Numerical Simulation of Turbulent Flow Around a ...

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Influences of Boundary Conditions on Laminar Natural Convection of Bingham Fluids in Rectangular Enclosures With Differentially Heated Side Walls Osman Turan

a b

a

, Robert J. Poole & Nilanjan Chakraborty

c

a

School of Engineering , University of Liverpool , Liverpool , United Kingdom

b

Department of Mechanical Engineering , Karadeniz Technical University , Trabzon , Turkey

c

School of Mechanical and System Engineering , Newcastle University , Newcastle-UponTyne , United Kingdom Accepted author version posted online: 18 Oct 2013.Published online: 17 Dec 2013.

To cite this article: Osman Turan , Robert J. Poole & Nilanjan Chakraborty (2014) Influences of Boundary Conditions on Laminar Natural Convection of Bingham Fluids in Rectangular Enclosures With Differentially Heated Side Walls, Heat Transfer Engineering, 35:9, 822-849, DOI: 10.1080/01457632.2014.852870 To link to this article: http://dx.doi.org/10.1080/01457632.2014.852870

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Heat Transfer Engineering, 35(9):822–849, 2014 C Taylor and Francis Group, LLC Copyright ISSN: 0145-7632 print / 1521-0537 online DOI: 10.1080/01457632.2014.852870

Influences of Boundary Conditions on Laminar Natural Convection of Bingham Fluids in Rectangular Enclosures With Differentially Heated Side Walls OSMAN TURAN,1,2 ROBERT J. POOLE,1 and NILANJAN CHAKRABORTY3 1

School of Engineering, University of Liverpool, Liverpool, United Kingdom Department of Mechanical Engineering, Karadeniz Technical University, Trabzon, Turkey 3 School of Mechanical and System Engineering, Newcastle University, Newcastle-Upon-Tyne, United Kingdom 2

Two-dimensional steady-state simulations of laminar natural convection in rectangular enclosures with differentially heated side walls have been conducted for a range of different aspect ratios (height/length) for both constant wall temperature and constant heat flux boundary conditions. The rectangular enclosures are considered to be completely filled with a yield-stress fluid obeying the Bingham model. Yield-stress effects on heat and momentum transport are investigated for nominal values of Rayleigh number in the range 104–106 and the aspect ratio range 1/8 to 8 for a single nominal Prandtl number (= 500). It is found that the mean Nusselt number increases (decreases) with increasing values of Rayleigh (Bingham) number irrespective of the boundary condition. For the constant wall temperature boundary condition, the aspect ratio at which the maximum mean Nusselt number occurs is found to decrease with increasing Rayleigh number. In contrast, the value of mean Nusselt number increases monotonically with increasing aspect ratio in the case of the constant wall heat flux boundary condition. Detailed physical explanations are provided for these aspect ratio effects. New correlations are proposed for the mean Nusselt number in both the constant wall temperature and wall heat flux boundary conditions, which are shown to satisfactorily capture the simulation results.

INTRODUCTION Natural convection is an important and much-studied physical phenomenon. Typically, fundamental studies of this effect have concentrated on the flow initiated in two-dimensional enclosures for various thermal boundary conditions and enclosure aspect ratios AR (ratio of enclosure height to length). Here we are concerned with the case of differentially heated vertical side walls for a special class of non-Newtonian fluids called yieldstress fluids. The majority of previous works in this general area have been concerned with Newtonian fluids and with constant

Address correspondence to Professor Nilanjan Chakraborty, School of Mechanical and System Engineering, Newcastle University, Claremont Road, Newcastle-Upon-Tyne, NE1 7RU, United Kingdom. E-mail: [email protected]

wall temperature (CWT) thermal boundary conditions; the papers of Ganguli et al. [1] and Ostrach [2] nicely review the work in this area. Although as both water and air, the most abundant fluids on the planet, are Newtonian fluids, it is logical that most previous work has concentrated on such Newtonian fluids, virtually all synthetic fluids are non-Newtonian in character. Thus, from an engineering perspective, knowledge of natural convection in more rheologically complex fluids than water or air is essential. Perhaps the simplest deviations from Newtonian behavior are fluids that display a shear-rate-dependent viscosity but are otherwise inelastic and time independent. This class of fluids is called generalized newtonian fluids (GNF), and natural convection effects have been investigated in such fluids by both analytical and numerical means. Lamsaadi et al. [3, 4] used an analytical technique to investigate the effect of the degree of shear thinning/thickening, captured using the simple

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power-law model, for asymptotically tall (AR 1) and shallow (AR 1) enclosures in the limit of high Prandtl number using a constant wall heat flux (CWHF) boundary condition. Barth and Carey [5] used a more realistic GNF model incorporating limiting high- and low-shear-rate viscosities to study a modified three-dimensional version of the problem to match the experimental conditions of Leung et al. [6]. Rayleigh Bernard natural convection, that is, heating from below and cooling from above, has also been studied for different non-Newtonian fluids including power-law GNF fluids [7], fluids with a yield stress [8–10], and viscoelastic fluids [11]. For natural convection in yield-stress fluids the current paper builds on the work by Vola et al. [12] and on a recent series of papers by the present authors [13–15]. Vola et al. [12] were the first to demonstrate that the strength of convection weakens with increasing yield stress and, as a consequence, the mean Nusselt number decreases with increasing yield stress. At a critical value of the yield stress, convection can even be completely eliminated. The yield-stress effects were captured using a regularized Bingham model in Turan et al. [13–15]: essentially replacing the unyielding solid by a liquid of extremely high viscosity. The effects of yield stress on heat and momentum transport for a large range of Rayleigh numbers (103–106) and Prandtl numbers (0.1 ≤ Pr ≤ 100) in a square enclosure subject to CWT sidewall boundary conditions were investigated in Turan et al. [13] and correlations proposed to capture the Nusselt number variation in these parameter ranges. Aspect ratio effects, once again with CWT thermal boundary conditions, were studied numerically in Turan et al. [14] and analytically by Vikhansky [16]. Vikhansky [16] concentrated on the influences of AR on the critical condition at which the buoyancy force just overcomes yield stress effects to give rise to initiation of flow in a rectangular enclosure. The full numerical simulations of Turan et al. [14] were able, however, to reveal that the critical conditions defined by Vikhansky [16] are insufficient to enhance the convective heat transfer and the Nusselt number remains equal to 1 (i.e., although there is fluid motion below this critical condition it is insufficient to enhance the heat transfer); thus, more useful correlations were provided by Turan et al. [14] to enable the necessary yield stress required to just inhibit convection. In Turan et al. [15] the effects of side-wall thermal boundary condition were addressed for square enclosures subjected to CWHF boundary conditions. The effects of AR on natural convection of Bingham fluids in rectangular enclosures with vertical side walls subjected to CWHF boundary condition are yet to be addressed in detail, and the present paper addresses this gap in the existing literature. In this respect the main objectives of the present study are as follows: 1. To analyze the effects of aspect ratio on the natural convection of Bingham fluids in a rectangular enclosure with side walls subjected to CWHF boundary conditions. 2. To indicate the differences between the effects of aspect ratio for CWT and CWHF boundary conditions. heat transfer engineering

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3. To propose a correlation for mean Nusselt number Nu that accounts for natural convection of Bingham fluids in rectangular enclosures with various aspect ratios for the CWHF boundary condition. The preceding objectives are met by undertaking a parametric numerical investigation of steady two-dimensional simulations of laminar natural convection in rectangular enclosures with differentially heated side walls for a range of different aspect ratios AR. The rectangular enclosures are considered to be completely filled with a yield-stress fluid obeying the Bingham model, which for a one-dimensional flow is given by ˙ where τy is the yield stress and μ the plastic τ = τ y + μγ, viscosity. Yield-stress effects on heat and momentum transport are investigated for nominal values of Rayleigh number (Ra) in the range 104–106 and the aspect ratio range 1/8 to 8 for a single Prandtl number (Pr = 500) representative of an incompressible fluid. The simulations have been carried out for a range of Bingham number 0 ≤ Bn ≤ Bn max (where Bn max is the Bingham number above which the mean Nusselt number attains a value of unity) and the heat transfer takes place purely due to thermal conduction. The value of Bn max depends on both Ra and Pr, which was discussed in detail in Turan et al. [13]. Thus, the range of Bingham numbers used for systematic numerical experiments in this paper is determined by the values of nominal Rayleigh and Prandtl numbers. Although some yield-stress fluids may have different values of Pr than Pr = 500, previous results [13] suggested that all the findings of this paper are qualitatively valid for other values of Prandtl number. As Bn max is parameterized here in terms of Ra, Pr, and AR and the parameterization of Pr dependence of Bn max has been validated earlier, it can be expected that the correlations proposed in this paper (see Eqs. (41), (43), (48), (49), (50), and (51) later in this paper) are going to be valid for other values of Prandtl number within the range of Rayleigh number considered here (indeed, it has been confirmed during the course of the present study that they work well for Pr = 7). It is worth noting that the present analysis has been carried out in nondimensional form so that a broad range of Ra and Bn could be explored, so that the results of this analysis are valid for a wide rage of fluids with different operating conditions (without being specific to only one fluid). The rate of heat transfer rate of Bingham fluids in a rectangular enclosure can be compared with the equivalent Newtonian fluid heat transfer rate at the same nominal values of Rayleigh number. The Rayleigh number represents the ratio of the strengths of thermal transport due to buoyancy to that due to thermal conduction, which can be defined for the CWT and CWHF boundary conditions as: RaCWT =

RaCWHF = vol. 35 no. 9 2014

ρ2 c p gβT L 3 = GrCWT Pr; μk ρ2 c p gβq L 4 = GrCWHF Pr μk 2

(1)

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where GrCWT and GrCWHF are the Grashof number in the CWT and CWHF configurations, respectively, and Pr is the Prandtl number, which are defined as: μc p ρ gβT L ρ gβq L ; and Pr = ; GrCWHF = GrCWT = μ2 μ2 k k (2) The Prandtl number depicts the ratio of the strengths of momentum diffusion to thermal diffusion, while the Grashof number represents the ratio of the strengths of buoyancy and viscous forces. Alternatively, the Prandtl number can be taken to represent the ratio of the hydrodynamic boundary layer to thermal boundary layer thicknesses. The preceding definitions should be referred to as “nominal” values, as they are based upon the constant plastic viscosity μ of the yielded fluid and are not based on a viscosity representative of the flow. Dimensional analysis shows that for Bingham fluids in the CWT (CWHF) configuration, Nu = f 1 (RaCWT , Pr, BnCWT , AR) (Nu = f 1 (RaCWHF , Pr, BnCWHF , AR)) where Bingham numbers in CWT and CWHF (i.e., BnCWT and BnCWHF ) and the Nusselt number Nu are given by τy τy L k hL BnCWT = ; BnCWHF = ; and Nu = μ gβT μ gβq k


2

3

2

4

(3) where Nu represents the ratio of heat transfer rate by convection to that by conduction in the fluid in question and the heat transfer coefficient h is defined as ∂ T 1 (4) × h = −k ∂ x wf (Twall − Tref ) where subscript wf refers to the condition of the fluid in contact with the wall, Twall is the wall temperature, and Tref is the appropriate reference temperature, which can be taken to be the temperature of the hot (cold) wall, respectively. In Eq. (3) the quantity τ y is the yield stress in the Bingham model [17], which expressed in tensorial form is γ˙ = 0

τ ≤ τy ,

for

τy τ = μ+ . γ

τ > τy ,

for

(6)

where γ˙ i j = ∂u i /∂ x j + ∂u j /∂ xi are the components of the rate ˙ τ the stress tensor, and τ and γ˙ are evaluated of strain tensor γ, based on the second invariants of the stress and the rate of strain tensors respectively (in a pure shear flow), which can be defined as 1/2 1 , (7) τ= τ:τ 2

1 γ˙ = γ˙ : γ˙ 2

τ = μyield γ˙

for

γ˙

γ˙ ≤

τy , μyield

τ y γ˙ τ = τ y + μ γ˙ − γ˙ μyield γ˙

(9)

for

γ˙ >

τy , μyield

(10)

where μyield is the yield viscosity. This GNF model replaces the solid material by a fluid of high viscosity. O’Donovan and Tanner [18] originally stated that a value of μyield equal to 1000 μ mimics the true Bingham model in a satisfactory way, although our own simulations have shown a more conservative value of 104 or 105 μ is arguably more appropriate to ensure high accuracy in certain conditions. Other regularization methods are available, and to investigate the effect of this choice of regularisation, some limited simulations have been conducted using the Papanastasiou exponential model [17], τ = τ y (1 − e−my ) + μγ˙

(11)

where m is the stress growth exponent which has the dimensions of time. As the viscosity varies throughout with the local shear rate in a Bingham fluid flow, an effective viscosity that may be estimated as μeff = τ y /γ˙ + μ may be more representative of the viscous stress within the flow than the constant plastic viscosity μ [13]. Therefore the Rayleigh, Prandtl, and Bingham numbers could have been defined differently if μeff were used instead ofμ (or indeed using some other choice of effective or characteristic viscosity). However, γ˙ will exhibit local variations in the flow domain, so using a single characteristic value in the definitions of the nondimensional numbers may not offer any additional benefit in comparison to the definitions given by Eqs. (1)–(3). The rest of the paper is organized as follows. The necessary mathematical background is discussed in the next section of the paper, followed by a brief discussion of the numerical implementation. Following these sections, results are presented and subsequently discussed. The main findings are summarized and conclusions are drawn in the final section of the paper.

