Z angew Math Phys 43 (1992) 0044-2275/92/010028-08 $ 1.50 + 0.20 (ZAMP) �9 1992 Birkh~iuser Verlag, Basel

Buoyancy driven instabilities in rotating layers with parallel axis of rotation

By F. H. Busse and M. Kropp, Institute of Physics, University of Bayreuth, W-8580 Bayreuth, Germany

Herrn Professor Klaus Kirchgiissner zum 60. Geburstag gewidmet

1. Introduction

Fluid motions induced by thermal buoyancy in rotating systems are of considerable interest to fluid dynamicists and atmospheric scientists because of their obvious applications to problems of atmospheric circulations. In order to test and to supplement theoretical studies, scientists have started early to perform laboratory experiments for which the geometry of a cylindrical annulus has proven especially convenient. Hide [1] and Fultz et al. [2] have thus investigated baroclinic waves in a rotating annulus with cylindrical walls kept at different temperatures. Later it was shown that theories applicable to convection in the thick atmospheres of rotating planets and stars could be modelled in a rotating annulus with conical top and bottom boundaries and with the centrifugal acceleration as driving force (Busse [3]; Busse and Carrigan [4], [5]). Besides their potential geophysical and astrophysical applications the flows observed in those configurations have received much attention on their own right because of their interesting dynamical properties.

In this paper the problem of the instability of flow in a differentially heated rotating annulus is revisited and a new type of instability is pointed out which may be relevant for some of the laboratory experiments that have been performed in the past. In order to exhibit this instability in its most simple form the unrealistic assumption of partly stress-free boundaries will be employed. But computations with realistic no-slip conditions will also be reported.

The paper starts with the mathematical formulation of the problem in section 2. The analysis for the high Prandtl number limit is given in section 3. The low Prandtl number limit is described in section 4. The paper concludes with an outlook on more general aspects of the problem.

Vol. 43, 1992 Buoyancy driven instabilities in rotating layers

2. Mathematical formulation of the problem

29

We consider fluid motions between two vertical coaxial cylinders with the radii R~ and R 2 (R 1 < R2) which are kept at the temperatures 7'1 and T2. The height L of the fluid filled gap is assumed to be large compared with its width R 2 - R I. The system is mounted on a turntable rotating with the angular speed f~D. We u~e gap width R 2 - R~ as length scale, (R 2 --R1)2/7r

as time scale, where ~: is the diffusivity of the fluid, and T2 - T1 as scale of the temperature to obtain dimensionless expressions and assume the small gap approximation, ( R 2 - R1)/(R2 + R~) ,~ 1. In this case a Cartesian coor- dinate system can be introduced with the x, y, z-coordinates in the az- imuthal, vertical and outward radial directions as shown in Fig. 1. The basic state of the fluid is described by the velocity field U and the temperature distribution |

| = z, U = - j zR( z 2-�88 = jU (2.1a,b)

where j is the unit vector in the upward vertical direction and where the Rayleigh number R is defined by

R = 7(T2 - T~)g(R2 - R~)3/v~r (2.2)

In this expression g denotes the acceleration of gravity, 7 is the thermal expansivity and v is the kinematic viscosity. The Coriolis force vanishes for the flow field U since we have neglected the effects of the upper and lower boundaries of the fluid annulus. Close to those boundaries an azimuthal flow will be induced by the Coriolis force akin to the thermal wind observed in the thick laboratory annuli of Hide [1] and Fultz et al. [2]. By considering a thin annulus we expect that deviations from the solution (2.1) are

Figure 1 Geometrical configuration of the rotating cylindrical differ- entially heated annulus.

I

J

TI I //

Tz

6gk

30 F .H . Busse and M. Kropp ZAMP

localized close to the ends and that expressions (2.1) provide a good approximation throughout the bulk of the annular region.

In order to investigate the stability of the solution (2.1) we superimpose infinitesimal disturbances u, 0 with an exponential time dependence, exp{crt}. By using the general representation for the solenoidal vector field

u = V x (V x kq~) + V x k~r (2.3)

where k is the unit vector in the z-direction we satisfy the equation of continuity. By taking the radial components of the curl and the curl curl of the equation of motion we obtain two equations for cp and

= P - ~ ( U V 2 - U") �9 VA2~ + P-~VZA2~o (2.4a)

e V 2 A 2 ~ - R ~ x ~ + ~yA29

= P - I ( U . V + a)A2$ - P - l k " U' x VA2~o. (2.4b)

In addition the heat equation for the temperature disturbance ~ is needed,

V20 + A2q~ = U. V0 + a0. (2.4c)

We have introduced the rotation parameter f~ and the Prandtl number P,

~ = 2 ( R 2 - R I ) 2 ~ D / V , P =v/Ir

and used the definition A2 = V 2 - (k. V) 2. We also have allowed for the centrifugal force measured as the fraction fi of the gravitational acceleration g.

