Stirring olive oil and vinegar to make salad dressing creates an emulsion of vinegar droplets in oil. More vigorous stirring gives smaller droplets, while if left to sit the droplets will begin to coalesce and the two fluids will separate. In this vein, Dodd & Ferrante (J. Fluid Mech., vol. 806, 2016, pp. 356–412) present a new analysis of how homogeneous turbulence in a carrier fluid interacts with a suspension of droplets of an immiscible liquid. Based on a set of direct numerical simulations, the authors provide new insights on how turbulence affects the motion of the droplets, their shape and size; then in turn how the droplets alter the flow including effects of interfacial surface energy on the kinetic energy of the flow.

Roughness predominantly alters the near-wall region of turbulent flow while the outer layer remains similar with respect to the wall shear stress. This makes it a prime candidate for the minimal-span channel, which only captures the near-wall flow by restricting the spanwise channel width to be of the order of a few hundred viscous units. Recently, Chung et al. (J. Fluid Mech., vol. 773, 2015, pp. 418–431) showed that a minimal-span channel can accurately characterise the hydraulic behaviour of roughness. Following this, we aim to investigate the fundamental dynamics of the minimal-span channel framework with an eye towards further improving performance. The streamwise domain length of the channel is investigated with the minimum length found to be three times the spanwise width or 1000 viscous units, whichever is longer. The outer layer of the minimal channel is inherently unphysical and as such alterations to it can be performed so long as the near-wall flow, which is the same as in a full-span channel, remains unchanged. Firstly, a half-height (open) channel with slip wall is shown to reproduce the near-wall behaviour seen in a standard channel, but with half the number of grid points. Next, a forcing model is introduced into the outer layer of a half-height channel. This reduces the high streamwise velocity associated with the minimal channel and allows for a larger computational time step. Finally, an investigation is conducted to see if varying the roughness Reynolds number with time is a feasible method for obtaining the full hydraulic behaviour of a rough surface. Currently, multiple steady simulations at fixed roughness Reynolds numbers are needed to obtain this behaviour. The results indicate that the non-dimensional pressure gradient parameter must be kept below 0.03–0.07 to ensure that pressure gradient effects do not lead to an inaccurate roughness function. An empirical costing argument is developed to determine the cost in terms of CPU hours of minimal-span channel simulations a priori. This argument involves counting the number of eddy lifespans in the channel, which is then related to the statistical uncertainty of the streamwise velocity. For a given statistical uncertainty in the roughness function, this can then be used to determine the simulation run time. Following this, a finite-volume code with a body-fitted grid is used to determine the roughness function for square-based pyramids using the above insights. Comparisons to experimental studies for the same roughness geometry are made and good agreement is observed.

We study suspensions of oblate rigid particles in a viscous fluid for different values of the particle volume fractions. Direct numerical simulations have been performed using a direct-forcing immersed boundary method to account for the dispersed phase, combined with a soft-sphere collision model and lubrication corrections for short-range particle–particle and particle–wall interactions. With respect to the single-phase flow, we show that in flows laden with oblate spheroids the drag is reduced and the turbulent fluctuations attenuated. In particular, the turbulence activity decreases to lower values than those obtained by accounting only for the effective suspension viscosity. To explain the observed drag reduction, we consider the particle dynamics and the interactions of the particles with the turbulent velocity field and show that the particle–wall layer, previously observed and found to be responsible for the increased dissipation in suspensions of spheres, disappears in the case of oblate particles. These rotate significantly slower than spheres near the wall and tend to stay with their major axes parallel to the wall, which leads to a decrease of the Reynolds stresses and turbulence production and so to the overall drag reduction.

A system of boundary integral equations is derived for flows in domains composed of a porous medium of permeability $k_{1}$, surrounded by another porous medium of different permeability, $k_{2}$. The incompressible Brinkman equation is used to describe the flow in the porous media. We first apply a boundary integral representation of the Brinkman flow on each side of the dividing interface, and impose continuity of the velocity at the interface to derive the final formulation in terms of the interfacial velocity and surface forces. We discuss relations between the surface stresses based on the additional conditions imposed at the interface that depend on the porosity and permeability of the media and the structural composition of the interface. We present simulated results for test problems and different interface stress conditions. The results show significant sensitivity to the choice of the interface conditions, especially when the permeability is large. Since the Brinkman equation approaches the Stokes equation when the permeability approaches infinity, our boundary integral formulation can also be used to model the flow in sub-categories of Stokes–Stokes and Stokes–Brinkman configurations by considering infinite permeability in the Stokes fluid domain.

