Turbulence In Wall-bounded Shear Flows Research Articles

Slow-growth approximation for near-wall patch representation of wall-bounded turbulence

Wall-bounded turbulent shear flows are known to exhibit universal small-scale dynamics that are modulated by large-scale flow structures. Strong pressure gradients complicate this characterization, however. They can cause significant variation of the mean flow in the streamwise direction. For such situations, we perform asymptotic analysis of the Navier–Stokes equations to inform a model for the effect of mean flow growth on near-wall turbulence in a small domain localized to the boundary. The asymptotics are valid whenever the viscous length scale is small relative to the length scale over which the mean flow varies. To ensure the correct momentum environment, a dynamic procedure is introduced that accounts for the additional sources of mean momentum flux through the upper domain boundary arising from the asymptotic terms. Comparisons of the model's low-order, single-point statistics with those from direct numerical simulation and well-resolved large eddy simulation of adverse-pressure-gradient turbulent boundary layers indicate the asymptotic model successfully accounts for the effect of boundary layer growth on the small-scale near-wall turbulence.

Homoclinic bifurcation and switching of edge state in plane Couette flow.

We identify the presence of three homoclinic bifurcations that are associated with edge states in a system that is governed by the full Navier-Stokes equation. In plane Couette flow with a streamwise period slightly longer than the minimal unit, we describe a rich bifurcation scenario that is related to new time-periodic solutions and the Nagata steady solution [M. Nagata, J. Fluid Mech. 217, 519-527 (1990)]. In this computational domain, the vigorous time-periodic solution (PO3) with comparable fluctuation amplitude to turbulence and the lower branch of the Nagata steady solution are considered as edge states at different ranges of Reynolds number. These edge states can help in understanding the mechanism of subcritical transition to turbulence in wall-bounded shear flows. At the Reynolds numbers at which the homoclinic bifurcations occur, we find the creation (or destruction) of the time-periodic solutions. At a higher Reynolds number, we observe the edge state switching from the lower-branch Nagata steady solution to PO3 at the creation of this vigorous cycle due to the homoclinic bifurcation. Consequently, the formation of the boundary separating the basins of attraction of the laminar attractor and the time-periodic/chaotic attractor also switches to the respective stable manifolds of the edge states, providing a change in the behavior of a typical amplitude of perturbation toward triggering the transition to turbulence.

Patterns in transitional shear turbulence. Part 2. Emergence and optimal wavelength

Low Reynolds number turbulence in wall-bounded shear flows en route to laminar flow takes the form of oblique, spatially intermittent turbulent structures. In plane Couette flow, these emerge from uniform turbulence via a spatio-temporal intermittent process in which localised quasi-laminar gaps randomly nucleate and disappear. For slightly lower Reynolds numbers, spatially periodic and approximately stationary turbulent–laminar patterns predominate. The statistics of quasi-laminar regions, including the distributions of space and time scales and their Reynolds-number dependence, are analysed. A smooth, but marked transition is observed between uniform turbulence and flow with intermittent quasi-laminar gaps, whereas the transition from gaps to regular patterns is more gradual. Wavelength selection in these patterns is analysed via numerical simulations in oblique domains of various sizes. Via lifetime measurements in minimal domains, and a wavelet-based analysis of wavelength predominance in a large domain, we quantify the existence and nonlinear stability of a pattern as a function of wavelength and Reynolds number. We report that the preferred wavelength maximises the energy and dissipation of the large-scale flow along laminar–turbulent interfaces. This optimal behaviour is due primarily to the advective nature of the large-scale flow, with turbulent fluctuations playing only a secondary role.

