There have been numerous studies concerning the possibility of self-similar scaling laws in fully developed turbulent shear flows, driven over the past half-century or so by the early seminal work of Townsend (1956, The Structure of Turbulent Shear Flow. Cambridge University Press). His and nearly all subsequent analyses depend crucially on a hypothesis about the nature of the dissipation, ${\it\epsilon}$, of turbulence kinetic energy, $k$. It has usually been assumed (sometimes implicitly) that this is governed by the famous Kolmogorov relation ${\it\epsilon}=C_{{\it\epsilon}}k^{3/2}/L$, where $L$ is a length scale of the energy-containing eddies and $C_{{\it\epsilon}}$ is a constant. The paper by Dairay et al. (J. Fluid Mech. vol. 781, 2015, pp. 166–195) demonstrates, however, that, in the specific context of an axisymmetric wake, there can be regions where ${\it\epsilon}$ has a different behaviour, characterised by a $C_{{\it\epsilon}}$ that is not constant but depends on a varying local Reynolds number (despite the existence of a $-5/3$ region in the spectra). This leads to fundamentally different scaling laws for the wake.

Upper-ocean turbulence at scales smaller than the mesoscale is believed to exchange surface and thermocline waters, which plays an important role in both physical and biogeochemical budgets. But what energizes this submesoscale turbulence remains a topic of debate. Two mechanisms have been proposed: mesoscale-driven surface frontogenesis and baroclinic mixed-layer instabilities. The goal here is to understand the differences between the dynamics of these two mechanisms, using a simple quasi-geostrophic model. The essence of mesoscale-driven surface frontogenesis is captured by the well-known surface quasi-geostrophic model, which describes the sharpening of surface buoyancy gradients and the subsequent breakup in secondary roll-up instabilities. We formulate a similarly archetypical Eady-like model of submesoscale turbulence induced by mixed-layer instabilities. The model captures the scale and structure of this baroclinic instability in the mixed layer. A wide range of scales are energized through a turbulent inverse cascade of kinetic energy that is fuelled by the submesoscale mixed-layer instability. Major differences to mesoscale-driven surface frontogenesis are that mixed-layer instabilities energize the entire depth of the mixed layer and produce larger vertical velocities. The distribution of energy across scales and in the vertical produced by our simple model of mixed-layer instabilities compares favourably to observations of energetic wintertime submesoscale flows, suggesting that it captures the leading-order balanced dynamics of these flows. The dynamics described here in an oceanographic context have potential applications to other geophysical fluids with layers of different stratifications.

A series of direct numerical simulations of turbulent channel flow with spanwise rotation at fixed global friction Reynolds number is performed to investigate the rotation effects on the mean velocity, streamwise velocity fluctuations, Reynolds shear stress and turbulent kinetic energy. The global friction Reynolds number is chosen to be $Re_{{\it\tau}}=u_{{\it\tau}}^{\ast }h^{\ast }/{\it\nu}^{\ast }=180$ ($u_{{\it\tau}}^{\ast }$ is the global friction velocity, $h^{\ast }$ is the channel half-width and ${\it\nu}^{\ast }$ is the kinematic viscosity), while the global-friction-velocity-based rotation number $Ro_{{\it\tau}}=2{\it\Omega}^{\ast }h^{\ast }/u_{{\it\tau}}^{\ast }$ (${\it\Omega}^{\ast }$ is the dimensional angular velocity) varies from 0 to 130. In the previously reported $2{\it\Omega}^{\ast }$-slope region for the mean velocity, a linear behaviour for the streamwise velocity fluctuations, a unit-slope linear profile for the Reynolds shear stress and a $-2Ro_{{\it\tau}}$-slope linear profile for the production term of $\langle u^{\prime }u^{\prime }\rangle$ have been identified for the first time. The critical rotation number, which corresponds to the laminar limit, is predicted to be equal to $Re_{{\it\tau}}$ according to the unit-slope linear profile of the Reynolds shear stress. Our results also show that a parabolic profile of the mean velocity can be identified around the ‘second plateau’ region of the Reynolds shear stress for $Ro_{{\it\tau}}\geqslant 22$. The parabolas at different rotation numbers have the same shape of $1/Re_{{\it\tau}}$, the radius of curvature at the vertex. Furthermore, the system rotation increases the volume-averaged turbulent kinetic energy at lower rotation rates, and then decreases it when $Ro_{{\it\tau}}\gtrsim 16$.

