A new analytical model is developed for the prediction of noise from serrated trailing edges. The model generalizes Amiet’s trailing-edge noise theory to sawtooth trailing edges, resulting in a complicated partial differential equation. The equation is then solved by means of a Fourier expansion technique combined with an iterative procedure. The solution is validated through comparison with the finite element method for a variety of serrations at different Mach numbers. The results obtained using the new model predict noise reduction of up to 10 dB at 90$^{\circ }$ above the trailing edge, which is more realistic than predictions based on Howe’s model and also more consistent with experimental observations. A thorough analytical and numerical analysis of the physical mechanism is carried out and suggests that the noise reduction due to serration originates primarily from interference effects near the trailing edge. A closer inspection of the proposed mathematical model has led to the development of two criteria for the effectiveness of the trailing-edge serrations, consistent but more general than those proposed by Howe. While experimental investigations often focus on noise reduction at 90$^{\circ }$ above the trailing edge, the new analytical model shows that the destructive interference scattering effects due to the serrations cause significant noise reduction at large polar angles, near the leading edge. It has also been observed that serrations can significantly change the directivity characteristics of the aerofoil at high frequencies and even lead to noise increase at high Mach numbers.

The presence of endwalls in Taylor–Couette flows has far reaching effects, leading to dynamics that are qualitatively different from those associated with the idealized situation involving infinitely long cylinders. This is well known in the classical situation where the inner cylinder is rotating and the outer cylinder is stationary. The effects of endwalls in the centrifugally stable situation with stationary inner cylinder and rotating outer cylinder have not been previously considered in detail. The meridional flows induced by the endwalls lead to the formation of a thin sidewall boundary layer on the inner cylinder wall if the endwalls are rotating, or on the outer cylinder wall if they are stationary. At sufficiently high Reynolds numbers (non-dimensional rotation rate of the outer cylinder), the sidewall boundary layer has concentrated shear, the pressure gradient in the azimuthal direction (which is the streamwise direction for the boundary layer flow) is zero (the flow is axisymmetric) and the boundary layer thickness is constant. At a critical Reynolds number, the sidewall boundary layer loses stability at a subcritical Hopf bifurcation, breaking the axisymmetry of the basic state flow, and for Reynolds numbers slightly above critical, the basic state is unstable to a packet of Hopf modes with azimuthal wavenumbers clustered about the critical wavenumber. The early time evolution of the critical Hopf mode is a rotating wave whose behaviour is analogous to a Tollmien–Schlichting wave. As the Hopf modes grow with time, nonlinear interactions lead to modulations in the waves, localization of the disturbances and the evolution of concentrated streamwise vortical streaks which become very intense via vortex stretching.

Schooling behaviours among self-propelled animals can benefit propulsion. Inspired by the schooling behaviours of swimming jellyfish, flexible bodies that self-propel through a paddling-based motion were modelled in a tandem configuration. This present study explored the hydrodynamic patterns generated by the interactions between two flexible bodies and the surrounding fluid in the framework of the penalty immersed boundary method. The hydrodynamic patterns produced in the wake revealed flow-mediated interactions between two tandem propulsors, including vortex–vortex and vortex–body interactions. Two tandem flexible propulsors paddling with identical amplitude and frequency produced stable configurations as a result of the flow-mediated interactions. Both the upstream and downstream propulsors benefited from the tandem configuration in terms of the locomotion velocity and the cost, compared with an isolated propulsion system. The interactions were examined as a function of the initial gap distance and the phase difference in the paddling frequency. The equilibrium gap distance between two propulsors remained constant, regardless of the initial gap distance, although it did depend on the phase difference in the paddling frequency.

The kinematics of passive scalar mixing and its relation to local flow topology and dilatation in compressible turbulence are examined using direct numerical simulations (DNS) of decaying isotropic turbulence. The main objective of this work is to characterize the dependence of various evolution mechanisms of scalar dissipation on local streamline topology and normalized dilatation. The DNS results indicate that the topology has a stronger influence on the nonlinear amplification mechanism than the dilatation level of a fluid element. In its appropriately normalized form, the amplification mechanism is found to be fairly independent of the Mach and Reynolds numbers. Non-focal topologies (the so called unstable-node/saddle/saddle and stable-node/saddle/saddle) are found to be associated with more intense mixing than the focal topologies at almost all dilatation levels. Alignment tendencies (jointly conditioned upon topology and dilatation) between the scalar-gradient vector and the strain-rate eigenvectors are shown to play a key role in shaping the observed behaviour in compressible turbulence. Finally, some modelling implications of these findings are discussed.

