Topological classification of affine mappings from ℝ<Superscript>2</Superscript> into ℝ<Superscript>2</Superscript>

We investigate affine mappings from ℝ² into ℝ² and establish necessary and sufficient conditions for the topological conjugacy of these mappings. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 472–480, October–December, 2008.     

Entropy Solutions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">∞</Emphasis></Superscript> for the Euler Equations in Nonlinear Elastodynamics and Related Equations

A compactness framework is established for approximate solutions to the Euler equations in one-dimensional nonlinear elastodynamics by identifying new properties of the Lax entropies, especially the higher order terms in the Lax entropy expansions, and by developing ways to employ these new properties in the method of compensated compactness. Then this framework is applied to establish the existence, compactness, and decay of entropy solutions in L for the Euler equations in nonlinear elastodynamics with a more general stress-strain relation than those for the previous existence results. This compactness framework is further applied to solving the Euler equations of conservation laws of mass, momentum, and energy for a class of thermoelastic media, and the equations of motion of viscoelastic media with memory.     

Chaos control and synchronization of the Φ<Superscript>6</Superscript>-Van der Pol system driven by external and parametric excitations

The dynamical behavior of the Φ⁶-Van der Pol system subjected to both external and parametric excitation is investigated. The effect of parametric excitation amplitude on the routes to chaos is studied by numerical analysis. It is found that the probability of chaos happening increases along with the parametric excitation amplitude increases while the external excitation amplitude fixed. Based on the invariance principle of differential equations, the system is lead to desirable periodic orbit or chaotic state (synchronization) with different control techniques. Numerical simulations are provided to validate the proposed method.     

Patterns and heat transfer in Rayleigh-Bénard thermal gravitational convection in helium (<Superscript>3</Superscript>He) near the critical point

Thermal gravitational convection in a bottom-heated layer of near-critical ³He is considered. The range of criteria determining the convection parameters beyond the stability threshold is discussed. The specific features of 2D and 3D supercritical structures, the adiabatic compression effect, and heat transfer are considered.     

Unsteady PTV velocity field past an airfoil solved with DNS: Part 1. Algorithm of hybrid simulation and hybrid velocity field at <Emphasis Type="Italic">Re</Emphasis> ≈ 10<Superscript>3</Superscript>

We develop a hybrid unsteady-flow simulation technique combining direct numerical simulation (DNS) and particle tracking velocimetry (PTV) and demonstrate its capabilities by investigating flows past an airfoil. We rectify instantaneous PTV velocity fields in a least-squares sense so that they satisfy the equation of continuity, and feed them to the DNS by equating the computational time step with the frame rate of the time-resolved PTV system. As a result, we can reconstruct unsteady velocity fields that satisfy the governing equations based on experimental data, with the resolution comparable to numerical simulation. In addition, unsteady pressure distribution can be solved simultaneously. In this study, particle velocities are acquired on a laser-light sheet in a water tunnel, and unsteady flow fields are reconstructed with the hybrid algorithm solving the incompressible Navier–Stokes equations in two dimensions. By performing the hybrid simulation, we investigate nominally two-dimensional flows past the NACA0012 airfoil at low Reynolds numbers. In part 1, we introduce the algorithm of the proposed technique and discuss the characteristics of hybrid velocity fields. In particular, we focus on a vortex shedding phenomenon under a deep stall condition (α = 15°) at Reynolds numbers of Re = 1000 and 1300, and compare the hybrid velocity fields with those computed with two-dimensional DNS. In part 2, the extension to higher Reynolds numbers is considered. The accuracy of the hybrid simulation is evaluated by comparing with independent experimental results at various angles of attack and Reynolds numbers up to Re = 10⁴. The capabilities of the hybrid simulation are also compared with two-dimensional unsteady Reynolds-Averaged Navier–Stokes (URANS) solutions in part 2. In the first part of these twin papers, we demonstrate that the hybrid velocity field approaches the PTV velocity field over time. We find that intensive alternate vortex shedding past the airfoil, which is predicted by the two-dimensional DNS, is substantially suppressed in the hybrid simulation and the resultant flow field is similar to the PTV velocity field, which is projection of the three-dimensional velocity field on the streamwise plane. We attempt to identify the motion that originates three-dimensional flow patterns by highlighting the deviation of the PTV velocity field from the two-dimensional governing equations at each snapshot. The results indicate that the intensive spots of the deviation appear in the regions in which three-dimensional instabilities are induced in the shear layer separated from the pressure side.     

