Prandtl Number Effects on Mixed Convection Between Rotating Coaxial Disks

Prandtl number characterizes the competition of viscous and thermal diffusion effects and, therefore, is an influential factor in thermal-fluid flows. In the present study, the Prandtl number effects on non-isothermal flow and heat transfer between two infinite coaxial disks are studied by using a similarity model for rotation-induced mixed convection. To account for the buoyancy effects, density variation in Coriolis and centrifugal force terms are considered by invoking Boussinesq approximation and a linear density-temperature relation. Co-rotating disks (Ω2=Ω1) and rotor-stator system (Ω1≠Ω2=0) are considered to investigate the free and mixed convection flows, respectively. For Reynolds number, Re, up to 1000 and the buoyancy parameter, B=βΔT, of the range of |B|≤0.05, the flow and heat transfer characteristics with Prandtl numbers of 100, 7, 0.7, 0.1, and 0.01 are examined. The results reveal that the Prandtl number shows significant impact on the fluid flow and heat transfer performance. In the typical cases of mixed convection in a rotor-stator system with |B|=0.05, the effects in buoyancy-opposed flows B=0.05 are more pronounced than that in buoyancy-assisted ones.


INTRODUCTION
he rotating-disk flow is related to a number of fundamental issues in fluid dynamics as well as to the practice of a variety of rotating machinery.After the pioneering work with the similarity analysis for the free-disk flow by von Karman [1921], and the analyses concerned with flows between coaxial disks by Batchelor  [1951] and Stewartson [1953], numerous subsequent investigators aimed at the solutions of the simple models of similarity nature for the two-disk problems, e.g.Lance and Rogers [1962], Pearson [1965], Mellor et al. [1968]  and Barrett [1975] etc.
By considering non-uniformity of the fluid tempera- ture in a rapidly rotating device, rotation-induced buoyancy effect may become important for high rotational forces.The rotational buoyancy, centrifugal and/or Coriolis, have been included in some previous studies on rotating systems, e.g. the rotating closed cylinders by Busse and Carrigan [1974], Homsy and Hudson   [1969a,b], Hudson et al. [1978], Guo and Zhang [1992], gas centrifuges by Matsuda et al. [1976], Matsuda and   Hashimoto [1976], and radially rotating channels by Siegel [1985], Soong and Hwang [1990, 1993].In the presence of a through-flow in wheel-space of the finite coaxial disks, the rotation-induced buoyancy effects have been studied by solutions of bounday-layer equations (Soong and Yan [1993]) and Navier-Stokes equations (Soong and Tzong [1991]).For infinite disks without through-flow, Hudson [1968] has performed an analysis for flow between two co-rotating disks with consider- ation of the rotational buoyancy.Only the low Reynolds numbers (Re <-100) and very small buoyancy param- eters ([3AT _< 0.01) were involved in the analysis.Later, Chew 1981 developed a linearized model by neglecting radial viscous terms and nonlinear inertia terms in momentum equations.However, the solution is essen- tially a highly simplified one.Most recently, by solely considering centrifugal buoyancy effect, a unified simi- larity analysis for the free, forced, and mixed convection flow and heat transfer in two-disk problems has been developed [1995].
In thermal-fluid flows, Prandtl number characterizes the competition of the viscous and the thermal diffusion 162 CHYI-YEOU SOONG effects.Therefore, it can be expected that the Prandtl number plays an influential role in the free and mixed convection flow and heat transfer problems.In the present study, a similarity model of rotation-induced buoyancy is employed and the emphasis is placed on the Prandtl number effects on flow and heat transfer charac- teristics in this class of buoyancy-influenced rotating flows.Two rotational conditions, i.e. the co-rotating disks (fe fl) and the rotor-stator system (fl 4= fe 0) are considered.For Reynolds number up to 1000 and buoyancy parameter in the range of -=0.05 -< [3AT _< 0.05, the flow and heat transfer characteristics with Pr 100, 7, 0.7, 0.1, and 0.01 are examined.
pRf z FIGURE 2 Rotational forces acting on the moving fluid particle.

