In the traditional hybrid RANS/LES approaches for the simulation of wall-bounded fluid turbulence, such as detached-eddy simulation (DES), the whole flow domain is divided into an inner layer and an outer layer. Typically the Reynolds-averaged Navier–Stokes (RANS) equations are used for the inner layer, while large-eddy simulation (LES) is used for the outer layer. The transition from the inner-layer solution to the outer-layer solution is often problematic due to the lack of small-scale dynamics in the RANS region. In this paper, we propose to simulate the whole flow domain by large-eddy simulation while enforcing a Reynolds-stress constraint on the subgrid-scale (SGS) stress model in the inner layer. Both the algebraic eddy-viscosity model and the one-equation Spalart–Allmaras (SA) model have been used to constrain the Reynolds stress in the inner layer. In this way, we improve the LES methodology by allowing the mean flow of the inner layer to satisfy the RANS solution while small-scale dynamics is included. We validate the Reynolds-stress-constrained large-eddy simulation (RSC-LES) model by simulating three-dimensional turbulent channel flow and flow past a circular cylinder. Our model is able to predict mean velocity, turbulent stress and skin-friction coefficients more accurately in turbulent channel flow and to estimate the pressure coefficient after separation more precisely in flow past a circular cylinder compared with the pure dynamic Smagorinsky model (DSM) and DES using the same grid resolution. Furthermore, the computational cost of the RSC-LES is almost the same as that of DES.

Linear stability analysis is used to investigate instability mechanisms for a horizontally oriented hyperbolic tangent mixing layer with uniform stable stratification and coordinate system rotation about the vertical axis. The important parameters governing inviscid dynamics are maximum shear , buoyancy frequency , angular velocity of rotation and characteristic shear thickness . Growth rates associated with the most unstable modes are explored as a function of stratification strength and rotation strength . In the case of strong stratification, growth rates exhibit self-similarity of the form . In the case of rapid rotation we also observe self-similar scaling of growth rates with respect to the vertical wavenumber and rotation rate. The unstratified cases show dependence while the strongly stratified cases show dependence where represents the difference between the angular velocity of rotation and least stable anticyclonic angular velocity, . Stratification was found to stabilize the inertial instability for weak anticyclonic rotation rates. Near the zero absolute vorticity state, stratification and rotation couple in a destabilizing manner increasing the range of unstable vertical wavenumbers associated with barotropic instability. In the case of rapid rotation, stratification prevents the stabilization of low , high modes that occurs in a homogeneous fluid. The structure of certain unstable eigenmodes and the coupling between horizontal vorticity and density fluctuations are explored to explain how buoyancy stabilizes or destabilizes inertial and barotropic modes.

Linear and nonlinear impulse responses are computed, using three-dimensional numerical simulations, for an incompressible and variable density (inhomogeneous) Batchelor vortex at a moderately high Reynolds number, . In the linear framework, the computed wavepacket is decomposed into azimuthal modes whose growth rates are determined along each spatiotemporal ray, in the laboratory frame. It is found that the Batchelor vortex undergoes a convective/absolute transition when the density ratio (inner/ambient), , is varied solely (there is no need for an external counter flow to trigger this transition like that needed in the constant density case). More precisely, it is shown that the transition occurs for heavy vortices when the density ratio reaches a critical value, . For light vortices () no transition was found. It is also shown that the first azimuthal mode that transits have an azimuthal wavenumber and the transition occurs for a swirl number (a measure of the azimuthal to axial velocity ratio), . It is followed by , then by . When nonlinearities are allowed, it is found that they saturate the amplitude within the linear-response wavepacket, leaving the wavepacket fronts unaffected. The conclusions should thus be the same as those obtained in the linear case: the linear convective/absolute transition should coincide with the nonlinear one for the variable-density Batchelor vortex.

We present experimental results supporting physics-based ejecta model development, where our main assumption is that ejecta form as a special limiting case of a Richtmyer–Meshkov (RM) instability at a metal–vacuum interface. From this assumption, we test established theory of unstable spike and bubble growth rates, rates that link to the wavelength and amplitudes of surface perturbations. We evaluate the rate theory through novel application of modern laser Doppler velocimetry (LDV) techniques, where we coincidentally measure bubble and spike velocities from explosively shocked solid and liquid metals with a single LDV probe. We also explore the relationship of ejecta formation from a solid material to the plastic flow stress it experiences at high-strain rates () and high strains (700 %) as the fundamental link to the onset of ejecta formation. Our experimental observations allow us to approximate the strength of Cu at high strains and strain rates, revealing a unique diagnostic method for use at these extreme conditions.

