Acknowledgements The author would like to, first and foremost, express gratitude and appreciation to her supervisor, Associate Professor, Dr Yeo Khoon Seng His guidance, support and assistance have, without doubts, contributed significantly to the completion of the author’s research as well as to this thesis Professor Yeo has given the author the space and freedom of exploration and self-development while providing direction and invaluable feedback in times of need The author would also like to thank Dr Tsai Her Mann and Dr Lum Kai Yew, Principal Investigators at Temasek Laboratories, National University of Singapore (NUS) Dr Tsai has offered the author numerous ideas and suggestions to approach problems and have assisted the author in overcoming difficulties in her work Dr Lum has motivated the author in the formulation of the boundary implementation using the least-squares with constraint approach Lively discussions with Dr Lum have also provided the author with much insight and inspiration Sincere gratitude also goes out to the author’s colleagues at Temasek Laboratories, in particular, Dr Zhang Zhengke for his patience in imparting his CFD knowledge Useful and engaging discussions with PhD student, Chew Choon Seng, have also helped the author significantly i Nomenclature a, c Speed of sound Cp Pressure coefficient E Total energy H Total enthalpy M Mach number P Pressure R Local radius of curvature S Surface area s Entropy t Time (u, v) Velocity components in two dimensions, horizontal and vertical V Velocity vector α Angle of attack γ Ratio of specific heats ρ Density Ω Volume (area in dimensions) ii Table of Contents Acknowledgements i Nomenclature ii Table of Contents iii Summary v List of Figures vii Introduction Literature Review 2.1 Development of structured grid methods 2.2 Development of unstructured grid methods 2.3 Development of Cartesian grid methods 2.3.1 Boundary condition using finite volume cut-cell method 2.3.2 Boundary condition using finite difference ghost cells method 2.3.3 Boundary condition using immersed boundary method 2.4 Development of meshless methods 5 11 Numerical Schemes and Methodology 3.1 Introduction 3.2 Finite-volume formulation of 2-D Euler Equations 3.2.1 Governing equations 3.2.2 Spatial discretization 3.2.3 Artificial dissipation 3.2.4 Temporal discretization 3.2.5 Far field boundary condition 3.3 Gridless surface boundary implementation 3.3.1 Nodal definition 3.3.2 Selection of cloud points 3.3.3 Boundary node treatment 3.3.4 Implementation of surface boundary conditions 18 18 19 20 20 22 24 25 26 27 28 30 31 Results and Discussions 4.1 Solution of flow over single airfoil 4.1.1 Symmetric NACA 0012 airfoil Case 1: M ∞ = 0.5, α = 3.0° Case 2: M ∞ = 0.85, α = 0.0° Case 3: M ∞ = 0.8, α = 1.25° Case 4: M ∞ = 2.0, α = 0.0° 34 34 36 36 39 41 44 13 14 15 iii 4.1.2 4.2 4.3 Asymmetric RAE 2822 airfoil Case 1: M ∞ = 0.5, α = 3.0° Case 2: M ∞ = 0.75, α = 3.0° Solutions of flows over multi-component objects Case 1: Flow over dual NACA 0012 airfoils Case 2: Flow over NLR with flap airfoil Case 3: Flow over 3-element airfoil Additional remarks 47 47 50 52 53 56 59 62 Further Improvement and Development 5.1 Boundary implementation using least-squares with constraint 5.1.1 Formulation 5.1.2 Constraint equations for ρ, ρE & P 5.1.3 Constraint equations for velocity 5.2 2-D solution of flow over circular cylinder 5.3 3-D solution of flow over uniform wing 5.4 Additional remarks 63 63 64 65 67 69 73 77 Conclusion 79 List of References 81 iv Summary Mesh generation for complex geometries is a continuing obstacle in Computational Fluid Dynamics (CFD) Thus, there is a strong need for a fast and efficient numerical method that can handle these complex configurations In this work, a Cartesian method for the computation of steady-state solution for compressible Euler equations is presented The proposed method attempts to combine the advantages of the conventional Cartesian grid method and the gridless method while avoiding their shortcomings The Cartesian method is used over the bulk of the computational domain for its efficiencies while the gridless method is only employed in handling solid boundaries In this way, we arrive at a general solution method that is flexible and efficient for problems with complex geometries In this boundary implementation, two types of cloud points are used The first type of cloud points is set up for each boundary node to compute the Euler fluxes whereas the second