c Pleiades Publishing, Ltd., 2018. ISSN 0021-8944, Journal of Applied Mechanics and Technical Physics, 2018, Vol. 59, No. 1, pp. 22–31. c V.G. Kozlov, S.V. Subbotin. Original Russian Text
STEADY FLOW GENERATED BY A CORE OSCILLATING IN A ROTATING SPHERICAL CAVITY V. G. Kozlov and S. V. Subbotin
UDC 532.5.01; 532.51; 532.526
Abstract: Steady flow generated by oscillations of an inner solid core in a fluid-filled rotating spherical cavity is experimentally studied. The core with density less than the fluid density is located near the center of the cavity and is acted upon by a centrifugal force. The gravity field directed perpendicular to the rotation axis leads to a stationary displacement of the core from the rotation axis. As a result, in the frame of reference attached to the cavity, the core performs circular oscillation with frequency equal to the rotation frequency, and its center moves along a circular trajectory in the equatorial plane around the center of the cavity. For the differential rotation of the core to be absent, one of the poles of the core is connected to the nearest pole of the cavity with a torsionally elastic, flexible fishing line. It is found that the oscillation of the core generates axisymmetric azimuthal fluid flow in the cavity which has the form of nested liquid columns rotating with different angular velocities. Comparison with the case of a free oscillating core which performs mean differential rotation suggests the existence of two mechanisms of flow generation (due to the differential rotation of the core in the Ekman layer and due to the oscillation of the core in the oscillating boundary layers). Keywords: rotation, inner core, oscillation, steady flow, differential rotation, inertial waves. DOI: 10.1134/S0021894418010042 INTRODUCTION The study of flows in rotating cavities is an important geophysical problem related to the problem of fluid motion in the atmospheres and cores of planets [1, 2]. The problem of fluid flow in a spherical layer between two concentric spheres rotating about a common axis with different angular velocities is classical [3]. In [4–6], the internal sphere (core) is located on the axis of the cavity and the differential rotation of the core is given from the outside. The vibrational mechanism of generation of mean rotation of a free inner core was discovered and investigated in [7, 8]. If the core oscillates relative to the cavity, mean stresses arise in the oscillating boundary layers which lead to differential rotation of the core. Depending on the oscillation frequency, both lagging and advancing rotation of the core is generated, accompanied by intense steady fluid flow [9]. It is of interest to study the oscillatory fluid motion generating steady flow in rotating cavities [10]. Fluid oscillation caused by tidal deformations (periodic change in shape) [11] or librations (periodic change in rotation velocity) [12] of a cavity, and precession [13] have been considered. The structure of the flow generated by a free core oscillating in a rotating spherical cavity has been investigated [14]. It has been shown that quasi-two-dimensional azimuthal fluid motion is generated in the cavity; before instability, the axisymmetric flow structure has the form of coaxial cylindrical surfaces rotating with different
Perm State Humanitarian Pedagogical University, Perm, 614000 Russia;
[email protected]; subbotin
[email protected]. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 59, No. 1, pp. 28–38, January–February, 2018. Original article submitted October 20,1, 2016; revision submitted November 22, 2016. 22
c 2018 by Pleiades Publishing, Ltd. 0021-8944/18/5901-0022
rL >rs 1
Wrot
2
rs
z
R1
b
3
R2
g
Fig. 1. Experimental setup: (1) chamber; (2) laser knife; (3) fishing line.
angular velocities. The intensity of differential rotation of the fluid is proportional to the velocity of differential rotation of the core ΔΩ. Because the free core performs differential rotation in addition to vibrations, it is necessary to investigate the contribution to the resulting motion from the steady flows generated by the oscillating boundary layers and Ekman boundary layers. The aim of this study is to investigate the structure of the flows generated by circular oscillations of the core in the absence of its own differential rotation. For this, the core is connected to one of the poles of the cavity with a thin fishing line which provides free oscillations of the core in the equatorial plane, but it prevents its differential rotation. In order to determine the contribution of the Ekman boundary layers, we investigate the flow structure for the given differential rotation of the core located in the center of the cavity.
