Large-amplitude internal solitary waves in a two-layer fluid system with a free surface are investigated in this paper. Laboratory experiments on strongly nonlinear internal solitary waves with a free surface for a deep configuration are conducted. After comparing the experimental data with the results of the Miyata–Choi–Camassa model that includes the free-surface effects (MCC-FS), we find that the MCC-FS model does not calculate accurately the internal solitary waves with a free surface. Thus, we develop a strongly nonlinear model for a deep configuration, namely the two-layer high-level Green–Naghdi (HLGN-FS) model that includes the free-surface effects. Numerical results of the HLGN-FS model, including the wave profile, velocity field and wave speed, are presented for three cases. The first case is a shallow configuration with $\rho _{2}/\rho _{1}=0.977$ and $h_{2}/h_{1}=1/4.13$, where $\rho _{2}$ and $\rho _{1}$ are the densities of the upper-fluid layer and the lower-fluid layer, respectively, and $h_{2}$ and $h_{1}$ are the depths of the upper-fluid layer and the lower-fluid layer, respectively. The second case is also a shallow configuration, where $h_{2}/h_{1}=1/5$ while $\rho _{2}/\rho _{1}=0.859$. The third case is related to the present physical experiments, where $\rho _{2}/\rho _{1}=0.869$ and $h_{2}/h_{1}=1/15$. It is shown that the MCC-FS model can provide accurate results for the shallow configurations. Meanwhile, the HLGN-FS model is shown to be accurate for describing the internal solitary waves for both shallow and deep configurations.

The onset of natural convection in an infinite horizontal layer of a fractured-porous medium is investigated. The breakdown of local equilibrium between the low-permeability matrix and fractures embedded in the matrix is accounted for by applying the dual-porosity dual-permeability model. The symmetric case of impermeable and isothermal boundaries of the layer is examined in detail. By means of linear perturbation analysis, the dispersion equation is derived, and its solutions are investigated numerically as well as analytically in a few asymptotic cases. It is determined that the critical Rayleigh number depends only on the permeability, heat conductivity ratios and the coefficient of heat and mass transfer between fractures and matrix. It is shown that the convection exhibits a rich variety of flow patterns at near-critical conditions. Nine flow regimes can arise with co-rotating or counter-rotating convection cells in the fractures and matrix. These modes can bifurcate to the plane flow regime, in which only cross-medium convection occurs. The complete classification of the flow regimes is provided and plotted in a solution map. Finally, the theoretical analysis is supported by the numerical modelling of the convection by using a reservoir simulator.

We perform interface-resolved simulations to study the modulation of statistically steady-state homogeneous shear turbulence by neutrally buoyant finite-size particles. We consider two shapes, spheres and oblates, and various solid volume fractions, up to 20%. The results show that a statistically steady state is not exclusive to single-phase homogeneous shear turbulence as the production and dissipation rates of the turbulent kinetic energy are also statistically in balance in particle-laden cases. The turbulent kinetic energy shows a non-monotonic behaviour with increasing solid volume fraction: increasing turbulence attenuation up to a certain concentration of solid particles and then enhancement of the turbulent kinetic energy at higher concentrations. This behaviour is observed at lower volume fractions for oblate particles than for spheres. The attenuation of the turbulence activity at lower volume fractions is explained through the enhancement of the dissipation rate close to the surface of particles. At higher volume fractions, however, particle pair interactions induce regions of high Reynolds shear stress, resulting in the enhancement of the turbulence activity. We show that the oblate particles of the considered size have larger rotational rates than spheres with no preferential orientation. This is in contrast to previous studies in wall-bounded flows where preferential orientation close to the wall and reduced rotation rates result in turbulence attenuation and thus drag reduction. Our results shed some light on the effect of rigid particles, smaller than the near-wall turbulent structures but still comparable to the viscous length scale, on the dynamics of the equilibrium logarithmic layer in wall-bounded flows.

