thermal buoyancy effects in rotating non-isothermal flows · 2019. 8. 1. · alinear...

International Journal of Rotating Machinery, 7(6): 435-446, 2001Copyright (C) 2001 Taylor & Francis1023-621X/01 $12.00 + .00

Thermal Buoyancy Effects in RotatingNon-Isothermal Flows

C. Y. SoongDepartment ofAeronautical Engineering, Feng Chia University, Taiwan, Republic of China

The present paper is concerned with the non-isothermalflow mechanisms in rotating systems with emphasis on therotation-induced thermal buoyancy effects stemming fromthe coexistence of rotational body forces and the non-uniformity of the fluid temperature field. Non-isothermalflow in rotating ducts of radial and parallel modes androtating cylindrical configurations, including rotating cylin-ders and disk systems, are considered. Previous investiga-tions closely related to the rotational buoyancy are surveyed.The mechanisms of the rotation-induced buoyancy are man-ifested by the author’s recent theoretical results and scalinganalyses pertaining to the rotation-induced buoyancy in ro-tating ducts and two-disk systems. Finally, the open issuesfor future researches in this area are proposed.

Keywords Non-isothermal flow, Rotation-induced buoyancy, Scal-ing analysis, Rotating ducts, Rotating cylinders, Rotatingdisks

INTRODUCTIONRotating flow is an important branch of fluid dynamics and is

full of complex physics. In practical applications, rotating ther-mal flows occur frequently in a variety of rotating machinery.For a long time this intriguing branch of thermal flows has at-tracted much attention ofresearchers. In past decades, there haveappeared several books and review articles on hydrodynamicand heat transfer characteristics of rotating flows. For example,Greenspan (1968) published a monograph on the theory of rotat-ing fluids. Zandbergen and Dijkstra (1987) presented a reviewof swirling flow; Morris (1981) and Owen and Rogers (1989)

Received in final form 10 July 2001.Address correspondence to C. Y. Soong, Department of Aeronau-

tical Engineering, Feng Chia University, Seatwen, Taichung, Taiwan40745, ROC. E-mail: [email protected]

This study was partially supported by National Science Council ofthe Republic of China through grant no. NSC 89-2212-E-0t4-015.

provided up-to-date (through the year of publication) informa-tion of flow and heat transfer in rotating ducts and rotor-stator

systems, respectively. Hwang and Soong (1992) surveyed liter-ature of convective heat transfer in radial and parallel rotatingducts and dealt with the experimental techniques for heat trans-fer measurement. A more recent review article by Yang et al.(1994) comprehensively discussed fluid flow and heat transfercharacteristics inside rotating channels.

Different from the previous reviewed works mentioned above,the present work confines itself to the thermal buoyancy effectsin rotating non-isothermal flows, in which the coexistence ofthe rotational forces and fluid temperature gradients leads tothe emergence of rotation-induced buoyancy effects. Since therotational forces are spatially varying with the local flow infor-mation, the buoyancy effects induced are obviously more sophis-ticated than the gravitational buoyancy resulted from constantgravity in a conventional natural/mixed convection flow. Thecoupling nature of the thermal flow phenomena with the rota-tional buoyancy is very influential to transport phenomena inrotating systems.

The present paper deals with non-isothermal flow mecha-nisms in rotating systems with emphasis on the rotation-inducedthermal buoyancy effects. Non-isothermal flow in rotating ductsof radial and parallel modes and rotating cylindrical configura-tions, including rotating cylinders and disk systems, are consid-ered. Previous investigations closely related to rotational buoy-ancy are surveyed. The mechanisms of the rotation-inducedbuoyancy are manifested by reorganized results of the author’sprevious theoretical works and scaling analyses pertaining to theinfluences ofthe rotation-induced buoyancy in rotating ducts andtwo-disk systems.

ROTATION-INDUCED BUOYANCYIn a non-isothermal flow field, fluid density varies spatially

with local temperature. As the density or temperature gradient isimposed normal to the body force in the field, a buoyancy-drivenfluid motion would emerge and play a significant role in thetransport phenomena ofthe flow field. As an illustrative example,Fig. shows non-isothermal fluid flows between two boundaries

435

436 c.Y. SOONG

Flow

T2 > T

(Isothermal)

U (Non-Isothermal)

T1

FIGURE 1Thermal buoyancy effect with the presence of body force and

temperature/density gradient.

