Leonardo da Vinci’s turbulent tank in two dimensions

• Published in: European journal of mechanics, B, Fluids Vol. 44; pp. 1 - 9
• Main Authors: Monaghan, J.J., Kajtar, J.B.
• Format: Journal Article
• Language: English
• Published: Issy-les-Moulineaux Elsevier Masson SAS 01.03.2014; Elsevier Masson
• Subjects: Entrainment; Exact sciences and technology; Flow into a tank; Fluid dynamics; Fundamental areas of phenomenology (including applications); Hydrodynamic waves; Physics; Simulation; Simulation; Entrainment; Flow into a tank; Water; Turbulent flow; Digital simulation; Free surface flow; Reservoir; Modelling
• ISSN: 0997-7546; 1873-7390
• DOI: 10.1016/j.euromechflu.2013.09.005

As part of his study of turbulence Leonardo da Vinci, in the years 1508–1513, illustrated the flow patterns produced by water from a channel entering a body of water from above its surface. Amongst the drawings is a study of the velocity streamlines. This figure shows that the flow within the tank consists of fluid moving into the tank then rising to the surface together with what appear to be curved streamlines ending in tight vortices around the main flow. In this paper we describe our simulations of the same problem but in two dimensions. We find that when the fluid from the channel exits with constant speed Vinj, the flow from the channel first produces a clearly defined narrow jet within the tank. If the velocity of the fluid leaving the channel Vinj is sufficiently low, the jet flows deep then emerges to flow over the wall of the container, and a large vortex is formed. If Vinj is sufficiently high the jet is still narrow, but it is confined to the upper half of the tank, where the flow becomes turbulent and may involve waves breaking. In this case the flow below the jet appears laminar and is in the form of a large but weak vortex. The transition from one state to the other occurs at speed Vinj that is a linear function of gL where L is the length of the tank and g the gravitational acceleration. The momentum in the same direction as the flow in the channel has a complex behaviour that depends on whether the narrow jet has deep penetration or is near the surface. The angular momentum around the centre of the tank may remain in the same direction or not depending on Vinj. Similar results are obtained for some inflows with speed that varies across the channel.