Velocity Streaks Research Articles

Reynolds number effects in shock-wave/turbulent boundary-layer interactions

We investigate Reynolds number effects in strong shock-wave/turbulent boundary-layer interactions (STBLI) by leveraging a new database of wall-resolved and long-integrated large-eddy simulations. The database encompasses STBLI with massive boundary-layer separation at Mach $2.0$ , impinging-shock angle $40^{\circ }$ and friction Reynolds numbers ${\textit {Re}}_\tau$ $355$ , $1226$ and $5118$ . Our analysis shows that the shape of the reverse-flow bubble is notably different at low and high Reynolds number, while the mean-flow separation length, separation-shock angle and incipient plateau pressure are rather insensitive to Reynolds number variations. Velocity statistics reveal a shift in the peak location of the streamwise Reynolds stress from the separation-shock foot to the core of the detached shear layer at high Reynolds number, which we attribute to increased pressure transport in the separation-shock excursion domain. Additionally, in the high Reynolds case, the separation shock originates deep within the turbulent boundary, resulting in intensified wall-pressure fluctuations and spanwise variations associated with the passage of coherent velocity structures. Temporal spectra of various signals show energetic low-frequency content in all cases, along with a distinct peak in the bubble-volume spectra at a separation-length-based Strouhal number $St_{L_{sep}}\approx 0.1$ . The separation shock is also found to lag behind bubble-volume variations, consistent with the acoustic propagation time from reattachment to separation and a downstream mechanism driving the shock motion. Finally, dynamic mode decomposition of three-dimensional fields suggests a Reynolds-independent statistical link among separation-shock excursions, velocity streaks and large-scale vortices at low frequencies.

Turbulent spot formation in three-dimensional yukawa liquids using large-scale molecular dynamics simulation—effect of system size

We have performed classical ‘first principles’ 3D Molecular Dynamics (MD) simulation of plane-Couette flow (PCF) in a 3D Yukawa liquid. The main aim of the present investigation is to study the effect of stream-wise (flow-direction) and span-wise (transverse direction to the flow) width on the subcritical transition to turbulence in plane Couette flow (PCF), separately. In the past, taking very large-aspect ratio systems, subcritical transition to turbulence in PCF has been studied in great detail. However, the effect of stream-wise and span-wise width on the turbulent dynamics separately, have not been studied in detail. In the present work, we have investigated the effect of stream-wise and span-wise width or length and found that the stream-wise length enhances the large-scale energy, which gives rise to strong large-scale flow, and span-wise width enhances the small-scale energy, which gives rise to strong small-scale structures. In other words, topology of the turbulent spot is observed to change with the change in system size. The spectral separation between the modes governing the large and small-scale dynamics improves with the increase in system sizes. The connection between the stream-wise vortices and the stream-wise velocity streaks is investigated in details. We have found that the number stream-wise velocity streaks is one unit larger than the number of stream-wise vortices. A qualitative comparison between the results obtained from the MD simulation and the hydrodynamics experiment is presented in this article. The behavior of the system was observed to vary with the range of interactions among the dust grains.

Interaction of barn owl leading edge serrations with freestream turbulence

The silent flight of barn owls is associated with wing and feather specialisations. Three special features are known: a serrated leading edge that is formed by free-standing barb tips which appears as a comb-like structure, a soft dorsal surface, and a fringed trailing edge. We used a model of the leading edge comb with 3D-curved serrations that was designed based on 3D micro-scans of rows of barbs from selected barn-owl feathers. The interaction of the flow with the serrations was measured with Particle-Image-Velocimetry in a flow channel at uniform steady inflow and was compared to the situation of inflow with freestream turbulence, generated from the turbulent wake of a cylinder placed upstream. In steady uniform flow, the serrations caused regular velocity streaks and a flow turning effect. When vortices of different size impacted the serrations, the serrations reduced the flow fluctuations downstream in each case, exemplified by a decreased root-mean-square value of the fluctuations in the wake of the serrations. This attenuation effect was stronger for the spanwise velocity component, leading to an overall flow homogenization. Our findings suggest that the serrations of the barn owl provide a passive flow control leading to reduced leading-edge noise when flying in turbulent environments.

