Analytic Solutions for Flows Through Cascades
Noise reduction is a principal aim of the aviation industry. Particular attention is devoted to reducing turbomachinery noise, which is an important component of overall aero-engine noise during the take-off and landing stages. This thesis therefore presents analytic investigations into turbomachinery flows with the aim that the resulting solutions be used by aircraft designers to produce quieter aircraft. In order to facilitate exact solutions, the turbine is unwrapped onto the two-dimensional plane, resulting in a periodic array of blades commonly referred to as a “cascade”. Previous research has been restricted to the case where the cascade consists of a single impermeable row of flat plates at zero angle of attack. Consequently, this thesis considers three separate scenarios where the cascades consist of (i) blades with realistic geometry, (ii) blades with porosity gradients, and (iii) multiple blades per period window. In each case, we begin by solving the steady potential flow and proceed by investigating the effects of unsteady perturbations. This coupled approach provides analysis from both aerodynamic and aeroacoustic perspectives which is essential for achieving practical noise reductions. In order to find analytic solutions, sophisticated complex analysis is employed in the form of singular integral equations, Riemann--Hilbert problems, the Wiener--Hopf method and conformal mappings via the transcendental Schottky--Klein prime function. These methods are applied in the context of rigorous asymptotic expansions where the solution is expanded in terms of a small parameter such as the amplitude of an unsteady incident disturbance or the size of the blades. The aerodynamic analysis generates exact expressions for the surface velocity, drag, lift and deflection angle whilst the aeroacoustic solutions furnish exact expressions for the unsteady surface pressure, sound power output and far-field sound. These formulae are rapid to compute compared to CFD simulations currently used in industry and, moreover, they provide fresh insight into the roles played by blade spacing, geometry and porosity for turbomachinery noise and aerodynamics. Although the solutions in this thesis are applied to turbomachinery, they will also be useful in other applications such as solid mechanics, poroelasticity and biological fliers or swimmers operating in formation.