Tracer particles are the workhorse of the fluid dynamicist for visualizing flow in transparent liquids. Thus a tracer becomes useless if its signal disappears, which frequently happens in practice, for instance due to bleaching. The opposite occurs in a recent work by Kim & Stone (J. Fluid Mech., vol. 850, 2018, pp. 769–783): the fading signal of a dissolving particle may reveal the local composition in a mixture. Such information is highly valuable in the study of evaporating droplets. In virtually all realistic cases, droplets consist of multiple components, ranging from trace impurities to engineered cocktails, which, for instance, generate a desired deposit pattern for a printing process. Different components typically evaporate at different rates, which causes inhomogeneities in droplet composition. Determining the latter is one of the main challenges in the field.

The flow physics controlling the stabilisation of a methane/air laminar premixed flame in a narrow channel (internal width $\ell _{i}=5~\text{mm}$) is revisited from numerical simulations. Combustion is described with complex chemistry and transport properties, along with a coupled simulation of heat transfer at and within the wall. To conduct a thorough analysis of the flame–wall interaction, the steady flame is obtained after applying a procedure to find the inlet mass flow rate that exactly matches the flame mass burning rate. The response of the premixed flame shape to various operating conditions is then analysed in terms of flame propagation velocity and flow topology in the vicinity of the reactive front. We focus on the interrelations between the flame speed, the configuration taken by the flame surface, the flow deviation induced by the heat released and the fluxes at the wall. Compared to an adiabatic flame, the flame speed increases with edge-flame quenching at an isothermal cold wall in the absence of a boundary layer, decreases with a boundary layer, to increase again with heat-transfer coupling within the wall. A regime diagram is proposed to delineate between flame shapes in order to build a classification versus heat-transfer properties. Under a small level of convective heat transfer with the ambient air surrounding the channel, the larger the thermal conductivity in the solid, the faster the reaction zone propagates in the vicinity of the wall, leaving the centreline reaction zone behind. The premixed flame front is then concave towards the fresh gases on the axis of symmetry (so-called tulip flame) with a flame speed higher than in the adiabatic case. Increasing the heat loss at the wall through convection with ambient air, the flame shape becomes convex (mushroom flame) and the flame speed decreases below its adiabatic level. Scaling laws are provided for the flame speed under these various regimes. Mesh resolution was calibrated, with and without heat loss, from simulations of one-dimensional detailed chemistry flames, leading to mesh resolution of $12.5~\unicode[STIX]{x03BC}\text{m}$ for detailed chemistry and $25.0~\unicode[STIX]{x03BC}\text{m}$ with a skeleton mechanism. The quality of the resolution was also assessed from multi-physics budgets derived from first principles, involving upstream-flame heat retrocession by the wall leading to flow acceleration, budgets bringing physical insights into flame/wall interaction. Additional overall mesh convergence tests of the multi-physics solution would have been desirable, but were not conducted due to the high computing cost of these fully compressible simulations, hence also solving for the acoustic field with low convective velocities.

Based on a variable-coefficient Kadomtsev–Petviashvili (KP) equation, the topographic effect on the wave interactions between two oblique internal solitary waves is investigated. In the absence of rotation and background shear, the model set-up featuring idealised shoaling topography and continuous stratification is motivated by the large expanse of continental shelf in the South China Sea. When the bottom is flat, the evolution of an initial wave consisting of two branches of internal solitary waves can be categorised into six patterns depending on the respective amplitudes and the oblique angles measured counterclockwise from the transverse axis. Using theoretical multi-soliton solutions of the constant-coefficient KP equation, we select three observed patterns and examine each of them in detail both analytically and numerically. The effect of shoaling topography leads to a complicated structure of the leading waves and the emergence of two types of trailing wave trains. Further, the case when the along-crest width is short compared with the transverse domain of interest is examined and it is found that although the topographic effect can still modulate the wave field, the spreading effect in the transverse direction is dominant.

