Elevated wall temperatures and impinging shock interactions are prevalent features in hypersonic flight. Currently, there is a lack of literature regarding experimental studies examining both features in a flight-representative environment. This work details hot-wall, hypersonic, impinging shock/boundary-layer interaction experiments performed in the T4 Stalker Tube. The model configuration was a two-dimensional heated flat plate and a shock generator. The surface of the graphite flat plate was resistively heated to a mean temperature from $T_w=298\ \textrm {K}$ to $T_w\approx 675\ \textrm {K}$ during an experimental run. An oblique shock, generated by a plate that was inclined at $10^{\circ }$ or $12^{\circ }$ to the free stream, was impinged on the heated flat plate to induce boundary layer separation. The primary flow condition produced Mach 7 flight-equivalent nozzle-supply enthalpy with a unit Reynolds number of $4.93\times 10^6\ \textrm {m}^{-1}$. More flow conditions with lower unit Reynolds numbers and flow enthalpies were considered to examine flow separation characteristics. Schlieren and infrared thermography captured the flow field and the wall temperature distribution, respectively. The results showed that the size of the flow separation grew with a higher $T_w$ and a lower unit Reynolds number. Moreover, the scaled separation of the present data showed a high discrepancy with existing separation correlations developed from a supersonic impinging shock and a hypersonic compression ramp, mainly due to the higher shock strength. Instead, the present data followed a scaling law that includes the pressure ratio across the impinging shock with a slight dependence on the wall temperature ratio.

Equilibrium solutions for hollow vortices in straining flow in a corner are obtained by solving a free-boundary problem. Conformal maps from a canonical doubly connected annular domain to the physical plane combining the Schottky–Klein prime function with an appropriate algebraic map lead to a problem similar to Pocklington's propagating hollow dipole. The result is a two-parameter family of solutions depending on the corner angle and on the non-dimensional ratio of strain to circulation.

The celebrated wall-scaling works for many statistical averages in turbulent flows near smooth walls, but the streamwise velocity fluctuation, $u^{\prime }$, is thought to be among the few exceptions. In particular, the near-wall mean-square peak, $\overline {u'u'}^+_p$ – where the superscript $+$ indicates normalization by the friction velocity $u_\tau$, the subscript $p$ indicates the peak value and the overbar indicates time averaging – is known to increase with increasing Reynolds number. The existing explanations suggest a logarithmic growth with respect to $Re$, where $Re$ is the Reynolds number based on $u_\tau$ and the thickness of the wall flow. We show that this boundless growth calls into question the veracity of wall-scaling and so cannot be sustained, and we establish an alternative formula for the peak magnitude that approaches a finite limit $\overline {u'u'}^+_\infty$ owing to the natural constraint of boundedness on the dissipation rate at the wall. This new formula agrees well with the existing data and, in contrast to the logarithmic growth, supports the classical wall-scaling for turbulent intensity at asymptotically high Reynolds numbers.

Stability analyses based on the rates of change of perturbations were performed to study the growth mechanisms of second-mode instability in hypersonic boundary layers. The results show that the streamwise velocity perturbation is strengthened by the concurrence of the momentum transfer due to the wall-normal velocity fluctuation and the streamwise gradient of the pressure perturbation near the wall, while the wall-normal velocity perturbation is dominated by the wall-normal gradient of the pressure perturbation. Meanwhile, the change of fluctuating internal energy is sustained by the advection of perturbed thermal energy in the vicinity of the critical layer and by the dilatation fluctuation near the wall. The energy transport by the wall-normal velocity fluctuation accounts for the growth of second-mode instability, and the growth rate depends on the relative phase of the energy transport by the wall-normal velocity fluctuation to the total time rate of change of fluctuating internal energy in the vicinity of the critical layer. Moreover, this relative phase is associated with the mutual interaction between the critical-layer fluctuation and the near-wall fluctuation. Porous walls recast this mutual interaction by delaying the phase of the wall-normal energy transport near the wall, resulting in the stabilization of the second mode.

