Arab J Sci Eng (2014) 39:2235–2250 DOI 10.1007/s13369-013-0814-8

RESEARCH ARTICLE - MECHANICAL ENGINEERING

Numerical Investigation and POD-Based Interpolation of Natural Convection Cooling of Two Heating Blocks in a Square Cavity Fatih Selimefendigil

Received: 6 February 2012 / Accepted: 14 July 2013 / Published online: 7 September 2013 © King Fahd University of Petroleum and Minerals 2013

Abstract In this study, natural convection cooling of two heating blocks placed in a square cavity is numerically investigated. The cavity is cooled from vertical walls and the surface of the heating blocks are kept at constant temperature. The effect of placing the heating blocks at top, middle and bottom of the cavity, and the distance between the blocks on the heat transfer characteristic is numerically analyzed for the range of Grashof numbers between 103 and 106 . The results are presented in terms of streamlines, isotherm plots and averaged Nusselt number plots. A POD-based interpolation method is also employed to predict the flow field and heat transfer characteristic for the case where the heating blocks are placed in the middle of the cavity. Nusselt number variation with respect to Grashof number is captured well with this approach of POD-based interpolation compared to the CFD calculations. Keywords POD

CFD · Interpolation and natural convection ·

F. Selimefendigil (B) Mechanical Engineering Department, Celal Bayar University, Manisa 45140, Turkey e-mail: [email protected]; [email protected]

Abbreviations a, b, c Modal coefficients g Gravitational acceleration Gr Grashof number (=gβT L 3 ν −2 ) h Local heat transfer coefficient k Thermal conductivity of the fluid L Length of the enclosure n Unit normal vector on the surface Nu Local Nusselt number (=h L/k) p Pressure Pr Prandtl number (= αν ) q Data set T Temperature u, v x–y velocity components x, y Cartesian coordinates α Thermal diffusivity β Fluid thermal expansion coefficient θ Dimensionless temperature ν Kinematic viscosity ρ Density of the fluid

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σ

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Singular value POD mode

Subscripts c Cold wall h Hot wall

1 Introduction Natural convection cooling in enclosures is widely used in many industrial applications due to its simplicity and cost. Some of these applications include chemical reactors, heat exchangers, cooling of electronic devices, room ventilation, and solar collectors. Heat transfer and flow field characteristics have been investigated for the natural convection in enclosures by many researchers. A comprehensive review was made by Ostrach [1]. Most of these studies deal with the cases where a vertically or a horizontally temperature difference is imposed. The cooling and heating may be partial and non-uniformities in the heating and cooling may also be considered. Nithyadevi et al. [2] have numerically studied the natural convection in a rectangular cavity with partially active side walls for the Grashof numbers between 103 and 105 . They found that heat transfer rate is high for the bottom-top thermally active location and an increase in the heat transfer is obtained with an increase in the aspect ratio. Sharif and Mohammad [3] have investigated natural convection in a cavity with constant flux heating at the bottom wall and isothermal cooling from the sidewalls for the Grashof numbers varying from 103 to 106 . They studied the effect of the inclination angle and aspect ratio of the cavity on the heat transfer characteristic. They observed that at lower Grashof number, the average Nusselt number at the heated surface does not change significantly whereas it increases rather rapidly with higher Grashof number. Al-Bahi et al. [4] have studied the laminar natural convection in an air filled vertical square cavity differentially heated with a single isoflux discrete heater on one wall with top. They obtained correlations for the heater location for the maximum heat dissipation rate which is Rayleigh number dependent. Alam et al. [5] presented a study of natural convection in a rectangular enclosure due to partial heating and cooling at vertical walls for different Rayleigh numbers (product of Grashof and Prandtl numbers) and cavity aspect ratios. The reported that the average heat transfer rate increases as the aspect ratio increases from 0.5 to 1 and beyond that it is decreases. As the Rayleigh number increases penetration of the flow inside the cavity increases and as a result the average heat transfer rate increases. Saravanan et al. [6] have numerically studied the natural convection in a square cavity induced by two mutually orthogonal arbitrarily placed heated thin plates under differ-

