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108 Chapter 6: Boundary conditions for low Mach number acoustic codes evaluate the acoustic response of a system due to changes in the mean flow caused by section variations, but to study the influence of an entropy variation on the acoustics of a given system. Hence, a thin one-dimensional flame will be considered. This flame, which is characterized by a mean entropy jump, is located at the middle of a constant section duct (Fig.6.5). An incoming acoustic wave w + 1 is imposed at the inlet and the acoustic reflected wave w− 1 generated by the flame is evaluated. In addition, the acoustic transmitted wave w + 2 and the entropy wave w2 S created after the entropy jump are also computed. This study will be carried out both analytically (Eqs. 6.31, 6.32 and 6.33) and numerically by resolving the LEE system thanks to the SNozzle tool (section 6.2). 1D Flame w 1 + w 1 - M 1 M 2 w 2 + w 2 S Figure 6.5: 1D Flame Non Reflecting condition at the outlet 6.5.1 Analytic Solution : Building the linear system of equations After having neglected B 2 and E 1 , the sources due to the presence of a compressor ˆ F and Ŵ, and the unsteady source of energy ˆ˙q the system of equations to resolve become (1 + ¯M 1 )/ ¯c 1 A 1 − (1 − ¯M 1 )/ ¯c 1 B 1 = (1 + ¯M 2 )/ ¯c 2 A 2 − ρ 2 ¯c 2 ¯M 2 E 2 /c p (6.44) (1 + ¯M 1 ) 2 A 1 + (1 − ¯M 1 ) 2 B 1 = (1 + ¯M 2 ) 2 A 2 − ρ 2 ¯c 2 2 ¯M 2 2 E 2/c p (6.45) (1 + ¯M 1 )a 1 ¯c 1 A 1 + (1 − ¯M 1 )a 2 ¯c 1 B 1 = (1 + ¯M 2 )a 3 ¯c 2 A 2 − ρ 2 ¯c 3 2 ¯M 3 2 E 2/2c p (6.46) with a 1 = ¯M 1 + 1 2 ¯M 2 1 + 1 γ−1 , a 2 = ¯M 1 − 1 2 ¯M 2 1 − 1 γ−1 and a 3 = ¯M 2 + 1 2 ¯M 2 2 + 1 γ−1 . These equations (6.44, 6.45 and 6.46) are resolved for three unknowns: B 1 , A 2 and E 2 . The coefficient A 1 is the amplitude of the forced acoustic wave and must be given as an input to the problem. Note that for this particular case ˆ˙q was set to zero. This term could be also model as ˆ˙q = c p ( ¯T t2 − ¯T t1 )( ¯ρ 1 û 1 + ˆρ 1 ū 1 ) as done in [22].
6.5 When entropy does not remain constant through a duct 109 6.5.2 The mean flow in SNozzle Before performing any analysis on fluctuating quantities, it is necessary to define the baseline flow of the 1D Flame configuration. The1D flame is considered to produce a jump of temperature at x = 0. This temperature jump must be as smooth as possible so that gradients of all related quantities are properly computed. The temperature profile, shown in Fig. 6.6(a) is built from an hyperbolic tangent function. The mean value of the entropy ¯s is function of the mean density ¯ρ and the mean temperature ¯T. It reads [ ] R ¯T ¯s = c v ln ¯ρ γ−1 (6.47) Along this study the perturbations of the heat release ˆ˙q(t) = 0 are neglected. Nevertheless a profile of the mean heat release ¯˙q is present and defined as ¯˙q = ¯pū r ∂¯s ∂x (6.48) δ f δ f !!"# !!"$ ! !"$ !"# $% &'$!( X (m) $#!! ¯T (K ) $!!! '!! &!! %!! ¯˙q (J/m 3 s) $! % #!! ! !!"# !!"$ ! !"$ !"# X (m) (a) Mean Temperature ! (b) Mean Heat Release Figure 6.6: Typical Profiles The mean quantities ū, ¯ρ and ¯p are obtained in such a way that the Euler equations for steady flows are satisfied. Figures 6.6, 6.7 and 6.8 show the mean profiles of heat release, pressure, velocity, density and the Mach number for the case in which M 1 = 0.1.
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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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