EXPERIMENTAL SETUP AND TECHNIQUE A spherical body (core) of radius R1 = 1.77 cm and a mean density ρs = 0.22 g/cm3 is in a fluid-filled spherical cavity of radius R2 = 3.60 cm (Fig. 1). The cavity is milled in the center of a Plexiglas box with polished edges, through which observation is performed. A cuvette fixed in ball bearings rotates around the horizontal axis passing through its center. The inner diameter of one of the bearings is larger than the diameter of the cavity to make observation along the axis of rotation. The rotation of the cuvette is driven by a FL86STH118-6004A stepper motor, which is controlled with a Microstep Driver M542 driver using a Mastech HY5005E direct current source for power supply. The velocity is controlled with the alternating signal generator of the a Zet-210 Sigma USB unit connected to a computer. The angular velocity of rotation of the cavity is set to within 0.01 s−1 and is varied in the range Ωrot = 60–240 s−1 . Under the action of a centrifugal force in the laboratory frame, the core is in a stationary position near the axis of rotation, and under the action of gravity, the center of the body moves a distance b from the axis of rotation. In the frame of reference associated with the cavity, the core performs circular oscillation in the equatorial plane with amplitude b. The core is connected to one of the poles of the cuvette with a thin nylon fishing line of thickness d = 0.37 mm to prevent relative (differential) rotation. Aqueous glycerol solutions with different kinematic viscosity ν = 10−6 –12 · 10−6 m2 /s, but with the same density ρL = 1.18 g/cm3 are used as the working fluid. The fluid density is maintained constant to ensure neutral buoyancy of the imaging particles; when the viscosity (concentration of glycerol) is changed, the density is adjusted by changing the concentration of sodium salt. Fluid viscosity is measured with a VPZh-2 capillary viscometer up to 10−8 m2 /s, and density is measured with a hydrometer. The flow is visualized using plastic particles of neutral buoyancy (ρ ≈ 1.2 g/cm3 ) diameter d < 0.5 mm. 23
b, mm 6.0
1.0 1 2 3 4 5
0.1 50
100
200
300 Wrot, s-1
Fig. 2. Radial displacement of the body from the axis of the cavity versus the velocity of its rotation for ν = 3.3 · 10−6 (1), 5.3 · 10−6 (2) (free core) [14], 6.7 · 10−6 (3), 9.0 · 10−6 (4), and 11.1 · 10−6 m2 /s (5); the vertical segments shows the scatter of the values.
The experiments are carried out as follows. In order to avoid twisting of the fishing line and the formation of nodules on it, initially the cuvette is set in rapid rotation around the vertical axis, with the light core held in the center of the cavity with the fishing line. After that, the cuvette is put in the working position, i.e., rotated by an angle of 90◦ . In the experiment, the velocity of rotation of the cavity Ωrot is changed stepwise. At each step, the displacement of the light body from the center of the cavity b (see Fig. 1) and the flow structure are recorded using a photographic technique. The azimuthal velocity of the fluid is investigated using particle image velocimetry (PIV). For this, the fluid is illuminated by a 2 mm light knife generated by a Z-Laser Z500Q continuous laser. The beam direction is selected so that, in view of the refraction at the boundaries of the cavity, it dissected it in a plane perpendicular to the rotation axis. The coordinate of the light knife varies in the range z/R2 = 0.08–0.80 (z is the distance from the center of the cavity to the plane of the light knife). Video recording was carried out using a CamRecord CL600x2 high-speed video camera (resolution of 700 × 700 pixels, the frequency equal to the frequency of rotation of the cavity) stationary in the laboratory frame and a DVR express core image acquisition system. Pairs of photographs taken at a time interval equal to the period of rotation of the cavity are processed using the PIVLab program [15]. As a result, the mean fluid velocity over one period is calculated in the frame of reference associated with the cavity. It should be noted that the velocity of differential rotation of the fluid is low compared with the velocity of rotation of the cavity: ΔΩL /Ωrot 1, and one period of the fluid is displaced insignificantly relative to the cavity. Given that the differential motion of the fluid is axisymmetric and does not change with time, we performed an additional averaging of at least 100 measurements to increase the accuracy of the results. RESULTS AND DISCUSSION Experiments were performed to study the oscillations of the core and the steady fluid flow in the