From the analysis of the isentropic limit of weak compression shock waves, oblique shock waves in which the post-shock Mach number is larger than the pre-shock Mach number, named non-ideal oblique shocks, are admissible in substances characterized by moderate molecular complexity and in the close proximity to the liquid–vapour saturation curve. Non-ideal oblique shocks of finite amplitude are systematically analysed, clarifying the roles of the pre-shock thermodynamic state and Mach number. The necessary conditions for the occurrence of non-ideal oblique shocks of finite amplitude are singled out. In the parameter space of pre-shock thermodynamic states and Mach number, a new domain is defined which embeds the pre-shock states for which the Mach number increase can possibly take place. The present findings are confirmed by state-of-the-art thermodynamic models applied to selected commercially available fluids, including siloxanes and hydrocarbons currently used as working fluids in renewable energy systems.

We study periodically driven Taylor–Couette turbulence, i.e. the flow confined between two concentric, independently rotating cylinders. Here, the inner cylinder is driven sinusoidally while the outer cylinder is kept at rest (time-averaged Reynolds number is $Re_{i}=5\times 10^{5}$ ). Using particle image velocimetry, we measure the velocity over a wide range of modulation periods, corresponding to a change in Womersley number in the range $15\leqslant Wo\leqslant 114$ . To understand how the flow responds to a given modulation, we calculate the phase delay and amplitude response of the azimuthal velocity. In agreement with earlier theoretical and numerical work, we find that for large modulation periods the system follows the given modulation of the driving, i.e. the behaviour of the system is quasi-stationary. For smaller modulation periods, the flow cannot follow the modulation, and the flow velocity responds with a phase delay and a smaller amplitude response to the given modulation. If we compare our results with numerical and theoretical results for the laminar case, we find that the scalings of the phase delay and the amplitude response are similar. However, the local response in the bulk of the flow is independent of the distance to the modulated boundary. Apparently, the turbulent mixing is strong enough to prevent the flow from having radius-dependent responses to the given modulation.

A combined experimental and numerical approach to the analysis of the secondary stability of realistic swept-wing boundary layers is presented. Global linear stability theory is applied to experimentally measured base flows. These base flows are three-dimensional laminar boundary layers subject to spanwise distortion due to the presence of primary stationary crossflow vortices. A full three-dimensional description of these flows is accessed through the use of tomographic particle image velocimetry (PIV). The stability analysis solves for the secondary high-frequency modes of type I and type II, ultimately responsible for turbulent breakdown. Several pertinent parameters arising from the application of the proposed methodology are investigated, including the mean flow ensemble size and the measurement domain extent. Extensive use is made of the decomposition of the eigensolutions into the terms of the Reynolds–Orr equation, allowing insight into the production and/or destruction of perturbations from various base flow features. Stability results demonstrate satisfactory convergence with respect to the mean flow ensemble size and are independent of the handling of the exterior of the measurement domain. The Reynolds–Orr analysis reveals a close relationship between the type I and type II instability modes with spanwise and wall-normal gradients of the base flow, respectively. The structural role of the in-plane velocity components in the perturbation growth, topology and sensitivity is identified. Using the developed framework, further insight is gained into the linear growth mechanisms and later stages of transition via the primary and secondary crossflow instabilities. Furthermore, the proposed methodology enables the extension and enhancement of the experimental measurement data to the pertinent instability eigenmodes. The present work is the first demonstration of the use of a measured base flow for stability analysis applied to the swept-wing boundary layer, directly avoiding the modelling of the primary vortices receptivity processes.

