Probing turbulence intermittency via Auto-Regressive Moving-Average models
We suggest a new approach to probing intermittency corrections to the Kolmogorov law in turbulent flows based on the Auto-Regressive Moving-Average modeling of turbulent time series. We introduce a new index that measures the distance from a Kolmogorov-Obukhov model in the Auto-Regressive Moving-Average models space. Applying our analysis to Particle Image Velocimetry and Laser Doppler Velocimetry measurements in a von Kármán swirling flow, we show that is proportional to the traditional intermittency correction computed from the structure function. Therefore it provides the same information, using much shorter time series. We conclude that is a suitable index to reconstruct the spatial intermittency of the dissipation in both numerical and experimental turbulent fields.
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One of the few exact results known for isotropic, homogeneous and mirror-symmetric turbulence is the 4/5 - law derived by Kolmogorv in 1941. It links the longitudinal velocity increments to the mean rate of energy dissipation via:
where denotes averaging. This exact relation was then generalized by KolmogorovKolmogorov (1962) as a scaling law , where means has the same statistical properties. Should be a non stochastic constant, the scaling law would imply self-similar behavior for the structure functions of order , , that would scale like:
For , we recover the 4/5 - law. For , this equation predicts a second order structure function that varies like . By Fourier transform, this can be shown to be equivalent to a one dimensional energy spectrum scaling with wavenumber as : , also known as the Kolmogorov spectrum Kolmogorov (1941); Obukhov (1941). Both the 4/5 - law and the Kolmogorov spectrum have been measured and checked in many natural and laboratory isotropic turbulent flows Frisch (1996). More generally, eq. (2) predicts a linear law for the exponent of the structure functions . However, as pointed out by Landau and recognized by Kolmogorov Kolmogorov (1962), there is no reason to assume that is a constant over space and/or time, so that it should rather be viewed as a stochastic process, that depends upon the scale at which it is measured: . In such a case, the correct scaling of the structure function is rather
This modified law therefore predicts correction to the linear law , that are connected to the intermittent nature of the dissipation. For example, a log-normal model for of the dissipation (a suggestion by Landau and Obukhov) implies a quadratic correction for the . Other models have been suggested and lead to different corrections She and Leveque (1994); Dubrulle (1994); Benzi et al. (1984). Intermittency corrections up to have now been measured in a variety of experimental and numerical flows and appear to be robustly consistent from an experiment to another (see e.g. the review of Arneodo et al. (1996)). Corrections for larger values of are subject to resolution and statistical convergence issues: the larger the scaling exponent, the larger the statistical sampling must be in order to capture the rare events. There is therefore presently no general consensus about the behavior of intermittency corrections at large order. This hinders progress in the understanding of the statistical properties of the energy dissipation.
In this Letter, we suggest a new approach to probing intermittency corrections based on the Auto-Regressive Moving-Average (ARMA) modeling of turbulent time series. We introduce a new index that measures the distance from a Kolmogorov-Obukhov model in the ARMA space. Applying our analysis to velocity measurements in a von Kármán swirling flow, we show that this index is proportional to the traditional intermittency correction computed from the structure function and therefore provides the same information, using much shorter time series.
In most laboratory turbulent flows, available datasets are time series of values of a physical observable at a fixed point or obtained by tracking Lagrangian particles. This motivated the shift of paradigm from space velocity increments to time velocity increments defined as and motivated measurements of the time structure function and its local exponent . Of course, in situations where measurements are made on the background of a strong mean velocity , scale velocity increments and time velocity increments can be directly related through the Taylor hypothesis . In situations such that the fluctuations are of the same order than the mean flow, however, the Taylor hypothesis fails. A suggestion has been made by Pinton and Labbé (1994) to then resort to a local Taylor Hypothesis, in which where is the local rms velocity. This is equivalent to consider a scale such that and may be seen as equivalent to modifying the space Kolmogorov refined hypothesis into a time hypothesis , that leads to:
Such scaling is equivalent to the scaling obtained using the Lagrangian structure function. In any case, we may define the intermittency as the deviation of the local exponents (space increments) or (time increments) with respect to a linear behavior and may be quantified to first order by the parameter:
This factor is for example proportional to the log of the parameter of the log-Poisson model She and Leveque (1994); Dubrulle (1994), or to the parameter of the log-normal model Kolmogorov (1962). Note that it is also valid when the scaling exponents have been computed using the Extended Self-Similarity (ESS) Benzi et al. (1993), which is especially interesting in situations where turbulence is inhomogeneous and when the Taylor hypothesis does not hold. In the sequel, we compare this intermittency index with another one, built in a purely statistical framework.