High Reynolds number wall-bounded turbulent flows subject to buoyancy forces are fraught with complex dynamics originating from the interplay between shear generation of turbulence ($S$) and its production or destruction by density gradients ($B$). For horizontal walls, $S$ augments the energy budget of the streamwise fluctuations, while $B$ influences the energy contained in the vertical fluctuations. Yet, return to isotropy remains a tendency of such flows where pressure–strain interaction redistributes turbulent energy among all three velocity components and thus limits, but cannot fully eliminate, the anisotropy of the velocity fluctuations. A reduced model of this energy redistribution in the inertial (logarithmic) sublayer, with no tuneable constants, is introduced and tested against large eddy and direct numerical simulations under both stable ($B<0$) and unstable ($B>0$) conditions. The model links key transitions in turbulence statistics with flux Richardson number (at $Ri_{f}=-B/S\approx$$-2$, $-1$ and $-0.5$) to shifts in the direction of energy redistribution. Furthermore, when coupled to a linear Rotta-type closure, an extended version of the model can predict individual variance components, as well as the degree of turbulence anisotropy. The extended model indicates a regime transition under stable conditions when $Ri_{f}$ approaches $Ri_{f,max}\approx +0.21$. Buoyant destruction $B$ increases with increasing stabilizing density gradients when $Ri_{f}<Ri_{f,max}$, while at $Ri_{f}\geqslant Ri_{f,max}$ limitations on the redistribution into the vertical component throttle the highest attainable rate of buoyant destruction, explaining the ‘self-preservation’ of turbulence at large positive gradient Richardson numbers. Despite adopting a ‘framework of maximum simplicity’, the model results in novel and insightful findings on how the interacting roles of energy redistribution and buoyancy modulate the variance budgets and the energy exchange among the components.

The dimensionless dissipation coefficient $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D700}L/U^{3}$, where $\unicode[STIX]{x1D700}$ is the dissipation rate, $U$ the root-mean-square velocity and $L$ the integral length scale, is an important characteristic of statistically stationary homogeneous turbulence. In studies of $\unicode[STIX]{x1D6FD}$, the external force is typically isotropic and large scale, and its helicity $H_{f}$ either zero or not measured. Here, we study the dependence of $\unicode[STIX]{x1D6FD}$ on $H_{f}$ and find that it decreases $\unicode[STIX]{x1D6FD}$ by up to 10 % for both isotropic forces and shear flows. The numerical finding is supported by static and dynamical upper bound theory. Both show a relative reduction similar to the numerical results. That is, the qualitative and quantitative dependence of $\unicode[STIX]{x1D6FD}$ on the helicity of the force is well captured by upper bound theory. Consequences for the value of the Kolmogorov constant and theoretical aspects of turbulence control and modelling are discussed in connection with the properties of the external force. In particular, the eddy viscosity in large-eddy simulations of homogeneous turbulence should be decreased by at least 10 % in the case of strongly helical forcing.

The unsteady ascent of a buoyant, turbulent line plume through a quiescent, uniform environment is modelled in terms of the width-averaged vertical velocity and density deficit. It is demonstrated that for a well-posed, linearly stable model, account must be made for the horizontal variation of the velocity and the density deficit; in particular the variance of the velocity field and the covariance of the density deficit and velocity fields, represented through shape factors, must exceed threshold values, and that models based upon ‘top-hat’ distributions in which the dependent fields are piecewise constant are ill-posed. Numerical solutions of the nonlinear governing equations are computed to reveal that the transient response of the system to an instantaneous change in buoyancy flux at the source may be captured through new similarity solutions, the form of which depend upon both the ratio of the old to new buoyancy fluxes and the shape factors.

