HIGH RAYLEIGH NUMBER FLOWS IN A CUBICAL ENCLOSURE. A Thesis by. Balaraju Bende

Transcription

1 HIGH RAYLEIGH NUMBER FLOWS IN A CUBICAL ENCLOSURE A Thesis by Balaraju Bende B.Tech. Mechanical Engineering, Jawaharlal Nehru Technological University, India, 2006 Submitted to the Department of Mechanical Engineering and the faculty of the Graduate School of Wichita State University in partial fulfillment of the requirements for the degree of Master of Science December 2010

2 Copyright 2010 by Balaraju Bende All Rights Reserved

3 HIGH RAYLEIGH NUMBER FLOWS IN A CUBICAL ENCLOSURE The following faculty members have examined the final copy of this thesis for form and content, and recommend that it be accepted in partial fulfillment of the requirement for the degree of Master of Science with a major in Mechanical Engineering. Ikrammuddin Ahmed, Committee Chair David N. Koert, Committee Member Thomas K. DeLillo, Committee Member iii

4 DEDICATION My mother always loved me and supported me in my education. It is my father s dream to see me as an engineer. My uncle stood as an inspiration to me. Both my sister and my brother encouraged me to persist at tough times. Of course, my friends are my strengths. Finally, my professor who supported me to the core in all aspects is just as important as the above mentioned people and played a crucial role in not only helping with my thesis work but also stood as an inspiration to better understand the life. So, I dedicate this thesis to all the above mentioned people who helped me realize this dream of my father. iv

5 Never discourage anyone who continually makes progress, no matter how slow. v

6 ACKNOWLEDGEMENTS I take this opportunity to convey special thanks to my advisor, Dr. Ikram Ahmed, for his consistent support throughout my thesis work. He came along as a great guide throughout my course of studies by helping me make key decisions and also supporting me financially. I am very thankful to Dr. David N. Koert for his moral support and also for coming along as a good friend. I am also greatly thankful to Mr. John Matrow who helped me not only with the computational resources at High Performance Computing Center (HiPeCC) but also with the coding required for my simulations to be run. I acknowledge the allocation awarded for this work by TeraGrid High Performance Computing Center at National Center for Supercomputing Applications (NCSA). Last but not least, I thank Miss Connie Noble from graduate school and Lynda Cushman from Curriculum & Instruction for giving me a tremendous moral support as a friend and supporting me during tough times. vi

7 ABSTRACT The current study was undertaken with an intention to verify and extend the boundaries of the work produced by Xia & Murthy [24] and Vargas [23] on natural convective flows inside a cubical enclosure to observe the transition to turbulence. Their primary focus was to better understand the flows inside a cubical enclosure at higher Rayleigh numbers (Ra) with the help of Direct Numerical Simulation (DNS). This helps in understanding the mixing process that takes place inside the center wing tank of a commercial airplane. The motivation behind the current study was to stretch the limits of the work done by Vargas by adapting central differencing scheme (CDS) of discretization at three different grid sizes of 50 3, 75 3 and 90 3 with DNS. In order to achieve this task, we used the code developed by FLUENT for finite volume method. The cavity was created and meshed in GAMBIT with the bottom wall heated, top wall cooled and the side walls maintained adiabatic. Then, the Ra is varied between 2 x 10 4 and 10 9 maintaining the Prandtl number (Pr) to be constant at 2.5. Initially, the results were verified with those of Vargas and Xia & Murthy and similar observations were made with steady convective flows at Ra = 2 x 10 4, 10 5 and 2 x Periodic flows were captured at the critical Ra = 4.07 x 10 5 beyond which flows exhibited the chaotic nature until Ra = The work was then carried forward with the idea of employing the CDS of discretization at higher Ra up to 10 9 along with a finer grid density of 90 3 control volumes. The power spectrum slopes have been compared with the Kolmogorov s -5/3 rule for turbulent flows to observe the transition to turbulence. vii

8 TABLE OF CONTENTS Chapter Page 1. BACKGROUND Motivation Introduction and Literature Review METHODOLOGY Problem Description Governing Equations Boundary Conditions Numerical Method Numerical Accuracy Grid Size Estimation: Time Step Size Estimation: RESULTS AND DISCUSSION Grid Independence Study: Comparison with Previous Work Steady-State Results Unsteady-State Results High Rayleigh Number Results CONCLUSIONS AND FUTURE STUDIES Conclusions Future Studies LIST OF REFERENCES APPENDICES A. POWER SPECTRUM CALCULATIONS B. CALCULATION OF THE SMALLEST LENGTH SCALE viii

9 LIST OF TABLES Table Page 1. t max values for 10 6 Ra Average heat flux at the top and bottom walls for Ra = 10 6 to Percentage changes in heat fluxes (δq'') for top and bottom walls of cubical enclosure Power spectrum slopes for increasing Rayleigh numbers Average Nusselt numbers ix

10 LIST OF FIGURES Figure Page 1. Cubical geometry under consideration with side L Contours of Y-velocity at Y = 0.5 for Ra = 2 x 10 4 (Present Results) Contours of Y-velocity at Y = 0.5 for Ra = 2 x 10 4 (Vargas) Contours of Y-velocity at Y = 0.5 for Ra = 10 5 (Present Results) Contours of Y-velocity at Y = 0.5 for Ra = 10 5 (Xia & Murthy) Contours of Y-velocity at Y = 0.5 for Ra = 2 x 10 5 (Present Results) Contours of Y-velocity at Y = 0.5 for Ra = 2 x 10 5 (Vargas) D Contours of Y-velocity at Y = 0.5 for Ra = 2 x D Contours of Y-velocity at Y = 0.5 for Ra = 2 x Plot of U vs. t* at point P = (0.7, 0.7, 0.7) at Ra = 4.07 x 10 5 (short range) Plot of U vs. t* at point P = (0.7, 0.7, 0.7) at Ra = 4.07 x 10 5 in short range (Vargas) Plot of U vs. t* at point P = (0.7, 0.7, 0.7) at Ra = 4.07 x 10 5 (long range) Plot of U vs. t* at point P = (0.7, 0.7, 0.7) at Ra = 4.07 x10 5 in long range (Vargas) Power Spectrum plot for U vs. t* at P = (0.7, 0.7, 0.7) at Ra = 4.07 x Power Spectrum plot for 'U vs. t*' at P = (0.7, 0.7, 0.7) at Ra = 4.07 x 10 5 (Vargas) Sequence of time snap-shots of X-velocity contours on the X = 0.5 plane Sequence of time snap-shots of temperature contours on the X = 0.5 plane Stable convective rolls inside the Cubical Cavity at (a) Ra = 2 x 10 4 and (b) 2 x Plot of (U vs. V vs. W) at Ra = 4.07 x 10 5 (Short range) x