type of cloud points is set up for each surface node, for boundary condition implementation and surface value determination The spatial discretization for the Euler equations is based on the cell-centered finite-volume approach The discretized equations are then solved using the modified four-stage Runge-Kutta scheme Numerous 2-D test cases involving flow over a single airfoil and flow over multicomponent objects are computed The results compare well with referenced body-fitted curvilinear grid solutions and converge well for the wide range of Mach numbers tested v An alternative gridless implementation is suggested as an improvement to the above gridless approach This involves using least-squares with constraint via the Lagrange Multiplier principle to implement the boundary conditions The formulation is general and is easily extensible to three dimensions Although there is no significant improvement in terms of accuracy and conservation over the previous scheme, the least-squares with constraint approach is better in terms of implementation Some preliminary three-dimensional results using the new approach are presented vi List of Figures Figure 1: Control volume of a single cell 21 Figure 2: Classification of grid nodes 28 Figure 3: Cloud points for (a) boundary nodes; (b) boundary nodes at thin surfaces 29 Figure 4: Cloud points for surface nodes (Closest points to surface node) 30 Figure 5: Boundary implementation for surface nodes 33 Figure 6: Close-up view of a stretched grid for the NACA 0012 airfoil 35 Figure 7: Close up view of body-fitted grid with NACA 0012 35 Figure 8: Solution for NACA 0012 with M∞ = 0.5, α = 3° (a) Cp plot (b) Mach contour plot (Cartesian grid) (c) Mach contour plot (Body-fitted grid) (d) Convergence plot [(b) and (c) are plotted with contours Mmin=0.1, 38 Mmax=0.67, interval=0.019] Figure 9: Solution for NACA 0012 with M∞ = 0.85, α = 0° (a) Cp plot (b) Mach contour plot (Cartesian Grid) (c) Mach contour plot (Bodyfitted grid) (d) Convergence plot [(b) and (c) are plotted with contours Mmin=0.1, Mmax=1.1, interval=0.034.] 41 Figure 10: Solution for NACA 0012 with M∞ = 0.8, α = 1.25° (a) Cp plot (b) Mach contour plot (Cartesian grid) (c) Mach contour plot (Bodyfitted grid) (d) Convergence plot [(b) and (c) are plotted with contours Mmin=0.1, Mmax=1.1, interval=0.026.] 43 Figure 11: Solution for NACA 0012 with M∞ = 2.0, α = 0° (a) Cp plot (b) Convergence plot (c) Close up Mach contour plot (Cartesian grid) (d) Close up Mach contour plot (Body-fitted grid) [(c) and (d) are plotted with contours Mmin=0.25, Mmax=1.9, interval=0.12.] 46 vii Figure 12: Solution for RAE 2822 with M∞ = 0.5, α = 3° (a) Cp plot (b) Mach contour plot (Cartesian grid) (c) Mach contour plot (Body-fitted grid) (d) Convergence plot [(b) and (c) are plotted with contours Mmin=0.05, Mmax=0.8, interval=0.019.] 49 Figure 13: Solution for RAE 2822 with M∞ = 0.75, α = 3° (a) Cp plot (b) Mach contour plot (Cartesian grid) (c) Mach contour plot (Body-fitted grid) (d) Convergence plot [(b) and (c) are plotted with contours Mmin=0.1, Mmax=1.1, interval=0.025.] 52 Figure 14: Solution for dual NACA0012 airfoil with M∞ = 0.5, α = 0° (a) Close up view of stretched grid (b) Cp plot (c) Mach contour plot with contours Mmin=0.02, Mmax=1.0, interval=0.034 (d) Convergence plot 55 Figure 15: Solution for NLR airfoil with flap with M∞ = 0.2, α = 0° (a) Close up view of stretched grid (b) Cp plot (c) Mach contour plot with contours Mmin=0.025, Mmax=0.375, interval=0.012 (d) Convergence plot 58 Figure 16: Solution for three-element airfoil with M∞ = 0.2, α = 20° (a) Close up view of stretched grid (b) Cp plot (c) Mach contour plot with contours Mmin=0.048, Mmax=0.72, interval=0.052 (d) Convergence plot 61 Figure 17: Solution for circular cylinder using least-squares with constraints approach with M∞ = 0.38 (a) Close up view of stretched grid (b) Mach contour plot (Least-squares with constraint) (c) Mach contour plot (Previous method) (d) Mach contour plot (Body-fitted grid) (e) Comparison of Cp plots (f) Comparison of convergence plots [(b), (c) and (d) are plotted with contours Mmin=0.052, Mmax=0.714, interval=0.047.] 