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Li, Yanpeng, et Boping Tian. « Optimal non-asymptotic concentration of centered empirical relative entropy in the high-dimensional regime ». Statistics & ; Probability Letters 197 (juin 2023) : 109803. http://dx.doi.org/10.1016/j.spl.2023.109803.
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Pham, Kim, Agnès Maurel et Jean-Jacques Marigo. « Revisiting imperfect interface laws for two-dimensional elastodynamics ». Proceedings of the Royal Society A : Mathematical, Physical and Engineering Sciences 477, no 2245 (janvier 2021) : 20200519. http://dx.doi.org/10.1098/rspa.2020.0519.
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We study the interaction of in-plane elastic waves with imperfect interfaces composed of a periodic array of voids or cracks. An effective model is derived from high-order asymptotic analysis based on two-scale homogenization and matched asymptotic technique. In two-dimensional elasticity, we obtain jump conditions set on the in-plane displacements and normal stresses; the jumps involve in addition effective parameters provided by static, elementary problems being the equivalents of the cell problems in classical two-scale homogenization. The derivation of the model is conducted in the transient regime and its stability is guarantied by the positiveness of the effective interfacial energy. Spring models are envisioned as particular cases. It is shown that massless-spring models are recovered in the limit of small void thicknesses and collinear cracks. By contrast, the use of mass-spring model is justified at normal incidence, otherwise unjustified. We provide quantitative validations of our model and comparison with spring models by means of comparison with direct numerical calculations in the harmonic regime.
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Xiao, Lechao, Hong Hu, Theodor Misiakiewicz, Yue M. Lu et Jeffrey Pennington. « Precise learning curves and higher-order scaling limits for dot-product kernel regression * ». Journal of Statistical Mechanics : Theory and Experiment 2023, no 11 (1 novembre 2023) : 114005. http://dx.doi.org/10.1088/1742-5468/ad01b7.
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Abstract As modern machine learning models continue to advance the computational frontier, it has become increasingly important to develop precise estimates for expected performance improvements under different model and data scaling regimes. Currently, theoretical understanding of the learning curves (LCs) that characterize how the prediction error depends on the number of samples is restricted to either large-sample asymptotics ( m → ∞ ) or, for certain simple data distributions, to the high-dimensional asymptotics in which the number of samples scales linearly with the dimension ( m ∝ d ). There is a wide gulf between these two regimes, including all higher-order scaling relations m ∝ d r , which are the subject of the present paper. We focus on the problem of kernel ridge regression for dot-product kernels and present precise formulas for the mean of the test error, bias and variance, for data drawn uniformly from the sphere with isotropic random labels in the rth-order asymptotic scaling regime m → ∞ with m / d r held constant. We observe a peak in the LC whenever m ≈ d r / r ! for any integer r, leading to multiple sample-wise descent and non-trivial behavior at multiple scales. We include a colab (available at: https://tinyurl.com/2nzym7ym) notebook that reproduces the essential results of the paper.
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Ananiev, V., et A. L. Read. « Linear approximation to the statistical significance autocovariance matrix in the asymptotic regime ». Journal of Instrumentation 18, no 10 (1 octobre 2023) : P10018. http://dx.doi.org/10.1088/1748-0221/18/10/p10018.
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Abstract Approximating significance scans of searches for new particles in high-energy physics experiments as Gaussian fields is a well-established way to estimate the trials factors required to quantify global significances. We propose a novel, highly efficient method to estimate the covariance matrix of such a Gaussian field. The method is based on the linear approximation of statistical fluctuations of the signal amplitude. For one-dimensional searches the upper bound on the trials factor can then be calculated directly from the covariance matrix. For higher dimensions, the Gaussian process described by this covariance matrix may be sampled to calculate the trials factor directly. This method also serves as the theoretical basis for a recent study of the trials factor with an empirically constructed set of Asmiov-like background datasets. We illustrate the method with studies of a H → γγ inspired model that was used in the empirical paper.
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Chang, Xiangyu, Yingcong Li, Samet Oymak et Christos Thrampoulidis. « Provable Benefits of Overparameterization in Model Compression : From Double Descent to Pruning Neural Networks ». Proceedings of the AAAI Conference on Artificial Intelligence 35, no 8 (18 mai 2021) : 6974–83. http://dx.doi.org/10.1609/aaai.v35i8.16859.
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Deep networks are typically trained with many more parameters than the size of the training dataset. Recent empirical evidence indicates that the practice of overparameterization not only benefits training large models, but also assists – perhaps counterintuitively – building lightweight models. Specifically, it suggests that overparameterization benefits model pruning / sparsification. This paper sheds light on these empirical findings by theoretically characterizing the high-dimensional asymptotics of model pruning in the overparameterized regime. The theory presented addresses the following core question: ``should one train a small model from the beginning, or first train a large model and then prune?''. We analytically identify regimes in which, even if the location of the most informative features is known, we are better off fitting a large model and then pruning rather than simply training with the known informative features. This leads to a new double descent in the training of sparse models: growing the original model, while preserving the target sparsity, improves the test accuracy as one moves beyond the overparameterization threshold. Our analysis further reveals the benefit of retraining by relating it to feature correlations. We find that the above phenomena are already present in linear and random-features models. Our technical approach advances the toolset of high-dimensional analysis and precisely characterizes the asymptotic distribution of over-parameterized least-squares. The intuition gained by analytically studying simpler models is numerically verified on neural networks.
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Gerace, Federica, Bruno Loureiro, Florent Krzakala, Marc Mézard et Lenka Zdeborová. « Generalisation error in learning with random features and the hidden manifold model* ». Journal of Statistical Mechanics : Theory and Experiment 2021, no 12 (1 décembre 2021) : 124013. http://dx.doi.org/10.1088/1742-5468/ac3ae6.
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Abstract We study generalised linear regression and classification for a synthetically generated dataset encompassing different problems of interest, such as learning with random features, neural networks in the lazy training regime, and the hidden manifold model. We consider the high-dimensional regime and using the replica method from statistical physics, we provide a closed-form expression for the asymptotic generalisation performance in these problems, valid in both the under- and over-parametrised regimes and for a broad choice of generalised linear model loss functions. In particular, we show how to obtain analytically the so-called double descent behaviour for logistic regression with a peak at the interpolation threshold, we illustrate the superiority of orthogonal against random Gaussian projections in learning with random features, and discuss the role played by correlations in the data generated by the hidden manifold model. Beyond the interest in these particular problems, the theoretical formalism introduced in this manuscript provides a path to further extensions to more complex tasks.
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Touber, Emile. « Small-scale two-dimensional turbulence shaped by bulk viscosity ». Journal of Fluid Mechanics 875 (26 juillet 2019) : 974–1003. http://dx.doi.org/10.1017/jfm.2019.531.
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Bulk-to-shear viscosity ratios of three orders of magnitude are often reported in carbon dioxide but are always neglected when predicting aerothermal loads in external (Mars exploration) or internal (turbomachinery, heat exchanger) turbulent flows. The recent (and first) numerical investigations of that matter suggest that the solenoidal turbulence kinetic energy is in fact well predicted despite this seemingly arbitrary simplification. The present work argues that such a conclusion may reflect limitations from the choice of configuration rather than provide a definite statement on the robustness of kinetic-energy transfers to the use of Stokes’ hypothesis. Two distinct asymptotic regimes (Euler–Landau and Stokes–Newton) in the eigenmodes of the Navier–Stokes equations are identified. In the Euler–Landau regime, the one captured by earlier studies, acoustic and entropy waves are damped by transport coefficients and the dilatational kinetic energy is dissipated, even more rapidly for high bulk-viscosity fluids and/or forcing frequencies. If the kinetic energy is initially or constantly injected through solenoidal motions, effects on the turbulence kinetic energy remain minor. However, in the Stokes–Newton regime, diffused bulk compressions and advected isothermal compressions are found to prevail and promote small-scale enstrophy via vorticity–dilatation correlations. In the absence of bulk viscosity, the transition to the Stokes–Newton regime occurs within the dissipative scales and is not observed in practice. In contrast, at high bulk viscosities, the Stokes–Newton regime can be made to overlap with the inertial range and disrupt the enstrophy at small scales, which is then dissipated by friction. Thus, flows with substantial inertial ranges and large bulk-to-shear viscosity ratios should experience enhanced transfers to small-scale solenoidal kinetic energy, and therefore faster dissipation rates leading to modifications of the heat-transfer properties. Observing numerically such transfers is still prohibitively expensive, and the present simulations are restricted to two-dimensional turbulence. However, the theory laid here offers useful guidelines to design experimental studies to track the Stokes–Newton regime and associated modifications of the turbulence kinetic energy, which are expected to persist in three-dimensional turbulence.
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Reese, Daniel R., Francisco Espinosa Lara et Michel Rieutord. « Pulsations of rapidly rotating stars with compositional discontinuities ». Proceedings of the International Astronomical Union 9, S301 (août 2013) : 169–72. http://dx.doi.org/10.1017/s1743921313014270.
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AbstractRecent observations of rapidly rotating stars have revealed the presence of regular patterns in their pulsation spectra. This has raised the question as to their physical origin, and, in particular, whether they can be explained by an asymptotic frequency formula for low-degree acoustic modes, as recently discovered through numerical calculations and theoretical considerations. In this context, a key question is whether compositional/density gradients can adversely affect such patterns to the point of hindering their identification. To answer this question, we calculate frequency spectra using two-dimensional ESTER stellar models. These models use a multi-domain spectral approach, allowing us to easily insert a compositional discontinuity while retaining a high numerical accuracy. We analyse the effects of such discontinuities on both the frequencies and eigenfunctions of pulsation modes in the asymptotic regime. We find that although there is more scatter around the asymptotic frequency formula, the semi-large frequency separation can still be clearly identified in a spectrum of low-degree acoustic modes.
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Taheri, Hossein, Ramtin Pedarsani et Christos Thrampoulidis. « Sharp Guarantees and Optimal Performance for Inference in Binary and Gaussian-Mixture Models ». Entropy 23, no 2 (30 janvier 2021) : 178. http://dx.doi.org/10.3390/e23020178.
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We study convex empirical risk minimization for high-dimensional inference in binary linear classification under both discriminative binary linear models, as well as generative Gaussian-mixture models. Our first result sharply predicts the statistical performance of such estimators in the proportional asymptotic regime under isotropic Gaussian features. Importantly, the predictions hold for a wide class of convex loss functions, which we exploit to prove bounds on the best achievable performance. Notably, we show that the proposed bounds are tight for popular binary models (such as signed and logistic) and for the Gaussian-mixture model by constructing appropriate loss functions that achieve it. Our numerical simulations suggest that the theory is accurate even for relatively small problem dimensions and that it enjoys a certain universality property.
