The stability of inviscid flows over passive compliant walls

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Abstract
The linear temporal stability of incompressible semi-bounded inviscid parallel flows over passive compliant walls is studied. It is shown that some of the well-known classical results for inviscid parallel flows with rigid boundaries can, in fact, be extended in modified form to passive compliant walls. These include a result of Rayleigh (1880) which shows that the real part of the phase velocity of a non-neutral disturbance must lie within the range of the velocity distribution; the semi-circle theorem of Howard (1961) and a result of Høiland (1953) which places a bound on the temporal amplification rates of unstable disturbances. The bounds on the phase velocity and the temporal amplification rates of unstable two-dimensional disturbances provide useful guides for numerical studies.The results are valid for a large class of passive compliant walls. This generality is achieved through a variational-Lagrangian formulation of the essential dynamics of wall motion. A general treatment of the marginal stability of thin shear flows over general passive compliant walls is given. It represents a generalization of the analysis given by Benjamin (1963) for membrane and plate surfaces. Sufficient conditions for the stability of thin shear flows over passive compliant walls are deduced. The applications of the stability criteria to simple cases of compliant wall are described to illustrate the use and the effectiveness of these criteria.