On the generation of near-wall dilatational motions in hypersonic turbulent boundary layers

Dilatational motions in the shape of travelling wave packets have been identified recently to be dynamically significant in hypersonic turbulent boundary layers. The present study investigates the mechanisms of their generation and their association with the solenoidal motions, especially the well-recognized near-wall self-sustaining process of the regeneration cycle between the velocity streaks and quasi-streamwise vortices. By exploiting the direct numerical simulation databases and orchestrating numerical experiments, we explore systematically the near-wall flow dynamics in the processes of the formation and transient growth of low-speed streaks. We conclude via theoretical ansatz that the nonlinearity related to the parallel density and pressure gradients close to the wall due to the restriction of the isothermal boundary condition is the primary cause of the generation of the dilatational structures at small scales. In fully developed turbulence, the formation and the existence of healthy dilatational travelling wave packets require the participation of the turbulence at scales larger than those of the near-wall regeneration cycles, especially the occurrence of the bursting events that generate vortex clusters. This is proven by the less intensified dilatational motions in the numerical experiments in which the Orr mechanism is alleviated and the vortical structures and turbulent bursts are weakened.

Turbulent–turbulent transient concept in pulsating flows

The turbulence behaviour of current-dominated pulsating flows has been investigated. Direct numerical simulations have been carried out for Stokes lengths over a range of $l_s^+=5\unicode{x2013}26$ , and amplitudes spanning 90 % of the current-dominated regime, about a mean flow of $\overline {Re}=6275$ . The results show that the turbulence response in intermediate and low-frequency pulsations is governed by a multistage turbulent–turbulent transition process, which bears a strong similarity to the multistage response of non-periodic acceleration. During the early acceleration period, the flow enters a pretransition stage, in which a new laminar perturbation boundary layer forms at the wall, and the streamwise velocity streaks are stretched. If the low-speed streaks destabilise prior to the deceleration period, then the flow enters a transition stage in which the perturbation boundary layer undergoes a bypass-like transition process. A unique feature of pulsating flows is the ongoing mechanism of turbulence decay, which initiates during the deceleration period and constitutes the main transient turbulence mechanism for much of the cycle. For high-frequency pulsations, the perturbation boundary layer fails to reach the pretransition stage prior to the deceleration period. Instead, the flow alternates between two inertial stages which are characterised by two layers of amplified viscous force; one growing at the wall, and one detached and moving towards the core.

Turbulent spot formation in stably stratified three-dimensional Yukawa liquids

A plane-Couette flow (PCF) is considered in a stably stratified three-dimensional (3D) Yukawa liquid, perturbed initially with a finite amplitude 3D perturbation. Stable stratification in density is achieved by subjecting the medium to external gravity. We demonstrate turbulent spot formation in a stably stratified PCF. The dynamics of the system is shown to depend upon the value of κ, which is associated with the range of interaction. We have performed “first principles” classical 3D molecular dynamics (MD) simulations for “hypergravity” (1.3g0, g0 is Earth's gravitational force for unit mass in our normalized unit) and “milligravity” (9×10−3g0) cases for κ, 1.0 and 4.0, respectively. We extract relevant fluid quantities from MD data. For the hypergravity case, when the system is evolved in time under stable stratification, the kinetic energy is observed to deposit in the lower wave vectors (Kx, Kz), leading to “inverse cascade” in the plane perpendicular to the direction of stratification. As a result, we observe large-scale structure formation in velocity and streamwise vorticity fields. Nucleation of velocity streak is observed for the first time in the stably stratified case in our simulation. The coherent structures in the velocity and streamwise vorticity fields are found to sustain for longer period of time for the stably stratified cases as compared to the unstratified case. For the milligravity case, the large-scale dynamics is observed to enhance. Unlike unstratified PCF, a background flow in Y-averaged streamwise fluid velocity field is observed for the stably stratified case. We believe that our results using “first principles” classical MD simulations on subcritical turbulence in stably stratified Yukawa liquid, may have ramifications to wider class turbulence problems. Published by the American Physical Society 2024