This study examines the interactions of a compliant wall with a turbulent boundary layer as the deformation scale increases from submicron to several wall units (δν). The friction velocity Reynolds number ranges between 1435 and 5179, and $E/\rho U_0^2$, where E is the Young modulus, varies from 59 to 2.4, $\rho $ is fluid density and $ U_0$ is free-stream velocity. Time-resolved Mach–Zehnder interferometry is used for measuring the spatial distribution of the surface deformation, and two-dimensional (2-D) particle image velocimetry for measuring the velocity in the inner part of the boundary layer. Reynolds stresses and two-point correlations are measured in the log layer. The deformation amplitude increases from 0.02δν at $E/\rho U_0^2 = 59$ to 3.6δν at $E/\rho U_0^2 = 2.4$. Wavenumber–frequency and 2-D spatial spectra show that the deformations consist of two modes: The first is an advected mode that travels downstream at 66 % of U0, has a lattice-like structure and a preferential spanwise alignment. The amplitude and frequency of this mode agree with the Chase (J. Acoust. Soc. Am., vol. 89, no. 6, 1991, pp. 2589–2596) and Benschop et al. (J. Fluid Mech., vol. 859, 2019, pp. 613–658) model predictions. The second mode is a streamwise-aligned wave that travels at the material shear speed (Ct = 7.85 m s−1) in the spanwise direction and has a wavelength of three times the compliant layer thickness. With decreasing $E/\rho U_0^2$, the velocity profiles in the boundary layer increasingly deviate from those of a rigid smooth wall. Yet, these deviations begin when the deformation is 0.02δν. The most prominent features are a sharp decrease in velocity at y < 10δν and an increase in the near-wall turbulence, both consistent, for matching $E/\rho U_0^2$, with the direct numerical simulation results of Rosti and Brandt (J. Fluid Mech., vol. 830, 2017, pp. 708–735).

In this work, we perform weakly nonlinear analysis of the reflection process of a thin oscillating wave beam on a non-critical surface in a fluid rotating and stratified along the same vertical axis in the limit of weak viscosity, i.e. small Ekman number $E$. We assume that the beam has the self-similar viscous structure obtained by Moore & Saffman (Phil. Trans. R. Soc. A, vol. 264, 1969, pp. 597–634) and Thomas & Stevenson (J. Fluid Mech., vol. 54, 1972, pp. 495–506). Such a solution describes the viscous internal shear layers of width $O(E^{1/3})$ generated by a localized oscillating source. We first show that the reflected beam conserves at leading order the self-similar structure of the incident beam and is modified by an $O(E^{1/6})$ correction with a different self-similar structure. We then analyse the nonlinear interaction of the reflected beam with the incident beam of amplitude $\varepsilon$ and demonstrate that a second-harmonic beam and localized meanflow correction, both of amplitude $\varepsilon ^{2} E^{-1/3}$, are created. We further show that for the purely stratified case (respectively the purely rotating case), a non-localized meanflow correction of amplitude $\varepsilon ^{2} E^{-1/6}$ is generated, except when the boundary is horizontal (respectively vertical). In this latter case, the meanflow correction remains localized but exhibits a triple-layer structure with a large $O(E^{4/9})$ viscous layer.

The evaporation of multicomponent droplets is relevant to various applications but challenging to study due to the complex physicochemical dynamics. Recently, Li etal. (Phys. Rev. Lett., vol. 120, 2018, 224501) reported evaporation-triggered segregation in 1,2-hexanediol–water binary droplets. In this present work, we added 0.5 wt % silicone oil to the 1,2-hexanediol–water binary solution. This minute silicone oil concentration dramatically modifies the evaporation process, as it triggers an early extraction of the 1,2-hexanediol from the mixture. Surprisingly, we observe that the segregation of 1,2-hexanediol forms plumes, rising up from the rim of the sessile droplet towards the apex during droplet evaporation. By orientating the droplet upside down, i.e. by studying a pendent droplet, the absence of the plumes indicates that the flow structure is induced by buoyancy, which drives a Rayleigh–Taylor instability (i.e. driven by density differences and gravitational acceleration). From micro particle image velocimetry measurement, we further prove that the segregation of the non-volatile component (1,2-hexanediol) hinders the evaporation near the contact line, which leads to a suppression of the Marangoni flow in this region. Hence, on long time scales, gravitational effects, rather than Marangoni flows, play the dominant role in the flow structure. We compare the measurement of the evaporation rate with the diffusion model of Popov (Phys. Rev., vol. 71, 2005, 036313), coupled with Raoult's law and the activity coefficient. This comparison indeed confirms that the silicone-oil-triggered segregation of the non-volatile 1,2-hexanediol significantly delays the evaporation. With an extended diffusion model, in which the influence of the segregation has been implemented, the evaporation can be well described.