11 LIST OF FIGURES (continued) Figure Page 20. Plot of (U vs. V vs. θ) at Ra = 4.07 x 10 5 (Short range) Plot of (U vs. V vs. W) at Ra = 4.07 x 10 5 (Long range) Plot of (U vs. V vs. θ) at Ra = 4.07 x 10 5 (Long range) Plot of (U vs. W vs. θ) at Ra = 4.89 x 10 5 (Short range) Plot of (U vs. V vs. W) at Ra = 4.89 x 10 5 (Long range) Plot of U vs. t* at point P = (0.7, 0.7, 0.7) at Ra = 4.89 x 10 5 (Short range) Plot of U vs. t* at point P = (0.7, 0.7, 0.7) at Ra = 4.89 x 10 5 (Long range) Power spectrum plot for signal 'U vs. t*' at P = (0.7, 0.7, 0.7) at Ra = 4.89 x Power spectrum of X-velocity at P(0.5, 0.95, 0.5) for Ra = Power spectrum of X-velocity at P(0.5, 0.95, 0.5) for Ra = 5 x Power spectrum of X-velocity at P(0.5, 0.95, 0.5) for Ra = Power spectrum of X-velocity at P(0.5, 0.95, 0.5) for Ra = 5 x Power spectrum of X-velocity at P(0.5, 0.95, 0.5) for Ra = Power spectrum of X-velocity at P(0.5, 0.95, 0.5) for Ra = Nu vs. Ra for the Ra range (10 6 Ra 10 9 ) Comparison of Nu vs. Ra plot with that of Vargas xi

12 NOMENCLATURE Ar f Aspect ratio Frequency [Hz] f* Non dimensional frequency f max Maximum frequency of oscillation [Hz] g Gravitational acceleration [m/s 2 ] G Power Spectral Density [m 2 s -2 Hz -1 ] Gr L n N Nu P Grashof Number [Dimensionless] Side of the Cubical Cavity [m] Unit vector normal to the cavity walls Number of samples Nusselt Number [Dimensionless] Non dimensional pressure Average Power [m 2 s -2 ] Pr Ra t Prandtl Number [Dimensionless] Rayleigh Number [Dimensionless] time [s] t* Non dimensional time t max T T c T h u Maximum time-step size [s] Temperature [K] Temperature of the cold wall [K] Temperature of the hot wall [K] Velocity field component in the x-direction [m/s] xii

13 NOMENCLATURE (continued) U v V V w W x X y Y z Z α β η θ Non dimensional velocity field component in the x-direction Velocity field component in the y-direction [m/s] Non dimensional velocity field component in the y-direction Non dimensional velocity field Velocity field component in the z-direction [m/s] Non dimensional velocity field component in the z-direction Length in the x-direction [m] Non dimensional length in the x-direction Length in the y-direction [m] Non dimensional length in the y-direction Length in the z-direction [m] Non dimensional length in the z-direction Thermal Diffusivity [m 2 /s] Thermal Expansion Coefficient [1/K] Kolmogorov s length scale [m] Non dimensional Temperature ρ Density [kg/m 3 ] ν Kinematic Viscosity [m 2 /s] Abbreviations: CDS CFD Central Differencing Scheme Computational Fluid Dynamics xiii

14 NOMENCLATURE (continued) DFT DNS FFT FLUENT FTTC GAMBIT HIPECC LDV NCSA PIV QP n SIMPLE SOUS Discrete Fourier Transform Direct Numerical Simulation Fast Fourier Transform Numerical code for CFD problems Fuel Tank Test Cell Software used to mesh the geometry High - Performance Computing Center Laser Doppler Velocimetry National Center for Supercomputing Applications Particle Image Velocimetry Quasi-Periodic with n incommensurate frequencies Semi-Implicit Method for Pressure-Linked Equations Second Order Upwind Scheme TERAGRID National Supercomputing Center xiv

15 CHAPTER 1 BACKGROUND 1.1 Motivation A wide range of studies (both numerical and experimental) and reviews have been put forward by researchers over the years on natural convective flows inside cubical cavities subjected to a variety of boundary conditions and orientations but only a few investigators stepped ahead to examine the onset of turbulence at higher Rayleigh number flows inside a cubical enclosure heated from the bottom and cooled from the top with the side walls insulated. This is because of the equipment cost and maintenance difficulties such as data acquisition problems, controlling and reproducing the boundary conditions involved with experimental procedures and also due to the high computational cost associated with the numerical studies. The current study was motivated by the necessity to better understand the high Ra number flows inside the ullage of a commercial airplane. Putthawong [18] conducted an experimental study in a rectangular fuel tank test cell (FTTC) and proved the formation of fuel cloud inside the ullage of a center wing tank (CWT) of an aircraft under operating conditions. He used nitrogen instead of air to avoid explosion. The bottom surface of a CWT is heated with the help of an Environment Conditioning System (ECS) and top surface is cooled down by the cabin temperature creating a temperature difference. At higher Rayleigh numbers, the mixing of fuel and air becomes more rapid and turbulent and therefore increases the evaporation rate at the fuel surface of the ullage. The vapors accumulate at the top (cabin) surface and condense to form droplets. Once the non-dimensional droplet spacing (NDS) reaches the critical value, the flammability conditions inside the fuel tank will be enhanced. For this reason, a better 1

