Session L43: Turbulence: General I

Experimental Results in the Turbulent Dissipation Range

An intrinsic feature of turbulence is its scale hierarchy, famously illustrated by Richardson's energy cascade and formalized by Kolmogorov's phenomenological theory in 1941 (K41). One of the key points from K41 is that the smallest scales (below the Kolmogorov length scale) are universal for sufficiently turbulent flows, and only depend on the viscosity and the mean dissipation rate. Nevertheless, little is still known about the dynamics at these scales, mainly due to computational and experimental resolution limitations. We circumvent this problem by performing hot-wire experiments in the Göttingen Variable Density Turbulence Tunnel filled with helium, which is approximately 10 times lighter than air. Varying the density of the gas by setting different pressures and temperatures, we are able to obtain Kolmogorov length scales on the order of the millimeter, and Taylor scale Reynolds numbers between 100 and 300. We present results focused in the dissipation regime, reaching a resolution of 5 times the Kolmogorov length scale for the smaller Reynolds numbers, and compare with prior numerical work.   

The coherent structure of the energy cascade in isotropic turbulence

The energy cascade, i.e. the transfer of kinetic energy from large-scale to small-scale flow motions, has been the cornerstone of turbulence theories and models since the 1940s. However, understanding the spatial organization of the energy transfer has remained elusive. In this study, we utilize numerical data of isotropic turbulence to investigate the three-dimensional spatial structure of the energy cascade. Specifically, we focus on analyzing the averaged flow patterns associated with intense energy transfer events in the inertial range. Our findings indicate that forward energy-transfer events are predominantly confined in the high strain-rate region created between two distinct zones of elevated enstrophy. On average, these zones manifest in the form of two hairpin-like shapes with opposing orientations. The mean velocity field associated with the energy transfer exhibits a saddle point topology when observed in the frame of reference local to the event. The analysis also suggests that the primary driving mechanism for energy transfer involves strain-rate self-amplification among local scales, followed closely by non-local interactions that promote interscale vortex stretching and strain self-amplification.   

Decay and propagation of an isolated turbulent blob

We create and sustain an isolated blob of turbulence by repeatedly firing together vortex loops. In the steady state, our PIV and 3D PTV measurements reveal that the blob consists of a turbulent core (Re_λ = 50−300) surrounded by comparatively quiescent fluid. The properties of the vortex loops determine the turbulent intensity and the scales of motion within the blob. When the injection of vortex rings stops, a spherical front that separates the turbulent core from the quiescent surroundings, begins to propagate within the chamber, and the turbulence decays. This turbulence endures throughout the decay process, lasting more than fifteen minutes, as evidenced by the energy spectrum. Through experimental comparison of turbulence induced by different methods within the same chamber, we demonstrate that the large-scale turbulence motion dictates the decay law of energy. By using a simple low-order closure model, we construct a spatially-extended description of the turbulence propagation and decay, and compare its predictions of energy profile and non-diffusive dynamics with data.   

Production of uncertainty in 3D Navier-Stokes turbulence

We address the predictability problem and derive the evolution equation of the average uncertainty energy for periodic/homogeneous incompressible Navier-Stokes turbulence and show that uncertainty increases by strain rate compression and decreases by strain rate stretching. We use DNS of non-decaying periodic turbulence and identify a time range with five properties: (1) the production and dissipation rates of uncertainty grow together in time, (2) the parts of the uncertainty production rates accountable to average strain rate properties on the one hand and fluctuating strain rate properties on the other also grow together in time, (3) the average uncertainty energies along the three different strain rate principal axes remain constant as a ratio of the total average uncertainty energy. Furthermore, (4) the uncertainty energy spectrum's evolution is self-similar if normalised by the average uncertainty energy and uncertainty's characteristic length and (5) the uncertainty production rate is extremely intermittent and skewed towards extreme compression events even though the most likely uncertainty production rate is zero. Properties (1), (2) and (3) imply that the average uncertainty energy grows exponentially in this time range. The resulting Lyapunov exponent depends on both the Kolmogorov time scale and the smallest Eulerian time scale, indicating a dependence on randon large-scale sweeping of dissipative eddies.   

The vorticity field and coherent structures in turbulent flow through rapid contractions

Following the work of Mugundhan et al. [1] we measure the three-dimensional velocity and vorticity r.m.s in four different contractions: Three different 2-D, 4:1-contractions with different lengths and a more rapid 3-D 16:1-contraction. An active-grid generates turbulence which is advected through the contractions in a vertical gravity-driven water tunnel.  Grid-rotation is at 240 rpm, Re_l of 220 at the inlet. We use Tomo-PIV to obtain velocities in 1 to 3 regions along the streamwise direction inside contractions. Up to 50,000 time-resolved volumes of 3-D velocities are obtained by tracking ~140,000 particles with shake-the-box PTV algorithm. This allows us to compute velocity and vorticity through these contractions. This shows decay in streamwise velocity r.m.s, u_rms and increase in transverse r.m.s, v_rms.  In the strongest contraction, u_rms after an initial decay, increases tending towards isotropy. This has been characterized as an “anomaly” at high-contraction ratios in point measurements [2]. The streamwise vorticity r.m.s gets amplified by ~3 in the 3-D contraction. Visualization of coherent structures using vorticity magnitude criteria shows the existence of long, stretched tubular structures aligned with the mean flow. This alignment is quantified by PDFs of cosine of alignment angles reaffirmed their preferential alignment. With the strongest-straining 3-D contraction, the relavite peak in the PDF values grows from  ~5 near the contraction inlet to ~50 near its exit.

