Session J42: Turbulence: Wall-Bounded III - Structure

Investigation of wall-bounded turbulence dynamics using spatiotemporal tracking and network-centric methods

Wall-bounded turbulence has long been characterized by coherent structures or eddies such as hairpins and streaks (Robinson 1991 ARFM). The kinematics and time-averaged statistical properties of these eddies have been extensively studied using techniques such as proper orthogonal decomposition (POD), conditional averaging, and flow visualization. However, the dynamical properties and spatiotemporal interactions of eddies remain less explored. We employ a spatiotemporal tracking algorithm capable of simultaneously tracking multiple sets of eddies to study their dynamics and interactions. The algorithm results in a large network whose properties are analyzed using node-level and higher-order organizational measures, referred to as network motifs. The mechanistic importance of the motifs, e.g., contribution to production/dissipation cycles, are quantified. Both minimal flow units and large domain channel flows are examined, where the ejection/sweep and low/high-speed-streak side-by-side pairing emerged as the lowest form of spatiotemporal organization. The application of the methodology to higher Reynolds number flows and to the study of other sets of eddies is discussed.   

Dynamics of the self-sustaining process in the minimal flow unit and fully turbulent channel flow: comparison of integral quantities

Minimal flow units (MFUs) of channel flow have facilitated the understanding of the self-sustaining process (SSP) of wall-bounded turbulence. While their statistics are comparable to those of fully turbulent channel flow, it is unclear whether the dynamics of the SSP remain unchanged in the larger domain. We aim to identify the SSP in large-domain turbulent channel flow and quantify how it differs from the one observed in MFUs. To compare the MFU and turbulent channel flow, subdomains comparable in size to the MFU are tracked in time in the larger domain at Re_τ = 180. We compute the key integral quantities of interest to the SSP (turbulent production, dissipation, and energy contained in specific length scales) in the MFU and the tracked subdomain at each timestep and assemble them into a temporal trajectory in variable-space. We compare state-transition probabilities of the MFU and turbulent channel flow based on the temporal trajectories. In addition, the trajectories are broken into short sequences of connections, and the most frequently occurring sequences of connections, known as motifs, are identified. These two metrics suggest similar dynamical processes occur in MFUs and the larger domain turbulent channel flow, albeit over shorter time scales in the large domain.   

The spatio-temporal dynamics of momentum-carrying and vortical structures in transitional and fully-developed wall-bounded turbulence

The late stages of transition are characterized by the breakdown and coalescence of multiple structures. During this breakdown, these structures visually appear less complex and more ordered than their fully turbulent counterparts. However, in regions of intense momentum transfer, the single-point (Park et al., 2012; Sayadi et al., 2013) and geometrical metrics (Elnahhas et al., 2022) of these transitional structures are statistically similar to their fully turbulent counterparts. In this work we analyze the dynamical properties of these transitional structures. Utilizing a generalized multi-type spatio-temporal tracking algorithm (Lozano-Duran & Jimenez, 2014; Elnahhas et al., 2023), two sets of structures, namely momentum transfer and three-dimensional vortical structures, are tracked simultaneously in the transitional and fully-developed regions of an H-type transitional boundary layer. In both regions, we analyze the statistical properties throughout the lifetimes of the structures and the interactions between the two sets. While the geometric properties adjust to the turbulent scaling laws immediately upon breakdown, the dynamical properties appear to take a longer downstream distance to forget the transition process and adjust to the turbulent scaling laws.   

Perron-Frobenius analysis of burst regeneration in wall turbulence

Intermittent bursting in wall-bounded turbulence can be modelled by the transient amplification of the wall-normal velocity as it is tilted by the shear, and can be represented as a clockwise path along the top of a roughly triangular pdf of flow states in an inclination-amplitude space. How bursts are regenerated is less well understood. The Perron-Frobenius operator describes probability transfer in time, and allows the identification of precursors and effects of particular states. When applied to an Re_τ ≈10³ small-box channel, it shows that the precursors of regeneration are at the lower left corner of the triangle. It also provides an estimate of the regeneration time and identifies the state-space trajectories involved. Conditional averaging along these trajectories shows that the most likely regenerative state contains a low-shear region near the wall and a high-shear layer above it. The new burst is generated from the latter. This is confirmed by numerical experiments in which the effect of the bursts on the mean shear is modified.   

The shortest channels that sustain short-streak turbulence

It is often assumed that infinitely long streaks of the streamwise velocity play a significant role in the sustainment of the turbulent energy cycle in channel flow, particularly as drivers of bursting. However, some channels are known to remain turbulent even when a numerical torque is applied to suppress the infinitely long streaks, calling their necessity into question. Here, we aim to understand how channels damped in this way remain turbulent by studying the effect of damping on the size of the minimal computational box sustaining turbulence at Re_τ ≈ 200. It is found that damped channels need to be at least 400 wall units long, approximately twice as much as canonical ones, and that they are characterized by an enhanced energy of the first streamwise spectral mode of the streamwise velocity. This suggest that this first mode acts as a substitute of the infinitely long streaks in the cycle sustainment. Intriguingly, both damped and undamped minimally short channels need to be wider than classical longer ones, suggesting that a collective interaction of several turbulence units is involved.   

