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Wavelets: Application to turbulence
... Turbulent signals may contain transient, energetic bursts, which correspond to large deviations and can be seen in the wide tails of the signal's probability density function (PDF). Second-order statistics such as the energy spectrum are relatively insensitive to these rare events because their duration is small compared to the sampling period and consequently they do not dominate the integral (Farge and Schneider, 2004). ...
... Since (Farge and Schneider, 2004) introduced wavelets into turbulence, they have become pervasive in turbulent signal analysis. There are examples of wavelet methods being applied to study the turbulent boundary layer (Chen et al., 2019;Nan, 2016;Hu and Du, 2020) in experimental tanks and flume settings, and turbulence-rotor interactions for wind turbines (Kelley et al., 2000(Kelley et al., , 2005. ...
... The basic analysing 'mother' wavelet should exhibit a fast decay for tending to infinity, must be oscillating and its Fourier transform̂( ) must decay fast as wave numbers tend to infinity. For analysis of turbulent signals, Farge and Schneider (2004) recommend to use the continuous wavelet transform with complex valued wavelet such as the Morlet. ...
Understanding turbulence is crucial to the design of tidal energy converters as it influences loads, fatigue life and power production. Acoustic Doppler Current Profilers (ADCPs) are normally employed in measurement campaigns to analyse turbulence at a particular location. Coherent turbulent structures are known to lead to the highest turbulence-induced stresses and are typically analysed by Fourier transform. However, this method can only yield parameters which are averaged over the signal length or the analysing window, and hence may not fully represent real flow turbulence, which is often non-stationary. This study applies wavelet time–frequency analysis to ADCP data from an active tidal test site to examine the turbulence spectrum and coherence characteristics of the flow, comparing these to Fourier methods. Results show that the average quantities from wavelet analysis agree favourably with Fourier analysis. Moreover, wavelet analysis yields instantaneous spectra and coherence, which show that at this site, high energy bursts occur at frequencies known to be significant for TEC loading and have turbulence intensities up to 80% higher than the average. We show that these intermittent, coherent bursts are obscured by the averages associated with Fourier analysis, highlighting the need for methods such as wavelets to understand interactions between fluid flow and TEC.
... The most promising aspect of local information theoretic measures was that a low and high-pass filters on these derived data qualitatively resembled the coherent and incoherent contributions as described by Farge and Schneider [2006] (see Figure A.2). Farge and Schneider [2006] perform a wavelet decomposition on the vorticity field and then split the field into a coherent and incoherent contributions by applying a nonlinear thresholding to the wavelet coefficients. The hope was that these partitioned information theoretic measures might provide a means to independently validate or invalidate this concept. ...
... The most promising aspect of local information theoretic measures was that a low and high-pass filters on these derived data qualitatively resembled the coherent and incoherent contributions as described by Farge and Schneider [2006] (see Figure A.2). Farge and Schneider [2006] perform a wavelet decomposition on the vorticity field and then split the field into a coherent and incoherent contributions by applying a nonlinear thresholding to the wavelet coefficients. The hope was that these partitioned information theoretic measures might provide a means to independently validate or invalidate this concept. ...
... Based on several working sessions, domain scientists were unable to isolate features of interest using these partition-based measures in multidimensional transfer func- Figure A.2: High-pass (left) and low-pass (right) filters on information entropy of the Taylor-Green enstrophy are qualitatively similar to the coherent and incoherent contributions as described by Farge and Schneider [2006]. ...
One of the barriers to visualization-enabled scientific discovery is the difficulty in clearly and quantitatively articulating the meaning of a visualization, particularly in the exploration of relationships between multiple variables in large-scale data sets. This issue becomes more complicated in the visualization of three-dimensional turbu- lence, since geometry, topology, and statistics play complicated, intertwined roles in the definitions of the features of interest, making them difficult or impossible to precisely describe.
This dissertation develops and evaluates a novel interactive multivariate volume visualization framework that allows features to be progressively isolated and defined using a combination of global and feature-local properties. I argue that a progressive and interactive multivariate feature-local approach is advantageous when investigating ill-defined features because it provides a physically meaningful, quantitatively rich en- vironment within which to examine the sensitivity of the structure properties to the identification parameters. The efficacy of this approach is demonstrated in the analysis of vortical structures in Taylor-Green turbulence. Through this analysis, two distinct structure populations have been discovered in these data: structures with minimal and maximal local absolute helicity distributions. These populations cannot be distinguished via global distributions; however, they were readily identified by this approach, since their feature-local statistics are distinctive.
... The following section reviews the experimental methods and the relevant parameter ranges used in the study, and is followed by a description of the wavelet transform technique 16,17 which can investigate the evolutionary interactions between the fluid turbulence and the particles. Both are then applied to measure V s , u p Ј , w p Ј , and the temporal in- [6][7][8] in attempt to address the aforementioned discrepancy. ...
... Farge 16 and her co-workers 17 have used the method of wavelet transforms, as a new mathematical tool, to study statistically elementary structures of turbulence. In this study, we employ the same wavelet transform concept 16 to investigate the temporal behaviors of heavy particles and fluid turbulence for the first time. ...
... Thus, the wavelet transform at specific positions on the time signal tϭb and at a specific wavelet scale a which represents the period can be defined. The admissibility condition as used by Farge 16 is applied on (t) ...
A pair of vertically oscillating grids were applied to experimentally produce a region of nearly isotropic stationary turbulence in homogeneous fluids contained in a water tank. With the appropriate choice of grid mesh size (M), oscillating frequency (f) and stroke (S), and the distance between the grids (H), two distinct flow regions are generated: a highly turbulent region near each grid in the form of turbulent wakes and an approximately isotropic stationary turbulence located in the core region between the grids, as verified by extensive laser-Doppler velocimetry. The former is similar to that generated by a single vibrating grid, which was commonly used in mixed layer experiments. The latter flow region has essentially zero mean velocities, nearly equal magnitude of root-mean-square turbulent intensities in all three directions, and nearly −5/3 energy decay slopes, indicating that the turbulence has some properties of isotropic turbulence. This region of interest is chosen to be a domain where the variations among values of horizontal and vertical root-mean-square velocities are less than 15% and its height is found to be linearly proportional to H. In this region, the effective (overall) turbulent intensity may be represented by an empirical relation of the form q = CfS1.5M0.5H−n, where the experimental constant C ≈ 0.89 and the power constant n = 1.5 at least for 4 ≤ H/M ≤ 6. The present results confirm the expectation of Villermaux et al. [13] that the energy released in the system per unit time by each grid is additive. Other parameters of interest, such as the autocorrelation, the flow integral length scale, and the experimental uncertainties also are reported. Finally, the current flow apparatus can be conveniently adopted for many experimental studies such as particle or scalar dispersion in isotropic turbulence and simulation of premixed turbulent combustion [12].