Patterns in transitional shear turbulence. Part 1. Energy transfer and mean-flow interaction

Low Reynolds number turbulence in wall-bounded shear flows en route to laminar flow takes the form of spatially intermittent turbulent structures. In plane shear flows, these appear as a regular pattern of alternating turbulent and quasi-laminar flow. Both the physical and the spectral energy balance of a turbulent–laminar pattern in plane Couette flow are computed and compared to those of uniform turbulence. In the patterned state, the mean flow is strongly modulated and is fuelled by two mechanisms: primarily, the nonlinear self-interaction of the mean flow (via mean advection), and secondly, the extraction of energy from turbulent fluctuations (via negative spectral production, associated with an energy transfer from small to large scales). Negative production at large scales is also found in the uniformly turbulent state. Important features of the energy budgets are surveyed as a function of$Re$through the transition between uniform turbulence and turbulent–laminar patterns.

Onset of large-scale convection in wall-bounded turbulent shear flows

Large-scale coherent rolls are observed frequently in unstably stratified turbulent wall-bounded flows where they influence strongly the turbulent transport and the mean flow properties. Here we address the question of their genesis by means of a linear stability analysis of the turbulent mean flow where a model of turbulent Reynolds stresses is embedded in the linear stability operator. We use the unstably stratified turbulent channel flow as a testbed for the analysis. We show that the onset of large-scale convection is associated to the linear instability of the mean flow to large-scale streamwise-uniform coherent rolls of aspect ratio$A \approx 3\unicode{x2013}3.3$when the friction Richardson number exceeds$|Ri_{\tau,c}|=0.86$. This corresponds to critical Rayleigh numbers that increase with the Reynolds number approximately as$Ra_c \approx 0.04\,Re_b^{1.8}$. These results are consistent with those obtained in recent direct numerical simulations performed in the same setting. It is also found that if turbulent Reynolds stresses are not modelled in the linear operator used in the stability analysis, then predictions are not consistent with direct numerical simulations results.

Stochastic Model for Quasi-One-Dimensional Transitional Turbulence with Streamwise Shear Interactions.

The transition to turbulence in wall-bounded shear flows is typically subcritical, with a poorly understood interplay between spatial fluctuations, pattern formation, and stochasticity near the critical Reynolds number. Here, we present a spatially extended stochastic minimal model for the energy budget in transitional pipe flow, which successfully recapitulates the way localized patches of turbulence (puffs) decay, split, and grow, respectively, as the Reynolds number increases through the laminar-turbulent transition. Our approach takes into account the flow geometry, as we demonstrate by extending the model to quasi-one-dimensional Taylor-Couette flow, reproducing the observed directed percolation pattern of turbulent patches in space and time.

Towards identification of a reliable framework to predict the thermal field in turbulent wall-bounded shear flows

Heat transfer modeling plays an integral role in design and optimization of traditional, as well as modern emerging thermal-fluid systems. However, mostly available models, known as eddy diffusivity models, face challenges in prediction of second order statistics such as heat fluxes in homogeneous directions and temperature variance. Additionally, these models are developed targeting fluids with Prandtl (Pr) number around unity and thus, having difficulty to capture thermal fields of working fluids with Pr numbers significantly different than unity at an acceptable level of accuracy. In an attempt to take first step to address the existing shortcomings, this investigation aims to identify a reliable framework to predict the turbulent thermal field of fluids with different Pr numbers (0.025, 0.71 and 10) in wall-bounded shear flows. Towards this, most advanced models, i.e. implicit and explicit algebraic turbulent heat flux models that try to incorporate the anisotropic nature of the turbulent heat flux, have been applied to a turbulent attached boundary layer of various working fluids with significantly different Pr numbers. It turns out that the explicit framework based on the representation theory is potentially capable of dealing with complex turbulent thermal fields and to address shortcomings of currently available models. Moreover, it has been shown that thermal time scale plays an integral role to accurately predict thermal field of fluids with Pr numbers significantly different than unity, as well as high order statistical quantities (e.g. temperature variance) of fluids with Pr numbers around unity.

Input-output inspired method for permissible perturbation amplitude of transitional wall-bounded shear flows.