(5)

γ˙

O’Donovan and Tanner [18] used a bi-viscosity model to regularize the Bingham model:

MATHEMATICAL BACKGROUND Governing Equations For the present study steady-state laminar flow of an incompressible fluid is considered. For incompressible fluids the conservation equations for mass, momentum, and energy under steady state can be written using tensor notation (i.e., x1 = x is the horizontal direction and x2 = y is the vertical direction) as: Mass conservation equation: ∂u i =0 ∂ xi

1/2 .

(8)

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Momentum conservation equations: ρu j

∂τi j ∂u i ∂P =− + ρgβδi2 (T − T ∗ ) + ∂x j ∂ xi ∂x j

Energy conservation equation:


∂T ∂ ρu j c p = ∂x j ∂x j

∂T k ∂x j

(13)

825

so this problem can also be treated as conjugate heat transfer problem. Inclusion of a viscous dissipation term is not a necessity in low-speed conjugate heat transfer problems. Similarly, the viscous dissipation term can also be ignored for natural convection of Bingham fluids where the kinetic energy of the fluid remains much smaller than its thermal energy.

(14)

where the temperature at the geometrical center of the domain is taken to be the reference temperature T ∗ for evaluating the buoyancy term ρgβδi2 (T − T ∗ ) in the momentum conservation equations for the CWHF configuration. For the CWT configuration, the cold wall temperature TC serves as the reference temperature T ∗ . The Kronecker delta δi2 ensures that the buoyancy term ρgβδi2 (T − T ∗ ) remains operational only in the momentum equation for the vertical direction (i.e., x2 direction). Two possible nondimensional forms of Eqs. (12)–(14) are presented in the appendix for the sake of completeness. The viscous dissipation terms (i.e., τi j ∂u i /∂ x j ) in the energy conservation equation (Eq. (14)) are neglected in the current analysis following several previous studies on natural convection of both Newtonian and Bingham fluids [1, 2, 9, 12, 16, 19–25]. The viscous dissipation effects are expected to play a key role in natural convection under the condition where kinetic energy of the fluid becomes of the same order of the thermal energy [26], which can be realized when gβL/c p >> 1. However, Gebhart [26] demonstrated that gβ/c p remains extremely small for most common fluids (∼10−9 to 10−6 m−1) and thus requires an extremely large length scale L for which viscous dissipation starts to affect the heat and momentum transport [26]. As this analysis does not attempt to address extremely large-scale applications, the effects of viscous dissipation could be neglected here without much loss of generality following previous studies [1, 2, 9, 12, 16, 19–25]. Moreover, the unyielded regions in Bingham fluids acts as a solid as far as heat transfer is concerned (due to only thermal conduction within the unyielded regions),

Boundary Conditions The simulation domain is shown schematically in Figure 1, where the two vertical walls of a rectangular enclosure are subjected to either constant wall heat flux or constant wall temperature, whereas the other boundaries are considered to be adiabatic in nature. The velocity components (i.e., u 1 = u and u 2 = v) are identically zero on each boundary because of the no-slip condition and impenetrability of rigid boundaries. For the CWHF configuration, the heat fluxes for the cold and hot vertical walls are specified (i.e., − k(∂ T /∂ x1 )|x1 =0 = q and − k(∂ T /∂ x1 )|x1 =L = q). In contrast, for the CWT configuration, the temperatures for the hot and cold vertical walls are specified (i.e., T (x1 = 0) = TH and T (x1 = L) = TC ). The temperature boundary conditions for the horizontal insulated boundaries are given by ∂ T /∂ x2 = 0 at x2 = 0 and x2 = H . The nondimensional forms of the preceding boundary conditions can be found in the appendix.

Numerical Implementation A finite-volume code is used to solve the coupled conservation equations of mass, momentum, and energy using a commercial software package called FLUENT. A second-order central differencing scheme is used for the diffusive terms and a secondorder upwind scheme for the convective terms. Coupling of the

u1 = 0 , u2 = 0 ⁄ =0

H

TH u1 = 0 u2 = 0

g

u1 = 0 , u2 = 0 ⁄ =0

TC u1 = 0 u2 = 0

q u1 = 0 u2 = 0

g

q u1 = 0 u2 = 0

x2 x1

⁄ =0 u1 = 0 , u2 = 0

⁄ =0 u1 = 0 , u2 = 0

L (a)

(b)

Figure 1 Schematic diagram of the simulation domain: (a) CWT configuration, (b) CWHF configuration.


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Table 1 Variations of Bn max with aspect ratio (AR) in the case of Bingham fluids for RaCWT = RaCWHF = 104–106 and Pr = 500

AR


0.125 0.25 0.5 1.0 2 4 8

RaCWT = RaCWHF = 104



— — 0.28 1.05 1.49 1.90 1.92

— 0.26 1.18 3.53 4.99 6.00 6.97

0.19 1.10 3.99 11.6 16.5 19.9 21.8

pressure and velocity is achieved using the well-known SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm [27]. The convergence criteria were set to 10−9 for all the relative (scaled) residuals. The grid independency of the results has been established using five different nonuniform meshes M1 (40 × 40), M2 (80 × 80), M3 (160 × 160), M4 (80 × 160), and M5 (80 × 320), and the details of these grids and grid-independency are provided in Table 1 of reference [14] and thus are not repeated here. In addition to this grid-dependency study, the simulation results for square enclosures were compared with the benchmark data of de Vahl Davis [19]. Interested readers are referred to references [13]–[15] for further details on benchmarking and grid independency for both CWT and CWHF boundary conditions.

RESULTS AND DISCUSSION Effects of Aspect Ratio and Bingham Number The distributions of nondimensional temperature for CWT (i.e., θCWT = (T − Tcen )/(TH − TC )) and CWHF (i.e., θCWHF = (T − Tcen )k/q L) configurations along the horizontal mid-plane (i.e., at x2 /H = 0.5) are shown in Figure 2 for Bingham number values (i.e., Bn CWT and Bn CWHF ) equal to 0, 0.5, and 1.5 at RaCWT = RaCWHF = 106 and Pr = 500, where Tcen is the temperature at the center of the domain. The corresponding distributions of nondimensional vertical velocity component V = u 2 L/α for the corresponding cases are shown in Figure 3. The case for which the highest mean Nusselt num H ber Nu = 0 hdy/k = Numax is obtained is also indicated in Figures 2 and 3 by an asterisk. It can be seen from Figure 2 that the temperature gradient ∂ T /∂ x1 at the vertical wall in the CWT configuration increases with increasing AR up to an aspect ratio value A Rmax for which the maximum value of Nu is obtained. For A R > A Rmax the temperature gradient ∂ T /∂ x1 at the vertical wall decreases with increasing AR for the CWT configuration. In contrast, in the CWHF configuration the temperature gradient ∂ T /∂ x1 next to the wall must remain unchanged to satisfy the boundary condition, that is, −k(∂ T /∂ x1 ) = q. Moreover, the highest value of Nu is obtained for the highest value of AR for the CWHF boundary condition. heat transfer engineering

In the present configuration a pure conduction solution yields a linear temperature profile across the length of the enclosure. It is evident from Figure 2 that for small values of AR (e.g., A R 1.0) the temperature profile remains linear in nature indicating conduction-dominated thermal transport. The temperature distribution with x1 /L becomes nonlinear for high values of AR at a given value of RaCWT (RaCWHF ) in the CWT (CWHF) configuration as the effects of convection strengthen for higher values of AR. It can be seen from Figure 3 that the magnitude of the nondimensional vertical velocity component (i.e., V = u 2 L/α) remains small for small values of aspect ratio indicating weak convective transport at these aspect ratios. It is evident from Figure 3 that the magnitude of V monotonically increases with increasing AR for a given value of RaCWT in the CWT configuration. Although this trend is not strictly monotonic in the CWHF configuration for all values of Bn CWHF , the magnitude of V attains high values at large aspect ratios. It can be seen from Figure 2 that the temperature profile becomes increasing nonlinear for increasing values of AR for both CWT and CWHF configurations. Moreover, the temperature profile becomes increasingly linear with increasing Bn CWT (Bn CWHF ) for a given set of values of nominal Rayleigh and Prandtl numbers. This behavior is consistent with the increasing magnitude of V for increasing AR for the CWT configuration. Although a monotonic increase in the magnitude of V is not obtained for the CWHF configuration, especially for large values of AR, the magnitude of V generally assumes high values for large values of AR. The magnitude of V represents the strength of advection in the enclosure, and an increase in V indicates the strengthening of convective transport and vice versa. In Bingham fluids, buoyancy force needs to overcome the yield stress effects to give rise to flow in the enclosure, and thus the effects of the buoyancy force are relatively weaker in Bingham fluids than in Newtonian fluids (i.e., Bn = 0). As a result of this, the effects of advection weaken with increasing Bingham number for a given set of values of Rayleigh and Prandtl numbers. For large values of Bingham number the flow within the enclosure becomes too weak to impart any influence on heat transfer by advection, and under this circumstance the thermal transport takes place predominantly due to thermal conduction. As a result of this the temperature distribution becomes increasingly linear with increasing values of Bingham number for both CWT and CWHF configurations. It is evident from Figures 2 and 3 that the effects of buoyancy-induced convection are stronger in high AR cases and thus the effects of buoyancy can counter the effects of flow resistance for relatively high values of Bingham number for both CWT and CWHF configurations. This is reflected in the linear distributions of θCWT and θCWHF in Figure 2 for small values of AR in both CWT and CWHF configurations. The effects of AR and Bingham number can further be seen from inspection of Figures 4 and 5, where the isotherms and the contours of nondimensional stream function = ψ/α are shown for Bn CWHF = 0, 0.5, and 1.5 for RaCWHF = 106 . The apparently unyielded regions (AUR) are shown by gray shading vol. 35 no. 9 2014

O. TURAN ET AL.

0.50

0.2 AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 AR = 8*

0.0

-0.1

RaCWHF = 10

6

0.00

-0.25

Bn = 0

0.2

0.4

RaCWT = 106 Bn = 0 Constant Wall Temperature

Constant Wall Heat Flux

-0.2 0.0

AR = 0.25 AR = 0.5* AR = 1 AR = 2 AR = 4 AR = 8

0.25

θ

θ

0.1

0.6

0.8

-0.50 0.0

1.0

0.2

0.4

0.6

0.8

1.0

x/L

x/L

(a) 0.50

0.50 AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 AR = 8 *

0.00

RaCWHF = 106

-0.25

AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4* AR = 8

0.25

θ

θ

0.25

0.00

-0.25

RaCWT = 10

Bn = 0.5

-0.50 0.0

0.2

6

Bn = 0.5


0.4

Constant Wall Temperature

0.6

0.8

-0.50 0.0

1.0

0.2

0.4

x/L

0.6

0.8

1.0

x/L

(b) 0.50

0.50 AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 AR = 8 *

0.25

0.00

RaCWHF = 106

-0.25

AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 AR = 8 *

0.25

θ

θ


827

0.00

RaCWT = 10

-0.25

Bn = 1.5

Bn = 1.5



-0.50 0.0

0.2

6

0.4

0.6

x/L

0.8

1.0

(c)

-0.50 0.0

0.2

0.4

0.6

0.8

1.0

x/L

Figure 2 Variations of nondimensional temperature θ along the horizontal mid-plane (i.e., x2 /H = 0.5) for Bingham fluid case at RaCWT = RaCWHF = 106 and Pr = 500: (a) BnCWT = BnCWHF = 0, (b) BnCWT = BnCWHF = 0.5, and (c) BnCWT = BnCWHF = 1.5 (asterisk highlights the AR in which the maximum mean Nusselt number Nu occurs).


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60

800 AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 AR = 8 *

40

400 200

V

V

20

AR = 0.25 AR = 0.5 * AR = 1 AR = 2 AR = 4 AR = 8

600

0

0 -200

-20 RaCWHF = 106

RaCWT = 106

-400

Bn = 0

-40

Bn = 0

-600

0.2


0.4

0.6

0.8

-800 0.0

1.0

0.2

0.4

0.6

0.8

1.0

x/L

x/L (a) 300

10 AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 AR = 8 *

8 6 4

AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4* AR = 8

200 100

V

V

2 0

0

-2 -100

-4

RaCWHF = 10

-6

6

RaCWT = 10

6

Bn = 0.5

-200

Bn = 0.5

-8


-10 0.0

0.2


0.4

0.6

0.8

-300 0.0

1.0

0.2

0.4

0.6

0.8

1.0

x/L

x/L (b) 4

80 AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 AR = 8 *

3 2

AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 AR = 8*

40

V

1

V



-60 0.0

0

0

-1 RaCWHF = 106

-2

RaCWT = 10

-40

Bn = 1.5

Bn = 1.5

-3



-4 0.0

0.2

6

0.4

0.6

x/L

0.8

-80 0.0

1.0

0.2

(c)

0.4

0.6

0.8

1.0

x/L

Figure 3 Variations of nondimensional vertical velocity component V along the horizontal mid-plane (i.e., x2 /H = 0.5) for Bingham fluid case at RaCWT = RaCWHF = 106 and Pr = 500: (a) BnCWT = BnCWHF = 0, (b) BnCWT = BnCWHF = 0.5, and (c) BnCWT = BnCWHF = 1.5 (asterisk highlights the AR in which the maximum mean Nusselt number Nu occurs).