... Wavelets can also aid in partitioning the turbulent time series into coherent and non-coherent scales, facilitating in accounting their contribution to the total flux. Also, intermittency and non-Gaussianity of turbulence are often attributed to the presence of coherent structures [14,15]. Wavelets can be fruitful in separating the Gaussian and non-Gaussian structures and extraction of coherent events and waves [17,19,26]. ...
... The velocity probability distribution for the non-coherent part shows a narrower distribution, while for the coherent part, the distribution is nearly the same as the original velocity. The studies by Farge and Schneider [15,18], Farge et al. [16] also found that coherent and original velocities have a similar distribution, while the incoherent scales show a narrower velocity distribution. The flatness factor or kurtosis for the PDFs shown in Fig. 5 is 2.0893, 4.1390, 2.3028 for original flow velocity u, small scale velocity u S and large scale velocity u L respectively. ...
Wavelets are employed to study atmospheric turbulent data of three wind components, temperature and passive scalars CO2 and H2O. The multiresolution analysis (MRA) based on maximal overlap discrete wavelet transform (MODWT) is used to separate turbulent fluctuations from the mean flow. These turbulent fluctuations are further partitioned into small scales x′s and large scales x′L, and the fluxes are calculated by averaging over the given time interval. The large scales are responsible for much of the flux transport, while the small scales are fine scales consisting of non-transporting, nearly isotropic motions. The velocity spectrum for both small (non-coherent) and large scale (coherent) follow -5/3 scaling, and the transfer efficiency Rwa similarity laws are better satisfied for the large scales. The velocity probability distribution of partitioned signals shows a narrower distribution for small scales compared to large ones. However, the flatness factor indicates deviation from Gaussianity. The joint probability distribution for large scales is skewed, suggesting the dominance of ejections and sweeps. Despite their wave-like nature, the large scales are not linear waves as indicated by the phase spectrum. The large scales are subjected to the continuous wavelet transform (CWT) to detect and isolate the strong localized events. The Mexican Hat (MHAT) wavelet transform and zero-crossing method is used to estimate the duration, separation, and frequency of occurrence of the detected events.
... In particular, in the context of assessment of a temporal signal, we use the well-known location-scaledependent intermittency indicator introduced in [2,3], named as the 'local intermittecy measure', as well as its scale-dependent temporal average, called the 'flatness factor' by Meneveau [4]. The latter is a special case of sparsity wavelet-moment-based measure of intermittency; see, for instance, [5][6][7]. Among other contributions of interest in this context, see, for example, [6,[8][9][10][11][12][13] and references therein. ...
... The latter is a special case of sparsity wavelet-moment-based measure of intermittency; see, for instance, [5][6][7]. Among other contributions of interest in this context, see, for example, [6,[8][9][10][11][12][13] and references therein. ...
Intermittency represents a certain form of heterogeneous behavior that has interest in diverse fields of application, particularly regarding the characterization of system dynamics and for risk assessment. Given its intrinsic location-scale-dependent nature, wavelets constitute a useful functional tool for technical analysis of intermittency. Deformation of the support may induce complex structural changes in a signal. In this paper, we study the effect of deformation on intermittency. Specifically, we analyze the interscale transfer of energy and its implications on different wavelet-based intermittency indicators, depending on whether the signal corresponds to a ‘level’- or a ‘flow’-type physical magnitude. Further, we evaluate the effect of deformation on the interscale distribution of energy in terms of generalized entropy and complexity measures. For illustration, various contrasting scenarios are considered based on simulation, as well as two segments corresponding to different regimes in a real seismic series before and after a significant earthquake.
... Note that these results could be affected by the difficult estimation of the field high-order moments (and thus of the kurtosis) through the scaledependent field increments, in particular when the spectral index approaches β ≃ 3 (Abry et al., 1993;Farge & Schneider, 2006), or when the scaling ranges are not well extended, and suggests the need for more extended investigation. ...
... Finally, it should be noted that various uncertainties arise when estimating small-scale intermittency, related to the quality of the measurements, the length of the time series, and the spectral properties of the fluctuations (Abry et al., 1993;Farge & Schneider, 2006;Cho & Lazarian, 2009;Kiyani et al., 2013). It would, therefore, be of interest, in the future, to obtain higher quality data (e.g. ...
The solar wind is a highly turbulent medium, with a high level of field fluctuations throughout a broad range of scales. These include an inertial range where a turbulent cascade is assumed to be active. The solar wind cascade shows intermittency, which however may depend on the wind conditions. Recent observations have shown that ion-scale magnetic turbulence is almost self-similar, rather than intermittent. A similar result was observed for the high resolution measurements of proton density provided by the spacecraft Spektr-R. Intermittency may be interpreted as the result of the multifractal properties of the turbulent cascade. In this perspective, this paper is devoted to the description of the multifractal properties of the high resolution density measurements. In particular, we have used the standard coarse-graining technique to evaluate the generalized dimensions , and from these the multifractal spectrum , in two ranges of scale. A fit with the p-model for intermittency provided a quantitative measure of multifractality. Such indicator was then compared with alternative measures: the width of the multifractal spectrum, the peak of the kurtosis, and its scaling exponent. The results indicate that the small-scale fluctuations are multifractal, and suggest that different measures of intermittency are required to fully understand the small scale cascade.
