalized by using for time, ρU∞ (T − T∞ )/(TW − T∞ ), TW and T∞ are temperatures on the torus surface and at infinity. All equations and numerical values presented in this work are in dimensionless form. Here Re denotes the Reynolds cp μ 2U∞ a , the Prandtl number is given by Pr = , where μ is the viscosity of number defined by Re = ν k the fluid, ν = μ/ρ is the kinematic viscosity, cp is the heat capacity, and κ is the thermal conductivity. Boundary conditions for vξ , vη , and T include: • On the torus surface: the no-slip, impermeability and constant temperature conditions vξ = 0,

vη = 0,

and

T = 1,

ξ ∈ (0, 2π],

η = η0 .

(6)

• The periodicity conditions: vξ (ξ, η) = vξ (ξ + 2π, η) , vη (ξ, η) = vη (ξ + 2π, η) , p (ξ, η) = p (ξ + 2π, η) , T (ξ, η) = T (ξ + 2π, η) .

(7)

• At infinity: uniform flow condition:

v = (vr , vz ) = (0, 1) ,

p=

p∞ , 2 ρU∞

T = 0 as r 2 + z 2 → ∞.

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(8)

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Here, vr and vz are the components of the velocity vector in the cylindrical coordinate system with     h h (cosh η cos ξ − 1) vz , vξ = − sinh η sin ξ vr + a    a h h sinh η sin ξ vz . vη = − (cosh η cos ξ − 1) vr − a a

125

(9)

• On the axis of symmetry r = 0: the velocity components and pressure are satisfied by the following boundary condition ∂vξ = 0, ∂η

vη = 0,

∂p = 0, ∂η

∂T = 0. ∂η

and

The two most important hydrodynamic characteristics of the flow around the body are the net force and angular momentum. The net force is decomposed into components FL and FD perpendicular and parallel to the flow direction. The net torque and FL are equal to zero due to the symmetry of the flow. FD , where Af rontal is the projected frontal area of The drag coefficient is defined as CD = 2 0.5ρAf rontal U∞ the body. The drag coefficient comprises a pressure drag coefficient and a viscous (form) drag coefficient, i.e., CD = CDp + CDf . They are defined as: CD p

CD f

1 =− 2 2abπρU∞

−1 = 2 2abπρU∞

2π 2π

sinh2 η0 p n · iz h sinh η0 dφdξ = − b

2π

2

0

2π 2π

0

p 0

− sinh2 η0 μ ( n × ω

) · iz h sinh η0 dφdξ = b

2π

2

0

0

sin ξ h2 dξ, (cosh η0 − cos ξ)

0

2ω (cosh η0 − 1) hdξ, Re (cosh η0 − cos ξ)

where iz is the unit vector in the z-axes direction. The vorticity ω is defined by the following equation:   ∂ ∂ 1 (hvη ) − (hvξ ) . (10) ω= 3 h sinh η0 ∂ξ ∂η An important parameter of interest in heat transfer problems is the heat transfer rate per unit area from the torus wall to the ambient fluid. The local Nusselt number in the toroidal coordinate system based on the diameter of the torus is     1 ∂T cosh η − cos ξ ∂T ∂T = −∇T · n = − =− . (11) Nu = − ∂n h ∂η a ∂η The average Nusselt number is calculated by averaging the local Nusselt number over the surface of the torus:  sinh(η0 ) 2π 2 Nu = − h Nu dξ, ξ ∈ [0, 2π]. (12) 2πab 0 3. DISCRETIZATION OF THE GOVERNING EQUATIONS In the case of steady flow, time in Eqs. (2), (3), and (5) can be considered as an artificial (iterative) parameter. A staggered placement of variables is used with velocity components u = vη located on the vertical sides of each cell and components v = vξ on the horizontal sides of each cell. The pressure p and temperature T are represented at cell centers. A two-step time-split projection method is utilized to advance the flow field. First, the velocity components are advanced from time level “n” to an intermediate level “*” by solving Eqs. (2) and (3) explicitly without the pressure terms. In the advection-diffusion step, the spatial derivatives are approximated by the central finite differences. One-side finite differences are utilized near boundaries due to the staggered arrangement of variables. Then, the Poisson equation for the pressure is solved JOURNAL OF ENGINEERING THERMOPHYSICS Vol. 22

