Many practical applications, such as in blade cascades and turbomachinery, involve inhomogeneous turbulent shear flows subjected simultaneously to multiple strains. In principle, the applied strain can be combined to yield an effective strain. However, no simple stress–strain relation is capable of establishing turbulent stress or energy balance in the mean or on an instantaneous basis. In the current investigation, a turbulent boundary layer is examined in the presence of convex curvatures of different strengths combined with streamwise (favourable and adverse) pressure gradients, with various values of pressure gradient ratio, (∂P/∂s)/(∂P/∂n). Measurements of the mean and turbulent parameters and flux Richardson number show appreciable changes, mainly in the outer portion of the boundary layer (y+ > 100). The turbulent burst frequency, particularly at the location of application of the additional strain rate, also changes relative to its value with wall curvature alone.

Three primary observations from these experiments are as follows: (i) in all cases, the mean velocity profile and all of the measured Reynolds stresses collapse in the near-wall region using standard inner scaling; (ii) the applied strains combine nonlinearly, with one of the strains dominating the local flow during its development; (iii) the ratio of the radial to axial pressure gradient magnitude influences both classical turbulence correlations and mean flow, as well as the physical production cycle of turbulence; and (iv) application rate of newly introduced strain rates is at least as important as their magnitudes.

We investigate mechanisms governing the initial growth and nonlinear evolution of interfacial waves in horizontal two-fluid plane Couette–Poiseuille flows. Nonlinear coupling of the Kelvin–Helmholtz interfacial instability with resonant wave interactions has been shown to be capable of rapidly generating long waves through the transfer of energy from linearly unstable short waves to stable long-wave components within the context of potential flow theory. The objective of this work is to determine whether that coupled mechanism persists in laminar and turbulent viscous flows. Utilizing both theoretical and computational methods, we analyse the initial Orr–Sommerfeld instability to quantify the frequencies and growth/decay rates of each wave mode for two-fluid laminar and turbulent channel flows. The obtained dispersion relation allows for the identification of resonant and/or near-resonant triads among (unstable and damped) wave components in an interfacial wave spectrum. We perform direct numerical simulations (DNS) of the two-phase Navier–Stokes equations with a fully nonlinear interface to formally establish the validity of our theoretical predictions for viscous flows. DNS results show the existence of a nonlinear energy cascade from unstable short- to damped long-wavelength waves due to resonant subharmonic and/or triadic interactions in both laminar Couette and turbulent Poiseuille flows. Spectral analysis of the interfacial evolution confirms that the combined instability–resonance mechanism persists in the presence of viscosity despite being derived under the assumption of potential flow theory. Finally, we perform a detailed examination of experimentally measured wave power spectra from Jurman et al. (J. Fluid Mech., vol. 238, 1992, pp. 187–219) and carry out a numerical sensitivity study of the flow conditions to demonstrate and verify the existence of the coupled instability–resonance mechanism in physical systems. Our analysis accurately predicts the initial instability and the resulting nonlinear energy cascade through subharmonic and triadic interfacial wave resonances.

Certain medical treatments involve the introduction of exogenous liquids in the lungs. These liquids can form plugs within the airways. The plugs propagate throughout the branching network in the lungs being forced by airflow. They leave a deposited film on the airway walls and split at bifurcations. Understanding the resulting distribution of liquid throughout the lungs is important for the effective administration of the prescribed treatments. In this paper, we investigate numerically the splitting of a liquid plug by a two-dimensional pulmonary bifurcation under the influence of a transverse gravitational field. The splitting is characterized by the splitting ratio, which is the ratio of volume of the liquid plug in the daughter channels and depends on the capillary number and the orientation of the bifurcation plane with respect to a three-dimensional gravitational field. It is observed that gravity induces asymmetry in the splitting, causing the splitting ratio to be reduced. This effect is mitigated as the capillary number is increased. It is also observed that there exists a critical capillary number where the plug will not split and will instead propagate entirely into the gravitationally favoured daughter channel. We also compute the wall stresses on the bifurcation walls and observe the locations where stresses and their gradients are the highest in magnitude.

