In this paper, we address the problem of how to account for second-order statistics of turbulent flows using low-complexity stochastic dynamical models based on the linearized Navier–Stokes equations. The complexity is quantified by the number of degrees of freedom in the linearized evolution model that are directly influenced by stochastic excitation sources. For the case where only a subset of velocity correlations are known, we develop a framework to complete unavailable second-order statistics in a way that is consistent with linearization around turbulent mean velocity. In general, white-in-time stochastic forcing is not sufficient to explain turbulent flow statistics. We develop models for coloured-in-time forcing using a maximum entropy formulation together with a regularization that serves as a proxy for rank minimization. We show that coloured-in-time excitation of the Navier–Stokes equations can also be interpreted as a low-rank modification to the generator of the linearized dynamics. Our method provides a data-driven refinement of models that originate from first principles and captures complex dynamics of turbulent flows in a way that is tractable for analysis, optimization and control design.

We consider the segregation of spheres of equal size and different density flowing over an inclined plane, theoretically and computationally by means of distinct element method (DEM) simulations. In the first part of the work, we study the settling of a single higher-density particle in the flow of otherwise identical particles. We show that the motion of the high-density tracer particle can be understood in terms of the buoyancy and drag forces acting on it. The buoyancy force is given by Archimedes principle, with an effective volume associated with the particle, which depends upon the local packing fraction, $\phi $. The buoyancy arises primarily from normal forces acting on the particle, and tangential forces have a negligible contribution. The drag force on a sphere of diameter $d$ sinking with a velocity $v$ in a granular medium of apparent viscosity $\eta $ is given by a modified Stokes law, ${F}_{d} = c\pi \eta dv$. The coefficient ($c$) is found to decrease with packing fraction. In the second part of the work, we consider the case of binary granular mixtures of particles of the same size but differing in density. A continuum model for segregation is presented, based on the single-particle results. The number fraction profile for the heavy particles at equilibrium is obtained in terms of the effective temperature, defined by a fluctuation–dissipation relation. The model predicts the equilibrium number fraction profiles at different inclination angles and for different mass ratios of the particles, which match the DEM results very well. Finally, a complete model for the theoretical prediction of the flow and number fraction profiles for a mixture of particles of different density is presented, which combines the segregation model with a model for the rheology of mixtures. The model predictions agree quite well with the simulation results.

Close interactions between pairs of two-dimensional vortices of like sign were investigated in experiments with barotropic vortices and baroclinic vortices. The vortices were generated by sources or sinks in a rotating fluid which, respectively, was homogeneous or contained a two-layer density stratification. For two identical anticyclonic, unstratified vortices there was a critical separation distance beyond which the vortices coalesced to form a single larger anticyclone. The critical distance d*, scaled by the radius R of a core having non-zero relative vorticity, was d*/R = 3.3 ± 0.2. This value is in agreement with results of previous numerical simulations for finite-area vortices in non-rotating flows. The effects on vortex structure of Ekman pumping due to the presence of a rigid boundary caused cyclonic vortices to coalesee from larger distances. Baroclinic vortices in a two-layer stratification were also found to coalesce despite a potential-energy barrier. However, the critical separation distance depended on the internal Rossby radius. When the Rossby radius was large compared with the core radius, vortices coalesced from distances much greater than the critical distance for barotropic vortices. Coalescing of two vortices of equal size and strength led to two symmetric entwined spirals of water, while close interaction of unequal vortices caused the weaker vortex to be wrapped around the outer edge of the stronger. Implications of these results are discussed for ocean eddies and intense atmospheric cyclones.

The swimming of a flagellar micro-organism by the propagation of helical waves along its flagellum is analysed by a boundary-element method. The method is not restricted to any particular geometry of the organism nor does it assume a specific wave motion for the flagellum. However, only results for an organism with a spherical or ellipsoidal cell body and a helically beating flagellum are presented here.

With regard to the flagellum, it is concluded that the optimum helical wave (amplitude α and wavenumber k) has αk ≈ 1 (pitch angle of 45°) and that for the optimum flagellar length L/A = 10 (L being the flagellar length, A being the radius of the assumed spherical cell body) the optimum number of wavelengths Nλ is about 1.5. Furthermore there appears to be no optimal value for the flagellar radius a, with the thinner flagella being favoured. These conclusions show excellent quantitative agreement with those of slender-body theory.

