Instability Analysis of Vortex Systems in Lifting-Body Wakes

Abstract

The present work deals with vortical flow instabilities, in particular with those that appear in the wake left by aircraft, wind turbines or any other body surrounded by fluid, generating a force perpendicular to the flow, namely lift. The case of the aircraft wake destruction caused by the growth of vortex instabilities is of great importance, as the selection of safe distances between aircraft is in most of the cases very conservative due to the lack of knowledge of the phenomena involved in this process. Therefore, any improvement on their knowledge could help to reduce these distances without concerning the safety of the flight, which is of great importance when large aircraft, like Airbus 380 or Boeing 747, are flying followed by small aeroplanes. Other relevant case where the analysis of vortical wakes is crucial, is that of the wind farms, where the interaction of the wake left from a turbine upwind could degrade the performance of the wind turbine downstream by a great amount or even concern its structural safety. A wide variety of methodologies are discussed and employed in the present thesis to perform the stability analysis of such flows. Two groups can be distinguished among these methods. These belonging to one of them, designed vortex methods, are, in most of the cases, incompressible and inviscid, in contrast with standard stability analysis methods, that are usually viscous. However, the main difference between these two approaches is the fact that vortex methods are grid-less (Lagrangian description), in contrast to standard methods, which need a grid (Eulerian description). Both, vortex and grid methods, could be used to solve the fluid motion as an evolution problem (Initial Value Problems, IVP). This approach is usually denominated DNS (Direct Numerical Simulation), although here this denomination will be used only for the grid IVP. The study of the flow behaviour to instabilities could be done using IVP of the two types, where imposed perturbations or random noise will activate the instabilities, whose growth or decay will be observed. However, in many cases, it is more interesting to perform the stability analysis by means of linearisation of the equations and analysis of the eigenvalues and eigenmodes of the resulting EVP (EigenValue Problem). This second approach could be also used with both of the aforementioned methods, based on the Lagrangian or the Eulerian descriptions of the flow equations, which constitutes the core of the present research work. Some canonical cases are studied first, and the complexity is increased gradually to add more realistic features. However, even the results of the most simple cases have their application to explain phenomena related to the behaviour of vortices in the real world. The case of the counter-rotating vortex pair is a clear example. Despite its simplicity, the results extracted from it are of great relevance, as it is the instability discovered by Crow [1]. In this thesis, the movement of the vortices in a two-dimensional plane is studied: from the simplification of point vortices by a vortex simulation, to the scenario where the vortices are close and there is viscosity in the flow by means of a two-dimensional DNS. Homogeneous instabilities in the axial direction are also studied, first for two point vortices, following the idea of Crow [1], and later for the viscous vortex dipole, using the BiGlobal methodology. Comparisons of these two approaches could be found in Tendero et al. [3, 4, 5]. In addition, axial diffusion could be added and the resulting problem studied employing the three dimensional Parabolised Stability Equations (PSE-3D) analysis concept. All the methodologies considered have their range of validity. Nevertheless, for a true counter-rotating vortex pair with a great amount of axial diffusion, only the PSE-3D analysis proves that can give trustworthy results, as the axial diffusion leads to saturation of the modal growth, in contrast to parallel vortex results. This thesis collects results from several publications: Paredes et al. [2], Tendero et al. [3, 4, 5], He et al. [6], and Suryanarayanan et al. [7], as well as some original work.