The collective hydrodynamics in fish schools and bird flocks, which includes self-organization of multiple dynamic bodies, is complex and lacks sufficient exploration. In this paper, we study the performance of multiple self-propelled foils in tandem formation, whose flapping motions are asynchronous with a phase difference. It is shown that a compact formation, in which all of the foils perform like a complete anguilliform swimmer, can be spontaneously formed by multiple foils via hydrodynamic interactions. Both velocity enhancement and energy saving can be achieved by multiple foils in anguilliform-like swimming. Furthermore, such anguilliform-like swimming behaviour can be observed over a wide range of parameters, including the number of foils, the phase difference, the initial distance, the heaving amplitude and the pitching amplitude. The results obtained here may provide some light on understanding the self-organization behaviour of biological collectives.

The Richtmyer–Meshkov instability of a helium layer surrounded by air is studied in a semi-annular convergent shock tube by high-speed schlieren photography. The gas layer is generated by an improved soap film technique such that its boundary shapes and thickness are precisely controlled. It is observed that the inner interface of the shocked light gas layer remains nearly undisturbed during the experimental time, even after the reshock, which is distinct from its counterpart in the heavy gas layer. This can be ascribed to the faster decay of the perturbation amplitude of the transmitted shock in the helium layer and Rayleigh–Taylor stabilization on the inner surface (light/heavy) during flow deceleration. The outer interface first experiences ‘accelerated’ phase inversion owing to geometric convergence, and later suffers a continuous deformation. Compared with a sole heavy/light interface, the wave influence (interface coupling) inhibits (promotes) growth of instability of the outer interface.

Nonlinear three-dimensional dynamo equilibrium solutions of viscous-resistive magneto-hydrodynamic equations are continued to formally infinite magnetic and hydrodynamic Reynolds numbers. The external driving mechanism of the dynamo is a uniform shear, which constitutes the base laminar flow and cannot support any kinematic dynamo. Nevertheless, an efficient subcritical nonlinear instability mechanism is found to be able to generate large-scale coherent structures known as streaks, for both velocity and magnetic fields. A finite amount of magnetic field generation is identified at the self-consistent asymptotic limit of the nonlinear solutions, thereby confirming the existence of an effective nonlinear dynamo action at astronomically large Reynolds numbers.

Shock–shock interaction structures and a newly discovered dynamic instability in granular streams resulting from such interactions are presented. Shock waves are generated by placing two similar triangular wedges in a gravity-driven granular stream. When the shock waves interact, grains collapse near the centre region of the wedges and a slow-moving concentrated diamond-shaped streak of grains is formed that grows as the inclination of the channel is increased. The diamond streak, under certain geometric conditions, is found to become unstable and start oscillating in the direction transverse to the mainstream. When the wedges are placed too close to each other, the granular flux of the incoming stream is unable to pass through the small gap, resulting in the formation of a single bow shock enveloping both the wedges. Experiments are performed for a wide range of flow speeds, wedge angles and wedge separations to investigate the interaction zone. We discuss a possible mechanism for the formation of the central streak and the associated dynamic instability observed for specific physical parameters.

Observational and computational studies of inertia-dominated wall turbulence with unstable thermal stratification have demonstrated that the inclination angle of large-scale motions (LSMs) increases with increasing buoyancy (as characterized by the Monin–Obukhov stability variable $\unicode[STIX]{x1D701}_{z}$). The physical implications of this structural steepening have received relatively less attention. Some authors have proposed that LSMs thicken – yet remain attached to the wall – with increasing buoyancy (Salesky & Anderson, J. Fluid Mech., vol. 856, 2018, pp. 135–168), while others have presented evidence that the upstream edge of an LSM remains anchored to the wall while its downstream edge lifts away from the wall (Hommema & Adrian, Boundary-Layer Meteorol., vol. 106, 2003, pp. 147–170). Using a suite of large-eddy simulations (LES) of unstably stratified turbulent channel flow, we demonstrate that buoyancy acts to lift LSMs away from the wall, leaving a wedge of fluid beneath with differing momentum. We develop a prognostic model for LSM inclination angle that accounts for this observed structure, where the LSM inclination angle $\unicode[STIX]{x1D6FE}$ is the sum of the inclination angle observed in a neutrally stratified wall-bounded turbulent flow, $\unicode[STIX]{x1D6FE}_{0}\approx 12^{\circ }{-}15^{\circ }$, and the stability-dependent inclination angle of the wedge $\unicode[STIX]{x1D6FE}_{w}(\unicode[STIX]{x1D701}_{z})$. Reported values of $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D701}_{z})$ from the literature, LES results and atmospheric surface layer observations are found to be in good agreement with the new model for $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D701}_{z})$.

