The dynamics of modulated point-vortex couples on a β-plane is investigated for arbitrary ratios of the vortex strength. The motion is analysed in terms of an angle- and a location-dependent potential and the structural changes in their shape. The location-dependent potential is best suited for understanding different types of vortex orbits. It is shown to be two-valued in a range of parameters, a feature which leads to the appearance of orbits with spikes, in spite of the integrability of the problem. The advection dynamics in this modulated two-vortex problem is chaotic. We find a transition from closed to open chaotic advection, implying that the transport properties of the flow might be drastically altered by changing some parameters or the initial conditions. The open case, characterized by permanent entrainment and detrainment of particles around the vortices, is interpreted in terms of an invariant chaotic saddle of the Lagrangian dynamics, while the dynamics of the closed case, with a permanently trapped area of the fluid, is governed by a chaotic region and interwoven KAM tori. The transition from open to closed chaotic advection is quantified by monitoring the escape rate of advected particles as a function of the vortex energy.

Up-current-rotated, shoreface-connected ridges are found in various coastal areas around the world. An often-quoted conjecture is that these ridges form during storm conditions through free instabilities in the erodible bed. Under these conditions both waves and currents are expected to play a significant role in the hydrodynamics. Although some existing models have included the effects of waves parametrically in their bottom friction terms and sediment equations, the dynamical effects of wave–current interaction have not been explored. In this paper we begin to rectify this by considering the effects of wave–current interaction on the bed-form instabilities of a simple model. This raises the possibility of unsteady alongshore flows and questions about the roles of wave parameters and boundary conditions, which we address here. We show that the flow is stable under the wave forcing; however the waves do affect the bed-form instability. The main dynamical effect of the waves is in altering the shapes of the unstable modes. Under various conditions, however, waves may enhance or suppress the instability or introduce new unstable modes. They also affect the celerity of the ridges. In addition, we investigate the mechanisms whereby the waves affect the instability. We also show a potential problem with the parameterization in terms of wave orbital velocity.

The linear stability of the axisymmetric flow past a fixed-shape spheroid with free-slip boundary conditions is studied numerically to gain some insight into the path instability of bubbles rising in liquids. Qualitatively, the results are similar to those for a solid sphere. The m = 1 mode gives rise to a double-threaded wake and proves to be the most unstable mode, with a first regular bifurcation followed by a Hopf bifurcation. The importance of the base-flow vorticity is highlighted by a stability analysis of the axisymmetric base flow ‘frozen’ before reaching steady state. Setting viscosity to zero in the perturbation equations results in a faster growth of the primary instability, which indicates its root in inertial effects.

Computational experiments based on the direct numerical simulation of turbulent channel flow reveal that the skin friction can be reduced as much as 70% by the action of a localized steady force acting against the flow close to the wall. In addition, the excessive shear stresses observed during the laminar-to-turbulence transition can be substantially reduced. For a sustained reduction in the skin friction, the control force has to act within a distance of 20 wall units (scaling with the location of the maximum Reynolds stress gradient); otherwise a transient drag reduction is observed or even an increase in drag. The forcing leads to the formation of a shear layer close to the wall that reduces the skin friction and limits the development of the Reynolds shear stresses. As the amplitude of the forcing is increased, the shear layer breaks down and generates its own turbulence, setting an upper limit to the level of drag reduction. This transition of the shear layer is correlated with a Reynolds number based on the forcing amplitude and length scale.

When a stably stratified density interface is embedded in a region of strong velocity shear, hydrodynamic instabilities result. Here we generalize the stratified shear layer to allow an offset between the centre of the shear layer and the density interface. By including this asymmetry, and keeping the density interface thin with respect to the shear layer, the asymmetric Holmboe (AH) instability emerges. This study examines the evolution and mixing behaviour of AH instabilities, and compares the results to the well-known Kelvin–Helmholtz (KH) and Holmboe instabilities. This is done by performing a series of direct numerical simulations (DNS). The simulation results show that there are two different mixing mechanisms present. The first is a feature of KH instabilities and leads to the mixing and production of intermediate density fluid. The second mixing mechanism is found in AH and Holmboe instabilities and consists of regions of mixing and turbulence production that are located on one or both sides of the density interface. Since the Holmboe-type instabilities do not generate a large-scale overturning of the central isopycnal, the density interface is able to retain its identity throughout the mixing event. The amount of mixing that takes place is found to be strongly dependent on the degree of asymmetry in the flow.