16 understanding of the flows inside the tank is essential in order to explain how a flammable fuel vapor-droplet-air mixture is formed from low volatility fuels. Apart from experimental studies, investigations have also been done with the numerical methods with Ra numbers ranging up to Xia and Murthy [24] have carried out the numerical work in a rectangular enclosure with Ra numbers up to 5 x 10 5 at various aspect ratios and found the flow to be chaotic. Vargas [23] carried forward the work up to Ra = 10 8 to predict the turbulence but transition to turbulence was not observed which is expected to be around Ra ~ The objective of the current work is to stretch the limits of the study on natural convective flows up to Ra = 10 9 with the central differencing scheme (CDS) of discretization using direct numerical simulation (DNS). 1.2 Introduction and Literature Review Convection is a heat transfer mechanism which may be classified according to the nature of the flow forced convection and free convection. The latter is also known as natural convection in which the basic driving force arises due to buoyancy effects. The temperature variation causes a difference in density, which then results in a buoyancy force due to the presence of gravity [5]. Buoyancy is due to the combined presence of a fluid density gradient and a body force that is proportional to density. In practice, the body force is usually gravitational, although it may be a centrifugal force in rotating fluid machinery or an electromagnetic force. The buoyancy driven flows are generally associated with a non-dimensional parameter called Rayleigh number (Ra) which is defined as the ratio of buoyancy forces and the product of both viscous and thermal diffusivities. Heat transfer is in the form of conduction when Ra is below the critical value Ra c for that fluid; otherwise, heat transfer is primarily in the form of convection. For example, in Rayleigh-Bénard convection [1], with top and bottom surfaces as 2

17 rigid, the onset of motion is observed at a critical Rayleigh number given by Ra c Rayleigh number can also be expressed as the product of Grashof number (Gr) and Prandtl number (Pr). The Grashof number is expressed as the ratio of buoyancy to viscous forces acting on a fluid and is analogous to Reynolds number (Re) in forced convection, with Gr ~ Re 2. The Prandtl number is again a non-dimensional number which is defined as the ratio between viscous and thermal diffusivities in a fluid. Prandtl number is held constant at 2.5 for the purpose of this study. The present work is focused on the natural convective flows inside a cubical enclosure with an aspect ratio (Ar) of one. By definition, enclosures are finite spaces bounded by walls with fluid media filled inside and in which internal partitions or obstacles may be present [7]. The bottom of the cavity is heated and the top is cooled while the side walls are maintained adiabatic. As a result of this temperature difference, density gradients exist which develops the buoyancy driven flows inside the enclosure. The significance of natural-convection phenomena inside an enclosure can best be appreciated by noting several important applications such as design of furnaces, operation of solar collectors, cooling of electronic equipment and devices such as circuit boards and chips [7]. Another practically important class of natural-convection phenomena in enclosures concerns the spread of fire and smoke in rooms, corridors, and other confined spaces. A proper understanding of the associated turbulent buoyant flows is essential in the development of countermeasures against the hazards of unwanted fires. Both numerical and experimental studies on the natural convective flows have been put forth by investigators. Experimental work is more reliable than the numerical work in terms of accuracy but poses a lot of difficulties such as disturbance of the flow field, equipment handling, reproduction of initial and boundary conditions, high cost and data acquisition problems. By far 3

18 most experimental studies on temperature and velocity fluctuations employ only point measurements, resulting in temperature and velocity time series acquired at a single point in the flow. The basic methods at hand are thermometry or bolometry for the temperature time series and later laser Doppler velocimetry (LDV) for the velocity time series [8]. However, with the maturing of particle image velocimetry (PIV) for the measurement of full velocity field, the experimental limitations of single-point measurements became less of a problem. On the other hand, the advantage of numerical simulations of natural convective flow is that in principle all data are available. However, due to computational costs, simulations at only limited Ra and Pr can be achieved, and only in the past 15 years or so have turbulent 3D simulations become possible. Nonetheless, one difficulty that pervades comparison of experimental and computational results is the lack of exact boundary conditions. For example, it is impossible to produce an adiabatic wall experimentally, yet this is a commonly employed boundary condition in computational works. Investigators have been studying natural convective flows inside the enclosures over a wide range of domains. Starting with simpler geometries, James and Webb [6] conducted a numerical study inside an air-filled square to predict different heat transfer quantities using the available turbulence models and then compared them with the experimental work at Ra = 1.58 x Although not closely linked, there have been several studies on natural convective flows with three dimensional configurations as well as orientations. For example, Olson and Glicksman [12] conducted an experimental demonstration to study transient natural convection in rectangular space (building) or enclosure at Ra as high as Niemela and Sreenivasan [11] carried out an experimental work in a cylindrical cell over a wide range of Ra from 10 6 to to investigate thermal transport using cryogenic helium gas as the working fluid. Further, a wide 4

19 range of boundary conditions were also employed. For example, Valencia & Frederick [21] studied natural convection of air inside a square with partially active vertical walls at Ra ranging from 10 3 to 10 7 and examined the heat transfer and fluid flow patterns at five different relative positions of the hot and cold wall regions. A few of the many other boundary conditions adopted by the investigators are heated from the sides, heated from the top and one side, heated from the bottom, cooled from the top, adiabatic side walls. Finally, a wide range of aspect ratios and orientations have been considered. For example, Sharif and Mohamed [19] studied the natural convection inside a rectangular enclosure to analyze the heat transfer process at various aspect ratios ranging from 0.5 to 2 and inclination angles from 0 to 30. In general, a number of studies on natural convective flows have been made but only a few investigators devoted their studies on cubical enclosures. Pallares et al. [14] studied natural convection numerically inside an air filled cubical enclosure heated from the bottom and cooled from the top at low Ra (3500 Ra 10 4 ). Four different stable convective structures S1-S4 were reported of which S1-S3 are depicted to be roll-type cells with axis of rotation horizontal and parallel to two opposing vertical walls at 3500 Ra < 8000 and the fourth structure at Ra 8000 is a toroidal roll with descending motion aligned with the four verticals edges and the single ascending current along the vertical axis of the enclosure. Later, numerical simulations of laminar flows were reported in the range of Rayleigh numbers from 7 x 10 3 to 10 5 [13] and flow structures inside a perfectly conducting cubical cavity were discussed for a fluid with Pr = 0.7. In the laminar regime, two single roll structures and a four-roll structure in which the axis of each roll is perpendicular to one sidewall were found to be stable [13]. Recently, both experimental and numerical works on natural convection inside a cubical cavity heated from the bottom and cooled from the top (10 7 Ra 10 8 and Pr = 6) were carried out using water as the fluid [22]. 5