We present a direct numerical simulation (DNS) study of pseudo-turbulence in buoyancy-driven bubbly flows for a range of Reynolds ($150\leqslant Re\leqslant 546$) and Atwood ($0.04\leqslant At\leqslant 0.9$) numbers. We study the probability distribution function of the horizontal and vertical liquid velocity fluctuations and find them to be in quantitative agreement with the experiments. The energy spectrum shows a $k^{-3}$ scaling at high $Re$ and becomes steeper on reducing $Re$. To investigate spectral transfers in the flow, we derive the scale-by-scale energy budget equation. Our analysis shows that, for scales smaller than the bubble diameter, the net transfer because of the surface tension and the kinetic energy flux balances viscous dissipation to give $k^{-3}$ scaling of the energy spectrum for both low and high $At$.

Experimental and numerical studies over the past two decades indicate that as the Reynolds number becomes large the turbulent boundary layer is increasingly composed of zones of uniform streamwise momentum, segregated by narrow regions of high shear. Recent experimental evidence suggests that passive scalar fields (for example, temperature) in turbulent boundary layers at high Reynolds number show similar characteristics; namely, large uniform temperature zones (UTZs) separated by narrow regions of high gradient, which we term thermal fissures (TFs). Herein, a model informed by analysis of the mean scalar transport equation, and that leverages the dynamical model recently developed by the authors (Cuevas Bautista et al., J. Fluid Mech., vol. 858, 2019, pp. 609–633), is formulated to predict passive scalar transport using the UTZ/TF concept. First, a finite number of TFs are distributed across the boundary layer. In analogy with the aforementioned dynamical model, the wall-normal positions of the TFs and their characteristic temperatures are then perturbed to generate independent ensembles, from which statistical moments are computed. The model successfully reproduces the statistical profiles of the temperature field as well as the streamwise turbulent heat flux. Lastly, the Prandtl number dependency of the empirically chosen parameters is investigated. It is concluded that the higher-order statistics, especially the kurtosis, produced by the model are sensitive to the Prandtl number, while the mean temperature and turbulent heat flux do not show noticeable Prandtl number dependency.

An important facet of water wave dynamics is the fact that Stokes’ 1847 approximate theory of water waves predicts mean Lagrangian velocities even when mean Eulerian velocities are zero. This motion, known as Stokes drift, is important to a wide variety of oceanic processes. Reflecting the difficulty of avoiding effects associated with the boundaries in wave tanks, the first convincing experimental evidence confirming this behaviour has only recently been given in van den Bremer et al. (J. Fluid Mech., vol. 879, 2019, pp. 168–186). This is an important result given prior evidence that the exact rotational waves first studied by Gerstner in 1802 may exist. Nonetheless, despite more than 200 years of work on the theory of water waves, much remains to be discovered.

