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Energy, Vorticity and Enstrophy Conserving Mimetic Spectral Method for the Euler Equation

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Energy, Vorticity and Enstrophy Conserving Mimetic Spectral Method for the Euler Equation Master of Science Thesis Energy, vorticity and enstrophy conserving mimetic spectral method for the Euler equation D.J.D. de Ruijter, BSc 5 September 2013 Faculty of Aerospace Engineering · Delft University of Technology Energy, vorticity and enstrophy conserving mimetic spectral method for the Euler equation Master of Science Thesis For obtaining the degree of Master of Science in Aerospace Engineering at Delft University of Technology D.J.D. de Ruijter, BSc 5 September 2013 Faculty of Aerospace Engineering · Delft University of Technology Copyright ⃝c D.J.D. de Ruijter, BSc All rights reserved. Delft University Of Technology Department Of Aerodynamics, Wind Energy, Flight Performance & Propulsion The undersigned hereby certify that they have read and recommend to the Faculty of Aerospace Engineering for acceptance a thesis entitled \Energy, vorticity and enstro- phy conserving mimetic spectral method for the Euler equation" by D.J.D. de Ruijter, BSc in partial fulfillment of the requirements for the degree of Master of Science. Dated: 5 September 2013 Head of department: prof. dr. F. Scarano Supervisor: dr. ir. M.I. Gerritsma Reader: dr. ir. A.H. van Zuijlen Reader: P.J. Pinto Rebelo, MSc Summary The behaviour of an inviscid, constant density fluid on which no body forces act, may be modelled by the two-dimensional incompressible Euler equations, a non-linear system of partial differential equations. If a fluid whose behaviour is described by these equations, is confined to a space where no fluid flows in or out, the kinetic energy, vorticity integral and enstrophy integral within that space remain constant in time. Solving the Euler equations accompanied by appropriate boundary and initial conditions may be done analytically, but more often than not, no analytical solution is available. The solution, however, can always be approximated and usually a computer is used to provide a numerical approximation. However, so far no numerical technique exists that yields a solution which actually keeps both the kinetic energy, the vorticity integral and the enstrophy integral constant. This means that for long-time simulations, the numerical approximation of the velocity or vorticity field will not at all accurately represent the solution to the problem at hand. In this thesis we develop a numerical method that approximates the two-dimensional incompressible Euler equations such that all of those invariants are conserved within a fixed space where no fluid flows in or out. We first rewrite the Euler equations and the corresponding vorticity equation in terms of differential geometry and then discretize the resulting equations using a structure preserving spectral method on a square domain. The numerical tests performed show conservation of kinetic energy, vorticity integral and enstrophy integral. Deviations from their initial values, which are of the order one, were always less than 1:2 · 10−10 and may be attributed to the accuracy of the implementation of the method instead of the method itself. Future work may consist of applying the method to a benchmark flow problem like the Taylor-Green vortex, creating a spectral element method based on the described spectral method and adding the viscous terms from the Navier-Stokes equations to our method. v vi Summary Acknowledgements Creating a thesis takes a lot of individual effort but it cannot be done without the help and support of others. It took my supervisor Marc little effort to intrigue me and convince me of graduating on such an interesting topic, the field of mimetic discretization techniques. However, after that he spent many hours in helping me getting to grips with the topic, discussing about the next steps to persue, guiding me in the right direction and answering many questions that he usually answered with striking ease, for which I am very grateful. Many thanks go out to Andrea, Artur, Blair, Dmitry, Guido, Jasper, Jouke, Mick, Pedro, Peter, Ren´eand Valentin, who all did or are doing research in mimetic discretizations. Every meeting that we had was instructive and the many talks we had enhanced my understanding of the topic. My gratitude goes to Artur, Deepesh, Jasper and Pedro for saving me many hours of programming. Jeanette and Ruud have shown an incredible amount of support during the creation of this thesis. More than ever I realise I really could not have wished for better parents. Finally I cannot thank my love Ann enough for her truly unbelievable support and for never losing faith. Delft, The Netherlands D.J.D. de Ruijter, BSc 5 September 2013 vii viii Acknowledgements Contents Summary v Acknowledgements vii 1 Introduction 1 2 The Euler equations 3 3 Differential geometry 9 3.1 Vectors ..................................... 