Indeed, Thomson Thomson (1987) showed that, in the Lagrangian framework, the ”time” refined Kolmogorov hypothesis is in fact equivalent to a stochastic description in terms of an Ornstein-Uhlenbeck process with suitable drift and noise term:
where is a decorrelation timescale, a universal constant and is the mean dissipation. Indeed, taking into account the definition of the particle position , we get a scaling of the time averages of velocity and position as:
The second property is the Richardson law. Then, defining and , we get from eq.(7) which leads to the space refined Kolmogorov hypothesis and spectrum.
The discrete time version of eq. (7) can be written as:
where is a discrete time label, are the increments of a Brownian motion, and are independent variables, normally distributed. Eq. (8) is the expression of an auto-regressive process of order one, noted AR. To be able to account for intermittency or memory effects that are present in real flows, it is then convenient to consider a projection of the velocity data on higher order ARMA models
and define the intermittency as a distance with respect to the Kolmogorov AR(1) model in this space.
Intermittency as a distance in ARMA space.
A stationary time series is said to follow an ARMA process if it satisfies the discrete equation:
with - where stands for white noise - and the polynomials and , with , have no common factors. Notice that the noise term will be assumed to be a white noise, which is a very general condition Brockwell and Davis (1990). We ensure unicity by applying the Box-Jenkis procedure Box and Jenkins (1970): we choose the lowest and such that the residuals of the series filtered by the process ARMA() are not correlated. To define a suitable distance in the space of ARMA models, we introduce the Bayesian information criterion (), measuring the relative quality of a statistical model, as:
where is the likelihood function for the investigated model and in our case and the length of the sample. The variance is computed from the sample and is a series-specific quantity. The normalized distance between the fit ARMA and the Kolmogorov AR(1) model is then defined as the normalized difference between the () and the AR(1) ():
with : it goes to zero if the dataset is well described by an AR(1) model and tends to one in the opposite case. When applied to time series of velocity increments, it is a measure of the deviations from the Kolmogorov model. We introduce the correction to magnify small values.
Application to turbulent data.
We apply the index defined in eq. 11 to time series of velocity fields obtained in a von Kármán turbulent swirling flow. The experimental set-up consists of two sets of blades mounted on two counter-rotating co-axial impellers at the top and bottom of a cylindric vessel of diameter m. The operating fluid is water, the rotation frequency of the impellers can reach Hz, resulting in large Reynolds numbers (). A detailed description of the experiment can be found in Ravelet et al. (2008); Cortet et al. (2010, 2009). Two techniques are used to measure the fluid velocity on a grid: the Particle Interferometry Velocimetry (PIV) and the Laser Doppler Velocimetry (LDV), mapped on a regular sampling time applying a sample-and-hold algorithm. The stereoscopic PIV measures the three components of the velocity field into a plane, while the LDV measurements reported have given the out-of-plane velocity component into a plane. The PIV produces regularly sampled time series at intervals of s over a sample size at most of order and a spatial resolution of order of mm, i.e. 10 to 100 times larger than the dissipation scale. The LDV time-series are sampled over time-scale of the order of s, producing sample size up to data on a grid of spatial resolution of the order of cm. Given these resolutions constraints, we compute spatial (resp. temporal) velocity increments for the PIV (resp. LDV) data. The idea is to compute at each spatial grid location the classical intermittency index , compare it to , and see how they vary spatially. All the analyses presented in this letter are done using the three components for the PIV and for the LDV.
