Development of rigorous methods in fluid mechanics and theory of water waves

The thesis consists of five chapters where the following problems are considered - the transformation of long linear waves in an ocean with a variable depth; long wave scattering in a canal with a rapidly varying cross-section; long linear surface waves on stationary currents in a canal of constant depth and variable width; and a resonant interaction of a solitary wave with external pulse-type perturbations within the framework of forced Korteweg-de Vries equation. Chapter 1 reviews the history of these problems, and notes previous literature and research in this area. In Chapter 2, the transformation of long linear waves in an ocean of a variable depth is studied. The transformation coeffcients are considered as functions of frequency and the total depth drop for three typical models of bottom profile variation: piecewise linear, piecewise quadratic and hyperbolic tangent profiles. The results obtained for all these profiles are analysed and compared from the physical point of view. In Chapter 3, long wave scattering in a canal with a rapidly varying cross-section is studied. The scattering coeffcients are calculated for all possible incident wave orientations (background current downstream and upstream with respect to the background flow); and current types subcritical, transcritical, and supercritical. It is shown that when the over-reflection or over-transmission occurs, negative energy waves can appear in the flow. A spontaneous wave generation can happen in a transcritical accelerating flow, resembling a spontaneous wave generation on the horizon of an evaporating black hole due to the Hawking effect in astrophysics. In Chapter 4, long wave transformation is studied in the canal with abrupt variation of width and depth. Again, all possible wave orientation with respect to the background current is considered, and all types of a current is studied (subcritical, transcritical, and supercritical). The transformation coefficients are calculated and analysed as functions of wave frequency, Froud number, and depth drop. In Chapter 5, we revise the solutions of the forced Korteweg{de Vries equation for the resonant interaction of a solitary wave with the various external pulse-type perturbations. In contrast to the previous studies, we consider an arbitrary relationship between the width of a solitary wave and external forcing. In many cases, exact solutions of the forced Korteweg{de Vries equation can be obtained for the specific forcings of arbitrary amplitude. The theoretical outcomes obtained by asymptotic method are in good agreement with the results of direct numerical modelling within the framework of forced Korteweg-de Vries equation. In Conclusion, we summarise the results obtained within the various models and equations; discuss the future applications and innovation of the results; and suggest further perspectives for the future research

Local stability of a plate with a circular inclusion under tensile stress

The paper deals with the problem of the local buckling caused by uniaxial stretching of an infinite plate with a circular inclusion from a different material. The effect of elastic modulus of the inclusion on the value of the critical load is investigated. In order to find the first critical load a variational principle is applied. The comparison of numerical results that were obtained in the Maple 18 and the results obtained by the finite element method in the ANSYS 13.1. The influence is analysed the ratio of the elastic properties of the inclusion and plate on the value of the critical load and the form of the loss of stability

Scattering of long water waves in a canal with rapidly varying cross-section in the presence of a current

The analytical study of long-wave scattering in a canal with a rapidly varying cross-section is presented. It is assumed that waves propagate on a stationary current with a given flow rate. Due to the fixed flow rate, the current speed is different in the different sections of the canal, upstream and downstream. The scattering coefficients (the transmission and reflection coefficients) are calculated for all possible orientations of incident wave with respect to the background current (downstream and upstream propagation) and for all possible regimes of current (subcritical, transcritical, and supercritical). It is shown that in some cases negative energy waves can appear in the process of waves scattering. The conditions are found when the over-reflection and over transmission phenomena occur. In particular, it is shown that a spontaneous wave generation can arise in a transcritical accelerating flow, when the background current enhances due to the canal narrowing. This resembles a spontaneous wave generation on the horizon of an evaporating black hole due to the Hawking effect

Soliton interaction with external forcing within the Korteweg–de Vries equation

We revise the solutions of the forced Korteweg–de Vries equation describing a resonant interaction of a solitary wave with external pulse-type perturbations. In contrast to previous work where only the limiting cases of a very narrow forcing in comparison with the initial soliton or a very narrow soliton in comparison with the width of external perturbation were studied, we consider here an arbitrary relationship between the widths of soliton and external perturbation of a relatively small amplitude. In many particular cases, exact solutions of the forced Korteweg–de Vries equation can be obtained for the specific forcings of arbitrary amplitude. We use the earlier developed asymptotic method to derive an approximate set of equations up to the second-order on a small parameter characterising the amplitude of external force. The analysis of exact solutions of the derived equations is presented and illustrated graphically. It is shown that the theoretical outcomes obtained by the asymptotic method are in a good agreement with the results of direct numerical modeling within the framework of forced Korteweg–de Vries equation

Development of an analytical model for a cyclorotor wave energy device

We present a new analytical model for a horizontal axis cyclorotor-based wave energy converter (WEC). A number of cyclorotor-based WEC concepts and models, with different numbers of hydrofoils, have previously been studied. Our model is derived for a horizontal cyclorotor with 2 hydrofoils. The governing equations are optimised and converted to the polar coordinate system. The mechanical model is based on Newton’s second law for rotation. Rotation is considered in two-dimensional potential flow, for both monochromatic and panchromatic waves, including waves generated by the rotating rotor, and viscous losses. The developed model is very convenient for modelling, analysis and control design for a cyclorotor based WEC. The authors of this work have derived new, exact, analytic functions for the free surface perturbation and induced fluid velocity field caused by hydrofoil rotation. These new formulae significantly decrease the model calculation time, compared to previous models, and increase the accuracy of the results. We present the results of free rotation simulations for the rotor in monochromatic and panchromatic waves obtained with the use of the newly derived equations