A number of recent studies have demonstrated the existence of so-called large- and very-large-scale motions (LSM, VLSM) that occur in the logarithmic region of inertia-dominated wall-bounded turbulent flows. These regions exhibit significant streamwise coherence, and have been shown to modulate the amplitude and frequency of small-scale inner-layer fluctuations in smooth-wall turbulent boundary layers. In contrast, the extent to which analogous modulation occurs in inertia-dominated flows subjected to convective thermal stratification (low Richardson number) and Coriolis forcing (low Rossby number), has not been considered. And yet, these parameter values encompass a wide range of important environmental flows. In this article, we present evidence of amplitude modulation (AM) phenomena in the unstably stratified (i.e. convective) atmospheric boundary layer, and link changes in AM to changes in the topology of coherent structures with increasing instability. We perform a suite of large eddy simulations spanning weakly ($-z_{i}/L=3.1$) to highly convective ($-z_{i}/L=1082$) conditions (where $-z_{i}/L$ is the bulk stability parameter formed from the boundary-layer depth $z_{i}$ and the Obukhov length $L$) to investigate how AM is affected by buoyancy. Results demonstrate that as unstable stratification increases, the inclination angle of surface layer structures (as determined from the two-point correlation of streamwise velocity) increases from $\unicode[STIX]{x1D6FE}\approx 15^{\circ }$ for weakly convective conditions to nearly vertical for highly convective conditions. As $-z_{i}/L$ increases, LSMs in the streamwise velocity field transition from long, linear updrafts (or horizontal convective rolls) to open cellular patterns, analogous to turbulent Rayleigh–Bénard convection. These changes in the instantaneous velocity field are accompanied by a shift in the outer peak in the streamwise and vertical velocity spectra to smaller dimensionless wavelengths until the energy is concentrated at a single peak. The decoupling procedure proposed by Mathis et al. (J. Fluid Mech., vol. 628, 2009a, pp. 311–337) is used to investigate the extent to which amplitude modulation of small-scale turbulence occurs due to large-scale streamwise and vertical velocity fluctuations. As the spatial attributes of flow structures change from streamwise to vertically dominated, modulation by the large-scale streamwise velocity decreases monotonically. However, the modulating influence of the large-scale vertical velocity remains significant across the stability range considered. We report, finally, that amplitude modulation correlations are insensitive to the computational mesh resolution for flows forced by shear, buoyancy and Coriolis accelerations.

A weakly nonlinear stability analysis of an axisymmetric Newtonian liquid jet is presented. The calculation is based on a small-amplitude perturbation method and performed to second order in the perturbation parameter. The obtained solution includes terms derived from a polynomial approximation of a viscous contribution containing products of Bessel functions with different arguments. The use of such an approximation is not needed in the inviscid case and the planar case, since the equations of those problems can be solved in an exact form. The developed model depends on three dimensionless parameters: the initial perturbation amplitude, the perturbation wavenumber and the liquid Ohnesorge number, the latter being the dimensionless liquid viscosity. The influence of the approximate terms was shown to be relatively small for a large range of Ohnesorge numbers so that they can be ignored. This simplification provides a jet model as simple to use as the previous ones, but taking into account the liquid viscosity and the cylindrical geometry. The jet model is used to reveal the effect of both the wavenumber and the Ohnesorge number on the formation of satellite drops, which is known as a nonlinear effect. Results are found in good agreement with direct numerical simulations and forced liquid jet experiments for wavenumbers lower than a threshold value. Satellite drop formation is retarded with increasing Ohnesorge number and wavenumber, as expected by the damping and size effects of viscosity. The threshold number corresponds to the maximum wavenumber for which satellite drop formation is predicted before jet breakup, and for which volume conservation is satisfied within a certain amount. The volume conservation criterion is imposed to ensure that the conclusions inferred by our model are safe.

