This paper presents advances towards the data-based control of periodic oscillator flows, from their fully developed regime to their equilibrium stabilized in closed loop, with linear time-invariant (LTI) controllers. The proposed approach directly builds upon the iterative method of Leclercq et al. (J. Fluid Mech., vol. 868, 2019, pp. 26–65) and provides several improvements for an efficient online implementation, aimed at being applicable in experiments. First, we use input–output data to construct an LTI mean transfer functions of the flow. The model is subsequently used for the design of an LTI controller with linear quadratic Gaussian synthesis, which is practical to automate online. Then, using the controller in a feedback loop, the flow shifts in phase space and oscillations are damped. The procedure is repeated until equilibrium is reached, by stacking controllers and performing balanced truncation to deal with the increasing order of the compound controller. In this article, we illustrate the method for the classic flow past a cylinder at Reynolds number $Re=100$. Care has been taken such that the method may be fully automated and hopefully used as a valuable tool in a forthcoming experiment.

Isolated-roughness-induced transitions controlled by local wall heating strips are studied via direct numerical simulation and BiGlobal linear stability analysis. The transition mechanisms are studied first with different wall temperatures. The separated shear layer–counter-rotating vortex system is found to be the main source for transitions. Symmetric and antisymmetric modes are observed in the wake, and the former is dominant. The local wall heating strip can delay the transition, and this effect is enhanced with higher heating temperature, wider strip and a combination of upstream and downstream control strips. The upstream strip lifts up the inlet flow and weakens the wake system in an indirect manner. The antisymmetric mode gradually vanishes, while the symmetric mode always exists but becomes weaker. The downstream strip exhibits a more effective transition delay by directly weakening the separated shear layer and vortex system in the wake. Vorticity transport analysis suggests that the downstream strip increases dissipation for streamwise vorticity and transfers it into wall-normal and spanwise vorticity. BiGlobal analyses indicate that the downstream strip shows less influence on the peak growth rate of the symmetric mode but significantly shrinks its unstable region. Analyses of the disturbance energy production indicate that the upstream strip wakens the wall-normal and spanwise shear at the same time, but the downstream strip mainly wakens the wall-normal one. More simulations are performed with different roughness heights, point-source disturbance and different roughness shapes. The results show that the current method remains effective enough in delaying transitions at a wide range of conditions.

This work examines the control of cross-flow instabilities (CFIs) and laminar–turbulent transition on a swept wing, through the plasma-based base flow modification (BFM) technique. The effect of experimentally derived plasma body forces on the steady boundary layer base flow is explored through numerical simulations. Linear stability theory is subsequently used to predict the net BFM effect on CFIs. Based on these preliminary predictions, experiments are conducted in a low-turbulence wind tunnel where a spanwise-invariant plasma actuator is installed near the wing leading edge and operated at constant input voltage and frequency. Various flow parameters governing the plasma-based BFM technique are investigated, namely the Reynolds number, angle of attack and wavelength of excited stationary CFI modes. Stationary and travelling CFIs are quantified by planar particle image velocimetry while the transition topology and location are recorded by infrared thermography. The results confirm the stabilising effect of BFM on the swept-wing boundary layer. However, the plasma-based BFM is found to render the boundary layer more susceptible to travelling CFIs. In the presence of both net BFM effect and intrinsic plasma unsteady perturbations, the plasma-based BFM technique achieves transition delay with specific combinations of Reynolds number, angle of attack and wavelength of excited stationary CFI modes. The present findings provide insights into the fundamental principles of operating plasma actuators within the context of BFM control.

The asymmetric instability in two streamwise orthogonal planes for three-dimensional flow-induced vibration (FIV) of an elastically mounted cube at a moderate Reynolds number of 300 is numerically investigated in this paper. The full-order computational fluid dynamics method, data-driven stability analysis via the eigensystem realization algorithm and the selective frequency damping method and total dynamic mode decomposition (TDMD) are applied here to explore this problem. Due to the unsteady non-axisymmetric wakefield formed for flow passing a stationary cube, the FIV response was found to exhibit separate structural stability and oscillations (including lock-in and galloping behaviour) in the two different streamwise orthogonal planes while the body is released. The initial kinetic energy accompanying the release of the cube could destabilize the above-mentioned structural stability. The observed FIV asymmetric instability is verified by the root trajectory of the structural mode obtained via data-driven stability analysis. The stability of the structural modes dominates regardless of whether the structural response oscillates significantly in various (reduced) velocity ranges. Further TDMD analysis on the wake structure, accompanied by the time–frequency spectrum of time-history structural displacements, suggested that the present FIV unit with galloping behaviour is dominated by the combination of the shifted base-flow mode, structure modes and several harmonics of the wake mode.