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ent boundary conditions. They have reported that for the better overall heat transfer one of the plates should be placed far away from the center of the cavity for isothermal boundary condition and should be placed near the center of the cavity for isoflux boundary condition. Saeid [7] has numerically investigated the two-dimensional unsteady natural convection flow in a square cavity filled with a porous medium and with sidewall heating. The temperature of the hot sidewall oscillates in time. He reported a resonance phenomenon when the heating frequency coincides with the frequency of the flow wheel circulating within the cavity. Hakeem et al. [8] have studied the natural convection in a square cavity in the presence of heated plate for the Grashof numbers varying from 103 to 105 and different aspect ratios and positions of the plate. They noted that for the vertical situation of thin plate heat transfer becomes more enhanced than for horizontal situation. Aydin and Yang [9] have numerically analyzed the buoyancy driven convection with a localized heating from below and symmetrical cooling from the sides. They found that the average Nusselt number at the heated part of the lower wall increases with an increase in the Rayleigh number and non-dimensional heat source thickness. Varol et al. [10] have studied the natural convection in inclined enclosure with a corner heater for a range of Rayleigh number, Prandtl number, inclination angle and lengths of the heater in x–y directions. They reported that the maximum or minimum heat transfer is achieved depending on the inclination angle and the length of the corner heaters. Natural convection in cavities containing blocks or baffles has gained considerable attention in recent years because of its practical engineering applications as in solar collectors and cooling of electronic components [11]. Bhave et al. [12] have numerically studied the effect of centrally placed solid block inside a cavity with differentially heated vertical walls. The effects of the Rayleigh number, Prandtl number and size of the block on the heat transfer are investigated. They reported that the wall heat transfer increases, with increase in the size of the block, until it reaches a critical value, where the wall heat transfer attains its maximum. Varol [13] has investigated the heat transfer and fluid flow due to free convection in a porous triangular enclosure with a centered conducting body. He observed that both height and width of the body and thermal conductivity ratio play an important role on heat and fluid flow inside the cavity. Saravanan et al. [14] have studied the natural convection in a square enclosure induced by two mutually orthogonal placed thin plates under isothermal and isoflux boundary conditions. They numerically showed that overall heat transfer is better for the case when placing one of the plates far away from the center of the cavity for isothermal boundary condition and near the center of the cavity for isoflux boundary condition. Mahmoodi and Sebdani [15] have numerically investigated the natural convection inside a square cavity having an adiabatic square body positioned

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at its center with nanofluid. They investigate the effects of pertinent parameters such as Rayleigh number, size of the adiabatic body, and volume fraction of the Cu nanoparticles on the fluid flow and thermal fields and heat transfer inside the cavity. Based on the above literature, a study of natural convection in a square enclosure containing two isothermal blocks has never been reported in the literature. One of the aims of the present study is to highlight the fluid flow and thermal characteristics of such systems and to determine numerically the effects of pertinent parameters related to the fluid flow and geometry on the thermal performance of the system. On the other hand, the bouncy driven convection studies with computational fluid dynamics (CFD) may require substantially larger computational time and storage for largescale cases and also for the cases when one has to deal with a large number of parameters. Proper orthogonal decomposition (POD) can be used to obtain reduced order models from “simulated” data sets generated by a CFD code for the natural convection case. Reduced order or low fidelity models obtained from POD method can be utilized to predict the thermal performance of the systems in an efficient and fast way. POD method is usually employed to obtain a reduced order models for the unsteady case by projecting the governing partial differential equations (PDEs) for the system onto the POD modes. The result of this projection is an ordinary differential equations of order of the number of POD modes. In another approach, POD method is employed for the steady case, where one has to vary one or many parameters. In this study, the latter approach is employed where Grashof number is the varying parameter of interest. POD is widely used in capturing the coherent structures in turbulent flow [16,17]. It has a statistical basis and is equivalent to principal component analysis or the Karhunen– Loeve method used in statistics. The equivalence of singular value decomposition, principal component analysis and Karhunen–Loeve decomposition is discussed by Wu et al. [18]. Reduced order models for the flow past bluff bodies were obtained by POD. The interesting features of this type of flows have been successfully captured by POD [19–21]. The application of POD to non-isothermal flow has also been studied by many researchers. Hasan and Sanghi [22] performed POD analysis of 2D flow in a thermally driven rotating cylinder and obtained a reduced order model. A hybrid approach to estimate the flow field for an intermediate Reynolds number using the hybrid POD modes was used. Brenner et al. [23] have obtained reduced order models based on POD for non-isothermal flow. They have discussed the sampling strategies for collecting the snapshots and computing the autocorrelation matrix. Haider et al. [24] have presented a reduced order modeling strategy for shipboard power-electronics cabinet which is of interest to naval applications with computational fluid dynamics in conjunction with proper orthogonal decomposition. They reported that