The precise set of parameters governing transition to turbulence in wall-bounded shear flows remains an open question; many theoretical bounds have been obtained, but there is not yet a consensus between these bounds and experimental or simulation results. In this work, we focus on a method to provide a provable Reynolds-number-dependent bound on the amplitude of perturbations a flow can sustain while maintaining the laminar state. Our analysis relies on an input-output approach that partitions the dynamics into a feedback interconnection of the linear and nonlinear dynamics (i.e., a Luré system that represents the nonlinearity as static feedback). We then construct quadratic constraints of the nonlinear term that is restricted by system physics to be energy-conserving (lossless) and to have bounded input-output energy. Computing the region of attraction of the laminar state (set of safe perturbations) and permissible perturbation amplitude are then reformulated as linear matrix inequalities, which allows more computationally efficient solutions than prevailing nonlinear approaches based on the sum of squares programming. The proposed framework can also be used for energy method computations and linear stability analysis. We apply our approach to low-dimensional nonlinear shear flow models for a range of Reynolds numbers. The results from our analytically derived bounds are consistent with the bounds identified through exhaustive simulations. However, they have the added benefit of being achieved at a much lower computational cost and providing a provable guarantee that a certain level of perturbation is permissible.

The mean logarithm emerges with self-similar energy balance

Abstract

Drag Reduction in Turbulent Flows by Polymer and Fiber Additives

This article provides a review of the recent progress in understanding and predicting additives-induced drag reduction (DR) in turbulent wall-bounded shear flows. We focus on the reduction in friction losses by the dilute addition of high-molecular weight polymers and/or fibers to flowing liquids. Although it has long been reasoned that the dynamical interactions between polymers/fibers and turbulence are responsible for DR, it was not until recently that progress was made in elucidating these interactions in detail. Advancements come largely from numerical simulations of viscoelastic turbulence and detailed measurements in turbulent flows of polymer/fiber solutions. Their impact on current understanding of the mechanics and prediction of DR is discussed, and perspectives for further advancement of knowledge are provided.

A scheme to correct the influence of calibration misalignment for cross-wire probes in turbulent shear flows

Velocity fluctuations, measured via multi-wire probes, are very sensitive to misalignment between the calibration coordinate system and that of the wind tunnel. The present study proposes a scheme to correct the erroneous velocity fluctuations processed from a misaligned calibration while investigating a wall-bounded turbulent shear flow. The scheme is based on the premise that the viscous-scaled spectral energy distribution in the small-scales is invariant with Reynolds number and solely depends on the viscous-scaled spatial resolution of the sensor. Energy spectra processed from the misaligned calibration, in this small-scale range, are compared with the ‘expected’ spectra obtained via synthetic experiments on a direct numerical simulation data set. The erroneous lateral velocity spectra is found to be either relatively amplified or attenuated, by almost the same factor, at all wall-normal distances across the shear flow. A unique gain, defined to be the correction ratio, is thus obtained by forcing the erroneous spectra onto the reference spectra in this scale range. This ratio is further used to rectify the time series of the lateral velocity fluctuations, acquired across the shear flow, via Fourier analysis. The scheme is shown to be effective for experiments conducted across a decade of Reynolds number and using probes of varying spatial resolution.

A comprehensive assessment of the Reynolds Analogy in predicting heat transfer in turbulent wall-bounded shear flows

Heat transfer modeling plays a major role in design and optimization of modern and efficient thermal-fluid systems. However, currently available models suffer from a fundamental shortcoming: their development is based on the general notion that accurate prediction of the flow field will guarantee an appropriate prediction of the thermal field, known as the Reynolds Analogy. This investigation presents a comprehensive assessment of the capability of the Reynolds Analogy in predicting turbulent heat transfer when applied to turbulent shear flows of fluids with different Prandtl numbers. It turns out that the Reynolds Analogy is able to provide acceptable results for first order statistics only when fluids with Prandtl number close to unity are considered. Further, it is shown that unsteady simulations could provide acceptable results on second order statistics concerning fluids with different Prandtl numbers, if appropriate grid design/resolution is provided that allows to resolve essential dynamics of the thermal field. However, accurate prediction of higher order statistics close to solid surface requires more advanced heat transfer models that can provide accurate information on thermal time scales, in case the grid is too coarse to support accurate resolving of the essential thermal dynamics in these regions.