We present an experimental study of reactive control of turbulent jets, in which we target axisymmetric coherent structures, known to play a key role in the generation of sound. We first consider a forced jet, in which coherent structures are amplified above background levels, facilitating their detection, estimation and control. We then consider the more challenging case of an unforced jet. The linear control targets coherent structures in the region just downstream of the nozzle exit plane, where linear models are known to be appropriate for description of the lowest-order azimuthal modes of the turbulence. The control law is constructed in frequency space, based on empirically determined transfer functions. And the Wiener–Hopf formalism is used to enforce causality and to provide an optimal controller, as opposed to the sub-optimal control laws provided by simpler wave-cancellation methods. Significant improvements are demonstrated in the control of both forced and unforced jets. In the former case, order-of-magnitude reductions are achieved; and in the latter, turbulence levels are reduced by up to 60 %. The results open new perspectives for the control of turbulent flow at high Reynolds number.

The efficacy of steady large-amplitude blowing/suction on instability and transition control for a hypersonic flat plate boundary layer with Mach number 5.86 is investigated systematically. The influence of the blowing/suction flux and amplitude on instability is examined through direct numerical simulation and resolvent analysis. When a relatively small flux is used, the two-dimensional instability critical frequency that distinguishes the promotion/suppression mode effect closely aligns with the synchronisation frequency. For the oblique wave, as the spanwise wavenumber increases, the suppression effects would become weaker and the mode suppression bandwidth diminishes/increases in general in the blowing/suction control. Increasing the blowing/suction flux can effectively broaden the frequency bandwidth of disturbance suppression. The influence of amplitude on disturbance suppression is weak in a scenario of constant flux. To gain a deeper insight into disturbance suppression mechanism, momentum potential theory (MPT) and kinetic energy budget analysis are further employed in analysing disturbance evolution with and without control. When the disturbance is suppressed, the blowing induces the transport of certain acoustic components along the compression wave out of the boundary layer, whereas the suction does not. The velocity fluctuations are derived from the momentum fluctuations of the MPT. Compared with the momentum fluctuations, the evolutions indicated by each component's velocity fluctuations greatly facilitate the investigations of the acoustic nature of the second mode. The rapid variation of disturbance amplitude near the blowing is caused by the oscillations of the acoustic component and phase speed differences between vortical and thermal components. Kinetic energy budget analysis is performed to address the non-parallel effect of the boundary layer introduced by blowing/suction, which tends to suppress disturbances near the blowing. Moreover, viscous effects leading to energy dissipation are identified to be stronger in regions where the boundary layer is rapidly thickening. Finally, it is demonstrated that a flat plate boundary layer transition triggered by a random disturbance can be delayed by a blowing/suction combination control. The resolvent analysis further demonstrates that disturbances with frequencies that dominate the early transition stage are dampened in the controlled base flow.

The present study aims to demonstrate the application of the oblique temperature gradient (OTG) to manipulate the buoyancy instabilities and resulting mixing. To accomplish this, we consider a model consisting of a liquid layer supported by a perfectly conducting bottom wall from below and the upper surface is exposed to ambient inert gas in a shallow container. The stability analysis employs the pseudospectral method and perturbation energy budget approach. The imposed OTG consists of a negative vertical temperature gradient (VTG), which leads to unstable density stratification, and a negative horizontal temperature gradient (HTG) component responsible for the Hadley circulation, which can be utilised to control instabilities. Without the HTG, a Rayleigh–Bénard mode of instability exists due to the imposed VTG termed as a VTG mode. The imposed HTG induces a linear VTG term and a quintic polynomial VTG term in the bottom wall-normal coordinate $y$. The induced linear VTG term reinforces the imposed VTG, while the induced quintic polynomial VTG term opposes the imposed VTG. For the vanishing Biot number ($Bi$), the induced quintic VTG term stabilises the VTG mode for the Prandtl number, $Pr=7$. However, for $Pr=0.01$, the Reynolds stresses associated with the Hadley circulation destabilise the VTG mode. For non-zero $Bi$, new streamwise and spanwise instability modes arise for $Pr=7$ due to the induced linear VTG term. Physical arguments show that the unstable density stratification near the gas–liquid interface caused by the synergistic effect of the imposed VTG and the induced linear VTG term lead to the existence of the predicted new modes. The present study thus shows that the strength of the HTG can be utilised to control the instability modes and critical parameters, thereby demonstrating the utility of the OTG in manipulating the buoyancy instabilities.

