Air curtains (AC) are contemporary devices that are used to minimize buoyancy-driven exchange flow across the doorway of a building. Under optimal conditions, they restrict up to 85% of outdoor fluid from infiltrating the indoor region. In the present study, we use large-eddy simulations and a Lagrangian analysis framework to investigate the spatial origin and transport mechanisms of fluid that penetrate an air curtain operating at Reynolds number $\mathrm{Re}=4000$ and Froude number ${\mathrm{Fr}}_0=28.4$. We further explain the saturation of sealing effectiveness at 85% by comparing transport characteristics of an AC with those of nonbuoyant jets and gravity currents. We observe that prior to the curtain establishment, most infiltration occurs via a gravity current, where particles initially near the wall are transported indoors along near-straight trajectories. Once the AC establishes, an entrainment-detrainment pathway becomes dominant. The outdoor fluid is first entrained into the turbulent core and subsequently detrained into the indoor region, either before reaching the wall or after its impingement. The present study thus highlights that it is the inherent entrainment of the outdoor fluid by the curtain which results in a loss of sealing by about 15 to 20%.

Accurate scaling of heat energy is critical for compressible turbulence modeling. In addition to the temperature-velocity relation, temperature transformations have received increasing interest recently, but their overall accuracy is still inferior to the velocity counterparts. In this work, we propose to use total enthalpy $H$ instead of temperature to construct the compressible law of the wall for heat energy. A critical justification is the structural resemblance between the equations of compressible $H$ and incompressible passive scalars. Also, $H$ reduces to static enthalpy at zero Mach number, meeting the requirement of self-consistency for transformations. To build the transformation, we first define and uncover the scalings of laminar and turbulent Prandtl numbers for $H$, and then reveal the strong analogy of heat flux budgets between compressible $H$ and incompressible passive scalars. Finally, a semilocal total enthalpy transformation of no fitted parameters is constructed for channel flows, which is structurally analogous and comparable in accuracy to the Trettel-Larsson velocity transformation. This new transformation can potentially serve as a wall model for energy equations to accurately predict the enthalpy or temperature in compressible turbulence.

The offshore wind industry is rapidly expanding to meet clean and secure energy needs. New developments are now increasingly constrained to deeper waters, where the water column is seasonally stratified. Here flows past offshore wind infrastructure will increase water column mixing, although such processes and their extent are poorly understood. Studies have so far been limited to field-scale simulations, which make sweeping assumptions regarding flow-structure interactions and fine-scale stratified turbulence, and field observations, which are limited by the sparsity of measurement campaigns and data captured. To isolate and quantify the key processes governing water column mixing by infrastructure, we present fully structure-resolved direct numerical simulations of two-layer stratified flow past a vertical cylinder. We identify two wake regimes by systematically varying the flow Reynolds and Richardson numbers: (1) a weakly stratified regime, characterized by a narrow but highly energetic wake dominated by horizontal shear and (2) a strongly stratified wake, characterized by the emergence of a thermocline-spanning recirculation cell attached to the cylinder. Here strong vertical motions develop from the recirculation cell which are responsible for the formation of large-scale stationary internal waves. These waves account for up to 10% of the total energy budget and provide a mechanism for far-field energy propagation. The weakly stratified wake regime is characteristic of existing offshore wind sites where temperature gradients are relatively weak; the identified strongly stratified regime describes the dynamics to be expected in future deep water offshore wind sites. This difference between the two regimes offers an explanation for the previously enigmatic discrepancies in field observations regarding wake persistence and detectability. Future work must focus on narrowing the gap between idealized simulations and field-scale flows, for which the datasets herein will provide a critical benchmark for validation.

Nonaxisymmetric droplet shape oscillations are investigated by extending the numerical framework developed in Akbari et al. [Phys. Rev. Fluids 11, 033602 (2026)], which is based on a Galerkin method using spherical harmonics, to fully three-dimensional configurations. The study focuses on the role of shape symmetry and mode excitation of inviscid droplet shape oscillation. The numerical results confirm that the oscillation frequency is independent of the azimuthal wave number, in agreement with classical theoretical predictions. The discrete geometric symmetries of both the initial droplet deformation and the initial velocity potential are demonstrated to be preserved throughout the oscillation. Consequently, only modes compatible with these symmetries are excited. When the initial velocity potential breaks the symmetries of the droplet deformation, leading to a reduction in these symmetries, additional modes are excited compared to those for a zero initial velocity. A similar result is observed in the droplet shape oscillations of a single initial deformation mode and a superposition of two initial deformation modes. In the former case, only modes that satisfy the symmetries of the initial deformation are excited. In the latter case, the superposition reduces the overall symmetry of the droplet shape, leading to additional mode excitation through symmetry breaking. These results highlight the fundamental role of symmetry preservation in nonlinear droplet oscillations. By exploiting these symmetry properties, the computational effort could be reduced without compromising numerical accuracy, as modes that do not occur can be excluded from the solution approach prior to the simulation.

