Date

2018

Document Type

Dissertation

Degree

Doctor of Philosophy

Department

Mechanical Engineering

First Adviser

Jaworski, Justin W.

Abstract

The aerodynamic, aeroelastic, and acoustic implications of a prescribed porosity distribution on a thin airfoil or panel in a steady, two-dimensional, incompressible flow are formulated and solved in four distinct model problems. In pursuit of the steady aerodynamic loads on a porous airfoil, a Darcy porosity condition on the airfoil surface furnishes a Fredholm integral equation for the pressure distribution. This singular integral equation is solved exactly and generally as a Riemann-Hilbert problem provided that the porosity distribution is H{\"o}lder-continuous. The comparison between the new steady aerodynamic theory and experimental measurements of integrated lift forces on porous SD7003 airfoils in the literature shows good agreement for sufficiently small values of a dimensionless porosity parameter identified in the theoretical analysis.The non-circulatory fluid forces are then derived on an oscillating porous panel or airfoil in a uniform incompressible flow. The fundamental integral equation for these unsteady loads resulting from a Darcy-type boundary condition with H{\"o}lder-continuous spatial distribution of porosity is formulated and solved in closed form as a Liouville-Neumann series. To demonstrate these analytical results, the non-circulatory pressure distributions for vibrating panels on simple or clamped supports with either uniform or variable chordwise porosity distributions are presented and compared.These presented non-circulatory fluid forces are applied to aeroelastic stability predictions for vibrating porous panels or liners fixed at both ends.Porous panels fixed at both ends lose aeroelastic stability by divergence, which is in agreement with the classical result for non-porous panels. However, the effect of porosity act to suppress divergence onset until higher flow speeds.Finally, the acoustic far-field pressure is determined for a finite-chord panel with uniform porosity. The free space Green's function for the two-dimensional Helmholtz equation propagates the unsteady non-circulatory forces on the panel into the acoustic field.Results from this analysis identify the effects of varying the magnitude of a Darcy-type porosity condition on the acoustic emission of a vibrating panel in comparison to its non-porous counterpart. It is shown that the sound pressure produced by a uniformly-porous airfoil depends on the reduced frequency, Mach number, and the dimensionless porosity parameter. At low Mach numbers, increasing the magnitude of a Darcy-type porosity parameter leads to a reduction in the acoustic emission from a vibrating panel at high frequencies, while the introduction of porosity does not reduce the produced sound pressure at lower frequencies.Furthermore, it is demonstrated that, even at high frequencies, porosity does not always reduce the sound pressure; as the Mach number increases, larger values of the porosity parameter are required to reduce the sound generated from vibrating panels in all directions.

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