scholarly journals Modelling nonlinear hydroelastic waves

2011 ◽  
Vol 369 (1947) ◽  
pp. 2942-2956 ◽  
Author(s):  
P. I. Plotnikov ◽  
J. F. Toland
Keyword(s):  
Wave Speed ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Lagrangian Description ◽  
Two Dimensional ◽  
Cosserat Theory ◽  
Elastic Sheet ◽  
Eulerian Description ◽  
Eulerian Coordinates ◽  
Special Case

This paper uses the special Cosserat theory of hyperelastic shells satisfying Kirchoff’s hypothesis and irrotational flow theory to model the interaction between a heavy thin elastic sheet and an infinite ocean beneath it. From a general discussion of three-dimensional motions, involving an Eulerian description of the flow and a Lagrangian description of the elastic sheet, a special case of two-dimensional travelling waves with two wave speed parameters, one for the sheet and another for the fluid, is developed only in terms of Eulerian coordinates.

Three-dimensional disturbances to a mixing layer in the nonlinear critical-layer regime

1995 ◽  
Vol 291 ◽  
pp. 57-81 ◽  
Author(s):  
S. M. Churilov ◽  
I. G. Shukhman
Keyword(s):  
Travelling Waves ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Critical Layer ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
New Type ◽  
Two Stages

We consider the nonlinear spatial evolution in the streamwise direction of slightly three-dimensional disturbances in the form of oblique travelling waves (with spanwise wavenumber kz much less than the streamwise one kx) in a mixing layer vx = u(y) at large Reynolds numbers. A study is made of the transition (with the growth of amplitude) to the regime of a nonlinear critical layer (CL) from regimes of a viscous CL and an unsteady CL, which we have investigated earlier (Churilov & Shukhman 1994). We have found a new type of transition to the nonlinear CL regime that has no analogy in the two-dimensional case, namely the transition from a stage of ‘explosive’ development. A nonlinear evolution equation is obtained which describes the development of disturbances in a regime of a quasi-steady nonlinear CL. We show that unlike the two-dimensional case there are two stages of disturbance growth after transition. In the first stage (immediately after transition) the amplitude A increases as x. Later, at the second stage, the ‘classical’ law A ∼ x2/3 is reached, which is usual for two-dimensional disturbances. It is demonstrated that with the growth of kz the region of three-dimensional behaviour is expanded, in particular the amplitude threshold of transition to the nonlinear CL regime from a stage of ‘explosive’ development rises and therefore in the ‘strongly three-dimensional’ limit kz = O(kx) such a transition cannot be realized in the framework of weakly nonlinear theory.

On analyticity of travelling water waves

2005 ◽  
Vol 461 (2057) ◽  
pp. 1283-1309 ◽  
Author(s):  
David P Nicholls ◽  
Fernando Reitich
Keyword(s):  
Water Waves ◽  
Bifurcation Theory ◽  
Amplitude Ratio ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Perturbation Parameter ◽  
Boundary Perturbation ◽  
Classical Boundary ◽  
Boundary Model ◽  
Special Case

In this paper we establish the existence and analyticity of periodic solutions of a classical free-boundary model of the evolution of three-dimensional, capillary–gravity waves on the surface of an ideal fluid. The result is achieved through the application of bifurcation theory to a boundary perturbation formulation of the problem, and it yields analyticity jointly with respect to the perturbation parameter and the spatial variables. The travelling waves we find can be interpreted as resulting from the (nonlinear) interaction of two two-dimensional wavetrains, giving rise to a periodic travelling pattern. Our analyticity theorem extends the most sophisticated results known to date in the absence of resonance; ‘short crested waves’, which result from the interaction of two wavetrains with unit amplitude ratio are realized as a special case. Our method of proof also sheds light on the convergence and conditioning properties of classical boundary perturbation methods for the numerical approximation of travelling surface waves. Indeed, we demonstrate that the rather unstable numerical behaviour of these approaches can be attributed to the strong but subtle cancellations in the formulas underlying their classical implementations. These observations motivate the derivation and use of an alternative, stable, formulation which, in addition to providing our method of proof, suggests new stabilized implementations of boundary perturbation algorithms.

