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42 Chapter 3: Development of a numerical tool for combustion noise analysis, AVSP-f 3.1 Discretizing the Phillips’ equation When neglecting convection, the Phillips’ wave operator reads ( ∂ ¯c 2 ∂ ) = ∇ · ¯c 2 ∇ (3.1) ∂x i ∂x i This operator may be discretized by different strategies, e. g.: finite differences, finite elements or finite volumes. The method chosen is the finite volume method. There is one main reason for doing so: three-dimensional and complex geometries are aimed. It means that it is indispensable to consider unstructured meshes, which are clearly much easier accounted in the finite volume discretization technique rather than finite differences. Also, the fact that finite volume methods have been conceived under the conservative philosophy, conservation equations are satisfactorily fulfilled as long as the method is well applied in each of the control volumes. Furthermore, finite volume methods are both less expensive and much easier to understand and to program than finite elements. Each step within the finite volume algorithm has a physical meaning which is clearly a great advantage. There is however one main drawback when using finite volumes: it is difficult to develop higher order schemes than those of second order when dealing with three-dimensional geometries. The reason is that finite volumes have to be well optimized in each of their steps (numerical interpolation, differentiation and integration) which is clearly a not easy task. Within the finite volume method several strategies arise. There are two common formulations: the cell-centered and cell-vertex method. In the first, the discrete values of the conserved variables are stored at the center of the control volume (the grid cell) and values at the nodes are obtained by averaging between the values of the neighboring cells. In the cell-vertex formulation the discrete conservative quantities are stored at vertices of the control volumes (or grid nodes) and values of mean fluxes are obtained by averaging all along the faces of the grid cell. The cell-vertex formulation is the one used in this study and has been inherited by the numerical scheme used in the AVBP solver [3]. N i+li+1 n n e k N i n N i−li−1 Figure 3.1: A triangular element For simplicity let us consider a triangular cell although a similar analysis can be straightfor-
3.1 Discretizing the Phillips’ equation 43 wardly carried out for tetrahedra. The notation used here is shown in Fig. 3.1. The index e k stands for a cell element k that is composed by three nodes N i±l . Each face of this triangle is characterized by its normal vector n. The gradient operator for a quantity ˆp associated to a cell element e k is discretized following Green’s theorem. It reads ∇ ˆp ⏐ = 1 ∮ ˆp ek ndS (3.2) ek V ek ∂e k where ˆp ek and V ek stands respectively for the pressure and the volume associated to the cell e k . Let us remind that in the cell-vertex method all quantities are stored in the vertices of the cell. As a result, ˆp ek must be expressed as an interpolation function g of the neighboring nodes N o . Hence ˆp ek = g( ˆp o ) where o = ..., i − l i−2 , i − l i−1 , i, i + l i+1 , i + l i+2 , ... is a vector that contains the neighbor nodes for a given node i and l is a vector that contains the spatial distribution of matrix elements between the node i and its neighbors 1 . Equation (3.2) is then expressed as: ∇ ˆp ⏐ = 1 ∮ g( ˆp o )ndS = h ek ( ˆp o ) (3.3) ek V ek ∂e k The triangular cell of Fig. 3.1 makes part of a whole set of triangular elements as shown in Fig. 3.2. Following Green’s theorem, the divergence of ¯c 2 ∇ ˆp for the node N i is discretized as ( ∇ · ¯c 2 ∇ ˆp ) ⏐ ⏐⏐Ni = 1 ∮ ¯c 2 N V i ∇ ˆp Ni ⏐ · ndS (3.4) ∂N i ek+α ( ∇ · ¯c 2 ∇ ˆp ) ⏐ ⏐ ⏐⏐Ni = 1 V Ni ∮ ∂N i ¯c 2 N i h ek+α ( ˆp o ) · ndS (3.5) where α is a vector that contains the spatial distribution between the cell e k and its the neighbors. Note that a volume V Ni is associated to the node i and that this volume is enclosed by surfaces ∂V Ni 2 . This volume V Ni is shown in the shaded region of Fig. 3.2 and ∂N i is represented by the thick black line enclosing it. Let us consider now what happens when this node N i is placed in a boundary of the computational domain. Such a case is represented by Fig. 3.3. The analysis is similar to that of the Eq. (3.5). Nevertheless, the volume of the node N i has been truncated. The surface enclosing V Ni is still represented by ∂V Ni as before but in the present case ∂V Ni is splitted in S 1,i and S 2,i , S 2,i being the surface corresponding to the node N i adjacent 1 Let us recall that for unstructured meshes a cell can be composed by nodes that are not numerate consecutivelly (e.g.: if a node i = 5 has adjacent nodes numbered 9 and 13, then l is 4 and 8 respectively) 2 We talk about a ‘volume’ V Ni being enclosed by ‘surfaces’ ∂V Ni . Of course for the present two dimensional case the volume and the surface become a surface and an edge respectively.
Page 1: UNIVERSITE MONTPELLIER II SCIENCES
Page 5: An expert is a man who has made all
Page 8 and 9: Y gracias familia mía!!, que así
Page 11: Abstract Today, much of the current
Page 14 and 15: 4 CONTENTS 3.4.1 Preconditioning .
Page 16 and 17: List of Figures 1.1 Main sources of
Page 18 and 19: 8 LIST OF FIGURES 5.19 Acoustic ene
Page 20: List of Tables 1.1 EU recommended r
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Page 30 and 31: 2 Computation of noise generated by
Page 32 and 33: 22 Chapter 2: Computation of noise
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