58 / J. Comput. Fluids Eng. Vol.20, No.4, pp.58-62, 2015. 12 1. Introduction Researchers have turned to dynamical systems theory in studying transition to turbulence and this helped in gaining a fresh perspective in understanding the phenomenon. In dynamical systems theory, the presence of coherent structures in turbulent flow may be seen as low-dimensional invariant sets in phase space[4]. These coherent structures, which are spatiotemporally organized, appear when a turbulent state visits the neighborhood of such an invariant set for a substantial fraction of time. The Navier-Stokes equation, which governs turbulent motion of viscous fluids, is an example of an infinite- dimensional dynamical system which may be approximately reduced to finite dimension[4]. In this equation, the simplest invariant solution can be an equilibrium or a time-periodic one. Researchers have searched for invariant solutions and analyzed numerically the long-term behavior on their stable and unstable manifolds. The computation of stable and unstable manifolds in state space is relevant in the study of the transition process. In wall-bounded shear flows, where laminar state is often linearly stable during transition to turbulence, an invariant set with only a single unstable direction in phase space is called an ‘edge state’[10]. The edge state has a special property in which its stable manifold separates the laminar and turbulent flows. State points on this laminar- turbulent boundary, known as the ‘edge of chaos’[1,10], is attracted to the edge state. For initial conditions just exceeding a critical value, corresponding state points will escape out of the laminar basin along the unstable manifold of the edge state. We study a periodic edge state to the incompressible Navier-Stokes equation in plane Couette flow. It is shown that computation of its unstable manifold leads to several orbits which return to the edge state along its stable manifold. This suggests the formation of a homoclinic connection, i.e., the intersection of the unstable and stable manifolds for the same saddle-type invariant solution. Two of the homoclinic orbits found move closer to each other for decreasing Reynolds until they eventually collide. 2. Transitional Plane Couette Flow Shown above is the flow configuration of plane Couette flow that is employed in this study. This is composed of two parallel flat plates that are separated by a distance 2h. The wall-parallel directions are the streamwise x- and spanwise z- directions while the wall-normal direction is the y- direction. The parallel plates move with the same velocity in the direction opposite each other, i.e., the upper plate is moving with velocity +U and the lower plate is moving with velocity -U. The velocity field u = (u,v,w) for an incompressible viscous flow is governed by the Navier-Stokes equation Received: July 14, 2015, Revised: October 29, 2015, Accepted: October 29, 2015. * Corresponding author, E-mail: juliusrhoanlustro@gmail.com DOI http://dx.doi.org/10.6112/kscfe.2015.20.4.058 Ⓒ KSCFE 2015 HOMOCLINIC ORBITS IN TRANSITIONAL PLANE COUETTE FLOW Julius Rhoan T. Lustro, *1 Genta Kawahara, 2 Lennaert van Veen 3 and Masaki Shimizu 2 1 College of Engineering and Agro-Industrial Technology, University of the Philippines Los Baños 2 Graduate School of Engineering Science, Osaka University 3 Faculty of Science, University of Ontario Institute of Technology Recent studies on wall-bounded shear flow have emphasized the significance of the stable manifold of simple nonlinear invariant solutions to the Navier-Stokes equation in the formation of the boundary between the laminar and turbulent regions in state space. In this paper we present newly discovered homoclinic orbits of the Kawahara and Kida(2001) periodic solution in plane Couette flow. We show that as the Reynolds number decreases a pair of homoclinic orbits move closer to each other until they disappear to exhibit homoclinic tangency. Key Words : subcritical transition to turbulence, periodic orbit, homoclinic orbit