Two-dimensional dynamical problems for incompressible isotropic linear elastic solids with time dependent moduli and variable density

1968 ◽  
Vol 5 (2) ◽  
pp. 118-130 ◽  
Author(s):  
J. Golecki ◽  
A. Jeffrey
Keyword(s):  
Variable Density ◽  
Time Dependent ◽  
Two Dimensional ◽  
Elastic Solids ◽  
Linear Elastic

Optimal perturbations in time-dependent variable-density Kelvin–Helmholtz billows

2016 ◽  
Vol 803 ◽  
pp. 466-501 ◽  
Author(s):  
Adriana Lopez-Zazueta ◽  
Jérôme Fontane ◽  
Laurent Joly
Keyword(s):  
Saddle Point ◽  
Shear Flows ◽  
Longitudinal Velocity ◽  
Three Dimensional ◽  
Variable Density ◽  
Energy Gain ◽  
Time Dependent ◽  
Two Dimensional ◽  
Energy Growth ◽  
Atwood Number

We analyse the influence of the specific features of time-dependent variable-density Kelvin–Helmholtz (VDKH) roll-ups on the development of three-dimensional secondary instabilities. Due to inertial (high Froude number) baroclinic sources of spanwise vorticity at high Atwood number (up to 0.5 here), temporally evolving mixing layers exhibit a layered structure associated with a strain field radically different from their homogeneous counterpart. We use a direct-adjoint non-modal linear approach to determine the fastest growing perturbations over a single period of the time-evolving two-dimensional base flow during a given time interval $[t_{0},T]$. When perturbations are seeded at the initial time of the primary KH mode growth, i.e. $t_{0}=0$, it is found that additional mechanisms of energy growth are onset around a bifurcation time $t_{b}$, a little before the saturation of the primary two-dimensional instability. The evolution of optimal perturbations is thus observed to develop in two distinct stages. Whatever the Atwood number, the first period $[t_{0},t_{b}]$ is characterised by a unique route for optimal energy growth resulting from a combination of the Orr and lift-up transient mechanisms. In the second period $[t_{b},T]$, the growing influence of mass inhomogeneities raises the energy gain over the whole range of spanwise wavenumbers. As the Atwood number increases, the short spanwise wavelength perturbations tend to benefit more from the onset of variable-density effects than large wavelength ones. The extra energy gain due to increasing Atwood numbers relies on contributions from spanwise baroclinic sources. The resulting vorticity field is structured into two elongated dipoles localised along the braid on either side of the saddle point. In return they yield two longitudinal velocity streaks of opposite sign which account for most of the energy growth. This transition towards three-dimensional motions is in marked contrast with the classic streamwise rib vortices, so far accepted as the paradigm for the transition of free shear flows, either homogeneous or not. It is argued that the emergence of these longitudinal velocity streaks is generic of the transition in variable-density shear flows. Among them, the light round jet is known to display striking side ejections as a result of the loss of axisymmetry. The present analysis helps to renew the question of the underlying flow structure behind side jets, otherwise based on radial induction between pairs of counter-rotating longitudinal vortices (Monkewitz & Pfizenmaier, Phys. Fluids A, vol. 3 (5), 1991, pp. 1356–1361). Instead, it is more likely that side ejections would result from the convergence of the longitudinal velocity streaks near the braid saddle point. When the injection time is delayed so as to suppress the initial stage of energy growth, a new class of perturbations arises at low wavenumber with energy gains far larger than those observed so far. They correspond to the two-dimensional Kelvin–Helmholtz secondary instability of the baroclinically enhanced vorticity braid discovered by Reinaud et al. (Phys. Fluids, vol. 12 (10), pp. 2489–2505), leading potentially to another route to turbulence through a two-dimensional fractal cascade.

Deformation of Inhomogeneous Elastic Solids With Two-Dimensional Damage

2001 ◽  
Vol 68 (4) ◽  
pp. 528-536
Author(s):  
J. J. Luo ◽  
I. M. Daniel
Keyword(s):  
Periodic Structures ◽  
Elastic Behavior ◽  
Two Dimensional ◽  
Elastic Solids ◽  
Inhomogeneous Materials ◽  
Linear Elastic ◽  
Reinforced Composite ◽  
Macroscopic Deformation ◽  
Microscopic Damage ◽  
Microscopic Deformation

