UNIVERSITY OF CALGARY. Computational Modeling of Cascade Effect on Blade Elements with an Airfoil Profile. Haoxuan Yan A THESIS

Transcription

1 UNIVERSITY OF CALGARY Computational Modeling of Cascade Effect on Blade Elements with an Airfoil Profile by Haoxuan Yan A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE GRADUATE PROGRAM IN MECHANICAL AND MANUFACTURING ENGINEERING CALGARY, ALBERTA January, 2016 Haoxuan Yan 2016

2 Abstract The aim of this project is to investigate the effect of cascade on lift and drag coefficients and the angle of attack at which lift to drag ratio is maximized using the method of Computational Fluid Dynamics (CFD). The effect of cascade includes two aspects: the spacing ratio between blades and the change of pitch angle. The blades of wind turbines are separated by a finite distance in the azimuthal direction. The Blade Element Theory (BET) assumes a blade element to be an airfoil and the blade element is independent to each other; and conducting experiments for low solidity blades is difficult. Thus, CFD would be an appropriate method to investigate how the cascade effect would impact the aerodynamics of wind turbines. The NACA 4415 was modeled and meshed in ANSYS ICEM 15.0 and then was calculated using transition SST (shear stress transport) 4-equation model in ANSYS Fluent 16.0 at Reynolds number of The computational results were compared with experimental results (Hoffmann 1996). User Defined Functions (UDF) of transition SST model were also tested in order to find the most optimal correlations. The typical range for a three-blade horizontal wind turbine varies from 0.3 near the hub to 0.1 at the tip (Bak 2013). The pitch angles considered in this study are 0, 20, and 90. The results show that the lift and drag coefficients are influenced by the solidity and pitch angle. ii

3 Acknowledgements I would like to express my sincere thanks to my supervisor Dr. David Wood for giving me the opportunity and guiding me through this project. I appreciate his endless support and patience. My gratitude also goes to Dr. Mohamed Arif bin Mohammed for his support and guidance in solving problems encountered in the project. I also give my sincere thanks to Dr. Mazharul Islam for providing beneficial advice and help through the completion of the work. I would also like to give my thanks to the Fluids group (Arman Hemmati, Jaime Wong, John Fernando, Kamaldeep Singh, Meraj Mohebi, Mohammed Hamman, Phillip du Plessix, Tahsina Hossain Loba, Tong Liu, Yunjian Li) at the University of Calgary for their supports. I also express my thanks my dad Feng Yan and mom Honghong Zhang for supporting me through the study in Canada. This project was supported by the National Science and Engineering Research Council (NSERC) under the Industrial Research Chairs program. iii

4 Table of Contents Abstract... ii Acknowledgements... iii Table of Contents... iv List of Tables... vi List of Figures... vii Chapter 1 Introduction...1 Chapter 2 Literature Review Introduction Experimental Study of the Airfoil Computational Study of the Airfoil Scope of the Project...15 Chapter 3 Computational Formulation and Numerical Modelling Introduction Governing Equations Conservation of Mass Conservation of Momentum Turbulence Modelling Modeling the Reynolds Shear Stress The Transition SST Model The Transport Equations of Standard SST k-ω Model The Transport Equation of Intermittency The Transport Equation of Reθt Coupling with the Transport Equation of k...28 Chapter 4 Aerodynamics of the Airfoil Introduction The Aerodynamics of Airfoil Isolated Airfoil Effect of Solidity Effect of Blade Pitch (Twist) Geometry of the Airfoil...39 Chapter 5 Test Cases Introduction The Isolated Airfoil Meshing Boundary Conditions Mesh Independence Test Results and Discussions The Effect of Solidity Meshing Boundary Conditions...63 iv

5 5.3.3 Mesh Independence Test Results and Discussions The Effect of Blade Pitch Meshing Boundary Conditions Mesh Independence Test Results and Discussions...80 Chapter 6 Conclusion...93 Reference...97 Appendix A: Computational Fluid Dynamics (CFD) A.1 Finite Volume Method A.2 Central Differencing Scheme A.3 Upwind Differencing Scheme A.4 Power Law Scheme and QUICK Scheme A.5 Properties of Discretization Schemes A.6 Evaluation of Gradients A.7 Schemes for Pressure-Velocity Coupling Appendix B: User-Defined-Function (UDF) Codes B.1 Correlation by Menter B.2 Correlation by Sørensen B.3 Correlation by Malan v

6 List of Tables Table 5.1 Computational results of mesh independence test at α = Table 5.2 GCI results of the mesh independence test Table 5.3 Comparison between computational and experimental results Table 5.4 Boundary conditions for cascaded blades Table 5.5 Details for inlet boundary conditions Table 5.6 Computational results of mesh independence test for solidity of Table 5.7 GCI results for the case of solidity of Table 5.8 Comparison between cascaded blades and isolated airfoil at α = Table 5.9 Results of mesh independence test for pitch angle of 20 and solidity of Table 5.10 GCI results for pitch angle of 20 and solidity of Table A.1 Coefficients for various differencing schemes vi

7 List of Figures Figure 2.1 Lift coefficient for the NACA 4415 (Miley 1982)... 6 Figure 2.2 Drag coefficient for the NACA 4415 (Miley 1982)... 7 Figure 2.3 Lift coefficient for the NACA 4415 (Hoffman 1996)... 9 Figure 2.4: Drag coefficient for the NACA 4415 (Hoffman 1996)... 9 Figure 2.5: Pressure distribution for the NACA 4415 (Hoffman 1996) Figure 2.6 Natural transition on a 2D flat plate in a water channel (Aupoix et al. 2011) Figure 2.7 Natural transition process (Schlichting 1979) Figure 4.1 Forces exerting on the airfoil Figure 4.2 Forces decomposition of lift and drag Figure 4.3 Schematic of local solidity Figure 4.4 Schematic of the cascade effect Figure 4.5 Schematic of the pitch angle Figure 4.6 Two-dimensional geometry of the NACA Figure 4.7 (a) Leading edge of the NACA 4415 in ANSYS ICEM Figure 4.7 (b) Trailing edge of the NACA 4415 in ANSYS ICEM Figure 5.1 H-type grid used for the meshing of airfoil Figure 5.2 Close up of the same grid near the wall of airfoil Figure 5.3 Number of nodes at each edge of the blockings of airfoil Figure 5.4 Boundaries for the isolated airfoil computation Figure 5.5 Decay of the turbulence intensity Figure 5.6 (a) Variation of lift coefficient Figure 5.6 (b) Variation of drag coefficient Figure 5.7 (a) Lift coefficient of the computational and experimental results Figure 5.7 (b) Drag coefficient of the computational and experimental results vii

8 Figure 5.7 (c) Lift to drag ratio of the computational and experimental results Figure 5.8 (a) Pressure distribution at α = Figure 5.8 (b) Pressure distribution at α = Figure 5.8 (c) Pressure distribution at α = Figure 5.9 (a) Velocity vectors (m/s) over the airfoil at α = Figure 5.9 (b) Close view of the velocity vector nears the wall of airfoil Figure 5.10 Skin friction coefficient of the isolated airfoil at α = Figure 5.12 Mesh domain of cascaded blades Figure 5.13 Schematic of boundaries for cascaded blades Figure 5.14 (a) Variation of lift coefficient Figure 5.14 (b) Variation of drag coefficient Figure 5.15 (a) Lift coefficient of various solidities Figure 5.15 (b) Drag coefficient of various solidities Figure 5.15 (c) Lift to drag ratio of various solidities Figure 5.16 Pressure distribution of cascaded blades at αm Figure 5.17 Wall shear stress distribution of cascaded blades at αm Figure 5.18 (a) Velocity (m/s) contours of cascaded blades at αm Figure 5.19 (a) Turbulence kinetics energy (m2/s2) contours of cascaded blades at αm Figure 5.20 (a) Mesh domain used for P = 0 (b) Rotated mesh domain for P = Figure 5.21 (a) Mesh domain used for the P = 20 (b) Rotated mesh domain for P = Figure 5.22 Close up of mesh domain for P = Figure 5.23 (a) Variation of lift coefficient Figure 5.23 (b) Variation of drag coefficient viii

9 Figure 5.24 (a) Lift coefficient of pitched blades with the solidity of Figure 5.24 (b) Drag coefficient of pitched blades with the solidity of Figure 5.24 (c) Lift to drag ratio of pitched blades with the solidity of Figure 5.25 (a) Lift coefficient of pitched blades with the solidity of Figure 5.25 (b) Drag coefficient of pitched blades with the solidity of Figure 5.25 (c) Lift to drag ratio of pitched blades with the solidity of Figure 5.26 (b) Drag coefficient of pitched blades with the solidity of Figure 5.26 (c) Lift to drag ratio of pitched blades with the solidity of Figure 5.27 (a) Lift coefficient of cascaded blades at P = Figure 5.27 (b) Drag coefficient of cascaded blades at P = Figure 5.27 (c) Lift to drag ratio of cascaded blades at P = Figure 5.28 (a) Lift coefficient of cascaded blades at P = Figure 5.28 (b) Drag coefficient of cascaded blades at P = Figure 5.28 (c) Lift to drag ratio of cascaded blades at P = Figure 5.29 (a) Lift coefficient of cascaded blades at P = Figure 5.29 (b) Drag coefficient of cascaded blades at P = Figure 5.29 (c) Lift to drag ratio of cascaded blades at P = Figure 5.30 Contour of velocity (m/s) of three pitched blades at solidity of 0.3 and α = Figure 5.31 Close up of velocity (m/s) contour for pitch angle of 0, 20, and 90 at solidity of 0.3 and α = Figure 5.32 Contour of turbulence kinetic energy (m2/s2) of pitched blades at solidity of 0.3 and α = Figure A.1 Discretization of finite control volume method (Versteeg and Malalasekera 2007) 102 Figure A.2 (a) The flow is in positive direction (Versteeg and Malalasekera 2007) Figure A.3 Flux through four control volumes (Versteeg and Malalasekera 2007) ix

10 Figure A.4 Distribution of a property ϕ in a vicinity of a source at different Peclet numbers (Versteeg and Malalasekera 2007) Figure A.5 Evaluation of the gradient at the cell centroid (ANSYS 2013) x

11 Chapter 1 Introduction In the past few decades, wind energy has been evolved dramatically to become one of world s most significant renewable energies due to the very large wind resources, low cost of installation, and trivial impacts to the environment. Wind power harnessed from the nature can be converted and generated into electricity through wind turbines. A wind turbine harvests the kinetic energy in the wind and then converts it to electrical energy. But until the early twentieth century, all wind turbines were used to generate the torques such as water pumping and grain milling rather than producing electricity (Wood 2011). Nowadays, wind turbines are mainly used for generating electricity, especially large wind turbines, which can reach the power level of megawatts. Consisted a group of wind turbines in the same location, wind farms can produce even much more electricity. Lots of the large-scale wind farms are built in Germany, China and United States. For example, the largest wind farm in the world, Gansu Wind Farm in China has a capacity of over 6,000 MW of power in 2012 with a goal of 20,000 MW by On the contrast, the large nonrenewable power stations in China and Taiwan have an average capacity of 5000 megawatts (Zhao et al. 2014). Wind power almost has an equivalent capability of producing electricity as conventional power stations while has slight impact on the environments rather than having severe issues such as air and aquatic pollutions produced by fire- power plants. Therefore wind energy plays an important role solving the global energy crisis and has a bright future to take the place of conventional power stations. 1

12 The study of airfoil aerodynamics is critical to wind turbine design, especially the lift and drag coefficient, which are needed to predict the power efficiency of a wind turbine. The Conducting experiments in the wind tunnel is the traditional way to measure the lift and drag coefficient data for airfoils. However, experimental method is time consuming and requires abundant funding support while the reliability of experimental results is also based on the quality and turbulence level of wind tunnels. On the other hand, the method of Computational Fluid Dynamics (CFD) can accurately simulate the flow passing through the airfoil and calculate the aerodynamic parameters, for example, lift and drag coefficient and wall shear stress. In the past few decades significant amount of progress has been made in the improvement of reliable turbulence models and CFD codes that can precisely simulate a wide range of engineering problems. The achievements by different researchers have resulted in a number of turbulence models ranging from Spalart- Allmaras one equation model to Transition SST four equation model that can be used in many industrial and academic applications, while balancing the accuracy of results and the computational expenses available to a CFD user (ANSYS 2013). Figure 1.1 shows the blades of a wind turbine intersected with a annular ring, which can be unfolded into an infinite cascade. In the figure, r is distance from rotor to blade element; N is number of blades; c is chord length; s is distance between blades; α is angle of attack. The Blade Element Theory (BET) assumes a blade element to be an airfoil and also assume the blade element is independent to each other. Conventional experimental measurements regarding to lift and drag coefficients assume zero solidity of the blades of a wind turbine, where the effect between blades is not considered. The solidity defines the spacing ratio between blades (Dixon 2005). It should be 2

