AN IMPLICIT-EXPLICIT FLOW SOLVER FOR COMPLEX UNSTEADY FLOWS

Transcription

1 AN IMPLICIT-EXPLICIT FLOW SOLVER FOR COMPLEX UNSTEADY FLOWS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF AERONAUTICAL ASTRONAUTICAL ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY John Ming-Jey Hsu December 4

2 c Copyright by John Ming-Jey Hsu 5 All Rights Reserved ii

3 I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Antony Jameson Principal Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Robert MacCormack I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Juan Alonso Approved for the University Committee on Graduate Studies. iii

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5 Abstract Current calculations of complex unsteady flows are prohibitively expensive for use in real engineering applications. Typical flow solvers for unsteady integration employ a fully implicit time stepping scheme, in which the equations are solved by an inner iteration. In order to achieve convergence within each physical time step, a substantial number of pseudo-time steps (typically between 3-, depending on the case) are required. Another unfavorable characteristic of the dual time stepping method is that there are no available error estimates for time accuracy available unless the inner iterations are fully converged, although numerical experiments have demonstrated second order accuracy in time. The approach in this thesis is to construct hybrid type schemes by combining implicit and explicit schemes in a manner that guarantees second order accuracy in time. An initial time accurate ADI step is introduced, followed by a small number of cycles of the dual-time stepping scheme augmented by multigrid. The formal second order accuracy in time should be retained without the need for large numbers of inner iterations. The number of inner iterations required for convergence can thus be reduced while maintaining the same overall error levels. To investigate the effectiveness of the proposed scheme, several pitching airfoil test cases were examined, offering a close look at possible reductions in computational cost by adopting the present approach. v

6 Acknowledgments I am grateful to all the people who have supported me during my graduate studies at Stanford. Without their help, this research would never have been possible. First, I would like to give my greatest appreciation to my advisor, Professor Antony Jameson. Not only has he been open-minded to my ideas, very patient with my progress, financially supportive to my research, his understanding to my struggles allowed me to continue this research. His profound teachings and his re-known accomplishments definitely drove my research to its end. In addition, I would like to thank my co-advisor Professor Robert W. MacCormack whose mentorship helped my investigations in this field. Not only did Professor MacCormack and Professor Jameson contributed, in every facet, to the scientific content of my research and studies, they made me feel like I was part of a family while at Stanford. I would like to extend my, my most sincerest thanks to Professor Juan J. Alonso, and Professor Thomas Pulliam. They have encouraged me throughout the entire process, were model figures for me to accelerate. I thank them for their participation in my thesis committee. I owe a special thanks to a select group of individuals: Peggy Huang, Peter Sturdza, Joaquim Martins, Kihwan Lee, Nawee Butsuntorn and everyone in the Aerospace Computing Laboratory. These people reinforced the importance of friendship in my life and provided me with many fond memories during my stay at Stanford. Finally, I would like to dedicate this work to my parents. I would be lost without their unconditional acceptance of who I am. vi

7 Contents Abstract Acknowledgments v vi Introduction. Practical Considerations and Motivations Background of CFD Stability Analysis of Numerical Schemes Levels of Approximations for Flow Simulations Steady Versus Unsteady Fluid Flows Characteristics Current Unsteady Flow Solvers Stanford s ASCI Project Objectives Of This Dissertation Preliminary Considerations CFD Code Design Aspects CFD Programming Style and Efficiency Considerations Governing Equations. Lagrangian and Eulerian Frames of Reference Conservation of Mass Conservation of Momentum Conservation of Energy Closure relationships for Navier-Stoke s Equation Navier-Stokes Equation Coordinate Transformation vii

8 .7. Time Related Unsteady Metrics Spatial Metrics Transformed Equations Spatial Discretization 3 3. The Semidiscrete Form, Method of Lines Inviscid Fluxes Artificial Dissipation Flux Difference Splitting Flux Vector Splitting JST Scheme CUSP-Type Schemes Comparison of E-CUSP and H-CUSP Schemes Explicit Schemes Multistage Runge-Kutta (RK) Schemes Implicit Schemes Fully Implicit Backward Difference Formula Linearized Implicit Operators Numerical Dissipations for Implicit Operators Linearized Scheme Alternate Direction Implicit (ADI) and ADI Scheme with 3-4- Backward Difference Formula (ADI-BDF) Diagonally Dominant Alternate Direction Implicit (DDADI) DDADI with BDF Hybrid Implicit-Explicit Scheme Iterative Dual-Time Stepping Scheme Explicit Unsteady Dual Time Stepping Implicit Unsteady Dual Time Stepping Point Implicit Treatment of Time Difference Source Term Point Implicit Treatment of Source Term for Multi-stage Schemes Point Implicit Treatment for Implicit Pseudo-Time Iterations Matrix Representation of ADI-BDF viii