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Loureiro, Bruno, Cédric Gerbelot, Maria Refinetti, Gabriele Sicuro et Florent Krzakala. « Fluctuations, bias, variance and ensemble of learners : exact asymptotics for convex losses in high-dimension * ». Journal of Statistical Mechanics : Theory and Experiment 2023, no 11 (1 novembre 2023) : 114001. http://dx.doi.org/10.1088/1742-5468/ad0221.
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Abstract From the sampling of data to the initialisation of parameters, randomness is ubiquitous in modern Machine Learning practice. Understanding the statistical fluctuations engendered by the different sources of randomness in prediction is therefore key to understanding robust generalisation. In this manuscript we develop a quantitative and rigorous theory for the study of fluctuations in an ensemble of generalised linear models trained on different, but correlated, features in high-dimensions. In particular, we provide a complete description of the asymptotic joint distribution of the empirical risk minimiser for generic convex loss and regularisation in the high-dimensional limit. Our result encompasses a rich set of classification and regression tasks, such as the lazy regime of overparametrised neural networks, or equivalently the random features approximation of kernels. While allowing to study directly the mitigating effect of ensembling (or bagging) on the bias-variance decomposition of the test error, our analysis also helps disentangle the contribution of statistical fluctuations, and the singular role played by the interpolation threshold that are at the roots of the ‘double-descent’ phenomenon.
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Bellec, Pierre C. « Out-of-sample error estimation for M-estimators with convex penalty ». Information and Inference : A Journal of the IMA 12, no 4 (18 septembre 2023) : 2782–817. http://dx.doi.org/10.1093/imaiai/iaad031.
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Abstract A generic out-of-sample error estimate is proposed for $M$-estimators regularized with a convex penalty in high-dimensional linear regression where $(\boldsymbol{X},\boldsymbol{y})$ is observed and the dimension $p$ and sample size $n$ are of the same order. The out-of-sample error estimate enjoys a relative error of order $n^{-1/2}$ in a linear model with Gaussian covariates and independent noise, either non-asymptotically when $p/n\le \gamma $ or asymptotically in the high-dimensional asymptotic regime $p/n\to \gamma ^{\prime}\in (0,\infty )$. General differentiable loss functions $\rho $ are allowed provided that the derivative of the loss is 1-Lipschitz; this includes the least-squares loss as well as robust losses such as the Huber loss and its smoothed versions. The validity of the out-of-sample error estimate holds either under a strong convexity assumption, or for the L1-penalized Huber M-estimator and the Lasso under a sparsity assumption and a bound on the number of contaminated observations. For the square loss and in the absence of corruption in the response, the results additionally yield $n^{-1/2}$-consistent estimates of the noise variance and of the generalization error. This generalizes, to arbitrary convex penalty and arbitrary covariance, estimates that were previously known for the Lasso.
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TIMOSHIN, S. N. « Instabilities in a high-Reynolds-number boundary layer on a film-coated surface ». Journal of Fluid Mechanics 353 (25 décembre 1997) : 163–95. http://dx.doi.org/10.1017/s0022112097007337.
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A high-Reynolds-number asymptotic theory is developed for linear instability waves in a two-dimensional incompressible boundary layer on a flat surface coated with a thin film of a different fluid. The focus in this study is on the influence of the film flow on the lower-branch Tollmien–Schlichting waves, and also on the effect of boundary-layer/potential flow interaction on interfacial instabilities. Accordingly, the film thickness is assumed to be comparable to the thickness of a viscous sublayer in a three-tier asymptotic structure of lower-branch Tollmien–Schlichting disturbances. A fully nonlinear viscous/inviscid interaction formulation is derived, and computational and analytical solutions for small disturbances are obtained for both Tollmien–Schlichting and interfacial instabilities for a range of density and viscosity ratios of the fluids, and for various values of the surface tension coefficient and the Froude number. It is shown that the interfacial instability contains the fastest growing modes and an upper-branch neutral point within the chosen flow regime if the film viscosity is greater than the viscosity of the ambient fluid. For a less viscous film the theory predicts a lower neutral branch of shorter-scale interfacial waves. The film flow is found to have a strong effect on the Tollmien–Schlichting instability, the most dramatic outcome being a powerful destabilization of the flow due to a linear resonance between growing Tollmien–Schlichting and decaying capillary modes. Increased film viscosity also destabilizes Tollmien–Schlichting disturbances, with the maximum growth rate shifted towards shorter waves. Qualitative and quantitative comparisons are made with experimental observations by Ludwieg & Hornung (1989).
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Pang, Guodong, et Yuhang Zhou. « Functional Limit Theorems for Shot Noise Processes with Weakly Dependent Noises ». Stochastic Systems 10, no 2 (juin 2020) : 99–123. http://dx.doi.org/10.1287/stsy.2019.0051.
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We study shot noise processes when the shot noises are weakly dependent, satisfying the ρ-mixing condition. We prove a functional weak law of large numbers and a functional central limit theorem for this shot noise process in an asymptotic regime with a high intensity of shots. The deterministic fluid limit is unaffected by the presence of weak dependence. The limit in the diffusion scale is a continuous Gaussian process whose covariance function explicitly captures the dependence among the noises. The model and results can be applied in financial and insurance risks with dependent claims as well as queueing systems with dependent service times. To prove the existence of the limit process, we employ the existence criterion that uses a maximal inequality requiring a set function with a superadditivity property. We identify such a set function for the limit process by exploiting the ρ-mixing condition. To prove the weak convergence, we establish the tightness property and the convergence of finite dimensional distributions. To prove tightness, we construct two auxiliary processes and apply an Ottaviani-type inequality for weakly dependent sequences.
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Chakraborty, S., P. K. Nguyen, C. Ruyer-Quil et V. Bontozoglou. « Extreme solitary waves on falling liquid films ». Journal of Fluid Mechanics 745 (24 mars 2014) : 564–91. http://dx.doi.org/10.1017/jfm.2014.91.
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AbstractDirect numerical simulation (DNS) of liquid film flow is used to compute fully developed solitary waves and to compare their characteristics with the predictions of low-dimensional models. Emphasis is placed on the regime of high inertia, where available models provide widely differing results. It is found that the parametric dependence of wave properties on inertia is highly non-trivial, and is satisfactorily approximated only by the four-equation model of Ruyer-Quil & Manneville (Eur. Phys. J. B, vol. 15, 2000, pp. 357–369). Detailed comparison of the asymptotic shapes of upstream and downstream tails is performed, and inherent limitations of all long-wave models are revealed. Local flow reversal in front of the main hump, which has been previously discussed in the literature, is shown to occur for an inertia range bounded from below and from above, and the boundaries are interpreted in terms of the capillary origin of the phenomenon. Computational results are reported for the entire range of Froude numbers, providing benchmark data for all wall inclinations.
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Priede, Jānis, Thomas Arlt et Leo Bühler. « Linear stability of magnetohydrodynamic flow in a square duct with thin conducting walls ». Journal of Fluid Mechanics 788 (22 décembre 2015) : 129–46. http://dx.doi.org/10.1017/jfm.2015.709.
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This study is concerned with the numerical linear stability analysis of liquid-metal flow in a square duct with thin electrically conducting walls subject to a uniform transverse magnetic field. We derive an asymptotic solution for the base flow that is valid for not only high but also moderate magnetic fields. This solution shows that, for low wall conductance ratios $c\ll 1$, an extremely strong magnetic field with Hartmann number $\mathit{Ha}\sim c^{-4}$ is required to attain the asymptotic flow regime considered in previous studies. We use a vector streamfunction–vorticity formulation and a Chebyshev collocation method to solve the eigenvalue problem for three-dimensional small-amplitude perturbations in ducts with realistic wall conductance ratios $c=1$, 0.1 and 0.01 and Hartmann numbers up to $10^{4}$. As for similar flows, instability in a sufficiently strong magnetic field is found to occur in the sidewall jets with characteristic thickness ${\it\delta}\sim \mathit{Ha}^{-1/2}$. This results in the critical Reynolds number and wavenumber increasing asymptotically with the magnetic field as $\mathit{Re}_{c}\sim 110\mathit{Ha}^{1/2}$ and $k_{c}\sim 0.5\mathit{Ha}^{1/2}$. The respective critical Reynolds number based on the total volume flux in a square duct with $c\ll 1$ is $\overline{\mathit{Re}}_{c}\approx 520$. Although this value is somewhat larger than $\overline{\mathit{Re}}_{c}\approx 313$ found by Ting et al. (Intl J. Engng Sci., vol. 29 (8), 1991, pp. 939–948) for the asymptotic sidewall jet profile, it still appears significantly lower than the Reynolds numbers at which turbulence is observed in experiments as well as in direct numerical simulations of this type of flow.
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Weng, Haolei, et Arian Maleki. « Low noise sensitivity analysis of -minimization in oversampled systems ». Information and Inference : A Journal of the IMA 9, no 1 (27 janvier 2019) : 113–55. http://dx.doi.org/10.1093/imaiai/iay024.
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Abstract The class of $\ell _q$-regularized least squares (LQLS) are considered for estimating $\beta \in \mathbb{R}^p$ from its $n$ noisy linear observations $y=X\beta + w$. The performance of these schemes are studied under the high-dimensional asymptotic setting in which the dimension of the signal grows linearly with the number of measurements. In this asymptotic setting, phase transition (PT) diagrams are often used for comparing the performance of different estimators. PT specifies the minimum number of observations required by a certain estimator to recover a structured signal, e.g. a sparse one, from its noiseless linear observations. Although PT analysis is shown to provide useful information for compressed sensing, the fact that it ignores the measurement noise not only limits its applicability in many application areas, but also may lead to misunderstandings. For instance, consider a linear regression problem in which $n>p$ and the signal is not exactly sparse. If the measurement noise is ignored in such systems, regularization techniques, such as LQLS, seem to be irrelevant since even the ordinary least squares (OLS) returns the exact solution. However, it is well known that if $n$ is not much larger than $p$, then the regularization techniques improve the performance of OLS. In response to this limitation of PT analysis, we consider the low-noise sensitivity analysis. We show that this analysis framework (i) reveals the advantage of LQLS over OLS, (ii) captures the difference between different LQLS estimators even when $n>p$, and (iii) provides a fair comparison among different estimators in high signal-to-noise ratios. As an application of this framework, we will show that under mild conditions LASSO outperforms other LQLS even when the signal is dense. Finally, by a simple transformation, we connect our low-noise sensitivity framework to the classical asymptotic regime in which $n/p \rightarrow \infty$, and characterize how and when regularization techniques offer improvements over ordinary least squares, and which regularizer gives the most improvement when the sample size is large.