Free-stream turbulence (FST) and its effect on boundary-layer transition is an intricate problem. Elongated unsteady streamwise streaks of low and high speed are created inside the boundary layer and their amplitude and spanwise wavelength are believed to be important for the onset of transition. The transitional Reynolds number is often simply correlated with the turbulence intensity (${Tu}$), and the characteristic length scales of the FST are often considered to have a small to negligible influence on the transition location. Here, we present new results from a large experimental measurement campaign, where both the ${Tu}$ and the integral length scale ($\Lambda _x$) are varied ($1.8\,\% < {Tu}< 6.2\,\%$; $16\ \textrm {mm}< \Lambda _x < 26\ \textrm {mm}$). In the current experiments it has been noted that on the one hand, for small $Tu$, an increase in $\Lambda _x$ advances transition, which is in agreement with established results. On the other hand, for large $Tu$, an increase in $\Lambda _x$ postpones transition. This trend can be explained by the fact that an optimal ratio between FST length scale and boundary-layer thickness at transition onset exists. Furthermore, our results strengthen the fact that the streaks play a key role in the transition process by showing a clear dependence of the FST characteristics on their spanwise scale. Our measurements show that the aspect ratio of the streaky structures correlates with an FST Reynolds number and that the aspect ratio can change by a factor of two at the location of transition. Finally, we derive a semi-empirical transition prediction model, which is able to predict the influence of $\Lambda _x$ for both small and high values of ${Tu}$.

Mixing of binary fluids by moving stirrers is a commonplace process in many industrial applications, where even modest improvements in mixing efficiency could translate into considerable power savings or enhanced product quality. We propose a gradient-based nonlinear optimisation scheme to minimise the mix-norm of a passive scalar. The velocities of two cylindrical stirrers, moving on concentric circular paths inside a circular container, represent the control variables, and an iterative direct–adjoint algorithm is employed to arrive at enhanced mixing results. The associated stirring protocol is characterised by a complex interplay of vortical structures, generated and promoted by the stirrers’ action. Full convergence of the optimisation process requires constraints that penalise the acceleration of the moving bodies. Under these conditions, considerable mixing enhancement can be accomplished, even though an optimum cannot be guaranteed due to the non-convex nature of the optimisation problem. Various challenges and extensions of our approach are discussed.

The Fokker–Planck approximation to the Boltzmann equation has emerged as an efficient alternative to the discrete simulation Monte Carlo method for various flow simulations. This method has been mostly limited to simulating single-component rarefied gas flows. In the present paper, we propose two models based on the Fokker–Planck equation and quasi-equilibrium models that are capable of describing the dynamics of rarefied binary gas mixtures over a large range of Schmidt numbers. We first prove that these models satisfy the necessary conservation laws and the $H$-theorem. We validate the model by simulating three benchmark problems – Graham's law for effusion, Couette flow and binary diffusion.

Thermal plumes are the energy-containing eddy motions that carry heat and momentum in a convective boundary layer. The detailed understanding of their structure is of fundamental interest for a range of applications, from wall-bounded engineering flows to quantifying surface–atmosphere flux exchanges. We address the aspect of Reynolds stress anisotropy associated with the intermittent nature of heat transport in thermal plumes by performing an invariant analysis of the Reynolds stress tensor in an unstable atmospheric surface layer flow, using a field-experimental dataset. Given the intermittent and asymmetric nature of the turbulent heat flux, we formulate this problem in an event-based framework. In this approach, we provide structural descriptions of warm-updraft and cold-downdraft events and investigate the degree of isotropy of the Reynolds stress tensor within these events of different sizes. We discover that only a subset of these events are associated with the least anisotropic turbulence in highly convective conditions. Additionally, intermittent large-heat-flux events are found to contribute substantially to turbulence anisotropy under unstable stratification. Moreover, we find that the sizes related to the maximum value of the degree of isotropy do not correspond to the peak positions of the heat-flux distributions. This is because the vertical velocity fluctuations pertaining to the sizes associated with the maximum heat flux transport a significant amount of streamwise momentum. A preliminary investigation shows that the sizes of the least anisotropic events probably scale with a mixed length scale ($z^{0.5}\lambda ^{0.5}$, where $z$ is the measurement height and $\lambda$ is the large-eddy length scale).