at different wall temperatures and T2 > T1. The fluid near thecold wall is denser than that adjacent to the hot wall. Therefore,the buoyancy due to body force produces an assisting effect onthe cooler fluid but an opposing effect on the hotter fluid. Toexplore the thermal buoyancy in the flow field, the baseline caseof incompressible fluid flow is employed. Governing equationsformulated with respect to a reference frame rotating at the samerate as the system, say f, are written as

mented in analyses. By invoking a Boussinesq approximation,density variation is considered only for the body force terms.A linear density-temperature relation, p pr[1 -/3(T Tr)],is employed, where the subscript r denotes a reference stateand 1 /pr (Op/0 T) t, is the thermal expansion coefficient.The reference state is of conditions V 0, T Tr, p p andP Pr. At the reference state, the original momentum equationcan be reduced to VP/Pr --1 (1 R) -+- g, which is theconservative part of the centrifugal and gravitational forces inflow field. Finally, a vector form of the Navier-Stokes equationwith explicit buoyancy terms can be formulated, viz.,

(V V)V -Vpc/Dr q- 1)V2V -Jc- fi(T Tr)’ x (’-1 x R)

-211 -/(r rr)]al V -t-/3(T rr)g, [4]

where R is the position vector, and Pd P P,- denotes thepressure departure from the reference state. For rapidly rotatingsystems, the gravity term can be neglected for the relatively smalleffect. Taking curl of equation (4) leads to the vorticity transportequation,

V.V-0, [1]

p(V V)V + P’]I x (’21 x R) + 2p1 x V-Vp + pg + #V2V, [2]

(V V) T cV2 T, [3]

where p g is the gravitational force, and pf x (f x R) and2/91 V, respectively, are the centrifugal and Coriolis forcesowing to system rotation. In a cylindrical problem, centrifugaland Coriolis forces due to curvilinear motion ofthe fluid, pV2/Rand pUV/R included implicitly in the inertial term p(V V)V,are also regarded as the body forces for buoyancy (Soong andChyuan, 1998a).

To explore thermal buoyancy effects, there are two approa-ches to constructing theoretical models. One is to use a fullcompressible version of Navier-Stokes equations, in which thedensity variation with temperature and pressure is automaticallytaken into account. The thermal buoyancy effect in a compress-ible flow analysis is usually measured by Ap/pr or AT/Tr. An-other approach is to invoke a Boussinesq approximation in theincompressible or constant-property flow model, in which thebuoyancy effect is usually measured by Grashof, Rayleigh, orthermal Rossby number. In some early literature, e.g. Trefethen(1955), it was intended to express the effects of rotation by ananalogy to conventional non-rotating natural convection or tocurved duct flow. It is not simple, however. By inspecting therotational force terms mentioned above, it is obvious that, be-sides the density variation, the rotational body forces vary withthe local position R and the local fluid velocity . The situationis definitely more complicated than the buoyancy produced in aconstant gravitational force field.

To facilitate an analysis of rotation-induced buoyancy, theassumption of Boussinesq fluids is useful and usually imple-

(V V)( (( V)V + vV2( + V x [/3(T Tr)fx (2 x R) 2V x {[1 -/(T T)]f x V}

[5]

The last two terms are vorticity generation by centrifugal buoy-ancy and Coriolis effect, respectively. It is noted that for isother-mal flow the centrifugal force becomes a conservative forceand has no contribution to the vorticity generation; whereas theCoriolis force works no matter if the flow is isothermal or not.

NON-ISOTHERMAL FLOWS IN ROTATING DUCTS

Literature SurveyRotating Ducts ofRadial Mode

Considering the orientation, the rotation of the duct can beof the radial, parallel, slant, and axial modes. The duct could bestraight, curved, spiral, and of various cross-section shapes. Inthe monograph (Morris, 1981) and the review articles (Hwangand Soong, 1992; Yang et al., 1994), a comprehensive discussionof the effects of rotational forces can be found. There is nointent to perform an exhaustive survey of the subject but onlyto mention the previous studies closely related to the rotation-induced buoyancy effects.

Schmidt’s work (1951) on heat transfer in a rapidly rotatingturbine blade might be the earliest study dealing with centrifu-gal buoyancy. Morris and Ayhan (1979, 1982) first proposedexperimental evidence of the significance of centrifugal buoy-ancy in radially rotating ducts. Later, with a model of a radiallyrotating pipe, Siegel (1985) performed a perturbation analysis to

investigate the centrifugal buoyancy effect on fully developedflow and heat transfer in the ducts. Soong and Hwang (1990,1993) developed a similarity analysis of mixed convection in

THERMAL BUOYANCY EFFECTS IN ROTATING NON-ISOTHERMAL FLOWS 437

a rotating flat-channel with and without wall transpiration. Fortriangular ducts, Clifford et al. (1985) measured the heat transferperformance at various rotation rates. The buoyancy effects onboth radially outward and inward main flows were considered insome investigations (Iskakov and Trusin, 1985; Harasgama andMorris, 1988; Guidez, 1989; Wagner et al., 1991 a). By changingthe wall-to-fluid temperature difference but with other condi-tions fixed, Soong et al. (1991) explored the influences of cen-trifugal buoyancy on heat transfer performance. All ofthe studiesdemonstrated that, in laminar and transitional flows, centrifugalbuoyancy degrades heat transfer performance in a radially out-ward flow and enhances it in a radially inward flow. As theReynolds number increases, however, the adverse effect of thecentrifugal buoyancy on heat transfer in radially outward flowis alleviated, and the situation may even be reversed in turbulentflows.

