Showing papers in "European Journal of Mechanics B-fluids in 2008"

Global two-dimensional stability measures of the flat plate boundary-layer flow

Abstract: The stability of the two-dimensional flat plate boundary-layer is studied by means of global eigenmodes. These eigenmodes depend both on the streamwise and wall-normal coordinate, hence there are no assumptions on the streamwise length scales of the disturbances. Expanding the perturbation velocity field in the basis of eigenmodes yields a reduced order model from which the stability characteristics of the flow, i.e. the initial condition and forcing function leading to the largest energy growth, are extracted by means of non-modal analysis. In this paper we show that, even when performing stability analysis using global eigenmodes, it is not sufficient to consider only a few of the least damped seemingly relevant eigenmodes. Instead it is the task of the optimization procedure, inherent in the non-modal analysis, to decide which eigenmodes are relevant. We show that both the optimal initial condition and the optimal forcing structure have the form of upstream tilted structures. Time integration reveals that these structures gain energy through the so called Orr mechanism, where the instabilities extract energy from the mean shear. This provides the optimal way of initiating Tollmien–Schlichting waves in the boundary layer. The optimal initial condition results in a localized Tollmien–Schlichting wavepacket that propagates downstream, whereas the optimal forcing results in a persistent Tollmien–Schlichting wave train.

Large-amplitude steady rotational water waves

Abstract: Two-dimensional, finite-depth periodic water waves with general vorticity and large amplitude are computed The mathematical formulation and numerical method that allow us to compute a continuum of such waves with arbitrary vorticity are described The problems of whether extreme waves exist, where their stagnation points occur, and what qualitative features such waves possess are addressed here with particular emphasis on constant vorticity

Gas flow through an elliptical tube over the whole range of the gas rarefaction

Abstract: A rarefied gas flow trough a long tube with an elliptical cross section is studied on the basis of the BGK kinetic model equation in the whole range of the Knudsen number varying from the free molecular regime to the hydrodynamic one. A wide range of the aspect ratio is considered. The mass flow rate is calculated as a function of the pressures on the tube ends.

Direct Numerical Simulation of turbulent Taylor-Couette flow

Abstract: The direct numerical simulation (DNS) of the Taylor–Couette flow in the fully turbulent regime is described. The numerical method extends the work by Quadrio and Luchini [M. Quadrio, P. Luchini, Eur. J. Mech. B/Fluids 21 (2002) 413–427], and is based on a parallel computer code which uses mixed spatial discretization (spectral schemes in the homogeneous directions, and fourth-order, compact explicit finite-difference schemes in the radial direction). A DNS is carried out to simulate for the first time the turbulent Taylor–Couette flow in the turbulent regime. Statistical quantities are computed to complement the existing experimental information, with a view to compare it to planar, pressure-driven turbulent flow at the same value of the Reynolds number. The main source for differences in flow statistics between plane and curved-wall flows is attributed to the presence of large-scale rotating structures generated by curvature effects.

Similarity stagnation point solutions of the Navier-Stokes equations - review and extension

Abstract: Stagnation regions exist on all blunt bodies moving in a viscous fluid. In certain stagnation flow problems, the Navier–Stokes equations reduce to nonlinear ordinary differential equations through a similarity transform. This paper reviews the existing steady similarity stagnation flow solutions and discusses a new area of research, stagnation flow with slip.

Double layer overlap in ac electroosmosis

Abstract: The frequency-dependent flow of electrolytes between pairs of parallel plate micro-electrodes is analyzed in this paper, for the cases in which electric double layers formed in vicinity of the solid boundaries may strongly interact with each other. Closed form expressions for the potential distributions are first developed under certain simplifying assumptions, depicting the interactions between the oscillating electric field and charge density distribution within the double layer. It is revealed that the impact of double layer overlap on ac electroosmotic flows turns out to be more predominant at frequencies of the order of relaxation frequency of the electrode–electrolyte system. At higher frequencies, potential drop across the double layer tends to zero, due to polarization of the electrode-solution interface, and virtually no electroosmotic flows can be obtained in such cases.