A self-preservation (SP) analysis is carried out for a zero pressure gradient (ZPG) rough-wall turbulent boundary layer with a view to establishing the requirements of complete SP (i.e. SP across the entire layer) and determining if these are achievable. The analysis shows that SP is achievable in certain rough-wall boundary layers (irrespectively of the Reynolds number $Re$), when the mean viscous stress is zero or negligible compared to the form drag across the entire boundary layer. In this case, the velocity scale $u^{\ast }$ must be constant, the length scale $l$ should vary linearly with the streamwise distance $x$ and the roughness height $k$ must be proportional to $l$. Although this result is consistent with that of Rotta (Prog. Aeronaut. Sci., vol. 2 (1), 1962, pp. 1–95), it is derived in a more rigorous manner than the method employed by Rotta. Further, it is noted that complete SP is not possible in a smooth-wall ZPG turbulent boundary layer. The SP conditions are tested against published experimental data on both a smooth wall (Kulandaivelu, 2012, PhD thesis, The University of Melbourne) and a rough wall, where the roughness height increases linearly with $x$ (Kameda et al., J. Fluid Sci. Technol., vol. 3 (1), 2008, pp. 31–42). Complete SP in a ZPG rough-wall turbulent boundary layer seems indeed possible when $k\propto x$.

In this study, a new centrifugal instability mode, which dominates within the boundary-layer flow over a slender rotating cone in still fluid, is used for the first time to model the problem within an enforced oncoming axial flow. The resulting problem necessitates an updated similarity solution to represent the basic flow more accurately than previous studies in the literature. The new mean flow field is subsequently perturbed, leading to disturbance equations that are solved via numerical and short-wavelength asymptotic approaches, yielding favourable comparisons with existing experiments. Essentially, the boundary-layer flow undergoes competition between the streamwise flow component, due to the oncoming flow, and the rotational flow component, due to effect of the spinning cone surface, which can be described mathematically in terms of a control parameter, namely the ratio of streamwise to axial flow. For a slender cone rotating in a sufficiently strong axial flow, the instability mode breaks down into Görtler-type counter-rotating spiral vortices, governed by an underlying centrifugal mechanism, which is consistent with experimental and theoretical studies for a slender rotating cone in otherwise still fluid.

Direct numerical simulations of the Taylor–Couette (TC) problem, the flow between two coaxial and independently rotating cylinders, have been performed. The study focuses on TC flow with mild curvature (small gap) with a radius ratio of ${\it\eta}=r_{i}/r_{o}=0.909$, an aspect ratio of ${\it\Gamma}=L/d=2{\rm\pi}/3$, and a stationary outer cylinder. Three inner cylinder Reynolds numbers of $1\times 10^{5}$, $2\times 10^{5}$ and $3\times 10^{5}$ were simulated, corresponding to frictional Reynolds numbers between $Re_{{\it\tau}}\approx 1400$ and $Re_{{\it\tau}}\approx 4000$. An additional case with a large gap, ${\it\eta}=0.5$ and driving of $Re=2\times 10^{5}$ was also investigated. Small-gap TC was found to be dominated by spatially fixed large-scale structures, known as Taylor rolls (TRs). TRs are attached to the boundary layer, and are active, i.e. they transport angular velocity through Reynolds stresses. An additional simulation was also conducted with inner cylinder Reynolds number of $Re=1\times 10^{5}$ and fixed outer cylinder with an externally imposed axial flow of comparable strength to the wind of the TRs. The axial flow was found to convect the TRs without any weakening effect. For small-gap TC flow, evidence was found for the existence of logarithmic velocity fluctuations, and of an overlap layer, in which the velocity fluctuations collapse in outer units. Profiles consistent with a logarithmic dependence were also found for the angular velocity in large-gap TC flow, albeit in a very reduced range of scales. Finally, the behaviour of both small- and large-gap TC flow was compared to other canonical flows. Small-gap TC flow has similar behaviour in the near-wall region to other canonical flows, while large-gap TC flow displays very different behaviour.