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the compact model runs about 350 times faster with a mean prediction error less than 5 percent for velocity, temperature and pressure fields. Samadiani and Joshi [25] have presented a new POD-based reduced order modeling procedure for temperature calculation in multi-scale convective systems. Comparing with computational fluid dynamics/heat transfer, their approach is 250 times faster. Recently, Wang et al. [26] have compared the POD interpolation and POD projection methods for heat conduction problems. They reported that under the same sampling conditions POD projection method can obtain accurate results for four-variable and six-variable heat conduction problem while POD interpolation method losses accuracy for the six-variable heat conduction problem. Bui-Thanh et al. [27] have applied a strategy to prediction of steady aerodynamic loads, using POD and interpolation. Normally, one can obtain global POD modes by taking all the data in the snapshot matrix corresponding to different parameter sets into account. But, this may not capture the influence of parameter changes on the modes (not only the modal coefficients, but the modes themselves may depend on the parameters!). As an alternative, one can build different POD bases corresponding to distinct parameter values from snapshots at different instances of time. Then, these two sets of modes can be combined with direct interpolation to get a new basis. But direct interpolation of the modes provides a set of non-orthogonal modes with poor representation of the dynamics in the transonic regime for aerodynamic application. Lieu and Farhat [28] developed a subspace angle interpolation method for getting a new basis from the POD bases that were calculated for different parameter values. They interpolated the angles between the POD modes rather than direct interpolation of the modes themselves, which ensures orthogonality of the new basis. They reported improved accuracy in the transonic flow for aerodynamic application. Amsallem and Farhat [29] proposed to use the concept of Grassmann Manifolds and to interpolate in the tangent space using mapping from a Grassmann manifold to a tangent space followed by inverse mapping. They showed that subspace angle interpolation is a special case of the Grassmann Manifold approach where the number of the bases is two. Siegel et al. [30] have developed the double POD approach, which uses the modes with double indices. The second set of modes is derived from the orthogonalization of the first POD modes with the idea of adding shift modes in order to account for the slightly changes of the modes with the changing flow conditions. The low order model (modal coefficients which are representative of the system’s governing equations) is obtained using artificial neural networks instead of Galerkin projection or interpolation. Rowley et al. [31] developed a framework for the low order modeling of the compressible Navier–Stokes equations with POD. In their approach, they used an energy-based inner product to ensure the stability of the Galerkin projection around the

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pendent except for the density in the bouncy force according to Boussinesq approximation. For a two dimensional, incompressible, laminar flow, the continuity, momentum and energy equations can be expressed in the non-dimensional form as in the following: ∂V ∂U + = 0, (1) ∂X ∂Y   2 ∂U ∂ 2U ∂V ∂P ∂ U U , (2) + +V =− + Pr ∂X ∂Y ∂X ∂ X2 ∂Y 2   2 ∂V ∂ V ∂2V ∂V ∂P U + +Gr Pr 2 θ, +V =− + Pr ∂X ∂Y ∂Y ∂ X 2 ∂Y 2 (3)   2 2 ∂θ ∂θ ∂ U ∂ U (4) U + +V = ∂X ∂Y ∂ X 2 ∂Y 2 Fig. 1 Geometry and boundary conditions for a square cavity cooled from vertical sides and including two heating blocks