The development of three-dimensional wave-packets in unbounded parallel flows

1968 ◽  
Vol 32 (4) ◽  
pp. 801-808 ◽  
Author(s):  
M. Gaster ◽  
A. Davey
Keyword(s):  
Wave Packet ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Mean Flow ◽  
Physical Plane ◽  
Parallel Flows ◽  
Flow Velocity Profile ◽  
The Stability ◽  
Special Case ◽  
Dimensional Wave

In this paper we examine the stability of a two-dimensional wake profile of the form u(y) = U∞(1 – r e-sy2) with respect to a pulsed disturbance at a point in the fluid. The disturbed flow forms an expanding wave packet which is convected downstream. Far downstream, where asymptotic expansions are valid, the motion at any point in the wave packet is described by a particular three-dimensional wave having complex wave-numbers. In the special case of very unstable flows, where viscosity does not have a significant influence, it is possible to evaluate the three-dimensional eigenvalues in terms of two-dimensional ones using the inviscid form of Squire's transformation. In this way each point in the physical plane can be linked to a particular two-dimensional wave growing in both space and time by simple algebraic expressions which are independent of the mean flow velocity profile. Computed eigenvalues for the wake profile are used in these relations to find the behaviour of the wave packet in the physical plane.

Crystallography of quasi-crystals

1986 ◽  
Vol 42 (4) ◽  
pp. 261-271 ◽  
Author(s):  
T. Janssen
Keyword(s):  
Three Dimensional ◽  
Three Dimensions ◽  
Crystal Phase ◽  
Two Dimensional ◽  
Space Group ◽  
Quasi Crystals ◽  
Special Case

The symmetry of quasi-crystals, a class of materials that has recently aroused interest, is discussed. It is shown that a quasi-crystal is a special case of an incommensurate crystal phase and that it can be described by a space group in more than three dimensions. A number of relevant three-dimensional quasi-crystals is discussed, in particular dihedral and icosahedral structures. The symmetry considerations are also applied to the two-dimensional Penrose patterns.

scholarly journals Thermocapillary instabilities in a horizontal liquid layer under partial basal slip

2018 ◽  
Vol 855 ◽  
pp. 839-859 ◽  
Author(s):  
Katarzyna N. Kowal ◽  
Stephen H. Davis ◽  
Peter W. Voorhees
Keyword(s):  
Liquid Layer ◽  
Basal Slip ◽  
Marangoni Number ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Partial Slip ◽  
Two Dimensional ◽  
Slip Parameter ◽  
Rigid Surface ◽  
Selection Of

We investigate the onset of three-dimensional hydrothermal waves in a low-capillary-number liquid layer of arbitrary depth, bounded by a free liquid–gas interface from above and a partial slip, rigid surface from below. A selection of two- and three-dimensional hydrothermal waves, longitudinal rolls and longitudinal travelling waves, form the preferred mode of instability, which depends intricately on the magnitude of the basal slip. Partial slip is destabilizing for all modes of instability. Specifically, the minimal Marangoni number required for the onset of instability follows $M_{m}\sim a(\unicode[STIX]{x1D6FD}^{-1}+b)^{-c}$ for each mode, where $a,b,c>0$ and $\unicode[STIX]{x1D6FD}^{-1}$ is the slip parameter. In the limit of free slip, longitudinal travelling waves disappear in favour of longitudinal rolls. With increasing slip, it is common for two-dimensional hydrothermal waves to exchange stability in favour of longitudinal rolls and oblique hydrothermal waves. Two types of oblique hydrothermal waves appear under partial slip, which exchange stability with increasing slip. The oblique mode that is preferred under no slip persists and remains near longitudinal for small slip parameters.