We use Bayesian inference, accelerated by adjoint methods, to construct a quantitatively accurate model of the thermoacoustic behaviour of a conical flame in a duct. We first perform a series of automated experiments on a ducted flame rig. Next, we propose several candidate models of the rig's components and assimilate data into each model to find the most probable parameters for that model. We rank the candidate models based on their marginal likelihood (evidence) and select the most likely model for each component. We begin this process by rigorously characterizing the acoustics of the cold rig. When the flame is introduced, we propose several candidate models for the fluctuating heat release rate. We find that the most likely flame model considers velocity perturbations in both the burner feed tube and the outer duct, even though studies in the literature typically neglect either one of these. Using the most likely model, we infer the flame transfer functions for 24 flames and quantify their uncertainties. We do this with the flames in situ, using only pressure measurements. We find that the inferred flame transfer functions render the model quantitatively accurate, and, where comparable, broadly consistent with direct measurements from several studies in the literature.

First-order sensitivities and adjoint analysis are used widely to control the linear stability of unstable flows. Second-order sensitivities have recently helped to increase accuracy. In this paper, a method is presented to calculate arbitrary high-order sensitivities based on Taylor expansions of the incompressible base flow and its eigenproblem around a scalar parameter. For the incompressible Navier–Stokes equations, general expressions for the sensitivities are derived, into which parameter-specific information can be inserted. The computational costs are low since, for all orders, a linear equation system has to be solved, of which the left-hand-side matrix stays constant and thus its preconditioning can be exploited. Two flow scenarios are examined. First, the cylinder flow equations are expanded around the inverse of the Reynolds number, enabling the prediction of the two-dimensional cylinder base flow and its leading eigenvalue as a function of the Reynolds number. This approach computes accurately the base flow and eigenvalue even in the unstable regime, providing, when executed subsequently, a mean to calculate unstable base flows. This case gives a clear introduction into the method and allows us to discuss its constraints regarding convergence behaviour. Second, a small control cylinder is introduced into the domain of the cylinder flow for stabilization. Higher-order sensitivity maps are calculated by modelling the small cylinder with a steady forcing. These maps help to identify stabilizing areas of the flow field for Reynolds numbers within the laminar vortex shedding regime, with the required number of orders increasing as the Reynolds number rises. The results obtained through the proposed method align well with numerically calculated eigenvalues that incorporate the cylinder directly into the grid.

Theoretical analysis and numerical results have shown that frequency lock-in in vortex-induced vibration (VIV) is caused by the instability of the structural mode rather than a resonant response to external excitations. However, there is a lack of experimental evidence supporting relevant theoretical research findings. This study investigates VIV suppression with a passive modal controller (PMC) for a circular cylinder at Reynolds numbers $Re = 60$ and $Re = 40$, using experiments to distinguish the effects of stable and unstable wake modes. Comparative analysis before and after the implementation of the PMC reveals significant reduction in the vibration amplitude and the disappearance of the lock-in phenomenon at $Re = 60$. The vibration frequency closely follows the vortex shedding frequency after control, while dynamic mode decomposition of the flow field indicates that the wake mode is dominant. For $Re = 40$, the vibration is eliminated and the flow becomes steady. Additionally, the root loci of the coupled system are investigated before and after the PMC implementation via linear stability analysis. The results indicate that the PMC can alter the dynamic characteristics of the original system, causing the structural mode and PMC mode to couple when approaching the PMC frequency. Then, the interaction typically improves the stability of the structural mode. Finally, a parametric study is conducted in the experiment, as well as a linear stability analysis. The study provides experimental evidence that stability control of the structural mode is the key to suppressing VIV and eliminating the lock-in phenomenon.