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1.4 8 0.

Bn = 0.5

0.6

1

0.2

-0.4

0

-0.2

0.2

0.4

0.4

0

0.4

0.2

-0.2

-1

0.4

0 -0.4

0. 2

-0.2

1

-0.4

0

-0.2

2

Bn = 1.5

0.4

AR = 4

AR = 2

AR = 1

AR = 0.5

-0.4

0

-0.2

0.4

0.2

-0.4

-0.2

-0.6

0

-0. 4 -1

AR = 8

0.2

-0 .

2

-0 .6

0

-1

.2

0.2

0. 4

-0 .

2

0.6

0.

2

0

-0.4

0.2

0. 4

1.2

-0.2

0.6

1.6

-0.

0.4

0

.4 -0

-0.8

-0.4

-1.4

0.2

-0.6

-1. 6


Bn = 0

AR = 0.25

Figure 4 Contours of nondimensional temperature θ for Bingham fluids in the case of CWHF boundary condition at Pr = 500 for RaCWHF = 106.


vol. 35 no. 9 2014

O. TURAN ET AL.


830

Figure 5 Contours of nondimensional stream functions (ψ/α) for Bingham fluids in the case of CWHF boundary condition at Pr = 500 for RaCWHF = 106. The apparently unyielded regions (AUR) are shown by gray shading.


vol. 35 no. 9 2014


O. TURAN ET AL.

(zones of fluid where |τ| ≤ τ y according to the criterion used by reference [28]) in Figure 5 for Bingham fluids. It is worth noting that these zones are not really “unyielded” in the true sense, as pointed out by Mitsoulis and Zisis [29]. In the present study a bi-viscosity approximation is used to model the Bingham fluid so flow will always be present within these essentially very-high-viscosity regions, which can alternatively be viewed as regions of extremely slowly moving fluid. It is important to stress that the islands of AUR within the center of the enclosure alter significantly with increasing values of μ yield (shown in Figure 5 for μ yield = 104 μ) while the mean Nusselt number, the stream function, and the zones of AUR at the corners of the enclosure are independent of μ yield for μ yield ≥ 103 μ. For a given value of τ y the zones with very low shear rate that satisfy |τ| ≤ τ y are expected to shrink with increase in μ yield . Therefore, the AUR zones are somewhat dependent on the choice of μ yield , while the velocity and temperature distributions (i.e., qualitative and quantitative distributions of stream function and isotherms) remain independent of the value of μ yield . Thus, the precise shape and size of AURs do not have any major influence on the mean Nusselt number Nu in the present configuration and any in-depth discussion of their significance is unnecessary given the scope of this paper. It is evident from Figure 5 that the size of AURs increases with increasing Bingham number, which demonstrates the weakening of convection in the enclosure. It can be seen from Figure 4 that the isotherms remain parallel to the vertical walls for large values of Bn CWHF (e.g., Bn CWHF = 1.5) in the Bingham fluid case, which indicates that the thermal transport is diffusion driven (i.e., conduction driven). The isotherms become increasingly curved with decreasing Bn when convection plays a significant role in thermal transport. Comparing the Bn CWHF = 0, 0.5, and 1.5 cases at RaCWHF = 106 reveals that for Bingham fluids the isotherms are parallel to the vertical walls, whereas the isotherms are curved in the case of Newtonian fluid flow for small values of AR. This difference essentially suggests that the aspect ratio AR above which convection effects are felt is greater for Bingham fluids than in the case of Newtonian fluids because of stronger viscous effects in the Bingham fluid at the same nominal values of Rayleigh number and aspect ratio. Although Figures 4 and 5 are shown for the CWHF boundary condition, the same qualitative behavior has been observed for the CWT boundary condition. The aforementioned behavior can be explained using orderof-magnitude scaling arguments. Equating the order of magnitudes of the inertial and buoyancy forces in the vertical direction yields ϑ2 /H ∼ gβTchar

and

ϑ∼

gβTchar H

(15)

where ϑ is a characteristic velocity scale and Tchar is a characteristic temperature difference that can be taken as the temperature difference between the vertical walls T in the CWT configuration. For the CWHF configuration Tchar can be estimated as: Tchar ∼ qδth /k and thus ϑ can be estimated heat transfer engineering

as

831

ϑ ∼ (μ/ρL) (RaCWT /Pr )A R (for CWT) and ϑ ∼ (μ/ρL) (RaCWHF /Pr )A R(δth /L) (for CWHF) (16)

Equation (16) clearly suggests that the velocity magnitude in both CWT and CWHF configurations increases with increasing AR for the same numerical values of Rayleigh number and aspect ratio. Moreover, Eq. (16) indicates that the V values in the CWHF configuration are likely to be smaller than those obtained in the CWT configuration, which is consistent with the observations made from Figure 3 (as δth /L ≤ 1). In the case of CWHF configuration, the temperature difference between the vertical walls remains equal to q L/k for conduction-driven heat transfer. This suggests that θCWHF scales as |θCWHF | ∼ O(δth /L), whereas θCWT scales as |θCWT | ∼ O(1) in the CWT case, which in turn implies that θCWHF is expected to decrease with decreasing δth . It can be seen from Figure 2 that the thermal boundary layer thickness δth increases with increasing Bingham number for both CWT and CWHF boundary conditions. This effect is a consequence of the thermal transport becoming increasingly diffusion driven (i.e., conduction driven) with increasing Bn CWHF for a given set of values of RaCWHF , Pr, and AR. An increase in the magnitude of θCWHF with increasing Bn CWHF in Figure 2 is a manifestation of this effect. It can be seen from Eq. (16) that the velocity scale ϑ is expected to weaken with decreasing AR, which indicates that the thermal transport is expected to be conduction driven for small values of AR, which is again consistent with the observations made in the context of Figures 2 and 3. The strengthening of convective transport with increasing AR was demonstrated earlier in several previous studies for Newtonian fluids in the CWT configuration by analytical [20–24], numerical [1, 30–36], and experimental [37–40] means. The weakening of convective transport for Bingham fluids at small values of AR for the CWT configuration has been demonstrated by Turan et al. [14] and Vikhansky [16] by numerical and analytical means respectively. As already noted, for Bingham fluids an effective viscosity μeff can be estimated as μeff = τ y /γ˙ + μ and scaling γ˙ ∼ ϑ/δ and one obtains the following expression using Eq. (16): √ μeff ∼ μ[1 + Bn CWT (δ/L)/ AR] (for CWT); √ μe f f ∼ μ[1 + Bn CWHF (δ/L)/ AR] (for CWHF) (17) The preceding expressions indicate that the effective viscosity increases with decreasing AR for a given value of Bingham number. This enhanced viscosity is reflected in the weaker convection for smaller values of AR (see Figures 2–5). Moreover, the effective viscosity increases with increasing Bingham number and thus the convection strength weakens with increasing Bingham number in both CWT and CWHF configurations (see Figures 2–5). As δ/L ≤ 1, the magnitude of μeff is likely to be higher in the CWHF configuration than in the CWT configuration, which is evident in the weaker convection strength vol. 35 no. 9 2014

832

O. TURAN ET AL.

in the CWHF configuration than in the CWT configuration for the same numerical values of Bingham, Prandtl, and Rayleigh numbers, as observed in Figures 2 and 3. The thermal transport due to advection within the boundary layer adjacent to the vertical wall can be scaled in the following way: δ ρc p u 2 T d x1 ∼ (kT ) RaCWT PrAR T1 = 0

× (δ/L) (for CWT) T1 =

(18a)

ρc p u 2 T d x1 ∼ (qL) RaCWHF PrAR(δth /L)3/2

δ


0

× (δ/L) (for CWHF)

(18b)

The maximum magnitude of thermal diffusion contribution in the vertical direction can be scaled as: L ∂T −k d x1 ∼ (kT )AR −1 (for CWT) (19a) T2 = ∂ x2 0 T2 = 0

L

−k

∂T d x1 ∼ (kT )AR −1 ∼ (qL)AR−1 ∂ x2

× (δth /L) (for CWHF)

(19b)

From the preceding scaling results it can be seen that the convective transport strengthens with increasing aspect ratio AR, while the diffusive transport weakens with increasing AR. Turan et al. [41] demonstrated, based on a similar scaling argument, that δ/L in Newtonian fluids (i.e., Bn = 0) scales as δ/L ∼ (A R)0.2 (Pr/RaCWHF )0.2 f 30.2 and δ/L ∼ (A R)0.25 (Pr/RaCWT )0.25 in the CWHF and CWT configurations, respectively, where f 3 depicts the ratio of δ and δth in the CWHF configuration (i.e., f 3 = δ/δth ). Equating the order of magnitudes of inertial and viscous forces for Bingham fluids gives rise to ϑ2 ϑ 1 ∼ τy + μ (20) H δ δ Substituting Eq. (16) in Eq. (20) yields the following relation for the CWHF configuration: √ (δth /L)5/2 RaCWHF /Pr ∼ A R/ f 32 + Bn CWHF (δth /L)1/2 / f 3 (21) whereas the following expression can be obtained for the CWT configuration [14]:

1 BnCWT 1 + δth /L ∼ √ Bn 2CWT 2 2 f 4 RaCWT /Pr + 4A R 1/2

RaCWT Pr

1/2 (22)


where f 4 depicts the ratio of δ and δth in the CWT configuration (i.e. f 4 = δ/δth ). Equation (22) clearly suggests that δth /L increases with AR (BnCWT ) for a given set of values of Rayleigh and Prandtl numbers. It is worth noting that an exact analytical solution to Eq. (21) does not exist but the limiting behaviors can be obtained. For example, for small values of BnCWHF the contribution√of BnCWHF (δth /L)1/2 / f 3 can be ignored in comparison to AR/ f 32 , which leads to for Bingδth /L ∼ (A R)0.2 (Pr/RaCWHF )0.2 f 3−0.8 . Moreover, √ 2 ham numbers where the contribution of AR/ f 3 remains comparable to BnCWHF (δth /L)1/2 / f 3 , one obtains δth /L ∼ 1/2 BnCWHF /[ f 30.5 (RaCWHF /Pr )0.25 ]. The limiting conditions just described suggest that δth /L also increases with AR (BnCWHF ) for a given set of values of Rayleigh and Prandtl numbers in the CWHF configuration, although the quantitative variations are likely to be different from the CWT boundary condition. In Newtonian fluids (i.e., Bn CWT = 0 and Bn CWHF = 0), 0.25 Pr 0.75 A R 0.75 and T1 ∼ T1 scales as T1 ∼ (kT )RaCWT (q L)Pr A R f 3−1 for CWT and CWHF boundary conditions, respectively [41]. The quantities kT and q L remain unchanged for the CWT and CWHF configurations, and thus the convective transport in the CWHF configuration strengthens more rapidly with increasing AR than in the CWT configuration for a given set of values of Rayleigh and Prandtl numbers in Newtonian fluids. Moreover, in Newtonian fluids the maximum magnitude of T2 scales as T2 ∼ (kT )AR −1 (T2 ∼ (q L)AR −0.8 (Pr/RaCWHF )0.2 f 3−0.8 ) in the CWT (CWHF) boundary condition [41]. Qualitatively similar behavior has been observed even in the case of Bingham fluids. The preceding scaling estimates suggest that the strengthening of convective transport can be eclipsed by the weakening of diffusive transport, and thus the value of Nu attains its maximum value at a particular aspect ratio AR max in the CWT configuration, and for AR > AR max the mean Nusselt number starts to decrease with increasing AR. The value of A Rmax at which Nu attains its maximum value depends on the relative strengths of thermal advection and diffusion mechanisms, and Eqs. (18) and (19) suggest that ARmax is expected to change with Bn CWT for a given set of values of Rayleigh and Prandtl numbers and Figure 3 indeed demonstrates that the value of AR max increases with increasing BnCWT . As the strengthening (weakening) of thermal advection (diffusion) in the CWHF configuration with increasing AR is more (less) rapid than in the CWT configuration (e.g., see the scaling estimates of T1 and T2 for Newtonian fluids shown above), the mean Nusselt number Nu increases monotonically with increasing AR in the CWHF configuration for the range of aspect ratios (i.e., 1/8 ≤ AR ≤ 8) considered in this study. Nonmonotonic aspect ratio dependence of Nu in the present configuration for CWT boundary condition is well known for Newtonian fluids, which was reported in several previous studies [1, 22, 30–41]. The nonmonotonic aspect ratio dependence of Nu for CWT boundary condition in the case of Bingham and Newtonian fluids was discussed in detail in references vol. 35 no. 9 2014