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MacDonald, M., L. Chan, D. Chung, N. Hutchins et A. Ooi. « Turbulent flow over transitionally rough surfaces with varying roughness densities ». Journal of Fluid Mechanics 804 (8 septembre 2016) : 130–61. http://dx.doi.org/10.1017/jfm.2016.459.
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We investigate rough-wall turbulent flows through direct numerical simulations of flow over three-dimensional transitionally rough sinusoidal surfaces. The roughness Reynolds number is fixed at $k^{+}=10$, where $k$ is the sinusoidal semi-amplitude, and the sinusoidal wavelength is varied, resulting in the roughness solidity $\unicode[STIX]{x1D6EC}$ (frontal area divided by plan area) ranging from 0.05 to 0.54. The high cost of resolving both the flow around the dense roughness elements and the bulk flow is circumvented by the use of the minimal-span channel technique, recently demonstrated by Chung et al. (J. Fluid Mech., vol. 773, 2015, pp. 418–431) to accurately determine the Hama roughness function, $\unicode[STIX]{x0394}U^{+}$. Good agreement of the second-order statistics in the near-wall roughness-affected region between minimal- and full-span rough-wall channels is observed. In the sparse regime of roughness ($\unicode[STIX]{x1D6EC}\lesssim 0.15$) the roughness function increases with increasing solidity, while in the dense regime the roughness function decreases with increasing solidity. It was found that the dense regime begins when $\unicode[STIX]{x1D6EC}\gtrsim 0.15{-}0.18$, in agreement with the literature. A model is proposed for the limit of $\unicode[STIX]{x1D6EC}\rightarrow \infty$, which is a smooth wall located at the crest of the roughness elements. This model assists with interpreting the asymptotic behaviour of the roughness, and the rough-wall data presented in this paper show that the near-wall flow is tending towards this modelled limit. The peak streamwise turbulence intensity, which is associated with the turbulent near-wall cycle, is seen to move further away from the wall with increasing solidity. In the sparse regime, increasing $\unicode[STIX]{x1D6EC}$ reduces the streamwise turbulent energy associated with the near-wall cycle, while increasing $\unicode[STIX]{x1D6EC}$ in the dense regime increases turbulent energy. An analysis of the difference of the integrated mean momentum balance between smooth- and rough-wall flows reveals that the roughness function decreases in the dense regime due to a reduction in the Reynolds shear stress. This is predominantly due to the near-wall cycle being pushed away from the roughness elements, which leads to a reduction in turbulent energy in the region previously occupied by these events.
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Coley, A. A. « The dynamics of brane-world cosmological models ». Canadian Journal of Physics 83, no 5 (1 mai 2005) : 475–525. http://dx.doi.org/10.1139/p05-035.
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Brane-world cosmology is motivated by recent developments in string/M-theory and offers a new perspective on the hierarchy problem. In the brane-world scenario, our Universe is a four-dimensional subspace or brane embedded in a higher-dimensional bulk spacetime. Ordinary matter fields are confined to the brane while the gravitational field can also propagate in the bulk, and it is not necessary for the extra dimensions to be small, or even compact, leading to modifications of Einstein's theory of general relativity at high energies. In particular, the RandallSundrum-type models are relatively simple phenomenological models that capture some of the essential features of the dimensional reduction of eleven-dimensional supergravity introduced by Hořava and Witten. These curved (or warped) models are self-consistent and simple and allow for an investigation of the essential nonlinear gravitational dynamics. The governing field equations induced on the brane differ from the general relativistic equations in that there are nonlocal effects from the free gravitational field in the bulk, transmitted via the projection of the bulk Weyl tensor, and the local quadratic energy-momentum corrections, which are significant in the high-energy regime close to the initial singularity. In this review, we investigate the dynamics of the five-dimensional warped RandallSundrum brane worlds and their generalizations, with particular emphasis on whether the currently observed high degree of homogeneity and isotropy can be explained. In particular, we discuss the asymptotic dynamical evolution of spatially homogeneous brane-world cosmological models containing both a perfect fluid and a scalar field close to the initial singularity. Using dynamical systems techniques, it is found that, for models with a physically relevant equation of state, an isotropic singularity is a past-attractor in all orthogonal spatially homogeneous models (including Bianchi type IX models). In addition, we describe the dynamics in a class of inhomogeneous brane-world models, and show that these models also have an isotropic initial singularity. These results provide support for the conjecture that typically the initial cosmological singularity is isotropic in brane-world cosmology. Consequently, we argue that, unlike the situation in general relativity, brane-world cosmological models may offer a plausible solution to the initial conditions problem in cosmology. PACS Nos.: 98.89.Cq/Jk, 04.20q
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Cohen, Nimrod, Boris Galperin et Semion Sukoriansky. « Zonons Are Solitons Produced by Rossby Wave Ringing ». Atmosphere 15, no 6 (14 juin 2024) : 711. http://dx.doi.org/10.3390/atmos15060711.
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Along with the familiar Rossby–Haurwitz waves, two-dimensional flows on the surface of a rotating sphere in the regime of zonostrophic turbulence harbor another class of waves known as zonons. Zonons are wave packets produced by energetic large-scale Rossby–Haurwitz wave modes ‘enslaving’ other wave modes. They propagate westward with the phase speed of the enslaving modes. Zonons can be visualized as enslaving modes’ ‘ringing’ in the enslaved ones with the frequencies of the former, the property that renders zonons non-dispersive. Zonons reside in high-shear regions confined between the opposing zonal jets yet they are mainly attached to westward jets and sustained by the ensuing barotropic instability. They exchange energy with the mean flow while preserving their identity in a fully turbulent environment, a feature characteristic of solitary waves. The goal of this study is to deepen our understanding of zonons’ physics using direct numerical simulations, a weakly non-linear theory, and asymptotic analysis, and ascertain that zonons are indeed isomorphic to solitary waves in the Korteweg–de Vries framework. Having this isomorphism established, the analysis is extended to eddies detected in the atmospheres of Jupiter and Saturn based upon the observed mean zonal velocity profiles and earlier findings that circulations on both planets obey the regime of zonostrophic macroturbulence. Not only the analysis confirms that many eddies and eddy trains on both giant planets indeed possess properties of zonons, but the theory also correctly predicts latitudinal bands that confine zonal trajectories of the eddies.
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SHANKAR, V., et V. KUMARAN. « Stability of fluid flow in a flexible tube to non-axisymmetric disturbances ». Journal of Fluid Mechanics 407 (25 mars 2000) : 291–314. http://dx.doi.org/10.1017/s0022112099007570.
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The stability of fluid flow in a flexible tube to non-axisymmetric perturbations is analysed in this paper. In the first part of the paper, the equivalents of classical theorems of hydrodynamic stability are derived for inviscid flow in a flexible tube subjected to arbitrary non-axisymmetric disturbances. Perturbations of the form vi = v˜i exp [ik(x − ct) + inθ] are imposed on a steady axisymmetric mean flow U(r) in a flexible tube, and the stability of mean flow velocity profiles and bounds for the phase velocity of the unstable modes are determined for arbitrary values of azimuthal wavenumber n. Here r, θ and x are respectively the radial, azimuthal and axial coordinates, and k and c are the axial wavenumber and phase velocity of disturbances. The flexible wall is represented by a standard constitutive relation which contains inertial, elastic and dissipative terms. The general results indicate that the fluid flow in a flexible tube is stable in the inviscid limit if the quantity Ud[Gscr ]/dr [ges ] 0, and could be unstable for Ud[Gscr ]/dr < 0, where [Gscr ] ≡ rU′/(n2 + k2r2). For the case of Hagen–Poiseuille flow, the general result implies that the flow is stable to axisymmetric disturbances (n = 0), but could be unstable to non-axisymmetric disturbances with any non-zero azimuthal wavenumber (n ≠ 0). This is in marked contrast to plane parallel flows where two-dimensional disturbances are always more unstable than three-dimensional ones (Squire theorem). Some new bounds are derived which place restrictions on the real and imaginary parts of the phase velocity for arbitrary non-axisymmetric disturbances.In the second part of this paper, the stability of the Hagen–Poiseuille flow in a flexible tube to non-axisymmetric disturbances is analysed in the high Reynolds number regime. An asymptotic analysis reveals that the Hagen–Poiseuille flow in a flexible tube is unstable to non-axisymmetric disturbances even in the inviscid limit, and this agrees with the general results derived in this paper. The asymptotic results are extended numerically to the moderate Reynolds number regime. The numerical results reveal that the critical Reynolds number obtained for inviscid instability to non-axisymmetric disturbances is much lower than the critical Reynolds numbers obtained in the previous studies for viscous instability to axisymmetric disturbances when the dimensionless parameter Σ = ρGR2/η2 is large. Here G is the shear modulus of the elastic medium, ρ is the density of the fluid, R is the radius of the tube and η is the viscosity of the fluid. The viscosity of the wall medium is found to have a stabilizing effect on this instability.
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Dentz, M., M. Icardi et J. J. Hidalgo. « Mechanisms of dispersion in a porous medium ». Journal of Fluid Mechanics 841 (1 mars 2018) : 851–82. http://dx.doi.org/10.1017/jfm.2018.120.
