It was only a matter of time before deep neural networks (DNNs) – deep learning – made their mark in turbulence modelling, or more broadly, in the general area of high-dimensional, complex dynamical systems. In the last decade, DNNs have become a dominant data mining tool for big data applications. Although neural networks have been applied previously to complex fluid flows, the article featured here (Ling et al., J. Fluid Mech., vol. 807, 2016, pp. 155–166) is the first to apply a true DNN architecture, specifically to Reynolds averaged Navier Stokes turbulence models. As one often expects with modern DNNs, performance gains are achieved over competing state-of-the-art methods, suggesting that DNNs may play a critically enabling role in the future of modelling complex flows.

In countless biological and technological processes, the flow of Newtonian liquids with a non-Newtonian interface is a common occurrence, such as in monomolecular films in ‘solid’ phases atop of aqueous bulk fluid. There is a lack of models that can predict the flow under conditions different from those used to measure the rheological response of the interface. Here, we present a model which describes interfacial hydrodynamics, including two-way coupling to a bulk Newtonian fluid described by the Navier–Stokes equations, that allows for shear-thinning response of the interface. The model includes a constitutive equation for the interface under steady shear that takes the Newtonian functional form but where the surface shear viscosity is generalized to be a function of the local shear rate. In the limit of a highly viscous interface, the interfacial hydrodynamics is decoupled from the bulk flow and the model can be solved analytically. This provides not only insight into the flow but also a means to validate the numerical technique for solving the two-way coupled problem. The numerical results of the coupled problem shed new light on existing experimental results on steadily sheared monolayers of dipalmitoylphosphatidylcholine (DPPC), the primary constituent of lung surfactant and the bilayers of mammalian cell walls. For low packing density DPPC monolayers, a Newtonian shear-independent surface shear viscosity model can reproduce the interfacial flows, but at high packing density, the shear-thinning properties of the new model presented here are needed.

The effect of viscosity on the longwave Kelvin–Helmholtz instability of two immiscible incompressible fluids under horizontal vibrations is considered. The linear stability boundaries are found analytically using series expansion in terms of small wavenumber. The values of parameters, at which a transition from the longwave to finite-wavelength instability takes place, are determined. It has been shown that for high-frequency vibrations a viscous dissipation has just a weak destabilizing effect. At vibrations of moderate frequencies, destabilization is more significant, especially in the systems with large viscosity contrast. In contrast to that, at low frequencies the viscosity stabilizes the basic flow by suppressing the longwave perturbations.

A two orthogonal view, holographic cinematography system (volume of $17\times 17\times 17~\text{mm}^{3}$ ) was used to measure three-dimensional fibre translational velocities, orientations and rotation rates in near homogeneous isotropic air turbulence (HIT). Flow characteristics were determined from temporally resolved particle image velocimetry measurements. Two sets of rigid, nylon fibres having the same nominal length (0.5 mm) but different diameters (13.7 and $19.1~\unicode[STIX]{x03BC}\text{m}$ ), were released in near HIT at a Taylor microscale Reynolds number of $Re_{\unicode[STIX]{x1D706}}\approx 130$ and tracked at more than five times the Kolmogorov frequency. The ratio of fibre length to the Kolmogorov length scale was 2.8 and the two sets were characterized by Stokes numbers of 1.35 and 2.44, respectively. As a result of increased inertia, the probability density functions (PDFs) of the fluctuating fibre translational velocities were narrower than the ones of the air and the fibre velocity autocorrelations decreased at a decreasing rate. While fibre orientations in the cameras’ frame of reference were random as a result of the strong turbulence, it was shown that fibres align with the flow to minimize drag. PDFs of the fibre rotation rates indicated the occurrence of extreme rotation rate events. Furthermore, increasing inertia lowered the normalized, mean squared fibre rotation rates in comparison to results obtained for neutrally buoyant fibres having the same aspect ratio and including the effect of preferential alignment. The present results compare well to direct numerical simulations including the effect of fibre inertia.