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Fig. 2. Staggered arrangement of u, v, and p.

fully implicitly by the method of stabilizing correction (see Yanenko [11]). The equation for pressure is derived by using the mass conservation requirement for each computational cell. Once the pressure is updated, the final level is computed with a pressure-correction step. When the steady solution of Navier– Stokes equations (2)–(4) is computed, the iterative method of stabilizing correction is used to find steady distribution of temperature field given by Eq. (5). Figure 2 shows the computational domain, sketch of the grid, and location of the unknowns. Far-field boundary conditions (8) are shifted on the boundary of domains Ω1 and Ω2 , which are defined as Ω1 = { (ξ, η) | 0 ≤ ξ ≤ εξ , 0 ≤ η ≤ εη } , Ω2 = { (ξ, η) | 2π − εξ ≤ ξ ≤ 2π, 0 ≤ η ≤ εη } ,

(13)

where εη = Kη and εξ = M ξ , K and M are integer numbers, and η and ξ are the size of computational cell in the η and ξ directions, respectively. In the physical space (x, y, z), the boundaries of domains Ω1 and Ω2 are located sufficiently far from the torus and these boundaries are the coordinate surfaces that are convenient for the implementation of a finite difference method. 4. RESULTS AND VALIDATIONS The present problem involves an effect of the aspect ratio and Reynolds number on the flow characteristics and additionally Prandtl number effects on the heat transfer characteristics. The grid refinement test has been carried out for the two Reynolds numbers Re = 20 and 40, and the three aspect ratios Ar = 2, 3, and 20. Three grids with refinement factor hcoarse /hf ine = 1.5 are used. The grid convergence indices (GCIs) are estimated in the standard way with a factor 1.25 (see, for example, Roache [12]). Dependence of the drag coefficient, its pressure and friction components, and average Nusselt number Nu on the grid resolution as well as GCI is illustrated in Table 1. To reach steady solution, the computations were advanced in time (artificial time) until the drag coefficient had reached a constant value. The main simulation was carried out with dimensionless time steps of 0.0001, chosen in agreement with stability requirements. Second, the flow around torus has been simulated at large aspect ratio. We expect that the results of flow over a torus at large Ar are similar to the results of flow over two circular cylinders with a large gap spacing between the cylinder surfaces (see Moshkin and Sompong [8], Moshkin and Suwannasri [10]). Since for a large gap spacing between the surfaces of cylinders the mutual influence of the cylinders on each other is negligible, we can assume that the flow and heat transfer will be similar to the flow and heat transfer over a single cylinder as well. The comparisons with flow and heat transfer around a single cylinder can be viewed as a partial validation of the developed algorithm. JOURNAL OF ENGINEERING THERMOPHYSICS Vol. 22 No. 2 2013

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Table 1. Effect of grid refinement on CD , CDp , and CDf Grid

CD

GCI%

40 × 156

1.484

Re = 40

60 × 234

1.345

2.08

0.873

1.9

0.472

Ar = 2

80 × 312

1.324

0.30

0.860

0.28

0.464

40 × 122

1.584

Re = 40

60 × 184

1.433

3.33

0.921

3.39

0.512

Ar = 3

80 × 244

1.400

0.80

0.899

0.79

0.501

20 × 40

2.255

Re = 20

40 × 80

2.120

3.62

1.269

3.05

0.851

Ar = 20

80 × 160

2.076

1.25

1.246

0.96

0.830

CDp

GCI%

0.971

GCI%

CDf

NuPr = 0.7

NuPr = 10

3.097

8.331

2.42

3.008

8.098

0.51

2.988

8.026

3.286

8.390

3.27

3.205

8.239

0.87

3.181

8.151

2.565

6.135

3.05

2.490

5.978

0.96

2.465

5.854

0.513

1.026

0.557

1.346

0.909

Table 2. Comparison of the average Nusselt numbers on torus surface with constant temperature for Ar = 20 for using a grid size of 40 × 80 Re

