, AD-All? 776 RENSSELAER POLYTECHNIC INST TROY NY F/ G 20/4INEATIN F ------- ADNN-XSMMT C ITRBNE -ECU

I1982 R C DI PRIMA. J S I JRAND OAAG29-79-C-0146

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I REPORT NUMBER 12. 3OVT ACCESSION NO. J.NIECIPIENI"S LATAL0)9. NUMLBN

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Interactions of Axisymmetric and Non-Axisymmetric ReprintDisturbances in the Flow between Concentric 6. PERFORMING ORG. REPORT NUMBERRotating Cylinders: Bifurcations near Multiple N/A

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R. C. i Prima DAAG29 79 C 0146~J. Sijbrand

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Rensselaer Polytechnic Institute AE OKUI UBR

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LJ_IJ

D j ,~'°*1 1473 EDITION or ' I OVY&S', OBSOLETE .fUnclassifie

Interactions of Axisymmetric and Non-AxisymmetricDisturbances in the Flow between ConcentricRotating Cylinders: Bifurcations near MultipleEigenvalues+

R.C. Di Prima and J. Sijbrand*

Rensselaer Polytechnic Institute. Troy, N.Y., U.S.A.University of Utrecht, Netherlands

We consider the instability of Couette flow between rotating concentric

cylinders of infinite length. Let R, R2 and D 1, P2 denote the radii and

angular velocities of the inner and outer cylinders, respectively; and let

= I /R2 and P = f2 /a. We also introduce the Taylor number T - -Afl d 1/v2

22 - 01 R 12)/(R22

- Bi 2).

It is known from the linear stability nalyis of Krueger, Gross, and DiPrima

(1966) for n near one that (i) for u - 0 and increasing T, Couette flow"

first becomes unstable to an axisy=netric disturbance at T = T (G = 0, n),c

(ii) in the absence of an axisym--etric disturbance Couette flow would become

unstable to a non-a-xisyrnetric disturbance at a value of T just slightly

greater than T c, and-(iii) at u sufficiently negative the critical value of

T occurs for a non-axisymetric disturbance. Thus, for fixed n near one there

are parameter values (u1 . T 1) for which axisymnetric and non-axisymmetric

disturbances are simultaneously critical for Couette flow. We wish to study

the bifurcation problem in the neighborhood of Cu TI ) where we note thatinstability to an axisy1etric disturbance gives a steady bifurcation while

instability to a non-axisymmetric disturbance gives a Hopf bifurcation.

In order to develop a mathematically tractable problem, we assume that the

axial vavenumber of the axisymmetric and non-axisymmetric disturbances are

the same, and we choose the vavenumber to be that corresponding to critical

conditions for an axisymmetric disturbance. We let A denote the dimensionless

axial wavenumber (scaled with respect to R2R-) and m denote the number of

waves in the azimuthal direction. Dr. Peter Eagles has -carried out the

necessary calculations for n - 0.95 to determine the points (pl. T ) for

This research was partially supported by the Army Research Office and theFluid "lechanics Branch of the Office of "Naval Research, and by the Nether-lands Organization for the Advancement of ?ure Research (Z.W.O.).

Present address: Koninklijke/Shell Laboratoriun, Amsterdam

Stability in the Mechanics of Continua, 383-386, 1982 Springer-

Verlag, ed. F.H. Schroeder (Proceedings of a IUTAM Symposium,

NUmbrecht, Germany, August 31 - September-4, 1981)

82 o ", 094

384

simultaneous instability to an axisymmetric disturbance (AO) and a non-axi-

syzmetric disturbance (1,m) vith the results given in the Table.

Table

1 -0.73976 11.973.7 3-182

2 -0.75123 12,319.01 3.507

1. -0.80284 14,o15.01 3.628

Thus, for example, for Yj = 0.95 and s = -0.73976, Couette flow is simultaneous-

ly unstable to an axisymetric disturbance ith vavenumbers (3.82,0) and a

non-axisyi.metric disturbance with wavenumbers (3.482,1). We also note that at

(pit TI) there are 6 critical modes vith axial (Z) and azimuthal (0) depend-

ence as follows: cos 'AZ, sin AZ, exp (I imO) cos AZ and exp (I imO) sin AZ.