Instabilities of a two-dimensional quasigeostrophic circular flow around a rigid circular wall (island) with radial offshore bottom slope are studied analytically. The basic flow is composed of two concentric, uniform potential-vorticity (PV) rings with zero net vorticity attached to the island. Linear stability analysis for perturbations in the form of azimuthal modes leads to a transcendental eigenvalue equation. The non-dimensional governing parameters are beta (associated with the steepness of the bottom slope, hence taken to be negative), the PV in the inner ring and the radii of the inner and outer rings. This setting up of the problem allows us to derive analytically the eigenvalue equation. We first analyse this equation for weak slopes to understand the asymptotic first-order corrections to the flat-bottom case. For azimuthal modes 1 and 2, it is found that the conical topographic beta effect stabilizes the counterclockwise flows, but destabilizes clockwise flows. For a clockwise flow, the beta effect gives rise to the mode-1 instability, contrary to the flat-bottom case where this mode is always stable. Moreover, however small the slope steepness (beta) is, it leads to the mode-1 instability in a large region in the parameter space. For steep slopes, the beta term in the PV expression may dominate the relative vorticity term, causing stabilization of the flow, as compared to the flat-bottom case, for both directions of the basic flow. When the flow is counterclockwise and the slope steepness is increased, mode 2 turns out to be entirely stable and modes 3, 4 and 5 enlarge their stability regions. In a clockwise flow, when the slope steepness is increased, mode 1 regains its stability in the entire parameter space, and mode 2 becomes more stable than mode 3. The bifurcation of mode 1 from stability to instability is discussed in terms of the Rossby waves at the contours of discontinuity of the basic PV and outside the uniform-PV rings.

Motivated by the importance of stratified shear flows in geophysical and environmental circumstances, we characterize their energetics, mixing and spectral behaviour through a series of direct numerical simulations of turbulence generated by Holmboe wave instability (HWI) under various initial conditions. We focus on circumstances where the stratification is sufficiently ‘strong’ so that HWI is the dominant primary instability of the flow. Our numerical findings demonstrate the emergence of self-organized criticality (SOC) that is manifest as an adjustment of an appropriately defined gradient Richardson number, $Ri_{g}$, associated with the horizontally averaged mean flow, in such a way that it is continuously attracted towards a critical value of $Ri_{g}\sim 1/4$. This self-organization occurs through a continuously reinforced localization of the ‘scouring’ motions (i.e. ‘avalanches’) that are characteristic of the turbulence induced by the breakdown of Holmboe wave instabilities and are developed on the upper and lower flanks of the sharply localized density interface, embedded within a much more diffuse shear layer. These localized ‘avalanches’ are also found to exhibit the expected scale-invariant characteristics. From an energetics perspective, the emergence of SOC is expressed in the form of a long-lived turbulent flow that remains in a ‘quasi-equilibrium’ state for an extended period of time. Most importantly, the irreversible mixing that results from such self-organized behaviour appears to be characterized generically by a universal cumulative turbulent flux coefficient of $\unicode[STIX]{x1D6E4}_{c}\sim 0.2$ only for turbulent flows engendered by Holmboe wave instability. The existence of this self-organized critical state corroborates the original physical arguments associated with self-regulation of stratified turbulent flows as involving a ‘kind of equilibrium’ as described by Turner (1973, Buoyancy Effects in Fluids, Cambridge University Press).

The onset and development of orbital motion of a trailing vortex from a wing undergoing small amplitude heaving motion is investigated using stereo particle image velocimetry in conjunction with three-dimensional reconstruction techniques. The effect of Strouhal number is examined via space–time representations of axial and azimuthal vorticity, axial velocity deficit and swirl ratio. At low Strouhal number, the undulation of the vortex remains unidirectional with no amplification in the streamwise direction. In contrast, at high Strouhal number, the amplitude of vortex undulation can increase by up to a factor of ten in the streamwise direction. These large amplitudes occur during orbital motion of the vortex. Irrespective of the value of either the Strouhal number of excitation or the streamwise location along the undulating vortex, generic physical mechanisms occur. Changes in curvature along the vortex are closely related to changes in the axial velocity deficit, extreme values of axial vorticity and swirl ratio and the onset and attenuation of pronounced azimuthal vorticity.