Various expressions have been proposed previously for the rise velocity of gas bubbles in homogeneous steady bubbly flows, generally a monotonically decreasing function of the bubble volume fraction. For suspensions of freely moving bubbles, some of these are of the form expected for ordered arrays of bubbles, and vice versa, as they do not reduce to the behaviour expected theoretically in the dilute limit. The microstructure of weakly inhomogeneous bubbly flows not being known generally, the effect of microstructure is an important consideration. We revisit this problem here for bubbly flows at small to moderate Reynolds number values for deformable bubbles, using direct numerical simulation and analysis. For ordered suspensions, the rise velocity is demonstrated not to be monotonically decreasing with volume fraction due to cooperative wake interactions. The fore-and-aft asymmetry of an isolated ellipsoidal bubble is reversed upon increasing the volume fraction, and the bubble aspect ratio approaches unity. Recent work on rising bubble pairs is used to explain most of these results; the present work therefore forms a platform of extending the former to suspensions of many bubbles. We adopt this new strategy also to support the existence of the oblique rise of ordered suspensions, the possibility of which is also demonstrated analytically. Finally, we demonstrate that most of the trends observed in ordered systems also appear in freely evolving suspensions. These similarities are supported by prior experimental measurements and attributed to the fact that free bubbles keep the same neighbours for extended periods of time.

Very long waves are generated when a ship moves across an appreciable depth change $\unicode[STIX]{x0394}h$ comparable to the average and relatively shallow water depth $h$ at the location, with $\unicode[STIX]{x0394}h/h\simeq 1$. The phenomenon is new and the waves were recently observed in the Oslofjord in Norway. The 0.5–1 km long waves, extending across the 2–3 km wide fjord, are observed as run-ups and run-downs along the shore, with periods of 30–60 s, where a wave height up to 1.4 m has been measured. The waves travelling with the shallow water speed, found ahead of the ships moving at subcritical depth Froude number, behave like a mini-tsunami. A qualitative explanation of the linear generation mechanism is provided by an asymptotic analysis, valid for $\unicode[STIX]{x0394}h/h\ll 1$ and long waves, expressing the generation in terms of a pressure impulse at the depth change. Complementary fully dispersive calculations for $\unicode[STIX]{x0394}h/h\simeq 1$ document symmetries of the waves at positive or negative $\unicode[STIX]{x0394}h$. The wave height grows with the ship speed $U$ according to $U^{n}$ with $n$ in the range 3–4, for $\unicode[STIX]{x0394}h/h\simeq 1$, while the growth in $U$ is only very weak for $\unicode[STIX]{x0394}h/h\ll 1$ (the asymptotics). Calculations show good agreement with observations.

The three-dimensional vortex clusters, and the structures based on the quadrant classification of the intense tangential Reynolds stress (Qs), are studied in direct numerical simulations of statistically stationary homogeneous shear turbulence (HST) at Taylor microscale Reynolds number $Re_{\unicode[STIX]{x1D706}}\approx 50{-}250$, with emphasis on comparisons with turbulent channels (CHs). The Qs and vortex clusters in HST are found to be versions of the corresponding detached (in the sense of del Álamo et al. (J. Fluid Mech., vol. 561 (2006), pp. 329–358)) structures in CHs, although statistically symmetrised with respect to the substitution of sweeps by ejections and vice versa. In turn, these are more symmetric versions of the corresponding attached Qs and clusters. In both flows, only co-gradient sweeps and ejections larger than the local Corrsin scale are found to couple with the shear. They are oriented anisotropically, and are responsible for carrying most of the total Reynolds stress. Most large eddies in CHs are attached to the wall, but it is shown that this is probably a geometric consequence of their size, rather than the reason for their dynamical significance. Most small Q structures associated with different quadrants are far from each other in comparison to their size, but those that are close to each other tend to form quasi-streamwise trains of groups of a sweep and an ejection paired side by side in the spanwise direction, with a vortex cluster in between, generalising to three dimensions the corresponding arrangement of attached eddies in CHs. These pairs are organised around an inclined large-scale conditional vortex ‘roller’, and it is shown that the composite structure tends to be located at the interface between high- and low-velocity streaks, as well as in strong ‘co-gradient’ shear layers that separate streaks of either sign in which velocity is more uniform. It is further found that the conditional rollers are terminated by ‘hooks’ reminiscent of hairpins, both upright and inverted. The inverted hook weakens as the structures approach the wall, while the upright one changes little. At the same time, the inclination of the roller with respect to the mean velocity decreases from $45^{\circ }$ in HST to quasi-streamwise for wall-attached eddies. Many of these observations are generalised to intense Reynolds stresses formed with different pairs of velocity components, and it is shown that most properties of the small structures can be traced to their definitions, rather than to their dynamics. It is concluded that the larger Reynolds-stress structures are associated with shear turbulence, rather than with the presence of a wall, while the smaller ones are generic to turbulence in general, whether sheared or not.