The physical mechanism governing the onset of transonic shock buffet on swept wings remains elusive, with no unequivocal description forthcoming despite over half a century of research. This paper elucidates the fundamental flow physics on a civil aircraft wing using an extensive experimental database from a transonic wind tunnel facility. The analysis covers a wide range of flow conditions at a Reynolds number of around $3.6\times 10^{6}$. Data at pre-buffet conditions and beyond onset are assessed for Mach numbers between 0.70 and 0.84. Critically, unsteady surface pressure data of high spatial and temporal resolution acquired by dynamic pressure-sensitive paint is analysed, in addition to conventional data from pressure transducers and a root strain gauge. We identify two distinct phenomena in shock buffet conditions. First, we highlight a low-frequency shock unsteadiness for Strouhal numbers between 0.05 and 0.15, based on mean aerodynamic chord and reference free stream velocity. This has a characteristic wavelength of approximately 0.8 semi-span lengths (equivalent to three mean aerodynamic chords). Such shock unsteadiness is already observed at low-incidence conditions, below the buffet onset defined by traditional indicators. This has the effect of propagating disturbances predominantly in the inboard direction, depending on localised separation, with a dimensionless convection speed of approximately 0.26 for a Strouhal number of 0.09. Second, we describe a broadband higher-frequency behaviour for Strouhal numbers between 0.2 and 0.5 with a wavelength of 0.2 to 0.3 semi-span lengths (0.6 to 1.2 mean aerodynamic chords). This outboard propagation is confined to the tip region, similar to previously reported buffet cells believed to constitute the shock buffet instability on conventional swept wings. Interestingly, a dimensionless outboard convection speed of approximately 0.26, coinciding with the low-frequency shock unsteadiness, is found to be nearly independent of frequency. We characterise these coexisting phenomena by use of signal processing tools and modal analysis of the dynamic pressure-sensitive paint data, specifically proper orthogonal and dynamic mode decomposition. The results are scrutinised within the context of a broader research effort, including numerical simulation, and viewed alongside other experiments. We anticipate our findings will help to clarify experimental and numerical observations in edge-of-the-envelope conditions and to ultimately inform buffet-control strategies.

We identify mechanisms through which open-loop control of thermoacoustic instability is achieved in a laminar combustor and characterize them using synchronization theory. The thermoacoustic system comprises two nonlinearly coupled damped harmonic oscillators – acoustic and unsteady heat release rate (HRR) field – each possessing different eigenfrequencies. The frequency of the preferred mode of HRR oscillations is less than the third acoustic eigenfrequency where thermoacoustic instability develops. We systematically subject the limit-cycle oscillations to an external harmonic forcing at different frequencies and amplitudes. We observe that forcing at a frequency near the preferred mode of the HRR oscillator leads to a greater than 90 % decrease in the amplitude of the limit-cycle oscillations through the phenomenon of asynchronous quenching. Concurrently, there is a resonant amplification in the amplitude of HRR oscillations. Further, we show that the flame dynamics plays a key role in controlling the frequency at which quenching is observed. Most importantly, we show that forcing can cause asynchronous quenching either by imposing out-of-phase relation between pressure and HRR oscillations or by inducing period-2 dynamics in pressure oscillations while period-1 in HRR oscillations, thereby causing phase drifting between the two subsystems. In each of the two cases, acoustic driving is very low and hence thermoacoustic instability is suppressed. We show that the characteristics of forced synchronization of the pressure and HRR oscillations are significantly different. Thus, we find that the simultaneous characterization of the two subsystems is necessary to quantify completely the nonlinear response of the forced thermoacoustic system.

The fluid–structure interactions (FSI) of compliant lifting surfaces is complicated by free-surface and multiphase flows such as cavitation and ventilation. This paper describes the dynamic FSI response of a flexible surface-piercing hydrofoil in dry, wetted, ventilating and cavitating conditions. Experimental modal analysis is used to quantify the resonant frequencies and damping ratios of the fluid–structure system in each flow regime. The generalized hydrodynamic stiffness, fluid damping and fluid added mass are also determined as ratios to the corresponding structural modal forces. Added mass increases with increasing partial immersion of the hydrofoil and decreases in the presence of gaseous cavities. In particular, modal frequencies were observed to increase significantly in fully ventilated flow compared to fully wetted flow. The modal frequencies varied non-monotonically with speed in fully wetted flow. Gaseous cavities reduced the modal added mass and reduced the fluid disturbing force. Modal damping increases non-monotonically with increasing immersion depth. Forward speed causes the fluid damping force to increase with an approximately quadratic functional behaviour, consistent with a series expansion of the Morison equation, although damping identification became increasingly difficult at high flow speeds. The results indicate that fluid damping is greater than the associated structural damping in a quiescent liquid, and increasingly so with increasing immersion, suggesting viscous dissipation as a dominant mechanism. A preliminary investigation of modal vibration as a means of controlling the size and stability of ventilated cavities indicates that low-order modes encourage the formation of ventilation, while higher-order modes encourage the washout and elimination of ventilation.

