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Nearly all analytical models of lock-exchange flow are based on the shallow-water approximation. Since the latter approximation fails at the leading edges of the mutually intruding fluids of lock-exchange flow, solutions to the shallow-water equations can be obtained only through the specification of front conditions. In the present paper, analytic solutions to the shallow-water equations for non-Boussinesq lock-exchange flow are given for front conditions deriving from free-boundary arguments. Analytic solutions are also derived for other proposed front conditions – conditions which appear to the shallow-water system as forced boundary conditions. Both solutions to the shallow-water equations are compared with the numerical solutions of the Navier–Stokes equations and a mixture of successes and failures is recorded. The apparent success of some aspects of the forced solutions of the shallow-water equations, together with the fact that in a real fluid the density interface is a free boundary, shows the need for an improved theory of lock-exchange flow taking into account non-hydrostatic effects for density interfaces intersecting rigid boundaries.
We consider the dam-break initial stage of propagation of a gravity current released from a lock of length x0 and height h0 into an ambient fluid in a channel of height H*. The system contains heavy and light fluids, of densities ρH and ρL, respectively. When the Reynolds number is large, the resulting flow is governed by the parameters R = ρL/ρH and H = H*/h0. We focus attention on non-Boussinesq effects, when the parameter R is not close to 1; in this case significant differences appear between the ‘light’ (top surface) current and the ‘heavy’ (bottom) current. We use a shallow-water two-layer formulations. We show that ‘exact’ solutions of the thickness and speed of the current and ambient can be obtained by the method of characteristics. However, this requires a careful matching with the conditions at the front (ambient) and the back (reservoir). We show that a jump, instead of a rarefaction wave, propagates into the reservoir when H < Hcrit(R), and solutions for these jumps are presented and discussed. The theory is applied to the full-depth lock-exchange H = 1 problem, and the results are compared with previous hydraulic models. The application of the theory is also illustrated for more general cases H > 1, including comparisons with the one-layer model results of Ungarish (J. Fluid Mech., vol. 579, 2007, p. 373).
Overall, the shallow-water two-layer theory yields consistent, self-contained, and physically acceptable analytical solutions for the dam-break problem over the full physical range of the R and H parameters, for both light-into-heavy and heavy-into-light gravity currents. The solution can be closed without adjustable constants or predetermined properties of the flow field. The thickness solution is formally valid until the jump, or rarefaction wave, hits the backwall; the speed of propagation prediction is valid until this reflected wave hits the nose, i.e. until the end of the slumping stage. This theory is a significant extension of the Boussinesq problem (recovered by the present solution for R = 1), which elucidates the non-Boussinesq effects during the first stage of propagation of lock-released gravity currents.
Motivated by the problem of gravity segregation in an inclined porous layer, we present a theoretical analysis of interface evolution between two immiscible fluids of unequal density and mobility, both in two and three dimensions. Applying perturbation theory to the appropriately scaled problem, we derive the governing equations for the pressure and interface height to leading order, obtained in the limit of a thin gravity tongue and a slightly dipping bed. According to the zeroth-order approximation, the pressure profile perpendicular to the bed is in equilibrium, a widely accepted assumption for this class of problems. We show that for the inclined bed two-dimensional problem, in the reference frame moving with the mean gravity-induced advection velocity, the interface motion is dictated by a degenerate parabolic equation, different from those previously published. In this case, the late-time behaviour of the gravity tongue can be derived analytically through a formal expansion of both the solution and its two moving boundaries. In three dimensions, using a moving coordinate along the dip direction, we obtain an elliptic–parabolic system of partial differential equations where the fluid pressure and interface height are the two dependent variables. Although analytical results are not available for this case, the evolution of the gravity tongue can be investigated by numerical computations in only two spatial dimensions. The solution features are identified for different combinations of dimensionless parameters, showing their respective influence on the shape and motion of the interface.