O. TURAN ET AL.

[14] and [41] by the present authors, and interested readers are referred to references [14] and [41] for a more detailed discussion. It is worth noting that Eqs. (21) and (22) are valid when δth /L < 1, and for large values of Bingham number δth ultimately becomes of the order of L and under that condition the heat transfer takes place predominantly due to thermal conduction and in that circumstance the nondimensional temperature variation along x1 direction becomes linear for both CWT and CWHF configurations; thus, the definitions of Rayleigh number and Bingham number for these two boundary conditions ultimately become equivalent to each other (i.e. RaCWT = RaCWHF and Bn CWT = Bn CWHF ). For shallow enclosures (i.e., AR 1) the limiting condition given by RaCWT AR 3 → 0 (RaCWHF A R 3 → 0) is referred to as the parallel-flow regime [20, 23]. In the parallel-flow regime the vertical velocity component at the core of the enclosure disappears and the fluid flow in the enclosure consists of two counterflowing horizontal streams, and the temperature gradient in the horizontal direction K = ∂ T /∂ x1 ∼ T /L ∼ q/k remains constant. Under this condition the definitions of Rayleigh number and Bingham number for these two boundary conditions are also ultimately equivalent to one other (i.e., RaCWT = RaCWHF and Bn CWT = Bn CWHF ). In this shallow AR limit where the definitions are identical, the equilibrium of vorticity generation/destruction by buoyancy and the molecular diffusion of vorticity at the middle of the domain yields ρgβK ∼ −

∂ 2 τ12 ∂ x22

(23a)

833

ergy transport equation gives ρc p u 1

∂T ∂2T ∼k 2 ∂ x1 ∂y

ρc p u c

T1 T ∼k 2 f 2 (for CWT) L FCWT H 2 5

or ρc p u c

T1 q ∼k 2 f 2 (for CWHF) k FCWHF H 2 5

−2 5 T1 ∼ RaCWT FCWT A R 5 T 1 − FCWT Bn CWT /[A R 2 (RaCWT /Pr)1/2 ] / f 5 (for CWT)

or ρgβ

τy q μu c ∼ 3 (for CWHF) (23b) + k FCWHF H FCWHF H 3

which leads to the following scaling for the horizontal velocity component at the core of the enclosure: −2 3 u c ∼ FCWT RaCWT A R 3 (α/L)[1 − FCWT Bn CWT

× (Pr/RaCWT )1/2 /A R 2 ] (for CWT)

(24a)

−2 3 u c ∼ FCWHF RaCWHF A R 3 (α/L)[1 − FCWHF Bn CWHF

× (Pr/RaCWHF )1/2 /A R 2 ] (for CWHF)

(24b)

where the hydrodynamic boundary layer thickness on horizontal surfaces is scaled as δ ∼ FCWT H (δ ∼ FCWHF H ) for CWT (CWHF) boundary condition, with FCWT (FCWHF ) being an appropriate fraction (i.e., 0 < FCWT < 1 and 0 < FCWHF < 1). Using the balance of convective and diffusive terms of the enheat transfer engineering

(26a)

−2 5 T1 ∼ RaCWHF FCWHF A R 5 (q L/k)[1 − FCWHF Bn CWHF

/[A R 2 (RaCWHF /Pr)1/2 ]]/ f 5 (for CWHF)

(26b)

where T1 is the characteristic temperature difference between the horizontal adiabatic walls and the thermal boundary layer thickness adjacent to the horizontal walls is scaled as δth ∼ FH/ f 5 . For Newtonian fluids (i.e., BnCWT = 0), T1 scales as T1 ∼ RaCWT F 5 A R 5 T / f 5 , which is consistent with the analytical results of Cormack et al. [23] (i.e., T1 = RaCWT A R 5 (TH − TC )/720). Bejan and Tien [20] argued that T1 ≤ (TH − TC )/10 in the parallel-flow regime (i.e., RaCWT A R 3 → 0), which yields the following criterion for this regime: RaCWT < 72(A R)−5

τy μu c T ∼ 3 (for CWT) + L FCWT H FCWT H 3

(25b)

which yields

or ρgβ

(25a)

(27)

Similarly a criterion for the parallel-flow regime for Newtonian fluids with the CWHF boundary condition can be constructed as follows: T1 < 0.1q L/k

(28a)

which leads to the following condition: RaCWHF < 72(AR)−5

(28b)

The other extreme convection condition is referred to as the “boundary-layer regime” [20], where the Rayleigh number attains large values and T1 remains comparable to T = (TH − TC ) (i.e., T1 ∼ T ). Under this condition, high values of temperature gradient are confined to two thin boundary layers adjacent to the vertical walls. Bejan and Tien [20] argued that the inception of the boundary-layer regime can be indicated by, K < 0.1(T /L) which gives rise to the following criterion for the CWT boundary condition: RaCWT > 4.4 × 104 AR−14/3

(29a)

Similarly, if the inception of boundary-layer regime for the CWHF boundary condition is taken to be K < 0.1q/k, one vol. 35 no. 9 2014

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O. TURAN ET AL.

obtains the following criterion: RaCWHF > 4.4 × 104 A R −14/3

(29b)

Some of the characteristics of both the boundary layer and parallel-flow regimes are observed if the Rayleigh number falls in the range 72(A R)−5 < RaCWT < 4.4 × 104 A R −14/3 (for CWT)

(30a)


72(A R)−5 < RaCWHF < 4.4 × 104 A R −14/3 (for CWHF) (30b) Bejan and Tien [20] termed this region as the “intermediate regime.” The convection regimes according to Eqs. (27)–(29) for the cases considered here are provided in Table 3 of reference [14]. Most AR ≥ 1 cases for Rayleigh numberRa = 104 , 105 , and 106 belong to the boundary-layer regime except for the A R = 1.0 case at Ra = 104 , which belongs to the intermediate flow regime. All the A R < 1 cases considered here represent either the parallel-flow regime or the intermediate flow regime.

zone with an almost zero horizontal temperature gradient at the core of the enclosure. As the temperature gradient at the wall needs to be constant in the CWHF configuration, the increase in the size of the core with zero horizontal temperature gradient leads to a smaller temperature difference between the vertical walls for higher values of Rayleigh number in this configuration. This observation is also consistent with the order of magnitude estimation of nondimensional temperature in the CWHF configuration (i.e., |θCWHF | ∼ O(δth /L)). It can be seen from Figure 7 that V assumes smaller values in the CWHF configuration than in the CWT configuration for the same numerical values of Rayleigh number at a given set of values of Pr and AR. This behavior is expected from the scaling estimates provided in Eq. (16). The smaller temperature difference in the CWHF configuration induces relatively weaker buoyancy effects than in the case of CWT configuration, which is reflected in the smaller value of velocity magnitude.

Behavior of Mean Nusselt Number Nu Effects of Rayleigh Number The distributions of nondimensional temperature for CWT (i.e., θCWT = (T − Tcen )/(TH − TC )) and CWHF (i.e., θCWHF = (T − Tcen )k/q L) configurations along the horizontal mid-plane (i.e., at x2 /H = 0.5) are shown for Bn CWT = 0.5(Bn CWHF = 0.5) in Figures 6a, 6b, and 6c for Rayleigh number RaCWT = RaCWHF = 104 , 105 , and 106 , respectively, at different values of aspect ratio AR ranging from 1/8 to 8.0. The corresponding distributions of V are shown in Figure 7. Once again, the case for which the highest mean Nusselt number Nu is obtained is also indicated in Figures 6 and 7 by an asterisk in the legend. It is apparent from Figures 6 and 7 that the nondimensional temperature profile becomes increasingly nonlinear and the velocity magnitude increases with increasing Rayleigh number for a given value of AR, indicating strengthening of convective transport. This behavior is consistent with the scaling estimates presented in Eq. (16) and Eq. (18) for both CWT and CWHF boundary conditions. It can be seen from Figure 7 that the magnitude of V remains negligible for small values of AR (e.g., A R = 1/8), which is consistent with the behavior in the parallel-flow regime. As the effects of convection strengthen with increasing Rayleigh number, the temperature distribution becomes nonlinear and V attains nonnegligible values for enclosures with small aspect ratios where heat transfer takes place predominantly due to thermal conduction for lower values of Rayleigh number for both CWT and CWHF boundary conditions. It can be seen from Figure 6 that the temperature difference between vertical walls decreases with increasing Rayleigh number in the CWHF configuration, whereas the temperature difference remains unaltered in the CWT boundary condition. An increase in Rayleigh number results in a decrease in thermal boundary-layer thickness (see Eq. (22) and the limiting behavior of δth /L from Eq. (21)), and this increases the size of the heat transfer engineering

The variation of mean Nusselt number Nu with Bn CWT (Bn CWHF ) for different values of RaCWT (RaCWHF ) is presented in Figure 8. It is evident from Figure 8 that Nu decreases with increasing Bingham number, and for large values of Bingham number Nu ultimately settles to unity (i.e., Nu = 1.0). It has been discussed earlier that the effective viscosity increases with increasing Bingham number (see Eq. (17)), and thus convection effects progressively weaken with increasing Bingham number; at a value of Bingham number Bn max the flow in the enclosure becomes too weak to impart any influence on thermal transport, and under this condition heat transfer takes place principally due to thermal conduction, which is reflected in the unity value of mean Nusselt number (i.e., Nu = 1.0). This suggests that the effects of convection are important for Bn < Bn max , and for Bn ≥ Bn max the heat transfer takes place predominantly due to conduction. That heat transfer is predominantly due to conduction is reflected in the unity value of Nu for Bn ≥ Bn max . Moreover, it can be seen from Figure 9 that Nu increases with increasing Rayleigh number for a given set of values of Pr , Bingham number and AR. The variation of Nu in response to Rayleigh and Bingham numbers for different aspect ratios is found to be qualitatively consistent with previous studies for the CWT configuration [12–14]. The effects of convection strengthen with increasing Rayleigh number, which in turn gives rise to a greater value of Nu for both Newtonian and Bingham fluids for a given value of aspect ratio AR. As convection strengthens with increasing Rayleigh number for a given set of values of Pr and AR, the effects of buoyancy-driven flow can counter viscous effects up to a larger value of Bingham number, which is reflected in the increase in Bn max with increasing Rayleigh number. As the Bingham number approaches Bn max the temperature distributions for both the CWT and CWHF configurations approach the profile governed by the pure-conduction solution. As a result of this, the definitions of Rayleigh and Bingham vol. 35 no. 9 2014

O. TURAN ET AL.

0.50

0.00

RaCWHF = 104

-0.25

0.2

RaCWT = 10

-0.25

Bn = 0.5

4

Bn = 0.5 Constant Wall Temperature

0.4

0.6

0.8

-0.50 0.0

1.0

0.2

0.4

x/L

0.6

0.8

1.0

x/L

(a) 0.50

0.00

RaCWHF = 105

-0.25

0.00

Bn = 0.5


0.2

RaCWT = 105

-0.25

Bn = 0.5

-0.50 0.0

AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 * AR = 8

0.25

θ

θ

0.50

AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 AR = 8 *

0.25

0.4


0.6

0.8

-0.50 0.0

1.0

0.2

0.4

x/L

0.6

0.8

1.0

x/L

(b) 0.50

0.50

AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 AR = 8 *

0.25

0.00

RaCWHF = 106

-0.25

AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 * AR = 8

0.25

θ

θ


0.00


-0.50 0.0

AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 AR = 8

0.25

θ

θ

0.50

AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 AR = 8

0.25

835

0.00

RaCWT = 106

-0.25

Bn = 0.5

Bn = 0.5 Constant Wall Heat Flux

-0.50 0.0

0.2

0.4


0.6

0.8

-0.50 0.0

1.0

x/L

0.2

0.4

0.6

0.8

1.0

x/L

(c)

Figure 6 Variations of nondimensional temperature θ along the horizontal mid-plane (i.e., x2 /H = 0.5) for Bingham fluid case at BnCWT = BnCWHF = 0.5 and Pr = 500: (a) RaCWT = RaCWHF = 104, (b) RaCWT = RaCWHF = 105, and (c) RaCWT = RaCWHF = 106 (asterisk highlights the AR in which the maximum mean Nusselt number Nu occurs).


vol. 35 no. 9 2014

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O. TURAN ET AL.