9 6 Boundary Conditions Far field Boundary Conditions Internal Boundary Conditions C-mesh Wake Implicit Boundary Implementation Implicit Periodic Boundary Condition for the O-mesh Solid Wall Boundary Conditions Inviscid Unsteady Wall Boundary Condition for Explicit Operators Inviscid Unsteady Wall Boundary Condition for Implicit Operators Viscous Flow Boundary Condition Convergence Acceleration Techniques Implicit Residual Averaging Multigrid Implementation of Multigrid Variable Local Time Stepping Determining Time Step Size Calculating the Pseudo-Time Step Size Accuracy Analysis 7 8. Accuracy of the Linearized Unfactored Equation Accuracy of the ADI-BDF Scheme Accuracy of the DDADI-BDF Scheme Stability Analysis Stability Analysis of Time Stepping Scheme General Form of Explicit and Implicit Stability Analysis Equivalence of the implicit and explicit schemes Advantages of 3-4- BDF over Trapezoidal Rule Choice of Model Equation Jameson s MultiStage Scheme ADI Equation ADI-BDF Equation Diagonally Dominant ADI Scheme DDADI with BDF ix

10 Unsteady Results 4. Description of Test Case Summary of Test Cases A Note on Numerical Experiments Notation Effects of Artificial Dissipation Overview of the Hybrid Scheme for Unsteady Flow Alternative Considerations Inviscid Unsteady Results Inviscid Unsteady AGARD 7 Test Case 3 CT Inviscid Unsteady AGARD 7 Test Case CT Viscous Unsteady Results Viscous Unsteady AGARD 7 Test Case 3 CT Viscous Unsteady AGARD 7 Test Case CT Viscous Unsteady NASA TP Conclusion 58. Future Work A Periodic Block Tridiagonal Matrix 6 B Periodic Block Pentadiagonal Matrix 65 C CUSP Scheme Derivation 67 C. E-CUSP Scheme C. H-CUSP Splitting D Two Dimensional Inviscid Euler Jacobians Derivation 73 D. Two Dimensional Inviscid Euler Jacobian in UFLO E Implicit Operator for Navier-Stokes Equations 76 E. Viscous D Jacobians Derivation E.. Derivation for Spatial Derivatives F Inviscid NACA CT Test Case 8 F. CT Inviscid O-mesh Grids x

11 G Inviscid NACA64A CT6 Test Case 85 G. CT6 Inviscid O-mesh Grids H Viscous NACA CT Test Case 9 H. CT Viscous C-mesh Grids I Viscous NACA64A CT6 Test Case 93 I. CT6 Viscous C-mesh Grids Bibliography 96 xi

12 List of Tables 9. Comparison of implicit and explicit schemes Test Cases Combination Schemes CFL for Different Time Step Sizes CFL for Different Time Step Sizes Meshes used for viscous unsteady calculations for the CT test case Meshes used for viscous unsteady calculations for the CT6 test case xii

13 List of Figures 3. Fluxes through a cell in the computational domain Perform matrix inversion along the strips across the wake Explicit unsteady Euler boundary condition explicit unsteady Euler boundary condition indices at a solid wall boundary Stability Region of 34 Backward Difference Formula (BDF) Stability Region of Trapezoidal Rule Stage RK scheme Stage RK scheme with Point Implicit Treatment of Time Difference Stage RK scheme with Separate Evaluation of Convection and Dissipation Terms Stage RK scheme with Point Implicit Treatment of Time Difference, Separate Evaluation of Convection and Dissipation Terms Stability region of RK scheme for steady state case, λ t = RK scheme for steady state case, contour plot of amplification vs. modified frequencies Stability region of RK scheme without point implicit treatment, λ t = Stability region of RK scheme with point implicit treatment, λ t = Stability region of RK scheme with point implicit treatment and tweaked stage coefficients. λ t = RK scheme without point implicit treatment at different ω t, for λ t = RK scheme with point implicit treatment at different ω t, for λ t = Amplification factor of RK scheme, point implicit treatment and tweaked stage coefficients. λ t = xiii