We investigate the influence of density inhomogeneities on the merger of two corotating two-dimensional vortices at infinite Froude number. In this situation, buoyancy effects are negligible, yet density variations still affect the flow by pure inertial effects through the baroclinic torque. We first re-address the effects of a finite Reynolds number on the interaction between two identical Gaussian vortices. Then, by means of direct numerical simulations, we show that vortices transporting light fluid in a heavier counterpart merge from further distances than vortices in a uniform density medium. On the other hand, heavy vortices only merge from small separation distances. We measure the critical distance a/b0 of the vortex radii to their initial separation distance. It departs from the homogeneous threshold of 0.22 in response to increasing density contrasts between the vortices and their surroundings. An analysis of the contribution of the baroclinic vorticity to the dynamics of the flow is detailed and explains the observed behaviour. This analysis is completed by a simple model based on point vortices that mimics the flow. It is concluded that vortices carrying light fluid are more likely to generate large-scale structures than heavy ones in an inhomogeneous fluid.

We investigate aspects of hovering insect flight by finding the optimal wing kinematics which minimize power consumption while still providing enough lift to maintain a time-averaged constant altitude over one flapping period. In particular, we study the flight of three insects whose masses vary by approximately three orders of magnitude: fruitfly (Drosophila melanogaster), bumblebee (Bombus terrestris), and hawkmoth (Manduca sexta). Here, we model an insect wing as a rigid body with three rotational degrees of freedom. The aerodynamic forces are modelled via a quasi-steady model of a thin plate interacting with the surrounding fluid. The advantage of this model, as opposed to the more computationally costly method of direct numerical simulation via computational fluid dynamics, is that it allows us to perform optimization procedures and detailed sensitivity analyses which require many cost function evaluations. The optimal solutions are found via a hybrid optimization algorithm combining aspects of a genetic algorithm and a gradient-based optimizer. We find that the results of this optimization yield kinematics which are qualitatively and quantitatively similar to previously observed data. We also perform sensitivity analyses on parameters of the optimal kinematics to gain insight into the values of the observed optima. Additionally, we find that all of the optimal kinematics found here maintain the same leading edge throughout the stroke, as is the case for nearly all insect wing motions. We show that this type of stroke takes advantage of a passive wing rotation in which aerodynamic forces help to reverse the wing pitch, similar to the turning of a free-falling leaf.

We report the results of an experimental study of the finite-amplitude thresholds for transition to turbulence in a constant mass flux pipe flow. The flow was perturbed using small impulsive jets and push–pull disturbances from holes in the pipe wall. The flux of the disturbance is used to define an amplitude for the perturbation and the critical value required to cause transition scales in proportion to Re−1 for jets. In this case, the transition is catastrophic and the scaling suggests a simple balance between inertia and viscosity. On the other hand, the threshold scales as Re−1.3 or Re−1.5 for push–pull disturbances with the precise value depending on the orientation of the perturbation. Further, the amplitudes required to cause transition are typically an order of magnitude smaller than for jets. When the push–pull perturbation was applied in the oblique direction, streaks and hairpin vortices appeared during the growth phase of the disturbance. The scaling of the threshold and the growth of structures are both consistent with ideas associated with temporary algebraic growth.

Low- and high-frequency acoustic resonances are computed numerically via a high-order finite-element code for a generic two-dimensional high-lift configuration with a leading-edge slat. Zero mean flow is assumed, approximating the low-Mach-number situation at aircraft landing and approach. To avoid unphysical reflections at the boundaries of the truncated computational domain, perfectly matched layer absorbing boundary conditions are implemented in the form of the complex scaling method of atomic and molecular physics. It is shown that two types of resonance exist: resonances of surface waves which scale with the total airfoil length and longitudinal cavity-type resonances which scale with the slat cove length. Minima exist in the temporal decay rate which can be associated with the slat cove resonances and depend on the slat cove geometry. All resonances are damped owing to radiation losses. However, if coherent noise sources exist, as observed in low-Reynolds-number experiments, these sources can be enhanced acoustically by the above resonances if the source frequency is close to a resonant frequency.

Laboratory experiments have been performed on resonantly forced interfacial waves in a circular cylindrical basin containing a two-layer stratified fluid. The results of this shallow-water study exhibit a number of similarities to previous shallow-water studies performed in single-layer fluids, such as the generation of a large-amplitude response over a frequency bandwidth offset from the primary resonance, generation of a swirling mode at the observed resonant condition, and the significant contribution of higher harmonics. The two-layer experiments also produce results that are unique to stratified domains. In particular, the observed negative nonlinearity of the resonant condition at shallow water depth, mixing of the density interface resulting in detuning the forced response from the resonant condition, the enhanced role of viscous dissipation, and an alternative pathway for the nonlinear generation of higher-frequency waves when the layer depths are disparate. The results of this study are considered with regard to their implications for enclosed basins at the geophysical scale that are subject to near resonant forcing.

