Free surface deformation due to a source or a sink in Stokes flow

Free surface shape and cusp formation are analyzed by considering two-dimensional viscous flow due to a line source or a line sink below the free surface where the strength of source/sink is given arbitrarily. In the analysis, the Stokes' approximation is used and surface tension effects are included, but gravity is neglected. The solution is obtained analytically by using conformal mapping and complex function theory. From the solution, shapes of the free surface are shown and the formation of a cusp on the free surface is discussed. As the capillary number decreases in negative, the free surface shape becomes singular and in a real fluid a cusp should form on the free surface below some negative critical capillary number. Typically, streamline patterns for some capillary numbers are also shown. As the small capillary number vanishes, the solution is reduced to a linearized potential flow solution.     

Free surface deformation due to a source and a sink of equal strength in Stokes flow

Two-dimensional Stokes flow due to a source and a sink of equal strength below the free surface is analyzed and free surface shape and cusp formation are discussed. The source-sink pair below the free surface are aligned vertical to the free surface. In the analysis, the Stokes' approximation is used and surface tension effects are included, but gravity is neglected. The solution is obtained by using conformal mapping and complex function theory. From the solution, typical free surface shapes are shown and formation of a cusp on the free surface is discussed. As the capillary number increases, the converging free surface shape becomes singular and tends to form a cusp for sufficiently large capillary number. Typically, streamline patterns for some capillary numbers are also shown. As the capillary number vanishes, the solution is reduced to the linearized potential flow solution.     

Problem of heavy fluid flow above a plate in the presence of a vortex

The free-surface flow of a heavy incompressible inviscid ideal fluid along a semi-infinite plate with the formation of a vortex near its edge is investigated. Solitary wave type flows are considered. A unique solution with a free vortex is obtained.     

Three-dimensional isothermal lava flows over a non-axisymmetric conical surface

Within the thin-layer approximation for a highly-viscous heavy incompressible fluid, a hydrodynamicmodel of a 3D isothermal lava flow over a non-axisymmetric conical surface is constructed. Using analytical methods, a self-similar solution for the law of leading-edge propagation is obtained in the case of a flow from a non-axisymmetric source located at the apex of a conical surface with smoothly varying properties. In the case of a flow over a substantially non-axisymmetric surface, it is shown that there exists a self-similar solution for the free-surface shape and the law of leading-edge motion. This solution is studied numerically for particular examples of the substrate surface and the source. In the general case of a non-self-similar flow over a substantially non-axisymmetric conical surface, a local analytical solution is obtained for the free-surface shape and the velocity field near the leading flow front.     

Experiments on elongational flow of dilute polymer solutions Part I: Jet reaction and excess pressure drop for the flow through small apertures

Experiments have been made with dilute polymer solutions on the reaction of jets issuing from small orifices and the excess pressure drop for orifice and capillary flows.Under the flow conditions with vortices occurring upstream of the aperture, the jet reaction is nearly zero below some mean velocity for PEO solutions and similarly zero below some generalized Reynolds number for Separan solutions. The normalized jet reactions, when they possess positive values, are correlated with the generalized Reynolds number irrespective of the aperture diameters for both kinds of solution.In most cases, the pressure is higher than in the corresponding water flow, but for some flows with no vortex it is lower. For the vortex flow of PEO solutions the normalized excess pressure drop is inversely proportional to the Reynolds number for both orifices and capillaries, while for Separan solutions this quantity is not correlated with the generalized Reynolds number for orifice flow but is correlated with it for capillary flow.     

Interactions between a rectangular cylinder and a free-surface flow

The salient features of the interaction between a free-surface flow and a cylinder of rectangular cross-section are investigated and discussed. Laboratory-scale experiments are performed in a water channel under various flow conditions and elevations of the cylinder above the channel floor. The flow field is characterized on the basis of time-averaged and fluctuating local velocity measurements. Dynamic loadings on the cylinder are measured by two water-insulated dynamometers placed inside the cylinder structure. Starting from frequency and spectral analyses of the force signals, insights on the relationship between force dominant frequencies and the Strouhal number of the vortex shedding phenomenon are provided. Experimental results highlight the strong influence of the asymmetric configuration imposed by the two different boundary conditions (free surface and channel floor) on (i) the mean force coefficients and (ii) the vortex shedding frequencies. We provide an analysis of the nature of the dependence of average force coefficients on relevant dimensionless groups, i.e., the Reynolds number, normalized flow depth and cylinder submersion.     

