Session S10: Turbulence: Wall-Bounded Flows (5:45pm - 6:30pm CST)

Spectral energetics of a quasilinear approximation in uniform shear turbulence

The spectral energetics of a quasilinear (QL) model is studied in uniform shear turbulence. For the QL approximation, the velocity is decomposed into a mean averaged in the streamwise direction and the remaining fluctuation. The equations for the mean are fully considered, while the equations for the fluctuation are linearised around the mean. The QL model exhibits an energy cascade in the spanwise direction, but this is mediated by highly anisotropic small-scale motions unlike that in direct numerical simulation mediated by isotropic motions. In the streamwise direction, the energy cascade is shown to be completely inhibited in the QL model, resulting in highly elevated spectral energy intensity residing only at the streamwise integral length scales. It is also found that the streamwise wavenumber spectra of turbulent transport, obtained with the classical Reynolds decomposition, statistically characterizes the instability of the linearised fluctuation equations. Further supporting evidence of this claim is presented by carrying out a numerical experiment, in which the QL model with single streamwise Fourier mode is found to generate the strongest turbulence for $L_x/L_z=1\sim 3$, consistent with previous findings.   

The structure of the non-linear feedback term in resolvent analyses of turbulent wall-bounded flows

In resolvent analyses of turbulent channel flows it is common practice to neglect or model the non-linear forcing term which forms the input of the resolvent. However, the structure of this term is mostly unknown. Here, this non-linear forcing term is quantified. The cross-spectral density (CSD) of this term is computed. The CSD is evaluated for two channel flows at friction Reynolds numbers 180 and 550 via direct numerical simulations (DNS). It is found that the forcing is structured, and that it is the combination of oblique streamwise vortices and a streamwise component which counteract each other, as in a destructive interference. It is shown that a rank-2 approximation of the forcing, based on the spectral proper orthogonal decomposition (SPOD) modes, leads to the bulk of the response. Moreover, it is found that the non-linear forcing term has a non-negligible projection onto the linear sub-optimal forcings of resolvent analysis, which demonstrates that the linear optimal forcing is not representative of the non-linear forcing.   

Toward a Transfer Function Model of Scale Interactions in Wall-Bounded Turbulence

The phase relationship between isolated, streamwise large-scales and their corresponding stress fluctuations is described using a transfer-function approach with semi-empirical mode shapes. The dynamical equations for the isolated scales and stresses are obtained by Fourier decomposition of the Navier Stokes equations and then simplified to obtain a transfer-function relating the stresses to their isolated scales, where the mode shapes are modeled as critical-layer (resolvent) modes that scale with the critical layer thickness. The transfer function is used to identify the phase lag between the scales and stresses. This lag relates directly to the amplitude modulation coefficient used to study scale interactions in wall-bounded flows, where the fluctuating stress can be taken to represent the envelope of small-scale fluctuations. Consistent with experiments, the transfer function predicts that the zero-crossing height of the amplitude modulation coefficient corresponds to a spatial lead of the small-scale fluctuations at the location of the peak spectral energy of the very large-scale motions.   

Biphase, Scale Interactions, and the Turbulent Energy Cascade

The phase of the bispectrum of a turbulent velocity signal is used to relate the geometry and energetics of interactions between large- and small-scale motions in wall-bounded turbulence. Because the normalized bispectrum naturally describes non-linear, triadic interactions, it is ideally suited for measuring the coupling between the different scales of motion in turbulence, without the use of filtering procedures. Its corresponding biphase represents the spatial delay between triadic scales imposed by convective coupling, and is shown to relate directly to the amplitude modulation coefficient used in previous studies of scale interactions. The biphase also indicates the direction of the turbulent, streamwise energy cascade between the interacting scales. The bispectrum and biphase are calculated from experimental measurements in a turbulent boundary layer and used to provide a unified energetic and geometric interpretation to the phase lag between large- and small-scales previously measured by correlation techniques.   