scholarly journals Nonlinear dynamics of surfactant-laden two-fluid Couette flows in the presence of inertia

2016 ◽  
Vol 802 ◽  
pp. 5-36 ◽  
Author(s):  
A. Kalogirou ◽  
D. T. Papageorgiou
Keyword(s):  
Stokes Equations ◽  
Travelling Waves ◽  
Type Theorem ◽  
Three Dimensional ◽  
Two Dimensional ◽  
One Dimensional ◽  
Wide Range ◽  
Time Periodic ◽  
Two Fluid ◽  
Permanent Form

The nonlinear stability of immiscible two-fluid Couette flows in the presence of inertia is considered. The interface between the two viscous fluids can support insoluble surfactants and the interplay between the underlying hydrodynamic instabilities and Marangoni effects is explored analytically and computationally in both two and three dimensions. Asymptotic analysis when one of the layers is thin relative to the other yields a coupled system of nonlinear equations describing the spatio-temporal evolution of the interface and its local surfactant concentration. The system is non-local and arises by appropriately matching solutions of the linearised Navier–Stokes equations in the thicker layer to the solution in the thin layer. The scaled models are used to study different physical mechanisms by varying the Reynolds number, the viscosity ratio between the two layers, the total amount of surfactant present initially and a scaled Péclet number measuring diffusion of surfactant along the interface. The linear stability of the underlying flow to two- and three-dimensional disturbances is investigated and a Squire’s type theorem is found to hold when inertia is absent. When inertia is present, three-dimensional disturbances can be more unstable than two-dimensional ones and so Squire’s theorem does not hold. The linear instabilities are followed into the nonlinear regime by solving the evolution equations numerically; this is achieved by implementing highly accurate linearly implicit schemes in time with spectral discretisations in space. Numerical experiments for finite Reynolds numbers indicate that for two-dimensional flows the solutions are mostly nonlinear travelling waves of permanent form, even though these can lose stability via Hopf bifurcations to time-periodic travelling waves. As the length of the system (that is the wavelength of periodic waves) increases, the dynamics becomes more complex and includes time-periodic, quasi-periodic as well as chaotic fluctuations. It is also found that one-dimensional interfacial travelling waves of permanent form can become unstable to spanwise perturbations for a wide range of parameters, producing three-dimensional flows with interfacial profiles that are two-dimensional and travel in the direction of the underlying shear. Nonlinear flows are also computed for parameters which predict linear instability to three-dimensional disturbances but not two-dimensional ones. These are found to have a one-dimensional interface in a rotated frame with respect to the direction of the underlying shear and travel obliquely without changing form.

Gradients of Distortion Seen in the Context of the Ponzo Illusion and Other Contours

1968 ◽  
Vol 20 (2) ◽  
pp. 212-217 ◽  
Author(s):  
Gerald H. Fisher
Keyword(s):  
Three Dimensional ◽  
Angular Size ◽  
Ponzo Illusion ◽  
Two Dimensional ◽  
Spatial Distortion ◽  
Special Case

It is suggested that the spatial distortion evident in the Ponzo figure is a special case of a more general illusion in which a gradient of attenuation appears within areas bounded by angular brackets. The magnitude of this gradient is measured in five lines seen against a number of angular contexts. A similar gradient appears also in the presence of single oblique lines. Accordingly, it is suggested that the distortions seen in the figures usually referred to as “the angle illusions” depend upon the presence of contours which do not necessarily define angles. The implications of these findings for certain existing theories which suggest that some illusions depend upon apparent-distortion of angular size and that they contain features usually associated with two-dimensional perspective projections of typical three-dimensional scenes are discussed.