... A wavelet function ψ(η) is a signal that has zero mean and can be localized in both time and frequency space (Farge (1992); Misiti et al. (1996)). Torrence and Compo (1998) stated that the types of chosen wavelet functions rely on the objectives of the analysis and the nature of the time series, orthogonality of the basis and features of width and shape as follows. ...
... The wavelet function with an orthogonal basis can be used with a discrete wavelet transform whilst a nonorthogonal basis is appropriate for either the discrete or continuous wavelet transform (Farge, 1992). In orthogonal wavelet transforms, the number of convolutions at each scale is proportional to the width of the wavelet basis at that scale, leading to discrete blocks of wavelet power which is useful for signal processing (Torrence and Compo, 1998). ...
... Although the influence of the latter on radon emission and time evolution is rather well known, in this context we will focus on the relationship between the scales of occurrence of the various meteorological events and those of radon dispersion in the ASL. Towards this purpose, a time series analysis is conducted by means of wavelets (Farge, 2000;Farge and Schneider, 2002). Wavelet analysis is a technique for the determination of localized (in time or in space) power spectra which adapts very well to the analysis of signals with a strong time or space variability. ...
... where η is a non-dimensional time parameter and ω 0 is the non-dimensional frequency (assumed equal to 6 to guarantee admissibility of the function; Farge, 2000). Given the time series of radon concentrationc i with i=0, . . . ...
A one-year time series of 222Rn measured in a rural area in the North of Italy in 1997 is analyzed. The scope of the investigation is to better understand the behavior of this common atmospheric tracer in relation to the meteorological conditions at the release site. Wavelet analysis is used as one of the investigation tools of the time series. The measurements and scalograms of 222Rn are compared to those of wind-speed, pressure, relative humidity, temperature and NOx. The use of wavelet analysis allows the identification of the various scales controlling the influence of the meteorological variables on 222Rn dispersion in the surface layer that are not visible through classical Fourier analysis or direct time series inspection. The analysis of the time series has identified specific periods during which the usual diurnal variation of radon is superimposed to a linear growth thus indicating the build up of concentration at the measurement level. From these specific cases an estimate of the surface flux of 222Rn is made. By means of a simple model these special cases are reproduced.
... Doing so, a large fraction of the information contained in the snapshots is lost, namely the flame dynamics and particularly the flame interaction with large scale coherent structures. To retain this information, one needs a numerical microscope, in other words to perform a distillation [8] enabling to sort out the information and only retain the fraction of interest in the form of a statistical quantity. Few techniques are available to do so including conditional averaging, wavelet analysis [8] and Proper Orthogonal Decomposition (POD) [9,10]. ...
... To retain this information, one needs a numerical microscope, in other words to perform a distillation [8] enabling to sort out the information and only retain the fraction of interest in the form of a statistical quantity. Few techniques are available to do so including conditional averaging, wavelet analysis [8] and Proper Orthogonal Decomposition (POD) [9,10]. Presently, we focus on the POD technique and its application to flames. ...
The aim of this work is to present the use of proper orthogonal decomposition (POD) and extended proper orthogonal decomposition
(EPOD) for revealing flame dynamics as a set of statistical quantities referred as modes. The flame fluctuations are used
to derive an empirical functions base representing the most important features of the flame. The capabilities of the technique
are exemplified in the case of an unsteady laminar flame. The flame is naturally unsteady and can be excited to amplify the
fluctuations. The data base consists of synchronous Particle Image Velocimetry (PIV) and OH-Planar Laser Induced Fluorescence
(OH-PLIF) measurements. It was found that the POD based upon the PIV vectors only reveals flow features when the excitation
is added. On contrary, the OH-PLIF based POD performs well in any case and constitutes a suitable base for the EPOD analysis.
... For instance, consider the nonlinear wavelet decomposition (see e.g. Farge & Schneider, 2006), or more recent data-driven methods, such as convolutional neural networks (CNNs) POD (see Guastoni et al., 2021;Güemes, Discetti, & Ianiro, 2019;Nakamura, Fukami, & Fukagata, 2022). A CNN POD method considers the same spatial basis as traditional POD; however, the time coefficients are estimated using a CNN, making it able to account for nonlinear features in the flow. ...
Proper orthogonal decomposition (POD) is applied to three-dimensional (3-D) velocity fields collected from large-eddy simulations (LES) of a baffled stirred tank. In the LES, the tank operates with a Rushton-type impeller under turbulent conditions (at least in the near-impeller region) and the working fluid exhibits either Newtonian or shear-thinning rheology. The most energetic POD modes are analysed, and a POD reconstruction based on the higher modes is proposed to approximate the fluctuating component of the velocity field. Subsequently, the POD reconstruction is used to identify vortical structures and characterise them in terms of their shape. The structures are identified by considering a frame-invariant formulation of a popular, Eulerian, local-region-type method: the Q -criterion. Statistics of shape-related parameters are then investigated to address the morphology of the structures. It is found that: (i) regardless of the working fluid rheology, it seems feasible to decompose the 3-D field into its mean, most energetic periodic and fluctuating components using POD, allowing, for instance, reduced-order modelling of the energetic periodic motions for mixing enhancement purposes, and (ii) vortical structures related to turbulence are mostly tubular. Finding (ii) implies that, as starting point, phenomenological models for the interaction between fluid particles (drops and bubbles) and vortices should consider the latter as cylindrical structures rather than of spherical shape, as classically assumed in these models.
... where a ∈ R + the scale parameter and b ∈ R is the position parameter, all wavelets being normalized in L 2 -norm (Farge and Schneider, 2005). ...