A novel algorithm is developed to calculate the nonlinear optimal boundary perturbations in three-dimensional incompressible flow. An optimal step length in the optimization loop is calculated without any additional calls to the Navier–Stokes equations. The algorithm is applied to compute the optimal inflow eddies for the flow around a wind turbine to clarify the mechanisms behind wake meandering, a phenomenon usually observed in wind farms. The turbine is modelled as an actuator disc using an immersed boundary method with the loading prescribed as a body force. At Reynolds number (based on free-stream velocity and turbine radius) $Re=1000$ , the most energetic inflow perturbation has a frequency $\unicode[STIX]{x1D714}=0.8$ –2, and is in the form of an azimuthal wave with wavenumber $m=1$ and the same radius as the actuator disc. The inflow perturbation is amplified by the strong shear downstream of the edge of the disc and then tilts the rolling-up vortex rings to induce wake meandering. This mechanism is verified by studying randomly perturbed flow at $Re\leqslant 8000$ . At five turbine diameters downstream of the disc, the axial velocity oscillates at a magnitude of more than 60 % of the free-stream velocity when the magnitude of the inflow perturbation is 6 % of the free-stream wind speed. The dominant Strouhal number of the wake oscillation is 0.16 at $Re=3000$ and keeps approximately constant at higher $Re$ . This Strouhal number agrees well with previous experimental findings. Overall the observations indicate that the well-observed stochastic wake meandering phenomenon appearing far downstream of wind turbines is induced by large-scale (the same order as the turbine rotor) and low-frequency free-stream eddies.

Under slightly supersaturated conditions, bubbles need many minutes to grow due to the low gas diffusivity in liquids. When coalescence occurs, the fact that the bubbles have diffusively grown on top of a surface allows for control with precision of the location and the timing at which the coalescence takes place. Numerous coalescences of two $\text{CO}_{2}$ microbubbles in water are recorded at a frame rate of ${\sim}65\,000~\text{fps}$ . The evolution of the coalescing process is analysed in detail, differentiating among three phases: neck formation, wave propagation along the bubble surface and bubble detachment. First of all, the formation of the collapsing neck between both bubbles is compared to a capillary–inertial theoretical model. Afterwards, the propagating deformation along the surface is characterised measuring its evolution, velocity and dominant wavelength. Once bubbles coalesce, the perturbing waves and the final shape of the new bubble breaks the equilibrium between buoyancy and capillary forces. Consequently, the coalesced bubble detaches and rises due to buoyancy, oscillating with its natural Minnaert frequency. In addition to the experiments, a boundary integral code has been used to obtain numerical results of the coalescence under similar conditions, showing excellent agreement with the experimental data.

The interaction between droplet dispersion and evaporation in an acetone spray evaporating under ambient conditions is experimentally studied with an aim to understand the physics behind the spatial correlation between the local vapour mass fraction and droplets. The influence of gas-phase turbulence and droplet–gas slip velocity of such correlations is examined, while the focus is on the consequence of droplet clustering on collective evaporation of droplet clouds. Simultaneous and planar measurements of droplet size, velocity and number density, and vapour mass fraction around the droplets, were obtained by combining the interferometric laser imaging for droplet sizing and planar laser induced fluorescence techniques (Sahu et al., Exp. Fluids, vol. 55, 1673, 2014b, pp. 1–21). Comparison with droplet measurements in a non-evaporating water spray under the same flow conditions showed that droplet evaporation leads to higher fluctuations of droplet number density and velocity relative to the respective mean values. While the mean droplet–gas slip velocity was found to be negligibly small, the vaporization Damköhler number ( $Da_{v}$ ) was approximately ‘one’, which means the droplet evaporation time and the characteristic time scale of large eddies are of the same order. Thus, the influence of the convective effect on droplet evaporation is not expected to be significant in comparison to the instantaneous fluctuations of slip velocity, which refers to the direct effect of turbulence. An overall linearly increasing trend was observed in the scatter plot of the instantaneous values of droplet number density ( $N$ ) and vapour mass fraction ( $Y_{F}$ ). Accordingly, the correlation coefficient of fluctuations of vapour mass fraction and droplet number density ( $R_{n\ast y}$ ) was relatively high ( ${\approx}0.5$ ) implying moderately high correlation. However, considerable spread of the $N$ versus $Y_{F}$ scatter plot along both coordinates demonstrated the influence on droplet evaporation due to turbulent droplet dispersion, which leads to droplet clustering. The presence of droplet clustering was confirmed by the measurement of spatial correlation coefficient of the fluctuations of droplet number density for different size classes ( $R_{n\ast n}$ ) and the radial distribution function (RDF) of the droplets. Also, the tendency of the droplets to form clusters was higher for the acetone spray than the water spray, indicating that droplet evaporation promoted droplet grouping in the spray. The instantaneous group evaporation number ( $G$ ) was evaluated from the measured length scale of droplet clusters (by the RDF) and the average droplet size and spacing in instantaneous clusters. The mean value of $G$ suggests an internal group evaporation mode of the droplet clouds near the spray centre, while single droplet evaporation prevails near the spray boundary. However, the large fluctuations in the magnitude of instantaneous values of $G$ at all measurement locations implied temporal variations in the mode of droplet cloud evaporation.

