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04 Artigos Antigos

(Parte **1** de 2)

Proceedings of COBEM 2005 18th International Congress of Mechanical Engineering Copyright © 2005 by ABCM November 6-1, 2005, Ouro Preto, MG

Rogério Fernandes Brito Federal University of Itajuba – UNIFEI – BPS, Av., #1303, Zip Code: 37500-903, Pinheirinho, Itajubá, Brazil. rogbrito@unifei.edu.br

Paulo Mohallem Guimarães Federal University of Itajuba – UNIFEI – BPS, Av., #1303, Zip Code: 37500-903, Pinheirinho, Itajubá, Brazil. paulomgui@uol.com.br

Aristeu da Silveira Neto Federal University of Uberlandia – UFU – Joao Pinheiro Av., #565, Zip Code: 38400-902, Uberlandia, Brazil. aristeus@mecanica.ufu.br

Genésio José Menon Federal University of Itajuba – UNIFEI – BPS, Av., #1303, Zip Code: 37500-903, Pinheirinho, Itajubá, Brazil. genesio@unifei.edu.br

Abstract. Turbulent natural convection of air that happens into inner square cavity with localized heating from horizontal bottom surface has been numerically investigated. Localized heating is simulated by a centrally located heat source on the bottom wall, and two values of the dimensionless heat source length are considered in this present work. Solutions are obtained for several Rayleigh numbers with Prandtl number 0.7. The horizontal top surface is thermally insulated and the vertical surfaces are assumed to be the cold isothermal surfaces whereas the heat source on the bottom wall is isothermally heated. In this study, the Navier-Stokes equations are used considering a two-dimensional and turbulent flow in the unsteady state. The Finite Element Method (FEM) with a Galerkin scheme is considered for solving the conservation equations. The formulation of the conservation equations is carried out for turbulent flow and the turbulence is modeled using Large-Eddy Simulation (LES). The stream function and temperature distributions are determined as functions of thermal and geometrical parameters. The average Nusselt number is shown to increase with an increase in the Rayleigh number as well as in the dimensionless heat source length. The results of this work can be applied to the design of electronic components.

Keywords: cavities, finite element method, turbulence, natural convection, LES.

1. Introduction

Natural convection in enclosures is an area of interest due to its wide application and great importance in engineering. Transient natural convection flows occur in many technological and industrial applications. Therefore, it is important to understand the heat transfer characteristics of natural convection in an enclosure.

Along the years, researchers have looked for more flows with the objective to approximate the real case found in geophysical or industrial means. Then, we can define four basic types of boundary conditions. They are: the natural convection due to a uniformly heated wall (with a temperature or a constant heat flux); the natural convection induced by a local heat source; the natural convection under multiple heat sources with the same strength and type; and the natural convection conjugated with inner heat-generating conductive body or conductive walls. The boundary conditions mentioned previously are based on a single temperature difference between the differentially heated walls. Most of the previous studies have addressed natural convection in enclosures due to either a horizontally or vertically imposed temperature difference. However, departures from this basic situation are often encountered in fields such as electronics cooling. The cooling of electronic components is essential for their reliable performance.

The characteristics of fluid flow and heat transfer under the multiple temperature differences are more complicated and have a practical importance in thermal management and design.

In the present work, a two-dimensional numerical simulation in a cavity is carried out for a turbulent flow. The turbulence study is a complex and challenging assumption. There are few works in literature that deal with natural convection in closed cavities using the turbulence model LES. The motivation to accomplish this work relies on the fact that there are a great number of problems in engineering that can use this geometry. One turbulence model is implemented here together with the finite element method.

A large eddy simulation (LES) seems a promising approach for the analysis of the high Grashof number turbulence that contains three-dimensional and unsteady characteristics. A direct simulation of turbulence gives us more accurate and precise data than experiments; it is essentially unsuitable for high Grashof number flows because of computational limitations. It is known that the LES enables an accurate prediction of turbulence, but spends much less CPU time than the direct simulation.

In literature, a large number of theoretical and experimental investigations are reported on natural convection in enclosures.