20 The averaged flow structure at Ra = 10 7 was reported to consist two main counter rotating vortex rings located near the horizontal walls and eight small vortex tubes near the vertices of the cavity and at Ra = 10 7, the flow structure primarily consists two main descending and ascending motions that occur close to two diagonally opposed vertical edges. Xia and Murthy [24] conducted the numerical investigation on natural convective flows inside deep cavities and reported two critical Rayleigh numbers Ra 1 (below which the fluid is at rest and heat transfer is purely due to conduction) and Ra 2 (beyond which the flow is in transition from steady state to unsteady chaotic flow) at different aspect ratios (Ar = 1, 2 and 5) and also depicted the path followed by the flow from periodic regime to chaos. For Ar = 1, Ra 1 has been found to be 3583 beyond which the flow is steady until it reaches a critical Rayleigh number Ra c = 4.07 x 10 5 at which the transition from steady to periodic flow is seen. Finally, the flow becomes chaotic at Ra 2 > 4.89 x 10 5 with the route to chaos being P (Periodic) QP 2 (Quasi- Periodic with two incommensurate frequencies) QP 3 (Quasi-Periodic with three incommensurate frequencies) N (Chaotic) with the Rayleigh number ranging from 4.07 x 10 5 Ra 4.89 x Feigenbaum [2] traced the route to chaos as S (Steady) P (Periodic) P 2 (Period doubling) P 4 N (Chaotic). McLaughlin et al. [9] numerically investigated the flow transition from periodic to chaotic regime and postulated the role of symmetry-breaking perturbations in the production of chaos. Vargas [23] first verified his results with Xia and Murthy s work at Ar = 1 and then extended the work at higher Rayleigh numbers ranging from 10 6 to 10 8 and Pr = 2.5 using DNS. The flow at this range was reported to have shown characteristics typically attributed to chaotic flow. However, the transition to fully turbulent flow has not been observed and is expected to be around Ra ~

21 Nevertheless, as mentioned earlier, the goal of the current study was to first verify the existing work done by Vargas followed by an attempt to observe the transition to turbulence with the application of CDS of discretization using DNS. 7

22 CHAPTER 2 METHODOLOGY 2.1 Problem Description The present work is a numerical study of natural convective flows at high Rayleigh numbers inside a cubical cavity. The orientation of the enclosure is horizontally aligned with respect to the Cartesian coordinate system as shown in the Figure 1 with the gravitational force acting vertically downward, i.e., opposite to the positive y-direction. The cavity is maintained at a constant temperature difference with the bottom wall heated and the top wall cooled keeping the vertical walls adiabatic. The working fluid inside the cavity was maintained at Pr = Governing Equations The fluid is assumed to be Newtonian and incompressible. The Boussinesq model of approximation [20] is employed which allows us to consider the density to be constant in all the terms but the gravity term in momentum equations, to account for buoyancy effects due to density gradients as a result of temperature variation. Two non-dimensional parameters Ra and Pr are obtained from the non-dimensionalization of N-S equations as shown below. Continuity: (1) X-Momentum: (2) 8

23 Y-Momentum: (3) Z-Momentum: (4) Energy: (5) where V in the Equations (1) (5) is the velocity vector and X, Y, Z, t*, U, V, W, P, θ in the Equations (2) (5) are non-dimensionalized as shown below Spatial Non-dimensionalization: (6) Temporal Non-dimensionalization: where (7) Non-dimensional Flow Variables: (8) where (9) where T ref = T c in this current study (10) 9

24 and U ref, T ref & t ref are reference velocity, temperature and time respectively Boundary Conditions The geometry under consideration is subjected to the following boundary conditions: At the Bottom Wall (Y = 0): (11) At the Top Wall (Y = 1): (12) At the Vertical Walls (X = Z = 0 and X = Z = 1): (13) 2.3 Numerical Method The three dimensional Navier-Stokes equations are solved at each control volume using a finite volume method without the use of turbulence modeling. Initially, a study to verify the results of Vargas [23] was performed using the computational fluid dynamics (CFD) code FLUENT. After verification, we extended the boundaries of the work done by Vargas at higher Rayleigh numbers along with the use of CDS of discretization. The cubical geometry under consideration was meshed into uniform hexahedral cells with the help of GAMBIT. While the first approach was based on the second order upwind scheme (SOUS) of discretization, the second approach was based on CDS for discretization of the momentum and energy equations. Since the solution to the governing equations is complicated due to the lack of an independent equation for pressure, pressure-velocity coupling is utilized. For this purpose, the SIMPLE 10

