Literatures Review

In this study, the behavior of flow motion for the following cases is investigated: (1) rotation of an inner cylinder, which remains static in an outer cylinder, (2) the inner and outer cylinders rotate at different speeds, and (3) the rotation of the inner cylinder is

periodic and modulated.

Donnelly[3] and Simon and Donnelly[4] utilized the torque produced in a cylinder during viscous flow motion to measure the circular Couette flow at the critical point between stable and unstable flow. As the rotational speed of the inner cylinder increases, the torque suddenly changes when the fluid status changes from stable to unstable.

From this, the Reynolds number of the critical point can be acquired. The critical point marks the transition from one-dimensional stable Couette flow to two-dimensional Taylor vortex flow. Koschmieder[5], Burkhalter and Koschmieder[6], and Swinney and Gollub[7] obtained the same result by observing flows. For inner and outer cylinders with the same rotational speed and direction, Taylor[2] experimentally demonstrated that when the rotational speed increases gradually to a speed exceeding the critical speed, the flow still retains its one-dimensional flow status. In other words, the rotating outer cylinder has inhibition to stable status. Coles[8], Schwarz et al.[9], and Nissan et al.[10] obtained the same experimental result as Taylor. Moreover, Walsh and Donnelly[11] demonstrated that when the rotational speed of the inner cylinder is fixed and the outer cylinder undergoes periodic motion, the flow is stabilized. Gollub and Swinney[12] and Walden and Donnelly[13] used the Laser Doppler Anemometer (LDA) to investigate circular Taylor-Couette flow by measuring the radial temperature of flow measurement points.

Through power-spectrum analysis, the time domain is transformed into a frequency domain. The advantage of power spectrum analysis is that different characteristics of a spectrum represent different flow states. When flow is periodic, peaks of a certain size appear in the spectrum at relative and harmonic frequencies. When flow transitions into quasi-periodic flow, a new frequency appears that is not associated with the original frequency. The power spectrum analysis is an efficient approach for studying the

transformation between quasi-periodic and chaotic flows. Cole[14] experimentally investigated the effect of cylinder height on flow stability and demonstrated that cylinder height does not influence the critical point for transformation from a Couette flow to a Taylor vortex flow, unless the aspect ratio between cylinder height and interval is <8. Additionally, according to the study by Hall and Blennerhasset[15], when the aspect ratio L/d ≥12 12, the numerical and experimental results are not significantly different, indicating that the effect of cylinder height on flow stability is negligible. Barenghi and Jones[16] and Murray et al.[17] obtained the same result numerically. Walowit et al.[18] applied linear theory to derive the critical value for different radius ratios and the ratio between inner and outer rotational speeds. To examine the stability of a Couette flow between cylinders with different radial temperatures, Snyder and Karlsson[19] experimentally determined the critical Taylor number (Ta =2d³R₁(Ω/ν)²) under different temperatures with η=0.958. They found that the result is stable over a small temperature range. Outside of this range, the positive and negative values of the critical Taylor number become unbalanced, and the flow typically becomes unstable.

The present study focuses primarily on the influence of modulated amplitude and frequency on flow stability between the cylinders. The rotational speed of the cylinder is

modulated by the factor Ω⁻(1+εcosω^'t) to investigate the effects of flow stabilization and destabilization. Donnelly[20] experimentally analyzed modulated flow stability and found that when the outer cylinder remains static and the inner cylinder rotates periodically, parameters such as interval, rotational frequency, and modulation amplitude of the two cylinders can be modified to determine how the circular Couette flow is affected by modulated rotation. Hall[21] utilized linear theory to determine low and high frequencies and used non-linear theory to analyze the flow for high frequency

modulations. Hall demonstrated that flow is slightly destabilized regardless of amplitude for low frequency modulations. Within a certain frequency range, the flow becomes increasingly stable. When the rotation is modulated at high frequency, the stability approaches the critical value of the non-modulated situation. Riley and Laurence[22] utilized Galerkin expansion and Floquet theory for stability analysis and investigated flow stability under modulated conditions with a narrow inter-cylinder gap.

