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Combustion dynamics Lecture 1b S. Candel, D. Durox , T. Schuller! CentraleSupélec Université Paris-Saclay, EM2C lab, CNRS
SC DD TS Copyright ©2019 by [Sébastien Candel]. This Princeton summer school, June 2019 material is not to be sold, reproduced or distributed without prior written permission of the owner, [Sébastien Candel].
Elements of acoustics
Basic equations of linear acoustics Plane waves in one dimension Harmonic waves Plane modes in a duct Harmonic spherical waves Acoustic energy balance
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Fundamentals of linear acoustics
Assumptions of the following analysis The thermodynamic state is determined by two thermodynamic variables
The fluid is ideal so that its viscosity and heat conductivity may be taken equal to zero
Chemical reactions are absent and there is no volumetric addition of mass or heat
There are no volume forces (volume forces are negligible)
Begin with the Euler set of equations
Mass @⇢ + ⇢v =0 @t r · Momentum @v ⇢( + v v)= p @t · r r Energy in entropy form @ ⇢T ( + v s)=0 @t · r
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Because the fluid is bivariant the entropy s may be expressed in terms of two other thermodynamic variables. For example s = s ( p, ⇢ ) or equivalently one may write p = p(⇢,s)
For example, in the case of a perfect gas the state equation takes the forms