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A spectral method for time modulated Taylor-Couette flow
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING NORTH-HOLLAND
80 (1990) 223-227
A SPECTRAL METHOD FOR TIME MODULATED TAYLOR-COUETTE
FLOW
Carlo F. BARENGHI Department of Mathematics, The University, Newcastle upon Tyne NE1 7RU, England
Received 26 June 1989 Revised manuscript received December 1989
Flows which are driven by periodic forces occur frequently in nature and range from the motion of the earth’s atmosphere to the blood circulation. They also have important technical applications. The recognition of these facts, together with the progress in the study of dynamical systems, has caused a recent interest in modulated hydrodynamics and in the problem of the stability of flows which vary periodically in time. The aim of this paper is to present some spectral methods to study temporally modulated Taylor-Couette flow as a prototype problem of modulated hydrodynamics [l, 21. The Taylor-Couette problem is the study of the motion of an incompressible fluid contained between two concentric cylinders which rotate at assigned angular velocities. It is useful to distinguish between a ‘steady’ and a ‘modulated’ Taylor-Couette problem. In the steady problem the cylinders’ velocities are held constant in time. For the sake of simplicity we suppose that the outer cylinder is fixed and we measure the velocity of the inner cylinder by means of the Reynolds number Re,. If we start from Re, = 0 and we consider higher and higher Reynolds numbers we have a series of bifurcations from azimuthal motion (called circular Couette flow) to axisymmetric toroidal motion (called Taylor vortex flow), to non axisymmetric vortices (called wavy modes), and so on towards more complicated structures and turbulence. These bifurcations, in particular the first ones, have been intensively studied. On the contrary much less is known about the ‘modulated’ Taylor-Couette problem, in which the Reynolds numbers Re, and Re, of the inner and the outer cylinder vary periodically in time with assigned mean values, amplitudes and frequencies: Re,(t) = E,(l
+ E~sin mlt) ,
Re,(t) = &(l
+ Edsin w$)
(I)
The interesting question is whether the modulation makes the flow more or less stable to the onset of toroidal motion (Taylor vortices) than the steady rotation of the cylinders. To answer this question and to study the nature of the flow after the onset we have developed a nonlinear initial value code and a linear Floquet theory code, both based on spectral methods. These methods have to be efficient for two reasons: first of all the parameter space which we want to investigate is much larger than in the usual steady Taylor-Couette problem; secondly we expect to compute long time integrations. This second feature arises from the fact that the oscillatory motion of a cylinder penetrates into the fluid only as far as a Stokes layer of 00457825/90/$03.50
0
1990 - Elsevier Science Publishers B.V. (North-Holland)
224
C. F. Barenghi, A spectral method for Taylor-Couette
flow
thickness proportional to w-l/2, and thus the modulation produces more important effects when 6.1is small. There is experimental evidence that the first bifurcation into modulated toroidal motion is axisymmetric, and this leads us to consider the following dimensionless incompressible Navier-Stokes equations for the potential vorticity 2, the stream function + and the azimuthal velocity u : g
=
m>+WC 9,u),
(2)
g = i,(u) + fi2(u,I)) + F(t) ,
(3)
z-i,(l))=O,
(4)
together with the boundary u=$=,=O The operators
conditions
w
atx=O
at the inner and the outer cylinder
(5)
and x=1.
I$, e, and I$ are linear and they are defined as
t, = -$
+ 3(1-7)/s
i,
+ (1 -
= $
-$ + -$
q)ls $
,
- (1 - 7?)‘/s2 -$
1
i,=-$-$+(l-q,is3&,-i,
)
a2
(8)
s al-
where s = (1 - 77)~ + q and n is the radius ratio. The operators are defined as r;l,(Z, 4, u) = 2(1-
r;;<%
44
=
-
1
w
;
a(x
1 w4
7
+(l-~)IS(UIS+24
w
ag )
7
where u” = a(t)s + b(t) /s is the instantaneous 40 = [Re2(t) - r7 Re,(W(l b(t) = [q
z>
- ; 8(x s) 7
77)ls2(u0 + u> $ $1
fir and fi2 are nonlinear
- v2)
and
(9) (10)
Couette flow with ,
Re,(t) - q2 Re,(t)] /(l - q2) .
