A Higher-Order Accurate Unstructured Finite Volume Newton-Krylov Algorithm for Inviscid Compressible Flows

Transcription

1 A Higher-Order Accurate Unstructured Finite Volume Newton-Krylov Algorithm for Inviscid Compressible Flows by AMIR NEJAT B.Sc. (Aerospace Engineering), AmirKabir University of Technology, 1996 M.Sc. (Aerospace Engineering), AmirKabir University of Technology, 1998 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Mechanical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA April 2007 c Amir Nejat, 2007

2 Abstract A fast implicit (Newton-Krylov) finite volume algorithm is developed for higher-order unstructured (cell-centered) steady-state computation of inviscid compressible flows (Euler equations). The matrix-free Generalized Minimal Residual (GMRES) algorithm is used for solving the linear system arising from implicit discretization of the governing equations, avoiding expensive and complicated explicit computation of the higher-order Jacobian matrix. An Incomplete Lower-Upper factorization technique is employed as the preconditioning strategy and a first-order Jacobian as a preconditioning matrix. The solution process is divided into two phases: start-up and Newton iterations. In the start-up phase an approximate solution of the fluid flow is computed which includes most of the physical characteristics of the steady-state flow. A defect correction procedure is proposed for the start-up phase consisting of multiple implicit pre-iterations. At the end of the start-up phase (when the linearization of the flow field is accurate enough for steady-state solution) the solution is switched to the Newton phase, taking an infinite time step and recovering a semi-quadratic convergence rate (for most of the cases). A proper limiter implementation for higher-order discretization is discussed and a new formula for limiting the higher-order terms of the reconstruction polynomial is introduced. The issue of mesh refinement in accuracy measurement for unstructured meshes is revisited. A straightforward methodology is applied for accuracy assessment of the higher-order unstructured approach based on total pressure loss, drag measurement, and direct solution error calculation. The accuracy, fast convergence and robustness of the proposed higher-order unstructured Newton-Krylov solver for different speed regimes are demonstrated via several test cases for the 2nd, 3rd and 4th-order discretization. Solutions of different orders of accuracy are compared in detail through several investigations. The possibility of reducing the computational cost required for a given level of accuracy using high-order discretization is demonstrated. ii

3 Contents 1 Introduction Motivation Background Mesh Generation and Spatial Discretization Higher-Order Discretization Implicit Method and Convergence Acceleration Objective Contributions Outline Flow Solver Governing Equations Implicit Time-Advance Upwind Flux Formulation The Godunov Approach Roe s Flux Difference Splitting Scheme Boundary Flux Treatment iii

4 CONTENTS iv Wall Boundary Condition Inlet/Outlet Boundary Conditions Flux Integration Reconstruction and Monotonicity Introduction K-Exact Least-Square Reconstruction Conservation of the Mean K-Exact Reconstruction Compact Support Boundary Constraints Dirichlet Boundary Constraint Neumann Boundary Constraint Constrained Least-Square System Accuracy Assessment for a Smooth Function Monotonicity Enforcement Flux Limiting Slope Limiting Flux Jacobian What is the Jacobian? Flux Jacobian Formulation Roe s Flux Jacobian

5 CONTENTS v Boundary Flux Jacobians Wall Boundary Flux Jacobian Subsonic Inlet Flux Jacobian Subsonic Outlet flux Jacobian Supersonic Inlet/Outlet Flux Jacobians Finite Difference Differentiation Numerical Flux Jacobian Linear Solver and Solution Strategy Introduction GMRES Linear Solver The Basic GMRES Algorithm Matrix-Vector Products Computation in GMRES Preconditioning GMRES with Right Preconditioning Solution Strategy Start-up Phase Newton Phase Results(I): Verification Cases Reconstruction Test Cases Square Test Case Annulus Test case

6 CONTENTS vi 6.2 Subsonic Flow Past a Semi-Circular Cylinder Supersonic Vortex Numerical Solution Solution accuracy measurement Results(II): Simulation Cases Subsonic Airfoil, NACA 0012, M = 0.63, α = Solution Process Transonic Airfoil, NACA 0012, M = 0.8, α = Supersonic flow, Diamond airfoil, M = 2.0, α = Concluding Remarks Summary and Contributions Conclusions Recommended Future Work Start-up Preconditioning Reconstruction Extension to 3D Extension to Viscous Flows Bibliography 175

7 List of Tables 1.1 Qualitative illustration of research on solver development Ratio of non-zero elements in factorized matrix nd-order error norms for the square case rd-order error norms for the square case th-order error norms for the square case nd-order error norms for the annulus case rd-order error norms for the annulus case th-order error norms for the annulus case Sizes and ratios of the control volumes for circular cylinder meshes Error norms for total pressure, 2nd-order solution Error norms for total pressure, 3rd-order solution Error norms for total pressure, 4th-order solution Solution error norms, 2nd-order discretization Solution error norms, 3rd-order discretization Solution error norms, 4th-order discretization vii

