New reduced-order models (ROMs) are derived for sinusoidal shear flow (also known as Waleffe flow) and plane Couette flow in small periodic domains. A first derivation for Waleffe flow exploits Fourier modes that form a natural orthonormal basis for the problem. A ROM for such basis is obtained by a Galerkin projection of the Navier-Stokes equation. A large basis was reduced to 12 modes that contribute significantly in maintaining chaotic, turbulent dynamics. A key difference from earlier ROMs is the inclusion of two roll-streak structures, with spanwise wavelengths equal to $L_z$ and $L_z/2$, where $L_z$ is the spanwise length of the computational box. The resulting system was adapted to Couette flow by rewriting the Galerkin system for the same 12 modes, modified so as to satisfy no-slip conditions on the walls. The resulting dynamical systems lead to turbulence with finite lifetimes, in agreement with earlier ROMs and simulations in small domains. However, the present models display lifetimes that are much longer than in earlier ROMs, with differences of more than an order of magnitude. The Couette-flow model is compared to results of direct numerical simulation (DNS), with statistics displaying fair agreement. The inclusion of the $L_z$ and $L_z/2$ lengthscales is seen to be a key feature for longer turbulence lifetimes: neglecting any of the roll modes, or their non-linear interaction, leads to drastic reductions of turbulence lifetimes. The present ROMs thus highlight some of the dominant nonlinear interactions that are relevant in maintaining turbulence for long lifetimes.
Recent experimental and computational studies indicate that near wall turbulent flows can be characterized by universal small scale autonomous dynamics that are modulated by large scale structures. We formulate numerical simulations of near wall turbulence in a small domain localized to the boundary, whose size scales in viscous units. To mimic the environment in which the near wall turbulence evolves, our formulation accounts for the flux of mean momentum through the upper boundary of the domain. Comparisons of the model's two dimensional energy spectra and low order single-point statistics w…
Turbulence intensity profiles are compared for smooth- and rough-wall pipe flow measurements made in the Princeton Superpipe. The profile development in the transition from hydraulically smooth to fully rough flow displays a propagating sequence from the pipe wall towards the pipe axis. The scaling of turbulence intensity with Reynolds number shows that the smooth- and rough wall level deviates with increasing Reynolds number. We quantify the correspondence between turbulence intensity and the friction factor.
By using new results from direct simulations of turbulent channels at moderate friction Reynolds numbers (Retau <= 1900) and in very large numerical boxes, we examine the corrections to the similarity assumptions in the overlap and outer regions of wall-bounded turbulence.
Turbulence is omnipresent in Nature and technology, governing the transport of heat, mass, and momentum on multiple scales. For real-world applications of wall-bounded turbulence, the underlying surfaces are virtually always rough; yet characterizing and understanding the effects of wall roughness for turbulence remains a challenge, especially for rotating and thermally driven turbulence. By combining extensive experiments and numerical simulations, here, taking as example the paradigmatic Taylor-Couette system (the closed flow between two independently rotating coaxial cylinders), we show how…
The observability of the flow field away from the wall in turbulent channel flow is studied using only wall observations. Reconstructions are generated from noiseless but limited wall data using linear stochastic estimation, with emphasis on the quality of the reconstruction. All the velocities are well reconstructed in the buffer layer, but only relatively large "wall-attached" eddies are observable farther from the wall. In particular, the large structures accounting for approximately 40% of the total kinetic energy and of the tangential Reynolds stress are captured accurately. It is argued …
This paper presents a new theory of turbulence in time-independent non-Newtonian fluids. The wall layer is modelled in terms of unsteady exchange of viscous momentum between the wall and the main stream, following the classic visualisation of inrush-sweep-ejection/burst. The thickness of the wall layer is found to be the same for Newtonian and purely viscous non-Newtonian fluids, when normalised with the instantaneous wall parameters at the onset of bursting. The results indicate that the mechanisms of turbulence in Newtonian and time-independent fluids are identical when structural similarity…
For wall turbulence, moments of velocity fluctuations are known to be logarithmic functions of the height from the wall. This logarithmic scaling is due to the existence of a characteristic velocity and to the nonexistence of any characteristic height in the range of the scaling. By using mathematics of random variables, we obtain its necessary and sufficient conditions. They are compared with characteristics of a phenomenological model of eddies attached to the wall and also with those of the logarithmic scaling of the mean velocity.
