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Competing instabilities of three-dimensional boundary layer flow over spinning bodies

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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 [1] and theoretical investigations [2] in the literature. Meanwhile, further details will be provided of potential comparisons with 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. 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 [2], visualised in terms of co-rotating spiral vortices, whereas slender rotating cones have transition characteristics governed by a centrifugal instability [3], which is visualised by the appearance of counter-rotating Gortler 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.

References [1] R. Kobayashi and H. Izumi, 1983 Boundary-layer transition on a rotating cone in still fluid. J. Fluid Mech. 127, 353–64. [2] S. J. Garrett, Z. Hussain and S. O. Stephen, 2009 The crossflow instability of the boundary layer on a rotating cone. J. Fluid Mech. 622, 209–232. [3] Z. Hussain, S. J. Garrett and S. O. Stephen, 2014 The centrifugal instability of the boundary-layer flow over slender rotating cones. J. Fluid Mech. 755, 274–293.

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