THEORETICAL ANALYSIS
equations can be depicted in a similar form as that in the work of Homsy and Hudson [1969]" Problem Statement Two infinite coaxial disks separated by a spacing S as shown in Figure is the physical model of the problem.
Two disks rotating at rotational rates "1 and f2 are of uniform temperatures T1 and Te, respectively.A cynlin- drical coordinate (R, q, Z) is fixed on the disk and its origin lies at the disk center.With respect to the rotating frame shown in Figure 2, the fluid particle locating at a radial distance R away from the axis of rotation and rotating at Ok encounters three rotational body forces, i.e. the centrifugal force, pR2, radial component, 2pIV, and circumferential component, 2pRfU, of the Coriolis force.The fluid flow is assumed to be steady, laminar, axi-symmetric and of constant-property; and the Boussinesq approximation is invoked to take into account the rotation-induced buoyancy effect.The stress-work ef- fects are all ignored.The gravitational force term is also neglected by comparing with the centrifugal force in rapidly rotating systems.In the present study, the wall condition of disk is used as the reference state, at which the fluid confined by the disks lies at the temperature Tr T1 and rotates with the reference frame as a solid body, therefore, U V W -= 0 and -V Pr/Pr -X X R. Furthermore, by considering a linear density-tempera- .
in which P' P Pr is the pressure departure from the reference condition, f 121k, R R i, and k and are the unit vectors in the axial and radial directions, respectively.By using the following dimensionless vari- ables and parameters 3AT (4)   where U, V and W, respectively, represent velocity components in R, q and Z directions, and AT is the characteristic temperature difference.Equations (1-3) can be recast into the following dimensionless form: 0" PrRe HI?.
The relationship between axial and radial velocities" H'+ Rel/2F 0 (8) has been introduced to the above system.The Z-component of momentum equation, with a dimensionless pressure parameter H P'/PrS2I 2, can be written as I-I Re-3/2H'-Re-ill2/2 (9) Obviously, Equation ( 9) is not coupled with the system of Equations (5-7).After solving the axial velocity solution H('q), the pressure function H can be evaluated.
The boundary conditions for the velocity and tempera- ture functions are where the parameter (122 -1)/-1 denotes the dimensionless rotation rate of the disk 2. Note that the boundary conditions for F(rl), F(0) F(1) 0, have been already replaced by H'(0) H'(1) 0 through Equation ( 8).

Governing Parameters
Four parameters are involved in the problem, they are Pr, Re, B, and ".The Prandtl number indicates the relative importance of viscous to thermal diffusion effects.The Reynolds number Re characterizes the rotational effect and the thermal Rossby number B measures the buoy- ancy effect.The last parameter / denotes the relative rotation rate of the disk 2 with respect to that of the disk 1.For example, the values of / 0 and -1 correspond to the cases of co-rotating disks (1) 2 11) and rotor- stator (121 4:2 0), respectively.Note that, in this two-disk flow configuration, the cases of /= 0 and B 4: 0 are the pure free-convection.While the forced convec- tion is characterized by -4:0 and B 0. For the non-zero B as well as , the problem becomes a mixed convection one, in which Re can be used to characterize the forced flow effect.In the conventional free-convec- tion study, for the validity of Boussinesq approximation, [3AT was usually small, for example, the magnitude less than 0.1 in the study of Gray and Giorgini [1976].The positive value of B implies T < T 2 and the temperature of the fluid adjacent to the disk is higher than T 1, the flow is designated as buoyancy-opposed flow."On the contrary, the flow with B < 0 (or T > T2) is a buoyancy-assisted one.In the present study, the param- eter B is restricted in the range of IBI -< 0.05, the Reynolds number based on the disk spacing lies up to 1000, and the rotation parameter / 0 and -1 are considered.
Heat transfer performance is characterized by Nusselt number defined as Nu -(0T/0n)w/(T2 T1).By this definition the positive and negative values of Nu denote the heat transferred from and to the wall, respectively.The heat transfer rates on the two disks are Nu and Nu2, viz.Nu 0'(0), Nu 2 0'(1) (13) NUMERICAL PROCEDURE The system of Equations (5-7) with boundary conditions (10) consist of a nonlinear eighth-order two-point bound- ary value problem.A typical shooting method can be started with the guessed missing conditions: H"(0) a, H'"(0) b, G'(0) c and 0'(0) d.In an iterative procedure, the values of a, b, c and d are updated continuously using Newton's method until the boundary conditions at xl 1, i.e.H(1) H'(1) G(1) /= 0(1) 0, are met.The iteration is regarded as convergent if the stopping criterion, max(Aa, Ab, Ac, Ad) -< 10-8, is satisfied.Low-Re solutions can be easily obtained using conventional shooting techniques.However, due to the stiffness of the system, the convergent solution is getting hard as Reynolds number increases.By applying non- uniform grid, under-relaxation, and the Aitkin accelera- tion technique, the convergent solutions at high Reynolds numbers can be attained.Through the numerical experi- ments, the grid-dependence of the numerical solutions were examined.In general, the grid of 201 points is sufficient for grid-independent solutions.For higher Re, small I1, and/or large IBI, the finer grids, e.g. 400points or more, were used for either high resolution and convergence of the solutions.

