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J. Fluid Mech. (2000), vol. 402, pp. 33–56. Printed in the United Kingdom
c
2000 Cambridge University Press
33
On the competition between centrifugal and
shear instability in spiral Couette flow
By A. MESEGUER
1AND F. MARQUES
2
1Oxford University Computing Laboratory (OUCL), Wolfson Building, Parks Road,
Oxford OX1 3QD, UK
2Dep. F´
ısica Aplicada, Universitat Polit `
ecnica de Catalunya, Jordi Girona Salgado s/n,
M`
odul B4 Campus Nord, 08034 Barcelona, Spain
(Received 4 May 1998 and in revised form 29 July 1999)
The linear stability of a fluid confined between two coaxial cylinders rotating in-
dependently and with axial sliding (spiral Couette flow) is examined. A wide range
of experimental parameters has been explored, including two different radius ratios.
Zeroth-order discontinuities are found in the critical surface; they are explained as a
result of the competition between the centrifugal and shear instability mechanisms,
which appears only in the co-rotating case, close to the rigid-body rotation region.
In the counter-rotating case, the centrifugal instability is dominant. Due to the com-
petition, the neutral stability curves develop islands of instability, which considerably
lower the instability threshold. Specific and robust numerical methods to handle these
geometrical complexities are developed. The results are in very good agreement with
the experimental data available, and with previous computations.
1. Introduction
We consider an incompressible viscous fluid which is contained in the gap between
two concentric cylinders that rotate independently about a common axis at constant
angular velocities. Forward motion is induced by an inertial sliding of the cylinders
relative to one another along the pipe axis. The basic motion whose linear stability
will be studied is, therefore, a combination of the Couette flow and the axial velocity
field induced by the relative sliding, the so-called spiral Couette flow (Joseph 1976).
This problem was first studied by Kiessling (1963) and Ludwieg (1964), who
obtained inviscid stability criteria in the narrow gap case. The experiments of Ludwieg
(1964) are, as far as we know, the only experiments on this problem until now.
The results showed the correctedness of the inviscid Ludwieg (1964) criterion (see
figure 12), later improved by Wedemeyer (1967). The general problem was studied
by Mott & Joseph (1968) and by Hung, Joseph & Munson (1972) with special
emphasis on energy methods; an excellent review can be found in the book Joseph
(1976, chap. VI). Recently, Ali & Weidman (1993) did a linear stability analysis of
spiral Couette flow, in the stationary outer cylinder case, for the so-called enclosed
geometry, which includes end effects. The more general problem of oscillatory sliding
has been recently considered by Hu & Kelly (1995) and Marques & Lopez (1997),
whose numerical simulations are in good agreement with the experiments of Weisberg,
Smits & Kevrekidis (1997). All of these works considered the infinite cylinders case,
assuming periodicity in the axial direction. As a result of the sliding, a non-zero mean
flow in the axial direction appears, which can only exist in open-ended configurations,
34 A. Meseguer and F. Marques
like in the experiments of Ludwieg (1964). The presence of lids enforces a zero axial
mean flow; this constraint is implemented by adding a suitable axial pressure gradient,
which mimics the lid effect, preserving the periodicity of the velocity field; this is the
so-called enclosed case.
An understanding of the stability of these flows could have applications in some
industrial processes like the purification of industrial waste water (Ollis, Pelizzetti &
Serpone 1991), the production of wire and cables (Tadmor & Bird 1974) and optical
fibre fabrication techniques (Chida et al. 1982). In all of them, axial sliding in a
cylindrical annulus takes place, and the rotation of one or both cylinders changes the
stability and properties of the flow.
This paper presents an extensive exploration of the linear stability of spiral Couette
flow mainly in the open ends case in order to compare with existing experimental
data, although some computations are performed for the enclosed configuration in
order to test our numerical code and quantify the effect of end caps. The exploration
covers a wide range of angular velocities of both cylinders, and two different radius
ratios are examined: one corresponds to the Ludwieg (1964) experiment, with a radius
ratio η=0.8 close to the narrow gap case (η→1). The other case (with a wide
gap η=0.5) has been considered because the instability appears at lower Reynolds
numbers than in the narrow gap case, and the change in the azimuthal wavenumbers
to be considered is also smaller, which permits a more detailed analysis.
It has long been known that whenever two or more control parameters representing
different physical mechanisms for instability compete, one can observe stability turning
points, islands of stability, multiple minima, and large changes in the critical azimuthal
wavenumber. Examples include the competition between buoyancy and shear in
inclined natural convection (Hart 1971), between buoyancy-induced shear and rotation
in radial Couette flow (Ali & Weidman 1990), between centrifugal and Kelvin–
Helmholtz instabilities in swirling jets (Martin & Meiburg 1994) and between rotation
and axial sliding in modulated Taylor–Couette flow (Marques & Lopez 1997). There
are also examples of complicated neutral curve topology in crystal–melt interface
problems as studied by McFadden et al. (1990). In the present problem the competition
between wall-driven shear and centrifugal instability mechanisms is responsible for
the geometrical complexities.
The paper is organized as follows. In §2, a complete description of the physical
system is given, and the analytical steady solutions are computed for the general case.
In §3 the linear differential equations which govern the stability of the first-order
perturbations are obtained, using a Petrov–Galerkin scheme. The symmetries of the
problem are considered in order to reduce the parameter space region to be explored.
The neutral stability curves in this problem may have multiple extrema and exhibit
sharp geometrical forms with sometimes disconnected parts. Specific and robust
methods for obtaining the neutral stability curves with such complex geometries are
designed. The results of our numerical method are checked with results obtained
previously by Ali & Weidman (1993). Section 4 is concerned with the wide gap case
η=0.5. For each pair of values of Roand Rz(outer rotation Reynolds number
and axial Reynolds number respectively) the neutral stability surface is computed.
Complex behaviour is found in the co-rotation zone, in particular as the axial
Reynolds number is increased. In fact, discontinuities in the critical inner rotation
Reynolds number surface have been observed. This phenomenon is explained in detail
as a competition between the centrifugal instability mechanism characteristic of the
Taylor–Couette problem and the shear instability mechanism induced by the axial
sliding. This interpretation is reinforced by examining the rigid rotation case with
Centrifugal versus shear instability 35
Uc
ri
ro
vz
B
vθ
B
¿i
¿o
Figure 1. Geometric sketch and parameters of the Taylor–Couette problem with axial sliding. The
basic flow vB,w
B, driven by the axial motion Ucand the angular rotations Ωi,Ω
o, is also depicted.
sliding. Section 5 deals with the narrow gap case η=0.8, where the same features are
present. The results are compared with the experimental results of Ludwieg (1964)
and the linear stability computations of Hung et al. (1972), obtaining a very good
agreement with both. A detailed analysis of the experimental data shows the presence
of hysteresis regions associated with the aforementioned discontinuities. Finally, §6
draws some conclusions.
2. Steady solutions
Taylor–Couette flow is the term used to describe fluid motion between two con-
centric rotating cylinders, whose radius and angular velocities are r∗
i,r∗
oand Ωi,Ωo
respectively. The annular gap between the cylinders is d=ro−ri. In addition, the
inner cylinder is moving parallel to the common axis with a constant velocity Uc
(see figure 1). The apparently more general flow with both cylinders moving axially
is reduced to the present case by Galilean invariance, changing to a reference frame
with constant axial speed.