We investigate the existence of short-time, local transient growth in the helical modes of a rapidly swirling, high-speed jet that has transitioned into an axisymmetric bubble breakdown state. The time-averaged flow consisting of the bubble and its wake downstream constitute the base state, which we show to exhibit strong transient amplification owing to the non-modal behaviour of the continuous eigenspectrum. A pseudospectrum analysis mathematically identifies the so-called potential modes within this continuous spectrum and the resultant non-orthogonality between these modes and the existing discrete stable modes is shown to be the main contributor to such growth. As the swirling flow develops post the collapsed bubble, the potential spectrum moves further toward the unstable half-plane, which along with the concurrent weakening of exponential growth from the discrete unstable modes, increases the dynamic importance of transient growth inside the wake region. The transient amplifications calculated at several locations inside the bubble and wake confirm this, where strong growths inside the wake far outstrip the corresponding modal growths (if available) at shorter times, but especially at the higher helical orders and smaller streamwise wavenumbers. The corresponding optimal perturbations at initial times consist of streamwise streaks of azimuthal velocity, which if concentrated inside the core vortical region, unfold via the classical Orr mechanism to yield structures resembling core (or viscous) Kelvin waves of the corresponding Lamb–Oseen vortex. However, in contrast to that in Lamb–Oseen vortex flow, where critical-layer waves are associated with higher transient gains, here, such core Kelvin modes with the more compact spiral structure at the vortex core are seen to yield the maximum transient amplifications.

It is known that in steady-state potential flows, the separation of a gravity-driven free surface from a solid exhibits a number of peculiar characteristics. For example, it can be shown that the fluid must separate from the body so as to form one of three possible in-fluid angles: (i) $180^{\circ }$, (ii) $120^{\circ }$ or (iii) an angle such that the surface is locally perpendicular to the direction of gravity. These necessary separation conditions were notably remarked upon by Dagan & Tulin (J. Fluid Mech., vol. 51 (3), 1972, pp. 529–543) in the context of ship hydrodynamics, but they are of crucial importance in many potential-flow applications. It is not particularly well understood why there is such a drastic change in the local separation behaviours when the global flow is altered. The question that motivates this work is the following: outside of a formal balance-of-terms argument, why must cases (i)–(iii) occur and furthermore, what are the connections between them? In this work, we seek to explain the transitions between the three cases in terms of the singularity structure of the associated solutions once they are extended into the complex plane. A numerical scheme is presented for the analytic continuation of a vertical jet (or alternatively a rising bubble). It will be shown that the transition between the three cases can be predicted by observing the coalescence of singularities as the speed of the jet is modified. A scaling law is derived for the coalescence rate of singularities.

Active drag reduction of an Ahmed body with a slant angle of $25^{\circ }$, corresponding to the high-drag regime, has been experimentally investigated at Reynolds number $Re=1.7\times 10^{5}$, based on the square root of the model cross-sectional area. Four individual actuations, produced by steady blowing, are applied separately around the edges of the rear window and vertical base, producing a drag reduction of up to 6–14 %. However, the combination of the individual actuations results in a drag reduction 29 %, higher than any previous drag reductions achieved experimentally and very close to the target (30 %) set by automotive industries. Extensive flow measurements are performed, with and without control, using force balance, pressure scanner, hot-wire, flow visualization and particle image velocimetry techniques. A marked change in the flow structure is captured in the wake of the body under control, including the flow separation bubbles, over the rear window or behind the vertical base, and the pair of C-pillar vortices at the two side edges of the rear window. The change is linked to the pressure rise on the slanted surface and the base. The mechanisms behind the effective control are proposed. The control efficiency is also estimated.

