scholarly journals Boundary-layer Features in Limit of Conservative Nonlinear Oscillators

2021 ◽  
Author(s):  
Yair Zarmi
Keyword(s):  
Boundary Layer ◽  
Phase Space ◽  
Turning Points ◽  
Nonlinear Oscillators ◽  
High Amplitude ◽  
Step Function ◽  
Tooth Shape ◽  
Oscillatory Systems ◽  
Double Limit

Abstract In the double limit of high amplitude (xmax → ∞) and high leading power (x2 N+1, N → ∞), (1+1) dimensional conservative nonlinear oscillatory systems exhibit characteristics akin to boundary layer phenomena. The oscillating entity, x(t), tends to a periodic saw-tooth shape of linear segments, the velocity, x′(t), tends to a periodic step-function and the x − x′ phase-space plot tends to a rectangle. This is demonstrated by transforming x and t into proportionately scaled variables, η and θ, respectively. η(θ) is (2-π) periodic in θ and bounded (|η(θ)| ≤ 1). The boundary-layer characteristics show up by the fact that the deviations of η(θ), η′(θ) and the η − η′ phase-space plot from the sharp asymptotic shapes occurs over a range in θ of O(1/N) near the turning points of the oscillations.

scholarly journals Effects of Vegetation and Topography on the Boundary Layer Structure above the Amazon Forest

2020 ◽  
Vol 77 (8) ◽  
pp. 2941-2957
Author(s):  
Marcelo Chamecki ◽  
Livia S. Freire ◽  
Nelson L. Dias ◽  
Bicheng Chen ◽  
Cléo Quaresma Dias-Junior ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Phase Space ◽  
Vertical Structure ◽  
Layer Structure ◽  
Roughness Sublayer ◽  
Large Contribution ◽  
Amazon Forest ◽  
Two Dimensional ◽  
Dimensional Phase Space ◽  
Field Campaigns

Abstract Observational data from two field campaigns in the Amazon forest were used to study the vertical structure of turbulence above the forest. The analysis was performed using the reduced turbulent kinetic energy (TKE) budget and its associated two-dimensional phase space. Results revealed the existence of two regions within the roughness sublayer in which the TKE budget cannot be explained by the canonical flat-terrain TKE budgets in the canopy roughness sublayer or in the lower portion of the convective ABL. Data analysis also suggested that deviations from horizontal homogeneity have a large contribution to the TKE budget. Results from LES of a model canopy over idealized topography presented similar features, leading to the conclusion that flow distortions caused by topography are responsible for the observed features in the TKE budget. These results support the conclusion that the boundary layer above the Amazon forest is strongly impacted by the gentle topography underneath.

On the generation of steady streamwise streaks in flat-plate boundary layers

2012 ◽  
Vol 698 ◽  
pp. 211-234 ◽  
Author(s):  
Jens H. M. Fransson ◽  
Alessandro Talamelli
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
High Speed ◽  
Control Method ◽  
Plate Boundary ◽  
High Amplitude ◽  
Base Flow ◽  
Stream Velocity ◽  
Bypass Transition ◽  
Streamwise Streaks

AbstractA study on the generation and development of high-amplitude steady streamwise streaks in a flat-plate boundary layer is presented. High-amplitude streamwise streaks are naturally present in many bypass transition scenarios, where they play a fundamental role in the breakdown to turbulence process. On the other hand, recent experiments and numerical simulations have shown that stable laminar streamwise streaks of alternating low and high speed are also capable of stabilizing the growth of Tollmien–Schlichting waves as well as localized disturbances and to delay transition. The larger the streak amplitude is, for a prescribed spanwise periodicity of the streaks, the stronger is the stabilizing mechanism. Previous experiments have shown that streaks of amplitudes up to 12 % of the free stream velocity can be generated by means of cylindrical roughness elements. Here we explore the possibility of generating streaks of much larger amplitude by using a row of miniature vortex generators (MVGs) similar to those used in the past to delay or even prevent boundary layer separation. In particular, we present a boundary layer experiment where streak amplitudes exceeding 30 % have been produced without having any secondary instability acting on them. Furthermore, the associated drag with the streaky base flow is quantified, and it is demonstrated that the streaks can be reinforced by placing a second array of MVGs downstream of the first one. In this way it is possible to make the control more persistent in the downstream direction. It must be pointed out that the use of MVGs opens also the possibility to set up a control method that acts twofold in the sense that both transition and separation are delayed or even prevented.