8 LIST OF TABLES viii 7.1 Mesh detail for NACA 0012 airfoil Convergence summary for NACA 0012 airfoil, M = 0.63, α = Lift and drag coefficients for all meshes and discretization orders, NACA 0012, M = 0.63, α = 2 0, far field size of 25 chords Effect of the far field distance on lift and drag coefficients, NACA 0012, M = 0.63, α = Convergence summary for NACA 0012 airfoil, M = 0.8, α = Lift and drag coefficients, NACA 0012, M = 0.8, α = Convergence summary for diamond airfoil, M = 2.0, α = Drag coefficient, diamond airfoil, M = 2.0, α =

9 List of Figures 1.1 Main approaches in fluid dynamics CFD overall algorithm Example of a structured and an unstructured mesh over a 2D airfoil Propagation of a linear wave in positive direction Shock-Tube problem Rounding the characteristic slope near zero Schematic illustration of Gauss quadrature points A typical cell center control volume and its reconstruction stencil, including three layers of neighbors Imposing boundary constraint at the Gauss boundary points Typical unlimited/limited linear reconstruction Using first neighbors for monotonicity enforcement Typical unlimited/limited quadratic reconstruction Defining σ as a function of φ Schematic of Direct Neighbors ix

10 LIST OF FIGURES x 4.2 Typical cell-centered mesh numbering Total numerical error versus perturbation magnitude Linearization of a sample function Unstructured meshes for a square domain Error-Mesh plot for the square case Unstructured meshes for a curved domain (annulus) Error-Mesh plot for the annulus case Circular domain over half a cylinder, Mesh 1 (1376 CVs) Circular cylinder, Mesh 1 (1376CVs) Circular cylinder, Mesh 2 (5539 CVs) Circular cylinder, Mesh 3 (22844 CVs) Convergence history for the coarse mesh (Mesh 1) Convergence history for the fine mesh (Mesh 3) nd-order pressure coefficient contours, Mesh rd-order pressure coefficient contours, Mesh th-order pressure coefficient contours, Mesh th-order pressure coefficient contours, Mesh Pressure coefficient along the axis Close up of the pressure coefficient along the axis (suction region) Pressure coefficient along the axis, Mesh Error in total pressure ratio, 2nd-order discretization, Mesh

11 LIST OF FIGURES xi 6.19 Error in total pressure ratio, 3rd-order discretization, Mesh Error in total pressure ratio, 4th-order discretization, Mesh Error in total pressure ratio, 4th-order discretization, Mesh Error-Mesh plot for the total pressure Drag coefficient versus mesh size Drag coefficient versus CPU time Annulus, Mesh 1 (108 CVs) Annulus, Mesh 2 (427 CVs) Annulus, Mesh 3 (1703 CVs) Annulus, Mesh 4 (6811 CVs) Annulus, Mesh 5 (27389 CVs) Convergence history for the coarse mesh (Mesh 1) Convergence history for the fine mesh (Mesh 5) nd-order Mach contours for the coarse mesh (Mesh 1) rd-order Mach contours for the coarse mesh (Mesh 1) th-order Mach contours for the coarse mesh (Mesh 1) nd-order density error for the coarse mesh (Mesh 1) rd-order density error for the coarse mesh (Mesh 1) th-order density error for the coarse mesh (Mesh 1) Density, 4th-order solution over the fine mesh (Mesh 5) Error-Mesh plot for the solution (Density) Error versus CPU Time

12 LIST OF FIGURES xii 7.1 NACA 0012, Mesh 1, 1245 CVs NACA 0012, Mesh 2, 2501 CVs NACA 0012, Mesh 3, 4958 CVs NACA 0012, Mesh 4, 9931 CVs NACA 0012, Mesh 5, CVs Cp over the upper surface after start-up, Mesh1 (1245 CVs) Cp over the upper surface after start-up, Mesh 5 (19957 CVs) CPU time versus the grid size, NACA 0012, M = 0.63, α = Total work unit versus the grid size, NACA 0012, M = 0.63, α = Newton phase work unit versus the grid size, NACA 0012, M = 0.63, α = Convergence history, NACA 0012, Mesh 1, M = 0.63, α = Convergence history, Mesh 5, M = 0.63, α = Non-linear residual versus linear system residual, Mesh 3, M = 0.63, α = Linear system residual dropping order, Mesh 3, M = 0.63, α = Eigenvalue pattern for the preconditioned system, Mesh 3, M = 0.63, α = Condition No. of the preconditioned system, Mesh 3, M = 0.63, α = Condition No. of the preconditioned system, Mesh 5, M = 0.63, α = Lift coefficient convergence history, NACA 0012, Mesh 1, M = 0.63, α = Drag coefficient convergence history, NACA 0012, Mesh 1, M = 0.63, α = Lift coefficient convergence history NACA 0012, Mesh 5, M = 0.63, α = Drag coefficient convergence history, NACA 0012, Mesh 5, M = 0.63, α = NACA 0012, Mesh 3 (4958 CVs)