(11) (12)
Finally F(t) is a known forcing term, F(t) = [-WrE1 Re,(?J/s - 77s)cos(w,t) - W2E2Re,(s - 7j2/s) cos(w2t)]l(l
- q2) . (13)
C.F. Barenghi, A spectral method for Taylor-Couette flow
225
We assume that the cylinders have infinite length and that the flow has axial periodicity with wavelength 27r/a. We expand the fields 2, + and u over truncated Chebyshev-Fourier series 2(x, 5, t”) = E
$(&
l9
l”)
=
k=l
M
K-+2
c m=l M+l
+,
5,
t”)
=
E Z”,, sin(mcud)T:_,(x)
m=l
c m=l
c k=l
$$k
,
(14)
sin(ma~)T~-l(x),
(15)
K c k=l
v;k
cos(m
-
l)(YlT;_l(X)
.
(16)
At the time t” = n At the flow is described by the set of coefficients { ZL,, $L,, uLk}. We perform the time integration of (2) and (3) by means of the Crank-Nicholson method for the linear terms and the Adams-Bashforth method for the nonlinear terms [3]. The forcing term F(t) is integrated exactly. The relation (4) between 2 and t,Gand the boundary conditions (5) are enforced at each time step. We evaluate (2) and (3) at the collocation points xi K - 2) and (4) at xi ( j = 1, . . . , K), where we define _xj= 4 (1 + cos njl(K + 1)) K). In this way we obtain the matrix equation FXn+l =
Ox”
+
yn,n-1
+
pn-1
.
(17)
The vector X” contains the Chebyshev-Fourier coefficients of 2, + and u at time t”, Y”7n-1 depends nonlinear_ly on 2, I(,and u, while PRV1 is a known vector. The key feature is that the matrices P and Q depend only on the geometry and the truncation parameters: we can compute F-‘Q and P-’ in advance of the time integration and store them in arrays. In conclusion we have an efficient algorithm: at each time step, to proceed from X” to X”+‘, we only have to find the vector Yn,“-l and perform the multiplication of precomputed matrices times a vector. A second tool to study the onset of toroidal motion is Floquet theory, shown in Fig. 1. Let FLOQUET THEORY
orbit P --P
p, -. ~c.3 I’
pert urbat P’ - P’
I on
p
‘\
8‘ \
‘\
.__-’
*’
\ \
: I’
Fig. 1. Floquet theory.
C. F. Barenghi, A spectral method for Taylor-Couette flow
226
the point P describe the flow in a space of finite dimensions, given by the truncation parameters M and K. The time modulation maps P into itself. We now consider perturbations P’ of the periodic orbit, given by an orthonormal base in the space. We integrate P’ around a cycle using a linearized version of the initial value code and obtain P”, which can be expressed by a linear combination of vectors of the original base. In this way the eigenvalues of the linear mapping from P’ into P” determine the stability of the flow. Both the initial value code and Floquet theory have been tested [4] for the truncation parameters and the time step, at both steady and modulated Reynolds numbers. Known results have been reproduced about the linear growth rates and the velocity components and the torque in steady Taylor vortex flow. In the modulated problem we need typically M = 8, K = 15 and about 1000 steps per cycle at low frequency of modulation. In Fig. 2 we compare Floquet theory results with the experiments of Ahlers [5] and Walsh and Donnelly [6]: it is found that the modulation of the inner cylinder with stationary outer cylinder causes a small destabilisation of the flow, while the modulation of the outer cylinder at constant velocity of the inner cylinder stabilises Taylor vortices. Nonlinear results are shown in Fig. 3: the occurrence of the sharp peak is due to the fact that during part of the cycle the Reynolds number is below critical and the vortex vanishes. The agreement with the measurements of Ahlers [5] is good. We plan to extend our work to the case in which the inner cylinder oscillate around zero mean and to cases in which E > 1 (counterrotation).