cavity. Position of the Core in the Cavity If the cavity is rotated around a horizontal axis at a high velocity, the core whose density is less than the fluid density is shifted a distance b/R1 < 0.3 from the rotation axis in the laboratory frame. With an increase in the rotational velocity, the displacement of the core b from the center of the cavity decreases as b ∼ Ω−2 rot (Fig 2). In this case, the results of experiments with fluids of different viscosity are in good agreement with the results obtained in experiments with a free core [14]. Therefore, the presence of a fishing line does not affect the position of the body in the cavity but only prevents its differential rotation. It is shown below that the deviation of the results obtained for low-viscosity fluids for rotation of the cavity at a low velocity (Ωrot < 100 s−1 ) is due to a change in the structure of the steady flows as a result of instability of axisymmetric motion. 24
y/R2 1.0
DvL, mm/s
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(b)
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0.5 _4 _6
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_8
1 2 3 4 5 6 7 8
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0.5 x/R2 1.0
0 6
8
10
12
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_16
0
0.5
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1.0
Fig. 3. Azimuthal velocity vector field at a distance from the equatorial plane z/R2 = 0.53 (a) and the azimuthal fluid velocity profile ΔvL in different cross sections z/R2 (b) for ν = 5.4 · 10−6 m2 /s, Ωrot = 188 s−1 , b = 0.32 mm: z/R2 = 0.17 (1), 0.22 (2), 0.33 (3), 0.39 (4), 0.50 (5), 0.56 (6), 0.67 (7), and 0.72 (8).
Steady Fluid Flow in the Cavity Oscillations of the core relative to the cavity lead to oscillatory motion of the fluid in it. Nonlinear effects in the oscillating boundary layers near the solid boundaries of the body and in the cavity generate steady flow that drives the fluid outside the boundary layers. The vector field at a distance from the equatorial plane z/R2 = 0.53 and the profile of the averaged azimuthal fluid velocity ΔvL for rotation of the cavity in a clockwise direction are shown in Fig. 3. Lagging differential rotation occurs in almost the entire volume of the fluid, and the azimuthal velocity ΔvL non-monotonically depends on the distance r/R2 from the rotation axis. The most rapid lagging rotation (first maximum) is observed in the central portion of the cavity at a distance r/R2 ≈ 0.1, the second the maximum at r/R2 ≈ 0.5, and the third maximum, larger than the previous one, is located at a distance r/R2 ≈ 0.9. The distribution of the azimuthal velocity in the cavity volume, except for the areas located in the vicinity of the solid boundaries of the cavity and the core is virtually unchanged with a change in the coordinate z/R2 , indicating that the flow has a two-dimensional structure. Figure 3b shows a near-equatorial region |z/R2 | < 0.4, r/R2 > 0.8 which is a kind of a pocket in which there is rapid lagging motion. Generally, the flow structure has the form of several coaxial cylindrical surfaces nested in each other and rotating with different angular velocities. Such steady fluid flow in the form of nested columns is generated not only in the case of circular oscillations of an inner core [14], but also in other types of periodic perturbations in rotating systems [10]. For example, in a precessing spheroidal cavity, along with the lagging motion in the fluid, intense advancing motion occurs at a distance r/R2 = 0.86 [13]. Similarly, the presence of librations [12] and periodic deformations of a rotating cavity [11] leads to generation of zonal flow in the form of fluid columns. Distribution of Heavy Particles at the Inner Boundary of the Cavity If the density of the imaging particles is greater than the fluid density, the particles settle on the inner surface of the cavity under the influence of a centrifugal force. The particles aggregate into axisymmetric annular structures (Fig. 4). In each hemisphere, three concentric rings at |z/R2 | = 0.78, 0.60, and 0.36 are formed (see 25
(a)
(b)
3 1
2
o
3 2
1
1 2
Fig. 4. Distribution of heavy particles on the wall of the cavity (ν = 5.2 · 10−6 m2 /s and Ωrot = 126 s−1 ) along the axis of rotation (a) and in a plane perpendicular to the axis of rotation (b); the solid and dotted curves are the conical surfaces along which internal and external inertial waves, respectively, propagate, the dashed curves are the lines of intersection of inertial waves with the surface of the cavity; (1) |z/R2 | = 0.78; (2) |z/R2 | = 0.60; (3) |z/R2 | = 0.36.
