A prediction tool for the thermal performance of a

TECHNICAL UNIVERSITY EINDHOVEN UNIVERSIDADE FEDERAL DA SANTA CATARINA SOLUZ ENERGIA

A prediction tool for the thermal performance of a hybrid photo-voltaic thermal solar collector

by Marcel van den Berg o Supervisor TU/e: dr. C.W.M. van der Geld o Supervisor UFSC: dr. L. Tachon

A thesis presented to the Technical University of Eindhoven in fulfillment of the Master of Science Degree in Mechanical Engineering

February 2016

“We don’t make mistakes, we just make happy accidents.”

Robert Norman ”Bob” Ross (1942-1995)

Abstract A recent surge of global renewable energy source exploration has led to the exploration of solar energy. A popular solar-to-electricity technology today is the hybrid photovoltaic thermal solar collector. In the situation that the solar collector is exposed to the sun, the photo-voltaic cell is able to convert a part of the intercepted radiation directly into electricity. The remainder of the intercepted radiation is transmitted to the thermal system and absorbed as heat. The thermal system is able discharge the heat, that will improve the electric efficiency of the photo-voltaic cell. Considering the variety of applications in a domestic household and the capabilities of the hybrid photovoltaic thermal solar collector, one is able to tweak the performance of the collector in a desirable way. As a consequence, the thermal system of the hybrid photo-voltaic thermal solar collector needs to be investigated. A numerical tool is developed that is able to predict the thermal performance of the thermal system in a hybrid photo-voltaic thermal solar collector for specified conditions. The thermal performance is classified by (1) the temperature of the photo-voltaic cells mounted on the top pane of the solar collector, (2) the amount of heat that is transferred through the bottom pane of the solar collector and (3) the quantity of heat that is absorbed by the fluid inside the thermal system and leaves through the exit of the solar collector. The specified conditions constitute the following non-dimensional quantities, the Reynolds, Grashof, Rayleigh, Prandtl and Biot number. The numerical tool is tested for a single geometry and specified conditions to show that it is able to predict the velocity flow and temperature field inside the thermal system. A qualitative comparison with similar studies from the literature revealed that the prediction is in-line with the results obtained from other authors.

Acknowledgements I have had the privilege to perform the presented work in Florianopolis, a beautiful island on the south-east coast of Brazil. I would like to thank Dr. Loic Tachon for providing me with the opportunity and guidance to perform this work. I would like to thank Dr. Cees van der Geld to grant me the possibility to perform this work. I would like to thank prof. Julio Cesar Passos for providing me with a workspace and the included facilities that I was able to use. I would like to thank Estevan Tavares, whom I had discussions with that helped me to developed a more profound understanding of subjects treated in this thesis. A special heartfelt appreciation towards Sheila Rabanal, for being a wonderful person and whose unconditional support has made this possible.

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Contents Abstract

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Acknowledgements

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List of Figures

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List of Tables

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List of Acronyms

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Physical Constants

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List of Symbols

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1 Introduction

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2 Validating the Velocity Field in the Lid-Driven Square Cavity. 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Lid-Driven Square Cavity Flow Problem . . . . . . . . . . . . . . . 2.3 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Modified Quadratic Upstream Interpolation for Convective Kinematics (QUICK) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Pressure-Velocity Coupling . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Mixed Convection in a Lid-Driven Square Cavity. 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 3.2 Governing Equations . . . . . . . . . . . . . . . . . . 3.3 Boundary Conditions . . . . . . . . . . . . . . . . . . 3.4 Discretization of the Energy-Equation . . . . . . . . 3.5 Results and Discussion . . . . . . . . . . . . . . . . . 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . .

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4 Natural Convection in a Square Cavity. 38 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 iv

Contents 4.2 4.3 4.4 4.5

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A Detailed Simulation Data A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Validation Model Velocity Flow Field . . . . . . . . . . . . . . . . . . . . . A.2.1 Central Differencing Scheme Discretization . . . . . . . . . . . . . A.3 The Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1 The Tested Pressure-Velocity Coupling Algorithms . . . . . . . . . A.3.1.1 The SIMPLE algorithm . . . . . . . . . . . . . . . . . . . A.3.1.2 The SIMPLEC algorithm . . . . . . . . . . . . . . . . . . A.3.1.3 The SIMPLER algorithm . . . . . . . . . . . . . . . . . . A.3.1.4 The SIMPLEX algorithm . . . . . . . . . . . . . . . . . . A.3.1.5 The PRIME algorithm . . . . . . . . . . . . . . . . . . . A.3.2 The Velocity Field Distribution Compared to a Literature Source . A.3.3 The Vortex Positions Compared to a Literature Source . . . . . . . A.3.4 Graphical Representation of the Uncertainty for the Five Algorithms A.4 Validation Model Mixed Convection . . . . . . . . . . . . . . . . . . . . . A.4.1 The Inner-Iterations in the Mixed Convection Model . . . . . . . . A.4.2 The Outer-Iterations in the Mixed Convection Model . . . . . . . A.4.3 The Velocity Flow Field Distribution in the Mixed Convection Model A.4.4 The Application of a Non-Uniform Grid . . . . . . . . . . . . . . . A.5 Validation Model Natural Convection . . . . . . . . . . . . . . . . . . . . A.5.1 The Inner-Iterations in the Natural Convection Model . . . . . . .

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Governing Equations . Boundary Conditions . Results and Discussion Conclusion . . . . . .

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Hybrid Photo-Voltaic Thermal Solar Collector Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Solar Collector . . . . . . . . . . . . . . . . . . . . . . . . Mixed Convection in the Solar Collector . . . . . . . . . . . . 5.3.1 Governing Equations . . . . . . . . . . . . . . . . . . . 5.3.2 Boundary Conditions . . . . . . . . . . . . . . . . . . Natural Convection in the Solar-Panel . . . . . . . . . . . . . 5.4.1 Governing Equations . . . . . . . . . . . . . . . . . . . 5.4.2 Boundary Conditions . . . . . . . . . . . . . . . . . . Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Mixed Convection . . . . . . . . . . . . . . . . . . . . 5.5.1.1 The Velocity Flow Field . . . . . . . . . . . . 5.5.1.2 The Temperature Field Along Top-Side Solar 5.5.1.3 Heat Transfer . . . . . . . . . . . . . . . . . 5.5.2 Natural Convection . . . . . . . . . . . . . . . . . . . 5.5.2.1 The Velocity Flow Field . . . . . . . . . . . . 5.5.2.2 The Temperature Field Along Top-Side Solar 5.5.2.3 Heat Transfer . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

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A.6 The Solar Collector . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6.1 Thermal Efficiency . . . . . . . . . . . . . . . . . . . . . . . . A.6.2 Flow Phenomena in Mixed Convection Flow Solar Collector . A.6.3 Flow Phenomena in Natural Convection Flow Solar Collector

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B Non-Uniform Grid 112 B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 B.2 Analytical Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 C Grid Dependency C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Uniform Grid . . . . . . . . . . . . . . . . . . . . . . . . C.2.1 RE in a Uniform Grid . . . . . . . . . . . . . . . C.2.2 RE for Case-Study I . . . . . . . . . . . . . . . . C.2.3 RE for Case-Study II . . . . . . . . . . . . . . . C.3 Non-Uniform Grid . . . . . . . . . . . . . . . . . . . . . C.3.1 RE in a Non-Uniform Grid . . . . . . . . . . . . C.3.2 RE for Test-Case II . . . . . . . . . . . . . . . . C.3.3 RE for Test-Case III . . . . . . . . . . . . . . . . C.3.4 RE for Mixed Convection in the Solar Collector . C.3.5 RE for Natural Convection in the Solar Collector

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117 . 117 . 118 . 118 . 119 . 121 . 124 . 124 . 125 . 127 . 129 . 130

D Analytical Solutions for the Solar Collector 132 D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 D.2 Two-Dimensional Laminar Flow Between Two Parallel Plates . . . . . . . 132 D.3 Two-Dimensional Convection-Diffusion Temperature Field in Parallel Plate Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 E Example MATLAB code

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Bibliography

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List of Figures 1.1

2.1 2.2

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2.4

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2.6

3.1 3.2

3.3

4.1

An artist impression of a hybrid photo-voltaic thermal solar collector that is exposed to solar radiation [1]. The photo-voltaic cell - attached to the top of the solar collector - will convert a part of the intercepted radiation in electricity, the remainder is absorbed as heat and transferred to the fluid, passing below through the thermal system. . . . . . . . . . . . . . . The lid-driven cavity flow problem and its boundary conditions. . . . . . A forward staggered 5x5 grid arrangement with artificial boundaries indicated with a light-blue color and the adjacent first physical boundary indicated by a light-red color. The black asterisks [∗] indicate the main control volume node, the red horizontal arrows [→] indicate the horizontal velocity node and the green vertical arrows [↑] indicate the vertical velocity node. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The laminar streamline solution for four different Reynolds numbers inside the square-cavity where the top-lid is driven to the positive horizontal direction. The flow field is obtained with the SIMPLE-algorithm and QUICK-interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The solution of the flow field for the x ˜-component of the Navier-Stokes through the geometric midsection of the cavity at x ˜ = 0.5. The blue triangles indicate the respective data of the velocity profiles taken from Ghia et al. [2]. Solution is obtained for a Re = 100 and QUICK-interpolation. ˜ -component of the Navier-Stokes The solution of the flow field for the y through the geometric midsection of the cavity at y˜ = 0.5. The blue triangles indicate the respective data of the velocity profiles taken from Ghia et al. [2]. Solution is obtained for a Re = 100 and QUICK-interpolation. Pressure distribution along the lid of the cavity for the five pressurevelocity coupling schemes and a Reynolds of Re = 1000. . . . . . . . . . .

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Boundary conditions for the mixed convection square-cavity flow. . . . . . 28 Isothermal contour-lines for different Richardson (Ri) numbers from l-r Re = 1, Re = 10, Re = 100 and Re = 1000 and a constant- Grasshof number Gr = 104 and Prandtl number P r = 0.71. Results are obtained with a 100x100 uniform computational grid, the SIMPLE-algorithm for pressure-velocity coupling and QUICK-scheme spatial interpolation. . . . 32 The Nusselt number (Nu) for different Richardson (Ri) numbers from l-r Re = 1, Re = 10, Re = 100 and Re = 1000 and a constant- Grasshof number Gr = 104 and Prandtl number P r = 0.71 along the top-side of the cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Boundary conditions for the natural convection square cavity flow. . . . . 41 vii

List of Figures

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4.2

The flow and temperature distribution for different Rayleigh (Ra) numbers from top to bottom Ra = 103 , Ra = 104 , Ra = 105 and Ra = 106 and a constant Prandtl number P r = 0.71. Shown results are obtained by the PRIME algorithm and QUICK spatial interpolation on a non-uniform (100x100) grid configuration. . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1

A typical flat plate analysis where a fluid enters at the base, is consequently driven through the channel due to an external pressure gradient or thermal imposed condition and is being pushed towards the exit of the channel. The domestic application of a series of hybrid solar collectors located inside an ordinary livingroom [1]. The hybrid solar collector is integrated into a double glazed window where the solar cells convert a portion of the solar energy into an electric current and the thermal system enables the redirection of any absorbed heat. This configuration of the solar collector has the potential to reinforce the air-conditioning system as it is able to discharge the incoming heat from the ambient by virtue of the incorporated thermal system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The geometry and dimensional boundary conditions for the mixed- and natural convection solar collector. The inclination angle γ is given parallel with respect to the direction of the acceleration of gravity. The prescribed velocity of the x-component at the inlet is distinguished for the two models. For mixed convection (MC) there is an imposed uniform inlet velocity u = Uin , while for the natural convection (NC) case there is a zero gradient active ∂u/∂x = 0 allowing in- and outflow of the fluid. The positions x1 , x2 , x3 and x4 indicate the location of the PV cells and L indicates the length of the channel where L = ϱH, where ϱ is the aspect ratio of the channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity vector field for a constant Grashof number Gr = 104 , two different Reynolds numbers Re = 1 and Re = 10, for three different inclination angles γ = 0, γ = π/4 and γ = π/2, a constant Biot number Bi = 5, an ambient temperature θ∞ = θc , heat flux intensity ξ = 0.1 and a geometric aspect ratio ϱ = 20. The top-side is marked with two red lines that indicate the position of the PV modules. The flow field is furthermore clarified with stream vertices - indicated by black lines - that emphasize the direction of the flow. . . . . . . . . . . . . . . . . . . . . . . . . . . The above figure presents the temperature distribution over the wall on the top-side of the solar collector. There are two PV modules mounted on the top-side of the collector inducing a local higher heat flux and as a result causing a local higher temperature at the position of the modules. Results are presented for three simulated Reynolds number Re = 1, Re = 10 and Re = 100, three inclination angles γ = 0, γ = π/4 and γ = π/2, two Grashof numbers Gr = 103 and Gr = 104 , a constant Biot number Bi = 5, an ambient temperature θ∞ = θc , heat flux intensity ξ = 0.1 and a geometric aspect ratio ϱ = 20. . . . . . . . . . . . . . . . . . . . . . .

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List of Figures 5.6

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Velocity vector field for a constant Grashof number Gr = 104 , two different Reynolds numbers Re = 1 and Re = 10, for three different inclination angles γ = 0, γ = π/4 and γ = π/2, a constant Biot number Bi = 5, an ambient temperature θ∞ = θc , heat flux intensity ξ = 0.1 and a geometric aspect ratio ϱ = 20. The top-side is marked with two red lines that indicate the position of the PV modules. The flow field is furthermore clarified with stream vertices - indicated by black lines - that emphasize the direction of the flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 The above figure presents the temperature distribution over the wall on the top-side of the solar collector. There are two PV modules mounted on the top-side of the collector inducing a local higher heat flux and as a result causing a local higher temperature at the position of the modules. Results are presented for two simulated Rayleigh numbers Ra = 103 and Ra = 104 , three inclination angles γ = 0, γ = π/4 and γ = π/2, a constant Biot number Bi = 5, an ambient temperature θ∞ = θc , heat flux intensity ξ = 0.1 and a geometric aspect ratio ϱ = 20. . . . . . . . . . . . 71 The cross-sectional horizontal velocity amplitude at x ˜ = 5 is presented, just underneath the first PV module. Results are presented for two simulated Rayleigh numbers Ra = 103 and Ra = 104 , three inclination angles γ = 0, γ = π/4 and γ = π/2, a constant Biot number Bi = 5, an ambient temperature θ∞ = θc , heat flux intensity ξ = 0.1 and a geometric aspect ratio ϱ = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

A.1 The solution of the flow field for the x ˜- and y ˜ component of the NavierStokes through the geometric midsection of the cavity indicated by blue and black coloring of the axis, respectively. The blue and black triangles indicate the respective data of the velocity profiles taken from the literature. Solution is obtained for a Re = 400 and QUICK-interpolation. . . . . . . A.2 The solution of the flow field for the x ˜- and y ˜ component of the NavierStokes through the geometric midsection of the cavity indicated by blue and black coloring of the axis, respectively. The blue and black triangles indicate the respective data of the velocity profiles taken from the literature. Solution is obtained for a Re = 1000 and QUICK-interpolation. . . . . . . A.3 The solution of the flow field for the x ˜- and y ˜ component of the NavierStokes through the geometric midsection of the cavity indicated by blue and black coloring of the axis, respectively. The blue and black triangles indicate the respective data of the velocity profiles taken from the literature. Solution is obtained for a Re = 10000 and QUICK-interpolation. (Laminar modelling). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 The mean absolute percentage uncertainty δE for every pressure-velocity coupling scheme compared to the literature benchmark model [2] for the solution of the Navier-Stokes inside the square-cavity. Results are given as a function of the Reynolds number using QUICK interpolation. The x ãnd y˜ of the Navier-Stokes are given on the left and right side, respectively. A.5 The mean absolute percentage uncertainty δE for every pressure-velocity coupling scheme compared to the literature benchmark model [2] for the solution of the Navier-Stokes inside the square-cavity. Results are given as a function of the Reynolds number using Power-Law interpolation. The x ãnd y˜ of the Navier-Stokes are given on the left and right side, respectively.

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List of Figures A.6 The outer-iterations (time-residual) for every simulated Richardson number. In the presented order of Richardson numbers the Reynolds varied from Re = 1, Re = 10, Re = 100 and Re = 1000 for a constant Grashof number Gr = 104 and a constant Prandtl number P r = 0.71. . . . . . . A.7 The velocity flow field distribution for different Richardson (Ri) numbers. In the presented order of Richardson numbers the Reynolds varied from Re = 1, Re = 10, Re = 100 and Re = 1000 for a constant Grashof number Gr = 104 and a constant Prandtl number P r = 0.71. . . . . . . A.8 The Nusselt number on the top-lid of the square cavity for a Richardson number of Ri = 100 (Re = 10, Gr = 104 , P r = 0.71). It is the equivalent of figure 3.3 except obtained in a non-uniform applied physical domain. The difference in percentages between a uniform and non-uniform applied domain with respect to the literature model is shown on the right. . . . . A.9 The inner-iterations for the pressure-velocity coupling algorithms during the convergence process for a Richardson number of Ri = 100 (Re = 10, Gr = 104 , P r = 0.71) in the non-uniform applied physical domain. Each hump during the convergence is the onset point of a new time-step and the convergence within that time-step until the set threshold value. . . . A.10 The inner-iterations that incorporate the averaged residual between the u ˜- and v˜-velocity and temperature residuals for the natural convection model until the set criteria for the averaged residual for these latter mentioned equations is obtained. From the top to the bottom it shows the Ra = 103 , 104 , 105 and 106 , respectively. The left and right show the inner-iterations of the predominantly implicit and explicit algorithms, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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B.1 The physical and computational plane in a 4x4 staggered grid arrangement. See also figure 2.2 for more detailed information regarding the individual symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 B.2 A cut-out part from the physical domain to clarify three types of control volumes in a staggered grid notation for a non-uniform grid. The position of the main control volume (I, J) is marked with a rectangular shaped box covered with diagonal hatched lines [///]. The main control volume is accompanied with its [u]-velocity control volume (i, J) in the rectangular box in red on the left-side and the [v]-velocity control volume (I, j) in the rectangular box in green on the right-side. The convective fluxes are positioned between two adjacent velocity control volumes indicated with bullets [•]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 C.1 The estimated absolute discretization error for the non-dimensional velocity at the midsection of the square cavity as a function of the y˜ coordinate and a Reynolds number of Re = 1000. . . . . . . . . . . . . . . . . . . . . 121 C.2 The estimated absolute discretization error of the local Nusselt number along the top-side of the square cavity as a function of the x ˜ coordinate and a Richardson number of Ri = 10000. . . . . . . . . . . . . . . . . . . 122 C.3 The maximum [△], averaged [] and minimum [▽] Nusselt number along the driven lid of the square cavity for a Richardson number of Ri = 100. Spatial interpolation performed with the QUICK scheme and pressurevelocity coupling with the SIMPLEC algorithm. Applied grid densities (25x25), (50x50), (100x100) and (200x200). . . . . . . . . . . . . . . . . . 123

List of Figures C.4 The representative local control volume size h for grid levels N1 , N2 and N3 being compressed near the borders and stretched in de mid-section of the physical domain. The minimum control volume size is for the three types of grid levels (h1 , h2 , h3 ) = (2.99e-3, 6.15e-3, 13e-3). The zeroderivative of the control volume size at the boundaries are the artificial boundaries, having the same size as the first adjacent control volume within the physical domain. . . . . . . . . . . . . . . . . . . . . . . . . . C.5 The estimated absolute discretization error of the local Nusselt number along the top-side of the square cavity as a function of the x ˜ coordinate and a Richardson number of Ri = 10000. . . . . . . . . . . . . . . . . . C.6 The estimated absolute discretization error of the local Nusselt number along the heated-side of the square cavity as a function of the y˜ coordinate and a Rayleigh number of Ra = 106 and a Prandtl number of P r = 0.71. C.7 The local Nusselt number on the heated side of the square cavity for Ra = 106 and P r = 0.71. The local Nusselt number is shown for a non-uniform (100x100) grid on the left, and a uniform (200x200) grid on the right. . C.8 The estimated absolute discretization error of the local Nusselt number along the bottom surface of the solar collector. The result is presented for a Rayleigh numbers of Ra = 104 , an inclination angle of γ = π/4, a constant Biot number Bi = 5, an ambient temperature θ∞ = θc , heat flux intensity ξ = 0.1 and a geometric aspect ratio ϱ = 20. . . . . . . . . . .

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D.1 The numerical and analytical solution (both at x ˜ = L) of the crosssectional velocity is presented. The numerical solution approximates a mixed convection flow where the Reynolds number and Grashof number are 102 and 104 , respectively. The ratio of dimensionless numbers dictate that forced convection is decisive, the aspect ratio of the channel induces a fully developed flow condition at x ˜ = L and the inclination angle determines the exclusive horizontal flow. The analytical solution of eq. D.5 and eq. D.7 are given. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 D.2 The cross-sectional temperature field distribution is shown for the parallel plate configuration at x = L/2. The solutions shown are obtained for two different Reynolds numbers, Re = 1 is shown on the left and Re = 100 is shown on the right. In both the analyzes the Prandtl number has been kept constant P r = 0.71, likewise for the aspect ratio of the channel that has been set to ϱ = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

List of Tables 2.1

The mean absolute percentage uncertainty δE for every pressure-velocity coupling scheme as a function of the Reynolds number compared to the benchmark model. Spatial interpolation performed with the QUICK and Power-Law scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1

The mean absolute percentage uncertainty δE for every pressure-velocity coupling scheme as a function of the Richardson number compared to the literature model [3]. Results obtained for the given Richardson numbers and in the same order the Reynolds number varied from Re = 1, Re = 10, Re = 100 and Re = 1000 with a constant- Grasshof number Gr = 104 and Prandtl number P r = 0.71. . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1

Shown results reveal five analyzed quantities, the maximum horizontal and vertical velocity defined on their respective perpendicular midline within the square cavity and the averaged, maximum and minimum Nusselt number on the heated wall of the square cavity. Results are obtained by QUICK interpolation on a non-uniform (100x100) grid where grid-refinement is applied near the boundaries. Two models from the literature De Vahl Davis [4] and Tian et al. [5] are shown in the last two columns for comparison. In their uniform applied grid they solved their equations with a (81x81) and (60x60) cells configuration, respectively. . . . . . . . . . . . . 43

5.1

The above table presents the averaged N u0 , maximum N umax and minimum N umin Nusselt number of the wall on the lower-side of the solar collector. Results are presented for three simulated Reynolds number Re = 1, Re = 10 and Re = 100, three Grashof numbers Gr = 102 , Gr = 103 and Gr = 104 , three inclination angles γ = 0, γ = π/4 and γ = π/2, a constant Biot number Bi = 5, an ambient temperature θ∞ = θc , heat flux intensity ξ = 0.1 and a geometric aspect ratio ϱ = 20. . . . . . . . . . . . 66 The thermal efficiency is shown for different inclination angles, Reynolds and Grashof numbers. The thermal efficiency is a quantity that indicates the capacity of the collector to discharge heat. The numerical results shown are given for a constant Prandtl number P r = 0.71, a constant Biot number Bi = 5 and an aspect ratio of ϱ = 20. . . . . . . . . . . . . . . . 67 The above table presents the averaged N u0 , maximum N umax and minimum N umin Nusselt number of the wall on the lower-side of the solar collector. Results are presented for two simulated Rayleigh numbers Ra = 103 and Ra = 104 , three inclination angles γ = 0, γ = π/4 and γ = π/2, a constant Biot number Bi = 5, an ambient temperature θ∞ = θc , heat flux intensity ξ = 0.1 and a geometric aspect ratio ϱ = 20. 73

5.2

5.3

xii

List of Tables 5.4

xiii

The thermal efficiency is shown for three inclination angles γ = 0, γ = π/4 and γ = π/2, and two Rayleigh numbers Ra = 103 and Ra = 104 . The thermal efficiency is a quantity that indicates the capacity of the collector to discharge heat. The numerical results shown are given for a constant Prandtl number P r = 0.71, a constant Biot number Bi = 5 and an aspect ratio of ϱ = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

A.1 The position, mean absolute error and the relative absolute mean uncertainty percentage of the centres of the primary vortex and the two secondary vortices for varying Reynolds number between Re = 100 − 1000 compared to literature data from Idris et al. [6]. . . . . . . . . . . . . . . . 91 A.2 The total amount of inner-iterations until convergence is obtained for all time-steps summed up for every pressure-velocity coupling algorithm as a function of the four simulated Richardson numbers. In the presented order of Richardson numbers the Reynolds varied from Re = 1, Re = 10, Re = 100 and Re = 1000 for a constant Grashof number Gr = 104 and a constant Prandtl number P r = 0.71. . . . . . . . . . . . . . . . . . . . . . 93 A.3 The total amount of inner-iterations until convergence is obtained for all time-steps summed up for every pressure-velocity coupling algorithm as a function of the four simulated Rayleigh numbers and a constant Prandtl number P r = 0.71. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 C.1 Grid densities in a uniform configuration applied to obtain the numerical solution in the lid-driven cavity flow. . . . . . . . . . . . . . . . . . . . . . 118 C.2 Discretization error estimates according to the Richardson extrapolation (RE) for a Reynolds number of Re = 1000. The local order of accuracy p is shown with the Richardson extrapolation discretization error REf21ine and the relative error e21 ˜ coordinate. a in percentages with respect to the y The absolute uncertainty of the non-dimensional velocity u ˜ is given by |e21 |.120 C.3 Discretization error estimates according to the Richardson extrapolation (RE) for a Richardson number of Ri = 10000. The local order of accuracy p is shown with the Richardson extrapolation discretization error REf21ine and the relative error e21 ˜ coordinate. a in percentages with respect to the x The absolute uncertainty of the Nusselt number (Nu) is given by |e21 |. . . 122 C.4 Three grid levels with its grid densities in a non-uniform configuration applied to obtain the numerical solution in the square cavity flow with the global representation of the control volume size h. . . . . . . . . . . . . . . 124 C.5 Discretization error estimates according to the Richardson extrapolation (RE) for a Richardson number of Ri = 10000 on a non-uniform grid. The local order of accuracy p is shown with the Richardson extrapolation discretization error REf21ine and the relative error e21 a in percentages with respect to the x ˜ coordinate. The absolute uncertainty of the Nusselt number (Nu) is given by |e21 |. . . . . . . . . . . . . . . . . . . . . . . . . . 126 C.6 Discretization error estimates according to the Richardson extrapolation (RE) for a Rayleigh number of Ra = 106 and a Prandtl number of P r = 0.71 on a non-uniform grid. The local order of accuracy p is shown with the Richardson extrapolation discretization error REf21ine and the relative ˜ coordinate. The absolute error e21 a in percentages with respect to the x uncertainty of the Nusselt number (Nu) is given by |e21 |. . . . . . . . . . . 128

List of Tables

xiv

C.7 The global representative control volume size h for three grid levels with their respective grid densities in a non-uniform configuration applied to obtain the numerical solution in the rectangular shaped domain of the solar collector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 C.8 The result of the Richardson extrapolation in terms of the local Nusselt number for the bottom-part of the solar collector. The result is obtained for a Rayleigh number of Ra = 104 and a Prandtl number of P r = 0.71 on a non-uniform rectangular grid. The quantities given are the local order of accuracy p, the Richardson extrapolation discretization error REf21ine in percentages, the relative error e21 a in percentages and the absolute uncertainty of the local Nusselt number given by |e21 |. . . . . . . . . . . . . . . 131

List of Acronyms General CV

Control Volume

Eq

Equation

NS

Navier Stokes

PRIME

Pressure Implicit Momentum Explicit

PVT

Photo Voltaic Thermal

QUICK

Quadratic Upwind Interpolation for Convective Kinematics

RE

Richardson Extrapolation

SIMPLE

Semi Implicit Method for Pressure Linked Equations

SIMPLEC

Semi Implicit Method for Pressure Linked Equations Consistent

SIMPLER

Semi Implicit Method for Pressure Linked Equations Revised

SIMPLEX

Semi Implicit Method for Pressure Linked Equations eXtrapolated

SOR

Successive Over Relaxation solver

Dimensionless

Numbers

Definition

Bi

Biot number

eq. 5.14

Gr

Grashof number

eq. 3.11

Nu

Nusselt number

eq. 3.30

Pe

Peclet number

eq. A.11 - A.12

Pr

Prandtl number

eq. 3.11

Ra

Rayleigh number

eq. 4.1

Re

Reynolds number

eq. 2.6

Ri

Richardson number

eq. 3.10

xv

Physical Constants Acceleration of gravity

xvi

g

=

9.81 ms−2

List of Symbols A

area

m2

aN,E,S,W,P

discretization coefficient main control volume

-

an,e,s,w,p

discretization coefficient velocity control volume

-

b

discretized source term

-

cp

specific heat

J(kgK)−1

D

diffusive conductance

-

du˜,˜v

d-coefficient

-

e21 a

relative error

%

F

convective mass flux

-

˜ G

total non-dimensional solar radiation on surface

-

H

characteristic length scale

m

h

non-dimensional representative control volume size

-

k

thermal conductivity

W (mK)−1

L

characteristic length scale

m

Min

incoming mass flux

kg s−1

Mout

outgoing mass flux

kg s−1

N

number of nodes

m

P

pressure

Pa

p

order of interpolation

-

p˜

non-dimensional pressure

-

˜ Q

collected non-dimensional heat exiting channel

-

q

heat flux

W m−2

REf21ine

Richardson extrapolation discretization error

%

S

source term

-

T

temperature

K xvii

Symbols

xviii

Tc

cold temperature

K

Th

hot temperature

K

Tp

temperature main control volume

m

T∞

Ambient temperature

K

t

time

s

t˜

non-dimensional time

-

U

velocity amplitude

m s−1

u

velocity

m s−1

u ˜

non-dimensional velocity

-

v

velocity

m s−1

v˜

non-dimensional velocity

-

x

dimensional length

m

x ˜

non-dimensional length

-

y

dimensional length

m

y˜

non-dimensional length

-

α

thermal diffusion coefficient

m s−2

α

constant that defines the location of grid refinement

-

β

stretching factor

-

β

thermal expansion coefficient

K −1

∆t

time step

s

δE

mean absolute percentage uncertainty

%

ϵmass

mass conservation criteria

-

ϵmass,s

mass source criteria

-

ϵmom

mass conservation criteria

-

ϵss

steady state residual

-

Γx,y

diffusion coefficient

-

γ

inclination angle

rad

ζ

heat intensity fraction

-

η

computational y˜ dimension

-

η˜th

thermal efficiency

-

θ

non-dimensional temperature

-

λ

fraction of interpolation

-

Symbols

xix

µ

dynamic viscosity

Ns (m−2 )

ν

kinematic viscosity

m2 s−1

ξ

computational x ˜ dimension

-

ρ

mass density

kg m−3

ϱ

aspect ratio L/H

-

ϕ

general transport variable

-

n

iteration counter

-

0

previous time step

-

∗

pseudo

-

′

correction

-

N, n

north

-

E, e

east

-

S, s

south

-

W, w

west

-

P, p

main

-

Superscript

Subscript

Chapter 1

Introduction The content of this work presents a research dedicated towards a new application of a hybrid PhotoVoltaic-Thermal (PVT) solar collector and has been carried out at the LEPTEN/Boiling laboratory at the faculty of Mechanical Engineering of the Federal University of Santa Catarina in Florianopolis, Brazil. The project originates from the company SoluzEnergia [1], a starting spin-off company from the Federal University of Santa Catarina seated in Florianopolis that focuses on the development of solar collector solutions for resident households throughout Brazil. The project is inspired by a recent surge of global renewable energy source exploration, providing so-called clean and environmental friendly alternatives instead of the conventional carbon-dioxide emitting fossil-fuel energy sources [7]. Among these renewable energy sources is solar energy that due to its abundance and even distribution in nature is a popular alternative [8]. The leading solar electric (solar-to-electricity) technology today is the solar photo-voltaic technology, consisting of about 98[%] of the global solar generation capacity in 2013 [9]. The photo-voltaic cell or module is an electric device that typically converts 6 - 20[%] of the intercepted radiation directly into electricity [10]. The remainder of the intercepted solar radiation absorbed by the photo-voltaic cell is transmitted as heat [11]. In a photo-voltaic solar collector application - typically a rectangular flat shaped surface where the photo-voltaic cells are mounted to the side of the collector exposed to the sun - the conversion of heat can lead to a significant increase of temperature of the solar cell [12]. As the performance in terms of the electrical efficiency of such a photo-voltaic module decreases with an increasing temperature [13], it is required to somehow control the temperature of these mounted modules to 1

Chapter 1. Introduction

2

optimize its electric efficiency. A measure to accomplish such thermal control is than by the utilization of a thermal system, integrated with the photo-voltaic solar collector into a single system, resulting in the foundation of the hybrid photo-voltaic thermal solar collector. An artist impression of the considered solar collector is given in figure 1.1.