Fish rheotaxis, or alignment into flow currents, results from intertwined sensory, neural and actuation mechanisms, all coupled with hydrodynamics to produce a behaviour that is critical for upstream migration and position holding in oncoming flows. Among several sensory modalities, the lateral-line sensory system is thought to play a major role in the fish ability to sense minute water motions in their vicinity and, thus, in their rheotactic behaviour. Here, we propose a theoretical model consisting of a fishlike body equipped with lateral pressure sensors in oncoming uniform flows. We compute the optimal sensor locations that maximize the sensory output. Our results confirm recent experimental findings that correlate the layout of the lateral-line sensors with the distribution of hydrodynamic information at the fish surface. We then examine the behavioural response of the fishlike model as a function of its orientation and swimming speed relative to the background flow. Our working hypothesis is that fish respond to sensory information by adjusting their orientation according to the perceived difference in pressure. We find that, as in fish rheotaxis, the fishlike body responds by aligning into the oncoming flow. These findings may have significant implications on understanding the interplay between the sensory output and fish behaviour.

The flow of a slipping fluid jet is examined theoretically as it emerges from a channel at moderate Reynolds number. The ratio of the slip length to the channel width $S$ is assumed to be of order one, one order of magnitude larger than the perturbation parameter ${\it\varepsilon}=Re^{-1/2}$, $Re$ being the Reynolds number. Poiseuille flow conditions are assumed to prevail far upstream from the exit. The problem is solved using the method of matched asymptotic expansions. A similarity solution is obtained in the inner layer of the free surface, with the outer layer extending to the jet centreline. The inner-layer thickness grows like $\sqrt{x/Re\,S}$. A slipping jet is found to contract like $x/Re$ very near and far from the channel exit, but does not have a definite behaviour in between compared to $(x/Re)^{1/3}$ for an adhering jet, $x$ being the distance from the channel exit. Eventually, the jet reaches uniform conditions far downstream. As in the case of entry flow, there is a rapid departure in flow behaviour for a slipping jet from the $S=0$ limit. This rapid change is notably observed in the drop of boundary-layer thickness, increase in exit and relaxation lengths as well as in jet width with slip length. Finally, the connections with microchannel and hydrophobic flows are highlighted.

A new nested-LES approach for computation of high Reynolds number, equilibrium, and non-equilibrium, wall-bounded turbulent flows is presented. The method couples coarse-resolution LES in the full computational domain with fine-resolution LES in a minimal flow unit to retain the accuracy of well-resolved LES throughout the computational domain, including in the near-wall region, while significantly reducing the computational cost. The two domains are coupled by renormalizing the instantaneous velocity fields in each domain dynamically during the course of the simulation to match the wall-normal profiles of single-time, ensemble-averaged kinetic energies of the components of ‘mean’ and fluctuating velocities in the inner layer to those of the minimal flow unit, and in the outer layer to those of the full domain. This simple renormalization procedure is shown to correct the energy spectra and wall shear stresses in both domains, thus leading to accurate turbulence statistics. The nested-LES approach has been applied to computation of equilibrium turbulent channel flow at $Re_{{\it\tau}}\approx 1000$, 2000, 5000, 10 000, and non-equilibrium, strained turbulent channel flow at $Re_{{\it\tau}}\approx 2000$. In both flows, nested-LES predicts the skin friction coefficient, first- and higher-order turbulence statistics, spectra and structure of the flow in agreement with available DNS and experimental data. Nested-LES can be applied to any flow with at least one direction of local or global homogeneity, while reducing the required number of grid points from $O(Re_{{\it\tau}}^{2})$ of conventional LES to $O(\log Re_{{\it\tau}})$ or $O(Re_{{\it\tau}})$ in flows with two or one locally or globally homogeneous directions, respectively.