The spreading of an incompressible viscous liquid over an isotropic homogeneous unsaturated porous substrate is considered. It is shown that, unlike the dynamic wetting of an impermeable solid substrate, where the dynamic contact angle has to be specified as a boundary condition in terms of the wetting velocity and other flow characteristics, the ‘effective’ dynamic contact angle on an unsaturated porous substrate is completely determined by the requirement of existence of a solution, i.e. the absence of a non-integrable singularity in the spreading fluid’s pressure at the ‘effective’ contact line. The obtained velocity dependence of the ‘effective’ contact angle determines the critical point at which a transition to a different flow regime takes place, where the fluid above the substrate stops spreading whereas the wetting front inside it continues to propagate.

This paper presents spatially averaged temperature equations for modelling macro-scale heat transfer in periodic solid structures such as fin and tube arrays. The governing equations for the periodically developed heat transfer regime in isothermal solids are derived. It is shown that the appropriate macro-scale temperature in the periodically developed heat transfer regime is obtained by averaging the temperature with a specific weighting function which is adapted to the temperature decay rate. This matched weighting function allows the representation of the macro-scale interfacial heat transfer and thermal dispersion source by means of a spatially constant interfacial heat transfer coefficient and thermal dispersion vector, which both can be calculated from the periodic rescaled temperature on a unit cell of the solid structures. Moreover, it is proved that for small temperature decay rates, the matched weighting function yields the same macro-scale description as repeated volume averaging. The theoretical framework of this paper is applied to a case study, describing the heat transfer between a fluid and an array of solid cylinders at constant temperature.

The physical picture of a fluid particle pair as a couple of material points rotating around their centre of mass is proposed to model turbulent relative dispersion in the inertial range. This scheme is used to constrain the non-uniqueness problem associated to the Lagrangian models in the well-mixed class and the properties of the stochastic process derived are analysed with respect to some turbulent velocity characteristics. A simple illustrative Markov model is developed in stationary homogeneous isotropic turbulence and the particle separation statistics are compared with direct numerical simulation data. In spite of the simplicity of the model, a consistent comparison is observed in the inertial range, supporting the formulation proposed.

Plane Poiseuille flow in which the pressure gradient has a small amplitude time-periodic component in addition to a constant component is considered. The velocity field close to the boundaries, arising from a small amplitude high frequency disturbance to the flow, is calculated to second order in the modulation amplitude. The energy-transfer integral for the disturbance is then calculated to the same order. It is found that, if the thickness of the disturbance shear wave relative to that of the modulation shear wave is greater than ½, the modulation inhibits energy transfer into the disturbance and so stabilizes the flow.

The impact of a wedge-shaped body on the free surface of a weightless inviscid incompressible liquid is considered. Both symmetrical and unsymmetrical entries at constant velocity are dealt with. The differential problem corresponds to the physicomathematical model of a distribution of potential singularities and, in particular, the flow singularities at the ends of the wetted regions are represented by sinks. A conformal transformation of the flow field is adopted and the unknown intensities of the discontinuities are found by an optimization procedure, together with the solution of the nonlinear free-surface problem. The flow separation at a sideslip is also considered.

This paper presents the extension of our previous investigation of confined round jets with large Reynolds numbers and large expansion ratios (Revuelta, Sánchez & Liñán 2002a) to the case of swirling jets with swirl numbers of order unity. In the absence of vortex breakdown, we encounter the four-region asymptotic structure identified earlier for the non-swirling jet, including a region of jet development where the azimuthal and axial velocity components are comparable. For the flow in the long recirculating eddy that forms downstream, where the pressure differences associated with the azimuthal motion become negligible, the jet is found to act as a point source with momentum flux equal to the flow force of the incoming jet, and angular momentum flux equal to that of the jet at the orifice. The solution for the weak circulation in this slender region, including the parameter-free leading-order description and the first-order corrections, is determined by integrating the azimuthal component of the momentum equation written in the boundary-layer approximation. The results are validated through comparisons with numerical integrations of the steady axisymmetric Navier–Stokes equations, which are also used to evaluate critical conditions for vortex breakdown.