THE DYNAMICS OF SYSTEMS OF COMPLEX NONLINEAR OSCILLATORS: A REVIEW

2004 ◽  
Vol 14 (11) ◽  
pp. 3821-3846 ◽  
Author(s):  
GAMAL M. MAHMOUD ◽  
TASSOS BOUNTIS
Keyword(s):  
Phase Space ◽  
Periodic Solutions ◽  
Chaotic Behavior ◽  
Nonlinear Oscillators ◽  
Complex Variables ◽  
Point Of View ◽  
Stability And Control ◽  
Ginzburg Landau Equations ◽  
The Stability ◽  
And Control

Dynamical systems in the real domain are currently one of the most popular areas of scientific study. A wealth of new phenomena of bifurcations and chaos has been discovered concerning the dynamics of nonlinear systems in real phase space. There is, however, a wide variety of physical problems, which, from a mathematical point of view, can be more conveniently studied using complex variables. The main advantage of introducing complex variables is the reduction of phase space dimensions by a half. In this survey, we shall focus on such classes of autonomous, parametrically excited and modulated systems of complex nonlinear oscillators. We first describe appropriate perturbation approaches, which have been specially adapted to study periodic solutions, their stability and control. The stability analysis of these fundamental periodic solutions, though local by itself, can yield considerable information about more global properties of the dynamics, since it is in the vicinity of such solutions that the largest regions of regular or chaotic motion are observed, depending on whether the periodic solution is, respectively, stable or unstable. We then summarize some recent studies on fixed points, periodic solutions, strange attractors, chaotic behavior and the problem of chaos control in systems of complex oscillators. Some important applications in physics, mechanics and engineering are mentioned. The connection with a class of complex partial differential equations, which contains such famous examples, as the nonlinear Schrödinger and Ginzburg–Landau equations is also discussed. These complex equations play an important role in many branches of physics, e.g. fluids, superconductors, plasma physics, geophysical fluids, modulated optical waves and electromagnetic fields.

scholarly journals Airfoil in a high amplitude oscillating stream

2016 ◽  
Vol 793 ◽  
pp. 79-108 ◽  
Author(s):  
C. Strangfeld ◽  
H. Müller-Vahl ◽  
C. N. Nayeri ◽  
C. O. Paschereit ◽  
D. Greenblatt
Keyword(s):  
Boundary Layer ◽  
High Frequency ◽  
Lift Coefficient ◽  
Closed Form Solution ◽  
Harmonic Oscillation ◽  
Vortex Sheet ◽  
High Amplitude ◽  
High Frequency Oscillation ◽  
Recirculation Bubble ◽  
Theoretical Approaches

A combined theoretical and experimental investigation was carried out with the objective of evaluating theoretical predictions relating to a two-dimensional airfoil subjected to high amplitude harmonic oscillation of the free stream at constant angle of attack. Current theoretical approaches were reviewed and extended for the purposes of quantifying the bound, unsteady vortex sheet strength along the airfoil chord. This resulted in a closed form solution that is valid for arbitrary reduced frequencies and amplitudes. In the experiments, the bound, unsteady vortex strength of a symmetric 18 % thick airfoil at low angles of attack was measured in a dedicated unsteady wind tunnel at maximum reduced frequencies of 0.1 and at velocity oscillations less than or equal to 50 %. With the boundary layer tripped near the leading edge and mid-chord, the phase and amplitude variations of the lift coefficient corresponded reasonably well with the theory. Near the maximum lift coefficient overshoot, the data exhibited an additional high-frequency oscillation. Comparisons of the measured and predicted vortex sheet indicated the existence of a recirculation bubble upstream of the trailing edge which sheds into the wake and modifies the Kutta condition. Without boundary layer tripping, a mid-chord bubble is present that strengthens during flow deceleration and its shedding produces a dramatically different effect. Instead of a lift coefficient overshoot, as per the theory, the data exhibit a significant undershoot. This undershoot is also accompanied by high-frequency oscillations that are characterized by the bubble shedding. In summary, the location of bubble and its subsequent shedding play decisive roles in the resulting temporal aerodynamic loads.

