Determination of the wave structure of closed flows with nonuniform rotation of the boundaries by the method of an instantaneous phase difference

Abstract

Unsteady axially symmetric flows of a viscous incompressible liquid in a spherical layer, which are formed by modulation of the rotation rate of one of the spherical boundaries, are considered. The wave struc tures of such flows are investigated by the method based on determination of the instantaneous difference in phases between the sphere velocity and the azimuthal velocity at each flow point. The steady state character of the distribution of the instantaneous phase difference in the meridional plane of flows is established. The possibility of applying the method at very low vibrational amplitudes is shown.

Nonlinear Interaction of Waves in Rotating Spherical Layers

“…In this work, such a method was used to calculate the phase of the azimuthal component u φ of the velocity at each point of the layer Ψ(t,r,θ) and phase of rotation of the oscillating sphere Ψ s (t). It was shown in [3] that the described method for analyzing the structure of the flow makes it possible to adequately reproduce inertial waves previously obtained in experiments and numerical calculations. It was also demonstrated in [3] that harmonic oscillations of the inner sphere with small amplitudes generate radially damping spherical waves (figure 1а).…”

“…It was shown in [3] that the described method for analyzing the structure of the flow makes it possible to adequately reproduce inertial waves previously obtained in experiments and numerical calculations. It was also demonstrated in [3] that harmonic oscillations of the inner sphere with small amplitudes generate radially damping spherical waves (figure 1а). The distance between phase jumps along radii coincides with half of the wavelength λ, which is calculated as the wavelength ( ) 1/ 2 2 2 f λ = πδ = π ν π in the Stokes problem on a flow generated by harmonic oscillations of an infinite plate at the frequency f [11], δ is the thickness of the dynamic boundary layer.…”

“…r t r t ψ θ = Ψ θ − Ψ at the point with the coordinates (r,θ) with respect to the phase of oscillations of the boundary (Ψ s (t) =2πft) is independent of the time [3]. Figure 2а shows the wave surfaces Ψ(t 3 ,r,θ) at a certain time instant t 3 (shown in figure 3с) at two-frequency small-amplitude oscillations.…”

“…The periodic (singlefrequency) modulation of the rotation velocity of one of the spherical boundaries in the case of their quite fast unidirectional rotation can result in the formation of inertial waves in a flow between these boundaries [2]. In the case of the absence of average rotation and small amplitudes, the rotational oscillations of the inner sphere at a single frequency produce damping spherical waves [3]. However, the time dependence of the rotation velocity of natural objects can be more complex.…”

“…Figure 2а shows the wave surfaces Ψ(t 3 ,r,θ) at a certain time instant t 3 (shown in figure 3с) at two-frequency small-amplitude oscillations. These surfaces are also attributed to a spherical wave, but the spherical layer in this case can be divided into three sublayers: a sublayer adjacent to the inner sphere where Ψ(t 3 ,r,θ) > 41, a sublayer adjacent to the outer sphere where Ψ(t 3 ,r,θ) < 9,5, and a narrow transient layer (shown in figure 2а in white) with a large phase gradient 9.5 < Ψ(t 3 ,r,θ) < 41, which increases in time (figure 2с shows the dependence of the phase of oscillations on the radius in the equatorial plane for three time instants, where the transient layer is shown in gray).…”