0.6

0.4 AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 AR = 8

0.3 0.2

0.2

V

V

0.1 0.0 -0.1

-0.2 RaCWT = 104

Bn = 0.5

-0.3

0.2


0.4

0.6

0.8

-0.6 0.0

1.0

x/L

0.2

0.4

0.6

0.8

1.0

x/L

(a)

4

60 AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 AR = 8 *

3 2 1

AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 * AR = 8

40 20

0

V

V

Bn = 0.5

-0.4


-0.4 0.0

-1

0 -20 RaCWT = 10

RaCWHF = 105

-2


0.2

0.4

5

Bn = 0.5

-40

Bn = 0.5

-4 0.0


0.6

0.8

-60 0.0

1.0

x/L

0.2

0.4

0.6

0.8

1.0

x/L

(b)

10

300 AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 AR = 8 *

8 6 4

AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 * AR = 8

200 100

2 0

V

V


0.0

RaCWHF = 104

-0.2

-3

AR = 0.25 AR = 0.5 AR = 1 AR = 2 AR = 4 AR = 8

0.4

0

-2 -100

-4 RaCWHF = 10

-6 -8

6

RaCWT = 10

Bn = 0.5

-200

0.2

0.4

Bn = 0.5 Constant Wall Temperature


-10 0.0

6

0.6

0.8

-300 0.0

1.0

0.2

0.4

0.6

0.8

1.0

x/L

x/L

(c) Figure 7 Variations of nondimensional vertical velocity component V along the horizontal mid-plane (i.e., x2 /H = 0.5) for Bingham fluid case at BnCWT = BnCWHF = 0.5 and Pr = 500: (a) RaCWT = RaCWHF = 104, (b) RaCWT = RaCWHF = 105, and (c) RaCWT = RaCWHF = 106 (asterisk highlights the AR in which the maximum mean Nusselt number Nu occurs).


vol. 35 no. 9 2014

O. TURAN ET AL.

RaCWHF (RaCWT) = 104

837



1.25

6

1.20

5

AR = 0.25

AR = 0.25 Nu

4

Nu

1.15 1.10

3

1.05

2

1.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30

1 0.0

0.3

0.6

Bn 1.30

4.0

1.25

3.5

0.9

1.2

1.5

4

5

Bn 10 9

Nu

1.15

2.5

1.10

2.0

1.05

1.5

1.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30

1.0 0.0

AR = 0.5

7

3.0

Nu

Nu

8

AR = 0.5

1.20

6 5 4 3 2 1

0.3

0.6

Bn

0.9

1.2

1.5

0

9

AR = 1

1.8 1.6

7

Nu

AR = 1 Nu

Nu

8

4

3

2

5

3

1.2

2 1 0.3

0.6

0.9

1.2

1.5

1 0

1

2

Bn

3

4

5

0

5

10

Bn

2.4

15

20

Bn

5

9 8

2.2

AR = 2

4

Nu

1.8 1.6

7

AR = 2

AR = 2

6

Nu

2.0

Nu

AR = 1

6

4

1.4

3

5 4

1.4

2

3

1.2

2 1 0.3

0.6

0.9

1.2

1.5

1 0

1

2

3

Bn

4

5

6

0

5

10

Bn

15

20

25

Bn

5

2.50

8 7

2.25 4

AR = 4

6

2.00

AR = 4

1.75

Nu

AR = 4 Nu

Nu

3

10

5

2.0

1.0 0.0

2

Bn

2.2

1.0 0.0

1

Bn

2.4

3

1.50

5 4 3

2 1.25

2

1.00 0.0

1 0.5

1.0

1.5

2.0

1 0

1

2

3

Bn

4

5

6

7

0

5

10

Bn

2.6

5

2.2

4

15

20

25

30

Bn 8 7 6

AR = 8

1.8

Nu

Nu

AR = 8 Nu


AR = 0.5

3

1.4

2

1.0 0.0

1

AR = 8

5 4 3 2

0.5

1.0

1.5

Bn

2.0

2.5

1 0

1

2

3

4

5

6

7

8

0

Bn

5

10

15

20

25

30

35

Bn

Figure 8 Variations of Nu with Bingham number for both (- - -) CWT, (—), and CWHF configurations for different values of aspect ratio at Pr = 500.


vol. 35 no. 9 2014

838

O. TURAN ET AL.

2.6

1.25

2.4

1.20

2.2

1.15

2.6 2.4

AR = 2

AR = 0.5

2.2

AR = 4

1.10

2.0

2.0

1.8

Nu

Nu

1.05 1.00 0.00 0.01 0.02 0.03 0.04 0.05

1.6

AR = 8

1.8 1.6

AR = 1- 2 - 4 - 8

1.4

1.4

1.2

1.2

1.0 0.00

0.05

0.10

0.15

0.20

AR = 1

1.0 0.00

0.25

0.05

0.10

Bn

0.15

0.20

0.25

Bn (a)

4.5 4.0

1.25

5.0

1.25

1.20

4.5

1.20

4.0

1.15

1.15

AR = 0.25

3.5 3.0

AR = 0.25

1.10

3.5

1.05

1.05

Nu

Nu

1.10

2.5

1.00 0.00

2.0

AR = 0.5 - 1 - 2 - 4 - 8

0.01

0.02

0.03

0.04

3.0

1.00 0.00

0.05

0.01

0.02

0.03

0.04

2.5 AR = 4

2.0

AR = 1 AR = 8

1.5

1.5

1.0 0.0

1.0 0.0

0.2

0.4

0.6

0.8

1.0

AR = 2

AR = 0.5

0.2

0.4

0.6

0.8

1.0

Bn

Bn (b) 8 7

3.0

10

6

2.5

9

5

8

4

7

3

6

2

AR = 0.25 2.0

6

AR = 0.25

1.5

5

1.0 0.00

4

0.04

0.08

0.12

Nu

Nu


AR = 0.5

0.16

AR = 0.5

5

0.08

0.12

0.16

AR = 2

3

AR = 0.5 - 1 - 2 - 4 - 8

0.04

AR = 1

4

3

1 0.00

AR = 4

2

2

1

1

0

1

2

3

4

AR = 8

0

5

1

2

3

4

5

Bn

Bn (c)

Figure 9 Variations of Nu with Bingham number for both CWHF (left column) and CWT (right column) cases for different values of aspect ratio at Pr = 500: (a) RaCWT = RaCWHF = 104, (b) RaCWT = RaCWHF = 105, and (c) RaCWT = RaCWHF = 106.


vol. 35 no. 9 2014

O. TURAN ET AL.

Nu = hL/k ∼ q f L/Tk ∼ kTL/Tk ∼ L/δth

(31)

This suggests that L/δth ∼ O(1) as the Bingham number approaches Bn max , which essentially gives rise to following expression for Bn max from Eq. (22): √ RaCWT AR Bn max ∼ f (Bn max , Pr, A R) − Pr f (Bn max , Pr, AR) (32a) Putting δth /L = 1 in Eq. (21) also yields the following expression: √ RaCWHF AR Bn max ∼ f (Bn max , Pr, A R) − Pr f (Bn max , Pr, A R) (32b) Although the critical Bingham number Bn crit under which the buoyancy force is just enough to overcome the yield stress leads to yielded fluid throughout the domain, Nu remains equal to unity even under this condition. The equilibrium of buoyancy and yield stress effects under the aforementioned critical condition gives rise to the following condition: τy τy ∼ . (33a) ρgβT ∼ ρgβq L/k ∼ δ L f (Bn crit , Pr, A R) The preceding relation can further be rewritten as RaCWT Bncrit ∼ f (Bn crit , Pr, AR) Pr and Bn crit ∼

RaCWHF f (Bn crit , Pr, AR) Pr

In Eq. (34) the Rayleigh number Ra can be taken to be RaCWT and RaCWHF in the CWT and CWHF boundary conditions, respectively. However, it is worth noting that Bn ≥ Bn crit ensures that Nu remains equal to unity but it does not imply that Nu > 1 when Bn is smaller than Bn crit . The mean Nusselt number Nu only attains values greater than unity when Bn is smaller than Bn max , and the fluid may remain in a yielded state under the condition Bn > Bn max even though Nu is equal to unity. Comparing Eqs. (32) and (33b) it is clear that Bn max is likely to be smaller than Bn crit and the difference between Bn crit and Bn max is likely to increase with increasing value of aspect ratio AR. These effects were confirmed by Turan et al. [14] based on simulation results and the variation of Bn crit and Bn max with AR reported by Turan et al. [14] and are not repeated here for conciseness. On comparing different aspect ratio cases for the CWT and CWHF boundary conditions it is evident that Nu increases monotonically with increasing AR for the CWHF boundary condition but the variation of Nu with AR is nonmonotonic for the CWT boundary condition. This behavior can be seen clearly from Figure 9, where the data in Figure 8 are replotted to bring out the aspect ratio effects more effectively. The nonmonotonic variation of Nu with Bn in the case of CWT originates due to the nonmonotonic variation of Nu with AR for Newtonian fluids, as shown in Figure 10. It has been discussed earlier that the strengths of convection and buoyancy forces weaken (strengthen) with decreasing (increasing) aspect ratio, and as a result of this, viscous effects dominate over the effects of buoyancy at a smaller (larger) value of Bingham number with a decrease (increase) in the value of aspect ratio AR. Using Eqs. (21) and (22) one can obtain the following Nu estimates for Newtonian fluids (i.e., Bn CWT = 0 and 12 11 10 9

CWT case Average of the prediction of Eq. (36) and (37c) Prediction of Eq. (38) (CWT) CWHF case Eq. (39)

Bejan Eq. (36)

8

Nu


numbers become equivalent to each other in this limit (i.e., RaCWT = RaCWHF and Bn CWT = Bn CWHF ). Therefore, the values of Bn max are found to be the same for both the CWT and CWHF boundary conditions. The values of Bn max are estimated here by carrying out simulations and identifying the Bingham number at which Nu obtains a value of 1.01 (i.e., Nu = 1.01), and the values of Bn max for various Ra and AR are listed in Table 1. Although this definition of Bnmax appears to be somewhat arbitrary, in this configuration convection is always present, so Nu = 1 is an asymptotic limit. In addition, uncertainties due to the level of mesh refinement and the choice of the regularization (and the exact value of μ yield ) are largest close to this situation and this precludes a more stringent criterion. The Nusselt number Nu can be scaled as

839

7 6

(33b)

5

As thermal transport takes place due to conduction heat transfer under the critical condition, the critical Bingham number expression remains exactly the same for both CWT and CWHF boundary conditions. The preceding relation is consistent with the recent analytical results by Vikhanisky [16] for the onset of natural convection in rectangular enclosures. According to Vikhanisky [16], Bn crit is given by √ 0.25 Ra/Pr . (34) Bn crit = 1 + 0.96/AR + 4/AR2

4


RaCWHF = 106

Bejan and Tien Eq. (37c)

Elsherbiny et al. Eq. (37a-b) 105

3 2 1 0.1

104

1

10

AR Figure 10 Variations of Nu with AR for Newtonian fluids in the case of CWT and CWHF boundary conditions at RaCWT = RaCWHF = 104, 105, and 106 at Pr = 500 along with the predictions of the correlations given by Eqs. (36)–(39). (Color figure available online.)

vol. 35 no. 9 2014


840

O. TURAN ET AL.

Bn CWHF = 0): Nu ∼ (RaCWT /Pr A R)1/4 f 4 (CWT) and Nu ∼ (RaCWHF /Pr A R)1/5 f 30.8 (CWHF), which suggests that Nu values are likely to be different between the CWT and CWHF configurations for the same numerical value of RaCWT and RaCWHF . For small values of RaCWT and RaCWHF the values of Nu for Newtonian fluid are comparable in both CWT and CWHF boundary conditions. For large values of Rayleigh number the difference between (RaCWT /PrAR)1/4 and (RaCWHF /PrAR)1/5 widens, and this difference is reflected in the smaller value of Nu of Newtonian fluids (i.e., Bn CWT = 0 and Bn CWHF = 0) in the CWHF case than that in the CWT case for the same numerical values of Rayleigh and Prandtl numbers for aspect ratios close to unity as can be observed from Figure 8. It was discussed earlier that the strength of advection is stronger in the CWT case than in the CWHF case for cases where the aspect ratio is approximately one for same values of Rayleigh and Prandtl numbers due to smaller velocity magnitude (see Eq. (16)), and this is reflected in the greater value of Nu in the CWT case than in the CWHF case. However, at large values of AR (i.e., A R > A Rmax ) Nu begins to decrease with increasing AR for the CWT configuration, whereas Nu increases monotonically with increasing AR for the CWHF configuration, and this eventually gives rise to higher values of Nu in the CWHF case than the corresponding value in the CWT case for the same numerical values of Rayleigh number (see Figure 8). The strength of convective transport weakens with increasing Bingham number, which is reflected in the reduction of velocity magnitude with increasing Bn CWT (Bn CWHF ) in Figure 3. This gives rise to a monotonic decrease in Nu with increasing Bingham number (see Figure 8). As Bn max remains the same for both CWT and CWHF boundary conditions, the value of Nu for Newtonian fluids (i.e., Bn CWT = 0 and Bn CWHF = 0) determines the nature of Nu variation with Bingham number. For the aspect ratios where Nu for Newtonian fluids in the CWT configuration is greater than in the CWHF configuration, the mean Nusselt number for Bingham fluids also assumes higher values for the CWT configuration than in the CWHF configuration for BnCWT < Bnmax and BnCWHF < Bnmax (see Figure 8). However, for the aspect ratios where Nu for Newtonian fluids in the CWHF configuration is greater than in the CWT configuration, the mean Nusselt number for Bingham fluids assumes a higher value for the CWHF (CWT) configuration than in the CWT (CWHF) configuration for small (i.e., Bn CWT ≈ 0 and Bn CWHF ≈ 0) (large (i.e., Bn CWT → Bn max and Bn CWHF → Bn max )) values of Bingham number (see Figure 8). Interested readers are referred to references [15] and [41] for more detailed discussion regarding the difference between the values of Nu for CWT and CWHF configurations. The variation of Nu with AR for both CWT and CWHF boundary conditions in the case of Newtonian fluids is compared in Figure 10. For tall enclosures (i.e., AR > > 1 but in practice usually 20 > A R > 2) the mean Nusselt number for the c2 A R c3 , CWT configuration is often expressed as Nu = c1 RaCWT and Bejan’s analysis [21] demonstrated that the correlation parameters c1 , c2 , and c3 are functions of RaCWT and AR. Bejan heat transfer engineering

[21] also showed that the analytical results of Gill [24] lead to the following expression of Nu for extremely large values of 1/7 aspect ratio (i.e., RaCWT A R → ∞): Nu = 0.364[RaCWT /(PrAR)]1/4