Figure 1.1: An artist impression of a hybrid photo-voltaic thermal solar collector that is exposed to solar radiation [1]. The photo-voltaic cell - attached to the top of the solar collector - will convert a part of the intercepted radiation in electricity, the remainder is absorbed as heat and transferred to the fluid, passing below through the thermal system.

The considered solar collector will be designed - in a parallel study - for a domestic household, where it will be employed as an ordinary double glazed window, with the difference of (1) having additional photo-voltaic cells placed on-top of the glass pane and (2) the space between the two glass panes that utilizes the thermal system. The photo-voltaic cells will provide electricity to the household, while the transparent spaces in between the solar cells will provide illumination for the room where the solar panel is employed. Considering the variety of applications in a domestic household and the capabilities of the studied solar collector, one is able to tweak the performance of the collector in a desirable way. As a result, three quantities in the thermal system need to be investigated. The first one is the temperature of the top pane on which the photovoltaic cells are mounted such that a certain electrical efficiency can be guaranteed. The second one is the amount of heat that is being transferred to the ambient through the bottom pane of the solar collector such that in a house, a room can either be heated up or cooled down. The third, and last one, is the amount of heat that passes through the exit of the channel that has the potential for re-usage in a variety of domestic applications. The goal of the present work is to set-up a numerical tool that is able to predict fluid


3

and thermal flow through the aforementioned thermal system for specified conditions in which an optimal geometry can be selected based upon the thermal performance of the system. The optimal geometry is determined by (1) the photovoltaic cell that operates within a certain temperature range, (2) the heat transfer through the bottom part of the thermal system and (3) the quantity of heat that is absorbed by the fluid going through the exit of the thermal system. In order to simulate fluid flow in the hybrid photo-voltaic thermal solar collector a numerical code is developed. Unfortunately, there is no means to validate this code using experimental data. However, there are ways to validate the code by a small reformulation of the actual problem towards known test cases that include the same type of flow phenomena occurring inside the thermal system. Due to the former reasoning the content of this work can be divided into two parts, it starts with, 1. the set-up and a step-by-step extension of a two-dimensional numerical code that attempts to validate the flow phenomena - similar to what is occurring inside the thermal system of the hybrid photo-voltaic thermal solar collector - by the utilization of a finite volume method [14] and, 2. the application of the former set-up numerical code towards the geometry and conditions of a specified hybrid Photo-Voltaic Thermal solar collector. The first part is spread out over chapter 2, 3 and 4. That part is dedicated towards three constituent processes of the numerical code - hence the three chapters - that when put together, is able to capture the flow phenomena that takes place in the solar collector. The three constituent processes that are considered are, (1) the coupling of the continuity and momentum equations, resulting in the velocity field, (2) the incorporation of a prescribed flow condition and a buoyancy force, resulting in a mixed convection flow, and (3) the incorporation of only a buoyancy force without the presence of a driven boundary, resulting in a pure natural convection flow. In chapter 2, the validation of the numerical code for the prediction of the velocity field is covered. The validation is accomplished by setting-up the numerical code to the well-known test case, the liddriven cavity flow problem [2][15]. Important concepts are introduced that cover, for example, the type of applied spatial interpolation scheme, method of coupling the continuity and momentum equations and convergence criteria. Chapter 3, will continue with the lid-driven cavity flow problem but incorporates the energy equation to predict


4

a mixed convection flow that will be predominantly characterized by the Reynolds (Re) and Grashof (Gr) number. During the validation it was found that the adopted uniform grid formation has difficulties describing essentially natural convection flows (Gr >> Re) near the boundaries. As a consequence, a non-uniform grid based on an analytical transformation has been adopted and to solve this introduced in the last case study, chapter 4. In that chapter an analysis is performed inside a buoyant driven cavity problem, where only natural convection is considered and the flow is predominantly characterized by the Rayleigh (Ra) number. The final chapter, chapter 5 is dedicated to the hybrid PVT solar collector. The analysis of the flow inside the solar collector incorporates two distinct numerical models - analogue to the flow phenomena described in chapter 3 and chapter 4 - and focuses on the mixed- and natural convection, separately. In the solar collector this represents the flow inside the solar collector driven with or without pump, respectively. The final chapter will - after a brief problem description - introduce the suggestion for the most-likely boundary conditions that should represent the real-life solar collector and an analysis very similar to a parallel plate flow is identified. Results will be shown for a single geometry, in terms of a defined aspect ratio (ratio length and height channel), to show that the numerical tool is working and are presented for various inclination angles, Reynolds, Grashof and Rayleigh numbers.

Chapter 2

Validating the Velocity Field in the Lid-Driven Square Cavity. 2.1

Introduction

In order to simulate fluid flow in a solar panel, a numerical tool is developed. As there is no means to validate this code using experimental data, the flow phenomena that occurs inside the thermal system of the PVT solar collector is divided into subproblems. In the current chapter the development of the numerical code is elucidated for the prediction of the velocity field. In the numerical tool that will be developed in this work, the effect of the buoyancy force is incorporated by means of the Boussinesq approximation [16]. As a result, a quasiincompressible formulation can be adopted for the prediction of the velocity field. A finite volume method is adopted [14], where the velocities are to be computed by solving a discretized form of the continuity equation, called the pressure(-correction) equation [14], that is coupled to the momentum equations. This chapter will introduce five different solution algorithms, so-called pressure-velocity coupling algorithms, that attempt to establish the coupling of the continuity and momentum equations. As a consequence the numerical tool will be equipped with the pressure-velocity coupling algorithm that showed (1) to require the least amount of iterations per simulation and (2) to be most in-line with benchmark model(s) from the literature 1 . 1

In this context, a benchmark model is considered a numerical model published in the literature that can be used as a point of reference.

5

Chapter 2. Validating the Velocity Field in the Lid-Driven Square Cavity.

6

A known and often examined two-dimensional stationary flow problem is the Lid-Driven Square Cavity flow [15]. Due to its rectangular domain and the no-slip boundary conditions it is known as a test or validation case for new codes or new solution methods [15]. The benchmark results from Ghia et al. [2] are compared to the results obtained from the developed numerical code for the prediction of the velocity field. Convergence criteria have been selected in the literature and have been kept equal for all algorithms, that are based on the mass and momentum conservation in order to enforce the same converged solution for all applied algorithms and making an equal comparison [17][18]. After a brief problem definition in section 2.2, the governing equations will be written in a non-dimensional form to reduce the number of variables in the equations in section 2.3. The Navier-Stokes and the major steps in the discretization will be shown throughout the sections 2.4 - 2.5. The applied pressure-velocity coupling algorithms will be presented in section 2.6, followed by the convergence criteria in section 2.7 and closing with results and discussion in section 2.8.

2.2

The Lid-Driven Square Cavity Flow Problem

The lid-driven cavity flow problem is a fluid filled cavity that is swirled by a uniformly translating lid [15], in this case the top-lid is moving in the positive horizontal direction. The problem is unambiguous with easily applied boundary conditions [19]. The cavity flow problem is defined by eight no-slip boundary conditions at the wall, all walls have no-slip boundary conditions u = v = 0 except for the top wall which has a u = U and v = 0 velocity condition. The initial conditions are u(t = 0) = 0 and v(t = 0) = 0 everywhere on the domain. The domain is furthermore square and a schematic of the problem is shown in figure 2.1. The governing equations will be modified that will enable to solve the governing equations in its unsteady form. This will contribute to numerical stability as time advances in steps towards the steady-state instead of one leap towards the steady-state. Time integration is done with a fully-implicit first-order forward differencing scheme being unconditionally stable [14]. This is a stable choice as both the explicit and Crank-Nicolson scheme are conditionally stable

2

and additionally the intermediate time solution is of

2 Time integration in this context is said to be unconditionally stable when in the discretization of the governing equations the term multiplied with the solution of the old time-step is always positive [14].


7

u = U, v = 0

u=v=0

u(t=0) = 0 v(t=0) = 0

u=v=0

u = 0, v = 0

L

y x

Figure 2.1: The lid-driven cavity flow problem and its boundary conditions.

no concern in this regard, omitting the need of a second order time integration. Spatial integration of the convection terms are performed by a first-order Power-Law scheme [14] and a third-order spatial modified quadratic upwind QUICK scheme [20]. Two different schemes are applied for the spatial integration of the convection terms, and the current study will point out and highlight the differences and as a result will select one, that showed to be in-line most with the benchmark model. The diffusion terms are interpolated with the second-order central differencing scheme [14]. Pressure-velocity coupling is accomplished with algorithms that are based on the SIMPLE -family [21]. The obtained linear system for every discretized equation is solved with an iterative solver, the so-called Successive-OverRelaxation (SOR) solver [18].

2.3

Governing Equations

The process of writing clear and detailed numerical work often goes hand-in-hand with a significant quantity of text and equations. For that reason the scalar ϕ is introduced, but replaced later, and used for now to prevent repetition. The scalar ϕ either represents the horizontal u or vertical v velocity or the temperature T , depending on the situation.


8

It is custom to adopt the general governing equation with transport variable ϕ [14], ∂ ∂ ∂ ∂ (ρϕ) + (ρuϕ) + (ρvϕ) = ∂t ∂x ∂y ∂x

( ) ( ) ∂ϕ ∂ ∂ϕ Γx + Γy + S, ∂x ∂y ∂y

(2.1)

where ρ is the density of the fluid, S is a source term and Γ a diffusion coefficient. Mass conservation is obtained for substituting ϕ = 1 and Γ = S = 0 in the above transport equation and the Navier-Stokes is obtained when the source term S would carry the pressure gradient and possible viscous dissipation terms. To reduce the number of variables in the system, the following non-dimensionalization is introduced, x ˜=

x , L

y˜ =

y , L

u ˜=

u , U

v˜ =

v , U

p˜ =

P , ρU 2

tU t˜ = , L

(2.2)

where L is the characteristic length scale. Substituting these values into eq. 2.1 and neglecting viscous dissipation, the total set of governing equations describing the conservation of mass and momentum in x ˜ and y˜ direction is than given by, ∂u ˜ ∂˜ v + = 0, ∂x ˜ ∂ y˜ ∂u ˜ ∂ ∂ 1 ∂ + (˜ uu ˜) + (˜ vu ˜) = ˜ ∂x ˜ ∂ y˜ Re ∂ x ˜ ∂t ∂˜ v ∂ ∂ 1 ∂ + (˜ uv˜) + (˜ v v˜) = ˜ ∂x ˜ ∂ y˜ Re ∂ x ˜ ∂t

( (

∂u ˜ ∂x ˜ ∂˜ v ∂x ˜

(2.3) ) )

1 ∂ + Re ∂ y˜ 1 ∂ + Re ∂ y˜

( (

∂u ˜ ∂ y˜ ∂˜ v ∂ y˜

) −

∂ p˜ , ∂x ˜

(2.4)

−

∂ p˜ , ∂ y˜

(2.5)

)

where the y˜ direction of the Navier-Stokes is obtained analogue to the x ˜ component and the transient term of the mass conservation is dropped out due to the assumption of the quasi-incompressible fluid. The Reynolds number is for this case given by, Re =

ρU L , µ

(2.6)

being defined by the characteristic velocity U and length L.

2.4

Boundary Conditions

This section will briefly discuss the boundary conditions that define the governing equations in the lid-driven square cavity problem. Boundary conditions need to be defined for the continuity and momentum equations. The boundary conditions for the cavity


9

flow as defined in figure 2.1 are written to a non-dimensional form using eq. 2.2 with respect to the origin (0, 0) at the left-lower corner of the domain. The new dimensionless boundary conditions become, (˜ x = 0, y˜),

u ˜ = 0,

v˜ = 0,

(2.7)

(˜ x = 1, y˜),

u ˜ = 0,

v˜ = 0,

(2.8)

(˜ x, y˜ = 0),

u ˜ = 0,

v˜ = 0,

(2.9)

(˜ x, y˜ = 1),

u ˜ = 0,

v˜ = 0,

(2.10)

(t = 0, x ˜, y˜),

u ˜ = 0,

v˜ = 0.

(2.11)

The boundary conditions are implemented by means of artificial (fictitious) boundaries, also known as ghost-cells [18]. To clarify this terminology a schematic is shown of the computational grid in figure 2.2. The outer light-blue area are the artificial boundaries, separated from the physical domain by a blue dashed line. Three different types of control volumes are considered that overlap each other, (1) the main control volume node recognized by an asterisk in the middle, (2) the u ˜-control volumes recognized by the shown red horizontal arrows and (3) the v˜-control volumes, recognized by the green vertical arrows. To implement boundary conditions for the following arrangement, a separate equation is written for every pair of nodes crossing the borders. For example, considering the west-side of the boundary, the boundary equation for the main control volume node defined at the midsection - that is between 0.4 < y < 0.6 - is written as [18], aP ϕP = aE ϕE + b.

(2.12)

The u ˜-velocity of the considered node is located exactly on the border, i.e., the blue dashed line. It is therefor not required to write an additional equation to prescribe the no-slip condition at this point. The v˜-velocity component is not defined on the boundary, indicating an interpolation is required to make sure the no-slip condition on the border is honored. This is obtained by the following expression, vf =

vP + vE , 2

(2.13)


10

Computational mesh with: NPI = 5, NPJ = 5 1.2 1 0.8

y [-]

0.6 0.4 0.2 0 −0.2 −0.2

0

0.2

0.4 0.6 x [-]

0.8

1

1.2

Figure 2.2: A forward staggered 5x5 grid arrangement with artificial boundaries indicated with a light-blue color and the adjacent first physical boundary indicated by a light-red color. The black asterisks [∗] indicate the main control volume node, the red horizontal arrows [→] indicate the horizontal velocity node and the green vertical arrows [↑] indicate the vertical velocity node.

where vf is the prescribed velocity on the border, in this case being equal to a zero value. Substituting into eq. 2.12 results in, aP = 1,

aE = −1,

b = 2vf .

(2.14)

Additionally, the continuity equation has to be defined on the borders. However, the mass conservation is rewritten to a pressure type equation and one may wonder which appropriate boundary conditions to use to define this equation. A standard approach is to prescribe homogeneous Neumann boundary conditions wherever no-slip boundary conditions are prescribed for the flow field [22]. Inappropriate implementation of the prescribed conditions that are not compatible to the physical problem will result in a non-convergent solution [23]. For the same considered boundary on the west-side the pressure and pressure-correction equation are defined by the zero derivative with respect


11

to the normal towards the boundary, ∂ p˜ ∂ p˜′ = = 0. ∂x ˜ ∂x ˜

(2.15)

Again, substituting this value into eq. 2.12 results in, aP = 1,

aE = 1,

b = 0.

(2.16)

The remaining equations that hold the different conditions are implemented but not included in this report to prevent repetition.

2.5

Modified Quadratic Upstream Interpolation for Convective Kinematics (QUICK)

This section will treat the implementation of a third-order interpolation scheme that is applied to the convection terms of the governing equations of eq. 2.4 - 2.5. The discretized equations with the third-order interpolation scheme are obtained in an equal manner as is done for the discretized equations that are interpolated with the central differencing scheme, shown in appendix section A.2.1. The QUICK interpolation scheme is originally set-up by Leonard [24] and uses a three-point upstream-weighted quadratic interpolation for cell face values. It can be shown that the scheme has a conservative nature as all nodes are quadratically interpolated [20]. The algorithm uses a three-point interpolation function making its truncation error in the Taylor series third-order accurate. The original QUICK-scheme is however not governed by boundedness 3 . Several authors have made attempts to improve the integration scheme [20]. Hayase et al. [25] compiled the work of some of these authors and managed to write a general form of the interpolation scheme. He additionally wrote his own, having superior stability and convergence characteristics compared to the earlier schemes [20]. The general form of a property ϕ defined at a control volume face is given, according to Hayase by [25], [ ] [ ] ϕw = αw b1 ϕW + b2 ϕP + b3 ϕE + Sw+ + (1 − αw ) a1 ϕW + a2 ϕP + a3 ϕE + Sw− , (2.17) [ ] [ ] ϕe = αe a1 ϕW + a2 ϕP + a3 ϕE + Se+ + (1 − αe ) b1 ϕW + b2 ϕP + b3 ϕE + Se− , (2.18) 3

A numerical property in which the coefficients of the discretized equations are always positive [14].


12

where ϕ is any scalar, in this case ϕ = (u, v). The coefficient αi is defined by,

αi =

  1,

if Fi > 0

 0,

if Fi < 0

for the index i being i = n, e, s, w and by that taking into account the numerical property of boundedness [14]. The source terms S of eq. 2.17 - 2.18 are defined by, Sw+

1 = − ϕW W + 8

(

) ( ) 3 3 − b1 ϕW + − b2 ϕP + (−b3 )ϕE , 4 8

( ) ( ) 1 3 3 = − ϕEE + − b1 ϕE + − b2 ϕP + (−b3 )ϕW , 8 4 8 ( ) ( ) ( ) 1 3 3 Sw− = − − a1 ϕE + − a 2 ϕP + − a 3 ϕW , 8 4 8 ( ) ( ) ( ) 1 3 3 + S e = − − a 1 ϕW + − a2 ϕP + − a 3 ϕE . 8 4 8

Se−

(2.19)

(2.20) (2.21) (2.22)

The coefficients a, b in the modified interpolation scheme of Hayase et al. are given by [25], a1 = a3 = b1 = b3 = 0,

a2 = b2 = 1.

(2.23)

The above result can be implemented as a first-order upwind scheme, with a third order correction source term [20]. For a two-dimensional problem substitution of the face velocities in the earlier obtained partially discretized equation of eq. A.1 leads to, aP ϕnP = aE ϕnE + aW ϕnW + aN ϕnN + aS ϕnS + +aop ϕoP + b − ∇P ϕ ∆V,

(2.24)

where n is the converged scalar value at level n. The source term b incorporating the correction terms is defined by the old converged scalar values at level (n - 1),

n−1 n−1 b = SP ∆V + bP ϕn−1 + bEE ϕn−1 + bW W ϕW P EE + bE ϕE W + ... n−1 n−1 n−1 + bS ϕSn−1 . (2.25) + bSS ϕSS ...bW ϕn−1 W + bN N ϕN N + bN ϕN

The coefficients can be shown to be given by, aE = De + ⟨−Fe , 0⟩,

aW = Dw + ⟨Fw , 0⟩,

(2.26)

Chapter 2. Validating the Velocity Field in the Lid-Driven Square Cavity. aN = Dn + ⟨−Fn , 0⟩,

aS = Ds + ⟨Fs , 0⟩,

13 (2.27)

where ⟨...⟩ determines the maximum value of the argument. The main coefficient aP is defined by eq. A.5 and the terms inside the source b consisting out of the third order correction terms are defined by, 2 1 3 bE = − ⟨Fe , 0⟩ − ⟨−Fe , 0⟩ + ⟨−Fw , 0⟩, 8 8 8

(2.28)

3 2 1 bW = − ⟨−Fw , 0⟩ − ⟨Fw , 0⟩ + ⟨Fe , 0⟩, 8 8 8

(2.29)

3 2 1 bN = − ⟨Fn , 0⟩ − ⟨−Fn , 0⟩ + ⟨−Fs , 0⟩, 8 8 8

(2.30)

3 2 1 bS = − ⟨−Fs , 0⟩ − ⟨Fs , 0⟩ + ⟨Fn , 0⟩, 8 8 8

(2.31)

and the coefficients further up- and downstream are given by, 1 bEE = − ⟨−Fe , 0⟩, 8 1 bN N = − ⟨−Fn , 0⟩, 8

1 aW W = − ⟨Fw , 0⟩, 8

(2.32)

1 aSS = − ⟨Fs , 0⟩. 8

(2.33)

Recognizing the advantage of the former introduction of the general variable ϕ, one can now substitute ϕ = (˜ u, v˜) and one obtains the discretized equation for the components of the Navier-Stokes, that are interpolated with the QUICK scheme. As the interpolation is done quadratically, three nodal-points are necessary to interpolate. This creates a scenario close to the border where extra attention is required, as conventional boundary conditions are defined by two-nodes instead of three in the present case. To enable the QUICK integration scheme the boundary itself is discretized as usual by writing a separate equation for every node on the boundary. Except for the adjacent node to the boundary, this node is solved with the Power-Law scheme as treated in section A.2.1.

2.6

Pressure-Velocity Coupling

This section will introduce the considered pressure-velocity coupling schemes.

To-

day the SIMPLE-family algorithm is one of the most popular algorithm for solving


14

(in)compressible Navier-Stokes equations with primitive variables [17]. Former comparison studies in the literature are mainly focused on the convergence characteristics and robustness of these algorithms and conclusions are mostly case-specific [17]. The current work will put the focus towards the performance of these schemes under conditions similar to those encountered in the solar collector, and thus in fact another specific case study. The five algorithms underneath the scope are known as the SIMPLE, SIMPLEC, SIMPLER, SIMPLEX and the PRIME algorithms. The differences of the applied algorithms are mainly found in the method of coupling the continuity equation to the momentum equations 4 . By considering the former algorithms, authors in the literature often refer to the SIMPLEfamily [17][21]. This is because all the algorithms are based upon a guess- and correct procedure of the pressures and/or velocities, and the only difference among the algorithms lies in the treatment of these guessed or corrected variables. Roughly among the tested algorithms the SIMPLE, SIMPLEC and SIMPLEX algorithms are referred to as (semi-) implicit algorithms as they include directly the pressures in the discretized equations. As opposed to the SIMPLER and PRIME algorithms that initially compute the pseudo-velocities i.e., the flow field solution of the discretized Navier-Stokes without the direct contribution of the pressure. Consequently, the pseudo-velocities are corrected either implicitly or explicitly by means of the computed pressure, respectively. All algorithms are based upon a guess- and correct procedure in a non-linear problem, and it may be necessary to under-relaxate the computed variables. This is necessary in the case when the guess is wrong and/or the correction is too fierce, which occurs every iteration until the accepted converged solution. In case of challenging problems

5

- in

this case for example higher Reynolds numbers (Re >> 1) - this creates the freedom of under-relaxating the algorithm until a convergence takes place. On the other hand, it is necessary to under-relaxate the variables even though one does not know beforehand what the right relaxation values should be, and this could lead to a tedious process. The specific details of each algorithm will be saved from the main text. However, the interested reader is advised to read more about it in appendix section A.3 where the continuity is discretized and appendix section A.3.1 where every algorithm is discussed 4

More specific details regarding these algorithms are given in appendix A section A.3.1 where also the implementation details are presented. 5 Considering that the implemented model is a laminar model simulating high Reynolds numbers (Re >> 1) is than said to be challenging.


15

separately in detail. The following section will cover the criteria that have been set-up in the current model to obtain a converged solution.

2.7

Convergence Criteria

Appropriate convergence criteria similar for all five algorithms forces the same converged solution. According to Zeng and Tao [17], an appropriate convergence condition should include both mass and momentum conservation requirements. An additional convergence criteria is added that considers the residual in the mass source term of the discretized pressure correction equation, that is suggested by Patankar [26]. Mass conservation can be obtained locally and globally. In the cavity problem global mass conservation is automatically obtained because all velocity components normal to the wall are defined zero and thus no flux in or out the domain can occur. Local mass conservation can be obtained by computing the divergence of the velocity vector at every control volume, the result should be approaching a numerical zero. The criterion is defined by a summation over every control volume, N 1 ∑ ue − uw vn − vs | + | ≤ ϵmass , N ∆˜ x ∆˜ y

(2.34)

i=1

where N refers to the number of nodes in the computational domain and ϵmass is a pre-specified value representing a numerical zero. As this criterion calculates the fluxes that enter and leave a control volume but are summed entirely over the domain, this criterion in fact is an averaged local mass conservation and additional criteria have to be set to guarantee an appropriate solution. The second criterion from Zeng and Tao requires a minimization of the the relative residual module in the momentum equations. The following condition is normalized to a reference momentum obtained within the first few values of the iterative process. The condition yields 6 , N ∑ 1 ∑ |aP ϕP − anb ϕnb + b + Ap (˜ pw − p˜e )| ≤ ϵmom , N ϕm i=1

6

(2.35)

nb

Unfortunately this criteria is unsuccessfully implemented for the SIMPLER algorithm. After spending a significant amount of time trying to incorporate it, it is chosen to drop this criteria for the SIMPLER and PRIME algorithm as both algorithms are very similar. Instead for the two (partly) explicit algorithms a form of the criteria given in eq. 2.37 is adopted replacing the momentum residual. Performance of the inner -iterations than are compared for the predominantly implicit and explicit algorithms, separately.


16

where ϕm refers to the largest amplitude occurring in the first ten iterations of the iterative process and ϵmom refers to a pre-specified value for momentum convergence. An additional criterion is added that is urged by Patankar [26] and covers the source term in the discretized pressure-correction equation. The source term in the discretized pressure correction equation represents a mass source for the pressure correction and must be annihilated when the guessed velocities satisfy the continuity equation. The criterion for the mass source yields for every control volume, N 1 ∑ ∗ |Fw − Fe∗ + Fn∗ − Fs∗ | ≤ ϵmass,s , N

(2.36)

i=1

where ϵmass,s is a pre-specified requirement for mass source convergence. Note that this is the source term of the pressure-correction equation, also shown in eq. A.19. Finally, the normalized time residual is determined by the difference of the quantity ϕ determined at the current time-step and the previous indicated by k [18], ϕk+1 − ϕkP P ≤ ϵss , ϕm

(2.37)

where ϕm is the largest value of ϕP obtained in the first ten converged time-steps and ϵss is a pre-specified value that defines the steady-state. All major steps have been treated now and correct implementation of all these steps will lead to a converged solution for every algorithm. The result of all above treated material is shown in the next section.

2.8

Results and Discussion

A numerical tool for two-dimensional laminar incompressible flow has been set-up that predicts the velocity field in a uniform square-shaped domain. Spatial interpolation of the advection terms is done with the first-order Power-Law scheme and the third-order QUICK scheme, while the diffusion terms are implemented with the second-order central differencing scheme. Additionally, five different algorithms are implemented that couple the Navier-Stokes and the continuity equation. A grid dependency study is performed based upon the velocity profile and elucidated in appendix C.2.2 for the interested reader. From the performed grid dependency study it is concluded that the maximum absolute discretization error that occurs for the velocity profile has found to be ±0.037. The flow region of interest for the solar panel is for a laminar flow in the order of


17

Reynolds of O(102 ). For that reason, the aim of literature comparison material has been specified to that range of Reynolds numbers. Four cases are specified around this region, namely Re = 100, Re = 400 and Re = 1000. Additionally, a fourth Reynolds number is simulated pushing the limits of the laminar model i.e., a Reynolds number of Re = 10000. Higher Reynolds numbers result in higher magnitudes of velocity and pressure (-corrections) both willing and unwilling. Indicating that it will reveal easily inconsistencies in the code, like contradicting defined velocity components on the boundary that would not have been a problem on lower Reynolds number simulations. Figure 2.3 shows streamlines for four Reynolds numbers. Re = 100

As observed, for higher Re = 400

0.8

0.8

0.6

0.6 y/L

1

y/L

1

0.4

0.4

0.2

0.2

0

0

0.5 x/L

0

1

0

Re = 1000

0.5 x/L

1

Re = 10000

0.8

0.8

0.6

0.6 y/L

1

y/L

1

0.4

0.4

0.2

0.2

0

0

0.5 x/L

1

0

0

0.5 x/L

Figure 2.3: The laminar streamline solution for four different Reynolds numbers inside the square-cavity where the top-lid is driven to the positive horizontal direction. The flow field is obtained with the SIMPLE-algorithm and QUICK-interpolation.

