Session F39: Turbulence: General I

Analysis of Lyapunov vectors of turbulent flows

In the perspective of predicting the formation of instabilities in turbulent flows, it is meaningful to characterize the formation and the propagation of disturbances. To this end, the Lyapunov theory provides a useful description of a turbulent flow field. On one hand, the computation of the Lyapunov spectrum provides the rate of growth of disturbances in a flow field and thereby quantifies the dimensionality of the strange attractor of the system. This is given by the Lyapunov exponents (LE) that have been the focus of numerous recent studies. On the other hand, the Lyapunov vectors (LV) can also be computed, but have been more overlooked since interpreting it is less obvious. In this work, the LV of several turbulent flows (1D Kuramoto Sivashinsky equation, homogeneous isotropic turbulence and turbulent channel flows) are computed, and their spatial structure is analyzed. It is shown that the LV exhibit non-trivial features for the vector associated with the null LE. In particular the concept of turbulent Lyapunov crisis is introduced and related to different turbulent forcing types.   

Taming turbulence via spectral and real space nudging

The technique of spectral nudging is commonly used to incorporate empirical data into a simulation in order to control its evolution and reproduce a given dynamical benchmark. We show how to do this in fully developed three dimensional turbulence. We give physical arguments for the choice of optimal parameters, based on the Reynolds number and on the amount of control provided by the nudging. We show how the algorithm can be used to extract unknown information from the data. Furthermore, we extend the technique to real space nudging, where the penalty is now performed in volumes in configuration space.   

Deep learning of turbulent velocity signals

We investigate the capability of a state-of-the-art deep neural model at learning features of turbulent velocity signals. Deep neural network (DNN) models are at the center of the present machine learning revolution. The set of complex tasks in which they over perform human capabilities and best algorithmic solutions grows at an impressive rate and includes, but it is not limited to, image, video and language analysis, automated control, and even life science modeling. Besides, deep learning is receiving increasing attention in connection to a vast set of problems in physics where quantitatively accurate outcomes are expected.

We consider turbulent velocity signals, spanning decades in Reynolds numbers, which have been generated via shell models for the turbulent energy cascade. Given the multi-scale nature of the turbulent signals, we focus on the fundamental question of whether a deep neural network (DNN) is capable of learning, after supervised training with very high statistics, feature extractors to address and distinguish intermittent and multi-scale signals. Can the DNN measure the Reynolds number of the signals? Which feature is the DNN learning?   

Computer Aided Image Segmentation and Classification of the Reynolds Stress Anisotropy Tensor

A combinatorial technique merging image segmentation via K-means clustering and colormap of the barycentric triangle to investigate the Reynolds stress anisotropy tensor is posed. The clustering aids in extracting the identical features from the spatial distribution of the anisotropy colormap images by minimizing the sum of squared error between cluster center and all data points, simultaneously minimizing the squared error over all clusters. Three data sets are used to investigate the applicability of the clustering technique including a converging-diverging channel flow, a supersonic jet flow, and the flow in a large wind farm array under three different thermal stratification cases (unstable, neutral and stable). The clustering technique improves pattern visualization and allows identifying different complex region of the turbulent flow. Helping to better understand the internal structure of the turbulent flow for different cases. The clustering images of the anisotropy colormap allow extracting the characteristic turbulence states in large three dimensional domains with clarity, revealing the natural grouping of the anisotropy stress tensor and illustrating the form and behavior of the turbulence in a particular region.   

Filtering techniques for statistically under sampled turbulent data sets

Extracting averaged values from low sample size turbulent data sets can be challenging due to the natural variation in the variables caused by the turbulent flows. Filtering techniques (Leonard 1974, Germano 1992) can be applied to these data sets to better match the statistics found from the larger sample size data. The experimental data from the turbulent mixing tunnel (TMT) (Charonko, Prestridge 2017) provides an excellent data set for testing these techniques as the sample size is large, the samples are high resolution, and variable density effects are tested. Containing 10,000 samples per location, a subset of the TMT data is chosen to represent a low sample size data set. Filtering techniques are then applied to the sample set and the statistics are compared to that of the full data set. These techniques have application to validation of simulations in which the domain has a very large number of spatial grid points, limiting the amount of time resolved data.   

A Non-Local Characterization of Coherent Regions in Turbulence: Disentangling Turbulence Eddies

The description of coherent motion in turbulent flows and the identification of eddies have been a topic of long standing debate. There exist numerous criteria based upon the velocity gradient, which can have their benefits, depending on the application, but suffer from a common shortcoming: they are all point-criteria, hence cannot account for the non-local structure of a coherent region, which we intuitively call an 'eddy'. We present a two-step method to overcome this. First, we introduce an instantaneous measure for spatial coherence, based upon a generalization of the two-point correlation: at each point in the velocity field, a correlation-vector is determined, by calculating a correlation-length along three orthogonal directions, using an invariant of the correlation-tensor. The iso-magnitude contours of this correlation-vector field represent a kernel of longer range coherent motions. Next, using Biot-Savart's law, the velocity field associated with isolated kernels can be reconstructed, which helps to isolate coherent motion in space, as well as scale. This procedure helps to visualize the dynamics of the eddies, possibly also giving a view into the ubiquitous eddy-breakup mechanism characteristic of the turbulence cascade, as summed up in Richardson's famous verse.
  

Energy Flow of Homogeneous Isotropic Turbulence at Sub-scale Eddies

Understanding the flow of energy at the smallest eddy scale can provide insights into energy cascade.  Much work has been done on investigating the energy flow at the level of the smallest eddies using different Navier-Stokes (NS) based techniques. However, resorting to NS based methods to understand the impact of the smallest eddies on the energy flow is tedious due to the strong coupling between the translational and rotational motion of the eddies. Therefore, a higher order morphing continuum theory (MCT) is employed. MCT treats the continuum not as a set of volume-less points as NS, but as a set of small-scale structures that possess their own rotational motion and inertia. The current study derives the MCT conservation laws under a translational symmetry. The resulting conservation laws reveal the existence of new routes for energy transfer among translation, rotation and internal energy. It broadens the view on the energy cascade phenomena in homogeneous isotropic turbulence as an example.  The energy analysis shows that the energy flux (positive or negative) is highly dependent on the rotation of the small-scale structure as well as their translational motion.