Unsteady PTV velocity field past an airfoil solved with DNS: Part 2. Validation and application at Reynolds numbers up to <Emphasis Type="Italic">Re</Emphasis> ≤ 10<Superscript>4</Superscript>

Hybrid unsteady-flow simulation combining particle tracking velocimetry (PTV) and direct numerical simulation (DNS) is introduced in the series of two papers. Particle velocities on a laser-light sheet acquired with time-resolved PTV in a water tunnel are supplied to two-dimensional DNS with time intervals corresponding to the frame rate of the PTV. Hybrid velocity fields then approach those representing the PTV data in the course of time, and the reconstructed velocity fields satisfy the governing equations with the resolution comparable to numerical simulation. In part 2, by extending the capabilities of the hybrid simulation to higher Reynolds numbers, we simulate flows past the NACA0012 airfoil over ranges of Reynolds numbers (Re ≤ 10⁴) and angles of attack (−5° ≤ α ≤ 20°) and validate the proposed technique by comparing with experimental results in terms of the lift and drag coefficients. We also compare the results with unsteady Reynolds-averaged Navier–Stokes (URANS) simulation in two-dimensions and show the advantages of the hybrid simulation against two-dimensional URANS.     

A New Approach to Counterexamples to <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> Estimates: Korn’s Inequality,Geometric Rigidity,and Regularity for Gradients of Separately Convex Functions

The derivation of counterexamples to L¹ estimates can be reduced to a geometric decomposition procedure along rank-one lines in matrix space. We illustrate this concept in two concrete applications. Firstly, we recover a celebrated, and rather complex, counterexample by Ornstein, proving the failure of Korns inequality, and of the corresponding geometrically nonlinear rigidity result, in L¹. Secondly, we construct a function f:² which is separately convex but whose gradient is not in BV_loc, in the sense that the mixed derivative ²f/x₁x₂ is not a bounded measure.Acknowledgement We thank BERND KIRCHHEIM for bringing the question of regularity of separately convex functions to our attention. This work was partially supported by the EU Research Training Network Hyperbolic and Kinetic Equations, contract HPRN-CT-2002-00282, and by the Deutsche Forschungsgemeinschaft through the Schwerpunktprogramm 1095 Analysis, Modeling and Simulation of Multiscale Problems.     

On the Cover of this Issue: Cyclic Strain Heterogeneity and Damage Formation in Rolled Magnesium by N. Shafaghi,E. Kapan,and C.C. Aydıner

Experimental Mechanics -     

Identification of Local Viscoplastic Properties in P91 Welds from Full Field Measurements at Room Temperature and 625 °C

A methodology to obtain visco-plastic laws in heterogeneous materials with digital image correlation (DIC) is proposed based on tensile and tensile-relaxation tests conducted at room temperature and at 625 °C. Tested samples are manufactured from a P91 weld in which a microstructural heterogeneity translates into graded mechanical properties. To simplify the problem, a classical decomposition of the weld into five different domains is considered. Strain field in each domain is obtained by means of digital image correlation applied to high magnification pictures recorded with an optical long distance microscope. The conducted identifications exhibit key features in the behaviour of each domain in terms of yield stress, ultimate tensile stress and hardening at both room temperature and 625 °C. Experimental fields are compared to the fields provided by finite element simulations. Eventually, the benefit of accounting for transverse strains in the identification procedure is examined, and the robustness of the procedure with addition of noise (representative of experimental conditions) in the measurement is characterized.     

Effect of perforation on flow past a conic cylinder at <Emphasis Type="Italic">Re</Emphasis> = 100: vortex-shedding pattern and force history

The flow past a circular-section cylinder with a conic shroud perforated with four holes at the peak was simulated numerically at \(Re=100\), considering two factors, viz. the angle of attack and the diameter of the holes. The effects of the perforated conic shroud on the vortex shedding pattern in the near wake was mainly investigated, as well as the time history of the drag and lift forces. In the investigated parameter space, three flow regimes were generally identified, corresponding to weak, moderate, and strong disturbance effects. In regime I, the wake can mainly be described by alternately shedding Kármán or Kármán-like vortices. In regime II, the spanwise vortices are obviously disturbed along the span due to the appearance of additional vorticity components and their interactions with the spanwise vortices, but still shed in synchronization along the spanwise direction. In regime III, the typical Kármán vortices partially or totally disappear, and some new vortex shedding patterns appear, such as \(\Omega \)-type, obliquely shedding, and crossed spanwise vortices with opposite sign. Corresponding to these complex vortex shedding patterns in the near wake, the fluid forces no longer oscillate regularly at a single vortex shedding frequency, but rather with a lower modulation frequency and multiple amplitudes. An overview of these flow regimes is presented.