A fully implicit combined field scheme for freely vibrating square cylinders with sharp and rounded corners Rajeev K. Jaiman ⇑ , Subhankar Sen, Pardha S. Gurugubelli Department of Mechanical Engineering, National University of Singapore, Singapore 119077, Singapore article info Article history: Received 28 September 2014 Received in revised form 25 January 2015 Accepted 4 February 2015 Available online 14 February 2015 Keywords: Fully implicit combined field Petrov–Galerkin Rounded square Lateral edge separation ALE Free vibration Galloping abstract We present a fully implicit combined field scheme based on Petrov–Galerkin formulation for fluid–body interaction problems. The motion of the fluid domain is accounted by an arbitrary Lagrangian–Eulerian (ALE) strategy. The combined field scheme is more efficient than conventional monolithic schemes as it decouples the computation of ALE mesh position from the fluid–body variables. The effect of corner rounding is studied in two-dimensions for stationary as well as freely vibrating square cylinders. The cylinder shapes considered are: square with sharp corners, circle and four intermediate rounded squares generated by varying a single rounding parameter. Rounding of the corners delays the primary separation originating from the cylinder base. The secondary separation, seen solely for the basic square along its lateral edges, initiates at a Reynolds number, Re between 95 and 100. Imposition of blockage lowers the critical Re marking the onset of secondary separation. For free vibrations without damping, Re range is 100–200 and mass ratio, m  of each cylinder is 10. The rounded cylinders undergo vortex-induced motion alone whereas motion of the basic square is vortex-induced at low Re and galloping at high Re. The flow is periodic for vortex-induced motion and quasi-periodic for galloping. The lower branch and desynchronization characterize the response of rounded cylinders. For the square cylinder, the compo- nents of response are the lower branch, desynchronization and galloping. Removal of the sharp corners of square cylinder drastically alters the flow and vibration characteristics. Ó 2015 Elsevier Ltd. All rights reserved. 1. Introduction The analysis of flow around a single body or multiple bluff bod- ies continues to attract the attention of the research community. The primary motivation behind the present work concerning an isolated obstacle stems from the need to optimize multi-column offshore structures subjected to ocean currents. In particular, there is a growing demand to reduce or control the vortex-induced motions of multi-column structures. Motion of such offshore struc- tures can be minimized by suitably altering the spacing between the columns and also by selecting appropriate column shapes such as circular, and square with sharp/rounded corners. The presence of sharp corners on a square cylinder largely alters the flow characteristics as compared to the ones with circular/el- liptical section having smooth contours. Besides the angle of inci- dence, the sharp corners appear as a major influencing factor in the body geometry, that affect the flow separation. The location of the separation points strongly depends on the body shape which in turn governs the wake dynamics and fluid loading. Removal of the sharp corners of a square cylinder and gradual transition to the circle through intermediate rounded squares generate cross- sections, that might be competitive both for a stationary and vibrating cylinder in various mechanical and civil engineering applications. It is therefore important to study the effect of gradually rounding the corners of a square cylinder at zero inci- dence on the flow till the circular section is reached. In correspondence with a steady or time-averaged flow, exis- tence of an even number of zero vorticity points on the surface of a symmetric or asymmetric obstacle was earlier suggested in [1]. For a separated flow, these singular points are alternate points of attachment and separation. In terms of streamlines, the schematics in Fig. 1 (upper row) illustrate various wake configura- tions of a square cylinder with sharp corners in low to moderate Reynolds number regime (Re 6 150). Also shown is the corre- sponding vorticity, x distribution along half the circumference of the cylinder. Here, h is the circumferential angle measured coun- terclockwise from the forward stagnation point. Points 1, 2, 6, 8 and 9 denote locations of attachment while 3, 4, 5 and 7 denote locations of separation. An attached laminar boundary layer as observed at very low Re, is represented by Fig. 1a where formation of wake does not take place. The wake configuration depicted by http://dx.doi.org/10.1016/j.compfluid.2015.02.002 0045-7930/Ó 2015 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail address: mperkj@nus.edu.sg (R.K. Jaiman). Computers & Fluids 112 (2015) 1–18 Contents lists available at ScienceDirect Computers & Fluids journal homepage: www.elsevier.com/locate/compfluid