Fig. 4b). It should be noted that in this case, the particle distribution coincides with the distribution in the case of a free oscillating core, i.e., a core which performs differential rotation [14]. This distribution is due to the presence of internal inertial waves (dotted curves in Fig. 4b) generated by an oscillating core [14, 16]. Inertial waves originate at critical latitudes on the core surface, where there is breaking of the boundary layer, and propagate along characteristic surfaces of a conical shape formed by oscillating shear layers. The angle between the generatrix of the cone and the axis of rotation depends on the ratio of the frequencies Ωosc /Ωrot (Ωosc is the oscillation frequency) as follows: θ = arcsin (Ωosc /(2Ωrot )) [16]. In the present case, |Ωosc | = Ωrot , θ = 30◦ , and the position of the critical latitudes on the surface of the core corresponds to the same angle θ = 30◦ [17]. It is evident from Fig. 4 that the position of the rings 1 and 2 is in satisfactory agreement with the position of the lines of incidence of the inertial wave on the inner boundary of the cavity. Presumably, in the neighborhood of the point of incidence in the oscillating boundary layers near the wall, steady flows occur which lead to a redistribution of the imaging particles. The presence of rings 3 in Fig. 4b may be due to the structure of steady flows in the pockets (see Fig. 3). Mechanism of Generation of Steady Flows The motion of a free cylindrical core in a cylinder rotating around a horizontal axis (the core performs circular oscillations relative to the cavity under gravity) has been theoretically and experimentally investigated [7]. It has been shown that the core performs differential rotation under the action of the averaged shear stresses generated in the oscillating boundary layers at the solid boundaries of the core and the cavity. A free spherical core in a rotating spherical cavity [14] behaves similarly. In both cases, the differential rotation velocity of the core |ΔΩ|/Ωrot is given by the ratio b2 |ΔΩ| , ∼ Ωrot R1 δ
where b is the amplitude of oscillations of the core relative to the cavity, δ ≡ 2ν/Ωrot is the thickness of the oscillating boundary layer. For a spherical core of size R1 /R2 = 0.49 in the region of stable axisymmetric motion, the relation |ΔΩ|/Ωrot = 0.17b2 /(R1 δ) holds. It has been found [14] that the velocity of the fluid ΔΩL ≡ ΔvL /r in a spherical layer is proportional to ΔΩ. In the case under consideration, the core does not perform differential 26
jDWLmaxj/Wrot 10-2
1 2 3 4 5 6 7 8
10-3
10-4
10-2
10-1
b2/(R1d)
Fig. 5. Differential fluid rotation velocity at the extremum at r/R2 = 0.5 (see Fig. 3b) versus parameter b2 /(R1 δ) at different kinematic viscosities: ν = 2.9 · 10−6 (1), 3.3 · 10−6 (2), 4.1 · 10−6 (3), 5.4 · 10−6 (4), 5.8 · 10−6 (5), 6.7 · 10−6 (6), 9.0 · 10−6 (7), and 11.1 · 10−6 m2 /s (8); the vertical segments show the scatter of values.
