数理解析研究所講究録
第 661 巻 1988 年 279-294

279

THREE DIMENSIONAL COMPUTATIONOF TAYLOR-COUETTE FLOW

Naoki Matsumoto (松本 直樹)
The Institute of Computational Fluid Dynamics, Tokyo, Japan

Susumu Shirayama (白山 晋)
The Institute of Computational Fluid Dynamics, Tokyo, Japan
Kunio Kuwahara (桑原 邦郎)
The Institute of Space and Astronautical Science, Tokyo, Japan

1. Abstract
Three dimensional incompressible Navier-Stokes equations are solved numerically for Taylor-Couette flow with the outer cylinder at rest. The wavelength of supercritical Taylor vortices created through an impulsive startof the inner cylinder is studied. The evolution of Taylor-vortex structure isvisualized and can be investigated precisely. The results are compared withexperimental results. Both results agree well qualitatively.

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2. Introduction
Experiments for flows between concentric cylinders were performed by Burkhalter&Koschmieder (1974). They measured the wavelengths of steadystate vortices that resulted ffom impulsively starting the inner cylinder froma state of rest with the outer cylinder held fixed. According to these experiments, the wavelength initially decreased up to $R/R_{c}\approx 4(R_{c}$ : Critical
Reynolds number from linear theory). Beyond $R/R_{c}\approx 4$ , the wavelength
increased with increasing $R$ . The objective of the present work is to examine
this problem. The visualization and the measurements of the 3-dimensional
flow are very difficult, and there seems to be no current theory available forsuch strongly nonlinear flows. Therefore it appears that numerical simulation is the only useful tool for our purpose. Neitzel (1984) performed theaxisymmetric computation of the incompressible Navier-Stokes equations
in finite-length concentric cylinder geometry. But the wavelength did notincrease for $R/R_{c}>4$ . So we have performed 3-dimensional computationand compared the results with the experiments.
3. Numerical Method
Consider a pair of concentric cylinders of radii and and height . We
assume the gap between the cylinders to be filled with a viscous incompressible fluid of kinematic viscosity . The entire system is assumed to
be in an initial state of rest. At time $t=0$ , the inner cylinder at radius
while the outer$r=a$ is impulsively set into rotation with angilar velocity
cylinder at $r=b$ is held fixed. The rigid endwalls at $z=0,$ are assumed
to be attached to the inner cylinder and therefore begin to rotate with it at$t=0.The$ variables are made dimensionless using the scales $d\equiv b-a,$for length, speed and time respectively.
andNumerical method is based on the MAC method except treating pressure. The incompressible Navier-Stokes equations are expressed as follows:
$a$

$h$

$b$

$\nu$

$\Omega$$h$$\Omega d$$\Omega^{-1}$$\frac{\partial v}{\partial t}+(v\cdot\nabla)v=-gradp+\frac{1}{R_{e}}\triangle v$(1)

divv $=0$(2)

These equations are written in a generalized coordinates system and solved
by finite-difference method. Applying the operator $grad\cdot div$ to the both

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sides of equation (l),we obtain the poisson equations for gradient of pres-sure:$\triangle P=-grad\cdot div(v\cdot\nabla)v+gradR+rot\cdot$

rotP$(3a)$

where$R=-\frac{\partial D}{\partial t}+\frac{1}{R_{e}}D$,$P=gradp$ ,$D=divv$$(3b)$and the formula of vector analysis $grad\cdot divX=\triangle X+rot\cdot$ rotX is used.
, is evaluated by forcing $D^{n+1}=0$ , i.e.,
The time derivative,$\partial D/\partial t$$\frac{\partial D}{\partial t}\simeq-\frac{D^{n}}{\triangle t}$

The boundary conditions for (1), (3) are as follows(dimensional variables):$u=V$,$v=0$ ,

$w=0$

$P_{r}=\frac{1}{R_{e}}\frac{\partial^{2}u}{\partial r^{2}}+\frac{V^{2}}{a}$,$P_{\theta}=\frac{1}{R_{e}}\frac{1}{a^{2}}\frac{\partial^{2}v}{\partial\theta^{2}}$$P_{z}=\frac{1}{R_{e}}\frac{\partial^{2}w}{\partial z^{2}}$at $r=a$$v=0$ ,

$u=0$ ,

$P_{r}=\frac{1}{R_{e}}\frac{\partial^{2}u}{\partial r^{2}}$$w=0$$P_{\theta}=\frac{1}{R_{e}}\frac{1}{b^{2}}\frac{\partial^{2}v}{\partial\theta^{2}}$$P_{z}=\frac{1}{R_{e}}\frac{\partial^{2}w}{\partial z^{2}}$at $r=b$$u=r\Omega$,$v=0$ ,

$P_{r}=\frac{1}{R_{e}}\frac{\partial^{2}u}{\partial r^{2}}+\frac{u^{2}}{r}$

$w=0$

,

$P_{\theta}=\frac{1}{R_{e}}\frac{1}{r^{2}}\frac{\partial^{2}v}{\partial\theta^{2}}$$P_{z}=\frac{1}{R_{e}}\frac{\partial^{2}w}{\partial z^{2}}$at $z=0,h$

are the velocity components in the directions given by the, and ( , P9, ) are the components ofcylindrical coordinatesin each direction. The Poisson equations for gradient of pressure are solved
by successive over relaxation. Dealing with gradient of pressure insteadof pressure,the Neumann problem is transformed to the Dirichlet problem$(u, v, w)$$(r, \theta, z)$$P_{r}$