A general correlation is derived between macroscopic stresses/strains and microscopic deformation on the damage surfaces for inhomogeneous elastic solids with two-dimensional damage. Assuming linear elastic behavior for the undamaged materials, the macroscopic deformation associated with nonlinear strains, or damage strains, is shown to be the weighted sum of the microscopic deformations on the damage surfaces. For inhomogeneous materials with periodic structures (laminated composites, for example) and various identifiable damage modes, simple relations are derived between the macroscopic deformation and microscopic damage. When the number of identifiable damage modes is less than or equal to the number of relevant measurable macroscopic strains, the correlation can be used to evaluate the damage progression from simple macroscopic stress and strain measurements. The simple case of a unidirectional fiber-reinforced composite under longitudinal load is used to show how the results can help detect and characterize the damage using macroscopic measurements, without resorting to assumptions of detailed microscopic deformation mechanisms.

Modeling of Performance of Single Elastic Capsule in Time-Dependent Two-Dimensional Microchannel Flow

2009 ◽  
Vol 74 ◽  
pp. 143-146
Author(s):  
Gang Ma ◽  
Hua Li ◽  
Teng Yong Ng
Keyword(s):  
Fluid Velocity ◽  
Simulation Method ◽  
Time Dependent ◽  
Two Dimensional ◽  
Microchannel Flow ◽  
Membrane Mechanics ◽  
Linear Elastic ◽  
Microscale Flow ◽  
Agricultural Industries ◽  
Volume Method

The mechanical behavior of capsule in microscale flow is fundamentally important in physiology, pharmaceutical or agricultural industries, fluid mechanics and physics. In the present paper, the performance of two-dimensional capsule in time-dependent microchannel flow is investigated by a numerical simulation method, which combines the finite volume method for solving the fluid problem with the front tracking technique for capturing and tracking the capsule membrane. Initially circular capsule is considered in the simulations, and the capsule is modeled as liquid medium enclosed by a thin membrane, for which linear elastic properties are taken into consideration. The membrane mechanics is adopted to calculate membrane-fluid interaction, based on which the solution of fluid problem is obtained. The capsule membrane is advected explicitly according to fluid velocity results. Results of capsule position, capsule moving velocity, capsule shape, velocity field of fluid inside and outside the membrane can be obtained from the numerical simulations. Based on these results, the effects of time-dependent background fluid on capsule behavior are studied.

A modified Fourier series-based solution with improved rate of convergence for two-dimensional rectangular isotropic linear elastic solids

2021 ◽  
pp. 108128652110255
Author(s):  
Néstor Darío Barulich ◽  
Aharon Deutsch ◽  
Moshe Eisenberger ◽  
Luis Augusto Godoy ◽  
Patricia Mónica Dardati
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Rate Of Convergence ◽  
Rectangular Domain ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Elastic Solids ◽  
Linear Elastic ◽  
Present Solution ◽  
Stress States

This paper presents a new displacement solution based on a Modified Fourier Series (MFS) for isotropic linear elastic solids under plane strain or plane stress states subject to continuous displacement and traction boundary conditions in a two-dimensional rectangular domain. In contrast with existing approaches that are restricted to Fourier series with a rate of convergence of second order O(m-2), the MFS allows increasing the rate of convergence of the solution. The governing Partial Differential Equations (PDEs) are satisfied exactly by two displacement solutions while the boundary conditions are approximated after solving a finite system of algebraic equations. Numerical results for a solution with an MFS with rate of convergence O(m-3) are compared with results from existing numerical and analytical methods, showing the enhanced behavior of the present solution.

scholarly journals Integrable unsteady motion with an application to ocean eddies

1998 ◽  
Vol 5 (3) ◽  
pp. 145-151
Author(s):  
A. D. Kirwan, Jr. ◽  
B. L. Lipphardt, Jr.
Keyword(s):  
Potential Vorticity ◽  
Particle Motion ◽  
Gulf Stream ◽  
Incompressible Flows ◽  
Unsteady Motion ◽  
Ring System ◽  
Time Dependent ◽  
Two Dimensional ◽  
Ocean Eddies ◽  
Multilayered Systems

Abstract. Application of the Brown-Samelson theorem, which shows that particle motion is integrable in a class of vorticity-conserving, two-dimensional incompressible flows, is extended here to a class of explicit time dependent dynamically balanced flows in multilayered systems. Particle motion for nonsteady two-dimensional flows with discontinuities in the vorticity or potential vorticity fields (modon solutions) is shown to be integrable. An example of a two-layer modon solution constrained by observations of a Gulf Stream ring system is discussed.

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