The Parker Solar Probe (PSP) satellite was launched by NASA in 2018 to study the Sun's environment from a closer distance than any spacecraft has ever reached before, revealing new insights about our star. Based on past few encounters data, numerous crucial findings have been already discovered; the data collected by the probe during each encounter is serving a unique opportunity to enhance our current understanding of solar behaviour. In the present work, we studied fluctuations on interplanetary magnetic field (IMF) (B) and its components, Bx, By, and Bz, from 0.50 AU (when PSP moved towards the Sun) to 0.17 AU (closest distance covered by PSP from the Sun) to 0.50 AU (when PSP moved back from the Sun) using continuous and discrete wavelet analysis. Continuous wavelet analysis presented localization of abrupt changes in both time and frequency domains. It revealed high power short-term fluctuations between 0.30 AU (when PSP moved towards the Sun) to 0.30 AU (when PSP moved back from the Sun) with key periodicity in the range of 4 to 64 hours. However, Bx showed high-power long-term fluctuation between 0.34 AU (when PSP moved towards the Sun) to 0.40 AU (when PSP moved back from the Sun) on a timescale of 128-256 hours, which was not reported before. Further, we decomposed the data into high and low-frequency signals using discrete wavelet transform (DWT), and the result revealed significant fluctuations on the IMF when the PSP was moving around the closest distance (0.17 AU). CWT and DWT emerged as sophisticated statistical methods capable of detecting variations in solar wind data in order to gain a better understanding of a specific solar event period and uncover scientific understanding about solar behavior near the Sun.
... They are also localised in space, which means that they can be used to work out the location of the signal with the greatest contribution to the power spectrum of the frame; oscillations can also be identified [4]. This approach is therefore used in a variety of sectors, from turbulence [12] to compression. ...
With the fast-paced developments in Artificial Intelligence, autonomous vehicles will
play an increasingly important part in people’s lives [16]. While very technologically
advanced, they make decisions using Artificial Neural Networks, which are notorious
for their opacity. This dissertation leverages the state-of-the-art in knowledge-based
modelling and proposes an alternative architecture towards an explainable robotic
system. More specifically, the Perception-Association-Hierarchy meta-model is programmed into a code for autonomous boats and tailored for water detection. Two
vision operators, the Fourier Transform and a Histogram of pixel intensities, are
used to transform the inputs from perception into knowledge and used to determine some of the system’s parameters, also known as the “magic numbers” [9].
Every frame is discretised into tiles, with a Decision function deciding whether to
associate each one to water or not. The results indicate optimal detection performance in wavy, well-illuminated water, especially at the immediate vicinity of the
ship, demonstrating a good capacity at identifying motion. Great interest is given
to the computational performance, which scales very well even at high discretisation levels. Despite the more unsatisfactory performance for quiescent, dark water,
the Perception-Association-Hierarchy shows excellent promise as the foundation for
creating an explainable robotic system capable of reasoning.
... The properties of signals were not taken into consideration in these methods; therefore, the universality is not good. The wavelet transforms (WT) are widely applied in many engineering fields for solving various real-life problems [4]- [6]. The Fourier transform of a signal contains the frequency content of the signal over the analysis window and, as such, lacks any time domain localization information. ...
Dynamical properties of mechanical systems can be obtained with the vibration signals from the systems. However, for the influence of noises, it is difficult to accurately acquire the features. Therefore, de-noising operation is significant for vibration signal in the practical engineering. In order to resolve this problem, discrete wavelet transform (DWT) with hard threshold is used to remove noises from vibration signal. We provide the principle of DWT at first. And then, a sample signal is constructed with white noise is processed by the technique. As the result shown, the random noise can be effectively eliminated. This reflects that the de-noising method based on DWT the hard threshold is effective.
... Wavelet analysis, as a time-frequency analysis method, overcomes the shortcomings of Fourier analysis, and it goes without saying that, except for periodic excellent signals and stationary signals. So wavelet is widely used in signal processing in engineering [1][2][3][4]. There are almost no other processing tools in signal processing is comparable to wavelet analysis [5][6]. ...
Vibration signals are often interfered by salt and pepper noise, impulse noise, Gaussian noise, and other high-frequency noise, affecting the effect of vibration control research. Therefore, de-noising operation is significant for vibration signal in the practical engineering. In order to resolve this problem, wavelet package is used to remove noises from vibration signal. We provide the principle of de-noising method based on wavelet package at first. And then, the threshold selection of the method is analyzed by a vibration signal with noises. As the result shown, the noise can be effectively eliminated. This reflects that the de-noising method based on wavelet package with the hard threshold is effective.
... A Ricker wavelet study is proceeded at these sweeping frequencies. The wavelet is able to analyse the hot-wire signals in time and scale [35] [36]. This process is similar to a sliding Fast Fourier Transform on a time window. ...
Sweeping jets are an emerging type of actuators that have gained interest due to their potential use in flow control applications. The working principle of these devices is based on the bi-stable attachment of a jet to adjacent walls. They are able to produce unsteady blowing within a wide range of operating frequencies. Nevertheless, the state of art shows a lack of space-time characterization of these actuators for high sweeping frequencies. This paper presents a conditional approach that reconstructs the spatial dynamic response of sweeping jets for sweeping frequencies above 500 Hz. The time-dependent velocity is measured with two single-hot-wire sensors: a reference one placed at the edge of the exit nozzle, and a flying one. The method is then tested to characterize the flow at the exit nozzle of an in-house sweeping jet actuator with 1 mm space resolution, and 50 μs time resolution. These measurements are performed with a sweeping frequency of 639 Hz. Overall this paper demonstrates that the conditional approach is very useful for understanding the physics of flow control actuators.
... Farge and Schneider [FS06] propose a similar scenario, but with very different implications. The coherent energy is injected at large scales and transferred towards small scales by non-linear interactions between vortex tubes. ...
The turbulent jet is a dominant source of acoustics in the automotive and aviation industries. When a gas flows into an ambient medium it creates an unstable boundary layer and thus pressure fluctuations that propagate into the far-field as acoustic waves. Since the turbulent jet is self-similar, one generally expects this behaviour from the radiated acoustics as well. However, there are a number of phenomena regarding hot jets that require further explanations, since so far no widely accepted physical mechanisms have been presented. The aim of this work is to quantify the influence of the temperature on jet noise and to investigate its causes.