We show how settling and phase change can combine to drive an instability, as a simple model for the formation of mammatus clouds. Our idealised system consists of a layer (an ‘anvil’) of air mixed with saturated water vapour and monodisperse water droplets, sitting atop dry air. The water droplets in the anvil settle under gravity due to their finite size, evaporating as they enter dry air and cooling the layer of air just below the anvil. The colder air just below the anvil thus becomes denser than the dry air below it, forming a density ‘overhang’, which is unstable. The strength of the instability depends on the density difference between the density overhang and the dry ambient, and the depth of the overhang. Using linear stability analysis and nonlinear simulations in one, two and three dimensions, we study how the amplitude and depth of the density layer depend on the initial conditions, finding that their variations can be explained in terms only of the size of the droplets making up the liquid content of the anvil and by the total amount of liquid water contained in the anvil. We find that the size of the water droplets is the controlling factor in the structure of the clouds: mammatus-like lobes form for large droplet sizes; and small droplet sizes lead to a ‘leaky’ instability resulting in a stringy cloud structure resembling the newly designated asperitas.

Highly stretched liquid drops, or filaments, surrounded by a gas are routinely encountered in nature and industry. Such filaments can exhibit complex and unexpected dynamics as they contract under the action of surface tension. Instead of simply retracting to a sphere of the same volume, low-viscosity filaments exceeding a critical aspect ratio undergo localized pinch-off at their two ends resulting in a sequence of daughter droplets – a phenomenon called endpinching – which is an archetype breakup mode that is distinct from the classical Rayleigh–Plateau instability seen in jet breakup. It has been shown that endpinching can be precluded in filaments of intermediate viscosity, with the so-called escape from endpinching being understood heretofore only qualitatively as being caused by a viscous mechanism. Here, we show that a similar escape can also occur in nearly inviscid filaments when surfactants are present at the free surface of a recoiling filament. The fluid dynamics of the escape phenomenon is probed by numerical simulations. The computational results are used to show that the escape is driven by the action of Marangoni stress. Despite the apparently distinct physical origins of escape in moderately viscous surfactant-free filaments and that in nearly inviscid but surfactant-covered filaments, it is demonstrated that the genesis of all escape events can be attributed to a single cause – the generation of vorticity at curved interfaces. By analysing vorticity dynamics and the balance of vorticity in recoiling filaments, the manner in which surface tension gradients and concomitant Marangoni stresses can lead to escape from endpinching is clarified.

Destratification of thermally stratified open-channel flow by surface cooling is investigated using direct numerical simulation. The initial states are the equilibrium states resulting from radiative heating. Using these states as initial conditions, a series of direct numerical simulations was run with radiative heating removed and a constant, uniform cooling flux applied at the upper surface. The flow evolves until the initial stable stratification is broken down and replaced by unstable stratification driven by surface cooling. The destratification process is described with reference to the evolution of the internal structure of the turbulent flow field. Based on these observations, we conclude that the dominant time scales in the flow from the perspective of destratification are the time scales associated with shear ${t}_{\tau }$, convection ${t}_*$ and stable density stratification ${t}_N$. Scaling arguments are then used to derive a scaling relationship for destratification rate as a function of a friction Richardson number $Ri_{\tau } = ( {t}_{\tau }/ {t}_N)^2$ and a convection Richardson number $Ri_* = ( {t}_*/ {t}_N)^2$. The relationship takes the form ${\mathcal {D}}_N = C_1Ri_{\tau }^{-1} + C_2Ri_*^{-1}$, where ${\mathcal {D}}_N$ is the destratification rate non-dimensionalised with respect to $ {t}_N$ and $C_1$ and $C_2$ are model coefficients. The relationship is compared with simulation results and is shown to accurately predict the destratification rate in the simulations across a range of parameters. This relationship is then integrated to give a formula for the time taken for the flow to destratify.

In this paper the configurations of shock wave–boundary layer interactions (SWBLI) are studied theoretically and experimentally in Mach number 2 and 2.5 flows on test models with various wedge angles ranging from $9^\circ$ to $21^\circ$. The proposed theoretical method couples the free interaction theory (FIT) with the minimum entropy production (MEP) principle to predict the appearance of separation shock, resulting in convex, straight and concave separation shock waves according to different solution combinations, which agree well with current experiments. Additionally, several influences on SWBLI are studied experimentally, in which the parameters related to theoretical solutions are found mostly determining the flow configuration, and SWBLI is much more sensitive to incident shock strength than incoming flow properties. Separation could be suppressed by incident shock when the MEP solution is smaller than the FIT, while it could be intensified when the MEP solution is larger than FIT; by contrast, the effects of separation position and model mounting height could be very weak.