To explain the large-scale hydrodynamic instability along a cavity-backed perforated plate in a flow duct, a two-dimensional multimodal analysis of flow disturbances is performed. First, a hole-by-hole description of the perforated plate shows a spatially growing wave with a wavelength close to the plate length, but much larger than the period of perforation. To better understand this problem and also cavity flow oscillations, we then combine the travelling mode and global mode analyses of the flow where the plate is represented by a homogeneous impedance. The spatially growing wave is, from a homogeneous point of view, essentially a Kelvin–Helmholtz instability wave, strongly distorted by evanescent acoustic waves near the cavity downstream edge. The phase difference of the unstable hydrodynamic mode at the two edges is found to be a bit larger than $2\unicode[STIX]{x03C0}$, whereas the upstream-travelling evanescent waves reduce the total phase change around the feedback loop, so that the phase condition of the global mode can still be satisfied. This particular case indicates the significant effects of those evanescent waves on both the amplitude and phase of cavity flow disturbances. The criterion of the global instability is discussed: the loop gain being larger or smaller than unity determines whether the global mode is unstable or stable. A global mode in the stable regime, which has so far received little attention, is explored by investigating the system response to external forcing. It is shown that sound can be produced when a lightly damped flow–acoustic resonance is excited by a vortical wave.

We investigate experimentally the turbulent flow through a two-dimensional contraction. Using a water tunnel with an active grid we generate turbulence at Taylor microscale Reynolds number $Re_{\unicode[STIX]{x1D706}}\sim 250$ which is advected through a 2.5 : 1 contraction. Volumetric and time-resolved tomographic particle image velocimetry and shake-the-box velocity measurements are used to characterize the evolution of coherent vortical structures at three streamwise locations upstream of and within the contraction. We confirm the conceptual picture of coherent large-scale vortices being stretched and aligned with the mean rate of strain. This alignment of the vortices with the tunnel centreline is stronger compared to the alignment of vorticity with the large-scale strain observed in numerical simulations of homogeneous turbulence. We judge this by the peak probability magnitudes of these alignments. This result is robust and independent of the grid-rotation protocols. On the other hand, while the pointwise vorticity vector also, to a lesser extent, aligns with the mean strain, it principally remains aligned with the intermediate eigenvector of the local instantaneous strain-rate tensor, as is known in other turbulent flows. These results persist when the distance from the grid to the entrance of the contraction is doubled, showing that modest transverse inhomogeneities do not significantly affect these vortical-orientation results.

We investigate experimentally the effect of double diffusion in the salt-fingering configuration on entrainment in turbulent plumes. Plumes over a range of source buoyancy fluxes $B_{0}$ and source density ratios $R_{\unicode[STIX]{x1D70C}}$ are examined. When the plumes are double diffusive ($R_{\unicode[STIX]{x1D70C}}>0$) the entrainment coefficient $\unicode[STIX]{x1D6FC}$ is not constant, with an up to 20 % reduction from the value found for single-diffusive plumes, that is, plumes with $R_{\unicode[STIX]{x1D70C}}=0$. The scale of reduction is found to be in direct relation to the source density ratio and is inversely related to the distance travelled by the plume, indicating that double-diffusive effects decrease as the plume evolves. We propose an explanation for the observed reduction in the entrainment coefficient on the basis of differential diffusion hindering large-scale engulfment at the edge of the plume.

Particle image velocimetry measurements have been carried out to assess the fully developed turbulence in square-sectioned porous duct flows. To the bottom duct wall, this study applies two types of porous media whose porosities are approximately 0.8 and ratios of wall-normal to streamwise permeabilities are 0.8 and 7.8. Both over- and under-surface turbulence of the porous layers are discussed at inlet flow Reynolds numbers of $Re\simeq 3500$ and 7500. Cross-sectional secondary flows are detected with an enhanced magnitude of approximately 6 % of the inlet bulk velocity. The secondary flow pattern consisting of four large vortices is observed to be insensitive to the porous structures. Over the porous wall, although turbulence is enhanced by the permeability, it is confirmed that turbulence over and under the porous surfaces is rather insensitive to the wall-normal permeability compared with the streamwise permeability as seen in porous-wall channel flows. In the present range of streamwise permeability Reynolds numbers of $Re_{K_{x}}=2.49{-}6.37$, the wall-normal fluctuations become dominant once underneath the porous surface while the streamwise ones become dominant again deep inside the porous layer. Applying streamwise–spanwise plane averaging, which covers a 52 % area in the middle of the duct, to the flow quantities, it is confirmed that the correlations between the pore-scale Reynolds number and the log-law parameters are similar to those seen in a wide range of porous-wall channels. The above characteristics are generally the same as those of porous-wall channels in the same range of porosities and permeability Reynolds numbers even with the enhanced secondary flows. However, from the spectral analysis of flows at the porous walls, it is found that, near the symmetry planes, the wavelengths of the Kelvin–Helmholtz waves become a little shorter than those in turbulent porous-wall channels possibly because of the sidewall boundary layers, particularly at low Reynolds numbers.