We investigate the linear stability analysis of rotating electro-osmotic flow in confined and unconfined configurations by appealing to the Debye–Hückel approximation. Pertaining to flow in confined and unconfined domains, the stability equations are solved using the Galerkin method to obtain the stability picture. Both qualitative and quantitative aspects of Ekman spirals are examined in stable and unstable scenarios within the unconfined domain. Within the confined domain, the variation of the real growth rate and the transition to instability are analysed using the modified Routh–Hurwitz criteria, employed for the first time in this context. The stability of the underlying flow, characterized by the number of roots with a positive real part, is determined by establishing a Routhian table. The inferences of this analysis show that the velocity plane produces intriguing closed Ekman spirals, which diminish in size with an increase in the rotation speed $\omega$. The Ekman spirals in the stable region exhibit a distinct discontinuity, indicating the dissipation of disturbances over time. In the confined domain, the flow appears consistently stable for a set of involved parameters pertinent to this analysis, such as electrokinetic parameter $K=1.5$ and rotational parameter $\omega$ approximately up to $6$. However, the flow instabilities become evident for $K=1.5$ and $\omega \geq 6$.

The axially symmetric deformation of a drop in a viscous fluid, under the influence of an externally imposed flow having simultaneous rotating and compressional or extensional components, is addressed. In the previous studies, two families of stationary drop shapes were constructed by simulating the dynamics of drop deformation: stable singly connected shapes with respect to axisymmetric disturbances, and unstable toroidal shapes. These two branches coexist at the same flow conditions, but were not connected. In this study, we obtain a new family of branches of unstable highly deformed stationary drops connecting with the stable flattened shapes and the toroidal ones. We use a method based on classical control theory. The controller is designed for a two-state dynamic model of the system and is employed on a high-order nonlinear dynamic model of the drop deformation. Through this feedback-control-centred approach, an extended collection of unstable stationary solutions is constructed, which spans the range from the loss of stability to the dimpled shapes almost collapsed at the centre. In the latter region, which was never obtained in previous studies, a multiplicity of solutions is identified.

We experimentally investigate the forced synchronization of a self-excited chaotic thermoacoustic oscillator with two natural frequencies, $f_1$ and $f_2$. On increasing the forcing amplitude, $\epsilon _f$, at a fixed forcing frequency, $f_f$, we find two different types of synchronization: (i) $f_f/f_1 = 1:1$ or $2:1$ chaos-destroying synchronization (CDS), and (ii) phase synchronization of chaos (PSC). En route to $1:1$ CDS, the system transitions from an unforced chaotic state (${\rm {CH}}_{1,2}$) to a forced chaotic state (${\rm {CH}}_{1,2,f}$), then to a two-frequency quasiperiodic state where chaos is destroyed ($\mathbb {T}^2_{2,f}$), and finally to a phase-locked period-1 state (${\rm {P1}}_f$). The route to $2:1$ CDS is similar, but the quasiperiodic state hosts a doubled torus $(2\mathbb {T}^2_{2,f})$ that transforms into a phase-locked period-2 orbit $({\rm {P2}}_f)$ when CDS occurs. En route to PSC, the system transitions to a forced chaotic state (${\rm {CH}}_{1,2,f}$) followed by a phase-locked chaotic state, where $f_1$, $f_2$ and $f_f$ still coexist but their phase difference remains bounded. We find that the maximum reduction in thermoacoustic amplitude occurs near the onset of CDS, and that the critical $\epsilon _f$ required for the onset of CDS does not vary significantly with $f_f$. We then use two unidirectionally coupled Anishchenko–Astakhov oscillators to phenomenologically model the experimental synchronization dynamics, including (i) the route to $1:1$ CDS, (ii) various phase dynamics, such as phase drifting, slipping and locking, and (iii) the thermoacoustic amplitude variations in the $f_f/f_1$–$\epsilon _f$ plane. This study extends the applicability of open-loop control further to a chaotic thermoacoustic system, demonstrating (i) the feasibility of using an existing actuation strategy to weaken aperiodic thermoacoustic oscillations, and (ii) the possibility of developing new active suppression strategies based on both established and emerging methods of chaos control.