1


18

Reynolds numbers the contribution of inertia becomes more evident and more eddies start to arise in the flow for higher Reynolds numbers. Three dominant vortices have been observed, a primary large vortex in the midsection and two smaller ones in the lower corners of the domain. The center of the primary vortex is offset for the rightupper corner at Re = 100. It moves towards the geometric center of the cavity for increasing Reynolds numbers, this is also confirmed by Ghia et al. [2]. That study appears to be a benchmark model for the lid-driven square cavity flow problem and the implemented algorithms for pressure-velocity coupling will hopefully be validated with help of that study. Ghia et al. numerically solved the vorticity-stream function formulation of the two-dimensional Navier-Stokes. They discretized the governing equations in a finite differencing fashion on a uniform divided (129x129) mesh where they applied a multigrid technique to allow local grid refinement. The spatial interpolation they applied consisted out of the first-order upwind scheme for the convection terms and a central differencing scheme for the diffusion terms. Ghia et al. provided benchmark data regarding the velocity profiles through the geometric midsection of the square-cavity. Velocity profiles of these lines through the cavity are shown in the figures 2.4 - 2.5 for a Reynolds number of Re = 100.

The velocity field comparison for the

Re = 400, Re = 1000 and Re = 10000 are to be found in appendix section A.3.2. When the numerical data generated by the present study is compared to the data from Ghia the present study seems to be in line and adequate and above all consistent for all algorithms. To determine the extend of uncertainty of this model, the difference between the present study and the study from Ghia is calculated at each provided point for each algorithm. The choice for the use of uncertainty terminology instead of an error arises in the fact that comparison is done with another numerical model, which is not necessarily true even if it is considered a benchmark model. Differences occur between the data-sets as Ghia applied a different methodology to solve the Navier-Stokes. To get an indication of uncertainty, the differences are quantified by means of a normalized form of the mean absolute percentage uncertainty δE and is calculated with the following expression, N 1 ∑ Ynum − Ylit δE (%) = ∗ 100[%], N YN

(2.38)

n=1

where Ynum is the data from the present study fitted on the exact position of the study of Ghia. The parameter Ylit is the data provided by Ghia, YN is a Frobenius norm of


19

1 0.9 0.8

y−coordinate

0.7 0.6 0.5 0.4 0.3 1: PRIME 2: SIMPLE 3: SIMPLER 4: SIMPLEC 5: SIMPLEX Ghia et al.

0.2 0.1 0

−0.2

0

0.2

0.4 u−velocity

0.6

0.8

1

Figure 2.4: The solution of the flow field for the x ˜-component of the Navier-Stokes through the geometric midsection of the cavity at x ˜ = 0.5. The blue triangles indicate the respective data of the velocity profiles taken from Ghia et al. [2]. Solution is obtained for a Re = 100 and QUICK-interpolation.

the present numerical data-set and N is the total number of data points. The following analyses obtains an impression of the uncertainty that occurs for each algorithm as compared to the benchmark model and allows an accessible comparison among the pressure-velocity methods. The results are presented for the QUICK and Power-Law interpolation in the upper and lower shared table 2.1 7 . The uncertainty is given for the u ˜- and v˜ component of the Navier-Stokes for four different Reynolds numbers for the QUICK scheme and three different Reynolds numbers for the Power-Law scheme. Simulations for the Power-Law scheme are performed up to a Reynolds number of Re = 1000. Above this first-order schemes are more likely to create numerical diffusion and data created above this level is no longer trusted. The third-order QUICK scheme shows a strong resemblance with the first-order Power-Law scheme for the three likewise tested Reynolds 7

The same result is also graphically shown by means of a three-dimensional bar-plot in appendix A.2 in figure A.5


20

0.3

0.2

v−velocity

0.1

0

−0.1

−0.2 1: PRIME 2: SIMPLE 3: SIMPLER 4: SIMPLEC 5: SIMPLEX Ghia et al.

−0.3

−0.4

0

0.2

0.4 0.6 x−coordinate

0.8

1

˜ -component of the Navier-Stokes Figure 2.5: The solution of the flow field for the y through the geometric midsection of the cavity at y˜ = 0.5. The blue triangles indicate the respective data of the velocity profiles taken from Ghia et al. [2]. Solution is obtained for a Re = 100 and QUICK-interpolation.

numbers. This indicates that that both schemes are appropriate for simulations in this range of Reynolds numbers, although small differences of the interpolation schemes occur in favor of the QUICK scheme. Note that the uncertainty for the v˜ velocity flow field is generally less-accurate compared to the u ˜ velocity flow field. This is - most likely - due to the natural physics of the problem since along the horizontal midsection the v˜component is exposed to higher occurring velocity gradients and amplitudes, aside from the boundary condition set upon the u ˜-component. This effect becomes more evident for higher Reynolds numbers, deteriorating the uncertainty. For lower Reynolds numbers the algorithms match up close with the benchmark model. For higher Reynolds however, a larger spread of uncertainty is observed with maximum uncertainties occurring in the first-order Power-Law scheme. An additional comparison of the positions of the occurring vortices inside the cavity

Chapter 2. Validating the Velocity Field in the Lid-Driven Square Cavity. Re

PRIME [%]

SIMPLE [%]

SIMPLER [%]

SIMPLEC [%]

SIMPLEX [%]

Results for QUICK interpolation 100 u ˜ 0.051 0.029 v˜ 0.227 0.187 400 u ˜ 0.054 0.031 v˜ 1.640 0.285 1000 u ˜ 0.674 0.099 v˜ 1.880 0.146 10000 u ˜ 0.773 0.958 v˜ 0.635 0.897

0.044 0.228 0.266 0.665 0.976 0.876 2.895 2.133

0.029 0.188 0.031 0.285 0.099 0.146 1.015 0.958

0.029 0.190 0.031 0.285 0.098 0.146 1.005 0.947

Results for Power-Law interpolation 100 u ˜ 0.056 0.025 v˜ 0.237 0.182 400 u ˜ 0.817 0.080 v˜ 2.087 0.369 1000 u ˜ 0.636 0.266 v˜ 1.610 0.453

0.042 0.227 0.281 0.661 0.561 0.536

0.025 0.182 0.080 0.369 0.266 0.453

0.025 0.185 0.080 0.369 0.267 0.454

21

Table 2.1: The mean absolute percentage uncertainty δE for every pressure-velocity coupling scheme as a function of the Reynolds number compared to the benchmark model. Spatial interpolation performed with the QUICK and Power-Law scheme.

with a different source from the literature is treated in appendix A.3.3. The maximum uncertainty describing the relative position of the vortices from that comparison study appeared to be 0.63 [%]. Concerning the individual pressure-velocity coupling methods, it is observed that the behavior of the SIMPLER and the PRIME algorithms are similar. This is not a coincidence as both algorithms solve and use the pressure equation to calculate the velocity field, instead of the pressure-correction equation. The similarity also extends to the pressure fields obtained from these two (semi-) explicit algorithms, see figure 2.6. The figure shown reveals the pressure distribution on the top-side of the cavity for a Reynolds number of Re = 1000. It is observed that the pressure field is over-estimated for the two (semi-) explicit algorithms compared to the three implicit algorithms. Over-estimated since the implicit algorithms seem to match more closely to the benchmark model, this over-estimation could explain why the uncertainty is worse for the explicit algorithms. This is also the point where first-order interpolation schemes become a concern as the over-estimation over the pressure-field combined with numerical diffusion in interpolation could result in difficult to interpreted behavior. A final observation for the figure


22

Relative Pressure for Re=1000 1: 2: 3: 4: 5:

1.4 1.2

0.8

2

p/(ρ U )

1

PRIME SIMPLE SIMPLER SIMPLEC SIMPLEX

0.6 0.4 0.2 0 −0.2

0

0.2

0.4 0.6 x/L,y/L=0.995

0.8

1

Figure 2.6: Pressure distribution along the lid of the cavity for the five pressurevelocity coupling schemes and a Reynolds of Re = 1000.

showing the pressure-field is made regarding the boundary conditions. Common and easily to implement boundary conditions for the pressure (-correction) equation are prescribed through homogeneous Neumann conditions on every wall [22], although looking at the right-side of the figure one can observe that it might not have been an ideal prescribed condition. Instead, implementing a non-homogeneous Neumann condition would have been a more suitable solution. For solely lower Reynolds numbers the choice of pressure-velocity algorithm will not significantly influence the solution of the flow field. Although the SIMPLER and the PRIME algorithm - as far as it concerns - do have a more defined pressure field distribution as compared to the implicit algorithms. The implicit pressure-velocity coupling algorithms however show an overall better performance regarding the flow field.

2.9

Conclusion

A numerical tool to solve the incompressible time-dependent Navier-Stokes in a twodimensional square-shaped uniform applied domain has been developed. It is based on


23

a finite volume method with a Power-law and QUICK interpolation scheme for the convection terms and a central differencing scheme for the diffusion terms. Fully-implicit time integration is applied to obtain the steady-state solution. The linear system is solved with a successive-overrelaxation (SOR) technique. Iteration convergence criteria takes into account mass conservation, continuity and momentum residual and the change of the flow field after each time-step. The SIMPLE, SIMPLEC, SIMPLER, SIMPLEX and PRIME algorithms were implemented for the pressure-velocity coupling and the performance of these algorithms have been tested against the benchmark model from Ghia et al. [2]. All algorithms show adequacy and consistency, but the SIMPLE, SIMPLEC and SIMPLEX have an overall better performance for the flow field. A maximum uncertainty is obtained for the SIMPLER algorithm at Re = 10000 of 2.895[%] for the horizontal velocity component. The tool for calculating the flow field is hereby hopefully validated with adequacy and consistency for a reasonable uncertainty and will be applied in the next validation test case for mixed convection flow.

Chapter 3

Mixed Convection in a Lid-Driven Square Cavity. 3.1

Introduction

The first case study in the lid-driven cavity flow aimed at verifying a correct implementation of the pressure-velocity coupling algorithms and a validation of the solution obtained for the flow field. The second case study will extend the numerical tool by incorporating the energy equation dedicated to the mixed convection that also occurs inside the solarpanel. A tradeoff between forced and natural convection is implemented by considering a driven lid - again on the top-side - and an active buoyancy-force. Compared to the first case study the flow will be driven by an additional temperature gradient within the square-cavity. The mixed convection model will be studied and verified with work found in the literature. As will become clear in the next section, the flow will be characterized by the Grashof, Reynolds, Prandtl and the Richardson number. The latter represents a ratio of the first two dimensionless numbers and is used to characterise the presence of natural to forced convection. A Richardson number equal to an unity order indicates an equal balance between natural and forced convection. Simulations in this chapter are performed with conditions where the Richardson number is in the order close to O(1), allowing the analyses of both types of convection. The chapter will start to cover the governing equations in section 3.2 where these dimensionless numbers appear after a scaling procedure is introduced and are implemented 24

Chapter 3. Case Study II: Mixed Convection in a Lid-Driven Square-Cavity.

25

accordingly. The problem is defined in the up-following section, section 3.3 covering the boundary conditions. These conditions are chosen according to popular benchmark models that can be found in the literature to allow a validation of the obtained results. As the energy-equation is introduced in this chapter, the chapter is followed-up by a brief discretization performed again by a fully-implicit time integration scheme and a QUICK spatial interpolation in section 3.4. The work is concluded with the results and the discussion about those results 3.5.

3.2

Governing Equations

An unsteady incompressible flow that is found to be inside a laminar mixed convective environment and flow conditions that occur within 0 < x < L and 0 < y < L for the y direction being defined in the vertical direction is governed by the following set of equations, ∂u ∂v + = 0, ∂x ∂y ( ) ( ) ∂ ∂ ∂ ∂ ∂u ∂ ∂u ∂p (ρu) + (ρuu) + (ρvu) = µ + µ − , ∂t ∂x ∂y ∂x ∂x ∂y ∂y ∂x ( ) ( ) ∂ ∂ ∂ ∂ ∂v ∂ ∂v ∂p (ρv) + (ρuv) + (ρvv) = µ + µ − + ρgβ(T − T∞ ), ∂t ∂x ∂y ∂x ∂x ∂y ∂y ∂y ( ) ( ) ∂ ∂ ∂ ∂ k ∂T ∂ k ∂T (ρT ) + (ρuT ) + (ρvT ) = + . ∂t ∂x ∂y ∂x cp ∂x ∂y cp ∂y

(3.1) (3.2) (3.3) (3.4)

Although the buoyancy effect occurs by means of density gradients in the flow, if the temperature difference between the reference state and fluid is small, an analyses for the velocities and temperatures can be done with a constant density, except in the body force term, where the density is a function of the temperature, this is also known as the Boussinesq approach [27][28]. The former governing equations are rewritten towards a dimensionless form where all quantities are defined according to the following definitions, x ˜=

x , L

y˜ =

y , L

u ˜=

u , U

v˜ =

v , U

Ut t˜ = , L

p˜ =

p , ρU 2

θ=

T − Tc , Th − Tc

(3.5)

where the dimensionless quantities are distinguished by a tilde. The resulting nondimensional governing equations for the mixed convection problem are then given by, ∂u ˜ ∂˜ v + = 0, ∂x ˜ ∂ y˜

(3.6)

Chapter 3. Case Study II: Mixed Convection in a Lid-Driven Square-Cavity. ( ) ∂u ˜ ∂ ∂ 1 ∂2u ˜ ∂2u ˜ ∂ p˜ + (˜ uu ˜) + (˜ vu ˜) = + 2 − , 2 ˜ ∂x ˜ ∂ y˜ Re ∂ x ˜ ∂ y˜ ∂x ˜ ∂t ) ( 2 ∂ 1 ∂ p˜ ρgβ(T − T∞ )L ∂ ∂ v˜ ∂ 2 v˜ ∂˜ v (˜ uv˜) + (˜ v v˜) = + 2 − + , + 2 ˜ ∂ y˜ Re ∂ x ˜ ∂ y˜ ∂ y˜ ρU 2 ∂ t˜ ∂ x ( 2 ) ∂θ ∂θ ∂θ 1 ∂ θ ∂2θ +u ˜ + v˜ = + . ∂x ˜ ∂ y˜ P rRe ∂ x ˜2 ∂ y˜2 ∂ t˜

26

(3.7)

(3.8)

(3.9)

The body-force term in the y˜-component of the Navier-Stokes is rewritten towards the Richardson number, [ ][ ][ 2 2 ] ρgβ(T − TC )L gβ(T − TC )L TH − TC µ L Gr = = = Riθ. ρU 2 U2 TH − TC ρ2 ν 2 L 2 Re2

(3.10)

Where the dimensionless numbers are defined as, Re =

ρU L , µ

Gr =

gβ(TH − TC )L3 , ν2

Pr =

µcp . k

(3.11)

The buoyancy force in the y˜-component of the Navier-Stokes is modelled as a source term interpolated at the vertical velocity node, which is further discussed at the end of the current section. Insert the Richardson number into the y˜-component of the momentum equation one obtains, ∂˜ v ∂ ∂ 1 + (˜ uv˜) + (˜ v v˜) = ˜ ∂ y˜ Re ∂ t˜ ∂ x

(

∂ 2 v˜ ∂ 2 v˜ + ∂x ˜2 ∂ y˜2

) −

∂ p˜ + Riθ, ∂ y˜

(3.12)

where the Richardson number is a measure for the balance between natural and forced convection. Note that the conductance coefficient D and the convective mass flux F of the discretized y˜-component of the Navier-Stokes are defined in this case by, D=

A , Re∆˜ x

F = vÃ.

(3.13)

For more information related to the discretization of the Navier-Stokes using QUICK interpolation see section 2.5, in the previous chapter. The above mentioned conductance and convective mass flux coefficients and the source term can be rewritten for the x ˜-component of the Navier-Stokes analogue to the y˜-component. The sole difference between the x ˜- and y˜-component is the source term of the y˜-component that takes into account the buoyancy force. Considering the source term of the discretized equation derived earlier by means of QUICK spatial interpolation (shown in previous chapter eq. 2.24) for the y˜ component of the Navier-Stokes, it is custom to apply a Picard


27

linearization in the following form [14][26], b = bp v˜p + bc ,

(3.14)

where, as a first implementation the coefficient bp equals zero. However, it should be possible to implement it as a non-zero (negative) value, making it contribute to a more diagonal linear system and thus a more stable system [18]. As simulations are performed with rather large Grashof numbers one can easily consider that bc will negatively effect the rate of convergence. In this case the contribution of the buoyancyforce will be entirely incorporated in the coefficient bc that consists out of the earlier defined Richardson number Ri and the dimensionless temperature θ (see the source term of eq. 3.12),

( Sc =

) θJ+1 − θJ Ri∆V, 2

(3.15)

where the temperature θ is interpolated on the forward staggered grid, necessary as velocities and temperature are not defined at the same location, see also figure 2.2. In the mixed convection model the relative high Grashof number in the source term will fortunately be balanced out to a certain extent by the presence of the Reynolds number, see also the definition of the Richardson number eq. 3.10. In the natural convection case that will be treated in the next chapter, the Rayleigh number is not balanced out causing a linear system where the system matrix is dominantly non-diagonal, leading to a more unstable and tedious convergence. How to deal with the occurrence of high Rayleigh numbers in the source terms will be discussed in the next chapter, chapter 4, covering natural convective flow phenomena.

3.3

Boundary Conditions

The boundary conditions related to the current mixed convection model are specified to commonly found conditions in the literature such that comparison of the results can be performed accordingly. Those involve no-slip velocity conditions for every wall in the square-cavity, adiabatic (zero heat flux) side-walls and prescribed hot and cold temperatures on the bottom- and top walls, respectively. The model including its boundary conditions is schematically shown in figure 3.1. The boundary conditions as presented


28

u = U, v = 0, T = TC g u=v=

u(t=0) = 0 v(t=0) = 0 T(t=0) = 0

∂T = 0 ∂x

u = v = ∂∂Tx = 0

y x

u = 0, v = 0, T = T H Figure 3.1: Boundary conditions for the mixed convection square-cavity flow.

in the figure are implemented in their non-dimensional state and are defined by, (˜ x = 0, y˜),

u ˜ = 0,

v˜ = 0,

∂θ/∂y = 0,

(3.16)

(˜ x = 1, y˜),

u ˜ = 0,

v˜ = 0,

∂θ/∂y = 0,

(3.17)

(˜ x, y˜ = 0),

u ˜ = 0,

v˜ = 0,

θ = 1,

(3.18)

(˜ x, y˜ = 1),

u ˜ = 0,

v˜ = 0,

θ = 0,

(3.19)

(t = 0, x ˜, y˜),

u ˜ = 0,

v˜ = 0,

θ = 0.

(3.20)

The boundary conditions regarding its implementation for the velocity field has been discussed earlier in chapter 2. However, some extra care is required for the yet untreated temperature boundary conditions. Two types of conditions are implemented, one of the Neumann type and another one of the Dirichlet type. The first type is the homogeneous temperature gradient forced upon the side-walls of the cavity. Considering eq. 2.12 then the discretized coefficients for the boundary equation defined by an adiabatic condition on the west-side for example yields, aP = 1,

aE = 1,

b = 0.

(3.21)


29

The east-side of the cavity can be written analogue to the west-side condition. The Dirichlet conditions for the north- and south-side can be implemented similar as shown in eq. 2.14 specified towards the temperature. The corner-point conditions are a different story and discussable. It is found that the most likely condition is of the Neumann type where the temperature gradients in the corners adjacent to the boundary are set equal, indicating,

∂θ ∂θ = , ∂x ˜ L ∂ x ˜ R

(3.22)

where L and R refer to the left- and right-side of the boundary, respectively, see also figure 2.2. Discretizing the former equation for the south-west corner results in, θE − θP θEE − θE = , ∆˜ xL ∆˜ xR

(3.23)

where the EE refers to the second node upstream. For x ˜L = x ˜R the above equation can be rewritten for the corner-node, aP θP = aE θE + aEE θEE + b,

(3.24)

where aP = 1, aE = 2, aEE = −1 and b = 0. The remaining boundary corner-nodes are written in a similar form. The careful reader may observe that the corner points do not take into account the contribution of the vertical direction. However, the vertical node adjacent to the corner-nodes are indirectly linked by means of the Dirichlet condition active on the north- and south-side boundaries.

3.4

Discretization of the Energy-Equation

The discretization of the components of the Navier-Stokes can be reviewed in chapter 2, the following section will briefly cover the discretization of the energy-equation. Considering a single control volume than integration of the energy-equation over time and space yields, ∫ t

t+∆t ∫ CV

{

} ∂θ ∂θ ∂θ +u ˜ + v˜ dV dt = ... ∂x ˜ ∂ y˜ ∂ t˜ ∫ t+∆t ∫ ... t

CV

{

1 P rRe

(

∂2θ ∂2θ + ∂x ˜2 ∂ y˜2

)} dV dt. (3.25)


30

Performing the actual integration over a control volume CV yields, (

θP − θPo ∆t˜

) + (˜ uAθ)e − (˜ uAθ)w + (˜ v Aθ)n − (˜ v Aθ)s = ... [ ( ) ( ) ] 1 θE − θP θP − θW ... Ae − Aw + ... P rRe ∆˜ x ∆˜ x e w ( [ ( ) ) ] θN − θP θP − θS 1 ... An − As . (3.26) P rRe ∆˜ y ∆˜ y n s

The velocities are defined at the interface of every main control volume and do not require any special attention as they are already at the desired position. Integrating the dimensionless temperature found within the convection terms at the interfaces requires a spatial interpolation. The interpolation is performed with the QUICK scheme and by doing so, one obtains the standard form of the discretized equation (see also section 2.5), n n n aP θPn = aE θE + aW θW + aN θN + aS θSn + +aop θPo + b,

(3.27)

where the source term b is given by eq. 2.25 and the coefficients by the equations shown in 2.26 - 2.27. The conductance coefficient D and the convective mass flux F are defined in this case by, D=

A , P rRe∆˜ x

F =u Ã.

(3.28)

When noting that eq. 3.27 is similar to the form of the momentum equations, one can determine its residual in the same way and is obtained with eq. 2.35 by excluding the contribution of pressure, leading to, N ∑ 1 ∑ |aP θP − anb θnb − b| ≤ ϵmom N θm i=1

(3.29)

nb

where ϵmom is kept equal to the criteria set to the velocity field and θm is again the normalization factor being the largest value for the residual obtained in the first ten iterations.

3.5


The aim of the current mixed convection model is to investigate the performance of the different pressure-velocity coupling algorithms when it is exposed to an additional


31

heat flux under standard test-operating conditions that is also to be expected in the solar-panel. Furthermore, it is desired to validate the model as a whole now the energyequation is added to the set of governing equations. The spatial interpolation of the advection terms is now solely done with the QUICK-scheme and the diffusion terms are again interpolated with the central differencing scheme. A uniform structured, forward staggered (100x100) grid is adopted, leading to square-shaped control volumes having a uniform size of 1e-2[-]. A grid dependency study regarding this second case study is performed in the appendix, it is given in section C.2.3. Rather large discretization errors are found, located a little offset from the left-boundary where the temperature gradient changes sign. It is believed that the current grid configuration may not be sufficient as discretization errors up to 9.1[%] are obtained inside the cavity. The current chapter will continue to present the results the way it is introduced to prevent confusion, a solution is presented to limit this discretization error at the end of this section. Four Richardson numbers have been simulated varying from Ri = 10000, Ri = 100, Ri = 1 and Ri = 0.001, the Grashof and Prandtl numbers have been kept constant having the designated values of 10000 and 0.71, respectively 1 . It has been found that difficulties to obtain a converged solution arise in the region where natural convection becomes more relevant (Ri >> 1). This was expected earlier in section 3.2 where the amplitude of the source term in the Navier-Stokes taking into account the Boussinesque approximation affected the linear system negatively creating numerical instability. A diverse behavior of convergence is noted among the different pressure-velocity coupling algorithms. The implicit algorithms have showed to disregard a change of simulated conditions concerning the amount of inner-iterations, while the (semi-) explicit algorithms show to require more- and more inner-iterations for lower Richardson numbers. Regarding the inner-iterations the SIMPLE-algorithm seemed to converge very constant throughout every simulation, while the SIMPLER-algorithm appeared to be most affected negatively. The SIMPLEX-algorithm has showed to require the least amount of inner-iterations where natural convection became dominant, while the SIMPLECalgorithm required the least amount of inner-iterations where forced convection became dominant. The different behavior in convergence for each algorithm per Richardson number is shown in table A.2 and figure A.6 and can be found in appendix A. Those table and figures present the inner- and outer-iterations (time-residual) for every applied 1

Appear to be common values in the literature.


32

algorithm in this case study. The velocity flow field distribution is presented for the four simulated Richardson numbers for the interested reader in appendix A figure A.7. Those figures reveal a strong similarity as compared to the earlier presented flow field figures of chapter 2 when forced convection becomes dominant (Ri 1) with a single cell in the midsection. The heat is distributed due to buoyancy-forces where the extend is determined by the amplitude of the Grashof number. The final regime (3) is an equilibrium between the latter two regimes and is observed for Richardson numbers reaching a unity value (Ri ∼ 1). An additional observation is made regarding the first two sub-figures of figure 3.2, where the similarity is striking even though the Richardson number is a factor 100 different between those two sub-figures. It is found that a dominant natural convective flow where there hardly is any bulk movement allows a vast and often complex interaction due to buoyancy-forces between hot and cold fluid inside the domain [3][31]. For the model to be able to record this it is believed that it is necessary to prioritize the computational accuracy around the edges of the domain where the temperature gradient is largest. The results that are obtained are compared to a model available in the literature, in this case study provided by Yapici et al. [3]. The author of this article applied a structured non-uniform grid (320x320) with a four-point fourth-order interpolation scheme (FPFOI) for the convection terms and the second-order central differencing scheme for the diffusion terms. Additionally, he used the SIMPLE-algorithm for the pressure-velocity coupling and the iterative tri-diagonal matrix algorithm (TDMA) to solve the algebraic equations. The comparison is done with the aim at heat transfer, for that reason the Nusselt number is computed and compared with the mentioned literature accordingly. De Vahl Davis defined the Nusselt number on the wall by means of the temperature gradient and is adopted here along the horizontal top-side of the square-cavity by [4], Nu = −

∂θ . ∂n

(3.30)

Yapici et al. published results for the Nusselt number along the top-side of the cavity at y˜ = 1 as a function of the x ˜ coordinate. The comparison is shown in figure 3.3 for the four Richardson numbers. The results of the individual pressure-velocity algorithms all show consistency for the simulated Richardson numbers. However, large differences

Chapter 3. Case Study II: Mixed Convection in a Lid-Driven Square-Cavity. Ri = 10000

Ri = 100 3.5

PRIME SIMPLE SIMPLER SIMPLEC SIMPLEX Yapici et al.

3

2.5

34


3 2.5

NuL

NuL

2

1.5

1.5

1

1

0.5

0

2

0.5

0

0.2

0.4

0.6

0.8

0

1

0

0.2

0.4

x/L

0.6

0.8

1

x/L

Ri = 1

Ri = 0.01 45 PRIME SIMPLE SIMPLER SIMPLEC SIMPLEX Yapici et al.

6 5


40 35 30

NuL

NuL

4

25 20

3

15 2 10 1

0

5

0

0.2

0.4

0.6 x/L

0.8

1

0

0

0.2

0.4

0.6

0.8

1

x/L

Figure 3.3: The Nusselt number (Nu) for different Richardson (Ri) numbers from l-r Re = 1, Re = 10, Re = 100 and Re = 1000 and a constant- Grasshof number Gr = 104 and Prandtl number P r = 0.71 along the top-side of the cavity.

occur between the data from the literature and the current model where the system is dominated by natural convection 2 . Although transition to a turbulent natural convective flow normally occurs at Ra ∼ 109 and the current flow is still far from that, turbulence is the amplification of initially small disturbances [31]. In all likelihood, as the domain is still rather coarse, it is the onset of micro-structures being in the order of the control volumes, and as such having difficulties to deal with the phenomena. If so, it would emphasize the limitations of the uniform applied grid and suggests local grid refinement. As the Reynolds number is increased and the flow enters the region 2

Note that this is also the point where discretization errors appeared to reach uncertainties up to 9.1[%].


35

of forced convection the current model matches the benchmark model more closely. A quantified comparison with the literature - according to eq. 2.38 - for the pressurevelocity algorithms and the simulated Richardson numbers have been summarized in table 3.1. The table shows that the absolute differences among the pressure-velocity Ri 10000 100 1 0.01

PRIME [%] 4.8579 4.9646 4.2050 2.4135

SIMPLE [%] 4.8081 4.9277 4.1822 2.4670

SIMPLER [%] 4.7485 4.8779 4.1293 2.3604

SIMPLEC [%] 4.8074 4.9272 4.1791 2.4647

SIMPLEX [%] 4.8077 4.9273 4.1820 2.4663

Table 3.1: The mean absolute percentage uncertainty δE for every pressure-velocity coupling scheme as a function of the Richardson number compared to the literature model [3]. Results obtained for the given Richardson numbers and in the same order the Reynolds number varied from Re = 1, Re = 10, Re = 100 and Re = 1000 with a constant- Grasshof number Gr = 104 and Prandtl number P r = 0.71.

algorithms are small and do not affect the heat transfer significantly. The SIMPLER algorithm showed to be slightly more in-line with the used model from the literature, compared to the remaining algorithms. As the differences among the applied pressurevelocity coupling algorithms are small, the result suggests to select an algorithm based on its characteristics of convergence. In that case, selecting a pressure-velocity algorithm when studying mixed convective heat transfer should be determined by the order of the simulated Richardson number that classifies the extent of natural convection over forced convection. A final note regarding the applied grid configuration throughout this chapter. The discretization error that occurred in the uniform grid was found to be significant in the case where natural convection dominated the flow inside the cavity. Additionally, the uncertainty that occurred when the same flow conditions were compared to the literature appeared to show similar significant differences. This confirms the suggestion implied earlier regarding local clustering of control volumes near the boundaries. In an attempt to resolve this issue a non-uniform grid is imposed and with the same method of determining the discretization error it was found that the uncertainty at the same position was reduced to 0.83[%], as opposed to the initial 9.1[%]. The non-uniform grid is officially introduced in the next chapter, regarding pure natural convection flows. The design and implementation details of the non-uniform grid are elucidated in appendix B and the derivation of the discretization error regarding the non-uniform grid is found in chapter C.3.2. A single (test-) simulation for the non-uniform grid is presented in figure


36

A.8 to be found in the appendix given for a (Ri = 100), the differences as compared to the benchmark model and the adopted uniform grid are shown on the right of that same figure. It reveals that although differences still occur, the (semi-) explicit algorithms seem to benefit most from the implementation of the non-uniform grid. The corresponding inner-iterations are presented on the same page, in figure A.9. It is noted that the non-uniform grid negatively affects the amount inner-iterations, where the (semi-) explicit algorithms suffer the most.