Electrohydrodynamic instabilities at liquid interfaces continue to defy our intuition, from the pioneering work of Taylor (Proc. R. Soc. Lond. A, vol. 280, issue 1382, 1964, pp. 383–397) on conical tips of electrified droplets to a recent numerical study by Wagoner et al. (J. Fluid Mech., vol. 904, 2020, R4). The problem studied by Wagoner et al. (2020) consists of a droplet immersed in a more conducting and more dielectric liquid medium, in a strong electrical field. When the droplet is more viscous than the outer medium, the droplet develops a biconcave shape which might eventually evolve to a torus shape (or doughnut). In contrast, when the droplet is less viscous, it adopts a lenticular shape and emits a thin fluid sheet from its equator which in turn breaks up into droplets. These droplets form a ring of satellites around the original droplet, which justifies its appellation ‘Saturnian droplet’. The numerical simulations bring light to this complex phenomenon and confirm the robustness of the leaky-dielectric framework (Melcher & Taylor, Annu. Rev. Fluid Mech., vol. 1, 1969, pp. 111–146).

Dense, small particles suspended in turbulent smooth-wall flow are known to migrate towards the wall. It is, however, not clear if the particle migration continues in a rough-wall flow and what the responsible mechanism is, especially with changing roughness parameters. Here, we address this using direct numerical simulation of a turbulent pipe flow of a fixed friction Reynolds numbers and changing the roughness size as well as the Stokes number of the particles. The transport and deposition mechanisms of particles are segregated into three different regimes dictated by the Stokes number. Particles with small Stokes number follow the carrier fluid and are affected by the turbulent structures of the rough wall. Flow separation in the wake of the roughness and stagnant flow in the trough of the roughness causes these particles to be trapped in the roughness canopy. Particles with very large Stokes number, on the other hand, are attracted to the wall due to turbophoresis and collide with the rough wall where the frequency of wall collision increases with increasing Stokes number. These ballistic particles are unaffected by the turbulent fluctuations of the flow and their trajectory is determined by the roughness topography. At intermediate Stokes numbers, the transport of the particles is influenced by both the wall collisions and also the turbulent flow. Particles in this range of Stokes number occasionally collide with the wall and are entrained by the turbulent flow. In this regime, the particles may have a mean streamwise velocity that is larger than the bulk flow rate of the fluid. Finally, we observe that bulk particle velocity scale better with a time scale based on the roughness elements rather than the usual viscous time scale.

We consider the resonant coupling of mode-1 and mode-2 internal solitary waves by topography. Mode-2 waves are generated by a mode-1 wave encountering variable topography, modelled by a coupled Korteweg–de Vries (KdV) system. Three cases, namely $(A)$ weak resonant coupling, $(B)$ moderate resonant coupling and $(C)$ strong resonant coupling, are examined in detail using a three-layer density-stratified fluid system with different stratification and topographic settings. The strength of the resonant coupling is determined by the range of values taken by the ratio of linear long-wave phase speeds ($c_2/c_1$, where $c_1$ is the mode-1 speed and $c_2$ the mode-2 speed) while the waves are above the slope. In case $A$ the range is from $0.42$ (ocean edge) to $0.48$ (shelf edge), in case $B$ from $0.58$ (ocean) to $0.72$ (shelf) and in case $C$ from $0.44$ (ocean) to $0.92$ (shelf). The feedback from mode-2 to mode-1 is estimated by comparing the coupled KdV system with a KdV model. In case $A$, a small-amplitude convex mode-2 wave is generated by a depression mode-1 wave and the feedback on the mode-1 wave is negligible. In case $B$, a concave mode-2 wave of comparable amplitude to that of the depression incident mode-1 wave is formed; strong feedback enhances the polarity change process of the mode-1 wave. In case $C$, a large-amplitude concave mode-2 wave is produced by an elevation mode-1 wave; strong feedback suppresses the fission of the mode-1 wave. Simulations for a wider range of topographic slopes and three-layer stratifications are then classified in terms of these responses.