Texte intégralRésumé :
This paper studies the mechanisms of dispersion in the laminar flow through the pore space of a three-dimensional porous medium. We focus on preasymptotic transport prior to the asymptotic hydrodynamic dispersion regime, in which solute motion may be described by the average flow velocity and a hydrodynamic dispersion coefficient. High-performance numerical flow and transport simulations of solute breakthrough at the outlet of a sand-like porous medium evidence marked deviations from the hydrodynamic dispersion paradigm and identify two distinct regimes. The first regime is characterised by a broad distribution of advective residence times in single pores. The second regime is characterised by diffusive mass transfer into low-velocity regions in the wake of solid grains. These mechanisms are quantified systematically in the framework of a time-domain random walk for the motion of marked elements (particles) of the transported material quantity. Particle transitions occur over the length scale imprinted in the pore structure at random times given by heterogeneous advection and diffusion. Under globally advection-dominated conditions, i.e., Péclet numbers larger than 1, particles sample the intrapore velocities by diffusion and the interpore velocities through advection. Thus, for a single transition, particle velocities are approximated by the mean pore velocity. In order to quantify this advection mechanism, we develop a model for the statistics of the Eulerian velocity magnitude based on Poiseuille’s law for flow through a single pore and for the distribution of mean pore velocities, both of which are linked to the distribution of pore diameters. Diffusion across streamlines through immobile zones in the wake of solid grains gives rise to exponentially distributed residence times that decay on the diffusion time over the pore length. The trapping rate is determined by the inverse diffusion time. This trapping mechanism is represented by a compound Poisson process conditioned on the advective residence time in the proposed time-domain random walk approach. The model is parameterised with the characteristics of the porous medium under consideration and captures both preasymptotic regimes. Macroscale transport is described by an integro-differential equation for solute concentration, whose memory kernels are given in terms of the distribution of mean pore velocities and trapping times. This approach quantifies the physical non-equilibrium caused by a broad distribution of mass transfer time scales, both advective and diffusive, on the representative elementary volume (REV). Thus, while the REV indicates the scale at which medium properties like porosity can be uniquely defined, this does not imply that transport can be characterised by hydrodynamic dispersion.
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Cerminara, M., T. Esposti Ongaro et L. C. Berselli. « ASHEE : a compressible, Equilibrium–Eulerian model for volcanic ash plumes ». Geoscientific Model Development Discussions 8, no 10 (19 octobre 2015) : 8895–979. http://dx.doi.org/10.5194/gmdd-8-8895-2015.
Texte intégralRésumé :
Abstract. A new fluid-dynamic model is developed to numerically simulate the non-equilibrium dynamics of polydisperse gas-particle mixtures forming volcanic plumes. Starting from the three-dimensional N-phase Eulerian transport equations (Neri et al., 2003) for a mixture of gases and solid dispersed particles, we adopt an asymptotic expansion strategy to derive a compressible version of the first-order non-equilibrium model (Ferry and Balachandar, 2001), valid for low concentration regimes (particle volume fraction less than 10−3) and particles Stokes number (St, i.e., the ratio between their relaxation time and flow characteristic time) not exceeding about 0.2. The new model, which is called ASHEE (ASH Equilibrium Eulerian), is significantly faster than the N-phase Eulerian model while retaining the capability to describe gas-particle non-equilibrium effects. Direct numerical simulation accurately reproduce the dynamics of isotropic, compressible turbulence in subsonic regime. For gas-particle mixtures, it describes the main features of density fluctuations and the preferential concentration and clustering of particles by turbulence, thus verifying the model reliability and suitability for the numerical simulation of high-Reynolds number and high-temperature regimes in presence of a dispersed phase. On the other hand, Large-Eddy Numerical Simulations of forced plumes are able to reproduce their observed averaged and instantaneous flow properties. In particular, the self-similar Gaussian radial profile and the development of large-scale coherent structures are reproduced, including the rate of turbulent mixing and entrainment of atmospheric air. Application to the Large-Eddy Simulation of the injection of the eruptive mixture in a stratified atmosphere describes some of important features of turbulent volcanic plumes, including air entrainment, buoyancy reversal, and maximum plume height. For very fine particles (St → 0, when non-equilibrium effects are negligible) the model reduces to the so-called dusty-gas model. However, coarse particles partially decouple from the gas phase within eddies (thus modifying the turbulent structure) and preferentially concentrate at the eddy periphery, eventually being lost from the plume margins due to the concurrent effect of gravity. By these mechanisms, gas-particle non-equilibrium processes are able to influence the large-scale behavior of volcanic plumes.
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Dobson, John F. « Towards efficient description of type-C London dispersion forces between low-dimensional metallic nanostructures ». Electronic Structure 3, no 4 (30 novembre 2021) : 044001. http://dx.doi.org/10.1088/2516-1075/ac3807.
Texte intégralRésumé :
Abstract This paper concerns one of the remaining difficulties encountered in efficient modelling schemes for dispersion forces between nanostructures. Dispersion (van der Waals, vdW) interactions between molecules and nanostructures can be reliably described in principle by computationally intensive high-level theory, but this is often not computationally feasible in practice, so more efficient methods are continually being developed. Progress has been made with nonlocal density functionals (vdW-DFs) and with atom-based schemes (D4, MBD, uMBD). In such efficient schemes, the effects beyond additive two-atom terms have been categorized as ‘type A’, ‘type B’ and ‘type C’ non-additivity (see Dobson (2014 Int. J. Quantum Chem. 114 1157)). Atom-based models using coupled-harmonic-oscillator theory (MBD) now deal adequately with type A and type B non-additivity, but type-C effects, related to gapless collective electronic excitations, can occur in low-dimensional metals, and these are not correctly described by the efficient schemes mentioned above. From analytic work within the direct random phase approximation (dRPA), type-C effects have long been known to cause the vdW interaction between well-separated low-d metals to fall off much more slowly with separation than is predicted by the above-mentioned efficient schemes. The slower decay means that type-C effects dominate in this asymptotic large-separation regime. It has not been clear, however, whether type-C physics contributes significantly to the vdW interaction of low-d metals near to contact, where the forces are much larger. The present work uses recent semi-analytic dRPA results to provide some evidence that type-C effects are indeed significant near to contact between metallic carbon nanotubes, and between doped graphene sheets. Some guidelines are therefore suggested for ways to combine the semi-analytic dRPA approach, here termed ‘SAM-dRPA’, with the existing efficient vdW algorithms described above.
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Kock, Anders Bredahl, et David Preinerstorfer. « Power in High‐Dimensional Testing Problems ». Econometrica 87, no 3 (2019) : 1055–69. http://dx.doi.org/10.3982/ecta15844.
Texte intégralRésumé :
Fan, Liao, and Yao (2015) recently introduced a remarkable method for increasing the asymptotic power of tests in high‐dimensional testing problems. If applicable to a given test, their power enhancement principle leads to an improved test that has the same asymptotic size, has uniformly non‐inferior asymptotic power, and is consistent against a strictly broader range of alternatives than the initially given test. We study under which conditions this method can be applied and show the following: In asymptotic regimes where the dimensionality of the parameter space is fixed as sample size increases, there often exist tests that cannot be further improved with the power enhancement principle. However, when the dimensionality of the parameter space increases sufficiently slowly with sample size and a marginal local asymptotic normality (LAN) condition is satisfied, every test with asymptotic size smaller than 1 can be improved with the power enhancement principle. While the marginal LAN condition alone does not allow one to extend the latter statement to all rates at which the dimensionality increases with sample size, we give sufficient conditions under which this is the case.
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Kim, Jeonglae, Scott Davidson et Ali Mani. « Characterization of Chaotic Electroconvection near Flat Inert Electrodes under Oscillatory Voltages ». Micromachines 10, no 3 (26 février 2019) : 161. http://dx.doi.org/10.3390/mi10030161.
Texte intégralRésumé :
The onset of electroconvective instability in an aqueous binary electrolyte under external oscillatory electric fields at a single constant frequency is investigated in a 2D parallel flat electrode setup. Direct numerical simulations (DNS) of the Poisson–Nernst–Planck equations coupled with the Navier–Stokes equations at a low Reynolds number are carried out. Previous studies show that direct current (DC) electric field can create electroconvection near ion-selecting membranes in microfluidic devices. In this study, we show that electroconvection can be generated near flat inert electrodes when the applied electric field is oscillatory in time. A range of applied voltage, the oscillation frequency and the ratio of ionic diffusivities is examined to characterize the regime in which electroconvection takes place. Similar to electroconvection under DC voltages, AC electroconvection occurs at sufficiently high applied voltages in units of thermal volts and is characterized by transverse instabilities, physically manifested by an array of counter-rotating vortices near the electrode surfaces. The oscillating external electric field periodically generate and destroy such unsteady vortical structures. As the oscillation frequency is reduced to O ( 10 − 1 ) of the intrinsic resistor–capacitor (RC) frequency of electrolyte, electroconvective instability is considerably amplified. This is accompanied by severe depletion of ionic species outside the thin electric double layer and by vigorous convective transport involving a wide range of scales including those comparable to the distance L between the parallel electrodes. The underlying mechanisms are distinctly nonlinear and multi-dimensional. However, at higher frequencies of order of the RC frequency, the electrolyte response becomes linear, and the present DNS prediction closely resembles those explained by 1D asymptotic studies. Electroconvective instability supports increased electric current across the system. Increasing anion diffusivity results in stronger amplification of electroconvection over all oscillation frequencies examined in this study. Such asymmetry in ionic diffusivity, however, does not yield consistent changes in statistics and energy spectrum at all wall-normal locations and frequencies, implying more complex dynamics and different scaling for electrolytes with unequal diffusivities. Electric current is substantially amplified beyond the ohmic current at high oscillation frequencies. Also, it is found that anion diffusivity higher than cation has stronger impact on smaller-scale motions (≲ 0.1 L).
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WU, XUESONG, S. J. LEIB et M. E. GOLDSTEIN. « On the nonlinear evolution of a pair of oblique Tollmien–Schlichting waves in boundary layers ». Journal of Fluid Mechanics 340 (10 juin 1997) : 361–94. http://dx.doi.org/10.1017/s0022112097005557.
Texte intégralRésumé :
This paper is concerned with the nonlinear interaction and development of a pair of oblique Tollmien–Schlichting waves which travel with equal but opposite angles to the free stream in a boundary layer. Our approach is based on high-Reynolds-number asymptotic methods. The so-called ‘upper-branch’ scaling is adopted so that there exists a well-defined critical layer, i.e. a thin region surrounding the level at which the basic flow velocity equals the phase velocity of the waves. We show that following the initial linear growth, the disturbance evolves through several distinct nonlinear stages. In the first of these, nonlinearity only affects the phase angle of the amplitude of the disturbance, causing rapid wavelength shortening, while the modulus of the amplitude still grows exponentially as in the linear regime. The second stage starts when the wavelength shortening produces a back reaction on the development of the modulus. The phase angle and the modulus then evolve on different spatial scales, and are governed by two coupled nonlinear equations. The solution to these equations develops a singularity at a finite distance downstream. As a result, the disturbance enters the third stage in which it evolves over a faster spatial scale, and the critical layer becomes both non-equilibrium and viscous in nature, in contrast to the two previous stages, where the critical layer is in equilibrium and purely viscosity dominated. In this stage, the development is governed by an amplitude equation with the same nonlinear term as that derived by Wu, Lee & Cowley (1993) for the interaction between a pair of Rayleigh waves. The solution develops a new singularity, leading to the fourth stage where the flow is governed by the fully nonlinear three-dimensional inviscid triple-deck equations. It is suggested that the stages of evolution revealed here may characterize the so-called ‘oblique breakdown’ in a boundary layer. A discussion of the extension of the analysis to include the resonant-triad interaction is given.