(35)

This analytical solution is consistent with the scaling estimate Nu ∼ L/δth ∼ [RaCWT /(PrAR)]1/4 for Newtonian fluids, which can be obtained by putting Bn CWT = 0 in Eq. (22). According to Bejan [21] Nu for tall enclosures is given by Nu = C B [RaCWT /(PrAR)]1/4

qe

(1 − q f )6 (1 + q f )2 (7 − q 2f )

−qe

(1 + q 2f )(1 + 3q 2f )14/3

dq f

(36)

where C B and qe are functions of Ra 1/7 A R and C B (qe ) is found to decrease (increase) from 1.0 to 0.912 (0.1 to 1.0) with 1/7 increase in RaCWT A R from 0 to 1000 [21]. Bejan [21] found that Nu for tall enclosures deviates from the asymptotic value when 1/7 RaCWT A R < 100, and the prediction of Eq. (36) approaches that of Eq. (35) for (RaCWT /A R)1/4 ≥ 10. Different mean Nusselt number correlations for the CWT configuration have been proposed for tall enclosures based on experimental [38–40] and computational [1, 30–36] studies, and interested readers are referred to Ganguli et al. [1] for an extensive review and the assumptions behind the respective correlations. One of the most used correlations for tall enclosures with A R > 5 was proposed by Elsherbiny et al. [39]: Nu = Max(Nu1c , Nu2c , N3c )

(37a)

where Nu1c , Nu2c and Nu3c are given by 1/3

Nu1c = 0.0605RaCWT ;

Nu2c

0.293 0.104RaCWT = 1+ 1 + (6310/RaCWT )1.36

and Nu3c = 0.242(RaCWT /A R)0.272

3 1/3

(37b)

Bejan and Tien [20] proposed the following correlation, which can be applied for the parallel flow and intermediate flow regimes: 2 Nu = 1 + (RaCWT A R 8 /362880)n + (0.623RaCWT A R −2/5 )n 1/5

1/n

where n = −0.386 (37c)

Figure 10 demonstrates that Eq. (37c) proposed in reference [20] satisfactorily captures the variation of Nu with AR for A R < 1 in the case of CWT boundary condition, and the agreement between the prediction of Eq. (37c) and the numerical results improves with decreasing aspect ratio. However, this expression underpredicts the value of Nu for aspect ratios of the order of unity (i.e., A R ∼ 1). The extent of this underprediction increases with increasing value of RaCWT . Equation vol. 35 no. 9 2014


O. TURAN ET AL.

(36), proposed by Bejan [21], satisfactorily predicts the mean Nusselt number Nu with AR for large values of aspect ratio. However, the expression by Bejan [21] overpredicts Nu for aspect ratio equal to unity and the extent of this overprediction increases with increasing RaCWT . The correlation (Eqs. (37a) and (37b)) by Elsherbiny et al. [39], although only proposed for A R > 5, exhibits satisfactory quantitative agreement with the present simulation data for tall enclosures with A R ≥ 2. However, the correlation by Elsherbiny et al. [39] (Eqs. (37a) and (37b)) also overpredicts the value of Nu for square enclosures for all the values of Rayleigh number considered in this study. The arithmetic mean of the predictions of Eqs. (37c) and (36) yields a satisfactory agreement with numerical prediction of Nu for A R = 1.0. Turan et al. [13] proposed an expression for Nu for square enclosures in the CWT configuration in the following manner: 0.091 Pr 0.293 Nu = 0.162RaCWT (38) 1 + Pr Equation (38) is consistent with the scaling estimation, which suggested Nu ∼ (RaCWT /Pr )1/4 f 4 , and is also shown in Figure 10, which indicates that Eq. (38) satisfactorily predict Nu for square enclosures for the CWT case. Recently, Turan et al. [41] proposed the following correlation for the CWHF boundary condition: 0.031 Pr 0.249 Nu = 0.209RaCWHF (c A ln A R + 1) 1 + Pr −0.189 with c A = 0.737RaCWHF

for 1/8 ≤ A R ≤ 8

(39a)

For 1/8 < A R < 1 the mean Nusselt number correlations were proposed as [41]

841

condition [14]: Nu ∼

⎡

⎤

Nu0 L f 6 ⎦ (40a) ∼ Max ⎣1.0, ∗ √ Bn 1 δth ∗2 + 4 + Bn 2 2

where f 6 is an appropriate function of RaCWT , Bn CWT , and Pr and the quantity Bn ∗ is given by Bn CWT (RaCWT A R/Pr )1/4

Bn ∗ =

(40b)

Based on the preceding scaling arguments, the following correlation for Nu is proposed here: Nu − 1 2[1 − (Bn ∗ /Bn ∗max )b0 ]b when = √ Nu0 − 1 Bn ∗ + Bn ∗2 + 4

Nu0 > 1, (41a)

and Nu = 1

when

Nu0 = 1,

(41b)

where b0 and b are the correlation parameters and Bn ∗max = Bn max (Ra A R/Pr )−1/4 , which can be expressed as b0 = 1.370,

0.131 b = 0.676RaCWT Pr 0.11

−0.21 Bn ∗max = C1 Ra0.31 AR−0.25 , CWT Pr

(41c) (41d)

where the parameter C1 is given by C1 = 0.019 + 0.010erf(2A R − 2).

(41e)

It is difficult to obtain a scaling estimate of Nu from Eq. (21), but Turan et al. [15] proposed a correlation for Bingham fluid flow in square enclosures for the CWHF boundary condition in the following manner: 1/2

Nu = N u A when RaCWHF A R 3 < 103 Nu = N u B when RaCWHF A R ≥ 10 3

Nu = 1 +

and

3

Bn CWHF 2

(39b)

where 2 N u A = 1 + RaCWHF A R 8 /362880

NuB =

0.249 0.209RaCWHF

Pr 1 + Pr

×

and

0.031

× (c B (1 − A R)cc + 1)

(39c)

where c B and cC are given by c B = −1.168

and

0.089 cC = 0.683RaCWHF

(39d)

Figure 10 shows that Eqs. (39b–d) satisfactorily predicts Nu obtained from simulation data in the Rayleigh number range 104 ≤ RaCWHF ≤ 106 in the CWHF configuration. Using Eq. (22) one can obtain the following scaling estimate of Nu in the boundary-layer regime for the CWT boundary heat transfer engineering

1−

A2 · RaCWHF RaCWHF 1/2 + 12 Bn 2CWHF + 4 Pr

Bn CWHF Bn max

b1 b2 (42)

where A2 , b1 , and b2 are the correlation parameters. This correlation is extended here to propose a correlation for (Nu−1)/(Nu0 − 1) to account for the aspect ratio effects in the boundary-layer regime in the case of CWHF boundary condition: b1 b2 2 1 − (Bn + /Bn + Nu − 1 max ) when Nu0 > 1, (43a) = √ Nu0 − 1 Bn + + Bn +2 + 4 and Nu = 1

when

Nu0 = 1,

(43b)

where A2 , b1 , and b2 are taken to be −0.001 A2 = 0.205RaCWHF

vol. 35 no. 9 2014

Pr −0.213 1 − ; 0.25 0.037 (1 + Pr ) RaCWHF Pr 0.25

842

O. TURAN ET AL.

b1 = 0.867;

b2 = 0.250Ra 0.207 Pr 0.062

(43c)

∗ In Eq. (43a) the quantity Bn + max is the same as Bn max for a given set values of Rayleigh, Prandtl, and aspect ratio, as Bn max is the same for both CWT and CWHF boundary conditions (see Figures 8 and 9). Thus, Bn + max is given by 0.31 −0.21 A R −0.25 , Bn + max = C 2 RaCWHF Pr

(43d)

where the parameter C2 is given by


(43e)

It can be seen from Figure 11 that Eqs. (41) and (43) predict both the qualitative and quantitative behaviors of (Nu − 1)/(Nu0 − 1) satisfactorily for all the AR ≥ 1 cases in both the CWT and CWHF configurations, respectively. For the parallel-flow regime the mean Nusselt number Nu can be estimated by the following integral at the middle of the domain: Nu = Nu1 + Nu2

(44a)

where L H kT

Nu2 = −

L H kT

H

ρc p u 1 T dy H

k 0

/ f 52 (for CWT)

∂T dy ∂ x1

(44b)

−2 2 8 Nu = 1 + aCWHF RaCWHF FCWT A R 8 (1 − FCWHF Bn ∗∗∗ )2

−2 × [1 − FCWT Bn ∗∗ ]2 / f 52 (for CWT)

where aCWT (aCWHF ) is an appropriate constant. Setting Bn ∗∗ = 2 A R 8 / f 52 for Newtonian 0 suggests that Nu = 1 + aCWT F 8 RaCWT fluids in the case of the CWT boundary condition, which is indeed found to be in good agreement with the asymptotic result of Cormack et al. [23] (i.e., Nu = 1 + Ra 2 A R 8 /362880), which essentially indicates F ∼ 1/5 for aCWT = 1 and f 5 ≈ 1.0. Using the Nu scaling of Newtonian fluids, Eq. (46) can be rewritten as follows: Nu − 1 ∼ (1 − F −2 Bn ∗∗ )2 (for CWT) Nu0 − 1

Nu = 1 when

Nu0 = 1

0

2 8 ρc p u 1 Tdy ∼ RaCWHF FCWHF A R8

−2 × [1 − FCWHF Bn ∗∗∗ ]2 / f 52 (for CWHF)

(45b)

whereas Nu2 can be scaled as H L ∂T H TLk =1 Nu2 = − k dy = H kT 0 ∂ x1 H TLk

(45c)

where Bn ∗∗ and Bn ∗∗∗ in Eqs. (45a) and (45b) are given by Bn

∗∗

Bn ∗∗∗

Bn CWT = (for CWT) (RaCWT /Pr )1/2 A R 2

Nu0 >1(47a) (47b)

(45a) Nu = 1 when

H

when

2 Nu − 1 1 −2 ∗∗ Bn when Nu0 > 1 and = 1− 4.55 Nu0 − 1

and

(46b)

Turan et al. [14] proposed the following correlation for Bingham fluid flow in the parallel-flow regime in the CWT configuration based on Eq. (47a):

Using Eqs. (24) and (26) Nu1 can be scaled as H L 2 8 Nu1 = ρc p u 1 T dy ∼ RaCWT FCWT A R8 H kT 0

L Nu1 = H kT

(46a)

Nu − 1 ∼ (1−F −2 Bn ∗∗∗ )2 (for CWHF) Nu0 − 1

and

0

−2 2 8 Nu = 1 + aCWT RaCWT FCWT A R 8 (1 − FCWT Bn ∗∗ )2

/ f 52 (for CWHF)

C2 = 0.019 + 0.010erf(2AR − 2).

Nu1 =

Using Eq. (44a) the mean Nusselt number Nu in the parallelflow regime can be given as

Nu0 = 1.

(48)

It can be seen from Figure 12 that the prediction of Eq. (48) is found to be satisfactory for the CWT cases, and a choice of FCWHF = 1/5.5 is found to yield satisfactorily agreement with the CWHF data. This suggests the following correlation for Bingham fluid flow in the parallel-flow regime in the CWHF configuration based on Eq. (47): 2 Nu − 1 1 −2 ∗∗∗ Bn when Nu0 > 1 and = 1− 5.5 Nu0 − 1 Nu = 1

when

Nu0 = 1.

(49)

and

Bn CWHF = (for CWHF) (RaCWHF /Pr )1/2 A R 2

(45d)


The correlations given by Eqs. (41) and (48) (Eqs. (43) and (49)) can be combined to yield the following correlations for CWT and CWHF configurations respectively: vol. 35 no. 9 2014

O. TURAN ET AL.


843

RaCWHF (RaCWT) = 105 1.2

1.2

1.2 AR = 1

0.6 0.4 0.2

0.02

0.04

0.06

0.08

0.8 0.6 0.4 0.2 0.0 0.00

0.10

0.04

Bn* (Bn+)

0.08

0.12

0.6 0.4 0.2

0.08

1.0 0.8 0.6 0.4 0.2 0.0 0.00

0.10

0.04

Bn* (Bn+)

0.08

0.12

0.4 0.2

0.04

0.06

0.08

1.0 0.8 0.6 0.4 0.2 0.0 0.00

0.10

0.04

0.08

0.12

0.6 0.4 0.2

0.08

0.10

0.3

0.5

0.8 0.6 0.4 0.2

0.1

0.2

0.3

0.4

0.5

1.2

1.0 0.8 0.6 0.4 0.2 0.0 0.00

0.4

1.0

0.0 0.0

0.16

AR = 8

(Nu-1)/(NuBn= 0 - 1)

0.8

0.2

AR = 8

(Nu-1)/(NuBn= 0 - 1)

1.0

Bn* (Bn+)

0.1

AR = 4

1.2

0.06

0.2

Bn* (Bn+)

AR = 8

0.04

0.4

Bn* (Bn+)

1.2

0.02

0.6

1.2

Bn* (Bn+)

0.0 0.00

0.8

0.0 0.0

0.16

(Nu-1)/(NuBn= 0 - 1)

0.6

0.4

1.0

AR = 4

(Nu-1)/(NuBn= 0 - 1)

(Nu-1)/(NuBn = 0-1)

0.8

0.3

AR = 2

1.2

1.0

0.2

Bn* (Bn+)

AR = 4

0.02

0.1

Bn* (Bn+)

1.2

0.0 0.00

0.2

1.2

(Nu-1)/(NuBn= 0 - 1)

0.8

0.06

0.4

AR = 2

(Nu-1)/(NuBn= 0 - 1)

(Nu-1)/(NuBn= 0-1)

1.0

0.04

0.6

0.0 0.0

0.16

1.2

0.02

0.8

Bn* (Bn+)

AR = 2

0.0 0.00

AR = 1 1.0

Bn* (Bn+)

1.2

(Nu-1)/(NuBn = 0-1)


CWHF case Eqs. 43 and 51 CWT case Eqs. 41 and 50

0.8

1.0

(Nu-1)/(NuBn= 0 - 1)

1.0

(Nu-1)/(NuBn= 0 - 1)

(Nu-1)/(NuBn= 0 - 1)

AR = 1

0.0 0.00


0.04

0.08

0.12

0.16

Bn* (Bn+)

1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.1

0.2

0.3

0.4

0.5

Bn* (Bn+)

Figure 11 Variation of (Nu − 1)/(Nu0 − 1) with Bn ∗ (Bn + ) for the A R ≥ 1 cases in the case of CWT (CWHF) boundary condition along with the predictions of Eqs. (41) and (50) (Eqs. (43) and (51)).


vol. 35 no. 9 2014

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O. TURAN ET AL.