In the context of waterflooding in oil recovery, the motion of an oil droplet through a capillary pore initially filled with aqueous liquid is considered. The droplet is affected by capillary and viscous forces, with a thin aqueous film being formed between the droplet and capillary wall. Moreover, the droplet surface and capillary wall surface have opposite and equal electrical charge. Attractive electro-osmotic interactions then tend to thin the film. A case is considered in which electro-osmotic interactions are strong and capillary forces are inherently weak, leading, in the first instance, to a viscous, electro-osmotic balance. Solutions are obtained for the droplet shape close to its front end. Whilst visco-electro-osmotic dominated solutions can indeed be found, additional solution classes are identified for which the film thickness oscillates with longitudinal position, and capillary forces regain importance. A parametric study is presented indicating that oscillation length scales along the film can be selected such that capillary effects never become negligible. Moreover gradients of electro-osmotic conjoining pressures are small in thinner parts of the film, even though electro-osmotic conjoining pressures themselves are not. Thus, rather than a visco-electro-osmotic balance being the norm, a capillary, viscous balance results in thinner parts of the film, giving way to a capillary, electro-osmotic balance in thicker parts. However, solutions are non-unique, and a given system can admit multiple solutions with films of various different thicknesses. Some of these solutions have films that increase monotonically in thickness with position, while others fall into the oscillatory class.

Three-dimensional flow fields of a rectangular synthetic jet with aspect ratio of 3 and its interaction with a laminar cross-flow are measured using time-resolved tomographic particle image velocimetry. The synthetic-jet to free-stream velocity ratio is fixed at 1, and two orifice orientations, normal to (spanwise configuration) and aligned with (streamwise configuration) the cross-flow direction, respectively, are investigated. The flow scenarios are composed of a tilted vortex ring, a secondary trailing vortex and a tertiary near-wall vortex. Compared to previous results for circular cases, the legs of the trailing structures are found to be jointed in the spanwise direction by two shear-layer vortices at different wall-normal heights. In addition, the near-wall vortex is a crescent-shaped spanwise vortex. For the first time, as the peculiar process of non-circular vortex rings in quiescent condition, axis switching is validated to exist in the cross-flow. Associated with axis switching, the spanwise case exhibits interaction between the leeward vortex of the vortex ring and the downstream shear-layer vortex, which causes slower penetration and recovery of vortex ring tilting, but faster jet transition. Furthermore, the trailing structures and vortex ring are weakened, leading to lower peaks of the velocity deficit and overshoot in the jet region than those for the streamwise case. On the other hand, the spanwise case can induce a stronger near-wall vortex, which transfers more momentum and initial disturbance into the boundary layer. Consequently, the spanwise case yields larger velocity gradient and velocity fluctuation peak inside the boundary layer with a lower shape factor, suggesting greater efficiency of promoting boundary-layer transition.

We study rapidly rotating turbulent flows in a highly elongated domain using an asymptotic expansion at simultaneously low Rossby number $Ro\ll 1$ and large domain height compared with the energy injection scale, $h=H/\ell _{in}\gg 1$. We solve the resulting equations using an extensive set of direct numerical simulations for different parameter regimes. As the parameter $\lambda = (h Ro)^{-1}$ is increased beyond a threshold $\lambda _c$, a transition is observed from a state without an inverse energy cascade to a state with an inverse energy cascade. For large Reynolds number and large horizontal box size, we provide evidence for criticality of the transition in terms of the large-scale energy dissipation rate.

