An investigation into the existence of hairpin vortices in turbulent channel flow is conducted using a database generated by the large-eddy simulation technique. It is shown that away from the wall the distribution of the inclination angle of vorticity vector gains its maximum at about 45° to the wall. Two-point correlations of velocity and vorticity fluctuations strongly support a flow model consisting of vortical structures inclined at 45° to the wall. The instantaneous vorticity vectors plotted in planes inclined at 45° show that the flow contains an appreciable number of hairpins. Vortex lines are used to display the three-dimensional structure of hairpins, which are shown to be generated from deformation (or roll-up) of sheets of transverse vorticity.

The hydrodynamic stability of flows over Kramer-type compliant surfaces is studied. Two main types of instability are considered. First, there are those which could not exist without viscosity, termed Tollmien–Schlichting Type Instabilities (TSI). Secondly, there are Flow-Induced Surface Instabilities (FISI), that depend fundamentally on surface flexibility and could exist with an inviscid fluid flow. Part 1, the present paper, deals mainly with the first type. The original Kramer experiments and the various subsequent attempts to confirm his results are reviewed, together with experimental studies of transition in flows over compliant surfaces and theoretical work concerned with the hydrodynamic stability of such flows.

The Kramer-type compliant surface is assumed to be an elastic plate supported by springs which are modelled by an elastic foundation. It is also assumed that the plate is backed by a viscous fluid substrate having, in general, a density and viscosity different from the mainstream fluid. The motion of the substrate fluid is assumed to be unaffected by the presence of the springs and is determined by solving the linearized Navier–Stokes equations. The visco-elastic properties of the plate and springs are taken into account approximately by using a complex elastic modulus which leads to complex flexural rigidity and spring stiffness. Values for the various parameters characterizing the surface properties are estimated for the actual Kramer coatings.

The boundary-layer stability problem for a flexible surface is formulated in a similar way to that of Landahl (1962) whereby the boundary condition at the surface is expressed in terms of an equality between the surface and boundary-layer admittances. This form of the boundary condition is exploited to develop an approximate theory which determines whether a particular change to the mechanical properties of the surface will be stabilizing or destabilizing with respect to the TSI. It is shown that a reduction in flexural rigidity and spring stiffness, an increase in plate mass, and the presence of an inviscid fluid substrate are all stabilizing, whereas viscous and visco-elastic damping are destabilizing.

Numerical solutions to the Orr–Sommerfeld equation are also obtained. Apart from Kramer-type compliant surfaces, solutions are also presented for the rigid wall, for the spring-backed tensioned membrane with damping, previously considered by Landahl & Kaplan (1965), and for some of the compliant surfaces investigated experimentally by Babenko and his colleagues. The results for the Kramer-type compliant surfaces on the whole confirm the predictions of the simple theory. For a free-stream speed of 18 m/s the introduction of a viscous substrate leads to a complex modal interaction between the TSI and FISI. A single combined unstable mode is formed in the case of highly viscous substrate fluids and in this case increased damping has a stabilizing effect. When the free-stream speed is reduced to 15 m/s the modal interaction no longer occurs. In this case the effects of combined viscous and visco-elastic damping are investigated. It is found that damping tends to have a strong stabilizing effect on the FISI, in the form of travelling-wave flutter, but a weaker destabilizing effect on the TSI. The opposing effects of damping on the two modes of instability forms the basis of a possible explanation for Kramer's empirical observation of an optimum substrate viscosity. Results obtained using the e9 method also indicate that a substantial transition delay is theoretically possible for flows over Kramer's compliant coatings.

This paper describes the calculation of the added-mass and damping coefficients of a submerged toroidal body that is undergoing a forced, periodic heaving motion. The velocity potential of the motion is expressed as an infinite sum of toroidal multipole potentials, and the problem is solved in a manner analogous to Ursell's classical solution for a submerged circular cylinder in two dimensions. When the torus is ‘slender’, in the sense that its tubular radius is small compared with its overall diameter, relatively simple closed-form asymptotic approximations for the addedmass and damping coefficients are obtained. This work is motivated by the proposed RS-35 design of ring-hulled semisubmersible platform.