20

30

40

Pr

Nu Present study

Moshkin [8]

Bharti [14, 15]

Juncu [17]

Soares [16]

0.7

2.490

2.478

2.465

1

2.802

2.787

2.857

7

5.308

5.286

5.214

10

5.978

6.041

5.857

0.7

2.934

2.936

2.929

1

3.309

3.310

3.357

7

6.446

6.397

6.286

10

7.322

7.294

7.071

0.7

3.322

3.333

3.283

3.200

1

3.757

3.773

3.714

3.570

7

7.506

7.533

7.214

10

8.589

8.727

8.214

2.430 2.730

Lange [13] 2.408

2.696

5.729

3.229

3.171

6.937 3.280

The case of torus with Ar = 20, Re = 20, 30, 40, and Pr = 0.7 − 10.0 was selected for the test. We used the grid with 40 × 80 nodes in computational domain. Far boundary conditions were set up for K = M = 2 (see Eqs. (13)). The distance between the torus centerline and the boundary of the domains Ω1 and Ω2 was greater than 40 inner torus diameters in the physical space. A comparison of average Nusselt number is provided in Table 2. In Table 2, the present results for convective heat transfer from torus are compared with study available for flow and heat transfer around two circular cylinders with a large gap spacing and a single cylinder. The results of Moshkin and Sompong [8] correspond to the flow past two circular cylinders with a large aspect ratio, all other results [13–17] represent average Nusselt number for convective heat transfer from a single cylinder. An examination of Table 2 reveals that the present results are within ±5% of the literature values. Deviations of this order are easily explained by different topology, numerical methods, etc. Figure 3 shows the variation of the local Nusselt number Nu(θ) on the torus surface in azimuthal cross section for Re = 20 (a) and Re = 40 (b) at Ar = 20 and Pr = 0.7 − 10. Due to the symmetry, only JOURNAL OF ENGINEERING THERMOPHYSICS Vol. 22

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Fig. 3. Local Nusselt number variation on the torus surface at Re = 20, 45 and Ar = 20 for various Prandtl number.

Fig. 4. Temperature contours over torus with aspect ratio Ar = 20 (1st row) and over two cylinders at g = 14 (2nd row) at Re = 20, 40, Pr = 0.7, 10.0.

variation along half cross section is demonstrated, 0 ≤ θ ≤ 180◦ . The value θ = 0 corresponds to the front stagnation point. In the same figure, the data of Bharti [14, 15] for Nu(θ) on the surface of a single cylinder are shown for comparison, and due to the large aspect ratio they are close to present results. At a small Prandtl number of order 1, the local values of the Nusselt numbers demonstrate small variation over the surface from θ = 0 to θ = π. This is an expected result because the heat transfer occurs mostly by conduction, with a little convection. As the Prandtl number is increased, the effect of convection increases and the Nusselt number is seen to vary significantly over the surface of the torus. The value of the local Nusselt number in the front of the torus (θ = 0) increases with an increase of Reynolds and Prandtl numbers. The Nusselt number decreases to θ = θs , the separation angle, and for θ > θs the Nusselt number increases. Representative plots of the isotherms for both the torus with Ar = 20 and the two cylinders with the same aspect ratio at Re = 20 and 40, and Pr = 0.7 and 10 are shown in Fig. 4. These figures show that isotherm patterns for the torus and the circular cylinder are almost the same in the considered cases, which became additional evidence of reasonable accuracy of the numerical algorithm. An increase in JOURNAL OF ENGINEERING THERMOPHYSICS Vol. 22 No. 2 2013