3.6

Conclusion

The numerical tool developed initially for predicting the velocity-field without heat transfer is extended to include mixed convection. Mixed convection in the presented model takes into account the contribution of natural- and forced convection by means of the Richardson number. Forced convection is incorporated by means of the moving lid while natural convection is implemented in the code by means of the Boussinesq approximation. The contribution of heat transfer is analyzed by the calculation of the local Nusselt number on the lid located on the top-side of the cavity. The results obtained for the local Nusselt number are compared to a benchmark model found in the literature. The numerical tool has found to predict forced- and natural convection heat transfer in a consistent manner for every velocity-coupling algorithm. However, differences with the literature of the local Nusselt number occurred where natural convection became dominant. The differences with the literature were largest at the position inside the cavity where the discretization error appeared to be largest as well, due to that it was believed that this occurred because of the limitations given by the size of the control volumes. The differences among the different tested pressure-velocity coupling algorithms were found to be small for the prediction of heat transfer. The latter suggested to select an algorithm based upon the speed of convergence rather than the accuracy of heat transfer in case of a mixed convection flow. In that matter it was found that the SIMPLEalgorithm showed a very constant quantity of inner-iterations throughout all the tested simulations. While the SIMPLEC-algorithm showed to require the least amount of inneriterations in the case of a dominant forced convection flow, the SIMPLEX-algorithm appeared to require the least amount of iterations in the case of a natural convection


37

flow. Furtermore, the PRIME- and SIMPLER-algorithm required the most amount of inner-iterations throughout the tested simulations.

Chapter 4

Natural Convection in a Square Cavity. 4.1

Introduction

The last step towards the validation of the numerical tool is dedicated towards a pure natural convection flow where the buoyancy-force is dominantly present. Flows of this caliber occur in the solar panel whenever the pump is either deactivated or not present at all. Two major differences are present compared to the last treated square-cavity problem those are (1) there is no longer a driven-lid in the square cavity, affecting the flow physically and (2) a non-uniform grid is applied and implemented in the sourcecode to obtain the scalar solutions, providing a more numerical detailed local resolution. The latter modification was inspired because of the results obtained in the second case study that appeared to require local grid refinement near the boundaries in simulations where natural convection dominated the flow. The current case study will no longer cover forced convection, the natural convection inside the square-cavity will now be characterized by the amplitude of the dimensionless numbers the Rayleigh number and the Prandtl number. In the current chapter flow conditions are modelled within the thermal laminar region for Rayleigh numbers in the range of Ra ∼ 103 - 106 and a constant Prandtl number of P r = 0.71. The problem will be introduced by introducing the governing equations in section 4.2,

38

Chapter 4. Case Study III: Natural Convection in a Square Cavity.

39

that are different from the previous test case of mixed convection, due to the applied nondimensional variables. In the same section it additionally introduces an enhanced linear interpolation specified to the source term incorporating the Boussinesq approach that is necessary to appropriately interpolate on a non-uniform grid. The natural convection problem is defined in section 4.3, covering the boundary conditions. As no further major differences are imposed on the discretization the follow-up section directly covers the results and the associated discussion in section 4.4.

4.2

Governing Equations

Although flow conditions that occur are still far from the thermal turbulent region (Ra ∼ 109 ) Bagchi et al [32] already observed thermal disturbances in the form of plumes from Rayleigh numbers on starting at Ra ∼ 106 . This indicates that the modelled conditions will be near the transition zone of laminar and turbulent thermal flow. Transition in the natural convection boundary layer depends on the relative magnitude of the buoyancy and viscous forces in the fluid. Due to that it is customary to correlate the occurrence of natural convection in terms of the Rayleigh number [33], as opposed to the Grashof number used earlier in the mixed convection model. The Rayleigh number is defined as the product of the Grashof and Prandtl number, applied to the cavity flow this yields, RaL = GrL P r =

gβ(TH − TC )L3 , να

Pr =

ν . α

(4.1)

As lower velocities are associated with natural convection and there is no longer an imposed velocity, the thermal diffusion coefficient α is used to non-dimensionalize the governing equations shown earlier in eqs. 3.1 - 3.4. The following non-dimensional quantities are considered, x ˜=

x , L

y˜ =

y , L

u ˜=

uL , α

v˜ =

vL , α

αt t˜ = 2 , L

p˜ =

pL2 , ρα2

θ=

T − Tc . Th − Tc

(4.2)

The resulting non-dimensional governing equations for a pure natural convection problem yields, ∂u ˜ ∂˜ v + = 0, ∂x ˜ ∂ y˜ ( 2 ) ∂u ˜ ∂ ∂ ∂ u ˜ ∂2u ˜ ∂ p˜ + (˜ uu ˜) + (˜ vu ˜) = P r + 2 − , 2 ˜ ∂ y˜ ∂x ˜ ∂ y˜ ∂x ˜ ∂ t˜ ∂ x

(4.3) (4.4)

Chapter 4. Case Study III: Natural Convection in a Square Cavity. ) ∂ 2 v˜ ∂ 2 v˜ ∂ p˜ + 2 − + P rRaL θ, 2 ∂x ˜ ∂ y˜ ∂ y˜ ( 2 ) ∂θ ∂ θ ∂2θ ∂θ ∂θ + v˜ = + . +u ˜ ∂x ˜ ∂ y˜ ∂x ˜2 ∂ y˜2 ∂ t˜

∂˜ v ∂ ∂ + (˜ uv˜) + (˜ v v˜) = P r ˜ ∂x ˜ ∂ y˜ ∂t

40

(

(4.5)

(4.6)

The source term in the vertical component of the Navier-Stokes is rewritten similar as shown in eq. 3.10 with an additional enhanced linear interpolation added due to the application of the non uniform grid, that is - for the interested reader - also extensively treated in appendix B. The enhanced linear interpolation within the source term takes into account the vertical non-uniform distribution of the non-dimensional temperature. The source term from eq. 5.22 and the previous treated source term in the mixed convection case where a midpoint rule was applied eq. 3.15 are rewritten and implemented for the natural convection case in the following form, ( ) Sc = θJ+1 λT − θJ (1 − λT ) P rRa∆V,

(4.7)

where λT takes into account the fraction of the temperature θJ+1 as opposed to θJ being defined similar to eq. B.7 and B.9. A more extensive explanation is found in appendix B where the enhanced interpolation is also applied to a different quantity and shown graphically for clarification purposes.

4.3

Boundary Conditions

Validation of the model is hopefully obtained again by comparing it to common problems encountered in the literature. The problem as defined for the natural convection case is only slightly different from the mixed convection model and is shown in figure 4.1. The same type of conditions are applied as earlier in the mixed convection model except being defined at a different position. The boundary conditions in their non-dimensional state are defined by, (˜ x = 0, y˜),

u ˜ = 0,

v˜ = 0,

θ = 1,

(4.8)

(˜ x = 1, y˜),

u ˜ = 0,

v˜ = 0,

θ = 0,

(4.9)

(˜ x, y˜ = 0),

u ˜ = 0,

v˜ = 0,

∂θ/∂y = 0,

(4.10)


41

u = v = ∂∂Ty = 0 g u(t=0) = 0 v(t=0) = 0 T(t=0) = 0

u = 0, v = 0, T = T H

u = 0, v = 0, T = T C

y x

u = v = ∂∂Ty = 0 Figure 4.1: Boundary conditions for the natural convection square cavity flow.

(˜ x, y˜ = 1),

u ˜ = 0,

v˜ = 0,

∂θ/∂y = 0,

(4.11)

(t = 0, x ˜, y˜),

u ˜ = 0,

v˜ = 0,

θ = 0.

(4.12)

The implementation of the boundary conditions is rather straightforward and were treated extensively in the previous test cases. A little extra care is taken for the cornernodes, similar for the mixed convection case. In the mixed convection model the corner nodes were defined by eq. 3.22, i.e., the gradients were kept equal in the direction of the imposed temperature which was the horizontal direction. In this case the temperatures are imposed on the vertical walls. Applying the same treatment this implies that the coefficients of the discretized boundary equation shown in eq. 2.12 now to be defined by, aP = 1,

aN = 2,

aN N = −1,

(4.13)

for the south-west corner-node. Similar treatment of this corner node is applied to the remaining three corner nodes. As discretization and convergence criteria for the additional energy-equation have already been treated in section 3.4 the following section will directly cover the results and discussion related to the model of natural convection.


4.4

42


Natural convection is investigated for four different Rayleigh numbers varying in the range from Ra ∼ 103 - 106 for a constant Prandtl number P r = 0.71. Simulations are performed for the five known pressure-velocity coupling algorithms.

The alge-

braic equations are solved yet again with successive over-relaxation (SOR) where overrelaxation is applied for lower Rayleigh numbers and under-relaxation is applied for higher Rayleigh numbers. The spatial interpolation of the advection terms is done with the QUICK scheme and the diffusion terms are again interpolated with the central differencing scheme. A non-uniform structured, and forward staggered grid is adopted having (100x100) elements leading to rectangular-shaped control volume elements spread out in the domain. Near the boundary a grid-refinement is applied and implemented according to an algebraic transformation that is covered in appendix B where the minimum square-shaped control volume is 3e-3[-]. Note that this is almost a factor ten times smaller than the previous presented uniform applied control volumes in the (100x100) configuration of chapter 3. A grid dependency study is performed and presented in the appendix section C.3.3. From that study it is found that the maximum discretization uncertainty percentage that occurs when determining the Nusselt number is 0.7[%], that corresponded to an absolute error of ±0.129[-]. The acquired converged solutions for the different Rayleigh numbers are again compared to recognized models from the literature. For that purpose, the obtained results according to the former implementation settings are compared by means of (1) the maximum u ˜-velocity along the vertical midline and (2) the maximum v˜-velocity along the horizontal midline of the square cavity and (3),(4) and (5) the averaged, maximum and minimum Nusselt number along the heated side of the square cavity, respectively. The averaged Nusselt number

1

is computed by the application of a trapezoidal rule on the

local Nusselt number defined earlier in eq. 3.30. Table 4.1 shows the five introduced quantities being compared to two models from the literature. The first model taken from the literature is from De Vahl Davis [4]. By means of a stream function-vorticity formulation he solved the governing equations on a square-shaped mesh with the finite differencing method. He defined the problem in a steady-state formulation, although he incorporated a ”false transient” which he solved with a first-order implicit method to 1

The averaged Nusselt number N u0 ∫is obtained by∫ the integration of the local Nusselt number over L 1 the characteristic length L, N u0 = 1/L 0 N uL dL = 0 N uL d˜ y.


43

create an artificial control of stability. Additionally, he applied the second-order central differencing scheme for all the spatial derivatives. De Vahl Davis recognized that its formulation in itself was not particularly accurate due to the second-order spatial interpolated solution. However, as a post-processing tool he improved its results by the application of a Richardson extrapolation leading to the eventual benchmark results. The second model from the literature is from Tian et al. [5]. Similar to De Vahl Davis they applied a stream vorticity-function formulation on a square-shaped mesh. However, instead of the second-order spatial interpolation they managed to incorporate a fourth-order spatial interpolation scheme in a solely steady-state defined system solved with a finite differencing technique. The five tested pressure-velocity coupling algoRa

PRIME

SIMPLE

SIMPLER SIMPLEC SIMPLEX [4]

[5]

103 umax vmax N u0 N umax N umin 104 umax vmax N u0 N umax N umin 105 umax vmax N u0 N umax N umin 106 umax vmax N u0 N umax N umin

3.6472 3.6705 1.1201 1.5038 0.6940 17.097 19.924 2.2694 3.5644 0.6211 40.565 70.809 4.5574 7.7658 0.8023 70.722 216.94 8.6998 17.274 1.1424

3.6499 3.6965 1.1211 1.5064 0.6913 16.180 19.623 2.2508 3.5319 0.5850 34.756 68.610 4.5346 7.7272 0.7279 64.901 220.49 8.8589 17.605 0.9780

3.6601 3.6671 1.1193 1.4898 0.7172 16.505 19.452 2.2164 3.4535 0.7447 43.020 66.736 4.2668 7.1438 1.1055 70.722 216.94 8.6998 17.274 1.1424

3.6481 3.6958 1.1176 1.5058 0.6912 16.184 19.628 2.2441 3.5295 0.5847 34.742 68.638 4.5195 7.7121 0.7275 64.831 220.57 8.8216 17.509 0.9787

3.6499 3.6965 1.1211 1.5064 0.6913 16.180 19.623 2.2508 3.5319 0.5850 34.756 68.610 4.5346 7.7272 0.7279 64.901 220.49 8.8589 17.605 0.9780

3.6499 3.6965 1.1211 1.5064 0.6913 16.180 19.623 2.2508 3.5319 0.5850 34.756 68.610 4.5346 7.7272 0.7279 64.901 220.49 8.8589 17.605 0.9780

3.6490 3.6970 1.1170 1.5050 0.6920 16.178 19.617 2.2380 3.5280 0.5860 34.730 68.590 4.5090 7.7170 0.7290 64.630 219.36 8.8170 17.925 0.9890

Table 4.1: Shown results reveal five analyzed quantities, the maximum horizontal and vertical velocity defined on their respective perpendicular midline within the square cavity and the averaged, maximum and minimum Nusselt number on the heated wall of the square cavity. Results are obtained by QUICK interpolation on a non-uniform (100x100) grid where grid-refinement is applied near the boundaries. Two models from the literature De Vahl Davis [4] and Tian et al. [5] are shown in the last two columns for comparison. In their uniform applied grid they solved their equations with a (81x81) and (60x60) cells configuration, respectively.

rithms show consistency for all simulated Rayleigh numbers and no particular problems


44

were encountered during the convergence process. Although, underrelaxation parameters have been used down to 0.05 for the velocity corrections. Additionally, an oscillating convergence of the inner-iterations is observed already at lower Rayleigh numbers and became more pronounced when Rayleigh numbers increased. For the implicit algorithms the SIMPLEC algorithm showed to require the least amount of inner-iterations and the SIMPLEX seemed to require in general the most iterations. For the explicit algorithms the SIMPLER scheme outperformed the PRIME in the number of inner-iterations in every simulation. As the Rayleigh number increased the SIMPLER-algorithm showed to require relatively less- and less inner-iterations compared to the other tested algorithms. The details of the inner-iterations are shown in the appendix in figure A.10 and table A.3. The predominantly implicit algorithms (SIMPLE, SIMPLEC and SIMPLEX) show hardly no difference - in the prediction of the five analyzed quantities - with the literature under the current convergence criteria, while the more governed (semi-) explicit schemes (PRIME and SIMPLER) do show differences. The differences of the PRIME and SIMPLER algorithm become more evident with Rayleigh numbers starting at Ra = 105 and increase for higher Rayleigh numbers. The temperature and flow distribution is shown in figure 4.2. A symmetrical form of the flow and temperature field is observed, consistent with earlier performed studies [3][34]. While the flow field at lower Rayleigh numbers is characterized by a large vortex, from a Rayleigh number of Ra = 105 the large vortex is split into two vortices. Increasing furthermore the Rayleigh number leads to the onset of thermal disturbances in the form of plumes as can be observed at Rayleigh number Ra = 106 , also recognized by Bagchi [32].


Ra = 103

0

0

0.5 x/L

0

1

0.57

0.14

43 0.

0.28

0.43

0

0.2 8

2

0. 57

0.7

0.72

0.57

0.5

0.86

0.5

0.86

y/L

y/L

1

0.43

1

0.14

Ra = 103

45

0.5 x/L

Ra = 104

1

Ra = 104 0.14

0.

86

1

0.43 0.57

1

0.5 x/L

0

1

3 0.28

0.28

14

0

0.

0.5 x/L

Ra = 105

1

Ra = 105 1

7

0.43 0.57

1

0.8

0.13

0

0. 4 0.43

0.72

0

0.57

0.57

0.28 0.14

0.86 0.7 2

0.5

0.86

0.5

0.57 0.43

y/L

y/L

0.72

0.72

0.5

0.57 0.43

0.57

0.43

0.4 3 0.28 0.13

0.5

0. 0.7287 0.5 7

y/L

y/L

0.72

0.87 0.72 0.5 0.437 0.28

0.28

0

0.5 x/L

0

1

3

0.1

0

0.5 x/L

Ra = 106

1

Ra = 106 1

0.88

0

0

0.5 x/L

1

0.73

0.73

0.5

8

0.58

0.58

0

0

0.42 0.27

0.42 0.0.12 27

0.5 0.88 0.73 00..458 2

0.5

0.88

y/L

y/L

0.7

3

1

0.73 0.5 82 0.4 0.27 0.12

0

0.28

0.42

0.27

0.27

0.5 x/L

2 0.1

1

Figure 4.2: The flow and temperature distribution for different Rayleigh (Ra) numbers from top to bottom Ra = 103 , Ra = 104 , Ra = 105 and Ra = 106 and a constant Prandtl number P r = 0.71. Shown results are obtained by the PRIME algorithm and QUICK spatial interpolation on a non-uniform (100x100) grid configuration.


4.5

46

Conclusion

The numerical tool is once more extended to incorporate besides mixed convection, an additional flow solely governed by the natural convection in a square-shaped domain. An additional grid enhancement is implemented that clusters control volumes near the boundaries. Spatial interpolation is again performed with the QUICK scheme for the convective terms and central differencing is applied for the diffusion terms. Five quantities have been analyzed and compared to the literature that determine the temperature and flow field. Those quantities are implemented for the five considered pressure-velocity algorithms and are consequently compared to two sources from the literature. The previous obtained results from the mixed convection model indicated that the prediction of flow is consistent for every algorithm and that selection of those algorithms should be advised to do on a speed of convergence basis instead of an accuracy based one within the tested conditions. This is confirmed for simulations with lower Rayleigh numbers. However, it has been found that at higher Rayleigh numbers the accuracy is deteriorating and differences arise for the predominantly explicit algorithms. The implicit algorithms have been shown to be more consistent with the literature throughout the entire range of tested Rayleigh numbers. In all tested Rayleigh numbers the SIMPLEC algorithm required the least amount of iterations and showed similar results to the SIMPLE and SIMPLEX algorithm. Although the SIMPLER-algorithm required a large quantity of inner-iterations at lower Rayleigh numbers, it showed to converge more rapidly as compared to the other algorithms on higher Rayleigh numbers. The PRIME-algorithm seemed to be the slowest algorithm as it required in every simulation the most amount of inner-iterations. Considering all test cases it is found that there is not an ideal pressure-velocity coupling algorithm for the simulated problems, also recognized by other authors, among others [21][35][36]. However, the predominantly implicit algorithms have been shown to be more consistent in obtaining their scalar solutions as compared to the (semi-) explicit algorithms with respect to the applied benchmark models from the literature. Additionally, for pushing conditions the PRIME-algorithm in the natural convection model required more inner-iterations than any other tested algorithms and showed a tedious convergence, while a similar observation is made for the SIMPLER-algorithm in the mixed convection model. In the general case the SIMPLEC-algorithm required


47

the least amount of inner-iterations although the SIMPLE-, SIMPLER- and SIMPLEXalgorithms had specific simulations in which they outperformed any algorithm. For the former reasons, summarizing the three analyzed test cases to validate the numerical tool it can be concluded that an appropriate choice of a pressure-velocity algorithm takes into account (1) the dimensionless numbers (e.g., Reynolds, Grashof or Rayleigh number), (2) the preferred speed of convergence and (3) the desired accuracy one would like to obtain.

Chapter 5

The Hybrid Photo-Voltaic Thermal Solar Collector 5.1

Introduction

This chapter will present the application of a developed numerical tool for a specific solar collector. Having set-up two numerical models that separately showed the prediction of the flow and temperature field distribution in a mixed- and natural convective environment, this chapter will elucidate the application of these models to a specified geometry of the solar collector. The analysis in the geometry of the solar collector to predict the flow phenomena is very similar to what is encountered in the cooling of electronic devices or building segments, some examples are a cavity wall or building skins [37]. The analysis usually consists of two parallel flat-plates making a channel being inclined with respect to a certain reference, visualized in figure 5.1. Fluid enters at the base of the channel, while the plates impose a thermal condition on the fluid. The fluid is driven by an external pressure gradient (e.g., due to a pump or hydrostatic pressure) and an additional buoyancy force due to the imposed thermal conditions. Fluid is discharged at the top section of the channel and new fluid is sucked in from the bottom. Depending on the thermal contribution of the imposed buoyancy force - where the density is considered only a function of the temperature 1

1

- an induced flow rate may be produced inside the channel where

The Boussinesq approximation is considered [16].

48

Chapter 5. The Hybrid Solar-Panel

49

γ

Figure 5.1: A typical flat plate analysis where a fluid enters at the base, is consequently driven through the channel due to an external pressure gradient or thermal imposed condition and is being pushed towards the exit of the channel.

the extent of these conditions determine possible flow reversal downstream the channel, affecting the total energy transfer between the walls and the fluid [38]. A literature review of previous studies that focused closely on related problems reveal that a large variety of conditions have been studied before. Choi et al. [39] studied the effect of a laminar buoyancy-induced mixed convective flow. In a parallel plate configuration with a single local discrete heat source and prescribed temperatures they studied the effect of different flow inlet conditions and inclination angles for a variety of buoyancy-induced flow conditions. Rheault et al. [38] analyzed laminar mixed convection with asymmetric isothermal boundary conditions. Again, for different inclination angles and buoyancy-induced conditions they studied the effect on the local Nusselt number. Mokheimer et al [40] provided a numerical and analytical solution in the limiting case of a hydrodynamical and thermal fully developed flow with vertical parallel isothermal plates. A large amount of authors have studied similar problems that are a combination of the former elucidated analyzes with more recently - among other phenomena - the added effect of porous media, an induced magnetic flux or the analyzes on the micro-scale where the additional interaction of molecules is considered. However, the aim of these studies are dominantly dedicated at acquiring insight of flow phenomena that occurs in the cooling of electronic devices. The analyzes of flow phenomena in a solar panel is very similar to those encountered in the cooling systems applied in electronic devices, although there is a difference in the boundary conditions, that has to take into account the ambient. The evaluated solar panel basically consists out of two transparent rectangular shaped panes that have been put together with a small uniform gap in between where fluid is able to move from one side to the other, permitting the conventional thermal system. Additional solar cells


50

are mounted on-top of the pane hence in combination with the thermal system a hybrid solar collector. Due to the high absorptivity of the solar cells and the transparent spaces in between the solar cells there will be a non-uniform distributed radiation heat flux imposed to the thermal section of the solar panel. Moreover, solar panels are essentially always positioned outside or in an ambient environment indicating the presence of a convection heat transfer that occurs from the bottom part of the pane to the ambient air. This chapter will start by briefly introducing the solar collector itself and the considered application of the solar panel, in section 5.2. The problem is interpreted as a flow problem and the boundary conditions are clarified in a schematic representation. The analysis is subdivided into two separate models, (1) a mixed convection model where the inlet of the channel is directed with a prescribed inlet velocity condition and (2) a pure natural convection flow model that allows free flow at the entrance and exit of the channel. The governing equations are introduced and the number of physical quantities has been reduced by the application of a non-dimensionalization - for each model - with respect to the characteristics of the flow, in section 5.3 and 5.4. The final section, section 5.5 will cover the results of both the models and the entailing discussion. The mixed convection model will show results for Reynolds and Grashof numbers varying between Re = 1 - 100 and Gr = 102 - 104 , respectively. The pure natural convection model will provide results that are aimed at Rayleigh numbers varying between Ra = 103 - 104 . For both the models the effect of different inclination angles, flow and thermal conditions are explored by considering the cross-sectional flow field distribution, heat transfer rate on the lower-side of the panel and the thermal performance of the solar collector.

5.2

The Solar Collector

This section will introduce a flow transport analysis that will be performed on the inside of a hybrid photo-voltaic thermal (PVT) solar collector. The PVT system in general integrates the photovoltaic (PV) and thermal solar system for the co-generation of electrical and thermal power from solar energy [41]. The purpose of the thermal system is said to be twofold [13], it serves as (1) a method to optimize the mentioned electric efficiency of the solar cell and (2) the collection of thermal heat that otherwise would have been lost to the environment. The thermal system is commonly applied to


51

the back of the photo-voltaic collector with a small gap in between - typically 0.1 - 2[cm] - that will enable the flow of a working fluid, typically water, glycol, mineral oil or air [42]. The employment of an imposed mass flow or low inlet temperature fluid will enable the thermal system to extract - either by forced or natural ventilation - the absorbed heat that has not been converted into electricity and that otherwise would deteriorate the performance of the PV module [43]. Figure 5.2 shows the application of such a PVT solar collector in a domestic household. The installed solar collectors in the figure appear as ordinary double glazed windows that are placed underneath a certain angle. The PV modules are attached on the top-side of the windows allowing the absorption of sunlight and in between the double glazed layered window a working fluid is active enabling the discharge of any absorbed heat. The structured formation of the PV modules leave clear transparent spaces in between that will allow sunlight to pass through and illuminate the domestic unit. A

Figure 5.2: The domestic application of a series of hybrid solar collectors located inside an ordinary livingroom [1]. The hybrid solar collector is integrated into a double glazed window where the solar cells convert a portion of the solar energy into an electric current and the thermal system enables the redirection of any absorbed heat. This configuration of the solar collector has the potential to reinforce the air-conditioning system as it is able to discharge the incoming heat from the ambient by virtue of the incorporated thermal system.

characteristic of the solar cell is that it contains a rather large absorptivity of radiative sunlight due to its dark coloring. The capacity of the solar cell to absorb solar radiation reaches in fact so far that - although the solar cell converts a portion of its absorbed energy into electricity - the thermal system, just underneath the cell, will experience a


52

larger heat flux as opposed to the transparent spaces in between the solar cells. The transparent spaces located on the top-side of the solar panel will transmit most sunlight and will allow the fluid to absorb a portion of it. The combined effect of the solar cells and the transparent spaces will result in an imposed non-uniform distributed heat flux as perceived by the thermal system. The aforementioned conditions need to be prescribed carefully such that they endeavor reality. The boundary conditions as prescribed for the mixed- and natural convection are identical, with one small difference. There is no prescribed velocity for the natural convection case, instead a free in- and outflow is considered at both the entrance and exit of the channel. The upper- and lower-wall of the thermal system will prescribe the no-slip condition and the exit will establish a developed velocity condition. The thermal conditions take into account a cold prescribed inlet temperature and a thermal fully developed flow at the exit. The bottom-part of the solar-panel is exposed to a convective environment and an ambient temperature. The transmitted heat towards the environment is modelled by setting-up a heat balance at the bottom-wall between the conductive and convective heat flux. The upper-part of the solar-panel is exposed to sunlight and takes into account the radiation heat flux. Two solar cells with three transparent spaces in between have been considered and positioned on the upper-part of the solar collector. To cover these phenomena into a single thermal condition a nonuniform distributed constant discrete heat flux q = f (x) is adopted, being defined as,

q(x) =

  C,

if

x1 ≤ x ≤ x2 ,

x3 ≤ x ≤ x4 ,

 ζC,

if

0 ≤ x ≤ x1 ,

x2 ≤ x ≤ x3 ,

x4 ≤ x ≤ L.

where C is a constant, ζ is a heat intensity constant with a value between [0, 1] and x1 , x2 , x3 , x4 and L are specified positions. These boundary conditions are schematically shown in figure 5.3. The next section will briefly introduce and define separately the governing equations and boundary conditions for the mixed- and natural convection model.

53

∂u

∂x

t Ou tle

q= q=

-T

∞

x 4

H

(T k ∂T ∂y = h

0, q v=

T

u=

∞

g

MC

{

x 1

I y nlet x

x 2

=-

L

x 3

+γ

NC : u = v = :∂x ∂u= Uin 0, T 0 =T in

)

0, v=

u=

Tra nsp

are n

t sp

ace

C,

PV

cel l

ζ

C

= ∂v ∂x = ∂ ∂x T= 0


Figure 5.3: The geometry and dimensional boundary conditions for the mixed- and natural convection solar collector. The inclination angle γ is given parallel with respect to the direction of the acceleration of gravity. The prescribed velocity of the x-component at the inlet is distinguished for the two models. For mixed convection (MC) there is an imposed uniform inlet velocity u = Uin , while for the natural convection (NC) case there is a zero gradient active ∂u/∂x = 0 allowing in- and outflow of the fluid. The positions x1 , x2 , x3 and x4 indicate the location of the PV cells and L indicates the length of the channel where L = ϱH, where ϱ is the aspect ratio of the channel.

5.3 5.3.1

Mixed Convection in the Solar Collector Governing Equations

The governing equations incorporate mass, momentum and energy conservation and are shown earlier in a similar dimensional form in the eqs. 3.1 - 3.4. Note that the current system requires the source term of the Navier-Stokes to be rewritten in regard to the specified geometry (e.g., the specified angle γ

2

and the characteristic length H).

Additionally, there is no longer a hot and cold prescribed temperature. Instead, there is a prescribed cold inlet temperature and a prescribed heat flux, the latter thus replacing 2

This indicates that the standard acceleration due to gravity is in this case defined by ⃗g = (gx , gy ) = (gcos(γ), gsin(γ)), where g is a constant being equal to 9.81[m/s2 ] .


