The Effect of Finite Compliant Panels on the Development of Linear Disturbances in the Rotating-Disk Boundary Layer

Abstract

The rotating disk boundary layer has long served as a fundamental model in fluid dynamics for studying the transition from laminar to turbulent flow, where both convective and absolute instabilities play key roles. While previous studies (e.g., Cooper & Carpenter [16]) have theoretically shown that infinite compliant panels can delay or suppress absolute instability, experimental validation remains challeng- ing due to the impracticality of infinite panels. This thesis extends earlier theoretical work by conducting a comprehensive numerical simulation of three-dimensional dis- turbance evolution over a boundary layer as it passes over finite-length compliant panels. Unlike prior research that utilised a single-layer viscoelastic model, this work em- ploys a spring-backed compliant panel model (Carpenter [12]) to more realistically represent experimental conditions. The analysis is performed using Davies’ [20] vorticity-velocity formulation, an advanced numerical approach that provides a pre- cise analysis of flow stability. To establish a reliable framework, standard linear stability theory (SLST) is first applied to determine critical parameters – such as Reynolds number (Re) and azimuthal mode number (n) – ensuring consistency with established studies like Malik [45]. Subsequent numerical simulations investigate both convective and absolute insta- bilities under two types of external disturbances: periodic forcing, which selectively excites modes at fixed frequencies, and impulsive forcing, which excites a broad spectrum of modes with the most unstable mode eventually dominating the re- sponse. The results indicate that finite-length compliant panels can reduce dis- turbance growth rates and, in certain cases, influence the direction of instability propagation. Furthermore, the stabilising benefits are enhanced as Re, n, and the compliant panel length increase. By providing the first full numerical analysis of finite-length compliant panels in this context, this thesis advances the understanding of passive flow control strategies for rotating boundary layers, offering practical insights that could inform experimental validation and the design of engineering systems with improved flow stability.