13 noted that in fact a wind turbine has not a zero solidity, but a typical range is from 0.1 to 0.3, (Bak 2013), which is in the range of low solidity (lower than 1.0). Thus the solidity effect should be considered when design the blades for a wind turbine. Current experimental data of lift and drag coefficients are only based on the isolated airfoils but ignore the solidity effect because it would be difficult to perform accurate wind tunnel tests at low solidity. Therefore the investigation of effect of cascade is critical to evaluate the aerodynamic performances of airfoils for a wind turbine. 2πr N Figure 1.1 Unfolding the blade elements of a wind turbine into an infinite cascade The main objective of this study is to investigate the effect of cascade on lift and drag coefficients; and the angle of attack at which the lift to drag ratio is maximized. The airfoil used in this project is NACA 4415, which is a four-digit airfoil developed by the National Advisory Committee for Aeronautics (NACA). The NACA 4415 is typically used in aircraft and has the similar aerodynamic performances to wind turbines. And the experimental data are easily available. Chapter 2 gives a review of relevant literature regarding to previous experimental results and CFD 3

14 simulations of airfoils. In Chapter 3, the approach of CFD simulations such as turbulence model and numerical modelling are described. The Chapter 4 describes the aerodynamics of airfoil and geometry creation. The results and discussions are included in each test case in Chapter 5. The isolated airfoil, effect of cascade, and effect of pitch are discussed in test cases. Finally, conclusions and outlook for future work are discussed in Chapter 6. 4

15 Chapter 2 Literature Review 2.1 Introduction The purpose of this chapter is to review previous experiments and researches that have been conducted on airfoils, especially on the NACA The first section of this chapter discusses the wind tunnel experiments conducted on the NACA 4415 in the following section. The Ohio State University (OSU) Aeronautical and Astronautical Research Laboratory conducted a series of steady and unsteady state wind tunnel tests on a series of airfoils that have been used for horizontal axis wind turbines, which include NACA 4415 (Hoffmann 1996). The turbulence intensity of the OSU wind tunnel is very low, smaller than 0.3% according to Hoffmann (1996). The experimental results from Hoffmann (1996) and Miley (1982) were used for CFD validation in this thesis. The second section describes previous CFD work on airfoils and turbulence modelling. Various transition models will also be summarized in this section. The major emphasis of this section will be on evaluating the transition models and selecting the most appropriate one for the airfoil simulations in this thesis. At low Reynolds number, laminar separation bubbles caused by adverse pressure will influence the results significantly. The last section describes the scope of this project. 5

16 Cl 2.2 Experimental Study of the Airfoil The most accurate and reliable method to study the aerodynamics over an airfoil is to conduct the experiments in a wind tunnel. By measuring related experimental parameters on the airfoil, lift and drag coefficients can be obtained as well as the pressure distribution on the airfoil surface, which is the essential data to validate the CFD calculations. There are plenty of experimental data of airfoils obtained by various research facilities but the reliable experimental results of airfoil NACA 4415 is of lack. There was a NACA 4415 experiment performed in Stuttgart in 1900 (Miley 1982). The wind tunnel has a dimension of 3x5m with the turbulence intensity of 0.1%. The Reynolds number ranges from 0.1 million to 3 million. The lift and drag coefficients are plotted in Figure 2.1 and Figure 2.2. However, pressure distribution is not included in the report Stuttgart Re=1.0 million Stuttgart Re=1.5 million Angle of Attack Figure 2.1 Lift coefficient for the NACA 4415 (Miley 1982) 6

17 Cd Stuttgart Re=1.0 million Stuttgart Re=1.5 million Angle of Attack Figure 2.2 Drag coefficient for the NACA 4415 (Miley 1982) It can be seen from the Stuttgart experimental data, the lift coefficient curve of Re=1.5 million is slightly higher than that of Re=1.0 million while the drag coefficient of Re=1.5 million is reduced. In 1995, the National Renewable Energy Laboratory (NREL), funded by the US Department of Energy, approved a contract to the Ohio State University (OSU) to perform a series of wind tunnel experiments (Hoffman 1996). Under this program OSU tested various popular wind turbine airfoils such as NACA 4415 and S809. Each airfoil was tested under a standard test matrix to assure the same test conditions, which include airfoil performance at high angles of attack, clean and rough surfaces, steady and unsteady angles of attack. All the experimental reports can be found on the OSU website (2015), which provides raw data of the experiments. In the experiment of NACA 4415, a 457mm constant-chord model of the NACA 4415 airfoil was used for testing under two- 7

18 dimensional steady state conditions in a 3 x5 Subsonic Wind Tunnel at the Ohio State University (Hoffman 1996). The wind tunnel is an open-circuit and has a velocity range of 0-55m/s with a turbulence intensity under 0.3%. In this experiment, data were acquired and processed from 60 surface pressure taps placed on the airfoil, four separate tunnel pressure transducers used for atmospheric pressure measurement, an angle of attack potentiometer, a wake probe position potentiometer, and a tunnel thermocouple. The pressure angle of attack were calibrated by a water manometer and potentiometer to determine their sensitivities and offsets before the tests. Then the data were entered into the data acquisition and reduction program. Lift and drag characteristics and surface pressure data were obtained in this experiment with the Reynolds number of 0.75, 1, 1.25, and 1.5 million. Figure 2.3 and Figure 2.4 show the lift and drag coefficients of the NACA 4415 with different Reynolds numbers. As can be seen from the plot of lift coefficient, the stall angle from Re = 0.75 million to 1.5 million is delayed with the decreasing of the Reynolds number, which happens at 15.3, 14.3, 13.3, and 12.2 respectively. However, the magnitude of Cl has no significant differences, especially below the α = 13. The plot of drag coefficient illustrates that Cd remains at around below angle of α = 6 and increases rapidly from 8 to 13. 8

19 Cd Cl OSU Re=0.75 million OSU Re=1.00 million OSU Re=1.25 million OSU Re=1.50 million Angle of Attack Figure 2.3 Lift coefficient for the NACA 4415 (Hoffman 1996) OSU Re=0.75 million OSU Re=1.00 million OSU Re=1.25 million OSU Re=1.50 million Angle of Attack Figure 2.4: Drag coefficient for the NACA 4415 (Hoffman 1996) 9

20 Cp The data of pressure coefficient are also included in the experiment report. The pressure distribution is one of the most crucial results to compare the experimental data and simulation results, especially to observe the laminar separation bubble on the suction side. Figure 2.5 shows the pressure coefficient of Re=1million at α = 4.1. It should be noted that Cp does not reach the point of 1.0 on the pressure side but there must be a stagnation point which exists between two pressure taps OSU Re=1 million x/c Figure 2.5: Pressure distribution for the NACA 4415 (Hoffman 1996) Traditional wind tunnel experiments can only provide the data of isolated airfoil. It is difficult to perform the experiments on cascaded airfoils with low solidity, which requires very large wind tunnel because the space between each airfoil becomes large for low solidity cascade and the experiment needs at least three airfoils. If the solidity is 0.1 and the chord of the airfoil model is 0.5m, the outlet height of the wind tunnel would at least be 12 meters. The purpose of wind tunnel 10

21 tests is used to test the airfoil calculation and then provide a validation to assess the changes due the effect of cascade. 2.3 Computational Study of the Airfoil In addition to wind tunnel experiments, computational method is another approach to investigate the aerodynamics of the airfoil with higher efficiency and easier way to obtain various results. For instance, it is convenient for computational study to obtain the velocity and pressure distribution and the wall shear stress, which are difficult to determine accurately in wind tunnel experiments. There are some computational studies that have been conducted on the airfoil regarding to turbulence models and cascade of the blades since the computer was able to run the CPU-expensive calculations. Rizzetta and Visbal investigated two turbulence models for use in calculating airfoil stall. They compared Baldwin Lomax model and k-epsilon model and found that k-epsilon had better results (Rizzetta et al. 1993). Zingg et al. (1991) conducted a computational study of the use of higher order viscous equations in the flow fields with a viscid-inviscid interaction scheme. The solution obtained by the first and second differencing schemes was compared with the experimental results. They found that the first order procedure underestimated the boundary layer displacement thickness and momentum thickness on the upper surface near the trailing edge. Higher order terms did not impact the lift significantly but had a more accurate prediction the drag (Zingg et al. 1991). Jonnavithula et al. (1990) investigated the stall propagation in axial compressors. In this study, compressor blades were placed as an isolated translational cascade of blades with an airfoil shape and stall propagation was calculated. Detailed parametric studies were 11

22 performed to analyze the effects of flow parameters such as inlet angle and blade solidity. The stall propagation velocity decreased with the increasing of the inlet angle but increased with the increasing of blade solidity. Ahmed (1995) conducted computational study of the viscous incompressible flow through a cascade of the NACA 0012 airfoil at the Reynolds number of An H-type grid was used in his study. In the parametric study, the angle of attack varied from 0 to 24 and solidity ranged from 0.55 to 0.83 and stagger angle changed from 10 to 30. He found that the incidence where the maximum lift coefficient is obtained increases as the increase of the solidity. At high incidences, reducing the stagger angle of the airfoil in the cascade can increase the lift coefficient at high incidences. The transition process from laminar to turbulent is critical in this study as it has a significant effect on predicting the lift and drag coefficients. There are two transition modes by which the transition is generally believed to occur (Aupoix et al. 2011). The first mode is Natural Transition, which happens when the freestream turbulence level is smaller than 1% (Mayle 1991). In natural transition, there are three successive steps which are Receptivity, Linear and Non-linear (Aupoix et al. 2011). Receptivity happens close to the leading edge where the forced disturbances enter the laminar boundary layer. In the second stage, periodic waves are generated and convert their energy in the streamwise direction. During this process, some of the waves are increased and will be responsible for transition. In the third stage, nonlinear interactions occur and the turbulence takes place when the wave amplitude reaches beyond a critical point. Figure 2.6 describes the visualization of natural transition on a 2D flat plate in a water channel. 12

23 Figure 2.6 Natural transition on a 2D flat plate in a water channel (Aupoix et al. 2011) In 2D flows, the linear waves are known as Tollmien-Schlichting (TS) waves where a laminar boundary layer becomes linearly unstable beyond a critical Reynolds number (Schlichting 1979). In 3D flows, such as on a swept wing, TS waves are not the only source of disturbances that cause the transition but also the cross-flow (CF) waves which are generated by the cross flow velocity component. A diagram of the natural transition process from is shown in Figure 2.7. Figure 2.7 Natural transition process (Schlichting 1979) 13

24 When the turbulence intensity of freestream is greater than 1%, the transition mode becomes the bypass transition where the first and possibly second and third phases of the natural transition process are skipped such that turbulent flows are directly produced in the boundary layer (Mayle 1991). In the bypass transition, the laminar to turbulent transition is triggered earlier than the natural transition if the energy of flows grows dramatically, which means the natural transition process is short-circuited. It should be noted that not only high turbulence level can cause the bypass transition but also the surface roughness where the disturbances are generated from the wall instead of from the freestream turbulence. Therefore, the predicting of transition process properly or turbulence modelling plays an important role in the Computational Fluid Dynamics because it determines how the laminar to turbulent transition process will be calculated. Thus developing effective turbulence models which can be widely used in different applications has been the essential subject in the computational study. The most popular method for predicting transition is the e N method, developed by van Ingen (2008). The N factor is the total growth rate of the most unstable disturbances (Aupoix et al. 2011). The laminar to turbulent transition is expected to occur for some specified value of N, for instance, N ranges from 8 to 10 on 2D airfoils in the wind tunnels with low turbulence level. The e N method is simply based on the linear theory which is the second stage of the natural transition but does not take the receptivity and the non-linear stages into consideration. The well-known airfoil design and analysis program XFOIL uses the e N method to predict the transition point and also uses a viscous-inviscid flow panel method. However, due to the reason that the e N method uses non-local quantities such as momentum thickness the model cannot be properly adapted to large parallel 14