Superhydrophobic surfaces dramatically reduce skin friction of overlying liquid flows. These surfaces are complex and numerical simulations usually rely on models to reduce this complexity. One of the simplest consists of finding an equivalent boundary condition through a homogenisation procedure, which in the case of channel flow over oriented riblets, leads to the presence of a small spanwise component in the homogenised base flow velocity. This work aims at investigating the influence of such a three-dimensionality of the base flow on stability and transition in a channel with walls covered by oriented riblets. Linear stability for this base flow is investigated: a new instability region, linked to cross-flow effects, is observed. Tollmien–Schlichting waves are also retrieved but the most unstable are three-dimensional. Transient growth is also affected as oblique streaks with non-zero streamwise wavenumber become the most amplified perturbations. When transition is induced by Tollmien–Schlichting waves, after an initial exponential growth regime, streaky structures with large spanwise wavenumber rapidly arise. Modal mechanisms appear to play a leading role in the development of these structures and a secondary stability analysis is performed to retrieve successfully some of their characteristics. The second scenario, initiated with cross-flow vortices, displays a strong influence of nonlinearities. The flow develops into large quasi-spanwise-invariant structures before breaking down to turbulence. Secondary stability on the saturated cross-flow vortices sheds light on this stage of transition. In both cases, cross-flow effects dominate the flow dynamics, suggesting the need to consider the anisotropicity of the wall condition when modelling superhydrophobic surfaces.

Pressure-gradient-driven flows through sinusoidal channels have been studied. The analysis was carried out up to the formation of secondary nonlinear states and spanned a range of low and moderate Reynolds numbers. Direct numerical simulations were used to identify and determine the properties of steady as well as non-stationary, two-dimensional (2-D) and three-dimensional secondary flows. Our results indicate the existence of several distinct solution types. Two-dimensional, stationary flows with periodicity determined by the corrugation represent the first type. The second type is associated with the appearance of 2-D oscillatory flows arising from the onset of unstable travelling waves. Such oscillatory solutions are generally out of phase with the wall corrugation but could be in phase in special cases determined by the ratio of the critical disturbance wavelength and the channel corrugation wavelength. Consequently, several distinct types of time-dependent solutions are possible. The third type of solution results from the centrifugal effect caused by wall curvature and leads to three-dimensionalization of the flow through the onset of stationary streamwise vortices. Finally, various states resulting from the interaction of different solution types are possible. We examine those states and present a bifurcation diagram illustrating the formation of some of them. The results presented in this paper might help with the development of small-scale flow measurement and detection devices operating at low and moderate Reynolds numbers, as well as in the use of wall topographies for the intensification of mixing in flows with moderate, subturbulent Reynolds numbers.

Natural flyers are capable of producing excessive lift via a stabilized leading-edge vortex (LEV), which appears to linger above the wing for a longer duration than it could in an equivalent two-dimensional flow. Previous studies found this stabilization behaviour closely related to a spanwise flow along the LEV axis; however, it is still debatable how the spanwise flow influences the LEV stability. In this work, potential flow theory is adopted to model an LEV attached to a flat-plate wing. To account for the spanwise flow effect, we propose a finite-area sink (FAS) model which allows the dynamical interaction between co-located LEV and spanwise flow. Through linear stability analysis of the dynamical system associated with the LEV movement, we arrive at a stable spiral-sink type of equilibrium, which is the first mathematical evidence supporting LEV stabilization by spanwise flow. It is further concluded that the LEV stability can be enhanced by either increasing the strength or decreasing the cross-section area of the spanwise flow.

The influence of surface roughness on transition to turbulence in a Mach 4.5 boundary layer is studied using direct numerical simulations. Transition is initiated by the nonlinearly most dangerous inflow disturbance, which causes the earliest possible breakdown on a flat plate for the prescribed inflow energy and Mach number. This disturbance primarily comprises two normal second-mode instability waves and an oblique first mode. When localized roughness is introduced, its shape and location relative to the synchronization points of the inflow waves are confirmed to have a clear impact on the amplification of the second-mode instabilities. The change in modal amplification coincides with the change in the height of the near-wall region where the instability wave speed is supersonic relative to the mean flow; the net effect of a protruding roughness is destabilizing when placed upstream of the synchronization point and stabilizing when placed downstream. Assessment of the effect of the roughness location is followed by an optimization of the roughness height, abruptness and width with the objective of achieving maximum transition delay. The optimization is performed using an ensemble-variational (EnVar) approach, while the location of the roughness is fixed upstream of the synchronization points of the two second-mode waves. The optimal roughness disrupts the phase of the near-wall pressure waves, suppresses the amplification of the primary instability waves and mitigates the nonlinear interactions that lead to breakdown to turbulence. The outcome is a sustained non-turbulent flow throughout the computational domain.

