Vertically forced surface wave in weakly viscous fluids bounded in a circular cylindrical vessel

Two-time scale perturbation expansions were developed in weakly viscous fluids to investigate surface wave motions by linearizing the Navier-Stokes equation in a circular cylindrical vessel which is subject to a vertical oscillation. The fluid field was divided into an outer potential flow region and an inner boundary layer region. A linear amplitude equation of slowly varying complex amplitude, which incorporates a damping term and external excitation, was derived for the weakly viscid fluids. The condition for the appearance of stable surface waves was obtained and the critical curve was determined. In addition, an analytical expression for the damping coefficient was determined and the relationship between damping and other related parameters (such as viscosity, forced amplitude, forced frequency and the depth of fluid, etc.) was presented. Finally, the influence both of the surface tension and the weak viscosity on the mode formation was described by comparing theoretical and experimental results. The results show that when the forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when the forcing frequency is high, the surface tension of the fluid is prominent.

Damping of Vertically Excited Surface Wave in Weakly Viscous Fluid

In a vertically oscillating circular cylindrical container, singular perturbation theory of two-time scale expansions is developed in weakly viscous fluids to investigate the motion of single free surface standing wave by linearizing the Navier-Stokes equation. The fluid field is divided into an outer potential flow region and an inner boundary layer region. The solutions of both two regions are obtained and a linear amplitude equation incorporating damping term and external excitation is derived. The condition to appear stable surface wave is obtained and the critical curve is determined. In addition, an analytical expression of damping coefficient is determined. Finally, the dispersion relation, which has been derived from the inviscid fluid approximation, is modified by adding linear damping. It is found that the modified results are reasonably closer to experimental results than former theory. Result shows that when forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when forcing frequency is high, the surface tension of the fluid is prominent.

Numerical simulation of a bubble rising in shear-thinning fluids

The motion of a single bubble rising freely in quiescent non-Newtonian viscous fluids was investigated experimentally and computationally. The non-Newtonian effects in the flow of viscous inelastic fluids are modeled by the Carreau theological model. An improved level set approach for computing the incompressible two-phase flow with deformable free interface is used. The control volume formulation with the SIMPLEC algorithm incorporated is used to solve the governing equations on a staggered Eulerian grid. The simulation results demonstrate that the algorithm is robust for shear-thinning liquids with large density (rho(1)/rho(g) up to 10(3)) and high viscosity (eta(1)/eta(g) up to 10(4)). The comparison of the experimental measurements of terminal bubble shape and velocity with the computational results is satisfactory. It is shown that the local change in viscosity around a bubble greatly depends on the bubble shape and the zero-shear viscosity of non-Newtonian shear-thinning liquids. The shear-rate distribution and velocity fields are used to elucidate the formation of a region of large viscosity at the rear of a bubble as a result of the rather stagnant flow behind the bubble. The numerical results provide the basis for further investigations, such as the numerical simulation of viscoelastic fluids. (C) 2010 Elsevier B.V. All rights reserved.

Instability Analysis of Nonlinear Surface Waves in a Circular Cylindrical Container Subjected to a Vertical Excitation

Singular perturbation theory of two-time scale expansions was developed both in inviscid and weak viscous fluids to investigate the motion of single surface standing wave in a liquid-filled circular cylindrical vessel, which is subject to a vertical periodical oscillation. Firstly, it is assumed that the fluid in the circular cylindrical vessel is inviscid, incompressible and the motion is irrotational, a nonlinear evolution equation of slowly varying complex amplitude, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from solvability condition of high-order approximation. It shows that when forced frequency is low, the effect of surface tension on mode selection of surface wave is not important. However, when forced frequency is high, the influence of surface tension is significant, and can not be neglected. This proved that the surface tension has the function, which causes free surface returning to equilibrium location. Theoretical results much close to experimental results when the surface tension is considered. In fact, the damping will appear in actual physical system due to dissipation of viscosity of fluid. Based upon weakly viscous fluids assumption, the fluid field was divided into an outer potential flow region and an inner boundary layer region. A linear amplitude equation of slowly varying complex amplitude, which incorporates damping term and external excitation, was derived from linearized Navier-Stokes equation. The analytical expression of damping coefficient was determined and the relation between damping and other related parameters (such as viscosity, forced amplitude and depth of fluid) was presented. The nonlinear amplitude equation and a dispersion, which had been derived from the inviscid fluid approximation, were modified by adding linear damping. It was found that the modified results much reasonably close to experimental results. Moreover, the influence both of the surface tension and the weak viscosity on the mode formation was described by comparing theoretical and experimental results. The results show that when the forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when the forcing frequency is high, the surface tension of the fluid is prominent. Finally, instability of the surface wave is analyzed and properties of the solutions of the modified amplitude equation are determined together with phase-plane trajectories. A necessary condition of forming stable surface wave is obtained and unstable regions are illustrated. (c) 2005 Elsevier SAS. All rights reserved.