Characterization of low-drag events at a moderate Reynolds number of $Re_{\tau}=700$

Low-drag events are intriguing, intermittent events in wall-bounded turbulent flows that are a natural target for flow control strategies. Sometimes referred to as hibernating turbulence, these events are described by extended periods ($\sim$ 3 eddy turnover times) where the skin friction of the system is considerably lower than its mean value ($\sim$ 90\% of the mean). Characterization of low-drag events can provide a better understanding of how and why these events manifest. While these events have been characterized in transitional and turbulent flows up to $Re_{\tau} \sim 100$, we extend the analysis to higher Reynolds numbers. In this talk, we discuss the characteristics of low-drag events at a moderate Reynolds number of $Re_{\tau} = 700$. We compare direct numerical simulations (DNS) of a turbulent channel flow with experimental data obtained by stereoscopic particle image velocimetry (SPIV) for a turbulent boundary layer at the same friction Reynolds number. Near-wall low-drag events are observed in both DNS and SPIV data, and flow characteristics of events found in each method are in good agreement. Turbulent statistics of low-drag events are also presented. Lastly, Reynolds number dependence of low-drag events is discussed.   

Controlling secondary flows in Taylor-Couette flow using inhomogeneous boundary conditions

Taylor-Couette (TC) flow, the flow between two independently rotating and co-axial cylinders, is known to have pinned secondary flows known as Taylor rolls. We study the possibility of affecting these secondary structures using one- and two-dimensional patterns of stress-free and no-slip boundary conditions on the inner cylinder. For this, we perform direct numerical simulations of TC flow with pure inner cylinder rotation at three different shear Reynolds numbers up to $Re_s=10^4$, fixing the radius ratio to $\eta=0.909$. We find that one-dimensional streamwise patterns do not have a significant effect on the flow, whereas one-dimensional spanwise patterns disrupt the rolls and decrease the torque substantially. Two-dimensional spiral inhomogeneities lie somewhere between the previous two cases, affecting the torque and moving the pinned secondary flows. We quantify the roll's movement for various angles and the widths of the spiral pattern, and find that the maximum speed occurs at a certain angle and width of the spiral pattern. Finally, we find that two-dimensional checkerboard patterns do not affect the flow or the torque substantially.   

Wall-attached structures of velocity fluctuations in turbulent Couette-Poiseuille flows

Direct numerical simulations (DNSs) of turbulent Couette-Poiseuille flows (CP-flows) under the moving wall conditions in the opposite direction to the main flow are performed to examine the turbulent characteristics of asymmetric wall-bounded flows. As the moving wall velocity increases, the positions for the maximum mean velocity and zero mean shear rates are shifted to the stationary wall, creating asymmetric shear layers. Although the friction Reynolds numbers on both walls increase, the logarithmic layer is elongated and shortened on the moving and stationary walls respectively. Furthermore, inspection of the turbulent intensities shows that the turbulent activity increases and decreases near the moving and stationary walls. The wall-attached structures in the CP-flows are self-similar with respect to their heights ($l_{y})$ and population density is inversely proportional with $l_{y}$ in the logarithmic layer. The turbulent characteristics of the asymmetric wall-bounded flows are closely associated with difference of the wall-attached structures between the moving and stationary walls.   

Influence of surface roughness on co-supporting cycle in a turbulent plane Couette-Poiseuille flow

Direct numerical simulations of fully developed turbulent plane Couette-Poiseuille flows (C-P flows) with and without a two-dimensional rod-roughened wall are performed to investigate the influence of the surface roughness on the flows. The Reynolds number based on the centerline laminar velocity and channel half-height is \textit{Re}$=$7200. When the surface roughness is imposed in the C-P flow with the smooth wall, the magnitude of the Reynolds shear stress decreases in the outer layer due to weakened very-large-scale motions (VLSMs) and roll-cell mode near the channel centerline Although the congregation of the near-wall small-scale motions by the roll-cell mode contributes to the formation of the VLSMs and the VLSMs, in turn, generate the roll-cell mode in the smooth-wall C-P flow, the weakened congregation motions of the small-scales by influence of the surface roughness reduces the strength of the VLSMs and roll-cell pattern.   

Degeneracy of turbulent states in 2D channel flow

Flows outside the laminar regime in 2D channels have received only limited attention until recently. In 2018, Falkovich and Vladimirova (Phys. Rev. Lett. 121,164501, 2018) studied pressure-gradient-driven 2D channel flows which reached fully turbulent regimes up to a Reynolds number Re $=$ 3.2x10$^{\mathrm{5}}$ and suggested a power law relationship between the pressure gradient and volume flux of the flow. We revisit the problem but now with the flow driven by constant volume flux. Our DNS study reveals a degeneracy of turbulent states over the range Re $\in $ [ 2.1x10$^{\mathrm{4}}$, 7.2x10$^{\mathrm{4}}$ ) with symmetric and asymmetric states (based on the mean shear on each of the channel walls) co-existing for at least 1.2x10$^{\mathrm{4}}$ channel transit times. The symmetric states correspond to the remarkably simple travelling wave structure observed by Falkovich and Vladimirova while the asymmetric states first transition to turbulence near one of the channel walls only. The bistability of these states is established by finding an unstable edge state which separates the two turbulent attractors. In this talk, the asymmetric path to turbulence is presented and the implications to the bistability of the states are discussed.   