Unsteady pressure gradients in turbulent flows not only influence the mean, but also affect the higher statistical moments of turbulence. In these flows, it is important to understand if and when turbulence is in quasi-equilibrium with the mean in order to better capture the dynamics and develop effective closure models. Therefore, this study aims to elucidate how turbulence decays or develops relative to a time-varying mean flow, and how the turbulent kinetic energy (TKE) production, transport and dissipation respond to changes in the imposed pressure forcing. The focus is on the neutral unsteady Ekman boundary layer, where pressure-gradient, Coriolis and turbulent friction forces interact, and the analyses are based on a suite of large-eddy simulations with unsteady pressure forcing. The results indicate that the dynamics is primarily controlled by the relative magnitudes of three time scales: the inertial time scale (characterized by Coriolis frequency ${\sim}12$ hours at mid-latitudes), the turbulent time scale (${\sim}2$ hours for the largest eddies in the present simulations) and the forcing variability time scale (which is varied to reflect different (sub)meso to synoptic scale dynamics). When the forcing time scale is comparable to the turbulence time scale, the quasi-equilibrium condition becomes invalid due to highly complex interactions between the mean and turbulence, the velocity profiles manifestly depart from the log-law and the normalized TKE budget terms vary strongly in time. However, for longer, and surprisingly for shorter, forcing times, quasi-equilibrium is reasonably maintained. The analyses elucidate the physical mechanisms that trigger these dynamics, and investigate the implications on turbulence closure models.

We analyse the linear stability of stably stratified fluids whose density depends on two scalar fields where one of the scalar fields is unstably stratified and involves a settling velocity. Such conditions may be found, for example, in flows involving the transport of sediment in addition to heat or salt. A linear stability analysis for constant-gradient base states demonstrates that the settling velocity generates a phase shift between the perturbation fields of the two scalars, which gives rise to a novel, settling-driven instability mode. This instability mechanism favours the growth of waves that are inclined with respect to the horizontal. It is active for all density and diffusivity ratios, including for cases in which the two scalars diffuse at identical rates. If the scalars have unequal diffusivities, it competes with the elevator mode waves of the classical double-diffusive instability. We present detailed linear stability results as a function of the governing dimensionless parameters, including for lateral gradients of the base state density fields that result in predominantly horizontal intrusion instabilities. Highly resolved direct numerical simulation results serve to illustrate the nonlinear competition of the various instabilities for such flows in different parameter regimes.

When particles settle through a stable temperature or salinity gradient they can drive an instability known as sedimentary fingering convection. This phenomenon is thought to occur beneath sediment-rich river plumes in lakes and oceans, in the context of marine snow where decaying organic materials serve as the suspended particles or in the atmosphere in the presence of aerosols or volcanic ash. Laboratory experiments of Houk & Green (Deep-Sea Res., vol. 20, 1973, pp. 757–761) and Green (Sedimentology, vol. 34(2), 1987, pp. 319–331) have shown sedimentary fingering convection to be similar to the more commonly known thermohaline fingering convection in many ways. Here, we study the phenomenon using three-dimensional direct numerical simulations. We find evidence for layer formation in sedimentary fingering convection in regions of parameter space where it does not occur for non-sedimentary systems. This is due to two complementary effects. Sedimentation affects the turbulent fluxes and broadens the region of parameter space unstable to the $\unicode[STIX]{x1D6FE}$-instability (Radko, J. Fluid Mech., vol. 497, 2003, pp. 365–380) to include systems at larger density ratios. It also gives rise to a new layering instability that exists in $\unicode[STIX]{x1D6FE}$-stable regimes. The former is likely quite ubiquitous in geophysical systems for sufficiently large settling velocities, while the latter probably grows too slowly to be relevant, at least in the context of sediments in water.