In this paper we introduce a new method for computations of two-dimensional magnetohydrodynamic (MHD) turbulence at low magnetic Prandtl number . When , the magnetic field dissipates at a scale much larger than the velocity field. The method we utilize is a novel hybrid contour–spectral method, the ‘combined Lagrangian advection method’, formally to integrate the equations with zero viscous dissipation. The method is compared with a standard pseudo-spectral method for decreasing for the problem of decaying two-dimensional MHD turbulence. The method is shown to agree well for a wide range of imposed magnetic field strengths. Examples of problems for which such a method may prove invaluable are also given.

The unidirectional shear flow driven by buoyancy effects in a vertical channel when a temperature difference is maintained between the walls of the channel is considered. The flow is unstable to waves which interact to reinforce the original flow and make it ‘streaky’. Such ‘vortex–wave’ interactions have been the subject of much recent research but little is yet known about what happens when the wavelength of the roll/streak flow becomes large. An asymptotic structure for long-wavelength interactions is derived and the tendency of the fluid to resist this state and the flow to become localized is revealed. Here the high-Grashof-number limit is considered and it is shown how a self-sustained process can occur with vortices interacting with a wave system in a manner similar to that discussed by Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666) and Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). The work is closely related to numerical simulations of self-sustained processes in for example Couette flow but the fact that the basic flow here is generated by buoyancy effects enables us to make analytical progress. It is shown that the wave part of the interaction process has a flat critical layer and its wavelength is twice that of the streaky flow which supports it. Such subharmonic vortex–wave/self-sustained process interactions have not been previously identified.

A drop on a circular support spontaneously spreads upon contact with a substrate. The motion is driven by a loss of surface energy. The loss of recoverable energy can be expressed alternatively as work done at the liquid–gas interface or dissipation through viscosity and sliding friction. In this paper we require consistency with the energy lost by dissipation in order to infer details of the contact-line region through simulations. Simulations with the boundary integral method are used to compute the flow field of a corresponding experiment where polydimethylsiloxane spreads on a relatively hydrophobic surface. The flow field is used to calculate the energy dissipation, from which slip lengths for local slip and Navier slip boundary conditions are found. Velocities, shear rates and pressures along the interface as well as interface shapes in the microscopic region of the contact line are also reported. Angles, slip length and viscous bending length scale allow a test of the Voinov–Hocking–Cox model without free parameters.

We investigate the time-harmonic small-amplitude motion of the mechanical system that consists of water and a body freely floating in it; water occupies a half-space, whereas the body is either surface-piercing or totally submerged. As a mathematical model of this coupled motion, we consider a spectral problem (the spectral parameter is the frequency of oscillations), for which the following results are obtained. The total energy of the water motion is finite and the equipartition of energy holds for the whole system. For any value of frequency, infinitely many eigensolutions are constructed and each of them consists of a non-trivial velocity potential and the zero vector describing the motion of the body; the latter means that trapping bodies (infinitely many of them are found) are motionless although they float freely. They are surface-piercing, have axisymmetric submerged parts and are obtained by virtue of the so-called semi-inverse procedure. We also prove that certain restrictions on the body geometry (which are violated for the constructed trapping bodies) guarantee that the problem has only a trivial solution for frequencies that are sufficiently large being measured in terms of a certain dimensionless quantity.

Aeroacoustic power generation due to a self-sustained oscillation by an axisymmetric compact cavity exposed to a low-Mach-number grazing flow is studied both experimentally and numerically. The feedback effect is produced by the velocity fluctuations resulting from a coupling with acoustic standing waves in a coaxial pipe. A numerical methodology that combines incompressible flow simulations with vortex sound theory is used to predict the time-averaged acoustic source power generated by the cavity. The effect of cavity depth on the whistling is addressed. It is observed that the whistling occurs around a peak-whistling Strouhal number which depends on the cavity depth to width ratio. The proposed numerical method provides excellent predictions of the peak-whistling Strouhal number as a function of cavity depth. Given the oscillation amplitude, the numerical method predicts the time-averaged acoustic source power within a factor of two for moderate fluctuation amplitudes. For deep cavities the time-averaged acoustic source power appears to be independent of the cavity depth.

The flow field around an asymmetric dielectric-barrier-discharge (DBD) plasma actuator in quiescent air is studied using particle image velocimetry (PIV) and smoke-flow visualization. On initiation of DBD plasma a starting vortex is created, which rolls up to form a coherent structure. The starting vortex becomes self-similar when the maximum velocity induced by the DBD plasma actuator reaches a steady state. Here, the plasma jet momentum increases linearly with time, suggesting that the DBD plasma actuator entrains and accelerates the surrounding fluid with a constant force. The wall-parallel and wall-normal distances of the vortex core are observed to scale with ${t}^{2/ 3} $ as it travels at $3{1}^{\circ } $ to the wall. The velocity of the starting vortex is found to scale with ${t}^{- 1/ 3} $, while the circulation induced by the plasma actuator scales with ${t}^{1/ 3} $.

