PHYSICAL REVIEW E 84, 037302 (2011) Azimuthal solitary surface wave in cylindrical tank Hamid Ait Abderrahmane * Department of Mechanical Engineering, McGill University, Montr´ eal, Qu´ ebec, Canada H3A 2K6 Mustapha Amaouche Laboratoire de Physique Nonlineaire, Universit´ e de B´ ejaia, Algeria Mohamed Fayed, Hoi Dick Ng, and Georgios H. Vatistas Department of Mechanical and Industrial Engineering, Concordia University, Montr´ eal, Qu´ ebec, Canada H3G 1M8 Kamran Siddiqui Department of Mechanical and Materials Engineering, University of Western Ontario, London, Ontario, Canada N6A 5B9 (Received 5 July 2011; revised manuscript received 12 August 2011; published 22 September 2011) This Brief Report is devoted to the study of the solitary surface wave rotating in the azimuthal direction, arising during water drainage from a cylindrical reservoir, when shallow flow conditions are reached. The linear dependence between the wave speed and its amplitude is shown to be similar to that expected from the classical Korteweg–de Vries equation. DOI: 10.1103/PhysRevE.84.037302 PACS number(s): 47.35.Fg Following its discovery in 1844 by Russell [1], the solitary wave has been the subject of intense research because of its manifestations in several fields of physics such as acoustic waves, magnetoacoustics, and several others; see [2–4]. Solitary waves were also observed in rotating flow and it is known that the background rotation significantly influences the propagation of both surface and internal gravity solitary waves. It is also found to be responsible for the creation of the inertial waves. This dual impact of rotation can be illustrated through two relevant physical cases. The first concerns internal gravity waves developed in a shallow rotating channel where the influence of rotation de- pends on its strength; see [5]. In the case of intense rotation, the influence is distinguishable from those of weak nonlinearity and dispersion; here the result is a transverse exponential decay of the solitary wave amplitude. However, when the rotation level becomes of the same order of magnitude as the nonlinearity and dispersion, the relevant governing equation is the rotation-modified Kadomstev-Petriashvili (KP) instead of the classical Korteweg–de Vries (KdV); see [5–7]. At the limit of infinitely large channel width, the KP equation reduces to the Ostrovsky equation, which was first derived to account for the effects of the earth’s rotation on oceanic internal solitary waves [8]. The second relates to the situation where rotation induces inertial waves as in the case of a rotating homogeneous fluid in a rigid long tube [9]. The last flow configuration indicates that solitary wave solutions of the KdV equation are possible in any swirling flow in which the angular velocity is nonuniformly distributed [5,9,10]. Inertial solitary waves have also been observed within the swirling flow produced in a large vertical cylindrical container. This wave is seen to propagate up and down between the bottom of the container and the free liquid surface, along the vortex core [9]. * haitabd@hotmail.com In this communication we report on yet another type of solitary wave within rotating flow. This wave is observed during the liquid drainage through a centrally located circular opening on the bottom of the container, when shallow water conditions are reached. The senior author initially reported the existence of this wave (only parenthetically) twenty-one years ago [11]. This wave manifestation is examined systematically via both theory and experiment. The experiments were performed in a Plexiglas cylindrical container of diameter D equal to 285 mm. The drainage hole of diameter d equal to 31.7 mm was located at the bottom center of the container. The tank was filled with water and two initial water heights, Hi = 435 and Hi = 270 mm, were considered. A blue water soluble food color was added to the fluid prior to the experiments for better visualization. A schematic of the experimental setup is shown in Fig. 1. Water within the tank was stirred with a rod, in a way similar to that of [9], at known rotations per minute (i.e., ω = 33, 60, 70, 92, 100, and 120 rpm), producing different initial swirls. Soon after, the rod was lifted and the bottom manifold was opened for water drainage. As the water level decreased, the amplitude of the oscillations of the free surface increased; these oscillations matured into a single surface bulge (solitary wave) when the shallow-water conditions were reached. At this time, we replugged the hole; therefore the water depth remained once again constant, and the solitary wave revolved around the cylindrical wall till it vanished. It is worth noting that a solitary wave could be observed only when either an initial swirl was imparted to the liquid or when a residual vorticity (to reservoir filling-up process) was present in the initial stages of the draining process. In fact, if the tank is filled and then left undisturbed for a while to calm down prior to draining (i.e., almost no residual vorticity), the solitary wave may not appear at all. The event was recorded using a CCD camera; the images were acquired at a rate of 90 Hz, respectively. The camera resolution was 1600 × 1200 and the shutter speed set equal to 1/500 seconds to avoid blurring effects. The image acquisition 037302-1 1539-3755/2011/84(3)/037302(4) ©2011 American Physical Society