The effects of superhydrophobic surfaces (SHSs), which consist of microgrates oriented transverse to the flow direction, on the onset of three-dimensional instability of flow past a bluff body were studied using Floquet analysis. The SHS was modelled on an air–water interface with a shear-free condition. The results showed that SHSs increased the vortex shedding frequency. Floquet analysis revealed that modes B $^{\prime }$ and S $^{\prime }$ were suppressed dramatically by the partial-slip condition compared with a regular no-slip body; however, mode A was less affected. Correspondingly, the critical spanwise wavelengths were not significantly affected by SHSs. A similar phenomenon was observed in flow past a circular cylinder coated by SHSs. The results also revealed that modes B $^{\prime }$ and S $^{\prime }$ were collapsed into mode A due to the increased width of the air–water region for flow past an elongated body. Furthermore, the critical Reynold numbers of different modes were diversely affected by gas fraction (GF) variations. The unstable modes with short wavelengths, such as modes B $^{\prime }$ and S $^{\prime }$ , stabilized with increasing GF. Conversely, the opposite was seen for the unstable mode A with a longer wavelength. The exact critical Reynolds number depended on the geometric configuration, which should be between the critical values of the two extreme cases. The application of SHSs could modify the transition route from two- to three-dimensionality by alternating different unstable modes. As the wavelength of the unstable mode decreases, the inhibition of three-dimensional instability becomes more efficient by SHSs.

While neutral atmospheric boundary layers are rare over land, they occur frequently over sea. In these cases they are almost always of the conventionally neutral type, in which the neutral boundary layer is capped by a strong inversion layer and a stably stratified atmosphere aloft. In the current study, we use large-eddy simulations (LES) to investigate the interaction between a large wind farm that has a fetch of 15 km and a conventionally neutral boundary layer (CNBL) in typical offshore conditions. At the domain inlet, we consider three different equilibrium CNBLs with heights of approximately 300 m, 500 m and 1000 m that are generated in a separate precursor LES. We find that the height of the inflow boundary layer has a significant impact on the wind farm flow development. First of all, above the farm, an internal boundary layer develops that interacts downwind with the capping inversion for the two lowest CNBL cases. Secondly, the upward displacement of the boundary layer by flow deceleration in the wind farm excites gravity waves in the inversion layer and the free atmosphere above. For the lower CNBL cases, these waves induce significant pressure gradients in the farm (both favourable and unfavourable depending on location and case). A detailed energy budget analysis in the turbine region shows that energy extracted by the wind turbines comes both from flow deceleration and from vertical turbulent entrainment. Though turbulent transport dominates near the end of the farm, flow deceleration remains significant, i.e. up to 35 % of the turbulent flux for the lowest CNBL case. In fact, while the turbulent fluxes are fully developed after eight turbine rows, the mean flow does not reach a stationary regime. A further energy budget analysis over the rest of the CNBL reveals that all energy available at turbine level comes from upwind kinetic energy in the boundary layer. In the lower CNBL cases, the pressure field induced by gravity waves plays an important role in redistributing this energy throughout the farm. Overall, in all cases entrainment at the capping inversion is negligible, and also the work done by the mean background pressure gradient, arising from the geostrophic balance in the free atmosphere, is small.

We consider the water entry of horizontal cylinders with vertical impact velocity, either kept constant or freely falling, without and with spin, into quiescent water under the effect of gravity. We focus on the flow and cavity forming stages with non-dimensional submergence time $t$ , Froude numbers $Fr$ , spin ratios $\unicode[STIX]{x1D6FC}$ and mass ratios $m$ , all of $O(1)$ . We develop numerical simulations using a modified smoothed particle hydrodynamics method to obtain predictions for the impact kinematics and dynamics. These are in detailed agreement with available experiments. We elucidate the evolutions of the free surface, contact point positions, flow field, forces and trajectories and their dependence on $Fr$ , $\unicode[STIX]{x1D6FC}$ and $m$ . We define and quantify the contact point location $\unicode[STIX]{x1D703}(t)$ as a function of $Fr$ , clarifying the qualitative difference between sub- and supercritical $Fr$ and the observed absence of air-entrained trailing cavities at low $Fr$ . By subtracting the buoyancy associated with $\unicode[STIX]{x1D703}(t)$ , we show that, unlike the total drag, the remaining dynamic components are qualitatively similar for all $Fr$ . For a freely falling cylinder, we show that the total drag can be predicted from the constant velocity case with the same instantaneous velocity, providing a simple way to predict its trajectory based on the latter. The presence of spin results in lift, even when the asymmetry in $\unicode[STIX]{x1D703}$ is small. For fixed $\unicode[STIX]{x1D6FC}$ , lift increases with subcritical $Fr$ . For a freely falling cylinder, the lateral motion causes an appreciable asymmetry in $\unicode[STIX]{x1D703}$ and a reduction in lift.

