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160 Chapter B: Publications SPL (dB) − micro 7 180 160 140 120 100 80 3 Millions Cells − LES 10 Millions Cells − LES Experimental Measurements . 0 500 1000 1500 2000 2500 Frequency (Hz) Figure 4. Sound Pressure Level a computation with 50 million cells is currently being conducted. Although the estimation of combustion noise through direct computation does not match experimental measurements, it is still a good exercise of analysis to compare direct computation results to those given by hybrid methods. For this comparison to take place, it is assumed that pressure fluctuations from LES contain a negligible amount of hydrodynamics. This comparison is carried out for the 10 million cells mesh and is independent from experimental data. Overall good agreement is found between both direct and hybrid approaches, as shown if Figs. 5 and 6 which shows the sound pressure levels obtained for microphones 5, 6 and 7 (see the location of M5, M6 and M7 in fig. 1) . SPL (dB) − micro 5 180 160 140 120 100 80 Hybrid Computation Direct Computation: LES SPL (dB) − micro 6 180 160 140 120 100 80 Hybrid Computation Direct Computation: LES 60 0 500 1000 1500 2000 2500 Frequency (Hz) 60 0 500 1000 1500 2000 2500 Frequency (Hz) Figure 5. Sound Pressure Levels from the direct and hybrid approaches SPL (dB) − micro 7 180 160 140 120 100 80 Hybrid Computation Direct Computation: LES 60 0 500 1000 1500 2000 2500 Frequency (Hz) Figure 6. Sound Pressure Levels from the direct and hybrid approaches 5 of 9 American Institute of Aeronautics and Astronautics
161 Observing with attention fig. ??, it is noticeable that there are some zones of the spectrum in which an important gap is present between hybrid and direct computations. It is probable, considering the direct computation, that the fluctuations of pressure coming from LES are composed by both acoustic and hydrodynamic contributions. On the other hand, presure fluctuations coming from the hybrid computation are totally due to acoustics. A suitable comparison is then not carried out, and it becomes important to be able to extract acoustics from LES computations in order to evaluate in a proper way the results from the hybrid computation. V. Filtering a LES pressure field to find the corresponding acoustic field Velocity fluctuations obtained by LES are composed by both hydrodynamics and acoustics Appling the operator ∂/∂t to Eq. 5 leads to u ′ i,LES = u ′ i,hyd + u ′ i,ac (5) ∂u ′ i,LES = ∂u′ i,hyd + ∂u′ i,ac ∂t ∂t ∂t From linear acoustics, the momentum equation is given by (6) ρ 0 ∂u ′ i,ac ∂t = − ∂p′ ac ∂x i (7) where [] 0 and [] ′ represent respectively the mean and fluctuating flow. Combining this term of eq. 7 into Eq. 6 − 1 ρ 0 ∂p ′ ac ∂x i + ∂u′ i,hyd ∂t Finally the divergence operator to this equation is applied = ∂u′ i,LES ∂t − ∂ ( 1 ∂p ′ ) ac + ∂ ( ∂u ′ ) i,hyd = ∂2 u ′ i,LES ∂x i ρ 0 ∂x i ∂x i ∂t ∂x i ∂t (8) (9) A. Finding ∂u i,hyd ∂x i Neglecting viscosity, species diffusion and heat conduction the Navier-Stokes equations for reacting flows read ∂ρ ∂t + ρ∂u j ∂x j + u j ∂ρ ∂x j = 0 (10) ρ ∂u i ∂t + ρu ∂u i j = − ∂p (11) ∂x j ∂x i ρc p ∂T ∂t + ρc pu j ∂T ∂x j = ˙q (12) In the low-Mach number approximation, the thermodynamic pressure P 0 only depends on temperature. The equation of state is simply Replacing Eq. 13 in the left hand side of the Eq. 12 leads to K 0 = ρT (13) ρc p ∂K 0 /ρ ∂t + ρc p u j ∂K 0 /ρ ∂x j = ˙q (14) 6 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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100 Chapter 6: Boundary conditions
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