steady-state (or linearization) point of the non-linear system. With the appropriate choice of the variables for the isentropic flow at moderately Mach numbers, they derived a set of quadratic equations. They considered a vectorial form of the POD modes (all modes have one common modal amplitude). They also derived low order model equations for the compressible Navier–Stokes equations which results in a mass matrix (requires the inversion during time integration) and cubic equations (in the form of coupling between the modal coefficients.) Another aim of the present study is to explore the prediction capabilities of the POD-interpolation approach for the natural convection problem in a square enclosure containing isothermal heating blocks and help researchers seeking efficient, cheap models of the high fidelity CFD computations for this type of heat transfer problems.

2 Numerical Simulation 2.1 Problem Description, Governing Equations, Physical Parameters and Solution Method A schematic description of the problem is shown in Fig. 1. A square cavity with two heating blocks is considered. The length of the square cavity and height of the square blocks are L = 0.1 m and H = L/10, respectively. Vertical side walls of the cavity are cooled (kept at temperature Tc ) and all walls of the square blocks are heated (kept at temperature Th ). The top and bottom walls of the cavity are considered to be adiabatic. Working fluid is air with a Prandtl number of Pr = 0.71. The flow is assumed to be two dimensional, Newtonian, incompressible and in the laminar regime. The physical properties are assumed to be temperature inde-

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pL where U = uαL , V = vL α and P = ρα 2 represent the two nondimensionalized velocity components, and pressure, respectively. X = Lx , Y = Ly denote the non-dimensionalized coorc dinates. θ = TTh−T −Tc , ρ and α denote the non-dimensionalized temperature, density, and thermal diffusivity, respectively. 3 c )L , where g, β Grashof number is defined as Gr = gβ(Thν−T 2 and ν denote the gravity, volumetric thermal expansion coefficient and kinematic viscosity, respectively. Boundary conditions are defined as : Left and right vertical walls: U = V = 0, θ = 0. Bottom and top walls: ∂θ U = V = 0, ∂Y = 0 Walls of the heating blocks: U = V = 0, θ = 1. Local Nusselt number is defined as   ∂θ hx L =− . (5) N ux = k ∂n S

where h x represent the local heat transfer coefficient and k denote the thermal conductivity of air. n and S denote the surface normal component and walls of the heating blocks, respectively. Nusselt number is obtained after integrating the local Nusselt number along the wall of the heating blocks. Equations (1), (2), (3) and (4) along with the boundary conditions are solved with a commercial finite element solver. The body-adapted mesh consists of 11008 triangular elements and is refined close to the heating blocks. Mesh independence of the solutions has been confirmed. The global convergence for the continuity, momentum and energy residuals are set to 10−4 , 10−5 and 10−5 , respectively. 2.2 CFD Results In the present study, Grashof number is varied between 103 and 106 . Heating blocks are placed at three positions (referring to Fig. 1, Top : y1 = y2 /2, Middle : y1 = y2 and Bottom

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Fig. 2 Geometry and boundary conditions for a square cavity cooled from vertical sides and including two heating blocks

: y1 = 2y2 ) and distance between the heating blocks at each position is also varied. Due to the lack of an experimental study for the current configuration studied in this work, the numerical code with the current settings are tested for the natural convection problem with sidewall cooling and bottom heating in an air filledenclosure as in [9]. The streamlines and isotherm plots are shown in Fig. 2 for GrPr = 1000 (top) and GrPr = 10000 (bottom). The plots show that streamlines and isotherms show similar behavior computed with the current solver as compared to the ones as in [9]. The deviations for the Nusselt numbers along the heated part of the enclosure for the the chosen Grashof numbers is less than 5 percent with the current solver. Therefore, the present code is also valid for further calculations. Streamline and isotherm plots are shown in Figs. 3, 4, 5, 6, 7, and 8 for different positioning of the heating blocks, different Grashof numbers and various distances among the heating blocks. With the symmetrical boundary condition at