To do so, two large eddy simulations of a subsonic, isothermal and hot jet (at a temperature ratio of 2.28) at constant Mach and Reynolds number are performed, the acoustic source terms are calculated using Lilley's acoustic analogy, and the resulting far-field is calculated using Ffowcs Williams-Hawkings integration. It turns out that the source terms scale directly proportional to the change in their energy potential, so the jet retains its efficiency in transforming the energy into an acoustic potential. The temperature-linked source term increases in a frequency-independent manner with the Mach number to the power of eleven and the resulting sound power increases with the Mach number to the power of eight, leading to an increase of 15 decibel in the hot jet with respect to the isothermal one. At this point, it is not clear whether the increase of the temperature related acoustic potential is the cause of the elevated sound power. Due to the spatial expansion of the source region the sound field forms modes that act as a low-pass filter and generate the characteristic far-field level decrease over frequency despite a frequency-invariant source strength.
The combination of far-field phenomena and acoustic source potential suggests the existence of broadband source mechanisms that behave through propagation mechanisms in the far-field like separate sources with different scaling behaviour and directionality. The comparison of the sound power eliminates this problem and allows the quantification of the true, source-describing quantities as well as a frequency- and direction-invariant scaling. In the future, with a comprehensive database, it should be possible to find simple scaling dependencies for jet noise and thus to deduce the radiated acoustics from the flow parameters and vice versa. This will allow fast and cost-efficient acoustic predictions and creates space for innovations such as acoustic-based detection of engine malfunctions.
... We refer the reader to useful and interesting articles [1,8,20,4,7,5] and both theoretical and practical book [2]. ...
One reminds for all function f : n → the so-called (min, +)-wavelets which are lower and upper hulls build from (min, +) analysis [12, 13]. One shows that this analysis can be applied numerically to the Weierstrass and Weierstrass-Mandelbrot functions, and that one recovers their theoretical Hölder exponents and fractal dimensions.
... The advantage afforded by wavelets is the ability to perform local analysis. Because wavelets are localized in time and scale, wavelet coefficients are able to localize characteristic changes or differences in analyzed signals [3,4]. By shifting parameters of wavelets, they can be applied as a focus directed to interesting signal area described by time and scale related to frequencies. ...
The human gait analysis by using wavelets transform of signal obtained from six inertial ProMove mini sensors is proposed in this work. The angular velocity data measured by the gyro sensors were used to estimate the translational acceleration in the gait analysis. As a result, the flexion - extension of joint angles of the knees were calculated for healthy people and with impaired locomotion system. After measurements we propose to use one of wavelet transform (wavelet type) in order to analyze the signals, indicate a characteristic feature and compare them.
... Meanwhile, vortex [6][7] is an interesting path to approach turbulence [8], since wingtip vortices could become wingtip turbulence and cause damages in practice. With the wavelet applications, the vortex and turbulence may be linked [9][10]. ...
... Meanwhile, vortex [6][7] is an interesting path to approach turbulence [8], since wingtip vortices could become wingtip turbulence and cause damages in practice. With the wavelet applications, the vortex and turbulence may be linked [9][10]. ...
... We refer the reader to useful and interesting articles [1,8,20,4,7,5] and both theoretical and practical book [2]. ...
pre>For all function f : Rn to R one introduces (min; +)-wavelets which are lower and upper hulls build from (min; +) analysis. One shows at theoretical level and on numerical applications for the Weierstrass functions, that (min, +)-wavelets decomposition opens a non-linear branch to the multi-resolution analysis of a signal, in particular for the Hölder exponents calculation and Empirical Mode Decomposition (EMD).
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... Meanwhile, vortex67 is an interesting path to approach turbulence [8], since wingtip vortices could become wingtip turbulence and cause damages in practice. With the wavelet applications, the vortex and turbulence may be linked910. Since physics focus on the fundamental mechanism of the nature and the universe, physical theories have to explain all the realities of the world [11]. ...
A physical linked-measure is mathematically consisted of a complex scalar, a complex vector and a bivector and is geometrically equivalent to a vortex. When the complex scalar means mass, the complex vector implies directed momentum and the bivector rotated angular momentum, with using the least action principle to the linked-measure, yielding energy-mass-momentum-angular momentum joint conservation. Hamilton equations and Lagrange equation keep as the core of physics, leading fluid dynamics of energy way. As any periodic function can be expressed as a Fourier series, energy spectrum is suggested to be an analytical method. Combining the vortex dynamics with relativity and thermodynamics, Bekenstein–Hawking entropy and " no-hair theorem " of black hole are naturally derived. Applying to wingtip vortices, with adding wavelets, simplified ideal turbulence is described.
... To further investigate this transition, we apply the continuous wavelet transform (CWT) to the time evolution of the drag force. The CWT transforms the signal into the time-frequency domain [42]. We use the complex-valued Morlet wavelet with the wavenumber k w denoting the barycenter of the wavelet support in Fourier space. ...
... where η is a non-dimensional time parameter and w 0 is the non-dimensional frequency (assumed equal to 6 to guarantee admissibility of the function) (Farge 2000). Given the time series of aerosol particle number concentration x i with i = 0, . . . ...
Continuous measurements of the aerosol particle number concentration (PNC) in the size range from 4.5 nm to 2 μm were performed at the Preila marine background site during 2008–2009. The concentration maxima in summer was twice the average (2650 ± 850 cm− 3). A trajectory-based approach was applied for source identification. Potential Source Contribution Function (PSCF) analysis was performed to estimate the possible contribution of long-range and local PNC transport to PNC concentrations recorded at the marine background site. The PSCF results showed that the marine boundary layer was not seriously affected by long-range transport, but that local transport of air pollution was recognized as an important factor. North Atlantic and Sea-Marine type clusters respectively represented 32.1% and 17.9% of the total PNC spectra and were characterized by the lowest PNCs (1 080 ± 1 340 and 1 210 ± 1 040 cm− 3 respectively) among all clusters.
... time series), expressed as a function of the time variable t, into frequency components using wavelets (Boggess and Narcowich, 2001). Wavelets are small waves defined by a function 0 (), where is a nondimensional time parameter that has zero mean and is localised in both time and frequency space (Farge 1992;Percival and Walden 2000). The wavelet is shifted forward and backward in time, along the localized time index , to filter or compress signals. ...