14 9.5 λ x = λ y = 4, λ t =, unsmoothed residuals λ x = λ y = 4, λ t =, smoothed residuals (after ADI), ɛ x = ɛ y = λ x = λ y = 4, λ t =, unsmoothed residuals λ x = λ y = 4, λ t =, smoothed residuals (after ADI), ɛ x = ɛ y = λ x = λ y = 4, λ t =, unsmoothed residuals λ x = λ y = 4, λ t =, smoothed residuals (after ADI-BDF step), ɛ x = ɛ y = λ x = λ y =, λ t =, unsmoothed residuals λ x = λ y =, λ t =, smoothed residuals (after ADI-BDF step), ɛ x = ɛ y = λ x = λ y = 4, λ t =, unsmoothed residuals λ x = λ y = 4, λ t =, smoothed residuals (after ADI-BDF step), ɛ x = ɛ y = λ x = λ y = 4, λ t =, unsmoothed residuals λ x = λ y = 4, λ t =, smoothed residuals (after ADI-BDF step), ɛ x = ɛ y = λ x = λ y = 4, λ t =, unsmoothed residuals λ x = λ y = 4, λ t =, smoothed residuals (after ADI-BDF step), ɛ x = ɛ y = λ x = λ y = 4, λ t =, unsmoothed residuals λ x = λ y = 4, λ t =, smoothed residuals (after ADI-BDF step), ɛ x = ɛ y = λ x = λ y = 4, λ t =, unsmoothed residuals λ x = λ y = 4, λ t =, smoothed residuals (after ADI-BDF step), ɛ x = ɛ y = λ x = λ y = 4, λ t =, unsmoothed residuals λ x = λ y = 4, λ t =, smoothed residuals (after ADI-BDF step), ɛ x = ɛ y = λ x = λ y = 4, λ t =, unsmoothed residuals λ x = λ y = 4, λ t =, smoothed residuals (after ADI-BDF step), ɛ x = ɛ y = λ x = λ y = 4, λ t =, unsmoothed residuals xiv

15 9.38 λ x = λ y = 4, λ t =, smoothed residuals (after ADI-BDF step), ɛ x = ɛ y = λ x = λ y = 4, λ t =, unsmoothed residuals λ x = λ y = 4, λ t =, smoothed residuals (after ADI-BDF step), ɛ x = ɛ y = CFL distribution on the airfoil surface for P ST EP = C P distribution on the airfoil surface with CFL at T.E. of about 6, C L versus α for P ST EP = 36 and NCY C = C M versus α for P ST EP = 36 and NCY C = C D versus α for P ST EP = 36 and NCY C = C L versus time step for P ST EP = 36 and NCY C = C L versus oscillation period for fixed α with P ST EP = 36 and NCY C = 4.8 C L versus oscillation period for fixed α with P ST EP = 36 and NCY C = 4.9 C D versus oscillation period for fixed α with P ST EP = 36 and NCY C = 4.C M versus oscillation period for fixed α with P ST EP = 36 and NCY C = 4.C L versus oscillation period for fixed α with P ST EP = 36 and NCY C = 5.C L versus oscillation period for fixed α with P ST EP = 36 and NCY C = 5.3C D versus oscillation period for fixed α with P ST EP = 36 and NCY C = 5.4C L versus oscillation period for fixed α with P ST EP = 36 and NCY C = 5.5convergence history during an inner iteration step NCY C = C L history at each time step C D history at each time step C M history at each time step C D history of hybrid scheme at each time step C M history of hybrid scheme at each time step C D history of DTSS at each time step C M history of DTSS at each time step Averaged absolute C L error through one pitching cycle for hybrid and DTSS multigrid scheme Averaged absolute C M error through one pitching cycle for hybrid and DTSS multigrid scheme RMS of pressure error in entire flow field for ADI-BDF without inner iterations. xv

16 .6C L percentage error in entire flow field for ADI-BDF without inner iterations..7rms of pressure error in entire flow field for hybrid and DTSS multigrid scheme c L percentage error through one pitching cycle for hybrid and DTSS multigrid scheme rms of pressure error in entire flow field for ADI-BDF and DTSS multigrid scheme c L percentage error through one pitching cycle for ADI-BDF and DTSS multigrid scheme comparison of C L versus α of Different grid sizes for P ST EP = 36 and NCY C = Comparison of C L versus α of Different grid sizes for P ST EP = 36 and NCY C = Comparison of C L versus α of Different grid sizes for P ST EP = 36 and NCY C = Comparison of C L versus α of Different grid sizes for P ST EP = 36 and NCY C = Comparison of C L error of Different P ST EP for the Inviscid CT: (a) P ST EP = 9, (b) P ST EP = 8, (c) P ST EP = 36 and (d) P ST EP = Comparison of C L error of Different NCY C for the Inviscid CT: (a) NCY C =, (b) NCY C =, (c) NCY C = 3 and (d) NCY C = CFL distribution over NACA64A airfoil with leading edge at x = O-mesh Comparison of C P Distributions of Different Numerical Schemes for the Inviscid Case for P ST EP = Comparison of C P Distributions of Different Numerical Schemes for the Inviscid Case for P ST EP = C L versus α of the Hybrid Scheme C L versus time of the Hybrid Scheme C D versus α of the Hybrid Scheme C M versus α of the Hybrid Scheme C L versus oscillation period for fixed α with P ST EP = 36 and NCY C = 3.46C L versus oscillation period for fixed α with P ST EP = 36 and NCY C = 3 xvi

