Wave drag of an ellipsoid of revolution rapidly moving in a horizontal direction in a uniformly stratified fluid

The problem of the wave drag of ellipsoids moving in a uniformly stratified ideal fluid is considered by means of modeling the bodies by surface distributions of mass sources. Analytical results are obtained using the distributions known from the theory of a uniform fluid, which make it possible to describe the emission of internal waves by rapidly moving ellipsoids of revolution (spheroids) in the limit of large Froude numbers. An asymptotically simplified form of the dependence of the wave drag on the Froude number and the spheroid axis ratio is found. In the particular case of a sphere, the result confirmed earlier by Greenslade by making comparisons with a numerical calculation and experimental data is obtained.     

Surface characters of internal waves generated by Rankine ovoid

A linear theory on the internal waves generated in the stratified fluid with a pycnocline is presented in this paper. The internal wave fields such as the velocity fields in the stratified fluid and velocity gradient fields at the free surface are also investigated by means of the theoretical and numerical method. From the numerical results, it is shown that the internal wave generated by horizontally moving Rankine ovoid is a sort of trapped wave which propagates in a wave guide, and its waveform is a kind of Mach front-type internal wave in the pycnocline. Influence of the internal wave on the flow fields at the free surface is represented by the velocity gradient fields resulted from the internal waves generated by motion of the Rankine ovoid. At the same time, it is also shown that under the hypothesis of inviscid fluid, the synchronism between the surface velocity gradient fields at the free surface and the internal wave fields in the fluid is retained. This theory opens a possibility to study further the modulated spectrum of the Bragg waves at the free surface.The project supported by the National Natural Science Foundation of China (40576010). The English text was polished by Keren Wang.     

A correction of the 2D KdV equation of Djordjevic ＆ Redekopp in exponentially stratified fluid＊

To further study the fission laws of initial internal solitons on the continental shelf/slope, we rederive and correct the 2D KdV equation of Djordjevic & Redekopp for exponentially stratified fluid (or ocean) and with two-dimensional topography. Through a combination of theoretical study and numerical experiments, we show that solitons in the odd vertical modes can fission. However, because of the corrections, the fission conditions are different from those of Djordjevic & Redekopp. The even modes cannot fission unless the initial internal solitons propagate from shallow sea to deep sea. This conclusion is entirely opposite to that of Djordjevic & Redekopp.The project supported by the National Natural Science Foundation of China (40276008) and the Grant of Key Laboratory of Marine Science and Numerical Modeling, SOA (0201(2003))The English text was polished by Keren Wang.     

Instability of a vertical columnar vortex in a stratified fluid

In this article, we study experimentally the evolution of a vertical columnar vortex in a stratified fluid. Three different measurement techniques are used. Particle image velocimetry allows us to monitor the time evolution of the characteristics of the vortex (Froude and Reynolds numbers). Dye visualizations reveal the existence of an instability for Froude numbers smaller than one, which creates an undulation of the vortex centerline. Synthetic schlieren visualization shows that the density structure of the unstable mode is very similar to the structure found recently numerically for the radiative instability of a Lamb–Oseen vortex (Riedinger et al. in J Fluid Mech, 2010). The experimental stability diagram and unstable wavelengths are compared with these numerical results. A secondary instability associated with the presence of critical layers is also observed for Froude numbers larger than one.     

Flow pattern behind a submerged body near a pycnocline

A flow past an ellipsoid immersed in a flow of a viscous stratified fluid is studied using a Large Eddy Simulation (LES) method for different locations of the body relative to a density discontinuity. It is shown that, even when the internal Froude number is large, for small angles of attack of the body the stratification affects its drag force. When the body is located above the pycnocline, the presence of the discontinuity results in the ascending of a vortex filament. A spectral analysis showed that the internal waves in the body wake have a multimode structure.     

Vortex motions in stratified wake flows

The vortex wakes of obstacles (circular cylinder and sphere) moving through a linearly stratified fluid have been investigated, by means of flow visualization, at Reynolds numbers smaller than 800 and non-dimensional buoyancy frequencies smaller than 6. Vortex shedding from a horizontally suspended circular cylinder is suppressed when the fluid is stratified. The wake of a sphere is affected by lee waves when the Reynolds number exceeds about 200.     