25 (Semi-Implicit Method for Pressure-Linkage Equations) scheme [3, 15] is employed which uses a guessed pressure field to solve the momentum equations. A pressure correction differential equation, deduced from the continuity equation, is then solved to obtain a pressure correction field, which in turn is used to update the velocity and pressure fields. These guessed fields are progressively improved through the iteration process until convergence is achieved for the velocity and pressure fields. Finally, a second order implicit formulation was adopted for the unsteady flow simulations. Here, the solution is considered to converge if the normalized residuals summed over the whole domain dropped by six orders of magnitude from initial conditions. Computations were performed using the supercomputing clusters at HIPECC and TERAGRID. 2.4 Numerical Accuracy Although there is a wide spectrum of turbulence models available for solving the turbulent flows, their practical applications are limited due to the approximations involved. DNS on the other hand does not involve approximations [20], other than those due to discretization, which are inherent in any numerical solution of differential equations. Since turbulence in fluids is a nonlinear phenomenon comprising a wide spectrum of spatial and temporal scales, employing DNS may not yield satisfactory results until a sufficiently refined mesh is adopted which is capable of capturing most of the dissipation [10]. The following sections estimate the size of the grid and time step required for the current work Grid Size Estimation: The accuracy of a given numerical method depends on the size of the grid. The errors due to discretization are reduced with the increase in grid resolution. For DNS, the highest possible 11

26 accuracy is obtained if the size of the grid is refined to a length scale at which the smallest flow field fluctuations can be captured. DNS, at least in theory, should provide acutely detailed information and a better understanding of the flow physics. A wide range of scales are exhibited in turbulent flows. For example, the smallest length scale [4] denoted by l b was calculated to be l b ~ 1.68 x 10-4 m (Appendix B) based on our computations at Ra = (14) below. Similarly, we estimate the smallest length scale [16] in fully turbulent flows as shown (15) where η is Kolmogorov s microscale For our cubical cavity of side L = 1 m, at Ra = 10 9 and Pr = 2.5, we get Re ~ O(20,000). Therefore, η is calculated to be 5.95 x 10-4 m which means that our cubical geometry should be refined to approximately control volumes. However, it is practically impossible to simulate the flow at this grid resolution with the computational resources available at hand. Hence, we have performed a grid independence study (detailed explanation given in chapter 3) and determined the optimal grid resolution to be 90 3 cells for the current study which yields the smallest side of the control volume to be m Time Step Size Estimation: We have conducted an a posteriori estimate based on our numerical simulations to check if the size of the time interval that we adopted was adequate for the purpose of the current study. 12

27 From the plots of power spectrum at different Ra ranging from 10 6 to 10 9, we have captured the readings of maximum frequency up to which the power decreases linearly using the mesh density of 90 3 cells. Using these readings, we have determined the maximum time step size (shown in Table 1) that can be allowed for our numerical simulations according to Nyquist criteria [17]. Nyquist criterion gives maximum time step size as, (16) where f max is the maximum frequency obtained at individual Ra number case. Table 1: t max values for 10 6 Ra 10 9 Ra 1e6 5e6 1e7 5e7 1e8 1e9 t max We adopted a time step size of t = 1 sec for Ra > 10 7 and t = 10 sec for the Ra range 4.07 x 10 5 Ra 10 7 which are within the permissible limits according to the time step values listed in Table 1. 13

28 CHAPTER 3 RESULTS AND DISCUSSION 3.1 Grid Independence Study: For any given CFD problem, the accuracy of the numerical method relies on the amount of numerical errors involved, e.g. round-off errors and discretization errors. In our simulations, the former type of errors is controlled as we employed double precision numerical calculations. In summary, the discretization errors can be minimized upon obtaining the grid independence by noting the changes in the flow quantities with mesh size refinement, i.e. increasing the spatial resolution [20]. We shall begin our discussion of results by first presenting the grid independence study performed at three different grid resolutions, i.e. at 50 3, 75 3 & 90 3 finite control volumes inside the cubical domain shown in Figure 1, inside which the flow field is solved using DNS. It should be understood that solving a flow field using DNS requires resolving the smallest length scale to the order of O(η) but not exactly equal to η (Kolmogorov s length scale) which is fine enough to accurately capture most of the dissipation [10]. According to Kolmogorov s theory of 1941, energy cascades down from larger to smaller eddies in the form of kinetic energy down to the smallest length scale at which further cascade of energy is not possible and it is dissipated in the form of heat due to the viscosity of the fluid. As mentioned earlier in our discussion, we simulated the fluid flow at higher Ra numbers ranging from 10 6 to 10 9 to observe the transition to fully turbulent flow. We measured the values of total surface heat flux at both the top and bottom walls of the cubical geometry. As seen from Table 3, the relative change in heat flux at bottom wall, after refining from 50 3 to 75 3 cells, is observed to vary from as low as 1.2 % (for Ra = 10 6 ) to a maximum of % (for Ra = 10 9 ). 14

29 Upon further refining from 75 3 to 90 3 control volumes, it varies between 0.92 % (for Ra = 10 6 ) and 3.11 % (for Ra = 10 9 ) as shown in Table 3. A similar trend was observed at the top wall as shown in Table 3 with a variation from 0.81 % (for Ra = 10 6 ) to % (for Ra = 10 9 ) after refining from 50 3 to 75 3 cells and falls down to a minimum of 0.16 % (for Ra = 10 6 ) and maximum of 3.75 % (for Ra = 10 9 ) upon refining from 75 3 to 90 3 cells. It is therefore concluded that our highest grid resolution of 90 3 was reasonable for the purpose of this study, considering the cost of computational calculations and resources at hand. Hence, we employed a grid of density 90 3 cells for the Ra number flows from 10 6 to 10 9 and also discussed the results at this size. To be consistent, we have simulated the flows at other smaller Ra numbers at the same mesh density of 90 3 and noted some minor changes upon comparison with the work done by the investigators in the past as described in the following sections. Table 2: Average heat flux at the top and bottom walls for Ra = 10 6 to 10 9 Grid Independence Study Average Heat Flux (W/m 2 ) Mesh Density Ra Wall Wall Wall Bottom Top Bottom Top Bottom Top 1e e e e e e