For the case of an ellipsoidal cell body, the optimum aspect ratios B/A and C/A of the ellipsoid are about 0.7 and 0.3 respectively; A, B and C are the principal radii of the ellipsoid. These and all of the above conclusions show good qualitative agreement with experimental observations of efficiently swimming micro-organisms.

The steady-state flow generated by a rotating bottom in a closed cylindrical container and the resulting vortex breakdown bubbles have been studied experimentally. By comparing the flow inside two different container geometries, one with a rigid cover and the other with a free surface, we examined the way in which the formation and structure of the breakdown bubbles depend on the surrounding flow. Details of the flow were visualized by means of the electrolytic precipitation technique, whereas a particle tracking technique was used to characterize the whole flow field. We found that the breakdown bubbles inside the container flow are in many ways similar to those in vortex tubes. First, the bubbles are open with in- and outflow and second, their structure is, like in the case of vortex breakdown in pipe flows, highly axisymmetric on the upstream side of the bubble and asymmetric on their rear side. However, and surprisingly, we observed bubbles which are open and stationary at the same time. This shows that open breakdown bubbles are not necessarily the result of periodic oscillations of the recirculation zone. The asymmetry of the flow structure is found to be related to the existence of asymmetric flow separations on the container wall. If the angular velocity of the rotating bottom is increased the evolution of the breakdown bubbles is different in both configurations: in the rigid cover case the breakdown bubbles disappear but persist in the free surface case.

It is widely accepted that both ripples and dunes form in rivers by primary linear instability; the wavelength of the former scaling on the grain size and that of the latter being controlled by the water depth. We revisit here this problem in a theoretical framework that allows to give a clear picture of the instability in terms of dynamical mechanisms. A multi-scale description of the problem is proposed, in which the details of the different mechanisms controlling sediment transport are encoded into three quantities: the saturated flux, the saturation length and the threshold shear stress. Hydrodynamics is linearized with respect to the bedform aspect ratio. We show that the phase shift of the basal shear stress with respect to the topography, responsible for the formation of bedforms, appears in an inner boundary layer where shear stress and pressure gradients balance. This phase shift is sensitive to the presence of the free surface, and the related effects can be interpreted in terms of standing gravity waves excited by topography. The basal shear stress is dominated by this finite depth effect in two ranges of wavelength: when the wavelength is large compared to the flow depth, so that the inner layer extends throughout the flow, and in the resonant conditions, when the downstream material velocity balances the upstream wave propagation. Performing the linear stability analysis of a flat sand bed, the relation between the wavelength at which ripples form and the flux saturation length is quantitatively derived. It explains the discrepancy between measured initial wavelengths and predictions that do not take this lag between flow velocity and sediment transport into account. Experimental data are used to determine the saturation length as a function of grain size and shear velocity. Taking the free surface into account, we show that the excitation of standing waves has a stabilizing effect, independent of the details of the flow and sediment transport models. Consequently, the shape of the dispersion relation obtained from the linear stability analysis of a flat sand bed is such that dunes cannot result from a primary linear instability. We present the results of field experiments performed in the natural sandy Leyre river, which show the formation of ripples by a linear instability and the formation of dunes by a nonlinear pattern coarsening limited by the free surface. Finally, we show that mega-dunes form when the sand bed presents heterogeneities such as a wide distribution of grain sizes.

A laboratory experiment is used to study the transient flow in an initially isothermal cavity at temperature T0 following the rapid change of the two vertical endwalls to temperatures T0 ± ΔT respectively. Individual temperature records are taken and the transient flow in the entire cavity is visualized with the aid of a tracer technique. It is shown that an oscillatory approach to final steady-state conditions exists for certain flow regimes, although the form of the oscillatory response is different to that suggested by previous work. It is argued that this oscillatory behaviour is due to the inertia of the flow entering the interior of the cavity from the sidewall boundary layers, which may lead to a form of internal hydraulic jump if the Rayleigh number is sufficiently large.