25 Reynolds-Averaged Navier Stokes (RANS) computations. Menter et al. developed a new correlation-based transition model which is based strictly on local variables (Langtry and Menter 2009). This model is based on the k ω SST (Shear Stress Transport) model with two additional transport equations, intermittency which can be used to activate transition locally and transition momentum thickness Reynolds number which is necessary to detect the non-local influence of the turbulence intensity. This turbulence model can be entirely calibrated with exclusive transition onset and transition length correlations. The details of this model is discussed in Chapter 3. The laminar separation bubble also plays an important role in airfoil calculation. It is generated due to the adverse pressure on the boundary layer. The laminar separation bubble could occur when Reynold number is lower than 700,000 (Johansen 1997). If the laminar separation bubble is large, it would affect the transition onset and pressure distribution significantly. 2.4 Scope of the Project From the literature review relevant to this project, there are abundant computational and experimental studies regarding to the aerodynamics of the isolated airfoils, which can provide detailed references to the present study. However, the effect of low solidity on the airfoil has not been well studied. For the blades of wind turbines, they are placed with the low solidity ranging from 0.1 to 0.3 (Bak 2013). It is difficult to conduct experiments of blades with low solidity because this requires large wind tunnel with high mean flow uniformity and low turbulence intensity. On the other hand, conventional Blade Element Theory (BET) only assumes aerodynamic characteristics of isolated airfoil. Thus, performing computational simulations to investigate the effect of solidity on airfoils is a proper method. The calculations of flow field through an isolated airfoil is also conducted to provide a validation with experimental results and 15

26 also a reference for the assessment of the cascade effect. This project is expected to broaden the knowledge of computational simulations and extend the data base of NACA The results is believed to be essential and beneficial to the aerodynamic design of airfoils and wind turbines. The objective of this study is aimed to investigate how the low solidity and the variation of pitch angle affect the aerodynamics characteristics of NACA 4415 airfoil. To accomplish this study, the Computational Fluid Dynamics (CFD) method is used. By this way, the airfoil is meshed in ANSYS ICEM and then calculated in ANSYS FLUENT with a transition SST turbulence model, developed by Menter and Langtry (Langtry and Menter 2009). 16

27 Chapter 3 Computational Formulation and Numerical Modelling 3.1 Introduction The first section of this chapter describes the governing equations used in the Computational Fluid Dynamics (CFD). In the second section, turbulence modelling and the turbulence model used in this study are discussed. The finite volume method and discretization schemes with the emphasis on the algorithms used in ANSYS Fluent 16.0 are discussed in Appendix. A. 3.2 Governing Equations Conservation of Mass The governing equations of fluid flow describe mathematical equations of the conservation laws of physics. The first equation describes that the mass of a fluid is conserved, which is named the continuity equation and is given by (Versteeg and Malalasekera 2007): ρ + div(ρu) = 0 (3.1) t or in Cartesian co-ordinate notation ρ t + (ρu) x + (ρv) y + (ρw) = 0 (3.2) z 17

28 where ρ is the density of the fluid, t is time, u is velocity vector, u, v, w are the velocity components. Equation (3.1) is the conservation form of the continuity equation. By simplifying the divergence term, such as div(ρu) = ρ u + u ρ (3.3) and substitute Equation (3.3) into Equation (3.1): ρ + ρ div u + u div ρ = 0 (3.4) t The first two terms on the left side in the Equation (3.4) are the substantial derivative of the density. Thus, Equation (3.4) becomes Dρ + ρ u = 0 (3.5) Dt For an incompressible fluid, the density ρ is constant and Equation (3.5) can be simplified to be u = 0 (3.6) or in longhand notation u x + v y + w z = 0 (3.7) 18

29 3.2.2 Conservation of Momentum The second governing equation gives the mathematical statement for the conservation of momentum, which applies Newton s Second Law. This equation is also known as Navier-Stokes equation, which is given by x momentum y momentum (ρu) t (ρv) t + div(ρuu) = p x + div(μ grad u) + S Mx + div(ρvu) = p y + div(μ grad v) + S My (3.8a) (3.8b) z momentum (ρw) t + div(ρwu) = p z + div(μ grad w) + S Mz (3.8c) or using Einstein s notation (ρu i ) t + (ρu iu j ) x j = p x i + μ x i ( u i x j + u j x i ) + S Mxi (3.9) where p is the pressure term and μ is dynamic viscosity of the fluid, S M is the momentum source. For the fluid without body forces, the momentum source term S M equals to zero. If a general variable φ conserved, a general form of the transport equations can be written as: (ρφ) t + div(ρφu) = div(γ grad φ) + S φ (3.10) Rate of Net rate of Rate of increase Rate of increase increase of φ flow of φ out of φ due to of φ due to 19

30 where Γ is the diffusion constant and φ is the conservative form of all fluid equations including scalar quantities such as temperature, and S φ is the source term of φ. Equation (3.10) is the transport equation for property φ, which indicates that the rate of change and the convective terms are equal to the diffusive and the source terms. This equation is the key to use the finite volume method in CFD, which is discussed in the following section. 20

31 3.3 Turbulence Modelling Modeling the Reynolds Shear Stress Turbulent flows exist in most CFD problems and have irregular fluctuations of velocity in all three directions, which have certain spatial structures known as eddies. The turbulence once generated in a flow usually continues and perpetuates itself without vanishing such that the turbulent flows are self-sustaining but sometimes turbulent flows can relaminarize. Turbulent flows have fluctuating vorticity, and are also diffusive and dissipative (Panton 2005). Turbulent flows are difficult to predict so that turbulence modeling is one of challenging tasks in CFD. The first step in turbulence modeling is Reynolds averaging where the solution variables in Navier-Stokes equations are decomposed into the mean and fluctuating components (Wilcox 2006). For the velocity components, u j = U j + u j (3.11) where u j is the decomposed velocity, U j is the mean velocity component and u j is the fluctuating component. Equation (3.40) is known as Reynolds Decomposition. Similarly, the pressure and other scalar quantities can be expressed using the same decomposition, such that φ j = Φ j + φ j (3.12) where φ denotes a scalar property such as pressure, energy, or temperature. Substituting Equation (3.11) into the momentum equations and taking a time average gives the ensemble-averaged momentum equations, which can be written as U j t + U U j i = p + (2μS x i x j x ij ρu i u j ) (3.13) i 21

32 Equation (3.42) is the Reynolds-Averaged Navier-Stokes (RANS) mean momentum equation. S ij is the mean strain-rate tensor given by S ij = U i x j + U j x i (3.14) And the Reynolds-stress tensor is denoted by τ ij = ρu i u j (3.15) The Transition SST Model The turbulence model used in this study is transition shear stress transport (SST) model is also known as the γ Re θ model, based on the SST k - ω two-equation model where k is the turbulence kinetic energy and ω is the specific dissipation rate, coupling with two other transport equations, one for the intermittency γ and one for the transition onset momentum thickness Reynolds number Re θ (Langtry and Menter 2009). γ is the probability that the flow is turbulent and Re θ controls the point where the flow starts to become turbulent. Empirical correlations can be embedded into the model to modify the transition onset and transition length. The default empirical correlation in Fluent was developed by Langtry and Menter (Langtry and Menter 2009). There are four main transport equations in this model. The transition model is needed because the flow has laminar-turbulent transition on the boundary layer of airfoil. The prediction of transition onset plays an important role in calculating the aerodynamic characteristics of the airfoil such as lift and drag coefficients. Therefore, precisely predicting the transition is the key to obtain the accurate result in this study. In Fluent, there are k-kl-ω three-equation model and transition SST 22

33 four-equation model that can predict the transition process. By comparing the results with experimental data (Hoffmann 1996) the transition SST four-equation model has the better match with experimental results than k-kl-ω model. Therefore, transition SST 4-equation model was selected as the turbulence model in this study The Transport Equations of Standard SST k-ω Model The equations in this and following sections are mainly from ANSYS Fluent theory guide (ANSYS 2013). The transport equations for the turbulence kinetic energy (k) and the specific dissipation rate (ω) of original SST k- ω model are given as: (ρk) + (ρku t x i ) = k (Γ i x k ) + G j x k Y k + S k (3.16) j and (ρω) + (ρωu t x i ) = ω (Γ i x ω ) + G j x ω Y ω + S ω (3.17) j where G k is the generation of turbulence kinetic energy due to mean velocity gradients, G ω is the generation of ω. While Γ k and Γ ω represent the effective diffusivity of k and ω, respectively. Y k and Y ω are the dissipation of k and ω due to turbulence. S k and S ω represent the user-defined source terms, which are not used in this study. All of the above terms are defined as below. The effective diffusivities are given by Γ k = μ + μ t σ k (3.18) 23

34 Γ ω = μ + μ t σ ω (3.19) Where σ k and σ ω are the turbulent Prandtl numbers for k and ω, respectively. And μ t is the turbulent viscosity, which is computed as the following: μ t = α ρk ω (3.20) Where α is a damping coefficient used to adjust the dissipation of k and ω in the viscous layer. In Fluent, the default value of α equals 1, which is recommended for medium or high Reynolds number calculations (ANSYS 2013) The Transport Equation of Intermittency The equation in terms of intermittency (γ) is given as: (ργ) + (ργu t x i ) = [(μ + μ t ) γ ] + P i x j σ γ x γ1 E γ1 + P γ2 E γ2 (3.21) j where P γ1 and E γ1 are the transition sources terms and P γ2 and E γ2 are the destruction sources terms. P γ1 and E γ1 are defined as the following: P γ1 = C a1 F length ρs[γf onset ] C γ3 (3.22) E γ1 = C e1 P γ1 γ (3.23) where F length is an experiment-based correlation that controls the length of the transition region, and S is the strain rate magnitude. The empirical correlation is based on a significant amount of flat plate experiments where a correlation of transition length was created. The transition 24

35 correlation can be exchanged according to the experimental information. F onset controls the position where the transition is activated. F onset is determined by the following functions: F onset1 = Re v 2.193Re θc (3.24) F onset2 = min (max(f onset1, F onset1 4 ), 2.0) (3.25) F onset3 = max (1 ( R T 25 ) 3, 0) (3.26) F onset = max(f onset2 F onset3, 0) (3.27) where Re θc is the critical Reynolds number where the intermittency initial starts to increase in the boundary layer. Re v and R T are the vorticity Reynolds number and viscosity ratio and they are defined as the following Re v = ρy2 S μ R T = ρk μω (3.28) where y is the distance from the nearest wall and μ is the dynamic viscosity. The vorticity Reynolds number is a local property and can be calculated at each grid node in the unstructured mesh (Langtry and Menter 2009). The relaminarization sources terms P γ2 and E γ2 in Equation (3.49) are given as follows: P γ2 = C a2 ρωγf turb (3.29) E γ2 = C e2 P γ2 γ (3.30) 25

36 where Ω is the vorticity magnitude. These terms are used to control the intermittency to remain zero in the laminar boundary layer and also allows the model to predict when the flow may become back to laminar (Langtry and Menter 2009). F turb is used to disable the destruction sources outside of a laminar boundary layer or in the viscous sublayer, which is defined as the following: F turb = e (R T 4 )4 (3.31) The constants for the intermittency equation and related equations are: C a1 = 0.5; C e1 = 1.0; C a2 = 0.03; C e2 = 50; σ f = The Transport Equation of Re θt The transport equation for the transition momentum thickness Reynolds number Re θt is given as t (ρre θt ) + (ρre θt U x i ) = i x j [(μ + μ t ) Re θt σ γ x j ] + P θt (3.32) where P θt is the source term and is used to force the transported scalar Re θt to match the local value of Re θt, which is calculated from the empirical correlation. The source term P θt is given as ρ P θt = C θt t (Re θt Re θt )(1.0 F θt ) (3.33) t = 500μ ρu 2 (3.34) where t is a time scale which is used to scale the convective and diffusive terms in the transport equation. 26