Flow over a single dimple recessed in a flat plate

Direct numerical simulations have been conducted to investigate a zero-pressure-gradient boundary layer flow over a single shallow dimple. Here, the dimple depth to dimple diameter ratio (d/D) as well as the Reynolds number (based on D and free-stream velocity) are fixed at 0.05 and 20 000, respectively. The effect of inlet boundary layer thickness δ on a given dimple is investigated by considering δ/D∈[0.023,0.1]. The flow within the dimple exhibits either a horseshoe vortex (a continuous core line through the two spirals within the dimple) or a tornado-like vortex pair (discontinuous core line). For the given parameter range, four different flow patterns have been identified within the single dimple: (i) a steady symmetric horseshoe vortex pattern for δ/D∈[0.053,0.1], (ii) a steady asymmetric horseshoe vortex pattern for δ/D=0.04, (iii) a quasi-periodic asymmetric horseshoe vortex pattern for δ/D=0.033, and (iv) a mixed horseshoe and tornado-like vortex pattern for δ/D=0.023. The growth of the streamwise vorticity, mainly caused by the tilting of the vertical vorticity, plays a key role in the transition between the different flow patterns. Dimple-induced velocity streaks above the single dimple have been investigated in detail for the first time, showing four different streaks: (i) a high-speed streak above the dimple, (ii) two side-low-speed streaks located outside the dimple span, (iii) two side-high-speed streaks, and (iv) a mid-low-speed streak in between them. These are mainly caused by a flow acceleration effect and a flow diffuser effect over the dimple, as well as a “lift-up” mechanism within the downstream part of the dimple, tilting the boundary layer upward.

Amplitude modulation in turbulent boundary layer over anisotropic porous wall

In this study, the amplitude modulation effect in a turbulent boundary layer over anisotropic porous walls is investigated experimentally at the Reynolds number based on friction velocity of Reτ = 236–319. The streamwise and wall-normal velocity fields were measured using time-resolved particle image velocimetry. To clarify the coherent structures related to the amplitude modulation over the porous wall with skin friction reduction effect, the large-scale structures are extracted from the low-pass filtered streamwise velocity fluctuations. The small-scale events related to high fluctuation energy are detected by the variable-interval space-averaging technique. Over the porous wall, the induced upwash and downwash motion leads to a notable suppression of large-scale structures. The small-scale motions in the near-wall region are mainly caused by the ejection events, while the sweep events are significantly suppressed. The amplitude modulation effects indicate that the positive and negative large-scale velocity streaks produce suppression and enhancement effects to the near-wall small-scale turbulence, respectively, which is contrary to the conventional phenomenon over the smooth wall case. The interaction between outer large-scale and inner small-scale structures is significantly weakened by the porous wall, contributing to the overall skin friction reduction.

Turbulent boundary layer over porous media with wall-normal permeability

Porous walls are a widely used passive flow control technique, which shows potential in reducing skin friction and mitigating flow-introduced noise. In the present study, porous media with wall-normal permeability is applied to a flat plate to investigate its interaction with the turbulent boundary layer at the Reynolds number based on friction velocity of Reτ=225. Time-resolved planar and tomographic particle image velocimetry were employed to identify the impact on mean statistics and coherent structures. An overall skin friction reduction of 22% is achieved. The porous wall induces counter-rotating streamwise vortex pairs at the spanwise sides of each pore, leading to momentum transport and the generation of alternative low- and high-speed regions close to the wall. Slip velocity is obtained, associated with the reduction in turbulent fluctuations and Reynolds shear stress. The streamwise velocity streaks and the hairpin vortices are significantly distorted compared with the smooth wall condition due to the downwash and upwash motion, featuring a notable reduction in the number and scale of the coherent structures, in which the skin friction reduction mechanism is related to. The proper orthogonal decomposition analysis returns the most energetic unsteady modes. Although the wall-coherent mode type remains to dominate the production of turbulent fluctuations, the scale and energy content of wall-incoherent modes increase, confirming the modification of the distribution and scale of near-wall turbulent structures.

Nonlinear optimal perturbation of turbulent channel flow as a precursor of extreme events

This work aims at studying the mechanisms behind the occurrence of extreme dissipation events in a channel flow, identifying nonlinear optimal perturbations as potential precursors of these events. Nonlinear optimal perturbations with respect to a generic turbulent instantaneous snapshot are computed for the first time using a direct-adjoint algorithm in the channel flow at $Re_{\tau }\approx 180$ . The resulting initial perturbation displays the upstream tilting characteristic of Orr's mechanism and is positioned along the interfaces between two opposite-sign velocity streaks of the pre-existing turbulent field. Such a perturbation induces a sudden breakdown of the pre-existing structures and a heavier tail in the dissipation probability density function distribution. Different mechanisms are at play during this process: the high shear present at the interface between coherent low- and high-momentum regions is exploited to break down the larger structures and drive energy to small scales. This energy cascade is fed by an enhanced lift-up effect that produces intense streaks near the wall. It is found that the optimal perturbation grows exponentially during the first phase of its evolution reflecting the existence of a secondary modal instability of the streaks. To corroborate the results, the conditional spatiotemporal proper orthogonal decomposition (POD) analysis of Hack & Schimdt (J. Fluid Mech., vol. 907, 2021, A9) is performed both in the perturbed and in the unperturbed flow, showing a clear agreement between the two cases and with the reference study. Thus, the optimal perturbation at initial time can be considered as a precursor of extreme events.