The price volatility of recycled materials exposes many different organisations to financial and regulatory risk. These risks can be partially mitigated by improved understanding of price volatility using econometric models, although these have tended to be sophisticated autoregressive models, beyond the analytical capability or cost structure of the average market participant. In this context, this study explores the use of a simple linear regression model to understand the behaviour of prices for recycled plastic, recovered paper and glass, based on the price of their primary inputs, with a specific focus on the UK market. The results of the simple regression model are compared with a wavelet analysis to determine what information is lost in the trade off with simplicity. Both models found significant relationships between the price of crude oil/ethylene/naphtha and recycled plastics and it appears that there is no loss in model performance by using oil as the explanatory variable, rather than direct inputs (ethylene/naphtha). However, few correlations were found for recovered paper and glass. Initial findings from the wavelet analysis suggest that since 2004 the oil price has become more closely linked to the price of recycled plastics. It must be noted that this study is exploratory and particular dynamics of UK markets for recycled materials may affect price development in a way that would not be repeated on international markets.
... . It is evident that every mode has a decaying behaviour depending on γ 1i and γ 2i . By using a Morlet wavelet, it is possible both to calculate the wavelet transform of η (e x, t) and to obtain the energy spectra E (t, ω ji ) corresponding to the eigenfrequencies ω ji of the i th mode, where j = 1, 2 indicates the high and the low eigenfrequency value (Farge, 1992). At last, by assuming negligible the nonlinear interactions among the modes during free oscillations, it is reasonable to interpolate log E (t, ω ji ) with the curve −2γ ji t and then to deduce the experimental logarithmic decrement coefficient γ ji . ...
In this paper a theoretical and experimental investigation on the slosh- ing of two liquid layers with a free-surface is performed. The mathemati- cal model is obtained by applying the Lagrangian variational formulation to the potential formulation of the fluid motion and a dynamical sys- tem which describes the dynamics of motion of the separation surface is derived. In order to account the damping of the motion, generalised dissipative forces are considered. To this aim, the logarithmic decrement coefficients are estimated by means of a wavelet analysis performed on the experimental free oscillations of the fluid system. The numerical integra- tion of the mathematical model gives results which are in good agreement with the experimental results. 1I ntroduction The aim of this paper consists in investigating the motion of the separation surface between two layers of immiscible liquids inside of a prismatic container which can rotate around a horizontal axis. Colored water and transparent vase- line oil are used. The upper liquid is limited by a free surface. The theoretical problem is analysed assuming a potential formulation for the absolute fluid mo- tion, moreover suitable expansions for the velocity potentials, the separation surface and the free surface are adopted so that the variational formulation, described in La Rocca et al.(2000, 2002), can be applied to the present case. In order to take into account for the damping of the motion, a dissipative function expressed in terms of logarithmic decrement coefficients is considered. To this aim, a wavelet analysis is applied to the decaying oscillations of the fluid system which permits to experimentally detect the decrement coefficients linked to each eigenfrequency of the considered mode. For different values of the forcing frequency and amplitude an experimental and numerical investigation is performed. The agreement between numerical
... In a broader view, better knowledge is also necessary to improve turbulent flow modelling (Jeong & Hussain 1995). A decomposition of flow fields into active (coherent structures) and inactive components has been advocated as an alternative to the more classical decomposition into average (large-scale) and fluctuating (subgridscale or SGS) components (Katul et al. 1998;Farge & Schneider 2006). ...
Large coherent structures over vegetation canopies are responsible for a substantial part of the turbulent transfer of momentum, heat and mass between the canopy and the atmosphere. As forested landscapes are often fragmented, edge regions may be of importance in turbulent transfer. The development of coherent structures from the leading edge of a forest is investigated here for the first time. For this purpose, the turbulent flow over a clearing–forest pattern is simulated using the Advanced Regional Prediction System (ARPS). In previous studies the code has been modified so as to simulate turbulent flows at very fine scale (0.1h, where h is the mean canopy height) within and above heterogeneous vegetation canopies, using a large-eddy simulation (LES) approach. Validations have also been performed over homogeneous forest canopies and over a simple forest–clearing–forest pattern, against field and wind-tunnel measurements. Here, a schematic picture of the development of coherent eddies downstream from the leading edge of a forest is extracted from the mean vorticity components, the Q-criterion field, the cross-correlation of the wind velocity components and the length and separation length scales of coherent structures, determined by using a wavelet transform. This schematic picture shows strong similarities with the development of coherent structures observed in a mixing layer, with four different regions: (i) close to the edge, Kelvin–Helmholtz instabilities develop when a strong wind gust hits the canopy; (ii) these instabilities roll over to form transverse vortices from around 3h downstream from the edge, characterized by a length scale close to the depth of the internal boundary layer that develops from the canopy edge; (iii) secondary instabilities destabilize these rollers and increase the vertical and streamwise vorticity components from around 6h, and two counter-rotating streamwise vortices appear; (iv) at about 9h the initial rollers have become complex three-dimensional coherent structures, with spatially constant mean length and separation length scales. These four stages of development occur closer to the edge with increasing canopy density. While this average picture of the development of coherent structures is similar to that observed in a mixing layer, the analysis of instantaneous fields shows that coherent structures behind the leading edge appear as resulting from the ‘branching’ of tubes localized in regions of low pressure, where their cores are characterized by high values of enstrophy and Q-criterion.
... In this paper, the continuous wavelet transform (CWT) following Torrence and Compo (1998) is used in this text. We assume that x n is a time series with equal time spacing δt and n = 0, …, N − 1. ψ 0 (η) is a wavelet function which depends on a non-dimensional 'time' parameter η, with zero mean and is localized in both frequency and time (Farge, 1992;Torrence and Compo, 1998). Because Morlet wavelet provides a good balance between time and frequency localization, here we applied the Morlet wavelet defined as ...