Within wall turbulence, there is a sublayer where the mean wall-normal flux of the streamwise momentum is constant and related to the logarithmic wall-normal profile of the mean streamwise velocity. This relation, i.e., the law of the wall, has been used to estimate the mean stress at the wall surface. However, the momentum flux exhibits large temporal fluctuations. To relate them theoretically to those of the streamwise velocity at the same position from the wall, we consider an orthogonal decomposition of the fluctuations on a plane of the streamwise and wall-normal velocities. Since a large…
Within wall turbulence, there is a sublayer where the mean velocity and the variance of velocity fluctuations vary logarithmically with the height from the wall. This logarithmic scaling is also known for the mean concentration of a passive scalar. By using heat as such a scalar in a laboratory experiment of a turbulent boundary layer, the existence of the logarithmic scaling is shown here for the variance of fluctuations of the scalar concentration. It is reproduced by a model of energy-containing eddies that are attached to the wall.
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 detai…
This article accompanies a fluid dynamics video entered into the Gallery of Fluid Motion of the 66th Annual Meeting of the APS Division of Fluid Dynamics.
Wall turbulence has a sublayer where one-point statistics, e.g., the mean velocity and the variances of some velocity fluctuations, vary logarithmically with the distance from the wall. This logarithmic scaling is found here for two-point statistics or specifically two-point cumulants of those fluctuations by means of experiments in a wind tunnel. As for corresponding statistics of the rate of the energy dissipation, the scaling is found to be not logarithmic. We reproduce these scaling laws with some mathematics and also with a model of energy-containing eddies that are attached to the wall.
Constructing wall turbulence using hierarchical hairpin vortices
State Key Laboratory for Turbulence and Complex Systems, School of Mechanics and Engineering Science, Peking University（北京大学力学与工程科学系湍流与复杂系统国家重点实验室）；Max Planck Institute for Solar System Research（马普太阳系研究所）；HEDPS-CAPT, Peking University（北京大学高能量密度物理教育部/科技部重点实验室）
The effect of damping the longest streaks in wall-bounded turbulence is explored using numerical experiments. It is found that long streaks are not required for the self-sustenance of the bursting process, which is relatively little affected by their absence. In particular, there are turbulence states in which the fluctuations of the streamwise velocity have approximately the same length as the bursts, and are thus presumably associated with the bursts themselves, while the burst structure is essentially indistinguishable from flows in which longer velocity fluctuations are present. This sugge…
We study streamwise turbulence intensity definitions using smooth- and rough-wall pipe flow measurements made in the Princeton Superpipe. Scaling of turbulence intensity with the bulk (and friction) Reynolds number is provided for the definitions. The turbulence intensity scales with the friction factor for both smooth- and rough-wall pipe flow. Turbulence intensity definitions providing the best description of the measurements are identified. A procedure to calculate the turbulence intensity based on the bulk Reynolds number (and the sand-grain roughness for rough-wall pipe flow) is outlined.
