18th Australasian Fluid Mechanics ConferenceLaunceston, Australia3-7 December 2012

Nusselt-number scaling and azimuthal velocity profiles in a rotating cylindrical tank with aradial horizontal convection imposed to model atmospheric polar vortices

Wisam K. Hussam 1,2, Martin P. King, 3 Luca Montabone 4 and Gregory J. Sheard 1

1Department of Mechanical and Aerospace Engineering, Monash University, Adelaide, VIC 3800, Australia2Electromechanical Department, University of Technology, Baghdad, Iraq

3Bjerknes Centre for Climate Research, Uni Research, NO 5007, Bergen, Norway4Atmospheric, Oceanic and Planetary Physics, University of Oxford, Parks Road, Oxford OX13PU, United Kingdom

Abstract

A system idealizing the polar atmosphere is constructed featur-ing a rotating cylindrical container with radial fluid flux drivenby a linear radial temperature profile along the base. This sys-tem is investigated numerically using a spectral-element solveremploying a Boussinesq approximation to model buoyancy.The study focuses on the scaling of mean Nusselt number withRayleigh number over a range of Reynolds numbers character-izing the tank rotation rate.

It is found that forRa . 104, Nusselt number is independent ofReynolds number and the flow is dominated by diffusion. Atintermediate Rayleigh numbers, mixed convection is observedwith Nusselt number decreasing with increasing Reynolds num-ber. BeyondRa ≈ 106 convective flow begins to dominate, firstat low Reynolds numbers, with higher Reynolds number dataprogressively collapsing onto a universal power-law trendthatis independent of Reynolds number. This trend is found to fol-low theoretical and observed scalings for horizontal convectionin rectangular enclosures, i.e.Nu ∝ Ra1/5.

Introduction

The convection of a fluid contained between two parallel platesand heated from below (Rayleigh–Benard convection, RBC)has received much attention over the years because of its impor-tance in understanding the flow dynamics and thermally driventurbulence [14, 13, 8]. The key parameter characterizing RBCis Rayleigh number, which quantifies the strength of thermalforcing on the flow.

Horizontal convection defines flows that are driven by tempera-ture differentials imposed along a horizontal boundary [6]. Thisis in contrast to RBC in which the temperature differential isin the vertical direction. The effect of rotation on convectionflows is important in many industrial applications as well asinastrophysical and geophysical flows, including meridionalover-turning circulation in the ocean [9], Earth’s core [4], and solarand mantle convections [5, 10]. The motivation for the presentpaper is the stability of atmospheric polar vortices; particularlytowards understanding the striking geometric patterns adoptedby polar vortices across various planets [3], and to better un-derstand the sudden configuration changes that on Earth affectweather at latitudes impacting on Australia.

Laboratory fluid models have proved useful for the study of po-lar vortex instability (e.g. [1]), though these models havetyp-ically featured mechanical forcing mechanisms dissimilartoatmospheric mechanisms. In the present paper, cylinder rota-tion mimics Earth’s rotation, and radial horizontal convectiondrives an annular radial flux circulating outward near the baseand returning poleward at the top surface. Conservation of an-gular momentum accelerates the angular velocity of poleward-

moving fluid, spinning it into a vortex in an analogous mannerto the generation of atmospheric polar vortices in the polarcon-vection cell.

As a first approximation of the atmospheric system, this modeldisregards gamma-plane effects associated with the changeinthe Coriolis effect with latitude in the vicinity of the pole[1].Moreover, while the controlling parameters for this systemareReynolds and Rayleigh number, the parameters of interest whenconsidering swirling atmospheric flows are typically the Rossbynumber (relating inertial to Coriolis forces) and the Ekmannumber (relating viscous to Coriolis forces). The Ekman num-ber is related to the reciprocal of the Reynolds number, but theRossby number, which relates the angular velocity of the modelpolar vortex to the background rotation, is flow-dependent.Agoal of the present study is to understand the effects of Reynoldsnumber and Rayleigh number on this flow, and how the Rossbynumber varies with these parameters.

To consider the suitability of this model for implementation ina laboratory setting, a wide range of Rayleigh number (3.2 ≤

Ra ≤ 1010) and Reynolds number (3.2≤ Re ≤ 103) are consid-ered in this study.