We numerically examine a kinematic dynamo driven by precession in a cylindrical geometry, with particular emphasis on the narrow range of Poincaré numbers where dynamo action is most likely at low and moderate magnetic Reynolds numbers. Using time-averaged velocity fields obtained from hydrodynamic simulations, we analyze the symmetry, oscillation frequency, and onset of the leading magnetic eigenmodes. Two competing families of magnetic fields are identified: a centrosymmetric, higher-frequency quadrupolar mode and a centroantisymmetric, lower-frequency dipolar mode. We then quantify how the dynamo threshold depends on the electromagnetic boundary conditions, including pseudovacuum versus true vacuum treatment and the presence of conducting and/or magnetically permeable vessel walls. We show that simplified vanishing-tangential-field boundary conditions systematically underestimate the critical magnetic Reynolds number, whereas realistic outer layers can either promote or suppress dynamo action depending on their electrical conductivity and on the selected hydrodynamic mean state. These results clarify the role of wall properties and mode selection for the forthcoming DRESDYN precession dynamo experiment.

We present a comprehensive framework for gravity-driven, surfactant-laden thin films flowing over slippery substrates, elucidating how wall slip modifies the coupled hydrodynamics and interfacial transport. A long-wave model is formulated with a conservative bulk-surface mass balance and a Navier slip condition. The Orr-Sommerfeld eigenvalue problem governs the linear regime, while a weighted-residual model captures the nonlinear evolution over a range of equilibrium surfactant coverages, Marangoni strengths, and adsorption kinetics. The analysis predicts a nonmonotonic variation of the critical Reynolds number with equilibrium coverage, exhibiting a maximum at intermediate ${\mathrm{\Gamma }}_e$, revealing the dual role of soluble surfactants: Increasing ${\mathrm{\Gamma }}_e$ stabilizes the flow up to an optimal coverage, whereas further enrichment weakens this stabilizing effect and promotes destabilization. Nonlinear simulations show that wall slip and Marangoni stresses modify the amplitude, phase, and capillary-ripple structure of localized traveling pulses. Under fast adsorption kinetics, the bulk-surface exchange rapidly reequilibrates the surfactant field, preserving the mean film shape and flux while strongly flattening the bulk inventory $\chi +h\varphi$ and only weakly modifying the surface concentration $\mathrm{\Gamma }$. A spurious interfacial mass growth reported by Pascal et al. [Phys. Rev. Fluids 4, 054004 (2019)] and D'Alessio et al. [J. Fluid Mech. 887, A20 (2020)] is resolved through a revised surface balance ensuring strict conservation. Wall slip thus emerges as a key control parameter, reducing viscous resistance and mitigating Marangoni back-stress. The slip parameter $\beta$ therefore provides a useful control parameter for surfactant-laden films, linking substrate slip to wave amplitude, ripple damping, and the bulk-surface surfactant balance.

This study investigates the nonlinear evolution of capillary-gravity waves in deep water under wind forcing and dissipation. When the wind energy input exceeds the wave energy dissipation, previous studies on wind-driven wave evolution have primarily established two distinct stages: the initial growth of wave components and the frequency downshift [T. Hara and C. C. Mei, J. Fluid Mech. 267, 221 (1994); A. Zavadsky, D. Liberzon, and L. Shemer, J. Phys. Oceanogr. 43, 65 (2013); T. Li and L. Shen, J. Fluid Mech. 934, A41 (2022)]. We reveal a stage that bridges these two previously recognized stages. This intermediate stage begins with the emergence of new wave components and ends with the onset of frequency downshift, characterized by coupled three-wave and four-wave near-resonant interactions that generate new components and broaden the wave spectrum. Furthermore, due to the interplay among nonlinear wave-wave interactions, potential wave-breaking dissipation, and wind forcing, the total wave energy may decrease during this stage.

