J. Fluid Mech. (2016), vol. 800, pp. 595–612. c  Cambridge University Press 2016 doi:10.1017/jfm.2016.364 595 The flow dynamics of the garden-hose instability Fangfang Xie 1 , Xiaoning Zheng 1 , Michael S. Triantafyllou 1 , Yiannis Constantinides 2 and George Em Karniadakis 3, † 1 Department of Mechanical Engineering, Massachusetts Institute Technology, Cambridge, MA 02139, USA 2 Chevron Energy Technology Company, Houston, TX 77002, USA 3 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA (Received 9 September 2015; revised 2 April 2016; accepted 27 May 2016; first published online 12 July 2016) We present fully resolved simulations of the flow–structure interaction in a flexible pipe conveying incompressible fluid. It is shown that the Reynolds number plays a significant role in the onset of flutter for a fluid-conveying pipe modelled through the classic garden-hose problem. We investigate the complex interaction between structural and internal flow dynamics and obtain a phase diagram of the transition between states as function of three non-dimensional quantities: the fluid-tension parameter, the dimensionless fluid velocity and the Reynolds number. We find that the flow patterns inside the pipe strongly affect the type of induced motion. For unsteady flow, if there is symmetry along a direction, this leads to in-plane motion whereas breaking of the flow symmetry results in both in-plane and out-of-plane motions. Hence, above a critical Reynolds number, complex flow patterns result for the vibrating pipe as there is continuous generation of new vorticity due to the pipe wall acceleration, which is subsequently shed in the confined space of the interior of the pipe. Key words: bifurcation, flow–structure interactions, vortex flows 1. Introduction A flexible, fluid-conveying pipe constitutes a simple flow–structure interaction system with intriguingly complex dynamical properties. Such systems are extensively used in the oil and gas industry and in nuclear engineering but are also of great interest in biomechanics, e.g. the blood flow in veins or air flow in pulmonary alveoli. A flutter instability arises if the fluid velocity in the pipe is sufficiently high, resulting in a pipe motion whose form is close to a sinusoidal one at lower velocity values, while it appears totally erratic at higher velocities. This is an instability that we can observe in everyday life, such as when watering the garden with a hose (hence the name garden-hose or water-hose instability), or watching ‘sky dancers’ – long flexible tubes dancing above air blowers in the streets to advertise a product (Doaré & De Langre 2002; Cros, Romero & Flores 2012). Bourrières (1939) was one of the first to conduct experiments to determine the flutter of a cantilevered pipe; a similar study was undertaken later by Ashley & † Email address for correspondence: george_karniadakis@brown.edu Downloaded from https://www.cambridge.org/core . Brown University Library , on 28 Jun 2018 at 19:07:38, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms . https://doi.org/10.1017/jfm.2016.364