## Publications

44 results found

Hwang Y, Myoungkyu L, 2020, The mean logarithm emerges with self-similar energetics, *Journal of Fluid Mechanics*, Vol: 903, Pages: R6-1-R6-11, ISSN: 0022-1120

The attached eddy hypothesis of Townsend (The structure of turbulent shear flow, 1956,Cambridge U. Press) states that the logarithmic mean velocity would admit self-similarenergy-containing eddies which scales with the distance from the wall. Over the pastdecade, there has been significant amount of evidence supporting the hypothesis, placingit to be the central platform for the statistical description of the general organisationof coherent structures in wall-bounded turbulent shear flows. Nevertheless, the mostfundamental question, namely why the hypothesis has to be true, remains unansweredover many decades. Under the assumption that the integral length scale is proportionalto the distance from the wally, in the present study, we analytically demonstrate thatthe mean velocity is a logarithmic function ofyif and only if the energy balance atintegral length scale is self-similar with respect toy, providing a theoretical groundfor the attached eddy hypothesis. The analysis is subsequently verified with the datafrom direct numerical simulation of incompressible channel flow at the friction ReynoldsnumberReτ'5200 (Lee & Moser,J. Fluid Mech., vol. 774, 2015, 395–415).

Fung L, Bearon R, Hwang Y, 2020, Bifurcation and stability of downflowing gyrotactic micro-organism suspensions in a vertical pipe, *Journal of Fluid Mechanics*, Vol: 902, Pages: 1-34, ISSN: 0022-1120

In the experiment that first demonstrated gyrotactic behaviour of bottom-heavy swimming microalgae (e.g. Chlamydomonas), Kessler (Nature, vol. 313, 1985, pp. 218-220) showed that a beam-like structure, often referred to as a gyrotactic plume, would spontaneously appear from a suspension of gyrotactic swimmers in a downflowing pipe. Such a plume is prone to an instability to form blips. This work models the gyrotactic plume as a steady parallel basic state and its subsequent breakdown into blips as an instability, employing both the Generalised Taylor Dispersion (GTD) theory and the Fokker-Planck model for comparison. Upon solving for the basic state, it is discovered that the steady plume solution undergoes sophisticated bifurcations. When there is no net flow, there exists a non-trivial solution of the plume structure other than the stationary uniform suspension, stemming from a transcritical bifurcation with the average cell concentration. When a net downflow is prescribed, there exists a cusp bifurcation. Furthermore, there is a critical concentration, at which the cell concentration at the centre would blow up for the GTD model. The subsequent stability analysis using the steady plume solution shows that the Fokker-Planck model is inconsistent with what was experimentally observed, as it predicts stabilisation of axisymmetric blips at high concentration of the plume and destabilisation of the first non-axisymmetric mode at low flow rates.

Fung L, Hwang Y, 2020, A sequence of transcritical bifurcations in a suspension of gyrotactic microswimmers in vertical pipe, *Journal of Fluid Mechanics*, Vol: 902, Pages: R2-1-R2-11, ISSN: 0022-1120

Kessler (Nature, vol. 313, 1985, pp. 218–220) first showed that plume-like structures spontaneously appear from both stationary and flowing suspensions of gyrotactic microswimmers in a vertical pipe. Recently, it has been shown that there exist multiple steady, axisymmetric and axially uniform solutions to such a system (Bees & Croze, Proc. R. Soc. A, vol. 466, 2010, pp. 2057–2077). In the present study, we generalise this finding by reporting that a countably infinite number of such solutions emerge as the Richardson number increases. Linear stability, weakly nonlinear and fully nonlinear analyses are performed, revealing that each of the solutions arises from the destabilisation of a uniform suspension. The countability of the solutions is due to the finite flow domain, while the transcritical nature of the bifurcation is because of the cylindrical geometry, which breaks the horizontal symmetry of the system. It is further shown that there exists a maximum threshold of achievable downward flow rate for each solution if the flow is to remain steady, as varying the pressure gradient can no longer increase the flow rate from the solution. All of the solutions found are unstable, except for the one arising at the lowest Richardson number, implying that they would play a role in the transient dynamics in the route from a uniform suspension to the fully developed gyrotactic pattern.