0.92
4
1 0.0
0.2
0.4
0.6
0.8
1.0
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
l-L.&
1.2
Fig. 2a. Onset of vortices for modulation of the inner cylinder with stationary outer cylinder. Plot of %,I El0 vs eIr where Re, is the critical Reynolds number in the modulated case and Re,, is the critical Reynolds number in the steady case. Bars: experiment of Ahlers [5] at w, = 6, 77= 0.75. Solid line: Floquet theory.
Fig. 2b. Onset of vortices for modulation of_the outer cylinder with angular velocity L&(t) = 0.5 R, sin o,t, where d, is the constant angular velocity of the inner Plot of critical Reynolds numbers cylinder. i%& vs log,, -yzwhere o2 = 2~:. Note that in this case at 1) = 0.88 the steady state critical Reynolds number is Re,, = 120.5. Bars: experiment of Walsh and Donnelly [6]. Solid line: Floquet theory.
C. F. Barenghi, A spectral method for Taylor-Couette flow
227
0.8
0.4
0. 4
0.2
0.0 0.0
0.4
0.8
1.2
1.6
t
Fig. 3. Modulated Taylor vortex flow: normalized axial velocity component vs time. Squares: experiment [5] at 17= 0.75, o, = 4, Re, = 94.35 and e1 = 0.36. Solid line: initial value code.
of Ahlers
Acknowledgment The author is indebted to Dr. C.A. Jones for help and suggestions during this work, and to Prof. G. Ahlers for providing experimental data in advance of publication. This research was supported by SERC Grant GR/D/30433.
References [l] H. Kuhlmann, D. Roth and M. Lucke, Taylor vortex flow under harmonic modulation of the driving, Phys. Rev. A39 (1988) 745-762. [2] C.F. Barenghi and C.A. Jones, Modulated Taylor-Couette flow, J. Fluid Mech. 208 (1989) 127-160. [3] D. Gottlieb and S.A. Orsxag, Numerical Analysis of Spectral Methods (SIAM, Philadelphia, 1977). [4] C.F. Barenghi, Computations of transitions and Taylor vortices in temporally modulated Taylor-Couette flow, to be published. [5] G. Ahlers, private communication. [6] T.J. Walsh and R.J. Donnelly, Couette flow with periodically corotated and counterrotated cylinders, Phys. Rev. Letters 60 (1988) 700-703.
80 (1990) 223-227
A SPECTRAL METHOD FOR TIME MODULATED TAYLOR-COUETTE
FLOW
Carlo F. BARENGHI Department of Mathematics, The University, Newcastle upon Tyne NE1 7RU, England
Received 26 June 1989 Revised manuscript received December 1989
Flows which are driven by periodic forces occur frequently in nature and range from the motion of the earth’s atmosphere to the blood circulation. They also have important technical applications. The recognition of these facts, together with the progress in the study of dynamical systems, has caused a recent interest in modulated hydrodynamics and in the problem of the stability of flows which vary periodically in time. The aim of this paper is to present some spectral methods to study temporally modulated Taylor-Couette flow as a prototype problem of modulated hydrodynamics [l, 21. The Taylor-Couette problem is the study of the motion of an incompressible fluid contained between two concentric cylinders which rotate at assigned angular velocities. It is useful to distinguish between a ‘steady’ and a ‘modulated’ Taylor-Couette problem. In the steady problem the cylinders’ velocities are held constant in time. For the sake of simplicity we suppose that the outer cylinder is fixed and we measure the velocity of the inner cylinder by means of the Reynolds number Re,. If we start from Re, = 0 and we consider higher and higher Reynolds numbers we have a series of bifurcations from azimuthal motion (called circular Couette flow) to axisymmetric toroidal motion (called Taylor vortex flow), to non axisymmetric vortices (called wavy modes), and so on towards more complicated structures and turbulence. These bifurcations, in particular the first ones, have been intensively studied. On the contrary much less is known about the ‘modulated’ Taylor-Couette problem, in which the Reynolds numbers Re, and Re, of the inner and the outer cylinder vary periodically in time with assigned mean values, amplitudes and frequencies: Re,(t) = E,(l
+ E~sin mlt) ,
Re,(t) = &(l
+ Edsin w$)
(I)
The interesting question is whether the modulation makes the flow more or less stable to the onset of toroidal motion (Taylor vortices) than the steady rotation of the cylinders. To answer this question and to study the nature of the flow after the onset we have developed a nonlinear initial value code and a linear Floquet theory code, both based on spectral methods. These methods have to be efficient for two reasons: first of all the parameter space which we want to investigate is much larger than in the usual steady Taylor-Couette problem; secondly we expect to compute long time integrations. This second feature arises from the fact that the oscillatory motion of a cylinder penetrates into the fluid only as far as a Stokes layer of 00457825/90/$03.50
0
1990 - Elsevier Science Publishers B.V. (North-Holland)
224
C. F. Barenghi, A spectral method for Taylor-Couette
flow
thickness proportional to w-l/2, and thus the modulation produces more important effects when 6.1is small. There is experimental evidence that the first bifurcation into modulated toroidal motion is axisymmetric, and this leads us to consider the following dimensionless incompressible Navier-Stokes equations for the potential vorticity 2, the stream function + and the azimuthal velocity u : g
=
m>+WC 9,u),
(2)
g = i,(u) + fi2(u,I)) + F(t) ,
(3)
z-i,(l))=O,
(4)
together with the boundary u=$=,=O The operators
conditions
w
atx=O
at the inner and the outer cylinder
(5)
and x=1.
