Synthetic velocity gradient tensors and the identification of statistically significant aspects of the structure of turbulence

A method is presented for deriving random velocity gradient tensors given a source tensor. These synthetic tensors are constrained to lie within mathematical bounds of the non-normality of the source tensor, but we do not impose direct constraints upon scalar quantities typically derived from the velocity gradient tensor and studied in fluid mechanics. Hence, it becomes possible to ask hypotheses of data at a point regarding the statistical significance of these scalar quantities. Having presented our method and the associated mathematical concepts, we apply it to homogeneous, isotropic turbulence to test the utility of the approach for a case where the behavior of the tensor is understood well. We show that, as well as the concentration of data along the Vieillefosse tail, actual turbulence is also preferentially located in the quadrant where there is both excess enstrophy ( Q > 0 ) and excess enstrophy production ( R < 0 ) . We also examine the topology implied by the strain eigenvalues and find that for the statistically significant results there is a particularly strong relative preference for the formation of disklike structures in the ( Q < 0 , R < 0 ) quadrant. With the method shown to be useful for a turbulence that is already understood well, it should be of even greater utility for studying complex flows seen in industry and the environment

Brief Communication: Dimensionality Reduction in Total Dynamic Mode Decomposition Using A Simple Geometric Method

Dynamic mode decomposition (DMD) and its variants have emerged as popular methods for the post-processing of fluid dynamics' simulations in order to visualize dominant coherent structures and to reduce the practical degrees of freedom to a restricted set of ``modes''. In this brief communication we provide a geometric method for choosing the number of modes for the Total DMD technique and test its efficacy using a synthetic example (to examine the effect of noise) and a cylinder wake case.Comment: 8 pages, 3 figure

A multifractal surrogate data generation algorithm that preserves pointwise Holder regularity structure, with initial applications to turbulence

An algorithm is described that can generate random variants of a time series or image while preserving the probability distribution of original values and the pointwise Holder regularity. Thus, it preserves the multifractal properties of the data. Our algorithm is similar in principle to well-known algorithms based on the preservation of the Fourier amplitude spectrum and original values of a time series. However, it is underpinned by a dual-tree complex wavelet transform rather than a Fourier transform. Our method, which we term the Iterated Amplitude Adjusted Wavelet Transform (IAAWT) method can be used to generate bootstrapped versions of multifractal data and, because it preserves the pointwise Holder regularity but not the local Holder regularity, it can be used to test hypotheses concerning the presence of oscillating singularities in a time series, an important feature of turbulence and econophysics data. Because the locations of the data values are randomized with respect to the multifractal structure, hypotheses about their mutual coupling can be tested, which is important for the velocity-intermittency structure of turbulence and self-regulating processes

Gradual multifractal reconstruction of time-series: Formulation of the method and an application to the coupling between stock market indices and their Hoelder exponents

A technique termed gradual multifractal reconstruction (GMR) is formulated. A continuum is defined from a signal that preserves the pointwise Hoelder exponent (multifractal) structure of a signal but randomises the locations of the original data values with respect to this (φ=0), to the original signal itself(φ=1). We demonstrate that this continuum may be populated with synthetic time series by undertaking selective randomisation of wavelet phases using a dual-tree complex wavelet transform. That is, the φ=0 end of the continuum is realised using the recently proposed iterated, amplitude adjusted wavelet transform algorithm (Keylock, 2017) that fully randomises the wavelet phases. This is extended to the GMR formulation by selective phase randomisation depending on whether or not the wavelet coefficient amplitudes exceeds a threshold criterion. An econophysics application of the technique is presented. The relation between the normalised log-returns and their Hoelder exponents for the daily returns of eight financial indices are compared. One particularly noticeable result is the change for the two American indices (NASDAQ 100 and S & P 500) from a non-significant to a strongly significant (as determined using GMR) cross-correlation between the returns and their Hoelder exponents from before the 2008 crash to afterwards. This is also reflected in the skewness of the phase difference distributions, which exhibit a geographical structure, with Asian markets not exhibiting significant skewness in contrast to those from elsewhere globally

The turbulence velocity gradient tensor formed additively by normal and non-normal tensors