We study the evolution of unidirectional water waves from a randomly forced input condition with uncorrelated Fourier components. We examine the kurtosis of the linearised free surface as a convenient proxy for the probability of a rogue wave. We repeat the laboratory experiments of Onorato et al. (Phys. Rev. E, vol. 70, 2004, 067302), both experimentally and numerically, and extend the parameter space in our numerical simulations. We consider numerical simulations based on the modified nonlinear Schrödinger equation and the fully nonlinear water wave equations, which are in good agreement. For low steepness, existing analytical models based on the nonlinear Schrödinger equation (NLS) are found to be accurate. For cases which are steep or have very narrow bandwidths, these analytical models over-predict the rate at which excess kurtosis develops. In these steep cases, the kurtosis in both our experiments and numerical simulations peaks before returning to an equilibrium level. Such transient maxima are not predicted by NLS-based analytical models. Above a certain threshold of steepness, the steady-state value of kurtosis is primarily dependent on the spectral bandwidth. We also examine how the average shape of extreme events is modified by nonlinearity over the evolution distance, showing significant asymmetry during the initial evolution, which is greatly reduced once the spectrum has reached equilibrium. The locations of the maxima in asymmetry coincide approximately with the locations of the maxima in kurtosis.

We derive a Noether current for the Eulerian variational principle of ideal non-barotropic magnetohydrodynamics (MHD). It was shown previously that ideal non-barotropic MHD is mathematically equivalent to a five function field theory with an induced geometrical structure in the case that field lines cover surfaces and this theory can be described using a variational principle. Here we use various symmetries of the flow to derive topological constants of motion through the derived Noether current and discuss their implication for non-barotropic MHD.

A reduced model is presented for the dynamics of a slender sheet of a viscoelastic fluid. Starting with the Oldroyd-B constitutive model and exploiting an asymptotic analysis in the small aspect ratio of the sheet, equations are derived for the evolution of a ‘visco-elastica’. These depend on an elastic modulus, a creep viscosity and a solvent viscosity. They resemble standard equations for an elastica or a viscida, to which they reduce under the appropriate limits. The model is used to explore the effects of viscoelasticity on the dynamics of a curling ribbon, a drooping cantilever, buckling sheets, snap-through and a falling catenary. We then incorporate a yield stress, for a fluid that deforms by creep only above a critical stress, revisiting the curling and cantilever problems. This model generalises a number of previous theories for viscoelastic and viscoplastic ribbons.

As far as plastron is sustained, superhydrophobic (SHPo) surfaces are expected to reduce skin-friction drag in any flow conditions including large-scale turbulent boundary-layer flows of marine vessels. However, despite many successful drag reductions reported using laboratory facilities, the plastron on SHPo surfaces was persistently lost in high-Reynolds-number flows on open water, and no reduction has been reported until a recent study using certain microtrench SHPo surfaces underneath a boat (Xu et al., Phys. Rev. Appl., vol. 13, no. 3, 2020, 034056). Since scientific studies with controlled flows are difficult with a boat on ocean water, in this paper we test similar SHPo surfaces in a high-speed towing tank, which provides well-controlled open-water flows, by developing a novel $0.7\ \textrm {m} \times 1.4\ \textrm {m}$ towing plate, which subjects a $4\ \textrm {cm} \times 7\ \textrm {cm}$ sample to the high-Reynolds-number flows of the plate. In addition to the 7 cm long microtrenches, trenches divided into two in length are also tested and reveal an improvement. The skin-friction drag ratio relative to a smooth surface is found to be decreasing with increasing Reynolds number, down to 73 % (i.e. 27 % drag reduction) at $Re_x\sim 8\times 10^6$, before starting to increase at higher speeds. For a given gas fraction, the trench width non-dimensionalized to the viscous length scale is found to govern the drag reduction, in agreement with previous numerical results.