The irrotational flow past two slender bodies of revolution at angles of yaw, translating in parallel paths in very close proximity, is analysed by extending the classical slender body theory. The flow far away from the two bodies is shown to be a direct problem, which is represented in terms of two line sources along their longitudinal axes, at the strengths of the variation rates of their cross-section areas. The inner flow near the two bodies is reduced to the plane flow problem of the expanding (contracting) and lateral translations of two parallel circular cylinders with different radii, which is then solved analytically using conformal mapping. Consequently, an analytical flow solution has been obtained for two arbitrary slender bodies of revolution at angles of yaw translating in close proximity. The lateral forces and yaw moments acting on the two bodies are obtained in terms of integrals along the body lengths. A comparison is made among the present model for two slender bodies in close proximity, Tuck & Newman's (1974) model for two slender bodies far apart, and VSAERO (AMI)–commercial software based on potential flow theory and the boundary element method (BEM). The attraction force of the present model agrees well with the BEM result, when the clearance, h0, is within 20% of the body length, whereas the attraction force of Tuck & Newman is much smaller than the BEM result when h0 is within 30% of the body length, but approaches the latter when h0 is about half the body length. Numerical simulations are performed for the three typical manoeuvres of two bodies: (i) a body passing a stationary body, (ii) two bodies in a meeting manoeuvre (translating in opposite directions), and (iii) two bodies in a passing manoeuvre (translating in the same direction). The analysis reveals the orders of the lateral forces and yaw moments, as well as their variation trends in terms of the manoeuvre type, velocities, sizes, angles of yaw of the two bodies, and their proximity, etc. These irrotational dynamic features are expected to provide a basic understanding of this problem and will be beneficial to further numerical and experimental studies involving additional physical effects.

Direct numerical simulations (DNS) of steady and pulsatile flow through 75% (by area reduction) stenosed tubes have been performed, with the motivation of understanding the biofluid dynamics of actual stenosed arteries. The spectral-element method, providing geometric flexibility and high-order spectral accuracy, was employed for the simulations. The steady flow results are examined here while the pulsatile flow analysis is dealt with in Part 2 of this study. At inlet Reynolds numbers of 500 and 1000, DNS predict a laminar flow field downstream of an axisymmetric stenosis and comparison to previous experiments show good agreement in the immediate post-stenotic region. The introduction of a geometric perturbation within the current model, in the form of a stenosis eccentricity that was 5% of the main vessel diameter at the throat, resulted in breaking of the symmetry of the post-stenotic flow field by causing the jet to deflect towards the side of the eccentricity and, at a high enough Reynolds number of 1000, jet breakdown occurred in the downstream region. The flow transitioned to turbulence about five diameters away from the stenosis, with velocity spectra taking on a broadband nature, acquiring a -5/3 slope that is typical of turbulent flows. Transition was accomplished by the breaking up of streamwise, hairpin vortices into a localized turbulent spot, reminiscent of the turbulent puff observed in pipe flow transition, within which r.m.s. velocity and turbulent energy levels were highest. Turbulent fluctuations and energy levels rapidly decayed beyond this region and flow relaminarized. The acceleration of the fluid through the stenosis resulted in wall shear stress (WSS) magnitudes that exceeded upstream levels by more than a factor of 30 but low WSS levels accompanied the flow separation zones that formed immediately downstream of the stenosis. Transition to turbulence in the case of the eccentric stenosis was found to be manifested as large temporal and spatial gradients of shear stress, with significant axial and circumferential variations in instantaneous WSS.