73 Figure 18: Cartesian grid for NACA 0012 uniform wing surface 74 Figure 19: Close up view of cross-sectional X-Z plane with surface nodes defined 75 Figure 20: Solution for uniform NACA 0012 wing using least-squares with constraints approach with M∞ = 0.85, α = 0° (a) Cp contour plot with contours Cpmin=-0.78, Cpmax=0.66, interval=0.103 (b) Mach contour plot with contours Mmin=0.1, Mmax=1.1 interval=0.034 (c) Convergence plot 77 viii Chapter Introduction Chapter Introduction Throughout most of the twentieth century the study and practice of fluid dynamics involved the use of pure theory and pure experiment However, the advent of the high speed digital computer combined with the development of accurate numerical algorithms for solving physical problems on these computers has revolutionized the way fluid dynamics is studied today Computational Fluid Dynamics (CFD) has become fundamentally important in the study of fluid dynamics It nicely and synergistically complements the other two approaches of pure theory and pure experiment in the study and development of the whole discipline of fluid dynamics As technology becomes more widely available in industry and academia, CFD is used to provide insights into many aspects of fluid motion Nowadays, CFD methodologies are routinely employed in the fields of aircraft, turbomachinery, car and ship designs It is also applied in meteorology, oceanography, astrophysics, in oil recovery and also in architecture Hence, CFD is becoming an increasingly important design tool in engineering and also a substantial research tool in certain physical sciences The future advancement of fluid dynamics will depend on a proper balance of all three approaches, with CFD helping to interpret and understand the results of theory and experiment, and vice versa Chapter Introduction Over the course of last decade, significant progress has been made on developing numerical methods for the solution of the compressible Euler and Navier-Stokes equation In general, these numerical methods can be classified by the mesh they use, which falls under the category of structured grid or unstructured grid methods Structured grids and unstructured grids each have their own specific advantages and shortcomings Examples of structured grids are body-fitted hexahedral grids and Cartesian grids Body-fitted grid has the advantage of ease in boundary implementation due to the body-aligned nature of the grid However, the major drawback of this is the difficulty of mesh generation for complex geometry Cartesian grid, on the other hand, does not encounter any problems with mesh generation However, as the grid is not body-aligned the cells near the boundary are cut by the surface which makes accurate implementation of boundary conditions complicated The success of Cartesian method depends greatly on having an accurate means of representation for the boundary Unstructured meshes are typically constructed from triangles in two-dimensional or tetrahedral cells in three-dimensional The main advantage of unstructured grids is the ease of grid generation about complex configuration since the cells may be oriented in any arbitrary way to conform to the geometry However, the computational time and cost for unstructured mesh computations are generally higher which makes it inefficient when applying to large scale three-dimensional problems In general, structured grids are favored for its simpler data structure, which leads to smaller computing times since no indirect addressing is required while unstructured grids are favored for its flexibility in mesh generation when handling arbitrarily complex geometries Chapter Further Improvement and Development Looking at three-dimensional computations, the solution of flow over a uniform NACA 0012 wing is computed As a preliminary test, the present Cartesian method using the gridless boundary condition implementation is employed on a 100x5x100 uniform grid with a far field chord lengths away from the airfoil as shown in Figure 18 The wing is uniform in the y-direction with a constant X-Z plane cross section as shown in Figure 19 Figure 19 also shows the surface nodes defined on the body The symmetric boundary condition is set for the planar boundaries normal to the y-axis while the rest of the planar boundaries are set as far field conditions A cloud of 16 points is used for the least-squares approximation Figure 18: Cartesian grid for NACA 0012 uniform wing surface 74 Chapter Further Improvement and Development Figure 19: Close up view of cross-sectional X-Z plane with surface nodes defined The solution for transonic flow with Mach = 0.85 and the angle of attack, α = 0° is computed The pressure and Mach contour plots at y = 0.1 are shown in Figure 20a and 20b respectively From the contour plot, it can be seen