Entropy Generation Assessment for Wall-Bounded Turbulent Shear Flows Based on Reynolds Analogy Assumptions

Heat transfer modeling plays a major role in design and optimization of modern and efficient thermal-fluid systems. Further, turbulent flows are thermodynamic processes, and thus, the second law of thermodynamics can be used for critical evaluations of such heat transfer models. However, currently available heat transfer models suffer from a fundamental shortcoming: their development is based on the general notion that accurate prediction of the flow field will guarantee an appropriate prediction of the thermal field, known as the. In this work, an assessment of the capability of the in predicting turbulent heat transfer when applied to shear flows of fluids of different Prandtl numbers will be given. Towards this, a detailed analysis of the predictive capabilities of the concerning entropy generation is presented for steady and unsteady state simulations. It turns out that the provides acceptable results only for mean entropy generation, while fails to predict entropy generation at small/sub-grid scales.

Self-sustaining process in Taylor-Couette flow

The transition from Tayor vortex flow to wavy-vortex flow is revisited. The Self-Sustaining Process (SSP) of Waleffe [Phys. Fluids 9, 883-900 (1997)] proposes that a key ingredient in transition to turbulence in wall-bounded shear flows is a three-step process involving rolls advecting streamwise velocity, leading to streaks which become unstable to a wavy perturbation whose nonlinear interaction with itself feeds the rolls. We investigate this process in Taylor-Couette flow. The instability of Taylor-vortex flow to wavy-vortex flow, a process which is the inspiration for the second phase of the SSP, is shown to be caused by the streaks, with the rolls playing a negligible role, as predicted by Jones [J. Fluid Mech. 157, 135-162 (1985)] and demonstrated by Martinand et al. [Phys. Fluids 26, 094102 (2014)]. In the third phase of the SSP, the nonlinear interaction of the waves with themselves reinforces the rolls. We show this both quantitatively and qualitatively, identifying physical regions in which this reinforcement is strongest, and also demonstrate that this nonlinear interaction depletes the streaks.

The mechanism by which nonlinearity sustains turbulence in plane Couette flow

Turbulence in wall-bounded shear flow results from a synergistic interaction between linear non-normality and nonlinearity in which non-normal growth of a subset of perturbations configured to transfer energy from the externally forced component of the turbulent state to the perturbation component maintains the perturbation energy, while the subset of energy-transferring perturbations is replenished by nonlinearity. Although it is accepted that both linear non-normality mediated energy transfer from the forced component of the mean flow and nonlinear interactions among perturbations are required to maintain the turbulent state, the detailed physical mechanism by which these processes interact in maintaining turbulence has not been determined. In this work a statistical state dynamics based analysis is performed on turbulent Couette flow at R = 600 and a comparison to DNS is used to demonstrate that the perturbation component in Couette flow turbulence is replenished by a non-normality mediated parametric growth process in which the fluctuating streamwise mean flow has been adjusted to marginal Lyapunov stability. It is further shown that the alternative mechanism in which the subspace of non-normally growing perturbations is maintained directly by perturbation-perturbation nonlinearity does not contribute to maintaining the turbulent state. This work identifies parametric interaction between the fluctuating streamwise mean flow and the streamwise varying perturbations to be the mechanism of the nonlinear interaction maintaining the perturbation component of the turbulent state, and identifies the associated Lyapunov vectors with positive energetics as the structures of the perturbation subspace supporting the turbulence.

Description and detection of burst events in turbulent flows

A mathematical and computational framework is developed for the detection and identification of coherent structures in turbulent wall-bounded shear flows. In a first step, this data-based technique will use an embedding methodology to formulate the fluid motion as a phase-space trajectory, from which state-transition probabilities can be computed. Within this formalism, a second step then applies repeated clustering and graph-community techniques to determine a hierarchy of coherent structures ranked by their persistencies. This latter information will be used to detect highly transitory states that act as precursors to violent and intermittent events in turbulent fluid motion (e.g., bursts). Used as an analysis tool, this technique allows the objective identification of intermittent (but important) events in turbulent fluid motion; however, it also lays the foundation for advanced control strategies for their manipulation. The techniques are applied to low-dimensional model equations for turbulent transport, such as the self-sustaining process (SSP), for varying levels of complexity.