Nonsteady sink and source in a rotating fluid

The effect of the Coriolis force on the nonsteady two-dimensional sink or source flow has been studied theoretically. With the assumptions of incompressibility and invisddity of fluid, the equations governing the flow are still non-linear. However, a set of analytical and exact solutions is obtained for the velocity and pressure distributions. The result shows that no significant vortex flow can be induced by a source flow, the circulation produced there mainly is because the observer is moving with the rotating system; however, a sink will always induce a vortex, the strength of circulation in the sink flow is simply in proportion to the rotating speed of the system and the time integrated sink strength. As the system is under a constant rotating speed, the circulation will vanish only when the time integrated sink strength vanishes. Several special cases of sink strength are examined and their results are presented in this paper. The purpose of this study is mainly to relate the strength of circulation with the strength of sink and source. Consequently, it- can be applied to the weather forecast for hurricanes or tornadoes.     

Plane vortex flow induced by a mass source (sink) in a rotating viscous fluid

A solution of the self-similar type, describing the development with time of a plane vortex flow excited by an axisymmetric mass source (sink) in a rotating viscous fluid, is obtained. Sources of two kinds — impulsive and of constant strength — are considered. The solutions for the velocity and vorticity fields are expressed in the form of functions similar to incomplete gamma functions and are presented in the form of graphs for various flow Reynolds numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 172–175, July–August, 1991.     

Incompressible flow in a rapidly rotating cylinder with a source/sink distribution in the lateral wall and a free inner boundary

The flow of an incompressible fluid in a rapidly rotating right circular cylinder is considered. A source/sink mass distribution at the lateral wall, which is azimuthally uniform and symmetric across the midplane, causes a deviation from wheel flow. The container is only partially full and the inner free surface is allowed to deviate slightly from the vertical. A finite-difference solution of the full axisymmetric, non-linear governing equations was used to obtain the flow field. A special implicit technique for the Coriolis terms which maintains geostrophy was developed and is described. The results obtained for a low Rossby number flow compare quite favourably with the linearized solution. Results are also presented for a case wherein the non-linear terms are important.     

Cusped ripples at the plane surface of a viscous liquid

Summary We study the time-evolution of periodical ripples of a viscous liquid at the plane free surface under the action of a distant pure straining flow. We neglect inertial forces (Stokes flow) and include surface tension effects. The solutions for a contracting surface and constant strain rate show that the ripples may develop near-cusps during a stage of the evolution, though later the free surface inevitably asymptotically tends to a smooth plane with vanishing ripples due to the action of capillarity. We obtain the condition for cusp formation in this intermediate stage in terms of the initial capillary number and aspect ratio. If the capillary number is kept constant, the surface tends to shrink through a succession of self-similar trochoidal shapes, whose aspect ratio is given by the capillary number. Received 23 March 1998, accepted for publication 23 July 1998     

Experimental study on swirling flow of dilute surfactant solution with deformed free-surface

An experimental investigation was performed on a swirling flow of dilute surfactant solution with deformed free-surface in a cylindrical container driven by the constantly rotating bottom wall. The purpose of the experiment was to estimate weak viscoelasticity in the tested surfactant solutions as well as to investigate the flow characteristics. The tested fluid was an aqueous solution of CTAC (CTAC: cetyltrimethyl ammonium chloride), which is a cationic surfactant. Water, 40 ppm, 60 ppm and 200 ppm CTAC solution flows were tested at Froude numbers ranging from 2.59 to 16.3. Particle image velocimetry (PIV) was used to measure the secondary velocity field in the meridional plane. The deformed free-surface level was extracted from the PIV images. At a similar Froude number, the depth of the dip formed at the center region of the free surface was decreased for CTAC solution flow compared with water flow. The inertia-driven vortex at the up-right corner in the meridional plane becomes more and more weakened with increase of the solution concentration or viscoelasticity. Through analyzing the overall force balance compared with water flow, the first normal stress difference characterizing the viscoelasticity was estimated for the dilute CTAC solution flows. The result supports the viscoelasticity-based turbulent drag-reduction mechanism of surfactant solution flow.     

Nonlinear interaction of a vortex pair with clean and surfactant-covered free surfaces

Two-dimensional unsteady viscous-flow problem associated with the normal incidence of a counter-rotating vortex pair on a free surface is analyzed. Effects of surface tension and insoluble surfactants on the generation of free-surface vorticity and surface waves are investigated. A recently developed finite-difference method based on boundary-fitted coordinates is used to solve the fully-nonlinear problem. Results show that in the absence of surfactants and at low Froude number (based on circulation strength and initial separation distance of the vortex pair), waves of short lengths are generated. However, secondary vorticity generated in this case is not strong enough to affect the outward translation of the primary vortices. At intermediate Froude number, a transient wave developing outboard of the primary vortex becomes steep, and eventually breaks because of local instability. Consequently, free-surface vorticity inhibits the outward translation of the primary vortices. Surface tension in a clean free surface dampens the steep short waves, hence also the generation of free-surface vorticity. However, variation in surface tension induced by surfactants intensifies the generation of surface vorticity, thereby causing the primary vortices to rebound. The increase in the rotational part of wave motion results in the dampening of overall free-surface deformations. However, it is found that the shear stress associated with a large gradient of surfactant concentration could cause local steepening of the short wave generated outboard of the primary vortex.     