17 .47C D versus oscillation period for fixed α with P ST EP = 36 and NCY C = 3.48C M versus oscillation period for fixed α with P ST EP = 36 and NCY C = 3.49C L versus oscillation period for fixed α with P ST EP = 36 and NCY C = 33.5C L versus oscillation period for fixed α with P ST EP = 36 and NCY C = 33.5C D versus oscillation period for fixed α with P ST EP = 36 and NCY C = 33.5C L versus oscillation period for fixed α with P ST EP = 36 and NCY C = 33.53Averaged C L Error vs. NCY C for P ST EP = Averaged C L Error vs. NCY C for P ST EP = Averaged C L Error vs. NCY C for P ST EP = Averaged C L Error vs. NCY C for P ST EP = Comparison of C L versus α of Different Numerical Schemes for the Inviscid Case Comparison of C L versus α of the Hybrid Scheme with Different Number of Inner Iterations for the Inviscid Case Pressure Distribution on the NACA64A Airfoil at T = 576, α =. and M = Pressure Distribution on the NACA64A Airfoil at T = 69, α =.883 and M = Pressure Distribution on the NACA64A Airfoil at T = 63, α =.883 and M = Pressure Distribution on the NACA64A Airfoil at T = 64, α =. and M = Comparison of C L versus α of the Hybrid Schemes for Different Grid Sizes Comparison of C M versus α of the Hybrid Scheme with Different Grid Sizes 38.65Comparison of C L versus α of Different Numerical Schemes for the Inviscid Case Comparison of C L versus α of the Hybrid Scheme with Different Number of Inner Iterations for the Inviscid Case Comparison of C L versus α of Different Numerical Schemes for the Inviscid Case Comparison of C L versus α of the Hybrid Scheme with Different Number of Inner Iterations for the Inviscid Case Averaged C L Error vs. NCY C for various P ST EP s xvii

18 .7Averaged Total Pressure Error vs. NCY C for various P ST EP s CFL distribution over airfoil surface for 9 64 c-mesh Comparison of C M versus α of the Hybrid Scheme with Different Grid Sizes 4.73C L versus α for Hybrid scheme C M versus α for Hybrid scheme C D versus α for Hybrid scheme RT RMS versus ST EP for Hybrid scheme C L history for fixed α C L history for fixed α C D history for fixed α C M history for fixed α C L history for fixed α C L history for fixed α C D history for fixed α C M history for fixed α C L Convergence history C M Convergence history C D Convergence history Density residual C L error for P ST EP = C L error for P ST EP = CFL distribution over airfoil surface for 9 64 c-mesh Comparison of C M versus α of the Hybrid Scheme with Different Grid Sizes 47.93C L versus α for hybrid scheme and DTSS C M versus α for hybrid scheme and DTSS C L history for fixed α C M history for fixed α C L history for fixed α C L history for fixed α C L Convergence history Density residual C L Convergence history C M Convergence history xviii

19 .3Pressure Distribution on the NACA64A Airfoil at T = 576, α =. and M =.796 for the Viscous Calculation with the Hybrid Scheme Pressure Distribution on the NACA64A Airfoil at T = 576, α =. and M =.796 for the Viscous Calculation Demonstration of Temporal Accuracy for the Viscous Case Pressure Distribution on the NACA64A Airfoil at T = 576, α =. and M = Pressure Distribution on the NACA64A Airfoil at T = 69, α =.883 and M = Pressure Distribution on the NACA64A Airfoil at T = 63, α =.883 and M = Pressure Distribution on the NACA64A Airfoil at T = 64, α =. and M = Comparison of C L versus α of Different Numerical Schemes for the Viscous Case Comparison of C L versus α of the Hybrid Scheme with Different Number of Inner Iterations for the Viscous Case Mach Contour of Dynamic Stall/Vortex Shedding Frame Mach Contour of Dynamic Stall/Vortex Shedding Frame Mach Contour of Dynamic Stall/Vortex Shedding Frame Mach Contour of Dynamic Stall/Vortex Shedding Frame Mach Contour of Dynamic Stall/Vortex Shedding Frame Mach Contour of Dynamic Stall/Vortex Shedding Frame Several Force Coefficients History E. two dimensional compact stencil for viscous calculation F. O-mesh Grid 6X3 Cells Inviscid Calculation F. O-mesh Grid 6X3 Cells Inviscid Calculation F.3 O-mesh Grid 6X3 Cells Inviscid Calculation G. O-mesh Grid 6X3 Cells Inviscid Calculation G. O-mesh Grid 6X3 Cells Inviscid Calculation G.3 O-mesh Grid 6X3 Cells Inviscid Calculation xix

20 G.4 CL of CT6 7 ADIBDF fv= G.5 CM of CT6 7 ADIBDF fv= G.6 CL of CT6 8 ADIBDF fv= G.7 CM of CT6 8 ADIBDF fv= G.8 CL of CT6 6 ADIBDF fv= G.9 CM of CT6 6 ADIBDF fv= G. CL of CT6 54 ADIBDF fv= G. CM of CT6 54 ADIBDF fv= H. C-mesh Grid 9 64 Cells Viscous Calculation H. C-mesh Grid Cells Viscous Calculation H.3 C-mesh Grid 5 96 Cells Viscous Calculation I. C-mesh Grid 9 64 Cells Viscous Calculation I. C-mesh Grid Cells Viscous Calculation I.3 C-mesh Grid 5 96 Cells Viscous Calculation xx