I$, e, and I$ are linear and they are defined as
t, = -$
+ 3(1-7)/s
i,
+ (1 -
= $
-$ + -$
q)ls $
,
- (1 - 7?)‘/s2 -$
1
i,=-$-$+(l-q,is3&,-i,
)
a2
(8)
s al-
where s = (1 - 77)~ + q and n is the radius ratio. The operators are defined as r;l,(Z, 4, u) = 2(1-
r;;<%
44
=
-
1
w
;
a(x
1 w4
7
+(l-~)IS(UIS+24
w
ag )
7
where u” = a(t)s + b(t) /s is the instantaneous 40 = [Re2(t) - r7 Re,(W(l b(t) = [q
z>
- ; 8(x s) 7
77)ls2(u0 + u> $ $1
fir and fi2 are nonlinear
- v2)
and
(9) (10)
Couette flow with ,
Re,(t) - q2 Re,(t)] /(l - q2) .
(11) (12)
Finally F(t) is a known forcing term, F(t) = [-WrE1 Re,(?J/s - 77s)cos(w,t) - W2E2Re,(s - 7j2/s) cos(w2t)]l(l
- q2) . (13)
C.F. Barenghi, A spectral method for Taylor-Couette flow
225
We assume that the cylinders have infinite length and that the flow has axial periodicity with wavelength 27r/a. We expand the fields 2, + and u over truncated Chebyshev-Fourier series 2(x, 5, t”) = E
$(&
l9
l”)
=
k=l
M
K-+2
c m=l M+l
+,
5,
t”)
=
E Z”,, sin(mcud)T:_,(x)
m=l
c m=l
c k=l
$$k
,
(14)
sin(ma~)T~-l(x),
(15)
K c k=l
v;k
cos(m
-
l)(YlT;_l(X)
.
(16)
At the time t” = n At the flow is described by the set of coefficients { ZL,, $L,, uLk}. We perform the time integration of (2) and (3) by means of the Crank-Nicholson method for the linear terms and the Adams-Bashforth method for the nonlinear terms [3]. The forcing term F(t) is integrated exactly. The relation (4) between 2 and t,Gand the boundary conditions (5) are enforced at each time step. We evaluate (2) and (3) at the collocation points xi K - 2) and (4) at xi ( j = 1, . . . , K), where we define _xj= 4 (1 + cos njl(K + 1)) K). In this way we obtain the matrix equation FXn+l =
Ox”
+
yn,n-1
+
pn-1
.
(17)
The vector X” contains the Chebyshev-Fourier coefficients of 2, + and u at time t”, Y”7n-1 depends nonlinear_ly on 2, I(,and u, while PRV1 is a known vector. The key feature is that the matrices P and Q depend only on the geometry and the truncation parameters: we can compute F-‘Q and P-’ in advance of the time integration and store them in arrays. In conclusion we have an efficient algorithm: at each time step, to proceed from X” to X”+‘, we only have to find the vector Yn,“-l and perform the multiplication of precomputed matrices times a vector. A second tool to study the onset of toroidal motion is Floquet theory, shown in Fig. 1. Let FLOQUET THEORY
orbit P --P
p, -. ~c.3 I’
pert urbat P’ - P’
I on
p
‘\
8‘ \
‘\
.__-’
*’
\ \
: I’
Fig. 1. Floquet theory.