New Measuring Method of Three-Dimensional Residual Stresses in Long Welded Joints Using Inherent Strains as Parameters—Lz Method

1989 ◽  
Vol 111 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Yukio Ueda ◽  
Keiji Fukuda
Keyword(s):  
Residual Stress ◽  
Residual Stresses ◽  
Welded Joint ◽  
Present Report ◽  
Three Dimensional ◽  
Longitudinal Direction ◽  
Measuring Method ◽  
Residual Stress Distribution ◽  
Two Dimensional ◽  
Special Case

In this paper, a new measuring method of three-dimensional residual stresses induced in a butt welded joint is presented. The proposed method is based on a general approach developed by the authors, in which inherent strains (the source of residual stresses) are dealt with as parameters. In the present report, three-dimensional residual stresses in a long body, in which the residual stress distribution is uniform in longitudinal direction, is considered as a special case. It is shown that the measurement of the three-dimensional residual stress, in this case, can be reduced to a combination of two sets of measurements of two-dimensional residual stresses. This method is applied to determine the residual stresses in an actual welded joint and its reliability and practicability are also demonstrated.

New results in rotating Hagen–Poiseuille flow

2000 ◽  
Vol 417 ◽  
pp. 103-126 ◽  
Author(s):  
D. R. BARNES ◽  
R. R. KERSWELL
Keyword(s):  
Hopf Bifurcation ◽  
Couette Flow ◽  
Poiseuille Flow ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Axial Rotation ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Rotation Rates ◽  
Supercritical Hopf Bifurcation

New three-dimensional finite-amplitude travelling-wave solutions are found in rotating Hagen–Poiseuille flow (RHPF[Ωa, Ωp]) where fluid is driven by a constant pressure gradient along a pipe rotating axially at rate Ωa and at Ωp about a perpendicular diameter. For purely axial rotation (RHPF[Ωa, 0]), the two-dimensional helical waves found by Toplosky & Akylas (1988) are found to become unstable to three-dimensional travelling waves in a supercritical Hopf bifurcation. The addition of a perpendicular rotation at low axial rotation rates is found only to stabilize the system. In the absence of axial rotation, the two-dimensional steady flow solution in RHPF[0, Ωp] which connects smoothly to Hagen–Poiseuille flow as Ωp → 0 is found to be stable at all Reynolds numbers below 104. At high axial rotation rates, the superposition of a perpendicular rotation produces a ‘precessional’ instability which here is found to be a supercritical Hopf bifurcation leading directly to three-dimensional travelling waves. Owing to the supercritical nature of this primary bifurcation and the secondary bifurcation found in RHPF[Ωa, 0], no opportunity therefore exists to continue these three-dimensional finite-amplitude solutions in RHPF back to Hagen–Poiseuille flow. This then contrasts with the situation in narrow-gap Taylor–Couette flow where just such a connection exists to plane Couette flow.

Exact connections between polycrystal and crystal properties in two-dimensional polycrystalline aggregates

1994 ◽  
Vol 447 (1929) ◽  
pp. 1-22 ◽  
Keyword(s):  
Three Dimensional ◽  
Local Fields ◽  
Field Analysis ◽  
Uniform Field ◽  
Two Dimensional ◽  
Special Cases ◽  
Crystal Properties ◽  
Polycrystalline Aggregates ◽  
Special Case ◽  
Effective Constants

Exact connections are shown to exist between the properties of two-dimensional polycrystalline aggregates and those of its constituent elongated crystals. The analysis is given for piezoelectric crystals and polycrystals. Both the crystal and the polycrystal are assumed to belong to the 2 mm class of the orthorhombic system. Classes that are special cases of 2 mm crystals are also admitted. The corresponding results for purely elastic aggregates, hitherto unknown, are obtained as a special case. The majority of the derived results are an outcome of uniform fields in the polycrystals considered, whose existence is established in this paper. In addition, these fields allow the derivation of certain correspondence relations between the pointwise local fields in the polycrystal, when it is subjected to certain electromechanical loadings. Exact connections for a subclass of the effective constants which are not amenable to the uniform field analysis are obtained by a matrix diagonalization formalism. It is shown that uniform fields and local correspondence relations exist also in three-dimensional elastic polycrystalline aggregates with tetragonal, hexagonal or trigonal crystals.

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