54

the hot temperature treated in the test cases. This will result in a subtle difference compared to the test cases and is introduced by means of the non-dimensional defined quantities that yield for the current case, x ˜=

x , H

y˜ =

y , H

u ˜=

u , Uin

v˜ =

v , Uin

tUin t˜ = , H

p˜ =

p 2 , ρUin

θ=

T − Tc . (5.1) ∆T

The careful reader may have observed that the non-dimensional temperature θ is defined by means of a temperature difference ∆T . This temperature difference previously incorporated the hot reference temperature and should be rewritten for this model to include the prescribed flux. In that regard, the application of the Fourier’s law turned out to be helpful and is rewritten with a scaling procedure enabling the temperature difference to correspond with ∆T ∼ q ′′ H/k. Substituting the dimensional quantities of the governing equations for these non-dimensional quantities, ∂u ˜ ∂˜ v + = 0, ∂x ˜ ∂ y˜ ∂ 1 ∂u ˜ ∂ (˜ uu ˜) + (˜ vu ˜) = + ˜ ∂ x ˜ ∂ y ˜ Re ∂t ∂ ∂ 1 ∂˜ v + (˜ uv˜) + (˜ v v˜) = ˜ ∂ y˜ Re ∂ t˜ ∂ x

( (

∂2u ˜ ∂2u ˜ + 2 ∂x ˜2 ∂ y˜ ∂ 2 v˜ ∂ 2 v˜ + ∂x ˜2 ∂ y˜2

(5.2)

) −

∂ p˜ ρgβ∆T H + cos(γ), ∂x ˜ ρU 2

(5.3)

)

∂ p˜ ρgβ∆T H + sin(γ), ∂ y˜ ρU 2 ( 2 ) ∂θ ∂θ ∂θ 1 ∂ θ ∂2θ +u ˜ + v˜ = + . ∂x ˜ ∂ y˜ P rRe ∂ x ˜2 ∂ y˜2 ∂ t˜ −

(5.4)

(5.5)

The governing equations are characterized by the non-dimensional numbers known as the Reynolds, Grashof and Prandtl numbers being defined in this case by, ReH =

ρUin H , µ

GrH =

gβ∆T H 3 , ν2

Pr =

µcp . k

(5.6)

Be aware that the temperature difference in the above presented Grashof number and the non-dimensional temperature are defined by means of the heat flux and can be written as, GrH =

gβqH 4 , kν 2

θ=

T − Tc . q ′′ H/k

(5.7)


5.3.2

55

Boundary Conditions

The dimensional boundary conditions given in the previous section (figure 5.3) are rewritten according to the non-dimensional parameters and yield for the current model, (˜ x = 0, y˜),

u ˜ = 1,

v˜ = 0,

θ = 0,

(5.8)

∂u ˜/∂ x ˜ = 0,

∂˜ v /∂ x ˜ = 0,

∂θ/∂ x ˜ = 0,

(5.9)

(˜ x, y˜ = 0),

u ˜ = 0,

v˜ = 0,

q˜k = q˜h ,

(˜ x, y˜ = 1),

u ˜ = 0,

v˜ = 0,

(˜ x=

L , y˜), H

(t = 0, x ˜, y˜),

u ˜ = 0,

v˜ = 0,

∂θ/∂ y˜ = f˜(˜ x), θ = 0.

(5.10) (5.11) (5.12)

The discretization of the general boundary conditions at this stage should be familiar to the reader and can be implemented rather straightforwardly without going into detail of separate conditions. There is one exception to the latter notion, as one condition requires an additional derivation, i.e., the non-dimensional diffusion flux q˜k at the bottom of the panel being set equal to the non-dimensional convection flux q˜h towards the ambient. They are introduced in the former defined non-dimensional boundary conditions by means of non-dimensional fluxes because the derivation of the discretization is done from a dimensional starting point and require some extra attention. For the considered condition, if one would like to couple the diffusion heat transfer through the boundary to the outside one commonly writes in dimensional form [18], ( q = ks

Tf − Tp ∆y

) = h(T∞ − Tf ),

(5.13)

where ks , h, Tf and T∞ are the solid heat conduction coefficient, the ambient convective heat transfer coefficient, the temperature at the border and the temperature of the environment, respectively. In order to implement the former relation, it is necessary to rewrite the relation to the standard form of the boundary equation (shown earlier in eq. 2.12). Having said that, the prescribed condition is rewritten such that the main boundary temperature Tp is written on one side of the equation, while the remaining temperatures are written on the other side similar to, ( h∆y ) h∆y Tp = 1 − Tf + T∞ . ks ks

(5.14)


56

Note that ∆y corresponds with the characteristic length scale H and consequently introducing the dimensionless Biot number Bi, providing a measure of the temperature drop in the solid relative to the temperature difference between the surface and the fluid outside [33]. The non-dimensional temperature (eq. 5.1) is introduced in the boundary equation by rewriting it for each temperature in the form of T = θ∆T + Tc and additionally the Biot number is substituted, after some rewriting the boundary equation appears to be,

( ) θp ∆T = θf ∆T 1 − Bi + θ∞ ∆T Bi.

(5.15)

For ∆T ̸= 0 and the non-dimensional temperature θf being expressed as a function of the temperatures found in the two adjacent control volumes (according to eq. 2.13) the standard form of the boundary equation is adopted and results in, ( θP = θN

1 − Bi 1 + Bi

)

(

) Biθ∞ +2 . 1 + Bi

(5.16)

Having obtained the standard form of the discretized equation, the coefficients are found accordingly,

( aP = 1,

aN =

) 1 − Bi , 1 + Bi

(

) Biθ∞ b=2 . 1 + Bi

(5.17)

Note that θ∞ is a prescribed value between [0, 1] and that a prescribed zero value means that θ∞ is set equal to incoming non-dimensional temperature of the channel, according to the definition of the non-dimensional temperature. Furthermore, Versteeg et al. mentioned that the SIMPLE-family does not guarantee global mass conservation. The previous test cases did not incorporate velocity fluxes normal to the boundary, that changed in the present model. To force a global mass conservation the in- and outlet velocity normal to the boundary is corrected by means of the ratio between the inner and outer mass flux. Versteeg et al. defined the outlet velocities by means of, u Ñ P I,J = u Ñ P I−1,J

Min , Mout

(5.18)

where N P I is the final defined velocity control volume in x ˜-direction and Min and Mout correspond to the inner and outer mass fluxes3 . This mass flux correction is computed ∫H The mass flux is obtained by computing the integral M = 0 ρ˜ u(˜ y )dA for each control volume, dA is said to be the cross-sectional area perpendicular to the flow of a single control volume and u ˜(˜ y ) the velocity x ˜-component in and out of the domain. 3


57

at the end of every inner-iteration and corrected at the new iteration step by means of the set boundary conditions.

5.4 5.4.1

Natural Convection in the Solar-Panel Governing Equations

The second model written for the solar panel takes into account solely a natural convective flow. This situation naturally happens when the pump attached to the system as a whole is deactivated or simply not present. It implies that there is no longer a prescribed velocity at the inlet and hence the non-dimensional parameters are written differently as opposed to the mixed convection model. The following non-dimensional parameters are adopted and the model is implemented accordingly, x ˜=

x , H

y˜ =

y , H

u ˜=

uH , α

v˜ =

vH , α

αt t˜ = 2 , H

p˜ =

pH 2 , ρα2

θ=

T − Tc . (5.19) ∆T

Substituting these non-dimensional parameters in the governing equations (shown earlier in eqs. 3.1 - 3.4) the resulting non-dimensional governing equations for the pure natural convection problem in the solar-panel yields, ∂u ˜ ∂˜ v + = 0, ∂x ˜ ∂ y˜ ∂u ˜ ∂ ∂ + (˜ uu ˜) + (˜ vu ˜) = P r ˜ ∂ y˜ ∂ t˜ ∂ x

(

∂2u ˜ ∂2u ˜ + 2 2 ∂x ˜ ∂ y˜

(5.20)

) −

∂ p˜ + P rRaH θcos(γ), ∂x ˜

) ∂ 2 v˜ ∂ 2 v˜ ∂ p˜ + P rRaH θsin(γ), + 2 − 2 ∂x ˜ ∂ y˜ ∂ y˜ ( 2 ) ∂θ ∂θ ∂θ ∂ θ ∂2θ +u ˜ + v˜ = + . ∂x ˜ ∂ y˜ ∂x ˜2 ∂ y˜2 ∂ t˜

∂˜ v ∂ ∂ + (˜ uv˜) + (˜ v v˜) = P r ˜ ∂x ˜ ∂ y˜ ∂t

(5.21)

(

(5.22)

(5.23)

The governing equations are characterized by the non-dimensional numbers known as the Rayleigh and Prandtl numbers being defined in this case by, RaH = GrH P r =

gβ∆T H 3 , να

Pr =

ν . α

(5.24)

Be aware that the temperature difference in the above presented Rayleigh number and the non-dimensional temperature are defined by means of the heat flux and can be

Chapter 5. The Hybrid Solar-Panel written as, RaH =

5.4.2

gβq ′′ H 4 , kνα

58

θ=

T − Tc . qH/k

(5.25)

Boundary Conditions

The boundary conditions that are implemented for the natural convection case are identical to the mixed convection model except for one, the prescribed flow condition at the inlet of the channel. The boundary conditions are presented regardless and yield, (˜ x = 0, y˜),

∂u ˜/∂ x ˜ = 0,

v˜ = 0,

θ = 0,

(5.26)

L , y˜), H

∂u ˜/∂ x ˜ = 0,

∂˜ v /∂ x ˜ = 0,

∂θ/∂ x ˜ = 0,

(5.27)

(˜ x, y˜ = 0),

u ˜ = 0,

v˜ = 0,

q˜k = q˜h ,

(5.28)

(˜ x, y˜ = 1),

u ˜ = 0,

v˜ = 0,

(˜ x=

(t = 0, x ˜, y˜),

u ˜ = 0,

v˜ = 0,

∂θ/∂ y˜ = f˜(˜ x), θ = 0.

(5.29) (5.30)

The non-dimensional conductional and convectional heat flux q˜k and q˜h have been elucidated in the previous section covering mixed convection and can be implemented analogue to that methodology. The following section will directly present the results of the latter two covered mixed- and natural convection models.

5.5


In this section the results obtained for the mixed and natural convection models are presented systematically. The two models will be presented in separate subsections and will follow below, where mixed convection is treated first in subsection 5.5.1 and natural convection will follow and is found in subsection 5.5.2. The simulations are captured with the SIMPLEC algorithm for the pressure-velocity coupling. That algorithm was shown in chapter 4 to perform on average with the least amount of iterations for the cavity-flow problem and omits the need to under-relaxate the pressure throughout the iterative process. The algebraic equations are solved yet again with successive overrelaxation (SOR) where under-relaxation was necessary for all simulated conditions. The spatial interpolation of the advection terms is solely done with the QUICK-scheme


59

and the diffusion terms are again interpolated with the central differencing scheme. A non-uniform structured, and forward staggered grid is adopted having (300x15) elements leading to rectangular-shaped control volume elements spread out in the domain. Near the boundary a grid-refinement is applied and implemented according to an algebraic transformation that is covered in appendix B where the minimum rectangular-shaped control volume is (2.0 x 2.3)·e−2 [-]. A grid dependency study is performed and presented in appendix section C.3.3. From that study it is found that the maximum discretization uncertainty percentage that occurs when determining the Nusselt number is 156.86[%], that corresponded to a value close to zero and an absolute error of ±0.37[-]. Two analytical solutions are furthermore presented in appendix D for the case where forced convection became decisive as opposed to the natural convection. The numerical model for mixed convection is compared to an analytical solution for two Reynolds numbers, Re = 1 and Re = 100. The maximum uncertainty of the numerical model compared to the analytical solution occurs at the Re = 100 case and is shown for the non-dimensional temperature to be 202[%] that corresponded to a value close to zero and appears to have an absolute error in nondimensional temperature of ±0.075.

5.5.1

Mixed Convection

Mixed convection is investigated in the specified rectangular shaped geometry of the hybrid PVT solar collector for Reynolds numbers varying in the range of Re = 1, Re = 10 and Re = 100, Grashof numbers Gr = 102 , Gr = 103 and Gr = 104 with a constant Prandtl number P r = 0.71, corresponding to that of air. Simulations are performed for three different inclination angles γ = 0, γ = π/4 and γ = π/2, where in the mixed convection model it is said that a flow is to be aiding when the external flow and buoyancy force are in the same hemispherical direction, in the current study γ = 0 and γ = π/4 are said to be aiding flows, while γ = π/2 is said to be a horizontal flow. The analyzes will start with the presentation of the velocity flow field, that will clarify the type of flow pattern that will occur under certain conditions. The analyzes continues with the temperature distribution along the top-side of the solar collector, that will provide information regarding the temperature of the solar cells. Having analyzed the velocity flow field and temperature along the top-side of the collector, attention is


60

dedicated towards the heat transfer through the bottom-part of the collector and the thermal performance of the collector.

5.5.1.1

The Velocity Flow Field

Figure 5.4 shows the effect of the Reynolds number and inclination angle on the recirculating flow pattern for a constant Grashof number (Gr = 104 ). The figure reveals two Reynolds numbers, the Re = 1 and the Re = 10 case with three different inclination angles γ = 0, γ = π/4 and γ = π/2. Although not shown in the figure, the length of hydrodynamical development for the Re = 100 case appears to be shorter than the length of the start of the first PV module, for any simulated inclination angle, indicating a prevailing Poiseuille flow throughout the solar collector and presentation of these figures is for that reason omitted to appendix A.6.2. In aiding flows (γ = 0, γ = π/4) a single recirculating cell is observed for the Re = 10 case, while a bicellular flow pattern is observed for the Re = 1 case provoked by local buoyancy forces. The recirculation that occurs in the channel is due to the imposed heat source on the top-side of the collector and a heat sink on the opposite wall as a result of the convective ambient. Choi and Ortega [39] investigated numerically a similar flow problem with a single discrete heat source on one side of the channel and a prescribed temperature on the opposite side and demonstrated the same observed flow patterns. Considering the rotational direction of the recirculation that occurs inside the channel, for aiding flows (γ = 0, γ = π/4) a clockwise rotation is witnessed, while a more complex flow is noticed for the horizontal geometry (γ = π/2) and is treated further down below. It is furthermore observed that the external flow in aiding geometries pushes the cell vortex towards the colder wall, that as a result allows the external flow to pass along the hot wall and perhaps more importantly prompts a flow reversal at the exit of the channel. The latter phenomena - in fact - could be beneficial forasmuch as cooling is concerned, as the colder external flow is in direct contact with the PV module and the external flow enables a discharge of heat. An increase of the Reynolds number will gradually fade out the existence of recirculating cells and enforce a Poiseuille-like velocity flow field. Finally, regarding the complex flow in the horizontal geometry (γ = π/2), a quadricellular flow pattern is observed for the Re = 1 case, while the Re = 10 case shows a


61

single recirculating cell at the entrance. In both cases an anti-clockwise recirculating cell occurs close to the entrance of the channel that appears to be trapped in front of the PV module. This feature is preventing the external flow to be in direct contact with the module that prevents potential cooling that could be essential to maintain a certain temperature for the solar cell. As the external flow remains active at the entrance, a continues cool flow will sweep the surface of the collector, contributing to a discharge of heat through the exit of the channel. The quadricellular flow pattern in the Re = 1 case also appeared in the pure natural convection model, where due to the symmetrical velocity boundary conditions is compared - although it is not the same - to the classical Bénard problem [44] and as such did provide more insight. More simulations are performed with equal input conditions (a constant Biot number Bi = 5, an ambient temperature θ∞ = θc , heat flux intensity ξ = 0.1 and a geometric aspect ratio ϱ = 20) but will not be covered in the main-text. All simulated velocity vector fields, also those that are not shown in this section, are presented with the corresponding isothermal lines in appendix A, section A.6.2.


62

4

o

Re = 10, Gr = 10 , γ = 0 1 0.5 0

0

2

4

6

8 10 12 14 4 o Re = 10, Gr = 10 , γ = 45

16

18

20

0

2

4

6

8 10 12 14 4 o Re = 10, Gr = 10 , γ = 90

16

18

20

0

2

4

6

8 10 12 14 4 o Re = 1, Gr = 10 , γ = 0

16

18

20

0

2

4

6

8 10 12 14 3 o Re = 1, Gr = 10 , γ = 45

16

18

20

0

2

4

6

8 10 12 14 4 o Re = 1, Gr = 10 , γ = 90

16

18

20

0

2

4

6

16

18

20

1 0.5 0 1 0.5 0 1 0.5 0 1 0.5 0 1 0.5 0

8

10

12

14

Figure 5.4: Velocity vector field for a constant Grashof number Gr = 104 , two different Reynolds numbers Re = 1 and Re = 10, for three different inclination angles γ = 0, γ = π/4 and γ = π/2, a constant Biot number Bi = 5, an ambient temperature θ∞ = θc , heat flux intensity ξ = 0.1 and a geometric aspect ratio ϱ = 20. The top-side is marked with two red lines that indicate the position of the PV modules. The flow field is furthermore clarified with stream vertices - indicated by black lines - that emphasize the direction of the flow.

Chapter 5. The Hybrid Solar-Panel 5.5.1.2

63

The Temperature Field Along Top-Side Solar Collector

Figure 5.5 shows the temperature distribution on the top-side of the collector for two Grashof numbers Gr = 103 and Gr = 104 . The two presented sub-figures show for a constant kept Grashof number the effect of the inclination angle and the Reynolds number on the temperature field. It is observed that an increase of Reynolds number will have a suppressing effect on the amplitude of the temperature, where the highest temperatures are reached for the Re = 1 case and the lowest temperatures for the Re = 100 case. In the former flow field analyzes it was already observed that recirculation patterns were gradually fading out for higher Reynolds number. In that sense the temperature field is acting intuitively, the external flow with high Reynolds number conditions will prevent recirculation and any absorbed heat by the fluid will only remain for a little time in the channel until it is discharged through the exit of the channel. For lower Reynolds number conditions the opposite is noticed, only a part of the external flow is recirculated while the remainder of the fluid is recirculated in the channel allowing the fluid to be heated up over a longer period of time and as a consequence an increased temperature of the PV module is observed. Considering the inclination angle, the largest amplitudes of temperature on the upperwall are observed in the horizontal configuration (γ = π/2). The latter observation suggests that the buoyancy force is acting on the fluid, driving and pushing the fluid against the upper-wall resulting in an increased observed temperature of the top surface. Analogue for the leaned-over configuration (γ = π/4), but in a minor extent due to the inclination that is favoring buoyant driven flow. The latter remark is a reference to the flow field, the external flow will in any simulated inclined configuration - except for the horizontal case, where trapped recirculation patterns were observed - be aided by the buoyancy force and thus favoring cooling. Out of the three simulated inclination angles, the vertical configuration (γ = 0) showed consistently the lowest temperature amplitudes on the top surface. A final observation is made regarding the thermal development length. It is noted that the thermal development length in any of the simulated conditions for the Gr = 103 case is shorter when compared to the simulations for the Gr = 104 case. A step-like behavior is observed for the Gr = 103 case, indicating a relative short thermal development length. Takahashi et al. [45] investigated the thermal development length in a laminar natural


64 Gr = 103

1 0.9 0.8

θ(x/H,y/H=1)

0.7 0.6 0.5 0.4 γ = π/2 γ = π/4 γ=0 Re = 1 Re = 10 Re = 100

0.3 0.2 0.1 0

0

5

10 x/H

15

20

Gr = 104 1 0.9 0.8

θ(x/H,y/H=1)

0.7 0.6 0.5 0.4 γ = π/2 γ = π/4 γ=0 Re = 1 Re = 10 Re = 100

0.3 0.2 0.1 0

0

5

10 x/H

15

20

Figure 5.5: The above figure presents the temperature distribution over the wall on the top-side of the solar collector. There are two PV modules mounted on the top-side of the collector inducing a local higher heat flux and as a result causing a local higher temperature at the position of the modules. Results are presented for three simulated Reynolds number Re = 1, Re = 10 and Re = 100, three inclination angles γ = 0, γ = π/4 and γ = π/2, two Grashof numbers Gr = 103 and Gr = 104 , a constant Biot number Bi = 5, an ambient temperature θ∞ = θc , heat flux intensity ξ = 0.1 and a geometric aspect ratio ϱ = 20.


65

convective flow for a vertical concentric pipe annuli with a uniform imposed heat flux over the length of the channel both experimental and numerical. They found a matching relation for the thermal development length that was almost proportional to the squareroot of the Grashof number (∼ Gr9/10 ). Although the situation is not identical, a similar type of behavior is observed in the figure where an increased Grashof number results in a longer thermal development length.

5.5.1.3

Heat Transfer

The heat transfer is analyzed by virtue of the Nusselt number and thermal efficiency of the solar collector, both quantities are presented in that segregated order. The Nusselt number is computed along the lower-side of the solar collector, representing a degree of heat transfer from the fluid to the wall, and by means of the boundary condition, from the wall to the convective ambient. Computation of the local Nusselt number is defined by the temperature gradient normal to the wall and is introduced earlier in chapter 3 (eq. 3.30). The analyzed Nusselt quantities are the averaged N u0 , maximum N umax and minimum N umin Nusselt number. The averaged Nusselt number is obtained by integrating the local Nusselt number over the length of the channel, defined earlier in chapter 4. The results are presented in table 5.1 for three simulated Reynolds Re = 1, Re = 10 and Re = 100, three Grashof numbers Gr = 102 , Gr = 103 and Gr = 104 and three inclination angles γ = 0, γ = π/4 and γ = π/2. The Nusselt number computed on the lower-wall is found to be most noticeable in the natural convection regime (Gr >> Re), while the lowest Nusselt numbers are found in the forced convection regime (Gr 10). The coefficients defined above are then rewritten to a following form [14], [ ] aW = Dw max 0, (1 − 0.1|P ew |)5 + max(Fw , 0),

(A.11)

[ ] aE = De max 0, (1 − 0.1|P ee |)5 + max(−Fe , 0).

(A.12)

Similar expressions are found for the South and North coefficients and the Peclet number is defined as the interfacial ratio between convection and diffusion P e = F/D. Note that equation A.4 is still valid with these new coefficients.

A.3

The Continuity Equation

The variables of interest in the current study are the velocities and the pressures or the pressure-gradients. However, until now only the Navier-Stokes in x ˜ and y˜ direction is treated and thus having mathematically an incomplete set of equations, for the three yet unknowns. Fortunately, there is still an equation waiting for the right moment to be solved. The SIMPLE-family applies the continuity equation to compute (correction) pressures at every node in the computational domain. The discretized continuity equation in a uniform grid yields, Fe − Fw + Fn − Fs = 0.

(A.13)

Appendix A. Detailed Simulation Data

80

Depending on the considered velocity-coupling scheme, two different discretized continuity equations are applied, the so-called pressure-correction equation and the actual pressure equation. The difference and derivation is now briefly shown. The principle of the pressure-correction equation is that the velocity u ˜ in the control volume is guessed by a velocity u ˜∗ and consequently corrected by the correction pressure p˜′ in the flow direction. Mathematically given by, u ˜P = u ˜∗P + u ˜′P = u ˜∗P + dP u (˜ p′w − p˜′e ),

(A.14)

v˜P = v˜P∗ + v˜P′ = v˜P∗ + dP v (˜ p′s − p˜′n ).

(A.15)

Substitution of the above defined velocities into the discretized continuity equation shown in eq. A.13 can be shown to be equal to, aP p˜′P = aE p˜′E + aW p˜′W + aN p˜′N + aS p˜′S + b′P ,

(A.16)

where in this dimensionless case the coefficients are defined by, aE = du˜,e Ae , aN = dv˜,n An ,

aW = du˜,w Aw ,

(A.17)

aS = dv˜,s As ,

(A.18)

and the source term is defined by, b′P = Fw∗ − Fe∗ + Fs∗ − Fn∗ .

(A.19)

The variable d in the above equations is the so-called d-coefficient and is defined by ratio of the flow directional cross-section of the control volume and the element node its main coefficient d = A/aP . The former source term b′P is very similar to the discretized continuity shown earlier, also observed by Patankar [26]. When the source term is sufficiently annihilated the pressure-corrections for the velocity field have done their job and reaches a zero value everywhere for which the solution satisfies mass conservation up to the last value of bP in the iterative process. For the actual pressure equation, things are only slightly different. The principle of the pressure equation is to solve the discretized Navier-Stokes equations with an initial or last converged velocity field without the pressure contribution. Additionally, as the


81

explicit determined velocities u ˆ and vˆ should honor mass conservation the obtained deficit that will occur in the discretized continuity determines the pressure field p˜ and is solved accordingly. Mathematically given by, u ˜P =

u ˆ∗P

∑ + dP u (˜ pw − p˜e ) =

v˜P = vˆP∗ + dP v (˜ ps − pñ ) =

ñb nb anb u aP u

+ dP u (˜ pw − p˜e ),

(A.20)

+ dP v (˜ ps − pñ ).

(A.21)

∑

ñb nb anb v aP v

Implementing the velocities in the discretized continuity equation shown in eq. A.13 results in the pressure equation, aP p˜P = aE p˜E + aW p˜W + aN pÑ + aS p˜S + bP ,

(A.22)

where the coefficients are exactly the same as defined earlier for the pressure-correction equation. The source term is only differently defined, it exists solely out of the so-called pseudo (explicitly determined) velocities u ˆ and vˆ and is given by, bP = Fˆw − Fê + Fˆs − Fˆn .

(A.23)

The next section will cover the application of these two different forms of the continuity equation and the differences and accessibility between the pressure-velocity schemes are emphasized.

A.3.1 A.3.1.1

The Tested Pressure-Velocity Coupling Algorithms The SIMPLE algorithm

This algorithm is developed by Patankar and Spalding [26] and the SIMPLE is an abbreviation for Semi-Implicit Method for Pressure-Linked Equations. It couples the pressure and velocities based on a guess and correct procedure. A guessed velocity and pressure is used to compute the fully discretized Navier-Stokes, the deficit that occurs when calculating the mass conservation determines the amount of correction required for the velocities. The velocities itself and pressures are corrected accordingly and a new iteration is initiated until the desired convergence is obtained. The operations in order of their execution are:


82

1. Guess a pressure field p˜∗ and an initial velocity field u ˜∗ and v˜∗ . 2. Set up the momentum and d-coefficients and solve the momentum equations. For Power-Law discretization one solves eq. A.4 and for QUICK discretization one ˜∗ and v˜∗ fields. solves eq. 2.24 with their known pressure p˜∗ and velocity u 3. Set up the coefficients for the pressure-correction equation and include the dcoefficients from the previous step. Solve the pressure-correction equation to obtain a pressure-correction p˜′ given in eq. A.16. 4. Correct the velocities u ˜∗ and v˜∗ with the pressure-correction p˜′ obtained in the previous step and shown in eq. A.14 - A.15 and correct the pressure p˜ by summing the guessed pressure p˜∗ and the pressure-correction p˜′ according to p˜ = p˜∗ + p˜′ . 5. In case of a coupled system, this would be the time to compute the other variables, for example the temperature field. 6. The last step, treat the corrected pressure p˜ as the new guessed pressure p˜∗ and set u ˜=u ˜∗ and v˜ = v˜∗ than return to step 2 of this routine until a converged solution is obtained.

For a transient problem the former sequence would cover a single time step ∆t and is repeated until the desired steady-state situation is obtained. As the problem is non-linear and a linearization is applied the solution will initially have troubles with converging. To deal with this constraint relaxation parameters are introduced to the pressure and velocity correction terms varying between values of 0 and 1. This is a crude method but will damp out oscillations occurring during the convergence process and could actually result in less computational effort. In this matter, the methodology given by Perić et al. [28] has found to be useful. By running multiple simulations to test the effectivity of relaxation parameters within the SIMPLE routine, they concluded that for any chosen relaxation dedicated to the velocities αu the optimum relaxation parameter for the pressure-correction is found to be αp = 1.1 − αu . They limited their analyses towards the SIMPLE routine, however it has been found that the SIMPLEX algorithms benefit from a similar approach.

Appendix A. Detailed Simulation Data A.3.1.2

83

The SIMPLEC algorithm

The second algorithm is the SIMPLEC algorithm, SIMPLE-consistent is developed by Van Doormaal and Raithby [36]. This algorithm is probably the algorithm that is most similar to the SIMPLE algorithm itself. The algorithm is introduced to omit the necessity of under-relaxating the pressure field. The only difference with the SIMPLEalgorithm is the computation of the velocity corrections. The u-velocity correction equation from the SIMPLE algorithm was given by u′p = dp (p′W − p′P ) where dp was defined earlier in equations A.14 - A.15. However, in the SIMPLEC routine the dcoefficient term is reformulated and given by Moukalled [21] et al., d˜u˜,p = ˜ is defined by, where H

du˜ , ˜ 1−H

∑ ˜ = H

anb u ñb . aP

(A.24)

(A.25)

This step has to be calculated between step 2 and 3 of the SIMPLE sequence given above so it can be used to calculate the pressure-correction p˜′ field i.e., step 3 of the routine. For the y-component of the momentum equation, u ˜ is replaced by v˜ in the above equations. A benefit of the SIMPLEC method that it is no longer necessary to under-relaxate the pressures and thus omitting the need of finding the correct under-relaxation parameter for the pressures, the velocities however still require an under-relaxation.

A.3.1.3

The SIMPLER algorithm

The third implemented algorithm is the SIMPLER algorithm, SIMPLE-revised is developed by Patankar [26]. The difference compared to the SIMPLE- and SIMPLEC algorithm is that this algorithm is not solving a pressure-correction equation, but an actual pressure equation for the pressure field. The result is a pressure field that can be applied directly to the momentum equations without the use of a pressure-correction. Velocities are however, still obtained through the pressure-correction equation and thus an extra equation has to be solved each iteration leading to more computational effort each iteration. The benefit of this procedure is a highly stable algorithm with excellent efficient convergence for both velocity and pressure field [17], making up for the extra


84

computational effort. The pressure equation is identical to the pressure-correction equation with the exception of the source term bp shown in eq. A.23. The assumption in the SIMPLE- and SIMPLEC algorithms is that the contribution of the neighbouring velocities is neglected in the derivation of the pressure-correction equation. In the SIMPLER algorithm this assumption is no longer neglected and is included in the pressure equation with the pseudo-velocities u ˆ and vˆ given by, ∑ u ˆp = ∑ vˆp =

anb u ñb + bP , aP

(A.26)

anb vñb + bP . aP

(A.27)

The pseudo-velocities are thus explicitly defined without the contribution of pressure, the remaining coefficients are defined identical to the earlier defined velocities u ˜ and v˜. Although the pressure equation is solved in the SIMPLER routine, the coefficients from the pressure-correction equation can be used to solve the pressure equation shown in eq. A.16. The operations for the SIMPLER algorithm is done in the following sequential order: 1. Guess the initial velocity field u ˜∗ and v˜∗ . In this routine it is not necessary to estimate a pressure p˜ at this point. 2. Set up the momentum coefficients and solve explicitly the pseudo-velocities u ˆ and vˆ according to eq. A.26 - A.27 and thus without the contribution of the pressure p˜! 3. Set up the coefficients from the pressure-correction equation p˜′ with the additional redefined source term for the pressure equation shown in eq. A.23. Solve the pressures p˜ by calculating the pressure equation with these coefficients and new defined source term. 4. With the explicit determined momentum coefficients incorporate the pressure p˜∗ obtained in the previous step and solve the momentum equations implicitly. Again, for Power-Law discretization one solves eq. A.4 and for QUICK discretization one solves eq. 2.24 with their known pressure p˜ and pseudo-velocity u ˆ and vˆ fields.