3$P_{z}$$P$

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and the convergence becomes good. The Euler semi-implicit scheme is used
for the time integration of velocity. (All but the convective velocity arecomputed implicitly.) All spatial derivatives except the nonlinear terms areapproximated by central differences. The nonlinear terms are $app_{f}roximated$by the third-order upwind scheme:$(u \frac{\partial u}{\partial x})_{i}=u;\frac{-u_{i+2}+8(u_{i+1}-u_{i-1})+u_{i-2}}{12h}$$+|u_{i}| \frac{u_{i+2}-4u_{i+1}+6u_{i}-4u_{i-1}+u_{i-2}}{4h}$We assume the flow to be symmetric about the midplane to reduce thesize of the computational domain. This restricts the flow to have an evennumber of vortices, which is the case normally observed in the laboratory.
A grid system is shown in fig.l. Constant grid spacing is used in eachdirection. The computations were done on Japanese supercomputer NEC
SX2.
4. Results
There are three nondimensional parameters:(Radius ratio)$\gamma=h/d$(Ratio of height and gap)
$R_{e}=\Omega d^{2}/\nu$(Reynolds number)$\eta=b/\grave{a}$$(R_{c}=31.03)$

cf)$Ta= \frac{2\eta^{2}}{1-\eta^{2}}R_{e}^{2}$(Taylor number)is fixed at 0.727, and is fixed at 23.35 to correspond with the experimentof Burkhalter &Koschmieder. The computations are performed for fourcases $R/R_{c}=2,3,4,6$ ( $R_{c}:Critica1$ Reynolds number from linear theory).
$\eta$$\gamma$Steady-state
Figure.2 shows instantaneous streamlines in the vertical surface for $R/R_{c}=$$2,3,4,6$ . Wavelengths of steady state results are plotted in fig.3 with the

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comparison of the experimental and other numerical results. The wavelength is defined as follows:$\lambda$$\lambda=\frac{\gamma-2\epsilon}{N}$

where is the length of the endcell,and $N$ is the number of vortex-rings $(=1/$number of cells) excluding endcells. The agreement is not good quantitatively. Probably the main reason is that the experiment is not a perfectimpulsive start. According to other experiments of them, the wavelength
depends on the history of the acceleration.(Table 1) And another reason isfrom the numerical error due to the discontinuity in the boundary conditions at the axial end plates. However wavelengths of computational results
are within the theoretical limit (Fig.4). The quantitative agreement is not
good, but the agreement is good qualitatively. Both wavelengths of thecomputational results and the experiments increase for $R/R_{c}>4$ .
$\epsilon$$2$

Evolution of Taylor-vortices
Why does the wavelength increase for $R/R_{c}>4$ ? We have investigated
the time-evolution of vortices.

(1) $R/R_{c}=3$
At first, a vortex develops from the end-boundary by Ekman pumping,
induces next vortex, and propagates up. Finally vortices fill the gap
between the cylinders and go to the steady-state. Flow is axisymmetricduring this procedure. Fig. $5a$ shows the instantaneous streamlines inthe vertical surface. Fig. $5b$ shows the contour of the vorticity normal
to the surface. It is found that the vorticity is supplied from the innerboundary.
(2) $R/R_{c}=6$
Vortices fill the gap between the cylinders by the same process withcase(l). However, this is not the steady-state. This state makes atransition to the state in which there are large Taylor vortices. Fig. $6a$shows the instantaneous streamlines in the vertical surface. During thetransition we found that wavelengths of vortices gradually increase by
vortex-connection. This feature is not found in the axisymmetric computation of incompressible Navier-Stokes equations by Neitze1(1984).

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So the transition are probably due to the non-axisymmetric effects.
Fig. $6b$ shows the contour of the vorticity normal to the surface. It isfound that the vortices which have the same sign are connecting.
5. Conclusion
The wavelength of Taylor vortices through an impulsive start increases for$R/R_{c}>4$ by vortex-connection, and this feature was not found by axisymmetric computation by Neitzel (1984). So this is the non-axisymmetriceffect.

REFERENCES[1] S.Chandrasekhar : Hydrodynamic and Hydromagnetic Stability (Oxford University Press,Oxford,1961) p.303[2] S.Kogelman&R.C.DiPrima : Stability of spatially periodic supercritical flows in hydrodynamics. Phys.Fluids 13 (1970)[3] J.E.Burkhalter&E.L.Koschmieder : Steady supercritical Taylor vortices after sudden starts. Phys.Fluids 17,1929 (1974)[4] G.P.Neitzel: Numerical computation of time-dependent Taylor vortex
flows in finite-length geometries. J.Fluid Mech.141,51 (1984)6

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$\Delta l(scc)$$30.9\pm 0.7165\pm 0.04$$1\overline{v_{m}\overline{\lambda}}$Table 1

3.16
$29.3\pm 0.91.74\pm 0.05$

11.8$27.8\pm 0.61.84\pm 004$

20.3
$27.8\pm 0.51.84\pm 003$

36.4.
$27.3\pm 0.61.87\pm 004$

3600

2.0425

Wavelengths after different acceleration periods at $T/T_{c}=5.44,raeting$ end plates

Fig.4

Stability diagram for supercritical Taylor vortex flow withthe experimental results and numerical results

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