Rarefied gas flow in a triangular duct based on a boundary fitted lattice

Abstract: The rarefied fully developed flow of a gas through a duct of a triangular cross section is solved in the whole range of the Knudsen number. The flow is modelled by the BGK kinetic equation, subject to Maxwell diffuse boundary conditions. The numerical solution is based on the discrete velocity method, which is applied for first time on a triangular lattice in the physical space. The boundaries of the flow and computational domains are identical deducing accurate results with modest computational effort. Results on the velocity profiles and on the flow rates for ducts of various triangular cross sections are reported and they are valid in the whole range of gas rarefaction. Their accuracy is validated in several ways, including the recovery of the analytical solutions at the free molecular and hydrodynamic limits. The successful implementation of the triangular grid elements is promising for generalizing kinetic type solutions to rarefied flows in domains with complex boundaries using adaptive and unstructured grids.

Rarefied gas flow in concentric annular tube : Estimation of the Poiseuille number and the exact hydraulic diameter

Abstract: The fully developed flow of rarefied gases through circular ducts of concentric annular cross sections is solved via kinetic theory. The flow is due to an externally imposed pressure gradient in the longitudinal direction and it is simulated by the BGK kinetic equation, subject to Maxwell diffuse-specular boundary conditions. The approximate principal of the hydraulic diameter is investigated for first time in the field of rarefied gas dynamics. For the specific flow pattern, in addition to the flow rates, results are reported for the Poiseuille number and the exact hydraulic diameter. The corresponding parameters include the whole range of the Knudsen number and various values of the accommodation coefficient and the ratio of the inner over the outer radius. The accuracy of the results is validated in several ways, including the recovery of the analytical solutions at the hydrodynamic and free molecular limits.

Database for flows of binary gas mixtures through a plane microchannel

Abstract: A binary mixture of rarefied gases between two parallel plates is considered. The Poiseuille flow, thermal transpiration (flow caused by a temperature gradient of the plates) and concentration-driven flow (flow caused by a gradient of concentration of the component species) are analyzed on the basis of the linearized model Boltzmann equation with the diffuse reflection boundary condition. The analyses are first performed for mixtures of virtual gases composed of the hard-sphere or Maxwell molecules and the results are compared with those of the original Boltzmann equation. Then, the analyses for noble gases (He–Ne, He–Ar and Ne–Ar) are performed assuming more realistic molecular models (the inverse power-law potential and Lennard-Jones 12,6 models). By use of the results, flux databases covering the entire ranges of the Knudsen number and of the concentration and a wide range of the temperature are constructed. The databases are prepared for the use in the fluid-dynamic model for mixtures in a stationary nonisothermal microchannel derived in [S. Takata, H. Sugimoto, S. Kosuge, Eur. J. Mech. B/Fluids 26 (2007) 155], but can also be incorporated in the generalized Reynolds equation [S. Fukui, R. Kaneko, J. Tribol. 110 (1988) 253] in the gas film lubrication theory. The databases constructed can be downloaded freely from Electronic Annex 2 in the online version of this article.

Particle grouping in oscillating flows

Abstract: An equation describing the dynamics of spherical particles in an oscillating Stokesian flow in the frame of reference moving with the phase velocity of the wave, and only taking into account the contribution of the drag force, is simplified in two limiting cases. Firstly, the case when Stokes numbers are small is considered. Secondly, the analysis focuses on the case when the initial location of the particles is close to the location where the particles are grouped (their velocities and accelerations in the wave frame of reference are equal to zero), x lim . This is followed by an analysis of the dynamics of non-Stokesian particles. In all cases, the analytical results are validated against the results of numerical solution of the equation of particle motion. Three types of trajectories are predicted when particles approach x lim : the trajectories describing the monotonic approach to x lim , the trajectories describing the approach to x lim with oscillations and trajectories repelled from x lim . These are identified with stable nodes, stable foci and saddles. The trajectories in the zone between stable nodes and foci are identified as stable stars. Using Dulac's criterion, it is pointed out that none of the particle trajectories in the position–velocity plane can be closed. This result is illustrated by the trajectories calculated using the numerical solution of the equation for particle dynamics for various parameter values.

