## Publications

60 results found

Holford J, Lee M, Hwang Y, 2023, Optimal white-noise stochastic forcing forlinear models of turbulent channel flow, *Journal of Fluid Mechanics*, ISSN: 0022-1120

Lorente-Macias J, Bengana Y, Hwang Y, 2023, Shape optimisation for a stochastic two-dimensional cylinder wake using ensemble variation, *Journal of Fluid Mechanics*, ISSN: 0022-1120

Bengana Y, Yang Q, Tu G,
et al., 2022, Exact coherent states in plane Couette flow under spanwise wall oscillation, *Journal of Fluid Mechanics*, Vol: 947, ISSN: 0022-1120

A set of several exact coherent states in plane Couette flow is computed under spanwise wall oscillation control, with a range of wall oscillation amplitudes and periods (Aw,T). It is found that the wall oscillation generally stabilises the upper branch of the equilibrium solutions and achieves the corresponding drag reduction, while it influences modestly the lower branch. The stabilisation effect is found to increase with the oscillation amplitude with an optimal time period around T+≈100. The exact coherent states reproduce some key dynamical behaviours of streaks observed in previous studies, while exhibiting the rich coherent structure dynamics that cannot be extracted from a phase average of turbulent states. Visualisation of state portraits shows that the size of the state space supporting turbulent solution is reduced by the spanwise wall oscillation, and the upper-branch equilibrium solutions become less repelling, with many of their unstable manifolds being stabilised. This change of the state space dynamics leads to a significant reduction in lifetime of turbulence. Finally, the main stabilisation mechanism of the exact coherent states is found to be the suppression of the lift-up effect of streaks, explaining why previous linear analyses have been so successful for turbulence stabilisation modelling and the resulting drag reduction.

Doohan P, Bengana Y, Yang Q,
et al., 2022, The state space and travelling-wave solutions in two-scale wall-bounded turbulence, *Journal of Fluid Mechanics*, Vol: 947, Pages: 1-36, ISSN: 0022-1120

The computation of invariant solutions and the visualisation of the associated state spacehave played a key role in the understanding of transition and the self-sustaining processin wall-bounded shear flows. In this study, an extension of this approach is sought for aturbulent flow which explicitly exhibits multi-scale behaviour. The minimal unit of multiscale near-wall turbulence, which resolves two adjacent spanwise integral length scalesof motion, is considered using a shear stress-driven flow model (Doohan et al., J. FluidMech., vol. 913, 2021, A8). The edge state, twenty-six travelling waves and two periodicorbits are computed, which represent either the large- or small-scale self-sustainingprocesses. Given that the spanwise length scales are not widely separated here, it could beenvisaged that turbulent trajectories visit these solutions in the state space. Consideringthe intra- and inter-scale dynamics of the flow, numerous phase portraits are examined,but the turbulent state is not found to approach any of these solutions. A detailedanalysis reveals that this is due to the lack of scale interaction processes captured bythe invariant solutions, including the mean-fluctuation interaction, the energy cascade inthe streamwise wavenumber space and the cascade-driven energy production discoveredrecently. There is a single solution that resembles turbulence much more than the others,which captures two-scale energetics and a scale interaction process involving energyfeeding from small to large spanwise scales through the subharmonic sinuous streakinstability mode. Based on these observations, it is conjectured that the state space viewof turbulent trajectories wandering between solutions would need suitable refinement tomodel multi-scale turbulence, when each solution does not represent multi-scale processesof turbulence. In particular, invariant solutions that are inherently multi-scale would berequired.

