ISSN 1028-3358, Doklady Physics, 2007, Vol. 52, No. 5, pp. 261–265. © Pleiades Publishing, Ltd., 2007. Original Russian Text © Yu.D. Chashechkin, Yu.V. Prikhod’ko, 2007, published in Doklady Akademii Nauk, 2007, Vol. 414, No. 1, pp. 44–48.
MECHANICS
Regular and Singular Flow Components for Stimulated and Free Oscillations of a Sphere in Continuously Stratified Liquid Yu. D. Chashechkin and Yu. V. Prikhod’ko Presented by Academician D.M. Klimov August 17, 2006 Received August 23, 2006
PACS numbers: 47.35.Lf, 47.55.Hd DOI: 10.1134/S1028335807050059
Studies of oscillations and waves in inhomogeneous liquids are traditional hydrodynamical problems [1]. In recent years, these studies have been extended significantly owing to investigating flow components that coexist with waves. Among these components, we can indicate upstream disturbances, boundary layers, finestructure vortices, and wakes [2]. When the waves are excited by freely oscillating bodies under the condition of neutral buoyancy in continuously stratified media, autocumulative jets are observed in the vicinity of a body’s-trajectory turning point. The jets are bounded by high-gradient envelopes [2]. These flows are similar to classical cumulative jets produced as a result of the drop fall or the cavern collapse near the liquid free surface [3, 4]. In the complete classification of infinitesimal threedimensional periodic motions in inhomogeneous media, the regular components (e.g., waves) coexist with two types of singular components [5]. Among them, there are internal and isopicnic boundary layers on a rigid surface contacting with the medium [6] and high-gradient “twinkled” envelopes of internal-wave beams [7]. Insofar as both regular and singular components of periodic motions are infinitesimal, a high sensitivity of instrumentations is a necessary condition of their registration. In addition, high spatial and time resolutions of the measuring instrumentation are also required. In this paper, we present for the first time the patterns visualizing singular components in flows generated by a sphere performing forced periodic oscillations in continuously stratified liquid. We also trace the transformation of the flow components into autocumulative jets
Institute of Problems in Mechanics, Russian Academy of Sciences, pr. Vernadskogo 101, Moscow, 117526 Russia e-mail:
[email protected]
accompanied by an increase in the amplitude of the body’s periodic motion. The basic characteristics of a stratified medium are the density ρ0(z) and its gradient dρ 0 ⁄ dz (both are dependent on the vertical coordinate z), the scale Λ = d ( ln ρ 0 ) –1 , the frequency N and the buoyancy period ------------------dz 2π Λ Tb = ------ = 2π ---- (g is the free-fall acceleration), the N g kinetic viscosity ν, and the diffusion coefficient κs of the stratifying agent (salt). (The equation of state, which connects the density and salinity, is assumed to be linear, with the coefficient of the salt contraction being included into the definition of salinity.) The source of the motion is a sphere of diameter D, which executes harmonic vertical oscillations at the frequency ω0, with the peak-to-peak value (double ampliHω tude) H and with the velocity magnitude U = -----------0 . In 2 the case of slow oscillations, the sphere produces a conic beam of internal waves at the same frequency and of the vertical-displacement amplitude A. The characteristic wavelength λ, as well as the entire beam width are determined by the diameter of the sphere and the peak-to-peak value of sphere oscillations [6]. The ultimate frequency of running infinitesimal periodic internal waves is limited by the buoyancy frequency ω ≤ N [1]. The transverse size of singular components of the periodic motions are characterized by the universal κ ν ---- and δs = -----s in the velocity field N N and density (or salinity) field, respectively [6, 7]. These microscales also describe the fine structure of the stratified flow, which is induced by diffusion on the sphere at rest [8]. microscales δN =
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The ratios of the characteristic scales determine the basic dimensionless problem parameters: the Stokes number St (the inverted traditional Reynolds number 2ν Re), St = Re–1 = ---------------- , and the internal Froude numH Dω 0 λ ber Fr = ----------- , as well as the family of relative scales: 2πD Λ of the buoyancy C = ---- , peak-to-peak value of oscillaD L H tions M = ---- , and the source size Mν = -----ν . These quanD D tities determine the structure and modality of the beam 3 gν [6] (Lν = ---------- is the viscous wave scale). The diversity N of length scales of different natures and of the corresponding ratios demonstrates the complexity of the pattern of fluid flow generated by the oscillating body. In addition, the field of each of the independent variables of the problem (the density, pressure, and three