We investigate the statistical properties of the kinetic $\unicode[STIX]{x1D700}_{u}$ and thermal $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D703}}$ energy dissipation rates in two-dimensional (2-D) turbulent Rayleigh–Bénard (RB) convection. Direct numerical simulations were carried out in a box with unit aspect ratio in the Rayleigh number range $10^{6}\leqslant Ra\leqslant 10^{10}$ for Prandtl numbers $Pr=0.7$ and 5.3. The probability density functions (PDFs) of both dissipation rates are found to deviate significantly from a log-normal distribution. The PDF tails can be well described by a stretched exponential function, and become broader for higher Rayleigh number and lower Prandtl number, indicating an increasing degree of small-scale intermittency with increasing Reynolds number. Our results show that the ensemble averages $\langle \unicode[STIX]{x1D700}_{u}\rangle _{V,t}$ and $\langle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D703}}\rangle _{V,t}$ scale as $Ra^{-0.18\sim -0.20}$ , which is in excellent agreement with the scaling estimated from the two global exact relations for the dissipation rates. By separating the bulk and boundary-layer contributions to the total dissipations, our results further reveal that $\langle \unicode[STIX]{x1D700}_{u}\rangle _{V,t}$ and $\langle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D703}}\rangle _{V,t}$ are both dominated by the boundary layers, corresponding to regimes $I_{l}$ and $I_{u}$ in the Grossmann–Lohse (GL) theory (J. Fluid Mech., vol. 407, 2000, pp. 27–56). To include the effects of thermal plumes, the plume–background partition is also considered and $\langle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D703}}\rangle _{V,t}$ is found to be plume dominated. Moreover, the boundary-layer/plume contributions scale as those predicted by the GL theory, while the deviations from the GL predictions are observed for the bulk/background contributions. The possible reasons for the deviations are discussed.

Non-ideal gases refer to deformable substances in which the speed of sound can decrease following an isentropic compression. This may occur near a phase transition such as the liquid–vapour critical point due to long-range molecular interactions. Isentropes can then become locally concave in the pressure/specific-volume phase diagram (e.g. Bethe–Zel’dovich–Thompson (BZT) gases). Following the pioneering work of Bethe (Tech. Rep. 545, Office of Scientific Research and Development, 1942) on shocks in non-ideal gases, this paper explores the refraction properties of stable compression shocks in non-reacting but arbitrary substances featuring a positive isobaric volume expansivity. A small-perturbation analysis is carried out to obtain analytical expressions for the thermo-acoustic properties of the refracted field normal to the shock front. Three new regimes are discovered: (i) an extensive but selective (in upstream Mach numbers) amplification of the entropy mode (hundreds of times larger than those of a corresponding ideal gas); (ii) discontinuous (in upstream Mach numbers) refraction properties following the appearance of non-admissible portions of the shock adiabats; (iii) the emergence of a phase shift for the generated acoustic modes and therefore the existence of conditions for which the perturbed shock does not produce any acoustic field (i.e. ‘quiet’ shocks, to contrast with the spontaneous D’yakov–Kontorovich acoustic emission expected in 2D or 3D). In the context of multidimensional flows, and compressible turbulence in particular, these results demonstrate a variety of pathways by which a supplied amount of energy (in the form of an entropy, vortical or acoustic mode) can be redistributed in the form of other entropy, acoustic and vortical modes in a manner that is simply not achievable in ideal gases. These findings are relevant for turbines and compressors operating close to the liquid–vapour critical point (e.g. organic Rankine cycle expanders, supercritical $\text{CO}_{2}$ compressors), astrophysical flows modelled as continuum media with exotic equations of state (e.g. the early Universe) or Bose–Einstein condensates with small but finite temperature effects.

