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Causality diagrams for (a) Δt = 2S −1 , (b) 5S −1 , and (c) 10S −1. Only connections with NT > 0.7 are included.
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Despite the large amount of information provided by direct numerical simulations of turbulent flows, their underlying dynamics remain elusive even in the most simple and canonical configurations. Most common approaches to investigate the turbulence phenomena do not provide a clear causal inference between events, which is essential to determine the...
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Context 1
... it could be anticipated given the complexity of turbulent flows, many causal relations are significant and there is no obvious dominant pattern. In order to attain a simplified physical interpretation of the results from Figure 4, the dominant cross-induced causal connectivities have been compiled in three causality diagrams ( Figure 5). Dominant causalities are defined as those with NT > 0.7. ...
Context 2
... addition to this constraint and to avoid spurious solutions due to statistical errors, those causalities which did not remain above 0.7 when considering half of the time history were excluded. The causality diagram for Δt = 2S −1 is shown in Figure 5 (a). The diagram reveals that most of the information flows from the streamwise streaks to the cross flow components whereas the mean flow remains inactive. ...
Context 3
... diagram reveals that most of the information flows from the streamwise streaks to the cross flow components whereas the mean flow remains inactive. At Δt = 5S −1 ( Figure 5 b), the straight streaks are still the dominant element with a strong causal effect on the mean flow, which begins to influence the meandering streaks. Finally, at Δt = 10S −1 , the causality diagram is reverted and the causal effect flows from the mean flow to the streamwise streaks, while the effect of the streamwise streaks are lessened. ...
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Despite the large amount of information provided by direct numerical simulations of turbulent flows, their underlying dynamics remain elusive even in the most simple and canonical configurations. Most common approaches to investigate the turbulence phenomena do not provide a clear causal inference between events, which is essential to determine the...
Citations
... A major drawback of Granger causality is its linearity, which may render it insufficient to expose causal behavior for nonlinear systems. On the other hand, transfer entropy causality analyses [10] have successfully identified relevant dynamics in the field of fluids [11][12][13]. One of the main strengths of transfer entropy is that it is agnostic to the form of the system dynamics, although a significantly large amount of data and a high computational cost are required to achieve converged results [14][15][16]. ...
This research focuses on the identification and causality analysis of coherent structures that arise in turbulent flows in square and rectangular ducts. Coherent structures are first identified from direct numerical simulation data via proper orthogonal decomposition (POD), both by using all velocity components, and after separating the streamwise and secondary components of the flow. The causal relations between the mode coefficients are analysed using pairwise-conditional Granger causality analysis. We also formulate a nonlinear Granger causality analysis that can account for nonlinear interactions between modes. Focusing on streamwise-constant structures within a duct of short streamwise extent, we show that the causal relationships are highly sensitive to whether the mode coefficients or their squared values are considered, whether nonlinear effects are explicitly accounted for, and whether streamwise and secondary flow structures are separated prior to causality analyses. We leverage these sensitivities to determine that linear mechanisms underpin causal relationships between modes that share the same symmetry or anti-symmetry properties about the corner bisector, while nonlinear effects govern the causal interactions between symmetric and antisymmetric modes. In all cases, we find that the secondary flow fluctuations (manifesting as streamwise vorticial structures) are the primary cause of both the presence and movement of near-wall streaks towards and away from the duct corners.
... Advantages and disadvantages of adsorbents, favorable conditions for particular adsorbate-adsorbent systems, and adsorption capacities of various low-cost adsorbents and commercial activated carbons as available in the literature are presented. [12][13][14][15][16] We are here to present the usefulness of MB. ...
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... One major obstacle to assess linear theories arises from the lack of tools in turbulence research that resolve the cause-and-effect dilemma and unambiguously attributes a set of observed dynamics to well-defined causes. This brings to attention the issue of causal inference, which is a central theme in many scientific disciplines but is barely discussed in turbulence research with the exception of a handful of works (Tissot et al. 2014;Liang & Lozano-Durán 2017;Bae, Encinar & Lozano-Durán 2018a;. It is via cause-and-effect relationships that we gain understanding of a given phenomenon, namely, that we are able to shape the course of events by deliberate actions or policies (Pearl 2009). ...
Despite the nonlinear nature of turbulence, there is evidence that part of the energy-transfer mechanisms sustaining wall turbulence can be ascribed to linear processes. The different scenarios stem from linear stability theory and comprise exponential instabilities, neutral modes, transient growth from non-normal operators and parametric instabilities from temporal mean flow variations, among others. These mechanisms, each potentially capable of leading to the observed turbulence structure, are rooted in simplified physical models. Whether the flow follows any or a combination of them remains elusive. Here, we evaluate the linear mechanisms responsible for the energy transfer from the streamwise-averaged mean flow (U) to the fluctuating velocities (u). To that end, we use cause-and-effect analysis based on interventions: manipulation of the causing variable leads to changes in the effect. This is achieved by direct numerical simulation of turbulent channel flows at low Reynolds number, in which the energy transfer from U to u is constrained to preclude a targeted linear mechanism. We show that transient growth is sufficient for sustaining realistic wall turbulence. Self-sustaining turbulence persists when exponential instabilities, neutral modes and parametric instabilities of the mean flow are suppressed. We further show that a key component of transient growth is the Orr/push-over mechanism induced by spanwise variations of the base flow. Finally, we demonstrate that an ensemble of simulations with various frozen-in-time U arranged so that only transient growth is active, can faithfully represent the energy transfer from U to u as in realistic turbulence. Our approach provides direct cause-and-effect evaluation of the linear energy-injection † A. Lozano-Durán and others mechanisms from U to u in the fully nonlinear system and simplifies the conceptual model of self-sustaining wall turbulence.
