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THREE DIMENSIONAL COMPUTATION

OF 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 nu-merically for Taylor-Couette flow with the outer cylinder at rest. The wave-length 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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数理解析研究所講究録第 661巻 1988年 279-294

箇 8 $U$

2. IntroductionExperiments for flows between concentric cylinders were performed by Bur-khalter&Koschmieder (1974). They measured the wavelengths of steady-state vortices that resulted ffom impulsively starting the inner cylinder froma state of rest with the outer cylinder held fixed. According to these exper-iments, the wavelength initially decreased up to $R/R_{c}\approx 4(R_{c}$ : CriticalReynolds number from linear theory). Beyond $R/R_{c}\approx 4$ , the wavelengthincreased with increasing $R$ . The objective of the present work is to examinethis problem. The visualization and the measurements of the 3-dimensionalflow are very difficult, and there seems to be no current theory available forsuch strongly nonlinear flows. Therefore it appears that numerical simula-tion is the only useful tool for our purpose. Neitzel (1984) performed theaxisymmetric computation of the incompressible Navier-Stokes equationsin 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 $a$ and $b$ and height $h$ . Weassume the gap between the cylinders to be filled with a viscous incom-pressible fluid of kinematic viscosity $\nu$ . The entire system is assumed tobe in an initial state of rest. At time $t=0$ , the inner cylinder at radius$r=a$ is impulsively set into rotation with angilar velocity $\Omega$ while the outercylinder at $r=b$ is held fixed. The rigid endwalls at $z=0,$ $h$ are assumedto be attached to the inner cylinder and therefore begin to rotate with it at$t=0.The$ variables are made dimensionless using the scales $d\equiv b-a,$ $\Omega d$

and $\Omega^{-1}$ for length, speed and time respectively.Numerical method is based on the MAC method except treating pres-

sure. The incompressible Navier-Stokes equations are expressed as follows:

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

divv $=0$ (2)

These equations are written in a generalized coordinates system and solvedby 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.The time derivative, $\partial D/\partial t$ , is evaluated by forcing $D^{n+1}=0$ , i.e.,

$\frac{\partial D}{\partial t}\simeq-\frac{D^{n}}{\triangle t}$

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 $r=a$

$u=0$ , $v=0$ , $w=0$

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

at $r=b$

$u=r\Omega$ , $v=0$ , $w=0$

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

at $z=0,h$

$(u, v, w)$ are the velocity components in the directions given by thecylindrical coordinates $(r, \theta, z)$ , and ( $P_{r}$ , P9, $P_{z}$ ) are the components of $P$

in each direction. The Poisson equations for gradient of pressure are solvedby successive over relaxation. Dealing with gradient of pressure insteadof pressure,the Neumann problem is transformed to the Dirichlet problem

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and the convergence becomes good. The Euler semi-implicit scheme is usedfor 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 NECSX2.

4. ResultsThere are three nondimensional parameters:

$\eta=b/\grave{a}$ (Radius ratio)$\gamma=h/d$ (Ratio of height and gap)

$R_{e}=\Omega d^{2}/\nu$ (Reynolds number)$(R_{c}=31.03)$

cf) $Ta= \frac{2\eta^{2}}{1-\eta^{2}}R_{e}^{2}$ (Taylor number)

$\eta$ is fixed at 0.727, and $\gamma$ 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).

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 wave-length $\lambda$ is defined as follows:

$\lambda=\frac{\gamma-2\epsilon}{N}$

where $\epsilon$ is the length of the endcell,and $N$ is the number of vortex-rings $(=1/$

$2$ number of cells) excluding endcells. The agreement is not good quanti-tatively. Probably the main reason is that the experiment is not a perfectimpulsive start. According to other experiments of them, the wavelengthdepends on the history of the acceleration.(Table 1) And another reason isfrom the numerical error due to the discontinuity in the boundary condi-tions at the axial end plates. However wavelengths of computational resultsare within the theoretical limit (Fig.4). The quantitative agreement is notgood, but the agreement is good qualitatively. Both wavelengths of thecomputational results and the experiments increase for $R/R_{c}>4$ .

Evolution of Taylor-vortices

Why does the wavelength increase for $R/R_{c}>4$ ? We have investigatedthe 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 gapbetween 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 normalto 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 byvortex-connection. This feature is not found in the axisymmetric com-putation 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. ConclusionThe wavelength of Taylor vortices through an impulsive start increases for$R/R_{c}>4$ by vortex-connection, and this feature was not found by ax-isymmetric computation by Neitzel (1984). So this is the non-axisymmetriceffect.

REFERENCES

[1] S.Chandrasekhar : Hydrodynamic and Hydromagnetic Stability (Ox-ford University Press,Oxford,1961) p.303

[2] S.Kogelman&R.C.DiPrima : Stability of spatially periodic supercrit-ical flows in hydrodynamics. Phys.Fluids 13 (1970)

[3] J.E.Burkhalter&E.L.Koschmieder : Steady supercritical Taylor vor-tices after sudden starts. Phys.Fluids 17,1929 (1974)

[4] G.P.Neitzel: Numerical computation of time-dependent Taylor vortexflows in finite-length geometries. J.Fluid Mech.141,51 (1984)

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$\Delta l(scc)$ 0.5 3.16 11.8 20.3 36.4. 3600

$1\overline{v_{m}\overline{\lambda}}$ $30.9\pm 0.7165\pm 0.04$ $29.3\pm 0.91.74\pm 0.05$ $27.8\pm 0.61.84\pm 004$ $27.8\pm 0.51.84\pm 003$ $27.3\pm 0.61.87\pm 004$ 2.0425

Table 1 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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