The flow of two stratified viscous immiscible perfect dielectric fluids in a channel with topographically structured walls is investigated. The flow is driven by a streamwise pressure gradient and an electric field across the channel gap. This problem is explored in detail by deriving and studying a nonlinear evolution equation for the interface valid for large-amplitude long waves in the Stokes flow regime. For flat walls, the electrified flow is long-wave unstable with a critical cutoff wavenumber that increases linearly with the magnitude of the applied voltage. In the nonlinear regime, it is found that the presence of pressure-driven flow prevents electrostatically induced interface touchdown that has been observed previously – time-modulated nonlinear travelling waves emerge instead. When topography is present, linearly stable uniform flows become non-uniform spatially periodic steady states; a small-amplitude asymptotic theory is carried out and compared with computations. In the linearly unstable regime, intricate nonlinear structures emerge that depend, among other things, on the magnitude of the wall corrugations. For a low-amplitude sinusoidal boundary, time-modulated travelling waves are observed that are similar to those found for flat walls but are influenced by the geometry of the wall and slide over it without touching. The flow over a high-amplitude sinusoidal pattern is also examined in detail and it is found that for sufficiently large voltages the interface evolves to large-amplitude waves that span the channel and are subharmonic relative to the wall. A type of ‘walking’ motion emerges that causes the lower fluid to wash through the troughs and create strong vortices over the peaks of the lower boundary. Non-uniform steady states induced by the topography are calculated numerically for moderate and large values of the flow rate, and their stability is analysed using Floquet theory. The effect of large flow rates is also considered asymptotically to find solutions that compare very well with numerical computations.

Small perturbation evolution in compressible Poiseuille flow is contrasted against the incompressible case using direct simulations and non-modal linear analysis. The onset of compressibility effects leads to a profound change in the behaviour of pressure and its interaction with the velocity field. Linear analysis shows that the most significant compressibility outcome is the harmonic coupling between pressure and wall-normal velocity perturbations. Oscillations in normal perturbations can lead to periods of negative production causing suppression of perturbation growth. The extent of the influence of compressibility can be characterized in terms of an effective gradient Mach number ( $M_{g}^{e}$ ). Analysis shows that $M_{g}^{e}$ diminishes as the angle of the perturbation increases with respect to the shear plane. Direct numerical simulations show that streamwise perturbations, which would lead to Tollmien–Schlichting instability in the incompressible case, are completely suppressed in the compressible case and experience the highest $M_{g}^{e}$ . At the other extreme, computations reveal that spanwise perturbations, which experience negligible $M_{g}^{e}$ , are entirely unaltered from the incompressible case. Perturbation behaviour at intermediate obliqueness angles is established. Moreover, the underlying pressure–velocity interactions are explicated.

A universal modelling approach of drop fragmentation after head-on drop collisions is presented. In this approach, the colliding drops are seen as liquid springs that coalesce, compress and relax, leading the merged drop to break up if it reaches a critical aspect ratio. Combining energetic balance of the compression and relaxation phases with a Rayleigh-like criterion, we deduce the fragmentation threshold velocity for the collision of two and three drops of the same liquid and of two drops of immiscible liquids. Predictions and experimental results obtained for these three kinds of collisions using various liquids and drop sizes are found to be in good agreement over a wide domain whose boundaries are discussed.

We compute the apparent hydrodynamic slip length for (laminar and fully developed) Poiseuille flow of liquid through a heated parallel-plate channel. One side of the channel is textured with parallel (streamwise) ridges and the opposite one is smooth. On the textured side of the channel, the liquid is in the Cassie state. No-slip and constant heat flux boundary conditions are imposed at the solid–liquid interfaces along the tips of the ridges, and the menisci between ridges are considered to be flat and adiabatic. The smooth side of the channel is subjected to no-slip and adiabatic boundary conditions. We account for the streamwise and transverse thermocapillary stresses along menisci. When the latter is sufficiently small, Stokes flow may be assumed. Then, our solution is based upon a conformal map. When, additionally, the ratio of channel height to half of the ridge pitch is of order 1 or larger, an accurate but less cumbersome solution follows from a matched asymptotic expansion. When inertial effects are relevant, the slip length is numerically computed. Setting the thermocapillary stress equal to zero yields the slip length for an adiabatic flow.