Natural convection of air in a two-dimensional rectangular enclosure with localized heating from below and symmetrical cooling from the sides was numerically investigated by Aydin and Yang (2000). Localized heating was simulated by a centrally located heat source on the bottom wall, and four different values of the dimensionless heat source length, 1/5, 2/5, 3/5 and 4/5 were considered. Solutions were obtained for Rayleigh number values from 103 to

106. The average Nusselt number at the heated part of the lower wall, Nu, was shown to increase with an increase of the Rayleigh number, Ra, or of the dimensionless heat source length, ∈.

were compared to experimental data and showed a stable thermal stratification under a low turbulence level |

Peng and Davidson (2001) studied the turbulent natural convection in a closed enclosure whose vertical lateral walls were maintained at different temperatures. Both the Smagorinsk and the dynamic models were applied to the turbulence simulation. Peng and Davidson (2001) modified the Smagorinsk model by adding the buoyancy term to the turbulent viscosity calculation. This model would be called the Smagorinsk model with buoyancy term. The computed results

It was performed in the work of Oliveira and Menon (2002), a numerical study of turbulent natural convection in square enclosures. The finite volume method together with LES was used. The enclosure lateral surfaces were kept to different isothermal temperatures and the upper and lower surfaces were isolated. The flow was studied for low

Rayleigh numbers Ra = 1.58 × 109. Three turbulence LES models were used. Ampofo and Karayiannis (2003) conducted an experimental study of low-level turbulence natural convection in an air filled vertical square cavity. The cavity was 0.75 m high × 1.5 m deep giving rise to a 2D flow. The hot and cold walls of the cavity were isothermal at 50 and 10 ºC respectively, that is, a Rayleigh number equals to 1.58 × 109. The experiments that were realized on Ampofo work and Karayiannis (2003) were conducted with very high accuracy and as such the results formed experimental benchmark data and were useful for validation of computational fluid dynamics codes.

In the present work, turbulent natural convection of air that happens into inner square cavity with localized heating from horizontal bottom surface has been numerically investigated. The objective of the analyses of heat transfer is to investigate the Nusselt number distribution on the vertical walls and heated lower horizontal surface. Another objective is to verify the effect of height variation I of the horizontal heated lower surface on the turbulent flow. Six cases are studied numerically. The Rayleigh number Ra is varied and so is the dimensionless length of the heat source∈, where (1 − ∈)/2 ≤ x ≤ (1 + ∈)/2 and x is the coordinate component in the x direction. For the cases 1, 2 and 3, the dimension ∈ is fixed in ∈ = 0.4 and the Rayleigh numbers Ra is varied (Ra = 107, 108, 109). For the cases 1, 2, and 3, it is used a nonstructured mesh of finite elements with 5,617 triangle elements with 2,908 nodal points. The other cases also used a non-structured mesh of finite elements with linear triangle elements. In cases 4, 5, and 6, ∈ is fixed in ∈ = 0.8. The cases 1 and 4, 2 and 5 and; 3 and 6 are simulated, respectively, for Ra = 107, 108 and 109. The cases 4, 5, and 6 are simulated with one mesh with 5,828 elements and 3,015 nodes. The turbulence model used in all cases is the Large- Eddy Simulation (LES) with the second-order structure-function sub-grid scale model (F2). It is adopted a geometry with an aspect ratio A = H/L = 1.0. Comparisons are made with experimental data and numerical results found in Tian and Karyiannis (2000), Oliveira and Menon (2002), Lankhorst (1991) and Cesini et al. (1999).

2. Problem Description

Figure 1 shows the geometry with the domain Ω. It will be considered a square cavity. The upper horizontal surface

S4 is thermally insulated and the vertical surfaces S1 and S3 are assumed to be the cold isothermal surfaces. The bottom horizontal surfaces S5 and S6 are also thermally insulated. Localized heating is simulated by a centrally located heat source on the bottom wall, S2. The initial condition in Ω is: T = 0 with ψ = ω = 0. All physical properties of the fluid are constant except the density in the buoyancy term where it obeys the Boussinesq approximation. It is assumed that the third dimension of the cavities is large enough so that the flow and heat transfer are two-dimensional.

The following hypotheses are employed in the present work: unsteady turbulent regime; incompressible twodimensional flow; constant fluid physical properties, except the density in the buoyancy terms. Figure 2 shows one of the meshes used in the numerical simulations of the present work.