Flood/drought series during the past 1000 yrs in the Yangtze Delta, China, was reconstructed based on historical documents and local chronologies. Continuous wavelet transform was applied to detect the periodicity and variability of the flood/drought series. Research results indicate that: (1) Larger fluctuations of climatic changes in the Tibetan Plateau result in higher wavelet variance of flood/drought in the Yangtze Delta, for example, during 1400–1700, the proxy indicators indicate that the annual temperature in Tibet experienced larger variability and that this time interval exactly corresponds to the time when the higher and significant wavelet variance occurred; (2) Periods featured by colder temperature in the Tibetan Plateau usually correspond to periods characterized by higher wetness with higher probability of flood events; (3) Variability of heating features of the Tibetan Plateau exerted great influences on intensity and onset of Indian monsoon and south Asian summer monsoon, and these atmospheric activities are in direct connection with precipitation in Eastern China. Current global warming may alter the snow mass of Tibetan Plateau and then alters the heating features of Tibetan Plateau, which may in turn impact flood/drought conditions in the Yangtze Delta.
... In laboratory experiments and in direct numerical simulation (DNS) one observes that small-scale contributions are not necessarily slaved by the large scales since there is some inverse energy transfer, called backscattering, which complicates modeling in LES. So far different multiscale decompositions have been applied for modeling and computing turbulent flows [6], [11], [17], [22]. The wavelet representation is an efficient tool to perform such decompositions since it uses well localized functions in space, scale, and direction. ...
Coherent vorticity simulation (CVS) is a multiscale method to compute incompressible turbulent flows based on the wavelet filtered Navier–Stokes equations. At each time step the vorticity field is decomposed into two orthogonal components using an orthogonal wavelet basis: the coherent vorticity, corresponding to the coefficients whose modulus is larger than a threshold, and the remaining incoherent vorticity. The threshold value only depends on the total enstrophy, which evolves in time, and on the maximal resolution, which remains constant. The induced coherent velocity is computed from the coherent vorticity using the Biot–Savart kernel. To compute the flow evolution one only retains the coherent wavelet coefficients and some of their neighbors in space, scale, and direction, which define the safety zone. Two different strategies are studied to minimize at each time step the number of degrees of freedom to be computed, either by increasing the threshold value or by reducing the width of the safety zone. Their efficiency is compared for a three-dimensional forced homogeneous isotropic turbulent flow at initial Taylor microscale Reynolds number R λ =153. The quality of the results is assessed in comparison to a direct numerical simulation of the same flow. It is found that, as long as a safety zone is present, CVS well preserves the statistical predictability of the turbulent flow (even the vorticity and velocity probability distribution functions) with a reduced number of degrees of freedom.
A physical linked-measure is mathematically consisted of a complex scalar, a complex vector and a bi-vector and is geometrically equivalent to a vortex. When the complex scalar means mass, the complex vector implies directed momentum and the bi-vector rotates angular momentum, with using the least action principle to the linked-measure, yielding energy-mass-momentum-angular momentum joint conservation.
Modern gas turbine spray combustors feature multiple swirlers with distributed fuel injection system for rapid fuel/air mixing and flame stabilization ensuring low NOx operations. In the present paper, we investigate the effects of different swirler designs on flame characteristics, stabilization, and behavior at lean blow out using a Triple Annular Research Swirler (TARS) burner. Simultaneous planar measurements using laser diagnostics, namely, Planar Laser Induced Fluorescence (LIF) of OH radicals indicating the reacting zone, LIF Acetone indicating unburnt fuel distribution and Particle Image Velocimetry (PIV) for flow field mapping, were applied to study the flow dynamics, fuel distribution and flame dynamics for different swirler geometries, air flow rates, and equivalence ratios. Both axial and nearly perpendicular to axis cross-sectional planes were investigated. The three swirler configurations allowed getting stable and repeatable flames over a wide range of different flow and fuel equivalence ratio conditions, confirming the good flexibility and operability of the TARS burner. Averaged fields are presented to compare the effect of different flow conditions using the same swirler configuration, and the effect of different swirler configurations at the same flow conditions. LIF and PIV instantaneous samples are also shown, both in axial and cross sectional planes, with structures captured in detail. Perfect matching is found between unburnt and burnt field, as well as agreement between axial and cross-sectional measurements. Particular attention has been placed on unstable flames and a highly unsteady flame near the lean blow out (LBO) is shown. Local extinctions are occasionally seen on instantaneous snapshots. Unsteadiness of such flame is suitable to exemplify the use of Proper Orthogonal Decomposition (POD) analysis that identifies the most “energetic” large scale structures or modes of the flame. In particular, rotational and helical modes are observed which can contribute to the swirling flame instability. The results show the effect of the strength and rotation direction of the swirlers can lead to strong flame stratification or to a more homogenous flames. Analysis of the flame dynamics, indicates that the flame can be stabilized dynamically without the presence of a Central Recirculation Zone (CRZ) through flame quenching and flame propagation.
In this paper a theoretical and experimental investigation is performed on the sloshing of a two-liquid system with both separation and free surface. The experimental configuration consists of an oscillating tank filled with two layers of immiscible liquids. The mathematical model is obtained by applying the Lagrangian variational approach to the potential formulation of the fluid motion, and a dynamical system which describes the dynamics of motion is derived. In order to account for the damping of the motion, generalized dissipative forces are considered. For this purpose, the logarithmic decrement coefficients are estimated by means of a wavelet analysis performed on the experimental free oscillations of the fluid system. Numerical integration of the mathematical model gives results which are in a fair agreement with the experimental results.
The Agulhas system (Agulhas Current, Retroflection, Agulhas Return Current (ARC)), south of Africa, constitutes a complex energetic system subject to high frequency mesoscale and low frequency variability. Using chlorophyll pigment concentration data collected by the SeaWiFS ocean color sensor and sea surface height satellite data, we demonstrate the utility of wavelet analysis in jointly analyzing ocean color and altimetric signals. Two distinct dynamical scenarios of the Agulhas system (October 1997 and January 1998) yield different dominant wavelengths (620 and 500 km) for the ARC Rossby wave. They are simultaneously retrieved from the chlorophyll and sea level anomaly signals. Longitudinal extension of the power, resulting from the wavelet analysis, also depends on the dynamical regime.
