On the plane problem of orthotropic quasi-static thermoelasticity

The plane displacement boundary value problem of quasi-static linear orthotropic thermoelasticity is discussed. The thermoelastic system on a bounded simply-connected domain is decoupled. The decoupled temperature equation is investigated by using an accurate estimate and the contractive mapping principle. Representation of solution of the field equation is obtained, and some solvability results are proved. The results are of both theoretical and numerical interest.     

Rankine combined vortex interaction with a rectangular prism

Large eddy simulation is utilised to study the three-dimensional interaction between a travelling Rankine combined vortex and a rectangular prism. The study examines the strength and the topology of a vortex during the interaction with a prism that is much wider than the vortex core diameter. The physics of the interaction is revealed for the straight (β = 0°) and the oblique (β = 45°) impacts. For both cases, the low-level portion of the vortex undergoes displacements in the streamwise and the lateral directions. Also the vortex shape and the core vorticity are substantially disrupted. Behind the prism the full vortex circulation is recovered after a considerable distance. This created a low-velocity region. The sheltering effect of the prism is noticed for both straight and oblique impacts. The flow velocities in the sheltering region, right behind the prism, are reduced by more than 42% compared to the maximum flow speeds before the interaction.     

On the interaction of a bubble and a vortex ring at high Reynolds numbers

The interaction of a bubble and a vortex ring at high Reynolds numbers could be considered a simplified model of the interaction of a bubble and a turbulent structure of similar size, with the possible subsequent bubble breakup. In this paper, some results from axisymmetric and 3D simulations of the interaction of a bubble and a vortex ring at high Reynolds numbers are presented for different values of the Weber number and vortex ring sizes. Some bubble breakup patterns that could not be obtained by previous axisymmetric boundary integral models are shown. Results obtained are discussed into the framework of the classical Kolgomorov–Hinze theory on bubble breakup and some recent experimental investigations.     

The interaction of a bluff body with a vortex wake

A theoretical, experimental and numerical study is presented of the interaction of a vortex–wake created by an upstream blade with a downstream prismatic block. The aim of the study is to investigate the fundamentals of force and noise generation for this type of flow and explain how inter-object spacing affects the far-field noise level. A theoretical model, based on a compact form of Curle's formulation, is developed and shows that acoustically constructive or destructive interference is determined by the amplitude and phase of the forces on each object. Experimental and two-dimensional, unsteady numerical results of the vortex–wake interaction case are presented for several blade–block separation distances. Using a combination of experimental and numerical data, the theoretical model is able to explain observed variations in far-field noise level with blade–block separation distance. The numerical model accurately predicts the phase relationship between the unsteady forces on each object.     

Nonlinear interaction of a large vortex with a boundary layer

The nonlinear problem of boundary layer instability under the influence of a plane vortex is investigated for high Reynolds numbers. The vortex occupies the entire thickness of the boundary layer and has a longitudinal dimension of the order of the Tollmien-Schlichting wavelength. The initial vortex is rapidly swept away by the flow, inducing a Stokes layer near the surface of the plate. Expanding, this layer reaches the dimensions of the viscous sublayer of free interaction theory, where wave packet generation takes place. In the case in question a feature of the nonlinear stage of development of the disturbances is the formation of a concentrated vortex, which arises in the Stokes layer and grows rapidly, whereas the wave packet propagated ahead of it remains linear. From the calculations there emerges a tendency for the new vortex to be formed above the wail, whereas the maximum vorticity of the vortex generated in the Stokes layer corresponds to the wall itself.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 70–77, January–February, 1993.The authors are grateful to V. V. Kozlov for his interest in their work.     

On the sign of the internal stresses in a problem of plane elasticity

An elastic plane body, shaped like a circular ring or a disk, is subjected to radial surface forces varying according to a sinusoidal law. The existence of a tensile region included in a purely compressive one is proven and its asymptotic behaviour studied when the surface forces converge to two concentrated loads acting along the same diameter. The results show that any maximum principle for the principal stress components would require conditions stronger than the simple negativity of both on the boundary of the domain.     

On the three-dimensional nature of the orthogonal blade–vortex interaction

An orthogonal blade–vortex interaction has been visualised using stereo particle image velocimetry. Significant changes to the vortex axial flow w component velocity are observed during the interaction, with a deceleration on the lower surface of the blade where the vortex axial flow is towards the blade surface. Over this surface the interaction process close to the blade surface spreads the vorticity out to the areas of oppositely signed blade w component, and the results suggest a non-uniform spreading over the leading edge region of the blade, with a tendency for a spanwise transport of vorticity. Over the upper surface of the blade, the vortex axial flow velocity increases and the vortex core shrinks slightly. During the lower surface interaction the vorticity and velocity vectors become significantly realigned with respect to one another, while this is not observed for the upper surface interaction.     