Characteristic scales of momentum-carrying eddies in wall turbulence

In our current understanding of wall turbulence, the logarithmic layer is populated by a collection of multi-scale momentum-carrying eddies attached to the wall (Townsend 1976). In this framework, the classic characteristic velocity and length scales of wall attached eddies are the friction velocity and distance to the wall, respectively. In the present work, we show that these classic scales can be thought of as a particular case, and that the momentum-carrying eddies are more generally controlled by the mean energy production and mean shear with no explicit reference to the wall. Consistent with this argument, we propose new characteristic velocity, length, and time scales for the momentum-carrying eddies. The validity of the new scaling is demonstrated by direct numerical simulations of modified turbulent channel flows in which the friction velocity and distance to the wall remain unaltered, but the mean energy production and mean shear change significantly. The results show that, under those conditions, the flow structures follow the new proposed velocity and length scales rather than the classic scales.   

The emergence of logarithmic mean velocity with self-similar spectral energy balance

The attached eddy hypothesis of Townsend (The structure of turbulent shear flow, 1956, CUP) states that the logarithmic mean velocity would admit self-similar energy-containing eddies which scale with the distance from the wall. Over the past decade, there has been a growing body of evidence supporting the hypothesis, placing it to be the central platform for the general organisation of coherent structures in wall-bounded turbulent shear flows. Despite this progress, the most fundamental question, namely why this hypothesis is true, remains unanswered over many decades. In this study, we analytically demonstrate that the mean velocity is a logarithmic function of the distance from the wall if and only if the energy balance at integral length scale is self-similar with the distance from the wall. This provides a direct theoretical ground for the attached eddy hypothesis. The analysis is verified with the DNS data of incompressible channel flow at the friction Reynolds number ReT= 5200 (Lee & Moser, 2015, J. Fluid Mech., 774:395-415).   

Improvements upon Characterizing the Thermal Boundary Layer in Turbulent, Transcritical Channel Flows

Recent studies have attempted to characterize the thermal boundary layer profile in flows with strong fluctuations in the thermodynamic transport variables. However, the successes of these collapses have been inconclusive. Failure in modelling has been observed especially in the regime of strong heat transfer and large ratios in thermodynamic variables -- conditions relevant to turbulent flows at transcritical conditions. We present DNS results and analysis for a series of turbulent channel flows at transcritical conditions, with mean density ratio approaching $O$(20). Through analysis of statistical results, we show that transformations and relations from the extensive literature of compressible flows cannot be directly applied without significant error. To this end, we propose recommendations and improvements toward more accurate characterization of the thermal boundary layer.   

Revisiting Townsend's attached eddy model with resolvent analysis: computations up to extremely high Reynolds number

The quasi-linear approximation subject to a stochastic forcing was recently found to be able to qualitatively replicate spanwise wavenumber spectra and turbulent intensities at a significantly lower cost compared to DNS. In this study the quasi-linear approximation is revisited in the resolvent analysis framework, further reducing the computational cost and hence making the latter a useful tool for high $Re_\tau$ regimes. The velocity field was decomposed such that the non-linear form of the mean was used whilst the fluctuating velocity was modelled by replacing the non-linear term with an eddy-viscosity-based turbulent diffusion and forcing. Under this model the fluctuating velocity equation allows the superposition of solutions, making the current approximation directly comparable to Townsend's attached eddy model. Thus the forcing was determined self-consistently by minimising the difference between the Reynolds shear stresses obtained from the mean and fluctuating velocity equations. The proposed quasi-linear approximation up to $Re_\tau=2\times10^5$ showed that the near-wall streamwise peak intensity scales favourably with the one proposed by Monkewitz \& Nagib (2015, J. Fluid Mech. 783:474-503). In the final presentation the result up to $Re_\tau=2\times10^6$ will be shown.   

