INTERNSHIP REPORT High order regularization of Dirac-delta sources in two space dimensions. Author: Supervisors: Wouter DE VRIES Prof. Guustaaf JACOBS s1010239 Jean-Piero SUAREZ HOST INSTITUTION: San Diego State University, San Diego (CA), USA Department of Aerospace Engineering and Engineering Mechanics Computational Fluid Dynamics laboratory Prof. G.B. JACOBS HOME INSTITUTION University of Twente, Enschede, Netherlands Faculty of Engineering Technology Group of Engineering Fluid Dynamics Prof. H.W.M. HOEIJMAKERS March 4, 2015 Abstract In this report the regularization of singular sources appearing in hyperbolic conservation laws is studied. Dirac-delta sources represent small particles which appear as source-terms in for example Particle-laden flow. The Dirac-delta sources are regularized with a high order accurate polynomial based regularization technique. Instead of regularizing a single isolated particle, the aim is to employ the regularization technique for a cloud of particles distributed on a non- uniform grid. By assuming that the resulting force exerted by a cloud of particles can be represented by a continuous function, the influence of the source term can be approximated by the convolution of the continuous function and the regularized Dirac-delta function. It is shown that in one dimension the distribution of the singular sources, represented by the convolution of the contin- uous function and Dirac-delta function, can be approximated with high accuracy using the regularization technique. The method is extended to two dimensions and high order approximations of the source-term are obtained as well. The resulting approximation of the singular sources are interpolated using a polynomial interpolation in order to regain a continuous distribution representing the force exerted by the cloud of particles. Finally the high-order approximation of the singular sources are used to solve the two-dimensional singular advec- tion equation. A Chebyshev spectral collocation method and a third order Total Variation Diminishing Runge-Kutta method are employed to approximate the spatial and temporal derivative. For a single isolated moving source and a stationary cloud of particles high-order accuracy is obtained outside the regularized zones. Inside the regularization zones the results are less satisfying and deserve more attention. The validation of these results and additional experi- ments to study the accuracy of the results, remain as future work. Keywords: Hyperbolic conservation laws, two-dimensional advection equation, regularization, Dirac-delta, singu- lar sources, Chebyshev spectral collocation. 1 Contents 1 Introduction 5 1.1 Background . 5 1.2 Objective . 7 2 Regularization, discretization and interpolation methods 9 2.1 Properties of Dirac-delta and Heaviside function . 9 2.2 Regularization of the Dirac-delta function . 10 2.3 Quadrature methods . 12 2.4 Polynomial Interpolation . 14 2.5 A Chebyshev spectral collocation method . 15 2.6 Time Integration . 17 2.7 Accuracy . 17 3 Regularization of singular sources in 1D 19 3.1 One-dimensional advection equation with singular sources . 19 3.2 Approximation of the singular sources in one dimension . 19 3.3 Conclusion on the one-dimensional results . 24 4 Regularization of singular sources in 2D 25 4.1 Two-dimensional advection equation with singular sources . 25 4.2 Approximation of the singular sources in two dimensions . 27 4.3 Interpolation of the singular sources . 33 4.4 Conclusions on the two-dimensional results . 35 5 Numerical results for the two-dimensional singular advection equation 36 5.1 The two-dimensional advection equation with one single isolated moving particle . 36 5.2 The 2D advection equation with a cloud of moving particles . 38 5.3 Conclusions . 40 6 Conclusions and future work 41 A Convergence order of quadrature rules 43 B Accuracy of the Chebyshev spectral collocation method 46 2 List of Figures 2.1 Two examples of a regularized Dirac-dela function . 11 2.2 The tensor-product of the regularized Dirac-delta functions . 12 3.1 Numerical approximation (top) and pointwise error (bottom) for case 3.1. 21 3.2 Numerical approximation (top) and pointwise error (bottom) for case 3.2. 22 3.3 Numerical approximation (top) and pointwise error (bottom) for case 3.3. 23 3.4 Numerical approximation (top) and pointwise error (bottom) for case 3.4. 24 4.1 Square grid with uniform nodes (left) and square grid with CGL-nodes (right). 26 4.2 Numerical approximation (left) and pointwise error on the square domain (right) for case 4.1. 28 4.3 Numerical approximation (left) and pointwise error on the interior domain (right) for case 4.1. 29 4.4 Numerical approximation (left) and pointwise error on the square domain (right) for case 4.2. 29 4.5 Numerical approximation (left) and pointwise error on the interior domain (right) for case 4.2. 