scholarly journals Geometrical Classification of Self-Similar Motion of Two-Dimensional Three Point Vortex System by Deviation Curvature on Jacobi Field

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Yuma Hirakui ◽  
Takahiro Yajima
Keyword(s):  
Point Vortex ◽  
Point Vortices ◽  
The Self ◽  
Fourth Power ◽  
Jacobi Field ◽  
Self Similarity ◽  
Vortex System ◽  
Geometric Singularity ◽  
Self Similar

In this study, we geometrically analyze the relation between a point vortex system and deviation curvatures on the Jacobi field. First, eigenvalues of deviation curvatures are calculated from relative distances of point vortices in a three point vortex system. Afterward, based on the assumption of self-similarity, time evolutions of eigenvalues of deviation curvatures are shown. The self-similar motions of three point vortices are classified into two types, expansion and collapse, when the relative distances vary monotonously. Then, we find that the eigenvalues of self-similarity are proportional to the inverse fourth power of relative distances. The eigenvalues of the deviation curvatures monotonically convergent to zero for expansion, whereas they monotonically diverge for collapse, which indicates that the strengths of interactions between point vortices related to the time evolution of spatial geometric structure in terms of the deviation curvatures. In particular, for collapse, the collision point becomes a geometric singularity because the eigenvalues of the deviation curvature diverge. These results show that the self-similar motions of point vortices are classified by eigenvalues of the deviation curvature. Further, nonself-similar expansion is numerically analyzed. In this case, the eigenvalues of the deviation curvature are nonmonotonous but converge to zero, suggesting that the motion of the nonself-similar three point vortex system is also classified by eigenvalues of the deviation curvature.

scholarly journals Instantaneous energy and enstrophy variations in Euler-alpha point vortices via triple collapse

2012 ◽  
Vol 702 ◽  
pp. 188-214 ◽  
Author(s):  
Takashi Sakajo
Keyword(s):  
Weak Solution ◽  
Euler Equations ◽  
Weak Solutions ◽  
Point Vortex ◽  
Critical Time ◽  
Point Vortices ◽  
The Self ◽  
Two Dimensional ◽  
Graphic System ◽  
Self Similar

AbstractIt has been pointed out that the anomalous enstrophy dissipation in non-smooth weak solutions of the two-dimensional Euler equations has a clue to the emergence of the inertial range in the energy density spectrum of two-dimensional turbulence corresponding to the enstrophy cascade as the viscosity coefficient tends to zero. However, it is uncertain how non-smooth weak solutions can dissipate the enstrophy. In the present paper, we construct a weak solution of the two-dimensional Euler equations from that of the Euler-$\ensuremath{\alpha} $ equations proposed by Holm, Marsden & Ratiu (Phys. Rev. Lett., vol. 80, 1998, pp. 4173–4176) by taking the limit of $\ensuremath{\alpha} \ensuremath{\rightarrow} 0$. To accomplish this task, we introduce the $\ensuremath{\alpha} $-point-vortex ($\ensuremath{\alpha} \mathrm{PV} $) system, whose evolution corresponds to a unique global weak solution of the two-dimensional Euler-$\ensuremath{\alpha} $ equations in the sense of distributions (Oliver & Shkoller, Commun. Part. Diff. Equ., vol. 26, 2001, pp. 295–314). Since the $\ensuremath{\alpha} \mathrm{PV} $ system is a formal regularization of the point-vortex system and it is known that, under a certain special condition, three point vortices collapse self-similarly in finite time (Kimura, J. Phys. Soc. Japan, vol. 56, 1987, pp. 2024–2030), we expect that the evolution of three $\ensuremath{\alpha} $-point vortices for the same condition converges to a singular weak solution of the Euler-$\ensuremath{\alpha} $ equations that is close to the triple collapse as $\ensuremath{\alpha} \ensuremath{\rightarrow} 0$, which is examined in the paper. As a result, we find that the three $\ensuremath{\alpha} $-point vortices collapse to a point and then expand to infinity self-similarly beyond the critical time in the limit. We also show that the Hamiltonian energy and a kinematic energy acquire a finite jump discontinuity at the critical time, but the energy dissipation rate converges to zero in the sense of distributions. On the other hand, an enstrophy variation converges to the $\delta $ measure with a negative mass, which indicates that the enstrophy dissipates in the distributional sense via the self-similar triple collapse. Moreover, even if the special condition is perturbed, we can confirm numerically the convergence to the singular self-similar evolution with the enstrophy dissipation. This indicates that the self-similar triple collapse is a robust mechanism of the anomalous enstrophy dissipation in the sense that it is observed for a certain range of the parameter region.

scholarly journals Onset of the Mach Reflection of Zel’dovich–von Neumann–Döring Detonations