A jet of granular material impinging on an inclined plane produces a diverse range of flows, from steady hydraulic jumps to periodic avalanches, self-channelised flows and pile collapse behaviour. We describe the various flow regimes and study in detail a steady-state flow, in which the jet generates a closed teardrop-shaped hydraulic jump on the plane, enclosing a region of fast-moving radial flow. On shallower slopes, a second steady regime exists in which the shock is not teardrop-shaped, but exhibits a more complex ‘blunted’ shape with a steadily breaking wave. We explain these regimes by consideration of the supercritical or subcritical nature of the flow surrounding the shock. A model is developed in which the impact of the jet on the inclined plane is treated as an inviscid flow, which is then coupled to a depth-integrated model for the resulting thin granular avalanche on the inclined plane. Numerical simulations produce a flow regime diagram strikingly similar to that obtained in experiments, with the model correctly reproducing the regimes and their dependence on the jet velocity and slope angle. The size and shape of the steady experimental shocks and the location of sub- and supercritical flow regions are also both accurately predicted. We find that the physics underlying the rapid flow inside the shock is dominated by depth-averaged mass and momentum transport, with granular friction, pressure gradients and three-dimensional aspects of the flow having comparatively little effect. Further downstream, the flow is governed by a friction–gravity balance, and some flow features, such as a persistent indentation in the free surface, are not reproduced in the numerical solutions. On planes inclined at a shallow angle, the effect of stationary granular material becomes important in the flow evolution, and oscillatory and more general time-dependent flows are observed. The hysteretic transition between static and dynamic friction leads to two phenomena observed in the flows: unsteady avalanching behaviour, and the feedback from static grains on the flowing region, leading to levéed, self-channelised flows.
An analytical theory is developed which illustrates the dynamics of the spontaneous generation of thermohaline intrusions in the stratified ocean with density compensated lateral temperature and salinity gradients. Intrusions in the model are driven by the interaction with the initially homogeneous field of salt fingers, whose amplitude and spatial orientation is weakly modulated by the long wavelength perturbations introduced into the system. The asymptotic multiscale analysis makes it possible to identify intrusive instabilities resulting from the positive feedback of salt fingers on large-scale perturbations and analyse the resulting patterns. The novelty of the proposed analysis is related to our ability to avoid using empirical double-diffusive flux laws – an approach taken by earlier models. Instead, we base our analytical explorations directly on the governing (Navier–Stokes) equations of motion. The model predictions of the growth rates and preferred slopes of intrusions are in general agreement with the laboratory and field measurements.
Euler's equations describe the dynamics of gravity waves on the surface of an ideal fluid with arbitrary depth. In this paper, we discuss the stability of periodic travelling wave solutions to the full set of nonlinear equations via a non-local formulation of the water wave problem, modified from that of Ablowitz, Fokas & Musslimani (J. Fluid Mech., vol. 562, 2006, p. 313), restricted to a one-dimensional surface. Transforming the non-local formulation to a travelling coordinate frame, we obtain a new formulation for the stationary solutions in the travelling reference frame as a single equation for the surface in physical coordinates. We demonstrate that this equation can be used to numerically determine non-trivial travelling wave solutions by exploiting the bifurcation structure of this new equation. Specifically, we use the continuous dependence of the amplitude of the solutions on their propagation speed. Finally, we numerically examine the spectral stability of the periodic travelling wave solutions by extending Fourier–Floquet analysis to apply to the associated linear non-local problem. In addition to presenting the full spectrum of this linear stability problem, we recover past well-known results such as the Benjamin–Feir instability for waves in deep water. In shallow water, we find different instabilities. These shallow water instabilities are critically related to the wavelength of the perturbation and are difficult to find numerically. To address this problem, we propose a strategy to estimate a priori the location in the complex plane of the eigenvalues associated with the instability.
Recently Stocchino & Brocchini (J. Fluid Mech., vol. 643, 2010, p. 425 have studied the dynamics of two-dimensional (2D) large-scale vortices with vertical axis evolving in a straight compound channel under quasi-uniform flow conditions. The mixing processes associated with such vortical structures are here analysed through the results of a dedicated experimental campaign. Time-resolved Eulerian surface velocity fields, measured using a 2D particle-image velocimetry system, form the basis for a Lagrangian analysis of the dispersive processes that occur in compound channels when the controlling physical parameters, i.e. the flow depth ratio (rh) and the Froude number (Fr) are changed. Lagrangian mixing is studied by means of various approaches based either on single-particle or multiple-particle statistics (relative and absolute statistics, probability density functions (p.d.f.s) of relative displacements and finite-scale Lyapunov exponents). Absolute statistics reveal that transitional macrovortices, typical of shallow flow conditions, strongly influence the growth in time of the total absolute dispersion, after the initial ballistic regime, leading to a non-monotonic behaviour. In deep flow conditions, on the contrary, the absolute dispersion displays a monotonic growth because the generation of transitional macrovortices does not take place. In all cases an asymptotic diffusive regime is reached.
Multiple-particle dynamics is controlled by rh and Fr. Different growth regimes of the relative diffusivity have been found depending on the flow conditions. This behaviour can be associated with different energy transfer processes and it is further confirmed by the p.d.f.s of relative displacements, which show a different asymptotical shape depending on the separation scales and the Froude number. Finally, an equilibrium regime is observed for all the experiments by analysing the decay of the finite-scale Lyapunov exponents with the particle separations.
