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  • 1
    Electronic Resource
    Electronic Resource
    The dynamics of spatiotemporal modulations (1995)
    Woodbury, NY : American Institute of Physics (AIP)
    Chaos 5 (1995), S. 578-588 
    Details
    ISSN: 1089-7682
    Source: AIP Digital Archive
    Topics: Physics
    Notes: The modulational instability of traveling waves is often thought to be a crucial point in the mechanism of transition to space–time disorder and turbulence. The aim of this paper is to study the effect of spatiotemporal modulations on some dynamics u0(x,t), which may occur as an instability process when a control parameter varies, for instance. We analyze the properties of the modulated dynamics of the form g1(x)g2(t)u0(x,t) compared to those of the reference dynamics u0(x,t), using operator theory. We show that, if the reference dynamics is invariant under some space–time symmetry in the sense of Ref. [J. Nonlinear Sci. 2, 183 (1992)], the modulation has the effect of either deforming this symmetry or breaking it, depending on whether the corresponding operator remains unitary or not. We also demonstrate that the smallest Euclidean space containing the modulated dynamics has a dimension smaller than or equal to the smallest Euclidean space containing u0(x,t). The previous results are then applied to the case of modulated uniformly traveling waves. While the spatiotemporal translation invariance of the wave never persists in the presence of a modulation, the existence of a spatiotemporal symmetry depends on the resonance of the Fourier sidebands due to the modulation. In case of nonresonance, a spatiotemporal symmetry exists and is explicitly determined. In this situation, the modulated wave and the carrier wave have the same spectrum (up to a normalization factor), the same entropy, and the spatial (resp., temporal) two-point correlation is deformed only by the spatial (resp., temporal) modulation. © 1995 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.166127
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  • 2
    Electronic Resource
    Electronic Resource
    Transition to turbulence on a rotating flat disk (1994)
    [S.l.] : American Institute of Physics (AIP)
    Physics of Fluids 6 (1994), S. 2800-2814 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: Experimental data from one-point measurements obtained in a transitional flow on a rotating flat disk are presented and analyzed by using biorthogonal decomposition techniques. The analysis is performed at various Reynolds numbers from slightly above the onset of the first instability to the transition to turbulence. As Reynolds number increases, biorthogonal spectra become broader and the entropy characterizing the distribution of energy among the various biorthogonal modes increases. Details of this increase are studied by analyzing local entropy maxima corresponding to eigenvalue degeneracies. At these values of the Reynolds number, internal bifurcations, responsible for a lack of smoothness in the dependence of the flow with Reynolds number, are shown to occur.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.868168
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  • 3
    Electronic Resource
    Electronic Resource
    Hierarchical order in wall-bounded shear turbulence (1996)
    [S.l.] : American Institute of Physics (AIP)
    Physics of Fluids 8 (1996), S. 1061-1075 
    Details
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: Since turbulence at realistic Reynolds numbers, such as those occurring in the atmosphere or in the ocean, involve a high number of modes that cannot be resolved computationally in the foreseeable future, there is a strong motivation for finding techniques which drastically decrease the number of such required modes, particularly under inhomogeneous conditions. The significance of this work is to show that wall-bounded shear turbulence, in its strongly inhomogeneous direction (normal to the wall), can be decomposed into one (or a few) space–time mother mode(s), with each mother generating a whole family of modes by stretching symmetry. In other words, the generated modes are similar, dilated copies of their mother. In addition, we show that the nature of all previous modes strongly depends on the symmetry itself. These findings constitute the first scaling theory of inhomogeneous turbulence. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.868881
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  • 4
    Electronic Resource
    Electronic Resource
    Erratum: The effect of modeled drag reduction on the Wall Region (1996)
    Springer
    Theoretical and computational fluid dynamics 8 (1996), S. 449-450 
    Details
    ISSN: 1432-2250
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00455995
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  • 5
    Electronic Resource
    Electronic Resource
    Erratum: The Effect of Modeled Drag Reduction on the Wall Region1 (Theoret. Comput. Fluid Dynamics, 1:229—248, 1990) (1996)
    Springer
    Theoretical and computational fluid dynamics 8 (1996), S. 449-450 
    Details
    ISSN: 1432-2250
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00455995
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  • 6
    Electronic Resource
    Electronic Resource
    On the hidden beauty of the proper orthogonal decomposition (1991)
    Springer
    Theoretical and computational fluid dynamics 2 (1991), S. 339-352 
    Details
    ISSN: 1432-2250
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Abstract The proper orthogonal decomposition theorem (Loève, 1955) of probability theory has been proposed by Lumley (1967, 1972, 1981) for detection of spatial coherent patterns in turbulent flows. More specifically, the decomposition extracts deterministic functions from second-order statistics of a random field and converges optimally fast in quadratic mean (i.e., in energy). The technique can be made completely deterministic in the sense that it can be applied to spatially and temporally evolving flows. The remarkable property of this deterministic decomposition is not only in its optimal convergence (as emphasized before) but also in its space/time symmetry which permits access to the spatiotemporal dynamics. The flow is decomposed into both spatial and temporal orthogonal modes which are coupled: each space component is associated with a time component partner. The latter is the time evolution of the former and the former is the spatial configuration of the latter. This generalizes the notion of spatial and temporal structures which can be followed through the various instabilities that the flow undergoes as Reynolds number increases. It also provides a nonlinear dynamics tool for spatiotemporal dynamical systems and can be used for bifurcation detection and analysis as well as dimension and degree of complexity estimates.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00271473
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  • 7