While the wake of a circular cylinder and, to a lesser extent, the normal flat plate have been studied in considerable detail, the wakes of elliptic cylinders have not received similar attention. However, the wakes from the first two bodies have considerably different characteristics, in terms of three-dimensional transition modes, and near- and far-wake structure. This paper focuses on elliptic cylinders, which span these two disparate cases. The Strouhal number and drag coefficient variations with Reynolds number are documented for the two-dimensional shedding regime. There are considerable differences from the standard circular cylinder curve. The different three-dimensional transition modes are also examined using Floquet stability analysis based on computed two-dimensional periodic base flows. As the cylinder aspect ratio (major to minor axis) is decreased, mode A is no longer unstable for aspect ratios below 0.25, as the wake deviates further from the standard Bénard–von Kármán state. For still smaller aspect ratios, another three-dimensional quasi-periodic mode becomes unstable, leading to a different transition scenario. Interestingly, for the 0.25 aspect ratio case, mode A restabilises above a Reynolds number of approximately 125, allowing the wake to return to a two-dimensional state, at least in the near wake. For the flat plate, three-dimensional simulations show that the shift in the Strouhal number from the two-dimensional value is gradual with Reynolds number, unlike the situation for the circular cylinder wake once mode A shedding develops. Dynamic mode decomposition is used to characterise the spatially evolving character of the wake as it undergoes transition from the primary Bénard–von Kármán-like near wake into a two-layered wake, through to a secondary Bénard–von Kármán-like wake further downstream, which in turn develops an even longer wavelength unsteadiness. It is also used to examine the differences in the two- and three-dimensional near-wake state, showing the increasing distortion of the two-dimensional rollers as the Reynolds number is increased.

A detailed analysis of the predictability of observed Monin–Obukhov (MO) similarity within the near-ground region of near-neutral to moderately convective atmospheric boundary layers (ABL) from large-eddy simulation (LES) fields is reported in this work. High-resolution LES predictions of means, variances, budgets of turbulent kinetic energy and temperature variance, and the velocity and temperature spectra from three ABL states (−zi/L=0.44, 3 and 8) are analysed under MO scaling. The resolution in the near-ground region is increased by using ‘nested meshes.’ For the close-to-neutral case (−zi/L=0.44) the relative roles of grid resolution and subgrid-scale scale (SGS) parameterization on the predictability of MO-similarity are also studied. The simulated temperature field is found to satisfy the MO hypothesis and agree well with observations. The simulated velocity field, on the other hand, shows significant departures. Except for the horizontal variance, MO scales are the appropriate normalizing scales for the near-ground-layer statistics. However, the LES suggest that the boundary layer depth zi has an ‘indirect’ influence on all near-ground-layer variables except temperature, and the LES-predicted MO-scaled variables exhibit a functional dependence on both z/L and z/zi. The simulated two-dimensional spectra of velocity and temperature fluctuations, however, suggest that while large scales deviate from MO-similarity, inertial subrange scales are MO-similar. Discrepancies with field observations raise important questions of the non-dimensional depth z/zi over which MO-similarity holds for a particular variable. Surface-layer field studies generally do not document zi. It is also not clear to what extent these discrepancies are due to approximations made in LES. Measurements are needed designed specifically for comparing with LES predictions.

We investigate the scale invariance of combustion noise generated from turbulent reacting flows in a confined environment using complex networks. The time series data of unsteady pressure, which is the indicative of spatiotemporal changes happening in the combustor, is converted into complex networks using the visibility algorithm. We show that the complex networks obtained from the low-amplitude, aperiodic pressure fluctuations during combustion noise have scale-free structure. The power-law distributions of connections in the scale-free network are related to the scale invariance of combustion noise. We also show that the scale-free feature of combustion noise disappears and order emerges in the complex network topology during the transition from combustion noise to combustion instability. The use of complex networks enables us to formalize the identification of the pattern (i.e. scale-free to order) during the transition from combustion noise to thermoacoustic instability as a structural change in topology of the network.

Laser-induced uorescence (LIF) and laser Doppler velocimetry (LDV) are used to explore the structure of a turbulent boundary layer over a wall made up of two-dimensional square cavities placed transversely to the flow direction. There is strong evidence of occurrence of outflows of fluid from the cavities as well as inflows into the cavities. These events occur in a pseudo-random manner and are closely associated with the passage of near-wall quasi-streamwise vortices. These vortices and the associated low-speed streaks are similar to those found in a turbulent boundary layer over a smooth wall. It is conjectured that outflows play an important role in maintaining the level of turbulent energy in the layer and enhancing the approach towards self-preservation. Relative to a smooth wall layer, there is a discernible increase in the magnitudes of all the Reynolds stresses and a smaller streamwise variation of the local skin friction coefficient. A local maximum in the Reynolds shear stress is observed in the shear layers over the cavities.