velocity components) is characterized by its own geometry and scales [7]. The smallest of the basic scales (δρ or δs) determines the spatial resolution of the instrument providing the registration of all flow components. At the same time, the most rapidly evolving component (the beam singular envelopes [7]) determines the time resolution. In the case when the stratified medium is formed by the aqueous solution of common salt (ν = 0.01 cm2/s, κs = 1.41 × 10–5 cm2/s, N ~ 1 s–1), only a high-technology schlieren instrument can satisfy the above requirements. The experiments were carried out in a laboratory tank 70 × 25 × 70 cm3 in size. The tank was supplied with transparent windows made of optical glass. Using the traditional replacement method, we filled the tank with a linearly stratified aqueous solution of common salt. The stratification homogeneity was checked by the schlieren instrument. The buoyancy period was determined by a contact electric-conductivity sensor. The measurement method is based on the variation of the period of proper oscillations excited immediately after a density mark has been immersed [9]. In the experiments being described, the buoyancy period was Tb = 7.3 and 11.2 s. The motion of the liquid was excited by a sphere 4−5 cm in diameter, which had been suspended on a thin wire in the tank center. The wire was connected with a crank mechanism installed above the tank. The structure of the mechanism allowed us to vary within a wide range the oscillation frequency ω0 and the amplitude (from 0.5 to 3 cm). The flow pattern was observed by means of the IAB-458 schlieren instrument with a vertical slit of the
size 10 × 0.03…0.1 mm. A Foucault knife or a 16-µm wire was used as a visualizing diaphragm. The image registration was carried out by photo and video cameras having a spatial resolution of 0.1 mm. The images obtained were transmitted to a personal computer and then transformed and processed by specified codes that allowed successive-frame photographic images to be obtained and the geometric characteristics of flow components to be measured. After the tank had been filled with the stratified solution and had been held for 24 h in order to decay all stratification inhomogeneities arising in filling in, the buoyancy frequency was determined. Then, the sphere and the contact electric-conductivity sensors intended to measure internal waves were immersed. The sphere was pushed into the motion after the attenuation of all disturbances introduced (the state of the solution was monitored by optical methods). The sphere had oscillated at a constant frequency for a sufficiently long time (longer than 10 periods) until the stationary pattern of periodic waves arose. The experiments were carried out for different amplitudes (0.5, 0.7, 1, 1.2, 1.4, 1.7, 2, 2.7 cm) and oscillation frequencies (ω/N = 0.43, 0.51, 0.56, 0.6, 0.68, 0.75, 1, 1.05, 1.1). After the registration and measurement cycle were completed, the sphere was stopped in the tank. Each consequent experiment started 1 to 2 h later than the ceasing of the motion and the decay of all disturbances in the tank, which had been registered by optical and contact methods. Typical schlieren images of stationary patterns of flows around the sphere oscillating at a constant frequency ω < N are shown in Fig. 1. In the right-hand side of the frame, the sensitive component of the singleelectrode electric-conductivity sensor is seen. The sensor is installed in the inclined position and makes it possible to measure both in the wave beam and inside other flow components beneath the sphere. The dotted lines in Fig. 1 mark the horizontal levels of the body’s trajectory turning points. The basic image components are four inclined bimodal beams of periodic internal waves. In Fig. 1, they are represented by broad diffuse dark and light bands. By virtue of the axial symmetry, only central cross sections of conic internal-wave beams are visualized. This is explained by the fact that deviations of light rays in the medium are mutually compensated everywhere except the vertical plane. In it, the direction of the light-ray propagation coincides with tangents to wave phase surfaces. The internal-wave beams 5.7 cm in width, which are presented in Fig. 1a, are inclined at an angle θ = 47° to the horizon, which is consistent with the dispersion ω equation sin θ = ------0 = 0.73 [1]. A part of the liquid with N DOKLADY PHYSICS
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(a)
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Fig. 1. Schlieren patterns of flow induced by the sphere (D = 4.5 cm) performing forced periodic vertical oscillations in stratified liquid (the patterns are obtained by the slit-wire method): (a) H = 1 cm, Tb = 11.2 s, ω0 = 0.73 N, St = 0.01, Fr = 0.2, C = 700, M = 0.22, Mν = 0.85; (b) H = 2.8 cm, Tb = 7.3 s, ω0 = 0.8 N, St = 0.002, Fr = 0.62, C = 300, M = 0.62, Mν = 0.55.