An improved method for calculation of interface pressure force in PLIC-VOF methods

Abstract: A method for the application of interface force in the computational modeling of free surfaces and interfaces which uses PLIC-VOF methods is developed. This method is based on evaluation of the surface tension force only in the interfacial cells with out using the neighboring cells. The normal and the interface surface area needed for the calculation of the surface tension force are calculated more accurately. This method is applied on a staggered grid and it is referred to as Staggered Grid Interfere Pressure calculation method or SGIP. The present method is applied to a two-dimensional motionless liquid drop and a gas bubble. In addition, oscillations of a non-circular two-dimensional drop and a bubble due only to the surface tension forces are considered. It is shown that the new method predicts the pressure jump at the interface more accurately and produces less spurious currents compared to CSF, CSS and Meier's methods when applied to the same cases.

Variable density and viscosity, miscible displacements in capillary tubes

Abstract: The displacement of a more viscous fluid by a miscible, less viscous one of lower density in a horizontal capillary tube is studied by means of Stokes flow simulations. Both axisymmetric and three-dimensional simulations are conducted at Peclet numbers up to 104, in order to resolve discrepancies between earlier simulations by [C.Y. Chen, E. Meiburg, Miscible displacements in capillary tubes. Part 2. Numerical simulations, J. Fluid Mech. 326 (1996) 57] and corresponding experiments of [P. Petitjeans, T. Maxworthy, Miscible displacements in capillary tubes. Part 1. Experiments, J. Fluid Mech. 326 (1996) 37] and [J. Kuang, T. Maxworthy, P. Petitjeans, Miscible displacements between silicone oils in capillary tubes, Eur. J. Mech. B Fluids 22 (2003) 271–277]. An initial set of simulations addresses the influence of different viscosity–concentration relations on the quasisteady finger tip velocity. The results indicate that steeper relations generally result in a higher tip velocity. However, the effect is too small to explain the above discrepancies. Further simulations show that a concentration-dependent diffusion coefficient results in a slight reduction of the tip velocity at moderate Pe, but again the effect is too small to fully account for the observed differences. Three-dimensional simulations that include gravitational forces yield a much more significant effect. Consistent with the experiments of [P. Petitjeans, T. Maxworthy, Miscible displacements in capillary tubes. Part 1. Experiments, J. Fluid Mech. 326 (1996) 37], at moderate Pe the tip slows down as the gravity parameter increases, an effect that becomes more pronounced as Pe decreases. However, the three-dimensional simulations do not produce the longitudinal splitting phenomenon observed by [P. Petitjeans, T. Maxworthy, Miscible displacements in capillary tubes. Part 1. Experiments, J. Fluid Mech. 326 (1996) 37]. In order to check for the existence of gravitational instabilities that might cause such a splitting, additional two-dimensional simulations are conducted in cross-sections of the tube. A comparison of these two-dimensional results with corresponding three-dimensional simulations demonstrates that for a wide range of parameters the evolution of the trailing finger sections is governed by a two-dimensional balance between gravitational and viscous forces. However, a gravitational instability along the lines suggested by [P. Petitjeans, T. Maxworthy, Miscible displacements in capillary tubes. Part 1. Experiments, J. Fluid Mech. 326 (1996) 37] was not observed. On the other hand, for some parameter combinations the evolution of a 'dimple' is observed on the lower side of the finger, and close to its tip. This dimple may signal the evolution of a splitting phenomenon after long times, which are beyond the reach of the current simulations. Taken together, the two- and three-dimensional simulations suggest that the splitting phenomenon observed by [P. Petitjeans, T. Maxworthy, Miscible displacements in capillary tubes. Part 1. Experiments, J. Fluid Mech. 326 (1996) 37] likely is caused by the gravity-induced modification of the flow around the tip of the finger, rather than by a gravitational instability per se.