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the Prandtl and/or Reynolds numbers compacted the isotherms lines, which resulted in an increased temperature gradient. The increase in Prandtl and/or Reynolds numbers increased the compactness of the isotherms toward the upstream direction, it is also consistent with the corresponding variation in local Nusselt number (Fig. 3). The comparisons of our numerical results with data from the literature (presented in Figs. 3, 4, and in Table 2) allowed us to conclude that the numerical method and computer code are well suited and can be used to simulate the flow and convective heat transfer over a torus. More details on validation tests can be found in Moshkin and Suwannasri [10].

Streamline Patterns and Isotherms Representative plots showing the dependence of the streamline patterns, the temperature contours in the vicinity of the torus on the Prandtl number (Pr) and aspect ratio are presented in Fig. 5 for the Reynolds number Re = 30. Because of the axial symmetry, we present the flow and temperature patterns only in a cross section of torus. For Ar = 2 in Fig. 5 we observe a frontal stagnation point on the surface of the torus. There are two stagnation points on the axis of symmetry, r = 0. The toroidal separation bubble in the wake of torus does not attach to the torus surface. This is different from the pattern of flow past bluff bodies. The stream passing through the hall of the torus hits this toroidal vortex bubble, turns back, and flows around the vortex separation babble. When Re increases this toroidal vortex zone becomes smaller and it is shifted in the downstream direction. The flow pattern transforms to the flow pattern past the circular cylinder as the aspect ratio increases (see second and third columns in Fig. 5). In the vortex region behind the torus velocities are small. Owing to weak advection this in turn leads to the formations of local “constant-temperature” regions in the wake behind the torus. Rows 2–4 in Fig. 5 show isotherms for Pr = 0.7, 5, and 10, respectively. It is seen that the front surface has the maximum density of temperature isotherms, which indicates large temperature gradients (and, thus, the highest local Nusselt number), as compared to the other points on the surface of the torus. The clustering of isotherms increases in the Reynolds number and/or Prandtl number, or both.

Local Nusselt Number Representative variations of the local Nusselt number (Nu(θ)) over the torus surface with the Reynolds and Prandtl numbers, and aspect ratios are shown in Figs. 6–8. The Nusselt number increases with increasing Reynolds number and/or Prandtl number. These figures show that a maximum value in Nu(θ) was located close to the front stagnation point (θ = 0). A more significant shift of the maximum value was found for a small aspect ratio Ar = 1.4 (see Fig. 6a). Figure 6 shows the variation of the local Nusselt number, Nu(θ), on the surface of the torus with Prandtl number at Re = 25 and Ar = 1.4 (Fig. 6a), Re = 30, 40 and Ar = 2.0 (Figs. 6b and 6c). At a low Prandtl number (Pr = 0.7, 1.0), the local values of the Nusselt number demonstrate relatively small variation over the surface of the torus from θ = 0 to θ = π (location θ = π/2 corresponds to the outer part of the torus surface). The values of the local Nusselt number decrease along the surface of the torus from the maximum value near the front stagnation point to a minimum value in the rear part of the torus (θ ≈ 260◦ for Ar = 1.4 and θ ≈ 220◦ for Ar = 2.0). As the Prandtl number is increased, the contribution of convection increases and the Nusselt number is seen to vary more significantly over the surface of the torus. The maximum value of the local Nusselt number near stagnation point increases almost two times as the Prandtl number increases from 0.7 to 10.0. Figure 6 shows that for Pr = 5 and Pr = 10, there are two local minima at θ ≈ 150◦ , 230◦ for Ar = 1.4 and θ ≈ 140◦ , 220◦ for Ar = 2.0. Such behavior of the local Nusselt number in the rear part of the torus indicates the existence or appearance of a vortex zone (see first rows in Fig. 5). It worth to note that the minimum values of the local Nusselt number in the rear part of the torus change comparatively slightly with increasing Prandtl number. Figures 7 and 8 show the variation of the local Nusselt number on the surface of the torus with aspect ratio at Re = 20 and 40 and Pr = 0.7, 5.0, 10.0. These figures show qualitatively similar behavior of the Nusselt number over the surface of the torus. For the large aspect ratio Ar = 20, graphics of Nu(θ) are similar to the case of the isolated circular cylinder. As expected, the relatively large values of the Nusselt number seen at the front stagnation point (θ ≈ 0) decrease gradually along the surface of the torus, to a minimum value near the point of separation. JOURNAL OF ENGINEERING THERMOPHYSICS Vol. 22

