J. of the Braz. Soc. of Mech. Sci. & Eng. July-September 2009, Vol. XXXI, No. 3

J. of the Braz. Soc. of Mech. Sci. & Eng. July-September 2009, Vol. XXXI, No. 3

(Parte 1 de 3)

J. of the Braz. Soc. of Mech. Sci. & EngCopyright  2009 by ABCM July-September 2009, Vol. XXXI, No. 3 / 199

Turbulent Natural Convection in Enclosures Using Large-Eddy Simulation with…

Rogerio Fernandes Brito

Member, ABCM rogbrito@unifei.edu.br

Genésio José Menon rogbrito@unifei.edu.br

Federal University of Itajuba – UNIFEI Department of Mechanical Engineering 37500-176 Itajubá, MG, Brazil

Marcelo José Pirani

Member, ABCM pirani@ufba.br

Federal University of Bahia – UFBA

Department of Mechanical Engineering 40 210-630 Salvador, BA, Brazil

Turbulent Natural Convection in Enclosures Using Large-Eddy Simulation with Localized Heating from Horizontal Bottom Surface and

Cooling from Vertical Surfaces

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 the present work. Solutions are obtained for several Rayleigh numbers with Prandtl number Pr = 0.70. 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 unsteady state. The Finite Element Method (FEM) with a Galerkin scheme is utilized for solving the conservation equations. The formulation of conservation equations is carried out for turbulent flow and the implementation of turbulent model is made by Large-Eddy Simulation (LES). The distributions of the stream function and of the temperature are determined as functions of thermal and geometrical parameters. The average Nusselt number Num is shown to increase with an increase in the Rayleigh number Ra 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, turbulence, natural convection, LES

Introduction

1Natural 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 found 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 the 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 is a great number of problems in

Paper accepted December, 2006. Technical Editor: Francisco R. Cunha engineering that can use this geometry. One turbulence model is implemented here together with the finite element method.

A Large Eddy Simulation (LES) seems as 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 ∈.

Peng and Davidson (2001) studied the turbulent natural convection in a closed enclosure in which 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 were compared to experimental data and showed a stable thermal stratification under a low turbulence level (Ra = 1.58 x 109).

Deng et al. (2002) studied numerically a two-dimensional laminar natural convection in a rectangular enclosure with discrete heat sources on walls in the unsteady regime. A new combined temperature scale was suggested to nondimensionalize the

Rogerio Fernandes Brito et al.

200 / Vol. XXXI, No. 3, July-September 2009 ABCM governing equations of natural convection induced by multiple temperature differences. The Rayleigh numbers used were Ra = 103

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 x 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 x 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 x 109 . The experiments that were carried out 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.

Martorell et al. (2003) work dealt with the natural convection flow and heat transfer from a horizontal plate cooled from above. Experiments were carried out for rectangular plates having aspect ratios between φ = 0.036 and 0.43 and Rayleigh numbers in the range of 290 ≤ Raw ≤ 3.3×105 . These values of Raw and φ were selected to the design of printed circuit boards. The results showed that such a low Raw effect could be accounted for in a physically consistent manner by adding a constant term to the heat transfer correlation.

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 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, in Ra = 107 , 108 and 109 .

For the cases 1, 2, and 3, it is used a non-structured mesh of finite elements with 5,617 triangle elements and 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, 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).

Nomenclature

Cθ j = Crossing turbulent flux di = Distance di from the target point

Tuj = Filtered variable products that describe the turbulent heat transport jiuu = Filtered variable products that describe the turbulent momentum transport Li j = Leonard Tensor

Lθ j = Leonard turbulent flux Ri j = Reynolds sub-grid tensor

"ijS = Deformation tensor rater

A = Dimensionless constant c = Enclosure aspect ratio

Cij = Crossing tensor g = Gravity acceleration, m/s2

H = Characteristic dimension of cavity I = Length of the heated horizontal lower surface L = Characteristic dimension of cavity ℓ = Scale lengths and the velocity N = Number of points from the neighborhood n = Unit vector normal to the surface or boundary Nu = Nusselt number p = Pressure, Pa Pr = Prandtl number q = Velocity, m/s r = Distance between two points, m Ra = Rayleigh number S = Source term, Surface T = Temperature, ºC t = Time, s tCPU = CPU processing times, s u = Velocity in x direction, m/s v = Velocity in y direction, m/s x = Coordinate component in x direction y = Coordinate component in y direction

Greek Symbols δij = Kronecker delta ϕ = Large eddy component τ ij = Reynolds tensor θj = Sub-grid turbulent flux νT = Turbulent kinematic viscosity ε = Dissipation of the turbulent kinetic energy ε φ = Aspect ratio ∈ = Dimensionless heat source length Ω = Studied domain ρ = Fluid density β = Fluid volumetric expansion coefficient ϕ = General variable

∆ = Geometric mean of distances di from neighboring elements to the point where vT is calculated ∆1 = Filter length in x direction

∆2 = Filter length in y direction ν = Kinematic viscosity ψ = Stream function α = Thermal diffusivity ω = Vorticity

Subscripts m relative to mean i relative to i directions j relative to j directions k relative to k directions T relative to turbulent c relative to cold h relative to hot w Wall 1,2,3,4,5 relative to surfaces 1,2,3,4,5

J. of the Braz. Soc. of Mech. Sci. & EngCopyright  2009 by ABCM July-September 2009, Vol. XXXI, No. 3 / 201

Turbulent Natural Convection in Enclosures Using Large-Eddy Simulation with…

Problem Description and Hypothesis

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.

Figure 1. Cavity geometry

Theory of Sub-Grid Scale Modelling The governing conservation equations are:

ijji jij jii TTg xux u v xxpx uut u δβρ 01

∂∂ j xT x Tut 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 kinematic viscosity, g is the gravity acceleration, β is the fluid volumetric expansion coefficient, δij 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 ϕ is the small eddy component.

Figure 2 shows one of the meshes used in the numerical simulations of the present work.

Figure 2. Mesh arrangement for cases 1, 2 and 3.

The following hypotheses are employed in the present work: unsteady turbulent regime; incompressible two-dimensional flow; constant fluid physical properties, except the density in the buoyancy terms.

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

ijji jij jii TTgxux u v xxpx uutu 201 δβρ j xT x Tut

In the Eqs. (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.

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202 / Vol. XXXI, No. 3, July-September 2009 ABCM

According to Oliveira and Menon (2002), the products jiuuand

Tuj are split into other terms by including the Leonard tensor Lij, the Crossing tensor Cij, the Reynolds sub-grid tensor Rij, the Leonard turbulent flux Lθj, the Crossing turbulent flux Cθ j 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:

j i ij jii TTg x uxpx uut u δβτ νρ 0 j j xxT x TutT ∂ where Pr is the Prandtl number with α = ν / Pr. The tensors τij and θj that appear in Eqs. (9) and (10) are modeled in the forthcoming topics.

Sub-grid scale model

Many sub-grid scale models use the diffusion gradient hypothesis similar to the Boussinesq one that expresses the subgrid Reynolds tensor in function of the deformation rate and kinematic energy. According to Silveira-Neto (1998), the Reynolds tensor is defined as:

(Parte 1 de 3)

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