The so-called Landau–Levich–Deryaguin problem treats the coating flow dynamics of a thin viscous liquid film entrained by a moving solid surface. In this context, we use a simple experimental set-up consisting of a partially immersed rotating disc in a liquid tank to study the role of inertia, and also curvature, on liquid entrainment. Using water and UCON$^{\textrm{TM} }$ mixtures, we point out a rich phenomenology in the presence of strong inertia: ejection of multiple liquid sheets on the emerging side of the disc, sheet fragmentation, ligament formation and atomization of the liquid flux entrained over the disc's rim. We focus our study on a single liquid sheet and the related average liquid flow rate entrained over a thin disc for various depth-to-radius ratio $h/R < 1$. We show that the liquid sheet is created via a ballistic mechanism as liquid is lifted out of the pool by the rotating disc. We then show that the flow rate in the entrained liquid film is controlled by both viscous and surface tension forces as in the classical Landau–Levich–Deryaguin problem, despite the three-dimensional, non-uniform and unsteady nature of the flow, and also despite the large values of the film thickness based flow Reynolds number. When the characteristic Froude and Weber numbers become significant, strong inertial effects influence the entrained liquid flux over the disc at large radius-to-immersion-depth ratio, namely via entrainment by the disc's lateral walls and via a contribution to the flow rate extracted from the three-dimensional liquid sheet itself, respectively.

The present work investigates the excitation process by which free-stream disturbances are transformed into vortex-core perturbations. This problem of receptivity is modelled in terms of the resolvent in frequency space as the linear response to forcing. This formulation of receptivity suggests that non-normality of the resolvent is necessary to allow free-stream disturbances to excite the vortex core. Considering a local (in frequency) measure of non-normality, we show that vortices are frequency-selectively non-normal in a narrow frequency band of retrograde perturbations while the rest of the range is governed by an effectively normal operator, thus not contributing to receptivity. Canonical decomposition of the resolvent reveals that vortices are most susceptible to coiled filaments localised about the critical layer that induce bending waves on the core. Considering Lamb–Oseen, Batchelor and Moore–Saffman vortices as reference-flow models, we find free-stream receptivity to be essentially generic and independent of the axial wavelength on the considered range. A stochastic interpretation of the results could be a model for trailing-vortex meandering.

Evolution of the two-dimensional single-mode Richtmyer–Meshkov (RM) instability in a cylindrical geometry is numerically investigated through direct numerical simulation. A proper decomposition of the measured initial perturbation amplitude is found to be crucial for a comparative study between the numerical simulation and benchmark experiment. A refined compressible model is proposed based on the Bell equation by taking the premixed width of the initial interface into consideration. The modified model can accurately reproduce the development history of a single-mode perturbed gaseous interface between the first shock-interface interaction and reshock based on the evolution data of the unperturbed interface under the same premixing condition. The detailed effects of the RM instability, Rayleigh–Taylor stabilization and compressibility coupled with the Bell–Plesset effect are also specified with the aid of this model. It turns out that the refined Bell model can be further applied to the post-reshock stage of the RM instability before the appearance of strong nonlinearity.

The present study investigates the fundamental mechanisms of interaction between the propeller wake vortices and an untipped non-lifting wing. The study consists of a comprehensive experimental survey of a reference propeller–wing configuration with a high thickness parameter and is based on time-resolved visualisations and detailed flow and wall-pressure measurements. The experiment was designed to investigate the dynamics of the propeller blade vortices during the approach, encounter and penetration phases of the interaction and downstream of the body. To this end, three different models of the wing were manufactured including a transparent Perspex model that was crucial to simultaneously visualise the evolution of the vortex branches on the pressure and suction side of the body during the penetration phase. The study gains insight into the fundamental underlying mechanisms of the complex interaction between the propeller tip and blade trailing vortices and the wing for different propeller loadings. It is found that, during the encounter and the early penetration phases, tip vortex behaviour is strongly influenced by its interaction with the boundary layer of the wing that is manifested by a non-symmetrical evolution and breakdown of the vortex portions travelling along the pressure and suction sides of the wing. Reconnection between the vortex lines originating within the vortex core and the wing boundary layer maintains the linkage between the pressure and suction side portions of the vortex during the penetration phase and drives their rejoining downstream of the wing.

