Fractal features of the turbulent/non-turbulent interface (TNTI) in shock wave/turbulent boundary-layer interaction (SWBLI) flows are essential in understanding the physics of the SWBLI and the supersonic turbulent boundary layer, yet have received almost no attention previously. Accordingly, this study utilises a high spatiotemporal resolution visualisation technique, ice-cluster-based planar laser scattering (IC-PLS), to acquire the TNTI downstream of the reattachment in a SWBLI flow. Evolution of the fractal features of the TNTI in this SWBLI flow is analysed by comparing the parameters of the TNTI acquired in this study with those from a previous result (Zhuang et al.J. Fluid Mech., vol. 843, 2018a).

Thermodynamic fluctuations of pressure, density, temperature or entropy $\{p^{\prime },\unicode[STIX]{x1D70C}^{\prime },T^{\prime },s^{\prime }\}$ in compressible aerodynamic turbulence, although generated by the flow, are fundamentally related to one another by the thermodynamic equation of state. Ratios between non-dimensional root-mean-square (r.m.s.) levels ($\text{CV}_{p^{\prime }}:=\bar{p}^{-1}\,p_{rms}^{\prime }$, $\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}:=\bar{\unicode[STIX]{x1D70C}}^{-1}\,\unicode[STIX]{x1D70C}_{rms}^{\prime }$, $\text{CV}_{T^{\prime }}:=\bar{T}^{-1}\,T_{rms}^{\prime }$), along with all possible 2-moment correlation coefficients $\{c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }},c_{p^{\prime }\unicode[STIX]{x1D70C}^{\prime }},c_{p^{\prime }T^{\prime }},c_{s^{\prime }\unicode[STIX]{x1D70C}^{\prime }},c_{s^{\prime }T^{\prime }},c_{s^{\prime }p^{\prime }}\}$, represent, in the sense of Bradshaw (Annu. Rev. Fluid Mech., vol. 9, 1977, pp. 33–54), the thermodynamic turbulence structure of the flow. We use direct numerical simulation (DNS) data, both for plane channel flow and for sustained homogeneous isotropic turbulence, to determine the range of validity of the leading-order, formally $O(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }})$, approximations of the exact relations between thermodynamic turbulence structure parameters. Available DNS data are mapped on the $(\text{CV}_{\unicode[STIX]{x1D70C}^{\prime }}^{-1}\,\text{CV}_{T^{\prime }},c_{\unicode[STIX]{x1D70C}^{\prime }T^{\prime }})$-plane and their loci, identified using the leading-order approximations, highlight specific behaviour for different flows or flow regions. For the particular case of sustained compressible homogeneous isotropic turbulence, it is shown that the DNS data collapse onto a single curve corresponding to $c_{s^{\prime }T^{\prime }}\approxeq 0.2$ (for air flow), while the approximation $c_{s^{\prime }p^{\prime }}\approxeq 0$ fits reasonably well wall turbulence DNS data, providing building blocks towards the construction of simple phenomenological models.

In order to make the extraction of tidal current energy economically viable, the power production per turbine must be optimised in each tidal array. Furthermore, the impact of power extraction on the marine flow environment must be understood. These two aims mean that designers must be able to model different configurations of a tidal array in order to create the most efficient, least invasive arrangement. In this paper, an analytical model is developed for array design in idealised rectangular tidal channels with idealised turbines. The model includes the effects of (i) local blockage, (ii) surface deformation and (iii) added drag due to the installation of the array. While these effects have been accounted for individually in past work, the model presented here is the first to include all three such that the interaction between different effects can be understood. Results are presented for optimal local blockage and turbine resistance as functions of inherent channel drag coefficient, channel length and Froude number at various global blockage values. It will be shown that it is necessary to model the effects of local blockage and added drag simultaneously in order to obtain the design parameters of a tidal array (global blockage, local blockage and turbine resistance), which will maximise the power extraction per turbine. Neglecting either effect will lead to an array design with lower power extraction than the optimum, the addition of unnecessary extra turbines and higher lost power from the array.