Hele-Shaw flows with an interface are known to give rise to complex pattern formation, whereby the Saffman-Taylor instability triggers a viscous fingering process accompanied by tip splitting and branching. The most popular of these experiments involves a radial configuration with a less viscous fluid injected into a more viscous fluid. In an attempt to characterize the resulting complexity in such an experiment, T. H. Beeson-Jones et al. [Sci. Rep. 9, 7780 (2019)] have proposed a type of simple empirical model that aims to predict the saturation profile of the invading fingers as a function of a radial coordinate. We revisit the proposed saturation model and test its validity over a broad parameter range using fully nonlinear numerical simulations computed with a level-set method. We find that the saturation model is very effective at predicting some near-universal properties of the viscous fingering patterns for one-phase flows, where the invading fluid is neglected, with a sufficiently small surface tension parameter. For larger values of this parameter and for two-phase flows, there are discrepancies between the model and our observations. We explain these differences by studying the morphology of the advancing fingers, including pinching at the base and the rate of tip splitting. Overall, our study shows that the Beeson-Jones–Woods saturation model serves as a valid description of diffusion-limited aggregation–like patterns but is not universal over two-phase flows, where surface tension and viscosity ratio substantially alter finger morphology and the resulting saturation profile.

This study investigates the spatiotemporal evolution of hypopycnal intrusive particle-laden flows to determine how particle concentration and size regulate flow instabilities, horizontal propagation, and vertical transport. These processes are directly related to the effectiveness of ocean alkalinity enhancement (OAE), where alkaline mineral particles are added to the ocean surface. Using high-resolution pseudospectral simulations, we systematically vary particle concentration and diameter to quantify the entrained area and effective settling velocity. Following particle release, the flow dynamics is governed by the competition between horizontal spreading and vertical sedimentation processes. We find that the terminal propagation distance decreases monotonically with increasing the particle concentration and diameter. This terminal distance is limited by collective vertical sedimentation driven by convective instabilities at the base of the sediment layer. Consequently, the effective vertical particle settling speed can exceed the Stokes settling speed of an individual particle by one to two orders of magnitude. The settling enhancement factor, defined as the ratio of the effective settling velocity to the Stokes settling velocity, reaches up to 100 for fine particles (i.e., particle diameter $5$µm) and increases with particle concentration. However, a large fraction of particles in the upper part of the particle-laden flow still settle according to the Stokes settling velocity, as they are not influenced by convective instabilities. This leads to a bulk centroid velocity that is much smaller than that of the vertical sedimentation front driven by convective instabilities. We categorize the flow evolution into three morphological regimes, governed by the balance between spreading and settling. We find that intermediate particle concentration and size can maximize the surface entrainment area and homogeneity, offering an optimal configuration for OAE. These results demonstrate that collective instabilities, rather than individual Stokes settling, control particle residence times, potentially limiting the carbon removal efficiency of surface-release OAE strategies.

Small-scale intermittency is studied as the deviation of the probability distributions of pseudodissipation, dissipation, and enstrophy in turbulence from those of a Gaussian random velocity field. This deviation is quantified using Kullback-Leibler (KL) divergence between the two distributions, directly measuring turbulence-induced intermittency separated from purely kinematic effects. Using direct numerical simulation data of forced isotropic turbulence over a wide range of Taylor Reynolds numbers ${Re}_{\lambda }$, we characterize the ${Re}_{\lambda }$ dependence of small-scale intermittency via KL divergence and uncertainty via Shannon entropy, identifying distinct behavioral regimes. Small-scale uncertainty exhibits a nonmonotonic dependence on ${Re}_{\lambda }$: Despite continuously growing variability, entropy decays above a certain Reynolds number, suggesting a fundamental change in the statistical nature of the small scales. Turbulence-induced intermittency, as characterized by KL divergence, grows logarithmically with Reynolds number in contrast to the commonly reported power-law scaling of individual moments, implying that turbulence shows a diminishing growth rate of intermittency at higher Reynolds numbers. We uncover an emergent symmetry: Turbulence dynamics is shown to generate nearly equal intermittency in dissipation rate and enstrophy, challenging the prevailing assumption of asymmetry between strain-rate and vorticity dynamics.