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 a…
In the present work, an attempt is made to map the sensitivity of the existing zero pressure gradient (ZPG) turbulent boundary layer (TBL) wall-pressure spectrum models with different TBL parameters, and eventually, with different Reynolds Averaged Navier Stokes (RANS) turbulence models, simulated in OpenFOAM and ANSYS Fluent solvers. This study will help future researchers to choose a particular RANS turbulence model vis-à-vis a particular wall-spectrum model in order to obtain a reasonably accurate wind tunnel result predicting capability. First, the best-predicting pressure spectrum models …
New scaling relations for the mean velocity and Reynolds shear stress in viscous sublayer were proposed based on the application of matched asymptotic expansion method to the mean momentum balance. It was shown that the new parameter $Λ_+$ is relevant for the wall turbulence in addition to the friction velocity. From the proposed new scaling of the viscous sublayer the power law for the mean velocity in the overlap region was derived. The complete or incomplete similarity of the parameter $Λ_+$ was shown to be the key factor, which determines whether the mean velocity obeys the power law or lo…
For the constant-stress layer of wall turbulence, two-point correlations of velocity fluctuations are studied theoretically by using the attached-eddy hypothesis, i.e., a phenomenological model of a random superposition of energy-containing eddies that are attached to the wall. While the previous studies had invoked additional assumptions, we focus on the minimum assumptions of the hypothesis to derive its most general forms of the correlation functions. They would allow us to use or assess the hypothesis without any effect of those additional assumptions. We also study the energy spectra and …
This paper extends the resolvent formulation proposed by McKeon & Sharma (2010) to consider turbulence-compliant wall interactions. Under this formulation, the turbulent velocity field is expressed as a linear superposition of propagating modes, identified via a gain-based decomposition of the Navier-Stokes equations. Compliant surfaces, modeled as a complex wall-admittance linking pressure and velocity, affect the gain and structure of these modes. With minimal computation, this framework accurately predicts the emergence of the quasi-2D propagating waves observed in recent direct numerical s…
Direct numerical simulations are performed to contrast turbulent boundary layers over a concave wall without and with free-stream turbulence. Adverse pressure gradient near the onset of curvature leads to sharp decrease in skin friction and intermittent separation. The presence of free-stream turbulence reduces the probability of reverse flow, accelerates the recovery of the boundary layer in the downstream zero-pressure gradient region, and leads to a sustained and appreciable increase in the skin friction. The forcing also promotes the amplification of coherent Görtler structures in the loga…
The influence of near-wall density and viscosity gradients on near-wall turbulence in a channel are studied by means of Direct Numerical Simulation (DNS) of the low-Mach number approximation of the Navier--Stokes equations. Different constitutive relations for density and viscosity as a function of temperature are used in order to mimic a wide range of fluid behaviours and to develop a generalised framework for studying turbulence modulations in variable property flows. Instead of scaling the velocity solely based on local density, as done for the van Driest transformation, we derive an extens…
In the present study, we use a fifth-order ordinary differential equation, a generalization of the Blasius equation derived from the Leray-alpha model, to examine the transition to turbulence in rough-wall boundary-layer flows. This equation, combined with a weaker formulation of the von Karman log law modified to include the effects of surface roughness, provides a family of turbulent velocity profiles with two free parameters: the momentum thickness based Reynolds number and the roughness function. As a result, a skin-friction correlation is obtained and predictions of the transitional Reyno…
In this paper, we derive mathematical formulas for the skin friction coefficient in wall-bounded turbulence based on the Reynolds averaged streamwise momentum equation and the total stress. Specially, with the theoretical or empirical relation of the total stress, the skin friction coefficient is expressed in terms of the mean velocity and the Reynolds shear stress in an arbitrary wall-normal region $[h_0, h_1]$. The formulas are validated using the direct numerical simulation data of turbulent channel and boundary layer flows, and the results show that our formulas estimate the skin friction …
We study the effect of particle shape on the turbulence in suspensions of spheroidal particles at volume fraction $ϕ= 10\%$ and show how the near-wall particle dynamics deeply changes with the particle aspect ratio and how this affects the global suspension behavior. The turbulence reduces with the aspect ratio of oblate particles, leading to drag reduction with respect to the single phase flow for particles with aspect ratio $\mathcal{AR}\leq1/3$, when the significant reduction in Reynolds shear stress is more than the compensation by the additional stresses, induced by the solid phase. Oblat…