Hernandez C, Hwang Y, Spectral energetics of a quasilinear approximation in uniform shear turbulence, *Journal of Fluid Mechanics*, ISSN: 0022-1120

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 meanare 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 cascadeis 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 wave number 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 Lx/Lz= 1∼3, consistent with previous findings (Lx and Lz are the streamwise and spanwise computational domains, respectively). Finally, the QL model is shown to completely ignore the role of slow pressure in the fluctuations, resulting in a significant damage of pressure-strain transport at all length scales. This explains the anisotropic turbulence of the QL model throughout the entire wavenumber space as well as the inhibited nonlinear regeneration of streamwise vortices in the self-sustaining process.

Hwang Y, Eckhardt B, 2020, Attached eddy model revisited using a minimal quasi-linear approximation, *Journal of Fluid Mechanics*, Vol: 894, ISSN: 0022-1120

Townsend’s model of attached eddies for boundary layers is revisited within a quasi-linear approximation. The velocity field is decomposed into a mean profile and fluctuations. While the mean is obtained from the nonlinear equations, the fluctuations are modelled by replacing the nonlinear self-interaction terms with an eddy-viscosity-based turbulent diffusion and stochastic forcing. Under this particular approximation, the resulting fluctuation equations remain linear, enabling solutions to be superposed, the same theoretical idea used in the original attached eddy model. By leveraging this feature, the stochastic forcing is determined self-consistently by solving an optimisation problem which minimises the difference between the Reynolds shear stresses from the mean and fluctuation equations, subject to a constraint that the averaged Reynolds shear-stress spectrum is sufficiently smooth in the spatial wavenumber space. The proposed quasi-linear approximation is subsequently applied to channel flow for Reynolds number Re𝜏 ranging from 500 to 20 000. The best result is obtained when the Reynolds stress is calculated by retaining only the two leading proper orthogonal decomposition modes, which further filters out the modelling artefact caused by the unphysical stochastic forcing. In this case, the resulting turbulence intensity profile and energy spectra exhibit the same qualitative behaviour as direct numerical simulation (DNS) data throughout the entire wall-normal domain, while reproducing the early theoretical predictions of the original attached eddy model within a controlled approximation to the Navier–Stokes equations. Finally, the proposed quasi-linear approximation reveals that the peak streamwise and spanwise turbulence intensities may deviate slightly from the logarithmic scaling with the Reynolds number for Re𝜏 ≳ 10 000, and the supporting evidence is presented using the existing DNS data.

Fung L, Hwang Y, 2020, Linear instability of tilted parallel shear flow in a strongly stratified and viscous medium, *JMST Advances*, Vol: 2, Pages: 37-51, ISSN: 2524-7905

A linear stability analysis is performed on a tilted parallel wake in a strongly stratified fluid at low Reynolds numbers. A particular emphasis of the present study is given to the understanding of the low-Froude-number mode observed by the recent experiment (Meunier in J Fluid Mech 699:174–197, 2012). In the limit of low Froude number, the linearised equations of motion can be reduced to the Orr–Sommerfeld equation on the horizontal plane, except the viscous term that contains vertical dissipation. Based on this equation, it is proposed that the low-Froude-number mode would be a horizontal inflectional instability and should remain two-dimensional at small tilting angles as long as the Reynolds number is sufficiently low. To support this claim, the asymptotic regime where this analysis is strictly valid is subsequently discussed in relation to previous work on the proper vertical length scale. The absolute and convective instability analysis of parallel wake is further performed, showing qualitatively good agreement with the experimental result. The low-Froude-number mode is found to be stabilised on increasing Froude number, as in the experiment. It is shown that the emergence of small vertical velocity at finite Froude number, the size of which is proportional to the square of Froude number, plays the key role in the stabilisation by modifying the inflectional instability and paradoxically creating stabilising buoyancy effect with the increase of Froude number.

Lakshmi MV, Fantuzzi G, Fernández-Caballero JD,
et al., 2020, Finding extremal periodic orbits with polynomial optimization, with application to a nine-mode model of shear ﬂow, *SIAM Journal on Applied Dynamical Systems*, Vol: 19, Pages: 763-787, ISSN: 1536-0040

Tobasco et al. [Phys. Lett. A, 382:382–386, 2018] recently suggested that trajectories of ODE systems that optimize the inﬁnite-time average of a certain observable can be localized using sublevel sets of a function that arise when bounding such averages using so-called auxiliary functions. In this paper we demonstrate that this idea is viable and allows for the computation of extremal unstable periodic orbits (UPOs) for polynomial ODE systems. First, we prove that polynomial optimization is guaranteed to produce auxiliary functions that yield near-sharp bounds on time averages, which is required in order to localize the extremal orbit accurately. Second, we show that points inside the relevant sublevel sets can be computed eﬃciently through direct nonlinear optimization. Such points provide good initial conditions for UPO computations. As a proof of concept, we then combine these methods with a single-shooting Netwon–Raphson algorithm to study extremal UPOs for a nine-dimensional model of sinusoidally forced shear ﬂow. We discover three previously unknown families of UPOs, one of which simultaneously minimizes the mean energy dissipation rate and maximizes the mean perturbation energy relative to the laminar state for Reynolds numbers approximately between 81.24 and 125.