9 3.2 Differential forms ................................ 10 3.3 Exterior derivative ............................... 12 3.4 Interior product ................................. 13 3.5 Lie derivative .................................. 14 3.6 Lie bracket ................................... 15 3.7 Hodge star operator .............................. 16 3.8 Stokes' theorem ................................. 16 3.9 Global scalar products ............................. 16 3.10 Rewriting the Euler equations ......................... 17 4 Numerical modelling 19 4.1 Basis functions and basis vector fields .................... 21 4.2 Commutator of two basis vector fields .................... 23 4.3 Approximating the momentum equation ................... 24 4.4 Kinetic energy conserving numerical time integration ............ 26 4.5 Conservation of vorticity and enstrophy ................... 29 4.5.1 Conservation of enstrophy ....................... 32 4.5.2 Conservation of vorticity ........................ 34 ix x Contents 5 Results 39 6 Conclusions and recommendations 43 6.1 Conclusions ................................... 43 6.2 Recommendations ............................... 44 References 45 A Gauss-Legendre quadrature rule 47 B Boundary conditions 51 C Edge function 57 D Large equations 59 Chapter 1 Introduction In many branches of physics, partial differential equations are used to model nature's behaviour, and the branch of aerodynamics is no exception. Solving a set of partial differential equations accompanied by appropriate boundary or initial conditions may be done analytically, but more often than not, no analytical solution is available. The solution, however, can always be approximated and usually a computer is used to provide a numerical approximation. Many numerical techniques exist, such as the finite difference method, the finite volume method and the finite element method. The behaviour of an inviscid, constant density fluid on which no body forces act, may be modelled by the two-dimensional incompressible Euler equations. If a fluid whose behaviour is described by these equations, is confined to a space where no fluid flows in or out, the kinetic energy, vorticity integral and enstrophy integral within that space remain constant in time. However, so far no numerical technique exists that yields a solution which actually keeps all of those invariants constant. This means that for long- time simulations, the numerical approximation of the velocity or vorticity field will not at all accurately represent the solution to the problem at hand. Furthermore, Arakawa [1] shows that only if both kinetic energy and vorticity integral are conserved, there is a constraint on the change in spectral energy, namely that there is an average wave number that should remain constant. Perot [11] and Zhang et al. [12] have shown conservation of both kinetic energy and vorticity integral for unstructured staggered mesh methods using the rotational form of the Navier-Stokes equation, but the enstrophy integral is not conserved in their method. Elcott et al. [3] consider vorticity to be a 2-form and, using concepts from algebraic topology and differential geometry, develop a structure-preserving method that conserves the vorticity integral along any discrete loop in the mesh. Employing ideas from those branches of mathematics as well, Mullen et al. [9], with the purpose of fluid animation in computer graphics in mind, have developed a kinetic energy conserving method for the Euler equations. Pavlov et al. [10] have combined concepts from differential geom- etry and variational mechanics into a structure-preserving discretization method for the Euler equations. Their result shows good, but not exact, long-term energy conservation. 1 2 Introduction Based on this work, Gawlik et al. [5] have extended the application of their method to magnetohydrodynamics and complex fluids. The purpose of our research is to develop a numerical method that approximates the two- dimensional incompressible Euler equations such that kinetic energy, vorticity integral and enstrophy integral are conserved within a fixed space where no fluid flows in or out. We will do so by first rewriting the Euler equations and the corresponding vorticity equation in terms of differential geometry and then discretizing the resulting equations using a structure preserving spectral method on a square domain. Chapter 2 will introduce the Euler equations and the corresponding vorticity equation. We will at the continuous level derive the conservation laws that we want to satisfy discretely and we will mention how differential geometry and algebraic topology relate to our problem. A short introduction to differential geometry is then given in chapter 3, which concludes with a restatement of the Euler equations and the vorticity equation in terms
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