13 LIST OF FIGURES xiii nd-order Mach contours for NACA 0012 airfoil, Mesh 3, M = 0.63, α = rd-order Mach contours for NACA 0012 airfoil, Mesh 3, M = 0.63, α = th-order Mach contours for NACA 0012 airfoil, Mesh 3, M = 0.63, α = Mach profile, upper side, NACA 0012 airfoil, Mesh 1, M = 0.63, α = Mach profile close up, upper side, NACA 0012 airfoil, Mesh 1, M = 0.63, α = Mesh 1, close-up at the leading edge region Mach profile, lower side, NACA 0012 airfoil, Mesh 1, M = 0.63, α = Mach profile, upper side, NACA 0012 airfoil, Mesh 5, M = 0.63, α = Mach profile close up, upper side, NACA 0012 airfoil, Mesh 5, M = 0.63, α = Mach profile, lower side, NACA 0012 airfoil, Mesh 5, M = 0.63, α = Pt P t, upper side, NACA 0012 airfoil, Mesh 1, M = 0.63, α = Pt P t, lower side, NACA 0012 airfoil, Mesh 1, M = 0.63, α = Pt P t, upper side, NACA 0012 airfoil, Mesh 5, M = 0.63, α = Pt P t, lower side, NACA 0012 airfoil, Mesh 5, M = 0.63, α = Mach profile at the end of start-up process, M = 0.8, α = Convergence history for NACA 0012, M = 0.8, α = nd-order Mach contours, NACA 0012, M = 0.8, α = rd-order Mach contours, NACA 0012, M = 0.8, α = th-order Mach contours, NACA 0012, M = 0.8, α = limiter φ (3rd-order), NACA 0012, M = 0.8, α = limiter σ (3rd-order), NACA 0012, M = 0.8, α = Mach profile, NACA 0012, M = 0.8, α =

14 LIST OF FIGURES xiv 7.45 Mach profile in shock regions, NACA 0012, M = 0.8, α = Mesh 7771 CVs, diamond airfoil, M = 2.0, α = Cp at the end of start-up process, diamond airfoil, M = 2.0, α = Convergence history for diamond airfoil, M = 2.0, α = nd-order Mach contours, diamond airfoil, M = 2.0, α = rd-order Mach contours, diamond airfoil, M = 2.0, α = th-order Mach contours, diamond airfoil, M = 2.0, α = nd-order Cp, diamond airfoil, M = 2.0, α = rd-order Cp, diamond airfoil, M = 2.0, α = th-order Cp, diamond airfoil, M = 2.0, α =

15 List of Symbols Roman Symbols a A speed of sound area, Jacobian matrix b righr hand side (Ax = b) B C C L C D C p C v D e E F G h H I fixed iteration matrix center lift coefficient drag coefficient specific heat at constant pressure, pressure coefficient specific heat at constant volume reconstruction solution vector (derivatives) specific internal energy total energy, error flux vector Gauss (integration) point specific enthalpy, mesh length scale Hessenberg matrix identity matrix xv

16 LIST OF SYMBOLS xvi J k K l L m M n Jacobian matrix number of subspace size order of accuracy, polynomial order, constant in Venkatakrishnan limiter length of the control volume face, norm norm, left eigenvector number of restarts Mach number, moments, preconditioner matrix, coefficient matrix in reconstruction normal vector N total number of... p P r R S t T U polynomial order static pressure, polynomial residual, distance gas constant, residual, radius, right eigenvector slope in higher-order limiter time, exponent of the weighting in reconstruction static temperature solution vector, conservative variables u, v velocity components V primitive variables, velocity, subspace x, y Cartesian coordinates, unknown vectors z preconditioning vector

17 LIST OF SYMBOLS xvii Greek Symbols α γ ε λ ρ σ φ ω angle of attack specific heat ratio perturbation parameter eigenvalue, wave speed Density higher-order limiter slope limiter relaxation factor Superscripts i iteration index k, K order, iteration number, subspace number m, n polynomial exponent Subscripts b c CD CV DB F D G i boundary center central difference control volume Dirichlet boundary forward difference Gauss point inner, control volume index

18 LIST OF SYMBOLS xviii in L m inlet left side magnitude min, max minimum and maximum n N N B o out ref R RB SubIn SubOut t W normal, normalized neighbor Neumann boundary outer outlet reference right side reconstructed at the boundary subsonic inlet subsonic outlet total wall x, y Cartesian directions

19 Acknowledgments I would like to express my deep appreciation to my research supervisor, Dr. Carl Ollivier- Gooch for all of his guidance, support and patience. His remarkable feedbacks throughout the course of this research were very helpful. I also would like to thank Dr. Chen Greif from the Computer Science Department. Attending his valuable lectures on sparse matrix solvers and having the opportunity to engage in discussions with him, have greatly benefited my research. I am grateful to my all colleagues in the ANSLab research group in the Mechanical Engineering Department, especially Chris Michalak, Serge Gosselin, and Harsha Perera, for their computer assistance on many occasions. Finally, my most sincere and deepest appreciation is for my great family, especially my mother Homa and my father Mehdi, whose constant support, encouragement, patience and love I have enjoyed all through my life. xix