Centrifugal buoyancy effects on heat transfer in more compli-cated situations have been studied in a number of investigations,e.g. uneven wall heating conditions (Han and Zhang, 1992), a

multi-pass duct with and without rib turbulators (Wagner et al.,1991b; Johnson et al., 1994; Fann et al., 1994; Ekkadand Han,1997), a serpentine duct with compressed air flow (Hwanget al., 2001), etc. Numerical computations were carried out forfully developed flow and heat transfer (Hwang and Jen, 1990),a developing thermal flow in entrance region (Jen and Lavine,1992; Jen et al., 1992; Yan and Soong, 1995), a flow with walltranspiration effects (Yah, 1994, 1995) and radiation effects (Yanet al., 1999). Dutta et al. (1996) used previous experimentaland numerical data to explore buoyancy induced flow separa-tion in radially rotating ducts with radially outward and inwardflows. Most recently, by using a large eddy simulation technique,Murata and Mochizuki (2001) studied aiding and opposing ef-fects of the centrifugal buoyancy on turbulent heat transfer ina radially rotating square duct with transverse or angled ribturbulators.

Rotating Ducts ofParallel ModeFor rotating ducts of parallel mode, Morris’ analysis (1965)

for low rates of heating was the first to consider centrifugalbuoyancy. Mori and Nakayama (1967) and Nakayama (1968),respectively, employed a boundary layer approach to analyzelaminar and turbulent convection with centrifugal buoyancy ina circular pipe. An experiment with parameter ranges coveringlaminar and turbulent flow regimes in a parallel rotating pipewas conducted (Humphreys et al., 1967) and their data corrobo-rated the heat transfer enhancement due to centrifugal buoyancy.Skiadarresis and Spalding (1976) and Majumdar et al. (1976)computed laminar and turbulent heat transfer in circular andsquare ducts by finite difference solutions of a model withrotation-induced buoyancy. Based on their numerical predic-tions, they claimed that the rotation-induced buoyancy effectsare more remarkable in laminar cases and the Coriolis effectis negligibly small in the parameter range they studied. Withconsideration of compressibility, Neti et al. (1985) investigated

convective heat transfer in a parallel rotating rectangular duct ofaspect ratio 2 by numerical computations. Although the densityvariation was included, only limited data on the centrifugal buoy-ancy effects were offered. Developing heat transfer rates andpressure drop with centrifugal buoyancy in a rotating duct ofpar-allel mode were measured by Levy et al. (1986). Mahadevappaet al. (1996) performed computations for fully developed flowsin rectangular and elliptic ducts and found that the friction factorand heat transfer in an elliptic duct are both higher than that in a

rectangular duct. By imposing various thermal boundary condi-tions, Soong and Yan (1999) investigated the axial evolution ofsecondary motion and its effects on heat transfer performanceduring the consideration of centrifugal buoyancy.

Rotating Curved DuctsRecently, mixed convection and flow transitions in a rotating

curved circular tube were analyzed by a regular perturbationmethod (Wang and Cheng, 1996). For the rotating curved squareduct, buoyancy-force-driven flow mode transition (Wang, 1997)and rotational instability (Wang, 1999) were both investigatedby finite volume solutions. In this flow configuration, centrifugalbuoyancy is also exerted in a cross flow plane similar to that ina parallel mode.

Scaling Analysis of Rotational Buoyancy in Duct FlowsThe non-isothermal flow considered is steady and laminar in

heated rotating ducts at a constant angular speed fl about anaxis normal or parallel to the duct axis. In a most recent work,Soong and Yan (2001) performed a scaling analysis by using theunified governing equations for the rotating ducts of radial andparallel modes with the coordinate systems defined in Figs. 2and 3, respectively. The Cartesian coordinate system attachesat the center of the inlet plane which is a distance Zo from theaxis of rotation. The eccentricity E of the duct is define as EZo + L/2, where L is the duct length. The inlet mean velocityVm is Vm Wo for orthogonal or radial mode and Vm Vo forparallel mode, and the inlet fluid lies at a uniform temperature To.Assume the gravitational effect is negligibly small in a rapidlyrotating case. By using the definitions of the rate vector ff21j, the velocity vector V U + Vj + Wk, and the positionvector R Xi + Yj + (Z + Zo)k with respect to the axis ofrotation for calculating the centrifugal force, unified Navier-Stokes/Boussinesq equations for thermal flow in the rotatingduct ofradial and parallel modes can be depicted in the followingscalar form:

OU OV OW0-- + + 0---- 0, [6]

OV OV OV V2 10Pdo- + v-ff + w o--2 v

Pr OY

[7]