In this study, a series of scale-resolving numerical simulations of mixed convection at a Richardson number of $\text{Ri}=1$ for heavy liquid metals is conducted to investigate the flow organization. The large-scale longitudinal rollers produced by the combination of shear and buoyancy effects are observed in mixed convection, and they occupy the whole channel height. Turbulence coherence and transportation are enhanced by longitudinal rollers, which are quantitatively supported by two-point spatial autocorrelations and quadrant analysis. Thermal stripes are captured, revealing the footprints of longitudinal rollers at wall vicinity. The multiscale characteristic of flow structures at wall vicinity is resolved. The velocity patches and streaks inside patches correspond to large- and small-scale motions, respectively, which are quantitatively identified by the typical separation wavelength ${\lambda }_z^{+}\approx 1000$. Mean velocity and temperature profiles are well predicted by the Businger-Dyer relationship with exponent $-1$ based on Monin-Obukhov similarity theory. As well, the phenomenological logarithmic behavior with slope modulation of mean velocity is observed. We utilize the linear coherence spectrum and an improved wall-attached eddy model to give an intuitive physical interpretation of this phenomenon. The equivalence between slope modulation coefficient and fraction of space occupied by wall-attached eddy structures is established, and it has been further scrutinized under several extended operating conditions.

Wall-pressure fluctuations beneath turbulent boundary layers induce structural vibration and acoustic radiation, especially in underwater and aerospace systems. Accurate prediction of their wave number-frequency spectra is critical for effective noise mitigation and design, yet empirical/analytical models rely on simplifying assumptions and miss the full spatiotemporal complexity, while high-fidelity simulations are prohibitive at high Reynolds numbers. Experimental measurements, though accessible, typically provide only pointwise signals and lack the resolution to recover full spatiotemporal fields. We propose a probabilistic generative framework that couples a patchwise (domain-decomposed) conditional neural field with a latent diffusion model to synthesize spatiotemporal wall-pressure fields under varying pressure-gradient conditions. The model conditions on sparse surface-sensor measurements and a low-cost mean-pressure descriptor, supports zero-shot adaptation to new sensor layouts, and produces ensembles with calibrated uncertainty. Validation against reference data shows accurate recovery of instantaneous fields and key statistics.

Collision of inwardly propagating axisymmetric gravity currents was investigated by means of three-dimensional high-resolution simulations. Our goal is to deepen our understanding of the flow structures, energy budgets, and the mean flow and turbulence characteristics in the axisymmetric collision in light of our understanding of the collision of two planar counterflowing gravity currents. The lifetime of axisymmetric collision is approximately $4\tilde{H}/{\tilde{u}}_{N,\text{max}}$, where $\tilde{H}$ is the depth of heavy and ambient fluids and ${\tilde{u}}_{N,\text{max}}$ is the maximum inward front speed before the collision, and the lifetime of axisymmetric collision can be divided into three phases. During Phase I, $0\le (\tilde{t}-{\tilde{t}}_c){\tilde{u}}_{N,\text{max}}/\tilde{H}\lesssim 0.7$, where $\tilde{t}$ is the time and ${\tilde{t}}_c$ is the time instance at which the “averaged” vertical acceleration of the density front from collision reaches its maximum value, a circular array of radially elongated vortices surrounding the center of the flow domain develop close to the bottom boundary and create strong turbulence and dissipation of turbulent kinetic energy. The radial vorticity is generated from the tilting of azimuthal vorticity due to the axisymmetric geometry and this mechanism for generation of vorticity is essentially different from the collision of two planar counterflowing gravity currents. During Phase II, $0.7\lesssim (\tilde{t}-{\tilde{t}}_c){\tilde{u}}_{N,\text{max}}/\tilde{H}\lesssim 1.26$, the deflected updraught of collided heavy fluid expands radially outwards and there exists another circular array of radially elongated vortices close to the top boundary. Both baroclinic production of vorticity and diffusion of vorticity act to stabilize the flow from further intensifying radial vorticity close to the top boundary. During Phase III, $1.26\lesssim (\tilde{t}-{\tilde{t}}_c){\tilde{u}}_{N,\text{max}}/\tilde{H}\lesssim 4$, the radially elongated vortices observed in Phase II break up into smaller structures while the collided heavy fluid slumps away from the center of the flow domain. In Phase I and Phase II, the primary source of turbulent kinetic energy is production of turbulent kinetic energy, while in Phase III, the primary source is due to turbulent buoyancy flux. Our space-time averaged eddy diffusivity data support the experimental data, numerical simulation data and the guide for parameterization of eddy diffusivity proposed for the collision of two planar counterflowing gravity currents of equal strengths in the literature.