Solving for unsteady airflow in a glottal model with immersed moving boundaries

Abstract: This work builds on previous efforts to characterize the dynamic development of the airflow in the glottis from a fluid mechanical point of view. A multigrid finite-difference method with immersed boundaries is implemented to solve the Navier–Stokes equations in a channel constricted by a vibrating rigid structure with a shape conforming to the human vocal folds. For the dynamically evolving boundaries we apply a forced oscillation glottal model. The large scale deformations of the boundaries are handled without regridding and tracheal input velocity is either set to a constant value or synchronized with wall motion. Particular attention is paid to the mobility of the point where the airflow detaches from the flapping walls. Results illustrate the relevance and the diversity of flow separation dynamics within the constriction standing for the glottis, while flow instabilities past the constriction are not found to affect flow behavior between the moving walls significantly. A comparison between static and dynamic numerical experiments show that mobility of the flow separation point is nontrivial in general and only rarely quasi-static.

Stability of the laminar boundary layer flow encountering a row of roughness elements: Biglobal stability approach and DNS

Abstract: The stability of the laminar boundary layer developing on a flat plate in the presence of a periodic row of roughness elements is investigated. A Direct Numerical Simulation is performed to compute the steady flow downstream of the roughness elements, which contains a pair of two counter-rotating streamwise vortices per element, which can be considered as a “pre-streaky” structure. The linear stability of this base flow is analyzed by means of the so-called “biglobal” stability approach. Three-dimensional eigenmodes are found, which are shown to be the continuation of the Tollmien–Schlichting waves present in the case of an unperturbed boundary layer. Moreover, a stabilizing effect due to the roughness-induced vortices is found. A Direct Numerical Simulation of the interaction between a two-dimensional Tollmien–Schlichting wave and the roughness array is also performed. The computed perturbation traveling downstream of the roughness elements is shown to be a linear combination of the biglobal eigenmodes.

Numerical solutions of the Boltzmann equation: comparison of different algorithms

Abstract: In the paper we compare different algorithms for numerical solutions of the Boltzmann equation. For this comparison we have taken the standard problem of the shock wave structure in a mono-atomic rarefied gas. Different parameters characterizing the shock structure have been calculated by a Monte Carlo simulation (DSMC), a second order time-relaxed Monte Carlo method (TRMC2), a fully deterministic discrete velocity method (DV), a discrete velocity method with Monte Carlo calculations of collision integral (DVMC) and a molecular dynamics method (MD). Results of these calculations have been compared with the shock wave structure obtained in experiments in a shock tube. The results of the comparison are not conclusive. We have observed general agreement between numerical and experimental results but there is no single numerical method which fits best to the experimental measurements.

Influence of pitching angle of incidence on the dynamic stall behavior of a symmetric airfoil

Abstract: This study presents the influence of pitch angle of an airfoil on its near-field vortex structure as well as the aerodynamic loads during a dynamic stall process. Dynamic stall behavior in a sinusoidally pitching airfoil is usually analyzed at low to medium reduced frequencies and with the maximum angle of attack of the airfoil not exceeding 25°. In this work, we study dynamic stall of a symmetric airfoil at medium to high reduced frequencies even as the maximum angle of attack goes from 25° to 45°. The evolution and growth of the laminar separation bubble, also known as a dynamic stall vortex, at the leading edge and the trailing edge are studied as the pitch cycle goes from the minimum to the maximum angle of attack. The effect of reduced frequencies on the vortex structure as well as the aerodynamic load coefficients is investigated. The reduced frequency is shown to be a bifurcation parameter triggering period doubling behavior. However, the bifurcation pattern is dependent on the variation of the pitch angle of incidence of the airfoil.

Viscous boundary layers in flows through a domain with permeable boundary

Abstract: The effect of small viscosity on nearly inviscid flows of an incompressible fluid through a given domain with permeable boundary is studied. The Vishik–Lyusternik method is applied to construct a boundary layer asymptotic at the outlet in the limit of vanishing viscosity. Mathematical problems with both consistent and inconsistent initial and boundary conditions at the outlet are considered. It is shown that in the former case, the viscosity leads to a boundary layer only at the outlet. In the latter case, in the leading term of the expansion there is a boundary layer at the outlet and there is no boundary layer at the inlet, but in higher order terms another boundary layer appears at the inlet. To verify the validity of the expansion, a number of simple examples are presented. The examples demonstrate that asymptotic solutions are in quite good agreement with exact or numerical solutions.