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Bernardini, Matteo, Sergio Pirozzoli et Paolo Orlandi. « Velocity statistics in turbulent channel flow up to ». Journal of Fluid Mechanics 742 (21 février 2014) : 171–91. http://dx.doi.org/10.1017/jfm.2013.674.
Texte intégralRésumé :
AbstractThe high-Reynolds-number behaviour of the canonical incompressible turbulent channel flow is investigated through large-scale direct numerical simulation (DNS). A Reynolds number is achieved ($Re_{\tau } = h/\delta _v \approx 4000$, where $h$ is the channel half-height, and $\delta _v$ is the viscous length scale) at which theory predicts the onset of phenomena typical of the asymptotic Reynolds number regime, namely a sensible layer with logarithmic variation of the mean velocity profile, and Kolmogorov scaling of the velocity spectra. Although higher Reynolds numbers can be achieved in experiments, the main advantage of the present DNS study is access to the full three-dimensional flow field. Consistent with refined overlap arguments (Afzal & Yajnik, J. Fluid Mech. vol. 61, 1973, pp. 23–31; Jiménez & Moser, Phil. Trans. R. Soc. Lond. A, vol. 365, 2007, pp. 715–732), our results suggest that the mean velocity profile never achieves a truly logarithmic profile, and the logarithmic diagnostic function instead exhibits a linear variation in the outer layer whose slope decreases with the Reynolds number. The extrapolated value of the von Kármán constant is $k \approx 0.41$. A near logarithmic layer is observed in the spanwise velocity variance, as predicted by Townsend’s attached eddy hypothesis, whereas the streamwise variance seems to exhibit a shoulder, perhaps being still affected by low-Reynolds-number effects. Comparison with previous DNS data at lower Reynolds number suggests enhancement of the imprinting effect of outer-layer eddies onto the near-wall region. This mechanisms is associated with excess turbulence kinetic energy production in the outer layer, and it reflects in flow visualizations and in the streamwise velocity spectra, which exhibit sharp peaks in the outer layer. Associated with the outer energy production site, we find evidence of a Kolmogorov-like inertial range, limited to the spanwise spectral density of $u$, whereas power laws with different exponents are found for the other spectra. Finally, arguments are given to explain the ‘odd’ scaling of the streamwise velocity variances, based on the analysis of the kinetic energy production term.
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Liang, Tengyuan, Subhabrata Sen et Pragya Sur. « High-dimensional asymptotics of Langevin dynamics in spiked matrix models ». Information and Inference : A Journal of the IMA 12, no 4 (18 septembre 2023) : 2720–52. http://dx.doi.org/10.1093/imaiai/iaad042.
Texte intégralRésumé :
Abstract We study Langevin dynamics for recovering the planted signal in the spiked matrix model. We provide a ‘path-wise’ characterization of the overlap between the output of the Langevin algorithm and the planted signal. This overlap is characterized in terms of a self-consistent system of integro-differential equations, usually referred to as the Crisanti–Horner–Sommers–Cugliandolo–Kurchan equations in the spin glass literature. As a second contribution, we derive an explicit formula for the limiting overlap in terms of the signal-to-noise ratio and the injected noise in the diffusion. This uncovers a sharp phase transition—in one regime, the limiting overlap is strictly positive, while in the other, the injected noise overcomes the signal, and the limiting overlap is zero.
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Morris, Wallace J., et Zvi Rusak. « Stall onset on aerofoils at low to moderately high Reynolds number flows ». Journal of Fluid Mechanics 733 (24 septembre 2013) : 439–72. http://dx.doi.org/10.1017/jfm.2013.440.
Texte intégralRésumé :
AbstractThe inception of leading-edge stall on stationary, two-dimensional, smooth, thin aerofoils at low to moderately high chord Reynolds number flows is investigated by a reduced-order, multiscale model problem via numerical simulations. The asymptotic theory demonstrates that a subsonic flow about a thin aerofoil can be described in terms of an outer region, around most of the aerofoil’s chord, and an inner region, around the nose, that asymptotically match each other. The flow in the outer region is dominated by the classical thin aerofoil theory. Scaled (magnified) coordinates and a modified (smaller) Reynolds number $(R{e}_{M} )$ are used to correctly account for the nonlinear behaviour and extreme velocity changes in the inner region, where both the near-stagnation and high suction areas occur. It results in a model problem of a uniform, incompressible and viscous flow past a semi-infinite parabola with a far-field circulation governed by a parameter $\tilde {A} $ that is related to the aerofoil’s angle of attack, nose radius of curvature, thickness ratio, and camber. The model flow problem is solved for various values of $\tilde {A} $ through numerical simulations based on the unsteady Navier–Stokes equations. The value ${\tilde {A} }_{s} $ where a global separation zone first erupts in the nose flow, accompanied by loss of peak streamwise velocity ahead of it and change in shedding frequency behind it, is determined as a function of $R{e}_{M} $. These values indicate the stall onset on the aerofoil at various flow conditions. It is found that ${\tilde {A} }_{s} $ decreases with $R{e}_{M} $ until some limit $R{e}_{M} $ (${\sim }300$) and then increases with further increase of Reynolds number. At low values of $R{e}_{M} $ the flow is laminar and steady, even when stall occurs. The flow in this regime is dominated by the increasing effect of the adverse pressure gradient, which eventually overcomes the ability of the viscous stress to keep the boundary layer attached to the aerofoil. The change in the nature of stall at the limit $R{e}_{M} $ is attributed to the appearance of downstream travelling waves in the boundary layer that shed from the marginal separation zone and grow in size with either $\tilde {A} $ or $R{e}_{M} $. These unsteady, convective vortical structures relax the effect of the adverse pressure gradient on the viscous boundary layer to delay the onset of stall in the mean flow to higher values of ${\tilde {A} }_{s} $. Computed results show agreement with marginal separation theory at low $R{e}_{M} $ and with available experimental data at higher $R{e}_{M} $. This simplified approach provides a universal criterion to determine the stall angle of stationary thin aerofoils with a parabolic nose.
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Li, Jian, Yong Liu et Weiping Wang. « High-Dimensional Analysis for Generalized Nonlinear Regression : From Asymptotics to Algorithm ». Proceedings of the AAAI Conference on Artificial Intelligence 38, no 12 (24 mars 2024) : 13500–13508. http://dx.doi.org/10.1609/aaai.v38i12.29253.
Texte intégralRésumé :
Overparameterization often leads to benign overfitting, where deep neural networks can be trained to overfit the training data but still generalize well on unseen data. However, it lacks a generalized asymptotic framework for nonlinear regressions and connections to conventional complexity notions. In this paper, we propose a generalized high-dimensional analysis for nonlinear regression models, including various nonlinear feature mapping methods and subsampling. Specifically, we first provide an implicit regularization parameter and asymptotic equivalents related to a classical complexity notion, i.e., effective dimension. We then present a high-dimensional analysis for nonlinear ridge regression and extend it to ridgeless regression in the under-parameterized and over-parameterized regimes, respectively. We find that the limiting risks decrease with the effective dimension. Motivated by these theoretical findings, we propose an algorithm, namely RFRed, to improve generalization ability. Finally, we validate our theoretical findings and the proposed algorithm through several experiments.
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WU, XUESONG, P. A. STEWART et S. J. COWLEY. « On the catalytic role of the phase-locked interaction of Tollmien–Schlichting waves in boundary-layer transition ». Journal of Fluid Mechanics 590 (15 octobre 2007) : 265–94. http://dx.doi.org/10.1017/s002211200700804x.
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This paper is concerned with the nonlinear interaction between a planar and a pair of oblique Tollmien–Schlichting (T-S) waves which are phase-locked in that they travel with (nearly) the same phase speed. The evolution of such a disturbance is described using a high-Reynolds-number asymptotic approach in the so-called ‘upper--branch’ scaling regime. It follows that there exists a well-defined common critical layer (i.e. a thin region surrounding the level at which the basic flow velocity equals the phase speed of the waves to leading order) and the dominant interactions take place there. The disturbance is shown to evolve through several distinctive stages. In the first of these, the critical layer is in equilibrium and viscosity dominated. If a small mismatching exists in the phase speeds, the interaction between the planar and oblique waves leads directly to super-exponential growth/decay of the oblique modes. However, if the modes are perfectly phase-locked, the interaction in the first instance affects only the phase of the amplitude function of the oblique modes (so causing rapid wavelength shortening), while the modulus of the amplitude still evolves exponentially until the wavelength shortening produces a back reaction on the modulus (which then induces a super-exponential growth). Whether or not there is a small mismatch or a perfect match in the phase speeds, once the growth rate of the oblique modes becomes sufficiently large, the disturbance enters a second stage, in which the critical layer becomes both non-equilibrium and viscous in nature. The oblique modes continue to experience super-exponential growth, albeit of a different form from that in the previous stages, until the self-interaction between them, as well as their back effect on the planar mode, becomes important. At that point, the disturbance enters a third, fully interactive stage, during which the development of the disturbance is governed by the amplitude equations with the same nonlinear terms as previously derived for the phase-locked interaction of Rayleigh instability waves. The solution develops a singularity, leading to the final stage where the flow is governed by fully nonlinear three-dimensional inviscid triple-deck equations. The present work indicates that seeding a planar T-S wave can enhance the amplification of all oblique modes which share approximately its phase speed.
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Yang, Weihua, et David Rideout. « High Dimensional Hyperbolic Geometry of Complex Networks ». Mathematics 8, no 11 (23 octobre 2020) : 1861. http://dx.doi.org/10.3390/math8111861.
Texte intégralRésumé :
High dimensional embeddings of graph data into hyperbolic space have recently been shown to have great value in encoding hierarchical structures, especially in the area of natural language processing, named entity recognition, and machine generation of ontologies. Given the striking success of these approaches, we extend the famous hyperbolic geometric random graph models of Krioukov et al. to arbitrary dimension, providing a detailed analysis of the degree distribution behavior of the model in an expanded portion of the parameter space, considering several regimes which have yet to be considered. Our analysis includes a study of the asymptotic correlations of degree in the network, revealing a non-trivial dependence on the dimension and power law exponent. These results pave the way to using hyperbolic geometric random graph models in high dimensional contexts, which may provide a new window into the internal states of network nodes, manifested only by their external interconnectivity.