    Electronic Resource
    Electronic Resource
    Spatiotemporal analysis of complex signals: Theory and applications (1991)
    Springer
    Journal of statistical physics 64 (1991), S. 683-739 
    Details
    ISSN: 1572-9613
    Keywords: Spatiotemporal complexity ; spatiotemporal chaos ; signal analysis ; Liapunov exponents ; coupled map lattices ; turbulence ; transition to turbulence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We present a space-time description of regular and complex phenomena which consists of a decomposition of a spatiotemporal signal into orthogonal temporal modes that we call chronos and orthogonal spatial modes that we call topos. This permits the introduction of several characteristics of the signal, three characteristic energies and entropies (one temporal, one spatial, and one global), and a characteristic dimension. Although the technique is general, we concentrate on its applications to hydrodynamic problems, specifically the transition to turbulence. We consider two cases of application: a coupled map lattice as a dynamical system model for spatiotemporal complexity and the open flow instability on a rotating disk. In the latter, we show a direct relation between the global entropy and the different instabilities that the flow undergoes as Reynolds number increases.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01048312
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  • 8
    Electronic Resource
    Electronic Resource
    Spatiotemporal and statistical symmetries (1995)
    Springer
    Journal of statistical physics 81 (1995), S. 793-828 
    Details
    ISSN: 1572-9613
    Keywords: Spatiotemporal complexity ; spatiotemporal symmetries ; statistical symmetries ; two-point correlations ; biorthogonal decomposition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The notion of symmetries, either statistical or deterministic, can be useful for the characterization of complex systems and their bifurcations. In this paper, we investigate the connection between the (microscopic) spatiotemporal symmetries of a space-time functionu(x, t), on the one hand, and the (macroscopic) symmetries of statistical quantities such as the spatial (resp. temporal) two-point correlations and the spatial (resp. temporal) average, on the other hand. We show, how, under certain conditions, these symmetries are related to the symmetries of the orbits described byu(x, t) in the characteristic (phase) spaces. We also determine the largest group of spatiotemporal symmetries (in the sense introduced in our earlier work) satisfied by a given space-time functionu(x, t) and indicate how to extract the subgroups of point symmetries, namely those directly implemented on the space and time variables. Conversely, we determine all the functions invariant by a given space-time symmetry group. Finally, we illustrate all the previous points with specific examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02179258
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  • 9
    Electronic Resource
    Electronic Resource
    Turbulence spectra (1992)
    Springer
    Journal of statistical physics 67 (1992), S. 203-228 
    Details
    ISSN: 1572-9613
    Keywords: Turbulence ; biorthogonal decomposition ; self-similarity ; fractals ; multifractals ; wavelets
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The scaling invariance of the Navier-Stokes equations in the limit of infinite Reynolds number is used to derive laws for the inertial range of the turbulence spectrum. Whether the flow is homogeneous or not, the spectrum is chosen to be that given by a well-chosen biorthogonal decomposition. If the flow is hoogeneous, this spectrum coincides with the classical Fourier (energy) spectrum which exhibits Kolmogorov's k−5/3 power law if the scaling exponent is assumed to be 1/3. In the more general case where the homogeneity assumption is relaxed, the spectrum is discrete and decays exponentially fast under the assumption that the flow is invariant (in a deterministic or statistical sense) under only one subgroup of the scaling coefficientλ of one scaling group of the equations (corresponding to one value of the scaling exponent). If the flow is invariant under two subgroups of scaling coefficientsλ andλ′, the spectrum becomes maximal, equal toR +. Finally, when a full symmetry, namely an invariance under a whole group, is assumed and the spectrum becomes continuous, the decaying law for the spectral density is derived and found to be independent of the specific value ofh These ideas are then applied to locally self-similar flows with multiple dilation centers (localized in space and time) and multiple scaling exponents, extending the concept of multifractals to space and time.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01049031
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  • 10
    Electronic Resource
    Electronic Resource
    Wave propagation phenomena from a spatiotemporal viewpoint: Resonances and bifurcations (1994)
    Springer
    Journal of statistical physics 76 (1994), S. 1005-1043 
    Details
    ISSN: 1572-9613
    Keywords: Wave propagation ; spatiotemporal bifurcation theory ; biorthogonal decomposition ; Fourier analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract By using biorthogonal decompositions, we show how uniformly propagating waves, togehter with their velocity, shape, and amplitude, can be extracted from a spatiotemporal signal consisting of the superposition of various traveling waves. The interaction between the different waves manifests itself in space-time resonances in case of a discrete biorthogonal spectrum and in resonant wavepackets in case of a continuous biorthogonal spectrum. Resonances appear as invariant subspaces under the biorthogonal operator, which leads to closed sets of algebraic equations. The analysis is then extended to superpositions of dispersive waves for which the (Fourier) dispersion relation is no longer linear. We then show how a space-time bifurcation, namely a qualitative change in the spatiotemporal nature of the solution, occurs when the biorthogonal operator is a nonholomorphic function of a parameter. This takes place when two eigenvalues are degenerate in the biorthogonal spectrum and when the spatial and temporal eigenvectors rotate within each eigenspace. Such a scenario applied to the superposition of traveling waves leads to the generation of additional waves propagating at new velocities, which can be computed from the spatial and temporal eigenmodes involved in the process (namely the shape of the propagating waves slightly before the bifurcation). An eigenvalue degeneracy, however, does not necessarily lead to a bifurcation, a situation we refer to as being self-avoiding. We illustrate our theoretical predictions by giving examples of bifurcating and self-avoiding events in propagating phenomena.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02188696
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