A family of new solutions on the wall jet

Abstract: The fluid flow problem within the wall jet created by fluid hitting on a solid surface at the right angle is solved based on the homotopy analysis method (HAM). A new family of solutions for jet with injection/suction which has been overlooked so far are obtained. Numerical evidence seems to suggest that these solutions decay algebraically far away from the wall. © 2007 Elsevier Masson SAS. All rights reserved.

Continuum analytical modelling of thermal creep

Abstract: A thermal creep process is studied in quite wide rectangular micro channels, sufficiently wide so that it is possible to consider this configuration as two parallel plates. The inlet and outlet reservoirs are maintained at the same constant pressure. A constant temperature gradient exists along the walls of the channel joining the two tanks. Thus a gas flow is induced and thermally sustained until steady conditions are reached. A complete analytical solution is derived in slip regime, yielding all the flow parameters, for Knudsen numbers smaller than 0.25. The analytical results are in good agreement with the numerical “exact” solution of the continuum equation system. Furthermore our continuum approach data are compared to those deduced from approaches based on Boltzmann equation model treatments: these various methods lead generally to a satisfactory agreement between their respective mean parameters. Nevertheless significant differences appear on the transversal velocity profiles and are further discussed.

Simulation of an optically induced asymmetric deformation of a liquid–liquid interface

Abstract: Deformations of liquid interfaces by the optical radiation pressure of a focused laser wave were generally expected to display similar behavior, whatever the direction of propagation of the incident beam. Recent experiments showed that the invariance of interface deformations with respect to the direction of propagation of the incident wave is broken at high laser intensities. In the case of a beam propagating from the liquid of smaller refractive index to that of larger one, the interface remains stable, forming a nipple-like shape, while for the opposite direction of propagation, an instability occurs, leading to a long needle-like deformation emitting micro-droplets. While an analytical model successfully predicts the equilibrium shape of weakly deformed interface, very few work has been accomplished in the regime of large interface deformations. In this work, we use the Boundary Integral Element Method (BIEM) to compute the evolution of the shape of a fluid-fluid interface under the effect of a continuous laser wave, and we compare our numerical simulations to experimental data in the regime of large deformations for both upward and downward beam propagation. We confirm the invariance breakdown observed experimentally and find good agreement between predicted and experimental interface hump heights below the instability threshold.

Catastrophic transition to instability of evaporation front in a porous medium

Abstract: Instability of a water layer located over an air-vapor layer in a horizontally infinite two dimensional domains of a porous medium is considered. A new mechanism of transition to instability of vertical flows developed in such a system is treated when the most unstable normal mode is affiliated with the zero wave number. Secondary structures bifurcating from the vertical base flow in a neighborhood of the threshold of instability obey the Kolmogorov–Petrovsky–Piscounov (KPP) diffusion-type equation. For the transition in question the KPP equation represents the analogue of the Ginzburg–Landau equation for the transition when the most unstable mode has a nonzero wave number. It is shown that in some neighborhood of the critical parameters there exist two different plane phase transition interfaces coinciding at the threshold of instability and ceasing to exist when the threshold is overcome. One of these interfaces is unstable, whereas the other is stable. It is shown nevertheless, that even the stable interface is destroyed by some perturbations of the unstable one due to nonlinear interplay of disturbances.

Short crested wave-current forces around a large vertical circular cylinder

Abstract: An analytical solution for the diffraction of short crested incident wave along positive x-axis direction on a large circular cylinder with uniform current is derived. The important influences of currents on wave frequency, water run-up, wave force, inertia and drag coefficients on the cylinder profiles are investigated for short-crested incident wave. Based on the numerical results, we find wave frequency of short crested wave system is affected by incident angle and the strength of the currents. The wave frequency increases or decreases with increasing current speed following or opposing wave propagating direction. It shows that the effects of current speeds, current directions on water run-up on the circular cylinder with different radius for different wave numbers are very conspicuous when the incident wave changes from long crested plane waves to short-crested waves. With the increase of current speed, the water run-up on the cylinder becomes more and more high, and will exceed that of long crested plane wave and short crested wave case without currents even though the current speed is small. The total wave loads, inertia coefficient and drag coefficient exerted on a cylinder with currents would be larger compared to the wave loads exerted pure short-crested waves. Therefore, ocean engineers should consider the short crested wave–current load on marine constructs carefully.