Two types of boundary conditions on the cylindrical walls z = ___ 1 will be considered. In the case of rigid walls which have been used for the derivation of expression (2.1b) for U we find

3 ~0=~z 9 = ~ 0 = 0 = 0 a t z = + _ � 8 9 (2.5)

The other case of partially stress-free boundary conditions

3 2 cp =~-~z2 ~0 =~O = 0 ---0 at z = +�89 (2.6)

is mainly introduced for mathematical convenience. It can be realized approximately by the introduction of baffles along the walls which are parallel to the tangential component of V3q9/Oz for the preferred solution cp. Without loss of generality equations (2.4) can be solved by expressions of

Vol. 43, 1992 Buoyancy driven instabilities in rotating layers 31

the form

((P, O, O) = (Cpo(Z), ~b0(z), Oo(Z)) exp{i0~x + i~y + at}. (2.7)

We shall make use of this simplification of the problem in the following discussion of special cases of equat ions (2.4).

3. The high Prandtl number limit

In the limit P ~ m the right hand side of equat ions (2.4a) and (2.4b) can be neglected and the variables 0 and ~q can be eliminated to yield the following equat ion for q)0(z)

+ Ro:]}cpo(z)=O. (3.1)

We have in t roduced the addit ional assumpt ion /~ ~ e such that all terms multiplied by /3 can be neglected except for the combina t ion f~/~ since f] typically assumes large values in the applications of the problem, i.e. we are considering the limit

f l~- l_~ = 0 with f~flc~ -1 - z = c o n s t a n t . (3.2)

We also have restricted the at tent ion to the neutral case o- = 0. By repeating the analysis below for a = ie) it can easily be shown that the onset of instability must occur tbr co = 0.

The assumpt ion (3.:2) can be justified by the P r o u d m a n - T a y l o r - t h e o r e m which applied to the present case states that nearly steady mot ions must be nearly y- independent in the limit of high rota t ion rates. In fact f rom the previous analysis [3] of the limit of large 6 and f rom experimental observa- tions [4] it is well known that rolls aligned with the axis of rota t ion represent the preferred form of instability. For values of ~ of the order unity or less, however, small values of fl may have an impor tan t influence as is shown below.

In the case of stress-free boundaries equat ion (3.I) and condit ions (2.6) are satisfied by

Cpo(Z) = sin ~(z + �89 (3.3)

and the algebraic expression

R=(aSc~-2+zZa2)(6a2+z)-I with a2--0c2+7c 2 (3.4)

is found. Since ~ is a free parameter of the problem depending on the inclination of the disturbance rolls with respect to the vertical, the min imum

32 F. H. Busse and M. Kropp ZAMP

of R as a function of z is of primary interest. The minimizing z and the corresponding expression Ro for the Rayleigh number are given by

To = --a26 +__ (a462 -I- a6~-2 ) 1/2, Ro(o 0 = 2a4[(a2~-2 --I- 62) 1/2 -- 3] (3.5)

where the positive sign in the expression for % applies. To obtain the critical Rayleigh number, the function R0(e) must also be minimized with respect to the wavenumber ~. The resulting expression is not very illumi- nating in the general case. Since the critical value e~ varies smoothly from the limit 6 = 0 to the limit of large 6, we present the critical values only in those two limits,

6--*0" Rc.=ZSg4N/5/8, O~c=rC/2, "ro~= 5x//5~2/4 (3.6)

6 --* oo �9 R~ = 27~4/46, ~c = rex//2, ~0~ = 9rc2/46. (3.7)

As must be expected, the limit (3.7) reproduces the result of Rayleigh- B6nard convection with the centrifugal force replacing gravity.