Flapping wings of insects can follow various complex-motion waveforms, influencing the flow structures over a wing and consequently the aerodynamic performance. However, most studies of insect-wing models incorporate either simple harmonic or robofly-like motion waveforms. The effects of other waveforms appear to be under-explored. Motivated by this, the present study investigates the individual and combined effects of the sweep- and pitch-motion waveforms for fixed flapping frequency and amplitude of a fruit-fly wing planform. Physical experiments are conducted to directly measure the forces and torques acting on the wing. Interestingly, the sweep waveform is observed to influence the overall variation in the lift coefficient ($C_{L}$), whereas the pitch waveform is observed to influence only the instantaneous $C_{L}$ during stroke reversal. Carefully validated three-dimensional numerical simulations reveal that a change in the strength of the large-scale vortex over the wing as the sweep profile parameter is varied is responsible for the observed variations in $C_{L}$. An exploration over wide ranges of the sweep and the pitch profile parameters shows that the waveforms maximising the mean lift coefficient are different from those maximising the power economy. Consistent with some previous experiments on robotic insects, the possibility of passive pitch motion is observed at slower pitching rates. Contours of the mean lift coefficient and power economy mapped on the planes of the sweep and the pitch profile parameters can help designers of flapping-wing micro air vehicles in selecting the waveforms appropriate for their design criteria.

We analyse the electrophoresis of a weakly charged particle with a thin double layer in a dilute polymer solution. The particle velocity in polymer solutions modelled with different constitutive equations is calculated using a regular perturbation in the polymer concentration and the generalized reciprocal theorem. The analysis shows that the polymer is strongly stretched in two regions, the birefringent strand and the high-shear region inside the double layer. The electrophoretic velocity of the particle always decreases with the addition of polymers due to both increased viscosity and fluid elasticity. At a small Weissenberg number ($Wi$), which is the product of the polymer relaxation time and the shear rate, the polymers inside the double layer contribute to most of the velocity reduction by increasing the fluid viscosity. With increasing $Wi$, viscoelasticity decreases and shear thinning increases the particle velocity. Polymer elasticity alters the fluid velocity disturbance outside the double layer from that of a neutral squirmer to a puller-type squirmer. At high $Wi$, the strong extensional stress inside the birefringent strand downstream of the particle dominates the velocity reduction. The scaling of the birefringent strand is used to estimate the particle velocity.

Large-scale coherent structures are present in compressible free shear flows, where they are known to be a main source of aerodynamic noise. Previous studies showed that these structures may be treated as instability waves or wavepackets supported by the underlying turbulent mean flow. By adopting this viewpoint in the framework of triple decomposition of the instantaneous flow into the mean field, coherent motion and small-scale turbulence, a strongly nonlinear dynamical model was constructed to describe the formation and development of coherent structures in incompressible turbulent free shear layers (Wu & Zhuang, J. Fluid Mech., vol. 787, 2016, pp. 396–439). That model is now extended to compressible flows, for which the coherent structures are extracted through a density-weighted (Favre) phase average. The nonlinear non-equilibrium critical-layer theory for instability waves in a laminar compressible mixing layer is adapted to analyse coherent structures in its turbulent counterpart. The strong non-parallelism associated with the fast spreading of the turbulent mean flow is taken into account and found to be significant. The model also accounts for the effect of fine-scale turbulence on coherent structures via a gradient type of closure model which now allows for a phase lag between the phase-averaged small-scale Reynolds stresses and the strain rates of coherent structures. The analysis results in an evolution system comprising of an amplitude equation, the critical-layer temperature and vorticity equations along with the appropriate initial and boundary conditions. The physical processes of acoustic radiation from the coherent structures are described by examining the far-field asymptote of the hydrodynamic fluctuations. We demonstrate that the nonlinearly generated slowly breathing mean-flow distortion radiates low-frequency sound waves. The true physical sources are identified. Equivalent sources in a Lighthill type of acoustic analogy context also arise, but they cannot be fully determined before the acoustic field is calculated, in which sense the radiated sound waves act back on the source. The numerical solutions to the evolution system show that coherent structures attenuate nonlinearly and their vorticity field rolls up to form the characteristic rollers. A study is also made of coherent structures represented by modulated wavetrains consisting of sideband modes, in which case nonlinear interactions generate components with frequencies that are combinations of those of the dominant modes. These components, especially the difference-frequency one, acquire significant amplitudes. Finally, the directivity and spectrum of the emitted acoustic field are calculated for both cases where the coherent structures consist of discrete, and a continuum of, sideband modes.