Experiments are reported on the formation and migration of isolated dunes in a turbulent channel flow. These dunes have a very robust crescentic shape with horns pointing downstream, very similar to that of the barchan dunes observed in deserts at a much larger scale. Their main geometrical and dynamical properties are studied in detail, for four types of grains: the conditions for their formation, their morphology, the threshold shear stress for their motion, their velocity, erosion rate, minimum size and the longitudinal stripes of grains hollowed by fluid streaks in the boundary layer. In particular, the law for the dune velocity is found to involve two dimensionless parameters, the Shields number and the sedimentation Reynolds number, in contrast with predictions based on classical laws for particle transport. As the dune migrates, its size slowly decreases because of a small leakage of particles at the horn tips, and the erosion law is given. A minimum size is evidenced, which is shown to increase with the friction velocity and scale with a settling length.
The stability of oscillatory two-layer flows is investigated with a linear perturbation analysis. An asymptotic case is considered where the oscillation amplitude is small when compared to the perturbation wavelength. The focus of the analysis is on the influence of viscosity and its contrast at the interface. The flows are unstable when the relative velocity of the layers is larger than a critical value. Depending on the oscillation frequency, the flows are in different dynamical regimes, which are characterized by the relative importance of the capillary wavelength and the thicknesses of the Stokes boundary layers developed on the interface. A particular regime is found in which instability occurs at a substantially lower critical velocity. The mechanism behind the instability is studied by identifying the velocity- and shear-induced components in the disturbance growth rate. They interchange dominance depending on the frequency and the viscosity contrast. Results of the analysis are compared with the experiments in the literature. Good agreement is found with the experiments that have a small oscillation amplitude. The validity condition of the asymptotic theory is estimated.
The stability of two-layer oscillatory flows was studied experimentally in a cylindrical container with a vertical axis. Two superposed immiscible liquids, differing greatly in viscosity, were set in relative oscillatory motion by alternating container rotation. Waves arising beyond a threshold were observed in detail for small oscillation frequencies ranging from 0.1 to 6 Hz. Measurements were performed on the growth rate and the wavenumber of these waves. The instability threshold was determined from the growth rate data. It was found that the threshold and the wavenumber varied with the frequency. In particular, significantly lower thresholds and longer waves were found than those predicted by the inviscid theory of the oscillatory Kelvin–Helmholtz instability. Favourable agreement with the predictions of an existing viscous theory for small oscillation amplitude flows indicates the important role of viscosity, even at the highest frequency, and suggests a similar mechanism behind the instability as that for the short wave instability in steady Couette flows. A semi-numerical stability determination for finite amplitude flows was also performed to improve the prediction in experiments with a frequency lower than 1 Hz.
Typical approaches to manipulation of flow separation employ passive means or active techniques such as blowing and suction or plasma acceleration. Here it is demonstrated that the flow can be significantly altered by making small changes to the shape of the surface. A proof of concept experiment is performed using a very simple time-dependent perturbation to the surface of a sphere: a roughness element of 1% of the sphere diameter is moved azimuthally around a sphere surface upstream of the uncontrolled laminar separation point, with a rotational frequency as large as the vortex shedding frequency. A key finding is that the non-dimensional time to observe a large effect on the lateral force due to the perturbation produced in the sphere boundary layers as the roughness moves along the surface is
A particle mesh Ewald (PME) Stokesian dynamics algorithm has been developed to model hydrodynamic interactions in suspensions of non-spherical dicolloidal particles. Dicolloids, which have recently been synthesized by a number of independent research groups (Johnson, van Kats & van Blaaderen (Langmuir, vol. 21, 2005, p. 11510), Mock et al. (Langmuir, vol. 22, 2006, p. 4037), Kim, Larsen & Weitz (J. Am. Chem. Soc., vol. 128, 2006, p. 14374)), consist of two intersecting spheres of varying radii and centre-to-centre separation. One-body resistance tensors and disturbance velocity fields are computed for general linear flows using a superposition of Stokes singularities along the symmetry axis of the dicolloid particles. The coefficients and the locations of the singularities are optimized to minimize the norm of the velocity error on the particle surface. The one-body solution provides all coefficients required for the far-field many-body interactions in the Stokesian dynamics algorithm. These generalize the analytical results for spheres employed in the classic algorithm. Modified lubrication interaction tensors are developed for dicolloids for the singular near-field lubrication interactions. Accuracy of the one-body solutions and two-body generalized Stokesian dynamics solutions are validated by comparison with high-precision numerical solutions computed with the spectral boundary element method of Muldowney & Higdon (J. Fluid Mech., vol. 298, 1995, p. 167). The newly developed PME Stokesian dynamics algorithm was used to study transport properties in dicolloidal suspensions over a range of volume fractions (φ ≤ 0.5). The effects of the degree of anisotropy on the properties of the suspension are discussed. For these mildly anisotropic particles, the transport properties remain close to those of spheres, however certain interesting trends emerge, with non-monotonic viscosity dependence as a function of increasing aspect ratio. The minimum viscosity in concentrated suspensions is lower than that for spheres with equal volume fraction over a range of volume fractions.