37 F θt is the blending function used to turn off the source term in the boundary layer and its value is 1.0 in the boundary layer and zero in the freestream. F θt is defined as the following: F θt = min {max [F wake e (y δ )4, 1.0 ( γ 1 C e C e2 2 ) ], 1.0} (3.35) θ BL = Re θt ρu δ BL = 15 2 θ BL δ = 50Ωy U δ BL (3.36) Re ω = ρωy2 μ (3.37) F wake = e ( Re ω 1E+5 )2 (3.38) where F wake is used to ensure that the blending functions are not activated in the wake regions (i.e. the downstream of an airfoil). The default boundary condition for Re θt at a wall is set to be zero. The boundary condition for Re θt at inlet is recommended to be calculated according to the empirical correlation based on the inlet turbulence intensity. The constants for the transport equation of transition momentum thickness Reynolds number are given as: C θt = 0.03; σ θt = 10.0 The transition SST 4-equation model contains three empirical correlations: Re θt, F length, and Re θc, which are based on the experiments. The default correlations of present version of ANSYS are provided by Menter and Langtry such that the natural transition has been improved (Langtry and Menter 2009). These three variables are given as: Re θt = f(tu, λ) 27

38 F length = f(re θt ) (3.39) Re θc = f(re θt ) It should be noted that there are other four empirical correlations available. These correlations were developed by Sørensen (2009), Malan (2009), Suluksna (2009), and Tomac (2013). Because all of the correlations were calibrated with flat plates but not airfoils so all of the correlations were tested in order to find the most accurate one to match the experimental data of airfoil. An example of Menter correlation for Re θc and F length is shown below: If Re θt > 1870, Re θc = Re θt ( (Re θt 1870)) (3.40) If Re θt < 400, F length = e e 4 Re θt e 6 2 Re θt (3.41) The entire correlation codes of Menter, Sørensen, and Malan are shown in Appendix. B Coupling with the Transport Equation of k Because there are two other transport equations added into the original SST k- ω model, the original transport equation of k needs to be coupled with the new transport equations of γ and Re θt, as the following: (ρk) + (ρku t x i ) = k (Γ i x k ) + G j x k Y k + S k (3.42) j 28

39 G k = γ eff G k Y k = min(max(γ eff, 0.1), 1.0) Y k (3.43) where G k is the new generation term while Y k is the new destruction term. It should be noted that the generation and destruction terms involving with ω is not modified. And γ eff is the effective intermittency, which is defined as the following: Re v γ sep = min {C s1 max [( ) 1.0] F 3.235Re reattach, 2} F θt C s1 = 2 (3.44) θc F reattach = e (R T 20 ) 4 (3.45) γ eff = max(γ, γ sep ) (3.46) where the constant C s1 can control the size of separation bubble. This modification is designed to improve the prediction of separation-induced transition. 29

40 Chapter 4 Aerodynamics of the Airfoil 4.1 Introduction This chapter describes the aerodynamics of airfoil, especially the lift and drag coefficient, pressure coefficient, and wall shear stress. The first section of this chapter discusses the aerodynamic characteristics of isolated airfoils, and cascades in terms of the effects of blade pitch and spacing ratio. The second section describes the procedure of airfoil model creation. MATLAB R2014.0a was used to generate the coordinate of airfoil and ANSYS ICEM 15.0 was used to create the twodimensional airfoil model. 4.2 The Aerodynamics of Airfoil Isolated Airfoil The most important parameters in this study are the lift and drag of the airfoil, which are the forces on a body in the direction parallel and normal to the flow direction respectively (Burton et al. 2011). The lift and drag can be obtained by splitting the resultants of the pressure and shear forces exerted on the airfoil, which are shown in the Figure

41 Figure 4.1 Forces exerting on the airfoil In Figure 4.1, α is the angle of attack, the angle between the direction of flow and the chord of airfoil; L and D are the lift and drag on the airfoil, which can also be decomposed into F x and F y. Such that the lift and drag can be related to F x and F y, as: L = F y cos α F x sin α (4.1) D = F y sin α + F x cos α (4.2) Equation (4.1) and (4.2) can be deducted directly from the relationship of the forces, which is shown in Figure 4.2. The marked edges are F x sin α and F y sin α respectively. 31

42 Figure 4.2 Forces decomposition of lift and drag The horizontal and vertical forces F x and F y can be obtained from the solution of the ANSYS Fluent, which are computed by summing the dot product of the pressure and viscous forces on each face with the specified force vector. This gives as F = a F p + a F v (4.3) where F represents the actual forces which are F x and F y, a is the surface normal and F p and F v are the pressure and viscous force vectors (ANSYS 2013). Once lift and drag are obtained, lift and drag coefficients can be determined by the following equations: C l = L 1 2 ρu2 c C d = D 1 2 ρu2 c (4.4) (4.5) 32

43 where ρ is the density of the fluid, U is the flow speed and c is the chord length of the airfoil. The lift and drag coefficients are the most fundamental characteristic of the airfoil performance but critically important for the design of the wind turbine blades, especially the C L C D ratio. The pressure coefficient, C p, is the third parameter considered in this study, which describes the pressure distribution around the airfoil. The pressure coefficient is defined as C p = P P 1 2 ρu2 (4.6) where P is the pressure at any point on the airfoil surface, P is the freestream pressure, and ρ is the density of flow. The pressure coefficient is the difference between static pressure and freestream pressure, divided by the freestream dynamic pressure. For incompressible flows, the pressure coefficient reaches 1.0 at the stagnation point where the velocity is zero. The transition from laminar to turbulent plays an important role in airfoil simulation and is critical to the accuracy of the computational results. The skin friction coefficient, C f, is one of parameters that can indicate the state of the boundary layer, which is defined as C f = where τ w is the wall shear stress that is defined as τ w 1 2 ρu2 (4.7) 33

44 τ w = μ ( u y ) y=0 (4.8) where μ is the dynamic viscosity, u is the flow velocity parallel to the wall and y is the distance to the wall Effect of Solidity For wind turbines, there is a solidity or cascade effect between the blades, which could impact their aerodynamic characteristics. When the power and thrust calculations of the blade for a wind turbine are performed, it would be more accurate if the blades of a wind turbine are considered not as isolated airfoils but to have relative influences on each other depending on the location of the adjacent blade cross section. The blades of wind turbine are separated azimuthally by an equal distance, denoted as s (Dixon 2005). As shown in Figure 4.3, this interaction between blades will depend on the local solidity, which is the ratio between the chord length of blade at a given radius and the circumferential length at that radius (Burton et al In addition to local solidity, there is also rotor solidity or global solidity which is total blade area divided by the rotor disc area. In this project, solidity refers to the local solidity, which is an important parameter in determining rotor performance for a wind turbine. High solidity turbines such as marine propellers or wind mills have solidity greater than 1.0 and can generate high torques. Lower solidity turbines such as wind turbines have the advantage of harvesting more energy from the wind resources. The typical solidity of a three-blade wind turbine ranges from 0.1 at the tip to 0.3 at the hub (Bak 2013). Figure 4.3 shows the unfolding of the blade elements into an infinite cascade, which is the same way that BET assumes a blade element to be an airfoil. 34

45 Figure 4.3 Schematic of local solidity The local solidity of blades or spacing ratio is defined as: σ = c s = Nc 2πr (4.9) where c is the chord length of airfoil and s is the distance between the adjacent blades. The solidity can also be expressed in terms of the number of blades (N), chord length, and the radius of the blade element (r), which is shown on the right hand side of Equation (4.9). This Equation (4.9) implies that the relationship between the axisymmetric wind turbine and Figure 4.3 is the 35

46 unfolding of the blade elements into an infinite cascade. This is the same way that BET assumes a blade element to be an airfoil. In ANSYS Fluent, the lift and drag coefficients of cascade blades cannot be obtained directly from the solution but can be calculated by substituting the results acquired from Fluent into a series of equations, which are used to correct the mean angle of attack and mean velocity due to the cascade effect. Thus the lift and drag coefficients for cascade blades can be obtained. As shown in the Figure 4.4, the mean angle of attack should be used during the calculation of cascade effect. The U x and U y are the horizontal and vertical velocities at the outlet. Figure 4.4 Schematic of the cascade effect 36

47 The mean angle of attack α m for cascade blades is expressed as the following: tan α m = 1 2 (tan α + U y U x ) (4.10) where is the angle of attack in terms of the inlet velocities and U x and U y are the outlet velocities of x and y directions, which can be obtained from the Fluent solution. By substituting the mean α into Equation (4.1) and (4.2), the lift and drag for cascade blades can be obtained, which gives L = F y cos α m F x sin α m (4.11) D = F y sin α m + F x cos α m (4.12) The mean velocity is required for computing the lift and drag coefficients, which can be obtained by the following equation U m = U x cos U m (4.13) Substituting the Equation (4.11), (4.12), and (4.13) into Equation (4.4) and (4.5) can give the expressions for lift and drag coefficients for cascade blades L C L = 1 2 ρu m 2 c D C D = 1 2 ρu m 2 c (4.14) (4.15) 37

48 4.2.3 Effect of Blade Pitch (Twist) The blade pitch or twist represents the angle between the plane of rotation of the blade and the blade s chord line and is denoted as θ p, which is shown in Figure 4.5. The pitch or twist angle varies with the blade radius from around 20 at the hub to approximately 0 at the tip. The term pitch angle is used in this study. The change of pitch angle can have an influence on the power output as well as the lift and drag coefficients of the blade. This project investigates the effect of blade pitch at 0, 20, and 90 as 0 and 20 are typical values at tip and hub for a wind turbine while 90 is the value for unpitched blade (Burton et al. 2011). Figure 4.5 Schematic of the pitch angle 38

49 4.3 Geometry of the Airfoil The airfoil used in this study is the NACA 4415, which is one of NACA four digit airfoil family and was developed by the National Advisory Committee for Aeronautics (NACA) in around 1940 (Abbott et.al 1949). The NACA airfoil series were designed for aircraft wings, which like wind turbines, need high lift and low drag. The thickness of the NACA 4415 is typical of airfoils used near the tip of large turbines and the experimental data of lift and drag coefficients is convenient to be obtained. Thus it is sensible to use NACA 4415 for this study. The geometry of NACA 4415 is shown in Figure 4.6. Figure 4.6 Two-dimensional geometry of the NACA 4415 The first digit of NACA four-digit airfoil denotes the maximum value of the camber line in percent of the chord (4% for NACA 4415); the second digit describes the location of maximum camber from the airfoil leading edge in tenths of the chord (0.4 for NACA 4415); and last two digits indicates the maximum thickness of the airfoil in percent of the chord (0.15 for NACA 4415). If the first two digits are both zero, the airfoil will be symmetrical (Abbott et.al. 1949). 39

50 The geometry of the NACA 4415 can be generated precisely according to the Cartesian coordinates, which is described by the following equation y t = t 0.2 ( x 0.126x x x x 4 ) (4.16) where y t is the thickness of the airfoil (coordinate of y) and x is the coordinate of x. This equation represents the airfoil with a blunt trailing edge. If the coefficient of x 4 is modified to , the equation will represent the airfoil with a sharp trailing edge. The equation of mean camber line gives m x (2p x), 0 x p p2 y c = m 1 x (4.17) { (1 p) 2 (1 + x 2p), p x 1 where y c is the coordinate of mean camber line, p is the location of maximum camber, and m is the value of maximum camber. For NACA 4415, p is equal to 0.4 and m is equal to Applying the thickness distribution perpendicularly to the camber line gives the final coordinates of the airfoil with upper and lower surfaces, which is expressed as the following upper surface: { x u = x y t sin θ y u = y c + y t cos θ (4.18) lower surface: { x l = x + y t sin θ y l = y c y t cos θ (4.19) where (x u, y u ) is the coordinate for the upper surface, (x l, y l ) is the coordinate for the lower surface, and θ is given as θ = tan 1 dy c dx (4.20) 40

51 where dy c 2m (p x ), dx = { p 2 c 0 x p 2m (1 p) (p x ), p x 1 (4.21) 2 c The coordinates of NACA 4415 were generated in MATLAB R2014.0a with 1000 points in total. Particular attention was paid to the leading and trailing edge, which are shown in the Figure 4.7 (a) and 4.7 (b). Smoothness over the entire airfoil is critical to obtain good quality of computational results from ANSYS Fluent. Figure 4.7 (a) Leading edge of the NACA 4415 in ANSYS ICEM