A millimetric droplet can bounce indefinitely on a vertically vibrating bath and generate a Faraday pilot-wave field that propels it forward. This hydrodynamic pilot-wave system, in which a walking droplet is accompanied by its quasi-monochromatic wave field, exhibits several features previously thought to be exclusive to the quantum realm. For steady dynamical states, the energy input from the vibrational forcing and output from viscous dissipation must balance, so one can consider the partitioning between the droplet and wave energy. The energetics of the hydrodynamic pilot-wave system have been considered only in the special case where there is resonance between the bouncing drop and its pilot wave, so one may neglect the drop's vertical dynamics. We here deepen our understanding of pilot-wave energetics by simulating these vertical dynamics, allowing us to assess the energetics of both resonant and nonresonant walking states. Specifically, we characterize the dependence of the system energetics on the droplet's bouncing or walking mode, and assess whether resonance has a significant influence on the partitioning between drop and wave energy. In all instances, the horizontal kinetic energy of the drop is negligible relative to its gravitational potential energy. At high memory, where the pilot waves are most persistent, the pilot-wave energy dominates the total system energy, exceeding the droplet energy by an order of magnitude. When the resonance between drop and wave is broken, the wave energy is slightly diminished due to wave interference, but remains dominant.

Statically stable stratifications of a less dense solution above a denser one can be destabilized in the gravity field when differential diffusion effects come into play. Specifically, if the stratification involves two solutes that diffuse at different rates, nonmonotonic density profiles can build up. Local convective zones develop then in the areas where, locally, the density decreases along the gravity field. Previous theoretical works have shown that for both double-diffusive (DD) and diffusive-layer-convection (DLC) regimes, the onset time and the mixing velocity of these buoyancy-driven instabilities are controlled by the dynamic density jump across these unstable zones. We provide here an experimental validation of these scalings by analysis of convective dynamics induced by differential diffusion around horizontal stratifications confined within a Hele-Shaw cell. We analyze a wide range of solute combinations and concentrations to vary the two important parameters of the problem which are the diffusion coefficients ratio $\delta$ and the buoyancy ratio $R$. By measuring the mixing length evolution, we quantify in each case the mixing velocity and the instability onset time. We show that, in both DD and DLC regimes, the convective dynamics are indeed governed by the dynamically generated local adverse density difference. In the DD case, the mixing velocity increases linearly with the dynamically induced density difference, while the onset time decreases accordingly. In the DLC regime, these properties are controlled by the local adverse density jump developing independently within each layer. The experimentally obtained scalings confirm the previous theoretical predictions and provide a quantitative framework for controlling buoyancy-driven mixing in stratified two-solute systems.

Classical shell models of turbulence do not display dual cascade—inverse for energy and direct for enstrophy—because they fail to reproduce the right thermal spectra. We propose here a multi-branch shell model, including a geometry hierarchically organized across scales, in order to overcome this limitation. For this model, we demonstrate numerically both the agreement of the thermal spectra with those of two-dimensional fluid equations and the emergence of a statistically stationary dual cascade. This construction also allows us to study local transfers and to investigate both self-similarity and non-Gaussianity.

In this study, we report direct experimental observations of self-sustaining ${\mathrm{CH}}_4$-air rotating flames formed spontaneously in an unheated, open, circular Hele-Shaw cell. These flames are observed under fuel-rich conditions and exhibit stable traveling-wave patterns, with edge velocities that can significantly exceed the nominal flame speed of the unburned mixture. Planar laser-induced fluorescence (PLIF) measurements across the central plane reveal that the flame front consists of a bibrachial structure, with a diffusion branch gliding along the side edges of the cell and a premixed branch extending into the interior. Complementary numerical simulations suggest that the formation of rotating flames is driven by a dynamic balance between local flame speed and unburned-gas velocity near the cell edges, where both wall heat loss and flow expansion play critical roles in stabilizing the rotation pattern. A parametric study is conducted for various equivalence ratios, flow rates, and gap distances, from which the regime diagrams of flame modes and rotation frequencies are obtained. At low flow rates, the rotating state is characterized by a single rotating flame wave, whose rotation frequency increases with flow rate. For this type of flame, a semiempirical model is established to predict their rotation frequencies and shapes as functions of mass flow rate and surface temperature. At elevated flow rates, multiple rotating waves appear with approximately equal azimuthal spacing, and the product of the wave number and rotation frequency increases with flow rate. Mode transition from rotating flames to steady ring-shaped flames anchored at the burner edges occurs at sufficiently high flow rates, while at sufficiently low flow rates, flame extinction occurs due to thermal quenching. These findings can provide useful guidance for the advancement of microcombustion technologies.