[81

438 c.Y. SOONG

Radially Rotating Mode pnx pr(Z+Zo)

LW: Leading Wall J //

TW Zo/iof Rotation

FIGURE 2Rotating duct of radial mode.

aw aw aw aPaev__ +w -vwO----OX OY p,. OZ

fl(T Tr)E22(Z + Zo) + 211 fl(T Tr)]IU, [9]

OT OT OTU a-- -I- V + W 0--- eV2T" [I01

Let the duct have a large enough eccentricity E, i.e. EZo >> H D. The Z-component of the centrifugal buoyancy

is dominant since

Fz,c/Fx,. fl(T Tr)"d2(Z -t- Zo)/fl(T Tr)ff22X(z + Zo)/X e >> 1, [11]

where s E/D is the dimensionless eccentricity of the duct.

Rotating Ducts ofRadial ModeThe mean velocity Vm of the main flow is the inlet velocity

Wo, i.e. Vm Wo. The characteristic x-component of velocityon a cross-flow plane is U. < Wo, then

Parallel Rotating ModP

E =Zo

FIGURE 3Rotating duct of parallel mode.

Axis of Rotation

Fz,co/Fx,co U/W Uc/Wm. [12]

In general, main flow is stronger than the cross flow, i.e. U./Vm < 1, and thus the Coriolis effect in the X-direction is moreimportant. Since the secondary flow is characterized by the fluidmotion in the central region (Y 0) of the cross flow plane,influences of pressure gradient and viscous forces in the crossflow plane are neglected. Therefore, the inertial force is balancedwith the summation of the Coriolis force and the centrifugalbuoyancy, i.e.

U2. /X, [-2(1 flAT.)E2W ZXZcS2ax.l. [13]

In a radial duct, the characteristic velocity Wc is selected as themain flow velocity, W. Vm. With the present definition ofa thermal Rossby number, B =- /3 ATc, and Rotation number,Ro =_ E2D/Vm, one has the velocity of the secondary flow incross-flow plane is

Uc/Vm [2(1 B)Ro + BRo2] 1/2. [14]

Considering the validity ofthe Boussinesq approximation, B hasto be small enough, say B < O(10-1). Using a typical case of


Ro O(10-1) and B O(10-1) as an illustrative example,one has

Uc/Vm 0.4 0(10-1), [15]

in which the Coriolis effect is the major contribution to the gen-eration of the secondary motion in cross section of the duct and,for the parameter ranges considered above, the secondary flowvelocity can be estimated by

Uc/Vm V/2(1 B)Ro. [16]

The influence of the Centrifugal buoyancy fl(T Tr)22(Z + Zo) BE in a radially rotating duct is mainly onthe main flow direction and modifies the longitudinal flow andheat transfer directly.

Rotating Duct ofParallel ModeFor the parallel mode, the main-flow mean velocity Vm is the

entrance velocity Vo. In the entrance region, the cross flow is notyet built up, i.e. U, W << Vm, the Coriolis effects are small. Thecentrifugal buoyancy is the major driving force in the cross flowplane. The order ofthe inertial term W0W/0Z Wc2/D and thecentrifugal buoyancy term fi(T Tr)Q2(Z + Zo) fiATcEare comparable. Therefore, we have

Wc/Vm RoBe. [17]

For the case of B O(10-1), Ro O(10-1) and e O(10),the cross flow velocity is of the order Wc/ Vm O (10-1). Oncethe secondary motion emerges, with the Coriolis force in thecross flow plane, the secondary flow velocity in the Z-directioncan be expressed as

Wc/Vm BRo2e(Uc/Vm)-1 + 2(1 B)Ro. [18]

From the X-momentum equation, the scale of Uc/Vm can bederived as

the entrance region. Once the secondary flow is built up, theZ-component of the Coriolis force, 2pr f21 U, becomes signifi-cant to the contribution of the secondary motion. Additionally,secondary flow in the parallel mode is more complicated thanthat in radial mode, due to resultant forces Fx and Fz in Fig. 3,by which the secondary vortices may become asymmetric andirregular.

Vorticity Generation in Rotating DuctsRotational forces usually generate vorticity in flow fields.

With consideration ofCoriolis buoyancy, the dimensionless formof centrifugal and Coriolis generation of vorticity in a recent

analysis (Soong and Yan, 2001) can be expressed as

Be V [w ( r)0] B{[-(z + Zo)OO/Oy]i

+[(z + Zo)OO/Ox xOO/Oz]j + x(OO/Oy)k}, [22]

and

-2ROY [(1 BO)w v]

-2Ro[(l BO)Ov/Oy B(OO/Oy)v],

where w l/lll j is the unit vector of the rotating speedand the lower case letters denote dimensionless variables definedin the nomenclature. It is observed that the vorticity generationdue to centrifugal buoyancy and Coriolis effects depend stronglyon the development of the velocity and temperature fields. In aduct of uniform wall temperature, temperature gradients tendto vanish at the fully-developed condition, while all vorticitygeneration due to temperature nonuniformity diminishes. For aduct rotating in a parallel mode, the contribution ofCoriolis forcevanishes, since velocity gradients in the main flow (y-) directionall approach zero in a fully developed flow region. More detaileddiscussion on secondary vortices in rotating ducts can be foundin the author’s recent works (Soong and Yan, 1999, 2001).