A numerical study of flows driven by a rotating magnetic field in a square container

Abstract: The magnetically induced fluid flow in a square container is investigated by means of numerical simulations. Low frequency/ low induction conditions are assumed. The effect of the rotating magnetic field gives rise to a time-independent magnetic body force, computed via the electrical potential equation and Ohm's law and a time-dependent part that is neglected due to the low-interaction parameter. The magnetic body force calculation is verified successfully by comparison with the exact solution. The behavior of the fluid flow in the square container reveals similar features to the flow in the cylindrical container, for instance, in the dependence on the intensities of the magnetic field. However, we did find differences in the velocity field distribution. Particularly, in the finite as well as infinite geometry, the velocity field is influenced by the corner of the container and remains non-axisymmetric in a wide range of Taylor numbers.

Asymptotic theory of an oscillating wing section in weak ground effect

Abstract: This study addresses unsteady aerodynamic forces acting on a wing section oscillating in a steady incompressible (and inviscid) uniform flow in presence of a distant flat ground. Three fundamental dimensionless parameters characterize the magnitude of those forces: the ratio δ of the wing transversal displacement to its chord, the ratio e of the wing chord to its average distance from the ground, and the ratio k of the wing chord to the distance traveled by the flow during one oscillation period. With the first two serving as small parameters, asymptotic series of the form δ f 0 ( k ) + δ e 2 f 1 ( k ) + δ e 2 g ( k / e ) f 2 ( k ) + ⋯ have been constructed for the wing lift and pitching moment. In the case of heave oscillations, three-terms-series for the lift fits nicely the available numerical data for wide range of δ , e and k values.

Conformal mapping and efficient boundary element method without boundary elements for fast vortex particle simulations

Abstract: In this paper, a revitalization of conformal mapping methods applied to fluid flows in two dimensions is proposed. The present work addresses several important issues concerning their application for vortex particle flow solvers. Difficulties of past conformal based method are reviewed. One difficulty concerns the ability of a mapping procedure to represent complicated shapes. The present paper improves past algorithms to be able to map new shapes, including multiply connected domains. A new fast procedure allows transferring a set of points in the mapped simplified plane to the complicated domain and vice versa. After a mapping construction, it is demonstrated how basic exact solutions to potential flow problems with vortices can be put in a new form which provides a faster and more accurate computation than with distributed singularity methods.

Simulation of particles in fluid: a two-dimensional benchmark for a cylinder settling in a wall-bounded box

Abstract: This paper presents numerical experiments inspired by the theoretical work of Faxen for predicting the terminal velocity of a cylinder, settling halfway between two parallel walls at low Reynolds numbers. It is demonstrated that unexpected correlations exist between Faxen's results and the relaxation of a rigid disk initially suspended in a wall-bounded square box. To this end, the 1-Fluid (1F) method is used within a frame of Direct Numerical Simulation (DNS). In first place, the assessment of 1F method in two dimensions is presented. Simulations are in good agreement with Faxen's approach in half-bounded domains, and with simulation data from literature as well. Numerical experiments are then designed in order to investigate the transient behavior of a circular disk in a wall-bounded square box. Significant ranges of particle-to-wall containment ratios, density ratios and Galileo numbers were used in simulations. In the case where the aspect ratio belongs to the range [0.005,0.4] and the Galileo number is smaller than 1, it is found that the wall correction factor based on the maximum settling velocity could be correlated directly with the Faxen's correction factor based on the terminal settling velocity. For extreme values of containment, Faxen's theory gives irrelevant predictions, and alternative approaches based on 1F simulations are suggested. Finally, an original benchmark is designed as an efficient and inexpensive tool for validating numerical approaches to fluid/particle systems.