Numerical model and methodology

The system comprises an open rotating cylindrical tank, filledwith fluid, and with a radial temperature distribution applied atthe base. The tank radiusR and heightH combine to define anaspect ratioAR = H/R, which in this study is fixed atAR = 0.4.The system is depicted in figure 1.

Given an angular velocity of the cylinder,Ω, the azimuthalvelocity imposed on the impermeable base and side wall isuθ = rΩ, wherer is the radial coordinate. To model a freesurface, a stress-free condition is imposed on the top boundary(uz = 0,∂ur/∂z = ∂uθ/∂z = 0). The side wall is thermally insu-lated by imposition of a zero normal temperature gradient, andto simplify the computational model, no heat loss is permittedthrough the stress-free top surface, which is also approximatedas being thermally insulated. A linear temperature profile in-creasing byδT from r = 0 to r = R is imposed along the baseto drive the horizontal convection in thez–r plane

A Boussinesq approximation for fluid buoyancy is employed,in which density differences in the fluid are neglected exceptthrough the gravity term in the momentum equation. Un-der this approximation the energy equation reduces to a scalaradvection-diffusion equation for temperature which is evolvedin conjunction with the velocity field. The fluid temperatureisrelated linearly to the density via a thermal expansion coeffi-cientα.

The dimensionless Navier–Stokes equations governing a

(a)

(b)

z

r

g

Axis

of sym

metr

y

Sid

e w

all

Top surface

Rotating base

Figure 1: (a) A schematic diagram of the system under inves-tigation, with temperature differenceδT varying radially alongthe base, and (b) the meridional semi-plane of the cylindricaltank on which computations are performed.

Boussinesq fluid may be written as

∂u∂t

=−(u ·∇)u−∇p+1

Re∇2u+ gT

Ra

Re2Pr, (1)

∇ ·u = 0, (2)

∂T∂t

=−(u ·∇)T +1

PrRe∇2T, (3)

whereu, p, t, Re, Ra, Pr, g andT are the velocity vector, kine-matic static pressure, time, Reynolds number, Rayleigh num-ber, Prandtl number, a unit vector in the direction of gravity,and temperature, respectively. Here lengths, velocities,timeand temperature are scaled byR, Rω, ω−1, δT , respectively.The Reynolds number, Rayleigh number, Prandtl number andNusselt number are respectively defined as

Re =R2Ω

ν, (4)

Ra =gαδT R3

νkT, (5)

Pr =ν

kT, (6)

Nu =∂T∂z

, (7)

whereg is the gravitational acceleration,ν is the kinematic vis-cosity andkT is the thermal diffusivity of the fluid. The meanNusselt number is calculated in the present configuration bycomputing the average temperature gradient normal to the bot-tom wall. Throughout this studyPr= 6.14, which approximateswater at laboratory conditions.

The governing flow equations (1)-(3) are computed on a two–dimensional axisymmetric domain using a high-order in-house

solver, which employs a spectral element method for spatialdis-cretzation and a third-order time integration scheme basedonbackwards-differencing. The solver has been validated andem-ployed widely (e.g. Refs. [12, 13, 7]).

A rectangular mesh comprising 1560 was constructed to dis-cretize the meridional (r–z) semi-plane. Care was taken to en-sure that the flow was resolved in the vicinity of the walls, andparticularly the heated boundary, with coarser mesh spacing inthe interior. A grid independence study determined that inte-grated Nusselt numbers were independent of resolution to bet-ter than 0.1% with an element polynomial degree of 5, which isused hereafter.

Results and discussion

Nusselt number variation with Rayleigh number for differentReynolds numbers is plotted on a log-log plot in Fig. 2(a). Theplot shows that atRa . 104, the Nusselt number is independentof Reynolds number (the diffusion dominated regime). AsRais increased, the Nusselt-number curves converge onto a singletrend, which is nearly linear on the log-log plot, demonstratinga power-law dependence betweenNu andRa. The gradient isvery close to 1/5, which is in agreement with theory and ob-servation for horizontal convection in a rectangular enclosure[11, 6, 13]. The scaling in the convection-dominated regimeisfurther illustrated by the plot of the gradients of the trends inFig. 2(b).