the oscillating body at its center is the source of the waves; it is enveloped by a double gray rhomb-shaped line. Inside the light domain, the sphere surface is adjoined by residues of separated boundary layers. They are convolved near the poles into small vortices 1 to 2 mm in diameter, which can be seen in magnified images. The dark horizontal line attaining the center of the wave beam is seen under the sphere. Two other horizontal lines reside near the sphere equatorial plane. The traces of one further line are seen above the upper sphere pole. When the sphere moves, these lines periodically appear and disappear without changing the character of the density profile. All these components relate to migrating infinitesimal types of motion; after they have passed by, residual disturbances are not accumulated in the liquid. Double gray lines visualizing vertical tubular structures whose external edges lie on horizons of the wavebeam separation adjoin the sphere’s upper and lower poles (Fig. 1a). The external edges of these structures are located at horizons of the wave-beam separation. In the experiments carried out, these components always were observed and identified in all phases of the motion. The length of the tubular structures residing near the poles increases with the amplitude, and their external part begins to expand. This geometry corresponds to both the structure and the character of the motion in the domain of the singular-component intersection at the edges of the beam of internal waves characterized by the proper time and scale [7]. Thus, even for small amplitudes of the sphere motion, the schlieren instrument visualizes fine-structure components in the domain of the internal-wave beam intersection only directly on the line of motion of the body’s center [7]. The expressiveness of fine-structure motions, as well as the length and the diameter of central tubular DOKLADY PHYSICS
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structures, increases with the amplitude of the body’s oscillations. The external part of tubular structures becomes funnel-shaped and contacts with the wave cone. Novel structural components appear in the flow pattern at large stimulated-oscillation amplitudes (Fig. 1b). These components are similar to those observed previously for free sphere oscillations at the neutral-buoyancy horizon [2]. In the flow pattern presented in Fig. 1b, there exists a constant, extended light domain of a complicated shape between the oscillating body and four wave beams (i.e., cross sections of two wave cones, at an inclination angle of θ = 53° to the horizon). Inside it, two vertical gray lines that connect the particular phase surfaces of the conic beams can be traced. Immediately beneath the sphere, the rapid autocumulative jet is located. The jet has a high-gradient central core and a convex mushroom-shaped envelope similar to that observed in [2]. The jet altitude counted off from the sphere’s lower pole is 3.8 cm, and the envelope external diameter is 2 cm. The jet-envelope external edges transform into the particular phase surfaces of internal waves. The central tubular structures (double gray line in Fig. 1b) penetrate through the envelope of the mushroom-shaped autocumulative jet. The external edge of the autocumulative jet is formed in the wave field, which occurs when the intersection point of the conic phase surfaces moves to the oscillating body at the external boundary of the finestructure disturbances formed previously. For one oscillation period, the envelope shape is consequently transformed from the convex to concave one, the external edge of the envelope rapidly moving to the body. In the next oscillation cycle, the flow pattern is reproduced: a more extended autocumulative jet is formed and equally rapidly flows to the sphere surface. As was observed for free sphere oscillations, the jet length noticeably exceeds the double body’s oscillation amplitude [2]. For long times, the autocumulative jet loses its
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(a)
(b)
Fig. 2. Autocumulative jets arising in the case of sphere vertical oscillations in stratified liquid (D = 4.5 cm, slit-wire method): (a) forced oscillations at a constant frequency ω0 = 0.68 N, Tb = 11.2 s, H = 5.4 cm, St = 0.002, Fr = 0.2, C = 700, M = 1.2, Mν = 0.85; (b) Free damped oscillations of the sphere at the neutral buoyancy horizon: Tb = 8 s, H0 = 11.9 cm.