The independent non-dimensional parameters appearing in this problem are: the
radius ratio η=r∗
i/r∗
o, which fixes the geometry of the annulus; the Couette flow
Reynolds numbers Ri=driΩi/ν and Ro=droΩo/ν of the rotating cylinders and the
axial Reynolds number Rz=dUc/ν measuring the translational velocity of the inner
cylinder.
Henceforth, all variables will be rendered dimensionless using d,d2/ν,ν2/d2as
units for space, time and the reduced pressure (p∗/ρ∗). The Navier–Stokes equation
and the incompressibility condition for this scaling become
∂tv+(v·∇)v=−∇p+∆v,∇·v=0.(2.1)
Let (u, v, w) be the physical components of the velocity vin cylindrical coordinates
(r, θ, z). The boundary conditions for the flow described above are
u(ri)=u(ro)=0,(2.2)
36 A. Meseguer and F. Marques
v(ri)=Ri,v(ro)=Ro,(2.3)
w(ri,t)=Rz,w(ro)=0,(2.4)
where ri=η/(1 −η), ro=1/(1 −η).
In order to compare with experiments and also with some previous work, two
different situations are considered in this paper. In both, the basic flow velocity field
is independent of the axial direction, but in one case the axial pressure gradient is
zero (open flow) and non-zero in the other (enclosed flow). The non-zero axial pressure
gradient in the enclosed flow case represents the presence of endwalls and allows us
to enforce a net zero axial mass flux, not only for the base flow but also for the
perturbed flow. The only experiments on the Taylor–Couette flow with axial sliding
of the inner cylinder known to us are those of Ludwieg (1964) which were carried out
in an annulus with open endwalls. The use of an axial pressure gradient to include
the large-scale endwall effects has been implemented by Ali & Weidman (1993) in the
linear analysis of Taylor–Couette flow with axial sliding, in the enclosed case and with
the outer cylinder at rest. This effect was also taken into account by Edwards et al.
(1991) and Sanchez, Crespo & Marques (1993) in Couette flow without sliding, where
the bifurcation to spirals in the counter-rotating case develops weak axial flows. The
axial pressure gradient is fixed by the zero axial mass flow condition
Zz=0
wrdrdθ=0.(2.5)
The steady velocity field vBindependent of the axial and azimuthal variables that
verifies the previous condition is
uB=0,v
B=Ar +B/r, wB=Cln(r/ro)+P(r2−r2
o)/4,(2.6)
as can be seen in Joseph (1976). The constants A,B,Care given by
A=Ro−ηRi
1+η,B=η(Ri−ηRo)
(1 −η)(1 −η2),C=1
ln ηRz+P(1+η)
4(1 −η)(2.7)
and Pis the non-dimensional Poiseuille number P=(dp∗/dz∗)d3/(ρ∗ν2) measuring
the axial pressure gradient imposed. In the open flow case, P= 0; in the enclosed
case, the mass conservation condition gives Pas a function of Rz:
P=−4Rz
(1 −η)(2η2ln η+1−η2)
(1+η)[(1 + η2)lnη+1−η2].(2.8)
3. Linear stability of the basic flow
In the preceding section the basic flow was obtained. We now perturb this basic
state by a small disturbance which is assumed to vary periodically in the azimuthal
and axial directions:
v(r, θ, z, t)=vB(r)+e
i(nθ+kz)+λtu(r),(3.1)
p(r, θ, z, t)=pB(r, z)+p0(r)ei(nθ+kz)+λt ,(3.2)
where vB=(0,v
B,w
B) is given by (2.6) and the boundary conditions for uare
homogeneous, u(ri)=u(ro)=0. Linearizing the Navier–Stokes equations about the
basic solution, we obtain the eigenvalue problem
λu=−∇p0+∆u−vB·∇u−u·∇vB.(3.3)
Centrifugal versus shear instability 37
The spatial discretization of the problem, in order to solve (3.3) numerically, is
accomplished by projecting (3.3) onto a suitable basis. The space of divergence-free
vector fields satisfying the boundary conditions of the problem is
V={u∈(L2(ri,r
o))3|∇·u=0,u(ri)=u(ro)=0},(3.4)
where (L2(ri,r
o))3is the Hilbert space of square-integrable vectorial-functions defined
in the interval (ri,r
o), with the inner product
hu,vi=Zro
ri
u∗·vrdr, (3.5)
where ∗denotes the complex conjugate. For any u∈Vand any function p, we have
hu,∇pi= 0. Therefore expanding uin a suitable basis of V
u=X
α
aαuα,uα∈V, (3.6)
and projecting the linearized equations (3.3) onto Vthe pressure term disappears,
and we get a linear system for the coefficients aα:
λX
βh˜
uα,uβiaβ=X
βh˜
uα,∆uβ−vB·∇uβ−uβ·∇vBiaβ.(3.7)
We implement a Petrov–Galerkin scheme, where the basis used to expand the unknown
velocity, {uα}, differs from that used to project the equations, {˜
uα}. A comprehensive
analysis of the method can be found in Moser, Moin & Leonard (1983) or Canuto
et al. (1988). The divergence-free condition for a velocity field of the form (3.1) is
D+u+inv/r +ikw = 0, and a basis for Vis obtained by taking
u1
j=(0,−rkhj(r),nh
j(r)),(3.8)
u2
j=(−ikfj(r),0,D+fj(r)),(3.9)
where D = ∂r,D
+=D+1/r. The functions fjand hjmust satisfy the homogeneous
boundary conditions fj=f0
j=hj=0onriand ro.
Introducing the new radial coordinate x=2(r−ri)−1, x∈[−1,+1] and using
Chebyshev polynomials Tj, a simple choice for fjand hj, which satisfies the homo-
geneous boundary conditions, is
fj(r)=(1−x2)2Tj−1(x),h
j(r)=(1−x2)Tj−1(x),(3.10)
where jranges from 1 to M, the number of Chebyshev polynomials used. In order
to preserve the orthogonality relationships between the Chebyshev polynomials, and
to avoid 1/r factors in the inner products in (3.7), a suitable choice for the projection
basis ˜
uis
˜
fj(r)=r2(1 −x2)3/2Tj−1(x),˜
hj(r)=r2(1 −x2)1/2Tj−1(x).(3.11)
With this choice, all the inner products in (3.7) involve polynomials, except those
containing the logarithmic term in wB, and therefore can be numerically computed
exactly using Gauss–Chebyshev quadrature (Isaacson & Keller 1966). Finally we
obtain a generalized eigenvalue system of the form
λGx=Hx,(3.12)
where the vector xcontains the real and imaginary parts of the coefficients aαin (3.6),
and G,Hare constant matrices, with Gpositive definite. The explicit expressions for
the matrix elements of Gand Hare given in the Appendix A.
