Eckhaus boundary and wave-number selection in rotating Couette-Taylor flow

Optimal Taylor–Couette flow: radius ratio dependence

“…Instabilities (Swinney & Gollub 1981;Pfister & Rehberg 1981;Pfister et al 1988;Chandrasekhar 1981;Drazin & Reid 1981;Busse 1967), nonlinear dynamics and chaos (Lorenz 1963;Ahlers 1974;Behringer 1985;Dominguez-Lerma et al 1986;Strogatz 1994), pattern formation (Andereck et al 1986;Cross & Hohenberg 1993;Bodenschatz et al 2000), and turbulence (Siggia 1994;Grossmann & Lohse 2000;Kadanoff 2001;Lathrop et al 1992b;Ahlers et al 2009;Lohse & Xia 2010) have been studied in both TC and RB and both numerically and experimentally. The main reasons behind the popularity of these systems are, in addition to the fact that they are closed systems, as mentioned previously, their simplicity due to the high amount of symmetries present.…”

Boundary limitation of wave numbers in Taylor-vortex flow

“…One of the successes in the study of pattern-forming systems is the agreement between the calculated and measured k E (ǫ) for TVF which has been found with three different ratios of the cylinder radii. [4,7,9,10,14] Theoretically, phase pinning (and thus the Eckhaus instability) at small ǫ is expected when the boundary conditions at the system ends z = 0, L correspond to a large amplitude A(0) = A(L) = A 0 of the velocity field, say A 0 = O(1), while in the system interior the amplitude A(z) is small, say O(ǫ 1/2 ). This situation closely corresponds to the TVF system with rigid ends where the influence of the Ekman vortex can be approximated by this boundary condition.…”

“…The (more narrow) stable band of states is limited by the long-wavelength Eckhaus instability. [3,4,5,6,7,8,9,10,11,12,13,14] The Eckhaus instability is a bulk instability which manifests itself in the system interior where one pattern wavelength (one pair of Taylor vortices) is either gained or lost, depending on whether k is larger or smaller than the stable band limited by k E (ǫ). One of the successes in the study of pattern-forming systems is the agreement between the calculated and measured k E (ǫ) for TVF which has been found with three different ratios of the cylinder radii.…”

“…This situation closely corresponds to the TVF system with rigid ends where the influence of the Ekman vortex can be approximated by this boundary condition. [15,7,10] The Eckhaus-stable band then has the same width as the band of stable states for the infinite system, [16] but the number of states is finite since only discrete states with an integer number of vortices N = 2L/λ = Lk/π can occur.…”

“…Using rigid, non-rotating ends, we reproduced the well known Eckhaus mechanism and stability range. [7,8]. We then investigated a system with a free liquid-air interface which terminated the fluid axially in an apparatus oriented vertically with its axis.…”