We present results on the global and local characterisation of heat transport in homogeneous bubbly flow. Experimental measurements were performed with and without the injection of ${\sim}2.5~\text{mm}$ diameter bubbles (corresponding to bubble Reynolds number $Re_{b}\approx 600$) in a rectangular water column heated from one side and cooled from the other. The gas volume fraction $\unicode[STIX]{x1D6FC}$ was varied in the range 0 %–5 %, and the Rayleigh number $Ra_{H}$ in the range $4.0\times 10^{9}{-}1.2\times 10^{11}$. We find that the global heat transfer is enhanced up to 20 times due to bubble injection. Interestingly, for bubbly flow, for our lowest concentration $\unicode[STIX]{x1D6FC}=0.5\,\%$ onwards, the Nusselt number $\overline{Nu}$ is nearly independent of $Ra_{H}$, and depends solely on the gas volume fraction $\unicode[STIX]{x1D6FC}$. We observe the scaling $\overline{Nu}\,\propto \,\unicode[STIX]{x1D6FC}^{0.45}$, which is suggestive of a diffusive transport mechanism, as found by Alméras et al. (J. Fluid Mech., vol. 776, 2015, pp. 458–474). Through local temperature measurements, we show that the bubbles induce a huge increase in the strength of liquid temperature fluctuations, e.g. by a factor of 200 for $\unicode[STIX]{x1D6FC}=0.9\,\%$. Further, we compare the power spectra of the temperature fluctuations for the single- and two-phase cases. In the single-phase cases, most of the spectral power of the temperature fluctuations is concentrated in the large-scale rolls/motions. However, with the injection of bubbles, we observe intense fluctuations over a wide range of scales, extending up to very high frequencies. Thus, while in the single-phase flow the thermal boundary layers control the heat transport, once the bubbles are injected, the bubble-induced liquid agitation governs the process from a very small bubble concentration onwards. Our findings demonstrate that the mixing induced by high Reynolds number bubbles ($Re_{b}\approx 600$) offers a powerful mechanism for heat transport enhancement in natural convection systems.

Control of laminar-to-turbulent transition on a swept-wing is achieved by base-flow modification in an experimental framework, up to a chord Reynolds number of 2.5 million. This technique is based on the control strategy used in the numerical simulation by Dörr & Kloker (J. Phys. D: Appl. Phys., vol. 48, 2015b, 285205). A spanwise uniform body force is introduced using dielectric barrier discharge plasma actuators, to either force against or along the local cross-flow component of the boundary layer. The effect of forcing on the stability of the boundary layer is analysed using a simplified model proposed by Serpieri et al. (J. Fluid Mech., vol. 833, 2017, pp. 164–205). A minimal thickness plasma actuator is fabricated using spray-on techniques and positioned near the leading edge of the swept-wing, while infrared thermography is used to detect and quantify transition location. Results from both the simplified model and experiment indicate that forcing along the local cross-flow component promotes transition while forcing against successfully delays transition. This is the first experimental demonstration of swept-wing transition delay via base-flow modification using plasma actuators.

High-resolution particle image velocimetry measurements were performed on laminar and transitional oblique shock wave reflections for a range of Mach numbers ($M=1.6{-}2.3$), Reynolds numbers ($Re_{x_{sh}}=1.4\times 10^{6}{-}3.5\times 10^{6}$) and flow deflection angles ($\unicode[STIX]{x1D703}=1^{\circ }{-}5^{\circ }$ or $p_{3}/p_{1}=1.11{-}1.64$). The laminar interactions revealed a long, flat and triangular shaped separation bubble. For relatively strong interactions ($p_{3}/p_{1}>1.2$), the bubble grows linearly in the upstream direction with increasing shock strength. Under these conditions, the boundary layer keeps an on average laminar velocity profile up to the shock impingement location, followed by a quick transition and subsequent reattachment of the boundary layer. For weaker interactions ($p_{3}/p_{1}<1.2$), the boundary layer is able to remain laminar further downstream of the bubble, which consequently results in a later reattachment of the boundary layer. The pressure distribution at the interaction onset for all laminar cases shows excellent agreement with the free-interaction theory, therefore supporting its validity even for incipiently separated laminar oblique shock wave reflections.