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BUTTON, BRADLY K., LEO RODRIGUEZ, CATHERINE A. WHITING et TUNA YILDIRIM. « A NEAR HORIZON CFT DUAL FOR KERR–NEWMAN AdS ». International Journal of Modern Physics A 26, no 18 (20 juillet 2011) : 3077–90. http://dx.doi.org/10.1142/s0217751x11053663.
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We show that the near horizon regime of a Kerr–Newman AdS (KNAdS) black hole, given by its two-dimensional analogue a là Robinson and Wilczek (Phys. Rev. Lett.95, 011303 (2005)), is asymptotically AdS2 and dual to a one-dimensional quantum conformal field theory (CFT). The s-wave contribution of the resulting CFT's energy–momentum tensor together with the asymptotic symmetries, generate a centrally extended Virasoro algebra, whose central charge reproduces the Bekenstein–Hawking entropy via Cardy's formula. Our derived central charge also agrees with the near extremal Kerr/CFT correspondence (Phys. Rev. D80, 124008 (2009)) in the appropriate limits. We also compute the Hawking temperature of the KNAdS black hole by coupling its Robinson and Wilczek two-dimensional analogue (RW2DA) to conformal matter.
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Bezrodnykh, S. I., V. B. Zametaev et Te Ha Chzhun. « Singularity Formation in an Incompressible Boundary Layer on an Upstream Moving Wall under Given External Pressure ». Журнал вычислительной математики и математической физики 63, no 12 (1 décembre 2023) : 2081–93. http://dx.doi.org/10.31857/s0044466923120074.
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The two-dimensional laminar flow of a viscous incompressible fluid over a flat surface is considered at high Reynolds numbers. The influence exerted on the Blasius boundary layer by a body moving downstream with a low velocity relative to the plate is studied within the framework of asymptotic theory. The case in which a small external body modeled by a potential dipole moves downstream at a constant velocity is investigated. Formally, this classical problem is nonstationary, but, after passing to a coordinate system comoving with the dipole, it is described by stationary solutions of boundary layer equations on the wall moving upstream. The numerically found solutions of this problem involve closed and open separation zones in the flow field. Nonlinear regimes of the influence exerted by the dipole on the boundary layer with counterflows are calculated. It is found that, as the dipole intensity grows, the dipole-induced pressure acting on the boundary layer grows as well, which, after reaching a certain critical dipole intensity, gives rise to a singularity in the flow field. The asymptotics of the solution near the isolated singular point of the flow field is studied. It is found that, at this point, the vertical velocity grows to infinity, viscous stress vanishes, and no solution of the problem exists at higher dipole intensities.
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Li, Kuan, Andrew Jackson et Philip W. Livermore. « Taylor state dynamos found by optimal control : axisymmetric examples ». Journal of Fluid Mechanics 853 (29 août 2018) : 647–97. http://dx.doi.org/10.1017/jfm.2018.569.
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Earth’s magnetic field is generated in its fluid metallic core through motional induction in a process termed the geodynamo. Fluid flow is heavily influenced by a combination of rapid rotation (Coriolis forces), Lorentz forces (from the interaction of electrical currents and magnetic fields) and buoyancy; it is believed that the inertial force and the viscous force are negligible. Direct approaches to this regime are far beyond the reach of modern high-performance computing power, hence an alternative ‘reduced’ approach may be beneficial. Taylor (Proc. R. Soc. Lond. A, vol. 274 (1357), 1963, pp. 274–283) studied an inertia-free and viscosity-free model as an asymptotic limit of such a rapidly rotating system. In this theoretical limit, the velocity and the magnetic field organize themselves in a special manner, such that the Lorentz torque acting on every geostrophic cylinder is zero, a property referred to as Taylor’s constraint. Moreover, the flow is instantaneously and uniquely determined by the buoyancy and the magnetic field. In order to find solutions to this mathematical system of equations in a full sphere, we use methods of optimal control to ensure that the required conditions on the geostrophic cylinders are satisfied at all times, through a conventional time-stepping procedure that implements the constraints at the end of each time step. A derivative-based approach is used to discover the correct geostrophic flow required so that the constraints are always satisfied. We report a new quantity, termed the Taylicity, that measures the adherence to Taylor’s constraint by analysing squared Lorentz torques, normalized by the squared energy in the magnetic field, over the entire core. Neglecting buoyancy, we solve the equations in a full sphere and seek axisymmetric solutions to the equations; we invoke $\unicode[STIX]{x1D6FC}$- and $\unicode[STIX]{x1D714}$-effects in order to sidestep Cowling’s anti-dynamo theorem so that the dynamo system possesses non-trivial solutions. Our methodology draws heavily on the use of fully spectral expansions for all divergenceless vector fields. We employ five special Galerkin polynomial bases in radius such that the boundary conditions are honoured by each member of the basis set, whilst satisfying an orthogonality relation defined in terms of energies. We demonstrate via numerous examples that there are stable solutions to the equations that possess a rapidly decreasing spectrum and are thus well-converged. Classic distributions for the $\unicode[STIX]{x1D6FC}$- and $\unicode[STIX]{x1D714}$-effects are invoked, as well as new distributions. One such new $\unicode[STIX]{x1D6FC}$-effect model possesses oscillatory solutions for the magnetic field, rarely before seen. By comparing our Taylor state model with one that allows torsional oscillations to develop and decay, we show the equilibrium state of both configurations to be coincident. In all our models, the geostrophic flow dominates the ageostrophic flow. Our work corroborates some results previously reported by Wu & Roberts (Geophys. Astrophys. Fluid Dyn., vol. 109 (1), 2015, pp. 84–110), as well as presenting new results; it sets the stage for a three-dimensional implementation where the system is driven by, for example, thermal convection.
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ENQVIST, K., S. MOHANTY et D. V. NANOPOULOS. « ASPECTS OF SUPERSTRING QUANTUM COSMOLOGY ». International Journal of Modern Physics A 04, no 04 (février 1989) : 873–95. http://dx.doi.org/10.1142/s0217751x89000406.
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We consider the quantum cosmology of a D-dimensional Universe using the point-like field theory effective action of closed bosonic strings. We find that in order to apply the Hartle-Hawking boundary condition, the Universe must have a product topology (like R × Sn × Sm). This rules out four-dimensional theories (unless they are supplemented by extra internal degrees of freedom). We also explain how the Universe evolves from the Euclidean regime (where the space and time are on equal footing) to a macroscopic Lorentzian space Mn+1 and a compact space Km. We find that by demanding the compactified space to be Ricci flat and of the order of Planck scale, we are led uniquely to the conditions D = 10, n = 3, m = 6. We also study the behavior of the gauge coupling constant with time and calculate its asymptotic value.
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Lin, Rui-Hui, et Xiang-Hua Zhai. « Thermal Casimir effect for rectangular cavities inside (D+1)-dimensional Minkowski space–time revisited ». International Journal of Modern Physics A 29, no 08 (24 mars 2014) : 1450043. http://dx.doi.org/10.1142/s0217751x14500432.
Texte intégralRésumé :
We reconsider the thermal scalar Casimir effect for p-dimensional rectangular cavity inside (D+1)-dimensional Minkowski space–time and clarify the ambiguity in the regularization of the temperature-dependent part of the free energy. We derive rigorously the regularization of the temperature-dependent part of the free energy by making use of the Abel–Plana formula repeatedly and get the explicit expression of the terms to be subtracted. In the cases of D = 3, p = 1 and D = 3, p = 3, we precisely recover the results of parallel plates and three-dimensional box in the literature. Furthermore, for D>p and D = p cases with periodic, Dirichlet and Neumann boundary conditions, we give the explicit expressions of the Casimir free energy in both low temperature (small separations) and high temperature (large separations) regimes, through which the asymptotic behavior of the free energy changing with temperature and the side length is easy to see. We find that for D>p, with the side length going to infinity, the Casimir free energy tends to positive or negative constants or zero, depending on the boundary conditions. But for D = p, the leading term of the Casimir free energy for all three boundary conditions is a logarithmic function of the side length. We also discuss the thermal Casimir force changing with temperature and the side length in different cases and find that when the side length goes to infinity, the force always tends to be zero for different boundary conditions regardless of D>p or D = p. The Casimir free energy and force at high temperature limit behave asymptotically alike that they are proportional to the temperature, be they positive (repulsive) or negative (attractive) in different cases. Our study may be helpful in providing a comprehensive and complete understanding of this old problem.
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Ruiz, Matias, et Ory Schnitzer. « Slender-body theory for plasmonic resonance ». Proceedings of the Royal Society A : Mathematical, Physical and Engineering Sciences 475, no 2229 (septembre 2019) : 20190294. http://dx.doi.org/10.1098/rspa.2019.0294.
Texte intégralRésumé :
We develop a slender-body theory for plasmonic resonance of slender metallic nanoparticles, focusing on a general class of axisymmetric geometries with locally paraboloidal tips. We adopt a modal approach where one first solves the plasmonic eigenvalue problem, a geometric spectral problem which governs the surface-plasmon modes of the particle; then, the latter modes are used, in conjunction with spectral-decomposition, to analyse localized-surface-plasmon resonance in the quasi-static limit. We show that the permittivity eigenvalues of the axisymmetric modes are strongly singular in the slenderness parameter, implying widely tunable, high-quality-factor, resonances in the near-infrared regime. For that family of modes, we use matched asymptotics to derive an effective eigenvalue problem, a singular non-local Sturm–Liouville problem, where the lumped one-dimensional eigenfunctions represent axial voltage profiles (or charge line densities). We solve the effective eigenvalue problem in closed form for a prolate spheroid and numerically, by expanding the eigenfunctions in Legendre polynomials, for arbitrarily shaped particles. We apply the theory to plane-wave illumination in order to elucidate the excitation of multiple resonances in the case of non-spheroidal particles.
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Ward, T. J., O. E. Jensen, H. Power et D. S. Riley. « High-Rayleigh-number convection of a reactive solute in a porous medium ». Journal of Fluid Mechanics 760 (4 novembre 2014) : 95–126. http://dx.doi.org/10.1017/jfm.2014.594.