Turbulent mixing layers emanating from slanted trailing edges or nozzles evolve in a manner that is explainable by applying the independence principle to boundary layer flows. Although measurements downstream of a planar chevron splitter plate validate the concept, the intent of this short article is to re-examine the broader ramifications of this observation. Turbulent boundary layer growth on a yawed flat plate is re-examined as is the attached flow direction near the trailing edge of a highly swept-back wing.
We present experimental results for the collapse of rectangular columns of sand down rough, inclined, parallel-walled channels. Results for basal inclination θ varying between 4.2° and 25° are compared with previous results for horizontal channels. Shallow-water theory can be usefully combined with scaling relationships obtained by dimensional analysis to yield analytical functions of the maximum runout distance, the maximum deposit height and the time to reach the maximum runout. While the theory excellently predicts the maximum lengths of the deposit it generally overestimates the runout time. The inertial flows are characterized by a moving internal interface separating upper flowing and lower static regions of material. In an initial free-fall phase of collapse the deposited area (= volume per unit width) below the internal interface varies with the square-root of time, independent of the initial height of the column and channel inclination. In the subsequent, lateral spreading phase the deposition rate decreases with increasing basal inclination or with decreasing initial height. The local deposition rate at any fixed distance is a constant, dependent on the column aspect ratio, the channel inclination and the longitudinal position, but invariant with flow velocity and depth. In the lateral spreading phase, vertical velocity profile in the flowing layer take a universal form and are independent of flow depth and velocity. They can be characterized by a shear rate as a function of channel inclination and a length scale describing the fraction of the column involved in flow.
Linear three-dimensional modal instability of steady laminar two-dimensional states developing in a lid-driven cavity of isosceles triangular cross-section is investigated theoretically and experimentally for the case in which the equal sides form a rectangular corner. An asymmetric steady two-dimensional motion is driven by the steady motion of one of the equal sides. If the side moves away from the rectangular corner, a stationary three-dimensional instability is found. If the motion is directed towards the corner, the instability is oscillatory. The respective critical Reynolds numbers are identified both theoretically and experimentally. The neutral curves pertinent to the two configurations and the properties of the respective leading eigenmodes are documented and analogies to instabilities in rectangular lid-driven cavities are discussed.
The global stability of confined uniform density wakes is studied numerically, using two-dimensional linear global modes and nonlinear direct numerical simulations. The wake inflow velocity is varied between different amounts of co-flow (base bleed). In accordance with previous studies, we find that the frequencies of both the most unstable linear and the saturated nonlinear global mode increase with confinement. For wake Reynolds number Re = 100 we find the confinement to be stabilising, decreasing the growth rate of the linear and the saturation amplitude of the nonlinear modes. The dampening effect is connected to the streamwise development of the base flow, and decreases for more parallel flows at higher Re. The linear analysis reveals that the critical wake velocities are almost identical for unconfined and confined wakes at Re ≈ 400. Further, the results are compared with literature data for an inviscid parallel wake. The confined wake is found to be more stable than its inviscid counterpart, whereas the unconfined wake is more unstable than the inviscid wake. The main reason for both is the base flow development. A detailed comparison of the linear and nonlinear results reveals that the most unstable linear global mode gives in all cases an excellent prediction of the initial nonlinear behaviour and therefore the stability boundary. However, the nonlinear saturated state is different, mainly for higher Re. For Re = 100, the saturated frequency differs less than 5% from the linear frequency, and trends regarding confinement observed in the linear analysis are confirmed.