Hernandez C, Yang Q, Hwang Y, 2022, Generalised quasilinear approximations of turbulent channel flow: Part 2. Spanwise triadic scale interactions, *Journal of Fluid Mechanics*, Vol: 944, ISSN: 0022-1120

Continuing from Part 1 (Hernández et al., J. Fluid Mech., vol. 936, A33, 2022), a generalized quasilinear (GQL) approximation is studied in turbulent channel flow using a flow decomposition defined with spanwise Fourier modes: the flow is decomposed into a set of low-wavenumber spanwise Fourier modes and the rest high-wavenumber modes. This decomposition leads to the nonlinear low-wavenumber group that supports the self-sustaining process within the given integral length scales, whereas the linearised high-wave number group is not able to do so, unlike the GQL models in Part 1 which place a minimal mathematical description for the self-sustaining process across all integral scales. Despite the important physical difference, it is shown that the GQL models in this study share some similarities with those in Part 1: i.e. the reduced multi-scale behaviour and anisotropic turbulent fluctuations. Furthermore, despite not being able to support the self-sustaining process in the high-wavenumber group, the GQL models in the present study are found to reproduce some key statistical features in the high-wavenumber group solely through the ‘scattering’ mechanism proposed by previous studies. Finally, using the nature of the GQL approximation, a set of numerical experiments suppressing certain triadic nonlinear interactions are further carried out. This unveils some key roles played by the certain types of triadic interactions including energy cascade and inverse energytransfer in the near-wall region. In particular, the inhibition of inverse energy transfer in the spanwise direction leads to suppression of the near-wall positive turbulent transport at large scales.

Jiao Y, Chernyshenko S, Hwang Y, 2022, A driving mechanism of near-wall turbulence subject to adverse pressure gradient in a plane Couette flow, *Journal of Fluid Mechanics*, Vol: 941, ISSN: 0022-1120

The effect of adverse pressure gradient (APG) on near-wall turbulence is studied, with a particular attention to the turbulence production mechanism. A plane turbulent Couette flow is considered for several values of constant APG in the lower wall region. A direct numerical simulation (DNS) in a large computational domain shows that turbulence near the lower wall continues to exist even at sufficiently large APGs. On increasing the APG, the cross-streamwise turbulence intensities increase, and the near-wall streaks gradually disappear. A linear analysis using the optimal transient growth indicates that the APG inhibits the generation of near-wall streaks due to the significant reduction of the mean shear in the region near the lower wall. The turbulent fluctuation dynamics beyond the linear regime is studied with a DNS in a minimal flow unit. The near-wall self-sustaining process involving streaks is significantly weakened or destroyed as APG increases, while the turbulent fluctuations become more isotropic and localised. Using a conditional averaging analysis, a new mechanism of near-wall turbulence production under strong APG is uncovered. This mechanism is initiated by the wall-normal nonlinear transport of an outer wall-normal velocity fluctuation to the near-wall region. The transported wall-normal velocity fluctuation is subsequently amplified via the Orr mechanism, resulting in the non-zero turbulence production involving spatially localised vortical structures. This mechanism is also confirmed by DNS of the flow in a large computational domain, where strong correlation between the wall-normal nonlinear transport and turbulence production is observed.

Khoo ZC, Chan CH, Hwang Y, 2022, A sparse optimal closure for a reduced-order model of wall-bounded turbulence, *Journal of Fluid Mechanics*, Vol: 939, ISSN: 0022-1120

In the present study, a set of physics-informed and data-driven approaches are examined towards the development of an accurate reduced-order model for a turbulent plane Couette flow. Based on the utilisation of the proper orthogonal decomposition (POD), a particular focus is given to the development of a reduced-order model where the number of POD modes are not large enough to cover the full dynamics of the given turbulent state, the situation directly relevant to the reduced-order modelling for turbulent flows. Starting from the conventional Galerkin projection approach ignoring the truncation error, three approaches enhanced by both physics and data are examined: (1) sparse regression of the POD-Galerkin dynamics; (2) Galerkin projection with an empirical eddy-viscosity model; (3) Galerkin projection with an optimal eddy viscosity obtained from a newly proposed sparse regression – an idea applying the sparse identification of nonlinear dynamics framework to an eddy-viscosity model. The sparse regression of the POD-Galerkin dynamics does not perform well, as the number of POD modes for the given chaotic dynamics appears to be too small. While the unsatisfactory performance of the Galerkin projection model with an empirical eddy viscosity is observed, the newly proposed approach, which combines the concept of an optimal eddy-viscosity closure with a sparse regression, more accurately approximates the chaotic dynamics than the other reduced-order models considered. This is demonstrated with the mean and time scales of the POD mode amplitudes as well as the first- and second-order turbulence statistics.