We study the mechanism of energy injection from the mean flow to the fluctuating velocity necessary to maintain wall turbulence. This process is believed to be correctly represented by the linearized Navier--Stokes equations, and three potential linear mechanisms have been considered, namely, modal instability of the streamwise mean cross-flow $U(y,z,t)$, non-modal transient growth, and non-modal transient growth supported by parametric instability. We have designed three numerical experiments of plane turbulent channel flow with additional forcing terms aiming to neutralize one or various lin…
Using the Lagrangian transport of momentum, the Reynolds stress can be expressed in terms of basic turbulence parameters. The Reynolds stress gradient represents the lateral transport of stream-wise momentum, balanced by the u2 transport, pressure and shear force terms in the momentum equation. Data from direct numerical simulations (DNS) have been used to validate this approach, with a good degree of consistency and agreement. This expression for the Reynolds stress gradient adds a transport equation to the Reynolds-averaged Navier-Stokes (RANS) equation, so that the turbulence structure can …
The majority of practical flows, particularly those flows in applications of importance to transport, distribution and climate, are turbulent and as a result experience complex three-dimensional motion with increased drag compared with the smoother, laminar condition. In this study, we describe the development of a simple model that predicts important structural and scaling features of wall turbulence. We show that a simple linear superposition of modes derived from a forcing-response analysis of the Navier-Stokes equations can be used to reconcile certain key statistical and structural descri…
We use the recent frame-indifferent theory of diffusive momentum transport to identify internal barriers in wall-bounded turbulence. Formed by the invariant manifolds of the Laplacian of the velocity field, the barriers block the viscous part of the instantaneous momentum flux in the flow. We employ the level sets of single-trajectory Lagrangian diagnostic tools, the trajectory rotation average and trajectory stretching exponent, to approximate both vortical and internal wall-parallel momentum transport barrier (MTB) interfaces. These interfaces provide frame-indifferent alternatives to classi…
Previous work has established the usefulness of the resolvent operator that maps the terms nonlinear in the turbulent fluctuations to the fluctuations themselves. Further work has described the self-similarity of the resolvent arising from that of the mean velocity profile. The orthogonal modes provided by the resolvent analysis describe the wall-normal coherence of the motions and inherit that self-similarity. In this contribution, we present the implications of this similarity for the nonlinear interaction between modes with different scales and wall-normal locations. By considering the nonl…
Contrasting with free shear flows presenting velocity profiles with inflection points which cascade to turbulence in a relatively mild way, wall bounded flows are deprived of (inertial) instability modes at low Reynolds numbers and become turbulent in a much wilder way, most often marked by the coexistence of laminar and turbulent domains at intermediate Reynolds numbers, well below the range where (viscous) instabilities can show up. There can even be no unstable mode at all, as for plane Couette flow (pCf) or for Poiseuille pipe flow (Ppf) that currently are the subject of intense research. …
We study single-phase and particulate turbulent channel flows, bounded by two incompressible hyper-elastic walls. Different wall elasticities are considered with and without a 10% volume fraction of finite-size rigid spherical particles, while elastic walls are modelled as a neo-Hookean material. We report a significant drag increase and an enhancement of the turbulence activity with growing wall elasticity for both single-phase and particulate cases in comparison with the single-phase flow over rigid walls. A drag reduction and a turbulence attenuation is obtained for the particulate cases wi…
We conduct direct simulations of turbulent channels imposing different virtual origins for all three velocities using Robin, slip-like boundary conditions to study the effect of displacing the origins perceived by different flow components, a mechanism common to small-textured surfaces. We also explore this effect using opposition control. For riblets, Luchini et al. (1991) proposed that their effect could be reduced to the offset between the streamwise- and spanwise-velocity origins, the latter being the origin perceived by turbulence. Later results on superhydrophobic surfaces suggest that t…
Townsend's attached-eddy hypothesis (AEH) provides a theoretical description of turbulence statistics in the logarithmic region in terms of coherent motions that are self-similar with the wall-normal distance (y). Here, we show the self-similar behaviour of turbulence motions contained within wall-attached structures of streamwise velocity fluctuations using the direct numerical simulation dataset of turbulent boundary layer, channel, and pipe flows ($Re_τ\approx 1000$) The physical sizes of the identified structures are geometrically self-similar in terms of height, and the associated turbule…
Direct numerical simulations of turbulent channels with rough walls are conducted in the transitionally rough regime. The effect that roughness produces on the overlying turbulence is studied using a modified triple decomposition of the flow. This decomposition separates the roughness-induced contribution from the background turbulence, with the latter essentially free of any texture footprint. For small roughness, the background turbulence is not significantly altered, but merely displaced closer to the roughness crests, with the change in drag being proportional to this displacement. As the …