We propose an analytical model based on the solution of the magnetohydrodynamics (MHD) equations for studying intrinsic magnetospheres. For this purpose, we introduce a new gauge condition for the electromagnetic vector potential, which simplifies the solution of this complex system of nonlinear equations. Using this model, we analyse the deformation of the terrestrial magnetic field due to the presence of the solar wind. By comparing the results with experimental observations, we find that our model reproduces with good agreement the geometrical configuration of the magnetosphere, and that the solar wind should have a finite conductivity. This model could also be used to perform linear stability analysis of fluid and magnetic instabilities. Finally, our solution is not limited to magnetospheric configurations but also applies to a steady-state incompressible and irrotational flow with large plasma parameter and small velocity fluctuations.

This study is concerned with the numerical linear stability analysis of liquid-metal flow in a square duct with thin electrically conducting walls subject to a uniform transverse magnetic field. We derive an asymptotic solution for the base flow that is valid for not only high but also moderate magnetic fields. This solution shows that, for low wall conductance ratios $c\ll 1$, an extremely strong magnetic field with Hartmann number $\mathit{Ha}\sim c^{-4}$ is required to attain the asymptotic flow regime considered in previous studies. We use a vector streamfunction–vorticity formulation and a Chebyshev collocation method to solve the eigenvalue problem for three-dimensional small-amplitude perturbations in ducts with realistic wall conductance ratios $c=1$, 0.1 and 0.01 and Hartmann numbers up to $10^{4}$. As for similar flows, instability in a sufficiently strong magnetic field is found to occur in the sidewall jets with characteristic thickness ${\it\delta}\sim \mathit{Ha}^{-1/2}$. This results in the critical Reynolds number and wavenumber increasing asymptotically with the magnetic field as $\mathit{Re}_{c}\sim 110\mathit{Ha}^{1/2}$ and $k_{c}\sim 0.5\mathit{Ha}^{1/2}$. The respective critical Reynolds number based on the total volume flux in a square duct with $c\ll 1$ is $\overline{\mathit{Re}}_{c}\approx 520$. Although this value is somewhat larger than $\overline{\mathit{Re}}_{c}\approx 313$ found by Ting et al. (Intl J. Engng Sci., vol. 29 (8), 1991, pp. 939–948) for the asymptotic sidewall jet profile, it still appears significantly lower than the Reynolds numbers at which turbulence is observed in experiments as well as in direct numerical simulations of this type of flow.

A nonlinear spectral model in terms of spherically averaged descriptors is derived for the prediction of homogeneous turbulence dynamics in the presence of arbitrary mean-velocity gradients. The governing equations for the tensor $\hat{\unicode[STIX]{x1D619}}_{ij}(\boldsymbol{k},t)$, the Fourier transform of the two-point second-order correlation tensor, are first closed by an anisotropic eddy-damped quasinormal Markovian procedure. This closure is restricted to turbulent flows where linear effects induced by mean-flow gradients have no essential qualitative effects on the dynamics of triple correlations compared with the induced production effects in the equations for second-order correlations. Truncation at the first relevant order of spectral angular dependence allows us to derive from these equations in vector $\boldsymbol{k}$ our final model equations in terms of the wavenumber modulus $k$ only. Analytical spherical integration results in a significant decrease in computational cost. Besides, the model remains consistent with the decomposition in terms of directional anisotropy and polarization anisotropy, with a spherically averaged anisotropic spectral tensor for each contribution. Restriction of anisotropy to spherically averaged descriptors, however, entails a loss of information, and realizability conditions are considered to quantify the upper boundary of anisotropy that can be investigated with the proposed model. Several flow configurations are considered to assess the validity of the present model. Satisfactory agreement with experiments on grid-generated turbulence subjected to successive plane strains is observed, which confirms the capability of the model to account for production of anisotropy by mean-flow gradients. The nonlinear transfer terms of the model are further tested by considering the return to isotropy (RTI) of different turbulent flows. Different RTI rates for directional anisotropy and polarization anisotropy allow us to correctly predict the apparent delayed RTI shown after axisymmetric expansion. The last test case deals with homogeneous turbulence subjected to a constant pure plane shear. The interplay between linear and nonlinear effects is reproduced, yielding the eventual exponential growth of the turbulent kinetic energy.