52 Figure 4.7 (b) Trailing edge of the NACA 4415 in ANSYS ICEM

53 Chapter 5 Test Cases 5.1 Introduction This chapter discusses test cases including isolated airfoil, cascade blades, and pitched blades. The results of the isolated airfoil are used to validate the computation methodology such that the further investigation can proceed. In each case, the meshing of the model is described first, followed by the boundary conditions and mesh independence test. The results including lift and drag coefficients, pressure coefficient, and wall shear stress and etc. are discussed at the end of each test case. All of the meshes were generated in the ANSYS ICEM 15.0 and computational simulations were completed in the ANSYS Fluent The Isolated Airfoil Meshing The coordinate file of the airfoil were firstly imported into SolidWorks 2013 to generate the geometry, which was then imported into the ANSYS ICEM CFD 15.0 for mesh generating. The chord length of airfoil is 1 m. H-type grid meshing is used for the calculation, which is shown in Figure 5.1 and

54 Figure 5.1 H-type grid used for the meshing of airfoil Figure 5.2 Close up of the same grid near the wall of airfoil As it can be seen in Figure 5.1, the H-type grid has the size of 10m 30m where the distance between inlet boundary and the airfoil is 10 times of the chord length of the airfoil while the distance between outlet boundary and the airfoil is 20 times of the chord length. Hexahedral (hex) 44

55 unstructured mesh is used, which is generated by the block topology. This method is beneficial for complex geometry, especially at the leading and trailing edge of the airfoil. Hex can automatically generates model-fitted internal and external grids to fit the block topology to the geometry so that the good quality of mesh can be ensured (ANSYS 2013). The arrangement of nodes numbers is shown in Figure 5.3. Figure 5.3 Number of nodes at each edge of the blockings of airfoil Figure 5.3 shows the number of nodes at each edge around the airfoil. All the parallel edges have the same number of nodes to ensure the good mesh quality. The block near airfoil is separated by two blocks with equal length, which have 120 nodes of the stream wise direction. The block on the leading and trailing edge has 50 nodes. Because the blocks far away from the airfoil has no significant influence on the airfoil, the density of nodes is not large as the boundary layer. The increase ratio between each node of the boundary layer blocks is 1.01 in order to ensure low aspect ratio. The aspect ratio is the ratio of longest to the shortest side in a cell. It should be ideally equal 45

56 or close to 1.0 to ensure best results. Local variations in cell size are required to be minimal, no more than 20%. If cells have large aspect ratio, an interpolation error of unacceptable magnitude will occur. For other blocks, the increase ratio is The most critical parameter of model meshing is the estimated y +, which is the dimensionless wall distance between the first cell and the wall of airfoil. It is defined as y + = y pu T ν (5.1) where y p is the distance to the nearest wall, u T is the friction velocity, and ν is the kinematic viscosity (ANSYS 2013). It should be noted that the actual value of y + can only be obtained after the calculation. But it needs to be estimated beforehand to generate the mesh. The appropriate y + is around 1.0 according to Langtry, Finally the y + is used to estimate the spacing of the first cell to the wall of airfoil, which is in this case. The total number of nodes for the calculation of isolated airfoil is , which is determined through the mesh independence test. Figure 5.3 shows the close up of mesh near the leading edge. Figure 5.3 Close up of the mesh near the leading edge 46

57 5.2.2 Boundary Conditions For the computation of isolated airfoil, the boundaries are: inlet boundary, outlet boundary and wall boundary, which are shown in the Figure 5.4. The type of inlet boundary is velocity-inlet, which has a constant value of 13.46m/s. The type of outlet boundary is outflow, which is typically used to simulate the flow exiting the computational domain where the details of the flow, such as velocity and pressure, are unknown before the calculation. The outflow boundary condition assumes a zero streamwise gradient for all flow variables except pressure (ANSYS 2015). The shear condition of wall boundary is no slip. The Figure 5.4 illustrates the boundaries for isolated airfoil. Figure 5.4 Boundaries for the isolated airfoil computation The velocity of flow is 13.46m/s at the inlet boundary to give the Reynolds number as The turbulence intensity is 0.1% in the OSU wind tunnel experiment but it has been observed that the turbulence intensity has a significant decay at an inlet depending on the inlet viscosity ratio in CFD calculations (Langtry 2006). Consequently, the local turbulence intensity near the airfoil can be much smaller than the specified value at the inlet boundary. Typically, the decay rate 47

58 decreases with the increasing of inlet viscosity ratio. Figure 5.5 shows a test case where the turbulence intensity decays from 5.5% at the inlet boundary to 0.5% at the leading edge of airfoil with the inlet viscosity ratio of 40. Figure 5.5 Decay of the turbulence intensity The desirable value of turbulence intensity can be predicted by estimating a relatively low (i.e. 1-10) inlet viscosity ratio with an estimated inlet value of turbulence intensity. According to Langtry (2006), the turbulence intensity for transition SST model can be corrected by the following equation Tu = (Tu 2 inlet (1 + 3ρ Uβ Tu inlet 2 ) 2μ(μ t μ) β β ) 0.5 (5.2) where Tu is the desirable turbulence intensity, Tu inlet is the turbulence intensity at the inlet boundary, μ t μ is the inlet viscosity ratio, β and β are constants in SST transition model, which are equal to and 0.09 respectively. Through trial and error tests and the calculation 48

59 according to Equation 5.2, the turbulence intensity at inlet is determined to be 12 at inlet boundary and inlet viscosity ratio is set to 40. As a result, the turbulence intensity will decay to 0.1% at the location of airfoil such that the same turbulence level of the OSU experiment can be met Mesh Independence Test The Grid Convergence Index (GCI) was used to test the mesh independence (Roache 1994). GCI provides an estimate of the discretization error without knowing the exact solution and also indicates the rate of convergence of the mesh. The GCI can be calculated by the following equation GCI = F s ε r p 1 (5.3) where F s is the safety factor, ε is the relative error, r is the refinement ratio, p is the convergence rate. The safety factor was selected as 1.25 in this study (Wilcox 2006). The other parameter can be determined by the following equations ε 21 = f 2 f 1 f 1 and ε 32 = f 3 f 2 f 2 (5.4) where f 1, f 2, and f 3 are the results from calculations and ε 21 and ε 32 are the relative error between these results. Thus r = h 2 h 1 = h 3 h 2 (5.5) where h is the number of nodes for each mesh case and the refinement ratio is required to be same for a three-case mesh independence test. 49

60 p = ln (1/R) ln r (5.6) where R is the convergence ratio, which is determined by R = ε 21 ε 32 (5.7) If R is between 0 and 1, the solution will be monotonic converging. If R is smaller than 0, the solution is oscillating. If R is greater than 1, the mesh will be divergence. In the case of isolated airfoil, three grid sizes are tested, which are 94376, , and respectively. They represent coarse, medium, and fine meshes with a constant refinement ratio of The test was conducted for α = 6.2. Table 5.1 shows the results of this test. Table 5.1 Computational results of mesh independence test at α = 6.2 Number of nodes Cl Cd Coarse mesh (3) Medium mesh (2) Fine mesh (1) The variation of lift and drag coefficients are shown in the Figure 5.6 (a) and (b). 50

61 Cd Cl Numer of nodes Figure 5.6 (a) Variation of lift coefficient Number of nodes Figure 5.6 (b) Variation of drag coefficient As a result, by substituting the test results into Equation 4.24 to 4.28, the relative error, convergence rate, and GCI for lift and drag coefficients of isolated airfoil can be obtained, which are shown in the Table 4.2 where GCI 32 represents the convergence index from coarse to medium mesh while GCI 21 indicates the convergence index from medium to fine mesh. The smaller GCI means the refined mesh has a more convergence trend such that further refinement is not needed. That is also to say that the mesh of domain is independent to the computation. 51

62 Table 5.2 GCI results of the mesh independence test ε 32 (%) ε 21 (%) R p GCI 32 (%) GCI 21 (%) Cl Cd The exact solution of Cl and Cd estimated using Richardson extrapolation, Equation 5.8 (Roache 1994), are 1.09 and respectively. This is a second order estimation of the exact solution, which is also used in the CFD calculation. f exact f 1 + f 1 f 2 r 2 1 (5.8) It can be seen from Table 5.2 that there is a decreased value of GCI for the fine mesh, which means GCI 21 is smaller than GCI 32. This indicates that the dependency of the computation on the grid size is reduced through refining the mesh. Additionally, as the GCI from medium to fine mesh is very low, it can be concluded that the grid independence was effectively achieved. The exact solution agrees with the GCI results that medium mesh is appropriate for the calculation of isolated airfoil; thus, the medium size mesh is used in the simulations for isolated airfoil. The mesh independence test for cascaded blades are discussed in the section of Effect of Blade Cascade. 52

63 5.2.4 Results and Discussions In this section, the results for the isolated airfoil are discussed, which include lift coefficient, drag coefficient, pressure coefficient, wall shear stress, contours of velocity and turbulence energy, and the validation with experimental results. The region around the angle of maximum lift to drag ratio is particularly concerned in this study. In addition, for the SST transition model the correlations of F length and Re θc developed by Menter (2009), Sørensen (2009), and Malan (2009) are compared and discussed. The Mentor s correlation is used as the default option in ANSYS Fluent. The Reynolds number is in this study for all the simulations. The solution was iterated until all of the residuals of variables decreases to 10 7 where the solution is considered to be converged (Versteeg and Malalasekera 2007). The residuals are the absolute error in the solution of a particular variable. In the first place, the coefficients of lift and drag are the most important parameters considered in this study, which also plays an important role in the aerodynamic performance of blades for wind turbines. The plots of lift and drag coefficients and lift to drag ratio are illustrated in Figure 4.14 (a), (b), and (c) where three computational results are compared with two experimental data. There are two experimental data used in this case: Hoffman (1996) and Miley (1982). The details of these two experiments are described in Chapter 2. 53

64 Cd Cl OSU Stuttgart 0.4 Menter Sørensen 0.2 Malan Angle of Attack Figure 5.7 (a) Lift coefficient of the computational and experimental results OSU Stuttgart Menter Sørensen Angle of Attack Figure 5.7 (b) Drag coefficient of the computational and experimental results 54

65 Cl/Cd OSU Stuttgart Menter Sørensen Angle of Attack Figure 5.7 (c) Lift to drag ratio of the computational and experimental results It can be seen from Figure 5.7 (a) that the computational results for the lift have similar behaviors, which are higher than the experimental results. Results obtained from correlations of Menter and Sørensen both have the same stall angle as the experimental results but the result of Menter is slightly lower than that of Sørensen, which is the reason that the correlation obtained by Menter is used for the investigation of cascade effect and pitch effect. The drag coefficient and lift to drag ratio have a good agreement with experimental result where the relative error is within 3% for drag coefficient and 10% for lift coefficient while 10% is typical in CFD validation (Oberkampf and Trucano 2000). Table 4.4 illustrates the comparison between computational and experimental results. The angle of attack at maximum lift coefficient of computational results agrees with OSU s result but Stuttgart s result has a difference with them. In terms of the stall angle, both the results 55

66 Cp of Menter and Sørensen correlations agree with OSU s results, which provides solid validation of this airfoil computation. Table 5.3 Comparison between computational and experimental results Max Cl α Max Cl/Cd α at max Cl OSU Stuttgart Menter Sørensen Moreover, the experimental report of OSU provides the pressure coefficients of airfoil, which are used to compare with computational result. The plots of pressure coefficient of airfoil at α = 4.1, 6.2, and 11.2 are shown in the Figure 5.8 (a), (b), and (c) Menter OSU x/c Figure 5.8 (a) Pressure distribution at α =

67 Cp Cp Menter OSU x/c Figure 5.8 (b) Pressure distribution at α = Menter OSU x/c Figure 5.8 (c) Pressure distribution at α =

68 As can be seen from Figure 5.8, the pressure distributions show a good agreement between the experimental and computational results, especially on the pressure side (lower side) of airfoil. However, for α = 4.1 and 6.2, there are regions of rapid change of C p (circled region in the plot) on the suction side (upper side) of airfoil, which indicates that extremely small laminar separation bubble may exits at those regions. This can be illustrated more clearly by showing the vector contours of velocity near the airfoil. Figure 5.9 (a) and (b) show the vector contour of velocity for α = 4.1 with fit view and close up of the airfoil. A laminar separation bubble is generated when an adverse pressure gradient occurs on the previously laminar boundary layer (Jahanmiri 2011). The effect of laminar separation bubble can increase the drag of airfoil. The laminar separation bubble should be avoided when Reynolds number reaches 1,000,000 (Lissaman 1983). In this case, the region of adverse gradient is extremely small such that the laminar separation bubble is not generated. The adverse gradient vanishes entirely after α = 6.2. Figure 5.9 (a) Velocity vectors (m/s) over the airfoil at α =