A modal stability analysis shows that pressure-driven pipe flow of an Oldroyd-B fluid is linearly unstable to axisymmetric perturbations, in stark contrast to its Newtonian counterpart which is linearly stable at all Reynolds numbers. The dimensionless groups that govern stability are the Reynolds number $Re = \rho U_{max} R /\eta$, the elasticity number $E = \lambda \eta /(R^2 \rho )$ and the ratio of solvent to solution viscosity $\beta = \eta _s/\eta$; here, $R$ is the pipe radius, $U_{max}$ is the maximum velocity of the base flow, $\rho$ is the fluid density and $\lambda$ is the microstructural relaxation time. The unstable mode has a phase speed close to $U_{max}$ over the entire unstable region in ($Re$, $E$, $\beta$) space. In the asymptotic limit $E (1-\beta ) \ll 1$, the critical Reynolds number for instability diverges as $Re_c \sim (E (1-\beta ))^{-3/2}$, the critical wavenumber increases as $k_c \sim (E (1-\beta ))^{-1/2}$, and the unstable eigenfunction is localized near the centreline, implying that the unstable mode belongs to a class of viscoelastic centre modes. In contrast, for $\beta \rightarrow 1$ and $E \sim 0.1$, $Re_c$ can be as low as $O(100)$, with the unstable eigenfunction no longer being localized near the centreline. Unlike the Newtonian transition which is dominated by nonlinear processes, the linear instability discussed in this study could be very relevant to the onset of turbulence in viscoelastic pipe flows. The prediction of a linear instability is, in fact, consistent with several experimental studies on pipe flow of polymer solutions, ranging from reports of ‘early turbulence’ in the 1970s to the more recent discovery of ‘elasto-inertial turbulence’ (Samanta et al., Proc. Natl Acad. Sci. USA, vol. 110, 2013, pp. 10557–10562). The instability identified in this study comprehensively dispels the prevailing notion of pipe flow of viscoelastic fluids being linearly stable in the $Re$–$W$ plane ($W = Re \, E$ being the Weissenberg number), marking a possible paradigm shift in our understanding of transition in rectilinear viscoelastic shearing flows. The predicted unstable eigenfunction should form a template in the search for novel nonlinear elasto-inertial states, and could provide an alternate route to the maximal drag-reduced state in polymer solutions. The latter has thus far been explained in terms of a viscoelastic modification of the nonlinear Newtonian coherent structures.

We present a mathematical description of turbulent entrainment that is applicable to free-shear problems that evolve in space, time or both. Defining the global entrainment velocity $\bar {V}_g$ to be the fluid motion across an isosurface of an averaged scalar, we find that for a slender flow, $\bar {V}_g=\bar {u}_\zeta - \bar {\textrm {D}}h_t/\bar {\textrm {D}}t$, where $\bar {\textrm {D}}/\bar {\textrm {D}} t$ is the material derivative of the average flow field and $\bar {u}_\zeta$ is the average velocity perpendicular to the flow direction across the interface located at $\zeta =h_t$. The description is shown to reproduce well-known results for the axisymmetric jet, the planar wake and the temporal jet, and provides a clear link between the local (small scale) and global (integral) descriptions of turbulent entrainment. Application to unsteady jets/plumes demonstrates that, under unsteady conditions, the entrainment coefficient $\alpha$ no longer only captures entrainment of ambient fluid, but also time-dependency effects due to the loss of self-similarity.