In order to compare the bubble dynamics of various quantum liquids, we performed the visual observation of a sound-induced bubble in a normal liquid 4He, pure superfluid 4He, and superfluid 3He–4He liquid mixtures of saturated and unsaturated 3He concentrations. When an acoustic wave pulse was applied to these liquids under saturated vapour pressure, a macroscopic bubble was generated on the surface of a piezoelectric transducer. For all liquids, the size of the bubble increased, as a higher voltage was applied to the transducer at a fixed temperature. In the normal 4He we observed a primary bubble surrounded with many small bubbles which ascended upward together. In contrast to normal phase, only one bubble was generated in the superfluid 4He, and its shape proved to be highly irregular with an ill-defined surface. In the 3He saturated superfluid mixture, we also observed a solitary bubble but with a nearly perfect spherical shape. The bubble in this mixture expanded on the transducer surface, grew to a maximum size of the order of 1 mm and then started shrinking. As the bubble detached from the transducer with shrinking, we clearly detected an origination of the upward jet flow which penetrated the bubble. The jet velocity in the liquid mixture was approximately 102–103 times smaller than in water. At the final stage of the process we could sometimes observe a vortex ring generation. It is interesting that, though the bubble size and time scale of the phenomenon differ from those in water, the behaviour in the collapsing process had much in common with the simulation study of the vortex ring generation in water. In addition, for the mixture with the unsaturated 3He concentration of about 25% at 600 mK, the shape of the upward jet was observed distinctly, using more precise measurement with shadowgraph method.

A non-perturbative nonlinear theory for moderately dense gas–solid suspensions is outlined within the framework of the Boltzmann–Enskog equation by extending the work of Saha & Alam (J. Fluid Mech., vol. 833, 2017, pp. 206–246). A linear Stokes’ drag law is adopted for gas–particle interactions, and the viscous dissipation due to hydrodynamic interactions is incorporated in the second-moment equation via a density-corrected Stokes number. For the homogeneous shear flow, the present theory provides a unified treatment of dilute to dense suspensions of highly inelastic particles, encompassing the high-Stokes-number rapid granular regime ($St\rightarrow \infty$) and its small-Stokes-number counterpart, with quantitative agreement for all transport coefficients. It is shown that the predictions of the shear viscosity and normal-stress differences based on existing theories deteriorate markedly with increasing density as well as with decreasing Stokes number and restitution coefficient.

The IUTAM Symposium Fundamental Aspects of Vortex Motion was held in Tokyo, Japan, from 31 August to 4 September 1987. We present an account of the technical sessions of that meeting. The main goals of this report are (i) to provide a widely accessible record of the four-and-one-half day meeting; (ii) to identify important new developments in the field of vortex dynamics of potential interest to a larger audience than the invited attendees; and (iii) to attempt some overview comments with the wisdom of hindsight that may be useful as a guide to specific papers in the proceedings and other current literature.

Using a physical space (i.e. non-modal) approach, we investigate interactions between fast inertio-gravity (IG) waves and slow balanced flows in a shallow rotating fluid. Specifically, we consider a train of IG waves impinging on a steady, exactly balanced vortex. For simplicity, the one-dimensional problem is studied first; the limitations of one-dimensionality are offset by the ability to define balance in an exact way. An asymptotic analysis of the problem in the small-amplitude limit is performed to demonstrate the existence of interactions. It is shown that these interactions are not confined to the modification of the wave field by the vortex but, more importantly, that the waves are able to alter in a non-trivial way the potential vorticity associated with that vortex. Interestingly, in this one-dimensional problem, once the waves have traversed the vortex region and have propagated away, the vortex exactly recovers its initial shape and thus bears no signature of the interaction. Furthermore, we prove this last result in the case of arbitrary vortex and wave amplitudes. Numerical integrations of the full one-dimensional shallow-water equations in strongly nonlinear regimes are also performed: they confirm that time-dependent interactions exist and increase with wave amplitude, while at the final state the vortex bears no sign of the interaction. In addition, they reveal that cyclonic vortices interact more strongly with the wave field than anticyclonic ones.

The error minimization technique is introduced as a method of obtaining the simultaneous solution to a set of partial differential equations. Because this numerical technique is at worst neutrally stable, it is believed to have fundamental advantages over existing techniques. The method is formulated in general for the steady fluid dynamic conservation equations and then applied to a specific case of irrotational isentropic flow of a perfect gas through a nozzle. Flow-field solutions were found in nozzle contours having a throat radius of curvature as low as 0·5. Comparison between experimental Mach number contours and the theoretical solution for a finite inlet nozzle with a conical exit is excellent.