21 Chapter Introduction. Practical Considerations and Motivations Computational fluid dynamics (CFD) has been a field of intense research in the past twenty years. CFD can be defined as a computer simulation of the flow of any fluid or gas. With the advent of today s computer technologies in terms of speed and data storage capacity, not only have CFD increased its efficiency and accuracy, but the goal of CFD has expanded. In the beginning of the computational era, CFD was restricted to only simplified versions of the conservation laws, e.g. potential equations and small disturbance theory utilized CFD as a tool. Today, with the exponential growth of the computing power in computer processors and the desire to simulate more complicated phenomenon in fluid flows, CFD algorithms are continuously improving so as to achieve greater capabilities, efficiency and robustness. The author of this dissertation believes that presently, numerical algorithms for steady state Euler calculations have been developed near the crux of their maturity. Some of the possible short term goals for further improvements of CFD lie in the areas including: addition of the viscous terms and/or turbulence modeling, grid generation techniques, parallel computation techniques due to the advent of parallel computers and unsteady simulations. Despite the ever developing relationship between physics of fluid and computer data processing, there are a few essential breakdowns of fundamentals behind the operations of CFD that have remained applicable. First, a mathematically well-posed problem must be defined if the solution is expected to make any sense. Second, a grid must be generated along with the proper boundary conditions defined. Third, a discrete formula constructed from the laws of physics must be applied to each data point. Lastly, the overall flow solution is

22 CHAPTER. INTRODUCTION obtained asymptotically by repeatedly running some preferred numerical algorithm -making educated guesses of the flow field at each grid point based on the discretized equations. All of these areas have been subjects of intense research in the past two and a half decades as they are interrelated and critical to the performance of CFD as a design tool for industrial applications. Depending on the ability of the numerical algorithm, CFD researchers strive to reduce CPU hour requirement for obtaining a reasonable solution of the flow field. During the 98 s, the introduction and efficient implementation of concepts such as Total Variation Diminishing (TVD) Scheme, multigrid and local preconditioning, have resulted in major gains in efficiency over a wide range of numerical disciplines. For CFD in particular, the CPU requirement for steady state Euler calculations have been reduced significantly. For example, one can obtain the steady state solution about a two dimensional inviscid transonic airfoil in a matter of seconds on a personal computer. A three dimensional inviscid transonic flow solution around an entire wing body configuration can be obtained in a matter of minutes on the same processor. Unsteady Euler calculations have been able to take advantage of this improvement through techniques such as Dual Time Stepping. As a result, phenomenon such as flutter and dynamic stall prediction for transonic airfoils can be easily simulated on a personal computer. In the aeronautical engineering design process, dynamics of unsteady fluid behavior can lead to unforeseen instabilities in gas or fluid flows. The instability coupled with elastic properties of the structure can trigger undesirable or even catastrophic behaviors in the flight vehicle. The ability to predict and avoid unwanted events such as rotating stalls in jet engines, buffeting or wing flutter should all depend on the ability of the flow solver to simulate unsteady flows with sufficient resolution and accuracy. Unsteady flow simulation has been historically challenging to the computational communities. The consideration of time dependence adds an extra dimension to the already taxing problem of computational fluid simulations. With the rapid advent of algorithms and computer hardware, the two subjects must grow in congruency to best achieve and utilize the numerical tools available today. As this thesis sets out to investigate the algorithmic side of the topic concerning numerical simulations, a more efficient and robust unsteady flow solver is proposed and analyzed in the subsequent text.

23 .. BACKGROUND OF CFD 3. Background of CFD Computational fluid dynamics (CFD) has become a required field of study for all aeronautical engineers. CFD is a numerical analysis of fluid or gaseous flows. In order to avoid expensive and time consuming wind tunnel experiments, aeronautical engineers need efficient CFD algorithms to predict steady or unsteady flows about any aerodynamic body..3 Stability Analysis of Numerical Schemes One of the most important concepts in practical numerical analysis is the CFL number, first introduced by Courant, Frederichs and Lewy. It was shown that the physics of the governing equations can be overshadowed by the instabilities of a numerical algorithm. Not all consistent numerical algorithms are able to yield reasonable solutions that approximate the exact solution of the governing Ordinary Differential Equation (ODE) or Partial Differential Equation (PDE) with arbitrary degree of accuracy. First, the differential equations and the representative difference equations have zonal dependencies due to the fact the fluid properties in the flow field such as density, pressure, convection velocities and viscosity travel at finite speeds (characteristic speeds). The numerical method must be carefully constructed to reflect the true zones of dependence for each characteristic speed. The CFL condition states that the zone of dependence of the numerical scheme must contain the zone of dependence of the flow. Obviously, if this statement is not satisfied, no realistic flow solution can come of the numerical computation. Upwinding is done by taking all the information necessary for evolving the solution from upstream of the flow. This is in disregard of the fact that some of the flow properties such as pressure in subsonic flow propagates in all directions rather than simply in the direction of the fluid flow. For supersonic flows, using flow variables from upstream of the flow leads to the correct zone of dependence. For transonic and subsonic flows, the term upstream depends on the characteristic speed of the flow variable in question. Upwinding of all flow variables in transonic flow provides stable results but leads to over-smoothing of flow features. To prevent the problem of over-smoothing of the solution, partial upwinding can be used. Some of the methods of partial upwinding are flux difference/vector splitting and the addition of artificial dissipation. These approaches appear to be quite different at first, but they can be shown to be rather similar. Some successful numerical schemes for fluid/gas dynamics require active modification