Proceedings of COBEM 2005 18th International Congress of Mechanical Engineering Copyright © 2005 by ABCM November 6-1, 2005, Ouro Preto, MG

Figure 1: Cavity geometry. Figure 2: Mesh arrangement for cases 1, 2 and 3. x

H 2S y g

1S Ω 4S

6S I 5S

3. Theory of Sub-Grid Scale Modeling The governing conservation equations are:

ijji jij jii δTTgβ xuxu x p1x uut

SxT x TutT

where xi are the axial coordinates x and y, ui are the velocity components, p is the pressure, T is the temperature, ρ is the fluid density, ν is the kinematics viscosity, g is the gravity acceleration, β is the fluid volumetric expansion coefficient, δ2j is the Kronecker delta, α is the thermal diffusivity, and S the source term. The last term in Eq. (2) is the Boussinesq buoyancy term where T0 is the reference temperature. In the large eddy simulation (LES), a variable decomposition similar to the one in the Reynolds decomposition is performed, where the quantity ϕ is split as follows:

where φ is the large eddy component and is the small eddy component. 'φ

The following filtered conservation equations are shown after applying the filtering operation to Eq. (1) to (3). This is done by using the volume filter function presented in Krajnovic (1998). The density is constant.

ijji jij jii δTTgβxuxu x p1x uut

SxT x TutT

where xi are the axial coordinates x and y, ui are the velocity components, p is the pressure, T is the temperature, ρ is the fluid density, ν is the kinematics viscosity, g is the gravity acceleration, β is the fluid volumetric expansion coefficient, δ2j is the Kronecker delta, α is the thermal diffusivity, and S the source term. The last term in Eq. (2) is the Boussinesq buoyancy term where T0 is the reference temperature. In Eq. (5) to (7), jiuu and Tuj are the filtered variable products that describe the turbulent momentum transport and the heat transport, respectively, between the large and sub-grid scales. According to Oliveira and Menon (2002), the products jiuu and Tujare split into other terms by including the

Leonard Lij tensor, the Crossing tensor Cij, the Reynolds sub-grid tensor Rij, the Leonard turbulent flux Lθj, the Crossing turbulent flux Cθj and the sub-grid turbulent flux θj. The Crossing and Leonard terms, according to Padilla (2000), can be neglected. After the development shown in Oliveira and Menon (2002), the following conservation equations are obtained:

iij j jii δTTgβ xτ x uνxpρ1x uut

j j uT θT α where, Pr is the Prandtl number with α = ν/Pr. 3.1 Sub-grid scale model According to Silveira Neto (1998), the Reynolds tensor is defined as:

kkijijTij Sδ3 uuS where νT is the turbulent kinematics viscosity, δij is the Kronecker delta and ijS is deformation tensor rate. Substituting ijS in τij and having some manipulation, it follows that the momentum and energy equations are:

ji j j j j i u u1 P u uνν gβ T δ

uTTT αα where αT, PrT, c, ℓ, q, ∆1 and ∆2 are, respectively, the turbulent thermal diffusivity, the turbulent Prandtl number, a dimensionless constant, the scale lengths, the velocity, the filter lengths in x and y directions, respectively.

3.2 The second-order structure-function sub-grid scale model (F2)

The turbulent viscosity (νT) and the geometric mean of distances di (∆) are calculated as follows:

Proceedings of COBEM 2005 18th International Congress of Mechanical Engineering Copyright © 2005 by ABCM November 6-1, 2005, Ouro Preto, MG where Ck = 1.4 is the Kolmogorov constant (Kolmogorov, 1941), the variable ∆ is the geometric mean of distances di from neighboring elements to the point where νT is calculated and ()t,,xF2∆r is the structure function of second order velocities. According to Kolmogorov (1941) law the structure function can be calculated as:

r r r r r , (15)

where ()iiiuxde,t+r and ()iiixde,tυ+r are the velocities at the point “i” of the neighboring centroid placed at a distance di from the target point, ()ux,tr and ()x,tυr are the velocities at this point of the element, N is the number of points from the neighborhood, t is the time and ier the vector on di direction. The turbulent thermal diffusion is estimated from the turbulent kinematics viscosity, by assuming. TPr0.4=

4. Initial and boundary conditions

From this section on, the upper bars that mean average values will be omitted. Figure 1 pictures the enclosure on which the initial boundary conditions are as follows: u(x,y,0) = 0, v(x,y,0) = 0, T(x,y,0) = 0 in Ω, u = v = 0, T = Tc = 0, on S1 and S3, u = v = 0, T = Th = 1 on S2, u = v = 0, ∂T/∂y = 0 on S4, S5 and S6. The flow field can be described by the stream function ψ and the vorticity ω distributions given by:

where u and υ are the velocity components in x and y directions , respectively. Hence, the continuity equation given by Eq. (1), is exactly satisfied. Working with dimensionless variables, it is possible to deal with Rayleigh number Ra, Prandtl number Pr and the enclosure aspect ratio A given by:

where Th and Tc are the surfaces temperatures S2 and S1-S3, respectively, H is the characteristic dimension of the cavity. 5. Numerical method

(Parte **1** de 2)