Linear response analysis of supersonic turbulent channel flows with a large parameter space

In this work, the linear responses of turbulent mean flow to both harmonic and stochastic forcing are investigated for supersonic channel flow. Well-established universal relations are utilized to obtain efficiently the mean profiles with a large parameter space, with the bulk Mach number up to 5 and the friction Reynolds number up to$10^4$, so a systematic parameter study is feasible. The most amplified structure takes the form of streamwise velocity and temperature streaks forced optimally by the streamwise vortices. The outer peak of the pre-multiplied energy amplification corresponds to the large-scale motion, whose spanwise wavelength ($\lambda _z^+$) is very insensitive to compressibility effects. In contrast, the classic inner peak representing small-scale near-wall motions disappears for the stochastic response with increasing Mach number. Meanwhile, the small-scale motions become much less coherent. A decomposition of the forcing identifies different effects of the incompressible counterpart and the thermodynamic components. Wall-cooling effects, arising with high Mach number, increase the spacing of the most amplified near-wall streaks; the spacing becomes nearly invariant with Mach number if expressed in semi-local units. Meanwhile, the coherence of stochastic response with$\lambda _z^+>90$is enhanced, but on the other hand, with$\lambda _z^+<90$it is decreased. The geometrical self-similarity of the response in the mid-$\lambda _z$range is still roughly satisfied, insensitive to Mach number. Finally, theoretical analyses of the perturbation equations are presented to help with understanding the scaling of energy amplification.

Resolvent-based analysis of hypersonic turbulent boundary layers with/without wall cooling

The ability of the low-rank approximation of hypersonic turbulent boundary layers with/without wall cooling is examined with the linear resolvent operator in a compressible form. The freestream Mach number of the base flow is 5.86, and the friction Reynolds number is 420. The wall-to-recovery temperature ratio is set as 1.0 and 0.25, respectively, corresponding to an adiabatic wall condition and a cold-wall condition. Different from the resolvent analysis of incompressible turbulent boundary layers, the optimal response mode in the wave-parameter space exhibits a relatively subsonic and a relatively supersonic region [Bae et al., “Resolvent-based study of compressibility effects on supersonic turbulent boundary layers,” J. Fluid Mech. 883, A29 (2020)], divided by the freestream relative Mach number of unity. The features of energy distribution of the optimal response mode in space and scales are examined, and the energy spectra of streamwise velocity and temperature fluctuations, carried by the optimal response mode, are discussed with typical subsets of streamwise and spanwise wavelengths. This reveals the dynamics of the near-wall small-scale and outer larger-scale motions and the distinction in the relatively subsonic/supersonic region. Moreover, the coherent structures, including the velocity and temperature streaks, quasi-streamwise vortices, and large-scale/very-large-scale motions, are identified in the optimal response mode. Special attention is paid to the effects of wall cooling.

Drag increase and turbulence augmentation in two-way coupled particle-laden wall-bounded flows

The exact regularized point particle method is used to characterize the turbulence modulation in two-way momentum-coupled direct numerical simulations of a turbulent pipe flow. The turbulence modification is parametrized by the particle Stokes number, the mass loading, and the particle-to-fluid density ratio. The data show that in the wide region of the parameter space addressed in the present paper, the overall friction drag is either increased or unaltered by the particles with respect to the uncoupled case. In the cases where the wall friction is enhanced, the fluid velocity fluctuations show a substantial modification in the viscous sub-layer and in the buffer layer. These effects are associated with a modified turbulent momentum flux toward the wall. The particles suppress the turbulent fluctuations in the buffer region and concurrently provide extra stress in the viscous sub-layer. The sum of the turbulent stress and the extra stress is larger than the Newtonian turbulent stress, thus explaining the drag increase. The non-trivial turbulence/particles interaction turns out in a clear alteration of the near-wall flow structures. The streamwise velocity streaks lose their spatial coherence when two-way coupling effects are predominant. This is associated with a shift of the streamwise vortices toward the center of the pipe and with the concurrent presence of small-scale and relatively more intense vortical structures near the wall.