Figs. 3 and 4 plot the temperature and relative azimuthal ve-locity for Re = 10 and 1000, respectively, at different Rayleighnumbers. It can be noted that for smallRe, the flow passes froma diffusion-dominated regime to a convection dominated regimeprior to Ra ≈ 106. On the other hand, forRe = 1000, the flowremains in the diffusion-dominated regime untilRa ≈ 106, andapproaches the convection-dominated regime beyondRa= 109.

It can be noted from Fig. 2 that at intermediate Rayleighnumbers (between the diffusion- and convection-dominatedregimes, Nussult number decreases with increasing Reynoldsnumber, which is consistent with the numerical finding of [14].To further investigate theNu − Re relationship, the effect ofReynolds number on Nusselt number at different Rayleigh num-ber is presented in Fig. 5. The figure demonstrates that Nusseltnumber is independent of Reynolds number forRa . 104, andat higher Rayleigh numbers, the Nusselt numbers are Reynolds-number independent until progressively higher Reynolds num-bers.

A Rossby number may be defined as the ratio of the relativeangular velocity of the polar vortex (ω) to the background ro-tation (Ω). An approximation based on the peak relative az-imuthal velocity (u′θ,peak) and its radial location (rpeak) is ω =

u′θ,peak/rpeak. This yields a Rossby number definition

Ro =ωΩ

=u′θ,peak/rpeak

Ω. (8)

Relative azimuthal velocity profiles are plotted in Fig. 6, forvarious Rayleigh numbers andRe = 10 and 1000. The gen-eral trend is for the relative velocity to remain unaffectedun-til Rayleigh numbers beyond the diffusion-dominated regime,whereby further Rayleigh number increases lead to an increasein peak relative azimuthal velocity. Coupled with this is a shiftin the radial location of the peak fromr = R/2 to slightly lowerradial distances. AtRe = 1000, the peak relative azimuthal ve-locity increases to a lesser extent than atRe = 10, by almosta factor of 2. Using the definition outlined here, atRe = 10

(a)

log10 Ra

log 10

Nu

0 2 4 6 8 100

0.4

0.8

1.2

1.6

2

Re = 3.2Re = 10Re = 32Re = 100Re = 320Re = 1000

1

5

(b)

log10 Ra

d(lo

g 10 N

u)/(

log 10

Ra)

2 4 6 8 10-0.1

0

0.1

0.2

0.3

0.4

0.5

Figure 2: (a) A plot of log10Nu against log10Ra for differentReynolds number as indicated. Akima splines are fitted to thedata for guidance. A dotted line shows the asymptotic log10Nu.(b) A plot of gradient of the curves in (a), calculated using finitedifference. A dotted line illustrates the theoretical gradient of1/5.

the Rossby number increases fromRo = 0 to 2.1 for Ra = 103

to 109, while atRe = 1000 the Rossby number increases fromRo= 0 to 1.0 for Ra= 105 to 109. These results suggest that thissystem may be useful for the study of polar vortex instability, aspolar vortices manifest at Rossby numbersRo < 1 [1].

Conclusions

Numerical simulations have considered the axisymmetric flowin an open cylindrical tank under constant rotation, and sub-jected to a thermal horizontal convection forcing invokinga ra-dial poleward flux near the free surface. The results demon-strate at low Rayleigh numbers (Ra. 104), the flow is diffusion-dominated with Nusselt number independent of Reynolds num-ber and producing a zero Rossby number. At higher Rayleighnumbers (upwards ofRa≈ 106) a convection-dominated regimeis reached, wherebyNu ∼ Ra1/5. Beyond the diffusion-dominated regime, higher Rayleigh numbers produce progres-sively higher Rossby numbers. AtRa= 109, Ro= 2.1 and 1.0 atRe = 10 and 1000, respectively, which encompasses the Rossby

Ra = 102

Ra = 105

Ra = 107

Ra = 109

Figure 3: Contour plots of temperature (left) and azimuthalve-locity relative to the tank rotation (uθ−rΩ, right) atRe= 10 andRayleigh numbers as indicated, plotted on a meridional cross-section through the centre of the tank (so that the frames meetat the symmetry axis). Thermal forcing is imposed along thebottom of each of these frames. Cold to hot temperatures aredenoted by contours ranging from blue through black throughred to yellow. Low to high relative velocities are denoted bycontours ranging from blue through to red.