A global analysis of steady states of low Mach number inviscid premixed reacting swirling flows in a straight circular finite-length open pipe is developed. We focus on modelling the basic interaction between the swirl and heat release of the reaction. For analytic simplicity, a one-step first-order Arrhenious reaction kinetics is considered in the limit of high activation energy and infinite Peclet number. Assuming a complete reaction with chemical equilibrium upstream and downstream of the reaction zone, a nonlinear partial differential equation is derived for the solution of the flow stream function downstream of the reaction zone in terms of the specific total enthalpy, specific entropy and circulation functions prescribed at the inlet. Several types of solutions of the nonlinear ordinary differential equation for the columnar flow case describe the outlet states of the flow in a long pipe. These solutions are used to form the bifurcation diagram of steady reacting flows with swirl as the inlet swirl level is increased at a fixed heat release from the reaction. The approach is applied to two profiles of inlet flows, the solid-body rotation and the Lamb–Oseen vortex, both with constant profiles of the axial velocity, temperature and mixture reactant mass fraction. The computed results provide theoretical predictions of the critical inlet swirl levels for the appearance of vortex breakdown states and for the size of the breakdown zone as a function of the inlet flow swirl level, Mach number and heat release of the reaction. For the inlet solid-body rotation, flow is decelerated to breakdown as the inlet swirl is increased above the critical swirl level, and there is a delay in the appearance of breakdown with the increase of the heat release of the reaction. For the inlet Lamb–Oseen vortex at low values of heat release, the critical swirl for breakdown is decreased with the increase of heat release while, at high values of heat release, the appearance of breakdown is delayed to higher incoming flow swirl levels with the increase of heat release. The analysis sheds light on the global dynamics of low Mach number reacting flows with swirl and vortex breakdown and on the interaction between vortex breakdown and heat release that affects the shape of the reaction zone in the domain.

Certain flows denominated as amplifiers are characterized by their global linear stability while showing large linear amplifications to sustained perturbations. As the forcing amplitude increases, a strong saturation of the response appears when compared to the linear prediction. However, a predictive model that describes the saturation of the response to higher amplitudes of forcing in stable laminar flows is still missing. While an asymptotic analysis based on the weakly nonlinear theory shows qualitative agreement only for very small forcing amplitudes, the linear response to harmonic forcing around the mean flow computed by direct numerical simulations presents a good prediction of the saturation also at higher forcing amplitudes. These results suggest that the saturation process is governed by the Reynolds stress and thus motivate the introduction of a simple self-consistent model. The model consists of a decomposition of the full nonlinear Navier–Stokes equations in a mean flow equation together with a linear perturbation equation around the mean flow, which are coupled through the Reynolds stress. The full fluctuating response and the resulting Reynolds stress are approximated by the first harmonic calculated from the linear response to the forcing around the aforementioned mean flow. This closed set of coupled equations is solved in an iterative manner as partial nonlinearity is still preserved in the mean flow equation despite the assumed simplifications. The results show an accurate prediction of the response energy when compared to direct numerical simulations. The approximated coupling is strong enough to retain the main nonlinear effects of the saturation process. Hence, a simple physical picture is formalized, wherein the response modifies the mean flow through the Reynolds stress in such a way that the correct response energy is attained.

The effect of a uniform through-surface flow (velocity $U_{b}$) on a rigid and stationary cylinder and sphere (radius $a$) fixed in a free stream (velocity $U_{\infty }$) is analysed analytically and numerically. The flow is characterised by a dimensionless blow velocity ${\it\Lambda}\,(=U_{b}/U_{\infty })$ and Reynolds number $Re\,(=2aU_{\infty }/{\it\nu}$, where ${\it\nu}$ is the kinematic viscosity). High resolution numerical calculations are compared against theoretical predictions over the range $-3\leqslant {\it\Lambda}\leqslant 3$ and $Re=1,10,100$ for planar flow past a cylinder and axisymmetric flow past a sphere. For $-{\it\Lambda}\gg 1$, the flow is viscously dominated in a thin boundary layer of thickness ${\it\nu}/|U_{b}|$ adjacent to the rigid surface which develops in a time ${\it\nu}/U_{b}^{2}$; the surface vorticity scales as $Re|{\it\Lambda}|U_{\infty }/a$ for a cylinder and sphere. A boundary layer analysis is developed to analyse the unsteady viscous forces. Numerical results show that the surface pressure and vorticity distribution within the boundary layer agrees with a steady state analysis. The flow downstream of the body is irrotational so the wake volume flux, $Q_{w}$, is zero and the drag force is $F_{D}=-{\it\rho}U_{\infty }Q_{b}$, where ${\it\rho}$ is the density of the fluid and $Q_{b}$ is the normal flux through the body surface. The drag coefficient is therefore $-2{\rm\pi}{\it\Lambda}$ or $-8{\it\Lambda}$ for a cylinder or sphere, respectively. A dissipation argument is applied to analyse the drag force; the rate of working of the drag force is balanced by viscous dissipation, flux of stagnation pressure and rate of work by viscous stresses due to sucking. At large $Re|{\it\Lambda}|$, the drag force is largely determined by viscous dissipation for a cylinder, with a weak contribution by the normal viscous stresses, while for a sphere, only $3/4$ of the drag force is determined by viscous dissipation with the remaining $1/4$ due to the flux of stagnation pressure through the sphere surface. When ${\it\Lambda}\gg 1$, the boundary layer thickness initially grows linearly with time as vorticity is blown away from the rigid surface. The vorticity in the boundary layer is weakly dependent on viscous effects and scales as $U_{\infty }/a{\it\Lambda}$ or $U_{\infty }/a{\it\Lambda}^{3/2}$ for a cylinder and sphere, respectively. For large blow velocity, the vorticity is swept into two well-separated shear layers and the maximum vorticity decreases due to diffusion. The drag force is related to the vorticity distribution on the body surface and an approximate expression can be derived by considering the first term of a Fourier expansion in the surface vorticity. It is found that the drag coefficient $C_{D}$ for a cylinder (corrected for flow boundedness) is weakly dependent on ${\it\Lambda}$ while for a sphere, $C_{D}$ decreases with ${\it\Lambda}$.