Small-span simulation of transient half-channel flow with application to riblets

This work tests the use of low-cost direct numerical simulation of half-channel flows with a small span to simulate non-equilibrium flow in response to step jump in bulk velocity. A setup similar to S. He and M. Seddighi \textbraceleft J. Fluid Mech.\textbraceright ,715,60--102(2013) is used with friction Reynolds number increasing from 180 to 418. Spanwise domain length is just sufficient to include near-wall structures within `healthy turbulence' region. Turbulent flow undergoes reverse transition towards quasi-laminar state, followed by a retransition phase, reaching new equilibrium state. Small-span captures the essential dynamics but shows slight delay in reaching final state, compared to full-span, which is attributed to slower streak transient growth due to exclusion of large attached eddies by limited span. Next, when small-span simulation is carried out in the presence of wall riblets, it was found that riblets delay retransition phase, by reducing streak-meandering and, consequently, weakening streak's transient growth. This work provides confidence on the use of small spanwise domain for extracting essential physics in a non-equilibrium accelerating flow.   

Approach to the Kolmogorov Inertial Range in Turbulent Pipe Flow

Kolmogorov's famous 4/5 (or, equivalently, 4/3) law follows from the notion that large scale separation (as represented by high Reynolds number) will eventually lead to the emergence of an ``inertial range" provided the effects of viscosity and the large-scale anisotopy are negligible. Within this range, the rate of interscale energy transfer is dictated by the mean dissipation rate of kinetic energy alone. Unlike other predicted features of the inertial range, such as the 2/3 or -5/3 laws, the emergence of the 4/5 (or 4/3) law may be characterized using a scale-by-scale energy budget equation derivable directly from the Navier-Stokes equations. In this presentation, we describe the evolution of the terms in this equation for a turbulent pipe flow and the resulting approach to the inertial range. To do so, we exploit several previously published single- and multi-component velocity measurements spanning a wide range of Reynolds numbers. It is found that the approach to the inertial range is slow at the centerline of the pipe, primarily due to the slow evolution with the Reynolds number of the turbulent diffusion related large scale term in the scale-by-scale energy budget.   

A Flow-structure-based wall-modeled large eddy simulation paradigm

A promising and cost-effective method for numerical simulation of high Re wall-bounded flows is wall-modeled large-eddy simulation (WMLES). Most wall models are formulated from the Reynolds-averaged Navier-Stokes equations (RANS). These RANS-based wall models are calibrated using mean turbulence data and make no use of the current vast knowledge on turbulent flow structure. Moreover, RANS-based models are limited to predicting the mean velocity profile and the mean wall shear stress, while higher-order statistics such as turbulent intensities and velocity spectra are not predicted near the wall. Here, we propose a near-wall model for LES which predicts subgrid-scale quantities such as the wall stress, velocity fluctuations, kinetic energy spectra, and flow structure across the entire near-wall layer. The model combines 1) a rescaling mapping which predicts the flow structure at different wall-normal distances and 2) a channel flow unit (CFU) with a domain-size fixed in inner units. The information from the mapping on how different flow structures scale with the distance to the wall is used to synthesize the top boundary condition for the CFU. This physics-based approach contrasts with other multiscale models where either the CFU size is linked to the LES grid size (Sandham, 2017), or the link between the two domains involves renormalization of velocity fields between time steps (Tang, 2016).   

Not Participating

Wall-shear stress fluctuations are of obvious importance for noise radiation, structural vibration, drag properties, and wall heat transfer mechanisms. A growing body of studies have reported that the generation of streamwise wall-shear stress fluctuations ($\tau_x'$) is linked to the large-scale motions. In the present study, we investigate the scale-based structures of $\tau_x'$ in turbulent channel flows at at $Re_{\tau}$ = 550, 950, and 2000. The wall-shear stress structures are identified using a two-dimensional clustering methodology. Depending on the sign of $\tau_x'$, these structures can be classified as positive-friction events ($PF_s$) and negative-friction events ($NF_s$). The statistical properties of the structures, including geometrical characteristics, spatial distribution, population density, fluctuating intensity, and correlations with outer motions are comprehensively investigated. Particular attention is paid to the asymmetries between $PF_s$ and $NF_s$, and their connection with wall-attached energy-containing eddies. In virtue of our results, only the large-scale $NF_s$ are the footprints of the inactive part of wall-attached eddies, and may serve as indicators for identifying Townsend's attached eddies in wall turbulence.   