Fung L, Hwang Y, Instability of tilted shear flow in a strongly stratified and viscous medium, *Journal of Mechanical Science and Technology*, ISSN: 1011-8861

A linear stability analysis is performed on a tilted parallel wake in astrongly stratified fluid at low Reynolds numbers. A particular emphasis of thepresent study is given to the understanding of the low-Froude-number modeobserved by the recent experiment (Meunier, J. Fluid Mech., vol. 699, 2012, pp.174-197). In the limit of low Froude number, the linearised equations of motioncan be reduced to the Orr-Sommerfeld equation on the horizontal plane, exceptthe viscous term that contains vertical dissipation. Based on this equation, itis proposed that the low-Froude-number mode would be a horizontal inflectionalinstability, and should remain two dimensional at small tilting angles. Tosupport this claim, the asymptotic regime where this equation would be strictlyvalid is subsequently discussed in relation to previous arguments on the propervertical length scale. Furthermore, the absolute and convective instabilityanalysis of parallel wake is performed, revealing qualitatively good agreementwith the experimental result. The low-Froude-number mode is found to bestabilised on increasing Froude number, as is in the experiment. The emergenceof small vertical velocity at finite Froude number, the size of which isproportional to the square of Froude number, plays the key role in thestabilisation. It modifies the inflectional instability and is also associatedwith the paradoxically stabilising buoyancy on increasing Froude number.Lastly, we proposed some possible behaviours of the base flow when the tiltingangle changes, and they may provide a better approximation to produce thebehaviour consistent with the experiment.

Hwang Y, 2019, Mesolayer of attached eddies in turbulent channel flow, *Physical Review Fluids*, Vol: 1, ISSN: 2469-990X

Recent experimental measurements have reported that the outer peak of the streamwisewavenumber spectra of the streamwise velocity depends on the Reynolds number. Starting fromthis puzzling observation, here it is proposed that the wall-parallel velocity components of eachof the energy-containing motions in the form of Towsnend’s attached eddies exhibit inner-scalingnature in the region close to the wall. Some compelling evidence on this proposition has been presentedwith a careful inspection of scaling of velocity spectra from DNS, a linear analysis with aneddy viscosity, and the recently computed statistical structure of the self-similar energy-containingmotions in the logarithmic region. This observation suggests that the viscous wall effect wouldnot be negligible at least below the peak wall-normal location of each of the energy-containingmotions in the logarithmic and outer regions, reminiscent of the concept of the ‘mesolayer’ previouslyobserved in the mean momentum balance. It is shown that this behavior emerges due to aminimal form of scale interaction, modeled by the eddy viscosity in the linear theory, and enablesone to explain the Reynolds-number-dependent behavior of the outer peak as well as the near-wallpenetration of the large-scale outer structures in a consistent manner. Incorporation of this viscouswall effect to Townsend’s attached eddies, which were originally built with an inviscid approximationat the wall, also reveals that the self-similarity of the wall-parallel velocity components of theenergy-containing motions would be theoretically broken in the region close to the wall.

Doohan P, Willis A, Hwang Y, 2019, Shear stress-driven flow: the state space of near-wall turbulence as Reτ →∞, *Journal of Fluid Mechanics*, Vol: 874, Pages: 606-638, ISSN: 0022-1120

An inner-scaled, shear stress-driven flow is considered as a model of independentnear-wall turbulence as Reτ → ∞. In this limit, the model is applicable to the nearwall region and the lower part of the logarithmic layer of various parallel shear flows,including turbulent Couette flow, Poiseuille flow and Hagen-Poiseuille flow. The modelis validated against damped Couette flow and there is excellent agreement between thevelocity statistics and spectra for y+ < 40. A near-wall flow domain of similar size tothe minimal unit is analysed from a dynamical systems perspective. The edge and fifteeninvariant solutions are computed, the first discovered for this flow configuration. Throughcontinuation in the spanwise width L+z, the bifurcation behaviour of the solutions overthe domain size is investigated. The physical properties of the solutions are exploredthrough phase portraits, including the energy input and dissipation plane, and streak,roll and wave energy space. Finally, a Reynolds number is defined in outer units and thehigh-Re asymptotic behaviour of the equilibria is studied. Three lower branch solutionsare found to scale consistently with vortex-wave interaction (VWI) theory, with waveforcing localising around the critical layer.