Maretvadakethope S, Hwang Y, Keaveny E, 2022, Synchronized states of hydrodynamically coupled filaments andtheir stability, *Physical Review Fluids*, ISSN: 2469-990X

Cilia and flagella are organelles that play central roles in unicellular locomotion, embryonicdevelopment, and fluid transport around tissues. In these examples, multiple cilia are often foundin close proximity and exhibit coordinated motion. Inspired by the flagellar motion of biflagellatecells, we examine the synchrony exhibited by a filament pair surrounded by a viscous fluid andtethered to a rigid planar surface. A geometrically-switching base moment drives filament motion,and we characterize how the stability of synchonized states depends of the base torque magnitude.In particular, we study the emergence of bistability that occurs when the anti-phase, breast-strokebranch becomes unstable. Using a bisection algorithm, we find the unstable edge-state that existsbetween the two basins of attraction when the system exhibits bistability. We establish a bifurcationdiagram, study the nature of the bifurcation points, and find that the observed dynamical systemcan be captured by a modified version of Adler’s equation. The bifurcation diagram and presenceof bistability reveal a simple mechanism by which the anti-phase breast stroke can be modulated, orswitched entirely to in-phase undulations through the variation of a single bifurcation parameter.

Fung L, Bearon RN, Hwang Y, 2022, A local approximation model for macro-scale transport of biased active Brownian particles in a flowing suspension, *Journal of Fluid Mechanics*, Vol: 935, ISSN: 0022-1120

A dilute suspension of motile microorganisms subjected to a strong ambient flow, such as algae in the ocean, can be modelled as a population of non-interacting, orientable active Brownian particles (ABPs). Using the Smoluchowski equation (i.e. Fokker–Planck equation in space and orientation), one can describe the non-trivial transport phenomena of ABPs such as taxis and shear-induced migration. This work transforms the Smoluchowski equation into a transport equation, in which the drifts and dispersions can be further approximated as a function of the local flow field. The new model can be applied to any global flow field due to its local nature, unlike previous methods such as those utilising the generalised Taylor dispersion theory. The transformation shows that the overall drift includes both the biased motility of individual particles in the presence of taxis and the shear-induced migration in the absence of taxis. In addition, it uncovers other new drifts and dispersions caused by the interactions between the orientational dynamics and the passive advection–diffusion of ABPs. Finally, the performance of this model is assessed using examples of gyrotactic suspensions, where the proposed model is demonstrated to be most accurate when the biased motility of particles (i.e. taxis) is weak.

Hernandez CG, Yang Q, Hwang Y, 2022, Generalised quasilinear approximations of turbulent channel flow. Part 1. streamwise nonlinear energy transfer, *Journal of Fluid Mechanics*, Vol: 936, ISSN: 0022-1120

A generalised quasilinear (GQL) approximation (Marston et al., Phys. Rev. Lett., vol. 116, 2016, 104502) is applied to turbulent channel flow at Reτ≃1700 ( Reτ is the friction Reynolds number), with emphasis on the energy transfer in the streamwise wavenumber space. The flow is decomposed into low- and high-streamwise-wavenumber groups, the former of which is solved by considering the full nonlinear equations whereas the latter is obtained from the linearised equations around the former. The performance of the GQL approximation is subsequently compared with that of a QL model (Thomas et al., Phys. Fluids, vol. 26, 2014, 105112), in which the low-wavenumber group only contains zero streamwise wavenumber. It is found that the QL model exhibits a considerably reduced multi-scale behaviour at the given moderately high Reynolds number. This is improved significantly by the GQL approximation which incorporates only a few more streamwise Fourier modes into the low-wavenumber group, and it reasonably well recovers the distance-from-the-wall scaling in the turbulence statistics and spectra. Finally, it is proposed that the energy transfer from the low- to the high-wavenumber group in the GQL approximation, referred to as the ‘scattering’ mechanism, depends on the neutrally stable leading Lyapunov spectrum of the linearised equations for the high-wavenumber group. In particular, it is shown that if the threshold wavenumber distinguishing the two groups is sufficiently high, the scattering mechanism can be completely absent due to the linear nature of the equations for the high-wavenumber group.