38 A. Meseguer and F. Marques
Let us consider the symmetries of our problem. The Navier–Stokes equations
are invariant with respect to the specular reflections {z→−z,w→−w}and
{θ→−θ, v →−v}. They are also invariant with respect to rotations around the axis,
axial translations and time translations. The boundary conditions break some of these
symmetries. Rior Rodifferent from zero breaks the specular reflection θ→−θ, and
Rz6= 0 breaks the specular reflection z→−z. In order to keep the invariance we must
change the sign of these Reynolds numbers, and of the corresponding wavenumbers
nand kin the solutions of the linearized system (3.12). Therefore the symmetries
allow us to restrict the computations to the cases Rz>0 and Ri>0. Furthermore,
since the Navier–Stokes equations are real, the complex conjugate of a perturbation
(3.1), (3.2) is also a solution, and we can change simultaneously the sign of n,kand
the imaginary part of λ. Then we can restrict the computations to the case k>0.
When nand kare non-zero, the eigenvector of the linear problem has the form of
a spiral pattern (see figure 12a, showing an experimentally observed spiral flow). The
wavenumbers nand k, together with the imaginary part of the critical eigenvalue,
ω=Imλ, fix the shape and speed of the spiral. The angle αof the spiral with a
constant-zplane is given by tan α=−n/(rok)=−(1 −η)n/k; the speed of the spiral in
the axial direction (on a constant-θline) is c=−ω/k, and in the azimuthal direction
(on a constant-zline) it is ωsp =−ω/n. In the n= 0 case the pattern is axisymmetric
and we have steady Taylor vortices if ω= 0 and travelling Taylor vortices if ω6=0,
with axial velocity c.
If Rz= 0, the symmetry z→−zis not broken, and at the bifurcation point, in the
n6= 0 case, we get two pairs of purely imaginary eigenvalues bifurcating at the same
time, representing spirals with opposite slope – or angle – (see Chossat & Iooss 1994).
These spirals have opposite values of n.ForRz6= 0, the corresponding eigenvalues
split apart, and one of the two spirals ±nbecomes dominant. Therefore we expect
mode competition and switching between +nand −nfor Rzclose to zero.
3.1. Computation of the neutral stability curves
Let σbe the real part of the first eigenvalue of the linear system (3.12) which crosses
the imaginary axis. The stability of the basic flow is determined by the sign of σ.
For negative values of σ, the basic flow is stable under perturbations. When σis
zero or slightly positive, the steady flow becomes unstable and bifurcated secondary
flows may appear. It should be remarked that σ(n, k, η, Ri,R
o,R
z) is a function of the
physical parameters which play an essential role in the dynamics of the system. For
fixed η,Ro,Rz, and given n,k, the inner Reynolds number Ric(n, k) such that σ=0is
computed. The critical inner Reynolds number is given by Ricrit = minn,k Ric(n, k), and
the corresponding values of n,kare the critical azimuthal and axial wavenumbers
ncrit,kcrit which will dictate the geometrical shape of the critical eigenfunction, which
may be a spiral flow or travelling Taylor vortices. Furthermore, the imaginary part of
the critical eigenvalue, ωcrit, gives the angular frequency of the critical eigenfunction.
Again, the critical values are functions of the parameters (η, Ro,R
z).
The curves in the (k, Ri)-plane given by σ(k, Ri) = 0 are commonly termed neutral
stability curves (NSC). The main goal at this stage is to compute the absolute
minimum of the NSC, which will give the critical parameters (kcrit,R
icrit) – in fact,
the absolute minimum of the set of the NSC corresponding to integer values of n
will be found. As will be seen later, the NSC curves for this problem may have
multiple extrema (maxima and minima), and exhibit disconnected parts and sharp
geometrical forms. Furthermore, these curves may exhibit multivalued branches as
functions of k, and these features can change abruptly in some parameter ranges
Centrifugal versus shear instability 39
NR
ic kc
8 495.9915 1881 3.0906 0924
16 496.6464 3476 3.3000 7171
24 496.6466 5308 3.3001 3843
32 496.6466 3840 3.3001 3305
40 496.6466 3825 3.3001 3014
48 496.6466 3825 3.3001 3300
Ta b le 1. Critical values as a function of the spectral approximation order. These values have been
evaluated for the specific case η=0.5, Ro= 250.0 and Rz=50.0, n=−1 being the dominant
azimuthal wavenumber.
(see figure 3). Standard methods applied to a regular grid in the plane (k, Ri) require
exorbitantly high-accuracy computations. Consequently, an alternative method has
been considered.
A local extremum (kc,R
ic) must satisfy the following conditions:
σ(kc,R
ic)=0,∂
kσ(kc,R
ic)=0.(3.13)
Using the implicit function theorem, it can be seen that the local extremum is a
minimum if, in addition, the inequality (∂2
k,kσ)(∂Riσ)<0 is satisfied. In order to
solve equation (3.13), a two-dimensional Newton–Raphson method is used. The
convergence of the method depends on the topological structure of the basin of
attraction of the different minima sought by the Newton method, and is strongly
dependent on the initial point of iteration in the plane (k, Ri). In order to optimize the
process, a predictor steepest-descent method has been employed. This gradient method
allows computations to reach the neighbouring zones where the convergence is almost
ensured. The predictor scheme is able to detect islands of instability independently of
their size and topological features.
In order to check the feasibility of the computational method, a numerical test was
carried out. For this purpose, specific critical values were computed for a different
number of spectral modes. The convergence of the numerical method is reflected
in table 1, where the critical inner Reynolds number Ric and the critical axial
wavenumber kcare presented as a function of the number of the spectral modes
(N) considered for their computation in each case. Other critical parameters, like ω,
behave similarly. The results presented correspond to the η=0.5 case, although we
have also checked the convergence in the η=0.8 case, particularly for the parameter
values corresponding to Ludwieg’s experiments (Ludwieg 1964), obtaining the same
behaviour. Throughout this work, the number of modes used for the computation
of the linear stability regime was N= 24, which provided seven exact figures in the
critical Reynolds number and five exact figures in the critical axial wavenumber.
3.2. Comparison between open and enclosed flows (Ro=0)
In order to check the numerical scheme, the linear stability of the open and enclosed
flows has been studied for η=0.4 and a stationary outer cylinder (Ro= 0).
For the enclosed flow case, the present computations are in complete agreement
with the results of Ali & Weidman (1993). The numerical results are displayed in
figure 2. For high axial sliding Reynolds number, the azimuthal dominant mode is
n= 4, as was predicted previously by Ali & Weidman (1993).
In order to study the effect of a non-zero mean flow, the same computations have
40 A. Meseguer and F. Marques
Enclosed
Open
0 100 200 300 400
80
70
60
50
40
30
0
1
2
3
n = 4
n = 3
2
1
0
Ric
Rz
45
41
37
150 550 950 1350
Rz
Enclosed, n = 3
Enclosed, n = 4
Open, n = 4
Open, n = 3
(a)(b)
Figure 2. Comparison between the open and enclosed cases for η=0.4 and Ro= 0; the critical Ric
is shown as a function of Rz.(a) The dominant azimuthal wavenumber is indicated in each curve
segment. (b) Asymptotic states for high axial Reynolds number Rz.
been performed for the open-flow configuration. The qualitative behaviour of the
system is similar to the zero mean flow case, but some quantitative differences can
be pointed out. First, the global end effects included in the enclosed case have a
stabilizing effect on the basic flow, an effect that increases at high axial Reynolds
number (see figure 2a). This enhancement of stability in the enclosed case is similar to
the one observed by Marques & Lopez (1997) when the inner cylinder undergoes axial
oscillations. Second, the asymptotic azimuthal wavenumber nis different, being the
dominant azimuthal wavenumber n= 3 in the open case, and n= 4 in the enclosed
one, in agreement with Ali & Weidman (1993) (see figure 2b). According to equation
(3.2), the azimuthal wavenumber in figure 2 should be negative, but we have adopted
here the Ali & Weidman prescription for comparison; they used ei(kz−nθ)instead of
(3.2).
