Cavitation in hydraulic machines causes different problems that can be related to its unsteady nature. An experimental and numerical study of developed cavitating flow was performed. Until now simulations of cavitating flow were limited to the self developed “in house” CFD codes. The goal of the work was to experimentally evaluate the capabilities of a commercial CFD code (Fluent) for simulation of a developed cavitating flow. Two simple hydrofoils that feature some 3D effects of cavitation were used for the experiments. A relatively new technique where PIV method combined with LIF technique was used to experimentally determine the instantaneous and average velocity and void ratio fields (cavity shapes) around the hydrofoils. Distribution of static pressure on the hydrofoil surface was determined. For the numerical simulation of cavitating flow a bubble dynamics cavitation model was used to describe the generation and evaporation of vapour phase. An unsteady RANS 3D simulation was performed. Comparison between numerical and experimental results shows good correlation. The distribution and size of vapour structures and the velocity fields agree well. The distribution of pressure on the hydrofoil surface is correctly predicted. The numerically predicted shedding frequencies are in fair agreement with the experimental data.
The intention of the “von Kármán sodium” (VKS) experiment is to study the hydromagnetic dynamo effect in a highly turbulent and unconstrained flow. Much effort has been devoted to the optimization of the mean flow and the lateral boundary conditions in order to minimize the critical magnetic Reynolds number and hence the necessary motor power. The main focus of this paper lies on the role of “lid layers”, i.e. layers of liquid sodium between the impellers and the end walls of the cylinder. First, we study an analytical test flow to show that lid layers can have an ambivalent effect on the efficiency of the dynamo. The critical magnetic Reynolds number shows a flat minimum for a small lid layer thickness, but increases for thicker layers. For the actual VKS geometry it is shown that static lid layers yield a moderate increase of by approximately 12 per cent. A more dramatic increase by 100 till 150 per cent can occur when some rotational flow is taken into account in those layers. Possible solutions of this problem are discussed for the real dynamo facility.
We propose a kinetic model which describes a mixture of reactive gases, in which a unique continuous internal energy parameter is present. This model enables to recover at the level of its hydrodynamical limit the Euler equations of a mixture of reactive polytropic gases.
This paper addresses the interaction of a slender structure and a sheared incident cross-flow. The oscillating wake of the structure is modeled using a distribution of van der Pol oscillators. Elementary configurations are first considered to assess the basic phenomena and structure dynamics allowing to reliably investigate more complex and realistic cases. The scope of the study thus ranges from a forced oscillating cylinder to a tensioned cable. This last configuration is found to experience wave-packets of vortex-induced motion due to a series of local lock-ins. A theoretical analysis is carried-out to predict the wave-packets amplitude and distribution. It is shown to be in reasonable agreement with the results of numerical simulations. This indicates that the system behavior can be described in terms of the interactions between the wake and the structure only.
The combination of an electric field and a moderate turbulent flow is a promising technique for separating stable water–oil emulsions. Field-induced charges on the water droplets will cause adjacent droplets to align with the field and attract each other. The present work describes the forces that influence the kinematics of droplets falling in oil when exposed to an electric field. Mathematical models for these forces are presented and discussed with respect to a possible implementation in a multi-droplet Lagrangian framework. The droplet motion is mainly due to buoyancy, drag, film-drainage, and dipole–dipole forces. Attention is paid to internal circulations, non-ideal dipoles, and the effects of surface tension gradients. Experiments are performed to observe the behavior of a droplet falling onto a stationary one. The droplet is exposed to an electric field parallel to the direction of the droplet motion. The behavior of two falling water droplets exposed to an electric field perpendicular to the direction of their motion is also investigated until droplet coalescence. The droplet motion is recorded with a high-speed CMOS camera. The optical observations are compared with the results from numerical simulations where the governing equations for the droplet motion are solved by the RK45 (Runge Kutta) Fehlberg method with step-size control and low tolerances. Results, using different models, are compared and discussed in detail. A framework is otlined to describe the kinematics of both a falling rigid spherical particle and a fluid droplet under the influence of an electric field.
Numerical solutions based on the method of fundamental solutions are discussed for Stokes flow inside a rectangular cavity in the presence of circular cylinders. The Stokeslets are used as the fundamental solutions to obtain the solution for the flow field by a linear combination of fundamental solutions. Flow results on the cellular structure of flow field resulting from the dynamics of cylinders and the horizontal walls of the cavity are reported for (i) one rotating cylinder in a rectangular cavity with two parallel horizontal sides moving in the same directions as well as in the opposite directions, (ii) two rotating cylinders kept apart in a rectangular cavity with two parallel horizontal sides moving in the same directions as well as in the opposite directions. The effect of aspect ratio of the rectangular cavity, direction of movement of the two parallel horizontal sides of the cavity and the diameter of the rotating cylinder on the flow structure are studied. The flow results obtained for the single cylinder case are in accordance with the results available in the literature. From the computational point of view, the present numerical procedure based on the method of fundamental solutions is efficient and simple to implement as compared to the mesh-dependent schemes, which needs complex mesh generation procedure for the multiply connected geometrical domains considered in this article.