Uc/Vm IBRo2 + 2(1 B)Ro(Wc/Vm)I 1/2. [19]

Then Eq. (18) becomes

Wc/Vm IBRo2e[BRo2 + 2(1 B)Ro(Wc/Vm)]-1/2

+2(1 B)Rol. [20]

For a typical case with B 0.1 and Ro O. 1, the estimate ofthe characteristic velocity of the secondary flow in cross-flowplane is

Wc/Vm 0.23 O(10-1). [21]

In the above scaling analysis, it is demonstrated that, in arotating duct of parallel mode, the centrifugal buoyancy effectplays a major role in the emergence of the secondary flow in

NON-ISOTHERMAL FLOWS IN ROTATINGCYLINDRICAL CONFIGURATIONS

The so-called rotating cylindrical configurations include ro-

tating cylindrical containers and rotating disk systems. Figure 4shows a rotating cylinder of radius Ro and height S. The rotat-ing disk systems can be as simple as a single disk or as com-plex as two co-axial disks rotating at different rates shown inFig. 5. Geometrically, for a large radius-to-height aspect ratio,Ro/S >> 1, the side-wall effect could be insignificant and theproblem is likely to be a special case of a two-disk system inFig. 5 with f21 f22. Actually, that is not always true because ofthe presence of the side wall. Therefore, in the following litera-ture survey, rotational convection in cylinders and disk systemsare given as two separate sections.

440 c.Y. SOONG

S Side Wall

/"Bottom Disk

Z

Core

Ekman

StewartsonLayer

R

FIGURE 4Rotating cylinder.

Literature SurveyRotating Cylinders

Ostrach and Braum (1958) first studied centrifugal-drivenconvection in a rotating container with fluid heated differentiallyin the vertical. Barcilon and Pedlosky (1967a, 1967b) employedlinear analysis to study the stratified fluid motion in a rotating

2 T2

nC T1

(a)

(b)

FIGURE 5Rotating disk systems. (a) Two co-axial disk rotating at

different rates; (b) Rotational forces on fluid particles at arate

cylinder. In a successive work (1967c), they performed a nonlin-ear analysis for stratified fluid heated from above in a rapidly ro-

tating cylinder with an insulated side-wall. They concluded thatthe buoyancy becomes important as PrBFr-1 >> Ek/2, whereFr f22S/g and Ek =_ 1)/’S2 are Froude number and Ekmannumber, respectively, and S denotes the height of the cylinder.In, this flow configuration with fluid heated from above, cen-

trifugal force pumping the colder (denser) fluid radially outwardalong the Ekman layer on the bottom disk, then through the ver-tical side-wall layers, momentum and heat are transported to theEkman layer on the upper disk. Barcilon and Pedloskyadvocated that this flow configuration is essentially distinctfrom the infinite disk flow without a vertical boundary.

Homsy and Hudson’s study (1969) considered buoyancyarose from centrifugal force and revealed that the side-wallthermal boundary condition (insulated, adiabatic, or perfectlyconducting) is very important. The critical parameters govern-ing the centrifugal driven convection are the radius- or diameter-to-height aspect ratio, Ro/S or D/S, and the dimensionless groupPrBEk. Their successive work (Homsy and Hudson, 1971a)further explored the effects of thermal boundary conditions ofside-wall and the bottom and top disks on the heat transfer per-formance. In a further analysis, Homsy and Hudson (1971b)demonstrated that the Nusselt numbers of the top and bottomdisks of a cylinder with a perfectly conducting side wall ap-proach that for the radially unbounded system as the cylinderradius increased. However, it is not the case for the cylinderswith an insulated side-wall. Hudson et al. (1978) conducted anexperiment for investigation of this centrifugal convection in a

rotating cylinder. Guo and Zhang (1992) numerically studiedfluid flow and heat transfer in a vertical rotating cylinder withand without gravity. To examine the validity of the similaritysolution in finite geometry, Brummell et al. (2000) numericallyinvestigated the effects of various side-wall conditions. Theircomparisons showed that the similarity solutions do describethe motions over three-quarters of the inner part of the cylinderprovided that the aspect ratio Ro/S > 10 or so.