Nevertheless, only the open-flow configuration will be considered henceforth. In
fact, all the previous experiments have been performed using the open axial circulation
configuration. We have computed the inner-cylinder critical rotation Reynolds number
Ricrit as a function of (Rz,R
o) for two different values of η, 0.5 and 0.8. This has been
done in the range 0 <R
z<150 and −250 <R
o<250. We have restricted the
computations to the cases Rz>0, Ri>0 and k>0, on the basis of the symmetries
of the physical problem.
4. Instability results for η=0.5
The computation of Ric(Rz,R
o) for the wide gap η=0.5, gives as a first striking
result the presence of a zeroth-order discontinuity in Ric, in the co-rotating case
(Ro>0). Although this behaviour has been considered possible by some authors
(Davis & Rosenblat 1977), specific examples showing this kind of discontinuity are
very unusual in the fluid mechanics literature.
For Ro= 200 the discontinuity appears for Rz=82.63. We show in figure 3(a)
the critical Rias a function of k.ForRz= 80 the dominant mode is n=−1,
Centrifugal versus shear instability 41
Ric
Rz
(a)
550
450
350
150
50
2
250
58
k
Rz = 80
258
k
Rz = 84
258
k
Rz = 120
258
k
Rz = 122
450
350
250
150
50
(b)
0 50 100 150
0–1
–2
–3
–3
–4
Figure 3. (a) Formation and evolution of an island of instability for η=0.5, Ro= 200. The solid
line corresponds to the n=−4 mode, and the dashed one to n=−1. (b) The corresponding critical
inner Reynolds number Ric as a function of Rz(solid line); the dashed line is a section (Ro= 200)
of the critical surface (figure 5). The labels refer to the dominant azimuthal mode number n; the
circles are the transitions between different n.Ric is discontinuous for Rz=82.64.
giving Ric = 373.43 and kc=1.68; but for Rz=82.63 the marginal stability curve of
the n=−4 mode develops an island of instability for a much lower Ric = 119.13,
introducing a discontinuity in Ric. Notice too that the change by ncrit is not ±1as
usual, but it changes by three units. The island of instability is very small (figure 3a,
Rz= 84), becoming larger when we move away of the discontinuity. All these features
make the numerical computation of the critical parameters very difficult from the
algorithmic point of view. For these reasons we have developed specific numerical
methods, outlined in §3.1, in order to detect the islands as soon as they appear.
Similar islands of instability have been found in McFadden et al. (1990) and Marques
& Lopez (1997).
Before crossing the Ric discontinuity, the marginal stability curve for n=−4 has
a single extremum, a minimum (figure 3a,Rz= 80), giving the critical parameter
values (Ric,kc). After crossing, and due to the appearance of the island, we have
three extrema, two minima and a maximum, and the marginal stability curve has
two disconnected branches. If we move to higher Rzvalues, the island grows until
it merges with the other branch (figure 3a,Rz= 120,122); the marginal curve has
now a single minimum. Plotting the position of all the extrema as a function of
Rz, we get an S-shaped curve, displayed in figure 3(b); the solid curve gives the
absolute minimum, and the dashed curve corresponds to the other extrema. The
critical Reynolds number Ric becomes discontinuous (zeroth-order discontinuity) as
soon as the island of instability appears for Rz=82.64; experiments with increasing
Riand Rzheld fixed would give the solid curve in figure 3(b). Nevertheless, this
curve is smooth except for the presence of discontinuities in the derivative (first-
order discontinuities), which appear when the critical azimuthal wavenumber changes
(bicritical points, marked with circles). The whole critical surface is multivalued and
continuous, but is folded in such a way that a cusp develops; figure 4(a) shows a
perspective view of the critical surface. Figure 4(b) shows the same critical surface
with the curves corresponding to a change in the critical azimuthal wavenumber
42 A. Meseguer and F. Marques
Rz
(a)
–200
–100
0
100
200
150
100
50
0
100
200
300
400
500
Ri
Ro
500
400
300
200
100
0
Ri
–250
–200 –150
–100
–50
0
50 100
150
200
250
(b)
Ro
Rz
100
50
0
150
Figure 4. (a) Perspective view of the critical surface Ric(Ro,R
z)forη=0.5. (b) Same view, explicitly
showing the changes in the dominant azimuthal mode nat criticality. The edges of the cusp region
are also plotted as thick lines.
n, where the surface is not smooth (the tangent plane is discontinuous along these
curves).
The discontinuity of the critical parameter depends on the experimental conditions.
If we had fixed Riand computed Rzc(Ri,R
o), we would have found a continuous
surface, formed by all three sheets in the cusp region. This is the way we have
followed in order to obtain the critical surface in the cusp region, because the
critical wavenumber ncan also change (see figure 3b). As stated in the introduction,
for dynamical systems depending on a sufficient number of parameters, the critical
surface (a manifold, in the general case) is likely to present discontinuities of the same
kind, or more complex. As we lack a priori knowledge of this possibility, the use of
robust strategies, like those we have implemented, to find the critical points becomes
necessary.
The projection of the curves corresponding to a change in the azimuthal wavenum-
ber nare plotted in figure 5. We see that along these curves, the change in ncrit
is always ±1, except very close to the Rz= 0 axis, where the competition between
Centrifugal versus shear instability 43
0
50
100
Rz
–200
Ro
150
–100 0 100 200
21 –1
0
–1
–2 –3 –4
–3
Figure 5. Dominant azimuthal mode nat criticality, as a function of Ro,Rz;η=0.5. The shaded
region corresponds to the fold, whose edges are plotted as thick dashed lines.
modes ±nis strong. As we have already mentioned, the symmetries of the problem for
Rz= 0 makes the eigenvalues corresponding to ±nbifurcate simultaneously. When
the symmetry breaking is small (Rz∼0), both eigenvalues are very close, and there is
switching between both critical surfaces close to the axis; see direct changes between
modes −2 and +1, and −1 and +1, close to the Rz= 0 axis in figure 5. In the region
of the cusp, near the discontinuity in Ric, the change in ncis also large, because we
jump between the different sheets of the critical surface; but if we continuously move
on the critical surface, the change is ncis also ±1. This is clearly seen in figure 3(b).
The edges of the cusp region are plotted as thick lines in both figures 4(b) and 5.
The discontinuity in Ric corresponds to the upper edge of the fold region, and inside
it the dominant azimuthal wavenumber is n=−4, except at the very end (Ro∼250)
where the mode n=−3 becomes dominant.