Through an Hamiltonian action we write down the system of equations of motions for a mixture of thermocapillary fluids under the assumption that the internal energy is a function not only of the gradient of the densities but also of the gradient of the entropies of each component. A Lagrangian associated with the kinetic energy and the internal energy allows to obtain the equations of momentum for each component and for the barycentric motion of the mixture. We obtain also the balance of energy and we prove that the equations are compatible with the second law of thermodynamics. Though the system is of parabolic type, we prove that there exist two tangential acceleration waves that characterize the interfacial motion. The dependence of the internal energy of the entropy gradients is mandatory for the existence of this kind of waves. The differential system is non-linear but the waves propagate without distortion due to the fact that they are linearly degenerate (exceptional waves).
The problem of strong shock-wave propagation through a dust-laden gas is studied as a limiting case of very intensive heat transfer. According to a potential law, the variable energy input is continuously supplied by a driving piston or a surface. A self-similar solution is found under isothermal condition of the flow field. The spherical case is worked out in detail to investigate to what extent the shock wave is influenced by the energy input as well as by the mass concentration of the solid particles in the medium and the ratio of density of the solid particles to the initial density of the medium. Three different cases are covered with respect to parameters describing the increase of energy or the piston velocity: One corresponds to a decelerated piston, the second to a constant piston velocity and the third to a continuously accelerated piston starting from rest. The dust-free flow is included in the numerical results as a limiting case.
A high Reynolds number flat plate turbulent boundary layer is investigated in a wind-tunnel experiment. The flow is subjected to an adverse pressure gradient which is strong enough to generate a weak separation bubble. This experimental study attempts to shed some new light on separation control by means of streamwise vortices with emphasize on the change in the boundary layer turbulence structure. In the present case, counter-rotating and initially non-equidistant streamwise vortices become and remain equidistant and confined within the boundary layer, contradictory to the prediction by inviscid theory. The viscous diffusion cause the vortices to grow, the swirling velocity component to decrease and the boundary layer to develop towards a two-dimensional state. At the position of the eliminated separation bubble the following changes in the turbulence structure were observed. The anisotropy state in the near-wall region is unchanged, which indicates that it is determined by the presence of the wall rather than the large scale vortices. However, the turbulence in the outer part of the boundary layer becomes overall more isotropic due to an increased wall-normal mixing and a significantly decreased production of streamwise fluctuations. The turbulent kinetic energy is decreased as a consequence of the latter. Despite the complete change in mean flow, the spatial turbulence structure and the anisotropy state, the process of transfer of turbulent kinetic energy to the spanwise fluctuating component seems to be unchanged. Local regions of anisotropy are strongly connected to maxima in the turbulent production. For example, at spanwise positions in between those of symmetry, the spanwise gradient of the streamwise velocity cause significant production of turbulent fluctuations. Transport of turbulence in the spanwise direction occurs in the same direction as the rotation of the vortices.
Recent progress in micro-fluid dynamics has identified an increased demand for efficient mixing of highly viscous fluids in small channels and cavities. One way to do this is through the steady streaming generated by the vibration of solid boundaries. In this paper we investigate the mixing properties of such streaming flows in an infinite channel. A Newtonian fluid is confined within flexible walls with transverse motion in the form of standing waves of small amplitude. The velocity field is determined using a perturbation approach with the slope of the wall as a small parameter [Phys. Fluids 16 (2004) 1822]. Streaming occurs at second order with the formation of cellular flow patterns in the channel. The Lagrangian velocities were found to mimic the Eulerian except for flows at large channel half-widths and low frequencies. Most effective mixing is observed for flows at channel half-widths of similar, or lower, order than the vibratory wavelength and for sufficiently high frequencies.
This experimental study is devoted to the transition to turbulence of the flow confined between a stationary and a rotating disk. Using visualization and video image analysis, we describe the different transitions occurring in the flow as the rotating velocity of the disk is varied. The space–time behavior of the wave patterns is analyzed using the Bi-Orthogonal Decomposition (BOD) technique. This decomposition of the experimental signals on proper modes permits to project the dynamics of the waves in a reduced embedding phase space. By this means, a torus doubling bifurcation is revealed before its complete destruction during the transition to a weak turbulence. Finally, a more classical 2D-Fourier analysis completes our description of the transition and shows for higher rotation rates, the appearance of a more developed turbulence issued from the former chaotic waves.