The above-mentioned works are all employed a Boussinesqapproximation in the analysis. Non-Boussineq compressibil-ity effects were examined in a series of studies by Sakuraiand Matsuda (1974), Matsuda et al. (1976) and Matsuda andHashimoto (1976). They found that the critical parameter forthis case is (g 1)PrGo Ek-/z/4v, where V is the ratio ofspecific heats and Go is the square of a rotational Mach num-ber. As the above mentioned parameter is larger than unity, thecoupled effect of the fluid compressibility and the thermal con-dition on the bottom and top disks suppresses the flow in thecylinder.

Rotating Disk SystemsRotating free-disk flow was first analyzed by von Karman

(1921), and two theoretical arguments on flow structure betweentwo co-axial disks were proposed (Batchelor, 1951; Sewartson,1953). After these classic works, a large number ofinvestigations


on rotating-disk flows have appeared in the past decades. Mostof the studies were concerned with hydrodynamic natures only.Non-isothermal flow effect between two rotating disks can betraced back to the work of Ostrach and Braum (1958). For co-rotation of two disks, they claimed that the Coriolis force wouldinhibit the radial flow and heat transfer. In the case of co-rotatingdisks with a large ratio of disk radius to spacing between disks,i.e. Ro/S >> 1, the isothermal fluid rotates at the same ratewith the disks and the whole system behaves as a solid body.Once an axial temperature gradient is imposed, the centrifugalbuoyancy effect leads to a natural convection flow between twodisks and the state of solid body rotation cannot be retainedanymore. This class of thermally induced flow related to a sin-gle disk rotating with its environment was also studied by us-

ing compressible boundary layer flow equations (Riley, 1967)as well as by invoking a Bousinnesq approximation (Hudson,1968a).

Hudson (1968b) performed an asymptotic analysis for two-disk problem with consideration of the rotational buoyancy ef-fects. He disclosed that significant convection occurs as the con-dition of PrB*Refa 1/2 >> and the second order expressionof Nusselt number is Nu (PrB*Real/2/2)(1 PrB*/2), inwhich B* /5(T2 TI)/2 and Rea =_ f21sZ/v. Chew (1981)studied non-isothermal flow between two co-rotating infinitedisks with a radial temperature distribution Tw Tr + CRn.It was claimed that, even with the presence of the radial tem-perature/density gradient in the flow field, there is no rotationalbuoyancy effect as long as the radial temperature distributionsof the two disks are the same. If there is an axial temperaturegradient between the disks, however, the rotational buoyancyemerges and alters the flow structure. Chew addressed that theforced, natural, and mixed convection are all the possible flowstructures between two heated disks. Soong (1996a), using asimilarity model of centrifugal buoyancy (i.e. only buoyancyinduced by centrifugal force was considered) to disclose thebuoyancy influences on the flow structure and heat transfercharacteristics between two disks at different thermal and ro-tational boundary conditions. Effects of unsteadiness (Soongand Ma, 1995) and Prandtl number (Soong, 1996b) also werestudied. To include more complete buoyancy effects, a similaritymodel with consideration of buoyancy effects stemming fromall rotational body forces was developed recently (Soong andChyuan, 1998a) and the scales of each body force componentwere analyzed. By using this new model, thermal and solutalbuoyancy effects were explored (Soong and Chyuan, 1998b;1998c).

As to mixed convection between two co-rotating finite diskswith effects of radial through-flow and centrifugal buoyancy,numerical computations were conducted for cases of symmet-ric heating (Soong and Yan, 1993), asymmetric heating andflow reversal phenomena (Soong and Yan, 1994), and a porousdisks with wall transpiration (Yan and Soong, 1997). For co-rotating disk systems with fixed and rotating enclosures. Herreroet al. (1994, 1999) explored by numerical computations the ther-

mal flow structure and heat transfer in the a rotating cavityheated differentially in the axial direction. They found that cen-trifugal buoyancy has a significant influence on the flow struc-ture, especially in symmetry-breaking bifurcation of the flowpattern.

Scaling Analysis of Rotational Buoyancy Effectsin Rotating Disk Systems

Figure 5 (a) shows schematically a system of two disks rotat-

ing independently. A cylindrical rotating frame (R, Z) is fixedon the disk 1 and rotating with it. Let the disk be a referenceand ’1 > 0 in the sense of the rotational speed vector. The rateof disk 2, f22, can be positive, zero, and negative for the con-ditions of co-rotation, rotor-stator, and counter-rotation of thesystem. Figure 5 (b) draws the rotational forces exerted on amoving fluid particle at a radial distance R and local rotatingrate f2. Under the assumptions of laminar, steady, axisymmet-ric, and constant-property flow, the governing equations can bedepicted as (Soong and Chyuan, 1998a):

00(RU)+ (RW)=0, [23]

10(RU) ++#- R OR -’ [24]

OV OVprV- + pFW o-- -pr[1- t(T T,)]

UV

O O(RV)+-pr[1 (T Tr)]2QU + t R OR

[25]ow ow

prU-Ol + prW a-- + pr(T T)gOZ

[26]