We reformulate shallow water theory to understand viscous shear induced natural hydraulic jumps in channels slightly deviated from the horizontal. One of the interesting contributions of the study is a modified expression for Froude number to predict jumps in inclined channels. The proposed Froude number is different from the conventional expression which incorporates channel inclination as a straight forward component of gravity. This highlights the complexity that a jump can generate even in single phase laminar flow. We also obtain an analytical expression for predicting jump strength and show that the scaling relationship originally proposed for jump location in horizontal channels is applicable for both upslope and downslope flows. As expected, upslope flow aids jump formation and beyond a critical adverse tilt, a submerged jump results in subcritical flow right from the entry. On the other hand, both Reynolds number and channel tilt suppress the tendency to jump in downslope flows and below a critical downslope inclination, the flow remains supercritical throughout the channel length. The film thickness for fully developed flow can be predicted from the exact solution of the Navier–Stokes equations. As the theory encounters a singularity in the jump region, numerical simulations and experimental results have been used to obtain additional insights into the physics of jump formation. They have revealed the existence of submerged jump, wavy jump, smooth jump and no jump conditions as a function of liquid Reynolds number, scaled channel length and channel inclination. Such a variety of jump geometries in planar laminar flow has not been reported earlier. Both theory and simulations also reveal that the linear free surface profile upstream of the jump is a function of Reynolds number only, while the downstream profiles can be tuned by changing both Reynolds number as well as the channel length and tilt over the range of parameters studied. We thus demonstrate that, despite the simplicity and the approximations involved, shallow water equations formulated assuming self-similar velocity profiles can elucidate the physics of planar laminar jumps over slight inclinations, difficult to avoid in practice. The analytical and simulated results have been extensively validated with experimental data obtained from a specially designed test rig which ensures laminar flow before and after the jump. To the authors’ knowledge, almost no experimental study has to date been reported on films ‘thin enough’ to remain laminar even after the planar jump.

The structure of streamwise vortices that arise due to secondary instabilities in the wake of a two-dimensional blunt body with a chord-to-thickness ratio of 12.5 was investigated using high-speed stereoscopic particle image velocimetry. Reynolds numbers spanning an order of magnitude from $Re(h)=2600$ to 25 800 were considered, where $h$ is the height of the blunt trailing edge. A modified two-dimensional $Q$-criterion ($Q^{\prime }=\unicode[STIX]{x1D714}_{x}Q/|\unicode[STIX]{x1D714}_{x}|$) was applied to identify the streamwise vortices. The wavelength of the streamwise vortices, defined as the spanwise distance between adjacent streamwise vortex pairs in the wake, was investigated by applying an autocorrelation algorithm to snapshots of $Q^{\prime }$. The most probable wavelength was found to range from $0.67h$ to $0.85h$ with increasing $Re$, and the mean wavelengths increased from $0.77h$ to $0.96h$. These wavelength values appeared to increase asymptotically. Visual inspection and cross-correlation analyses based on $Q^{\prime }$ showed that the streamwise vortices maintain their directions of rotation during primary shedding cycles. The latter analysis was carried out at low $Re$ because of a large amount of wake distortion and the absence of time-resolved data at high $Re$. The characteristics of the streamwise vortex structure found here match those of mode B, which, at similar $Re$, dominates the wakes of circular and square cylinders and has also recently been shown to exist in the wake of an elongated blunt body with a larger chord-to-thickness ratio of 46.5 (Gibeau et al., J. Fluid Mech., vol. 846, 2018, pp. 578–604).