Throughout the above derivation we have assumed that the Rayleigh number R is a positive quantity, i.e. that the outer cylinder is kept at a higher temperature than the inner one. The new mechanism of instability which can be "isolated in the limit (3.6) does not depend on this assump- tion. Expression (3.4) is valid for negative values of R as well in which case the negative square root in the expression (3.5) for T0 must be chosen such that a negative sign appears in the front of the square root in the expression for R0(~). Except for a change in sign, expression (3.6) for R~ thus remains valid.

An impression of the form of the instability in the case 6 ~ 0 can be gained from Fig. 2. The plane of motion within the rolls is strongly inclined with respect to the axis of the rolls such that potential energy can be gained from the field of gravity directed parallel to the axis of rotation. While the component of motion described by (p generates the temperature disturbances 0, the work done on the resulting buoyancy force by the nearly vertical component of motion described by ~ provides the energy for the growth of the instability.

In the case of the realistic no-slip conditions (2.5) equation (3.1) can be integrated numerically. Only the case 6 --0 has been considered which yields

Rc = 1331, ec = 1.88 corresponding to "c0c = 43.4. (3.8)

As to be expected the differences between the values (3.6) and (3.8) are much smaller than in the case of Rayleigh-B6nard convection with stress- free and rigid boundary conditions.

Vol. 43, 1992 Buoyancy driven instabilities in rotating layers 33

Figure 2 Sketch of the streamlines of the preferred mode of instability in the high Prandtl number limit.

4. The low Prandtl number limit

In the low Prandtl number limit of equations (2.4) a similar form of instability is obtained for large rotation rates f~ as in the high Prandtl number limit, although the physical mechanism is quite different. By introducing the Grashof number G - RP-~ and by considering the double limit

P ~ 0 with G =const . (4.1a)

~ o e , fi/c~-~O w i t h ~ - l f l f ~ - z = c o n s t a n t (4.1b)

we obtain from equations (2.4) after elimination of O0(z) the following equation for ~o0(z),

[ (J~2 -- ~2) 3 -- ~(Ti -~- l G(z2 --1))o~21(po(z) = O. (4.2)

As in the previous section we have restricted the attention to the monotonous onset of instability by assuming a = 0. The temperature distur- bance ,9 does not enter the problem in the limit (4.1 a) and the terms on the right hand of equations (2.4) have been neglected except for the term proportional to U'.

Equation (4.2) is formally identical with the Rayleigh-B6nard problem of the onset of convection in a layer with a z-dependent Rayleigh number. In order to obtain the critical Grashof number Go, the lowest eigenvalue of equation (4.2) together with the boundary conditions (2.5) must be mini- mized as a function of ~ and ~. We have carried out numerical integrations

34 F, H. Busse and M. Kropp ZAMP

of the problem using the shooting procedure with a Runge-Kut ta method and have obtained the result

Gc = 2867 corresponding to c~c = 3.12, % = 41.7. (4.3)

As in the case 6 = 0 of equation (3.1) the problem does not change when the temperature gradient is reversed. For a negative value of G the parameter becomes negative as well, and the result (4.3) applies for the absolute values of G and z. We thus conclude that the instability always occurs in such way that the component of the rotation vector perpendicular to the roll axis forms a negative scalar product with the torque generated by the basic temperature distribution. In this formulation the case of large Prandtl numbers is included and at the same time the close relationship to the generalized Taylor vortex instability (Ludwieg [10]; Busse [8]) is indicated.

The special role of roll like disturbances nearly aligned with the axis of rotation has been noticed originally by Pedley [6] and by Joseph and Carmi [7] in work on the stability of flows in rapidly rotating pipes and annuli. In those cases it has been shown that typically the energy stability limit agrees with the critical Reynolds number of the linear stability analysis. This agreement provides a proof for the property that the bifurcation must occur supercritically and that growing oscillatory disturbances can not precede the monotonous instability. This proof cannot be applied in the case of equa- tions (2.4), but the close relationship to the problem defined by equation (4.2) in the limit of low Prandtl numbers suggest that the assumption of vanishing imaginary part ai of the growth rate is a valid one.

5. Concluding remarks

Both mechanisms of instability considered in this paper demonstrate that small changes in the orientation of disturbances can have profound influences on their growth rates. The effect of rotation which often is regarded as stabilizing can promote shear flow instabilities as has been known since the work of Pedley [6] Joseph and Carmi [7] and Busse [8]. In this paper it has been shown that this promoting influence can work in the case of buoyancy driven instabilities as well. Indeed, in non-rotating vertical layers the onset of instability becomes oscillatory and the critical Rayleigh number increases in proportion to the square root of the Prandtl number (Gill and Kirkham [9]).