Ra = 102

Ra = 105

Ra = 107

Ra = 109

Figure 4: Contour plots of temperature (left) and azimuthalve-locity relative to the tank rotation (uθ − rΩ, right) atRe = 1000and Rayleigh numbers as indicated. Orientation and shadingareas per Fig. 3.

X X X X X X

log10 Re

log 10

Nu

0.5 1 1.5 2 2.5 30

0.4

0.8

1.2

1.6

2

Ra = 109

Ra = 108

Ra = 107

Ra = 106

Ra = 105

Ra = 104

Figure 5: A plot of log10Nu against log10Re for differentRayleigh numbers as indicated.

(a)

r/R

u’θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

(b)

r/R

u’θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

Figure 6: The relative azimuthal velocity along the free sur-face for different Rayleigh numbers at (a)Re = 10 and (b)Re = 1000.

number range relevant for polar vortex flows.

Acknowledgements

This work is supported by the Australian Research Councilthrough Discovery Grant DP120100153, by the National Com-putational Infrastructure (NCI) Merit Allocation Scheme,andby Monash University through a Faculty of Engineering SeedGrant.

References

[1] Aguiar, A.C.B., Read, P.L., Wordsworth, R.D., Salter,T. and Yamazaki, Y.H., A Laboratory Model of Saturn’sNorth Polar Hexagon,Icarus, 206, 2010, 755–763.

[2] Akima, H., A New Method of Interpolation and SmoothCurve Fitting Based on Local Procedures,J. Assoc. Com-put. Mach., 17 (4), 1970, 589–602.

[3] Fletcher, L.N., Irwin, P.G.J., Orton, G.S., Teanby, N.A.,Achterberg, R.K., Bjoraker, G.L., Read, P.L., Simon-Miller, A.A., Howett, C., de Kok, R., Bowles, N., Calcutt,S.B., Hesman, B. and Flasar, F.M., Temperature and Com-position of Saturn’s Polar Hot Spots and Hexagon,Science,319, 2008, 79–81.

[4] Glatzmaier, G.A., Coe, R.S., Hongre, L. and Roberts, P.H.,The role of the Earth’s mantle in controlling the frequencyof geomagnetic reversals,Nature, 401, 1999, 885–890.

[5] Glatzmaier, G.A., Coe, R.S., Hongre, L. and Roberts, P.H.,Convection driven zonal flows and vortices in the majorplanets,Chaos, 4, 1994, 123–134.

[6] Hughes, G.O. and Griffiths, R.W., Horizontal Convection,Annu. Rev. Fluid Mech., 40, 2008, 185–208.

[7] Hussam, W.K., Thompson, M.C. and Gregory J. Sheard,Dynamics and Heat Transfer in a Quasi-Two-DimensionalMHD Flow Past a Circular Cylinder in a Duct at High Hart-mann Number,Int. J. Heat Mass Trans., 54, 2011, 1091–1100.

[8] King, E.M., Stellmach, S. and Aurnou, J.M., Heat transferby rapidly rotating Rayleigh-Benard Convection,J. FluidMech., 691, 2012, 568–582.

[9] Marshall, J. and Schott, F., Open-ocean convection: Ob-servations, theory, and models,Rev. Geophys., 691, 1999,1–64.

[10] Miesch, M.S., The coupling of solar convection and rota-tionm Solar Physics, 192, 2000, 59–89.

[11] Mullarney, J.C., Griffiths, R.W. and Hughes, G.O., Con-vection driven by differential heating at a horizontal bound-ary,J. Fluid Mech., 516, 2004, 181–209.

[12] Sheard, G.J., Leweke, T., Thompson, M.C. and Hourigan,K., Flow around an impulsively arrested circular cylinder,Phys. Fluids, 19, 2007, 083601.

[13] Sheard, G.J. and King, M.P., The influence of height ra-tio on Rayleigh-number scaling and stability of horizontalconvection,Appl. Math. Model., 35, 2011, 1647–1655.

[14] Zhong, J.-Q., Stevens, R.J.A.M., Clercx, H.J.H., Verz-icco, R., Lohse, D. and Ahlers, G., Prandtl-, Rayleigh-,and Rossby-Number Dependence of Heat Transport in Tur-bulent Rotating Rayleigh-Benard Convection,Phys. Rev.Lett., 102, 2009, 44502.