The solution is obtained in general terms for the velocity fluctuations generated in a laminar boundary layer by an oscillating disturbance on the boundry wall. The form of excitation is chosen to represent a vibrating ribbon of the type used by Schubauer to force disturbance in boundary layers. The forced wave system generated by the ribbon is shown to be a spatially growing one, which is described far downstream by an eigenmode of the system which has a complex wave-number.

A numerical model based on a Lagrangian description has been developed for studying run-up of long water waves governed by a set of Boussinesq equations. The performance of the numerical scheme has been tested by comparing with analytical solutions and experimental data. Simulations of the run-up of solitary waves on relatively steep planes (inclination angle > 20°) show surface displacements and run-up heights in good agreement with experiments. For waves with relatively large amplitude the simulations reveal the development of a breaking bore during the backwash. Results for run-up heights in converging and diverging channels are also presented.

Sinusoidal modifications to the leading edge of a foil, or tubercles, have been shown to improve aerodynamic performance under certain flow conditions. One of the mechanisms of performance enhancement is believed to be the generation of streamwise vortices, which improve the momentum exchange in the boundary layer. This experimental and numerical study investigates the formation and evolution of these streamwise vortices at a low Reynolds number of $Re=2230$, providing insight into both the averaged and time-dependent flow patterns. Furthermore, the strength of the vortices is quantified through calculation of the vorticity and circulation, and it is found that the circulation increases in the downstream direction. There is strong agreement between the experimental and numerical observations, and this allows close examination of the flow structure. The results demonstrate that the presence of strong pressure gradients near the leading edge gives rise to a significant surface flux of vorticity in this region. As soon as this vorticity is created, it is stretched, tilted and diffused in a highly three-dimensional manner. These processes lead to the generation of a pair of streamwise vortices between the tubercle peaks. A horseshoe-shaped separation zone is shown to initiate behind a tubercle trough, and this region of separation is bounded by a canopy of boundary-layer vorticity. Along the sides of this shear layer canopy, a continued influx of boundary-layer vorticity occurs, resulting in an increase in circulation of the primary streamwise vortices in the downstream direction. Flow visualisation and particle image velocimetry studies support these observations and demonstrate that the flow characteristics vary with time, particularly near the trailing edge and at a higher angle of attack. Numerical evaluation of the lift and drag coefficients reveals that, for this particular flow regime, the performance of a foil with tubercles is slightly better than that of an unmodified foil.

Experiments have shown that bubbles approaching an air–water interface give rise to axisymmetric jets projected upwards into the air. Similar jets occur during the collapse of cavitation bubbles near a solid surface. In this paper we show that such jets are well modelled by a Dirichlet hyperboloid, a hyperbolic form of the better-known ellipsoid. The vertex angle of the hyperboloid is calculated as a function of time and found to agree with the observations of Blake & Gibson (1981) and others.

The jet is initiated, according to this model, when the vertex angle passes through 2 arctan √2, or 109·47°, at which instant the fluid accelerations become large. This compares with a vertical angle of 90° in the corresponding two-dimensional flow.

Further experiments demonstrate that an axisymmetric standing wave, when driven beyond its maximum amplitude, can break by throwing up a jet of the same hyperbolic form. Hyperbolic jets may occur commonly in free-surface flows.

The glancing interaction between an oblique shock wave and a turbulent boundary layer has been studied experimentally using a variable-incidence wedge mounted from the side wall of a supersonic wind tunnel. The Mach number was 2·3 and the Reynolds number 5 × 104, based on the 99·5 % thickness of the boundary layer just upstream of the interaction region. The study includes oil flow pictures, vapour and smoke-screen photographs, wall-pressure distributions and local heat-transfer measurements. The results suggest that the complicated interaction region involves two viscous layers: an induced layer formed from fluid initially in the boundary layer growing along the wedge surface near the root, and the thick turbulent layer on the tunnel side wall. The mutual interference between these layers is described, separation is defined and a discussion of incipient separation is included.

Small regions in a flow where the density is different from that of the surrounding gas do not exactly follow accelerated motions of the latter, but move faster or slower depending on whether their density is smaller or larger than that of the main flow. This behaviour cannot be quantitatively explained by treating a gas ‘bubble’ as a hypothetical solid particle of the same density, because a gas bubble cannot move relative to the surrounding gas without being transformed into a vortex which absorbs part of the energy of the relative motion.

To illustrate the acceleration effect, the flow velocity behind known pressure waves in a shock tube is compared with the observed velocity of a bubble produced by a spark discharge. The displacement of such a bubble by a wave exceeds that of a flow element by more than 20%, but the bubble density is not known. If the spark discharge is replaced by a small jet of another gas, a pressure wave cuts off a section of this jet which then represents a bubble of known density.