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the vertical side walls and heating blocks, the flow field is almost identical for the right and left half of the cavity. The flow first ascends through the surfaces of the heating blocks, faces the adiabatic wall, moves towards the cooled vertical sides and descends along the cooled walls. In Fig. 3, when the Grashof number increases, the core of the cells move upward. When the distance between the heating blocks increases, the distortion of the cells within the spacing between the blocks decreases, and at high Grashof numbers, two symmetrical vortices above the heating blocks are formed. In Fig. 4, at high Grashof numbers, isotherms are more clustered along the sides of the heating blocks, and the formation and thinning of the boundary layer is observed. At Gr = 103 and Gr = 104 , the clustering of the isotherms around the heating blocks change when changing the distance between the blocks, but at at Gr = 105 and Gr = 106 , there is a slight change of isotherms around the blocks. From these plots, we can observe that, the effect of the changing the distance between the blocks is more effective in the conduction mode, rather than the convection mode. In Fig. 5, when the heating blocks are placed in the middle, two symmetrical cells are formed around the blocks and when the distance between the blocks are increased, these cells coalesce and form a bigger cell surrounding each of the blocks at Gr = 103 . At Gr = 106 , with increasing the distances between the blocks, two cells are formed above the heating blocks and the streamlines above the heating blocks are distorted towards the left and right for the first and second heating blocks, respectively. At this position, the isotherm plots in Fig. 6 show the same trend for the case when the heating blocks are placed at the top. In Fig. 7, when the heating blocks are placed at the bottom of the cavity, for all distances among the heating blocks, vortices are formed at the top of the heating blocks and the size of the cells above the heating blocks increases compared to cases when the heating blocks placed at the top and in the middle at Gr = 106 . Calculated average Nusselt number plots are shown in Figs. 9, 10, and 11 for the top, middle and bottom positioning of the heat sources, respectively. In all three plots, in the conduction dominated mode (for Gr = 103 and Gr = 104 ), Nusselt number increases with the increase of distance among the heating blocks. At the top positioning of the heating blocks, for the convection dominated mode, Nusselt number does not change with the change of the distance among the heating blocks, but at Gr = 105 , when the heating blocks are placed at the middle and bottom positions, Nusselt number is maximum for the minimum distance among the heating blocks. A close inspection of the streamlines and isotherms for the case when the heating blocks are placed at the bottom for two different distances at Gr = 105 in Fig. 12, show that, convection is

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Fig. 3 Streamlines for top position of the heating blocks for various distances (1st column d/L = 0.16, 2nd column d/L = 0.27, 3rd column d/L = 0.40) at different Grashof numbers (1st row Gr = 103 , 2nd row Gr = 104 , 3rd row Gr = 105 , 4th row Gr = 106 )

more effective for the top and left side of the first heating block for the small distance case. 3 Proper Orthogonal Decomposition (POD)-Based Interpolation In this section, POD-based interpolation method for the natural convection heat transfer of the previously described model is detailed. The global POD approach, which collects data from different cases of CFD model, is presented. 3.1 Computing POD Modes An ensemble of data, either from numerical simulations or experiments, can be expressed in terms of a reduced order

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basis. The data set is projected onto the new basis and the difference between the original data and the projected one will be minimized in a least square sense. Let us denote by q such a data set obtained from CFD calculations for different parameters. Then it can be expanded in terms of basis functions as,  a ji i , j = 1, 2, . . . , M, i = 1, 2, . . . , N , qj = (6) where j is the parameter index and i is the index for the mode number and truncated at N . The modes will then be calculated by minimizing the distance between the original data and approximated (projected) data [32] ||q − Projection(q)|| → MIN.

(7)

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Fig. 4 Isotherms for top position of the heating blocks for various distances (1st column d/L = 0.16, 2nd column d/L = 0.27, 3rd column d/L = 0.40) at different Grashof numbers (1st row Gr = 103 , 2nd row Gr = 104 , 3rd row Gr = 105 , 4th row Gr = 106 )

This is equivalent to maximizing the inner product of ensemble average, normalized by the inner product of the basis vectors [32], (8) (q, ) / ( , ) → MAX. For two vector field variables r and s, the inner product is the defined as the integration of their scalar product over the domain V ,

 (r, s) =

ri si dV.