The coordinates of the cusp point are Ro=93.22, Rz=73.41, Ric = 107.63, inside
the region nc=−3, but very close to the border with n=−4. Although we could
think of this cusp point as being a bifurcation point of codimension higher than 1,
it is not. The cusp point is characterized by having a tangent plane parallel to the
Ri-axis, simultaneous with an inflection point in the Ro-constant section. But if we
look at the critical surface from another point of view (for example changing Rito a
linear combination of the Reynolds numbers, as for experimental purposes), the cusp
point changes its position on the surface. In fact, looking for the critical Rzwith Ri,
Rofixed, all the folding region is now single-valued, and Rzis continuous there. In this
case we also have discontinuities (in Rz) and a multivalued critical surface, but now
in a different region of the critical surface. Figure 7(a) shows that close to Rz=0,
when the n= 0 mode is dominant, Ric slightly increases with Rz, but decreases for
higher Rzwhen n6= 0. Therefore Rzis multivalued, and a discontinuity appears. These
discontinuities and fold structure may have important consequences which could be
detected experimentally, like hysteresis phenomena, as well as the discontinuity in Ric.
Figure 6 shows Ric,ωc,αand cas a function of Rofor different values of Rz. The
critical Reynolds number Ric (figure 6a) is almost independent of Rzin the counter-
rotating region Ro<0. But in the co-rotating region, where the cusp develops, we have
two well-separated kinds of behaviour. This figure is a front view of the cusp structure
(figure 4) along the Rz-axis. For small axial sliding Rz, before the discontinuity, Ric
is very close to the values without sliding (Taylor–Couette flow). For higher axial
sliding, after the discontinuity, Ric falls to much lower values. The axial sliding is
always destabilizing, but its effect becomes significant only in the co-rotating case,
after the discontinuity. The centrifugal instability seems the dominant mechanism (as
in Taylor–Couette, Rz= 0) except after the discontinuity, where a shear instability
44 A. Meseguer and F. Marques
Ric
(a)
400
0
–250
200
Rz
0 150 250–150 –50
150
120
90
60
30
0
ωc
(b)
300
–300
600
0
–600
(c)
1.0
–0.2
0.2
0.6
1.4
0
–400
–800
–1200
(d)
–250 0 150 250–150 –50
αc
RoRo
Figure 6. Critical parameters for η=0.5, as functions of the outer Reynolds number Ro.(a) Critical
inner Reynolds number Ric; the solid straight line is the rigid rotation line Ri=ηRo.(b) Imaginary
part of the critical eigenvalue ωc.(c) Angle of the spiral pattern αin radians. (d) Axial pattern
velocity c.
due to the axial sliding becomes dominant; the cuspidal zone can be thought as
the transition region between both mechanisms. This qualitative change can also be
noticed in the angle of the spiral pattern α(figure 6c), which jumps from values
less than 0.2 radians (10◦) to values close to 1.2 radians (70◦). This dramatic change
in shape is reflected too in the axial speed of the spirals, in figure 6(d). We have
presented our results in terms of αand cinstead of kbecause these variables are easy
to measure experimentally and better discriminate between the two mechanisms; the k
value can be immediately obtained by using the expression k=−ω/c. We also notice
that the shear-instability-dominated branch is very close to the solid-body rotation
line (see figure 6a), where the centrifugal instability does not play a significant role;
see detailed comments in §4.1.
The angular velocity of the spiral pattern, ωc, which is the imaginary part of the
critical eigenvalue, is displayed in figure 6(b). It changes in a linear way with Ro, except
for jumps when the azimuthal mode nchanges. Looking at figure 5, we see that the
azimuthal wavenumber ndecreases when Rzincreases, except in two regions: the first
Centrifugal versus shear instability 45
Ric
(a)
400
0
200
100 15050
ωc
(b)
300
500
100
–100
(c)
1.0
0
0.5
1.5
0
–400
–800
–1200
(d)
0 100 15050
αc
RzRz
Ro
50
100
150
200
250
600
Figure 7. Critical parameters for η=0.5, as functions of the axial Reynolds number Rzin the
co-rotating case Ro>0. (a) Critical inner Reynolds number Ric; the bars mark the change in the
critical azimuthal mode n.(b) Imaginary part of the critical eigenvalue ωc.(c) Angle of the spiral
pattern αin radians. (d) Axial pattern velocity c.
one, close to Rz= 0 in the counter-rotating area, displays competition between ±n
modes, due to the breaking of the reflectional symmetry z→−z, as described in §3;
the second region, after the discontinuity, shows a kind of saturation– the azimuthal
n=−4 mode is dominant in a very large area.
Figure 7 shows Ric,ωc,αand cas a function of Rzfor different values of Roin the
co-rotating case. In figure 7(a) sections of the cusp region are displayed; the critical
Ric is in fact the minimum of the values in the multivalued region, so we have a
discontinuity which grows when increasing Ro. The discontinuity has been displayed
in the remaining critical parameter plots, figure 7(b, c, d). The bicritical points where
the azimuthal wavenumber nchanges and two eigenvalues bifurcates simultaneously
are distinguished with a vertical bar. The dominant mode for small sliding (Rzclose
to zero) is axisymmetric n= 0, and when the imaginary part of the critical eigenvalue
is not zero (except for Ro= 0, see figure 7b), we have Taylor vortices travelling
axially with a speed cas shown in figure 7(d). The effect of the sliding on these
axisymmetric modes is slightly stabilizing, in contrast to their destabilizing effect on
46 A. Meseguer and F. Marques
Ric
130
40
70
100 15050
ωc
(b)
–200
0
–400
–600
(c)
0.5
0
0
1.0
100
60
20
–20
(d)
0 100 15050
αc
RzRz
160
100
(a)
Ro
0
–50
–100
–150
–200
–250
Figure 8. Critical parameters for η=0.5, as functions of the axial Reynolds number Rzin the
counter–rotating case Ro<0. (a) Critical inner Reynolds number Ric.(b) Imaginary part of the
critical eigenvalue ωc.(c) Angle of the spiral pattern α.(d) Axial pattern velocity c.
the non-axisymmetric modes, mainly in the co-rotating region, an effect also reported
by Ali & Weidman (1993).
Figure 8 shows Ric,ωc,αand cas a function of Rzfor different values of Ro
in the counter-rotating case. Here all the critical parameters change smoothly, in
an almost linear way. Figure 8(a) shows that the critical Reynolds number Ric is
almost independent of the axial sliding Rzfor Ro<0, suggesting that the centrifugal
instability is the dominant instability mechanism as has been noticed before.
4.1. Axial sliding with rigid rotation (Ri=ηRo)
In this section we analyse the rigid rotation case Ωi=Ωo(or equivalently R=Ri=
ηRo) with sliding because it is relevant to understanding the dominant instability
mechanism in the cusp region. This situation is also interesting because of its linear
stability for both limiting cases Rz=0,R= 0. The situation is very similar to
Mackrodt (1976) where it is shown that although the Poiseuille flow in a circular pipe
is linearly stable for any Reynolds number, adding a slow rotation of the pipe makes
the flow unstable at some finite Reynolds number. And conversely, the rigid rotation
Centrifugal versus shear instability 47
(a)
103(b)
log R
log Rz
102
102103
n = –4
n = –5
Rz
_ = 85.11
R_ = 33.24
0.8
kc
log Rz
0.2
102103
0.6
0.4
0
Figure 9. Sliding rigid rotation. (a) Critical rotation number R=Ri=ηRoand (b) critical
wavenumber kas a function of the axial speed Rz.
flow in a circular pipe is linearly stable, but adding a finite axial pressure gradient
makes the flow unstable.