The development of Marangoni convection induced by local addition of a surfactant

The development of concentration convection induced by local addition of a surfactant solution onto a horizontal free surface of water is studied experimentally and theoretically. The experiment revealed that the capillary motion develops in a threshold manner, with the threshold value depending on the degree of purification of the fluid, the initial concentration of the surfactant, and the area of the free surface. To describe the threshold mechanism of the concentration convection, a number of theoretical models is considered. Different rheological properties of the surface phase, including the nonlinear dependence of the surface shear stress on the surface velocity, are examined. In the numerical experiment, the convective-flow patterns are calculated for different free-surface boundary conditions, and the time dependence of the flow intensity is investigated.     

Free surface asymptotics for thermocapillary flow at high marangoni numbers

Formal asymptotic expansions of the solution of the steady-state problem of incompressible flow in an unbounded region under the influence of a given temperature gradient along the free boundary are constructed for high Marangoni numbers. In the boundary layer near the free surface the flow satisfies a system of nonlinear equations for which in the neighborhood of the critical point self-similar solutions are found. Outside the boundary layer the slow flow approximately satisfies the equations of an inviscid fluid. A free surface equation, which when the temperature gradient vanishes determines the equilibrium free surface of the capillary fluid, is obtained. The surface of a gas bubble contiguous with a rigid wall and the shape of the capillary meniscus in the presence of nonuniform heating of the free boundary are calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 61–67, May–June, 1989.     

Interaction of viscous wakes with a free surface

The interaction of laminar wakes with.free-surface waves generated by a moving body beneath the surface of an incompressible viscous fluid of infinite depth was investigated analytically. The analysis was based on the steady Oseen equations for disturbed flows.The kinematic and dynamic boundary conditions were linearized for the small-amplitude free-surface waves. The effect of the moving body was mathematically modeled as an Oseenlet.The disturbed flow was regarded as the sum of an unbounded singular Oseen flow which represents the effect of the viscous wake and a bounded regular Oseen flow which represents the influence of the free surface. The exact solution for the free-surface waves was obtained by the method of integral transforms. The asymptotic representation with additive corrections for the free-surface waves was derived by means of Lighthill‘s two-stage scheme. The symmetric solution obtained shows that the amplitudes of the free-surface waves are exponentially damped by the presences of viscosity and submergence depth.     

Nonlinear problem of a steady-state flow of a weighable liquid bounded by a free surface about a system of vortices

The problem of a fluid flow with a free surface about a system of two vortices of opposite intensity is considered within the framework of the nonlinear theory. The range of parameters of the problem in which there is no stationary solution is found. Results of the numerical experiment on the effect of vortex intensities and the Froude number on the shape of the free surface and the hydrodynamic reactions of the singularities are given. Omsk Department of the Sobolev Institute of Mathematics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 6, pp. 63–68, November–December, 1999.     

An experimental study on free surface vortex dynamics

In the present research, hydrodynamic behavior of free surface vortices including surface displacement of the vortex core as a function of the vortex stability, relationship between the intake hydraulic parameters and vortex strength and characteristics of the vortex vertical stretching as a function of the intake hydraulic parameters were experimentally investigated in a horizontal intake. Relationship between the vortex-induced air and vortex strength was also evaluated and compared with the previous researches. In this regard, by defining a non-dimensional parameter as intake number, relationships between intake hydraulic parameters and other investigated dynamic aspects of free-surface vortices were discussed.     

Structure of wake of a sharp-edged bluff body in a shallow channel flow

The flow field downstream of a bluff body in a typical open channel flow was explored by two-dimensional particle image velocimetry. Measurements are obtained in horizontal planes at the near-bed, mid-depth and near-surface locations downstream of the body up to a streamwise distance of 10D, where D is the width of the body. The dimensionless streamwise defect velocity profile of the wake flow matches well with the data of a previous investigation and does not reflect any dependency on the distance from the bed. However, the nature of development of the recirculation region is found to be different at the three vertical locations. The time-averaged streamline pattern indicates the existence of a unique nodal pattern close to the bed. The variation of the half-width is also found to be affected by the presence of the bed and the free surface. The bed friction arrests the transverse growth of the shear layer, and the free-surface helps to redistribute the turbulent kinetic energy in the streamwise and transverse directions. Swirling strength analysis is carried out to compare the behavior and statistics of the vortex population in the vertical direction. The prevailing magnitude of the swirling strength is found to be different at the three vertical locations. Bed friction assists to dissipate vorticity rapidly, and therefore reduces the probability of appearance of strong vortices close to the bed.