The interaction between flow inertia and elasticity in high-Reynolds-number, axisymmetric and near-critical swirling flows of an incompressible and viscoelastic fluid in an open finite-length straight circular pipe is studied at the limit of low elasticity. The stresses of the viscoelastic fluid are described by the generalized Giesekus constitutive model. This model helps to focus the analysis on low fluid elastic effects with shear thinning of the viscosity. The application of the Giesekus model to columnar streamwise vortices is first investigated. Then, a nonlinear small-disturbance analysis is developed from the governing equations of motion. It reveals the complicated interactions between flow inertia, swirl and fluid rheology. An effective Reynolds number that links between steady states of swirling flows of a viscoelastic fluid and those of a Newtonian fluid is revealed. The effects of the fluid viscosity, relaxation time, retardation time and mobility parameter on the flow development in the pipe and on the critical swirl for the appearance of vortex breakdown are explored. It is found that in vortex flows with either an axial jet or an axial wake profile, increasing the shear thinning by decreasing the ratio of the viscoelastic characteristic times from one (with fixed values of the Weissenberg number and the mobility parameter) increases the critical swirl ratio for breakdown. Increasing the fluid elasticity by increasing the Weissenberg number from zero (with a fixed ratio of the viscoelastic characteristic times and a fixed value of the mobility parameter) or increasing the fluid mobility parameter from zero (with fixed values of the Weissenberg number and the ratio of viscoelastic times) causes a similar effect. The results may explain the trend of changes in the appearance of breakdown zones as a function of swirl level that were observed in the experiments by Stokes et al. (J. Fluid Mech., vol. 429, 2001, pp. 67–115), where Boger fluids were used. This work extends for the first time the theory of vortex breakdown to include effects of non-Newtonian fluids.

The wake of a five-bladed marine propeller at design operating condition is studied using large eddy simulation (LES). The mean loads and phase-averaged flow field show good agreement with experiments. Phase-averaged and azimuthal-averaged flow fields are analysed in detail to examine the mechanisms of wake instability. The propeller wake consisting of tip and hub vortices undergoes streamtube contraction, which is followed by the onset of instabilities as evident from the oscillations of the tip vortices. Simulation results reveal a mutual-induction mechanism of instability where, instead of the tip vortices interacting among themselves, they interact with the smaller vortices generated by the roll-up of the blade trailing edge wake in the near wake. It is argued that although the mutual-inductance mode is the dominant mode of instability in propellers, the actual mechanism depends on the propeller geometry and the operating conditions. The axial evolution of the propeller wake from near to far field is discussed. Once the propeller wake becomes unstable, the coherent vortical structures break up and evolve into the far wake, composed of a fluid mass swirling around an oscillating hub vortex. The hub vortex remains coherent over the length of the computational domain.

In this paper, we analyse the curvature instability of a curved Batchelor vortex. We consider this short-wavelength instability when the radius of curvature of the vortex centreline is large compared with the vortex core size. In this limit, the curvature instability can be interpreted as a resonant phenomenon. It results from the resonant coupling of two Kelvin modes of the underlying Batchelor vortex with the dipolar correction induced by curvature. The condition of resonance of the two modes is analysed in detail as a function of the axial jet strength of the Batchelor vortex. In contrast to the Rankine vortex, only a few configurations involving $m=0$ and $m=1$ modes are found to become the most unstable. The growth rate of the resonant configurations is systematically computed and used to determine the characteristics of the most unstable mode as a function of the curvature ratio, the Reynolds number and the axial flow parameter. The competition of the curvature instability with another short-wavelength instability, which was considered in a companion paper (Blanco-Rodríguez & Le Dizès, J. Fluid Mech., vol. 804, 2016, pp. 224–247), is analysed for a vortex ring. A numerical error found in this paper, which affects the relative strength of the elliptic instability, is also corrected. We show that the curvature instability becomes the dominant instability in large rings as soon as axial flow is present (vortex ring with swirl).

Long-duration particle image velocimetry measurements in rough-bed open-channel flows (OCFs) reveal that the pre-multiplied spectra of the streamwise velocity have a bimodal distribution due to the presence of large- and very-large-scale motions (LSMs and VLSMs, respectively). The existence of VLSMs in boundary layers, pipes and closed channels has been acknowledged for some time, but strong supporting evidence for their presence in OCF has been lacking. The data reported in this paper fill this gap. Length scales of the LSMs and VLSMs in OCF exhibit different scaling properties; whereas the streamwise length of the LSM scales with the flow depth, the VLSM streamwise length does not scale purely with flow depth and may additionally depend on other scales such as the channel width, roughness height or viscous length. The transverse extent of the LSMs was found to increase with increasing elevation, but the VLSM transverse scale is anchored around two flow depths. The origin and nature of LSMs and VLSMs are still to be resolved, but differences in their scaling suggest that VLSMs in rough-bed OCFs form independently rather than as a spatial alignment of LSMs.