Interaction of Kortewegâ-de Vries solitons with external sources

We consider the problem of interaction of a solitary wave with a moving external source within the framework of Korteweg– de Vries (KdV) equation. We show that for certain profiles of external source the problem has exact solutions in the form of a stationary solitary waves coupled with the force. For the solitary waves which are not trapped by the external force of a small amplitude we obtain approximate solutions by means of the asymptotic method and analyse solutions with the arbi- trary relationship between the widths of forcing function and solitary wave. Results obtained agree well with the results of previous works where only the limiting cases of very narrow or infinitely wide forcing as compared with the width of soli- tary wave were studied. Several new regimes of soliton interac- tion with width the forcing have been revealed. The theoretical results have been validated by the direct numerical modelling within the framework of forced KdV equation

Description of vortical flows of incompressible fluid in terms of quasi-potential function

It has been shown [1, 2] that a wide class of 3D motions of in- compressible viscous fluid in Cartesian coordinates can be de- scribed by only one scalar function dubbed the quasi-potential. This class of fluid flows is characterized by three-component velocity field having two-component vorticity field; both these fields can depend of all three spatial variables and time, in gen- eral. Governing equations for the quasi-potential have been de- rived and simple illustrative examples of 3D flows have been presented. In this paper the concept of quasi-potential is fur- ther developed for fluid flows in cylindrical coordinates. It is shown that the introduction of a quasi-potential in curvilinear coordinates is non-trivial and may be a subject of additional restrictions. In the cases when it is possible, we construct il- lustrative examples which can be of interest for some practical applications

Mathematical modelling of applanation tonometry for intraocular pressure measurements

Intraocular pressure (IOP) is an important aspect in the evaluation of patients at risk from glaucoma. Applanation tonometry estimates intraocular pressure by quantifying the force needed to generate a defined amount of deformation of the cornea (Goldmann Tonometry) or by estimating the diameter of the circular contact area of the cornea and flat tonometer of defined load (Maklakoff Tonometery). The geometrical parameters of eyes essentially vary for different people and change with age, different pathologies of vision or after refractive surgery. The corneal responses are not fully understood and predictable. It is now clear that mathematical modelling plays an important role in the analysis of the overall mechanical interactions between cornea and sclera. Proposed model can elucidate how scleral properties may play a role in determining IOP. In the developed model, the shell of an eye is considered to consist of two segments with different mechanical properties. The two-segments shell is filled with incompressible liquid under the pressure. Nonlinear theory of shells that takes into account normal and shear stresses, and normal strain is used to analyze deformations since the deformation of the shell part, which models the cornea, is significant. The effect of the geometrical and physical properties of the shell on the results of the modelling is studied. The model shows that tonometry readings do not always reflect true IOP values. For example, if IOP is not very high and after refractive surgery the thickness of the cornea is small the cornea may buckle and detach from the tonometer. This model permits to estimate the effect of parameters of a sclera on the change of the IOP after injections into a vitreous body. Considering nonlinear shells model could be used to explain different results of elastotonometry

Experimental evaluation of phase and velocity control for a cyclorotor wave energy converter

The research presented in the paper is dedicated to the analysis of the 3D experimental testing results of a 1:20 scale prototype LiftWEC cyclorotor wave energy converter (WEC). The scaled prototype was built and tested in the Hydraulic and Offshore Engineering wave Tank (HOET) by Ecole Centrale Nantes (ECN) in 2022. The analysis is conducted using the analytical control-oriented point-vortex model. The presented research covers a range of tests, with particular focus on cases where positive mechanical power generation has been recorded. The analysis of such cases is important, in highlighting the conditions needed for optimum energy conversion, for future development of cyclorotor WEC technology. The study also reviews the results of tests where the rotor rotational speed is varied within each period of monochromatic waves. This is the first experimental test of such a control strategy for cyclorotor WECs

Validation of a control-oriented point vortex model for a cyclorotor-based wave energy device

Recently conducted analytical assessment of the potential performance of cyclorotor wave energy converters (WECs) have shown that such devices offer the best wave absorption behaviour, if energy capture can be optimised through suitable control. Such claims require additional investigation. This article is dedicated to validation of the control-oriented point vortex model of cyclorotor WECs against numerical and experimental assessments conducted by various research groups. The validation is conducted in terms of the traditional metrics for cyclorotor WECs: (a) cancellation of incoming waves; (b) generation of lift and drag forces (c) mechanical power generation. It is shown that the point vortex model generally confirms the previously conducted analytical assessment of device performance. However, accounting for the influence of the hydrofoil induced wakes decreases performance estimates to some extent. It is also shown that, overall, wave cancellation metrics are more optimistic than actual shaft power generation. Analysis of the lift and drag coefficients, which were derived from experimental data, reveal a range of hydrodynamic and mechanic effects which could influence actual device performance. It has been shown that, due to the complexity of hydrodynamic effects, lift and drag coefficients for the control-oriented model should be considered not only as functions of the Reynolds number and angle of attack, but also related to submergence of the foils and direction of their rotation with respect to the free surface. This method allows us to achieve the best validation against experimental results in terms of generation of tangential and radial forces