Coherent organizational states in turbulent pipe flow at moderate Reynolds numbers

Turbulent pipe flow is still an essentially open area of research, boosted in the last two decades by considerable progress achieved on both the experimental and numerical frontiers, mainly related to the identification and characterization of coherent structures as basic building blocks of turbulence. It has been a challenging task, however, to detect and visualize these coherent states. We address, by means of stereoscopic particle image velocimetry, that issue with the help of a large diameter (6 in.) pipe loop, which allowed us to probe for coherent states at various moderate Reynolds numbers (5300 < Re < 29 000) of the single-phase Newtonian flow. Although these states have been observed at flow regimes around laminar–turbulent transition (Re ≈ 2300) and also at high Reynolds number pipe flow (Re ≈ 35 000), at moderate Reynolds numbers, their existence had not been observed yet by experiment. By conditionally averaging the flow fields with respect to their dominant azimuthal wavenumber of streamwise velocity streaks, we have been able to uncover the existence of ten well-defined coherent flow patterns. It turns out, as a remarkable phenomenon, that their occurrence probabilities and the total number of dominant modes do not essentially change as the Reynolds number is varied. Their occurrence probabilities are noted to be reasonably well described by a Poisson distribution, which suggests that low-speed streaks are created as a Poisson process on the pipe circular geometry.

DNS of heat transfer in a plane channel flow with spatial transition

In the present work Direct Numerical Simulation of heat transfer in a spatial laminar-turbulent transitional channel flow is carried out to investigate the effects of the transitional phenomenon in the temperature field. The configuration is a spatially developing channel flow with two thermal boundary conditions: Uniform Wall Temperature (UWT) and Uniform Heat Flux (UHF). The set of simulations are carried out for Re=4200 and Pr=0.1,0.2,0.5,0.71 and 1. The spatial evolution of the Nusselt number (Nu) along the transition is computed to show five distinctive qualitative zones: a laminar development, the spike zone, where the quantities evolve to reach the peak transitional zone, the post-transitional zone and finally, the fully turbulent zone. This data is employed to develop a correlation that captures the space evolution of Nu along the transition. Moreover, the analysis of flow structures shows that near the wall and downstream the onset of the transition, streaks of velocity and temperature fluctuations have strong similarity. Different turbulent statistics are calculated along the transition and compared with fully turbulent reference data. It is shown that upstream the peak transitional zone, the existence of coherent hairpin vortices yields a high-intensity region for these quantities. Additionally, it is found that the thermal boundary condition exerts influence on the profile of the second-order parameters near the wall and in the logarithmic region all along the transition.

Streamwise Extent and Ramp Rate Effects on Laminar Boundary Layer Response to Plasma Actuator Vortex Generators

AbstractThe unsteady output of a spanwise array of plasma actuators, arranged to generate streaks of high and low streamwise velocity in a Blasius boundary layer, is modeled numerically. When the forcing by the actuators undergoes a step change from off to on, the flow response downstream of the actuators is initially inverted in terms of the streamwise vorticity, disturbance velocity, and wall shear stress before approaching the steady-state behavior. The present study considers the effect of the length of the actuator in the streamwise direction and the effect of the rate change of the output of the actuator with respect to the nonminimum phase behavior. For the different rates of a gradually applied force, the body force increases linearly from zero to the same maximum value of the step, which simulates a reduced slew rate or ramped output. It is shown that the inverse flow response remains for the gradually applied input; however, the peak magnitude is less, and the overall response appears more damped. As the actuator length is reduced, while forcing was compensated for the shorter convective time over the actuator, the peak inverse response was enhanced.