69 Figure 5.9 (b) Close view of the velocity vector nears the wall of airfoil In addition to the coefficients of lift, drag, and pressure, wall shear stress or skin friction coefficient is important because it can indicate the onset of transition. The transition SST model used in this study has two equations in terms of intermittency and momentum thickness Reynolds number, which can control the transition onset. The plot of skin friction coefficient at α = 6.2 is shown in the Figure 5.10, which is compared with the result obtained from XFOIL, which is a program based on e N method for the design and analysis of subsonic isolated airfoils (Drela 1989). 59

70 Cf Menter Xfoil x/c Figure 5.10 Skin friction coefficient of the isolated airfoil at α = 6.2 As shown in Figure 5.11, the wall shear stress decreases from the leading edge meaning that the gradient of the velocity parallel to the wall is reducing. When the wall shear stress reaches zero, the flow has separated. That is to say the transition onset occurs at the location of around 0.4 of x/c of the airfoil. Unfortunately, there is no experimental data that can be used to validate the plot of skin friction coefficient or the transition onset. However, XFOIL can be used to provide a consistency check for Fluent results. It can be concluded that they have good agreement, especially with the prediction of transition onset, which occurs at around 0.4 of x/c. Furthermore, the transition onset can be clearly located from the contour of turbulence kinetic energy, which is shown in the Figure The turbulence kinetic energy (TKE) has a significant increase at the location of 0.4 x/c of airfoil, which indicates that the transition occurs at that point. It can be concluded that the computational results have a good agreement with the experimental results. And the method used in this study is validated. Consequently, further investigations concerning 60

71 the effect of cascade and blade twist can be taken account into calculations. The next two sections describe the test cases of cascade effect and pitch effect including meshing, mesh independence test, boundary conditions, and results and discussions. Figure 5.11 Contour of the turbulence kinetic energy (m 2 /s 2 ) 61

72 5.3 The Effect of Solidity This section discusses the airfoil calculation with the effect of cascade. The solidities considered in this case are 0.1, 0.2, and 0.3, which range from low to high solidity as appropriate for modern wind turbines. The mesh schemes for each case of solidity are described first, followed by the boundary conditions and mesh independence test. The results are discussed at the end of this section Meshing The mesh methodology of cascaded blades is similar to that of isolated airfoil. The difference is that the grid size has been adjusted according to the value of solidity. Figure 5.12 shows the grid domain of cascaded blades with three periodicities. Figure 5.12 Mesh domain of cascaded blades 62

73 Although Figure 5.12 shows three regions of mesh domain, only one grid is used for computation, for economy. The grid has periodic boundary condition such that it represents an infinite cascade in the y direction. Three solidities are investigated in this case: 0.1, 0.2, and 0.3. According to Equation 4.9, the distances between blade elements of solidity are 10.0 m, 5.0 m, and 3.3 m respectively Boundary Conditions The boundaries of cascaded blades shown in Figure The type and details of boundary conditions are listed in Table 5.4 and 5.5. Figure 5.13 Schematic of boundaries for cascaded blades 63

74 Table 5.4 Boundary conditions for cascaded blades Inlet Outlet Periodic Wall of blades boundary Type Velocity inlet Outflow Translational No slip periodic Table 5.5 Details for inlet boundary conditions Re Flow density (ρ) kg/m 3 Dynamic viscosity (μ) N s/m 2 Intermittency (γ) 1 Turbulence intensity (%) 12 Turbulence viscosity ratio (μ t μ) 40 Inlet velocity magnitude (U) m/s 64

75 Cl Mesh Independence Test A Mesh independence test was conducted before performing the computations of cascaded blades. The test procedure is the same to isolated airfoil using the method of GCI. Due to the similarity of solidity cases, only one case of mesh domain was tested, which has the solidity of 0.3. The results of mesh independence test are listed in the following table. Table 5.6 Computational results of mesh independence test for solidity of 0.3 Number of nodes Cl Cd Coarse mesh (3) Medium mesh (2) Fine mesh (1) Moreover, the variation of lift and drag coefficients are shown in the Figure 5.14 (a) and (b) Numer of nodes Figure 5.14 (a) Variation of lift coefficient 65

76 Cd Number of nodes Figure 5.14 (b) Variation of drag coefficient Finally, the grid convergence index can be obtained through Equation The results of GCI are listed in the Table 5.7 Table 5.7 GCI results for the case of solidity of 0.3 ε 32 (%) ε 21 (%) R p GCI 32 (%) GCI 21 (%) Cl Cd It can be seen from Table 5.7 that for both Cl and Cd the GCI from medium to fine mesh is smaller than that from coarse to medium mesh, which is to say that the refinement of mesh has decreased the dependency of grid. Additionally, as the GCI from medium to fine mesh (GCI 21 ) is very low, it can be determined that the computation is independent of the grid and fine mesh has no significant influence on the computation compared with medium-size mesh. Therefore, the medium mesh is used in the simulations of cascaded blades. 66

77 Cl Results and Discussions This section discusses the lift coefficient, drag coefficient, pressure coefficient, wall shear stress, and contours of velocity and turbulence energy for the three solidities. Menter s correlation is used in the computations for this case. The OSU s experimental data of isolated airfoil is used as the reference to check the accuracy of results of cascade blades. S denotes the solidity and S=0 indicates the isolated airfoil. The distance between cascaded blades decreases with the increase of solidity. For wind turbines, the blade at hub location has highest solidity, which is around 0.2, while the blade at tip has lowest solidity, which is below 0.1. The solidity calculated for wind turbines is very low comparing with a windmill. Firstly, the plots of lift and drag coefficients and lift to drag ratio are illustrated in Figure 5.15 (a), (b), and (c), which shows the results of cascaded blades and the measurements for the isolated airfoil. In this test case, the pitch angle is OSU S=0 S= S=0.2 S= Angle of Attack 67

78 Cl/Cd Cd Figure 5.15 (a) Lift coefficient of various solidities OSU S=0 S=0.1 S=0.2 S= Angle of Attack Figure 5.15 (b) Drag coefficient of various solidities OSU S=0 S=0.1 S=0.2 S= Angle of Attack Figure 5.15 (c) Lift to drag ratio of various solidities 68

79 It can be seen from Figure 5.15, both of lift and drag coefficients are slightly reduced as the increase of solidity. Table 5.8 shows the results of comparison between cascaded blades and isolated airfoil at α = 11.2 as well as α where the Cl/Cd reaches the maximum value. The relative change is calculated as the reduction of cascaded blades. The reduction is considered between the increased solidity and solidity of zero. Table 5.8 Comparison between cascaded blades and isolated airfoil at α = 11.2 Cl Reduction of Cl Cd Reduction of Cd Angle of max Cl/Cd S= / / 6.20 S= % % 6.49 S= % % 6.93 S= % % 7.33 Table 5.8 shows the lift coefficient is slightly changed but the drag coefficient has dramatically reduced when the solidity increases. The drag coefficient change compared with isolated airfoil for solidity of 0.2 and 0.3 has decreased more than 50%. It should also be noted that the angle of attack where the maximum Cl/Cd occurs increases as the solidity increases. Figure 5.16 shows the pressure coefficients for solidities of 0, 0.1, 0.2, and 0.3 at the mean angle of attack (α m ) of 6.2 when the pitch angle is 90. However, the mean angles of attack for S=0.1, 0.2, and 0.3 are not precisely 6.2 as the mean angle of attack of cascaded blades is calculated after the computation. It can be seen that the pressure coefficient on the suction side of blade decreases 69

80 Cp as solidity increases. On the other hand, the change of pressure coefficient around 0.4 x/c diminishes as the solidity increases. This implies that the transition onset occurs closer to the leading edge, which has an agreement with the plot of wall shear stress Lower side Upper side x/c S=0 S=0.1 S=0.2 S=0.3 Figure 5.16 Pressure distribution of cascaded blades at α m

81 Cf Upper side S=0 S= S=0.2 Lower side S= x/c Figure 5.17 Wall shear stress distribution of cascaded blades at α m 6.2 Figure 5.17 shows the wall shear stress distribution for the cascaded blades. It can be seen from the position around 0.4 x/c that the transition onset shifts forward to the leading edge as the increase of solidity. The contours of velocity and turbulence kinetic energy are shown in the Figure 5.18 and The differences between various solidities are the maximum velocity near the leading edge and the maximum turbulence kinetic energy near the trailing edge. With the increase of solidity, the maximum velocity and turbulence kinetic energy decreases slightly. 71

82 Figure 5.18 (a) Velocity (m/s) contours of cascaded blades at α m

83 Figure 5.19 (a) Turbulence kinetics energy (m 2 /s 2 ) contours of cascaded blades at α m

84 5.4 The Effect of Blade Pitch This section describes the influence of blade pitch. The pitch angles P considered in this case are 0, 20, and and 0 are the typical pitch angle of the blades near the hub and on the tip for a wind turbine. A 90 pitch angle is used to shut down the wind turbine as this minimizes the rotor idling speed (Butron et al. 2011). The case study of pitch angle is based on the effect of cascade such that there is a correlation between solidity and pitch for wind turbines in that high solidity near the hub is normally associated with high pitch. The mesh schemes for each case of pitch are described first, followed by the boundary conditions and mesh independence test. The results are discussed at the end of this section Meshing The mesh methodology of pitched blades is different from the isolated airfoil or cascaded blades. The most critical challenge is that the number of nodes on the two sides of periodic boundary must be identical such that the periodic boundary can be generated in ANSYS Fluent However, having the same number of nodes on the boundaries for pitched blades makes it difficult to ensure that the quality of mesh can be satisfied, especially the aspect ratio near the leading and trailing edge. To solve this problem, the blade element is placed horizontally while the mesh domain is rotated according to the pitch angle; such that the requirement for boundary conditions and high mesh quality can both be satisfied. Figure 5.20 (a) illustrates the grid domain of P=0 with three periodicities, which is used in calculation. Figure 5.20 (b) shows the grid domain that the blade is twisted 0, which is the angle between the plane of rotation and the chord of blade. It should be noted that Figure 5.20 (b) could be obtained by rotating (a) clockwise 90. They are essentially identical but (a) is much convenient for the calculation procedure. 74

85 Figure 5.20 (a) Mesh domain used for P = 0 (b) Rotated mesh domain for P = 0 Figure 5.21 (a) Mesh domain used for the P = 20 (b) Rotated mesh domain for P = 20 75

86 The grid for pitch angle of 20 is shown in Figure 5.21 (a) and (b), which are the similar to the mesh domain for P=0 It should be noted that only one grid domain is used for computation, which can save the computational cost efficiently. The grid also has periodic boundary condition as marked in the Figure 5.20 and However, for the pitch of 20, the number of nodes on periodic boundaries must be identical. Figure 5.22 shows the close up of the mesh domain near the blade. Three pitch angles are investigated in this case, which are 0, 20, and 90. The pitch angle of 90 is the case with horizontal mesh domain, which are calculated during the investigation of cascade effect. Figure 5.22 Close up of mesh domain for P = 20 76

87 5.4.2 Boundary Conditions The boundary conditions of pitched blades are shown in the Figure 4.26 (a) and 4.27 (a) and the parameters are identical to that of cascaded blades; therefore, it is not discussed in this section Mesh Independence Test Mesh independence test is firstly conducted before performing the computations of cascaded blades. The test procedure is the same to isolated airfoil using the method of GCI. Due to the similarity of pitch-angle cases, only one case of mesh domain is tested, which has the pitch angle of 20 and solidity of 0.2. The results of mesh independence test are listed in the following table. Table 5.9 Results of mesh independence test for pitch angle of 20 and solidity of 0.2 Number of nodes Cl Cd Coarse mesh (3) Medium mesh (2) Fine mesh (1) The variation of lift and drag coefficients are shown in the Figure 5.23 (a) and (b). 77

