The purpose is to extend Vilenski - Rienstra’s [32] results concerning mainly the general aspects of the existence of non zero modes in annular lined ducts. The case, when the radial and circumferential components of the mean flow are equal to zero and the axial component depends only on the distance to the duct axis, is investigated. Conditions for the existence of non zero modes, which satisfy the linearized homogeneous Euler equations (obtained by linearization around the mean flow) and the boundary conditions, (corresponding to the perturbation - liner interaction of mass-spring-damper type) are found. The first condition, called dispersion relation, is expressed in terms of the solutions of two normalized initial value problems and is equivalent to the linear dependence of these solutions. It is shown that the set of non zero modes, corresponding to a given frequency and given axial and circumferential wave number, is either the null space, either is a one dimensional function space. It is shown also that if the mean flow is symmetric with respect to the “center of the ring”, then neither symmetric, nor anti-symmetric modes exist. This difference between the annular and rectangular or circular lined duct models explains while one of the boundary conditions can not be transferred in the center of symmetry. For symmetric flow, being constant in the “central part of the ring”, new dispersion relations are derived. The new relations beside the solutions of the two normalized initial value problems incorporate also modified Bessel functions or additional Bessel functions. The Lyapunov stability of the mean flow with respect to the initial value perturbation by mode type perturbations is discussed in terms of the zero’s of the dispersion relation.
Existence of non zero modes in an annular lined duct
9TH INTERNATIONAL CONFERENCE ON MATHEMATICAL PROBLEMS IN ENGINEERING, AEROSPACE AND SCIENCES: ICNPAA 2012 ; 2012 ; Vienna, Austria
AIP Conference Proceedings ; 1493 , 1 ; 1034-1042
2012-11-06
9 pages
Conference paper
Electronic Resource
English
Existence of non zero modes in an annular lined duct
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