20 Chapter 1 Introduction Prediction of fluid flow quantities, such as pressure, velocity, and temperature, and the study of flow behavior are the main goals in the field of fluid dynamics. Like other science and engineering disciplines, fluid dynamics has greatly benefited from the development of computing technology (numerical algorithms and computational tools) over the last four decades, resulting in the creation of a new born approach in the field known as Computational Fluid Dynamics (CFD), Fig (1.1). CFD has shown remarkable capability for fluid flow analysis both in academia and industry. CFD has not only made possible the simulation of flows (such as reentry of space vehicles) for which complete analysis (either by theory or experiment) was impossible before, but also has provided a valuable feedback and information source for improving theoretical and experimental fluid dynamics. Increasing computing power over the last two decades has resulted in the development of new computational techniques and algorithms, enhancing the versatility of CFD application. Nowadays, CFD is not just a research tool, and it is used extensively and successfully in industry throughout the design process, from preliminary design to shape optimization. 1.1 Motivation In the field of computational aerodynamics, the final goal is the accurate simulation of the flow field around (and/or inside) complex 3D geometries to compute aerodynamic force coefficients. In the mid 1980 s, Jameson [36] was the first person to compute the threedimensional flow over realistic aerodynamic configurations using the finite volume technique (a robust conservative numerical approach for discretization of the fluid flow equations over 1

21 CHAPTER 1. INTRODUCTION 2 general meshes [38]); since then tremendous progress has been made in this area and application of CFD for aerodynamic computations has revolutionized the process of aerodynamic design [27]. To simulate the flow field around a 3D complex geometry accurately, a CFD package should include three essential parts: 1. State of the art mesh generation capability. Mesh generation or domain discretization, is one of the most important parts (if not the most!) in CFD simulations. Since the discretization of the fluid flow equations is carried out on the mesh, without a good domain discretization the CFD solution can be very inaccurate. Furthermore generating an appropriate mesh, especially for complex geometries in practice is the most time consuming part for a CFD user and certainly is not a trivial task. According to a real aerodynamic case study in aerospace engineering, the mesh preparation time can be up to 45 times larger than the required computation time for the fluid flow simulation [47]. Therefore, it is both desirable and necessary to reduce the mesh preparation time and there is a large potential to gain by automating this process. Ideally, meshing software should be able to generate a geometrically and physically suitable mesh around or inside a complex 3D geometry with a reasonable user workload. Also the user should be able to refine the mesh according to geometric parameters and to adapt the mesh based on flow features without excessive effort. The unstructured mesh technique, among other types of mesh generation methods, is a very good candidate to address mesh generation issues due to its automation capability in generating meshes for complex geometries and its flexibility in refinement and adaptation. 2. Accurate physical modeling of high Reynolds number turbulent flow. Most practical engineering applications such as aircraft aerodynamics involve turbulent flows. The Direct Numerical Simulation (DNS) of turbulent flows for practical purposes is not feasible at least for the next couple of decades due to computing technology limitations (memory and speed). Therefore modeling the turbulent flow is the only viable approach in high Reynolds number CFD simulations and will remain a major active research area for the foreseeable future [7, 75]. Discussing the physical and numerical criteria for choosing an appropriate turbulence model and related issues are beyond the scope of this thesis, but accurate physical modeling is a key part of valid CFD simulation for practical engineering problems. It should be noted that the modeling of the physical phenomena in CFD simulations is not just limited to turbulence, but is also essential for combustion, multiphase flows, hydromagnetic flows, and other types of fluid flows.

22 CHAPTER 1. INTRODUCTION 3 Theoretical Fluid Dynamics Experimental Fluid Dynamics Applied Math Numerical Algorithms Computational Fluid Dynamics Computing Tools Figure 1.1: Main approaches in fluid dynamics 3. A robust, efficient, accurate flow solver for the generic mesh. Numerical solution of the fluid flow equations is what CFD is all about and this solution must be stable and converge to the correct answer at a reasonable cost. A solver algorithm includes three separate components: discretization of the fluid flow equations, numerical flux computation, and updating the solution mostly through time advance methods. Fig (1.2) shows overall schematic of a CFD algorithm. Current CFD flow solver algorithms still have some limitations both in terms of efficiency and accuracy especially for simulation of physically complicated flows. Specifically, the memory and speed of current computers (even using parallel techniques) do not yet allow us to simulate physically complex flows in realistic geometries, with sufficient accuracy, in a short time and at a reasonable cost. As a result, we have to simplify the physics of the fluid flow via approximate modeling, and neglect some of the numerical/physical issues caused by insufficient mesh density (especially for complex geometries), limiting the validity of the simulation and adversely affecting its application. Before proceeding forward with further details, it should be mentioned that this thesis is aimed only at the third component mentioned above improving the efficiency and accuracy of CFD algorithms. The numerical error in a simulation can be written in the form of h p, where h is the mesh length scale and p is the discretization order of accuracy. Clearly, then, improving the accuracy of a numerical simulation (modeling issues aside) is possible by means of increasing

