Computational Analysis of Turbulent Natural Convection in Water Filled Tall Enclosure

In this study, the turbulent buoyancy driven fluid flow and heat transfer in a differentially heated rectangular enclosure filled with water is quantified numerically. The two dimensional governing differential equations are discretized using the finite volume method. SIMPLE algorithm is employed to obtain stabilized solution for high Rayleigh numbers by a computational code written in FORTRAN language. A parametric study is undertaken and the effect of Rayleigh numbers (10 to 10), the aspect ratio (30, 40 and 50), and the tilt angle (10 to 170) on fluid flow and heat transfer are investigated. The results of the adopted model in the present work is compared with previously published results and a qualitative agreement and a good validation is obtained. Results show that the fluid circulation and temperature fields are strongly affected by the enclosure tilt angle and Rayleigh Number.


INTRODUCTION
Natural convection in tall enclosures with differentially heated sidewalls is importance in many engineering applications.These include energy transfer in rooms and buildings, nuclear reactor cooling, solar collectors, electronic equipment cooling, sloped or vertical double pan windows, photovoltaicthermal systems and ventilated facades which have been frequently applied in buildings.In these cavities, the turbulent flow is completely developed, i.e., hot and cold thermal boundary layers interact and show a turbulent core and laminar sublayers near the walls, (Albet-Chico et al 2008).
Several researchers have examined the influence of the cavity inclination on the nature of the flow and convection heat transfer.(Ozoe and Sayama 1975) dealt with the problem of natural convection in inclined rectangular channels heated on one side and cooled on the opposing side.Their results indicate that minimum and maximum heat transfer occurred as inclination angle was increased from 0 o to 180 o .They found that, the angle of inclination was a strong function of the aspect ratio and a weak function of the Rayleigh number.(Elsherbiny etal.1982)examined experimentally the influence of the tilt angle and the aspect ratio of air layer on the heat transfer rate for wide range of aspect ratios, 5-110.They developed a heat transfer correlation for angle 60 ο .In numerical study of (Kuyper et al. 1993) the Nusselt number shows a strong dependence on the orientation of the cavity and power law dependence on the Rayleigh number.(Lee and Lin 1995) have examined the flow structures, thermal fields, and stability conditions in vertical and inclined cubical enclosures with hot and cold side walls.This works are among the most significant treating the 3D features of the flow in closed cavities.(Elsherbiny et al. 1996) found that the average Nusselt number is decreased monotonically as the angle of tilt was increased from 0 to 180°.(Alvarado et al 2008) investigated numerically the interaction between two modes of heat transfer, natural convection and surface thermal radiation, in a tilted slender cavity of the aspect ratio (8 ≤AR≤16) and the inclination angle (15 ο -35 ο ).Their results indicated that the coupled transport processes modifies appreciably the flow patterns and the average heat transfer in the slender cavity.The total heat transfer increases when the inclination angle increases, except when the flow structure changes from the multi-cell to the unit-cell pattern.(Cooper et al. 2009) investigated numerically and experimentally the buoyant flow in a rectangular tall cavity of 28.6 aspect ratio.They show that for moderate angles of inclination i.e. under stable temperature stratification, the main difference from the vertical case is the reduced levels of fluctuations of the velocity and temperature fields.(Cooper et al. 2012) studied experimentally the effect of inclination angle on buoyancy driven flows inside air filled tall rectangular differentially heated cavity of 28.6 aspect ratio and Rayleigh number based on temperature difference and spacing between the long sides of 0.86×10 6 .
Turbulent natural convection flows in inclined, differentially heated tall cavities have received less attention than the corresponding vertical cavity flows.The objective of the present work is to study numerically the transient fluid flow and turbulent natural convection heat transfer in a tilted closed tall cavity.The momentum and energy equations are solved for turbulent flow considering natural convection between walls using finite volume technique and Low Reynolds Number LRN k-ω turbulence model of (Wilcox 1994).Parametric studies are conducted to determine the influence of each of: Rayleigh number (based on the height of cavity) with range of 10 10 to 10 14 , aspect ratios of 30, 40 and 50, and inclination angles of range of 10 ο to 170 ο on the turbulent natural convection in tall cavity.