THE 2-D SEXTIC HAMILTONIAN OSCILLATOR

2013 ◽  
Vol 23 (06) ◽  
pp. 1330019
Author(s):  
F. J. MOLERO ◽  
J. C. VAN DER MEER ◽  
S. FERRER ◽  
F. J. CÉSPEDES
Keyword(s):  
Phase Space ◽  
Elliptic Functions ◽  
Singular Values ◽  
Nonlinear Oscillators ◽  
Momentum Mapping ◽  
Dimensional Model ◽  
Integrable Hamiltonian Systems ◽  
One Dimensional ◽  
Dimensional Phase Space ◽  
The One

The 2-D sextic oscillator is studied as a family of axial symmetric parametric integrable Hamiltonian systems, presenting a bifurcation analysis of the different flows. It includes the "elliptic core" model in 1-D nonlinear oscillators, recently proposed in the literature. We make use of the energy-momentum mapping, which will give us the fundamental fibration of the four-dimensional phase space. Special attention is given to the singular values of the energy-momentum mapping connected with rectilinear and circular orbits. They are related to the saddle-center and pitchfork scenarios with the associated homoclinic and heteroclinic trajectories. We also study how the geometry of the phase space evolves during the transition from the one-dimensional to the two-dimensional model. Within an elliptic function approach, the solutions are given using Legendre elliptic integrals of the first and third kind and the corresponding Jacobi elliptic functions.

ON AUTOPARAMETRIC ROUTE LEADING TO CHAOS IN DYNAMICAL SYSTEMS

2002 ◽  
Vol 12 (04) ◽  
pp. 819-826 ◽  
Author(s):  
S. N. VLADIMIROV ◽  
V. V. NEGRUL
Keyword(s):  
Numerical Simulation ◽  
Phase Space ◽  
Dynamical Systems ◽  
Equilibrium State ◽  
Dynamic Systems ◽  
Strange Attractor ◽  
Oscillatory Systems ◽  
Physical Experiments ◽  
Bifurcation Phenomena ◽  
Infinite Dimensionality

Features of transition from regular types of oscillations to chaos in dynamic systems with finite and infinite dimensionality of phase space have been discussed. It has been found that for some types of nonlinearity, transition to the chaotic motion in these systems occurs according to the identical autoparametric scenario. The sequence of bifurcation phenomena looks as follows: equilibrium state ⇒ limit cycle ⇒ semitorus ⇒ strange attractor. On the basis of the results of numerical simulation a conclusion was made about the typical nature of such a scenario. The results of numerical calculations are confirmed by results of physical experiments carried out on the base of radiophysical self-oscillatory systems.

scholarly journals Investigation of the Equivalent Representation Form of Strongly Damped Nonlinear Oscillators by a Nonlinear Transformation Approach

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Alex Elías-Zúñiga ◽  
Oscar Martínez-Romero
Keyword(s):  
Equations Of Motion ◽  
Nonlinear Oscillators ◽  
Transformation Method ◽  
Nonlinear Transformation ◽  
Equivalent Representation ◽  
Oscillatory Systems ◽  
Representation Form ◽  
Engineering Problems ◽  
Linear And Nonlinear ◽  
Strongly Damped

We use a nonlinear transformation method to develop equivalent equations of motion of nonlinear homogeneous oscillatory systems with linear and nonlinear odd damping terms. We illustrate the applicability of our approach by using the equations of motion that arise in many engineering problems and compare their amplitude-time curves with those obtained by the numerical integration solutions of the original equations of motion.

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