Figure 9 shows the computed critical rotation number R=Ri=ηRoand criti-
cal wavenumber kas a function of the axial speed Rz. The critical regime has an
asymptotic value as Rzis increased, being the asymptotic rotation Reynolds number
R∞=33.24. In this limit, the critical azimuthal mode is n=−4. On the other
hand, as the rotation Reynolds number increases, Rzapproaches another asymp-
totic value which is R∞
z=85.11, with a critical azimuthal wavenumber n=−5.
The dependence of the critical axial wavenumber kcon the marginal curve is de-
picted in figure 9(b). We can observe the presence of a maximum for the values
Rz= 122.05, Ri=50.25, kmax =0.7638. Figures 9(a) and 9(b) are very similar to
figures 1 and 3 in Mackrodt (1976), where the same problem but without the inner
cylinder is considered. Figure 9(b) shows that the critical wavenumber kdecreases
to very low values (less than 0.1) as the axial Reynolds number Rzincreases; as a
result, the spiral’s slope grows. This behaviour completely agrees with the splitting
observed in figure 7(c) between the centrifugal (small α) and shear (large α) dominated
flows.
Therefore, in the co-rotation region, when we increase the inner Reynolds num-
ber Riand prior to the onset of centrifugal instability, the solid-body rotation line
(figure 6a) must be traversed. If also Rzis greater than 85.11, the shear instabil-
ity comes into play: the lower part of the cusp appears, giving a discontinuous
Reynolds number with shear dominating the centrifugal instability mechanism. The
corresponding eigenfunctions are clearly different from the centrifugally dominated
ones. Now the axial wavenumber kis very small, giving large spiral angles α, and an
almost constant azimuthal wavenumber n(equal to −4). The cuspidal zone, where
the critical surface is multivalued, corresponds to the competition between the cen-
trifugal instability mechanism (upper branch) and the shear instability mechanism
(lower branch), continuously connected by the intermediate sheet. All three branches
can be experimentally observed if we fix Riand steadily increase the axial Reynolds
number Rz.
48 A. Meseguer and F. Marques
150
100
50
0
Rz
–200 –100 0 100 200
Ro
2
1
0
–1
–2
–3 – 4 –5 –6 –7 –8 –9 –10 –11 –12 –11
–13
–3
Figure 10. Dominant azimuthal mode nat criticality, as a function of Ro,Rz;η=0.8.
5. Instability results for η=0.8
The qualitative features of the critical surface for η=0.8 are the same as those in
the wide gap case η=0.5 previously analysed, although there are some quantitative
differences. For example, note the rapid change of the critical helical mode structure
ncdisplayed in figure 10. The number of azimuthal modes to be considered in the
stability analysis increases substantially in the narrow gap geometry. A case in point is
circular Couette flow with radial heating (Ali & Weidman 1990) whereby the number
of critical modes increases to nc=52forη=0.959 at Prandtl number Pr =4.35.
Tracking all these modes requires significantly more computation time.
Figure 11 shows the critical parameters Ric,ωc,αand cas functions of Rzand Ro
for η=0.8. The critical surface develops a cusp, but for higher positive values of Rz,
outside the range plotted. The early stages of the cusp can be seen in figures 11(a), 11(c)
and 11(d), where the curves display the same splitting in two different mechanisms
as in the η=0.5 case. We also notice that the shear-instability-dominated branch is
very close to the solid-body rotation line (figure 11a). Additional numerical results
in the region where the cusp is present will be given in §5.1 when comparing with
experimental results.
The n= 0 axisymmetric mode is stabilized by the axial sliding, giving axially
travelling Taylor vortices. But now the dominance of the axisymmetric mode is
restricted to a very narrow range of Rzvalues as shown in figure 10. From the
numerical results, it could be asserted that the sliding has a global destabilizing effect
on the basic flow. Another curious feature is the presence of a small window of the
n=−3 critical mode between the regions n= 2 and 1. As it happens very close to
the Rz= 0 axis, we consider it as a side effect of the mode competition and switching
when the reflectional symmetry z→−zis broken.
5.1. Comparison with previous results
Some previous experimental studies have been reported on the stability of spiral
Couette flow. In fact, in an excellent study by Ludwieg (1964), both theoretical and
experimental, a stability analysis has been devoted to a specific zone in the parameter
space, inside the cusp region. The experimental apparatus has a gap η=0.8, with
open ends, corresponding to our open flow case. The rotational speed of the external
cylinder is held fixed at Ro≈750. Ludwieg’s experimental device needed high external
rotation speeds in order to avoid preturbulent stages induced by transients. The
unique design of the experimental apparatus enforced a linear dependence between
axial velocity and azimuthal rotation speed of the inner cylinder moving relative to
an outer stationary cylinder (without axial velocity but rotating) for each orientation
of the roller guides. As a result, the experimental paths in the parameter space (Ri,R
z)
Centrifugal versus shear instability 49
Ric
100
50 250–50
ωc
(b)
300
600
0
–300
(c)
0.6
–250
0
1.2
0
–400
–800
(d)
–250 50 250–150
αc
RoRo
300
200
(a)
Rz
150
120
90
60
30
0
–150 150 –200 100
Figure 11. Critical parameters for η=0.8. (a) Critical inner Reynolds number Ric(Ro); the solid
straight line is the rigid rotation line Ri=ηRo.(b) Imaginary part of the critical eigenvalue ωc.
(c) Angle of the spiral pattern αin radians. (d) Axial pattern velocity c.
were straight lines, as can be seen in both figures 12(b) and 14. Ludwieg’s experimental
results (figure 12) are given in terms of two non-dimensional parameters cφand cz
which describe the motion of the fluid. These parameters are functions of the radial
variable r,
cφ(r)= r
vB
dvB
dr,c
z(r)= r
vB
dwB
dr,(5.1)
where ri6r6ro. In some specific situations, these functions suffer only tiny
variations in the prescribed range of values of r, mainly in the narrow gap case. As a
consequence, Ludwieg (1964) considered mean values ˜
cφand ˜
czof these functions as
the control parameters; he took r=(ri+ro)/2, the arithmetic mean radius, for the r
factor in front of the definitions of czand cφ, but he did not specify which values of
the azimuthal and axial velocities and their derivatives were used. Hung et al. (1972)
used the values of ˜
cφ,˜
czat the geometric mean radius ¯
r=√riroto compare with
Ludwieg’s results. Since the difference between the arithmetic and geometric means
is about 0.6% for the η=0.8 case, and moreover, the expressions for ˜
cφand ˜
czare
simpler using the Hung et al. prescription, we adopt it; a more detailed discussion
50 A. Meseguer and F. Marques
(a)
(b)
3
2
1
0–2 –1 0 1 2
Ludwieg (nur rot. Stöc)Ludwieg (nur rot. Stöc)Ludwieg (nur rot. Stöc)
ChandrasekharChandrasekharChandrasekhar
Howard und GuptaHoward und GuptaHoward und Gupta
LudwiegLudwiegLudwieg
Exp StabilitätsgrenzeExp StabilitätsgrenzeExp Stabilitätsgrenze
c˜φ
c˜z
Figure 12. Ludwieg’s experiments. (a) Picture of the spirals, from Wedemeyer (1967).