regular character due to the accumulation of fine-structure disturbances at the flow axis. The external edge of the central perturbed domain resides at a significant distance from the body. The typical flow pattern observed in the case of high-amplitude long-time sphere oscillations is given in Fig. 2a. The size and the inclination of beams, as well as the position of the dark bands, are determined by the oscillation phase. In the vicinity of the trajectory turning points, the enveloping boundary layers are separated from the body. The edges of these layers are convolved into ring-shaped vortices. The residues of the vortex high-gradient envelopes fill in the spherical domain immediately adjoined the body’s surface (Fig. 2a). As in the case of sphere-free oscillations [2], the spear-shaped autocumulative jet has a complicated internal structure formed by envelopes of several small vortices embedded one inside the other. The high-gradient central part of the stimulated autocumulative jet is surrounded by a sequence of band structures adjoining the jet at an angle of 31° to the vertical direction. The maximum diameter of the bandstructure domain is 3.4 cm at a distance of 4.5 cm from the sphere’s lower pole. In the experiments described, the sphere vibrates in the upper part of the field of vision. This allows us to observe extended singular components of the flow pattern in the lower part of the tank, where possible disturbance sources (i.e., the supporting wire, sensors, etc.) are absent. In the upper part of the tank, the flow pattern is more complicated insofar as fine-structure components generated by the oscillating wire that supports the sphere are present in it. For the sake of comparison, in Fig. 2b, the schlieren image of the first autocumulative jet is shown. This jet is formed in the case of free sphere oscillations under the conditions of neutral buoyancy. In all flow patterns shown in Figs. 1 and 2, axisymmetric fine-structure components whose lower edges are situated outside the body’s oscillation domain are
present. The process of the formation of these components is closely related to the motion of the phase surfaces of internal waves. Their contact points move to the source (sphere) along the central vertical line insofar as the phase velocities for all the waves are directed to the central horizontal plane. In contrast, singular disturbances and blocked liquid move from the body toward the waves. These disturbances determine the localization of domains of the maximum kinetic-energy dissipation rate and also significantly affect the vorticity distribution and the substance transport since the impurities, as a rule, are accumulated by high-gradient flow components. Indirect confirmation of this tendency is the appearance of lumps of gel produced due to the corrosion of tank aluminum walls in the vicinity of autocumulative jets (see Figs. 1a, 2a). Singular disturbances including centered tubular structures and autocumulative jets were observed in all the experiments carried out. In repeated experiments, the flow pattern was reproduced in all its details. The shape, the degree of the expressiveness, the localization domain, and the character of generated singular structures varied depending on the motion parameters. The formation rate and the length of autocumulative jets attain their maxima when the frequency of the body’s stimulated oscillations coincides with the buoyancy frequency. With a further increase in the stimulated-oscillation frequency, when effects of the formation of smallscale and large-scale ring-shaped vortices become significant, the autocumulative jets are not visualized. ACKNOWLEDGMENTS This work was supported in part by the Russian Academy of Sciences (OÉMMPU RAN OÉ-15) Program “Dynamics and Acoustics of Inhomogeneous Liquids, Gas-Liquid Mixtures, and Suspensions” and by the Russian Foundation for Basic Research, project no. 05-05-64090. DOKLADY PHYSICS
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6. Yu. D. Chashechkin and A. Yu. Vasil’ev, Dokl. Phys. 49, 122 (2004) [Dokl. Akad. Nauk 394, 621 (2004)]. 7. Yu. D. Chashechkin, A. Yu. Vasil’ev, and R. N. Bardakov, Dokl. Akad. Nauk 397, 404 (2004). 8. V. G. Baœdulov, P. V. Matyushin, and Yu. D. Chashechkin, Dokl. Phys. 50, 195 (2005) [Dokl. Akad. Nauk 401, 613 (2005)]. 9. S. A. Smirnov, Yu. D. Chashechkin, and Yu. S. Il’inykh, Izmer. Tekh., No. 6, 15 (1998).
Translated by G. Merzon