23 CHAPTER 1. INTRODUCTION 4 Order Structured Structured Unstructured Unstructured Explicit Implicit Explicit Implicit Second-Order Higher-Order? Table 1.1: Qualitative illustration of research on solver development the mesh density (using smaller grid or decreasing h), and/or increasing the discretization order. Reducing mesh length scale can be achieved by global or local mesh refinement or adaptation. Nearly all modern CFD codes use second-order methods, which produce a diffusive error proportional to h 2 due to diffusive derivatives beside possible added artificial viscosity for stability purposes (in central difference schemes). For instance in a second-order 2D finite volume formulation in the Cartesian coordinates this numerical error for each control volume can be written in the following form: Numerical diffusive error = 2 U x 2 x U 2 x y x y + 2 U y 2 y 2 2 (1.1) This leading-order error term causes two significant numerical problems. First, it smears sharp gradients in convection dominated parts of the flow and spoils the conservation of total pressure in isentropic regions of the flow field as it acts like a (numerical) viscosity. Second, this numerical diffusivity produces parasitic error in viscous regions by adding extra diffusion, which is very grid dependent, to the solution. Therefore using a high resolution numerical scheme for discretization is quite desirable. Application of discretization orders larger than second-order both for structured and unstructured grids has been an area of ongoing research for the last two decades [35, 10], and will be the focus of this thesis. However, convergence of high resolution schemes is not as efficient as second-order schemes especially for unstructured grids [88, 46] due to the increased complexity of the discretization, decreased damping (lack of diffusive damping) and adding more error modes (which must be damped in the solution process). Consequently implementing a higher-order unstructured discretization within an implicit framework to achieve the efficient convergence is extremely helpful if not necessary! As shown in Fig(1.2), up to the flux integral computation, the overall CFD algorithm is the same, but the choice of the time advance technique in updating the solution changes the level of complexity of the solution process completely. Integration of the discretized equations in time can be done either explicitly or implicitly. In the explicit time integration the space discretization is performed at the previous time

24 CHAPTER 1. INTRODUCTION 5 Geometry & Solution Domain Mesh Generation Package Physics & Fluid Flow Equations Discretized Domain Boundary & Initial Conditions Discretization of the Fluid Flow Equations over the Discretized Domain Flux Integral Explicit Time Advance Implicit Time Advance Multistage Techniques Flux Integral Linearization Solution Update Preconditioning Sparse Matrix Solver Solution Update Figure 1.2: CFD overall algorithm

25 CHAPTER 1. INTRODUCTION 6 level using the known flow quantities found at the previous time iteration. In the implicit time integration both the space and the time discretizations are performed at the current time level where the flow quantities are needed as unknowns. Equation (1.2) shows a typical unsteady fluid flow PDE where the right hand side represents the spatial discretization and the left hand side shows the time derivative. For example, employing the first-order forward differencing time advance technique in the explicit and implicit forms leads to Eq. (1.3) and Eq. (1.4). du dt Explicit time advance: Implicit time advance: = R(U) (1.2) U n+1 i U n+1 i Ui n t Ui n t = R(U n i ) (1.3) = R(U n+1 i ) (1.4) While an explicit update just needs multistage integration of the flux integral using a Runge- Kutta type scheme, an implicit update requires linearization of the flux integral (Chapter 2) and constructing a large linear system which requires an efficient sparse matrix solver (discussed in Chapter 4 and 5). Efficiently solving a large linear system, especially with an ill conditioned matrix resulting from a higher-order discretization, demands effective preconditioning which adds to the complexity of the process. But the error reduction of each solution update in implicit integration is far larger than the explicit one, since implicit methods do not suffer from the stability issues of explicit methods and large time steps can be taken. On balance, therefore, it is preferable to bear the complexity of the algorithm and accelerate the solution toward the steady-state in a relatively small number of implicit iterations. Table 1.1 provides a qualitative summary on finite volume solver development research, where the number of symbols represents the approximate volume of the research on the solver development since early 80 s (based on the author s survey). Clearly the trend in solver development is moving toward: 1. Unstructured meshes to address the mesh generation issues for complex geometries 2. Higher-order discretizations to increase the global solution accuracy 3. Implicit techniques to improve the efficiency of the solution process This research is mainly intended to contribute to the development of efficient and accurate flow solvers filling the gap in high-order implicit methods, on unstructured meshes.