Smitha M, Keaveny E, Hwang Y, The instability of gyrotactically-trapped cell layers, *Journal of Fluid Mechanics*, ISSN: 0022-1120

Several meters below the coastal ocean surface there are areas of high ecological activitythat contain thin layers of concentrated motile phytoplankton. Gyrotactic trapping hasbeen proposed as a potential mechanism for layer formation of bottom-heavy swimmingalgae cells, especially in flows where the vorticity varies linearly with depth (Durham,Stocker & Kessler,Science, vol. 323, 2009, pp. 1067-1070). Using a continuum modelfor dilute microswimmer suspensions, we report that an instability of a gyrotacticallytrapped cell-layer can arise in a pressure-driven plane channel flow. The linear stabilityanalysis reveals that the equilibrium cell-layer solution is hydrodynamically unstabledue to negative microswimmer buoyancy (i.e. a gravitational instability) over a range ofbiologically relevant parameter values. The critical cell concentration for this instabilityis found to beNc'104cells/cm3, a value comparable to the typical maximum cellconcentration observed in thin layers. This result indicates that the instability may bea potential mechanism for limiting the layer’s maximum cell concentration, especiallyin regions where turbulence is weak, and motivates the study of its nonlinear evolution,perhaps, in the presence of turbulence.

Yang Q, Willis A, Hwang Y, 2019, Exact coherent states of attached eddies in channel flow, *Journal of Fluid Mechanics*, Vol: 862, Pages: 1029-1059, ISSN: 0022-1120

A new set of exact coherent states in the form of a travelling wave is reported in plane channel flow. They are continued over a range in Re from approximately 2600 up to 30 000, an order of magnitude higher than those discovered in the transitional regime. This particular type of exact coherent states is found to be gradually more localised in the near-wall region on increasing the Reynolds number. As larger spanwise sizes L + z are considered, these exact coherent states appear via a saddle-node bifurcation with a spanwise size of L + z ' 50 and their phase speed is found to be c + ' 11 at all the Reynolds numbers considered. Computation of the eigenspectra shows that the time scale of the exact coherent states is given by h/Ucl in channel flow at all Reynolds numbers, and it becomes equivalent to the viscous inner time scale for the exact coherent states in the limit of Re → ∞. The exact coherent states at several different spanwise sizes are further continued to a higher Reynolds number, Re = 55 000, using the eddy-viscosity approach (Hwang & Cossu, Phys. Rev. Lett., vol. 105, 2010, 044505). It is found that the continued exact coherent states at different sizes are self-similar at the given Reynolds number. These observations suggest that, on increasing Reynolds number, new sets of self-sustaining coherent structures are born in the near-wall region. Near this onset, these structures scale in inner units, forming the near-wall self-sustaining structures. With further increase of Reynolds number, the structures that emerged at lower Reynolds numbers subsequently evolve into the self-sustaining structures in the logarithmic region at different length scales, forming a hierarchy of self-similar coherent structures as hypothesised by Townsend (i.e. attached eddy hypothesis). Finally, the energetics of turbulent flow is discussed for a consistent extension of these dynamical systems notions to high Reynolds numbers.

Ibrahim J, Yang Q, Doohan P,
et al., 2019, Phase-space dynamics of opposition control in wall-bounded turbulent flows, *Journal of Fluid Mechanics*, Vol: 861, Pages: 29-54, ISSN: 0022-1120

We investigate the nonlinear phase-space dynamics of plane Couette flow and plane Poiseuille flow under the action of opposition control at low Reynolds numbers in domains close to the minimal unit. In Couette flow, the effect of the control is analysed by focussing on a pair of non-trivial equilibrium solutions. It is found that the control only slightly modifies the statistics, turbulent skin friction and phase-space projection of the lower-branch equilibrium solution, which, in this case, is in fact identical to the edge state. On the other hand, the upper-branch equilibrium solution and mean turbulent state are modified considerably when the control is applied. In phase space, they gradually approach the lower-branch equilibrium solution on increasing the control amplitude, and this results in an elevation of the critical Reynolds number at which the equilibrium solutions first occur via a saddle-node bifurcation. It is also found that the upper-branch equilibrium solution is stabilised by the control. In Poiseuille flow, we study an unstable periodic orbit on the edge state and find that it, too, is modified very little by opposition control. We again observe that the turbulent state gradually approaches the edge state in phase space as the control amplitude is increased. In both flows, we find that the control significantly reduces the fluctuating strength of the turbulent state in phase space. However, the reduced distance between the turbulent trajectory and the edge state yields a significant reduction in turbulence lifetimes for both Couette and Poiseuille flow. This demonstrates that opposition control greatly increases the probability of the trajectory escaping from the turbulent state, which takes the form of a chaotic saddle.