In the present study, we apply a proportional (P)–integral (I) feedback control to a turbulent channel flow for skin-friction reduction. The instantaneous wall-normal velocity at a sensing plane above the wall is measured as a sensing parameter, and blowing/suction is provided at the wall based on the PI control. The performance of PI controls is estimated by the change in the skin friction while varying the sensing plane location $y_{s}$ and the proportional and integral feedback gains ( $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ respectively). The opposition control proposed by Choi et al. (J. Fluid Mech., vol. 262, 1994, pp. 75–110) corresponds to a P control with $\unicode[STIX]{x1D6FC}=1$ . When the sensing plane is located close to the wall ( $y_{s}^{+}\lesssim 10$ ), PI controls result in greater skin-friction reductions than corresponding P controls. The root-mean-square (r.m.s.) sensing velocity fluctuations, considered as the control error, approach zero with successful PI controls, but do not with P controls. Successful PI controls reduce the strength of near-wall coherent structures and the r.m.s. velocity fluctuations above the wall apart from those near the wall due to the control input. The frequency spectra of the sensing velocity show that the I component of PI controls significantly reduces the energy at low frequencies, much more than P controls do. Proportional–integral controls are also applied to a linearized flow model having transient growth of disturbances. The performance of PI controls for a linearized flow model is very similar to that for a turbulent channel flow, i.e. the low-frequency components of disturbances are significantly reduced by the I component of PI controls, and the transient energy growth is suppressed more than by P controls.

Three-dimensional direct numerical simulations of a shearless mixing layer in a small fraction of the cloud–clear air interface are performed to study the response of an ensemble of cloud water droplets to the turbulent entrainment of clear air into a cloud filament. The main goal of this work is to understand how mixing of cloudy and clear air evolves as turbulence and thermodynamics interact through phase changes, and how the cloud droplets respond. In the main simulation case, mixing proceeds between a higher level of turbulence in the cloudy filament and a lower level of turbulence in the clear air environment – the typical shearless mixing layer set-up. Fluid turbulence is driven solely by buoyancy, which incorporates feedbacks from the temperature, the vapour content and the liquid water content fields. Two different variations on the core set of shearless mixing layer simulations are discussed, a simulation in a larger domain and a simulation with the same turbulence level inside the filament and its environment. Overall, it is found that, as evaporation occurs for the droplets that enter subsaturated clear air regions, buoyancy comes to dominate the subsequent evolution of the mixing layer. The buoyancy feedback leads initially to downdraughts at the cloudy–clear air interface and to updraughts in the bulk regions. The strength of the turbulence after initial transients depends on the domain size, showing that the range of scales is an important parameter in the shearless mixing layer set-up. In contrast, the level of turbulence in the clear air is found to have little effect on the evolution of the mixing process. The distributions of cloud water droplet size, supersaturation at the droplet positions and vertical velocity are more sensitive to domain size than to the details of the turbulence profile, suggesting that the evolution of cloud microphysics is more sensitive to large-scale as opposed to small-scale properties of the flow.

We generalize a recent method for computing optimal 2D convection cooling flows in a horizontal layer to a wide range of geometries, including those relevant for technological applications. We write the problem in a conformal pair of coordinates which are the pure conduction temperature and its harmonic conjugate. We find optimal flows for cooling a cylinder in an annular domain, a hot plate embedded in a cold surface, and a channel with a hot interior and cold exterior. With a constraint of fixed kinetic energy, the optimal flows are all essentially the same in the conformal coordinates. In the physical coordinates, they consist of vortices ranging in size from the length of the hot surface to a small cutoff length at the interface of the hot and cold surfaces. With the constraint of fixed enstrophy (or fixed rate of viscous dissipation), a geometry-dependent metric factor appears in the equations. The conformal coordinates are useful here because they map the problems to a rectangular domain, facilitating numerical solutions. With a small enstrophy budget, the optimal flows are dominated by vortices that have the same size as the flow domain.