Analysis, extensions and applications of the Finite-Volume Particle Method (FVPM) PN-II-RU-TE Synthesis of the technical report -

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1 Analysis, extensions and applications of the Finite-Volume Particle Method (FVPM) PN-II-RU-TE Synthesis of the technical report - Phase 1: Preparation phase Authors: Delia Teleaga, Eliza Munteanu Project Coordinator: Delia Teleaga December 2011

2 Technical report for octombrie decembrie 2011 (preparation phase) Our intended research is concerned with a relatively new meshless method initially designed for solving hyperbolic systems of conservation laws, called the Finite-Volume particle method (FVPM). The motivation to develop this new scheme was to unify the advantages of Finite-Volume methods and particle methods in one scheme by combining their generic features. Thus the FVPM uses both the concept of a numerical flux function and the flow description using moving particles. The method was firstly developed in 1998 in (Hietel, Steiner and Struckmeier 2000) for a system of conservation laws in the spatial domain Ω =. The basic FVPM scheme was further investigated and extended by various authors. The PI was one of the firsts who investigated this method in (Teleaga 2000). In (Teleaga 2005, 2008)) the FVPM has been extended to incorporate moving boundaries in inviscid compressible flow. Furthermore, a projection technique for incompressible flow (Keck 2002) and second-order accuracy and viscous flow (Nestor, et al. 2009) were proposed. The aim of this project is to build a team of researchers which will enhance the work done by the PI concerning the Finite-Volume Particle Method Method (Teleaga 2000, Teleaga 2005). Status of the project The first phase of the project was a preparatory one, aiming for the team members to get ready for working with and extending the existing FVPM method. The phase 1 consisted in the following activities: 1. Familiarization of the team members with the FVPM method; FVPM method applied to compressible and inviscid flows was studied; a short presentation of the method is given in Annex 1; 2. In-depth study of the recent publications in the field, with focus on utilizations of FVPM; 3. Analysis of the existent FVPM, implemented in C / Linux, as well as of the way the memory allocation is done, with the purpose of identifying necessary modifications to be made in order to apply it to large dimension problems, as for the case of using a large number of particles;

3 4. Market study for selecting hardware configurations that are capable to round large dimension problems; acquisition of the necessary equipements; installing the adequate software programmes (compiler, libraries etc.); 5. Familiarisation of the team members with the C code; 6. Comparative analysis with other CFD programmes / software; 7. Identifying aspects that need to be studied in-depth during the following 6 months for the development and application of the FVPM to solve compressible flows (whether viscous or inviscid): - Computation of geometrical coefficients, - Correction procedures for, - Viscosity modelling in FVPM, - Choice of benchmark problems, for compressible and viscous flows, in order to be solved by FVPM; 8. Determining the detailed tasks to be implemented for the following 3 months and distributing them to the team members.

4 Short presentation of FVPM method applied to compressible and inviscid flows The FVPM is a meshless numerical method for solving conservation laws written in the form with initial conditions (1) Annex 1 and with suitable boundary conditions, where Ω is a bounded domain in,, denotes the vector of conservative quantities, and denotes the flux function of the conservation law. A natural approach to discretize conservation laws is to evaluate the weak formulation of (1) with a discrete set of test functions In classical Finite-Volume methods, the test functions are taken as the characteristic functions of the non-overlapping control volumes. In FVPM, a different set of test functions is introduced. The conservative variables are approximated at each time step by a finite set of particles located in the spatial domain Ω, with particle positions irregularly spaced and moving. To each position function of the form which may be a compactly supported, overlapping test is associated, where is a kernel function with compact support 2h, centred at. h > 0 is called the smoothing length. In (teleagaphd) a quadratic kernel is used. An important propriety of the test functions is that they form a partition of unity: Each particle is associated with a time-dependent volume. and a discrete local average through the relation

5 In (teleagaphd), starting from the weak form of (1). the following semi-discrete form of the FVPM for inviscid flow is derived where are called geometrical coefficients since they carry geometrical information about the particles and their relative position, and is the boundary term. The discretization of is explained in (teleagaphd) for the general case of a moving boundary. The flux is approximated in terms of the discrete values with the help of a numerical flux function for the modified flux function, which consists of the flux of the given conservation law, as well as a contribution due to the particle movement with velocity. Thus the semi-discrete FVPM for inviscid flow is defined by (2). If the transient term in (2) is discretized using, for example, an explicit Euler approach, then a first-order scheme in time is obtained: First-order spatial accuracy is obtained if the numerical flux function is computed on the basis of a zero order extrapolation of the discrete particle values to the particle interfaces. The extension to

6 higher-order accuracy is one of the objectives of the proposed research and this should be achieved by higher-order extrapolation of the variables to the particle interfaces. The main objective of the proposed research is to develop the meshless FVPM scheme such that to simulate turbulent flows on time-dependent geometries, which is, by the best knowledge of the PI, a pioneering effort in the development of the meshless methods. In industrial and scientific applications, incompressible viscous flows are modeled by the well-known Navier-Stokes equations. Thus, the first step towards achieving the proposed objectives will be the formulation of the numerical scheme for solving Navier-Stokes equations. Our approach will use the Chorin s projection method. Project coordinator, Dr. Delia TELEAGA