The flow regime in the vicinity of oscillatory slender bodies, either an isolated one or a row of many bodies, immersed in viscous fluid (i.e. under creeping flow conditions) is studied. Applying the slender-body theory by distributing proper singularities on the bodies’ major axes yields reasonably accurate and easily computed solutions. The effect of the oscillations is revealed by comparisons with known Stokes flow solutions and is found to be most significant for motion along the normal direction. Streamline patterns associated with motion of a single body are characterized by formation and evolution of eddies. The motion of adjacent bodies results, with a reduction or an increase of the drag force exerted by each body depending on the direction of motion and the specific geometrical set-up. This dependence is demonstrated by parametric results for frequency of oscillations, number of bodies, their slenderness ratio and the spacing between them. Our method, being valid for a wide range of parameter values and for densely packed arrays of rods, enables simulation of realistic flapping of bristled wings of some tiny insects and of locomotion of flagella and ciliated micro-organisms, and might serve as an efficient tool in the design of minuscule vehicles. Its potency is demonstrated by a solution for the flapping of thrips.

The instability of an electrically conducting fluid of magnetic diffusivity λ and viscosity v in a rapidly rotating spherical container of magnetic diffusivity $\hat{\lambda}$ in the presence of a toroidal magnetic field is investigated. Attention is focused on the case of a toroidal magnetic field induced by a uniform current density parallel to the axis of rotation, which was first studied by Malkus (1967). We show that the internal ohmic dissipation does not affect the stability of the hydromagnetic solutions obtained by Malkus (1967) in the limit of small λ. It is solely the effect of the magnetic Hartmann boundary layer that causes instabilities of the otherwise stable solutions. When the container is a perfect conductor, $\hat{\lambda}$ = 0, the hydromagnetic instabilities grow at a rate proportional to the magnetic Ekman number of the fluid Eλ; when the container is a nearly perfect insulator, $\lambda/\hat{\lambda}\ll 1$, the hydromagnetic instabilities grow at a rate proportional to E1/2λ; when the container is a nearly perfect conductor, λ 1, the growth rates are proportional to λ, where λ is the magnetic Ekman number based on the diffusivity λ of the container. The main characteristics of the instabilities are not affected by varying magnetic properties of the container. In light of the destabilizing role played by the Hartmann boundary layer, we also examine the corresponding magnetoconvection in a rapidly rotating fluid sphere with the perfectly conducting container and stress-free velocity boundary conditions. Analytical magnetoconvection solutions in closed form are obtained and implications are discussed.

The retarded Green's function for the linearized version of the equation of the mixed type governing the potential flow around a rotating helicopter blade or a propeller (with no forward motion) is derived and is shown to constitute the unifying feature of the various existing approaches to rotor acoustics. This Green's function is then used to pinpoint the singularity predicted by the linearized theory of rotor acoustics which signals its experimentally confirmed breakdown in the transonic regime: the gradient of the near-field sound amplitude, associated with a linear flow which is steady in the blade-fixed rotating frame, diverges on the sonic cylinder at the dividing boundary between the subsonic and supersonic regions of the flow. Prom the point of view of the equivalent Cauchy problem for the homogeneous wave equation, this singularity is caused by the imposition of entirely non-characteristic initial data on a space—time hypersurface which, at its points of intersection with the sonic cylinder, is locally characteristic. It also emerges from the analysis presented that the acoustic discontinuities detected in the far zone are generated by the quadrupole source term in the Ffowcs Williams-Hawkings equation and that the impulsive noise resulting from these discontinuities would be removed if the flow in the transonic region were to be rendered unsteady (as viewed from the blade-fixed rotating frame).

This paper is an extension of work on separation from a downstream moving wall by Ruban et al. (J. Fluid. Mech., vol. 678, 2011, pp. 124–155) and is in particular concerned with the boundary-layer separation in unsteady two-dimensional laminar supersonic flow. In a frame attached to the wall, the separation is assumed to be provoked by a shock wave impinging upon the boundary layer at a point that moves downstream with a non-dimensional speed which is assumed to be of order where is the Reynolds number. In the coordinate system of the shock however, the wall moves upstream. The strength of the shock and its speed are allowed to vary with time on a characteristic time scale that is large compared to . The ‘triple-deck’ model is used to describe the interaction process. The governing equations of the interaction problem can be derived from the Navier–Stokes equations in the limit . The numerical solutions are obtained using a combination of finite differences along the streamwise direction and Chebyshev collocation along the normal direction in conjunction with Newton linearization. In the present study with the wall moving upstream, the evidence is inconclusive regarding the so-called ‘Moore–Rott–Sears’ criterion being satisfied. Instead it is observed that the pressure rise from its initial value is very slow and that a recirculation region forms, the upstream part of which is wedge-shaped, as also observed in turbulent marginal separation for large values of angle of attack.