Jin D, Hwang Y, Kampf N,
et al., 2021, Direct measurement of the viscoelectric effect in water, *Proceedings of the National Academy of Sciences of USA*, Vol: 119, Pages: 1-7, ISSN: 0027-8424

The viscoelectric effect concerns the increase in viscosity of a polar liquid in an electric field due to its interaction with the dipolar molecules, and was first determined for polar organic liquids more than 80 years ago. For the case of water, however, the most common polar liquid, direct measurement of the viscoelectric effect is challenging and has not to date been carried out, despite its importance in a wide range of electrokinetic and flow effects. In consequence, estimates of its magnitude for water vary by more than 3 orders of magnitude. Here we measure the viscoelectric effect in water directly using a surface force balance (SFB), by measuring the dynamic approach of two molecularly-smooth surfaces with a controlled, uniform electric field between them across highly-purified water. As the water is squeezed out of the gap between the approaching surfaces, viscous damping dominates the approach dynamics; this is modulated by the viscoelectric effect under the uniform transverse electric field across the water, enabling its magnitude to be directly determined as a function of the field. We find a value for this magnitude which differs respectively by 1 and by 2 orders of magnitude from its highest and lowest previously estimated values.

Hwang Y, Hutchins N, Marusic I, 2021, The logarithmic variance of streamwise velocity and k−1 conundrum in wall turbulence, *Journal of Fluid Mechanics*, Vol: 933, Pages: 1-24, ISSN: 0022-1120

The logarithmic dependence of streamwise turbulence intensity has repeatedly beenobserved in recent experimental and direct numerical simulation data. However, itsspectral counterpart, a well-developed k−1spectrum (k is the spatial wavenumber ina wall-parallel direction), has not been convincingly observed from the same data. Inthe present study, we revisit the spectrum-based attached eddy model of Perry andco-workers, who proposed the emergence of k−1spectrum in the inviscid limit, forsmall but finite z/δ and for finite Reynolds numbers (z is the wall-normal co-ordinateand δ the outer length scale). In the upper logarithmic layer (or inertial sublayer),a reexamination reveals that the intensity of the spectrum must vary with the wallnormal location at order of z/δ, consistent with the early observation argued with‘incomplete similarity’. The streamwise turbulence intensity is subsequently calculated,demonstrating that the existence of a well-developed k−1spectrum is not a necessarycondition for the approximate logarithmic wall-normal dependence of turbulence intensity– a more general condition is the existence of pre-multiplied power-spectral intensity ofO(1) for O(1/δ) < k < O(1/z). Furthermore, it is shown that the Townsend-Perryconstant must weakly be dependent on the Reynolds number. Finally, the analysis issemi-empirically extended to the lower logarithmic layer (or mesolayer), and a near-wallcorrection for the turbulence intensity is subsequently proposed. All the predictions ofthe proposed model and the related analyses/assumptions are validated with high-fidelityexperimental data (Samie et al., J. Fluid Mech., 2018, vol. 851, pp. 391–415).