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 314
Author(s):  
Tianyu Jing ◽  
Huilan Ren ◽  
Jian Li
Keyword(s):  
Hot Spot ◽  
Mach Reflection ◽  
Near Field ◽  
The Self ◽  
Small Scale ◽  
Mach Stem ◽  
Self Similarity ◽  
Von Neumann ◽  
Similarity Problem ◽  
Self Similar

The present study investigates the similarity problem associated with the onset of the Mach reflection of Zel’dovich–von Neumann–Döring (ZND) detonations in the near field. The results reveal that the self-similarity in the frozen-limit regime is strictly valid only within a small scale, i.e., of the order of the induction length. The Mach reflection becomes non-self-similar during the transition of the Mach stem from “frozen” to “reactive” by coupling with the reaction zone. The triple-point trajectory first rises from the self-similar result due to compressive waves generated by the “hot spot”, and then decays after establishment of the reactive Mach stem. It is also found, by removing the restriction, that the frozen limit can be extended to a much larger distance than expected. The obtained results elucidate the physical origin of the onset of Mach reflection with chemical reactions, which has previously been observed in both experiments and numerical simulations.

TUBE FORMULAS FOR GRAPH-DIRECTED FRACTALS

Fractals ◽  
2010 ◽  
Vol 18 (03) ◽  
pp. 349-361 ◽  
Author(s):  
BÜNYAMIN DEMÍR ◽  
ALI DENÍZ ◽  
ŞAHIN KOÇAK ◽  
A. ERSIN ÜREYEN
Keyword(s):  
The Self ◽  
Self Similarity ◽  
Complex Dimensions ◽  
Tubular Neighborhoods ◽  
Self Similar ◽  
Tube Formulas

Lapidus and Pearse proved recently an interesting formula about the volume of tubular neighborhoods of fractal sprays, including the self-similar fractals. We consider the graph-directed fractals in the sense of graph self-similarity of Mauldin-Williams within this framework of Lapidus-Pearse. Extending the notion of complex dimensions to the graph-directed fractals we compute the volumes of tubular neighborhoods of their associated tilings and give a simplified and pointwise proof of a version of Lapidus-Pearse formula, which can be applied to both self-similar and graph-directed fractals.

ECCENTRIC DISTANCE SUM OF SIERPIŃSKI GASKET AND SIERPIŃSKI NETWORK

Fractals ◽  
2019 ◽  
Vol 27 (02) ◽  
pp. 1950016 ◽  
Author(s):  
JIN CHEN ◽  
LONG HE ◽  
QIN WANG
Keyword(s):  
Asymptotic Formula ◽  
Complex Networks ◽  
Sierpinski Gasket ◽  
The Self ◽  
Self Similarity ◽  
Sierpiński Gasket ◽  
Similar Measure ◽  
Self Similar ◽  
Eccentric Distance

The eccentric distance sum is concerned with complex networks. To obtain the asymptotic formula of eccentric distance sum on growing Sierpiński networks, we study some nonlinear integral in terms of self-similar measure on the Sierpiński gasket and use the self-similarity of distance and measure to obtain the exact value of this integral.

Identification of Chaos-Periodic Transitions, Band Merging, and Internal Crisis Using Wavelet-DFA Method

2016 ◽  
Vol 26 (04) ◽  
pp. 1650065 ◽  
Author(s):  
Mahsa Vaghefi ◽  
Ali Motie Nasrabadi ◽  
Seyed Mohammad Reza Hashemi Golpayegani ◽  
Mohammad Reza Mohammadi ◽  
Shahriar Gharibzadeh
Keyword(s):  
Detrended Fluctuation Analysis ◽  
Period Doubling ◽  
Nonstationary Time Series ◽  
The Self ◽  
Fluctuation Analysis ◽  
Scaling Analysis ◽  
Analysis Method ◽  
Self Similarity ◽  
Self Similar ◽  
Detrended Fluctuation

Detrended Fluctuation Analysis (DFA) is a scaling analysis method that can identify intrinsic self-similarity in any nonstationary time series. In contrast, Wavelet Transform (WT) method is widely used to investigate the self-similar processes, as the self-similarity properties exist within the subbands. Therefore, a combination of these two approaches, DFA and WPT, is promising for rigorous investigation of such a system. In this paper a new methodology, so-called wavelet DFA, is introduced and interpreted to evaluate this idea. This approach, further than identifying self-similarity properties, enable us to detect and capture the chaos-periodic transitions, band merging, and internal crisis in systems that become chaotic through period-doubling phenomena. Changes of wavelet DFA exponent have been compared with that of Lyapunov and DFA through Logistic, Sine, Gaussian, Cubic, and Quartic Maps. Furthermore, the potential capabilities of this new exponent have been presented.