We report a Lagrangian study on the evolution of material surfaces in the Klebanoff-type temporal transitional channel flow. Based on the Eulerian velocity field from the direct numerical simulation, a backward-particle-tracking method is applied to solve the transport equation of the Lagrangian scalar field, and then the isosurfaces of the Lagrangian field can be extracted as material surfaces in the evolution. Three critical issues for Lagrangian investigations on the evolution of coherent structures using material surfaces are addressed. First, the initial scalar field is uniquely determined based on the proposed criteria, so that the initial material surfaces can be approximated as vortex surfaces, and remain invariant in the initial laminar state. Second, the evolution of typical material surfaces initially from different wall distances is presented, and then the influential material surface with the maximum deformation is identified. Large vorticity variations with the maximum curvature growth of vortex lines are also observed on this surface. Moreover, crucial events in the transition can be characterized in a Lagrangian approach by conditional statistics on the material surfaces. Finally, the influential material surface, which is initially a vortex surface, is demonstrated as a surrogate of the vortex surface before significant topological changes of vortical structures. Therefore, this material surface can be used to elucidate the continuous temporal evolution of vortical structures in transitional wall-bounded flows in a Lagrangian perspective. The evolution of the influential material surface is divided into three stages: the formation of a triangular bulge from an initially disturbed streamwise–spanwise sheet, rolling up of the vortex sheet near the bulge ridges with the vorticity intensification and the generation and evolution of signature hairpin-like structures with self-induced dynamics of vortex filaments.

The analysis of axial dispersion of solute is presented in a pulsatile flow of Casson fluid through a tube in the presence of interfacial mass transport due to irreversible first-order reaction catalysed by the tube wall. The theory of dispersion is studied by employing the generalized dispersion model proposed by Sankarasubramanian & Gill (Proc. R. Soc. Lond. A, vol. 333 (1592), 1973, pp. 115–132). This dispersion model describes the whole dispersion process in terms of three effective transport coefficients, i.e. exchange, convection and dispersion coefficients. In the present study, the effects of yield stress of Casson fluid ${\it\tau}_{y}$, wall absorption parameter ${\it\beta}$, amplitude of fluctuating pressure component $e$ and Womersley frequency parameter ${\it\alpha}$ on the dispersion process are discussed under the influence of pulsatile pressure gradient. In a pulsatile flow, the plug flow radius changes during the period of oscillation and it has an effect on the dispersion process. Even with the Casson fluid model also, in an oscillatory flow, for small values of ${\it\alpha}$, the dispersion coefficient $K_{2}$ is positive, but when the value of ${\it\alpha}$ is as large as 3, $K_{2}$ takes both positive and negative values due to the fluctuations in the velocity profiles. This nature becomes more predominant for ${\it\tau}_{y}$, $e$ and ${\it\beta}$. It is observed that initially, for small time, the amplitude and magnitude of fluctuations of $K_{2}$ becomes more rapid and increases with time but it decreases after certain time and reaches a non-transient state for large time. Like in the case of Newtonian model, double frequency period for $K_{2}$ is observed at small time for large values of ${\it\alpha}$ with the Casson model for blood. It is seen that critical time for which $K_{2}$ reaches a non-transient state is independent of ${\it\tau}_{y}$ and $e$ but is dependent on ${\it\alpha}$. It is also observed that the axial distribution of mean concentration $C_{m}$ of solute depends on ${\it\tau}_{y}$ and ${\it\beta}$. But the effect of $e$ and ${\it\alpha}$ on $C_{m}$ is not very significant. This dispersion model in non-Newtonian pulsatile flow can be applied to study the dispersion process in the cardiovascular system and blood oxygenators.