OT OT _kI O (OT) 02T ]pr cpU- + prcpW ---- R- +- [271

By using the variables and parameters: r/-- Z/S, F U/Rf2,G V/Rf2, H W/(vQ)/2, 0 (T T)/ATc, Rea$22/v, and B =/3ATc --/(T2 T1), this set of equations canbe transformed to a similarity form:

H’"’ Re/2HH’’’ + 4Re/2[(1 + G)G’ B(G’O + GO’)]3/22BRe3/20 2Braea G(2G’O + GO’), [28]

1/2G" ,sea [HG’-(1 + G)H’ + BH’O + BGH’O], [29]112 0 t.0"-- Prtxee H [30]

442 e. . sooy6

In the previous work (Soong and Chyuan, 1998a), thebuoyancy effects rising from body forces in radial, tangential,and axial directions were examined. The radial componentsof body forces consist of the centrifugal and Coriolisforces due to disk rotation, Frb,ce and Frb,co, and the centrifugalforce due to curvilinear motion of the fluids, F,u,

Frh,co -pr(T Tr)2f21V -prflATc2S21Vc.

[31]

[32]

[33]

buoyancy forces, we have

prUOU/OR prfi(T T,.)V2/R + prfi(T Tr)Rf2+ prfi(T Tr) 2Q1Vp,.BO(V2/R + Rf2 + 2f21V); [37]

prUOV/OR prfi(T Tr)UV/R + prfi(T Tr)2AU,

[38]

or in terms of the thermal Rossby number B and the similarityvariables,

Tangential components ofbuoyancy forces induced by the curvi-linear motion of the fluids, Fth,cu and the Coriolis force, F,,coare

Ft,cu prfi(T Tr)UV/R prfiATcUcVclR,

Ft,,co prfi(T Tr) 2f21U prATc2f21Uc.

[34]

[35]

The only buoyancy force in the axial direction is the gravita-tional buoyancy. In the following text, assume the gravity effectcan be neglected with little effect. Denote the starred quantitiesas the force components normalized by the radial centrifugalbuoyancy prflATcRf212, i.e. F* F/prflATcRf212 and con-sider V R(f2- f21) with respect to the rotating frame. Interms of the similarity variable for the local tangential velocity,G V/Rf21 (Rf2 Rf21)/RS21 (S2 f21)/f21, the scalesof the buoyancy components can be rewritten in the followingform:

[36]

Co-rotating Disks at the Same RateFor two infinite disks rotating at the same rate and the same

sense, the isothermal flow is relatively stationary with respectto the rotating frame and lies at a conductive state, i.e. 0and U V R"1 W 0 or F G H 0. As an ax-ial temperature gradient is imposed, the rotation-induced buoy-ancy emerges and drives the fluid to move radially outward inan Ekman layer on the cold disk and radially inward alongthe hot disk. The Ekman layer thickness 3E has the order of3E/S Re-1/2, e.g. a correlation 6F/S 0.782Re-1/2 forco-rotating disks was proposed by Soong (1996). For weak con-vection induced by rotational buoyancy, the temperature is ap-proximately at the conductive state, i.e. 0

As the rigid-body rotation (U V W 0) is modified inthe presence of the rotational buoyancy, from radial and tangen-tial momentum equations with the inetial force balanced by the

F -t-v/0(1 4- G), [39]

G 2BO/(1 BO). [40]

Using the values of F and G, the axial velocity gradient can beestimated by H’ 2Rela/ZF + 2Rela/2/-O(1 + G). Soongand Chyuan (1998a) used a case of Rea 300, Pr 0.7,and B 0.1 to show that the above estimates of 6e, F, G, andH’ are all of reasonable agreement with the numerical solutionsof similarity equations. Furthermore, with G O (10-2) in thistypical example, it is obvious that, compared to the radial cen-trifugal buoyancy Prfi AT.Rf2, the order-of-magnitudes of theother radial components F/*b,cu -G2 O (10-4) and F*b,co-2G O(10-2) are negligibly small. It implies that in theco-rotating disk systems the major contribution to the rota-tional convection is the centrifugal buoyancy induced by systemrotation.

Disks Rotating at Different RatesFor the two disks rotating independently, the rotational buoy-

ancy between two disks is more complicated. In the case of twodisks rotating at different rates, even if the fluid is isothermal,a flow motion can be driven. Therefore, between two indepen-dently rotating disks, the rotation-induced flow can be drivenby two mechanisms. One is the unbalanced pumping effect inthe Ekman layers of the two disks; the other is the rotationalbuoyancy effect. The tangential velocity of the fluid and thelocal buoyancy effect could be spatially different in the flowfield. The buoyancy forces for various rotation rates of the lo-cal fluids can be estimated by Eqn. (36). For example, as lo-cal rotation rate of the fluid is f21/2, G is -0.5 by definition.Therefore, with * -G2F/,c -1, we have F/,c -0.25and Fr*b,co 2G 1. The centrifugal buoyancy due to sys-tem rotation is no longer dominant and the rotational force ef-fects become more complicated. It should be noted that, in thecases of two disks rotating at different rates, all the rotationalforces should be regarded as the sources ofthe thermal buoyancyeffect.

