Taylor Vortex Flow

A classic pattern formation is Taylor vortex flow, which is obtained between two concentric rotating cylinders. In the simplest case only the inner cylinder is rotating. An experimental set-up is shown in the figure below. In this case a slight modification has been applied: one of the cylinders has been tapered. This reduces the local Taylor number, which measures the effective rotation rate, in the lower section of the system with the effect that only in the upper part vortices form.




Picture courtesy G. Ahlers, UCSB.

Comparison between the theoretical stability boundary of axisymmetric Taylor vortex flow with respect to long-wavelength perturbations (Eckhaus instability) and experimental results in a set of straight cylinders. The vertical axis gives the reduced Reynolds number and the horizontal axis the wavenumber q of the pattern. The dotted line indicates the theoretical neutral curve above which perturbations with the indicated wave number grow and form vortices. The two solid lines give the theoretical stability limits of these vortices as obtained in a weakly nonlinear theory and by a full numerical Galerkin procedure. The experimental results are indicated by plusses.
Experiment: M.A. Dominguez-Lerma, D.S. Cannell and G. Ahlers, Phys. Rev. A 34 (1986) 4956.
Theory: H. Riecke and H.-G. Paap, Phys. Rev. A 33 (1986) 547; Phys. Rev. Lett. 59 (1987) 2570.

The stability limit is given by the Eckhaus instability, which is a long-wave sideband instability. It usually does not saturate and eventually leads to the destruction or creation of a vortex pair. Such a process is shown in the sequence of figures below.


Photos: courtesy L. Ning and G. Ahlers, UCSB.

As time progresses from left to right a vortex pair located just below the thin white marking line is destroyed leaving behind a vortex pair with larger wavelength. In the usual set-up, eventually the pattern will equilibrate to a spatially constant wavelength.

The long-wave Eckhaus instability can also be captured by the phase diffusion equation
TF = D(q) X2F.
It can be derived arbitrarily far from threshold, i.e. even in the strongly nonlinear regime. Here X = ex and T = e2t are slow spatial and temporal variables with e << 1 characterizing the slowness of the spatial and temporal variations. The phase F gives the local wavenumber q(X,T) via
q = XF.
Axisymmetric Taylor vortex flow is described by the streamfunction y for the vortical motion and the azimuthal velocity v . The diffusion coefficient is given by
D(q) = B
A
 ,
where A and B denote large expressions in terms of the spatially periodic solutions y(F(X,T)/e) and v(F(X,T)/e) ,
A = < V1+rfv+V2+( 1
r
 q2f2+4 ~
D
 
 )fy >

B
=
< V1+ é
ê
ë 		- r
2
 (1+2qq)fv+ æ
ç
è 		h2-m
h(1+h)
 - 1-h
2r
 ö
÷
ø 		qy+yyqv-qyyv ö
÷
ø 		> +
+
< V2+ é
ê
ë 		æ
ç
è 		q2
r
 f2+4 ~
D
 
 ö
÷
ø 		(1+2qq)fy-2 q2
r
 f3y
+
ì
í
î 		Th2(1+h)
2(h2-m)
 v
r
 + Th3
2(h2-m)(1-h)
 æ
ç
è 		1-m
r2
 - h2-m
h2
 ö
÷
ø 		ü
ý
þ 		qv
+
yy ì
í
î 		æ
ç
è 		4
r
 ~
D
 
 + 3q2
r2
 f2 ö
÷
ø 		qy ü
ý
þ 		-qfyy 1
r2
 f- æ
ç
è 		2q2fyy 1
r2
 f+q2f2y 1
r2
 y+4y 1
r
 ~
D
 
 y ö
÷
ø 		qy ] > .
and < ... > denoting the average over one period of the solution (for details see H.-G. Paap and H. Riecke, Phys. Fluids 3 (1991) 1519).

When the gap between the two cylinders is wider (radius ratio=1/2) the stability limits become more asymmetrical due to a resonance between modes with wavenumber q and wavenubmer 2q. It results in a significant and relatively sudden deviation of the fully nonlinear theory (solid line) from the weakly nonlinear theory (dashed line) for low wavenumbers.

In a system like that depicted in the top figure, in which one or both of the cylinders are tapered providing a spatial ramp in the Taylor number the band of stable wavenumbers collapses to a single wavenumber. Experimental and theoretical results are given in the figure below (theory: solid line, experiment: plusses). The wave numbers selected by ramps employing different ratios of the tapering angles of the two cylinders are shown as dashed lines.
Theory: H. Riecke and H.-G. Paap, Phys. Rev. A 33 (1986) 547; Phys. Rev. Lett. 59 (1987) 2570.
Experiment: M.A. Dominguez-Lerma, D.S. Cannell and G. Ahlers, Phys. Rev. A 34 (1986) 4956.

When a wave number is selected which is Eckhaus-unstable (cf. the two right-most selection curves in the above figure) persistent dynamics arises in which vortex pairs are annihilated in the bulk and resupplied via the ramp. Thus the phaseslip process shown above occurs repeatedly and a drift of the pattern is generated. The plot below shows the measured and theoretically determined frequency of this dynamics.
Theory: H.-G. Paap and H. Riecke, Phys. Fluids 3 (1991) 1519.
Experiment: L. Ning, G. Ahlers and D.S. Cannell, Phys. Rev. Lett. 64 (1990) 1235.