26 CHAPTER 1. INTRODUCTION Background This section provides an overview of relevant aspects of current CFD solvers including discretization type and order as well as implicit algorithms. This overview is complemented by detailed discussion of previous work related to each part of the solver in the relevant chapters of the thesis Mesh Generation and Spatial Discretization Numerical simulation of fluid flow consists of two main parts: discretization of the flow field around or inside the geometry by a finite number of cells (grid generation) and solution of the fluid flow equations over the discretized domain (flow solver). Structured and unstructured meshes, Fig (1.3), are the most common types of the grids used in CFD applications. In a structured mesh, all cells and vertices have the same topology but in an unstructured mesh, elements can have irregular and variable topologies. The task of generating structured grids around complex configurations has proved to be a considerable challenge. Sophisticated structured approaches such as multi-block grid generation has resolved this issue by dividing the domain between the body and far field into simple geometrical blocks; structured grids are generated inside each block. However, automation of the blocking procedure is still a relatively difficult job [76]. Another structured approach is overlap or chimera grids. In this technique, the computational domain is divided into multiple zones and a suitable grid is generated in each zone. The chimera approach allows zones to overlap, and interpolation routines are used to transmit data between the overset grids in the flow solver. However, generalizing the grid generation and adaptation in this approach is still not an easy task and demands a high level of expertise as well as considerable effort. Furthermore, interpolation between the blocks and overlapped meshes has its own issues and can introduce additional error. The most powerful approach for complex geometries is unstructured grids (typically triangular in 2D and tetrahedral in 3D). Unstructured grids have a higher flexibility in refinement based on the geometry and adaptation based on the solution features and gradients. Therefore, unstructured meshes are one of the most suitable choices for complex geometries; associated solvers are becoming more common in modern CFD applications and promise to be more capable and successful for complex aerodynamic problems [88]. The fluid flow equations (PDEs) generally are discretized in one of the following forms: finite difference, finite element or finite volume. Finite difference is the point wise representation of the flow field where the flow equations are solved only for variables defined at

27 CHAPTER 1. INTRODUCTION 8 (a) Structured (b) Unstructured Figure 1.3: Example of a structured and an unstructured mesh over a 2D airfoil grid points. The finite difference scheme was the original approach to the CFD problems and it is well suited for structured grids. Therefore, its higher-order implementation can be easily achieved by employing higher-order differencing formula. However, the finite difference discretization does not conserve mass, momentum and energy of the flow which is an important issue for most of the practical applications such as shock capturing. Furthermore and more importantly the finite difference discretization can not be implemented on unstructured grids. The finite element method, another discretization technique, is one of the most complete and well established mathematical approach for numerical solution of PDEs. In this method, the flow equations are multiplied by a test function and then integrated over the discretized domain. The solution is represented by a local basis function (interpolation function) for each element. Finite element method is very flexible both in terms of theory and application and it can be easily used for unstructured meshes. Its high-order extension is also fairly common by employing higher-order basis and test functions. The challenging part in application of the finite element for CFD computation is again the conservation of the flow equations especially for non-smooth flows. Although conserving the mass, momentum and energy is possible in the finite element formulation but it is not an easy task, and finite element codes require considerable fine tuning in shock capturing. The finite volume approach is designed based on the conservation of mass, momentum

28 CHAPTER 1. INTRODUCTION 9 and energy. The solution is represented by control volume averages and equations are discretized over the volume integrals. Like the finite element method, it is very flexible for complex geometries and unstructured meshes. At the same time its robustness for nearly all CFD applications especially shock capturing problems is well established. Higherorder application of finite volume methods is possible by using a higher-order polynomial inside each control volume which the polynomial average integral over the control volume represents the control volume average. Jameson and Mavriplis [37] reported some of the earliest unstructured finite volume CFD results, solving two-dimensional inviscid flow on regular triangular grids obtained by subdividing quadrilateral grids; central differencing was used. Their approach was second-order (linear distribution). The artificial viscosity and the second-order truncation error made the mentioned approach relatively high diffusive for general irregular unstructured meshes. Second-order upwind schemes (discretization of the flow equations based on the physical waves propagation directions) have also been used on unstructured grids either through Green-Gauss gradient technique or least-squares linear reconstruction method. Applying upwind schemes for unstructured grids is more complicated than for structured grids especially for higher-order approximation. For the unstructured case, over each finite volume (triangle in 2D) a polynomial approximation to the solution is reconstructed with the help of the neighboring control volumes, and then the Riemann (shock tube) problem is solved approximately at the control volume interfaces. One of the most successful approaches in applying the upwind scheme was undertaken by Barth[10]. In this approach, Barth defined a general upwind formulation (in multi-dimensions), introducing the minimum energy leastsquares reconstruction procedure for flux calculation up to the desired order of accuracy. However, any upwind scheme higher than first-order (which is monotone by its nature), often causes oscillations in the vicinity of sharp gradients and discontinuities which can produce instability problems. For example, Agarwal and Halt [6] proposed a compact higher-order scheme for solution of Euler equations over unstructured grids using explicit time integration. Considerable over and undershoots in their transonic airfoil case (3rd-Order) were evident. A common solution to that is using limiters, which enforce monotonicity at the expense of adversely affecting both accuracy and convergence. Barth and Jespersen [13] introduced a multi-dimensional limiter to achieve the monotonic solution. Although that approach was quite successful in suppressing oscillations, reaching full convergence was not possible even after freezing the limiter because the limiter was not differentiable. Since then several attempts have been made to design a differentiable limiter for unstructured grid solvers, and Venkatakrishnan s limiter [87] seems to be one of the most robust. That limiter does not strictly enforce monotonicity but allows only small overshoots in the con-