30 Table 3: Percentage changes in heat fluxes (δq'') for top and bottom walls of cubical enclosure Change in heat flux with grid refinement Bottom Wall Top Wall e6 1.2 % 0.92 % 0.81 % 0.16 % 5e % 2.43 % 5.46 % 1.49 % 1e % 0.69 % 4.38 % 1.92 % 5e % 0.85 % 5.13 % 0.29 % 1e % 1.17 % 3.28 % 1.03 % 1e % 3.11 % % 3.75 % 3.2 Comparison with Previous Work The present study was initiated with an effort to first verify the work done by Vargas [23] followed by attempt to extend the boundaries of study up to Ra = 10 9 to observe the transition to turbulence using CDS of discretization. As stated by Xia and Murthy, at very small Ra numbers (Ra < 3583), the flow is nearly at rest due to the domination of viscous forces over buoyancy forces. Before the attempt to run the unsteady state simulations at higher Ra numbers, we initially carried out the steady state simulations at lower Ra numbers (Ra = 2 x 10 4, 10 5 and 2 x 10 5 ) to check the validity of our numerical approach and observed the results to be in good agreement with those obtained by Vargas [23]. We then employed a grid independence study to determine the mesh density for the geometry under consideration. The unsteady state flows at higher Ra numbers were simulated in order to observe the transition to turbulence. We gave a 16

31 new dimension to the study of high Ra number buoyancy driven flows by employing CDS of discretization up to Ra = 10 9 and the results were compared Steady-State Results As explained above, the numerical approach adopted for the current study is verified at three different numbers i.e. Ra = 2 x 10 4, 10 5 and 2 x We compared the contours of Y- velocity at the plane Y = 0.5 on the cubical geometry and the flow was found to be steady at all these Ra numbers. At Ra = 2 x 10 4, the fluid flow as depicted in Figures 2 and 3 is seen to produce a peak (denoted by +) rising up at one of the vertical faces and a valley (denoted by -) moving down at the other, thus forming a single cell. A three-dimensional (3-D) view of the Y- velocity contours is also shown in Figure 8 for better understanding. As discussed by Vargas, this type of flow configuration is referred to as 1T. Our results upon comparison with those of Vargas at Ra = 2 x 10 4 shows that the numerical schemes are in good agreement. Once again, our results at Ra = 2 x 10 5 were compared with the results obtained by Vargas as shown in Figures 6 and 7 depicting the Y-velocity contours at the horizontal plane Y = 0.5 inside the cubical geometry. Here, we see in Figure 6 that the fluid flow rises up along two adjacent vertical corners of the cube with two peaks of different amplitudes indicated by (+) sign which are in agreement with those obtained by Vargas as shown in Figure 7. Similarly, when it comes to comparison of the valleys, we have a single valley at the third vertical corner indicated by (-) sign as shown in Figure 6 which is similar to the results of Vargas as shown in Figure 7. A 3-D plot of Y-velocity contours on the horizontal plane at Y = 0.5 is also shown in the Figure 9 depicting the amplitudes of the crests and troughs discussed in this section. As described by 17

32 Vargas, this type of flow is considered to be of 2T configuration. Thus, we conclude that our steady state results are in good agreement with those obtained by Vargas. We then focused our attention onto the third Ra number (Ra = 10 5 ) and the results were compared with the work done by Xia & Murthy at Ar = 1. As shown in Figures 4 and 5, the contours of Y-velocity at the horizontal plane Y = 0.5 inside the cavity are depicted. The fluid flow rises up at two adjacent vertical corners and flows down at the other two adjacent vertical corners of the cube. Our results shown in Figure 4 were in good accordance with those produced by Xia and Murthy shown in Figure 5 and hence, we conclude that our numerical scheme is verified at Ra = Finally, the stable convective structures of fluid motion at Ra = 2 x 10 4 and 2 x 10 5 are also depicted in the Figures 18 (a) and 18 (b). It can be clearly understood and visualized from these Figures that at Ra = 2 x 10 4, the flow attains a steady motion in the form of convective rolls whose axes of rotation are parallel to the horizontal plane (1T flow configuration) and at Ra = 2 x 10 5, the flow regime remains steady with stable convective rolls but the axes of rotation are not parallel to the horizontal plane (2T flow configuration) Unsteady-State Results After verifying the numerical scheme at lower Ra numbers, we carried out unsteady state simulations for the fluid flow at Ra = 4.07 x 10 5 and compared our results with Vargas [23]. Here, we picked a point in Cartesian coordinates P = (0.7 m, 0.7 m, 0.7 m) in the flow regime inside the cubical geometry and plotted the variation of X-velocity with respect to nondimensional time t * as shown in Figure 10 (short range) and Figure 12 (long range). These plots depict the unsteady nature of the flow and indicate that the flow is periodic in both the short and 18

33 long ranges of time. In comparison, we observe that our results are in good agreement with those of Vargas as seen in Figure 11 for the short range and Figure 13 for the long range. Further, we also obtained the power spectrum for the X-component of the velocity signal by plotting the graph between Power P and frequency f. For a given signal, the power spectrum shows the distribution of the signal s power (energy per unit time). We generated the power spectrum plot with the help of fast Fourier transform (FFT) which is an algorithm to implement discrete Fourier transform [17]. From the graph shown in Figure 14, it can be seen that we have obtained seven frequency components: f 1 * = 3.43, f 2 * = 10.3, f 3 * = 20.6, f 4 * = 106.5, f 5 * = 147.5, f 6 * = 161.5, f 7 * = This describes the flow at Ra = 4.07 x 10 5 to be quasi-periodic (QP n ) with seven incommensurate frequencies (n = 7). In contrast, Figure 15 depicts the results obtained by Vargas which had only four frequency components: f 1 * = 0.4, f 2 * = 10.9, f 3 * = f 4 * = This variance in the number of incommensurate frequency components can be explained by the reason that we adopted CDS of discretization with a finer mesh density of 90 3 cells whereas Vargas adopted second order upwind scheme of discretization with a mesh density of 75 3 cells. The other reason for this discrepancy could have been due to the fact that we considered the solution to be converged if the normalized residuals summed over the whole domain dropped by six orders of magnitude from initial conditions whereas Vargas considered dropping by three orders of magnitude. Nonetheless, apart from the additional three frequencies (f 3 *, f 4 *, f 6 *) captured in the present study, our other four frequencies (f 1 *, f 2 *, f 5 *, f 7 *) were in good agreement with those obtained by Vargas (f 1 * through f 4 *). Also, the periodic behavior of the flow is captured with the help of the sequence of time snap-shots of the contours of X-velocity on the plane X = 0.5 as shown in Figure 16. Here, we observe that the first and last snap-shots look similar, thus demonstrating the 19