Numerical investigation of the interaction of a vortex dipole with a deformable plate

Energy harvesting from coherent fluid structures is a current research topic due to its application in the design of small self-powered sensors for underwater applications. The impact of a vortex dipole with a deformable cantilevered plate at the plate tip is herein studied numerically using a strongly coupled staggered fluid–structure interaction algorithm. Three dipole Reynolds numbers, Re=500, 1500, and 3000, are investigated for constant plate properties. As the dipole approaches the plate, positive vorticity is produced on the impact face, while negative vorticity is generated at the tip of the plate. Upon impact, the dipole splits into two, and two secondary dipoles are formed. The circulation and, therefore, the trajectories of these dipoles depend on both the Reynolds number and the elasticity of the plate, and these secondary dipoles may return for subsequent impacts. While the maximum deflection of the plate does not depend significantly on Reynolds number, the plate response due to subsequent impacts of secondary dipoles does vary with Reynolds number. These results elucidate the strong interdependency between plate deformation and vortex dynamics, as well as the effect of Reynolds number on both.     

Acoustic emission from interaction of a vortex ring with a sphere

Acoustic waves emitted by a vortex ring interacting with a fixed solid sphere are studied experimentally and theoretically. The experiments are carried out for two kindsof vortex-sphere arrangement: (A) a vortex ring passes over the sphere, and (B) a vortex ring passes by the sphere. The vortex motion is examined optically by means of a photosensor system, and the pressure signals of the emitted wave are detected by 1/2-inch microphones in the far field. In case A, the measured diameter of the vortex ring after passing the sphere increases from its initial diameter. The observed acoustic wave is dominated mainly by a dipole emission, and some contribution from a quadrupole radiation is present. In case B, the emitted wave is characterized by a rotating dipole emission in which the dipole axis rotates as the vortex position changes relative to the sphere.     

The nonlinear interaction of vortex rings with a free surface

Nonlinear interactions of vortex rings with a free surface are considered in an incompressible, ideal fluid using the vortex contour dynamics technique and the boundary integral equation method. The flow is axisymmetric and the vorticity is linearly distributed in the vortex. Effects of the gravity and the surface tension as well as the initial geometric parameter of the vortex on the interaction process are investigated in considerable detail. The interaction process may be divided into three major stages: the vortex free-traveling stage, the collision stage, and the vortex stretching and rebounding stage. Time evolutions of both the vortex and free surface under various conditions are provided and analyzed. Two kinds of waves exist on the free surface during interaction. In a special case where the gravity and surface tension are very weak or the vortex is very strong, an electric-bulb-like ‘cavity’ is formed on the free surface and the vortex is trapped in the ‘cavity’ for quite a long time, resulting in a large amount of fluid above the mean fluid surface. The project supported by the National Education Commission of China and NASA under cooperative grant agreement # NCC5-34     

On the solution singularity in the plane elasticity problem for a cantilever strip

The plane elasticity problem of bending of a cantilever strip whose material is assumed to be incompressible in the transverse direction is solved. It is shown that, in the classical statement of of the boundary condition for the fixed edge of the strip, the solution has a singularity at the corner points of the edge. Several cases of the strip fixation and loading characterized by the presence or absence of the solution singularity are considered. The strength of glass beams of three types, for which the theory of elasticity predicts whether the normal stress has a singularity, is studied experimentally. It is shown that the limit stresses for the beams of the types under study are practically the same, which testifies that the solution singularity does not have any physical nature.     

Approximate two-dimensional equations of vortex flow in a plane layer with solid boundaries

Approximate two-dimensional equations governing turbulent vortex flows in plane fluid layers are considered. The equations were derived by the author in his earlier studies using the shallow water approximation and neglecting circulatory flows in the layer cross-sections. It is shown that, due to the centrifugal effect in the vortex flow, return flows in the layer cross-sections have only a slight influence on the fluid flow in the plane layer and can be neglected.     

Propeller tip and hub vortex dynamics in the interaction with a rudder

In the present paper, the interaction mechanisms of the vortices shed by a single-screw propeller with a rudder installed in its wake are addressed; in particular, following the works by Felli et al. (Exp Fluids 6(1):1–11, 2006a, Exp Fluids 46(1):147–1641, 2009a, Proceedings of the 8th international symposium on particle image velocimetry: Piv09, Melbourne, 2009b), the attention is focused on the analysis of the evolution, instability, breakdown and recovering mechanisms of the propeller tip and hub vortices during the interaction with the rudder. To investigate these mechanisms in detail, a wide experimental activity consisting in time-resolved visualizations, velocity measurements by particle image velocimetry (PIV) and laser Doppler velocimetry (LDV) along horizontal chordwise, vertical chordwise and transversal sections of the wake have been performed in the Cavitation Tunnel of the Italian Navy. Collected data allows to investigate the major flow features that distinguish the flow field around a rudder operating in the wake of a propeller, as, for example, the spiral breakdown of the vortex filaments, the rejoining mechanism of the tip vortices behind the rudder and the mechanisms governing the different spanwise misalignment of the vortex filaments in the pressure and suction sides of the appendage.     

On mechanism of self-oscillatory interaction of plane subsonic jet with acoustic resonator

The interaction of a plane subsonic jet with an acoustic Helmholtz resonator is considered. Visualization of the flow pattern is conducted using a schlieren technique. Oscillations of pressure in the resonator cavity are measured. The dependence of frequency and amplitude of oscillations on the jet velocity is analyzed. A simple closed model of the self-oscillatory process is developed that makes it possible to calculate the frequency and amplitude of the self-oscillations and to determine the intervals within which induction of the proper mode of oscillation occurs. It is shown that the finite amplitude of oscillation is determined primarily by a convolution of vortices in the jet.