This paper is concerned with rapid distortion theory on transversely sheared mean flows that (among other things) can be used to analyse the unsteady motion resulting from the interaction of a turbulent shear flow with a solid surface. It expands on a previous analysis of Goldstein et al. (J. Fluid Mech., vol. 824, 2017, pp. 477–512) that uses a pair of conservation laws to derive upstream boundary conditions for planar mean flows and extends these findings to transversely sheared flows of arbitrary cross-section. The results, which turn out to be quite general, are applied to the specific case of a round jet interacting with the trailing edge of a flat plate and are used to calculate the radiated sound field, which is then compared with experimental data taken at the NASA Glenn Research Center.

A novel wavelet-based adaptive delayed detached eddy simulation (W-DDES) approach for simulations of wall-bounded compressible turbulent flows is proposed. The new approach utilizes anisotropic wavelet-based mesh refinement and its effectiveness is demonstrated for flow simulations using the Spalart–Allmaras DDES model. A variable wavelet thresholding strategy blending two distinct thresholds for the Reynolds-averaged Navier–Stokes (RANS) and large-eddy simulation (LES) regimes is used. A novel mesh adaptation on mean and fluctuating quantities with different wavelet threshold levels is proposed. The new strategy is more accurate and efficient compared to the adaptation on instantaneous quantities using a priori defined uniform thresholds. The effectiveness of the W-DDES method is demonstrated by comparing the results of the W-DDES simulations with results already available in the literature. Supersonic plane channel flow for two different configurations is tested as benchmark wall-bounded flows. Both the accuracy indicated by the threshold and efficiency in terms of degrees of freedom for the novel adaptation strategy are successfully gained compared with the wavelet-based adaptive LES method. Moreover, the newly proposed W-DDES resolves the typical log-layer match issue encountered in the conventional non-adaptive DDES method mainly due to the use of wavelet-based adaptive mesh refinement. The W-DDES capability for simulations of complex turbulent flows is validated by two other flow configurations – a subsonic channel flow with periodic hill constrictions and a supersonic flow over a compression ramp inducing the shock wave–turbulent boundary layer interaction. The current study serves as a crucial step towards construction of a unified wavelet-based adaptive hierarchical RANS/LES modelling framework, capable of performing simulations of varying fidelities from no-modelling direct numerical simulations to full-modelling RANS simulations.

The evolution of drift with time at a plane normal to the passage of a body from far right to far left is investigated. It is shown that the measurable part of the drift volume as a function of time is uniquely determined, and that the indeterminacy of Darwin's drift integral simply does not matter to the physical world, in which there is a Darwin theorem stating that the drift mass is equal to the added mass. The method for determining the shape of the drift surface is also given.

Grant (1973) showed that the expansion giving the profile of a steady Stokes wave near a 120° corner was more complicated than had previously been assumed. This paper gives further terms in such an expansion, and shows that generating them cannot introduce transcendental quantities beyond those noted by Grant.

