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158 Chapter B: Publications As a consequence, the acoustic wave equation for low Mach number reacting flows reads ∇ · (c 2 0∇p ′ ) − ∂2 p ′ ˙q′ ∂t 2 = −(γ − 1)∂ ∂t − γp 0∇v : ∇v + γp 0 ∂ 2 W ′ W 0 ∂t 2 (2) where c 0 , p, γ, ˙q, v, W represent respectivelly the speed of sound, the pressure, heat capacity ratio, the heat release rate, the velocity vector and the mixture molar weight. The symbols () 0 and () ′ define respectivelly mean quantities and fluctuation quantities. As it can be noticed in the left hand side of eq. 2, the speed of sound c is placed inside the divergence operator. This ensures to capture acoustic fluctuations with strong variation of the mean temperature as it occurs close to the flame front. In the combustion case exposed in this paper, the non-isomolar combustion noise does not play an important role since the reactant mixtures are highly dilluted in nitrogen. Further on, the aerodynamic source of noise is considered small with respect to the noise source associated with the perturbation of the heat release rate. 22 The inhomogeneous wave equation then reduces to ∇ · (c 2 0∇p ′ ) − ∂2 p ′ ∂t 2 ˙q′ = −(γ − 1)∂ ∂t Under the harmonic oscillation assumption, in which p ′ (x, t) = ˆp(x)e −iωt , Eq. 3 becomes ⎧ ⎨ ∇ · c 2 0∇ˆp + ω 2 ˆp = −iω(γ − 1)ˆ˙q in Ω ⎩+ Boundary Conditions on Γ (3) (4) where ω = 2πf. The quantities ˆp and ˆ˙q are complex amplitudes which depend on space only. III. Experimental Configuration This article describes the evaluation procedure of noise due to the combustion within a swirled premixed combustor 26, 27 (EC2 Combustor) performing both direct and indirect computations. The experimental study is carried out in the laboratory EM2C (École Centrale Paris). The EC2 combustor consists in two geometrical identical stages for air-fuel injection, a premixer and a combustion chamber. The flame is controlled by the Fuel-Air ratio imposed in each of the two stages and stabilized by a swirled premixed. The test rig accounts for 7 different measurement points of pressure (denoted M1 to M7 in fig.1) placed at equivalent distances along the combustor. Figure 1. Two staged swirled premixed combustor. (Courtesy of École Centrale Paris) IV. Combustion noise Analysis AVBP, developed by CERFACS, is the parallel solver used for the LES computations. 28 In this study, the full compressible Navier Stokes equations are solved on hybrid (structured and unstructured) grids with second order spatial and temporal accuracy. Subgrid stresses are described by the Smagorinksy model. The 3 of 9 American Institute of Aeronautics and Astronautics
159 flame/turbulence interactions are modeled by the Thickened Flame (TF) model. 29 The spatial discretization is based on the finite volume method with a cell-vertex approach, combined to a numerical scheme derived from the Lax-Wendroff scheme. AVBP has been validated/used for a considerable number of configurations. 30–32 In 19 the computation of the combustion noise from the EC2 combustor was carried out with two different grid resolutions: A ’coarse’ mesh of 3 million cells and a ’refined’ mesh of 10 million cells. It was found that both meshes reproduced sufficiently well PIV measurements 33 when considering mean quantities. On the other hand, when RMS values were considered only the refined mesh results agreed well with the experimental results. Figure 3 shows the axial velocity profiles along the cuts displayed in Fig. 2 7mm 17mm 27mm Figure 2. Section cuts for velocity profiles 50 x = 7 mm x = 17 mm 50 x = 27 mm 50 50 x = 7 mm x = 17 mm 50 x = 27 mm 50 40 40 40 40 40 40 30 30 30 30 30 30 20 20 20 20 20 20 z (mm) 10 0 −10 10 0 −10 10 0 −10 z (mm) 10 0 −10 10 0 −10 10 0 −10 −20 −20 −20 −20 −20 −20 −30 −30 −30 −30 −30 −30 −40 −40 −40 −40 −40 −40 −50 −20 0 20 40 −50 −50 −20 0 20 40 −20 0 20 40 −50 0 10 20 −50 0 10 20 −50 0 10 20 (a) Mean Axial Velocity (b) RMS Axial Velocity Figure 3. Velocity Profiles: ◦ Experimental PIV measurements – – – LES 3 million cells, —— LES 10 million cells Reproducing flow dynamics in a proper way (see RMS values for the refined mesh in Fig. 3) does not mean necessarily a good estimation of combustion noise. Computation of combustion noise is very challenging, since a good prediction of the rate of change in the flame surface area must be achieved. 34 Figure 4 shows the Sound Pressure Level (SPL) spectrum at microphone 7 (see the location of M7 in Fig. 1) for the computations with the refined and coarse meshes as well as the experimental measurements. 33 It has been observed that in order to correctly evaluate the dynamics of a flame and the acoustics generated by this one, the resolution of the computation (resolution of the mesh and order of the numerical scheme) is of significative importance. The gap seen in fig. 4 between experimental results of the acoustic spectrum and the refined LES is problably due to the lack of resolution of the computation. For this purpose, 4 of 9 American Institute of Aeronautics and Astronautics
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UNIVERSITE MONTPELLIER II SCIENCES
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An expert is a man who has made all
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Y gracias familia mía!!, que así
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Abstract Today, much of the current
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4 CONTENTS 3.4.1 Preconditioning .
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List of Figures 1.1 Main sources of
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8 LIST OF FIGURES 5.19 Acoustic ene
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List of Tables 1.1 EU recommended r
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14 Chapter 1: General Introduction
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2 Computation of noise generated by
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22 Chapter 2: Computation of noise
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