Texte intégralRésumé :
AbstractWe consider two-dimensional one-sided convection of a solute in a fluid-saturated porous medium, where the solute decays via a first-order reaction. Fully nonlinear convection is investigated using high-resolution numerical simulations and a low-order model that couples the dynamic boundary layer immediately beneath the distributed solute source to the slender vertical plumes that form beneath. A transient-growth analysis of the boundary layer is used to characterise its excitability. Three asymptotic regimes are investigated in the limit of high Rayleigh number $\mathit{Ra}$, in which the domain is considered deep, shallow or of intermediate depth, and for which the Damköhler number $\mathit{Da}$ is respectively large, small or of order unity. Scaling properties of the flow are identified numerically and rationalised via the analytic model. For fully established high-$\mathit{Ra}$ convection, analysis and simulation suggest that the time-averaged solute transfer rate scales with $\mathit{Ra}$ and the plume horizontal wavenumber with $\mathit{Ra}^{1/2}$, with coefficients modulated by $\mathit{Da}$ in each case. For large $\mathit{Da}$, the rapid reaction rate limits the plume depth and the boundary layer restricts the rate of solute transfer to the bulk, whereas for small $\mathit{Da}$ the average solute transfer rate is ultimately limited by the domain depth and the convection is correspondingly weaker.
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Luneau, Clément, Nicolas Macris et Jean Barbier. « Information theoretic limits of learning a sparse rule ». Journal of Statistical Mechanics : Theory and Experiment 2022, no 4 (1 avril 2022) : 044001. http://dx.doi.org/10.1088/1742-5468/ac59ac.
Texte intégralRésumé :
Abstract We consider generalized linear models in regimes where the number of nonzero components of the signal and accessible data points are sublinear with respect to the size of the signal. We prove a variational formula for the asymptotic mutual information per sample when the system size grows to infinity. This result allows us to derive an expression for the minimum mean-square error (MMSE) of the Bayesian estimator when the signal entries have a discrete distribution with finite support. We find that, for such signals and suitable vanishing scalings of the sparsity and sampling rate, the MMSE is nonincreasing piecewise constant. In specific instances the MMSE even displays an all-or-nothing phase transition, that is, the MMSE sharply jumps from its maximum value to zero at a critical sampling rate. The all-or-nothing phenomenon has previously been shown to occur in high-dimensional linear regression. Our analysis goes beyond the linear case and applies to learning the weights of a perceptron with general activation function in a teacher-student scenario. In particular, we discuss an all-or-nothing phenomenon for the generalization error with a sublinear set of training examples.
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Luneau, Clément, Nicolas Macris et Jean Barbier. « Information theoretic limits of learning a sparse rule ». Journal of Statistical Mechanics : Theory and Experiment 2022, no 4 (1 avril 2022) : 044001. http://dx.doi.org/10.1088/1742-5468/ac59ac.
Texte intégralRésumé :
Abstract We consider generalized linear models in regimes where the number of nonzero components of the signal and accessible data points are sublinear with respect to the size of the signal. We prove a variational formula for the asymptotic mutual information per sample when the system size grows to infinity. This result allows us to derive an expression for the minimum mean-square error (MMSE) of the Bayesian estimator when the signal entries have a discrete distribution with finite support. We find that, for such signals and suitable vanishing scalings of the sparsity and sampling rate, the MMSE is nonincreasing piecewise constant. In specific instances the MMSE even displays an all-or-nothing phase transition, that is, the MMSE sharply jumps from its maximum value to zero at a critical sampling rate. The all-or-nothing phenomenon has previously been shown to occur in high-dimensional linear regression. Our analysis goes beyond the linear case and applies to learning the weights of a perceptron with general activation function in a teacher-student scenario. In particular, we discuss an all-or-nothing phenomenon for the generalization error with a sublinear set of training examples.
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MacDonald, M., N. Hutchins et D. Chung. « Roughness effects in turbulent forced convection ». Journal of Fluid Mechanics 861 (19 décembre 2018) : 138–62. http://dx.doi.org/10.1017/jfm.2018.900.
Texte intégralRésumé :
We conducted direct numerical simulations of turbulent flow over three-dimensional sinusoidal roughness in a channel. A passive scalar is present in the flow with Prandtl number $Pr=0.7$, to study heat transfer by forced convection over this rough surface. The minimal-span channel is used to circumvent the high cost of simulating high-Reynolds-number flows, which enables a range of rough surfaces to be efficiently simulated. The near-wall temperature profile in the minimal-span channel agrees well with that of the conventional full-span channel, indicating that it can be readily used for heat-transfer studies at a much reduced cost compared to conventional direct numerical simulation. As the roughness Reynolds number, $k^{+}$, is increased, the Hama roughness function, $\unicode[STIX]{x0394}U^{+}$, increases in the transitionally rough regime before tending towards the fully rough asymptote of $\unicode[STIX]{x1D705}_{m}^{-1}\log (k^{+})+C$, where $C$ is a constant that depends on the particular roughness geometry and $\unicode[STIX]{x1D705}_{m}\approx 0.4$ is the von Kármán constant. In this fully rough regime, the skin-friction coefficient is constant with bulk Reynolds number, $Re_{b}$. Meanwhile, the temperature difference between smooth- and rough-wall flows, $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}$, appears to tend towards a constant value, $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}_{FR}^{+}$. This corresponds to the Stanton number (the temperature analogue of the skin-friction coefficient) monotonically decreasing with $Re_{b}$ in the fully rough regime. Using shifted logarithmic velocity and temperature profiles, the heat-transfer law as described by the Stanton number in the fully rough regime can be derived once both the equivalent sand-grain roughness $k_{s}/k$ and the temperature difference $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}_{FR}^{+}$ are known. In meteorology, this corresponds to the ratio of momentum and heat-transfer roughness lengths, $z_{0m}/z_{0h}$, being linearly proportional to the inner-normalised momentum roughness length, $z_{0m}^{+}$, where the constant of proportionality is related to $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}_{FR}^{+}$. While Reynolds analogy, or similarity between momentum and heat transfer, breaks down for the bulk skin-friction and heat-transfer coefficients, similar distribution patterns between the heat flux and viscous component of the wall shear stress are observed. Instantaneous visualisations of the temperature field show a thin thermal diffusive sublayer following the roughness geometry in the fully rough regime, resembling the viscous sublayer of a contorted smooth wall.
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Dimarco, Giacomo, et Giuseppe Toscani. « Social climbing and Amoroso distribution ». Mathematical Models and Methods in Applied Sciences 30, no 11 (octobre 2020) : 2229–62. http://dx.doi.org/10.1142/s0218202520500426.
Texte intégralRésumé :
We introduce a class of one-dimensional linear kinetic equations of Boltzmann and Fokker–Planck type, describing the dynamics of individuals of a multi-agent society questing for high status in the social hierarchy. At the Boltzmann level, the microscopic variation of the status of agents around a universal desired target, is built up introducing as main criterion for the change of status a suitable value function in the spirit of the prospect theory of Kahneman and Twersky. In the asymptotics of grazing interactions, the solution density of the Boltzmann-type kinetic equation is shown to converge towards the solution of a Fokker–Planck type equation with variable coefficients of diffusion and drift, characterized by the mathematical properties of the value function. The steady states of the statistical distribution of the social status predicted by the Fokker–Planck equations belong to the class of Amoroso distributions with Pareto tails, which correspond to the emergence of a social elite. The details of the microscopic kinetic interaction allow to clarify the meaning of the various parameters characterizing the resulting equilibrium. Numerical results then show that the steady state of the underlying kinetic equation is close to Amoroso distribution even in an intermediate regime in which interactions are not grazing.
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HALL, PHILIP, et DEMETRIOS T. PAPAGEORGIOU. « The onset of chaos in a class of Navier–Stokes solutions ». Journal of Fluid Mechanics 393 (25 août 1999) : 59–87. http://dx.doi.org/10.1017/s0022112099005364.
Texte intégralRésumé :
The flow between parallel walls driven by the time-periodic oscillation of one of the walls is investigated. The flow is characterized by a non-dimensional amplitude Δ and a Reynolds number R. At small values of the Reynolds number the flow is synchronous with the wall motion and is stable. If the amplitude of oscillation is held fixed and the Reynolds number is increased there is a symmetry-breaking bifurcation at a finite value of R. When R is further increased, additional bifurcations take place, but the structure which develops, essentially chaotic flow resulting from a Feigenbaum cascade or a quasi-periodic flow, depends on the amplitude of oscillation. The flow in the different regimes is investigated by a combination of asymptotic and numerical methods. In the small-amplitude high-Reynolds-number limit we show that the flow structure develops on two time scales with chaos occurring on the longer time scale. The chaos in that case is shown to be associated with the unsteady breakdown of a steady streaming flow. The chaotic flows which we describe are of particular interest because they correspond to Navier–Stokes solutions of stagnation-point form. These flows are relevant to a wide variety of flows of practical importance.
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Briard, A., T. Gomez, P. Sagaut et S. Memari. « Passive scalar decay laws in isotropic turbulence : Prandtl number effects ». Journal of Fluid Mechanics 784 (4 novembre 2015) : 274–303. http://dx.doi.org/10.1017/jfm.2015.575.
Texte intégralRésumé :
The passive scalar dynamics in a freely decaying turbulent flow is studied. The classical framework of homogeneous isotropic turbulence without forcing is considered. Both low and high Reynolds number regimes are investigated for very small and very large Prandtl numbers. The long time behaviours of integrated quantities such as the scalar variance or the scalar dissipation rate are analysed by considering that the decay follows power laws. This study addresses three major topics. First, the Comte-Bellot and Corrsin (CBC) dimensional analysis for the temporal decay exponents is extended to the case of a passive scalar when the permanence of large eddies is broken. Secondly, using numerical simulations based on an eddy-damped quasi-normal Markovian (EDQNM) model, the time evolution of integrated quantities is accurately determined for a wide range of Reynolds and Prandtl numbers. These simulations show that, whatever the values of the Reynolds and the Prandtl numbers are, the decay follows an algebraic law with an exponent very close to the value predicted by the CBC theory. Finally, the initial position of the scalar integral scale$L_{T}$has no influence on the asymptotic values of the decay exponents, and an analytical law predicting the relative positions of the kinetic and scalar spectra peaks is derived.
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Cerminara, M., T. Esposti Ongaro et L. C. Berselli. « ASHEE-1.0 : a compressible, equilibrium–Eulerian model for volcanic ash plumes ». Geoscientific Model Development 9, no 2 (18 février 2016) : 697–730. http://dx.doi.org/10.5194/gmd-9-697-2016.