Statistics and Scaling of Turbulent Couette-Poiseuille Flows at the Verge of Separation

Turbulent Couette-Poiseuille (C-P) flows with zero mean skin friction in adverse pressure gradients (APG) are investigated using direct numerical simulation (DNS). The aim of this study is to characterize the flow behaviour and investigate appropriate scaling laws when this flow is at the verge of separation. Four DNSs different Reynolds number were performed. All DNSs have approximately zero mean skin friction on the stationary bottom wall, while the moving top wall provides strong wall shear. Profiles of the mean streamwise velocity $U$ are presented using different velocity and length scales to investigate the appropriate scaling for the C-P flow The attached flow on the moving wall behaves similar to canonical channel flow. The friction-viscous scales, $u_\tau$ and $l_\tau$, provide the proper scaling of the mean velocity. In contrast to the flow on the shearless (stationary) wall, the friction-viscous scales are not applicable. Instead, pressure-viscous scales, $u_P$ and $l_P$, provide the appropriate scaling of the mean streamwise velocity with a square-root law observed. In the outer layer, collapsed profiles of the mean streamwise velocity are observed in outer scales that we developed using the approach similar to Zagarola & Smits (1998).   

A Three-Dimensional Extension of Minimal Quasi-Linear Approximation for Channel Flow

This paper extends a minimal quasi-linear approximation proposed in Hwang and Eckhardt (2020, J. Fluid Mech., 894, A23) to account for turbulent channel flow. A data-driven approach is applied to determine the optimal stochastic forcing for a linearized, eddy-viscosity based model. The streamwise forcing distribution is determined through an optimization problem, matching the two-dimensional spectra from a DNS at $Re_\tau = 5200$ to the linearised response, with the forcing subject to sufficient smoothness. Results are determined for fixed spanwise lengthscales and the self-similarity of energy-containing motions throughout the near-wall and logarithmic regions is exploited to determine a universal distribution. The spanwise forcing distribution is then determined self-consistently by matching the spanwise forcing distribution such that the Reynolds stress produced from the fluctuations matches that from the eddy-viscosity based mean flow. The two-dimensional spectra and turbulence intensities and quasi-linear approximation are then compared, with improvement found over the anisotropic previous results, qualitatively consistent with energetics of the self-sustaining process.   

12-mode reduced order models of Waleffe and Couette flow

New reduced-order models (ROMs) are derived for sinusoidal shear flow (Waleffe flow) and plane Couette flow. The derivation for Waleffe flow exploits Fourier modes that form an orthonormal basis for the problem, and a ROM 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 with the same 12 modes, modified 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 of the models: neglecting their interaction leads to drastic lifetime reductions.   

Near-Body Velocity and Turbulence Measurements of an Inclined 6:1 Prolate Spheroid at Low and Moderate Reynolds Numbers

Despite dramatic advances in computational power seen in the last decades, computational models are unable to predict transition, separation, and wake development for flow over three-dimensional bodies to the desired level of accuracy at acceptable computational cost. Without the ability to predict the forces and moments on the body, critical design parameters such as drag and loads on control surfaces for air- and water-borne vehicles cannot be predicted. The prolate spheroid is a popular body upon which to verify CFD models because of its simple geometry and complex, three-dimensional flow field. Advances in computational speed and experimental capabilities have prompted a renewed interest in related research. An experiment was conducted in the large recirculating water tunnel at the U.S. Naval Academy, using a 6:1 prolate spheroid measuring 0.43 m (18 in.) in length. The spheroid model was inclined at 2.5, 5, 10, and 20 degrees relative to the mean flow at flow speeds yielding length-based Reynolds numbers from 0.1-3x10$^{\mathrm{6}}$. Stereo particle image velocimetry (SPIV) was used to provide two-dimensional velocity maps in three spatial-dimensions (3C2D) and turbulence statistics. Additionally, several methods of boundary layer trip were employed and discussion of the resulting flow field presented.   