The motion of tracer particles is kinematically simulated around three elementary flow patterns; a Burgers vortex, a shear-layer structure with coincident vortices and a node-saddle topology. These patterns are representative for their broader class of coherent structures in turbulence. Therefore, examining the dispersion in these elementary structures can improve the general understanding of turbulent dispersion at short time scales. The shear-layer structure and the node-saddle topology exhibit similar pair dispersion statistics compared to the actual turbulent flow for times up to $3{-}10\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$, where, $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ is the Kolmogorov time scale. However, oscillations are observed for the pair dispersion in the Burgers vortex. Furthermore, all three structures exhibit Batchelor’s scaling. Richardson’s scaling was observed for initial particle pair separations $r_{0}\leqslant 4\unicode[STIX]{x1D702}$ for the shear-layer topology and the node-saddle topology and was related to the formation of the particle sheets. Moreover, the material line orientation statistics for the shear-layer and node-saddle topology are similar to the actual turbulent flow statistics, up to at least $4\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$. However, only the shear-layer structure can explain the non-perpendicular preferential alignment between the material lines and the direction of the most compressive strain, as observed in actual turbulence. This behaviour is due to shear-layer vorticity, which rotates the particle sheet generated by straining motions and causes the particles to spread in the direction of compressive strain also. The material line statistics in the Burgers vortex clearly differ, due to the presence of two compressive principal straining directions as opposed to two stretching directions in the shear-layer structure and the node-saddle topology. The tetrad dispersion statistics for the shear-layer structure qualitatively capture the behaviour of the shape parameters as observed in actual turbulence. In particular, it shows the initial development towards planar shapes followed by a return to more volumetric tetrads at approximately $10\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$, which is associated with the particles approaching the vortices inside the shear layer. However, a large deviation is observed in such behaviour in the node-saddle topology and the Burgers vortex. It is concluded that the results for the Burgers vortex deviated the most from actual turbulence and the node-saddle topology dispersion exhibits some similarities, but does not capture the geometrical features associated with material lines and tetrad dispersion. Finally, the dispersion around the shear-layer structure shows many quantitative (until 2–$4\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$) and qualitative (until $20\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$) similarities to the actual turbulence.

It is well known that deterministic two-dimensional marine ice sheets can only be stable if the grounding line is positioned at a sufficiently steep, downward sloping bedrock. When bedrock conditions favour instabilities, multiple stable ice sheet profiles may occur. Here, we employ continuation techniques to examine the sensitivity of a two-dimensional marine ice sheet to stochastic noise representing short time scale variability, either in the accumulation rate or in the sea level height. We find that in unique regimes, the position of the grounding line is most sensitive to noise in the accumulation rate and can explain excursions observed in field measurements. In the multiple equilibrium regime, there is a strong asymmetry in transition probabilities between the different ice sheet states, with a strong preference to switch to the branch with a steeper bedrock slope.

Both Schwartz (J. Fluid Mech., vol. 62 (3), 1974, pp. 553–578) and Cokelet (Phil. Trans. R. Soc. Lond., vol. 286 (1335), 1977, pp. 183–230) failed to gain convergent results for limiting Stokes waves in extremely shallow water by means of perturbation methods, even with the aid of extrapolation techniques such as the Padé approximant. In particular, it is extremely difficult for traditional analytic/numerical approaches to present the wave profile of limiting waves with a sharp crest of $120^{\circ }$ included angle first mentioned by Stokes in the 1880s. Thus, traditionally, different wave models are used for waves in different water depths. In this paper, by means of the homotopy analysis method (HAM), an analytic approximation method for highly nonlinear equations, we successfully gain convergent results (and especially the wave profiles) of the limiting Stokes waves with this kind of sharp crest in arbitrary water depth, even including solitary waves of extreme form in extremely shallow water, without using any extrapolation techniques. Therefore, in the frame of the HAM, the Stokes wave can be used as a unified theory for all kinds of waves, including periodic waves in deep and intermediate depths, cnoidal waves in shallow water and solitary waves in extremely shallow water.