Texte intégralRésumé :
Abstract. A new fluid-dynamic model is developed to numerically simulate the non-equilibrium dynamics of polydisperse gas–particle mixtures forming volcanic plumes. Starting from the three-dimensional N-phase Eulerian transport equations for a mixture of gases and solid dispersed particles, we adopt an asymptotic expansion strategy to derive a compressible version of the first-order non-equilibrium model, valid for low-concentration regimes (particle volume fraction less than 10−3) and particle Stokes number (St – i.e., the ratio between relaxation time and flow characteristic time) not exceeding about 0.2. The new model, which is called ASHEE (ASH Equilibrium Eulerian), is significantly faster than the N-phase Eulerian model while retaining the capability to describe gas–particle non-equilibrium effects. Direct Numerical Simulation accurately reproduces the dynamics of isotropic, compressible turbulence in subsonic regimes. For gas–particle mixtures, it describes the main features of density fluctuations and the preferential concentration and clustering of particles by turbulence, thus verifying the model reliability and suitability for the numerical simulation of high-Reynolds number and high-temperature regimes in the presence of a dispersed phase. On the other hand, Large-Eddy Numerical Simulations of forced plumes are able to reproduce the averaged and instantaneous flow properties. In particular, the self-similar Gaussian radial profile and the development of large-scale coherent structures are reproduced, including the rate of turbulent mixing and entrainment of atmospheric air. Application to the Large-Eddy Simulation of the injection of the eruptive mixture in a stratified atmosphere describes some of the important features of turbulent volcanic plumes, including air entrainment, buoyancy reversal and maximum plume height. For very fine particles (St → 0, when non-equilibrium effects are negligible) the model reduces to the so-called dusty-gas model. However, coarse particles partially decouple from the gas phase within eddies (thus modifying the turbulent structure) and preferentially concentrate at the eddy periphery, eventually being lost from the plume margins due to the concurrent effect of gravity. By these mechanisms, gas–particle non-equilibrium processes are able to influence the large-scale behavior of volcanic plumes.
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LEGENDRE, DOMINIQUE, et JACQUES MAGNAUDET. « The lift force on a spherical bubble in a viscous linear shear flow ». Journal of Fluid Mechanics 368 (10 août 1998) : 81–126. http://dx.doi.org/10.1017/s0022112098001621.
Texte intégralRésumé :
The three-dimensional flow around a spherical bubble moving steadily in a viscous linear shear flow is studied numerically by solving the full Navier–Stokes equations. The bubble surface is assumed to be clean so that the outer flow obeys a zero-shear-stress condition and does not induce any rotation of the bubble. The main goal of the present study is to provide a complete description of the lift force experienced by the bubble and of the mechanisms responsible for this force over a wide range of Reynolds number (0.1[les ]Re[les ]500, Re being based on the bubble diameter) and shear rate (0[les ]Sr[les ]1, Sr being the ratio between the velocity difference across the bubble and the relative velocity). For that purpose the structure of the flow field, the influence of the Reynolds number on the streamwise vorticity field and the distribution of the tangential velocities at the surface of the bubble are first studied in detail. It is shown that the latter distribution which plays a central role in the production of the lift force is dramatically dependent on viscous effects. The numerical results concerning the lift coefficient reveal very different behaviours at low and high Reynolds numbers. These two asymptotic regimes shed light on the respective roles played by the vorticity produced at the bubble surface and by that contained in the undisturbed flow. At low Reynolds number it is found that the lift coefficient depends strongly on both the Reynolds number and the shear rate. In contrast, for moderate to high Reynolds numbers these dependences are found to be very weak. The numerical values obtained for the lift coefficient agree very well with available asymptotic results in the low- and high-Reynolds-number limits. The range of validity of these asymptotic solutions is specified by varying the characteristic parameters of the problem and examining the corresponding evolution of the lift coefficient. The numerical results are also used for obtaining empirical correlations useful for practical calculations at finite Reynolds number. The transient behaviour of the lift force is then examined. It is found that, starting from the undisturbed flow, the value of the lift force at short time differs from its steady value, even when the Reynolds number is high, because the vorticity field needs a finite time to reach its steady distribution. This finding is confirmed by an analytical derivation of the initial value of the lift coefficient in an inviscid shear flow. Finally, a specific investigation of the evolution of the lift and drag coefficients with the shear rate at high Reynolds number is carried out. It is found that when the shear rate becomes large, i.e. Sr=O(1), a small but consistent decrease of the lift coefficient occurs while a very significant increase of the drag coefficient, essentially produced by the modifications of the pressure distribution, is observed. Some of the foregoing results are used to show that the well-known equality between the added mass coefficient and the lift coefficient holds only in the limit of weak shears and nearly steady flows.
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Kim, J. S., F. A. Williams et P. D. Ronney. « Diffusional-thermal instability of diffusion flames ». Journal of Fluid Mechanics 327 (25 novembre 1996) : 273–301. http://dx.doi.org/10.1017/s0022112096008543.
Texte intégralRésumé :
The diffusional–thermal instability, which gives rise to striped quenching patterns that have been observed for diffusion flames, is analysed by studying the model of a one-dimensional convective diffusion flame in the diffusion-flame regime of activation-energy asymptotics. Attention is focused principally on near-extinction conditions with Lewis numbers less than unity, in which the reactants with high diffusivity diffuse into the strong segments of the reaction sheet, so that the regions between the strong segments become deficient in reactant and subject to the local quenching that leads to the striped patterns. This analysis differs from other flame stability analyses in that the complete description of the dispersion relation is obtained from a composite expansion of the results of an analysis with the conventional convective-diffusive scaling and one with reaction-zone scaling. The results predict that striped patterns will occur, for flames sufficiently close to quasi-steady extinction, with a finite wavenumber that in convective–diffusive scaling is proportional to the cube root of the Zel'dovich number. The convective–diffusive response contributes to the stabilization of long-wavelength disturbances by through positive excess enthalpies by which the flame becomes more resistant to instability, while the reaction-zone response provides stabilization of short-wavelength disturbances by transverse diffusion, within the reactive inner layer, which relaxes the perturbed scalar fields towards their unperturbed states. As quasi-steady extinction is approached, marginal stability arises first at an intermediate range between these two scalings. Parametric results for this bifurcation point are obtained through numerical solutions of the associated generalized eigenvalue problems. Comparisons with measured pattern dimensions for different sets of reactants and diluents reveal excellent qualitative agreement.
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SELF, R. H., C. P. PLEASE et T. J. SLUCKIN. « Deformation of nematic liquid crystals in an electric field ». European Journal of Applied Mathematics 13, no 1 (février 2002) : 1–23. http://dx.doi.org/10.1017/s0956792501004740.
Texte intégralRésumé :
The behaviour of liquid crystal materials used in display devices is discussed. The underlying continuum theory developed by Frank, Ericksen and Leslie for describing this behaviour is reviewed. Particular attention is paid to the approximations and extensions relevant to existing device technology areas where mathematical analysis would aid device development. To illustrate some of the special behaviour of liquid crystals and in order to demonstrate the techniques employed, the specific case of a nematic liquid crystal held between two parallel electrical conductors is considered. It has long been known that there is a critical voltage below which the internal elastic strength of the liquid crystal exceeds the electric forces and hence the system remains undeformed from its base state. This bifurcation behaviour is called the Freedericksz transition. Conventional analytic analysis of this problem normally considers a magnetic, rather than electric, field or a near-transition voltage since in these cases the electromagnetic field structure decouples from the rest of the problem. Here we consider more practical situations where the electromagnetic field interacts with the liquid crystal deformation. Assuming strong anchoring at surfaces and a one dimensional deformation, three nondimensional parameters are identified. These relate to the applied voltage, the anisotropy of the electrical permittivity of the liquid crystal, and to the anisotropy of the elastic stiffness of the liquid crystal. The analysis uses asymptotic methods to determine the solution in a numerous of different regimes defined by physically relevant limiting cases of the parameters. In particular, results are presented showing the delicate balance between an anisotropic material trying to push the electric field away from regions of large deformation and the deformation trying to be maximum in regions of high electric field.
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CHUA, LEON O., VALERY I. SBITNEV et SOOK YOON. « A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE PART VI : FROM TIME-REVERSIBLE ATTRACTORS TO THE ARROW OF TIME ». International Journal of Bifurcation and Chaos 16, no 05 (mai 2006) : 1097–373. http://dx.doi.org/10.1142/s0218127406015544.
Texte intégralRésumé :
This paper proves, via an analytical approach, that 170 (out of 256) Boolean CA rules in a one-dimensional cellular automata (CA) are time-reversible in a generalized sense. The dynamics on each attractor of a time-reversible rule N is exactly mirrored, in both space and time, by its bilateral twin ruleN†. In particular, all 69 period-1 rules, 17 (out of 25) period-2 rules, and 84 (out of 112) Bernoulli rules are time-reversible. The remaining 86 CA rules are time-irreversible in the sense that N and N† mirror their dynamics only in space, but not in time. In this case, each attractor of N defines a unique arrow of time. A simple "time-reversal test" is given for testing whether an attractor of a CA rule is time-reversible or time-irreversible. For a time-reversible attractor of a CA rule N the past can be uniquely recovered from the future of N†, and vice versa. This remarkable property provides 170 concrete examples of CA time machines where time travel can be routinely achieved by merely hopping from one attractor to its bilateral twin attractor, and vice versa. Moreover, the time-reversal property of some local rules can be programmed to mimic the matter–antimatter "annihilation" or "pair-production" phenomenon from high-energy physics, as well as to mimic the "contraction" or "expansion" scenarios associated with the Big Bang from cosmology. Unlike the conventional laws of physics, which are based on a unique universe, most CA rules have multiple universes (i.e. attractors), each blessed with its own laws. Moreover, some CA rules are endowed with both time-reversible attractors and time-irreversible attractors. Using an analytical approach, the time-τ return map of each Bernoulli στ-shift attractor of all 112 Bernoulli rules are shown to obey an ultra-compact formula in closed form, namely,. [Formula: see text] or its inverse map. These maps completely characterize the time-asymptotic (steady state) behavior of the nonlinear dynamics on the attractors. In-depth analysis of all but 18 global equivalence classes of CA rules have been derived, along with their basins of attraction, which characterize their transient regimes. Above all, this paper provides a rigorous nonlinear dynamics foundation for a paradigm shift from an empirical-based approach à la Wolfram to an attractor-based analytical theory of cellular automata.