We investigate the vibration properties of a circular horizontal film that is bounded by a meniscus (or Plateau border) and suspended between two catenary films. The suspending films act as capillary springs, and the system is thus free to oscillate around its equilibrium position. We study its free and forced oscillations. In our experiments, we track simultaneously the positions of the Plateau border and the film. The model that we present predicts the eigenfrequency of the system and its resonance characteristics (in forced oscillations). In particular, we show that the dynamics of both the Plateau border and the film have to be taken into account in order to provide an accurate prediction of the oscillation frequency.

We reconcile two scaling laws that have been proposed in the literature for the slip length associated with a moving contact line in diffuse interface models, by demonstrating each to apply in a different regime of the ratio of the microscopic interfacial width $l$ and the macroscopic diffusive length $l_{D}=(M{\it\eta})^{1/2}$, where ${\it\eta}$ is the fluid viscosity and $M$ the mobility governing intermolecular diffusion. For small $l_{D}/l$ we find a diffuse interface regime in which the slip length scales as ${\it\xi}\sim (l_{D}l)^{1/2}$. For larger $l_{D}/l>1$ we find a sharp interface regime in which the slip length depends only on the diffusive length, ${\it\xi}\sim l_{D}\sim (M{\it\eta})^{1/2}$, and therefore only on the macroscopic variables ${\it\eta}$ and $M$, independent of the microscopic interfacial width $l$. We also give evidence that modifying the microscopic interfacial terms in the model’s free energy functional appears to affect the value of the slip length only in the diffuse interface regime, consistent with the slip length depending only on macroscopic variables in the sharp interface regime. Finally, we demonstrate the dependence of the dynamic contact angle on the capillary number to be in excellent agreement with the theoretical prediction of Cox (J. Fluid Mech., vol. 168, 1986, p. 169), provided we allow the slip length to be rescaled by a dimensionless prefactor. This prefactor appears to converge to unity in the sharp interface limit, but is smaller in the diffuse interface limit. The excellent agreement of results obtained using three independent numerical methods, across several decades of the relevant dimensionless variables, demonstrates our findings to be free of numerical artefacts.

We consider a sphere with a circular pore embedded in an unbounded viscous fluid, where the rim of the pore moves in such a way that the radius of the sphere is constant. Away from the pore, the surface area stretches or compresses uniformly. An exact form for the axisymmetric velocity field which describes the quasi-static motion of the bulk fluid is calculated. The resulting dissipation function yields an analytical value for the aqueous drag coefficient for the sphere with a shrinking pore. Additionally, we examine the small hole and small angle limits, which converge to the unsteady flow for the expansion of a hole in a plane wall, and for the contraction of a circular disk.

The forced radial expansion of a spherical liquid shell by an exothermic chemical reaction is a prototypical configuration for the explosion of cohesive materials in three dimensions. The shell is formed by the capillary pinch off of a thin liquid annular jet surrounding a jet of reactive gaseous mixture at ambient pressure. The encapsulated gas in the resulting liquid bubble is a mixture of hydrogen and oxygen in controlled relative proportions, which is ignited by a laser plasma aimed at the centre of the bubble. The strongly exothermic combustion of the mixture induces the expansion of the hot burnt gas, pushing the shell radially outwards in a violently accelerated motion. That motion triggers the instability of the shell, developing thickness modulations ultimately piercing it in a number of places. The capillary retraction of the holes concentrates the liquid constituting the shell into a web of ligaments, whose breakup leads to stable drops. We offer a comprehensive description of the overall process, from the kinematics of the shell initial expansion, to the final drop size distribution as a function of the composition of the gas mixture, the initial shell radius and thickness.