(9)

V

The expression in Eq. (8) can be then reformulated as an integral eigenvalue problem. The integral is a Fredholm integral of 1st type,

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Fig. 5 Streamlines for middle position of the heating blocks for various distances (1st column d/L = 0.16, 2nd column d/L = 0.27, 3rd column d/L = 0.40) at different Grashof numbers (1st row Gr = 103 , 2nd row Gr = 104 , 3rd row Gr = 105 , 4th row Gr = 106 )

 q(x)



q(x  ) (x  ) dx  = λ (x).

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The first integrand is the auto-correlation tensor. This integral can be solved numerically, or singular value decomposition can be used. The first mode captures the greatest fraction of the energy (associated with the norm that is defined), the second mode captures the second greatest fraction of energy, and so on. Once the POD modes are calculated, the modal coefficients a ji as in Eq. (6) can be obtained by projecting the data sets onto the POD modes, namely, a = (q, ) .

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(11)

3.2 Interpolation Among the Modal Coefficients The field variables u, v and T will be expressed as the superposition of Nu , Nv and N T POD modes as in the following,

u(Gr, x) =

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ak (Gr ) uk (x)

(12)

bk (Gr ) vk (x)

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ck (Gr ) kT (x)

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Fig. 6 Isotherms for middle position of the heating blocks for various distances (1st column d/L = 0.16, 2nd column d/L = 0.27, 3rd column d/L = 0.40) at different Grashof numbers (1st row Gr = 103 , 2nd row Gr = 104 , 3rd row Gr = 105 , 4th row Gr = 106 )

Parametric variation (Grashof number) is expressed in the modal coefficients. Basic procedure for the interpolation among the modal coefficients is outlined below: (1) Suppose that m snapshot data set (for each of the field variables, u, v and T ) are collected from CFD computations for a range of Grashof numbers Gr1 , Gr2 , . . . , Grm

  in the snapshot matrix MT = TGr1 , TGr2 , . . . , TGrm which has the dimension of (number of grid points) ×m. (2) POD modes are calculated for each of the field variables from the singular value decomposition (SVD) of the snapshot matrix. The number of the modes retained in the reconstruction is based on the 2-norm of the snapshot matrix. For example, for the u-velocity component,

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Fig. 7 Streamlines for bottom position of the heating blocks for various distances (1st column d/L = 0.16, 2nd column d/L = 0.27, 3rd column d/L = 0.40) at different Grashof numbers (1st row Gr = 103 , 2nd row Gr = 104 , 3rd row Gr = 105 , 4th row Gr = 106 )

number of the POD modes Nu is obtained from the singular values σi of the SVD of the snapshot matrix as,

F(Grt , x) =

NF 

sk (Grt ) kF (x)

(16)

k=1

Nu σk k=1 ≥ 0.9999 m k=1 σk

(15)

(3) Modal coefficients ak , bk and ck are obtained by projecting the data onto the POD modes as given in Eq. (11). For a Grashof number Grt which is not considered in / (Gr1 , . . . , Grm ), modal coeffisnapshot matrix, Grt ∈ cient at Grt is obtained with a cubic spline interpolation among the modal coefficients. (4) After obtaining the POD modes and the interpolated modal coefficient, field variables can be written as a superposition of the POD modes as

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where F denotes any of the field variables u, v or T and s represents the modal coefficients a, b, c as in Eqs. (12), (13) and (14). 3.3 POD Results The case when the heating blocks are placed in the middle of the cavity is considered for the range of Grashof numbers Gr = 103 , 5×103 , 104 , 5×104 , 105 , 5×105 , 106 . The reconstructed data (or interpolated coefficients) are considered at Gr = 3 × 103 , 7 × 103 , 3 × 104 , 7 × 104 , 3 × 105 , 7 × 105