29 CHAPTER 1. INTRODUCTION 10 verged solution. However, it preserves accuracy especially for smooth regions where there are local extrema. Although this limiter shows better convergence behavior, it still has some convergence issues with implicit solvers. Designing an appropriate limiter for higher-order unstructured grid solvers is a fairly unexplored topic so far and needs to be addressed for practical higher-order unstructured application. Another more sophisticated approach to cure oscillations in compressible flow computation is the essentially non-oscillatory (ENO) family of schemes. These schemes are uniformly accurate and prevent oscillations in the non-smooth regions by detecting discontinuity and modifying the reconstruction stencil from cell to cell and time level to time level [30]. ENO schemes are computationally expensive and sacrifice fast convergence because of their dynamic stencils [34]. Weighted ENO (WENO) schemes were developed to address the problems caused by dynamic stencils. Near discontinuities, weighted ENO schemes ([3, 57, 28]) remove the effect of non-smooth data in the reconstruction stencil by giving it an asymptotically small weight. However no comprehensive convergence analysis and/or computational cost studies are presented. Furthermore, performance of ENO/WENO schemes in the context of implicit time advance, which is one of the most efficient solution strategies, has not yet been studied Higher-Order Discretization For structured meshes, application of higher-order algorithms has progressed considerably and it has been shown that, for practical levels of accuracy, using a higher-order accurate method can be more efficient both in terms of solution time and memory usage. With higher-order accurate methods, the cost of flux computation, integration, and other associated numerical calculations increase per control volume. However, as we can use a coarser mesh, computation time and memory are saved overall and accuracy can be increased as well. De Rango and Zingg [66] applied a globally third-order accurate algorithm for steady turbulent flow over a 2-D airfoil using a structured grid. They showed this approach can lead to a dramatic reduction in numerical error in drag using relatively coarse grids, and the results provide a convincing demonstration of the benefits of higher-order methods for practical flows. Zingg et al. [96] compared different flux discretization techniques with higher-order accuracy for laminar and turbulent flows (including transition) both in subsonic and transonic speed regimes. Extending the conclusion of the previous research, it was shown that the higher-order discretization produces solutions of a given accuracy much more efficiently than the second-order methods. More aspects of implementation of higherorder methods have been discussed by De Rango and Zingg [67], and convergence behavior

30 CHAPTER 1. INTRODUCTION 11 of this approach has been studied in detail. Again a higher-order algorithm has been applied for calculation of the flow around the multi-element airfoil using the multi-block grid technique by the same researchers [68]. A grid convergence study in this research showed that the higher-order discretization produces a substantial reduction in the numerical errors in the flow field in comparison with the second-order algorithm. This smaller error has been achieved on a grid several times coarser than the grid which had been used for the secondorder algorithm. In summary, these studies show that achieving the desired accuracy in practical aerodynamic flows using higher-order algorithms not only is possible but also has some advantages. Research in high-order unstructured solvers is motivated by the desire to combine the accuracy and efficiency benefits seen in the application of high-order methods on structured meshes with the geometric and adaptive flexibility of unstructured meshes. The application of methods having higher than second-order accuracy for solving the compressible Euler and Navier-Stokes equations on unstructured meshes has not been thoroughly investigated yet and remains an active research topic. Several researchers have achieved higher-order accuracy by the use of the finite element method. Bey and Oden [17] have used a discontinuous Galerkin method to reach fourthorder accuracy for smooth flows. Bassi and Rebay [15] have used a new discontinuous element for discretization of Euler equations and have computed the compressible flow over a simple unstructured grid. The finite volume approach has received more attention, Barth and Fredrickson [12] derived a general condition for a scheme to be higher-order accurate, including a reconstruction procedure satisfying the properties of conservation of mean, K-exactness and compact support (these criteria are discussed in detail in Chapter 3). They also proposed a minimum energy (least-square) reconstruction to calculate the required polynomial coefficients. Delanaye and Essers [23] proposed a quadratic reconstruction finite volume scheme for compressible flows on unstructured adaptive grids. The overall accuracy of the scheme was second-order. The inviscid flux was computed directly from their quadratic polynomials; however, diffusive derivatives were obtained through a linear interpolation. For monotonicity enforcement, a discontinuity detector was introduced and higher-order terms in reconstructed polynomials were dropped in the vicinity of discontinuities. Ollivier-Gooch and Van Altena [60] have analyzed a new approach for higher-order accurate finite-volume discretization for diffusive fluxes that is based on the gradients computed during solution reconstruction; fourth-order accurate solution for the advection-diffusion problem has been computed. Recently Nejat and Ollivier-Gooch [48] developed an implicit higher-order unstructured solver for Poisson s