MATHEMATICAL MODEL
The geometry of interest for this study is shown in Fig. 1.The study is limited to two dimensional flow in a rectangular cavity of length H and width L. The two active walls, hot (x = 0) and cold (x = L), have equal dimensions and are opposite.They are maintained isothermally at high temperature T H and low temperature T C , respectively.The other surfaces, y=0 and y=H, are insulated.The enclosure is tilted at an angle φ measured from horizontal.
The fluid in the cavity is assumed to be incompressible, with constant physical properties and negligible viscous dissipation.
The buoyancy effects upon momentum transfer are taken into account through the Boussinesq approximation.
For two-dimensional incompressible turbulent natural convection flow, the timeaveraged governing equations, continuity, momentum and energy are: Continuity equation: (1) x and y momentum equations: ) ) The standard turbulence models like standard k-ε and standard k-ω which adopted the logarithmic wall function dose not hold near a wall for natural convection.These wall functions were originally derived for forced convection flows (Henkes et al. 1991).
A modification of standard turbulence models "the so-called Low-Reynolds Number turbulence models" were done to consider the low turbulent Reynolds number effect and the damping of the turbulence effects in the wall region.In this study LRN k-ω model by (Wilcox 1994) is employed to predict the turbulent viscosity (ν t =μ t /ρ) by the resolution of the turbulent kinetic energy (k) and the specific turbulent kinetic energy dissipation (ω).Hence, the additional set of equations that closes the problem is; The turbulent kinetic energy transport equation can be written as: The stress production term, P k , is modeled by: The buoyancy term, G k , can be written as: Damping functions and constants of this turbulence model are given below, Wilcox (1994): c 3 is calculated as, (Henkes et al. 1991), is the turbulence Reynolds number.

Boundary Conditions
The fluid velocities at solid walls are equal to zero; temperatures are specified at vertical walls as (T=T H for x =0 and T=T C for x=L, (T C <T H ); adiabatic conditions are given as (∂T/∂y=0 for y=0 and y=H).In the LRN k-ω turbulence model, the boundary conditions are: k=0 at walls and 2 6 p y β ν ω = at the first node nearest to the wall, as the value at the wall is theoretically infinity (where y p is the normal distance of the nearest node to the wall), (Albet-Chico et al. 2008).To obtain non-dimensional form of the governing differential equations, the following dimensionless variables are introduced: The governing equations become: Continuity equation: Energy equation: The dimensionless boundary conditions can be written as: The

NUMERICAL PROCEDURE
The general equation from which all the governing differential equations (eqs.10 to 15) can be extracted is: The above equations have been solved using finite-volume techniques suggested by (Patankar 1980).The flow field is discretized into cells forming a staggered grid arrangement Fig. 2 and the general equation eq. ( 17) is converted into an algebraic equation with the following form: Fully implicit time integration has been applied and a pressure based method of Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) is used to solve the velocity-pressure field coupling.Central differences are employed to evaluate the diffusion terms.The convective terms are discretized by means of hybrid scheme.The source terms for k and ω equations have been linearized in by adding unconditional positive values to avoid the numerical instabilities.An iterative through a line-by-line (LBL) application of the Thomas algorithm is adopted.Starting from specified initial values of the independent variables, i.e., specified initial temperature and velocity fields (as well as k and ω).Under-relaxation is introduced by means pseudo-transient for allowing and/or improving the rate of convergence.
A uniform time step and non-uniform grid has been used, with the grid clustered near walls to resolve the wall-damping effect.The spatial solution is considered to be fully converged when the maximum absolute values of both the mass source and the percent changes of the independent variables at each grid-node from iteration to iteration are smaller than a prescribed values, taken as 10 -4 .The above solution algorithm was implemented into a computer code in FORTRAN (refer to Appendix I).
The average Nusselt number Nu H of the hot wall is determined by: This integral is evaluated by the trapezoidal rule.

4.VALIDATION OF THE NUMERICAL CODE
In order to validate the present numerical model, the problem of turbulent natural convection of air in a differentially heated tall cavity with an aspect ratio of 30 has been solve for Rayleigh number (based on enclosure height) of 2.43×10 10 .A comparison of present results of vertical velocity component in the middle height of the enclosure (Y=0.5) with results of (Albets-Chico et al. 2008) for LRN k-ω , and the experimental results of (Dafa'alla and Betts 1996) is shown in Fig. 3.A qualitative agreement and a good validation is obtained as indicated in Table 1.The maximum deviations obtained were of (6.5% and 4.4%) between present results and previously published data of (Albet-Chico et al. 2008) and (Dafa'Alla and Betts 1996) respectively.
Further, a mesh testing procedure was conducted to guarantee a grid independent solution.Four different mesh combinations were explored for the case of Ra=2.43×10 10 , Pr=0.7.The present code was tested for grid independence by calculating the average Nusselt number on the hot wall as shown in Fig. 4 and Table 2.The results confirmed that the present numerical solution is a grid independent solution and grid size of 60×180 is used in this study.