Direct numerical simulation of axisymmetric wakes embedded in turbulence

2012 ◽  
Vol 710 ◽  
pp. 482-504 ◽  
Author(s):  
Elad Rind ◽  
Ian P. Castro
Keyword(s):  
Numerical Simulation ◽  
Direct Numerical Simulation ◽  
Decay Rate ◽  
Relative Strength ◽  
Decay Rates ◽  
The Self ◽  
Self Similarity ◽  
Self Similar ◽  
External Turbulence ◽  
Different Levels

AbstractDirect numerical simulation has been used to study the effects of external turbulence on axisymmetric wakes. In the absence of such turbulence, the time-developing axially homogeneous wake is found to have the self-similar properties expected whereas, in the absence of the wake, the turbulence fields had properties similar to Saffman-type turbulence. Merging of the two flows was undertaken for three different levels of external turbulence (relative to the wake strength) and it is shown that the presence of the external turbulence enhances the decay rate of the wake, with the new decay rates increasing with the relative strength of the initial external turbulence. The external turbulence is found to destroy any possibility of self-similarity within the developing wake, causing a significant transformation in the latter as it gradually evolves towards the former.

scholarly journals On the self-similarity of wind turbine wakes in complex terrain using large-eddy simulation

2019 ◽  
Author(s):  
Arslan Salim Dar ◽  
Jacob Berg ◽  
Niels Troldborg ◽  
Edward G. Patton
Keyword(s):  
Large Eddy Simulation ◽  
Complex Terrain ◽  
The Self ◽  
Self Similarity ◽  
Flat Terrain ◽  
Eddy Simulation ◽  
Atmospheric Stratification ◽  
Large Eddy ◽  
Self Similar ◽  
Turbine Wake

Abstract. We perform large-eddy simulation of flow in complex terrain under neutral atmospheric stratification. We study the self-similar behavior of a turbine wake as a function of varying terrain complexity and perform comparison with a flat terrain. By plotting normalized velocity deficit profiles in different complex terrain cases, we verify that self-similarity is preserved as we move downstream from the turbine. We find that this preservation is valid for a shorter distance downstream compared to what is observed in flat terrain. A larger spread of the profiles toward the tails due to varying levels of shear is also observed.

scholarly journals Self-similar cosmological solutions in f(R,T) gravity theory

2021 ◽  
pp. 2150206
Author(s):  
José Antonio Belinchón ◽  
Carlos González ◽  
Sami Dib
Keyword(s):  
Exact Solutions ◽  
The Self ◽  
Exact Form ◽  
Field Equations ◽  
Self Similarity ◽  
Similarity Hypothesis ◽  
Geometrical Quantity ◽  
Cosmological Solutions ◽  
Self Similar ◽  
Matter Collineation

We study the [Formula: see text] cosmological models under the self-similarity hypothesis. We determine the exact form that each physical and geometrical quantity may take in order that the field equations (FE) admit exact self-similar (SS) solutions through the matter collineation approach. We study two models: the case[Formula: see text] and the case [Formula: see text]. In each case, we state general theorems which determine completely the form of the unknown functions [Formula: see text] such that the FE admit SS solutions. We also state some corollaries as limiting cases. These results are quite general and valid for any homogeneous SS metric[Formula: see text] In this way, we are able to generate new cosmological scenarios. As examples, we study two cases by finding exact solutions to these particular models.

Coherent Structures and Correlation Fields in the Mixing Transition of a Turbulent Round Free Jet

2018 ◽  
Author(s):  
Benedikt Krohn ◽  
Sunming Qin ◽  
Victor Petrov ◽  
Annalisa Manera
Keyword(s):  
Coherent Structures ◽  
High Speed ◽  
Near Field ◽  
The Self ◽  
Turbulent Shear ◽  
Past Century ◽  
Self Similarity ◽  
Free Jet ◽  
Similar Region ◽  
Self Similar

Turbulent free jets attracted the focus of many scientists within the past century regarding the understanding of mass- and momentum transport in the turbulent shear field, especially in the near-field and the self-similar region. Recent investigations attempt to understand the intermediate fields, called the mixing transition or ‘the route to self-similarity’. An apparent gap is recognized in light of this mixing transition, with two main conjectures being put forth. Firstly the flow will always asymptotically reach a fully self-similar state if boundary conditions permit. The second proposes partial and local self-similarity within the mixing transition. We address the later with an experimental investigation of the intermediate field turbulence dynamics in a non-confined free jet with a nozzle diameter of 12.7 mm and an outer scale Reynolds number of 15,000. High speed Particle Image Velocimetry (PIV) is used to record the velocity fields with a final spatial resolution of 194 × 194 μm2. The analysis focuses on higher order moments and two-point correlations of velocity variances in space and time. We observed local self-similarity in the measured correlation fields. Coherent structures are present within the near-field where the turbulent energy spectrum cascades along a dissipative slope. Towards the transition region, the spectrum smoothly transforms to a viscous cascade, as it is commonly observed in the self-similar region.

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