30 4.6 Numerical approximation (left) and pointwise error on the square domain (right) for case 4.3. 30 4.7 Numerical approximation (left) and pointwise error on the interior domain (right) for case 4.3. 31 4.8 Numerical approximation (left) and pointwise error on the square domain (right) for case 4.4. 31 4.9 Numerical approximation (left) and pointwise error on the interior domain (right) for case 4.4. 32 4.10 Cross sections of the numerical approximation for case 4.4 . 33 4.11 One-dimensional polynomial interpolation of the cross sections . 34 4.12 One-dimensional polynomial interpolation of the diagonal . 34 4.13 Numerical approximation (left) and full 2D polynomial interpolation (right) for case 4.4. 35 5.1 Numerical approximation (top-left), regularized analytical solution (top-right), pointwise error (bottom- left), heatmap of the error (bottom-right) for case 5.1. 38 5.2 Numerical solution course grid (top left), numerical solution on fine grid (top right), pointwise error (bottom left), heatmap of the error (bottom right) for case 5.2. 40 B.1 Two-dimensional homogeneous advection equation . 47 B.2 Two-dimensional weigthed L2-norm as function of N .......................... 47 3 List of Tables 3.1 parameters for cases 3.1-3.4 . 20 4.1 Parameters for cases 4.1-4.4 . 27 4.2 L2-norm on the interior domain for cases 4.1-4.4. 28 5.1 parameters for case 5.1 . 37 5.2 L2-norms for case 5.1 . 38 5.3 parameters for case 5.2 . 39 5.4 L2-norms for case 5.2 . 40 A.1 Order of convergence for the one-dimensional Simpson’s rule . 44 A.2 Order of convergence for the two-dimensional Simpson’s rule . 44 A.3 Order of convergence for the two-dimensional Newton-Cotes 7-point rule . 45 4 Chapter 1 Introduction In this report the regularization of singular sources in hyperbolic conservation laws in two spatial dimensions is con- sidered. A high order regularization technique based on compactly-supported piecewise polynomials (presented in [1]) is used to regularize singular Dirac-delta sources. The present study is in continuation of the research discussed in [1], where the regularization of a single isolated particle is discussed. Instead of a single isolated particle, the aim is to regularize a dense, non-uniformly distributed cloud of particles for the simulation of particle-laden flow. The main objective is to improve the accuracy of a high-order and high-resolution particle-source-in-cell method in the simulation of particle-laden flows with shocks [3]. The aim of this chapter is to give an introduction to the problem at hand and discuss the need of a regularization method in order to approximate singular sources in particle-laden flow. Before diving into detail, a brief overview of the background of the research and a motivation for the present research is discussed. 1.1 Background In Engineering problems involving high-speed (compressible) gas flows, such as the flow over a supersonic aircraft and flows encountered in combustion problems, the flow is desribed by the non-linear hyperbolic conservation law: @ Q~ (~x;t) + r · F~ (Q~ ) = S~(Q;~ ~x;t) (1.1) @t Where Q~ are the conserved quantities and F~ is the flux function and S~ respresents the source terms. The hyperbolic nature of these equations allow for discontinuous solutions in the form of shock waves. Since hyperbolic conservation laws govern so many evolution problems in various type of scientific fields, the numerical evaluation of these problems has gained a lot of attention past few decades. The focus has been on the accurate capturing of shock waves. Several methods have been developed in order to capture non-linear shock solutions in numerical approximations accurately [16]. The non-homogenous case with (singular) sources-terms however, have been much less investigated. Particle-laden flow The source terms in equation 1.1 play an important role in engineering problems involving high-speed gas flow inter- acting with liquid or solid particles such as combustion. The flow model describing such a flow is Particle-laden flow. Particle-laden flow is a class of two-phase fluid flow where one of the phases represents the carier gas and the other phase typically represents one or more dilute particles. The carier gas flow is governed by Navier-Stokes equations, which reduce to the Euler equations in case viscosity and heat conduction are neglected (neglecting viscosity and heat-conduction is generally a reasonable assumption in compressible gas dynamics problems). The two-dimensional Euler equations (in Cartesian coordinates) are given by: @ @ @ Q~ (x; y; t) + F~ (Q~ ) + G~ (Q~ ) = S~(Q;~ x; y; t) (1.2) @t @x @y 5 where: Q~ = [ρ, ρu, ρv; ρE] F~ (Q~ ) = [ρu, ρu2 + P; ρuv; (E + P )u] G~ (Q~ ) = [ρv; ρuv; ρv2 + P; (E + P )u] ~ S~ = [0; ρfx; ρfy; ρf · ~u + Qv] And: 1 P = (γ − 1)E − ρu2 (1.3) 2 Clearly equation 1.2 is a hyperbolic conservation law of the same form as equation 1.1.