85

5. Calculate the source term for the pressure-correction equation and with the already determined coefficients for the pressure-equation solve the pressure-correction equation to obtain p˜′ for the velocities u ˜ and v˜. 6. Correct the velocities with the expressions shown in eq. A.14 - A.15 with previously determined p˜′ . Correcting the pressure field is not necessary as the actual pressure p˜ is already determined. 7. In case of a coupled system, this would be the time to compute the other variables, for example the temperature field. 8. Set u ˜=u ˜∗ and v˜ = v˜∗ and go back to step 2 until convergence is obtained.

The former sequence is for a single time-step ∆t and should be repeated until a defined steady-state solution is obtained. Due to the pseudo-velocities explicitly defined without the pressure field the SIMPLER algorithm has a unique convergence behavior that becomes evident when comparing it to other pressure-velocity coupling algorithms. Relaxation parameters are chosen identical to those of the SIMPLEC algorithm. This is due to that in step 4 of the above presented SIMPLER routine the momentum coefficients are not recalculated to incorporate the pressure, instead the pressure is added outside of the discretized equations. Although one could enforce a relaxation towards the added pressure in this step, it has been found not necessary. The velocities are still required to relaxate and thus being similar to the SIMPLEC routine.

A.3.1.4

The SIMPLEX algorithm

The fourth implemented algorithm is the SIMPLEX algorithm, SIMPLE-extrapolated [36]. The SIMPLEX algorithm takes care of the rate of convergence at grid refinement, it does so by considering the neighbouring pressure differences. This is accomplished by using an extrapolation technique to express all pressure differences in the domain in terms of the pressure difference local to the velocity [21]. The difference in implementation compared to the ordinary SIMPLE routine is the d-coefficient used in the pressure correction equation. This has to be solved separately and is defined by, ∑ dˆX u ˜,p

=

au˜,nb du˜,nb 1 + . au˜,p au˜,p

(A.28)


86

This dˆX u ˜,p -coefficient has to be determined right after the calculation of the momentum equations i.e., between step 2-3 of the SIMPLE-routine similar to the SIMPLEC routine. The coefficients au˜,nb and du˜,nb are already calculated in the momentum equation operation and will not have to be recalculated. The coefficient dˆX u ˜,p is than used to replace the former du˜,p to compute the pressure correction equation. The remaining steps are equal to the SIMPLE routine shown earlier. For the y˜-component of the momentum equation, u ˜ is replaced by v˜ in the above equation. Relaxation parameters are chosen similar to the SIMPLE algorithm, where the relaxation parameter for the pressure-corrections is defined as a function of the relaxation parameter for the velocity-corrections, see also the section regarding the SIMPLE routine.

A.3.1.5

The PRIME algorithm

The last implemented algorithm is the PRIME algorithm, the algorithm is developed by Maliska and Raithby [18]. This algorithm is very similar to the previous shown SIMPLER algorithm with the difference that the momentum equation is solely solved explicitly. That makes this algorithm probably out of the five implemented algorithms the most easy to implement. Additionally, as the PRIME algorithm omits the step of solving the momentum equations implicitly there is a reduction of computational effort. The velocities are explicitly corrected every iteration by the actual pressure. The operations for this algorithm are as follows: 1. Guess the initial velocity field u ˜∗ and v˜∗ . 2. Set up the momentum coefficients and solve explicitly the pseudo-velocities u ˆ and vˆ according to eq. A.26 - A.27 and thus without the contribution of the pressure p˜! 3. Set up the coefficients from the pressure-correction equation p˜′ with the additional redefined source term for the pressure equation shown in eq. A.23. Solve the pressures p˜ by calculating the pressure equation with these coefficients and new defined source term.


87

4. Correct the calculated pseudo-velocities u ˆ and vˆ from step 2 explicitly with the expressions shown in eq. A.20 and eq. A.21 with previously determined p˜. Correcting the pressures is not necessary as the actual pressure p˜ is already determined. 5. In case of a coupled system, this would be the time to compute the other variables, for example the temperature field. 6. Set u ˜=u ˜∗ and v˜ = v˜∗ and go back to step 2 until convergence is obtained.

When the PRIME algorithm is compared to the SIMPLER algorithm one may observe that the momentum equations are solely solved explicitly i.e., the velocities u ˜ and v˜ are obtained only by correcting the pseudo-velocities u ˆ and vˆ with the pressure p˜ contribution, while the SIMPLER algorithm does incorporate an implicit step in its routine. The computational effort of the PRIME algorithm each iteration is thus less compared to the SIMPLER algorithm.

A.3.2

The Velocity Field Distribution Compared to a Literature Source


88

x−coordinate 1

0

0.4

0.6

0.8

1

1: PRIME 2: SIMPLE 3: SIMPLER 4: SIMPLEC 5: SIMPLEX Ghia et al.

1

0.6

0.6

0.2

0.4

−0.2

0.2

−0.6

v−velocity

y−coordinate

0.8

0.2

0 −1

−0.6

−0.2 0.2 u−velocity

0.6

1

−1

Figure A.1: The solution of the flow field for the x ˜- and y ˜ component of the NavierStokes through the geometric midsection of the cavity indicated by blue and black coloring of the axis, respectively. The blue and black triangles indicate the respective data of the velocity profiles taken from the literature. Solution is obtained for a Re = 400 and QUICK-interpolation.


89

x−coordinate 1

0

0.4

0.6

0.8

1

1: PRIME 2: SIMPLE 3: SIMPLER 4: SIMPLEC 5: SIMPLEX Ghia et al.

1

0.6

0.6

0.2

0.4

−0.2

0.2

−0.6

v−velocity

y−coordinate

0.8

0.2

0 −1

−0.6


0.6

1

−1

Figure A.2: The solution of the flow field for the x ˜- and y ˜ component of the NavierStokes through the geometric midsection of the cavity indicated by blue and black coloring of the axis, respectively. The blue and black triangles indicate the respective data of the velocity profiles taken from the literature. Solution is obtained for a Re = 1000 and QUICK-interpolation.


90

x−coordinate 1

0

0.4

0.6

0.8

1

PRIME SIMPLE SIMPLEC SIMPLER SIMPLEX Ghia et al.

1

0.6

0.6

0.2

0.4

−0.2

0.2

−0.6

v−velocity

y−coordinate

0.8

0.2

0 −1

−0.6


0.6

1

−1

Figure A.3: The solution of the flow field for the x ˜- and y ˜ component of the NavierStokes through the geometric midsection of the cavity indicated by blue and black coloring of the axis, respectively. The blue and black triangles indicate the respective data of the velocity profiles taken from the literature. Solution is obtained for a Re = 10000 and QUICK-interpolation. (Laminar modelling).


A.3.3

91

The Vortex Positions Compared to a Literature Source

Idris et al. [6] provided numerical values in their study that was dedicated to the position of the primary and secondary vortex centers 1 . They used a non-uniform (50x50) mesh, where they clustered the elements near the boundaries. With a finite differencing method they solved the vorticity-stream function formulation of the Navier-Stokes. However, they did not mention the spatial interpolation scheme(s) they used to approximate the spatial terms and additionally did not mention any specific details regarding grid dependency nor minimum or maximum grid size in their non-uniform meshing. Furthermore, they validated their results based upon the observation that the primary vortex - that was offset for the geometric center at a Reynolds from Re = 100 - was moving towards the geometric midsection for increasing Reynolds numbers. They found out that the obtained results did not meet with their applied benchmark model, but concluded a successful implementation nonetheless. They did however, publish numerical values regarding the positions of the vortices, table A.1 shows these positions along with the results from the present study that are obtained with the SIMPLE-algorithm. A note to the used methodology to determine the position of the centers of the vortices. Re

100 400 1000

x y x y x y

Primary vortex

[6]

0.612 0.740 0.553 0.605 0.536 0.566

0.611 0.737 0.538 0.611 0.538 0.575

Secondary [6] vortex (upstream) 0.933 0.945 0.052 0.055 0.882 0.880 0.111 0.120 0.866 0.860 0.096 0.102

Secondary vortex (downstream) 0.033 0.038 0.055 0.049 0.088 0.087

[6]

0.032 0.032 0.055 0.043 0.085 0.085

Difference

Percentage

(absolute) 0.005

uncertainty 0.47

0.006

0.63

0.005

0.47

Table A.1: The position, mean absolute error and the relative absolute mean uncertainty percentage of the centres of the primary vortex and the two secondary vortices for varying Reynolds number between Re = 100 − 1000 compared to literature data from Idris et al. [6].

Idris et al. did not provide any information of how he determined these positions, making it harder to compare the current results. One could determine the centers of vortices throughout calculation of the curl of a velocity vector or field and taking the highest value for the curl, or one could determine the curvature density occurring in a streamline figure and again taking the highest resulting value. In the present study an interpolation has been applied to determine the centers of the vortices. All these methods however, affect the positions of the vortex centers slightly and by that affecting the uncertainty that occurs when comparing two different studies. A comparison between the current 1 The primary and secondary vortices are observed earlier in figure 2.3 that showed the streamline solutions for four different Reynolds numbers in the square-cavity.


92

study and that of Idris et al. lead to uncertainties that are smaller than 1% and will be considered reasonable.

A.3.4

Graphical Representation of the Uncertainty for the Five Algorithms x/L−component

y/L−component

2.5%

2.5

2%

2

1.5%

1.5

1%

1

0.5%

0.5

0%

0


PRIME SIMPLE SIMPLER Re=10000 SIMPLEC Re=1000 SIMPLEX Re=400 Re=100

Re=10000 Re=1000 Re=400 Re=100

Figure A.4: The mean absolute percentage uncertainty δE for every pressure-velocity coupling scheme compared to the literature benchmark model [2] for the solution of the Navier-Stokes inside the square-cavity. Results are given as a function of the Reynolds number using QUICK interpolation. The x ˜- and y˜ of the Navier-Stokes are given on the left and right side, respectively.


93

x/L−component

y/L−component

2.5%

2.5%

2%

2%

1.5%

1.5%

1%

1%

0.5%

0.5%

0%

0%



NaN Re=1000 Re=400 Re=100

NaN Re=1000 Re=400 Re=100

Figure A.5: The mean absolute percentage uncertainty δE for every pressure-velocity coupling scheme compared to the literature benchmark model [2] for the solution of the Navier-Stokes inside the square-cavity. Results are given as a function of the Reynolds number using Power-Law interpolation. The x ˜- and y˜ of the Navier-Stokes are given on the left and right side, respectively.

A.4 A.4.1

Validation Model Mixed Convection The Inner-Iterations in the Mixed Convection Model

Ri

PRIME

SIMPLE

SIMPLER

SIMPLEC

SIMPLEX

10000

23448

10920

26350

6348

6315

100

36559

11655

40093

10121

9986

1

57396

11034

46878

8228

9125

0.01

45486

10393

106583

8695

10691

Table A.2: The total amount of inner-iterations until convergence is obtained for all time-steps summed up for every pressure-velocity coupling algorithm as a function of the four simulated Richardson numbers. In the presented order of Richardson numbers the Reynolds varied from Re = 1, Re = 10, Re = 100 and Re = 1000 for a constant Grashof number Gr = 104 and a constant Prandtl number P r = 0.71.


A.4.2

94

The Outer-Iterations in the Mixed Convection Model t−Residual for Re = 10000

1

10 PRIME SIMPLE SIMPLER SIMPLEC SIMPLEX

0

10

0

t−Residual

t−Residual

−1

−2

10

−3

−2

10

10

−4

−4

0

5

10 Time−Steps

10

15

t−Residual for Ri = 1

1

0

5

10 15 Time−Steps

20

t−Residual for Ri = 0.01

1

10

10 PRIME SIMPLE SIMPLER SIMPLEC SIMPLEX

0

10


0

10

−1

−1

10

t−Residual

t−Residual

10

−3

10

−2

10

−3

10

−2

10

−3

10

10

−4

10


10

−1

10

10

t−Residual for Ri = 100

1

10

−4

0

10

20 30 Time−Steps

40

50

10

0

50

100 Time−Steps

150

Figure A.6: The outer-iterations (time-residual) for every simulated Richardson number. In the presented order of Richardson numbers the Reynolds varied from Re = 1, Re = 10, Re = 100 and Re = 1000 for a constant Grashof number Gr = 104 and a constant Prandtl number P r = 0.71.


A.4.3

95

The Velocity Flow Field Distribution in the Mixed Convection Model Ri = 10000

Ri = 100

0.8

0.8

0.6

0.6 y/L

1

y/L

1

0.4

0.4

0.2

0.2

0

0

0.5 x/L

0

1

0

Ri = 1

0.5 x/L

1

Ri = 0.01

0.8

0.8

0.6

0.6 y/L

1

y/L

1

0.4

0.4

0.2

0.2

0

0

0.5 x/L

1

0

0

0.5 x/L

Figure A.7: The velocity flow field distribution for different Richardson (Ri) numbers. In the presented order of Richardson numbers the Reynolds varied from Re = 1, Re = 10, Re = 100 and Re = 1000 for a constant Grashof number Gr = 104 and a constant Prandtl number P r = 0.71.

1


A.4.4

96

The Application of a Non-Uniform Grid Ri = 100 PRIME SIMPLE SIMPLER SIMPLEC SIMPLEX Yapici

3 2.5

5.57% −0.296%

NuL

2

1.32% 1.5

−0.346% 1

−0.368%

0.5 0

0

0.2

0.4

0.6

0.8

1

x/L

Figure A.8: The Nusselt number on the top-lid of the square cavity for a Richardson number of Ri = 100 (Re = 10, Gr = 104 , P r = 0.71). It is the equivalent of figure 3.3 except obtained in a non-uniform applied physical domain. The difference in percentages between a uniform and non-uniform applied domain with respect to the literature model is shown on the right. 0

0

10

10 SIMPLE SIMPLEC SIMPLEX

−2

−2

10 Averaged Residual

PRIME SIMPLER 10

−4

−4

10

10

−6

−6

10

10

−8

−8

10

10

2000 4000 6000 8000 1000012000 Iteration

1

2

3

4

5

6 4

x 10

Figure A.9: The inner-iterations for the pressure-velocity coupling algorithms during the convergence process for a Richardson number of Ri = 100 (Re = 10, Gr = 104 , P r = 0.71) in the non-uniform applied physical domain. Each hump during the convergence is the onset point of a new time-step and the convergence within that time-step until the set threshold value.


A.5 A.5.1

97

Validation Model Natural Convection The Inner-Iterations in the Natural Convection Model

Ra

PRIME

SIMPLE

SIMPLER

SIMPLEC

SIMPLEX

103

65459

23301

52594

23301

25627

104

45875

15835

34468

6022

6780

105

219342

47513

24824

47440

159782

106

649535

94206

75695

75561

99616

Table A.3: The total amount of inner-iterations until convergence is obtained for all time-steps summed up for every pressure-velocity coupling algorithm as a function of the four simulated Rayleigh numbers and a constant Prandtl number P r = 0.71.


0

0

Averaged Residual

10


PRIME SIMPLER

−5

−5

10

10

0.5

1 1.5 Iteration

2

2.5

Averaged Residual

4

6 4

x 10

0

x 10 0


PRIME SIMPLER

−5

−5

10

10

5000 10000 Iteration

15000

1

2

3

4 4

x 10

0

0


2

4

10


PRIME SIMPLER

−5

−5

10

10

5

10 Iteration

15

0.5

1

1.5

4

0

2 5

x 10

x 10 0


98


PRIME SIMPLER

−5

−5

10

10

2

4 6 Iteration

8

2 4

x 10

4

6 5

x 10

Figure A.10: The inner-iterations that incorporate the averaged residual between the u ˜- and v˜-velocity and temperature residuals for the natural convection model until the set criteria for the averaged residual for these latter mentioned equations is obtained. From the top to the bottom it shows the Ra = 103 , 104 , 105 and 106 , respectively. The left and right show the inner-iterations of the predominantly implicit and explicit algorithms, respectively.


A.6 A.6.1

99

The Solar Collector Thermal Efficiency

A method to characterize the performance for each and every simulation is by means of a total efficiency. Considering the PVT solar collector there are two embedded efficiencies, the thermal efficiency ηth and the electrical efficiency ηe of the PV cell where the summation of the two provides an indication for the total efficiency ηt , ηt = ηth + ηe .

(A.29)

The electrical efficiency of the PV cell is not considered in the current work as a discrete heat source is imposed, instead of the actual radiation heat flux computed through the solar cell. Notwithstanding the imposed heat flux, the literature is covered with authors that worked on the efficiency of specific PV cells, either by application of a different type of material or a different geometric design, some example references are [10][12][51]. Instead of the total efficiency, a more profound analyzes is conducted regarding the thermal efficiency of the numerical model. The thermal efficiency is a measure that will point out the capacity of the fluid - flowing through the solar collector - to absorb the incoming radiation heat flux. It is a relevant quantity for the engineer that is required to design a collector that can cool or discharge the most amount of heat. A high thermal efficiency will imply that the system is able to discharge heat sufficiently, while a low thermal efficiency will indicate the opposite. It is defined by the ratio of collected heat Q exiting the channel and the solar radiation flux G exposed to the surface of the collector, ηth =

Q . G

(A.30)

The collected heat is defined in its dimensional form by means of the in- and outgoing temperature, Q = mc ˙ p (Tout − Tc ).

(A.31)

Rewriting the latter expression for the collected heat Q towards a non-dimensional form ˜ will result in, of collected heat Q ˜MC = Q = u Q ˜θout P rRe, q0 Ac

ÑC = Q = u Q ˜θout , q0 Ac

(A.32)


100

for mixed- and natural convection, respectively. In the above equations Ac is the crosssectional area and θout is defined by the integral over the cross-sectional area θout = ∫ ˜ Additionally, in the above equation to obtain the given expression x = ϱ, y˜)dA. ˜c θ(˜ A one has to substitute the thermal diffusion coefficient by α = k/ρcp . The solar radiation flux G is covered with a discrete heat flux in the current model, the considered expression is the following, G = q0 A,

(A.33)

where A is the exposed surface directly to the sun. Again considering the non-dimensionalization the latter expression can be written as, ∂θ ˜ = q0 x H = x G ˜ . k∆T L ∂ y˜

(A.34)

The thermal efficiency defined by its non-dimensional parameters is obtained for substi˜ and G, ˜ tuting Q η˜th =

˜MC Q P rRe u ˜θout , = ˜ x ˜(∂θ/∂ y˜) G

η˜th =

ÑC Q 1 u ˜θout . = ˜ x ˜(∂θ/∂ y˜) G

(A.35)

Again, for mixed- and natural convection, respectively. The non-dimensional velocity and temperature are integrated over the exit of the channel, this is believed to be a more accurate solution than taking an average of the two variables. The expression for the integral over the cross-sectional area at the exit takes the following form for mixed convection,

∫ ˜ Q P rRe ˜ η˜th = = u ˜(˜ x = ϱ, y˜)θout (˜ x = ϱ, y˜)dA, ˜ x ˜(∂θ/∂ y˜) A˜c G

(A.36)

where both the length x ˜ and the non-dimensional temperature gradient (∂θ/∂ y˜) in the denominator are considered constant. The former expression is solved with a trapezoidal rule. The natural convection is written analogue to the former given expression for mixed convection.

A.6.2

Flow Phenomena in Mixed Convection Flow Solar Collector

The following section is presenting the velocity vector field with corresponding isothermal lines for simulations performed with a constant Biot number Bi = 5, an ambient temperature θ∞ = θc , heat flux intensity ξ = 0.1 and a geometric aspect ratio ϱ = 20.


101

The top-side of the figures are marked with two red lines that indicate the position of the PV modules. The flow field is furthermore clarified with stream vertices - indicated by black lines - that emphasize the direction of the flow. The order of figure presentation is as follows, every subsequent figure has a Grashof number increase for the corresponding Reynolds number and inclination angle. After the simulated Grashof number conditions are presented for the corresponding aforementioned conditions, the inclination angle is varied, after all the inclination angles are presented the Reynolds number is increased and the rotation starts over again. 2

o

Re = 1, Gr = 10 , γ = 0 1 0.5

2

0.17

4

0.26

8

10

0.1

7

18 0.82 0.73

16 0.92

0.7

3

5

4

0.08

6

14

0 0.26 .36

12 3

0.64 0.54 0.45

20

0.64 0. 0 26 .3 0.40.5 6 5 4

6

0.3

0.26

0.17

0.54

0.45

0.5

82

4 0.6

12 0.

82

3

10 0.64

0.

0.92

0.7

0.08

0

8 0.73

6

0.4

4

0.64 0.54

0.45

0.08

0.5 0

2

36

0 0.36 0.26 0.17 0.08

1

0.

0

7 0.1 0.08

14

16

14

16

18

20

18

20

o

Re = 1, Gr = 10 , γ = 0 1

2

12 0. 82

0.7 0.6 3 3

10

0.45

0.36 0.26 0.17

4

0.

0.2

6

0.08

6

8

10

36

0.91

0.7 03.6 3 0 0.4 .54 5

6 .3 60 0.2 0.08

0.17

12

0.7 0.4 05.50. 3 4 63

4

0.5

0.08

0

8 0.91

2

6 0.7 0.63 3

0.8

4 0.45

0.36

0.26 7 0.1

0.08

0.5 0

2

54

0 0.17 0.08

1

0.

0

0.82

0.5

14

16

14

16

18

20

18

20

Re = 1, Gr = 104 , γ = 0o 1 0.5 0

0

4

6

0.29 2 0.2

0.14 0.06

0.5 0

2

0.06

1

0. 14

8

10

0.5 0 9 0 .52 0. .44 37 0. 29 0. 22

0.67 0 0.5 .59 2 0.44 7 0.3 0.29 0.22 0.14 0.06

0.06

0

2

4

6

12

8

10

12

5 0.7

0 0 .52 0.3 .44 7 0.22 0.14

14

16

18

20


102 2

o

Re = 1, Gr = 10 , γ = 45 1 0.5

6

0.26

0.17

4

5

3

0.4

0.08

6

8

10

16

18

0.92

0.7

2

4

14

0 0.17 .26

0.36

12

0.64 0.54 0.45

20

0.64 0. 0 26 .3 0.40.5 6 5 4

8 0.

0.54

0.45

0.5

4

82

4 0.6

12 0.6

0.

3

0.3

0.26

2

10

0.92

0.7

0.08

0

8

0.8 0.732

6

0.73

4

0.64 0.54

0.45

0.17

0.08

0.5 0

2

36

0 0.36 0.26 0.17 0.08

1

0.

0

7 0.1 0.08

14

16

18

20

14

16

18

20

Re = 1, Gr = 103 , γ = 45o 1

0.91

54

0.

0.45 0.35

0.63

0.08

0

8

2

0.26 0.17

4

0.45

0.35

6

8

10

10

12 0.

82

0.91

0.7

2 0 .63 0.5 0 4 0.3 .45 5 0.2 6 0.17 0.08

12

14

0.

6 802 .72

3

4 0.

0.26 0.17

0.08

0.5 0

2

0.6

0 7 0.1 0.08

1

0.5 0.7 2 4

0

0.82 0.72 350 .405. 0 54.63

0.5

6 7 0.2 0.1 0.08

16

18

20

18

20

Re = 10, Gr = 104 , γ = 45o 1 0.5 0

0

2

4 30

0

2

10

14

22

07

4

6

12 0.5 0.45 3 0.38 0.30 0.22 0.14 0.07

0.

0.

0.

0

8 0.6 1

0.

0.5

6 0 0 .5 0. .45 3 38

2 0.2 4 0.1 0.07

1

8

10

12

14

16

0.6 0.68 1

6 0.7

0.53 0.4 0.38 5 0.30 0.22 0.14 0.07

14

16

18

20


103 2

o

Re = 1, Gr = 10 , γ = 90 1 0.5

8

10

16 0.91

0.7

3

0 0.17 .26

0.08

6

14 0.45

0.36

18

20 0. 0 0 0 26 .3 .4 .54 6 5

0. 45

2

36

4

0.26

0.17

0.8

0.45

0.

0.26

0.17

0.64 0.54

12

0.64 4 0.5

2

0.8

3

2

10

0.91

0.7

0.08

0

8

0.8 0.732

6

0.73

4

0.54 0.64

0.45

0.08

0.5 0

2

6

0 0.36 0.26 0.17 0.08

1

0.3

0

0.64 0.54

7

0.1 0.08

12

14

16

18

20

14

16

18

20

Re = 1, Gr = 103 , γ = 90o 1

6

4

0.17

6

82

6

5

0.5

4

3 0.

0.26

0.08

6

8

0.91

0.

0.3

0.17

0.26

0.63

0.54

12 3

3

0.4

2

10 0.6

82

0.

0.7

0.08

0

8 0.91

0.7 3

10

0.17

0.26

0.36

12

0.63 0.54 0.45

0.63 0. 0.4 0.5 36 5 4

4

0.63 0.54

0.45

0.08

0.5 0

2

0.73

0 0.36 0.26 0.17 0.08

1

0.4 5

0

0.8 0.732

0.5

26 07. 0.1 0.08

14

16

18

20

14

16

18

20

Re = 1, Gr = 104 , γ = 90o 1

0

0

4

2

6

8

0.

0.88

79

0.43 0.3 0.25 4 0.16 0.07

4

6

0.52

10

12 0.

79

1 0.6

0.4 0.34 3 0.25 0.16 0.07

8

10

12

0.88

0.70 0. 0.52 61

14

0. 0 52 .6 1

2 70 0. 1 0.6

0.5

0

0.52 0.43 0.34 0.16 0.25 0.07 0.16 0.07

1

0.7 00.79

0

0.7 9 0.70

0.5

43 0. .34 0 0.25 0.16 0.07

16

18

20


104 2

o

Re = 10, Gr = 10 , γ = 0 1 0.5 6

8

10

0.49 44 0.

12

4 0.2 9 0.1 9 4 .0 0.1 0

14

4

18

0.

0.14 0.09

2

16 0.49 4 0.4

0 0.2.34 0. 0.24 9 19

0.19

0.0

0.09

6

8

10

20

4

0.0

4

0.04

0

14

0. 39

0.39 0.1 04 .10. 0 0. 924.2934

4 0. 3 0.9 0.34 29

0.04

0.5 0

2

24 0.3 0.20 9 9.34

0 09 0. 0.04

1

0.

0

12

14

16

18

20

14

16

18

20

Re = 10, Gr = 103 , γ = 0o 1

0

4 0.

6

8 0.53 47 0.

420

.36

0.15 0.10

0.20

10

12 0.

42

6 0.2

0.0

2

4

0 0..336 0. 0.26 1 20 0.1 0

0.04

4

0.04

0

0.53 47 0.

15 0.

31 0. 6 0.2 0 0.2 5 0 0.1 0.1

0.04

0.5 0

2

10 0.0.04

1

0.3 0.42 01.3 6

0

6

8

10

0.42 0. 150 .200. 0.03.36 26 1

0.5

12

14

16

18

20

14

16

18

20

Re = 10, Gr = 104 , γ = 0o 1 0.5 0

0

2

4

6

57

0.

06

0

2

4

6

12 0.5 0.43 0 0.35 0.28 0.21 0.14 0.06

14

0.

0

10

0.

0.5

8

0 0.5 0 .43 0 0. .35 28 0. 21

1 0.2 4 0.1 0.06

1

8

10

12

2 0.7 0.57 0 0. .50 0.3 43 0.28 5 0.21 0.14 0.06

0.64

14

16

18

20


105 2

o

Re = 10, Gr = 10 , γ = 45 1 0.5 8 0.37 0.33 5 220.2 0.180. 0.14 0.10

9

10

0.06

0

2

4

6

8

14

10

12

16

18

0.37 0.33

9 0.0.2 22 5 0.1 8 0.10.14 0. 0 06 0.0 3

0.03

0

12 0.2

9

6

20 5 0.222 0.18 0.

0.2

4 0.2

0. 0 10.1 4

5 0.2 2 2 0.

0. 0.18 14

0 0.1.060.03 0

0.5

2

9

0 0.03

1

0.2

0

06 0.

3

0.0

14

16

18

20

18

20

Re = 10, Gr = 103 , γ = 45o 1

0

6

0.4.302 0 0.24

0

4

0.15 .07 0

0.5

2

0.

2

0. 15

24

8

0.5 0. 6 48 0.4 0. 0 32

0.81 0.65

10

0.07

4

6

8

12

14

16

0. 0 73 0.5 .65 0.48 6 0.40 0.32 0.24 0.15 0.07

3

0 0.07

1

0.7

0

10

12

0. 0.7 506 . 3 65

0.5

0.81

8

0.4

0 0.4 0.32 24 0. 0.15

14

16

18

20

18

20

Re = 10, Gr = 104 , γ = 45o 1 0.5 0

0

2

4 30

0

2

10

14

22

07

4

6

12 0.5 0.45 3 0.38 0.30 0.22 0.14 0.07

0.

0.

0.

0

8 0.6 1

0.

0.5

6 0 0 .5 0. .45 3 38

2 0.2 4 0.1 0.07

1

8

10

12

14

16

0.6 0.68 1

6 0.7

0.53 0.4 0.38 5 0.30 0.22 0.14 0.07

14

16

18

20


106 2

o

Re = 10, Gr = 10 , γ = 90 1 0.5 2

6

8

10

0.430.39

0.12

.2 0.17 0

0.08

2

4

16 0.430.39

.30 0.201.25 0. 0.17 12

18

08

0.0

8

10

12

3

0.0

3

6

20

0.

8

0

14

34 0

1

0.03

0

12 0.

0 0.

8 0.0 0.03

25 0. 1 0.2 7 0.1 12

34 0 .30

0.

0.5

4 0.

0.3 0 4 0.10 0 0 .30 2.17.21.25

0 0.03

1

0.20.3 05.3 4 0

0

14

16

18

20

16

18

20

Re = 10, Gr = 103 , γ = 90o 1

0

4

6

8

10

1 0. 39 0. 0.19

0.50 45 0.

0.3 0.24 0.29 4

9

4

0.0

0.04

0.5 0

2

0.14 09 0. 0.04

1

0. 0.39 1 0.1 90 0.34 4 .204.2 9

0

0.09

0.04

0

2

4

6

8

10

12

14

0.

39 0 .34 0.29 0.1 0.24 9 0.14 0.0 9 0.0 4

12

14

0.50 45 0.

0.3 0.349 0.10. 0.2 924 9

0.5

4

0.0

16

18

20

16

18

20

Re = 10, Gr = 104 , γ = 90o 1

0

0

2

6

8

0.3

0. 3 0.1 25 6 0.07

0.4 2

6

.77

0

0.5

4

10

0.86

68

0.7 7 0.5 9 0.51 0.42 0.33 0.25 0.16 0.07

90 .51

8

12

10

12

14

0.86 0.68

9

4 0.