Causally significant structures in fully developed wall turbulence

We investigated events with strong causal effects on the future state of wall turbulence by interventional way. Numerical experiments were conducted in a fully developed turbulent open channel flow at a friction Reynolds number of Re_τ=609 by introducing spatially localized perturbations. The experiments were repeated 38,000 times, varying the initial snapshots and locations of the perturbations. Causal effect of the perturbation was defined as the amplification of the perturbation velocity energy relative to its initial value. We examined common characteristics in the perturbed flow structure of causally significant and irrelevant samples. The analysis was performed at the time when the strength of causality showed the highest variation.

Near the wall, strong causality was found to be associated with high shear regions, namely high-speed streaks. Similarly, sweep regions were found to be associated with causality in the far wall. Taking those results into account, we conducted the second experiment by dividing the flow field into a grid-like structure and perturbing each cell, which allowed us to create a complete map of causal significance. Using the map, correspondence between canonical sweep & ejection pairs and causal & non-causal structures were studied in detail.   

Intermittent structures of intense inter-scale energy fluxes in turbulent channel flow

The phenomenology of turbulence energy cascade is believed to be well characterized by Kolmogorov’s theory of energy cascade. However, the theory becomes increasingly inaccurate at small scales because of small-scale intermittency. Recent studies have supported a causal influence of energy cascade on small-scale intermittency (Vela-Martín, A., 2022. J. Fluid Mech., 937, p.A13.). In this work, we characterize the spatial intermittency of inter-scale energy flux at various scales, down to the dissipative (Kolmogorov) scales, in a turbulent channel flow. A band-pass filter-based multiscale analysis is used for this purpose. We characterize the skewness and kurtosis of the energy flux distribution as a function of wall-normal distance and Reynolds number. Preliminary results show that kurtosis of inter-scale energy flux increases as we move towards smaller scales and also as we go away from the wall. As kurtosis is a measure of the ‘tailedness’ of distribution, this observation suggests an increased presence of intense intermittent events. We use a simple threshold to isolate structures of intense inter-scale energy flux and subsequently characterize their morphology using Minkowski functionals. We also try to establish a correlation between the intermittent structures identified across all scales.   

Transient growth of time-localized resolvent modes in a turbulent channel

To better understand transient growth in wall-bounded turbulence, this work studies the effect of the time-localized principal resolvent forcing mode on a fully-nonlinear turbulent channel flow. The principal resolvent forcing mode creates the largest energy amplification in the linearized Navier-Stokes system; this work aims to characterize how this mode affects the fully nonlinear flow and how predictable the resulting perturbation is using the optimal linear response trajectory. To obtain the time-localized resolvent forcing modes and its corresponding transient response, we formulate resolvent analysis in a wavelet basis in time for a turbulent channel flow at Re_τ = 180, and constrain the forcing to a compactly-supported wavelet by windowing the resolvent operator. We then test the effectiveness of the resolvent forcing modes in a fully-nonlinear turbulent channel flow by running an ensemble of direct numerical simulations in which we inject the computed resolvent forcing mode, then extract relevant statistics. The results show that the principal resolvent forcing mode follows the linear response for short times and close to the wall, and is more effective in changing the energy of the system compared to random forcing.   

History Effects in Adverse Pressure Gradient Turbulent Boundary Layers through Resolvent Analysis

Adverse pressure gradient (APG) turbulent boundary layers (TBL) are paramaterized by their Reynolds number, APG strength, and APG history. Resolvent analysis identifies a decomposition of the Navier-Stokes operator, linearized about a mean flow field, that has been shown to predict key features of energetic motions in shear-driven turbulence. Biglobal resolvent analysis is leveraged to account for the nonparallel terms in the APG TBL, specifying only the temporal frequency and spanwise wavenumber. The Bobke et al. (2017) LES datasets are used to test the effect of different APG strengths and histories on the resolvent amplification. A masking technique is used to identify small and large scale structures. Similar to the energization of large scale structures in experiment, the resolvent amplification of large scale structures increases with APG strength. Domains with a larger accumulated APG have increased amplification, similar to what is observed in the streamwise fluctuations when the APG history is varied. The history effect on the resolvent amplification shares similarities with a hybrid velocity scale that mitigates APG effects of the streamwise fluctuations in the outer region. The resolvent modes are then used to reconstruct the turbulent statistics using a model-based framework.