RESULTS AND DISCUSSION
A wide parametric numerical study was conducted for natural convection in enclosure filled with water for Rayleigh number range 10 10 to 10 14 , aspect ratio of (30, 40 and 50) and tilt angle range from 10 o to 170 o .The temperature difference between the hot and cold walls was fixed to 20 ο C.

Vertical Velocity Component
The distributions of dimensionless vertical velocity component V with the normalized distance (i.e.X/L) along the horizontal mid-plane (i.e.Y/H= 0.5) are shown in Fig. 5a-c for Rayleigh number Ra=10 12 at different values of aspect ratio AR 30, 40 and 50 respectively.The inclination angle was varied from 10 ο to 170 ο .There is a gradual decreasing in boundary layer thickness with increasing tilt angle ∅ from 10 ο to 90 ο while the velocity component V increases as shown in Fig. 3a.The reverse behavior is obtained with increasing tilt angle from 90 ο to 170 ο .
Enclosure of aspect ratio 30 has the highest maximum velocity about active walls (for tilt angle of 100 ο ), while for aspect ratios 40 and 50 this occur approximately at the same angle with lower values.The central part of the flow between the two wall regions (or between the two velocity peaks near the hot and cold walls) shows that the mean velocity profile here crosses the zero-velocity line at a single point for the aspect ratios 40 and 50, while in the enclosure of aspect ratio 30, a no-flow core regions where the turbulence level is low have been observed as shown in Fig. 3a.
The effect of Rayleigh number on the vertical velocity profile is shown in Fig. 6a-c for tilt angle ∅=33 ο (which is the latitude angle in Baghdad, Iraq).Fig. 6a-c indicates that the magnitude of V decreases with increasing Rayleigh number Ra for a given aspect ratio.The magnitude of V decreases with aspect ratio when Rayleigh number is held constant.Moreover as Ra number increases the fluid velocity was equal to zero in the central zone of the enclosure of aspect ratio=30.

Temperature Profile
Fig. 7a-c presents the effect of the tilt angle ∅ on the temperature distribution along the horizontal mid-plane for different aspect ratios.The Rayleigh number is held at Ra=10 12 .It is clear that the temperature distributions were nonlinear and symmetric for all the considered angles∅.These distributions essentially indicate a formation of thermal boundary layers adjacent to the hot and cold walls and an almost isothermal core at the center of the enclosure.As ∅ increases from 10 ο , the gradients of temperature become higher close to the active walls of the cavity until ∅=95 ο then the effect is reversed.The steepness of the temperature profile close to the vertical walls essentially determines the heat transfer rate and the maximum value of Nu is obtained for the steepest temperature gradient near-wall region.The steepness variation of the temperature profile close to the active walls decreases with increasing aspect ratio for the same Ra number as shown in Fig. 7.
Fig. 8 shows that as Rayleigh number increases, the temperature gradients at midplane increase for a given aspect ratio and inclination angle ∅.The temperature gradients decrease as aspect ratio increase.The temperature profile at Ra=10 10 essentially indicates conduction-driven thermal transport more than convection due to smallest distance between the active walls as shown in Fig. 8-c.

Heat Transfer Rate
The heat transfer rate increases gradually to a maximum value at tilt angle 95 ο and then decreases as reported in Fig. 9.Moreover, the average Nusselt number increases as aspect ratio increase.The variation of the average Nusselt number at enclosure hot wall for different values of aspect ratio and Rayleigh number is presented in Fig. 10.Higher average Nusselt number is indicated for higher aspect ratio due to the small distance between the hot and cold walls.It can also be seen that the higher heat transfer is obtained for the higher Rayleigh number.However, the average Nusselt number increases more rapidly for Ra>10 12 due to effects of turbulent convection mode.

CONCLUSIONS
Turbulent natural convection in twodimensional tall cavity has been studied numerically to examine the effect of the inclination angle and aspect ratio on the fluid V i,j j j-1 j+1 i i+1 i-1 Fig. 2 A typical control volume showing the locations of the principle variables -Scalar variables at point (i,j), P i,j , T i,j , k i,j ,ω i,j ,….
-U i,j and V i,j at staggered grids.
normalized boundary conditions for LRN kω model are: For kinetic energy k * =0 at walls and normalized specific dissipation near the wall ' and 'o' denote the coefficient for the neighbor grids and the old time step value, respectively.
Fig.1 Schematic diagram of the enclosure configurations

Fig
Fig.3 Comparison of vertical-velocity profiles of air at Y=0.5 in cavity of AR=30 and