Symmetric and asymmetric inertial instability of the westerly mid-latitude barotropic Bickley jet is analysed without the traditional approximation which neglects the vertical component of the Coriolis force, as well as the contribution of the vertical velocity to the latter. A detailed linear stability analysis of the jet at large Rossby numbers on the non-traditional $f$-plane is performed for long waves in both the two-layer rotating shallow-water and continuously stratified Boussinesq models. The dependence of the instability on both the Rossby and Burger numbers of the jet is investigated and compared to the traditional case. It is shown that non-traditional effects significantly increase the growth rate of the instability at small enough Burger numbers (weak stratifications) for realistic aspect ratios of the jet. The main results are as follows. (i) Two-layer shallow-water model. In the parameter regimes where the jet is inertially stable on the traditional $f$-plane, the symmetric inertial instability with respect to perturbations with zero along-jet wavenumber arises on the non-traditional $f$-plane. Both non-traditional symmetric and asymmetric (small but non-zero wavenumbers) inertial instabilities have higher growth rates than their traditional counterparts. (ii) Continuously stratified model. It is shown that by a proper change of variables the linear stability problem for the barotropic jet, on the non-traditional $f$-plane, can be rendered separable and analysed along the same lines as in the traditional approximation. Neutral, weak and strong background stratifications are considered. For the neutral stratification the jet is inertially unstable if the traditional approximation is relaxed, while its traditional counterpart is not. For a sufficiently weak stratification, both symmetric and asymmetric inertial instabilities have substantially higher growth rates than in the traditional approximation. The across-jet structure of non-traditional unstable modes is strikingly different, as compared to those under the traditional approximation. No discernible differences between the two approximations are observed for strong enough stratifications. The influence of dissipation and non-hydrostatic effects upon the instability is quantified.

The wavelet-based eddy-capturing approach with variable thresholding is extended to bluff-body flows, where the obstacle geometry is enforced through Brinkman volume penalization. The use of a spatio-temporally varying threshold allows one to perform adaptive large-eddy simulations with the prescribed fidelity on a near optimal computational mesh. The space–time evolution of the threshold variable is achieved by solving a transport equation based on the Lagrangian path-line diffusive averaging methodology. The coupled wavelet-collocation/volume-penalization approach with variable thresholding is illustrated for a turbulent incompressible flow around an isolated stationary prism with square cross-section. Wavelet-based adaptive large-eddy simulations supplied with the one-equation localized dynamic kinetic energy-based model are successfully performed at moderately high Reynolds number. The present study demonstrates that the proposed variable thresholding methodology for wavelet-based modelling of turbulent flows around solid obstacles is feasible, accurate and efficient.

We study how heat transport is affected by finitely extensible polymers in a laminar boundary layer flow within the framework of the Prandtl–Blasius–Pohlhausen theory. The polymers are described by the finitely extensible nonlinear elastic-Peterlin model with a parameter $b^{2}$, which is the ratio of the maximum to the equilibrium value of the trace of the polymer conformation tensor. For very large $b^{2}$, heat transport is reduced. When $b^{2}$ is small, heat transport is enhanced. We investigate the transition from heat reduction to heat enhancement as a function of the polymer relaxation time and concentration, and show that the transition can be explained in terms of the functional shape of the space-dependent effective viscosity due to the polymers.

We present a study of the turbulent and acoustic fields of subsonic jets, controlled by means of a novel actuator that introduces perturbations via steady-fluidic actuation from a rotating centrebody. The actuation can produce louder or quieter jets, and these are analysed using time-resolved stereoscopic particle image velocimetry and a hot-wire anemometer. We place the analysis in the framework of wavepackets and linear stability theory, whence we show, using solutions of the linear parabolised stability equations, that the quieter flows can be understood to result from a mean-flow deformation that modifies wavepacket dynamics, and in particular their phase velocities, which are significantly reduced. The mean-flow deformation is shown, by a triple decomposition, to be due to the generation of Reynolds stresses associated with incoherent turbulence (rather than coherent structures) which arises when the actuation energises the flow with a frequency–azimuthal wavenumber (${\it\omega}$–$m$) combination to which the mean flow is stable. When the actuation excites the flow with an ${\it\omega}$–$m$ combination to which the mean flow is unstable, the response is dominated by coherent structures, whose rapid growth takes them beyond the linear limit, where they undergo quadratic wave interactions and lead, consequently, to a louder flow.