Influence of non-uniform thermal boundary on flow and heat transfer characteristics in rectangular channel

In this paper, the flow behavior and heat transfer characteristic in a rectangular channel are numerically investigated. The non-uniform thermal boundary condition is arranged along the streamwise direction at the bottom of the rectangular channel. Furthermore, based on the flow field parameters obtained with numerical simulation, the dynamic modal decomposition (DMD) is carried out for viscous layer, buffer layer, and logarithmic region, respectively. The numerical results show that the hot bands of non-uniform thermal boundary affect the interaction of the velocity streaks along the streamwise direction, which reduces the vorticity of the buffer layer and the fluctuation of the velocity gradient vector. In the terms of entropy analysis, it can be found that the hot bands of non-uniform thermal boundary play a similar role of “riblets” and block the self-sustainment of the turbulent coherent structures. Moreover, the results of DMD manifest that the hot bands of non-uniform thermal boundary can improve the stability of viscous layer and buffer layer. The development of turbulent boundary layer is delayed by affecting the fluid characteristics in buffer layer. Compared to the channel without non-uniform thermal boundary condition, the maximum drag reduction rate of 8.35% can be achieved in considered cases, while a reduction in heat transfer performance of 2.74% occurs. In addition, the comprehensive performance coefficient increases slightly to 1.0013.

Wall-attached temperature structures in supersonic turbulent boundary layers

It is well known that low- and high-speed velocity streaks are statistically asymmetric. However, it is unclear how different the low- and high-temperature structures (T-structures) are even though they are strongly coupled with the streamwise velocity. Therefore, this paper identifies three-dimensional wall-attached temperature structures in supersonic turbulent boundary layers over cooled and heated walls (coming from direct numerical simulations) and separates them into positive and negative families. Wall-attached T-structures are self-similar; especially, the length and width of the positive family are linear functions of the height. The superposed temperature variance in both positive and negative families exhibits a logarithmic decay with the wall distance, while the superposed intensity of the wall-normal heat flux in the negative family shows a logarithmic growth. The modified strong Reynolds analogy proposed by Huang, Coleman, and Bradshaw [“Compressible turbulent channel flows: DNS results and modelling,” J. Fluid Mech. 305, 185–218 (1995)] is still valid in the negative family. The relative position between T-structures of opposite signs depends on the wall temperature and that in the cooled-wall case differs significantly from the relative position between low- and high-speed streaks, especially those tall ones. In the cooled-wall case, although positive temperature fluctuations below and above the maximum of the mean temperature can cluster to large-scale wall-attached structures, they are very likely dynamically unrelated.

Mechanistic study on modification of convective internal boundary layer by spanwise surface motion

Modification of a convective internal boundary layer (IBL) by spanwise motion of a warm surface is investigated by imposing different surface moving speeds in the present study. Our analysis shows that the spanwise surface motion reduces the Reynolds shear stress right after the increase in the surface temperature in the convective IBL. The maximum decreasing rate of the Reynolds shear stress is found to be approximately 75% at the largest moving speed of the warm surface considered in the manuscript. Due to the reduction of the Reynolds shear stress, the vertical momentum transport is fundamentally altered, and the mean flow accelerates immediately after the increase in the surface temperature. By scrutinizing the instantaneous and conditional averaged flow fields as well as the pre-multiplied energy spectra, we have attributed the reduction of the Reynolds shear stress to the suppression of the near-surface velocity streaks and quasi-streamwise vortices, and the delayed growth of the convective structures, such as thermal plumes. Our investigation suggests that the developments of the convective IBL can be influenced by a strong spanwise motion of the warm surface, which should be taken into consideration in the prediction model for practical applications.

A spectral inspection for turbulence amplification in oblique shock wave/turbulent boundary layer interaction

Turbulence amplification and the large-scale coherent structures in shock wave/turbulent boundary layer interaction flows have been studied at length in previous research, while the direct association between these two flow features is still lacking. In the present study, the transport equation of turbulent kinetic energy spectra is derived and utilized to analyse the scale-by-scale energy budget across the interaction zone, enabling us to reveal the association between the genesis of the large-scale motions and the turbulence amplification. For the presently considered flow with incipient shock-induced separation, we identified in turbulent kinetic energy spectra distribution that the most energetic motions are converted from the near-wall small-scale motions to large-scale motions consisting of velocity streaks and cross-stream circulations as they go through the interaction zone. The amplification of streamwise velocity fluctuation is triggered first, resulting in the emergence of large-scale velocity streaks, which is attributed to the adverse pressure gradient, as indicated by the spectra of the production term. The energy carried by large-scale velocity streaks is transferred to other velocity components by the pressure-strain term, producing large-scale cross-stream circulations. When large-scale motions are convected downstream, their energy is transferred via turbulent cascade to smaller scales and dissipated by viscosity. The spanwise uniform fluctuations, reminiscent of the unsteadiness of the separation bubble, are contributed primarily by the inter-scale energy transfer from the finite spanwise scale motions.