Direct numerical simulations (DNS) of stenotic flows under conditions of steady inlet flow were discussed in Part 1 of this study. DNS of pulsatile flow through the 75% stenosed tube (by area) employed for the computations in Part 1 is examined here. Analogous to the steady flow results, DNS predicts a laminar post-stenotic flow field in the case of pulsatile flow through the axisymmetric stenosis model, in contrast to previous experiments, in which intermittent disturbed flow regions and turbulent breakdown were observed in the downstream region. The introduction of a stenosis eccentricity, that was 5% of the main vessel diameter at the throat, resulted in periodic, localized transition to turbulence. Analysis in this study indicates that the early and mid-acceleration phases of the time period cycle were relatively stable, with no turbulent activity in the post-stenotic region. However, towards the end of acceleration, the starting vortex, formed earlier as the fluid accelerated through the stenosis at the beginning of acceleration, started to break up into elongated streamwise structures. These streamwise vortices broke down at peak flow, forming a turbulent spot in the post-stenotic region. In the early part of deceleration there was intense turbulent activity within this spot. Past the mid-deceleration phase, through to minimum flow, the inlet flow lost its momentum and the flow field began to relaminarize. The start of acceleration in the following cycle saw a recurrence of the entire process of a starting structure undergoing turbulent breakdown and subsequent relaminarization of the post-stenotic flow field. Peak wall shear stress (WSS) levels occurred at the stenosis throat, with the rest of the vessel experiencing much lower levels. Turbulent breakdown at peak flow resulted in a sharp amplification of instantaneous WSS magnitudes across the region corresponding to the turbulent spot, accompanied by large axial and circumferential fluctuations, even while ensemble-averaged axial shear stresses remained mostly low and negative. WSS levels dropped rapidly after the mid-deceleration phase, when the relaminarization process took over, and were almost identical to laminar, axisymmetric shear levels through most of the acceleration phase.

The two-dimensional flow through a constricted channel is studied. A semi-circular bump is located on one side of the channel and the extent of blockage is varied by adjusting the radius of the bump. The blockage is varied between 0.05 and 0.9 of the channel width and the upstream Reynolds number between 25 and 3000. The geometry presents a simplified blockage specified by a single parameter, serving as a starting point for investigations of other more complex blockage geometries. For blockage ratios in excess of 0.4, the variation of reattachment length with Reynolds number collapses to within approximately 15%, while at lower ratios the behaviour differs. For the constrained two-dimensional flow, various phenomena are identified, such as multiple mini-recirculations contained within the main recirculation bubble and vortex shedding at higher Reynolds numbers. The stability of the flow to three-dimensional perturbations is analysed, revealing a transition to a three-dimensional state at a critical Reynolds number which decreases with higher blockage ratios. Separation lengths and the onset and structure of three-dimensional instability observed from the geometry of blockage ratio 0.5 resemble results taken from backward-facing step investigations. The question of the underlying mechanism behind the instability being either centrifugal or elliptic in nature and operating within the initial recirculation zone is analytically tested.

The stability properties of round variable-density low-Mach-number jets are studied by means of direct numerical simulation (DNS) and linear stability analysis. Fully three-dimensional DNS of variable-density jets, with and without gravity, demonstrate that the presence of buoyancy causes a more abrupt transition to turbulence. This effect helps to explain differences between normal gravity and microgravity jet diffusion flames observed in the laboratory.

The complete spectrum of spatial eigenmodes of the linearized low-Mach-number equations is calculated using a global matrix method. Also, an analytic form for the continuous portion of this spectrum is derived, and used to verify the numerical method. The absolute instability of variable-density jets is confirmed using Brigg's method, and a comprehensive parametric study of the strength and frequency of this instability is performed. Effects of Reynolds number, the density ratio of ambient-to-jet fluid (S1), shear-layer thickness and Froude number are considered. Finally, a region of local absolute instability is shown to exist in the near field of the jet by applying linear stability analysis to mean profiles measured from DNS.

A new subgrid eddy-viscosity model is proposed in this paper. Full details of the derivation of the model are given with the assumption of homogeneous turbulence. The formulation of the model is based on the dynamic equation of the structure function of resolved scale turbulence. By means of the local volume average, the effect of the anisotropy is taken into account in the generalized Kolmogorov equation, which represents the equilibrium energy transfer in the inertial subrange. Since the proposed model is formulated directly from the filtered Navier–Stokes equation, the resulting subgrid eddy viscosity has the feature that it can be adopted in various turbulent flows without any adjustments of model coefficient. The proposed model predicts the major statistical properties of rotating turbulence perfectly at fairly low-turbulence Rossby numbers whereas subgrid models, which do not consider anisotropic effects in turbulence energy transfer, cannot predict this typical anisotropic turbulence correctly. The model is also tested in plane wall turbulence, i.e. plane Couette flow and channel flow, and the major statistical properties are in better agreement with those predicted by DNS results than the predictions by the Smagorinsky, the dynamic Smagorinsky and the recent Cui–Zhou–Zhang–Shao models.