that the solution is smooth on the whole except near the surface This poor resolution is due to the fact that a coarse grid is being used Despite the coarseness of grid, the 3-D Mach contour plot is reasonable compared to the 2-D Mach contour plot shown in Figure 9b The 3-D solution can be improved with a refinement of the grid As the current code runs only on a single grid, refining the Cartesian grid further to compute in three dimensions will be very costly It is advisable to implement acceleration techniques such as the multigrid scheme to accelerate the solution to convergence or to implement embedded grid computation, so as to reduce the computational time required Thus, the focus here is in testing the method in 3-D and less emphasis is placed on the accuracy of the solution The convergence plot for this case is presented in Figure 20c 75 Chapter Further Improvement and Development a) Cp contour plot b) Mach contour plot 76 Chapter Further Improvement and Development c) Convergence plot Figure 20: Solution for uniform NACA 0012 wing using least-squares with constraints approach with M∞ = 0.85, α = 0° (a) Cp contour plot with contours Cpmin= -0.78, Cpmax=0.66, interval=0.103 (b) Mach contour plot with contours Mmin=0.1, Mmax=1.1 interval=0.034 (c) Convergence plot 5.4 Additional remarks In the 2-D test case, the solution obtained using the least-squares with constraint approach appears to be very similar to that obtained using the previous gridless approach Although there is no significant improvement in terms of accuracy and conservation, the least-squares with constraint approach is better in terms of implementation Unlike the previous approach where the cloud points in each surface cloud has to be rotated to the normal and tangential direction of the respective surface node in order to implement the boundary conditions, the formulation for the leastsquares with constraint approach is more general The boundary conditions are rewritten in the global co-ordinate system and solved together with the system of equations Furthermore, boundary implementation through the use of constraint 77 Chapter Further Improvement and Development equations allows more than one surface node to be included in each cloud This means that the boundary conditions at these surface nodes can be satisfied simultaneously in each cloud, which is not achievable in the previous approach However, with more surface nodes, hence more constraint equations, the matrix becomes stiffer and more cloud points have to be included to solve the equations properly The 3-D implementation using least-squares with constraint is very similar to the 2-D implementation However it is noted that for 3-D computations, the direction of the tangential velocity is dependent on the fluid flow direction which changes as the flow computes Hence, the tangential velocity has to be determined at every iteration as is the inverse of the least-squares matrixes This is unlike 2-D computations, in which the direction of surface tangent is fixed and the inverse of the least-squares matrix need only be computed once at the start of the code Hence, while the gridless approach is especially suited for 2-D computations, further work may be needed to make it efficient for 3-D computations 78 Chapter Conclusion Chapter Conclusion A method that combines the advantages of efficiency, accuracy, and ease of grid generation of a Cartesian grid method together with the flexibility in handling complex geometries of a gridless method while avoiding their shortcomings is presented The approach uses the “gridless” or “meshless” method to address the boundary or interface while a standard structured grid method is used everywhere else Boundary conditions are implemented on the geometry surface and are automatically satisfied as the surface values are solved in a least-squares fashion In the first approach where two types of cloud points are employed, numerous 2-D test cases involving subsonic, transonic, and supersonic flows over the NACA 0012 and RAE 2822 airfoils are computed The results obtained compares well with referenced body-fitted curvilinear grid solutions In addition, three other test cases involving flow over multi-component objects are computed to demonstrate the capability of the method in handling complex configurations On the whole, the method is robust, stable and it converges well for a wide range of Mach numbers In an 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