We experimentally study the effect of having hinged leaflets at the jet exit on the formation of a two-dimensional counter-rotating vortex pair. A piston–cylinder mechanism is used to generate a starting jet from a high-aspect-ratio channel into a quiescent medium. For a rigid exit, with no leaflets at the channel exit, the measurements at a central plane show that the trailing jet in the present case is never detached from the vortex pair, and keeps feeding into the latter, unlike in the axisymmetric case. Passive flexibility is introduced in the form of rigid leaflets or flaps that are hinged at the exit of the channel, with the flaps initially parallel to the channel walls. The experimental arrangement closely approximates the limiting case of a free-to-rotate rigid flap with negligible structural stiffness, damping and flap inertia, as these limiting structural properties permit the largest flap openings. Using this arrangement, we start the flow and measure the flap kinematics and the vorticity fields for different flap lengths and piston velocity programs. The typical motion of the flaps involves a rapid opening and a subsequent more gradual return to its initial position, both of which occur when the piston is still moving. The initial opening of the flaps can be attributed to an excess pressure that develops in the channel when the flow starts, due to the acceleration that has to be imparted to the fluid slug between the flaps. In the case with flaps, two additional pairs of vortices are formed because of the motion of the flaps, leading to the ejection of a total of up to three vortex pairs from the hinged exit. The flaps’ length (${L}_{f} $) is found to significantly affect flap motions when plotted using the conventional time scale $L/ d$, where $L$ is the piston stroke and $d$ is the channel width. However, with a newly defined time scale based on the flap length ($L/ {L}_{f} $), we find a good collapse of all the measured flap motions irrespective of flap length and piston velocity for an impulsively started piston motion. The maximum opening angle in all these impulsive velocity program cases, irrespective of the flap length, is found to be close to 15°. Even though the flap kinematics collapses well with $L/ {L}_{f} $, there are differences in the distribution of the ejected vorticity even for the same $L/ {L}_{f} $. Such a redistribution of vorticity can lead to important changes in the overall properties of the flow, and it gives us a better understanding of the importance of exit flexibility in such flows.

A popular model for a generic fat-cored vortex ring or eddy is Hill’s spherical vortex (Phil. Trans. R. Soc. A, vol. 185, 1894, pp. 213–245). This well-known solution of the Euler equations may be considered a special case of the doubly infinite family of swirling spherical vortices identified by Moffatt (J. Fluid Mech., vol. 35 (1), 1969, pp. 117–129). Here we find exact solutions for such spherical vortices propagating steadily along the axis of a rotating ideal fluid. The boundary of the spherical vortex swirls in such a way as to exactly cancel out the background rotation of the system. The flow external to the spherical vortex exhibits fully nonlinear inertial wave motion. We show that above a critical rotation rate, closed streamlines may form in this outer fluid region and hence carry fluid along with the spherical vortex. As the rotation rate is further increased, further concentric ‘sibling’ vortex rings are formed.

The motivation for this work is the stability problem for short-crested Stokes waves. A new point of view is proposed, based on the observation that an understanding of the linear stability of short-crested waves (SCWs) is closely associated with an understanding of the stability of the oblique non-resonant interaction between two waves. The proposed approach is to embed the SCWs in a six-parameter family of oblique non-resonant interactions. A variational framework is developed for the existence and stability of this general two-wave interaction. It is argued that the resonant SCW limit makes sense a posteriori, and leads to a new stability theory for both weakly nonlinear and finite-amplitude SCWs. Even in the weakly nonlinear case the results are new: transverse weakly nonlinear long-wave instability is independent of the nonlinear frequency correction for SCWs whereas longitudinal instability is influenced by the SCW frequency correction, and, in parameter regions of physical interest there may be more than one unstable mode. With explicit results, a critique of existing results in the literature can be given, and several errors and misconceptions in previous work are pointed out. The theory is developed in some generality for Hamiltonian PDEs. Water waves and a nonlinear wave equation in two space dimensions are used for illustration of the theory.