A theory is developed which permits computing the response of such bubbles to accelerations. The ratio of the bubble velocity to the velocity of the surrounding gas depends on the density ratio for the two gases and on the shape of the bubble, but not on the acceleration. Experimental results with H2, He, and SF6 bubbles in air, accelerated by shock waves of various strength, are presented and agree well with the theoretical predictions. The results apply regardless of whether accelerations are produced by pressure waves in a non-steady flow or by curvature of streamlines in a steady flow. Various aspects of the experimental observations are discussed.

High-frequency fluctuations in temperature and velocity were measured at a height of 2 m above a harvested, nearly level field of rye grass. Conditions were both stably and unstably stratified. Reynolds numbers ranged from 370000 to 740000. Measurements of velocity were made with a hot-wire anemometer and measurements of temperature with a platinum resistance element which had a diameter of 0[sdot ]5 μm and a length of 1 mm. Thirteen runs ranging in length from 78 to 238 s were analysed.

Spectra of velocity fluctuations are consistent with previously reported universal forms. Spectra of temperature, however, exhibit an increase in slope with increasing wavenumber as the maximum in the one-dimensional dissipation spectrum is approached. The peak of the one-dimensional dissipation spectrum for temperature fluctuations occurs at a higher wavenumber than that of simultaneous spectra of the dissipation of velocity fluctuations. It is suggested that the change in slope of the temperature spectra and the dissimilarity between temperature and velocity spectra may be due to spatial dissimilarity in the dissipation of temperature and velocity fluctuations. The temperature spectra are compared with a theoretical prediction for fluids with large Prandtl number, due to Batchelor (1959). Even though air has a Prandtl number of 0[sdot ]7, the observations are in qualitative agreement with predictions of the theory. The non-dimensional wavenumber at which the increase in slope occurs is about 0[sdot ]02, in good agreement with observations in the ocean reported by Grant et al. (1968).

For the two runs for which the stratification was stable, the normalized spectra of the temperature derivative fall on average slightly below the mean of the spectra of the remaining runs in the range in which the slope is approximately one-third. Hence the Reynolds number may not have always been sufficiently high to satisfy completely the conditions for an inertial subrange.

Universal inertial-subrange constants were directly evaluated from one-dimensional dissipation spectra and found to be 0[sdot ]54 and 1[sdot ]00 for velocity and temperature, respectively. The constant for velocity is consistent with previously reported values, while the value for temperature differs from some of the previous direct estimates but is only 20% greater than the mean of the indirect estimates. This discrepancy may be explained by the neglect in the indirect estimates of the divergence terms in the conservation equation for the variance of temperature fluctuations. There is weak evidence that the one-dimensional constant, and hence the temperature spectra, may depend upon the turbulence Reynolds number, which varied from 1200 to 4300 in the observations reported.

An infinitesimal centre disturbance is imposed on a fully Ldveloped plane Poiseuille flow at a Reynolds number R slightly greater than the critical value Rc for instability. After a long time, t, the disturbance consists of a modulated wave whose amplitude A is a slowly varying function of position and time. In an earlier paper (Stewartson & Stuart 1971) the parabolic differential equation satisfied by A for two-dimensional disturbances was found; the theory is here extended to three dimensions. Although the coefficients of the equation are coinples, a start is made on elucidating the properties of its solutions by assuming that these coefficients are real. It is then found numerically and confirmed analytically that, for a finite value of (R-Rc)t, the amplitude A develops an infinite peak at the wave centre. The possible relevance of this work to the phenomenon of transition is discussed.

The phenomenon of vortex shedding from a heated/cooled circular cylinder has been investigated numerically in the mixed natural and forced convection regimes. Accuracy of the computation was achieved by the fourth-order Hermitian relation applied to the contravariant velocity components in the convection terms of the vorticity transport equation, and by the far-boundary stream-function condition of an integral-series form developed by the authors. Purely periodic flows at Re = 100, efficiently established through the use of a direct elliptic solver called the SEVP, was found to degenerate into a steady twin-vortex pattern at the critical Grashof number 1500, confirming an earlier experimental observation identified as ‘breakdown of the Kármán vortex street’. Various other buoyancy effects about the heated/cooled cylinder are discussed by means of the flow patterns, the Nusselt number and the drag coefficient curves.