Pausch M, Yang Q, Hwang Y,
et al., 2019, Quasilinear approximation for exact coherent states in parallel shearflows, *Fluid Dynamics Research*, Vol: 51, ISSN: 0169-5983

In the quasilinear approximation to the Navier–Stokes equation a minimal set of nonlinearities that is able to maintain turbulent dynamics is kept. For transitional Reynolds numbers, exact coherent structures provide an opportunity for a detailed comparison between full direct numerical solutions of the Navier–Stokes equation with their quasilinear approximation. We show here, for both plane Couette flow and plane Poiseuille flow, that the quasilinear approximation is able to reproduce many properties of exact coherent structures. For higher Reynolds numbers differences in the stability properties and the friction values for the upper branch appear that are connected with a reduction in the number of downstream wavenumbers in the quasilinear approximation. The results show the strengths and limitations of the quasilinear approximation and suggest modelling approaches for turbulent flows.

Maretvadakethope S, Keaveny EE, Hwang Y, 2019, The instability of gyrotactically trapped cell layers

© 2019 Cambridge University Press. Several metres below the coastal ocean surface there are areas of high ecological activity that contain thin layers of concentrated motile phytoplankton. Gyrotactic trapping has been proposed as a potential mechanism for layer formation of bottom-heavy swimming algae cells, especially in flows where the vorticity varies linearly with depth (Durham et al.Science, vol. 323(5917), 2009, pp. 1067-1070). Using a continuum model for dilute microswimmer suspensions, we report that an instability of a gyrotactically trapped cell layer can arise in a pressure-driven plane channel flow. The linear stability analysis reveals that the equilibrium cell-layer solution is hydrodynamically unstable due to negative microswimmer buoyancy (i.e. a gravitational instability) over a range of biologically relevant parameter values. The critical cell concentration for this instability is found to be, a value comparable to the typical maximum cell concentration observed in thin layers. This result indicates that the instability may be a potential mechanism for limiting the layer's maximum cell concentration, especially in regions where turbulence is weak, and motivates the study of its nonlinear evolution, perhaps, in the presence of turbulence.

Dunstan J, Lee K, Hwang Y,
et al., 2018, Evaporation-driven convective flows in suspensions of non-motile bacteria, *Physical Review Fluids*, Vol: 13, ISSN: 2469-990X

We report a novel form of convection in suspensions of the bioluminescent marine bacterium Photobacterium phosphoreum. Suspensions of these bacteria placed in a chamber open to the air create persistent luminescent plumes most easily visible when observed in the dark. These flows are strikingly similar to the classical bioconvection pattern of aerotactic swimming bacteria, which create an unstable stratification by swimming upwards to an air-water interface, but they are a puzzle since the strain of P. phosphoreum used does not express flagella and therefore cannot swim. When microspheres were used instead of bacteria, similar flow patterns were observed, suggesting that the convective motion was not driven by bacteria but instead by the accumulation of salt at the air-water interface due to evaporation of the culture medium. Even at room temperature and humidity, and physiologically relevant salt concentrations, the water evaporation was found to be sufficient to drive convection patterns. To prove this hypothesis, experiments were complemented with a mathematical model that aimed to understand the mechanism of plume formation and the role of salt in triggering the instability. The simplified system of evaporating salty water was first studied using linear stability analysis, and then with finite element simulations. A comparison between these three approaches is presented. While evaporation-driven convection has not been discussed extensively in the context of biological systems, these results suggest that the phenomenon may be broadly relevant, particularly in those systems involving microorganisms of limited motility.