Impingement of a streamwise-oriented vortex upon a fin, tail, blade or wing represents a fundamental class of flow–structure interaction that extends across a range of applications. It can give rise to unsteady loading known as buffeting and to changes of the lift to drag ratio. These consequences are sensitive to parameters of the incident vortex as well as the location of vortex impingement on the downstream aerodynamic surface, generically designated as a wing. Particle image velocimetry is employed to determine patterns of velocity and vorticity on successive cross-flow planes along the vortex, which lead to volume representations and thereby characterization of the streamwise evolution of the vortex structure as it approaches the downstream wing. This evolution of the incident vortex is affected by the upstream influence of the downstream wing, and is highly dependent on the spanwise location of vortex impingement. Even at spanwise locations of impingement well outboard of the wing tip, a substantial influence on the structure of the incident vortex at locations significantly upstream of the leading edge of the wing was observed. For spanwise locations close to or intersecting the vortex core, the effects of upstream influence of the wing on the vortex are to: decrease the swirl ratio; increase the streamwise velocity deficit; decrease the streamwise vorticity; increase the azimuthal vorticity; increase the upwash; decrease the downwash; and increase the root-mean-square fluctuations of both streamwise velocity and vorticity. The interrelationship between these effects is addressed, including the rapid attenuation of axial vorticity in presence of an enhanced defect of axial velocity in the central region of the vortex. When the incident vortex is aligned with, or inboard of, the tip of the wing, the swirl ratio decreases to values associated with instability of the vortex, thereby giving rise to enhanced values of azimuthal vorticity relative to the streamwise (axial) vorticity, as well as relatively large root-mean-square values of streamwise velocity and vorticity.

Steady free-surface flows can produce sudden changes in height and velocity, namely standing jumps, which demarcate supercritical from subcritical flows. Standing jumps have traditionally been observed and studied experimentally with water in order to mimic various hydraulic configurations, for instance in the vicinity of energy dissipators. More recently, some studies have emerged that investigate standing jumps formed in flows of dry granular materials, which are relevant to the design of protection dams against avalanches. In the present paper, we present a new explicit relation for the prediction of the height of standing jumps. We demonstrate the robustness of the new relation proposed by revisiting and cross-comparing a great number of data sets on standing jumps formed in water flows on horizontal and inclined smooth beds, in water flows on horizontal rough beds, and in flows of dry granular materials down smooth inclines. Our study reveals the limits of the traditional one-to-one relation between the sequent depth ratio of the jump and the Froude number of the incoming supercritical flow, namely the Bélanger equation. The latter is a Rankine–Hugoniot relation which does not take into account the gravitational and frictional forces acting within the jump volume, over the jump length, as well as the possible density change across the jump when the incoming fluid is compressible. The newly proposed relation, which is exact for grains and a reasonable approximation for water, can solve all of these issues. However, this relation can predict the height of the standing jump only if another length scale, namely the length of the jump, is known. We conclude our study by discussing empirical but simple closure relations to get a reasonable estimate of the jump length for water flows and dry granular flows. These closure relations can be used to feed the general jump relation and then predict with accuracy the heights of the jumps in a number of situations, provided that well-calibrated friction laws – described in the present study – are considered.

Mountain-generated inertia–gravity waves (IGWs) affect the dynamics of both the atmosphere and the ocean through the mean force they exert as they interact with the flow. A key to this interaction is the presence of critical-level singularities or, when planetary rotation is taken into account, inertial-level singularities, where the Doppler-shifted wave frequency matches the local Coriolis frequency. We examine the role of the latter singularities by studying the steady wavepacket generated by a multiscale mountain in a rotating linear shear flow at low Rossby number. Using a combination of Wentzel–Kramers–Brillouin (WKB) and saddle-point approximations, we provide an explicit description of the form of the wavepacket, of the mean forcing it induces and of the mean-flow response. We identify two distinguished regimes of wave propagation: Regime I applies far enough from a dominant inertial level for the standard ray-tracing approximation to be valid; Regime II applies to a thin region where the wavepacket structure is controlled by the inertial-level singularities. The wave–mean-flow interaction is governed by the change in Eliassen–Palm (or pseudomomentum) flux. This change is localised in a thin inertial layer where the wavepacket takes a limiting form of that found in Regime II. We solve a quasi-geostrophic potential-vorticity equation forced by the divergence of the Eliassen–Palm flux to compute the wave-induced mean flow. Our results, obtained in an inviscid limit, show that the wavepacket reaches a large-but-finite distance downstream of the mountain (specifically, a distance of order $(k_{\ast }\unicode[STIX]{x1D6E5})^{1/2}\unicode[STIX]{x1D6E5}$, where $k_{\ast }^{-1}$ and $\unicode[STIX]{x1D6E5}$ measure the wave and envelope scales of the mountain) and extends horizontally over a similar scale.