Combined free-stream disturbance measurements and receptivity studies in hypersonic wind tunnels were conducted by means of a slender wedge probe and direct numerical simulation. The study comprises comparative tunnel noise measurements at Mach 3, 6 and 7.4 in two Ludwieg tube facilities and a shock tunnel. Surface pressure fluctuations were measured over a wide range of frequencies and test conditions including harsh test environments not accessible to measurement techniques such as Pitot probes and hot-wire anemometry. A good agreement was found between normalized Pitot pressure fluctuations converted into normalized static pressure fluctuations and the wedge probe readings. Quantitative results of the tunnel noise are provided in frequency ranges relevant for hypersonic boundary-layer transition. Complementary numerical simulations of the leading-edge receptivity to fast and slow acoustic waves were performed for the applied wedge probe at conditions corresponding to the experimental free-stream conditions. The receptivity to fast acoustic waves was found to be characterized by an early amplification of the induced fast mode. For slow acoustic waves an initial decay was found close to the leading edge. At all Mach numbers, and for all considered frequencies, the leading-edge receptivity to fast acoustic waves was found to be higher than the receptivity to slow acoustic waves. Further, the effect of inclination angles of the acoustic wave with respect to the flow direction was investigated. An inclination angle was found to increase the response on the wave-facing surface of the probe and decrease the response on the opposite surface for fast acoustic waves. A frequency-dependent response was found for slow acoustic waves. The combined numerical and experimental approach in the present study confirmed the previous suggestion that the slow acoustic wave is the dominant acoustic mode in noisy hypersonic wind tunnels.

Macroscopic models based on moment equations are developed to describe the transport of mass and energy near the phase boundary between a liquid and its rarefied vapour due to evaporation and hence, in this study, condensation. For evaporation from a spherical droplet, analytic solutions are obtained to the linearised equations from the Navier–Stokes–Fourier, regularised 13-moment and regularised 26-moment frameworks. Results are shown to approach computational solutions to the Boltzmann equation as the number of moments are increased, with good agreement for Knudsen number ${\lesssim}1$, whilst providing clear insight into non-equilibrium phenomena occurring adjacent to the interface.

The generalised Kolmogorov equation is used to describe the scale-by-scale turbulence dynamics in the shear layer and in the separation bubble generated by a bulge at one of the walls in a turbulent channel flow. The second-order structure function, which is the basis of such an equation, is used as a proxy to define a scale-energy content, that is an interpretation of the energy associated with a given scale. Production and dissipation regions and the flux interchange between them, in both physical and separation space, are identified. Results show how the generalised Kolmogorov equation, a five-dimensional equation in our anisotropic and strongly inhomogeneous flow, can describe the turbulent flow behaviour and related energy mechanisms. Such complex statistical observables are linked to a visual inspection of instantaneous turbulent structures detected by means of the Q-criterion. Part of these turbulent structures are trapped in the recirculation where they undergo a pseudo-cyclic process of disruption and reformation. The rest are convected downstream, grow and tend to larger streamwise scales in an inverse cascade. The classical picture of homogeneous isotropic turbulence in which energy is fed at large scales and transferred to dissipate at small scales does not simply apply to this flow where the energy dynamics strongly depends on position, orientation and length scale.

We consider the problem of a rigid plate, inclined at an angle $\unicode[STIX]{x1D6FC}\in (0,\unicode[STIX]{x03C0}/2)$ to the horizontal, accelerating uniformly from rest into, or away from, a semi-infinite strip of inviscid, incompressible fluid under gravity. Following on from Gallagher et al. (J. Fluid Mech., vol. 841, 2018, pp. 109–145) (henceforth referred to as GNB), it is of interest to analyse the well-posedness and stability of the principal flow with respect to perturbations in the initially horizontal free surface close to the plate contact point. In particular we find that the solution to the principal unperturbed problem, denoted by [IBVP] in GNB, is well-posed and stable with respect to perturbations in initial data in the region of interest, localised close to the contact point of the free surface and the plate, when the plate is accelerated with dimensionless acceleration $\unicode[STIX]{x1D70E}\geqslant -\cot \,\unicode[STIX]{x1D6FC}$, while the solution to [IBVP] is ill-posed with respect to such perturbations in the initial data, when the plate is accelerated with dimensionless acceleration $\unicode[STIX]{x1D70E}<-\cot \,\unicode[STIX]{x1D6FC}$. The physical source of the ill-posedness of the principal problem [IBVP], when $\unicode[STIX]{x1D70E}<-\cot \,\unicode[STIX]{x1D6FC}$, is revealed to be due to the leading-order problem in the innermost region localised close to the initial contact point being in the form of a local Rayleigh–Taylor problem. As a consequence of this mechanistic interpretation we anticipate that, when the plate is accelerated with $\unicode[STIX]{x1D70E}<-\cot \,\unicode[STIX]{x1D6FC}$, the inclusion of weak surface tension effects will restore well-posedness of the problem [IBVP] which will, however, remain temporally unstable.