0. 5

0.51 0.42 0.33 0.25 0.160.07

0.5

2

8

0 0.07

1

0.6

0

0.77

0.5

2

0.4 0.33 0.25 0.16 0.07

14

16

18

20


107 2

o

Re = 100, Gr = 10 , γ = 0 1 0.5 0

0

2

4

0.5 0

0

6

8

0 0 0.2 0.26 .30 00.1.15.18 2 1 0. 0. 07 03

7.03 0.0

1

2

4

6

0.37

10

8

12

14

10

12

16

18

0.37 0.33

0 0.3 0.2.26 0 0.18 2 5 0.1 0.11 0.07 0.03

0.33

14

20

30 02. 6 .02. 2 018 0. 0.15 0.11 0.07 0.03

16

18

20

16

18

20

Re = 100, Gr = 103 , γ = 0o 1

0

2

0.5 0

0

4

6

8

0.10.19 0 0.13 6 0.00.08 .11 5 0.0 2

2 0.0

1

2

4

6

10

8

12

14

000..119 0.1.113 6 05.08 0.0 0.02

0.27 0.24 0.22

10

12

0.27 0.24 0.22

6 .13 00.11 0.080.1 0.05 0.02

14

0.

0

19

0.5

16

18

20

16

18

20

Re = 100, Gr = 104 , γ = 0o 1 0.5 0

0

2

4

0.5 0

0

6 0 0 0.20.24 .27 00.1.1.417 0 0 0.0 0. 7 03

7 0.0.03 0

1

2

4

6

8

10

0.34 0.31

8

12

14

0.340.31

0.204.27 0.20 7 0.1 4 0.1 0.10 0.07 0.03

10

12

27 .020.4 00.2 7 0.1 0.14 0.10

0.03

14

16

18

20


108 2

o

Re = 100, Gr = 10 , γ = 45 1 0.5 0

0

2

4

0.5 0

0

6

8

0 0 0.2 0.26 .30 00.1.15.19 2 1 0.0 0. 7 03

07 0. .03 0

1

2

4

6

0.38

8

10 .34

12

14

0

10

12

16

18

20

0.38 0.34

30 02. 6 .02. 2 019 0. 0.15 0.11

0 0.3 0.2.26 0 0.192 0.15 0.11 0.07 0.03

0.03

14

16

18

20

16

18

20

Re = 100, Gr = 103 , γ = 45o 1 0.5 0

0

2

4

0.5 0

6 0.12 0.11 0.09 0.07 0.05 0.0 3 0.01

0.01

1

0

2

4

6

8

10

8

12

14

00 00. .1.1 0.0.50079 12 0.03 0.0 1

0.18 0.16 0.14

10

0.18 0.16 0.14

12

14

16

2 ..9111 00.0.5007 .0 0.003 1 0.0

18

20

18

20

Re = 100, Gr = 104 , γ = 45o 1 0.5 0

0

2

4

0.5 0

0

6

8

10

2

4

6

12 0.3

0.20.30 00.2 5 0.1.150 0 0.0 5

5 0.0

1

5

0 0. .30 0.225 0.15 0 0.1 0 0.0 5

8

10

12

14

16

0.50 0.45 0.35 0 0 .30 0. .25 0.1520 0.10 0.05

0.4

0

14

16

18

20


109 2

o

Re = 100, Gr = 10 , γ = 90 1 0.5 0

0

2

4

0.5 0

0

6

8

0 0.34 0 0.20.26 .30 00.1.1.519 2 1 0.0 0. 7 03

7 0.00.03

1

2

4

10

6

8

12

14

0.34 0.02.3 60 0.22 0.19 0.15 0.11 0.07 0.03

0.38

10

12

18

20

4 00..3360 .22 00.2 9 0.1 0.15 0.11

0.38

0.03

14

3

16

16

18

20

16

18

20

o

Re = 100, Gr = 10 , γ = 90 1 0.5 0

0

2

0.5 0

0

4

6

8

2

4

10

12 00 0.1.2.225 0.16 9 09.12 0.0

0.32 0.28

0.25 0 0.10.22 0.00.12.16 9 9 0.0 0. 6 03

6 0.0.03 0

1

14

25 0..292 00.61 .1 0 0.12 0.09

0.32 0.28 0.06

0.03

0.03

6

8

10

12

14

16

18

20

18

20

Re = 100, Gr = 104 , γ = 90o 1 0.5 0

0

2

4

0.5 0

6

8

10

12 0.

0. 00.2 32 0..1217 6 0.0

11 0. 5 0.0

1

0. 0 43 0.2.32 7 0. 0.1621

5

0.1 1

0.0

0

A.6.3

2

4

14

16

0.54 0.49 0.43 0.38 0 0.732 0.2.2 0.16 1

38

6

8

10

0.1 1

5

12

0.05

14

16

18

20

Flow Phenomena in Natural Convection Flow Solar Collector

The following section is presenting the velocity vector field with corresponding isothermal lines for simulations performed with a constant Biot number Bi = 5, an ambient temperature θ∞ = θc , heat flux intensity ξ = 0.1 and a geometric aspect ratio ϱ = 20. The top-side of the figures are marked with two red lines that indicate the position of the PV modules. The flow field is furthermore clarified with stream vertices - indicated


110

by black lines - that emphasize the direction of the flow. The order of figure presentation is as follows, every page will contain three sub-figures with a constant Rayleigh number where the inclination angle will be given by γ = 0, γ = π/4 and γ = π/2. 3

o

10

12

Ra = 10 , γ = 0 1 0.5

2

0.3 3

0.2

4

4

0.4

2

0.6

6

8

10

0

9

20

0

2

0.4 0.33 0.24 0.14 0.05

12

3

18 0.89

0.7

0.5 2

0.4 0.33 2 0.24 0.14 0.05

0.05

4

9

1

16

0.7

0.6 01.5 2

14 0.7

0.7

0.89

70

0.1

0

8

61

6 0.

0.14

2 0.5 0.423 0.3 0.24

0

0.05

0.5

4

0.

2

0.7 0

0 0.05

1

0. 79

0

14

16

14

16

18

20

18

20

o

Ra = 10 , γ = 45 1

0.2

2

4

4

0.3 4

0.43

0.52

12

1

0.8

0.3 4 0.24

0.15

0.4

0.05

6

0

8

10

0.5 3

2

0.90

0 0.6 .71 2

43

. 40

0.3 0.24 0.05

0.15

12

0.50 0 2 .62.71

2

1

10 0.7

0.6

0.7

0.05

0

8 0.90

0

6

2

4

2 0.5 0.43 0.34

0.240.15

0.05

0.5 0

2

0.8

0 0.15 0.05

1

0.6

0

0.80

0.5

14

16

18

20

14

16

18

20

Ra = 103 , γ = 90o 1

6

2

4

0.16

6

0.25

2

4

0.5

12 0.8

5

0.3

0.25

0.53

0.4

3

0.91

0.7 2

0.63

4

0.63

3

2

10 0.6

2

0.7

0.06

0

8 0.91

5 0.3

0.06

8

10

0.16

12

0.25

0.35

0.44

14

0.53

0.63 0. 0. 0.5 35 44 3

4 0.8

0.63 0.53

0.16 0.06

0.5 0

2

0.72

0 0.44 0.35 0.25 0.16 0.06

1

0.4

0

0.8 2 0.72

0.5

25 0. 6 1 . 0 0.06

16

18

20


111 4

o

10

12

Ra = 10 , γ = 0 1 0.5 0

0

4

8

4

16

8

10

20

0. 0.3 40 0.26 3 0.19 0.12 0.06

0.06

6

18 0.67 0.60 0.53

53 0.

2

14

0.4 0.40 7 0.33 0.26 0.19 0.12

2

0. 06

0

6 0 0.4 0 .40 7 0..33 26 0. 19

1 0.

0.5 0

2 9 0.1 2 0.1 0.06

1

12

14

16

14

16

18

20

18

20

Ra = 104 , γ = 45o 1 0.5 0

0

2

6

8

10

0 0 .5 0 .43 1 0. .36 28 0. 21 0. 14

0.5 0.43 1 0.36 0.28 0.21 0.14 0.06

06

0

0

2

4

0.6

8

0.

12 5 0.

0.06

0.5

4

1 0.2 4 0.1

1

6

8

10

0.73 0.58

5

0.51 0.4 0.36 3 0.28 0.21 0.14 0.06

12

14

16

18

20

14

16

18

20

Ra = 104 , γ = 90o 1

2

6

8

10

0.87

78

12

9

0.41 0.3 0.23 2 0.14 0.04

4

6

0.50

0.5

0.41 0.32 0.23 0.14 0.04

8

10

12

0.87

0.69

0.50

0.59

14

0. 0 50 .5 9

4 0.

78 0.

0

69 0. 9 0.5

0

0.50 0.41 2 .14 0.3.2 0 0 3 0.04

0.5

2

9

0 0.14 0.04

1

0.6

0

0.7 8 0.69

0.5

1 .4 20 0.3 0.23 0.14 0.04

16

18

20

Appendix B

Non-Uniform Grid B.1

Introduction

Inspired by post-processing result that varied with the literature, especially in the vicinity near the borders of the domain it is chosen to implement a non-uniform grid. The nonuniform grid has hopefully been successful implemented. Several simulations throughout this work have already been performed by means of this non-uniform grid. The nonuniform grid is a so-called regular grid that is structured orthogonal with cartesian applied coordinates. The application of such a grid is accepted as the geometry of the encountered problems so far did not require a more complex coordinate system. It should be noted that any enhancement of geometry most likely results in the necessity of the application of an orthogonal curvilinear mesh or more complex coordinate grids. The current appendix covers the transformation done where some terminology is introduced as two separate domains are considered, a so-called computational domain that will be transformed to a physical domain. It continues with implementation details and an enhanced linear interpolation approximation required for the convective flux terms (Fu,v ) defined at the interfaces of the velocity control volumes due to the no longer square shaped control volumes but rectangular shaped ones.

112

Appendix B. Non-Uniform Grid

B.2

113

Analytical Transformation

In the post-processing phase it was observed that scalar solutions - especially near the boundary - differed compared with the literature and had an increased discretization error, see also appendix C. Due to that observation an attempt is made to tackle the problem such that future solutions obtained are in closer resemblance with the literature. Several methods could help to resolve the issue of uncertainty, think of improved higher order spatial interpolation schemes, in particular for the convective terms or multigrid techniques. However, it is believed that local grid refinement by means of an analytical or algebraic transformation will contribute, in the positive sense, to a more accurate solution with the least amount of implementational required changes. With the use of an algebraic expression to cluster grid points near solid boundaries it is desired to obtain a relation ⃗r = ⃗r(ξ, η) in a computational uniform structured grid that describes the transformation towards a physical non-uniform structured grid. The situation is shown in figure B.1. To generate computational domains using this technique, known functions NPI = 4, NPJ = 4

NPI = 4, NPJ = 4

1.2

1.2

1

1

0.8

0.8

y/L

η [-]

0.6 0.4

0.6 0.4 0.2

0.2

0

0

−0.2

−0.2 0

0.5 x/L

1

0

0.5 ξ [-]

1

Figure B.1: The physical and computational plane in a 4x4 staggered grid arrangement. See also figure 2.2 for more detailed information regarding the individual symbols.

are encountered throughout the literature to map an arbitrarily shaped physical region into a rectangular orthogonal computational domain. Literature provided by Maliska [18] and Tannehill [52] has been found useful in this matter, they treat several similar like


114

transformations throughout their work. One of these transformations is partly provided in Maliska [18] and rewritten to the purpose of this work, ( ξ(˜ x) = α + (1 − α)ln ( η(˜ y ) = α + (1 − α)ln

β − 2α + (2α + 1)˜ x/L β + 2α − (2α + 1)˜ x/L β − 2α + (2α + 1)˜ y /L β + 2α − (2α + 1)˜ y /L

)/

( ln

)/

( ln

) β+1 , β−1

(B.1)

) β+1 , β−1

(B.2)

where α and β are constants defining the position of grid-refinement and the factor of stretching, respectively. Having defined a computational domain, similar to the one shown in figure B.1 one can rewrite the former equations to define the physical domain. Rewriting the former equations for the x ˜ and y˜ coordinate yields, x ˜(ξ) =

(β(Υx (ξ) − 1) + 2α(Υx (ξ) + 1)) , (2α + 1)(Υx (ξ) + 1)

(B.3)

y˜(η) =

(β(Υy (η) − 1) + 2α(Υy (η) + 1)) , (2α + 1)(Υy (η) + 1)

(B.4)

where Υx,y is defined by, ( ( )( )) β+1 ξ−α Υx (ξ) = exp ln , β−1 1−α

( ( )( )) β+1 η−α Υy (η) = exp ln . (B.5) β−1 1−α

Successful implementation results in the physical domain shown in figure B.1. As every integration is defined separately over an individual control volume having its own surface area and cell faces, it should not be necessary to include the metrics of the mapping in the differential equations. As a staggered grid notation is adopted the velocities are defined at the faces of the main control volume, omitting the need for interpolation as required in a co-located grid. However, it is necessary to re-define the convective flux terms Fu,v that were defined earlier exactly between two adjacent velocity control volumes and this no longer is the case, made clear in figure B.2. In the previous uniform grid arrangement if one wanted to determine the convective flux components one could apply a simple midpoint rule, as the convective fluxes were positioned exactly at the midsection of the velocity control face. In the present case the convective fluxes are positioned outside the midsection and one has to apply a slight more advanced interpolation. The simplest enhancement one could apply is a linear interpolation. If one would like to determine the north component of the convective flux of the u-velocity control volume, assigned


115

i,J+1

I,j

i-1,J

I,J

i,J+1

I,j

I+1,j

i,J

i-1,J

I,j-1

I,J

I+1,j

i,J

I,j-1

Figure B.2: A cut-out part from the physical domain to clarify three types of control volumes in a staggered grid notation for a non-uniform grid. The position of the main control volume (I, J) is marked with a rectangular shaped box covered with diagonal hatched lines [///]. The main control volume is accompanied with its [u]-velocity control volume (i, J) in the rectangular box in red on the left-side and the [v]-velocity control volume (I, j) in the rectangular box in green on the right-side. The convective fluxes are positioned between two adjacent velocity control volumes indicated with bullets [•].

by (i, J), the linear interpolation takes the following form, Fn = FI,j (1 − λn ) + FI+1,j λn , where λn is defined by, λn =

xn − xI,J . xI+1,J − xI,J

(B.6)

(B.7)

and xn is the x-position of the convective flux on the north face. Note that midpoint interpolation for the u-velocity control volume is still allowed in this case for the considered west- and east convective fluxes. The opposite is true for the v-velocity control volume shown in the green rectangle on the right-side of figure B.2 assigned by the subscript (I,j). The component of the convective flux on the north face is determined by a midpoint rule, while the east-side for example is obtained by, Fe = Fi,J λe + Fi,J+1 (1 − λe ), where λe is defined by, λe =

ye − yi,J , yi,J+1 − yi,J

(B.8)

(B.9)

and ye is the y-position of the convective flux on the east face. In the mixed- and natural convection models treated in chapter 3 and 4, respectively, the buoyancy and gravity forces were acting parallel but opposite in the y-direction. This indicates that the source


116

term of the v velocity component requires an enhanced linear interpolation similar to the previous treatment of the convective fluxes and is treated separately in chapter 4 where the non-uniform grid is introduced officially for the first time. The advantage of the current grid arrangement is that the former defined boundary conditions can be kept exactly the same as compared to the uniform grid. Implementation of boundary conditions is done according to artificial boundaries, already introduced in 2 in figure 2.2. These artificial boundaries have the exact same size as the first adjacent control volume within the physical domain, omitting the necessity of re-defining the boundary conditions for a non-uniform grid. The former treated details conclude this section for the non-uniform grid. For the interested reader, appendix C will cover the discretization error that occurs for this type of grid in mixed and natural convection problems in square shaped geometries.

Appendix C

Grid Dependency C.1

Introduction

To guarantee a numerical solution that is precise up to the discretization error imposed by the numerical schemes, grid or individual size of the control volumes, a grid dependency study is performed and is covered in the current appendix. Error estimates are performed depending on the orientation of the applied physical domain. In the current work two types of domains have been applied, a uniform and a non-uniform domain. A recommended method for discretization error estimation is the Richardson extrapolation (RE), as its intricacies, shortcomings and generalizations have been widely investigated [50][49]. In the following section the RE will be elucidated by calculating the discretization uncertainty of a variable ϕ. The section continues with the application of the method accordingly and ϕ is replaced by a velocity component and the local Nusselt number for the first two test-cases in a uniform domain, respectively. The third section will cover the grid dependency study for a non-uniform grid, where briefly additional information is provided to adapt the RE to a non-uniform grid. The non-uniform domain has been applied to the second and third test-case, indicating an overlap of the second test-case that is also discussed in the former section. Grid dependency in these test-cases is determined by computing again the RE and consequently specifying to the local Nusselt number.

117

Appendix C. Grid Dependency

C.2 C.2.1

118

Uniform Grid RE in a Uniform Grid

In a stepwise procedure, highlighted in this section and also covered by Celik [50], the Richardson extrapolation (RE) is obtained. For the error estimation by means of a RE it is required to have simulations done on different grid levels that will be called N 1, N 2 and N 3. Those grid levels have a representative control volume size h that is defined in a two-dimensional case by,

[

N 1 ∑ h= ∆Ai N

]1/2 ,

(C.1)

i=1

where Ai is the area of the ith control volume and N is the total number of control volumes used in each computation. Note that the representative control volume size h in a uniform grid is also equal to ∆˜ x or ∆˜ y , applied earlier in the chapters 2 and 3. The different grid levels and densities with their respective representative control volume size are shown in table C.1. Next is to determine the apparent order of spatial interpolation Grid level [N] N1 N2 N3

Grid density [Nx xNy ] 100 x 100 50 x 50 25 x 25

Grid control-volume size [h] 0.01 0.02 0.04

Table C.1: Grid densities in a uniform configuration applied to obtain the numerical solution in the lid-driven cavity flow.

p = f (p) that is determined with the following expression [50],

p=

( ) log|ϵ32 /ϵ21 | + log (rp − s)/(rp − s) 21 32 log(r21 )

,

(C.2)

where the variables ϵ, r and s are defined by, ϵ21 = ϕ2 − ϕ1 ,

ϵ32 = ϕ3 − ϕ2 ,

r21

h2 = , h1

r32

h3 = , h2

( s = sign

ϵ32 ϵ21

) ,

(C.3)

and ϕk denotes the solution on the k th grid. It is emphasized that the order p is a function of itself and to solve the equation a fixed-point iteration is used to obtain the roots. In the application of this iterative method it is possible to write an equation for p where the root pi equals a new function g(pi ). For continuation of the iterative process -


119

after initiating an initial guess - one can converge up to a specified threshold. To clarify the former a dummy-code is given in the following example, iter = 0; threshold = 1e-6; %% Initial guess p1 = 1; p2 = g(p1); %% Iteration Loop while abs(p2-p1) > threshold p1 = p2; p2 = g(p1); iter = iter + 1; end

In the dummy-code the function g(p) is defined by the equation shown earlier for p in eq. C.2 and the initial guess has been set to p1 = 1. The difference between pi and pi+1 is set to the threshold value, leading to the end result pi+1 = g(pi ). Having obtained the order p one can compute the RE for each solution by means of the given expression [50], REf21ine =

1.25e21 a , p r21 −1

(C.4)

where the relative error ea is also estimated for each solution by means of the following expression, e21 a

ϕ1 − ϕ2 . = ϕ1

(C.5)

One can extend this method - rather easily - to obtain an extrapolated value for the solution ϕ that takes into account the discretization error. This is not treated within this work, although the interested reader is advised to look into the elucidation of Celik [50] or the application of De Vahl Davis [4].

C.2.2

RE for Case-Study I

Having highlighted the major steps in approximating the discretization error, the procedure is first applied to the first case-study, i.e., the validation of the Navier-Stokes solution in the lid-driven square cavity. No major differences were encountered during


120

post-processing of the different applied pressure-velocity algorithms and for that reason the discretization error is determined for a single arbitrary selected pressure-velocity scheme, in this case the SIMPLEX algorithm. The spatial discretization used in the current analyses is the modified quadratic upwind scheme (QUICK) for the convection terms and the central differencing scheme for the diffusion terms. Simulations are performed with a Reynolds number equal to Re = 1000. The applied physical domain is equal to the domain shown in figure 2.2, where grid levels of table C.1 are used in the computation. The discretization error is estimated for the u ˜ component of the NavierStokes at x ˜ = 0.5 along the vertical line y˜ = [0, 1], similar to the previous shown figures ??-A.3 in chapter 2. It is believed that the v˜ component of the Navier-Stokes does not differ significant from the u ˜ component and it is therefore assumed that grid dependency can be excluded up to the discretization error by solely analyzing the u ˜ component. The results of the former treated steps of the Richardson extrapolation and implementation details are shown in table C.2. y˜ p REf21ine e21 a |e21 | ±

[-] [-] [%] [%] [-]

0 0.586 7515 3013 0.000

0.1 0.863 12.70 8.300 0.037

0.2 2.172 1.200 3.400 0.005

0.3 2.714 0.300 1.300 0.001

The local order of accuracy p varied from 0.4 2.985 0.300 1.500 0.000

0.5 4.564 0.000 0.700 0.000

0.6 6.248 0.100 8.000 0.000

0.7 0.997 4.900 3.900 0.007

0.8 1.715 0.002 3.400 0.005

0.9 2.047 1.600 4.000 0.006

1 0.030 0.000 0.000 0.000

Table C.2: Discretization error estimates according to the Richardson extrapolation (RE) for a Reynolds number of Re = 1000. The local order of accuracy p is shown with the Richardson extrapolation discretization error REf21ine and the relative error e21 ˜ coordinate. The absolute uncertainty of the a in percentages with respect to the y non-dimensional velocity u ˜ is given by |e21 |.

0.030 − 6.248 and the average order of accuracy is found to be pavg = 2.3 which corresponds to the quadratic interpolation (QUICK) applied in the spatial discretization of the differential equations. The maximum discretization error by the RE and relative error is estimated to be 7515[%] and 3013[%], respectively. These points are relative to a non-dimensional velocity near zero and correspond to an absolute uncertainty in velocity of approximately ±0.000. The absolute error |e21 |, also shown in table C.2, is graphically shown in figure C.1. There it can be observed that the largest absolute discretization errors occur near the edge of the domain due to the velocity gradients. The largest absolute error occurs at y˜ = 0.1 and yields ±0.037.

121

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

y/L

y/L


0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

N1 N2 N3

0 0

0.5 u/U−velocity

1

0.1 N1 RE

0 0

0.5 u/U−velocity

1

Figure C.1: The estimated absolute discretization error for the non-dimensional velocity at the midsection of the square cavity as a function of the y˜ coordinate and a Reynolds number of Re = 1000.

C.2.3

RE for Case-Study II

The second case study analyzed the phenomena of mixed convection that was classified by the Richardson number (Ri), not to be confused with the Richardson extrapolation (RE). Similar to the first case study, a uniform structured physical domain has been applied to obtain the scalar solutions. However, in the present study an additional non-uniform grid was inspired due to de-similarities with the literature under conditions where natural convection was dominating the mixed convective flow. The grid dependency study - and only the grid dependency study - of mixed convection is therefore performed on a uniform grid and an additional non-uniform grid that will be covered in the next section. Having analyzed the earlier shown result of figure 3.3 where the local Nusselt number on the top-side of the cavity was compared to the literature, it is interesting to perform a Richardson extrapolation where the differences with the literature were biggest i.e., the Ri = 10000 case. Spatial interpolation is performed identical to the first case study with the difference that the PRIME algorithm now is used for

Appendix C. Grid Dependency x ˜ p REf21ine e21 a |e21 | ±

[-] [-] [%] [%] [-]

0 1.615 0.989 1.633 0.025

0.1 0.952 4.810 3.597 0.124

0.2 0.780 9.135 5.237 0.250

122 0.3 0.844 8.269 5.260 0.230

0.4 0.942 5.566 4.099 0.147

0.5 1.322 1.930 2.316 0.045

0.6 2.886 0.097 0.496 0.002

0.7 1.837 0.298 0.614 0.004

0.8 3.927 0.009 0.103 0.000

0.9 2.990 0.600 3.335 0.005

1 4.950 0.238 5.697 0.002

Table C.3: Discretization error estimates according to the Richardson extrapolation (RE) for a Richardson number of Ri = 10000. The local order of accuracy p is shown with the Richardson extrapolation discretization error REf21ine and the relative error e21 a in percentages with respect to the x ˜ coordinate. The absolute uncertainty of the Nusselt number (Nu) is given by |e21 |.

pressure-velocity coupling. The result of the grid dependency study by means of a RE

3

3

2.5

2.5

2

2

NuL

NuL

is summarized in table C.3. From the table it is concluded that the local order of accu-

1.5

1.5

1

1

N1 N2 N3 0

N1 RE 0.5 x/L

1

0

0.5 x/L

1

Figure C.2: The estimated absolute discretization error of the local Nusselt number along the top-side of the square cavity as a function of the x ˜ coordinate and a Richardson number of Ri = 10000.

racy p varied between 0.780 − 3.927, while the average was found to be pavg = 2.1. The largest discretization error as indicated by the Richardson extrapolation and the relative error are respectively 9.1[%] and 5.3[%]. The discretization uncertainty has been found


123

largest at the point where the Nusselt number also differed most from the literature, as observed earlier in chapter 3. The maximum absolute local Nusselt uncertainty as computed appears to be ±0.250. The local absolute local Nusselt number uncertainty is given in figure C.2. Large discretization errors are observed on the left-side of the domain indicating that further grid refinement would still be beneficial. One could wonder if grid independency would be more beneficial to obtain - considering (1) the time to obtain the converged solution and (2) the occurring discretization error - in a uniform configuration as opposed to a non-uniform grid by simple increasing the grid density. A quick review of the maximum, averaged and minimum Nusselt number along the lid of the cavity for the uniform grid densities (25x25), (50x50), (100x100) and

[Nu]

(200x200) reveals that this is not the case, see figure C.3. 3.1

2.15

0.55

3.05

2.1

0.5

3

4

10 [Nx x Ny]

2.05

4

10 [Nx x Ny]

0.45

4

10 [Nx x Ny]

Figure C.3: The maximum [△], averaged [] and minimum [▽] Nusselt number along the driven lid of the square cavity for a Richardson number of Ri = 100. Spatial interpolation performed with the QUICK scheme and pressure-velocity coupling with the SIMPLEC algorithm. Applied grid densities (25x25), (50x50), (100x100) and (200x200).

Significant discretization errors are to be expected as the difference of the Nusselt quantity in sequential grid densities is still clearly visible. It is concluded that further grid refinement beyond the finer (200x200) grid would still be recommendable within a uniform grid configuration.


C.3 C.3.1

124

Non-Uniform Grid RE in a Non-Uniform Grid

Expansion of the RE towards a structured non-uniform grid can be obtained in a similar fashion. The only major difference is the interpolation of the scalar ϕ on every grid level that requires a little enhancement. Each grid level is still represented by a control volume representative size h being defined earlier by eq. C.1 and shown in table C.4 with its applied grid densities for the non-uniform configuration. Grid level [N]

Grid density [Nx xNy ]

Grid control-volume size [h]

N1

100 x 100

0.0117

N2

50 x 50

0.0206

N3

25 x 25

0.0403

Table C.4: Three grid levels with its grid densities in a non-uniform configuration applied to obtain the numerical solution in the square cavity flow with the global representation of the control volume size h.

The representative control volume size h in the non-uniform grid is almost identical to the obtained h in the uniform grid treated in the previous section. Although the control volume size is a variable in a non-uniform grid, the definition of h is interpreted in its global representation of the control volume at each grid level i.e., being a constant. Table C.4 furthermore shows the same identical grid- levels (N1, N2 and N3) and densities as applied in the uniform grid but distinguished by its logarithmic distributed local control volume size, as discussed in appendix B. This local size distribution per control volume is shown for two imaginary diagonal lines going through the midsection of the square cavity in figure C.4. The figure shows the control volume size distribution in a structured non-uniform grid, the grid density on the horizontal axis shows the control volumes in x and y direction in their specified configuration. It indicates that the largest control volume sizes are to be found in the midsection, while the smallest control volume sizes are found near the edges. Having discussed the control volume size h, the remaining adaptation involves an enhanced interpolation for the variable ϕ. The variable ϕ is no longer defined at the position of the control volume node or exactly halfway but somewhere along the way


125

−1

Representative control volume size h [−]

10

25x25 50x50 100x100

−2

10

0

20

40 60 Grid density [Nx x Ny]

80

100

Figure C.4: The representative local control volume size h for grid levels N1 , N2 and N3 being compressed near the borders and stretched in de mid-section of the physical domain. The minimum control volume size is for the three types of grid levels (h1 , h2 , h3 ) = (2.99e-3, 6.15e-3, 13e-3). The zero-derivative of the control volume size at the boundaries are the artificial boundaries, having the same size as the first adjacent control volume within the physical domain.

at a position xn . The enhanced linear interpolation of eq. B.6 and eq. B.7 discussed in former appendix B is adopted and implemented. The next section will elucidate the application of the RE in a non-uniform grid in the same fashion as done in the previous treatment of a uniform grid.