Turbulent channel flows laden with particles are investigated using direct numerical simulation with a point-force approximation for small, heavy particles with a diameter smaller than the Kolmogorov length scale of the fluid. The Stokes numbers based on the wall units considered in our study are $St^+=0.5, 5, 35$ and 125. The main purpose of this study is to examine the effect of Stokes number on turbulence modification in a channel. We found that particles with $St^+=0.5$ enhance turbulence by increasing the occurrence of quasistreamwise vortices, while larger-Stokes-number particles attenua…
Modeling of wall-bounded turbulent flows is still an open problem in classical physics, with only modest progress made in the last few decades beyond the so-called `log law', which describes only the intermediate region in wall-bounded turbulence, i.e., $30-50 y^+ \text{ to } 0.1-0.2 R^+$ (in wall units) in a pipe of radius $R$. Here we propose a fundamentally new approach based on fractional calculus to model the {\em entire} mean velocity profile from the wall to the centerline of the pipe. Specifically, we represent the Reynolds stresses with a non-local fractional derivative of {\em variab…
We derive here Lagrangian fluctuation-dissipation relations for advected scalars in wall-bounded flows. The relations equate the dissipation rate for either passive or active scalars to the variance of scalar inputs from the initial values, boundary values, and internal sources, as those are sampled backward in time by stochastic Lagrangian trajectories. New probabilistic concepts are required to represent scalar boundary conditions at the walls: the boundary local-time density at points on the wall where scalar fluxes are imposed and the boundary first hitting-time at points where scalar valu…
General scenario of turbulence theory is proposed and applied to streaky wall-bounded turbulence. This scenario introduces a new field of transverse waves. Significance of the theory rests on a mathematical theorem associated with the conservation law of current flux, expressed in a form of 4d physical space-time representation, which predicts a system of Maxwell-type equation and supports transverse waves traveling with a phase speed c_t. In regard to the streaky wall flows, there exist both dynamical mechanism and energy channel which excite transverse waves and exchange energy between flow …
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 t…
We use interface-resolved simulations to study near-wall turbulence modulation by small inertial particles, much denser than the fluid, in dilute/semi-dilute conditions. We considered three bulk solid mass fractions, $Ψ=0.34\%$, $3.37\%$ and $33.7\%$, with only the latter two showing turbulence modulation. The increase of the drag is strong at $Ψ=3.37\%$, but mild in the densest case. Actually, two distinct regimes of turbulence modulation emerge: for smaller mass fractions, the turbulence statistics are weakly affected and the near-wall particle accumulation increases the drag so the flow app…
We consider linear feedback flow control of the largest scales in an incompressible turbulent channel flow at a friction Reynolds number of Re$_τ$ = 2000. A linear model is formed by linearizing the Navier-Stokes equations about the turbulent mean and augmenting it with an eddy viscosity. Velocity perturbations are then generated by stochastically forcing the linear operator. The objective is to reduce the kinetic energy of these velocity perturbations at the largest scales using body forces. It is shown that a control set-up with a well-placed array of sensors and actuators performs comparabl…
A theoretical description of the phenomenon of modulation of near-wall turbulence by large scale structures is investigated. The description given is simple in that the effect of large-scale structures is limited to a quasi-steady response of the near-wall turbulence to slow large-scale fluctuations of the skin friction. The most natural and compact form of expressing this mechanism is given by the usual Reynolds-number-independent representation of the total skin friction and velocity, scaled in wall variables, where the mean quantities are replaced by large-scale low-pass-filtered fluctuatin…
The statistical properties of velocity gradients in a wall-bounded turbulent channel flow are discussed on the basis of three-dimensional direct numerical simulations. Our analysis is concentrated on the trend of the statistical properties of the local enstrophy ${\bm ω}^2({\bm x},t)$ and the energy dissipation rate $ε({\bm x},t)$ with increasing distance from the wall. We detect a sensitive dependence of the largest amplitudes of both fields (which correspond with the tail of the distribution) on the spectral resolution. The probability density functions of each single field as well as their …
This paper reviews results from the study of wall-bounded turbulent flows 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 which isolates the interaction between the streamwise mean and the equivalent of the perturbation covariance. This closure restricts nonlinearity in the SSD to that explicitly retained in the streamwise constant mean together with nonlinear interactions between the mean and the perturbation covariance. …