34 periodicity. Similarly, the sequence of time snap-shots of temperature contours are also captured at the plane X = 0.5 as shown in Figure 17. Once again, the first and last snap-shots are observed to be similar proving the existence of periodicity. We have then plotted the path traced by the fluid properties on the graphs: (U vs. V vs. W) & (U vs. V vs. θ) in the short and long ranges of the fluid flow at Ra = 4.07 x 10 5 and observed (at both short and long ranges) the profile as shown in Figures from 19 to 22 to be following a closed circuit for any given period and closely overlapping the same loop for the consecutive cycles representing a periodic motion. At the next higher Ra = 4.89 x 10 5, we observed that the flow transitions from periodic regime to chaotic regime upon observing the plots of U vs. t* at a point P = (0.7m, 0.7m, 0.7m) in both the short and ranges of time as shown in Figures 25 and 26 respectively. Further, we have plotted the graphs between (U vs. W vs. θ) & (U vs. V vs. W) and observed that, in both the short and long ranges as shown in Figures 23 and 24, the path followed by the trajectory shows a number of closed loops which do not overlap each other but stay close within a well defined region showing the signs of chaotic behavior. Finally, the chaotic behavior is confirmed upon observing several incommensurate frequencies at which the power is distributed in the power spectrum plot as shown in figure 27. Hence, the flow is characterized to be chaotic or quasiperiodic with n incommensurate frequencies denoted as QP n. 3.3 High Rayleigh Number Results After successfully verifying the steady and unsteady-state results up to Ra = 4.89 x 10 5, we simulated the flow at higher Ra numbers in the range from 10 6 to Our work was initiated by an attempt to verify the work done by Vargas at the highest Ra = 10 8 by plotting the power 20

35 spectrum for the X-component of the velocity signal at a point P = (0.5, 0.95, 0.5) in the flow field and calculating the slope of the power signal obtained. As shown in Table 4, the value of slope from our results at Ra = 10 8 is in close agreement with that of Vargas but do not match exactly at the decimal places. As mentioned earlier in the previous section, the factors contributing for this discrepancy could be that we employed the CDS of discretization whereas Vargas adopted SOUS of discretization and also we used a finer grid resolution of 90 3 cells as compared to the mesh density of 75 3 cells by Vargas. Further, we also simulated the flow at a range of Ra numbers from 10 6 to 10 9 with the CDS of discretization at a mesh density of 90 3 cell volumes. The X-velocity fluctuations were captured at a point P = (0.5, 0.95, 0.5) inside the cubical geometry. This point was chosen based on the idea generated by Vargas [23] which says that turbulent boundary layers would be formed near the walls (i.e. away from a possibly inviscid core in the center) in the presence of turbulent flows. Then, we plotted the average power spectrum (Appendix A) obtained from a set of four different power signals, each based on a series of two thousand velocity readings measured at a constant time interval and compared the calculated slope with the Kolmogorov s slope of -5/3 to observe the transition to turbulence. The plots of average power spectra for all the Ra numbers from 10 6 to 10 9 are shown in Figures from 28 to 33 and the corresponding slopes are presented as shown in Table 4. 21

36 Table 4: Power spectrum slopes for increasing Rayleigh numbers Rayleigh Number (Ra) Power Spectrum Slope 1e e e e e e8 (Vargas work) e Kolmogorov s Spectrum It can be seen from the above Table that none of the values of the slopes at different Ra is close to the Kolmogorov s slope of -5/3. This means that the flow has still not reached the fully turbulent regime since the expected Kolmogorov s slope has not been attained. As it can be seen from Figures 28 to 33, the rate of power decay is continuous which makes the task of distinguishing the main frequency components difficult. In conclusion, we observe that the flow is subjected to chaos and can be characterized as chaotic or weakly turbulent. Finally, we have also attempted to observe the relationship between Nusselt (Nu) number and Ra number. The Nu number for the top and bottom walls was obtained and average Nusselt (Nu avg ) number for 10 6 Ra 10 9 as shown in Table 5 is calculated using the formula shown in Equation 17. The plot shown in Figure 34 depicts the variation of Nu number with the Ra number given by the power law where β = Further, we compared our plot (shown in Figure 35) of Nu vs. Ra with that obtained by Vargas which varied as agreement with that obtained by Vargas. where β = In conclusion, we say that our value of β is in close 22

37 (17) Table 5: Average Nusselt numbers Ra Nu avg 1e e e e e e

38 Figure 1: Cubical geometry under consideration with side L 24

39 Figure 2: Contours of Y-velocity at Y = 0.5 for Ra = 2 x 10 4 (Present Results) Figure 3: Contours of Y-velocity at Y = 0.5 for Ra = 2 x 10 4 (Vargas) 25

40 Figure 4: Contours of Y-velocity at Y = 0.5 for Ra = 10 5 (Present Results) Figure 5: Contours of Y-velocity at Y = 0.5 for Ra = 10 5 (Xia & Murthy) 26

41 Figure 6: Contours of Y-velocity at Y = 0.5 for Ra = 2 x 10 5 (Present Results) Figure 7: Contours of Y-velocity at Y = 0.5 for Ra = 2 x 10 5 (Vargas) 27

42 Figure 8: 3-D Contours of Y-velocity at Y = 0.5 for Ra = 2 x 10 4 Figure 9: 3-D Contours of Y-velocity at Y = 0.5 for Ra = 2 x

