Theoretical and numerical analysis of wake vortices

Abstract

The rst part of the paper analyses the linear dynamics of a vortex pair thanks to a normal mode analysis. The basic flow is a superposition of two Lamb-Oseen vortices of radius a and distance b. The long-wave Crow instability and the short-wave Widnall instabilities are fully characterized for dierent aspect ratios a=b of the vortex pair. In particular, as a=b increases, it is shown that the antisymmetric Widnall instabilities are favoured for short-wave perturbations. This constitutes an explanation for the results obtained by Leweke & Williamson (1998). The second part of the paper deals with turbulence in the Batchelor vortex. A 3D Large Eddy Simulation (LES) shows that, starting from a linearly unstable wake-type vortex, high levels of turbulence can be obtained. But, the mean-flow then rapidly evolves towards a stable state. These results are in accordance with those of Ragab & Sreedhar (1995). This paper analyses some linear and non-linear properties of the dynamics of vortices. Two problems will be discussed. The rst concerns the linear dynamics of two Lamb-Oseen vortices. The second deals with the turbulence eects on a Batchelor-type vortex. These constitute two dierent aspects of the general airplane vortex wake hazard problem. Linear stability of two counterrotating vortices, including the (long-wave) Crow instability (Crow 1970) and the (short-wave) Widnall instability (Tsai & Widnall 1976 and Moore & Saman 1975), may contribute to an aircraft vortex wake dispersion. In the present paper, these two instabilities are fully characterised in a unied frame by means of a normal mode analysis. The development of turbulence and its impact on aircraft vortices is still a matter of question. A temporal numerical simulation of an unstable Batchelor vortex is performed in order to get an insight into this problem. The rst part of this paper is devoted to linear dynamics. We rst obtain the bending wave oscillation frequencies of a single Lamb-Oseen vortex by means of a shooting method. It is shown that a single Lamb-Oseen vortex is stable. Then, a spectral collocation and matrix eigenvalue method is used to analyse the dynamics of a vortex pair. In particular, the Crow instability in the long-wave limit and the Widnall instabilities are recovered by this general method. We also prove that increasing the core radius makes the interaction between the two vortices stronger. It is shown, for example, that the antisymmetric mode of the Widnall instability is promoted as the core radius increases. The second part deals with turbulence in the Batchelor vortex. It is shown that high levels of turbulence can be obtained in the vortex. Starting from a linearly unstable wake-type vortex, the