  

Synchronization of large-scale coherent structures in vortex-gas free shear layers

Turbulent flows are chaotic systems characterized by sensitivity to initial conditions but can be ‘synchronized’ to a known solution by forcing the large scales in spectral space (Yoshida et al., Phys. Rev. Letters 94.1, 2005). Recently, this concept of synchronization in physical space for turbulent channel flow was studied by Wang & Zaki (J.Fluid Mech. 943, 2022) who showed that slabs of thickness of the local Taylor microscale can be synchronized.  This talk aims to address the question whether the chaotic motion of the large-scale structures in turbulence can also be synchronized under certain circumstances.  The vortex-gas shear layer is a chaotic Hamiltonian system relevant to the large-scale dynamics of a turbulent mixing layer. It has an interesting, underlying statistical mechanics (Suryanarayanan et al. Phys Rev. E 89.1, 2014), and provides a simpler setting to understand the concept of synchronization applied to free shear flows and large-scale coherent structures.  Extensive numerical simulations are performed to understand the circumstances under which a shear layer can recreate the correct coherent structures when the evolution of neighboring structures is forcibly synchronized with a given solution.  Implications for flow control are discussed.   

Causal decomposition of flow variables using information theory

We present a method for decomposing a source variable into its causal and non-causal contributions relative to a target variable. The concept of causality is defined within the framework of information theory as the time flux of information between variables. The latter can be interpreted as how much the knowledge about the past of one variable aids the understanding of another variable in the future. The decomposition of the source variable into its causal and non-causal components is formulated as an optimization problem that seeks to maximize the information flux of the causal part to the target variable while simultaneously minimizing the information flux of the non-causal part to the target variable. We demonstrate the method in two cases. The first case consists of a problem with a known analytical solution. In the second case, we analyze the velocity regions in a turbulent channel flow that are causal to the wall shear stress. Our method offers a new approach to study interactions within complex flow phenomena by disentangling causal and non-causal contributions.   

Non-homogeneous, non-isotropic turbulence

Turbulent flows are often studied in two opposite conditions: in the presence of confining boundaries as in the case of pipe or channel flows, and with periodic boundary conditions with the aim to realize homogeneous-isotropic turbulence. Between these two cases there is a range of possibilities among which flows in which isotropy and/or homogeneity is broken by the large scale forcing (or flow) but still in the absence of physical boundaries.

This talk will report a series of numerical results in two classical examples of these flows: the turbulent Kolmogorov flow and the turbulent cellular flow produced by a BC forcing. The effects of inhomogeneity, anisotropy and non-stationarity on the small scale statistics of the turbulent flow will be discussed.   

Localized resolvent-mode bases for turbulence statistics

Global resolvent analysis modes have been shown to accurately model the frequencies and spatial structure of the dominant coherent structures in several turbulent flows. However, to predict the amplitude of the structures, or other statistics of the flow, resolvent-mode forcing models must be devised. The present research seeks to enable data-driven approaches to this problem by fitting the mode coefficients to match lower-order statistics available from Reynolds-averaged Navier-Stokes (RANS) predictions.

As a first step towards this goal, we present a novel localized resolvent framework that breaks the resolvent forcing into a set of possibly overlapping spatial regions and reconstructs global quantities based on retaining the three highest-gain modes generated by the respective forcings. The localized modes are seen to provide a more flexible, robust basis by analogy with the general problem of approximating functions using global and local bases for both dominant and subdominant modes. By using the localized modes, lower-order flow statistics are reconstructed, such as Reynolds stresses, and compared to statistics computed using data from a large-eddy simulation.   

Third-order anomaly and the Zeroth Law of turbulence

A building block of any turbulence theory is that the mean kinetic energy dissipation rate is finite in the inviscid limit – known as the Zeroth Law of turbulence. In homogeneous and isotropic turbulence, this has been tested many times, both experimentally and computationally. Yet, since one cannot prove this result by data alone, several theoretical efforts have been made to put it on a firm ground. Such efforts are of fundamental importance. One of them is based on the relation between the mean energy dissipation rate and the scaling of the third-order absolute velocity increment. We will explore this relation in this talk using data from experiments and simulations of homogeneous and isotropic turbulence.