We quantify the variability of the characteristic length scales of isotropically forced Boussinesq flows with stratification and frame rotation, as functions of the ratio $N/f$ of the Brunt–Väisälä frequency to the Coriolis frequency. The parameter ranges $0<N<f$, domain aspect ratio $1\leqslant \unicode[STIX]{x1D6FF}_{d}\leqslant 32$ and Burger number $Bu=\unicode[STIX]{x1D6FF}_{d}N/f\leqslant 1$ are explored for two values of $f$, one resulting in linear potential vorticity and the other in nonlinear potential vorticity. Characteristic length scales of the wave and vortical linear eigenmodes are separately quantified using $n$th-order spectral moments in both horizontal and vertical directions, for integer $n\leqslant 3$. In flows with linear potential vorticity, the horizontal vortical length scale $L_{0}$, characterizing a typical width of columnar structures, grows as ${\sim}(N/f)^{1/2}$ at all orders of $n$, regardless of domain aspect ratio. In unit-aspect-ratio domains, when intermediate scales are measured by filtering out the largest scales and using higher-order moments $n>1$, the vortical-mode aspect ratio $\unicode[STIX]{x1D6FF}_{0}$ asymptotes to a scaling of ${\sim}(N/f)^{-1}$, in agreement with quasi-geostrophic estimates. In contrast, the $\unicode[STIX]{x1D6FF}_{0}$ in tall-aspect-ratio domain flows yields a decay rate of at most ${\sim}(N/f)^{-1/2}$ after large-scale filtering. Flows with nonlinear potential vorticity display consistently weaker dependence of the characteristic scales on $N/f$ than the corresponding ones with linear potential vorticity. The wave-mode aspect ratios for all flows are essentially independent of $N/f$. We highlight the differences of these flow structure scalings relative to those expected for quasi-geostrophic flows, and those observed in strongly stratified, non-quasi-geostrophic flows.

This paper investigates the effect of a finite Reynolds number (FRN) on the flatness factor ($F$) of the velocity derivative in decaying homogeneous isotropic turbulence by applying the eddy damped quasi-normal Markovian (EDQNM) method to calculate all terms in an analytic expression for $F$ (Djenidi et al., Phys. Fluids, vol. 29 (5), 2017b, 051702). These terms and hence $F$ become constant when the Taylor microscale Reynolds number, $Re_{\unicode[STIX]{x1D706}}$ exceeds approximately $10^{4}$. For smaller values of $Re_{\unicode[STIX]{x1D706}}$, $F$, like the skewness $-S$, increases with $Re_{\unicode[STIX]{x1D706}}$; this behaviour is in quantitative agreement with experimental and direct numerical simulation data. These results indicate that one must first ensure that $Re_{\unicode[STIX]{x1D706}}$ is large enough for the FRN effect to be negligibly small before the hypotheses of Kolmogorov (Dokl. Akad. Nauk SSSR, vol. 30, 1941a, pp. 301–305; Dokl. Akad. Nauk SSSR, vol. 32, 1941b, pp. 16–18; J. Fluid Mech., vol. 13, 1962, pp. 82–85) can be assessed unambiguously. An obvious implication is that results from experiments and direct numerical simulations for which $Re_{\unicode[STIX]{x1D706}}$ is well below $10^{4}$ may not be immune from the FRN effect. Another implication is that a power-law increase of $F$ with respect to $Re_{\unicode[STIX]{x1D706}}$, as suggested by the Kolmogorov 1962 theory, is not tenable when $Re_{\unicode[STIX]{x1D706}}$ is large enough.