The critical effect of the windward interior angles of elastically mounted trapezoidal bodies on a galloping instability is numerically investigated in this paper using two methodologies of high-fidelity computational fluid dynamics simulations and data-driven stability analysis using the eigensystem realization algorithm. A micro exploration of the dynamical response is processed to understand the mechanism underpinning the structural amplification at the initial stage of the galloping instability and the competition between wake and structural modes. It is observed that very small changes in the windward interior angle of an isosceles-trapezoidal body can provoke or suppress galloping – indeed, a small decrease or increase (low to $1^\circ$) of the windward interior angle from a right angle ($90^\circ$) can result in a significant enhancement and complete suppression, respectively, of the galloping oscillations. This supports our hypothesis that the contraction and/or expansion (viz., fore-aft tapering and/or widening) of the cross-section in the streamline direction has potential influences on galloping triggering from the geometrical perspective. The data-driven stability analysis is also applied to verify and analyse this phenomenon from the perspective of modal analysis. The experimental measurements are also conducted in the wind tunnel to support this hypothesis.

The effects of perturbation-based active flow control on supersonic rectangular twin jets (SRTJ) over a wide range of nozzle pressure ratios (NPR = 2.77 to 6.7, corresponding to fully expanded Mach numbers Mj = 1.3 to 1.9) were investigated. The aspect ratio and design Mach number for the bi-conic, converging-diverging nozzles were 2 and 1.5, respectively. The flow and acoustic fields of SRTJ are known to couple, often generating high near-field (NF) pressure fluctuations and elevated far-field (FF) noise levels. Large-scale structures (LSS), or equivalently instability waves or wave packets, are responsible for mixing noise, broadband shock-associated noise, screech and coupling. The primary objective of this research was to manipulate the development of LSS in this complex flow to better understand and mitigate their effects. The organization and passage frequency of the LSS were altered by excitation of instabilities over a wide range of frequencies and modes. Key findings include: (1) the screech mode of each jet was flapping along its minor axis; (2) the jets coupled, out-of-phase primarily in overexpanded cases and in-phase primarily in underexpanded cases, along the minor axis of the SRTJ; (3) coupling has significant effects on the NF pressure fluctuations, but only minor effect on the FF noise; (4) standing waves were observed only on the minor axis plane of the SRTJ; (5) altering or suppressing coupling can significantly reduce NF pressure fluctuations; (6) two high-frequency excitation methods proved effective in reducing the FF noise; and (7) nonlinear interactions between the screech tones and excitation input were observed in controlled cases in which screech was only partially suppressed.

The stability and sensitivity of two- and three-dimensional global modes developing on steady spanwise-homogeneous laminar separated flows around NACA 4412 swept wings are numerically investigated for different Reynolds numbers ${\textit {Re}}$ and angles of attack $\alpha$. The wake dynamics is driven by the two-dimensional von Kármán mode whose emergence threshold in the $\alpha \unicode{x2013}{\textit {Re}}$ plane is computed with that of the three-dimensional centrifugal mode. At the critical Reynolds number, the Strouhal number, the streamwise wavenumber of the von Kármán mode and the spanwise wavenumber of the leading three-dimensional centrifugal mode scale as a power law of $\alpha$. The introduction of a sweep angle attenuates the growth of all unstable modes and entails a Doppler effect in the leading modes’ dynamics and a shift towards non-zero frequencies of the three-dimensional centrifugal modes. These are found to be non-dispersive as opposed to the von Kármán modes. The sensitivity of the leading global modes is investigated in the vicinity of the critical conditions through adjoint-based methods. The growth-rate sensitivity map displays a region on the suction side of the wing, wherein a streamwise-oriented force has a net stabilising effect, comparable to what could have been obtained inside the recirculation bubble. In agreement with the predictions of the sensitivity analysis, a spanwise-homogeneous force suppresses the Hopf bifurcation and stabilises the entire branch of von Kármán modes. In the limit of small amplitudes, passive control via spanwise-wavy forcing produces a stabilising effect similar to that of a spanwise-homogeneous control and is more effective than localised spherical forces.