The free surface and flow field structure generated by the uniform acceleration (with dimensionless acceleration $\unicode[STIX]{x1D70E}$) of a rigid plate, inclined at an angle $\unicode[STIX]{x1D6FC}\in (0,\unicode[STIX]{x03C0}/2)$ to the exterior horizontal, as it advances ($\unicode[STIX]{x1D70E}>0$) or retreats ($\unicode[STIX]{x1D70E}<0$) from an initially stationary and horizontal strip of inviscid incompressible fluid under gravity, are studied in the small-time limit via the method of matched asymptotic expansions. This work generalises the case of a uniformly accelerating plate advancing into a fluid as studied by Needham et al. (Q. J. Mech. Appl. Maths, vol. 61 (4), 2008, pp. 581–614). Particular attention is paid to the innermost asymptotic regions encompassing the initial interaction between the plate and the free surface. We find that the structure of the solution to the governing initial boundary value problem is characterised in terms of the parameters $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D707}$ (where $\unicode[STIX]{x1D707}=1+\unicode[STIX]{x1D70E}\tan \unicode[STIX]{x1D6FC}$), with a bifurcation in structure as $\unicode[STIX]{x1D707}$ changes sign. This bifurcation in structure leads us to question the well-posedness and stability of the governing initial boundary value problem with respect to small perturbations in initial data in the innermost asymptotic regions, the discussion of which will be presented in the companion paper Gallagher et al. (J. Fluid Mech. vol. 841, 2018, pp. 146–166). In particular, when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times \mathbb{R}^{+}$, the free surface close to the initial contact point remains monotone, and encompasses a swelling jet when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times [1,\infty )$ or a collapsing jet when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times (0,1)$. However, when $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D707})\in (0,\unicode[STIX]{x03C0}/2)\times \mathbb{R}^{-}$, the collapsing jet develops a more complex structure, with the free surface close to the initial contact point now developing a finite number of local oscillations, with near resonance type behaviour occurring close to a countable set of critical plate angles $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{n}^{\ast }\in (0,\unicode[STIX]{x03C0}/2)$ ($n=1,2,\ldots$).

Radionuclide scanning images published in Nature by Di Chiro in 1964 showed a downward migration along the spinal canal of particle tracers injected in the brain ventricles while also showing an upward flow of tracers injected in the lumbar region of the canal. These observations, since then corroborated by many radiological measurements, have been the basis for the hypothesis that there must be an active circulation mechanism associated with the transport of cerebrospinal fluid (CSF) deep down into the spinal canal and subsequently returning a portion back to the cranial vault. However, to date, there has been no physical explanation for the mechanism responsible for the establishment of such a bulk recirculating motion. To investigate the origin and characteristics of this recirculating flow, we have analyzed the motion of the CSF in the subarachnoid space of the spinal canal. Our analysis accounts for the slender geometry of the spinal canal, the small compliance of the dura membrane enclosing the CSF in the canal, and the fact that the CSF is confined to a thin annular subarachnoid space surrounding the spinal cord. We apply this general formulation to study the characteristics of the flow generated in a simplified model of the spinal canal consisting of a slender compliant cylindrical pipe with a coaxial cylindrical inclusion, closed at its distal end, and subjected to small periodic pressure pulsations at its open entrance. We show that the balance between the local acceleration and viscous forces produces a leading-order flow consisting of pure oscillatory motion with axial velocities on the order of a few centimetres per second and amplitudes monotonically decreasing along the length of the canal. We then demonstrate that the nonlinear term associated with the convective acceleration contributes to a second-order correction consisting of a steady streaming that generates a bulk recirculating motion of the CSF along the length of the canal with characteristic velocities two orders of magnitude smaller than the leading-order oscillatory flow. The results of the analysis of this idealized geometry of the spinal canal are shown to be in good agreement not only with experimental measurements in an in-vitro model but also with radiological measurements conducted in human adults.