A direct-forcing fictitious domain method is employed to study the dynamics of an open triangle in a vertically oscillatory flow. The flow structures, the vertical force and the torque on the fixed body are analysed for the stable flow regime in which the flow structures form and evolve exactly in the same way in each period and the unstable regime, respectively. Our results indicate that in the stable flow regime for the body with upright orientation, the steady streaming structure mainly comprises two vortex pairs located respectively above and below the body. Due to up–down asymmetry of the body, the pair below the body produces a larger vertical force on the body than the upper pair, which is mainly responsible for the non-zero average force at relatively high Reynolds numbers. The average force increases with increasing Reynolds number or increasing dimensionless period for the parameter range studied, due to the vortex effects. In the unstable regime, a vortex pair is ejected downward from each body edge. The irregular motion of the emitted vortices below the body leads to the irregular fluctuation of the vertical force. Regarding the torque on a tilted body, in the stable regime, the body experiences a restoring torque when its vertex angle is larger than a critical value being close to (and smaller than) 60°, and otherwise a destructive torque, irrespective of the value of tilt angle. For a fixed vertex angle, the torque magnitude is largest when the tilt angle is around 45°. In the unstable regime, the persistent ejection of the vortex pair during upward flow and corresponding restoring torque are observed for a large tilt angle with one edge aligned close to the horizontal direction, as in the experiment of Liu et al. (Phys. Rev. Lett., vol. 108, 2012, 068103). For a relatively small tilt angle, the emission direction of the vortex pair has intermittency, leading to the intermittency in the direction of torque. The reasons for the above observations are discussed. The predictions on the stable orientation for a hovering body in the stable flow regime and the irregular orientation in the unstable regime are confirmed in the dynamic simulation of a freely moving body. The body with the stable horizontal orientation in case of small vertex angle migrates along the body-shape-diverging direction.

Stingrays, in contrast with many other aquatic animals, have flattened disk-shaped bodies with expanded pectoral ‘wings’, which are used for locomotion in water. To discover the key features of stingray locomotion, large-eddy simulations of a self-propelled stingray, modelled closely after the freshwater stingray, Potamotrygon orbignyi, are performed. The stingray’s body motion was prescribed based on three-dimensional experimental measurement of wing and body kinematics in live stingrays at two different swimming speeds of 1.5 and $2.5L~\text{s}^{-1}$ ($L$ is the disk length of the stingray). The swimming speeds predicted by the self-propelled simulations were within 12 % of the nominal swimming speeds in the experiments. It was found that the fast-swimming stingray (Reynolds number $Re=23\,000$ and Strouhal number $St=0.27$) is approximately 12 % more efficient than the slow-swimming one ($Re=13\,500$, $St=0.34$). This is related to the wake of the fast- and slow-swimming stingrays, which was visualized along with the pressure on the stingray’s body. A horseshoe vortex was discovered to be present at the anterior margin of the stingray, creating a low-pressure region that enhances thrust for both fast and slow swimming speeds. Furthermore, it was found that a leading-edge vortex (LEV) on the pectoral disk of swimming stingrays generates a low-pressure region in the fast-swimming stingray, whereas the low- and high-pressure regions in the slow-swimming one are in the back half of the wing and not close to any vortical structures. The undulatory motion creates thrust by accelerating the adjacent fluid (the added-mass mechanism), which is maximized in the back of the wing because of higher undulations and velocities in the back. However, the thrust enhancement by the LEV occurs in the front portion of the wing. By computing the forces on the front half and the back half of the wing, it was found that the contribution of the back half of the wing to thrust in a slow-swimming stingray is several-fold higher than in the fast-swimming one. This indicates that the LEV enhances thrust in fast-swimming stingrays and improves the efficiency of swimming.

Three spatially extended travelling wave exact coherent states, together with one spanwise-localised state, are presented for channel flow. Two of the extended flows are derived by homotopy from solutions to the problem of channel flow subject to a spanwise rotation investigated by Wall & Nagata (J. Fluid Mech., vol. 727, 2013, pp. 523–581). Both these flows are asymmetric with respect to the channel centreplane, and feature streaky structures in streamwise velocity flanked by staggered vortical structures. One of these flows features two streak/vortex systems per spanwise wavelength, while the other features one such system. The former substantially reduces the value of the lowest Reynolds number at which channel flow solutions ,other than the basic flow, are known to exist down to 665. The third flow has, in contrast, half-turn rotational symmetry about a streamwise axis through a point on the channel centreplane, and is found to be the flow from which one of the asymmetric flows bifurcates in a symmetry-breaking bifurcation. This flow is found to exist on an isolated bifurcation branch, whose upper and lower branches both lie on the boundary basin separating initial conditions that lead to turbulent events, and those that directly decay back to laminar flow. The structure of this flow, in which the disturbance to the basic flow is concentrated in a core region in a spanwise period, allowed the derivation of a corresponding spanwise-localised flow, which is also discussed.