Understanding the Role of Nonlinearity in the Dynamics of Turbulent Couette Flow by Comparing Quasilinear Simulations to DNS

Both linear non-normality and nonlinearity are essential to sustaining wall turbulence. Inspired by studies of the statistical state dynamics of wall turbulence, the nonlinearity in the Navier-Stokes equations can be partitioned into a quasilinear component and a perturbation-perturbation nonlinearity component. Using the streamwise mean in this partition optimally separates the quasilinear and nonlinear mechanisms in the turbulence. These mechanisms are parametric growth arising from straining of the perturbations in the turbulence by the fluctuating streamwise mean flow and transient growth of perturbations arising from perturbation-perturbation nonlinearity. Comparing turbulence diagnostics between a quasilinear simulation and its associated DNS separates the mechanisms of parametric growth of perturbations in the turbulence from transient growth of perturbations introduced into the turbulence by perturbation nonlinearity. It is found using this comparison that the parametric mechanism is primarily responsible for maintaining perturbations in both the quasilinear simulation and DNS. It is concluded that the mechanism supporting the turbulence in Couette flow is the parametric mechanism isolated by quasilinear dynamics.   

Isotropy and inertial range behavior at pipe centerline

Isotropic relations form a cornerstone of turbulence research, but the exact extent of their validity in different flow configurations is always elusive. An investigation of multiple isotropic relations for high Reynolds number turbulence has been performed along the centerline of the Princeton Superpipe, a fully-developed turbulent pipe flow facility. Simultaneous measurements of the streamwise and radial component of velocity were acquired with a nanoscale crosswire, and the consistency of several different isotropic dissipation estimates was evaluated. The estimations indicate that isotropic dissipation relations can work in limited regions, and no universality is found at the moderate Reynolds numbers of the experiments performed. A correction to Kolmogorov's 4/5ths law for radial inhomogeneity is shown to be necessary for increasing separation of the structure functions but is less prominent with increasing Reynolds number.   

Large Scale Motions and Friction Scaling in Pipes

The scaling of the friction factor in pipe flow is exactly known for low Reynolds number laminar flow and reasonably well understood for high Reynolds number turbulence, where the Prandtl-Karman law is followed. Arguably, the least understood regime is the intermediate one, from $Re \approx 3000$, where turbulence becomes space filling, to $Re \approx 100 000$. As shown in our experiments and direct numerical simulations here friction factors fall precisely onto the Blasius power law. However, the structural and dynamical changes related to the scaling transition that occurs at $Re \approx 65000$ and eventually gives rise to the Prandtl-Karman regime, remain elusive. Here, combining experiments and simulations, we argue that the regime change from Blasius to Prandtl-Karman is driven by a structural transition within turbulence. Using tools from information theory (mutual information), it is shown that the change of the friction law is connected to the appearance of large scale motions in the logarithmic layer. Subsequent to the transition these large scale motions exert an increasing dominance on the turbulent drag.   

Evolution of secondary cross-flow in turbulent duct flow

In addition to the primary axial flow, fully developed turbulent flow in a square duct differs from turbulent pipe flow in that it exhibits mean secondary Nikuradse cross-flow. In order to study the appearance and evolution of this cross-flow, we have performed a series of RANS-RSM simulations of this duct flow for Reynolds number Re \textless 10,000. At a critical Reynolds number Re$_{\mathrm{c}}=$704, the flow becomes turbulent, with the pressure drop, dp/dx, discontinuously increasing and a weak cross-flow discontinuously developing. For the square duct geometry, the cross-flow consists of alternating Nikuradse vortices located in each octant, with flow directed diagonally towards the corners and away along the sides. Associated with each vortex, opposite vorticity shear develops at the walls. We quantify the cross-flow by the circulation around each octant. For Re \textgreater 2500, the bulk Nikuradse vortex contributes $_{\mathrm{\sim }}$ 9/5, and the wall vorticity $_{\mathrm{\sim }}$ - 4/5 to the circulation. The octant circulation increases sub-linearly with Re (exponent $_{\mathrm{\sim }}$ 0.85); similarly, dp/dx$_{\mathrm{\thinspace \sim }}$ Re$^{\mathrm{1.64}}$, consistent with pipe flow scaling. Near the transition, Re \textless 750, both octant circulation and dp/dx exhibit downward curvature, characteristic of a second-order transition; however, the transition appears to occur abruptly.   