Cho M, Hwang Y, Choi H, 2018, Scale interactions and spectral energy transfer in turbulent channel flow, *Journal of Fluid Mechanics*, Vol: 854, Pages: 474-504, ISSN: 0022-1120

Spectral energy transfer in a turbulent channel flow is investigated at Reynolds number Re ≃1700 , based on the wall shear velocity and channel half-height, with a particular emphasis on full visualization of triadic wave interactions involved in turbulent transport. As in previous studies, turbulent production is found to be almost uniform, especially over the logarithmic region, and the related spanwise integral length scale is approximately proportional to the distance from the wall. In the logarithmic and outer regions, the energy balance at the integral length scales is mainly formed between production and nonlinear turbulent transport, the latter of which plays the central role in the energy cascade down to the Kolmogorov microscale. While confirming the classical role of the turbulent transport, the triadic wave interaction analysis unveils two new types of scale interaction processes, highly active in the near-wall and the lower logarithmic regions. First, for relatively small energy-containing motions, part of the energy transfer mechanisms from the integral to the adjacent small length scale in the energy cascade is found to be provided by the interactions between larger energy-containing motions. It is subsequently shown that this is related to involvement of large energy-containing motions in skin-friction generation. Second, there exists a non-negligible amount of energy transfer from small to large integral scales in the process of downward energy transfer to the near-wall region. This type of scale interaction is predominant only for the streamwise and spanwise velocity components, and it plays a central role in the formation of the wall-reaching inactive part of large energy-containing motions. A further analysis reveals that this type of scale interaction leads the wall-reaching inactive part to scale in the inner units, consistent with the recent observation. Finally, it is proposed that turbulence production and pressure–strain spectra supp

Yang Q, Willis AP, Hwang Y, 2017, Energy production and self-sustained turbulence at the Kolmogorov scale inCouette flow, *Journal of Fluid Mechanics*, Vol: 834, Pages: 531-554, ISSN: 0022-1120

Several recent studies have reported that there exists a self-similar form of invariant solutions down to the Kolmogorov microscale in the bulk region of turbulent Couette flow. While their role in a fully developed turbulent flow is yet to be identified, here we report that there exists a related mechanism of turbulence production at the Kolmogorov microscale in the bulk region of turbulent Couette flow by performing a set of minimal-span direct numerical simulations up to friction Reynolds number . This mechanism is found to essentially originate from the non-zero mean shear in the bulk region of the Couette flow, and involves eddy turn-over dynamics remarkably similar to the so-called self-sustaining process (SSP) and/or vortex–wave interaction (VWI). A numerical experiment that removes all other motions except in the core region is also performed, which demonstrates that the eddies at a given wall-normal location in the bulk region are sustained in the absence of other motions at different wall-normal locations. It is proposed that the self-sustaining eddies at the Kolmogorov microscale correspond to those in uniform shear turbulence at transitional Reynolds numbers, and a quantitative comparison between the eddies in uniform shear and near-wall turbulence is subsequently made. Finally, it is shown that turbulence production by the self-sustaining eddies at the Kolmogorov microscale is much smaller than that of full-scale simulations, and that the difference between the two increases with Reynolds number.

de Giovanetti M, Sung HJ, Hwang Y, 2017, Streak instability in turbulent channel flow: the seeding mechanism of large-scale motions, *Journal of Fluid Mechanics*, Vol: 832, Pages: 483-513, ISSN: 0022-1120

It has often been proposed that the formation of large-scale motion (or bulges) is aconsequence of successive mergers and/or growth of near-wall hairpin vortices. In thepresent study, we report our direct observation that large-scale motion is generated byan instability of an ‘amplified’ streaky motion in the outer region (i.e. very-large-scalemotion). We design a numerical experiment in turbulent channel flow up to Reτ '2000where a streamwise-uniform streaky motion is artificially driven by body forcing inthe outer region computed from the previous linear theory (Hwang & Cossu, J. FluidMech., vol. 664, 2015, pp. 51–73). As the forcing amplitude is increased, it is foundthat an energetic streamwise vortical structure emerges at a streamwise wavelength ofλx/h '1–5 (h is the half-height of the channel). The application of dynamic modedecomposition and the examination of turbulence statistics reveal that this structureis a consequence of the sinuous-mode instability of the streak, a subprocess of theself-sustaining mechanism of the large-scale outer structures. It is also found that thestatistical features of the vortical structure are remarkably similar to those of the largescalemotion in the outer region. Finally, it is proposed that the largest streamwiselength of the streak instability determines the streamwise length scale of very-largescalemotion.