The equations that govern, away from equilibrium and accounting for dissipation, the evolution of the contact line of an evaporating or condensing sessile drop on a rigid, planar substrate are derived. Aside from the normal and tangential components of the standard (Newtonian) force balance, these include a configurational force balance. At equilibrium, the normal component of the standard force balance reduces to the modified Young’s equation first mentioned by Gibbs. The remaining balances are purely dissipative and hence are vacuous in equilibrium. A complete description of contact-line dynamics generally involves all three equations. The theory is embedded in a thermodynamic framework that ensures consistency of all constitutive relations with the second law. In the linearly dissipative case, these involve six contact-line viscosities. When viscous coupling is neglected, only three viscosities remain. One is associated with stretching of the fluid along the contact line. The remaining two are related to dissipation that accompanies mass transfer between liquid and vapour phases during evaporation or condensation.

We study two-dimensional magnetohydrodynamic turbulence, with an emphasis on its energetics and inertial-range scaling laws. A detailed spectral analysis shows that dynamo triads (those converting kinetic into magnetic energy) are associated with a direct magnetic energy flux while anti-dynamo triads (those converting magnetic into kinetic energy) are associated with an inverse magnetic energy flux. As both dynamo and anti-dynamo interacting triads are integral parts of the direct energy transfer, the anti-dynamo inverse flux partially neutralizes the dynamo direct flux, arguably resulting in relatively weak direct energy transfer and giving rise to dynamo saturation. This result is consistent with a qualitative prediction of energy transfer reduction due to Alfvén wave effects by the Iroshnikov–Kraichnan theory (which was originally formulated for magnetohydrodynamic turbulence in three dimensions). We numerically confirm the correlation between dynamo action and direct magnetic energy flux and investigate the applicability of quantitative aspects of the Iroshnikov–Kraichnan theory to the present case, particularly its predictions of energy equipartition and spectra in the energy inertial range. It is found that for turbulence satisfying the Kraichnan condition of magnetic energy at large scales exceeding total energy in the inertial range, the kinetic energy spectrum, which is significantly shallower than , is shallower than its magnetic counterpart. This result suggests no energy equipartition. The total energy spectrum appears to depend on the energy composition of the turbulence but is clearly shallower than for , even at moderate resolutions. Here is the magnetic-to-kinetic energy ratio during the stage when the turbulence can be considered fully developed. The implication of the present findings is discussed in conjunction with further numerical results on the dependence of the energy dissipation rate on resolution.

The flow topologies of compressible turbulent boundary layers at Mach 2 are investigated by means of direct numerical simulation (DNS) of the compressible Navier–Stokes equations, and statistical analysis of the invariants of the velocity gradient tensor. We identify a preference for an unstable focus/compressing topology in the inner layer and an unstable node/saddle/saddle (UN/S/S) topology in the outer layer. The dissipation and dissipation production originate mainly from this UN/S/S topology. The enstrophy depends mainly on an unstable focus/stretching (UFS) topology, and the enstrophy production relies on a UN/S/S topology in the inner layer and on a UFS topology in the outer layer. The compressibility effect on the statistical properties of the topologies is investigated in terms of the ‘incompressible’, compressed and expanding regions. It is found that the locally compressed region tends to be more stable and the locally expanding region tends to be more dissipative. The compressibility is mainly related to unstable focus/compressing and stable focus/stretching topologies. Moreover, the features of the average dissipation, enstrophy, dissipation production and enstrophy production of the various topologies are clarified in the locally compressed and expanding regions.

Internal bores, or internal hydraulic jumps, arise in many atmospheric and oceanographic phenomena. The classic single-layer hydraulic jump model accurately predicts the bore height and propagation velocity when the difference between the densities of the expanding and contracting layers is large (i.e. water and air), but fails in the Boussinesq limit. A two-layer model, which conserves mass separately in each layer and momentum globally is more accurate in the Boussinesq limit, but it requires for closure an assumption about the loss of energy across a bore. It is widely believed that bounds on the bore speed can be found by restricting the energy loss entirely to one of the two layers, but under some circumstances, both bounds overpredict the propagation speed. A front velocity slower than both bounds implies that, somehow, the expanding layer is gaining energy. We directly examine the flux of energy within internal bores using two- and three-dimensional direct numerical simulations and find that although there is a global loss of energy across a bore, a transfer of energy from the contracting to the expanding layer causes a net energy gain in the expanding layer. The energy transfer is largely the result of turbulent mixing at the interface. Within the parameter regime investigated, the effect of mixing is much larger than non-hydrostatic and viscous effects, both of which are neglected in the two-layer analytical models. Based on our results, we propose an improved two-layer model that provides an accurate propagation velocity as a function of the geometrical parameters, the Reynolds number, and the Schmidt number.