We investigate the impact on the boundary-layer stability of spanwise-periodic, streamwise-elongated surface roughness elements. Our interest is in their effects on the so-called lower-branch Tollmien–Schlichting modes, and so the spanwise spacing of the elements is taken to be comparable with the spanwise wavelength of the latter, which is of $O(R^{-3/8} L)$, where $L$ is the dimensional length from the leading edge of the flat plate to the surface roughness, and $R$ is the Reynolds number based on $L$. The streamwise length is much longer, consistent with experimental set-ups. The roughness height is chosen such that the wall shear is altered by $O(1)$. From the generic triple-deck theory for three-dimensional roughness elements with both the streamwise and spanwise length scales being of $O(R^{-3/8 }L)$, we derived the relevant governing equations by appropriate rescaling. The resulting equations are nonlinear but parabolic because the pressure gradient in the streamwise direction is negligible while in the spanwise direction it is completely determined by the roughness shape. Appropriate upstream, boundary and matching conditions are derived for the problem. Owing to the parabolicity, the equations are solved efficiently using a marching method to obtain the streaky flow. The instability of the streaky flow is shown to be controlled by the spanwise-dependent (periodic) wall shear. Two- and weakly three-dimensional lower-frequency modes are found to be stabilised by the streaks, confirming previous experimental findings, while stronger three-dimensional and higher-frequency modes are destabilised. Among the three roughness shapes considered, the roughness elements in the form of hemispherical caps are found to be most effective for a given height. A resonant subharmonic interaction was found to occur for modes with spanwise wavelength twice that of the roughness elements.

We experimentally study the nonlinear flow-induced instability of an elastically mounted pitching wing in a circulating water tunnel. The structural parameters of the finite-span wing are simulated and regulated using a cyber-physical control system. At a small fixed damping, we systematically vary the stiffness of the wing for different inertia values to test for the stability boundaries of the system. We observe that, for a high-inertia wing, the system dynamics bifurcates from stable fixed points to small-amplitude oscillations followed by large-amplitude limit-cycle oscillations (LCOs) via a subcritical bifurcation, which features hysteretic bistability and an abrupt amplitude jump. Under this condition, the pitching frequency of the wing locks onto its structural frequency and the oscillation is dominated by the inertial force, corresponding to a structural mode. Force and flow field measurements indicate the presence of a secondary leading-edge vortex (LEV). As the wing inertia decreases, the width of the bistable region shrinks. At a sufficiently low inertia, the pitching amplitude changes smoothly with the stiffness without any hysteresis, revealing a supercritical bifurcation. Under this condition, no lock-in phenomenon is observed and the pitching frequency remains relatively constant at a value lower than the structural frequency. Force decomposition shows dominating fluid force, indicating a hydrodynamic mode. The secondary LEV is absent. We show that the onset of large-amplitude LCOs in both the structural mode and the hydrodynamic mode scales with the Cauchy number, and the LCOs in the structural mode collapse with the non-dimensional velocity. We examine the subcritical transition in detail; we find that this transition depends on the static characteristics of the wing, and the secondary LEV starts to emerge at the early stage of the transition. Lastly, we adopt an energy approach to map out the stability of the system and explain the existence of the two distinct types of bifurcations observed for different inertia values.

We experimentally study the unsteady forcing of the turbulent wake of a three-dimensional blunt body for drag reduction purposes. The forcing is provided by pulsed jets coupled to small flush-mounted curved surfaces and affects the dynamics of the shear layer at separation from the trailing edge of the model. The systematic analysis of the influence of various parameters (forcing frequency and amplitude, radius of curvature $r$ of the surfaces, free-stream velocity $U_0$) on the base drag reduction provides key ingredients to identify proper scaling laws of the mechanisms involved and to model them. The flow reattachment and separation on the curved surfaces result in a boat-tailing of the wake leading to drag reductions of up to $12\,\%$ and are noticeably influenced by the time scale of unsteadiness of the forcing. For high frequencies of the order of $O(U_0/r)$, strong vortical coherent structures produced by the interaction between the pulsed jets and the separating shear layer promote the interaction of the flow with the curved surfaces. Moreover, the local curvature and pressure gradients across the separating shear layer in the vicinity of flow separation are noticeably modified to result in a further pressure drag reduction for a given forcing amplitude. A simple inviscid-flow model illustrates the peculiar induced effect of these coherent structures on the flow, which explains both the curvature effects leading to additional drag decrease and the saturation in drag decrease for increasing forcing amplitude. The results point to the need for careful combination between forcing frequency and size of the curved surfaces to achieve all the potential in drag reduction of the unsteady Coanda effect. The effort to propose scaling laws and models of the unsteady Coanda effect is a step towards implementing this control strategy at an industrial scale or on different fluid dynamics problems.