... Isosurfaces of u/U b are plotted at the channel center for cases C0 and C6 in Fig. 2. The streamwise streaks in case C0 are trackers of the large-scale rolls [12,15]. Streaks and rolls are involved in a regeneration cycle (i.e., self-sustaining mechanism) in both the buffer [28] logarithmic layer [29]. In this process, a streak flanked by an attached Q2 and Q4 event occasionally meanders and destroys itself, generating a peak of wall-normal velocity, Reynolds stress, and dissipation. ...
The existence of the large-scale structures appearing in turbulent Couette flows is studied by means of a direct numerical simulation data set of active thermal Couette flows for different friction Richardson numbers, at the Prandtl number of air, Pr=0.71. The existence of these structures is linked to the nonexistence of an active thermal flow. As soon as the Richardson number is greater than 1.5, the structures are less energetic, and for a value of only 3, the structures have vanished. This is due to the reorganization of the intense Reynolds stress events. Thus, large-scale structures will hardly appear in real-life Couette flows of air with a stable wall-normal gradient of temperature.
... One major obstacle to assess linear theories arises from the lack of tools in turbulence research that resolve the cause-and-effect dilemma and unambiguously attributes a set of observed dynamics to well-defined causes. This brings to attention the issue of causal inference, which is a central theme in many scientific disciplines but is barely discussed in turbulence research with the exception of a handful of works (Tissot et al. 2014;Liang & Lozano-Durán 2017;Bae et al. 2018a;. It is via cause-and-effect relationships that we gain a sense of understanding of a given phenomenon, namely, that we are able to shape the course of events by deliberate actions or policies (Pearl 2009). ...
Despite the nonlinear nature of turbulence, there is evidence that the energy-injection mechanisms sustaining wall turbulence can be ascribed to linear processes. The different scenarios stem from linear stability theory and comprise exponential instabilities from mean-flow inflection points, neutral modes, transient growth from non-normal operators, and parametric instabilities from temporal mean-flow variations, among others. These mechanisms, each potentially capable of leading to the observed turbulence structure, are rooted in simplified theories and conceptual arguments. Whether the flow follows any or a combination of them remains unclear. Here, we devise a collection of numerical experiments in which the Navier-Stokes equations are sensibly modified to quantify the role of the different linear mechanisms at low Reynolds number (Re_τ ≈ 180). We achieve this by direct numerical simulation of turbulent channel flows, in which the energy transfer from the streamwise-averaged mean-flow (U) to the fluctuating velocities (u') is constrained to preclude a targeted linear mechanism. In contrast to other studies, our approach allows for direct evaluation of cause-and-effect links between the linear energy injection from U to u' in a fully nonlinear setup. Our results show that turbulence persists when exponential instabilities and neutral modes of the mean flow are suppressed. Removing all exponential instabilities from the flow only leads to a 10% reduction of turbulent velocity fluctuations. We also show that the energy transfer from U to u' via transient growth alone, without exponential and parametric instabilities, is sufficient for sustaining turbulence. Finally, we demonstrate that a collection of simulations with frozen-in-time U exclusively supported by transient growth provides a faithful representation of the actual time-varying energy transfer from U to u'.
... One major obstacle arises from the lack of a tool in turbulence research that resolves the cause-and-effect dilemma and unambiguously attributes a set of observed dynamics to well-defined causes. This brings to attention the issue of causal inference, which is a central theme in many scientific disciplines but has barely been discussed in turbulent flows with the exception of a handful of works (Cerbus & Goldburg 2013;Tissot et al. 2014;Liang & Lozano-Durán 2017;Bae et al. 2018a). Given that the events in question are usually known in the form of time series, the quantification of causality among temporal signals has received the most attention. ...
Turbulent flows in the presence of walls may be apprehended as a collection of momentum-and energy-containing eddies (energy-eddies), whose sizes differ by many orders of magnitude. These eddies follow a self-sustaining cycle, i.e., existing eddies are seeds for the inception of new ones, and so forth. Understanding this process is critical for the modelling and control of geophysical and industrial flows, in which a non-negligible fraction of the energy is dissipated by turbulence in the immediate vicinity of walls. In this study, we examine the causal interactions of energy-eddies in wall-bounded turbulence by quantifying how the knowledge of the past states of eddies reduces the uncertainty of their future states. The analysis is performed via direct numerical simulation (DNS) of turbulent channel flows in which time-resolved energy-eddies are isolated at a prescribed scale. Our approach unveils, in a simple manner, that causality of energy-eddies in the buffer and logarithmic layers is similar and independent of the eddy size. We further show an example of how novel flow control and modelling strategies can take advantage of such self-similar causality.
... One major obstacle arises from the lack of a tool in turbulence research that resolves the cause-and-effect dilemma and unambiguously attributes a set of observed dynamics to well-defined causes. This brings to attention the issue of causal inference, which is a central theme in many scientific disciplines but has barely been discussed in turbulent flows with the exception of a handful of works (Cerbus & Goldburg 2013;Tissot et al. 2014;Liang & Lozano-Durán 2017;Bae, Encinar & Lozano-Durán 2018a). Given that the events in question are usually known in the form of time series, the quantification of causality among temporal signals has received the most attention. ...
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