The numerical result obtained by Riley and Laurence is the same as that acquired by Hall[21]. Davis[23] developed the notion of the quasi-steady limit by showing that a modulated inner cylinder destabilizes flow. Under extremely low frequency, the critical stability value of a flow declines to 1/(1+ε) of the non-modulated situation. Carmi and Tustaniwskyj[24] examined modulated flow stability under small-gap conditions and the influence of axis symmetry on modulated flow. In their study, the unstable offset of the critical Reynolds number increases at low frequencies. At medium-to-high frequencies, no stable critical value appears. Walsh and Donnelly[11] analyzed the flow between inner and outer cylinders with different radii using the photo voltage of observation particles. They determined that the critical Reynolds number has a relatively large offset at low frequency. In conclusion, a critical Reynolds number lower than the theoretical value indicates that the flow is temporarily unstable, but not permanently unstable. Therefore, the stable critical value is lower than the theoretical value at low frequency.

Walsh et al.[25] measured the critical Reynolds number for concentric cylinders with different radius ratios and, multiplying the result by a factor, obtained roughly the same value as that calculated by Carmi and Tustaniwskyj[24]. Kuhlmann et al.[26]

utilized a low-dimensional model to examine the stability of circular Couette flow and Taylor vortex flow under instability. They found that a large modulated amplitude

causes subharmonic perturbation. Ganske et al.[27] used a static outer cylinder and a modulated vortex flow in the inner cylinder to investigate the influence on stability of different amplitudes. They demonstrated that the modulated effect causes the flow to become increasingly unstable, and that the effect of amplitude on flow stability is significantly more important than the effect of frequency. When the amplitude is large, the unstable effect increases and the critical stability value declines further. Meksyn[28]

used numerical methods to predict the occurrence of instability when the inner and outer cylinders rotated in either the same or opposite directions. Sparrow et al.[29] also applied numerical methods to investigate the same and opposite flows around inner and outer cylinders for a radius ratio of 0.95–0.1. Youd et al.[30], who analyzed zero-equivalent modulated flow around concentric cylinders with a radius ratio of

75 .

=0

η , identified the axis symmetry of the Taylor vortex.

After the Taylor vortex problem had approached nonlinearity for many years, Coles[31] brought it decisively into nonlinearity in reporting the non-uniqueness of the wavy flow in the Taylor-Couette flow. The entire pattern of wavy vortices moves with uniform velocity in the azimuthal direction. Because the term “wavy” is typically associated with a motion that has periodic vertical oscillation, this study emphasizes that wavy Taylor vortices move in the azimuthal direction as rings that have an integer number k₁ of fixed sinusoidal upward and downward deformations (k₁ is the integer number of azimuthal waves). An example of a wavy Taylor vortex flow, as observed by Taylor[2], Lewis[32], Coles[31], and Schultz-Grunow and Hein[33], was not recognized as a characteristic new feature of the flow. After Coles’ preliminary results were published, wavy vortices were also observed by Nissan et al.[10]. Schwarz et al.[9]

reported experiments in which they observed an asymmetrical mode with k₁ =1.

Burkhalter and Koschmieder[6] concluded that the wavelength of axisymmetrical

vortices with large radius ratios is independent of the Reynolds number in fluid columns of infinite length when the Reynolds number increases quasi-steadily. However, the wavelength of Taylor vortices is constant only as long as the flow is quasi-static.

Jones[34] reported the stability boundary for wave number α =3.13, the critical value of a quasi-static transition, for a wide range of η.

Although Taylor analyzed the flow under supercritical conditions, Stuart[35]

concluded that the vortex size remains unchanged above the critical Reynolds number.

However, numerous studies (see Ahlers et al.[36], Andereck et al.[37], Park et al.[38], Burkhalter and Koschmieder [6, 39], and Antonijoan[40]) have demonstrated the importance of acceleration/deceleration in determining the final state of the flow. These vortices have axial wavelengths that are shorter or longer than those obtained after a quasi-static transition. This study demonstrates that the lowest stability boundary occurs at the critical wavelength of a quasi-static transition and also in another. These solutions are connected with standard Taylor vortices and can be obtained quasi-statically for certain radius ratios when using a mechanism to modify the axial wavelength (see Ref.

[41]).