Instability of the vertically forced surface wave in a circular cylindrical container

The nonlinear amplitude equation, which was derived by Jian Yongjun employing expansion of two-time scales in inviscid fluids in a vertically oscillating circular cylindrical vessel, is modified by introducing a damping term due to the viscous dissipation of this system. Instability of the surface wave is analysed and properties of the solutions of the modified equation are determined together with phase-plane trajectories. A necessary condition of forming a stable surface wave is obtained and unstable regions are illustrated. Research results show that the stable pattern of surface wave will not lose its stability to an infinitesimal disturbance.

Computation of Viscous Incompressible Flow using Pressure Correction Method on Unstructured Chimera Grid

In this paper, a pressure correction algorithm for computing incompressible flows is modified and implemented on unstructured Chimera grid. Schwarz method is used to couple the solutions of different sub-domains. A new interpolation to ensure consistency between primary variables and auxiliary variables is proposed. Other important issues such as global mass conservation and order of accuracy in the interpolations are also discussed. Two numerical simulations are successfully performed. They include one steady case, the lid-driven cavity and one unsteady case, the flow around a circular cylinder. The results demonstrate a very good performance of the proposed scheme on unstructured Chimera grids. It prevents the decoupling of pressure field in the overlapping region and requires only little modification to the existing unstructured Navier–Stokes (NS) solver. The numerical experiments show the reliability and potential of this method in applying to practical problems.

Large-Eddy-Simulation of 3-Dimensional Rayleigh-Taylor Instability in Incompressible Fluids

The 3-dimensiqnal incompressible Rayleigh-Taylor instability is numerically studied through the large-eddy-simulation (LES) approach based on the passive scalar transport model. Both the instantaneous velocity and the passive scalar fields excited by sinu

Flow Characteristics of Liquids in Microtubes Driven by a High Pressure

The flow characteristics of liquids in microtubes driven by a high pressure ranging from 1 MPa to 30 MPa are studied in this paper. The diameter of the microtube is from 3 μm to 10 μm and liquids composed of simple small molecules are chosen as the working fluids. The Reynolds number ranges from 0. 1 to 24. The behavior of isopropanol and carbon tetrachloride under high pressure is found different from the prediction from conventional Hagen-Poiseuille (HP) equation. The normalized friction coefficient C* increases significantly with the pressure. From an analysis of the microtube deformation, liquid compressibility, viscous heating and wall slip, it may be seen that the viscosity at high pressure plays an important role here. An exponential function of viscosity vs pressure is introduced into the HP equation to counteract the difference between experimental and theoretical values. However, this difference is not so marked for di-water.

Three-dimensional instability of an oscillating viscous flow past a circular cylinder

A systematically numerical study of the sinusoidally oscillating viscous flow around a circular cylinder was performed to investigate vortical instability by solving the three-dimensional incompressible Navier-Stokes equations. The transition from two- to three-dimensional flow structures along the axial direction due to the vortical instability appears, and the three-dimensional structures lie alternatively on the two sides of the cylinder. Numerical study has been taken for the Keulegan-Carpenter( KC) numbers from 1 to 3.2 and frequency parameters from 100 to 600. The force behaviors are also studied by solving the Morison equation. Calculated results agree well with experimental data and theoretical prediction.