Evaporation and breakup of liquid droplets are common in many applications of the shock-driven multiphase instability (SDMI), such as in liquid-fuelled detonation engines, multiphase ejector pumps and turbines and explosive dispersal of liquid particles (i.e. chemical or biological agents). In this paper, the effects of evaporation and breakup of droplets on the mixing induced by the SDMI are considered through simulations and compared with experimental results. The evaporation model is validated against previous experimental data. The capabilities of the simulations and particle models are then demonstrated through a qualitative comparison with experimental results where breakup effects are negligible (i.e. small droplets). The simulation results are explored further to quantify the effects of evaporation (i.e. mixing enhancement) in the SDMI, providing further insight into the experimental results. A new breakup model, derived from previous works, is then presented for low Reynolds number (below 500), low Weber number (below 100) droplets in a shock-driven multiphase instability. The breakup model capabilities are then demonstrated through a comparison with experimental results where breakup effects are significant (larger droplet sizes). Finally, the simulation results are used to highlight the importance of breakup parameters on the evaporation rate and large-scale mixing in the SDMI. Overall, it is shown that evaporation is enhanced by the large-scale hydrodynamics instability, the SDMI, and that breakup of the droplets significantly increases the strength of the instability, and rate of droplet evaporation.

The paper is devoted to an experimental investigation of distributed receptivity of a laminar swept-wing boundary layer to unsteady freestream vortices with streamwise orientation of the vorticity vector. The experiments were performed on a model of a swept wing with sweep angle of $25^{\circ }$ at fully controlled disturbance conditions with freestream vortices generated by a special disturbance source. It is found that the unsteady streamwise vortices are able to provide very efficient excitation of non-stationary cross-flow instability modes without the necessity of the presence of any surface non-uniformities. The developed experimental approach provides the possibility for a detailed quantitative investigation of the mechanism of distributed excitation of unsteady boundary-layer disturbances due to scattering of freestream vortices on natural base-flow non-uniformity. This mechanism has been studied experimentally in detail. This paper (Part 1 of the present study) is devoted to description of: (a) the experimental approach and the base-flow structure; (b) the method of excitation of fully controlled streamwise-elongated freestream vortices; (c) the results of measurements of structure of these vortices; and (d) the experimental evidence of high efficiency of the distributed vortex receptivity mechanism under study. Part 2 of this study (see Borodulin et al., J. Fluid Mech., vol. 908, 2021, A15) is devoted to the theoretical background and experimental quantitative characteristics of the distributed vortex receptivity. Values of the corresponding receptivity coefficients are estimated there for their three different definitions as functions of the disturbance frequency, spanwise wavenumber and wave propagation angle.

This paper is devoted to an experimental investigation of the distributed receptivity of a laminar swept-wing boundary layer to unsteady freestream vortices with streamwise orientation of the vorticity vector. The experiments were performed on a model of a swept wing with a sweep angle of $25^{\circ }$ at fully controlled disturbance conditions with freestream vortices generated by a special disturbance source. It is found that the unsteady streamwise vortices are able to provide very efficient excitation of cross-flow instability modes without requiring the presence of any surface non-uniformities. The developed experimental approach is shown to allow us to perform a detailed quantitative investigation of the mechanism of distributed excitation of unsteady boundary-layer disturbances due to scattering of freestream vortices on natural base-flow non-uniformity. This mechanism has been studied experimentally in detail. Part 1 of the present investigation (Borodulin et al., J. Fluid Mech., vol. 908, 2021, A14) was devoted to the description of the experimental approach and the base-flow structure, the method of excitation of fully controlled streamwise-elongated freestream vortices, the results of measurements of structure of these vortices and the experimental evidence of high efficiency of the distributed vortex receptivity mechanism under study. Meanwhile, the present paper (Part 2) is devoted to: (a) theoretical background and definition of the distributed receptivity coefficients and (b) obtaining experimental quantitative characteristics of the distributed vortex receptivity including values of the corresponding receptivity coefficients for their three different definitions as functions of the disturbance frequency, spanwise wavenumber and wave propagation angle.