The reservoir technique: a way to make Godunov-type schemes zero or very low diffuse. Application to Colella–Glaz solver

Abstract: Although it is commonly thought that first order schemes are not accurate enough to approximate nonlinear hyperbolic problems, we here explore a conservative time integration with global time steps but local updates (see [F. Alouges, F. De Vuyst, G. Le Coq, E. Lorin, Un procede de reduction de la diffusion numerique des schemas a difference de flux d'ordre un pour les systemes hyperboliques non lineaires, C. R. Math. Acad. Sci. Paris, Ser. I 335 (7) (2002) 627–632. [1] ]; [F. Alouges, F. De Vuyst, G. Le Coq, E. Lorin, The reservoir scheme for systems of conservation laws, in: Finite Volumes for Complex Applications, III, Porquerolles, 2002, Lab. Anal. Topol. Probab. CNRS, Marseille, 2002, pp. 247–254 (electronic). [2] ]). This overall conservative method can be interpreted as a system of reservoirs at cell interfaces that fill up and empty when local CFL conditions are reached. For Euler equations, particularly good results are obtained when one uses this technique together with the Riemann solver proposed by Colella and Glaz.

Supercritical surface gravity waves generated by a positive forcing

Abstract: Forced surface waves on an incompressible, inviscid fluid in a two-dimensional channel with a small bump on a horizontal rigid flat bottom are studied. The wave motion on the free surface is determined by a nondimensional wave speed F, called Froude number, and F = 1 is a critical value of F. If F = 1 + λ ϵ with ϵ > 0 a small parameter, then a time-dependent forced Korteweg–de Vries (FKdV) equation can be derived to model the wave motion on the free surface. Here, the case λ ⩾ 0 (or F ⩾ 1 , called supercritical case) is considered. The steady FKdV equation is first studied both theoretically and numerically. It is shown that there exists a cut-off value λ 0 of λ. For λ ⩾ λ 0 there are steady solutions, while for 0 ⩽ λ λ 0 no steady solution of FKdV exists. For the unsteady FKdV equation, it is found that for λ > λ 0 , the solution of FKdV with zero initial condition tends to a stable steady solution, whilst for 0 λ λ 0 a succession of solitary waves are periodically generated and continuously propagating upstream as time evolves. Moreover, the solutions of FKdV equation with nonzero initial conditions are studied.

Dispersion of fine settling particles from an elevated source in an oscillatory turbulent flow

Abstract: The present paper examines the stream-wise dispersion of suspended fine particles with settling velocities in an oscillatory turbulent shear flow with or without a non-zero mean over a rough-bed surface when the particles are being released from an elevated continuous source. A finite-difference implicit method is employed to solve the unsteady turbulent convective-diffusion equation. A combined scheme of central and four-point upwind differences is used to solve the steady state equation and the Alternating Direction Implicit (ADI) method is adopted for unsteady equation. It is shown how the mixing of settling particles is influenced by the tidal oscillatory current and the corresponding eddy diffusivity when the initial distribution of concentration regarded as a line-source. The vertical concentration profiles of suspended fine particles with settling velocities are presented for different downstream stations for various values of settling velocity and the frequency of the oscillation in tidal flow. For two-dimensional unsteady dispersion equation, the behaviour of iso-concentration lines for different values of settling velocity, frequency of the oscillation, dispersion time and releasing height is studied in terms of the relative importance of convection and eddy diffusion.

Turbulence manipulation in air-water flows on a stepped chute: An experimental study

Abstract: In a stepped channel operating with large flow rates, the flow skims over the pseudo-bottom formed by the step edges as a coherent stream. Intense three-dimensional recirculation is maintained by shear stress transmission from the mainstream to the step cavities, while significant free-surface aeration takes place. The interactions between free-surface aeration and cavity recirculation are investigated herein with seven step cavity configuration. The experiments were conducted in a large stepped channel operating at large Reynolds numbers. For some experiments, triangular vanes, or longitudinal ribs, were placed across the step cavities to manipulate the flow turbulence to enhance the interactions between the mainstream flow and the cavity recirculation region. The results showed a strong influence of the vanes on the air-water flow properties in both free-stream and cavity flows. The findings demonstrate some passive turbulence manipulation in highly turbulent air-water flows.