(b) Experimental results, from Ludwieg (1964); η=0.8, Ro= 750.
about the parameters used by different authors is given in Appendix B. Thus, the
dependence between ˜
cφ,˜
czand our variables Ri,Ro,Rzare given by the following
equations (for η=0.8):
˜
cφ=1+η
1−η
Ro−Ri
Ro+Ri
,˜
cz=1+η
1−η
Rz
Ro+Ri
.(5.2)
For the η=0.8 case the narrow gap approximation is not clearly justified. In figure 13
we can see that the variation of the functions cz(r) and cφ(r) is about 10% with respect
to the mean values ˜
cz,˜
cφ. This can be a source of error in the experimental values
given by Ludwieg (1964). It would be necessary to know the original experimental
results in terms of the Reynolds numbers in order to work with the true control
parameters Rzand Ri.
The experimental results of Ludwieg are summarized in figure 12(b). The shaded
area is the error bandwidth experimentally obtained. These errors are very large
in the fold region of the critical surface, and the reasons will be analysed shortly.
Figure 12(b) also shows several stability criteria. Three of them, labelled Ludwieg
(nur rot. St¨
oc), Chandrasekhar and Howard und Gupta, were obtained assuming axi-
symmetric perturbations, and using physical considerations as in Rayleigh’s criterion
(labelled Chandrasekhar in figure 12b). All of them are in very poor agreement with the
experimental data. Instead, Ludwieg’s stability criterion, obtained by exactly solving
the linearized Euler equations in the narrow gap limit, is reasonably close to the
experimental data.
Centrifugal versus shear instability 51
(a)
0.1
cz (r) – c˜z
r
–0.1
4.0 5.0
Ri
0
4.5
550
600
750
937.5
(b)
0.1
cφ (r) – c˜φ
r
–0.1
4.0 5.0
0
4.5
Figure 13. Variation of (a)cz(r) and (b)cφ(r)forη=0.8 in the gap, compared with the mean
values ˜
cz,˜
cφ;ascz(r) is linear in Rz, we have plotted it only for the characteristic value Rz= 100.
500 700 900
400
200
0
B
A
Ri
Rz
Unstable
Stable
Uncertain
Inviscid (Ludwieg)
HJM
This work
Ludwieg
experiments
*
Figure 14. Comparison between the experimental and theoretical results of Ludwieg (1964),
HJM and the present work. Parameters: η=0.8, Ro= 750.
A linear stability analysis of the spiral Couette problem was reported by Hung et al.
(1972, referred to from now on as HJM), where only particular regions in parameter
space were considered. Their results are in good agreement with some of Ludwieg’s
results, although there were some unexplored zones that the present work has studied
in detail. We have computed the critical curve for Ro= 750, which is single-valued
considering Rz(Ri), but it is well within the cusp region. The global results of the three
analyses have been sketched in figure 14, which corresponds to the section Ro= 750
of the critical surface.
52 A. Meseguer and F. Marques
Our results fully coincide with the previous computations of HJM, except for two
points on the left of the minimum of our critical curve in figure 14, where the results
of HJM clearly diverge from the experimental results. It is apparent that the results
of HJM are confined to the intermediate branch of the critical surface fold, where
the changes in Rzcare small. The other branches shows very high slopes of Rzc(Ri);
furthermore, the change in the critical azimuthal wavenumber nis more than 15 units
in this range. This is an indication of the difficulties HJM encountered outside the
intermediate branch, which explains the discrepancy of their two computed points in
the high-slope region of the stability curve.
The experimental results of Ludwieg show remarkable agreement with our nu-
merical results. The best experimentally defined bifurcation points correspond to the
vertical branch (where shear is the dominant instability mechanism), and on this curve
the discrepancies with our results are less than 4%; we must mention that this is the
first time the vertical branch has been computed numerically. The biggest discrepancy
appears for high Rz, but for these parameter values, the splitting between the mean
values ˜
cφ,˜
czand the functions cφ(r), cz(r) has a maximum (see figure 13). In the
region close to the minimum of the critical curve, the onset of instability is in very
good agreement with the experiments, but some points on the right-hand side of the
minimum clearly deviate from the numerical predictions. In order to understand why,
we must look carefully at the experimental setting. In Ludwieg’s experiments, a long
rod (the inner cylinder) goes through the outer cylinder, with an axial movement and
simultaneous rotation; the rod accelerates from rest to the final desired inner rotation
and axial velocity. The experimental path follows the curves (containing the open
and filled circles) in figure 14 from left to right. The experiment lasts until the rod
has run through the outer cylinder, a short time interval in all cases. Coming back to
figure 14, if we look at the two experimental series for low Rz(the two lowest curves),
we see that to reach the point labelled A, when increasing the axial velocity of the
rod, we cross the minimum of the instability curve, so a spiral flow appears before
reaching A. Shortly thereafter, when the velocity continues growing, the basic flow
again becomes stable, but now the flow is in the spiral regime, so we have different
possibilities. If the spiral flow is also stable, the flow will not come back to the basic
flow; if the bifurcation is subcritical, the spiral flow will persist within the region where
the basic flow is stable (hysteresis effect). The spiral flow can also become unstable
when we cross to the other side of the critical curve (supercritical bifurcation), but
in this case too it will take a finite time for the spiral flow to decay into the basic
flow. If we are close to the bifurcation point, which is the case of concern here, this
time can be longer than the duration of the experiment. Making use of our linear
stability analysis, we cannot decide which of these possibilities really occurs; weakly
or fully nonlinear computations are required at this stage. These considerations help
to explain why the experimental points close to Ashow a discrepancy with the current
numerical computations. The experiment would have to be reconfigured using a path
in parameter space not crossing the (multivalued) neutral stability curve, in order
to be free of hysteresis and relaxation phenomena. Notice that the points Aand B,
marked with a black and white circle, where Ludwieg could not decide about their
stability, are very close to the hysteresis region, strongly suggesting that the bifurca-
tion could be subcritical in this region of parameter space. Ludwieg acknowledged the
experimental uncertainties in this parameter region; figure 12(b) shows the estimated
uncertainty as a dashed area.
Unfortunately, Ludwieg’s experimental data do not include information about the
azimuthal wavenumber n, or other raw order parameters such as the linear and
Centrifugal versus shear instability 53
angular speed of the inner cylinder, and the observed angle of spiral inclination.
Therefore our comparison is reduced to the analysis of stability boundaries in appro-
priate regions of (Rz,R
i)-space. Nevertheless, figure 12(a) shows a spiral mode with a
large angle, corresponding to the shear-dominated branch; as figure 12(b) shows, the
Ludwieg’s experimental paths cross the critical surface through the mentioned branch.
This feature is in complete agreement with the computed critical angle (figure 11c).