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Fig. 8 Isotherms for bottom position of the heating blocks for various distances (1st column d/L = 0.16, 2nd column d/L = 0.27, 3rd column d/L = 0.40) at different Grashof numbers (1st row Gr = 103 , 2nd row Gr = 104 , 3rd row Gr = 105 , 4th row Gr = 106 )

which are not included in the data set when obtaining the POD basis. Based on Eq. (15), the number of the modes retained for the temperature is N T = 3. The first three modal coefficients for the temperature is shown in Fig. 13. These coefficients keep the information on how the field variables change with respect to a change in the Grashof number. The recon-

structed temperature fields with POD at Gr = 3 × 103 , 7 × 103 , 3 × 104 , 7 × 104 are shown in Fig. 14. Figure 15 shows the reconstructed temperature field at Gr = 3×105 , 7×105 . On the left hand side of these figures, the results from the CFD computations are also shown. For the considered range of the Grashof numbers, POD-based interpolation captures well the variation of the temperature with respect to a

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Fig. 9 Nusselt number versus Grashof number for various distances among the heating blocks for top position

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Fig. 10 Nusselt number versus Grashof number for various distances among the heating blocks for middle position

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Fig. 11 Nusselt number versus Grashof number for various distances among the heating blocks for bottom position

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Fig. 12 Streamlines and isotherms for bottom positioning of the heating blocks for various distances (1st column d/L = 0.16, 2nd column d/L = 0.27) at Grashof number of 105

Fig. 13 First three modal coefficients of the temperature modes

20 10 0 1st mode 2nd mode 3rd mode

c

k

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change in the Grashof number. Nusselt number comparison for Gr = 3×103 , 7×103 , 3×104 , 7×104 , 3×105 , 7×105 is shown in Fig. 16. This plot shows that POD-based interpolating approximation with three modes captures well the Nusselt number variation with respect to a change in Grashof number.

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4 Discussion and Conclusion In this study, natural convection cooling of two heating blocks placed in a square cavity which is cooled from vertical side walls is numerically analyzed with a finite element code. The boundaries of the heating blocks are kept at constant

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Fig. 14 Comparison of the temperature contours calculated with CFD and predicted with POD at Gr = 3 × 103 , 7 × 103 , 3 × 104 , 7 × 104 from top to bottom, Left: CFD, Right: POD

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temperature. The effect of placing the heating blocks at top, middle and bottom of the cavity, and the distance between the blocks on the heat transfer characteristic is numerically analyzed for the range of Grashof numbers between 103 and 106 . Nusselt number increases with an increase in the Grashof number. For all positions of the heating blocks (top, middle

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and bottom), Nusselt number increases with an increase in the distance between the blocks for the conduction dominated case. When the heating blocks are placed at the top, in the convection dominated case, there is negligible influence of the distance among the heating blocks on the Nusselt number. At Gr = 105 , when the heating blocks are placed in the

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Fig. 15 Comparison of the temperature contours calculated with CFD and predicted with POD at Gr = 3 × 105 , 7 × 105 from top to bottom, Left: CFD, Right: POD

Fig. 16 Nusselt number versus Grashof number calculated from CFD and obtained with 1 POD mode and 3 POD modes

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middle and bottom positions, Nusselt number is maximum for the minimum distance among the heating blocks since convection is more effective for the top and left side of the first heating block. Furthermore, a POD-based interpolation is also employed to compute the flow field and heat transfer characteristic in an efficient and fast way for the case when the heating blocks are placed in the middle of the cavity. The change in the temperature field within the cavity with respect to a change in the Grashof number is captured well with this approach of POD-based interpolation. Nusselt number variation ver-

Gr

10

sus Grashof number is also obtained accurately with POD compared to CFD computations. References 1. Ostrach, S.: Natural convection in enclosures. J. Heat Transf. 110, 1175–1190. (1988) 2. Nithyadevi, N.; Kandaswamy, P.; Lee, J.: Natural convection in a rectangular cavity with partially active side walls. Int. J. Heat Mass Transf. 50, 4688–4697 (2007) 3. Sharif, M.A.; Mohammad, T.R.: Natural convection in cavities with constant flux heating at the bottom wall and isothermal cooling from the sidewalls. Int. J. Thermal Sci. 44, 865–878 (2005)

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