Self-sustaining processes at all scales in wall-bounded turbulent shear flows

We collect and discuss the results of our recent studies which show evidence of the existence of a whole family of self-sustaining motions in wall-bounded turbulent shear flows with scales ranging from those of buffer-layer streaks to those of large-scale and very-large-scale motions in the outer layer. The statistical and dynamical features of this family of self-sustaining motions, which are associated with streaks and quasi-streamwise vortices, are consistent with those of Townsend's attached eddies. Motions at each relevant scale are able to sustain themselves in the absence of forcing from larger- or smaller-scale motions by extracting energy from the mean flow via a coherent lift-up effect. The coherent self-sustaining process is embedded in a set of invariant solutions of the filtered Navier-Stokes equations which take into full account the Reynolds stresses associated with the residual smaller-scale motions.This article is part of the themed issue 'Toward the development of high-fidelity models of wall turbulence at large Reynolds number'.

A statistical state dynamics approach to wall turbulence

This paper reviews results obtained using statistical state dynamics (SSD) that demonstrate the benefits of adopting this perspective for understanding turbulence in wall-bounded shear flows. The SSD approach used in this work employs a second-order closure that retains only the interaction between the streamwise mean flow and the streamwise mean perturbation covariance. This closure restricts nonlinearity in the SSD to that explicitly retained in the streamwise constant mean flow together with nonlinear interactions between the mean flow and the perturbation covariance. This dynamical restriction, in which explicit perturbation-perturbation nonlinearity is removed from the perturbation equation, results in a simplified dynamics referred to as the restricted nonlinear (RNL) dynamics. RNL systems, in which a finite ensemble of realizations of the perturbation equation share the same mean flow, provide tractable approximations to the SSD, which is equivalent to an infinite ensemble RNL system. This infinite ensemble system, referred to as the stochastic structural stability theory system, introduces new analysis tools for studying turbulence. RNL systems provide computationally efficient means to approximate the SSD and produce self-sustaining turbulence exhibiting qualitative features similar to those observed in direct numerical simulations despite greatly simplified dynamics. The results presented show that RNL turbulence can be supported by as few as a single streamwise varying component interacting with the streamwise constant mean flow and that judicious selection of this truncated support or 'band-limiting' can be used to improve quantitative accuracy of RNL turbulence. These results suggest that the SSD approach provides new analytical and computational tools that allow new insights into wall turbulence.This article is part of the themed issue 'Toward the development of high-fidelity models of wall turbulence at large Reynolds number'.

Properties of the kinetic energy budgets in wall-bounded turbulent flows

The layer structure associated with the total kinetic energy for wall-bounded turbulent shear flows are examined using high-quality numerical simulation data.

Turbulent–laminar patterns in shear flows without walls

Turbulent–laminar intermittency, typically in the form of bands and spots, is a ubiquitous feature of the route to turbulence in wall-bounded shear flows. Here we study the idealised shear between stress-free boundaries driven by a sinusoidal body force and demonstrate quantitative agreement between turbulence in this flow and that found in the interior of plane Couette flow – the region excluding the boundary layers. Exploiting the absence of boundary layers, we construct a model flow that uses only four Fourier modes in the shear direction and yet robustly captures the range of spatiotemporal phenomena observed in transition, from spot growth to turbulent bands and uniform turbulence. The model substantially reduces the cost of simulating intermittent turbulent structures while maintaining the essential physics and a direct connection to the Navier–Stokes equations. We demonstrate the generic nature of this process by introducing stress-free equivalent flows for plane Poiseuille and pipe flows that again capture the turbulent–laminar structures seen in transition.