In considering solutions of the form (2.7) we have neglected the effect of the end boundaries of the vertical fluid layer. The aspect ratio L / ( R 2 -- R~)

needed in order to realize the conditions for the applicability of the theory may become rather large with increasing rate of rotation. The influence of the new mechanism of instability appears to be noticeable, however, in

Vol. 43, 1992 Buoyancy driven instabilities in rotating layers 35

previous experimental measurements with moderate values of L/ (R2 - R~). The fact that the wavelength of convection columns observed by Busse and Carrigan [4] was larger than expected on the basis of the theory in which laboratory gravity was neglected suggests an influence of the new instability. More detailed numerical and experimental studies will be attempted in the future.

The support of the research reported in this paper through the Deutsche Forschungsgemeinschaft under Bu 589/2-1 is gratefully acknowledged.

References

[ 1] R. Hide, An experimental study of thermal convection in a rotating liquid. Phil. Trans. Roy. Soc., London 250, 441-478 (1958).

[2] D. Fultz, R. R. Long, G. V. Owens, W. Bohan, R. Kaylor and L Weil, Studies of thermal convection in a rotating cylinder with some implications for large-scale atmospheric motions. Met. Monograph 4, 1-104 (1959).

[3] F. H. Busse, Thermal instabilities in rapidly rotating systems. J. Fluid Mech. 44, 441-460 (1970). [4] F. H. Busse and C. R. Carrigan, Convection induced by centrifugal buoyancy. J. Fluid Mech. 62,

579-592 (1974). [5] F. H. Busse and C. R. Carrigan, Laboratory simulation of thermal convection in rotating planets and

stars. Science 191, 81-83 (1976). [6] T. J. Pedley, On the instability of viscous flow in a rapidly rotating pipe. J. Fluid Mech. 35, 97-115

(1969). [7] D. D. Joseph and S. Carmi, Stability of Poiseuilleflow in pipes, annuli, and channels. Quart. Appl.

Math. 26, 575-599 (1969). [8] F. H. Busse, t)ber notwendige und hinreiehende Kriterien fiir die Stabilitiit yon Szr6mungen. Z.

Angew. Math. Mech. (ZAMM) 50, T173-T174 (1970). [9] A. E. Gill and C. C. Kirkham, A note on the stability of convection in a vertical slot. J. Fluid Mech.

42, 125-127 (1970). [I0] H. Ludwieg, Stabilitiit der Strdmung in einem zylindrischen Ringraum. Z. Flugwiss. 9, 359 (1960).

Summary

The stability of fluid flow in the gap between two co-axial cylinders kept at different temperatures is considered in the case when the configuration is rotating rigidly about its vertical axis of symmetry. Instabilities in the form of baroclinic waves and thermal convection driven by centrifugal buoyancy have long been studied experimentally and theoretically for this configuration. It is shown that there is still another mechanism of instability which can be isolated in the limit of large Prandtl numbers. Rolls slightly inclined with respect to the axis of rotation convert potential energy into kinetic energy of motion and viscous dissipation. Another simple solution of the stability problem is obtained in the low Prandtl number limit where a shear flow instability akin to the Taylor vortex mechanism is realized.

Zusammenfassung

Die Stabilit/it der Str6mung im Zwischenraum zwischen zwei ko-axialen Zylindern, die auf verschiedenen Temperaturen gehalten werden, wird betrachtet in dem Fall, wenn die Konfiguration um ihre vertikale Symmetrieachse start rotiert. Instabilit/iten in der Form yon baroklinen Wellen und thermische Konvektion angetrieben durch den zentrifugalen Auftrieb sind fiir diese Konfiguration seit langem experimentell und theoretisch untersucht worden. Es wird gezeigt, dab ein weiterer Instabili- t/itsmechanismus existiert, der im Fall groBer Prandtlzahlen isoliert werden kann. Rollen, die bez/iglich der Rotationsachse leicht geneigt sind, wandeln potentielle Energie um in kinetische Energie der Bewegung und viskose Dissipation. Eine andere einfache L6sung des Stabilit/itsproblems bekommt man im Limes kleiner Prandttzahlen, wobei eine dem Taylorwirbel-Mechanismus/ihnliche Scherstr6mungsin- stabilit/it realisiert wird.

(Received: November 20, 1990)