24 4 CHAPTER. INTRODUCTION of the difference method depending on the local flow characteristics. Implicit flow solvers are typically accompanied by some kind of characteristic splitting of the flux Jacobians. By choosing the zone of dependence from the numerical scheme to match that of the governing equation, shocks and discontinuities are solved in the numerical solution. More recently, artificial dissipation were used to indirectly pick the correct zone of dependence. For example, Jameson s JST scheme[?] reduces the central differencing in space to upwinding near regions of strong flow gradients. Jameson s explicit scheme uses a compact stencil and requires low computer storage needs. Explicit schemes tend to be locally optimized and they disregard anything that is too far away from a meshing perspective. Implicit schemes, on the other hand, seek to take into consideration the values of the majority of the domain of computation before updating each individual computational cell. This approach results in dramatically increased computational cost for each update, but has the potential of taking larger steps in time rather than an explicit scheme due to higher stability margins. The most successful convergence acceleration technique during the present study is undoubtedly the multigrid method. Extending the numerical domain of an explicit scheme through the use of coarser grids results in phenomenal convergence improvements. The multigrid approach gives explicit schemes the advantage of increased time step size previously possible only to implicit schemes, while keeping the algorithm compact and efficient. Multigrid has also been applied to implicit schemes, and has been demonstrated to have similar improvements in performance of the implicit algorithms. In general, implicit schemes are better behaved than explicit schemes while implicit schemes consume more computational time and power. The advantages of implicit over explicit schemes for stiff problems are largely motivated by the ability of the scheme to handle larger CFL numbers; leading to reduced number of iterations. For most of the time, the fluid dynamicist would choose the computational tool based on the flow physics, and the computer hardware resources available. It would seem that a switched implicit/explicit scheme can try and exploit the advantages of both worlds. To do so, one must first understand the advantages and disadvantages of explicit and implicit schemes. To perform a study on the differences, a few sample explicit methods and implicit methods were considered. Numerical experiments were performed for the selected schemes. While the example schemes used in this study may not be deemed the most ideal representation of explicit or implicit numerical scheme for the study, an attempt is made to compare the fundamental differences between the leading categories of schemes.

25 .4. LEVELS OF APPROXIMATIONS FOR FLOW SIMULATIONS 5.4 Levels of Approximations for Flow Simulations When CFD is used as a design tool for calculating flows, the degree of accuracy desired varies depending on the specific application. The details of the flow are not always computed exactly. Below is a list of the common approximations applied to the flow equations:. Small Disturbance Theory,. Potential Flow Calculation, 3. Inviscid Euler s Equations, 4. Thin Layer Navier Stokes Equations (TLNS), 5. Reynolds Averaged Navier Stokes Equations (RANS), and 6. Direct Numerical Simulation (DNS). The scope of analysis in this work ranges from inviscid Euler s equations up to fully viscous Reynolds averaged Navier Stokes equations..5 Steady Versus Unsteady Fluid Flows Characteristics Fluid flows for the majority of aerodynamics engineering applications are unsteady due to the inherent instabilities of the flow. Turbulence with increasing Reynolds number, shear layer, vortex shedding, flow separation and shock-wave boundary layer interactions are some of the examples of the instabilities of the fluid flow. It is important to study unsteady flows since an unsteady solution can have drastically different characteristics when compared with its quasi-steady counterpart. The non-trivial gradients of the state variables with respect to time cannot be ignored because the transient solutions carry important information such as the hysteresis effects. The complexity behind unsteady flows can often be captured by a good numerical flow solver. Several papers on studies of wing flutter, for example, were presented with experimental validation of the numerical solvers proposed [6]. The accuracy of flow solvers is vital to obtaining realistic flow solutions. Typical A-stable and O( t ) accurate flow solvers are capable of resolving most of the unsteady effects. Due to limited computational power and due to non-ideal convergence rates of numerical algorithms, the accuracy of flow solution is