This article reviews state-of-the-art adaptive, multiresolution wavelet methodologies for modeling and simulation of turbulent flows with various examples. Different numerical methods for solving the Navier-Stokes equations in adaptive wavelet bases are described. We summarize coherent vortex extraction methodologies, which utilize the efficient wavelet decomposition of turbulent flows into space-scale contributions, and present a hierarchy of wavelet-based turbulence models. Perspectives for modeling and computing industrially relevant flows are also given.
A variety of flow regimes may be observed in the hydrotransport of solids. Maintaining a consistent operation requires rapid detection of any change in flow regime. The objective of this study was to develop reliable, non-intrusive, and on-line methods for flow regime detection in a hydrotransport pipe based on acoustic measurements.
Slurries were transported in a 0.05 m diameter stainless steel hydrotransport pipe. Acoustic measurements were recorded by microphones along the length of the horizontal pipe. Two flow regimes were observed: suspended solids flow and flow over settled solids. Cycle and frequency analyses of the acoustic signals provided reliable identification of these flow regimes.
Plusieurs régimes d'écoulement peuvent être observés dans le transport hydraulique des solides. Le maintien d'un fonctionnement consistant nécessite la détection rapide de tout changement dans le régime d'écoulement. L'objectif de cette étude est de développer des méthodes fiables, non intrusives et en continu pour la détection des régimes d'écoulement dans une conduite de transport hydraulique reposant sur des mesures acoustiques. Les suspensions ont été transportées dans une conduite de transport hydraulique en acier inoxydable de 0,05 m de diamètre. Les mesures acoustiques ont été enregistrées par des microphones le long de la conduite horizontale. Deux régimes d'écoulement ont été observés : l'écoulement de solides suspendus et l'écoulement au-dessus des solides déposés. Les analyses de cycle et de fréquence des signaux acoustiques fournissent une identification fiable de ces régimes d'écoulement.
A method for extracting coherent vorticity sheets and current sheets out of three-dimensional homogeneous magnetohydrodynamic MHD turbulence is proposed, which is based on the orthogonal wavelet decomposition of the vorticity and current density fields. Thresholding the wavelet coefficients allows both fields to be split into coherent and incoherent parts. The fields to be analyzed are obtained by direct numerical simulation DNS of forced incompressible MHD turbulence without mean magnetic field, using a classical Fourier spectral method at a resolution of 512 3 . Coherent vorticity sheets and current sheets are extracted from the DNS data at a given time instant. It is found that the coherent vorticity and current density preserve both the vorticity sheets and the current sheets present in the total fields while retaining only a few percent of the degrees of freedom. The incoherent vorticity and current density are shown to be structureless and of mainly dissipative nature. The spectral distributions of kinetic and magnetic energies of the coherent fields only differ in the dissipative range, while the corresponding incoherent fields exhibit near-equipartition of energy. The probability distribution functions of total and coherent fields for both vorticity and current density coincide almost perfectly, while the incoherent fields have strongly reduced variances. Studying the energy flux confirms that the nonlinear dynamics is fully captured by the coherent fields only. © 2009 American Institute of Physics.
Coherent vortices are extracted from data obtained by direct numerical simulation DNS of three-dimensional homogeneous isotropic turbulence performed for different Taylor microscale Reynolds numbers, ranging from Re = 167 to 732, in order to study their role with respect to the flow intermittency. The wavelet-based extraction method assumes that coherent vortices are what remains after denoising, without requiring any template of their shape. Hypotheses are only made on the noise that, as the simplest guess, is considered to be additive, Gaussian, and white. The vorticity vector field is projected onto an orthogonal wavelet basis, and the coefficients whose moduli are larger than a given threshold are reconstructed in physical space, the threshold value depending on the enstrophy and the resolution of the field, which are both known a priori. The DNS dataset, computed with a dealiased pseudospectral method at resolutions N = 256 3 , 512 3 , 1024 3 , and 2048 3 , is analyzed. It shows that, as the Reynolds number increases, the percentage of wavelet coefficients representing the coherent vortices decreases; i.e., flow intermittency increases. Although the number of degrees of freedom necessary to track the coherent vortices remains small e.g., 2.6% of N = 2048 3 for Re = 732, it preserves the nonlinear dynamics of the flow. It is thus conjectured that using the wavelet representation the number of degrees of freedom to compute fully developed turbulent flows could be reduced in comparison to the standard estimation based on Kolmogorov's theory. © 2007 American Institute of Physics.
A wavelet-based method to extract coherent vortices is applied to data of three-dimensional incompressible homogeneous isotropic
turbulence with the Taylor micro-scale Reynolds number 471 in order to examine contribution of the vortices to statistics
on the turbulent flow. We observe a strong scale-by-scale correlation between the velocity field induced by them and the total
velocity field over the scales retained by the data. We also find that the vortices almost preserve statistics of nonlinear
interactions of the total flow over the inertial range.
A numerical simulation of kinetic plasma turbulence is performed to assess
the applicability of critical balance to kinetic, dissipation scale turbulence.
The analysis is performed in the frequency domain to obviate complications
inherent in performing a local analysis of turbulence. A theoretical model of
dissipation scale critical balance is constructed and compared to simulation
results, and excellent agreement is found. This result constitutes the first
evidence of critical balance in a kinetic turbulence simulation and provides
evidence of an anisotropic turbulence cascade extending into the dissipation
range. We also perform an Eulerian frequency analysis of the simulation data
and compare it to the results of a previous study of magnetohydrodynamic
turbulence simulations.
This paper proposes a new method to synthesize and inpaint geometric textures. The texture model is composed of a geometric layer that drives the computation of a new grouplet transform. The geometry is an orientation flow that follows the patterns of the texture to analyze or synthesize. The grouplet transform extends the original construction of Mallat and is adapted to the modeling of natural textures. Each grouplet atoms is an elongated stroke located along the geometric flow. These atoms exhibit a wide range of lengths and widths, which is important to match the variety of structures present in natural images. Statistical modeling and sparsity optimization over these grouplet coefficients enable the synthesis of texture patterns along the flow. This paper explores texture inpainting and texture synthesis, which both require the joint optimization of the geometric flow and the grouplet coefficients.
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