Hydrodynamic characteristics of a wing in a stratified fluid near the bottom

The problem of the motion of a thin wing in a stratified fluid near the bottom is considered. A solution is found using the logarithmic dynamic potential. The dependence of the hydrodynamic force and moment on the input parameters, namely, the Strouhal and Froude numbers and the distance to the bottom, is studied. An important feature of the amplitudes of nonstationary loads on the wing is their nonmonotonic character in the case where the frequency of vibrations is lower than the Brunt-Väisälä frequency, which is explained by the interaction between the wing's vibrations and the internal waves reflected from the bottom.     

Waves induced by a submerged moving dipole in a two-layer fluid of finite depth

The waves induced by a moving dipole in a twofluid system are analytically and experimentally investigated.The velocity potential of a dipole moving horizontally in the lower layer of a two-layer fluid with finite depth is derived by superposing Green‘s functions of sources (or sinks). The far-field waves are studied by using the method of stationary phase. The effects of two resulting modes, i.e. the surfaceand internal-wave modes, on both the surface divergence field and the interfacial elevation are analyzed. A laboratory study on the internal waves generated by a moving sphere in a two-layer fluid is conducted in a towing tank under the same conditions as in the theoretical approach. The qualitative consistency between the present theory and the laboratory study is examined and confirmed.     

Internal waves generated by a moving sphere and its wake in a stratified fluid

The internal gravity waves and the turbulent wake of a sphere moving through stratified fluid were studied by the fluorescent dye technique. The Reynolds number Re=U·2a/v was kept nearly constant at about 3 · 10³ and the Froude number F;U/a N ranged from 0.5 to 12.5. It is observed that waves generated by the body are dominant only when F<4 and are replaced by waves generated by the large scale coherent structures of the wake when F>4.     

Linear logistic regression with weight thresholding for flow regime classification of a stratified wake

A stratified wake has multiple flow regimes, and exhibits different behaviors in these regimes due to the competing physical effects of momentum and buoyancy. This work aims at automated classification of the weakly and the strongly stratified turbulence regimes based on information available in a full Reynolds stress model. First, we generate a direct numerical simulation database with Reynolds numbers from 10,000 to 50,000 and Froude numbers from 2 to 50. Order (100) independent realizations of temporally evolving wakes are computed to get converged statistics. Second, we train a linear logistic regression classifier with weight thresholding for automated flow regime classification. The classifier is designed to identify the physics critical to classification. Trained against data at one flow condition, the classifier is found to generalize well to other Reynolds and Froude numbers. The results show that the physics governing wake evolution is universal, and that the classifier captures that physics.     

Experimental Investigation of the Velocity Field near a Cylinder in a Contonuously Stratified Fluid

The velocity distributions in the upstream disturbance and lagging wake are measured using markers in the shadow flow pattern near a horizontal cylinder uniformly towed in a stratified fluid. The dimensions of the upstream total blocking zone and the velocity damping laws in the upstream disturbance are determined. On the basis of the experimental results the actual limits of applicability of existing methods of calculating the structure of stratified flow past obstacles are found for small Froude numbers.     

Effect of homogeneous stratification of the fluid on the dynamics of a kármán vortex street behind a circular cylinder

The experimental results of studying the effect of homogeneous stratification of the fluid on the conditions of generation of a Kárman vortex street [1] developing in the wake of a cylinder in steady horizontal motion are described. In a homogeneous medium at Reynolds numbers Re >5 two symmetrical regions of vorticity of opposite sign are formed behind the cylinder and move together with the latter. As the speed of the cylinder increases, the link between the vortices and the cylinder grows weaker, the vortices are stretched out along the flow and at Re > 40 begin to separate alternately, forming a vortex street in the wake. At first, the frequency of vortex separation increases sharply with increase in Re, but then levels off. It is found that in a uniformly stratified fluid the onset of vortex separation from the moving cylinder is delayed. The dependence of the critical Reynolds number (onset of vortex separation) on the internal Froude number is obtained. The effect of stratification of the fluid on the frequency of separation of the vortices in the Kármán street is investigated. The effect of the Froude number on the dependence of the Strouhal number on the Reynolds number is established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 83–86, January–February, 1986.In conclusion the authors wish to thank A. T. Onufriev for his interest in their work and useful discussions of the results.     