rotation, but oscillates relative to the cavity with an amplitude b. It can be assumed, that in the absence of differential rotation of the core, the parameter b2 /(R1 δ) also determines the dimensionless azimuthal velocity of the fluid flow. The azimuthal velocity of the lagging fluid motion at the extremum |ΔΩL max |/Ωrot at a distance r/R2 = 0.5 does not depend on its viscosity; the results of all the experiments are in good agreement in the plane of the dimensionless parameters b2 /(R1 δ) and |ΔΩL max |/Ωrot (Fig. 5). With increasing values of the parameter b2 /(R1 δ), the fluid velocity increases linearly. This indicates during oscillations of a core in a rotating cavity, the mechanism of generation of steady flow is determined by nonlinear effects in the oscillating boundary layers and does not depend on the differential rotation of the core. Similar quadratic dependence of the intensity of azimuthal flows on the oscillation amplitude is also typical of other types of periodic perturbations in rotating systems (precession, libration, elliptic deformation) [10]. Note that at a fixed value of the amplitude of oscillations of the core b/R1 , the 1/2 fluid velocity in geostrophic cylinders varies as |ΔΩ|/Ωrot ∼ Ωrot R1 /ν 1/2 ∼ E−1/2 [E ≡ ν/(Ωrot R12 ) is the Ekman number]. It has been shown [8, 18] that the stability of the steady axisymmetric flow generated by oscillations of a free core depends on the Reynolds number Re ≡ |ΔΩ|R12 /ν. Instability manifests itself in a series of critical transitions. The first threshold value at Re = 54 is due to the development of vortices inside the Taylor column [8]. Experiments are conducted for small Ekman numbers E < 10−3 . Given that the velocity ΔΩ is proportional to the fluid velocity ΔΩL [14], which is, in turn, determined by the amplitude of oscillation of the core ΔΩL ∼ Ωrot b2 /(R1 δ) (see Fig. 5) for the case of an oscillating but not rotating core, we introduce the Reynolds number in the form Reb ≡
b2 Ωrot R1 . δν
Since in the case of a free core, the amplitude of oscillations of the body is virtually not different from the oscillation amplitude in the case where the differential rotation is restrained by an elastic fishing line (see Fig. 2), the Reynolds numbers Re and Reb are related as Re = 0.17 Reb . For large values of Reb , the axial symmetry of the flow is violated; in this paper, this range of Reynolds numbers is not considered, and the results of stable axisymmetric motion of fluid at Reb < 320 are discussed.
27
DWL R1d/(Wrotb2 )
0
_0.1
_0.2
1 2 3 4 5
_0.3
_0.4
0
0.5
1.0 r/R2
Fig. 6. Dimensionless angular velocity of axisymmetric fluid motion in the cross section z/R2 = 0.56 versus r/R2 for different values of the kinematic viscosity and Reynolds number: (1) ν = 11.1 × 10−6 m2 /s and Reb = 105; (2) ν = 9.0 · 10−6 m2 /s and Reb = 135; (3) ν = 6.7 · 10−6 m2 /s and Reb = 162; (4) ν = 5.4 · 10−6 m2 /s and Reb = 256; (5) ν = 5.9 · 10−6 m2 /s and Reb = 283.
It follows from Fig. 5 that the angular velocity of differential rotation of the fluid is given by the formula Ωrot b2 /(R1 δ). The dimensionless angular fluid velocity profiles are shown in Fig. 6. In the plane of the selected parameters, the experimental points obtained for different values of the fluid viscosity and the amplitude of oscillations of the core are in good agreement with each other. The obtained profile is universal for stable axisymmetric fluid motion in the range of small Ekman numbers E < 10−3 . Effect of the Differential Rotation of the Core on the Flow Structure. The Principle of Superposition It has been found [8] that in the case of a free core, the oscillating boundary layers, on the one hand, generate two-dimensional azimuthal flow in the fluid [14] and, on the other case, cause differential rotation of the core itself. This rotation, in turn, should cause fluid flow (ΔΩE L ) in the form of a Taylor–Proudman column [19] and ). To estimate this contribution, we to make a contribution to the resultant flow generated by the free core (ΔΩFree L performed additional velocity measurement for the case of predetermined differential rotation of a core located in the center of a cavity. A spherical body of radius R1 = 1.27 cm is fixed in the geometric center of the cavity using a metal rod of diameter 4.0 mm located on the axis of rotation. The radius of the cavity is equal to R2 = 2.60 cm, and the relative size of the core has the same value as in the experiments with a body on a fishing line: R1 /R2 = 0.49. The rod mounted in sealing gaskets can freely rotate together with the core relative to the cavity. The rotation of the core Ωs with an accuracy of 0.01 s−1 is driven by a stepper motor, whose shaft is connected to the rod. The rotation velocity of the cuvette Ωrot with the same accuracy is set independently from the core rotation velocity by an additional stepper motor through belt transmission. The experiments are carried out as follows. Initially the cuvette and the core are set in uniform rotation at the same velocities (Ωrot = Ωs ). Next, at a constant value Ωrot , the rotation velocity of the core Ωs gradually decreases by a value |ΔΩ|. After the establishment of steady motion of the fluid, the azimuthal velocity of the fluid is determined using the above-described PIV-method. 28