Relaxation of turbulent pipe flow following an axisymmetric square bar roughness element

The relaxation of turbulent pipe flow following an axisymmetric square bar was studied. The upstream flow was fully developed with a bulk Reynolds number of 156,000. Three bar heights were investigated: $h/D$ = 0.02, 0.05, and 0.1 ($D$ is the pipe diameter). PIV data were collected at multiple downstream stations, with the farthest one at approximately 50$D$ downstream. It was found that the downstream pipe flow evolved in three phases. Immediately following the square bar is the development of the shear layer, where the turbulence intensity scales linearly with $Re_h$ (Reynolds number based on the bar height). The length of the first phase is a linear function of $h$, although the peak turbulence intensity occurs around $x/x_R = 1$ ($x$ is the downstream distance and $x_R$ is the flow reattachment length). The second phase features the convection of turbulence towards the pipe centerline, in which the triple correlation in the Reynolds stress transport equation plays a central role. Scaling laws in the second phase are given by ratios $h/D$ and $x_R/D$. The last phase is the long-lasting non-monotonic recovery to equilibrium, in which damped harmonic oscillation is observed. A predictive model derived from RANS equations are examined against the recovery behavior.   

Cause-and-effect of Linear Mechanisms in Wall Turbulence

A crucial element in closing the loop in the self-sustaining cycle in wall turbulence is the energy transfer from the large-scale mean flow to the turbulent fluctuations. There is consensus that this energy transfer is attributed to linear processes, but the mechanism by which this transfer occurs has been a subject of heated debates. Different scenarios stem from linear stability theory and comprise, among others, exponential instabilities, neutral modes, transient growth from non-normal operators, and parametric instabilities from temporal mean-flow variations. Here, we assess the role of the various linear mechanisms potentially responsible for the energy transfer from the streamwise-averaged mean-flow, $U(y, z, t)$, to the fluctuating velocities, $u'(x, y, z, t)$. We use cause-and-effect analysis based on interventions: manipulation of the causing variable leads to changes in the effect. Our main conclusion is that the dominant process responsible for this energy transfer is transient growth of fluctuations. Transient growth alone is able to sustain realistic wall turbulence. Furthermore, we demonstrate that the self-sustaining cycle of turbulence persists when either exponential instabilities, neutral modes, and parametric instabilities of the mean flow are suppressed.   

Resolvent-based estimation of turbulent channel flow

We employ a resolvent-based methodology to estimate time-domain velocity and pressure fluctuations within turbulent channel flows at friction Reynolds numbers of 550 and 1000 using only measurements of shear stress and pressure at the wall. The resolvent-based estimation method recovers fluctuations in the time domain by convolving the measurements with a transfer function formulated in terms of the resolvent operator obtained from the linearized Navier-Stokes equations as well as coloured statistics of the nonlinear terms, which are computed from DNS data. The estimation of buffer-layer structures is very accurate, with normalized correlation between the estimated flow and DNS fields higher than 0.95 for all variables. The accuracy is lower when log layer fluctuations are estimated from the wall; however, large-scale structures are still well estimated, and the normalized correlation between estimation and DNS is approximately 0.6 at $y^{+} = 200$. The energy spectra and variance of the filtered DNS and the estimated flow also exhibit good agreement. The use of coloured forcing statistics is crucial for obtaining accurate estimates; if white-noise forcing is considered, buffer-layer structures can still be estimated accurately, but the errors in the log layer become significant.   

Optimal Reynolds-stress Decomposition of the Velocity Field in a Channel Flow

Using a novel methodology, referred to as cross proper orthogonal decomposition (CPOD), cross-covariances of flow fluctuations (such as Reynolds stresses) are decomposed into modes that are optimal in representing an appropriately chosen inner product. This framework is applied to the representation of Reynolds shear stress in turbulent channel flow with friction Reynolds numbers of 179, 550, and 1000. Leading modes are shown to comprise streamwise vortices and streaks with phase opposition between streamwise ($u'$) and wall-normal ($v'$) velocities, representing ejections and sweeps, and higher-order modes show similar structures, but with $u'$ and $v'$ in phase. A combination of such structures leads to an accurate reconstruction of the Reynolds stress, and consequently of the mean flow, with a reasonable near-wall reconstruction with the leading CPOD mode pair (even and odd modes) for each considered wavenumber, and a close match of the profiles with the five leading CPOD mode pairs. For the Reynolds numbers of 550 and 1000, the leading modes for the wavenumber spectral peak present a self-similar behavior when scaled in outer units, highlighting the large-scale structures that determine the bulk of Reynolds stress.