Basso R, Hwang Y, Assi G,
et al., 2021, Instabilities and sensitivities in a flow over a rotationally flexible cylinder with a rigid splitter plate, *Journal of Fluid Mechanics*, Vol: 928, Pages: 1-32, ISSN: 0022-1120

This paper investigates the origin of flow-induced instabilities and their sensitivities ina flow over a rotationally flexible circular cylinder with a rigid splitter plate. A linearstability and sensitivity problem is formulated in the Eulerian frame by considering thegeometric nonlinearity arising from the rotational motion of the cylinder which is notpresent in the stationary or purely translating stability methodology. This nonlinearityneeds careful and consistent treatment in the linearised problem particularly whenconsidering the Eulerian frame or reference adopted in this study and not so widelyconsidered. Two types of instabilities arising from the fluid-structure interaction arefound. The first type of the instabilities is the stationary symmetry-breaking mode, whichwas well reported in previous studies. This instability exhibits a strong correlation withthe length of the recirculation zone. A detailed analysis of the instability mode andits sensitivity reveals the importance of the flow near the tip region of the plate for thegeneration and control of this instability mode. The second type is an oscillatory torsionalflapping mode, which has not been well reported. This instability typically emerges whenthe length of the splitter plate is sufficiently long. Unlike the symmetry breaking mode,it is not so closely correlated with the length of the recirculation zone. The sensitivityanalysis however also reveals the crucial role played by the flow near the tip region inthis instability. Finally, it is found that many physical features of this instability arereminiscent of those of the flapping (or flutter instability) observed in a flow over aflexible plate or a flag, suggesting that these instabilities share the same physical origin.

Fung L, Hwang Y, 2021, Instability of tilted shear flow in a strongly stratified and viscous medium, *IUTAM Bookseries*, Vol: 38, Pages: 379-389, ISSN: 1875-3507

It is well known that stratification can stabilise shear flow. In a vertical shear flow, the Miles-Howard’s criterion firmly indicates that flow should be stable if the local Richardson number is greater than one fourth. However, if shear is tilted with a non-zero angle from vertical, an instability can arise even under very strong stratification, and such an instability was recently observed in a titled wake flow at low Reynolds number (Meunier 2012, J. Fluid Mech., 699:174). In the present study, we showed that in the limit of low Froude number and low Reynolds number, the linearised equations of motion could 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 demonstrated that the low-Froude-number mode would be a horizontal inflectional instability, and should remain two dimensional at small tilting angles. It is further shown that the emergence of small vertical velocity at finite Froude number modifies the horizontal inflectional instability and leads to paradoxically stabilising buoyancy force on increasing Froude number. Finally, an absolute instability analysis is performed, revealing qualitatively good agreement with the experimental result.

Doohan P, Willis A, Hwang Y, 2021, Minimal multi-scale dynamics of near-wall turbulence, *Journal of Fluid Mechanics*, Vol: 913, ISSN: 0022-1120

Recent numerical experiments have shown that the temporal dynamics of isolated energy-containing eddies in the hierarchy of wall-bounded turbulence are governed by the self-sustaining process (SSP). However, high-Reynolds-number turbulence is a multi-scale phenomenon and exhibits interaction between the structures of different scales, but the dynamics of such multi-scale flows are poorly understood. In this study, the temporal dynamics of near-wall turbulent flow with two integral length scales of motion are investigated using a shear stress-driven flow model (Doohan et al., J. Fluid Mech., vol. 874, 2019, pp. 606–638), with a focus on identifying scale interaction processes through the governing equations and relating these to the SSPs at each scale. It is observed that the dynamics of the energy cascade from large to small scales is entirely determined by the large-scale SSP and the timing of the corresponding inter-scale turbulent transport coincides with the large-scale streak breakdown stage. Furthermore, the characteristic time scales of the resulting small-scale dissipation match those of the large-scale SSP, indicative of non-equilibrium turbulent dissipation dynamics. A new scale interaction process is identified, namely that the transfer of wall-normal energy from large to small scales drives small-scale turbulent production via the Orr mechanism. While the main outcome of this driving process appears to be the transient amplification of localised small-scale velocity structures and their subsequent dissipation, it also has an energising effect on the small-scale SSP. Finally, the feeding of energy from small to large scales is impelled by the small-scale SSP and coincides with the small-scale streak instability stage. The streamwise feeding process seems to be related to the subharmonic sinuous streak instability mode in particular and leads to the formation of the wall-reaching part of high-speed large-scale streaks.