CONCLUSIONSThe present work has presented a literature survey ofprevious

works related to the study of rotation-induced buoyancy effects


in rotating systems. With scaling analysis ofrotational buoyancyforces in rotating ducts and rotating disk systems, the authorhas tried to offer physical insights into the qualitative nature ofthe rotating non-isothermal flow behaviors. Although numerousstudies on the subject have been conducted on this complicatedflow with interaction of rotational forces and thermal effects,much is still incomplete and poorly understood. There are stilla number of questions open to future investigation. Especially,rotational buoyancy-induced instabilities, rotational buoyancy Freffects in turbulent flows, and turbulence modeling in numerical gsimulation of rotating flows are the topics most worthwhile to hbe investigated, k

Recently, thermal flow instability in a rotating configuration Nuattracted lots of attention. Hart (2000) investigated the interac- Ption ofcentrifugal convection and thermal instability in a rotating Pdcylinder. Turkyilmazoglu et al. (1999) performed a linear stabil- Prity analysis for the absolute and convective instabilities in the Recompressible boundary layer of a rotating disk. It is concludedthat the overall effect of compressibility is to reduce the extend of Reabsolute instability at higher Mach numbers. The wailheatingeffect enhances the absolute instability regime for theparam- Roeter range studied. A series of linear stability analyses (Soong R, Zand Tuliszka-Sznitko, 2000; Tuliszka-Sznitko and Soong, 2000; STuliska-Sznitko et al., 2001) ofmixed convection between rotor- Tstator disks have been performed for the basic flow states from ATe.Soong and Chyuan (1998a). It is found that rotational buoyancyindeed has significant influences on the instability of the non-isothermal flow between coaxial rotating disks. To this class ofcomplicated flow, however, much more effort must be made forbetter understanding of the underlying physics.

The influences of rotational buoyancy on the transport phe-nomena in turbulent flow are another important issue worthyto be investigated. The assisting and opposing effects on thethermal flow adjacent to the wall could interact with the wallturbulence and the resultant effects must be very distinct from othe buoyancy effects in laminar and transitional flows. The de-tails of the nature of the coupling are not completely understood.Numerical simulation of the rotating non-isothermal flow needsa reliable turbulence model with the inclusion of the effects ofrotational buoyancy forces. To develop a turbulence model andnumerical techniques would be significant in improvingnumerical prediction of rotating non-isothermal flows, vExperimental work for high confidence measurements of flow 0and heat transfer characteristics in rapidly rotating systemsis most needed for the usefulness in either understanding ofthe flow physics or development of a turbulence model.

NOMENCLATUREB thermal Rossby number,/3 ATe.B* alternative thermal Rossby number,/3(T2 T1)/2D hydraulic diameter of ductE eccentricity defined as the distance of duct center to

axis of rotation

F,G,H

Fb, Fr*

Fb, F*b

U,V,W

X,Y,Zx, y, z

Zo

dimensionless velocity functions in radial, azimuthaland axial directiondimensional and normalized radial buoyancy force,F;*b Frb/Prfl ATcRf2dimensional and normalized tangential buoyancyforce, F?bdimensional and normalized axial buoyancy force,

F*b Fza/pATFroude number, R/ggravitational accelerationheat transfer coefficientthermal conductivityNusselt number, hD/k

pressurepressure departure from the reference state, P PrPrandtl number, v/Reynolds number based on mean velocity of mainflow, VmD/vrotational Reynolds number, DZl/v for ducts andS2/v for disksrotation number, D / Vmcylindrical radial and axial coordinatescylinder height or spacing between two diskstemperaturecharacteristic temperature difference, T To or

T Tvelocity componentsmean velocity of main flow in ductsdimensionless velocity componentsCartesian coordinatesdimensionless coordinatesdistance from center of inlet plane to axis of rotation

Greek Symbolsthermal diffusivity, k/pcpthermal-expansion coefficient, -(1/Pr)(Op /OT)pratio of specific heatsboundary layer thicknessdimensionless eccentricity of the duct, E/Ddimensionless axial coordinate, Z/Sdynamic viscositykinematic viscositydimensionless temperature function, (T Tr)/ATe.rate of rotation

Subscriptsdisk

2 disk 2c characteristic quantityce centrifugal buoyancy component due to disk rotationco buoyancy component due to Coriolis forcecu buoyancy component due to curvilinear motion of

the fluidsE Ekman layer

444 c.Y. SOONG

inlet conditionreference statewall conditionrotation

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