Cassinelli A, de Giovanetti M, Hwang Y, 2017, Streak instability in near-wall turbulence revisited, *Journal of Turbulence*, Vol: 18, Pages: 443-464, ISSN: 1468-5248

The regeneration cycle of streaks and streamwise vortices plays a central role in the sustainment of near-wall turbulence. In particular, the streak breakdown phase in the regeneration cycle is the core process in the formation of the streamwise vortices, but its current understanding is limited particularly in a real turbulent environment. This study is aimed at gaining fundamental insight into the underlying physical mechanism of the streak breakdown in the presence of background turbulent fluctuation. We perform a numerical experiment based on direct numerical simulation, in which streaks are artificially generated by a body forcing computed from previous linear theory. Upon increasing the forcing amplitude, the artificially driven streaks are found to generate an intense fluctuation of the wall-normal and spanwise velocities in a fairly large range of amplitudes. This cross-streamwise velocity fluctuation shows its maximum at λ+ x ≈ 200 − 300 (λ+ x is the inner-scaled streamwise wavelength), but it only appears for λ+ x ≲ 3000 − 4000. Further examination with dynamic mode decomposition reveals that the related flow field is composed of sinuous meandering motion of the driven streaks and alternating cross-streamwise velocity structures, clearly reminiscent of sinuous-mode streak instability found in previous studies. Finally, it is shown that these structures are reasonably well aligned along the critical layer of the secondary instability, indicating that the surrounding turbulence does not significantly modify the inviscid inflectional mechanism of the streak breakdown via streak instability and/or streak transient growth.

Cossu C, Hwang Y, 2017, Self-sustaining processes at all scales in wall-boundedturbulent shear flows, *Royal Society of London. Philosophical Transactions A. Mathematical, Physical and Engineering Sciences*, Vol: 375, ISSN: 1364-503X

We collect and discuss the results of our recentstudies which show evidence of the existence ofa whole family of self-sustaining motions in wallboundedturbulent shear flows with scales rangingfrom those of buffer-layer streaks to those of largescaleand very-large-scale motions in the outer layer.The statistical and dynamical features of this familyof self-sustaining motions, which are associated withstreaks and quasi-streamwise vortices, are consistentwith those of Townsend’s attached eddies. Motionsat each relevant scale are able to sustain themselvesin the absence of forcing from larger- or smaller-scalemotions by extracting energy from the mean flow viaa coherent lift-up effect. The coherent self-sustainingprocess is embedded in a set of invariant solutions ofthe filtered Navier-Stokes equations which take intofull account the Reynolds stresses associated with theresidual smaller-scale motions.

de Giovanetti M, Hwang Y, Choi H, 2016, Skin-friction generation by attached eddies in turbulent channel flow, *Journal of Fluid Mechanics*, Vol: 808, Pages: 511-538, ISSN: 1469-7645

Despite a growing body of recent evidence on the hierarchical organization of the selfsimilarenergy-containing motions in the form of Townsend’s attached eddies in wallboundedturbulent flows, their role in turbulent skin-friction generation is currentlyknown very little. In this paper, the contribution of each of these self-similar energycontainingmotions to turbulent skin friction is explored up to Reτ ≃ 4000. Threedifferent approaches are employed to quantify the skin-friction generation by the motions,the spanwise length scale of which is smaller than a given cut-off wavelength: 1) FIKidentity in combination with the spanwise wavenumber spectra of the Reynolds shearstress; 2) confinement of the spanwise computational domain; 3) artificial damping ofthe motions to be examined. The near-wall motions are found to continuously lose theirrole in skin-friction generation on increasing the Reynolds number, consistent with theprevious finding at low Reynolds numbers. The largest structures given in the form ofvery-large-scale and large-scale motions are also found to be of limited importance: dueto a non-trivial scale-interaction process, their complete removal yields only 5 ∼ 8% ofskin-friction reduction at all the Reynolds numbers considered, although they are foundto be responsible for 20 ∼ 30% of total skin friction at Reτ ≃ 2000. Application of all thethree approaches consistently reveals that the largest amount of skin friction is generatedby the self-similar motions populating the logarithmic region. It is further shown thatthe contribution of these motions to turbulent skin friction gradually increases with theReynolds number, and that these coherent structures are eventually responsible for mostof turbulent skin-friction generation at sufficiently high Reynolds numbers.