RaCWHF (RaCWT) = 105 1.2

1.2

1.2 AR = 0.5

CWHF case Eqs. 49 and 51 CWT case Eqs. 48 and 50

0.8 0.6 0.4 0.2 0.0 0.00

0.01

0.02

0.03

0.04

0.8 0.6 0.4 0.2 0.0 0.00

0.05

AR = 0.5

1.0

(Nu-1)/(NuBn= 0 - 1)

1.0

(Nu-1)/(NuBn= 0 - 1)

(Nu-1)/(NuBn= 0-1)

AR = 0.5

0.01

Bn** (Bn***)

0.02

0.03

0.04

1.0 0.8 0.6 0.4 0.2 0.0 0.00

0.05

0.01

Bn** (Bn***)

0.02

0.03

0.05

1.2 AR = 0.25

AR = 0.25

(Nu-1)/(NuBn= 0 - 1)

1.0 0.8 0.6 0.4 0.2 0.0 0.00

0.04

Bn** (Bn***)

1.2

(Nu-1)/(NuBn= 0 - 1)



0.01

0.02

0.03

0.04

1.0 0.8 0.6 0.4 0.2 0.0 0.00

0.05

0.01

Bn** (Bn***)

0.02

0.03

0.04

0.05

Bn** (Bn***)

Figure 12 Variation of (Nu − 1)/(Nu0 − 1) with Bn ∗∗ (Bn ∗∗∗ ) for the A R < 1 cases in the case of CWT (CWHF) boundary condition along with the predictions of Eqs. (48) and (50) (Eqs. (49) and (51)).

For CWT:

⎡

Nu − 1 = ⎣ 1− Nu0 − 1 +

1 4.55

−2

when

Nu = 1

Bn ∗∗

2[1 − (Bn ∗ /Bn ∗max )b0 ]b √ Bn ∗ + Bn ∗2 + 4

n 1 = −0.02

and

2/n 1

Nu0 > 1

1/n 1 n 1 with

(50a)

and Nu = 1 when

Nu0 = 1

(50b)

For CWHF:

⎡ 2/n 1 −2 Nu − 1 1 Bn ∗∗∗ = ⎣ 1− 5.5 Nu0 − 1 1/n 1 ⎤n 1 b1 b2 2 1 − (Bn + /Bn + ) max ⎦ with + √ Bn + + Bn +2 + 4 n 1 = −0.02

when

Nu0 > 1

(51a)


when

Nu0 = 1

(51b)

It has been found that Eqs. (50) and (51) predict the behavior of (Nu − 1)/(Nu0 − 1) satisfactorily in the CWT and CWHF boundary conditions, respectively, for both the A R ≥ 1 and A R < 1 cases considered here, and it can be seen from Figures 11 and 12 that the prediction of Eq. (50) (Eq. (51)) is comparable to that of Eqs. (41) and (48) (Eqs. (43) and (49)) and in fact the respective predictions cannot be distinguished from each other in Figures 11 and 12. Although the correlations shown here were developed using data obtained for Pr = 500, we have confirmed that they work well even for a much lower value of Prandtl number (Pr = 7). Generally, the correlation quality is equally good for both Prandtl numbers except that the CWT correlation for the lower Pr case overpredicts (Nu − 1)/(Nu Bn=0 − 1) slightly for the AR = 0.5, Ra = 104 case.

CONCLUSIONS The influences of CWT and CWHF boundary conditions on the effects of aspect ratio (= H/L where H is the vol. 35 no. 9 2014


O. TURAN ET AL.

enclosure height and L is the enclosure length) on the heat transfer characteristics of steady laminar natural convection of yield-stress fluids obeying the Bingham model in a rectangular enclosure with differentially heated side walls have been numerically studied. It is found that the mean Nusselt number Nu follows a nonmonotonic trend with aspect ratio AR for a given set of values of the Rayleigh number and Prandtl number for both Newtonian and Bingham fluids for the CWT boundary condition, whereas a monotonic increase in Nu was obtained for increasing values of aspect ratio AR in the case of CWHF boundary condition. It is demonstrated that the effects of convection strengthen with increasing value of AR, whereas the strength of thermal conduction weakens with increasing AR in both CWT and CWHF boundary conditions. These competing effects of thermal convective and diffusive transports ultimately result in a nonmonotonic variation of Nu with aspect ratio AR for both Newtonian and Bingham fluids for the CWT boundary condition. The strengthening of convective transport with increasing AR dominates over the weakening of thermal diffusion in the CWHF boundary condition in the aspect ratio range considered here (i.e., 1/8 ≤ A R ≤ 8). For very small values of aspect ratio the thermal transport remains predominantly conduction dominated for both CWT and CWHF boundary conditions. The effects of convective transport are relatively weaker in Bingham fluids than in Newtonian fluids for the same nominal values of Rayleigh number Ra because of the stronger viscous forces in Bingham fluids. The effects of weaker convective transport in Bingham fluids than in Newtonian fluids are reflected in the smaller values of the mean Nusselt numbers for Bingham fluids than those obtained in the case of Newtonian fluids with the same values of nominal Rayleigh number. The Nusselt number was found to decrease with increasing Bingham number, and for large values of Bingham number (i.e., Bn CWT ≥ Bn max and Bn CWHF ≥ Bn max ) the value of mean Nusselt number settles to unity (i.e., Nu = 1) as the fluid flow becomes too weak to influence heat transfer due to strong flow resistance and heat transfer takes place principally due to conduction. The conductiondominated regime occurs at higher values of Bingham number for increasing values of Rayleigh number (aspect ratio) for a given value of aspect ratio (Rayleigh number). The aspect ratio A Rmax at which the maximum value of Nu is attained decreases with increasing value of Rayleigh number in both Newtonian and Bingham fluids in the CWT boundary condition. The value of A Rmax is found to increase with increasing value of Bingham number for a given value of nominal Rayleigh number for the CWT boundary condition. For the CWHF boundary condition the value of Nu is found to increase with increasing aspect ratio for both Bingham and Newtonian fluids. The variation of Nu with Bingham number for the CWT boundary condition does not exhibit monotonic behavior in terms of aspect ratio AR due to nonmonotonic AR dependence of Nu in Newtonian fluids (i.e., Nu at zero Bingham number). However, Nu/Nu0 reaches an asymptotic value (i.e., 1/Nu0 ) corresponding to the predominantly conduction mode of heat transfer (i.e., Nu = 1) when the Bingham number reaches a heat transfer engineering

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threshold value Bn max , and Bn max is found to increase with increasing AR. The strength of convection and buoyancy forces relative to viscous actions decreases (increases) with decreasing (increasing) aspect ratio, and as a result of this, viscous effects dominate over the buoyancy effects at a smaller (larger) value of Bingham number with decreasing (increasing) aspect ratio AR for the CWT boundary condition. The mean Nusselt number Nu is shown to increase monotonically with increasing AR for the CWHF boundary condition. The present simulation results are utilised to propose new correlations of Nu by accounting for aspect ratio AR effects in the case of convection in Bingham fluids for both CWT and CWHF boundary conditions. These correlations are shown to satisfactorily capture the variation of Nu with Rayleigh number, aspect ratio and Bingham number for all the cases considered in this study. Additional simulations for Pr = 7, not shown in detail here, confirm that the effect of Pr is also well captured by the proposed correlations. Natural convection of Bingham fluids have been investigated previously from purely analytical [9, 10, 16] and numerical [12–15] perspectives. Following this precedence, the present analysis has been carried out based on numerical simulations supported by scaling arguments, in the absence of experimental data in the available literature. However, experimental validation of the present findings will be necessary for more comprehensive analysis, which will be addressed in future investigations.

APPENDIX: NONDIMENSIONAL MASS, MOMENTUM, AND ENERGY CONSERVATION EQUATIONS The spatial coordinates, velocity components, pressure, and temperature can be nondimensionalised in the following manner: xi+ = xi /L ,

2 u i+ = u i /Uref , P + = P/ρUref

θ = (T − Tref )/Tref

and (52)

where Uref is the reference velocity scale and Tref is a reference temperature difference. For the CWT configuration Tref can be taken to be T = (TH − TC ), whereas Tref can be taken to be equal to q L/k (i.e., Tref = q L/k) for the CWHF configuration. The reference temperature Tref is taken to be the cold wall temperature TC in the CWT configuration, whereas the temperature at the centre of the domain Tcen is taken to be the reference temperature in the CWHF configuration. If Uref is taken to be equal to gβTref L (i.e., Uref = gβTref L) based on the equilibrium of inertial and buoyancy forces (see Eq. (15)), one obtains the following nondimensional forms of steady-state mass, momentum, and energy conservation equations (i.e., Eqs. (12)–(14)): vol. 35 no. 9 2014

846

O. TURAN ET AL.

For Uref = α/L one obtains √ γ˙ + = 0 for τ+ ≤ Bn Ra Pr ,

Nondimensional mass conservation equation: ∂u i+ =0 ∂ xi+

(53)

Nondimensional momentum conservation equations: u +j

+ ∂u i+ ∂ P+ 1 ∂τi j = − + δ θ + i2 Gr 1/2 ∂ x +j ∂ x +j ∂ xi+

+

τ = 1+ (54)

Nondimensional energy conservation equation:


u +j

∂θ ∂ 2θ 1 + = 1/2 PrGr ∂ x +j ∂ x +j ∂x j

(55)

In Eq. (54) τi+j is the nondimensional stress tensor, which is given by τi+j For Uref =

τi j L = μ gβTref L

gβTref L one obtains

+

γ˙ = 0 for +

(56)

τ = 1+

τ+ ≤ Bn,

Bn

(57a)

γ˙ +

. γ+

for

τ+ > Bn

(57b)

a A2 AR b,b0 ,b1 ,b2 Bn Bn∗ (Bn+) Bn∗∗ (Bn∗∗∗ )

Nondimensional mass conservation equation:

F

Nondimensional momentum conservation equations: u +j

∂τi+j ∂u i+ ∂ P+ = − + δ Ra Pr θ + Pr i2 ∂ x +j ∂ xi+ ∂ x +j

(59)

Nondimensional energy conservation equation: u +j

∂θ ∂ 2θ + = ∂x j ∂ x +j ∂ x +j

(60)

In Eq. (59) τi+j is the nondimensional stress tensor, which is given by τi+j =

τi j L μ(α/L)

(61)


γ˙ +

for

√ τ+ > Bn RaPr (62b)

NOMENCLATURE

cp c1 ,c2 ,c3 cA ,cB ,cC C1 , C2 CB e f1 ,f2 ,f3 ,f4 ,f5 ,f6

(58)

. γ+

˙ 2 /α is the nondimensional strain rate tensor. where γ˙ + = γL It is important to note that the numerical simulations of Eqs. (53)–(57) yield results identical to that of the solution of Eqs. (58)–(62), irrespective of the choice of Uref .