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Fig. 5. Stream line (first row) and temperature contours over torus at Re = 30 (rows 2–3), 2nd row: Pr = 0.7, 3d row: Pr = 5, 4th row: Pr = 10. 1st column: Ar = 2, 2nd column: Ar = 3, and 3rd column: Ar = 5.

Average Nusselt Number The dependence of the average Nusselt number on the Reynolds and Prandtl numbers and the aspect ratio is shown in Fig. 9. For a fixed value of the Reynolds number, the average Nusselt number increases with a gradual increase in the Prandtl number, irrespective of the aspect ratio, Ar. For a fixed value of the aspect ratio, the value of average Nusselt number increases with a gradual increase in Prandtl number and/or Reynolds number, and/or both. For instance, at Pr = 1, the value of the Nusselt number increases from 2.413 to 3.464 at Ar = 2 and from 2.756 to 3.666 at Ar = 5 as Re is increased from 20 JOURNAL OF ENGINEERING THERMOPHYSICS Vol. 22 No. 2 2013

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Fig. 6. Local Nusselt number variation on the surface of the torus for different Prandtl numbers: (a) Ar = 1.4 at Re = 25; (b) Ar = 2.0 at Re = 30; (c) Ar = 2.0, Re = 40.

Fig. 7. Local Nusselt number variation on the surface of the torus for different Ar at Re = 20: (a) Pr = 0.7; (b) Pr = 5; (c) Pr = 10.

Fig. 8. Local Nusselt number variation on the surface of the torus for different Ar at Re = 40: (a) Pr = 0.7; (b) Pr = 5; (c) Pr = 10.

to 40, whereas at Pr = 10, it increases from 5.760 to 8.081 and from 5.859 to 8.234 at Ar = 2 and 5, respectively. 5. CONCLUSIONS AND DISCUSSIONS In the present work, the forced convection heat transfer to viscous incompressible fluid from a heated torus in the steady cross-flow regime has been investigated numerically using a finite-difference scheme for the range of Reynolds number (20 ≤ Re ≤ 40), Prandtl number (0.7 ≤ Pr ≤ 10), and aspect ratio JOURNAL OF ENGINEERING THERMOPHYSICS Vol. 22

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Fig. 9. Dependence of the average Nusselt number (Nu) on Pr and Ar at (a) Re = 20, and (b) Re = 40.