Skouloudis N, Hwang Y, 2021, Scaling of turbulence intensities up to Reτ=10^6 with a resolvent-based quasi-linear approximation, *Physical Review Fluids*, Vol: 6, ISSN: 2469-990X

A minimal form of quasilinear approximation (QLA), recently proposed with a stochastic forcing and proper orthogonal decomposition modes [Hwang and Ekchardt, J. Fluid Mech. 894, A23 (2020)], has been extended by employing a resolvent framework. A particular effort is made to reach an extremely high Reynolds number by carefully controlling the approximation without loss of the general scaling properties in the spectra, while setting out the main limitations and accuracy of the proposed QLA with possibility of further improvement. The QLA is subsequently applied to turbulent channel flow up to Reτ=106 (Reτ is the friction Reynolds number). While confirming that the logarithmic wall-normal dependence in streamwise and spanwise turbulence intensities robustly appears, it reveals some nontrivial difference from the scaling of the classical attached eddy model based on inviscid flow assumption. First, the spanwise wave number spectra do not show any clearly visible inverse-law behavior due to the viscous wall effect prevailing in a significant portion of the lower part of the logarithmic layer. Second, the near-wall peak streamwise and spanwise turbulence intensities are found to deviate from lnReτ scaling for Reτ≳104. Importantly, the near-wall streamwise turbulence intensity is inversely proportional to 1/U+cl (U+cl is the inner-scaled channel centreline velocity), consistent with the scaling obtained from an asymptotic analysis of the Navier-Stokes equations [Monkewitz and Nagib, J. Fluid Mech. 783, 474 (2015)]. The same behavior was also observed for the streamwise turbulence intensity in the logarithmic region, as was predicted with the asymptotic analysis. Finally, the streamwise turbulence intensity in the logarithmic region is found to become greater than the near-wall one at Reτ≃O(105). It is shown that this behavior originates from the near-wall spectra associated with large-scale inactive motions, the intensity of which gradually decays as R

Jiao Y, Hwang Y, Chernyshenko S, 2021, The Orr mechanism in transition of parallel shear flow, *Physical Review Fluids*, Vol: 6, ISSN: 2469-990X

The Orr mechanism is revisited to understand its precise role in the transition of plane Couette flow. By considering homogeneous shear flow and plane Couette flow, it is identified that the Orr mechanism induces a lift-up effect which significantly amplifies spanwise velocity. An optimal perturbation analysis for an individual velocity component reveals that the amplification of spanwise velocity is most active at the streamwise length comparable to the given spanwise length of the perturbation. The relevance of this mechanism to transition is subsequently examined in plane Couette flow. To this end, a set of initial conditions, which combines the optimal perturbation for spanwise velocity with the one for all the velocity components, is considered by varying their amplitudes. Two representative transition scenarios are found: oblique and streak transitions. In the former, the spanwise velocity perturbation amplified with the Orr mechanism initiates both streak amplification and breakdown, whereas in the latter, its role is limited only to the streak breakdown at the late stage of transition. As such, the oblique transition offers a route to turbulence energetically more efficient than the streak transition, at least for the cases examined in the present paper. Finally, the oblique transition is found to share many physical similarities with the transition by the minimal seed.

Hernandez C, Hwang Y, 2020, Spectral energetics of a quasilinear approximation in uniform shear turbulence, *Journal of Fluid Mechanics*, Vol: 604, Pages: A11-1-A11-26, 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, 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.

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.

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.

Maretvadakethope S, Keaveny E, Hwang Y, 2019, The instability of gyrotactically-trapped cell layers, *Journal of Fluid Mechanics*, Vol: 868, ISSN: 0022-1120

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 Nc≃104 cells cm−3 , 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.

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.

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