Using theory and experiments, we investigate granular surface avalanching due to material outflow from a narrow silo. The assumed silo geometry is a deep rectangular box, of moderate spanwise width and small gap thickness between smooth front and back walls. A small orifice deep below the free surface lets grains drain out at a constant rate. The resulting granular flows can therefore be assumed quasi-two-dimensional and quasi-steady over most of the surface descent history. To model these flows, we couple a kinematic model of deep granular flow with a dynamic model of shallow surface avalanching. We then compare the calculated flow fields with detailed particle tracking measurements, letting the silo ascend relative to the high-speed camera to increase spatial resolution. The results show that the avalanching surface shape and near-surface flow are controlled by the spanwise gradient in subsidence velocity, and how this gradient is in turn controlled by the height above orifice and the gap thickness. Whereas the deep flow pattern is rate independent, shallow avalanching is paced by the granular rheology.

Direct numerical simulations are undertaken to investigate the nature of instability mechanisms induced by singular and distributed roughnesses on a blunt-capsule configuration. On the base of a capsule-like hemispherical forebody at wind-tunnel conditions ($M=5.9$), we analyse the development of unsteady disturbances behind a patch of two different roughness geometries. First, spanwise periodic roughness elements are considered and cross-validation with other methods of the stability analysis is achieved. Two main unstable modes are found in the roughness wake, corresponding to the symmetric and antisymmetric modes already known for single roughness elements. Second, the case of a patch of (pseudo-)randomly distributed roughness is presented. A new type of roughness-induced cross-flow-like instability is observed for the blunt-capsule configuration. The rapid growth of primary and secondary instabilities in the cross-flow-type vortex is analysed and quantified in both the linear and nonlinear stages up to the laminar–turbulent breakdown. Spatio-temporal Fourier analysis is performed to track the onset of secondary instabilities, whereas laminar–turbulent transition is identified by the steep increase of the wall heat flux.

Oceanic internal tides and other inertia–gravity waves propagate in an energetic turbulent flow whose length scales are similar to the wavelengths. Advection and refraction by this flow cause the scattering of the waves, redistributing their energy in wavevector space. As a result, initially plane waves radiated from a source such as a topographic ridge become spatially incoherent away from the source. To examine this process, we derive a kinetic equation which describes the statistics of the scattering under the assumptions that the flow is quasigeostrophic, barotropic and well represented by a stationary homogeneous random field. Energy transfers are quantified by computing a scattering cross-section and shown to be restricted to waves with the same frequency and identical vertical structure, hence the same horizontal wavelength. For isotropic flows, scattering leads to an isotropic wave field. We estimate the characteristic time and length scales of this isotropisation, and study their dependence on parameters including the energy spectrum of the flow. Simulations of internal tides generated by a planar wavemaker carried out for the linearised shallow-water model confirm the pertinence of these scales. A comparison with the numerical solution of the kinetic equation demonstrates the validity of the latter and illustrates how the interplay between wave scattering and transport shapes the wave statistics.

Synthetic jets are zero-net-mass-flux actuators that can be used in a range of flow control applications. For some applications, the scaling of the trajectory of the jet with actuation and cross-flow parameters is important. This scaling is investigated for changes in the friction Reynolds number, changes in the velocity ratio (defined as the ratio between the mean jet blowing velocity and the free-stream velocity) and changes in the actuation frequency of the jet. A distinctive aspect of this study is the high-Reynolds-number turbulent boundary layers (up to $Re_{\unicode[STIX]{x1D70F}}=12\,800$) of the cross-flow. To our knowledge, this is the first study to investigate the effect of the friction Reynolds number of the cross-flow on the trajectory of an (unsteady) jet, as well as the first study to systematically investigate the scaling of the trajectory with actuation frequency. A broad range of parameters is varied (rather than an in-depth investigation of a single parameter) and the results of this study are meant to indicate the relative importance of each parameter rather than the exact influence on the trajectory. Within the range of parameters explored, the critical ones are found to be the velocity ratio as well as a non-dimensional frequency based on the jet actuation frequency, the cross-flow velocity and the jet dimensions. The Reynolds number of the boundary layer is shown to have only a small effect on the trajectory. An expression for the trajectory of the jet is derived from the data, which (in the limit) is consistent with known expressions for the trajectory of a steady jet in a cross-flow.