Finally we must mention the effectiveness of the inviscid criterion of Ludwieg,
displayed as a dashed line in figure 14. The curve follows qualitatively the behaviour
of the numerically computed viscous curve, and predicts a multivalued critical surface.
The comparison between the inviscid criterion and our computations shows that
viscosity stabilizes the flow, delaying the instability, except for high Rz, in the shear-
dominated instability region, where the viscosity in fact destabilizes the basic solution,
as has been observed in other shear flows.
6. Conclusions
In this work we have made a comprehensive analysis on the effect of axial sliding
in the Taylor–Couette problem. We have developed specific and robust numerical
methods to deal with the geometrical complexities of the neutral stability curves. We
have tested our code by comparing with the results of Ali & Weidman (1993) for
the capped endwall geometry. In the open geometry, detailed computations of the
critical surface have been performed for two gap values. The wide gap η=0.5 has
been analysed in detail because the instability appears at lower Reynolds numbers
than in the narrow gap case, and the change in the azimuthal wavenumber is also
smaller. We have also considered the relatively narrow gap η=0.8, because as far
as we know, the only experimental data available on this problem correspond to this
value. We have found that the sliding has a global destabilizing effect on the helical
modes. By contrast, the n= 0 mode remains stabilized by the same effect, although
the range of dominance of this behaviour is quite limited. The bifurcation is mainly
to a spiral flow, but travelling Taylor vortices are also observed in small parameter
ranges.
Remarkable differences can be pointed out between the co-rotation and counter-
rotation zones. Counter-rotation configurations exhibit a regular behaviour in the
critical regime. Nevertheless, we find a sudden dominance of non-consecutive az-
imuthal modes for low Rzvalues. This phenomenon is due to the breaking of the
reflectional symmetry z→−z, which leads to mode competition and switching between
±nmodes.
The critical behaviour is radically different in the co-rotation zone. The critical
surface Ric =f(Rz,R
o) exhibits zeroth-order discontinuities which can only be detected
by making use of the specific numerical scheme for the computation of the neutral
stability curves. The discontinuity is due to the presence of a sudden dominant island
corresponding to a different azimuthal mode. This unusual phenomenon in hydro-
dynamical stability problems has been explained in terms of competition between two
independent instability mechanisms: in the current problem the centrifugal instability,
dominant in the counter-rotating regime and also for small axial sliding, competes
with the shear instability induced by the axial motion. The latter mechanism becomes
dominant near the solid-body rotation line, substantially lowering the onset of the
instability. The discontinuity of the critical surface is associated with the competition
between these mechanisms; the critical surface is folded into a cusp, and hysteresis
behaviour becomes possible. The eigenfunctions corresponding to each mechanism are
54 A. Meseguer and F. Marques
clearly different: spirals with large angles correspond to shear-type eigenfunctions and
show little variation of the azimuthal wavenumber n, while those corresponding to the
centrifugal instability exhibit small spiral inclination angles, and large variations of n.
We have found very good agreement with the computations of Hung et al. (1972), who
obtained one of the branches in the fold region. The agreement with the experimental
results of Ludwieg (1964) is also very good; in spite of the difficulties encountered with
the parameters defined by Ludwieg, the agreement in the shear-dominated branch,
computed for the first time, is better than 4%.
In order to precisely measure the bifurcation point in the region where hysteresis is
present, new experiments should be performed, trying to avoid unwanted crossings of
the critical surface, and designing a parameter path far from the tangencies exhibited
by the lower experimental series of Ludwieg in figure 14. These experiments could also
supply additional information about other critical parameters we have also computed.
The authors thanks Dr V. Iranzo for his help with the papers Ludwieg (1964),
Kiessling (1963) and Wedemeyer (1967) (in German). This work was supported by
the Direcci ´
on General de Investigaci ´
on Cient´
ıfica y T´
ecnica (DGICYT), under grant
PB97-0685.
Appendix A. Matrix elements (Petrov–Galerkin scheme)
The spectral projections of the Petrov–Galerkin formalism are described explicitly
by the following inner products:
G11
ij =Zro
ri
r˜
hi(r2k2+n2)hjdr;G12
ij =nZro
ri
r˜
hiD+fjdr,
G21
ij =nZro
ri
r(D+˜
fi)hjdr;G22
ij =Zro
ri
r[k2˜
fi+(D
+˜
fi)D+]fjdr,
H11
ij =Zro
ri
˜
hir2k2D+D−n2+1
r2−k2r+rn2D+D−n2
r2−k2hjdr
−iZro
ri
˜
hir2k2+n2n
rvB+kwBrhjdr,
H12
ij =nZro
ri
˜
hi2k2fj+rD+D−n2
r2−k2D+fjdr
−iZro
ri
˜
hihk2r2vB
r+∂rvB+nnvB
r+kwBD++kn (∂rwB)ifjdr,
H21
ij =nZro
ri2k2˜
fi+rD+˜
fiD+D−n2
r2−k2hjdr,
+i Zro
rih2k2rvB˜
fi−rD+˜
finvB
r+kwBihjdr,
H22
ij =Zro
rirk2˜
fiD+D−n2+1
r2−k2+rD+˜
fiD+D−n2
r2−k2D+fjdr,
−iZro
rihnvB
r+kwBk2˜
fi+D+˜
fiD++kD+˜
fi(∂rwB)ifjdr.
Centrifugal versus shear instability 55
Appendix B. Parameters from different authors
The functions cφ(r), cz(r) (5.1) introduced by Ludwieg (1964) are easily computed
from the expressions (2.6):
cφ(r)=Ar2−B
Ar2+B,c
z(r)= Cr
Ar2+B,(B 1)
where the constants A,B,Care given by (2.7). Evaluating these expressions at the
geometric mean radius ¯
r=√riro,weget
˜
cφ=1+η
1−η
Ro−Ri
Ro+Ri
,˜
cz=−1+η
√ηln (1/η)
Rz
Ro+Ri
.(B 2)
By Taylor expanding near η= 1 we obtain
√ηln (1/η)
1−η=1−(1 −η)2
24 +..., (B 3)
therefore (5.2) are the narrow gap approximations of the expressions (B 2). In fact the
expression for ˜
cφis exact, and the difference in ˜
czis only 0.2% for η=0.8, so we will
use the expressions (5.2) from now on. The difference in sign has been introduced for
better comparison with the experiments, because if we simultneously change the signs
of Rz,nand ωcthe marginal stability curve does not change (see §3 for a detailed
account of the system symmetries).
The variables χ,˜
Ω2and Rused by HJM are related to the present parameters by
Ri=Rη
1−η˜
Ω2+ sin χ,R
o=R˜
Ω2
1−η,R
z=Rcos χ. (B 4)
Their dependence with the Ludwieg parameters ˜
cφand ˜
czis
˜
Ω2=1+a˜
cφ
p(a+1)
2˜
c2
z+(1−˜
cφ)2,sin χ=1−˜
cφ
p(a+1)
2˜
c2
z+(1−˜
cφ)2,(B 5)
where a=(1−η)/(1+η) and Ro= 750 is held fixed. From (B 4), (B 5) we can easily
arrive at the same formulas (5.2), showing that HJM used the narrow-gap limiting
values of ˜
cφ,˜
cz, or equivalently their values at the geometric mean radius ¯
r=√riro.
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