Since the von Kármán flow is inhomogeneous and anisotropic with large fluctuations Cortet et al. (2009), we expect that the time and space velocity structure functions depend on the measurement points. This is illustrated in Fig. 1, for the second and fourth order spatial and time structure functions. For the spatial case, deviations from the Kolmogorov scaling (solid lines) are small for the spatial structure functions, near the symmetry plane . This plane is the location of an intense shear layer, and has traditionally been used to perform ”isotropic homogeneous” like measurements. Outside this plane, deviations from the Kolmogorov scaling are large. For the time case, one observes two distinct behaviors: outside the shear layer, where a mean velocity is well-defined, one observes close to Eulerian Kolmogorov scaling at the smallest time increments (). In the shear layer, where no Taylor hypothesis holds, the scaling is closer to Lagrangian scaling (). However, as already noted by Pinton and Labbé (1994) and shown in Fig. 2, the relative scaling exponents computed as (ESS method) are in most of the flow close to the universal scaling exponents found by Arneodo et al. (1996), in a variety of homogeneous turbulent flows even those with no obvious inertial range. Using these ESS scaling exponents to compute the index, we may then draw a map of the intermittency and compare it with . This is done in Fig. 3 for an LDV experiment at . The spatial patterns look indeed similar. Moreover, the plot of as a function of (lower panel of Fig. 3) evidences a linear relation between them; the linear regression represented by the red line leads to a linear correlation coefficient . This means that traces the same intermittency characteristics as the time structure functions.
The comparison of , with the intermittency index computed for spatial structure functions is also informative: because of convergency issues, we have to use a data set of about to data points to converge the estimate of , while only are needed to converge . To illustrate this, we use the longest data set available, a velocity fields of a PIV experiment performed at . At this value, the von Karman flow experiences the equivalent of a phase transition Cortet et al. (2011), with time wandering of the shear layer in between and . This corresponds to a very large time intermittency and is clearly detected by the index as shown in Fig. 4, under the shape of two patches at , and . This pattern is unique of the phase transition and is not present in other PIV experiments (see Faranda
et al. (2014a) for examples). Besides this, one observes a fairly symmetric structure, with maxima correspond to the four cells structure of the flow. However, the time intermittency prevents the convergency of the spatial structure functions, resulting in a lack of of symmetry of the field (not shown). As a result, fluctuates over a decade around a value of about 0.05, while spans several orders of magnitude, as can be seen in the lower panel of Fig. 4. This shows that is a more sensitive tool to detect intermittency than .
We have introduced an intermittency index that can be intuitively interpreted as a statistical distance between the best fit linear model for a turbulent time series and the simplest possible process, i.e. an AR(1). To this purpose, we have exploited a Bayesian information criterion, opportunely normalized.
We have compared the obtained values of with a classical intermittency index ; the two parameters show to be linearly related, with a coefficient of determination for the time structure function.
In the spatial case, while clearly catches important characteristics of the mean flow, the lack of convergence of prevents us from a comparison. Therefore, the main advantage of this new index is the applicability to cases in which no big datasets are available: while a robust estimation of the structure functions requires very long time series, an ARMA() process can be usually fitted even in case the series is of the order of data. In fact, as we have verified for the LDV data, the estimates of do not change signifcantly when resampling the original series at a frequency comparable to the PIV one. This opens the possibility to compute from the much shorter PIV time series, without requiring high-order moments computation.Our results reinforce the hypothesis firstly proposed by Laval et al. (2001) that intermittency propagates in direct interactions between large and small scales, rather than in cascades.
Moreover, several models of complex systems are available in terms of a stochastic differential equation Faranda et al. (2014b). The methodology presented here can be used to validate such models with respect to experimental data.
We acknowledge the other members of the VKE collaboration who performed the experiments: E. Herbert, P.F. Cortet, C. Wiertel and V. Padilla. DF acknowledges the support of a CNRS postdoctoral grant.
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