On the stability of boundary-layer flow over a rotating cone using new solution methods

Abstract

Abstract In this study, a new solution is applied to the model problem of boundary-layer flow over a rotating cone in still fluid. The mean flow field is perturbed leading to disturbance equations that are solved via a more accurate spectral numerical method involving Chebyshev polynomials, both of which are compared with previous numerical and analytical approaches. Importantly, favourable comparisons are yielded with existing experiments [17] and theoretical investigations [6] in the literature. Meanwhile, further details will be provided of potential comparisons with new experiments currently in the pipeline. Physically, the problem represents a model of airflow over rotating machinery components at the leading edge of a turbofan. In such applications, laminar-turbulent transition within the boundary layer can lead to significant increases in drag, resulting in negative implications for fuel efficiency, energy consumption and noise generation. Consequently, delaying transition to turbulent flow is seen as beneficial, and controlling the primary instability may be one route to achieving this. Ultimately, control of the input parameters of such a problem may lead to future design modifications and potential cost savings. Our results are discussed in terms of existing experimental data and previous stability analyses on related bodies. Importantly, broad-angled rotating cones are susceptible to a crossflow instability [6], visualised in terms of co-rotating spiral vortices, whereas slender rotating cones have transition characteristics governed by a centrifugal instability [9], which is visualised by the appearance of counter-rotating Görtler vortices. We investigate both parameter regimes in this study and comment on the accuracy of the new solution method compared with previous methods of solving the stability equations.