Lagrangian statistics of fluid-particle velocity and acceleration conditioned on fluctuations of dissipation, enstrophy and pseudo-dissipation representing different characteristics of local relative motion are extracted from a direct numerical simulation database of stationary (forced) homogeneous isotropic turbulence. The grid resolution in the simulations is up to 20483, and the Taylor-scale Reynolds number ranges from about 40 to 650, where characteristics of small-scale intermittency in the Eulerian flow field are well developed. A key joint statistic of the conditioning variables is the dissipation-enstrophy cross-correlation, which is asymmetric, but becomes less so at high Reynolds number. Conditional velocity autocorrelations are consistent with rapid changes in the velocity of fluid particles moving in regions of large velocity gradients. Examination of statistics conditioned upon enstrophy, especially in a local coordinate frame moving with the vorticity vector, and of the centripetal acceleration suggests the presence of vortex-trapping effects which persist for several Kolmogorov time scales. Further results on acceleration statistics and joint velocity-acceleration autocorrelations are also presented to help characterize in detail the properties of a joint stochastic process of velocity, acceleration and the pseudo-dissipation. Together with recent work on Eulerian conditional acceleration and Reynolds-number dependence of basic Lagrangian quantities, the present results are directly useful for the development of a new stochastic model formulated to account for intermittency and Reynolds-number effects as described in detail in a companion paper.

The modelling of fluid-particle acceleration in homogeneous isotropic turbulence in terms of stochastic models for the Lagrangian velocity, acceleration and a dissipation rate variable is considered. The basis for the Reynolds model (A. M. Reynolds, Phys. Rev. Lett. vol. 91, 2003, 084503) is reviewed and examined by reference to direct numerical simulations (DNS) of isotropic turbulence at Taylor-scale Reynolds number (Rλ) up to about 650. In particular, we show DNS data that support stochastic modelling of the logarithm of pseudo-dissipation as an Ornstein–Uhlenbeck process and reveal non-Gaussianity of the acceleration conditioned on fluctuations of the pseudo-dissipation rate. The DNS data are used to construct a new stochastic model that is exactly consistent with Gaussian velocity and conditionally cubic-Gaussian acceleration statistics. This model captures the effects of small-scale intermittency on acceleration and the conditional dependence of acceleration on pseudo-dissipation (which differs from that predicted by the refined Kolmogorov hypotheses). Non-Gaussianity of the conditionally standardized acceleration probability density function (PDF) is accounted for in terms of model nonlinearity. The large-time behaviour of the new model is that of a velocity-dissipation model that can be matched with DNS data for conditional second-order Lagrangian velocity structure functions. As a result, the diffusion coefficient for the new model incorporates two-time information and its Reynolds-number dependence as observed in DNS. The resulting model predictions for conditional and unconditional velocity autocorrelations and time scales are shown to be in very good agreement with DNS.

Direct numerical simulation is used to study the mixing of a passive scalar by a vortex ring issuing from a nozzle into stationary fluid. The ‘formation number’ (Gharib et al. J. Fluid Mech. vol. 360, 1998, p. 121), is found to be 3.6. Simulations are performed for a range of stroke ratios (ratio of stroke length to nozzle exit diameter) encompassing the formation number, and the effect of stroke ratio on entrainment and mixing is examined. When the stroke ratio is greater than the formation number, the resulting vortex ring with trailing column of fluid is shown to be less effective at mixing and entrainment. As the ring forms, ambient fluid is entrained radially into the ring from the region outside the nozzle exit. This entrainment stops once the ring forms, and is absent in the trailing column. The rate of change of scalar-containing fluid is found to depend linearly on stroke ratio until the formation number is reached, and falls below the linear curve for stroke ratios greater than the formation number. This behaviour is explained by considering the entrainment to be a combination of that due to the leading vortex ring and that due to the trailing column. For stroke ratios less than the formation number, the trailing column is absent, and the size of the vortex ring increases with stroke ratio, resulting in increased mixing. For stroke ratios above the formation number, the leading vortex ring remains the same, and the length of the trailing column increases with stroke ratio. The overall entrainment decreases as a result.

A large number of simulations have been performed to reveal how the occurrence of freak waves on deep water depends on the group and crest lengths for fixed steepness. It is found that there is a sharp qualitative transition between short- and long-crested sea, for a crest length of approximately ten wavelengths. For short crest lengths the statistics of freak waves deviates little from Gaussian and their occurrence is independent of group length (or Benjamin–Feir index, BFI). For long crest lengths the statistics of freak waves is strongly non-Gaussian and the group length (or BFI) is a good indicator of increased freak wave activity.