Hwang Y, Willis AP, Cossu C, 2016, Invariant solutions of minimal large-scale structures in turbulent channel flow for Reτ up to 1000, *Journal of Fluid Mechanics*, Vol: 802, ISSN: 1469-7645

Understanding the origin of large-scale structures in high Reynolds number wall turbulencehas been a central issue over a number of years. Recently, Rawat et al. (J.Fluid Mech., 2015, 782, p515) have computed invariant solutions for the large-scalestructures in turbulent Couette flow at Reτ ≃ 128 using an over-damped LES with theSmagorinsky model to account for the effect of the surrounding small-scale motions.Here, we extend this approach to an order of magnitude higher Reynolds numbers inturbulent channel flow, towards the regime where the large-scale structures in the formof very-large-scale motions (long streaky motions) and large-scale motions (short vorticalstructures) energetically emerge. We demonstrate that a set of invariant solutions canbe computed from simulations of the self-sustaining large-scale structures in the minimalunit (domain of size Lx = 3.0h streamwise and Lz = 1.5h spanwise) with midplanereflection symmetry at least up to Reτ ≃ 1000. By approximating the surrounding smallscales with an artificially elevated Smagorinsky constant, a set of equilibrium states arefound, labelled upper- and lower-branch according to their associated drag. It is shownthat the upper-branch equilibrium state is a reasonable proxy for the spatial structureand the turbulent statistics of the self-sustaining large-scale structures.

Hwang Y, Bengana Y, 2016, Self-sustaining process of minimal attached eddies in turbulent channel flow, *Journal of Fluid Mechanics*, Vol: 795, Pages: 708-738, ISSN: 1469-7645

It has been recently shown that the energy-containing motions (i.e. coherent structures)in turbulent channel flow exist in the form of Townsend’s attached eddies by a numericalexperiment which simulates the energy-containing motions only at a prescribed spanwiselength scale using their self-sustaining nature (Hwang, 2015, J. Fluid Mech., 767,p254). In the present study, a detailed investigation of the self-sustaining process of theenergy-containing motions at each spanwise length scale (i.e. the attached eddies) inthe logarithmic and outer regions is carried out with an emphasis on its relevance to‘bursting’, which refers to an energetic temporal oscillation of the motions (Flores &Jim´enez, 2010, Phys. Fluids, 22, 071704). It is shown that the attached eddies in thelogarithmic and outer regions, composed of streaks and quasi-streamwise vortical structures,bear the self-sustaining process remarkably similar to that in the near-wall region:i.e. the streaks are significantly amplified by the quasi-streamwise vortices via the liftupeffect; the amplified streaks subsequently undergo a ‘rapid streamwise meanderingmotion’, reminiscent of streak instability or transient growth, which eventually resultsin breakdown of the streaks and regeneration of new quasi-streamwise vortices. For theattached eddies at a given spanwise length scale λz between λ+z ≃ 100 and λz ≃ 1.5h,the single turn-over time period of the self-sustaining process is found to be T uτ /λz ≃ 2(uτ is the friction velocity), which corresponds well to the time scale of the bursting.Two additional numerical experiments, designed to artificially suppress the lift-up effectand the streak meandering motions, respectively, reveal that these processes are essentialingredients of the self-sustaining process of the attached eddies in the logarithmic andouter regions, consistent with several previous theoretical studies. It is also shown thatthe

Rawat S, Cossu C, Hwang Y,
et al., 2015, On the self-sustained nature of large-scale motions in turbulent Couette flow, *Journal of Fluid Mechanics*, Vol: 782, Pages: 515-540, ISSN: 1469-7645

Hogan B, Babataheri A, Hwang Y,
et al., 2015, Characterizing cell adhesion by using micropipette aspiration, *Biophysical Journal*, Vol: 109, Pages: 209-219, ISSN: 1542-0086

Gouget CLM, Hwang Y, Barakat AI, 2015, Model of cellular mechanotransduction via actin stress fibers, *Biomechanics and Modeling in Mechanobiology*, Vol: 15, Pages: 331-344, ISSN: 1617-7959

Hwang Y, 2015, Statistical structure of self-sustaining attached eddies in turbulent channel flow, *Journal of Fluid Mechanics*, Vol: 767, Pages: 254-289, ISSN: 1469-7645

Hwang Y, Kumar P, Barakat AI, 2014, Intracellular regulation of cell signaling cascades: how location makes a difference, *JOURNAL OF MATHEMATICAL BIOLOGY*, Vol: 69, Pages: 213-242, ISSN: 0303-6812

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Hwang Y, Pedley TJ, 2014, Stability of downflowing gyrotactic microorganism suspensions in a two-dimensional vertical channel, *Journal of Fluid Mechanics*, Vol: 749, Pages: 750-777, ISSN: 1469-7645

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