DWL /jDWj
0 _0.2 1 2 3 4 5
_0.4 _0.6 0
1.0 r/R2
0.5
Fig. 7. Fluid velocity profile at a given velocity of differential rotation of the core ΔΩ (in the absence of its oscillations) for ν = 2.1 · 10−6 m2 /s, Ωrot = 113 s−1 , and different values of the Reynolds numbers Re and the coordinate z/R2 : (1) Re = 27 and z/R2 = 0.56; (2) Re = 44 and z/R2 = 0.56; (3) Re = 65 and z/R2 = 0.56; (4) Re = 65 and z/R2 = 0.17; (5) Re = 100 and z/R2 = 0.56. Free/W ).102 (DWL rot
0
_0.5
_1.0
1 2
_1.5
_2.0
0
0.5
1.0 r/R2
Fig. 8. Averaged angular fluid velocity profiles in the case of a free oscillating core (1) [14] and in the case of −1 ≡ ΔΩL + ΔΩE = 11 000, Re = 44, and Reb = 283. calculation by the formula ΔΩFree L L (2) for z/R2 = 0.56, E
For |ΔΩ| = 0, the fluid, the core, and the cavity rotate as a rigid body. For small values of |ΔΩ|, the flow is axisymmetric and has the form of a Taylor–Proudman column. The dimensionless velocity profiles for various values of Re are shown in Fig. 7. Outside the column, the fluid rotates substantially as a rigid body, together with the cuvette inside the column, at a velocity ΔΩE L ≈ |ΔΩ|/2. The continuity of the azimuthal fluid velocity component inside and outside of the Taylor–Proudman column is provided by Stewartson shear layers. The obtained velocity profiles are in good agreement with the results of theoretical studies [19], which show that at low Ekman numbers E 1, the fluid inside the column is rotated at a velocity which is intermediate between the velocities of the outer and inner spheres. When using the differential rotation velocity of the core |ΔΩ| as the measurement unit for the fluid velocity ΔΩE L , the velocity profiles for different rotation velocities of the inner core agree with each other. The results are valid for the region of stability of the axisymmetric motion. 29
Comparison of the profiles shows that the flow generated by an oscillating core (see Fig. 6) is qualitatively different from the flow in the case of given differential rotation (see Fig. 7). In the first case, the steady flow has the form of nested coaxial columns rotating at different velocities, and in the second case, the flow structure has the form of a Taylor–Proudman column elongated along the rotation axis. It can be assumed that under the simultaneous action of the two mechanisms involved in the oscillation and differential rotation of the core, the resulting fluid motion is a linear superposition of the flows described above. This is true in the region of stable axisymmetric motion in the limiting case of low Ekman numbers. Then we represent the velocity ΔΩFree of the L fluid motion generated by a free oscillating core as the sum ΔΩL + ΔΩE . Figure 8 shows the velocity profile for a L free oscillating core [14] and the profile, resulting from the addition of the two fluid velocity components (points 2), where E−1 = 11 000, Re = 44, and Reb = 283. It is seen that the flow structures are similar and the velocities ΔΩFree are in good agreement over the entire range of r/R2 . In both cases, the differential rotation velocity of the L core ΔΩ is the same.
CONCLUSIONS The steady flow in a rotating spherical cavity generated by circular oscillations of a core in the absence of its own differential rotation was experimentally studied. It is found that the structure of the steady flow consists of several nested fluid columns rotating with different angular velocities. This flow is caused by the presence of nonlinear effects in the oscillating boundary layers. The intensity of the azimuthal motion of the fluid in the columns is proportional to the square of the amplitude of oscillations of the core. The findings support the hypothesis that the steady fluid motion generated by a free oscillating core [14] is a superposition of the motions generated by the differential rotation of the core (in the form of a Taylor– Proudman column under the action of the Ekman flow generation mechanism) and the steady motion generated in the oscillating boundary layers near the boundaries of the cavity and the core. Comparison with the results of studies for a free core [14] shows that the mechanisms of flow generation in the region of stability of axisymmetric motion are independent. This work was supported by the Russian Science Foundation (Project No. 14-11-00476).
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