43 -5 U vs t* at Ra = 4.07E5 (short range) U t* Figure 10: Plot of U vs. t* at point P = (0.7, 0.7, 0.7) at Ra = 4.07 x 10 5 (short range) Figure 11: Plot of U vs. t* at point P = (0.7, 0.7, 0.7) at Ra = 4.07 x 10 5 in short range (Vargas) 29

44 10 U Vs t* at Ra = 4.07E5 (long range) U t* Figure 12: Plot of U vs. t* at point P = (0.7, 0.7, 0.7) at Ra = 4.07 x 10 5 (long range) Figure 13: Plot of U vs. t* at point P = (0.7, 0.7, 0.7) at Ra = 4.07 x10 5 in long range (Vargas) 30

45 Power Spectrum plot at Ra = 4.07e5 at P(0.7,0.7,0.7) 10-4 Power f* Figure 14: Power Spectrum plot for U vs. t* at P = (0.7, 0.7, 0.7) at Ra = 4.07 x 10 5 Figure 15: Power Spectrum plot for 'U vs. t*' at P = (0.7, 0.7, 0.7) at Ra = 4.07 x 10 5 (Vargas) 31

46 Snap-shot 1 Snap-shot 2 Snap-shot 3 Snap-shot 4 Snap-shot 5 Snap-shot 6 Snap-shot 7 Snap-shot 8 Figure 16: Sequence of time snap-shots of X-velocity contours on the X = 0.5 plane 32

47 Snap-shot 1 Snap-shot 2 Snap-shot 3 Snap-shot 4 Snap-shot 5 Snap-shot 6 Snap-shot 7 Snap-shot 8 Figure 17: Sequence of time snap-shots of temperature contours on the X = 0.5 plane 33

48 (a) (b) Figure 18: Stable convective rolls inside the Cubical Cavity at (a) Ra = 2 x 10 4 and (b) 2 x

49 U vs V vs W W V U Figure 19: Plot of (U vs. V vs. W) at Ra = 4.07 x 10 5 (Short range) Figure 20: Plot of (U vs. V vs. θ) at Ra = 4.07 x 10 5 (Short range) 35

50 U vs. V vs. W W V U Figure 21: Plot of (U vs. V vs. W) at Ra = 4.07 x 10 5 (Long range) Figure 22: Plot of (U vs. V vs. θ) at Ra = 4.07 x 10 5 (Long range) 36

51 Figure 23: Plot of (U vs. W vs. θ) at Ra = 4.89 x 10 5 (Short range) U vs V vs W W V U Figure 24: Plot of (U vs. V vs. W) at Ra = 4.89 x 10 5 (Long range) 37

52 60 U vs t* U t* Figure 25: Plot of U vs. t* at point P = (0.7, 0.7, 0.7) at Ra = 4.89 x 10 5 (Short range) 100 U vs t* 50 0 U t* Figure 26: Plot of U vs. t* at point P = (0.7, 0.7, 0.7) at Ra = 4.89 x 10 5 (Long range) 38

53 Power f* Figure 27: Power spectrum plot for signal 'U vs. t*' at P = (0.7, 0.7, 0.7) at Ra = 4.89 x

54 10 1 Average Power Spectrum Plot for Ra = 1E Power Figure 28: Power spectrum of X-velocity at P(0.5, 0.95, 0.5) for Ra = 10 6 f 10 2 Average Power Spectrum Plot for Ra = 5E Power Figure 29: Power spectrum of X-velocity at P(0.5, 0.95, 0.5) for Ra = 5 x 10 6 f 40

55 10 2 Average Power Spectrum Plot for Ra = 1E Power Figure 30: Power spectrum of X-velocity at P(0.5, 0.95, 0.5) for Ra = 10 7 f 10 2 Average Power Spectrum Plot for Ra = 5E Power f Figure 31: Power spectrum of X-velocity at P(0.5, 0.95, 0.5) for Ra = 5 x

56 10 2 Average Power Spectrum Plot for Ra = 1E Power f Figure 32: Power spectrum of X-velocity at P(0.5, 0.95, 0.5) for Ra = Average Power Spectrum Plot for Ra = 1E Power f Figure 33: Power spectrum of X-velocity at P(0.5, 0.95, 0.5) for Ra =

57 Figure 34: Nu vs. Ra for the Ra range (10 6 Ra 10 9 ) Figure 35: Comparison of Nu vs. Ra plot with that of Vargas 43

58 CHAPTER 4 CONCLUSIONS AND FUTURE STUDIES 4.1 Conclusions An attempt to extend the boundaries of study on natural convective flows inside a cubical cavity at higher Ra up to 10 9 with CDS of discretization using DNS has been made and results are presented for a range of Ra numbers from 2 x 10 4 to At smaller Ra (2 x 10 4, 10 5 & 2 x 10 5 ), the fluid flow is mainly characterized by the steady convective structures, thus verifying the results attained by Xia & Murthy and Vargas. Then, we also reproduced the work done by Vargas at Ra = 4.07 x 10 5 and witnessed the periodic motion in both the short and long ranges of time with seven incommensurate frequencies, thus portraying the quasi-periodic flow (QP n ) where n = 7. Also, the evidence for the commencement of chaotic regime having n incommensurate frequencies was shown at Ra = 4.89 x Finally, the simulations were performed at higher Ra numbers ranging from 10 6 to 10 9 and the slopes of the power spectrum plots were calculated and compared with the Kolmogorov s -5/3 rule for turbulent flows as shown in Table 4. It is therefore concluded that the transition to fully turbulent flow has not been observed as the expected slope value of (according to Kolmogorov s study) is not attained. However, the transition from steady state to chaotic regime is depicted by the route S (Steady) P (Periodic) QP n (Quasi-Periodic with n incommensurate frequencies) N (Chaotic). In conclusion, although all our simulations at Ra numbers ranging from 10 6 to 10 9 have produced highly fluctuating velocity fields exhibiting the typical characteristics of turbulence, we 44