31 CHAPTER 1. INTRODUCTION 12 equation. They clearly showed the possibility of reducing computational cost required for a given level of solution accuracy using higher-order discretization over an unstructured stencil for certain fluid problems Implicit Method and Convergence Acceleration Flow features, especially in physically complicated flows, vary greatly in size. In particular, time and length scales associated with physical phenomena like turbulence and combustion can be very disparate. The use of millions of cells, which is common today in practical simulations, results in global length scales spanning the computational domain which are several orders of magnitude larger than the smallest scales resolved by neighboring grid points. With such disparate length scales, and such large numerically stiff problemes, efficient time integration and solution convergence are real challenges for the solution of the resulting discrete systems. Generally explicit integration methods (multi-stage Runge-Kutta family schemes), even with the help of acceleration techniques, such as local time stepping and residual smoothing, still show slow convergence behavior for steady-state solution of large and/or stiff CFD problems. Implicit methods are a fairly common and efficient approach for the steady-state solution of the fluid flow equations. Regardless of the space discretization technique, finding solutions to fluid flow problems requires solving a large linear system resulting from the linearization of fluid flow equations in time (temporal discretization). In the limit of very large time steps, implicit time advance schemes approach Newton s method. Newton methods, which will be discussed in Chapter 5, have been used in CFD since the late 80 s and are considered an attractive approach for solution convergence of steady flows due to their property of quadratic convergence (when starting from a good initial guess). In early attempts direct methods were employed for solving the linear system arising at each Newton iteration [85, 89, 8]. While direct solvers have been developed for stiff linear systems, the size of the systems of equations arising in CFD makes applying direct methods impossible in practice. Therefore, using iterative linear solvers with proper preconditioning, which is a crucial factor for complex problems, is the only reasonable choice for solving the linear system in each Newton iteration. At the same time, the cost of each iteration in terms of CPU time and memory usage for a pure Newton method is relatively high. Quasi-Newton methods can have satisfactory convergence behavior, lower memory usage and less cost per iteration at the expense of increasing total number of Newton iterations and losing quadratic convergence rate [62]. Quasi-Newton methods are generally categorized as Approximate Newton and Inexact Newton methods.

32 CHAPTER 1. INTRODUCTION 13 In Approximate Newton methods the flux Jacobian on the left hand side (arising from linearization) either is computed through some simplifications or is evaluated based on lower-order discretization, while the flux integral on the right hand side is evaluated up to the desired order of accuracy. In either case the linearization is done approximately and although the Jacobian matrix has simpler structure and is better conditioned (i.e. easier to invert), the overall convergence rate of the non-linear problem will be degraded. This approach is also known as the defect correction technique, and it is useful when there are memory limitations and storing the full Jacobian is impractical, specifically in 3D. If the true Jacobian is very stiff and solving the linear system of the true linearization is challenging, Approximate Newton may work better than the original Newton method. This is often true in the early stage of Newton iterations when a good starting solution is not yet available. Therefore, Approximate Newton is a very good candidate for the start-up process, especially if the solution process is started from a poor initial guess. In the second category, Inexact Newton methods [25], the complete linearization based on the flux integral on the right hand side is employed and the true Jacobian is calculated. However, the resultant linear system at each Newton iteration is solved approximately by an iterative linear solver. For highly non-linear problems such as compressible flows, the linearized system, especially at the initial iterations is not an accurate representation of the non-linear problem. As a result, completely solving the linear system does not improve the overall convergence rate. Instead, the linear system is solved up to some tolerance criteria which normally is chosen as a fraction (typically between 10 1 and 10 2 ) of the flux integral on the right hand side. Among iterative linear solvers, Krylov subspace methods are the most common, and amongst these, the Generalized Minimal Residual (GMRES) [72] algorithm (visited in section 5.2 in detail) has been developed mainly for non-symmetric systems such as those resulting from unstructured meshes. In the matrix-free GMRES, the matrix vector products required by the GMRES algorithm are computed without forming the matrix explicitly. Matrix-free GMRES [14] is a very attractive technique for dealing with complicated Jacobian matrices, because it reduces memory usage considerably and eliminates the problem of explicitly forming the Jacobian matrix. This is especially helpful for higher-order unstructured mesh solvers where full (analytic) Jacobian calculation is extremely costly and difficult, if not impossible. GMRES efficiency depends strongly on the conditioning of the linear system. This is especially important for higher-order discretization, which makes the Jacobian matrix more off-diagonally dominant and quite ill-conditioned, and for the Euler equations (compressible flow) with the non-linear flux function and possible discontinuities in the solution. Applying a good preconditioner for GMRES under these circumstances becomes a