˙ where γ˙ + = γL/ gβTref L is the nondimensional strain rate tensor. It is worth noting that it is equally valid to use α/L, or other suitable combination, as the reference velocity Uref (i.e., Uref = α/L). Using Uref = α/L in Eqs. (12)–(14) yield the following alternative forms of nondimensional mass, momentum and energy conservation equations:

∂u i+ =0 ∂ xi+

√ Bn Ra Pr

(62a)

g Gr h H k K L n,n1 , n2 nB

Nu Nu1 Nu2 Nu0 Nu1c Nu2c Nu3c Nu

correlation parameter correlation parameter aspect ratio (AR = H/L) correlation parameter Bingham number modified Bingham number for boundary-layer regime in the case of constant wall temperature (constant wall heat flux) boundary condition modified Bingham number for parallel-flow regime in the case of constant wall temperature (constant wall heat flux) boundary condition specific heat at constant pressure, J/kg-K correlation parameter correlation parameter correlation parameter correlation parameter relative error functions relating thermal and hydrodynamic boundary layers fraction determining the ratio of the hydrodynamic boundary layer thickness on horizontal surface to the height of the enclosure gravitational acceleration, m/s2 Grashof number heat transfer coefficient, W/m2-K height of the enclosure, m thermal conductivity, W/m-K thermal gradient in horizontal direction, K/m length of the enclosure, m correlation parameter exponent of aspect ratio for self similar variation of mean Nusselt number in the boundarylayer regime Nusselt number convective contribution to Nusselt number conduction contribution to Nusselt number Nusselt number for Newtonian fluids correlation parameter mean Nusselt number

vol. 35 no. 9 2014

O. TURAN ET AL.

P Pr qf qc q Ra T T∗ T

pressure, Pa Prandtl number general quantity correlation parameter heat flux, W/m2 Rayleigh number temperature, K reference temperature, K difference between hot and cold wall temperature, K the temperature difference between the horizontal walls, K characteristic temperature difference, K ith velocity component, m/s dimensionless horizontal (U = u1 L/ α) and vertical velocity (V = u2 L/ α) characteristic velocity in vertical direction, m/s coordinate in ith direction, m


T1 Tchar ui U, V ϑ xi

[2] [3]

[4]

[5]

[6]

Greek Symbols α β γ˙ δ,δth θ μ μyield ν ρ τ(τi,j ) τy φ ψ

thermal diffusivity, m2/s coefficient of thermal expansion, 1/K strain rate, 1/s hydrodynamic and thermal boundary layer thickness, m dimensionless temperature plastic viscosity, N-s/m2 yield viscosity, N-s/m2 kinematic viscosity, m2/s density, kg/m3 shear stress, N/m2 yield stress, N/m2 general primitive variable stream function, m2/s

Subscripts C cen crit CWHF CWT ext eff H max ref wall

[7]

[8]

[9]

[10]

[11]

[12]

cold wall center of the simulation domain critical value constant wall heat flux constant wall temperature extrapolated value effective value hot wall maximum value reference value wall value

[13]

[14]

[15] REFERENCES [1] Ganguli, A. A., Pandit, A. B., and Joshi, J. B., CFD Simulation of Heat Transfer in a Two-Dimensional Vertical heat transfer engineering

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Enclosure, Chemical Engineering Research and Design, vol. 87, pp. 711–727, 2009. Ostrach, S., Natural Convection in Enclosures, Journal of Heat Transfer, vol. 110, pp. 1175–1190, 1988. Lamsaadi, M., Na¨ımi, M., Hasnaoui, M., and Mamou, M., Natural Convection in a Vertical Rectangular Cavity Filled With a Non-Newtonian Power Law Fluid and Subjected to a Horizontal Temperature Gradient, Numerical Heat Transfer Part A, vol. 49, pp. 969–990, 2006. Lamsaadi, M., Na¨ımi, M., and Hasnaoui, M., Natural Convection Heat Transfer in Shallow Horizontal Rectangular Enclosures Uniformly Heated From the Side and Filled With Non-Newtonian Power Law Fluids, Energy Conversion and Management, vol. 47, pp. 2535–2551, 2006. Barth, W. L., and Carey, G. F., On a Natural-Convection Benchmark Problem in Non-Newtonian Fluids, Numerical Heat Transfer Part B, vol. 50, pp. 193–216, 2006. Leung, W. H., Hollands, K. G. T., and Brunger, A. P., On a Physically-Realizable Benchmark Problem in Internal Natural Convection, International Journal of Heat and Mass Transfer, vol. 41, pp. 3817–3828, 1988. Ozoe, H., and Churchill, S. W., Hydrodynamic Stability and Natural Convection in Ostwald–De Waele and Ellis Fluids: The Development of a Numerical Solution, AIChE Journal, vol. 18, pp. 1196–1207, 1972. Balmforth, N. J., and Rust, A. C., Weakly Nonlinear Viscoplastic Convection, Journal of Non-Newtonian Fluid Mechanics, vol. 158, pp. 36–45, 2009. Vikhansky, A., Thermal Convection of a Viscoplastic Liquid With High Rayleigh and Bingham Numbers, Physics of Fluids, vol. 21, pp. 103103, 2009. Zhang, J., Vola, D., and Frigaard, I. A., Yield Stress Effects on Rayleigh–Bénard Convection, Journal of Fluid Mechanics, vol. 566, pp. 389–419, 2006. Park, H. M., and Ryu, D. H., Rayleigh–Bénard Convection of viscoelastic fluids in finite domains, Journal of NonNewtonian Fluid Mechanics, vol. 98, pp. 169–184, 2001. Vola, D., Boscardin, L., and Latché, J. C., Laminar Unsteady Flows of Bingham Fluids: A Numerical Strategy and some Benchmark Results, Journal of Computational Physics, vol. 187, pp. 441–456, 2003. Turan, O., Chakraborty, N., and Poole, R. J., Laminar Natural Convection of Bingham Fluids in a Square Enclosure With Differentially Heated Side Walls, Journal of NonNewtonian Fluid Mechanics, vol. 165, pp. 901–913, 2010. Turan, O., Poole, R. J., and Chakraborty, N., Aspect Ratio Effects in Laminar Natural Convection of Bingham Fluids in Rectangular Enclosures With Differentially Heated Side Walls, Journal of Non-Newtonian Fluid Mechanics, vol. 166, pp. 208–230, 2011. Turan, O., Sachdeva, A., Poole, R. J., and Chakraborty, N., Laminar Natural Convection of Bingham Fluids in a Square Enclosure With Vertical Walls Subjected to Constant Heat Flux, Numerical Heat Transfer Part A, vol. 60, pp. 381–409, 2011.

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[16] Vikhansky, A., On the Onset of Bingham Liquid in Rectangular Enclosures, Journal of Non-Newtonian Fluid Mechanics, vol. 165, pp. 901–913, 2010. [17] Barnes, H. A., The Yield Stress—A Review or ‘παντα ρει’—Everything Flows?, Journal of Non-Newtonian Fluid Mechanics, vol. 81, pp. 133–178, 1999. [18] O’Donovan, E. J., and Tanner, R. I., Numerical Study of the Bingham Squeeze Film Problem, Journal of NonNewtonian Fluid Mechanics, vol. 15, pp. 75–83, 1984. [19] de Vahl Davis, G., Natural Convection of Air in a Square Cavity: A Bench Mark Numerical Solution, International Journal for Numerical Methods in Fluids, vol. 3, pp. 249–264, 1983. [20] Bejan, A., and Tien, C. L., Laminar Natural Convection Heat Transfer in a Horizontal Cavity With Different End Temperatures, Journal of Heat Transfer, vol. 100, pp. 641–647, 1978. [21] Bejan, A., Note on Gill’s Solution for Free Convection in a Vertical Enclosure, Journal of Fluid Mechanics, vol. 90, pp. 561–568, 1979. [22] Bejan, A., A Synthesis of Analytical Results for Natural Convection Heat Transfer Across Rectangular Enclosures, International Journal of Heat and Mass Transfer, vol. 23, pp. 723–726, 1980. [23] Cormack, D. E., Leal, L. G., and Imberger, J., Natural Convection in a Shallow Cavity With Differentially Heated End Walls. Part 1. Asymptotic Theory, Journal of Fluid Mechanics, vol. 65, pp. 209–229, 1974. [24] Gill, A. E., The Boundary Layer Regime for Convection in a Rectangular Cavity, Journal of Fluid Mechanics, vol. 26, pp. 515–536, 1966. [25] Cormack, D. E., Leal, L. G., and Seinfeld, J. H., Natural Convection in a Shallow Cavity With Differentially Heated End Walls. Part 2. Numerical Solution, Journal of Fluid Mechanics, vol. 65, pp. 231–246, 1974. [26] Gebhart, B., Effects of Viscous Dissipation in Natural Convection, Journal of Fluid Mechanics, vol. 14, pp. 225–232, 1962. [27] Patankar, S. V., Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC, 1980. [28] Mitsoulis, E., Flows of Viscoplastic Materials: Models and Computations, in Rheology Reviews, eds. D. M. Binding, N. E. Hudson, and R. Keunings, Glasgow, British Society of Rheology, pp. 135–178, 2007. [29] Mitsoulis, E., and Zisis, T., Flow of Bingham Plastics in a Lid-Driven Square Cavity, Journal of Non-Newtonian Fluid Mechanics, vol. 101, pp. 173–180, 2001. [30] Dong, Y., and Zhai, Q., Natural Convection Study in an Enclosure With Different Aspect Ratios, International Journal of Modern Physics C, vol. 18, no. 12, pp. 1903–1922, British Society of Rheology, Glasgow, U.K., 2007.


[31] Frederick, R. L., On the Aspect Ratio for Which the Heat Transfer in Differentially Heated Cavities Is Maximum, International Communications in Heat and Mass Transfer, vol. 26, no. 4, pp. 549–558, 1999. [32] Lartigue, B., Lorente, S., and Bourret, B., Multicellular Natural Convection in High Aspect Ratio Cavity: Experimental and Numerical Results, International Journal of Heat and Mass Transfer, vol. 43, pp. 3157–3170, 2000. [33] Lee, Y., and Korpela, S., Multicellular Natural Convection in a Vertical Slot, Journal of Fluid Mechanics, vol. 126, pp. 91–124, 1983. [34] Le Quéré, P., A Note on Multiple and Unsteady Solutions in Two-Dimensional Convection in a Tall Cavity, Trans. ASME Journal of Heat Transfer, vol. 112, pp. 965–973, 1990. [35] Wakitani, S., Development of Multicellular Solutions in Natural Convection in an Air-Filled Vertical Cavity, Trans. ASME Journal of Heat Transfer, vol. 119, pp. 97–101, 1997. [36] Zhao, Y., Curcija, D., and Gross, W. P., Prediction of Multicellular Flow Regime of Natural Convection in Fenestration Glazing Cavities, ASHRAE Trans., vol. 103, no. 1, pp. 1–12, 1997. [37] Elder, J. W., Laminar Free Convection in a Vertical Slot, Journal of Fluid Mechanics, vol. 23, pp. 77–98, 1965. [38] Yin, S. H., Wung, T. Y., and Chen, K., Natural Convection in an Air Layer Enclosed Within Rectangular Cavities, International Journal of Heat and Mass Transfer, vol. 21, pp. 307–315, 1978. [39] Elsherbiny, S. M., Raithby, G. D., and Hollands, K. G. T., Heat Transfer by Natural Convection Across Vertical and Inclined Air Layers, ASME Journal of Heat Transfer, vol. 104, pp. 96–102, 1982. [40] Wakitani, S., Formation of Cells in Natural Convection in a Vertical Slot at Large Prandtl Number, Journal of Fluid Mechanics, vol. 314, pp. 299–314, 1996. [41] Turan, O., Chakraborty, N., and Poole, R. J., Influences of Boundary Conditions on the Aspect Ratio Effects in Laminar Natural Convection in Rectangular Enclosures With Differentially Heated Side Walls, International Journal of Heat and Fluid Flow, vol. 33, pp. 131–146, 2012. Osman Turan is currently a research assistant at the Mechanical Engineering Department of Karadeniz Technical University (KTU), Turkey. He received his bachelor’s (2004) and master’s (2007) degrees in mechanical engineering from KTU. He is currently working toward his Ph.D. studies at the same university. In addition to this, he worked as an honorary research fellow in the University of Liverpool between 2009 and 2011. His research interests include natural convection of non-Newtonian fluids, and boundarylayer control.

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O. TURAN ET AL.


Rob Poole is currently a senior lecturer within the School of Engineering at the University of Liverpool. He has a Ph.D. in mechanical engineering from the same institution and an undergraduate degree in mechanical engineering from Loughborough University, UK. His research interests are primarily centred around the flow of complex fluids, including both experimental and numerical approaches. In addition to investigating natural convection effects on nonNewtonian fluids, including power-law and yieldstress fluids such as in the current paper, he has investigated viscoelastic fluids in both laminar and turbulent flows including so-called purely elastic flow instabilities and in “elastic” turbulence. He is currently the Honorary Bulletin Editor of the British Society of Rheology.


849

Nilanjan Chakraborty is presently a professor of fluid dynamics at the School of Mechanical and Systems Engineering of Newcastle University, UK. Previously he was a senior lecturer at the School of Engineering of the University of Liverpool. He gained industrial experience as a mechanical engineer in General Electric, India, before getting the prestigious Gates Cambridge Scholarship for pursuing a Ph.D. at Cambridge University. His research interests include direct numerical simulations of turbulent combustion, combustion modeling in the context of Reynolds averaged Navier–Stokes and large eddy simulations, melting/solidification-related heat transfer problems in classical manufacturing, and laser-aided manufacturing applications and natural convection of complex non-Newtonian fluids. In 2005, he and his co-authors were awarded the Gaydon Prize for the most significant UK contribution to the 30th International Symposium on Combustion. He was also awarded the Hinshelwood Prize for 2007 by the British Section of the Combustion Institute for his contribution to combustion science as a young member. Recently, the Combustion Institute judged a paper co-authored by him to be the most significant paper presented in the droplet combustion colloquium of the 32nd International Combustion Symposium.

vol. 35 no. 9 2014

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