Motion of a cylinder below the free surface of a fluid at large froude numbers

A method of solving the problem of the motion of a cylinder of given shape below the free surface of an infinitely deep heavy fluid is perfected for large Froude numbers. The motion of a circular cylinder is investigated at small distances from the free surface. Solutions of the problem are given for cylinders with noncircular cross-sections. Kazan. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 34–45, July–August, 2000. The work was carried out with financial support from the Russian Foundation for Basic Research (projects Nos. 99-01-00169 and 99-01-00173).     

Comparison of the drag coefficients of bodies moving in liquids with various stratification profiles

For bodies moving in liquids with various stratification profiles, the relation between the drag coefficients considered as functions of the Froude number is investigated. The problems of stratified liquid dynamics have not previously been studied from this viewpoint, either experimentally or theoretically. On the range of the Froude numbers F₁-1, the drag force coefficients obtained from bench measurements of the towing resistance to the uniform horizontal motion of models in two-layer and continuously stratified liquids are compared. The experimental data obtained in a thermocline are then compared with the results of [1,2] for linearly stratified and two-layer liquids.Nizhnii Novgorod. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 4–11, July–August, 1996.     

Theoretical analysis of the onset of liquid entrainment for slots of finite width

The onset of liquid entrainment during discharge from large reservoirs containing a stratified mixture of two immiscible fluids through a side slot of a finite width is considered theoretically. A previously reported analysis in which the slot was approximated as a two-dimensional line sink has been extended to account for the finite width of the slot. The model resulting from the present analysis is expressed in terms of two simple algebraic equations suitable for hand calculations. According to the present results, the ratio of the critical height to the slot width is dependent only on the Froude number. Numerical results show that the present model approaches the correct physical limits at low Froude numbers and it converges to the predictions of the previously reported simple model at high Froude numbers.     

Stokes flow past a porous spheroid embedded in another porous medium

A study is made with an analysis of an incompressible viscous fluid flow past a slightly deformed porous sphere embedded in another porous medium. The Brinkman equations for the flow inside and outside the deformed porous sphere in their stream function formulations are used. Explicit expressions are investigated for both the inside and outside flow fields to the first order in small parameter characterizing the deformation. The flow through the porous oblate spheroid embedded in another porous medium is considered as the particular example of the deformed porous sphere embedded in another porous medium. The drag experienced by porous oblate spheroid in another porous medium is also evaluated. The dependence of drag coefficient and dimensionless shearing stress on the permeability parameter, viscosity ratio and deformation parameter for the porous oblate spheroid is presented graphically and discussed. Previous well-known results are then also deduced from the present analysis.     

On the motion of a solid sphere through a viscous medium with a vertical temperature gradient

In this paper, we report experiments conducted on the temperature distribution around a solid sphere that settles slowly in a thermally stratified viscous fluid. The temperature field has been obtained using interferometry. In some cases thermochromic liquid crystal tracers have been used to simultaneously obtain the velocity and the temperature fields. We also describe a theoretical analysis of the problem and compare experimental and theoretical findings and discuss the results. we find that natural convection has a significant influence in determining the nature of the temperature field away from the sphere.     

The oscillation of a spheroid along its axis in a non-Newtonian fluid

Summary The present paper investigates the nature of the fluid flow when a spheroid is suspended in an infinitely extending elastico-viscous fluid defined by the constitutive equations given byOldroyd orRivlin andEricksen, and is made to perform small amplitude oscillations along its axis. The solution of the vector wave equation is expressed in terms of the solution of the corresponding scalar wave equation, without the use ofHeine's function or spheroidal wave functions. Two special cases (i) a sphere and (ii) a spheroid of small ellipticity, are studied in detail.Née MissRenuka Rajagopalan.