We combine experimental, theoretical and numerical efforts to investigate the turbulent wake far behind a surface ship at model scales. Experimental measurements using digital particle image velocimetry (DPIV) are performed for the wakes of three towed hulls with beam-to-draught ratios b/d = 1, 2, 6. Based on model speed and beam, the Reynolds and Froude numbers are O(103) and O(10−2) respectively. Distinct surface features associated with persistent surface-normal vorticity have been identified, which are characterized by large-scale meandering structures. Both lateral and longitudinal scales of the meandering are quantified, with the former found to increase as b/d decreases and the latter independent of b/d. Based on measurements at multiple horizontal and vertical planes, profiles of the mean flow and fluctuation intensity for each velocity component are obtained. To understand the turbulence transition mechanism, an Orr–Sommerfeld stability analysis (OS) is formulated for the wake flow with free-surface boundary conditions, and solved by using a fourth-order finite-difference scheme. Unstable modes antisymmetric to the wake centre-plane are identified. Consistent with the experimental results, the growth rates of unstable modes increase substantially as b/d decreases, while the dependence of meandering wavelengths on b/d is found to be weak. Finally, we perform direct numerical simulation (DNS) of Navier–Stokes equations for the wake flow. The growth rates of unstable modes agree well with the predictions by OS analysis. Compared with experiments, DNS accurately captures the surface-normal vorticity signatures, the meandering features, as well as statistics of turbulence intensity. We also obtain from DNS a detailed description of enstrophy, turbulence length scales, and vortex structures for the wake flow.

The interaction was observed between a supersonic free jet of partially ionized argon and the magnetic field of a coil concentric with the jet. Nominal values of the parameters were: Mach number, 3; Reynolds number, 1000; magnetic Reynolds number, 0·2; magnetic interaction parameter, 1; Hall parameter, 1. The jet was strongly channelled. Axial and radial electric fields were observed in the jet with a net rise in potential across the interaction.

These observations were consistent with predictions based on the single fluid, macroscopic equations, and a simple slug flow model. The current equation was solved to second order in the Hall parameter giving a closed form expression for the Hall potential which agreed with the experiments for weak fields. Flow perturbations were calculated for Mach numbers of 3.11, 6.08, and 10.05, neglecting Hall currents; the calculations are in qualitative agreement with the experiments and show Joule heating to be the main factor in perturbing the flow at high Mach numbers.

The nonlinear development of the interfacial-surfactant instability is studied for the semi-infinite plane Couette film flow. Disturbances whose spatial period is close to the marginal wavelength of the long-wave instability are considered first. Appropriate weakly nonlinear partial differential equations (PDEs) which couple the disturbances of the film thickness and the surfactant concentration are obtained from the strongly nonlinear lubrication-approximation PDEs. In a rescaled form each of the two systems of PDEs is controlled by a single parameter C, the ‘shear-Marangoni number’. From the weakly nonlinear PDEs, a single Stuart–Landau ordinary differential equation (ODE) for an amplitude describing the unstable fundamental mode is derived. By comparing the solutions of the Stuart–Landau equation with numerical simulations of the underlying weakly and strongly nonlinear PDEs, it is verified that the Stuart–Landau equation closely approximates the small-amplitude saturation to travelling waves, and that the error of the approximation converges to zero at the marginal stability curve. In contrast to all previous stability work on flows that combine interfacial shear and surfactant, some analytical nonlinear results are obtained. The Hopf bifurcation to travelling waves is supercritical for C < Cs and subcritical for C > Cs, where Cs is approximately 0.29. This is confirmed with a numerical continuation and bifurcation technique for ODEs. For the subcritical cases, there are two values of equilibrium amplitude for a range of C near Cs, but the travelling wave with the smaller amplitude is unstable as a periodic orbit of the associated dynamical system (whose independent variable is the spatial coordinate). By using the Bloch (‘Floquet’) disturbance modes in the linearized PDEs, it transpires that all the small-amplitude travelling-wave equilibria are unstable to sufficiently long-wave disturbances. This theoretical result is confirmed by numerical simulations which invariably show the large-amplitude saturation of the disturbances. In view of this secondary instability, the existence of small-amplitude periodic solutions (on the real line) bifurcating from the uniform flow at the marginal values of the shear-Marangoni number does not contradict the earlier conclusions that the interfacial-surfactant instability has a strongly nonlinear character, in the sense that there are no small-amplitude attractors such that the entire evolution towards them is captured by weakly-nonlinear equations. This suggests that, in general, for flowing-film instabilities that have zero wavenumber at criticality, the saturated disturbance amplitudes do not always have to decrease to zero as the control parameter approaches its value at criticality.