We decompose the velocity gradient tensor for turbulence into normal and non-normal parts, and condition our analysis on the strain eigenvector alignments between these tensors. We identify states that always enhance, and always counteract the axisymmetric expansion state, and give a rationale for decomposing the production balance term into its constituents: complex behavior arises when the dominant strain alignments involve the non-normal tensor. Finally, we develop a topological analysis framework where mathematical bounds on two of the three variables leads to an analysis in two planes

Multifractal surrogate-data generation algorithm that preserves pointwise Hölder regularity structure, with initial applications to turbulence

An algorithm is described that can generate random variants of a time series while preserving the probability distribution of original values and the pointwise Hoelder regularity. Thus, it preserves the multifractal properties of the data. Our algorithm is similar in principle to well-known algorithms based on the preservation of the Fourier amplitude spectrum and original values of a time series. However, it is underpinned by a dual-tree complex wavelet transform rather than a Fourier transform. Our method, which we term the iterated amplitude adjusted wavelet transform can be used to generate bootstrapped versions of multifractal data, and because it preserves the pointwise Hoelder regularity but not the local Hoelder regularity, it can be used to test hypotheses concerning the presence of oscillating singularities in a time series, an important feature of turbulence and econophysics data. Because the locations of the data values are randomized with respect to the multifractal structure, hypotheses about their mutual coupling can be tested, which is important for the velocity-intermittency structure of turbulence and self-regulating processes

The Schur decomposition of the velocity gradient tensor for turbulent flows

The velocity gradient tensor for turbulent flow contains crucial information on the topology of turbulence, vortex stretching and the dissipation of energy. A Schur decomposition of the velocity gradient tensor (VGT) is introduced to supplement the standard decomposition into rotation and strain tensors. Thus, the normal parts of the tensor (represented by the eigenvalues) are separated explicitly from non-normality. Using a direct numerical simulation of homogeneous isotropic turbulence, it is shown that the norm of the non-normal part of the tensor is of a similar magnitude to the normal part. It is common to examine the second and third invariants of the characteristic equation of the tensor simultaneously (the diagram). With the Schur approach, the discriminant function separating real and complex eigenvalues of the VGT has an explicit form in terms of strain and enstrophy: where eigenvalues are all real, enstrophy arises from the non-normal term only. Re-deriving the evolution equations for enstrophy and total strain highlights the production of non-normality and interaction production (normal straining of non-normality). These cancel when considering the evolution of the VGT in terms of its eigenvalues but are important for the full dynamics. Their properties as a function of location in space are characterized. The Schur framework is then used to explain two properties of the VGT: the preference to form disc-like rather than rod-like flow structures, and the vorticity vector and strain alignments. In both cases, non-normality is critical for explaining behaviour in vortical regions

Hypothesis testing for nonlinear phenomena in the geosciences using synthetic, surrogate data

©2018. The Authors. Studying nonlinear and potentially chaotic phenomena in geophysics from measured signals is problematic when system noise interferes with the dynamic processes that one is trying to infer. In such circumstances, a framework for statistical hypothesis testing is necessary but the nonlinear nature of the phenomena studied makes the formulation of standard hypothesis tests, such as analysis of variance, problematic as they are based on underlying linear, Gaussian assumptions. One approach to this problem is the method of surrogate data, which is the technique explained in this paper. In particular, we focus on (i) hypothesis testing for nonlinearity by generating linearized surrogates as a null hypothesis, (ii) a variant of this that is perhaps more appropriate for image data where structural nonlinearities are common and should be retained in the surrogates, and (iii) gradual reconstruction where we systematically constrain the surrogates until there is no significant difference between data and surrogates and use this to understand geophysical processes. In addition to time series of sunspot activity, solutions to the Lorenz equations, and spatial maps of enstrophy in a turbulent channel flow, two examples are considered in detail. The first concerns gradual wavelet reconstruction testing of the significance of a specific vortical flow structure from turbulence time series acquired at a point. In the second, the degree of nonlinearity in the spatial profiles of river curvature is shown to be affected by the occurrence of meander cutoff processes but in a more complex fashion than previously envisaged

Obituary for Norbert Peters

Obituary for Norbert Peter