We study the coupled problem of a liquid bridge connected to a porous surface and an impermeable surface, where the gap between the surfaces is an externally controlled function of time. The relative motion between the surfaces influences the pressure distribution and geometry of the liquid bridge, thus affecting the shape of liquid penetration into the porous material. Utilizing the lubrication approximation and Darcy’s phenomenological law, we obtain an implicit integral relation between the relative motion between the surfaces and the shape of liquid penetration. A method to control the shape of liquid penetration is suggested and illustrated for the case of conical penetration shapes with an arbitrary cone opening angle. We obtain explicit analytic expressions for the case of constant relative speed of the surfaces as well as for the relative motion between the surfaces required to create conical penetration shapes. Our theoretical results are compared with experiments and reasonable agreement between the analytical and experimental data is observed.

The two-dimensional, incompressible flow over a rounded backward-facing step at Reynolds number is characterized by a detachment of the flow close to the step followed by a recirculation zone. Even though the flow is globally stable, perturbations are amplified as they are convected along the shear layer, and the presence of upstream random noise renders the flow unsteady, leading to a broadband spectrum of excited frequencies. This paper is aimed at suppressing this unsteadiness using a controller that converts a shear-stress measurement taken from a wall-mounted sensor into a control law that is supplied to an actuator. A comprehensive study of various components of closed-loop control design – covering sensor placement, choice and influence of the cost functional, accuracy of the reduced-order model, compensator stability and performance – shows that successful control of this flow requires a judicious balance between estimation speed and estimation accuracy, and between stability limits and performance requirements. The inherent amplification behaviour of the flow can be reduced by an order of magnitude if the above-mentioned constraints are observed. In particular, to achieve superior controller performance, the estimation sensor should be placed upstream near the actuator to ensure sufficient estimation speed. Also, if high-performance compensators are sought, a very accurate reduced-order model is required, especially for the dynamics between the actuator and the estimation sensor; otherwise, very minute errors even at low energies and high frequencies may render the large-scale compensated linearized simulation unstable. Finally, coupling the linear compensator to nonlinear simulations shows a gradual deterioration in control performance as the amplitude of the noise increases.

The unstable dynamics of a transitional laminar separation bubble behind a two-dimensional bump geometry is investigated experimentally using dye visualizations and particle image velocimetry measurements. For Reynolds numbers above a critical value, the initially two-dimensional recirculation bubble is subject to modulations in the spanwise direction which can trigger vortex shedding. Increasing the Reynolds number further, the unstable behaviour is dominated by a low-frequency flapping motion, well known in transonic flows, and here investigated for the first time experimentally in an incompressible flow regime. These phenomena are characterized by non-intrusive measurements of the spatial structure and the frequencies of the unsteady motion. The results are in excellent agreement with previous numerical and theoretical predictions for the same geometry.

The anisotropy created by stratification and or rotation places restrictions, severe if viscosity is present, on the construction of analytic solutions to wave generation and scattering problems. Consequently, much literature is devoted to frequency space and so careful consideration of oscillatory motion generated from rest is advisable. Moreover, the use of complex coordinates in the inviscid case has been incompletely presented. The detailed inversion of the Fourier time transform for a breathing or heaving sphere demonstrates an expanded, perhaps more crucial, role for the complex coordinates and shows that the known phase changes in the energy propagation regions are present throughout the St Andrew’s cross that circumscribes the sphere. However, the different solution structure for the heaving disk requires and allows a more direct calculation. Though the inclusion of rotation does not affect the dynamics, it enables the significance of their relative magnitude to be identified and reference to rotation only results achieved.

The multimodal method requires analytical (exact or approximate) natural sloshing modes that exactly satisfy the Laplace equation and boundary condition on the wetted tank surface. When dealing with the nonlinear sloshing problem, the modes should also allow for an analytical continuation throughout the mean free surface. Appropriate analytically approximate modes were constructed by Faltinsen & Timokha (J. Fluid Mech., vol. 695, 2012, pp. 467–477) for the two-dimensional circular tank. The present paper extends this result to the three-dimensional, spherical tank shape and, based on that, establishes specific properties of the linear liquid sloshing.

We report high-speed optical and X-ray observations of jets formed during the impact of a drop with a deep pool of the same liquid. We show that a scaling that relies entirely on liquid properties, as is conventionally employed, is insufficient to determine the threshold for splashing. In order to determine if the gas properties could account for this deficit, we conducted experiments with different surrounding gases. We find that the splashing threshold depends on the gas’s dynamic viscosity, but not its density. We argue that these results are consistent with a thickening of the ejecta caused by the bubble trapped on impact between the drop and the pool. We also show that drop impact can generate a third jet, distinct from the lamella and the ejecta, that produces secondary droplets of an intermediate size.