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Register a new userCause-and-effect of linear mechanisms sustaining wall turbulence
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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 theoretical and conceptual arguments. 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 ($bf U$) to the fluctuating velocities ($bf u$). We use cause-and-effect analysis based on interventions. This is achieved by direct numerical simulation of turbulent channel flows at low Reynolds number, in which the energy transfer from $bf U$ to $bf u$ 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 $bf U$ arranged so that only transient growth is active, can faithfully represent the energy transfer from $bf U$ to $bf u$ as in realistic turbulence. Our approach provides direct cause-and-effect evaluation of the linear energy-injection mechanisms from $bf U$ to $bf u$ in the fully nonlinear system and simplifies the conceptual model of self-sustaining wall turbulence.
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This paper describes a study of the self-sustaining process in wall-turbulence based on a second order statistical state dynamics (SSD) model of Couette flow. SSD models with this form are referred to as S3T models and self-sustain turbulence with a mean flow and second order perturbation structure similar to that obtained by DNS. The use of a SSD model to study the physical mechanisms underlying turbulence has advantages over the traditional approach of studying the dynamics of individual realizations of turbulence. One advantage is that the analytical structure of SSD isolates and directly expresses the interaction between the coherent mean flow and the incoherent perturbation components of the turbulence. Isolation of the interaction between these components reveals how this interaction underlies both the maintenance of the turbulence variance by transfer of energy from the externally driven flow to the perturbation components as well as the enforcement of the observed statistical mean turbulent state by feedback regulation between the mean and perturbation fields. Another advantage of studying turbulence using SSD models is that the analytical structure of S3T turbulence can be completely characterized. For example, turbulence in the S3T system is maintained by a parametric growth mechanism. Furthermore, the equilibrium statistical state of the turbulence can be demonstrated to be enforced by feedback regulation in which transient growth of the incoherent perturbations episodically suppresses coherent streak growth preventing runaway parametric growth of the incoherent turbulent component. Using S3T to isolate these parametric growth and feedback regulation mechanisms allows a detailed characterization of the dynamics of the self-sustaining process in S3T turbulence with compelling implications for understanding the mechanism of wall-turbulence.
We use theory and Direct Numerical Simulations (DNS) to explore the average vertical velocities and spatial distributions of inertial particles settling in a wall-bounded turbulent flow. The theory is based on the exact phase-space equation for the Probability Density Function describing particle positions and velocities. This allowed us to identify the distinct physical mechanisms governing the particle transport. We then examined the asymptotic behavior of the particle motion near the wall, revealing the fundamental differences to the near wall behavior that is produced when incorporating gravitational settling. When the average vertical particle mass flux is zero, the averaged vertical particle velocity is zero away from the wall due to the particles preferentially sampling regions where the fluid velocity is positive, which balances with the downward Stokes settling velocity. When the average mass flux is negative, the combined effects of turbulence and particle inertia lead to average vertical particle velocities that can significantly exceed the Stokes settling velocity, by as much as ten times. Sufficiently far from the wall, the enhanced vertical velocities are due to the preferential sweeping mechanism. However, as the particles approach the wall, the contribution from the preferential sweeping mechanism becomes small, and a downward contribution from the turbophoretic velocity dominates the behavior. Close to the wall, the particle concentration grows as a power-law, but the nature of this power law depends on the particle Stokes number. Finally, our results highlight how the Rouse model of particle concentration is to be modified for particles with finite inertia.
Despite the nonlinear nature of wall 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, 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. In the present study, 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. This is achieved by direct numerical simulation of turbulent channel flows with constrained energy extraction from the streamwise-averaged mean-flow. We demonstrate that (i) transient growth alone is not sufficient to sustain wall turbulence and (ii) the flow remains turbulent when the exponential instabilities are suppressed. On the other hand, we show that (iii) transient growth combined with the parametric instability of the time-varying mean-flow is able to sustain turbulence.
Simulations of elastoinertial turbulence (EIT) of a polymer solution at low Reynolds number are shown to display localized polymer stretch fluctuations. These are very similar to structures arising from linear stability (Tollmien-Schlichting (TS) modes) and resolvent analyses: i.e., critical-layer structures localized where the mean fluid velocity equals the wavespeed. Computation of self-sustained nonlinear TS waves reveals that the critical layer exhibits stagnation points that generate sheets of large polymer stretch. These kinematics may be the genesis of similar structures in EIT.
There exists continuous demand of improved turbulence models for the closure of Reynolds Averaged Navier-Stokes (RANS) simulations. Machine Learning (ML) offers effective tools for establishing advanced empirical Reynolds stress closures on the basis of high fidelity simulation data. This paper presents a turbulence model based on the Deep Neural Network(DNN) which takes into account the non-linear relationship between the Reynolds stress anisotropy tensor and the local mean velocity gradient as well as the near-wall effect. The construction and the tuning of the DNN-turbulence model are detailed. We show that the DNN-turbulence model trained on data from direct numerical simulations yields an accurate prediction of the Reynolds stresses for plane channel flow. In particular, we propose including the local turbulence Reynolds number in the model input.
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