26 6 CHAPTER. INTRODUCTION often compromised to achieve reasonable computation times. Ideally, iterating the residual errors to machine zero at every time step of an unsteady calculation will be preferred if no time constraints were imposed. In reality however, in order to save computational costs, only enough iterations are performed to obtain an approximate ensemble average of the flow that captures the key components of physics. Methods of unsteady flow analysis also depend on the time scales of the unsteady flow structures and the amount of computational power allotted. Generally, in order to save CPU hours, if the influence of the unsteady portion of the flow is not significant, then an approximation technique is used to generalize the unsteady effect into steady flow. For example, in designing an airplane s cruise conditions, turbulence within the boundary layers is usually generalized by an ensemble averaging technique such as turbulence modeling. Turbulence modeling is used to avoid the large number of mesh points needed near regions of high gradients in the flow solution. Resolution of turbulent flow within the boundary layer is usually not necessary from the perspective of an engineer. A turbulence model that predicts the flow separation is adequate in most engineering applications. Although it is doubtful whether a universally valid turbulence model can be constructed [5], the turbulence modeling techniques may or may never reach a stage where a robust closure model can be developed. The alternative to turbulence modeling is to perform direct numerical simulations in order to resolve the details of the flow within the boundary layer. Given that the time scales of the flow structures being simulated is not unreasonably small, the unsteady time accurate flow solver should reproduce predominant characteristics of the flow field..6 Current Unsteady Flow Solvers Typical flow solvers for unsteady integration employ a fully implicit time stepping scheme in which the equations are solved by inner iterations. Some of the issues that need to be addressed in the development of an effective unsteady flow solver are:. consistency, stability, convergence criteria,. spatial and temporal discretization, 3. conservativeness of the schemes, 4. oscillation control: LED/ELED, ENO/WENO, TVD,

27 .7. STANFORD S ASCI PROJECT 7 5. implicit and explicit schemes, 6. convergence acceleration (residual averaging, multigrid, etc.), 7. parallel computing, and 8. turbulence modeling. Despite all the advances that have been made in increasing the efficiency of numerical algorithms for CFD, computation time required remains a major roadblock for making CFD a cost-efficient tool for aeronautical design..7 Stanford s ASCI Project The ultimate goal of the ASCI project is to fully resolve the unsteady flow inside a jet turbine engine using the most advanced CFD simulation tools available on a platform consisting of hundreds of thousands of CPUs working in parallel. A key objective is to test the envelope of CFD using the state of the art hardware architecture. Since CFD s best performance depends heavily on the type of unsteady flow solver used, it is necessary that a robust and efficient numerical algorithm be implemented so as to speed up the design process. The ASCI turbomachinery unsteady flow solver TFLO uses Jameson Dual-Time-Stepping- Scheme with Multistage Runge-Kutta (RK) time integration, implicit residual averaging, variable local time stepping and multigrid convergence acceleration techniques. The inner iterations advance to a steady state in fictitious time at every real time step. In the ASCI Project, an attempt has been made to compute the unsteady flow within an entire jet engine starting at the compressor through the combustion chamber and turbine, and then exiting the exhaust nozzle. To capture the inherent unsteadiness and the physical phenomenon that exist under such environments, from setting up the grid generation to converging the numerical results, the entire process takes much longer than desired. In one of the preliminary runs in the ASCI project, the unsteady flow through a complete turbine with 9 blade rows is computed using a mesh with approximately ninety-four million cells. Using the TFLO code, approximately twenty-five hundred time steps and two million CPU hours are required to reach a stationary periodic state. Using five hundred and twelve processors, the calculation requires approximately 8 months. With the present flow solver, there are no available error bound estimates for accuracy of the flow solution unless the

28 8 CHAPTER. INTRODUCTION inner iterations are fully converged. However, numerical experiments have demonstrated second order accuracy in time with approximately twenty-five iterations per real time step. The turn around time is too long, leaving design engineers the ability to run only a few design cycles in a year..8 Objectives Of This Dissertation This dissertation is motivated by the clear need to solve the time-accurate Navier-Stokes equations with greatly decreased computational cost compared to the state-of-the art solver used in the ASCI project. An attempt is made to facilitate the unsteady calculations while analyzing the differences between the various types of numerical schemes for the gas dynamics equations in two dimensions. The proposed new numerical method involves the introduction of an initial Alternating Direction Implicit (ADI) step, guaranteeing second order accuracy in time, followed by a small number of cycles of the dual time stepping algorithm for reduction of the factorization and linearization error. The second-order accuracy in time should be retained with arbitrary (and hopefully fewer than typically required for convergence of iterative methods) numbers of inner iterations following the initial ADI step. Details of this new scheme will be presented along with examples that demonstrate the second order accuracy and the convergence properties. Properties such as consistency, stability, conservativeness and convergence are also considered..9 Preliminary Considerations The large time requirements for fully unsteady simulations in turbomachinery signify the need to improve the efficiency of the numerical algorithms. Some approaches to be considered are:. To search for a more rapidly convergent inner iteration method; for this purpose, a Preconditioned Symmetric Gauss-Seidel Relaxation Method is being studied [9,53,8],. To reformulate the scheme so that it yields sufficient accuracy without the need for full convergence of the inner iterations, 3. To consider the alternative approach of representing the solution in the frequency domain; studies of a nonlinear frequency domain approach are described by McMullen