In plane Couette flow (PCF), turbulent coherent structures typically appear as turbulent spots or bands. In this study, we investigate the subcritical transition to turbulence in PCF in a three-dimensional Yukawa liquid using first-principles classical 3D molecular dynamics simulations, wherein a monopolar columnar vortex is initialized at the center of the simulation domain to trigger the transition. Some of the features of subcritical turbulence are found to depend on the type of perturbation used in the simulation. In unstratified cases, we observe the emergence of a tripolar vortex structure, whereas in stably stratified conditions, a stable quadrupolar (two stable dipoles) structure forms—a phenomenon rarely reported in prior experiments and simulations. Unlike in supercritical turbulence, the stable background PCF is observed to constrain the coherent vortex structures and prevent them from undergoing rotational motion. The growth of radial kinetic energy is found to be greater in the stratified case compared to the unstratified one. In addition to azimuthal modes 1 and 2, mode 0 emerges as the most energetically dominant in the system. Our results demonstrate that both tripole and quadrupole structures can arise as quasi-coherent modes during a transition to turbulence in PCF.

We use the FENE-P constitutive model to revisit the hoop-stress-driven linear instability in viscoelastic Taylor-Couette flow, considering both inertialess and inertial regimes. The FENE-P model is appropriate for dilute polymer solutions, with the finite extensibility of the polymer leading to shear thinning of both viscosity and the first-normal-stress coefficient. Shear thinning weakens elastic effects and is generally expected to stabilize the hoop-stress mode (HSM), as suggested by the Pakdel-McKinley scaling criterion. Our numerical results confirm this expectation for small values of the ratio of the gap width to the radius of the inner cylinder $(\epsilon <1)$. Unexpectedly, for $\epsilon >1$, finite-extensiblity-induced increase in shear rates at the inner rotating cylinder results in a destabilizing effect on the HSM. We compare our theoretical predictions with relatively recent experimental observations on viscoelastic Taylor-Couette flow for a range of elasticity numbers $\left(E\right)$ with $\epsilon \in [0.25-0.4]$. In this regime, the experimental results for the onset of instability are well predicted both by the Oldroyd-B and FENE-P (with experimentally realistic polymer extensibility parameter values) models, with finite extensibility playing only a secondary role. Our theory predicts that for very dilute solutions with $\beta =0.98$, the hoop-stress instability is suppressed for realistic values of extensibility parameters. Overall, we show that the predictions of the FENE-P model will be qualitatively different from those of the Oldroyd-B model both for very low $\epsilon \text{"s}$ (suppression of HSM) and for $\epsilon >1$ (destabilization of HSM). Further, we also show that the FENE-P predictions for the instability onset are in significant disagreement with experiments carried out using highly concentrated polymer solutions, underscoring the need for the use of more realistic rheological models appropriate for such systems.

The turbulent radial flow between parallel annular plates of finite span is studied with a focus on instabilities induced by a periodically modulated swirling source inflow. This investigation utilizes a combination of theoretical modeling along with simplified two- and three-dimensional numerical simulations. Different levels of complexity gradually evolve in the radial diffuser model to identify the different instabilities and to trace back the instabilities to different physical mechanisms. First, a linear stability analysis is performed for a two-dimensional core-flow instability mechanism on a one-dimensional basic state, considering the flow to be inviscid and homogeneous in the azimuthal $\theta$ direction. Following this, several two-dimensional numerical simulations are conducted, including viscous effects, nonlinear effects, and inflow modulation in the azimuthal direction. Finally, three-dimensional numerical simulations are carried out to evaluate the impact of the boundary layer on the instabilities.