237 Numerical investigation of a swirling flow under the optimal perturbation Cheng Chen 1 , De-jun Sun 1* 1 Department of Modern mechanics, University of Science and Technology of China Hefei , China * E-mail: dsun@ ustc.edu.cn ABSTRACT: The nonlinear evolution of 3-D instability of a viscous swirling flow, namely, the Oseen vortex, is addressed by direct numerical simulation with a Reynolds number equal to 5000. The global optimal perturbation is considered as the initial perturbation. In axisymmetric cases, three flow regimes are found: (1) the linear growth; (2) the decay of perturbation energy; (3) secondary energy growth. The linear energy growth, which is characterized by the amplification of radial perturbations, is arrested by the interaction between the vortex ring and the Oseen vortex core. The development of the vortex structure is also investigated for non-symmetric flows. KEY WORDS: Oseen vortex; optimal perturbation; transient growth; nonlinear evolution. 1 INTRODUCTION In recent years there has been strong interest in the so- called bypass transition to turbulence of parallel shear flows [1-3] and the linear transient growth of small disturbances is generally thought to be the starting point for bypass transition mechanisms. Mathematically, the phenomena of transient growth is due to the effect of non-normality in the linearized evolution operator and the non-orthogonality of eigenmodes. Namely, in a non-normal linear system transient growth can occur at short times, even if the flow system is linearly stable. Physically, a lift-up mechanism that spanwise vorticity in basic flow transfers energy to streamwise velocity has been universally accepted to explain this short-term growth in asymptotically stable flow regimes. More recently, transient growth has also been applied to swirling flows widely. Nolan & Farrell reported that two-dimensional spiral perturbations were enhanced to levels by transient growth in hurricane- like geophysical vortices. [4] Schmid et al. have studied the potential for transient growth in the Batchlor- vortex. [5] It is found that perturbations formed by superpositions of the stable eigenmodes are capable of growing to large levels. The concept of ε - pseudospectra is applied to show that the spectrum is highly sensitive to small perturbations, estimating the extent of transient growth; however, the physical mechanisms of growth was not investigated. Transient growth in the Oseen vortex has been studied in the work of Antkowiak & Brancher, [6] which is mainly on the bending waves (azimuthal wavenumber 1 n =± ) and the growth mechanisms in the two-dimensional situation. Pradeep & Hussain presented a excellent study of linear transient growth in an Oseen vortex by extracting ‘optimal modes’ of perturbations, [7] in which two distinct mechanisms for growth, corresponding to two-dimensional and three- dimensional disturbances respectively, were confirmed. Furthermore, these transient growth mechanisms were explained in terms of the combined effects of the strain and vorticity components of the mean flow, which play counteractive roles. Thus, the transient behavior of the Oseen vortex immediately raises an interesting question of the nonlinear evolution characters of the initial perturbations which is of significant amplification. Different from the study of Delbende et al., 8 which involves the velocity perturbation of a single unstable normal mode, the present work concentrates on this problem by direct numerical simulation of the dynamics of the global optimal perturbation obtained by solving the transient growth problem with given azimuthal and axial wavenumbers ( ,) nk . The remainder of the paper is organized as follows. In the next section, we state the numerical implementation, including a brief description of numerical method, initial flow conditions. Numerical results of the vortex dynamics of the Oseen vortex under the optimal perturbation in the axisymmetric and non-symmetric cases are presented respectively, and a further discussion on the effect of n and k is