29 .. CFD CODE DESIGN ASPECTS 9 et al [7], and 4. To increase the parallel efficiency of numerical algorithms across larger number of interconnected memories and CPUs. In this dissertation, the second approach is followed. The idea is to formulate a hybrid scheme which introduces an initial linearized ADI step. When compared with commonly used iterative schemes, the hybrid scheme does not need to iterate to full convergence at each real time step to ensure nd order accuracy. However, if the ADI scheme is used alone, the factorization error restricts the time step size for which sufficient stability and accuracy can be attained. In addition, the three point backward time difference formula is shown to be unconditionally unstable[84] without additional numerical dissipation. In order to overcome these deficiencies in ADI-type schemes, a number of dual-time-stepping iterations are added. The additional iteration reduces the errors due to both linearization and factorization and thus increases the upper bound of the feasible time step of the ADI scheme.. CFD Code Design Aspects In general, the formulation of numerical algorithms includes a few main areas of study:. Derivation of the governing equations; this is generally adapted from one of the levels of approximation from the previous section,. Construction of the grid/mesh; the present analysis uses O-meshes generated from Karmen-Treftz transformation with stretched grid in the normal direction for inviscid calculations. Hyperbolic mesh generators were used to create C-mesh for viscous calculations, 3. Numerical discretization of the governing equations; the main efforts in writing a new solver is consumed by this step. 4. Application of linear algebra; system of linearized difference equations can be solved using conjugate gradient solvers. In the present approach, approximated factorization is used to reduce the bandwidth of the implicit operator, resulting in inversion of block tridiagonal and pentadiagonal systems,

30 CHAPTER. INTRODUCTION 5. Parallelization strategies for multiple processor computer architectures; not examined in the present analysis, and 6. Software programming issues; briefly discussed the implementation issues.. CFD Programming Style and Efficiency Considerations In the era of parallel programming and the ever changing architecture of computer hardware, it is hard to decide what kind of software will last through more versions of compilers and CPUs. Although the CFD software largely depends on the core technique implemented in the flow solver, the programming methodology also plays a large role in the overall performance and life expectancy of the code. While different factors to be considered drive the software in different directions, the optimal balance of the main design issues are not trivial. Sometimes, individual subroutines, if designed simply and precisely, may outlast the main code. There are several predominant issues for developing software, including data structure of the solver for maximum operations speed and efficiency, clarity in arrangement of the data structure, modularity of the subroutines, and platform dependence of the computer code. Arrangement of the data structure is very important in terms of the resulting FLOPS. Typical large arrays of the CFD code must be sequenced so that simultaneously accessed elements are located closely in the hardware memory address for shortest data retrieval time during the calculations. Examining the access of data by the CFD solver on average reveals that in most cases, different state and geometric variables at adjacent nodes are accessed by finite differencing operators simultaneously; thus, we would like to have the data grouped in the hardware memory correspondingly. Access frequency of each set of variables need to be considered in order to design the source code accordingly. The higher the frequency of simultaneous access, (i.e. accessed in one pass of a loop), the closer the data should reside in the memory. This usually leads to an ordering of the hierarchy of the flow variables:. different state and geometric variables,. adjacent mesh nodes, 3. multiple block elements, and 4. multigrid levels.

31 .. CFD PROGRAMMING STYLE AND EFFICIENCY CONSIDERATIONS Besides data storage, modularity of subroutines is also a key factor to consider in designing a CFD code. Subroutines have modular dependency, i.e. they only operate on certain variables and can be loaded when the corresponding variables are accessed. In a single processor case, the most heavily used subroutines such as flux calculations, time step calculations, prolongation and restriction, implicit Jacobian setup and implicit Jacobian inversions should remain static in memory for shortest execution times. In practice, modularizing subroutines are necessary only if the computer source code needs to be clearly organized for auditing by more than one programmer. The source code organization remains a lower priority than the speed and efficiency of the executable. The decision on whether intermediate variables should be calculated on the fly or computed in advance and stored in the computer memory is not so obvious. In order to understand the trade-off, the data size and hardware performance are some of the crucial factors to consider. When the data size of the transient variables is large, storing the value needed may increase the range of memory access and hence, dramatically deteriorate the computational efficiency as the CPU must reload different memory pages into the cache repeatedly. On the other hand, calculating the transient variables on the fly will require a few extra FLOPS per element, resulting in increased calculation time by a factor of five to fifty percent depending on the algorithm. If the system cache is large enough, then it is desirable to store the needed variables. Otherwise, it is better to calculate them as needed.