C. F. Barenghi, A spectral method for Taylor-Couette flow
226
the point P describe the flow in a space of finite dimensions, given by the truncation parameters M and K. The time modulation maps P into itself. We now consider perturbations P’ of the periodic orbit, given by an orthonormal base in the space. We integrate P’ around a cycle using a linearized version of the initial value code and obtain P”, which can be expressed by a linear combination of vectors of the original base. In this way the eigenvalues of the linear mapping from P’ into P” determine the stability of the flow. Both the initial value code and Floquet theory have been tested [4] for the truncation parameters and the time step, at both steady and modulated Reynolds numbers. Known results have been reproduced about the linear growth rates and the velocity components and the torque in steady Taylor vortex flow. In the modulated problem we need typically M = 8, K = 15 and about 1000 steps per cycle at low frequency of modulation. In Fig. 2 we compare Floquet theory results with the experiments of Ahlers [5] and Walsh and Donnelly [6]: it is found that the modulation of the inner cylinder with stationary outer cylinder causes a small destabilisation of the flow, while the modulation of the outer cylinder at constant velocity of the inner cylinder stabilises Taylor vortices. Nonlinear results are shown in Fig. 3: the occurrence of the sharp peak is due to the fact that during part of the cycle the Reynolds number is below critical and the vortex vanishes. The agreement with the measurements of Ahlers [5] is good. We plan to extend our work to the case in which the inner cylinder oscillate around zero mean and to cases in which E > 1 (counterrotation).
0.92
4
1 0.0
0.2
0.4
0.6
0.8
1.0
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
l-L.&
1.2
Fig. 2a. Onset of vortices for modulation of the inner cylinder with stationary outer cylinder. Plot of %,I El0 vs eIr where Re, is the critical Reynolds number in the modulated case and Re,, is the critical Reynolds number in the steady case. Bars: experiment of Ahlers [5] at w, = 6, 77= 0.75. Solid line: Floquet theory.
Fig. 2b. Onset of vortices for modulation of_the outer cylinder with angular velocity L&(t) = 0.5 R, sin o,t, where d, is the constant angular velocity of the inner Plot of critical Reynolds numbers cylinder. i%& vs log,, -yzwhere o2 = 2~:. Note that in this case at 1) = 0.88 the steady state critical Reynolds number is Re,, = 120.5. Bars: experiment of Walsh and Donnelly [6]. Solid line: Floquet theory.
C. F. Barenghi, A spectral method for Taylor-Couette flow
227
0.8
0.4
0. 4
0.2
0.0 0.0
0.4
0.8
1.2
1.6
t
Fig. 3. Modulated Taylor vortex flow: normalized axial velocity component vs time. Squares: experiment [5] at 17= 0.75, o, = 4, Re, = 94.35 and e1 = 0.36. Solid line: initial value code.
of Ahlers
Acknowledgment The author is indebted to Dr. C.A. Jones for help and suggestions during this work, and to Prof. G. Ahlers for providing experimental data in advance of publication. This research was supported by SERC Grant GR/D/30433.
References [l] H. Kuhlmann, D. Roth and M. Lucke, Taylor vortex flow under harmonic modulation of the driving, Phys. Rev. A39 (1988) 745-762. [2] C.F. Barenghi and C.A. Jones, Modulated Taylor-Couette flow, J. Fluid Mech. 208 (1989) 127-160. [3] D. Gottlieb and S.A. Orsxag, Numerical Analysis of Spectral Methods (SIAM, Philadelphia, 1977). [4] C.F. Barenghi, Computations of transitions and Taylor vortices in temporally modulated Taylor-Couette flow, to be published. [5] G. Ahlers, private communication. [6] T.J. Walsh and R.J. Donnelly, Couette flow with periodically corotated and counterrotated cylinders, Phys. Rev. Letters 60 (1988) 700-703.