An explicit expression for the time-dependent force on a stationary, finite-sized spherical particle located in an unsteady inhomogeneous ambient flow is presented. The force expression accounts for both viscous and compressible effects. Towards this end, a time-harmonic plane travelling wave of a given frequency propagating in a viscous compressible flow over a sphere is considered. Linearized compressible Navier–Stokes equations are solved to obtain an analytical expression for the force exerted on the particle in the frequency domain. The force obtained in the Laplace space due to a travelling wave of a given frequency and wavenumber is then generalized to any arbitrary incoming flow. This is achieved by relating the radial and tangential velocity components in the Laplace space to the surface-averaged radial velocity and volume-averaged velocity vectors respectively in the time space. Moreover an expression relating the surface-averaged radial velocity and volume-averaged velocity vector has been provided. The total force is written as a summation of the undisturbed and disturbed force (quasi-steady, inviscid-unsteady and viscous-unsteady) contributions. The force contributions thus obtained are expressed as comprising of two parts – that arising due to spatial variation in the ambient flow and the other arising due to temporal variation. The current formulation is applicable to inhomogeneous ambient flows, however in the limit of negligible Reynolds and Mach numbers. The results are applicable even for particles of sizes larger than the acoustic wavelength. The accuracy of the explicit time-domain force expression is first tested by computing the force on an 80 mm diameter particle due to a weak planar expansion fan. Extension of this formulation when nonlinear effects become important is also proposed and tested by considering strong expansion fans. The results thus obtained are compared against corresponding axisymmetric numerical simulations.

Using simple kinematics, we propose a general theory of linear wave interactions between the interfacial waves of a two-dimensional (2D), inviscid, multilayered fluid system. The strength of our formalism is that one does not have to specify the physics of the waves in advance. Wave interactions may lead to instabilities, which may or may not be of the familiar ‘normal-mode’ type. Contrary to intuition, the underlying dynamical system describing linear wave interactions is found to be nonlinear. Specifically, a saw-tooth jet profile with three interfaces possessing kinematic and geometric symmetry is explored. Fixed points of the system for different ranges of a Froude number like control parameter $\unicode[STIX]{x1D6FE}$ are derived, and their stability evaluated. Depending upon the initial condition and $\unicode[STIX]{x1D6FE}$, the dynamical system may reveal transient growth, weakly positive Lyapunov exponents, as well as different nonlinear phenomena such as the formation of periodic and pseudo-periodic orbits. All these occur for those ranges of $\unicode[STIX]{x1D6FE}$ where normal-mode theory predicts neutral stability. Such rich nonlinear phenomena are not observed in a 2D dynamical system resulting from the two-wave problem, which reveals only stable and unstable nodes.

Electrostatic atomization of a liquid of finite electrical conductivity in the so-called cone-jet regime relies on the electric shear stresses that appear in a region of the liquid surface when a meniscus of the liquid is subjected to an intense electric field. An order of magnitude analysis is used to describe the flow induced by these stresses, which drive the liquid of the meniscus into a jet that issues from the tip of the meniscus and breaks into droplets at some distance from it. When the dielectric constant of the liquid is large, the electric shear stresses extend into the jet and cause a depression that sucks liquid from the meniscus. The induced flow rate is estimated and shown to represent approximately the minimum flow rate at which a cone-jet can be established. It is argued that the meniscus can be stabilized by the electric field that the charge of the jet induces on it. This stabilizing mechanism weakens when the flow rate supplied to the meniscus decreases, and its failure may determine an alternative minimum flow rate for the cone-jet regime. The instability of the jet and existing scaling laws for the size of the spray droplets are discussed.