(1.4 ≤ Ar ≤ 20). For a large aspect ratio variation of the local Nusselt number over the surface of the torus is similar to that in the case of a heated isolated cylinder. For a low Prandtl number the local value of the Nusselt number shows relatively small variation with only local minimum in the rear part of torus. The average Nusselt number increases with increasing Prandtl number and/or aspect ratio, and/or Reynolds number. ACKNOWLEDGMENTS This research was partially supported by the Center of Excellence in Mathematics, the Commission of Higher Education, Thailand. REFERENCES 1. Khan, W.A., Culham, J.R., and Yovanovich, M.M., Fluid Flow around and Heat Transfer from an Infinite Circular Cylinder, ASME J. Heat Transfer, 2005, vol. 127, no. 7, pp. 785–790. 2. Kramers, H., Heat Transfer from Spheres to Flowing Media, Phys., 1946, vol. 12, p. 61. 3. Brenner, H., Forced Convection Heat and Mass Transfer at Small Peclet Numbers from a Particle of Arbitrary Shape, Chem. Eng. Sci., 1963, vol. 18, no. 2, pp. 109–122. 4. Dennis, S.C., Walker, J.D., and Hudson, J.D., Heat Transfer from a Sphere at Low Reynolds Number, J. Fluid Mech., 1973, vol. 60, pp. 273–283. 5. Feng, Z.-G. and Michaelides, E.E., A Numerical Study on the Transient Heat Transfer from a Sphere at High Reynolds and Peclet Numbers, Int. J. Heat Mass Transfer, 2000, vol. 43, pp. 219–229. 6. Sungnul, S. and Moshkin, N.P., Numerical Simulation of Flow over Two Rotating Self-Moving Circular Cylinders, Recent Advances in Computational Sciences, Selected Papers from the Int. Workshop on Computational Sciences and Its Education, Jorgensen, P., Xiaoping Shen, Chi-Wang Shu, and Ningning Yan, Eds., Beijing, China, 2005, pp. 278–296. 7. Sungnul, S. and Moshkin, N.P., Numerical Simulation of Steady Viscous Flow Past Two Rotating Circular Cylinders, Suranaree J. Sci. Technol., 2006, vol. 13, no. 3, pp. 219–233. 8. Moshkin, N.P. and Sompong, J., Numerical Simulation of Heat Transfer and Fluid Flow over Two Rotating Circular Cylinders at Low Reynolds Number, Heat Transfer–Asian Res., 2010, vol. 39, no. 4, pp. 246–261. 9. Sheard, G.J., Hourigan, K., and Thompson, M.C., Computations of the Drag Coefficients for Low-ReynoldsNumber Flow Past Rings, J. Fluid Mech., 2005, vol. 526, pp. 257–275. 10. Moshkin, N.P. and Suwannasri, P., Self-Propelled Motion of a Torus Rotating about Its Centerline in a Viscous Incompressible Fluid, Phys. Fluids, 2010, vol. 22, p. 113602. 11. Yanenko, N.N., The Method of Fractional Steps: The Solution of Problems of Mathematical Physics in Several Variables, New York: Springer-Verlag, 1977. 12. Roache, P. J., Perspective: A Method for Uniform Reporting of Grid Refinement Studies, J. Fluids Eng., 1994, vol. 116, pp. 405–413. JOURNAL OF ENGINEERING THERMOPHYSICS Vol. 22 No. 2 2013

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13. Lange, C.F., Durst, F., and Breuer, M., Momentum and Heat Transfer from Cylinders in Laminar Crossflow at 10−4 ≤ Re ≤ 200, Int. J. Heat Mass Transfer, 1998, vol. 41, pp. 3409–3430. 14. Bharti, R.P., Chhabra, R.P., and Eswaran, V., A Numerical Study of the Steady Forced Convection Heat Transfer from an Unconfined Circular Cylinder, Heat Mass Transfer, 2007, vol. 43, pp. 639—648. 15. Bharti, R.P., Chhabra, R.P., and Eswaran, V., Steady Forced Convection Heat Transfer from a Heated Circular Cylinder to Power-Law Fluids, Int. J. Heat Mass Transfer, 2007, vol. 50, pp. 977–990. 16. Soares, A.A. and Ferreira, J.M., Flow and Forced Convection Heat Transfer in Crossflow of Non-Newtonian Fluid over a Circular Cylinder, Ind. Eng. Chem. Res., 2005, vol. 44, no. 15, pp. 5815–5827. 17. Juncu, G., A Numerical Study of Momentum and Forced Convection Heat Transfer around Two Tandem Circular Cylinders at Low Reynolds Numbers, pt. 2, Forced Convection Heat Transfer, Int. J. Heat Mass Transfer, 2007, vol. 50, pp. 3799–3808.

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