88 Cd Cl Numer of nodes Figure 5.23 (a) Variation of lift coefficient Number of nodes Figure 5.23 (b) Variation of drag coefficient The results of mesh independence test in the case of pitch angle of 20 and solidity of 0.2 using GCI method are shown in the Table

89 Table 5.10 GCI results for pitch angle of 20 and solidity of 0.2 ε 32 (%) ε 21 (%) R p GCI 32 (%) GCI 21 (%) Cl Cd It can be seen from Table 5.10 that for Cl and Cd the GCI from medium to fine mesh is smaller than that from coarse to medium mesh, which indicates that the refinement of mesh has decreased the dependency of grid. Besides, the convergence indexes of lift coefficient from two refinements are both very low, which means that even coarse mesh is independent to the computation. However, the GCI 21 of drag coefficient has a dramatic reduction from the coarse-to-medium refinement. Consequently, the coarse mesh cannot be used due to the GCI of drag coefficient. Therefore, the medium mesh is independent to the calculation and should be selected in the simulations of pitched blades. 79

90 Cl Results and Discussions This section discusses the results from the computation of pitched angle as well as cascaded blades. In order to compare the results in an organized way, the results are presented by groups of constant solidities and constant pitches. Firstly, the results of lift coefficient, drag coefficient, and lift to drag ratio for the solidity of 0.1, 0.2, and 0.3 as well as for the pitch angle of 0, 20, and 90 are presented below. They are grouped by the solidity and then pitch angle. It should be noted that the OSU in the following plots represents the experimental result for isolated airfoils. Solidity = OSU S=0.1 P=90 S=0.1 P=20 S=0.1 P= Angle of Attack Figure 5.24 (a) Lift coefficient of pitched blades with the solidity of

91 Cl/Cd Cd OSU S=0.1 P=90 S=0.1 P=20 S=0.1 P= Angle of Attack Figure 5.24 (b) Drag coefficient of pitched blades with the solidity of OSU S=0.1 P=90 S=0.1 P=20 S=0.1 P= Angle of Attack Figure 5.24 (c) Lift to drag ratio of pitched blades with the solidity of

92 Cd Cl Solidity = OSU S=0.2 P=90 S=0.2 P=20 S=0.2 P= Angle of Attack Figure 5.25 (a) Lift coefficient of pitched blades with the solidity of OSU S=0.2 P=90 S=0.2 P=20 S=0.2 P= Angle of Attack Figure 5.25 (b) Drag coefficient of pitched blades with the solidity of

93 Cl Cl/Cd OSU S=0.2 P=90 S=0.2 P=20 S=0.2 P= Angle of Attack Figure 5.25 (c) Lift to drag ratio of pitched blades with the solidity of 0.2 Solidity = OSU S=0.3 P=90 S=0.3 P=20 S=0.3 P= Angle of Attack Figure 5.26 (a) Lift coefficient of pitched blades with the solidity of

94 Cl/Cd Cd OSU S=0.3 P=90 S=0.3 P=20 S=0.3 P= Angle of Attack Figure 5.26 (b) Drag coefficient of pitched blades with the solidity of OSU S=0.3 P=90 S=0.3 P=20 S=0.3 P= Angle of Attack Figure 5.26 (c) Lift to drag ratio of pitched blades with the solidity of

95 Cd Cl Pitch angle = OSU S=0.1 P=0 S=0.2 P=0 S=0.3 P= Angle of Attack Figure 5.27 (a) Lift coefficient of cascaded blades at P = OSU S=0.1 P=0 S=0.2 P=0 S=0.3 P= Angle of Attack Figure 5.27 (b) Drag coefficient of cascaded blades at P = 0 85

96 Cl Cl/Cd OSU S=0.1 P=0 S=0.2 P=0 S=0.3 P= Angle of Attack Figure 5.27 (c) Lift to drag ratio of cascaded blades at P = 0 Pitch angle = OSU S=0.1 P=20 S=0.2 P=20 S=0.3 P= Angle of Attack Figure 5.28 (a) Lift coefficient of cascaded blades at P = 20 86

97 Cl/Cd Cd OSU S=0.1 P=20 S=0.2 P=20 S=0.3 P= Angle of Attack Figure 5.28 (b) Drag coefficient of cascaded blades at P = OSU S=0.1 P=20 S=0.2 P=20 S=0.3 P= Angle of Attack Figure 5.28 (c) Lift to drag ratio of cascaded blades at P = 20 87

98 Cd Cl Pitch angle = OSU S=0.1 P=90 S=0.2 P=90 S=0.3 P= Angle of Attack Figure 5.29 (a) Lift coefficient of cascaded blades at P = OSU S=0.1 P=90 S=0.2 P=90 S=0.3 P= Angle of Attack Figure 5.29 (b) Drag coefficient of cascaded blades at P = 90 88

99 Cl/Cd OSU S=0.1 P=90 S=0.2 P=90 S=0.3 P= Angle of Attack Figure 5.29 (c) Lift to drag ratio of cascaded blades at P = 90 It can be seen from the Figure 5.24 to 5.26 that for constant solidity, the lift and drag coefficients of blades do not change with the pitch angle. For the solidity of 0.1 and 0.2, the lift coefficient of P=0 and P=20 are lower than the un-pitched blades while Cd is higher. The P=20 has the lowest drag to lift ratio for solidity of 0.1 and 0.2. For the solidity of 0.3, the lift coefficient and lift to drag ratio both increase while the drag coefficient reduces with the increase of the pitch angle. The lift and drag coefficients reduce as the solidity increases. However, the lift to drag ratio of has nonuniform behavior. For the pitch angle of 0, the lift to drag ratio decreases with the increase of solidity. For the pitch angle of 20, the lift to drag ratio increases with the increase of solidity. And for the pitch angle of 90, there is no significant difference of lift to drag ratio when the solidity changes. Furthermore, the contours of velocity and turbulence kinetic energy for the pitched blades of solidity of 0.3 are shown in the following figures. The angle of attack for the inflow is

100 Figure 5.30 Contour of velocity (m/s) of three pitched blades at solidity of 0.3 and α = 13.2 Figure 5.30 shows the velocity decreases with the increase of pitch angle. The blade is influenced by the wake of adjacent blade for the pitch angle of 0 and 20, which is the reason why the P=0 and P=20 have higher velocity than P=90. Figure 5.31 shows the close up of the contour of velocity for the solidity of 0.3 and pitch angle of 0, 20, and

101 P=0 P=20 P=90 Figure 5.31 Close up of velocity (m/s) contour for pitch angle of 0, 20, and 90 at solidity of 0.3 and α =

102 Figure 5.32 Contour of turbulence kinetic energy (m 2 /s 2 ) of pitched blades at solidity of 0.3 and α = 13.2 As the pitch angle increases, the maximum turbulence kinetic energy near the region of trailing edge decreases. The drag coefficient decreases with the increase of solidity. For the pitch angle of 0, the turbulence kinetic energy pass through the grid domain and impacts the nearby blade, which is the reason that P=0 has highest turbulence kinetic energy. 92

103 Chapter 6 Conclusion In this project, the NACA 4415 was meshed in ANSYS ICEM 15.0 and then simulated in ANSYS Fluent The isolated airfoil and cascade effect including pitch change and spacing ratio were investigated as the test cases. The methodology of meshing was blocking method, which has the advantages of flexibility to adjust the mesh, especially around the boundary layer of airfoil. The Grid Convergence Index (GCI) was used to test the mesh independence for each test case. The transition SST turbulence model was selected in this study as it can predict the transition process reasonably based on the testing of turbulence models. The position of transition onset has significant influence on the wall shear stress as well as lift and drag coefficient, which are the major results considered in this study. Thus, selecting a turbulence model with the capability of predicting transition process is important in this study. All of the test cases were calculated at a Reynolds number of The isolated airfoil was analyzed first in order to validate the turbulence model and computational methodology against high quality wind tunnel tests. The transition SST model enables user-defined-functions to import the empirical correlations, which are used to control the transition onset and transition length. Three correlations conducted by Menter (2009), Sørensen (2009), and Malan (2009) were tested in order to find the best match with the experimental data (Hoffmann 1996). The results of isolated airfoil in Chapter 5 show that the correlation of Menter has the best match with the experimental 93

104 result in terms of distributions of lift as well as the stall angle, at which the lift coefficient starts to decrease. It can be seen that the C l has a reasonable match with the experimental result. When the angle of attack is lower than 8, all of the computational results are close to the experimental data but the deviation arises after α is higher than 10. This can be accepted because the emphasis of this study are focused on the maximum C l /C d, which occurs below α = 10 and the angle shifting under the effect of cascade. The C d of computational results matches well with the experimental data of OSU and Stuttgart with a deviation less than 5%. The maximum C l /C d of OSU is while it is for Menter where the stall angle are both 6.2. This indicates a good agreement between the computational and experimental result. The pressure distribution and wall shear stress were also analyzed at α = 4.1 and 6.2 respectively. The plot of C p has a reasonable match between OSU and Menter except that there exists a small region of reversed velocity, which could be the reason that the turbulence intensity is very low (0.1%) in this study. When the turbulence level increases to 1%, the reversed velocity will vanish under the same boundary conditions. The transition onset of suction side can be obtained from the plot of skin friction coefficient, C f, which decreases to zero from the leading edge. When the flow starts to become turbulent from laminar, the C f will have an explosive increase due to the transition. Unfortunately, there is no experimental data to validate the C f but the XFOIL was used to examine the consistency of the computational results. For α = 6.2, the transition occurs at around 0.4 of x/c on the suction side of airfoil. The effect of cascade and blade pitch in terms of spacing ratio and pitch angle were studied computationally as there are no experimental data available. Three solidities were used: 0.1, 0.2, and 0.3 and the isolated airfoil has a solidity of 0. The results in Chapter 5 indicate that the C l and 94

105 C d slightly decreases while the C l /C d increases as the solidity increases. The mean angle of attack where the max C l /C d occurs increases with the increasing of solidity. This result is shown in Table 5.8. The transition onset occurs also earlier as the solidity increases, which is shown in Figure For the implication for wind turbine aerodynamics, the results of solidity calculation provide optimized results, C l and C d, for the Blade Element calculation, which should take the change influenced by the spacing ratio into account. In particular, a wind turbine should be designed to work at the angle for maximum C l /C d. This project implies the angle of maximum C l /C d increases with the increase of solidity, which means the blades close the hub should have higher angle than the blades at the tip. For the effect of blade pitch, three pitch angles are considered: 0, 20, and 90. The test case of pitch angle was the second stage of investigating the effect of cascade. Based on the meshing used for solidity calculation, the blade was rotated according to the pitch angle in the mesh domain. The usual meshing method for the pitched blade with solidity is to capture the wake of blade within the boundary to ensure that the mesh could have good quality. However, that method is suitable for pitch angle of around 45. In this study, the pitch angle is too small or too large so that the special method that rotating the blade in the mesh domain was used. The momentum balance check proved that this method is reasonable. The results of pitched and cascaded blades have 9 groups of plots in terms of different solidities and pitch angles. Unlike solidity calculation, the aerodynamic behavior of pitch angle calculation is not monotonic with the change of pitch angle. The P = 90 has highest C l and lowest C d for all three solidities while the P = 0 and P = 20 have close results. When the pitch angle remains constant, the C l and C d decreases as the solidity increases. For P = 20, the C l has the best match with the OSU result for solidity of 0.2 and

106 For P = 90, the differences between solidities is not significant. It should be noted that the flow is influenced considerably by the interaction of the blades for solidity of 0.3, which is shown in Figure 5.30 and Based on the results in this study, the outlook for future work are recommended: 1) The experiment to measure the distribution of wall shear stress should be conducted to validate the computational C f, which can provide further information of transition onset. 2) The experiment of blades with low solidity is difficult but the high solidity experiment can be conducted to validate the computational results of cascade effect of high solidity first. Then the low solidity calculation can be examined through the validation of high solidity calculation. 3) The Blade Element calculation can be optimized according to the change of C l and C d from the solidity calculation such that a new design of wind turbine aerodynamics can be made. 96