Instabilities in the flow past a rounded square cylinder have been numerically studied in order to clarify the effects of rounding the sharp edges of a square cylinder on the primary and secondary instabilities associated with the flow. Rounding the edges was done by inscribing a quarter circle of radius $r$ in each edge of a square cylinder of height $d$. Nine cases of rounding were considered, ranging from a square cylinder ($r/d=0$) to a circular cylinder ($r/d=0.5$) with an increment of 0.0625. Each cross-section was numerically implemented in a Cartesian grid system by using an immersed boundary method. The key parameters are the Reynolds number (Re) and the edge-radius ratio ($r/d$). For low Re, the flow is steady and symmetric with respect to the centreline. Over the first critical Reynolds number ($Re_{c1}$), the flow undergoes a Hopf bifurcation to a time-periodic flow, termed the primary instability. As Re further increases, the onset of the secondary instability of three-dimensional (3D) nature is detected beyond the second critical Reynolds number ($Re_{c2}$). Rounding the sharp edges of a square cylinder significantly affects the flow topology, leading to noticeable changes in both instabilities. By employing the Stuart–Landau equation, we investigated the criticality of the primary instability depending upon $r/d$. The onset of the 3D secondary instability was detected by using Floquet stability analysis. The temporal and spatial characteristics of the dominant modes (A, B, QP) were described. The neutral stability curves of each mode were computed depending upon $r/d$.

The interaction of the barotropic tide with a tall, two-dimensional ridge is examined analytically and numerically at latitudes where the tide is subinertial, and contrasted to when the tide is superinertial. When the tide is subinertial, the energy density associated with the response grows with latitude as both the oscillatory along-ridge flow and near-ridge isopycnal displacement become large. Where $f\neq 0$, nonlinear processes lead to the formation of along-ridge jets, which become faster at high latitudes. Dissipation and mixing is larger, and peaks later in the tidal cycle when the tide is subinertial compared with when the tide is superinertial. Mixing occurs mainly on the flanks of the topography in both cases, though a superinertial tide may additionally generate mixing above topography arising from convective breaking of radiating waves.

We describe new experiments in which particle-laden turbulent fountains with source Froude numbers $20>Fr_{0}>6$ are produced when particle-laden fresh water is injected upwards into a reservoir filled with fresh water. We find that the ratio $U$ of the particle fall speed to the characteristic speed of the fountain determines whether the flow is analogous to a single-phase fountain ($U\ll 1$) or becomes a fully separated flow ($U\geqslant 1$). In the single-phase limit, a fountain with momentum flux $M$ and buoyancy flux $B$ oscillates about the mean height, $h_{m}=(1.56\pm 0.04)M^{3/4}B^{-1/2}$, as fluid periodically cascades from the maximum height, $h_{t}=h_{m}+{\rm\Delta}h$, to the base of the tank. Experimental measurements of the speed $u$ and radius $r$ of the fountain at the mean height $h_{m}$, combined with the conservation of buoyancy, suggest that $Fr(h_{m})=u(g^{\prime }r)^{-1/2}\approx 1$. Using these values, we find that the classical scaling for the frequency of the oscillations, ${\it\omega}\sim BM^{-1}$, is equivalent to the scaling $u(h_{m})/r(h_{m})$ for a fountain supplied at $z=h_{m}$ with $Fr=1$ (Burridge & Hunt, J. Fluid Mech., vol. 728, 2013, pp. 91–119). This suggests that the oscillations are controlled in the upper part of the fountain where $Fr\leqslant 1$, and that they may be understood in terms of a balance between the upward supply of a growing dense particle cloud, at the height where $Fr=1$, and the downward flow of this cloud. In contrast, in the separated flow regime, we find that particles do not reach the height at which $Fr=1$: instead, they are transported to the level at which the upward speed of the fountain fluid equals their fall speed. The particles then continuously sediment while the particle-free fountain fluid continues to rise slowly above the height of particle fallout, carried by its momentum.