C.3.2

RE for Test-Case II

The discretization error for the second test-case (mixed convection in a square cavity) has already been determined for a uniform grid configuration. It was found that further grid refinement, even further than the finest applied grid (100x100) would still be beneficial as to limit the discretization error, see also section C.2.3. The trade-off made in the introduction of this section is the time to reach the solution bounded by the convergence criteria on a uniform considered fine (100x100) or a finer (200x200) grid. Obviously the


126

finer grid would result in solutions for ϕ with a considerable smaller discretization error, but the time to reach that solution would be tedious which is why a non-uniform (100x100) configuration is adopted. x ˜ p REf21ine e21 a |e21 | ±

[-] [-] [%] [%] [-]

0 3.204 0.122 0.502 0.003

0.1 2.618 0.312 0.849 0.008

0.15 2.252 0.573 1.181 0.016

The current section than provides a comparison

0.2 2.055 0.831 1.462 0.024

0.25 1.928 1.053 1.665 0.031

0.3 1.786 1.295 1.810 0.038

0.35 1.666 1.484 1.861 0.043

0.4 1.691 1.360 1.745 0.038

0.7 2.990 0.108 0.383 0.002

0.9 3.319 0.047 0.208 0.000

1 2.318 0.137 0.298 0.001

Table C.5: Discretization error estimates according to the Richardson extrapolation (RE) for a Richardson number of Ri = 10000 on a non-uniform grid. The local order of accuracy p is shown with the Richardson extrapolation discretization error REf21ine and the relative error e21 ˜ coordinate. The absolute a in percentages with respect to the x uncertainty of the Nusselt number (Nu) is given by |e21 |.

with the earlier obtained discretization error for the uniform grid with an equal grid density distribution. The RE is performed - again - on the driven lid of the square cavity by calculating the local Nusselt number for the Richardson number Ri = 10000 case. The spatial interpolation is performed with the spatial interpolation QUICK scheme and pressure-velocity coupling is obtained with the PRIME algorithm, indicating once more that the only difference with the already performed RE on the driven lid is the grid configuration. The result of the grid dependency study is again presented in the form of a table, given in table C.5. The focus of the coordinate x ˜ along the lid is dedicated towards the area where the largest discretization errors occurred in the uniform grid configuration, as can be seen from the table and figure C.5 where the local Nusselt number is given. It is concluded that the local order of interpolation varied between 1.666-3.319 and an average of 2.4. The averaged order of interpolation again corresponded with the applied quadratic interpolation (QUICK) and compared to the uniform grid resulted in an increase in spatial interpolation effectivity of approximately 9.5[%]. The largest discretization error as indicated by the Richardson extrapolation and the relative error in the non-uniform configuration are respectively 1.5[%] and 1.9[%] and compared to the uniform grid 9.1[%] and 5.3[%] respectively are a significant improvement.

127

3

3

2.5

2.5

NuL

NuL


2

1.5

2

1.5

N1 N2 N3

1

0

1 N1 RE 0.5 x/L

1

0

0.5 x/L

1

Figure C.5: The estimated absolute discretization error of the local Nusselt number along the top-side of the square cavity as a function of the x ˜ coordinate and a Richardson number of Ri = 10000.

C.3.3

RE for Test-Case III

The third test-case involved a pure natural convection flow inside the square cavity. Simulations were performed under conditions where the Rayleigh varied from 103 -106 and the grid dependency will again put the focus on the most extreme simulated case, in this model the Ra = 106 . The discretization uncertainty is determined with QUICK interpolation and this time the SIMPLE algorithm for pressure-velocity coupling. The non-uniform grid is adopted having the known grid densities (N1, N2 and N3). The RE is performed on the heated (left) side of the cavity by computing the local Nusselt number. The results of the RE are given in table C.6. The absolute uncertainty of the local Nusselt number is given in figure C.6, along the heated side of the cavity. The local order of interpolation p varied from 1.806 - 2.325 and the averaged was found to be 2.0, again corresponding to the quadratic applied interpolation QUICK scheme. The largest discretization error as indicated by the Richardson extrapolation and the relative error in the non-uniform configuration are respectively 0.7[%] and 1.5[%] corresponding both

Appendix C. Grid Dependency y˜ p REf21ine e21 a |e21 | ±

[-] [-] [%] [%] [-]

0 2.213 0.482 1.360 0.080

0.025 1.845 0.740 1.494 0.129

128

0.050 1.987 0.442 1.017 0.077

0.075 2.012 0.324 0.764 0.055

0.100 2.021 0.258 0.613 0.042

0.150 2.052 0.174 0.425 0.026

0.300 1.951 0.073 0.162 0.009

0.400 2.325 0.010 0.031 0.001

0.700 1.812 0.121 0.236 0.007

0.900 1.840 0.325 0.651 0.008

1 1.806 0.140 0.271 0.001

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

y/L

y/L

Table C.6: Discretization error estimates according to the Richardson extrapolation (RE) for a Rayleigh number of Ra = 106 and a Prandtl number of P r = 0.71 on a nonuniform grid. The local order of accuracy p is shown with the Richardson extrapolation discretization error REf21ine and the relative error e21 a in percentages with respect to the x ˜ coordinate. The absolute uncertainty of the Nusselt number (Nu) is given by |e21 |.

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

N1 N2 N3

0 0

5

N1 RE

0 10 NuL

15

0

5

10 NuL

15

Figure C.6: The estimated absolute discretization error of the local Nusselt number along the heated-side of the square cavity as a function of the y˜ coordinate and a Rayleigh number of Ra = 106 and a Prandtl number of P r = 0.71.

to an absolute discretization error of ±0.129. Finally, it might be interesting to see a comparison of the outcome of a finer (200x200) uniform grid and a fine (100x100) non-uniform grid by showing the same computed local Nusselt number on the heated side of the cavity and is given in figure C.7. The figure shows that there is more consistency among the pressure-velocity coupling algorithms in


129

Ra = 1000000

Ra = 1000000

1

1

0.8

y/L

0.6

0.8 0.6

0.4

0.4

0.2

0.2

0

5

10 NuL

15

PRIME SIMPLE SIMPLER SIMPLEC SIMPLEX Yapici

y/L

PRIME SIMPLE SIMPLER SIMPLEC SIMPLEX Yapici

0

5

10 NuL

15

Figure C.7: The local Nusselt number on the heated side of the square cavity for Ra = 106 and P r = 0.71. The local Nusselt number is shown for a non-uniform (100x100) grid on the left, and a uniform (200x200) grid on the right.

a non-uniform configuration. The SIMPLER-algorithm in particular seemed to benefit from the application of a non-uniform grid. Additionally, a benchmark model given by Yapici [3] et al.

1

that computed the same Nusselt number along the heated side of the

cavity shows that the fine non-uniform configuration corresponds closer than the finer uniform grid.

C.3.4

RE for Mixed Convection in the Solar Collector

The discretization error for the mixed convection flow in the rectangular shaped thermal system of the solar collector is analyzed by means of the Richardson extrapolation. The discretization error will be analyzed for thermal conditions where the temperature gradients are expected to be significant, it is assumed that this will occur in the natural convection regime (Re = 1, Gr = 104 and P r = 0.71). The RE in the former test cases were computed on a square shaped domain, the current analyzes of the discretization error will compute the RE in a rectangular shaped domain. As a consequence, the representative control volume h defined in eq. C.1 will be different and is given for the three applied grid levels (N1 , N2 and N3 ) in table C.7. 1

Their numerical configuration has been introduced in chapter 3 where they applied a (320x320) non-uniform grid, a fourth order spatial interpolation scheme for the convection terms and a SIMPLE algorithm for the pressure-velocity coupling.


130

Grid level [N]

Grid density [Nx xNy ]

Grid control-volume size [h]

N1

300 x 15

0.0639

N2

140 x 7

0.1329

N3

60 x 3

0.3111

Table C.7: The global representative control volume size h for three grid levels with their respective grid densities in a non-uniform configuration applied to obtain the numerical solution in the rectangular shaped domain of the solar collector.

C.3.5

RE for Natural Convection in the Solar Collector

The discretization error for the natural convection flow in the rectangular shaped thermal system of the solar collector is analyzed. The discretization error will be analyzed for thermal conditions where the temperature gradients are expected to be significant, it is assumed that this will occur in the upper natural convection regime (Ra = 104 and P r = 0.71). The spatial interpolation of the advection terms is done with the QUICKscheme and the diffusion terms are interpolated with the central differencing scheme. The time interval for the governing equations are integrated with a fully implicit integration scheme, where the continuity and momentum equations are coupled with the SIMPLEC algorithm. The methodology described in section C.2.1 and section C.3.1 is adopted, where the representative control volume size is given in former section C.3.4. The Richardson extrapolation is computed for the local Nusselt number on the bottom-part of the solar collector and is given in table C.8. The local order of interpolation p varied from 0.0077 - 4.1660 and the averaged was found to be 2.1839, corresponding to that of the quadratic applied QUICK interpolation scheme. The largest discretization error, in terms of the Richardson extrapolation and the relative error have been found to be 165.86[%] and 39.22[%], corresponding to an absolute discretization error for the Nusselt number of ±0.37. The absolute discretization error for the local Nusselt number is presented in figure C.8.


131 Grid Dependency Result

x ˜[−] 0 3 6 9 10 11 12 13 18 18.5 19 20

REf21ine [%] 222.41 165.86 0.9958 0.1585 0.4578 2.8240 6.3508 6.0135 1.8668 1.8884 2.9559 6.4406

p[−] 0.0077 0.3537 3.0379 4.1660 3.4878 2.3404 1.8381 1.8649 2.5476 2.5364 2.2545 1.7716

e21 a [%] 0.1000 39.223 6.5715 2.5519 4.3430 10.279 14.438 14.038 8.1528 8.1683 9.9595 13.703

|e21 |[−] 0.0000 0.3677 0.0042 0.0009 0.0027 0.0167 0.0374 0.0366 0.0136 0.0140 0.0221 0.0476

Table C.8: The result of the Richardson extrapolation in terms of the local Nusselt number for the bottom-part of the solar collector. The result is obtained for a Rayleigh number of Ra = 104 and a Prandtl number of P r = 0.71 on a non-uniform rectangular grid. The quantities given are the local order of accuracy p, the Richardson extrapolation discretization error REf21ine in percentages, the relative error e21 a in percentages and the absolute uncertainty of the local Nusselt number given by |e21 |.

N1 N2 N3

0.7

0.6

0.6

0.5

0.5

0.4

0.4

NuH

NuH

0.7

0.3

0.3

0.2

0.2

0.1

0.1

0

0

−0.1

−0.1 0

5

10 x/H

15

20

N1 RE 0

5

10 x/H

15

20

Figure C.8: The estimated absolute discretization error of the local Nusselt number along the bottom surface of the solar collector. The result is presented for a Rayleigh numbers of Ra = 104 , an inclination angle of γ = π/4, a constant Biot number Bi = 5, an ambient temperature θ∞ = θc , heat flux intensity ξ = 0.1 and a geometric aspect ratio ϱ = 20.

Appendix D

Analytical Solutions for the Solar Collector D.1

Introduction

The commenced appendix will present analytical solutions limited in their case and are going to be compared to the corresponding numerical model. Attempted analytical solutions are obtained equal to the geometry found for the solar collector approximation. Two types of analytical solutions are considered, (1) will determine a velocity field in a parallel plate configuration where a Poiseuille flow is obtained and (2) will consider the temperature field in an equal geometry that is obtained for different Reynolds numbers. The obtained solutions are derived and presented in non-dimensional form, equal to the presented results throughout this report. The first section in this appendix will elucidate the velocity field and the second section will continue with the derivation of a temperature field.

D.2

Two-Dimensional Laminar Flow Between Two Parallel Plates

In this section the x ˜-component of the Navier-Stokes in the mixed convection model of chapter 5 is considered. In the approximation that forced convection starts to be

132

Appendix D. Analytical Solutions

133

decisive (Ri ≤ 1), the aspect ratio of the channel is very large (L >> H) and the solar collector is positioned horizontally (γ = π/2) the Navier-Stokes can be simplified such that an exact solution for the flow field is possible. The complete x ˜-component of the Navier-Stokes that is considered in the mixed convection model is given by, ∂u ˜ ∂ ∂ 1 + (˜ uu ˜) + (˜ vu ˜) = ˜ ∂x ˜ ∂ y˜ Re ∂t

(

∂2u ˜ ∂2u ˜ + 2 2 ∂x ˜ ∂ y˜

) −

∂ p˜ + Riθcos(γ). ∂x ˜

(D.1)

By taking into account the aforementioned assumptions the latter equation can be written according to, 0=

˜ ∂ p˜ 1 ∂2u − . Re ∂ y˜2 ∂x ˜

(D.2)

The combination of small Richardson numbers and large aspect ratios within a laminar flow will engender a fully developed flow and as a consequence will result in small amplitudes of the v˜-velocity (˜ v → 0), justifying the former simplified equation. The well-known solution of the simplified governing equation is obtained with the no-slip conditions (˜ u(˜ y = 0) = u ˜(˜ y = 1) = 0) on the wall and yields, ( ) Re ∂ p˜ u ˜(˜ y) = − y˜ 1 − y˜ . 2 ∂x ˜

(D.3)

One can couple the former solution to the maximum velocity that would occur in such a situation (typical Poiseuille flow), that would normally occur at exactly y˜ = 1/2, u ˜max = u ˜(˜ y = 1/2) = −

Re ∂ p˜ . 8 ∂x ˜

(D.4)

The former expression for the maximum occurring velocity can be substituted back in the solution for the cross-sectional velocity, ( ) u ˜(˜ y ) = 4˜ y 1 − y˜ u ˜max .

(D.5)

In the mixed convection model there is a prescribed uniform inlet velocity condition u ˜in , if one could couple the maximum occurring velocity u ˜max to the inlet velocity u ˜in one is able to easily compare and verify the solution of the numerical model. In the Boussinesq approximation there are no density variations - except for in the source term that appeared to be zero (γ = π/2) - and one can suppose a quasi-incompressible flow. In that case, whatever comes inside the channel also has to come out in an equal rate,


134

the mass flow is constant, ∫

1

m ˙ =

( ) 4˜ y 1 − y˜ u ˜max d˜ y=u ˜in 1.

(D.6)

0

The notorious solution is obtained that couples the inlet velocity to the maximum velocity, u ˜max = 3/2˜ uin .

(D.7)

Having a prescribed inlet condition of u ˜(˜ x = 0, y˜) = u ˜in = 1 the maximum velocity in its developed stage should be u ˜max = 3/2. 1 Numerical Solution Analytical Solution Analytical Max Velocity

0.8

y/H

0.6

0.4

0.2

0

0

0.5

1 u/U

1.5

2

Figure D.1: The numerical and analytical solution (both at x ˜ = L) of the crosssectional velocity is presented. The numerical solution approximates a mixed convection flow where the Reynolds number and Grashof number are 102 and 104 , respectively. The ratio of dimensionless numbers dictate that forced convection is decisive, the aspect ratio of the channel induces a fully developed flow condition at x ˜ = L and the inclination angle determines the exclusive horizontal flow. The analytical solution of eq. D.5 and eq. D.7 are given.

D.3

Two-Dimensional Convection-Diffusion Temperature Field in Parallel Plate Configuration

Having derived the velocity field in a laminar flow parallel plate configuration (γ = π/2), one can employ - in the limited case of having a uniform heat flux boundary condition over the entire length of the top-plate - the energy equation and obtain an analytical solution. To this regard, the methodology elucidated by Bird and Stewart [53] was found


135

to be helpful. This section will cover the derivation of the analytical solution for a specific form of the energy equation and the corresponding comparison with the numerical model. The thermal flow is furthermore defined by a cold inlet temperature, an adiabatic lowerplate and a thermal fully developed flow at the exit of the channel. Again, the mixed convection model of chapter 5 is considered, although this time exclusively the energy equation, ∂ ∂ ∂ ∂ (ρT ) + (ρuT ) + (ρvT ) = ∂t ∂x ∂y ∂x

(

k ∂T cp ∂x

)

∂ + ∂y

(

k ∂T cp ∂y

) .

(D.8)

In a laminar steady-state quasi-incompressible fluid flow situation where the crosssectional velocity component is fully developed and diffusive heating in the flow direction is neglected the latter energy-equation can be simplified towards the following, ∂ ∂ (ρuT ) = ∂x ∂x

(

k ∂T cp ∂x

) .

(D.9)

The problem is non-dimensionalized with the known parameters, shown in eq. 5.1. By doing so, the non-dimensional governing equation is obtained in the following form, ( ) ∂θ ∂ ( ∂θ ) 4˜ y P rRe 1 − y˜ u ˜max = , ∂x ˜ ∂ y˜ ∂ y˜

(D.10)

where the non-dimensional velocity profile u ˜(˜ y ) has been substituted (eq. D.5). Likewise, the known Reynolds and Prandtl numbers are introduced. The governing equation is defined by its boundary conditions that need to be non-dimensionalized in the same way and are given by,

1.

2.

3.

∂T k ∂y k ∂T ∂y

= q0

→

y=H

∂θ ∂ y˜

= 1,

y˜=1

=0

→

y=0

T (x, y) = Tc

∂θ ∂ y˜

= 0,

y˜=0

→

θ(˜ x, y˜) = 0.

If the temperature is considered to be linear dependent on the x ˜-direction, which is


136

reasonable for a fluid being sufficiently far downstream the channel, and that the crosssectional temperature field is not depending on the x ˜ coordinate than a solution can be imposed having the following form, θ(˜ x, y˜) = C0 x ˜ + ψ(˜ y ),

(D.11)

where C0 is a constant. The imposed solution is substituted into the non-dimensional governing equation (eq. D.10) and after some rewriting appears to be, d ( dψ ) = 4˜ y (1 − y˜)˜ umax P rRe. d˜ y d˜ y

(D.12)

The governing equation has turned into a second order ordinary differential equation, that allows it to be integrated twice rather straightforwardly, ψ(˜ y ) = 4C0 P rRe

( y˜3 6

−

y˜4 ) u ˜max + C1 y˜ + C2 . 12

(D.13)

The expression for ψ(˜ y ) is substituted back in the earlier imposed solution for the dimensionless temperature θ(˜ x, y˜), θ(˜ x, y˜) = C0 x ˜ + 4C0 P rRe

( y˜3 6

−

y˜4 ) u ˜max + C1 y˜ + C2 . 12

(D.14)

The application of boundary condition (1) and (2) will determine the constants C0 and C1 that after some rewriting appear to be, C0 =

1 2/3˜ umax P rRe

,

C1 = 0.

(D.15)

The third boundary condition is not that straightforward as substituting that condition will result in an arbitrary solution of the temperature field. It is however possible to determine an energy conservation condition that will balance the in- and outgoing heat. Disregarding the heat losses to the environment one can impose a condition that takes into account the constant heat flux entering the top-side and leaving the exit of the channel with respect to the inlet condition, ∫ xW q0 = W

H

ρcp (T − Tc )u(y)dy,

(D.16)

0

where W is the imagionairy width of the channel being unequal to zero. Rewriting this


137

to the non-dimensional form by substituting the non-dimensional parameters will give the following result after some reshaping, ∫ x ˜ = 4˜ umax P rRe

1

θ(˜ x, y˜)˜ y (1 − y˜)d˜ y.

(D.17)

0

One can solve this integral by substituting the expression found earlier for the nondimensional temperature in eq. D.14 (that one can be simplified by substituting the ( ) constants leading to θ(˜ x, y˜) = C0 x ˜ + y˜3 − y˜4 /2 + C2 , note that the first C0 could be substituted but is omitted to shorten up the text.). After some rewriting one will obtain the following expression, ( x ˜ = 4˜ umax P rRe

x ˜ C2 3 ) + + . 4˜ umax P rRe 6 140

(D.18)

In the former expression, the variable x ˜ will drop out of the equation and one is able to determine the constant C2 with some rewriting, 12 2 ˜max P rReC2 + u ˜max P rRe, 0= u 3 140

→

C2 = −

9 . 70

(D.19)

All constants have been determined for the earlier obtained solution for the non-dimensional temperature and everything put together appears to have the following form, θ(˜ x, y˜) =

( x ˜ y˜ ) 9 + y˜3 1 − − . P rRe 2 70

(D.20)

It should be noted that the obtained temperature distribution is not defined by an inlet condition and only becomes exact in the limit that x ˜ is sufficiently large, also mentioned by Bird and Stewart [53]. The same problem is considered for the numerical model that predicts flow phenomena in the geometry of the solar collector. However, the numerical model is defined by an additional prescribed inlet temperature and an additional fully developed flow at the exit of the channel. Hence the latter, a comparison would only make sense for temperature field approximations near the midsection of the channel at L/2. Figure D.2 shows such result for two different Reynolds numbers, Re = 1 and Re = 100 with a constant Prandtl number P r = 0.71 and a constant aspect ratio ϱ = 20.


138

Re = 1

Re = 100

0.8

0.8

0.6

0.6 y/H

1

y/H

1

0.4

0.4

0.2

0

Analytical Solution Numerical Solution 14

14.5 θ(x/H,y/H)

15

0.2

0

Analytical Solution Numerical Solution 0

0.5 θ(x/H,y/H)

1

Figure D.2: The cross-sectional temperature field distribution is shown for the parallel plate configuration at x = L/2. The solutions shown are obtained for two different Reynolds numbers, Re = 1 is shown on the left and Re = 100 is shown on the right. In both the analyzes the Prandtl number has been kept constant P r = 0.71, likewise for the aspect ratio of the channel that has been set to ϱ = 20.

Appendix E

Example MATLAB code clear all ; close all ; clc ; tic % Calculates Mixed convection for a % Two - Dimensional Transient State Advection - Diffusion with an Incom pressi ble Medium

%% Initializing LOAD =1;

% Update Changes

[ update ]= initializing ( LOAD );

% Will initialize ALL variables

load ( ’ folder \ Initializing . mat ’);

% Will load any new changes in the program

disp ( update );

% Will display what happened

% Plot the current mesh , on default is disabled . if plotmesh ==1 plot_mesh (x , x_u ,y , y_v , NPI , NPJ , XMAX , YMAX , RATIO ); end

%% Settings for the SIMPLE -[ Family ] Algorithm % 1 = Power - Law interpolation || 2 = QUICK scheme interpolation interpolation =2;

% Loading the Convergence Criteria [ all , convergence , criteria ]= co nver gen c edat a ( Re );

% % Setting the matrices for saving the pressures and velocities . % This will require a LOT of memory when enabled . uMAT = cell (1 , Re ); vMAT = uMAT ; pMAT = uMAT ; TMAT = uMAT ; DIVERGENCE_V = zeros (2 , Gr ); for i =1: Re uMAT {1 , i }(: ,:)= u ; vMAT {1 , i }(: ,:)= v ; pMAT {1 , i }(: ,:)= p ; TMAT {1 , i }(: ,:)= T ; end

139

Appendix E. Example MATLAB code

140

% Initiate Iteration Counters t =0;

% Initial time

n =1;

% Time counter - Outerloop

d =1;

% Used in Outerloop for ’ damping ’ any possible oscilating behaviour

z =1;

% Continuous counter inside the Innerloop

sumb =1; % Mass source initialize div_v =1;% Divergence of velocity source initialize

% A limiter set for the amount of shown covergence ticks , throughout the iterative process . % Will deteriorate speed for small values . kak =1; dkak =200;

figure (1) % For viewing live the time - residuals %% Start SIMPLE - sequence % Condition for time convergence , respectively 1: time - residual , 2: divergence of velocity % , 3: mass source , 4: criteria set to limit oscillations while steady_state < tRESuv

|| ( EpsilonMASS < abs ( div_v ) || EpsilonMASS < sumb ) || n Epsilon % Inner - iterations loop

% Initiating Velocity - Coupling Scheme %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Setting for SIMPLE velocity - pressure coupling . if scheme ==1 % Forcing global mass conservation after z iterations , z is conducted because % of non - linearity behavior in the beginning of convergence if z > t h r e s h o l d _ m a s s f l u x _ c o r r e c t i o n [ m_in , m_out ]= cont inu ity_c orr ( variables ); end % Set the boundary conditions [u ,v , T ]= BC ( variables ); % Set the convective flux per control volume [ F_u , F_v ]= conv_flux ( variables ); %% Navier Stokes % x - Velocity Navier - Stokes for Power - Law or QUICK scheme if interpolation ==1 [ aEu , aWu , aNu , aSu , aPu , aP0u , bu , d_u ]= ucoeffPL ( variables ); elseif interpolation ==2 [ aEu , aWu , aNu , aSu , aPu , aP0u , bu , d_u ]= ucoeffQU ( variables ); end % SOR solver


141

for i =1: K [ u ]= SOR_uPL (u , u0 , aEu , aWu , aNu , aSu , aPu , aP0u , bu , UIstart , UJstart , UIend , UJend , w ); end u_dU = u ; % Residuals for x - component NS . [ RESu ]= RESIDUALSvel (u , u0 , aEu , aWu , aNu , aSu , aPu , aP0u , bu , UIstart , UIend , UJstart , UJend ); if z L RESuNORM = RESu /( RESuN * NPI * NPJ ); else RESuNORM = RESu /(1 e -5* NPI * NPJ ); end

% y - Velocity Navier - Stokes for Power - Law or QUICK scheme if interpolation ==1 [ aEv , aWv , aNv , aSv , aPv , aP0v , bv , d_v ]= vcoeffPL ( variables ); elseif interpolation ==2 [ aEv , aWv , aNv , aSv , aPv , aP0v , bv , d_v ]= vcoeffQU ( variables ); end for i =1: K [ v ]= SOR_vPL (v , v0 , aEv , aWv , aNv , aSv , aPv , aP0v , bv , VIstart , VIend , VJstart , VJend , w ); end v_dV = v ; % Residuals for y - component NS . [ RESv ]= RESIDUALSvel (v , v0 , aEv , aWv , aNv , aSv , aPv , aP0v , bv , VIstart , VIend , VJstart , VJend ); if z L RESvNORM = RESv /( RESvN * NPI * NPJ ); else RESvNORM = RESv /(1 e -5* NPI * NPJ ); end

% Update Convective Terms [ F_u , F_v ]= conv_flux ( Istart , Jstart , Iend , Jend ,u ,v , F_u , F_v ); % Update BC [u ,v , T ]= BC ( variables );

% Pressure - Correction Equation , determine coefficients


142

[ aWpc , aEpc , aSpc , aNpc , aPpc , bpc , pc , sumb ]= pccoeff ( variables ); % Pressure - Correction Equation SOLVE for i =1: K [ pc ]= SOR_pc ( pc , aEpc , aWpc , aNpc , aSpc , aPpc , bpc , Istart , Iend , Jstart , Jend , w ); end % Apply the Corrections [p ,u , v ]= velcorr ( Istart , Iend , Jstart , Jend , relax_pc ,p ,u ,v , pc , d_u , d_v );

%% Energy - Equation for Power - Law or QUICK scheme if interpolation ==1 [ aET , aWT , aNT , aST , aPT , aP0T , bT ]= TcoeffPL ( variables ); elseif interpolation ==2 [ aET , aWT , aNT , aST , aPT , aP0T , bT ]= TcoeffQU ( variables ); end for i =1: K [ T ]= SOR_TPL (T , T0 , aET , aWT , aNT , aST , aPT , aP0T , bT , Istart , Iend , Jstart , Jend , w ); end T_dT = T ; % Residuals for Energy - Equation . [ REST ]= RESIDUALSvel (T , T0 , aET , aWT , aNT , aST , aPT , aP0T , bT , Istart , Iend , Jstart , Jend ); if z L RESTNORM = REST /( RESTN * NPI * NPJ ); else RESTNORM = REST /(1 e -5* NPI * NPJ ); end

l = l +1; % Inner - Counter z = z +1; % Inner - Counter - Global disp ([ ’ Iteration #: ’ num2str (l -1)

’ time - step # ’ num2str ( n )...

... ’ u - residual = ’ num2str ( RESuNORM ) ’ v - residual = ’ num2str ( RESvNORM )... ... ’ , T - residual = ’ num2str ( RESTNORM ) ’, t - residual = ’ num2str ( tRESuv )])

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Setting for SIMPLEC velocity - pressure coupling . elseif scheme ==2

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Setting for SIMPLER velocity - pressure coupling .


143

elseif scheme ==3

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Setting for SIMPLEX velocity - pressure coupling . elseif scheme ==4

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Setting for PRIME velocity - pressure coupling . elseif scheme ==5

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end RESmix =( RESuNORM + RESvNORM + RESTNORM )/3; % Sum ALL residuals if kak == dkak figure (2); axis ([0 z Epsilon 10]); semilogy (z , RESmix , ’o ’); axis ([0 z Epsilon 10]); hold on ; kak =1; end kak = kak +1; end % Inner - loop %% Time Residuals % Time - Residuals Ref Maliska pg 178 if n >1 [ RESut ]= RESIDUALStime (u , u0 , UIstart , UIend , UJstart , UJend ); [ RESvt ]= RESIDUALStime (v , v0 , VIstart , VIend , VJstart , VJend ); [ RESTt ]= RESIDUALStime (T , T0 , Istart , Iend , Jstart , Jend ); tRESuvAVG =( RESut + RESvt )/(2* NPI * NPJ ); if n < L tRES_NORM ( n )= tRESuvAVG ; tRESuv = tRESuvAVG /(1 e -4); elseif n == L tRES_NORM = mean ( tRES_NORM ); tRESuv = tRESuvAVG / tRES_NORM ; elseif n > L tRESuv = tRESuvAVG / tRES_NORM ; end tRES ( n )= tRESuv ; end

%% After - Treatment % Velocity update u0 = u ;


144

v0 = v ; T0 = T ; % Creating cells to compare the change of the velocity field u and v // % Quase - transient solution due to the transient boundary condition . uMAT {1 , n +1}(: ,:)= u ; vMAT {1 , n +1}(: ,:)= v ; pMAT {1 , n +1}(: ,:)= p ; TMAT {1 , n +1}(: ,:)= T ; % Mass conservation [ div_v , local_mass , sum_norm ]= ma s sco n ser v ati o n ( F_u , F_v , x_u , y_v , Istart , Iend , Jstart , Jend ); % Mass conservation over a local volume , averaged . DIVERGENCE_V (1 , n )= div_v ; % Weighted mass term averaged over domain . DIVERGENCE_V (2 , n )= sumb ;

% Time - Counter Update n = n +1;

% To ’ damp out ’ possible oscillations if steady_state > tRESuv d = d +1; % Part of Convergence criteria end % Live report of the time - residual figure (1) semilogy (n , tRESuv , ’o ’); xlabel ( ’ Time - Steps ’); ylabel ( ’t - Residual ’); title ([ ’ time = ’ num2str ( n * dt ) ]); hold on ; axis ([0 n steady_state 5]); end % Outer - Loop toc

disp ( ’ Solution converged for normalized mixed velocity and time residual , respectively yield : ’) disp ([ ’1: ’

num2str ( RESmix ) ]);

disp ([ ’2: ’

num2str ( tRESuv ) ]);

disp ([ ’3: Mass conservation yields : ’ num2str ( div_v ) ]); disp ([ ’4: Pressure correction source term yields : ’ num2str ( sumb ) ]);

save ([ ’ data_Re_ ’ num2str ( Re ) ’ NPI_NPJ_ ’ num2str ( NPI ) PVcoup ’. mat ’] , ’ - mat ’)

% Save the Averaged Inner - Iterations figure (2); % Calling the window of inner - iterations % Function to generate x - and ydata of the inner - iterations [ calmdown ]= i n n e r _ i t e r a t i o n s _ o b t a i n e r ( PVcoup , Re ); disp ( calmdown ) % If function is succesfully run this should display a text

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