The first step in the bifurcation analysis is to represent the components ofthe disturbance by

A ui (r) cosIZ 4 A u (r) sinlZ +

B v (r) exp (in$) cosAZ + -3 v (r) exp (ime) sinkZ + (1)c c 3 Sc c(r) exp (-i=O) cosXZ + I s; (r) exp (-ime) sinkZ 4 Y

Here the functions u c, u a, T c v sdenote the r adial dependqnce of the res-

pective modes. The quantities Ac . As, B % a are scalar functions of time

with Ac, A varying in the real numbers and B , B in the complex domain.o 5

The residual component Y is orthogonal to the six critical modes.

t We are now able to reduce the nonlinear partial differential equation to a

finite- dimensional center manifold of dimension six; this reduction is

performed by expressing 7 in terms of the coefficients Ae o As, Bc, B . The

nonlinear ordinary differential equations for these coefficients describe

the 'flo' in the center manifold and are similar to those derived by Davey.

i)Prima and Stuart (1968) for the case v - 0.

We give particular attention to the interaction of the three modes cosXZ

and exp (-i=m) sinkZ, vhich can give rise to Taylor-vortex and vavy-vortex

flows vhieb are often observed in experiments. So ve put As Be - 0 and

ve obtain the following equations vhere terms of the order 7 have been

omitted:

:!.

p!- - - - -

385

-dtc " A + a A3 + a 1AIBJ' + aA5 + a6A31B 12 + aAcIB " ()

dit I c c 2 A (2)-=bo bOB, l t1la 2 + b A% + be. I BDIAC + bTBI Bsl

dt Os i sc 6s c s

Complex conjugate equation for Bs -

The scalar quantities a. are real and the b. are complex numbers. In the.3 3

sequel ve shall denote by bjr and bji the real and imaginary parts of b .

After performing the transformation Ac . x, Bs = y exp i(bOt+#) the equation

for 4 uncouples from those for x and y and ve obtain

dx aX 34 2 5 3 2+ XYdt 0z ix +aiey +*a5 x + a 6 x y a.X (3)'

3 2 a 2 3 5AX-br dr b rx y+b y+ b~rSdt or4bizY" rXY 5r b 6 r UY

The coefficients a. and b.r depend on v and T, vhile a0 (0,1 T I ) b or(pi, T1 )

- 0.

It is now possible to study changes in the phase plane as a function of the

tvo parameters r-P1 and T-T 1 . It is natural first to analyze system (3) with

the fifth order terms left out. Keener (1976) and Langford and looss (1980)

consider pairs of coupled ordinary differential equations vith the same cubic

nonlinearities as (3) by taking &V a4, bir' b r fixed and varying a. and bOr.

The nature of the changes in the phase plane depends of course on the specific

values of the 3 rd order coefficients and the cubic systems of o.d.e. 'a can

be classified accordingly.

While the investigations by Keener and Langford-looss vere carried out for

arbitrary nonvanishing values of the 3rd order coefficients, it turns out

that in the problem vhich is presently under discussion some of these

coefficients are very small and change sign for values of u, T very close

to J"i T 1 .. For example, if ve let u take values close to. 1 aP -cpute

T - T (pn - .95, A - 3.482. a - 0) then a 1 (PT) - 0 for e - :.6 39, vhieh

is indeed very close to V 1 -. 73976. For this reason the analysis of (3)

has to be taken to fifth order and the dependence of sit a, b bir on 0and T has to be taken into account up to first order in V- 1 and T-T.

The numerical evaluation of the very complicated expressions for the fifth

order coefficients have been carried out by Dr. Eagles and the analysis of

the changes in the pbase plane is in progress.

A

306

The full system of six anplitude equations is also studied, but only at

cubic order in the amplitudes. For this case there does not appear to be a

transformation that ill reduce the order of the system.

For the case u = 0, we have noted that the axisymmetric and non-axisysmetric

modes are not simultaneously unstable. However, it is possible to ibed this

problem in alarger class of mathematical problems vhich we can analyze

using the techniques of center manifold theory. In this vay ve present a

rational basis (though admittedly complicated) for the analysis of the

successive bifurcations from Couette flow to Taylor-vortex flov and then to

vavy-vortex flow for increasing T with i = 0 as first studied by Davey,

Dilrima. and Stuart (1968).

Accession For.

DTIC TAB 0Unannounced 0justifloat io

By

Distribution/Availability Codes

AvailSpe and/or

Dist Special

References

1. .R, Krueger, A. Gross and B.C. DiPrima; 1966; On the relative importanceof Taylor-vortex and non-axisymmetric modes in flow between rotating cy-linders. J. Fluid Mech. 21., 521-538.

2. A. Davey, B.C. DiPrima, and J.T. Stuart; 1968; On the instability ofTaylorvortices. J. Fluid M4ech. 11, 17-52.

3. J.P. Keener; 1976; Secondary bifurcations in nonliear diffusion reaction

equations. Studies in -App. Math. U, 187-211.

4. V.F. Langford and G. looss; 1980; Interactions of Kopf and pitchfork

bifurcations.To appear-in Birkhluser Lecture Notes, N.D. ittel am ed.