This work investigates the three-dimensional, spatio-temporal flow development in the aft portion of a laminar separation bubble. The bubble is forming on a flat plate geometry, subjected to an adverse pressure gradient, featuring maximum reverse flow of approximately 2 % of the local free-stream velocity. Time-resolved velocity measurements are performed by means of planar and tomographic particle image velocimetry, in the vicinity of the reattachment region. The measurements are complemented with a numerical solution of the boundary layer equations in the upstream field. The combined numerical and measured boundary layer is used as a baseline flow for linear stability theory analysis. The results provide insight into the dynamics of dominant coherent structures that form in the separated shear layer and deform along the span. Stability analysis shows that the flow becomes unstable upstream of separation, where both normal and oblique modes undergo amplification. While the shear layer roll up is linked to the amplification of the fundamental normal mode, the oblique modes at angles lower than approximately $30^{\circ }$ are also amplified substantially at the fundamental frequency. A model based on the stability analysis and experimental measurements is employed to demonstrate that the spanwise deformations of rollers are produced due to a superposition of normal and oblique instability modes initiating upstream of separation. The degree of the initial spanwise deformations is shown to depend on the relative amplitude of the dominant normal and oblique waves. This is confirmed by forcing the normal mode through a controlled impulsive perturbation introduced by a spanwise invariant dielectric-barrier-discharge plasma actuator, resulting in the formation of spanwise coherent vortices. The findings elucidate the link between important features in the bubble shedding dynamics and stability characteristics and provide further clarification on the differences in the development of coherent structures seen in recent experiments. Moreover, the results present a handle on the development of effective control strategies that can be used to either promote or suppress shedding in separation bubbles, which is of interest for system performance improvement and control of aeroacoustic emissions in relevant applications.

Comprehensive experimental and computational investigations have revealed possible mechanisms underlying low-frequency unsteadiness observed in spanwise homogeneous shock-wave/turbulent-boundary-layer interactions (STBLI). In the present work, we extend this understanding by examining the dynamic linear response of a moderately separated Mach 2.3 STBLI to small perturbations. The statistically stationary linear response is analysed to identify potential time-local and time-mean linear tendencies present in the unsteady base flow: these provide insight into the selective amplification properties of the flow at various points in the limit cycle, as well as asymmetry and restoring mechanisms in the dynamics of the separation bubble. The numerical technique uses the synchronized large-eddy simulation method, previously developed for free shear flows, significantly extended to include a linear constraint necessary for wall-bounded flows. The results demonstrate that the STBLI fosters a global absolute linear instability corresponding to a time-mean linear tendency for upstream shock motion. The absolute instability is maintained through constructive feedback of perturbations through the recirculation: it is self-sustaining and insensitive to external forcing. The dynamics are characterized for key frequency bands corresponding to high–mid-frequency Kelvin–Helmholtz shedding along the separated shear layer $(St_{L}\sim 0.5)$, low–mid-frequency oscillations of the separation bubble $(St_{L}\sim 0.1)$ and low-frequency large-scale bubble breathing and shock motion $(St_{L}\sim 0.03)$, where the Strouhal number is based on the nominal length of the separation bubble, $L$: $St_{L}=fL/U_{\infty }$. A band-pass filtering decomposition isolates the dynamic flow features and linear responses associated with these mechanisms. For example, in the low-frequency band, extreme shock displacements are shown to correlate with time-local linear tendencies toward more moderate displacements, indicating a restoring mechanism in the linear dynamics. However, a disparity between the linearly stable shock position and the mean shock position leads to an observed asymmetry in the low-frequency shock motion cycle, in which upstream motion occurs more rapidly than downstream motion. This is explained through competing linear and nonlinear (mass depletion through shedding) mechanisms and discussed in the context of an oscillator model. The analysis successfully illustrates how time-local linear dynamics sustain several key unsteady broadband flow features in a causal manner.