In this paper we study the transition from a deep to a shallow water layer and the formation of quasi-two-dimensional vortex dynamics. Vortices are experimentally generated by a circular horizontal turbulent pulsed jet. The dimensional analysis gives two relevant dimensionless parameters: the jet Reynolds number Re and a number C which characterizes the vertical confinement. They are respectively defined by Re=√Q/ν and C=(√Q/H )t (H is the water depth, ν is the kinematical viscosity, Q the injected momentum flux and t the injection duration). Experimental results show a strong influence of C on the flow: when C2 we observe the formation of large vortex dipoles. However these dipoles are not strictly two-dimensional because of the presence of a vertical circulation in the front of the dipole. Results are independent of the jet Reynolds number in the range 1000
A Boussinesq method is derived that is fully dispersive, in the sense that the error of the approximation is small for all ( the magnitude of the wave number and the water depth). This is made possible by introducing the generalized (2D) Hilbert transform, which is evaluated using the fast Fourier transform. Variable depth terms are derived both in mild-slope form, and in augmented mild-slope form including all terms that are linear in derivatives of . A spectral solution is used to solve for highly nonlinear steady waves using the new equations, showing that the fully dispersive behavior carries over to nonlinear waves. A finite-difference–FFT implementation of the method is also described and applied to more general problems including Bragg resonant reflection from a rippled bottom, waves passing over a submerged bar, and nonlinear shoaling of a spectrum of waves from deep to shallow water.
The effect of convection on the heat generated by an exothermic reaction is considered within a square reactor bounded by constant temperature walls. The fluid flow and heat transfer equations are solved numerically for a range of values of the Rayleigh and Frank-Kamenetskii numbers. From these solutions, boundaries between bounded solutions and thermal runaway are determined, with the range of for bounded solutions increasing as is increased. The system is capable of sustaining both steady and oscillatory behaviour, including simple periodic as well as both chaotic and period-three responses. A parametric region where there is hysteresis is found with either oscillations or steady states seen at the same parameter values, dependent on the initial configuration of the system.
We revisit the problem of the stability of pulsatile pipe flow for axisymmetric perturbations. In contrast to the earlier approach based on the Chebyshev expansion for the spatial discretization [J. Appl. Mech. ASME 53 (1986) 187], we use the set of the eigenfunctions derived from the longwave limit of the Orr–Sommerfeld equation. We show that the Orr–Sommerfeld basis gives greater accuracy than the Chebyshev basis if fewer terms are used in the Galerkin expansion. For the time evolution of the flow perturbation, instead of the usual Floquet analysis, a different representation for the solution of the periodic system of linear differential equations is employed. We found that the flow structures corresponding to the largest energy growth are toroidal vortex tubes. They are stretched by the shear stress of the mean flow so that a maximum energy growth occurs. The flow perturbation subsequently decays due to viscous effects. The maximum energy growth is then evaluated over a range of Reynolds and Womersley numbers. Asymptotic solutions provided for the longwave limit as well as the limit of large Womersley numbers agree well with the numerical results, confirming the known linear stability of the flow.
Fully dispersive deterministic evolution equations for irregular water waves are derived. The equations are formulated in the complex amplitudes of an irregular, directional wave spectrum and are valid for waves propagating in directions up to ±90° from the main direction of propagation under the assumptions of weak nonlinearity, slowly varying depth and negligible reflected waves. A weak deviation from straight and parallel bottom contours is allowed for. No assumptions on the vertical structure of the velocity field is made and as a result, the equations possess exact second-order bichromatic transfer functions when comparing to the reference solution of a Stokes-type analysis. Introduction of the so-called ‘resonance assumption’ leads to the evolution equations of among others Agnon, Sheremet, Gonsalves and Stiassnie [Coastal Engrg. 20 (1993) 29–58]. For unidirectional waves, the bichromatic transfer functions of the ‘resonant’ models are found to have only small deviations in general from the reference solution. We demonstrate that the ‘resonant’ models can be solved efficiently using Fast Fourier Transforms, while this is not possible for the ‘exact’ models. Simulation results for unidirectional wave propagation over a submerged bar show that the new models provide a good improvement from linear theory with respect to wave shape. This is due to the quadratic terms, enabling a nonlinear description of shoaling and de-shoaling, including the release of higher harmonics after the bar. For these simulations, the similarity between the ‘exact’ and ‘resonant’ models is confirmed. A test case of shorter waves, however, shows that the amplitude dispersion can be quite over-predicted in the models. This behaviour is investigated and confirmed through a third-order Stokes-type perturbation analysis.
High-temperature geothermal reservoir in porous media is under consideration, consisting of two high-permeability layers, which are separated by a low-permeability stratum. The thermodynamic conditions are assumed to imply that the upper and lower high-permeability layers are filled in by water and by vapour, respectively. In these circumstances the low-permeability stratum possesses the phase transition interface, separating domains occupied by water and vapour. The stable stationary regimes of vertical phase flow between water and vapour layers in the low-permeability stratum may exist. Stability of such regimes where the heavier fluid is located over the lighter one is supported by a heat transfer, caused by a temperature gradient in the Earth's interior. We give the classification of the possible types of transition to instability of the vertical flows in such a system under the condition of smallness of the advective heat transfer in comparison with the conductive one. It is found that in the non-degenerate case there exist three different scenarios of the onset of instability of the stationary vertical phase transition flows. Two of them are accompanied by the bifurcations of the destabilizing vertical flow, leading to appearance of horizontally non-homogeneous regimes with non-constant shape of the interface. The bifurcations correspond to the simple resonance and -resonance, which typically arise in reversible systems.