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  • 1
    Electronic Resource
    Electronic Resource
    [S.l.] : American Institute of Physics (AIP)
    Physics of Fluids 9 (1997), S. 2023-2031 
    ISSN: 1089-7666
    Source: AIP Digital Archive
    Topics: Physics
    Notes: A low-dimensional model, using the proper orthogonal, or Karhunen–Loève decomposition, has been remarkably successful in representing the behavior of the wall region of a turbulent boundary layer. We briefly summarize this work. We may hope for similar success in other flows in which coherent structures play an important role, in particular flows with density fluctuations. We sketch an approach to such a decomposition for flows with density fluctuations, suggesting various alternatives which weigh the available information differently. In such a low-dimensional model, obtaining the empirical eigenfunctions poses a problem, since they can usually be determined only from extensive measurements or direct numerical simulations. However, recent work with energy method stability theory (modified by use of an anisotropic eddy viscosity and feedback to the mean profile) has been remarkably successful in predicting the form of the empirical eigenfunctions in the isothermal boundary layer. We present here preliminary results for sheared Rayleigh–Bénard convection; these results do not include anisotropic eddy viscosities and feedback, and do not predict directly the form of the POD eigenfunctions; however, a very satisfactory comparison can be made with the second order moments obtained from a DNS. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Theoretical and computational fluid dynamics 9 (1997), S. 269-280 
    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. We study numerical solutions of the reduced-gravity shallow-water equation on a beta plane, subjected to a sinusoidally varying wind forcing leading to the formation of a double gyre circulation. As expected the dynamics of the numerical solutions are highly dependent on the grid resolution and the given numerical algorithm. In particular, the statistics of the solutions are critically dependent on the scheme's ability to resolve the Rossby deformation radius. We present a method, applicable to any finite-difference scheme, which effectively increases the spatial resolution of the given algorithm without changing its temporal stability or memory requirements. This enslaving method makes use of properties of the governing equations in the absence of time derivatives to reduce the overall truncation error. By examining statistical measures of stochastic solutions at resolutions near the Rossby radius, we show that the enslaved schemes are capable of reproducing statistics of standard schemes computed at twice the resolution.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1573-1987
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Abstract We present brief précis of three related investigations. Fuller accounts can be found elsewhere. The investigations bear on the identification and prediction of coherent structures in turbulent shear flows. A second unifying thread is the Proper Orthogonal Decomposition (POD), or Karhunen-Loève expansion, which appears in all three investigations described. The first investigation demonstrates a close connection between the coherent structures obtained using linear stochastic estimation, and those obtained from the POD. Linear stochastic estimation is often used for the identification of coherent structures. The second investigation explores the use (in homogeneous directions) of wavelets instead of Fourier modes, in the construction of dynamical models; the particular problem considered here is the Kuramoto-Sivashinsky equation. The POD eigenfunctions, of course, reduce to Fourier modes in homogeneous situations, and either can be shown to converge optimally fast; we address the question of how rapidly (by comparison) a wavelet representation converges, and how the wavelet-wavelet interactions can be handled to construct a simple model. The third investigation deals with the prediction of POD eigenfunctions in a turbulent shear flow. We show that energy-method stability theory, combined with an anisotropic eddy viscosity, and erosion of the mean velocity profile by the growing eigenfunctions, produces eigenfunctions very close to those of the POD, and the same eigenvalue spectrum at low wavenumbers.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 12 (1996), S. 13-40 
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We show how the accuracy of a given finite difference scheme approximating a dissipative nonlinear PDE may be improved. The numerical solutions are decomposed into two parts that may be interpreted as approximating the large and small scales of the true solutions. By enslaving the small scales in terms of the larger ones, we derive a new difference scheme that is, in general, more accurate than the original scheme. The new scheme is also more computationally efficient, provided that the time derivatives of the problem are not too large. © 1996 John Wiley & Sons, Inc.
    Additional Material: 4 Ill.
    Type of Medium: Electronic Resource
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