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111 Appendix A: Computational Fluid Dynamics (CFD) This section describes how ANSYS Fluent 16.0 solves the general scalar transport equations. The finite volume method is used in Fluent to convert a transport equation into an algebraic equation, which can be solved mathematically (ANSYS 2013). This control volume technique requires integrating the transport equation over each control volume, representing a discrete equation that states the conservation law based a control volume. Discretization of the governing equations will be discussed in the following sections. One of advantages of finite volume method is its capacity to accommodate any type of grid. Unstructured grids can be generated that allow various options for the selections of the shape and the combination of two shapes. This type of mesh provides good flexibility for complex geometries. However, one of its disadvantages is that the developing of differencing approximations beyond the second order is difficult due to the requisite for two stages of approximation, which are interpolation and integration. A.1 Finite Volume Method This section describes the finite volume method used in ANSYS Fluent, especially for convectiondiffusion problem, which is relevant to this project. The steady convection-diffusion equation can be derived from the differential form of the general transport equation (Equation 3.11) by removing the transient term: div(ρφu) = div(γ grad φ) + S φ (A. 1) 101

112 Integrating Equation (3.11) gives the control volume integration, which forms the crucial step of the finite volume method that differentiates it from all other CFD methods, yields the following form: n (ρφu)da = n (Γ grad φ) da + S φ dv (A. 2) A A CV If the sources are absent and the scalar quantity φ of steady state diffusion is in a one-dimensional domain, which is illustrated in Figure 3.1, the Equation (3.12) can be written as: d dx (ρuφ) = d dx Also the continuity equation of the flow must be satisfied as well, so d φ (Γ ) (A. 3) dx d(ρu) dx = 0 (A. 4) Figure A.1 Discretization of finite control volume method (Versteeg and Malalasekera 2007) 102

113 Dividing the domain into discrete control volumes is the first step of finite volume method. The Figure A.1 shows a one-dimensional example of the discretization of finite control volume approximation, the control volume (shaded area) has a target point N and two boundaries a side and b side. Points A and B are adjacent nodes to node N, which is the neighboring point between two mesh elements. The distances between node N and face a and face b are marked as δx an and δx Nb respectively. Similarly the distances between node N and node A and node B are identified by δx AN and δx NB respectively. The width of control volume is x which is equal to δx ab. U a and U b represent the velocities entering and leaving the control volume respectively. The second and key step of the finite volume method is discretization, which is the approximation of derivatives from the governing equation over a control. In one-dimensional case, integrating the Equation (3.14) over the control volume of Figure 3.1 gives (ρuaφ) b (ρuaφ) a = (Γ A dφ dx ) (Γ A dφ b dx ) a (A. 5) And integrating the continuity equation (3.15) yields (ρua) b (ρua) a = 0 (A. 6) where A is the cross-section area of the control volume face and u is the velocity at the boundaries of the control volume. Equation (A.6) indicates that the net generation of φ is equal to the flux of φ leaving the b side minus the flux of φ entering the a side, which represents a conservation of φ over the control volume. 103

114 In order to obtain discretized equations, the terms in the Equation (3.16) must be approximated. It is convenient to define two variables F and D to denote the convective mass flux per unit area and diffusion conductance at cell faces: F = ρu and D = Γ δx (A. 7) The cell face values on a and b can be written as F a = (ρu) a F b = (ρu) b (A. 8a) D a = Γ a δx AN D b = Γ b δx NB (A. 9b) such that Equations (A. 5) and (A.6) become: F b φ b F a φ a = D b (φ B φ N ) D a (φ N φ A ) (A. 10) F b F a = 0 (A. 11) The velocity field in Equation (A.10) and (A.11) is assumed to be known in order to simplify the case. By stating the equations in this way, different schemes can be used to solve the Equation (A.10) and calculate the transport properties φ a and φ b at the a and b faces. The schemes of discretization are discussed in the following section. A.2 Central Differencing Scheme The central differencing approximation is a discretization method to approximate the diffusion terms which appear on the right hand side of Equation (A.10). It uses linear interpolation to 104

115 calculate the cell face values for the convective terms which are on the left hand side of this equation. For a uniform grid, the cell face values of property φ can be written as φ a = φ N + φ A 2 (A. 12a) φ b = φ B + φ N 2 (A. 12b) Substituting the above equations into Equation (3.20) and rearranging the terms gives [(D a + F a 2 ) + (D b F b 2 )] φ N = (D a + F a 2 ) φ A (D b + F b 2 ) φ B (A. 13) which follows the form: C N φ N = C A φ A + C B φ B (A. 14) where C N = C A + C B C A = D a + F a 2 C B = D b + F b 2 By applying the central differencing that approximates the diffusion term in the central, a set of algebraic equations can be obtained such that the distribution of the transported property φ can be obtained (Versteeg and Malalasekera 2007). In Fluent, central differencing scheme is only available when using the Large Eddy Simulation (LES) turbulence model and can produce second order accuracy. However, when F/D ratio is high the central differencing scheme can produce oscillations in the solution, which can lead to problems of divergence for the calculation. This problem can often be solved by using a deferred correction (ANSYS 2013). Another disadvantage 105

116 of central differencing scheme is its inability to identify flow direction. If the left hand side of the convective flow has more influence on the target node N, the face a should receive more impact but central differencing scheme still balance the two directions equally. Therefore, there is another differencing method to solve this problem. The following section discusses the Upwind Differencing Scheme A.3 Upwind Differencing Scheme Upwind differencing scheme takes the direction of convective flow into account to evaluate the transport property φ more accurately than central differencing scheme. Figure A.2 (a) and (b) illustrate the different scenarios of flow direction, which are positive and negative respectively. Figure A.2 (a) The flow is in positive direction (Versteeg and Malalasekera 2007) As shown in Figure A.2 (a), the values of each node are marked and the flow is in the positive direction, U a > 0, U b > 0 (F a > 0, F b > 0), and the upwind scheme gives φ a = φ A and φ b = φ N (A. 15) and the discretized Equation (A.10) becomes 106

117 F b φ N F a φ A = D b (φ B φ N ) D a (φ N φ A ) (A. 16) which can be manipulated to be [(D a + F a ) + D b + (F b F a )]φ N = (D a + F a )φ A + D b φ B (A. 17) Figure A.2 (b) The flow is in negative direction (Versteeg and Malalasekera 2007) Similarly, when the flow is in the negative direction, U a < 0, U b < 0 (F a < 0, F b < 0), the properties of each nodal point are φ a = φ N and φ b = φ B (A. 18) and the discretized equation can be written as [(D b F b ) + D a + (F b F a )]φ N = D a φ A + (D b F b )φ B (A. 19) By identifying the coefficients of φ N, φ A, and φ B as C N, C A, and C B, a summary of the upwind differencing scheme can be concluded in Table A

118 Table A.1 Coefficients for various differencing schemes Scheme Type C N C A C B Central Differencing C A + C B D a + F a 2 D b + F b 2 Upwind Differencing (positive flow) Upwind Differencing (negative flow) C A + C B + (F b F a ) D a + F a D b C A + C B + (F b F a ) D a D b F b In Fluent, the upwind differencing scheme includes the first and second order. The second order has higher accuracy than the first order but more difficult to obtain a converged solution. When the first order is used, the face value φ b equals to the constant value of φ of central cell in the upstream (ANSYS 2013). When the second order is used, the face value φ b is calculated using the following expression, which is the expansion of Taylor series φ b+ x = φ b + φ x (A. 20) x where φ b is the cell-centered value and x is the distance from the center of upstream cell to the center of face cell. This formulation requires the determination of the gradient of φ in each cell. The calculation of gradients is discussed in the section A

119 A.4 Power Law Scheme and QUICK Scheme In addition to central and upwind differencing schemes, there are other two approximation methods: the Power Law and Quadratic Upwind Interpolation for Convective Kinetics (QUICK) schemes, which have higher accuracy. The power law uses Peclet number which represents the relationship between F and D to evaluate the face value of the property φ. The power law differencing scheme is more suitable for one-dimensional problems because it uses the exact solution to a one-dimensional convection-diffusion equation (ANSYS 2013). On the other hand, QUICK is one of higher-order schemes which compute a higher-order value of the property φ at the face. However, higher-order discretization requires additional computational time and memory overheads. Considering the balance of accuracy and computational expense, the second order upwind differencing scheme is the most suitable discretization method in this project by testing and comparing the computational results for those schemes. A.5 Properties of Discretization Schemes The discretization schemes of the finite volume method have certain fundamental properties, which impact the accuracy of a solution. These factors also are used to determine which type of scheme is suitable for a particular problem. There are three of the most important properties of discretization schemes which are discussed in this section. They are Conservativeness, Boundedness, and Transportiveness. 109

120 Conservativeness This property represents the conservation of φ for the entire solution domain where the flux of φ exiting a control volume must be equal to the flux of φ entering the neighboring control volume through the same face. To obtain this the flux through a common face must be denoted in a consistent method by one and the same expression in adjacent control volumes. For example, consider the simple one-dimensional steady state diffusion case without source terms shown in Figure A. 3. Figure A.3 Flux through four control volumes (Versteeg and Malalasekera 2007) The fluxes through the domain boundaries are denoted as q in and q out, which enters and leaves the domain respectively. There are four control volumes and central differencing scheme is applied to compute the diffusive flux. By adding the net flux through each control volume and considering the boundary fluxes for the domain from node 1 to 4 [Γ b1 φ 2 φ 1 δx q in ] + [Γ b2 φ 3 φ 2 δx φ 2 φ 1 Γ a2 ] δx 110

121 + [Γ b3 φ 4 φ 3 δx φ 3 φ 2 φ 4 φ 3 Γ a3 ] + [q δx out Γ a4 ] δx = q out q in (A. 21) Subsequently the diffusion constants have same values, Γ b1 = Γ a2, Γ b2 = Γ a3, Γ b3 = Γ a4, the fluxes across the domain are expressed in a consistent manner such that the terms are eliminated in pairs and only boundary fluxes q in and q out remain. By this way, flux consistency guarantees the conservation of φ in the whole control volumes. However, if the gradients of φ between each node are different, the flux values at nodal face may be unequal and conservation cannot be satisfied. Boundedness The term boundedness represents a property of a numerical scheme in which the calculated values of any property φ are limited to be less than or equal to the values at the boundary of a finite volume. The converged solution cannot be obtained if the discretization scheme does not satisfy the boundedness requirement (Versteeg and Malalasekera 2007). The discretized equation at each node yields a set of algebraic equations that must be solved. Generally, iterative numerical methods are used to solve enormous equation sets for most problems. The solution process is started by guessing the values of the variable φ and then iterating the φ until a converged solution is obtained. Consequently, there is a sufficient condition for the computational method which can be expressed in terms of the coefficients of the discretized equations: C nb C N 1 at all nodes { < 1 at one node at least (A. 22) 111

122 where C N is the net coefficient of a central node N from C N to S N (the coefficient of the source tem), and the numerator C nb denotes for the summation of coefficient C b in Equation (A.14) for all the neighboring nodes (nb). If the coefficients produced from the discretization scheme are satisfied by the Equation (A.22), the resulting matrix of coefficients will be diagonally dominant, which is a desirable feature to satisfy the boundedness criterion. In order to obtain the diagonal dominance, large values (C N to S N ) of net coefficients are needed and S N should be always negative. If the requirement of boundedness cannot be satisfied by the differencing scheme, it is very possible that the numerical solution would not converge at all. Transportiveness The transportiveness is a property of numerical scheme that takes into account the direction of the flow. It is important to consider the relationship between the influence of flow direction and the magnitude of the Peclet number, which is defined as Pe = F D = ρu Γ/δx (A. 23) where F indicates the convective mass flux per unit area while D represents the diffusivity at cell faces. The most appropriate way to explain the transportiveness is through the diagram illustrated in Figure A

123 Figure A.4 Distribution of a property φ in a vicinity of a source at different Peclet numbers (Versteeg and Malalasekera 2007) As can be seen from Figure A.4 (a), when there is no mass flux so that the Peclet number is 0, the flow will be pure diffusion and the contours of φ will be concentric circles around A and B because the diffusion process spreads φ equally in all directions. With an increase of Pe, the contour changes its shape from circular to elliptical as shown in Figure A.4 (b) and finally shifts to the direction of the flow as shown in Figure A.4 (c). In this case, the Peclet number reaches infinity and there is no diffusion but only convection. Consequently, the property φ at N is completely influenced by the property at A, which means φ N = φ A. If the flow is in the opposite direction, the value of φ N will be equal to φ B. It is important that the transportiveness describing the relationship between the influence of flow direction and the magnitude of the Pe is inherent in the discretization 113