The group vaporization of a monodisperse fuel-spray jet discharging into a hot coflowing gaseous stream is investigated for steady flow by numerical and asymptotic methods with a two-continua formulation used for the description of the gas and liquid phases. The jet is assumed to be slender and laminar, as occurs when the Reynolds number is moderately large, so that the boundary-layer form of the conservation equations can be employed in the analysis. Two dimensionless parameters are found to control the flow structure, namely the spray dilution parameter λ, defined as the mass of liquid fuel per unit mass of gas in the spray stream, and the group vaporization parameter ϵ, defined as the ratio of the characteristic time of spray evolution due to droplet vaporization to the characteristic diffusion time across the jet. It is observed that, for the small values of ϵ often encountered in applications, vaporization occurs only in a thin layer separating the spray from the outer droplet-free stream. This regime of sheath vaporization, which is controlled by heat conduction, is amenable to a simplified asymptotic description, independent of ϵ, in which the location of the vaporization layer is determined numerically as a free boundary in a parabolic problem involving matching of the separate solutions in the external streams, with appropriate jump conditions obtained from analysis of the quasi-steady vaporization front. Separate consideration of dilute and dense sprays, corresponding, respectively, to the asymptotic limits λ ≪ 1 and λ ≫ 1, enables simplified descriptions to be obtained for the different flow variables, including explicit analytic expressions for the spray penetration distance.
We address the challenge of optimal incompressible stirring to mix an initially inhomogeneous distribution of passive tracers. As a quantitative measure of mixing we adopt the H−1 norm of the scalar fluctuation field, equivalent to the (square root of the) variance of a low-pass filtered image of the tracer concentration field. First we establish that this is a useful gauge even in the absence of molecular diffusion: its vanishing as t → ∞ is evidence of the stirring flow's mixing properties in the sense of ergodic theory. Then we derive absolute limits on the total amount of mixing, as a function of time, on a periodic spatial domain with a prescribed instantaneous stirring energy or stirring power budget. We subsequently determine the flow field that instantaneously maximizes the decay of this mixing measure – when such a flow exists. When no such ‘steepest descent’ flow exists (a possible but non-generic situation), we determine the flow that maximizes the growth rate of the H−1 norm's decay rate. This local-in-time optimal stirring strategy is implemented numerically on a benchmark problem and compared to an optimal control approach using a restricted set of flows. Some significant challenges for analysis are outlined.
We investigate the nonlinear wave signature of a translating and oscillating disturbance under the influence of ambient waves in a two-layer fluid. The main interests are the generation and features of the far-field waves due to nonlinear wave resonances. We show, using perturbation theory, that free waves on the surface and/or interface can be produced by triad-resonant interactions, a mechanism not obtained in a homogeneous fluid. These occur among the radiated waves due to the disturbance motion (disturbance waves); and between the disturbance waves and free ocean waves (ambient waves). Such resonance-generated waves can appear upstream or downstream, and may propagate away from or towards the disturbance. In realistic situations where ambient waves and disturbance oscillations contain multiple frequencies, numerous resonant and near-resonant interactions at second and higher orders may occur, making the theoretical analysis of the problem intractable. For this purpose, we develop a direct simulation capability using a high-order spectral method, which provides independent validation of the theoretical predictions. Our investigations show that, under specific but realistic conditions, resonance interactions may lead to significant far-field short waves that are more amenable to remote sensing. If the characteristics of the disturbance are known, we illustrate how nonlinear wave resonances provide a mechanism for more precise estimation of ocean stratification properties using surface wave measurements. Finally we show that when a moving disturbance oscillates at multiple frequencies, ensuing multiple resonances may lead to energy spreading across a broader spectrum, resulting in the loss of information about the body motion.
Inspired by the correlation between the propulsion efficiency of a flapping foil propeller and stability of the wake behind it (which leads to the optimal Strouhal number for propulsion), we numerically simulated a heaving/pitching foil in energy harvesting regime, and investigated the relation between wake stability and the energy harvesting efficiency. The base flow is computed using a Navier–Stokes algorithm and the stability analysis is performed via the Orr–Sommerfeld equation. The wake is found to be convectively unstable and the frequency of the most unstable mode fw is determined. The case when fw ~ f coincides with maximum energy harvesting efficiency of the system (f is the frequency of foil oscillation), suggesting that flow energy extraction is closely related to efficient evolution of the wake. This occurs at a frequency of f ~ 0.15 (f is normalized by the chord length and the flow speed), under the constraint that there is significant vortex shedding from the leading edge at sufficiently large effective angles of attack. Indeed, this ‘foil–wake resonance’ is usually associated with multi-vortex shedding from the leading edge. Furthermore, detailed examination of energy extractions from the heaving and the pitching motions indicates that near the optimal performance point the average energy extraction from the pitching motion is close to zero. This suggests the feasibility of achieving high-efficient energy harvesting through a simple fully passive system we proposed earlier in which no activation is needed.