Probe for Measurements of Density/Conductivity in Flows of Conducting Fluids

A probe utilizing the bipolar pulse method to measure the density of a conducting fluid has been developed. The probe is specially designed such that the concentration of a stream tube can be sampled continuously. The density was determined indirectly from the measurement of solution conductivity. The probe was calibrated using standard NaCl solutions of varying molarity and was able to rapidly determine the density of a fluid with continuously varying conductance. Measurements of the conductivity profiles, corresponding density profiles, and their fluctuation levels are demonstrated in a channel flow with an electrolyte injected from a slot in one wall.

Rossby Waves With Linear Topography In Barotropic Fluids

Rossby waves are the most important waves in the atmosphere and ocean, and are parts of a large-scale system in fluid. The theory and observation show that, they satisfy quasi-geostrophic and quasi-static equilibrium approximations. In this paper, solitary Rossby waves induced by linear topography in barotropic fluids with a shear flow are studied. In order to simplify the problem, the topography is taken as a linear function of latitude variable y, then employing a weakly nonlinear method and a perturbation method, a KdV (Korteweg-de Vries) equation describing evolution of the amplitude of solitary Rossby waves induced by linear topography is derived. The results show that the variation of linear topography can induce the solitary Rossby waves in barotropic fluids with a shear flow, and extend the classical geophysical theory of fluid dynamics.

Proper orthogonal decomposition of oscillatory Marangoni flow in half-zone liquid bridges of low-Pr fluids

Proper orthogonal decomposition (POD) using method of snapshots was performed on three different types of oscillatory Marangoni flows in half-zone liquid bridges of low-Pr fluid (Pr = 0.01). For each oscillation type, a series of characteristic modes (eigenfunctions) have been extracted from the velocity and temperature disturbances, and the POD provided spatial structures of the eigenfunctions, their oscillation frequencies, amplitudes, and phase shifts between them. The present analyses revealed the common features of the characteristic modes for different oscillation modes: four major velocity eigenfunctions captured more than 99% of the velocity fluctuation energy form two pairs, one of which is the most energetic. Different from the velocity disturbance, one of the major temperature eigenfunctions makes the dominant contribution to the temperature fluctuation energy. On the other hand, within the most energetic velocity eigenfuction pair, the two eigenfunctions have similar spatial structures and were tightly coupled to oscillate with the same frequency, and it was determined that the spatial structures and phase shifts of the eigenfunctions produced the different oscillatory disturbances. The interaction of other major modes only enriches the secondary spatio-temporal structures of the oscillatory disturbances. Moreover, the present analyses imply that the oscillatory disturbance, which is hydrodynamic in nature, primarily originates from the interior of the liquid bridge. (C) 2007 Elsevier B.V. All rights reserved.

Numerical simulation of hypersonic viscous flow around space shuttle with method of diffusion analogy

Hypersonic viscous flow around a space shuttle with M(infinity) = 7, Re = 148000 and angle of attack alpha = 5-degrees is simulated numerically with the special Jacobian matrix splitting technique and simplified diffusion analogy method. With the simplified diffusion analogy method the efficiency of computation and resolution of the shock can be improved.

Distribution, stability, bifurcations and catastrophe of steady rotation of a symmetrical heavy gyroscope with viscous-liquid-filled cavity

The number, the angles of orientation and the stability in Rumyantsev Movchan's sense of oblique steady rotations of a symmetric heavy gyroscope with a cavity completely filled with a uniform viscous liquid, possessing a fixed point 0 on its symmetric axis. are given for various values of the parameters. By taking the square of the upright component of the angular momentum M2 as a control parameter, three types of bifurcation diagrams of the steady rotations, two types of jumps and two kinds of local catastrophes, one being the symmetric reduced cusp type and the other being of the symmetric reduced butterfly type, are obtained. By taking account of the M2-damping owing to the moment of unavoidable faint friction, two different modes for the gyroscope, initially in a stable quasi-steady upright rotation with a nutation angle theta(s) equal to zero, to topple over are found.