Flows induced by vertical lifting of a rectangular beam partly immersed into shallow water in a rectangular prismatic channel with a horizontal bottom are studied within the framework of the long-wave approximation. The beam width coincides with the channel width and the lower and upper planes of the beam are parallel to the channel bottom. The lifting process in the general case consists of three stages. At the first stage, the lower surface of the beam is completely located in the liquid, which ascends following the beam under the action of hydrostatic pressure. At the second stage, the edges of the lower surface of the beam leave the water medium, the wetted part of the beam becomes smaller, and the liquid under this part of the beam move upward. At the beginning of the third stage, the beam is separated from water; as a result, liquid lifting that occurred at the second stage leads to the formation of two diverging waves. The liquid flow in the domain adjacent to the lower surface of the beam is calculated analytically, while the liquid flow outside this domain is obtained by means of numerical calculations by the CABARET (compact accurately boundary-adjusting high-resolution technique) scheme, which provides the second order of accuracy on smooth solutions.

The distinctive pendulum-like oscillation and pitching patterns of cubic and rectangular slung prisms were inspected for two aspect ratios at various Reynolds numbers $Re$ under two free-stream turbulence levels. Systematic experiments were performed using high-resolution telemetry and hotwire anemometry to quantitatively characterize the dynamics of the prisms and the wake fluctuation. The results show that the dynamics of the prisms can be characterized by two distinctive regions depending on the prism shape. Specifically, in the case of cubic prisms the regions are defined by the growth rate of the pitching amplitude; whereas the dynamics of the rectangular prisms is more sensitive to the angle of attack. In particular, when the large side initially faces the flow, the regions are defined by the synchronization between the vortex shedding and pure oscillations under very low turbulence. When the smaller side initially faces the flow, the regions are defined by the equilibrium pitching position. Regardless of the geometry of the prism and flow condition the dominant oscillation frequency resulted as being close to the natural frequency of the small-amplitude pendulum-like oscillation.

Suspensions of microswimmers are a rich source of fascinating new fluid mechanics. Recently we predicted the active pipe flow dispersion of gyrotactic microalgae, whose orientation is biased by gravity and flow shear. Analytical theory predicts that these active swimmers disperse in a markedly distinct manner from passive tracers (Taylor dispersion). Dispersing swimmers display non-zero drift and effective diffusivity that is non-monotonic with Péclet number. Such predictions agree with numerical simulations, but hitherto have not been tested experimentally. Here, to facilitate comparison, we obtain new solutions of the axial dispersion theory accounting both for swimmer negative buoyancy and a local nonlinear response of swimmers to shear, provided by two alternative microscopic stochastic descriptions. We obtain new predictions for suspensions of the model swimming alga Dunaliella salina, whose motility and buoyant mass we parametrise using tracking video microscopy. We then present a new experimental method to measure gyrotactic dispersion using fluorescently stained D. salina and provide a preliminary comparison with predictions of a non-zero drift above the mean flow for each microscopic stochastic description. Finally, we propose further experiments for a full experimental characterisation of gyrotactic dispersion measures and discuss the implications of our results for algal dispersion in industrial photobioreactors.

Local linear stability analysis is applied to the flow inside a cyclone separator to investigate the unsteady precession of the vortex core. The results of the stability analysis are compared with experimental measurements of the vortex oscillations using high-speed photography with particle seeding and hot-wire anemometry. The experiments reveal distinct spatial variation in the oscillation behaviour within the cyclone. The unsteady motion is focused at each end of the device, at both the narrow cone tip and just below the exhaust duct at the top of the cone, which is known as a vortex finder. The local stability analysis shows that an absolute instability is present throughout the flow for some non-zero azimuthal wavenumbers. The unsteady flow is observed to be driven by coupling between the shear layer and inertial waves confined within the vortex core. Comparison of the stability analysis with experiments shows the same frequency and mode shape behaviour and suggests that the local analysis accurately predicts the unstable modes of the system. The precessing vortex core is responsible for a narrow-band acoustic noise. Comparisons are also drawn with acoustic measurements made on cyclones in which the system is defined by key non-dimensional parameters, such as the swirl number and outlet diameter ratio. The results in this study demonstrate the applicability of local stability analysis to a complex swirling system and yield credible details about the underlying mechanisms of the unstable flow inside the cyclone.