RESULTS AND DISCUSSION
Flow and Temperature Fields In a former study (Soong [1995]), the buoyancy-free solutions of the present formulation have demonstrated CHYI-YEOU SOONG their reasonable agreement with the experimental results.
In the present work the emphasis is placed on the rotation-induced buoyancy effects.Figure 3 shows the free-coneection solutions of Re 300, / 0 and B 0.05.For higher Pr, e.g.Pr 7, the temperature distribution deviates from the conductive solutions due to the convection effect.For the increasing temperature gradient near the lisk 1, the cooler fluid is accelerated radially outward.This enhances the peak value of radial velocity which, in turn, alters the axial velocity distribu- tion H. Through the action of the buoyancy effect by Coriolis force the circumferential velocity G presents a noticeable change.
A forced convection (buoyancy-free) solutions for a rotor-stator system (y 1) with Re 500 and variable Prandtl numbers are presented in Figure 4. Since the velocity-temperature coupling has been broken by ignor- ing the buoyancy effect, the velocity profiles become independent of the Prandtl number.While the tempera- ture solutions are still a strong function of Pr.For .relatively higher Prandtl numbers, e.g.Pr 0.7 and 7 in Figure 4, thermal boundary layer emerges on the disk 1.The appearance of the thermal boundary layer is attrib- uted to the relatively strong convection effect.
Figures 5 and 6 show the buoyancy-influenced coun- terparts of the case in Figure 4.In Figure 5 with B 0.05, the Prandtl number effect significantly alters the flow fields.For large Prandtl number, Pr 7, the temperature function changes abruptly in the thin ther- mal boundary layer but remain uniform in large portion of the wheel space.As Pr decreases from 7 to 0.01, the 0.34 0.12 F o.oo Pr-effects on solutions of forced convection in rotor-stator FIGURE 4 system.
thermal diffusion is getting more and more important and, then, the temperature variation appears notably in the whole domain rather than confined in a narrow region of thermal boundary layer.For small Pr, the temperature gradient near the disk 1, i.e.Z 0, is alleviated.Therefore, the reduction in buoyancy-opposing effect enhances the radial velocity peak near the disk 1.The axial velocity distribution is modified with the variation of radial velocity.Due to coupling of the Coriolis- induced buoyancy in circumferential fluid motion and 0.005 the Prandtl number effects, salient Pr-dependence of the circumferential velocity is presented.While in the case of buoyancy-assisted flow with B -0.05, the velocity fields are only slightly altered by the change in Prandtl number, see Figure 6, although the temperature distribu- tion varies in the similar manner as that in buoyancy-free and buoyancy-opposed flows.
Friction Factors.and Heat Transfer Rates Figure 7, as a typical example, shows radial and tangen- tial friction factors in a rotor-stator system with B 0.05.In the region of Re < 100, friction factors have no significant change with Prandtl number.As the Reynolds number or the rotation-rate increases, the Pr-effect on the friction factors can be enhanced through the remarkable variation in temperature distribution.It is also .notewor-thy that the changes in radial friction factor is larger than that in tangential one.
Heat transfer performance is measured by the Nusselt number.Figure 8 presents Nusselt numbers on the disks and 2, i.e.Nu and Nu2, for buoyancy-assisted flows in a system of "y and B -0.05.As that mentioned in the last section, the Prandtl number significantly affects the temperature fields, especially, for moderate/ high Prandtl numbers.In the limit of very small Pr, the heat transfer solution approaches to'the conductive state, i.e.NUl Nu2 1.In the rotor-stator system, since the driven force of the mixed convection flow is the rotation of the disk 1, the temperature gradient as well as the heat transfer rate are more pronounced on the disk 1.This is the reason why the value of Nu changes drastically but the Nu2 curves appear flat even for a wide range of Pr, i.e. 0.01 -< Pr -< 100.Re FIGURE 6 Pr-effects on buoyancy-assisted mixed convection in FIGURE 8 Nusselt numbers on disks and 2 for mixed convection in rotor-stator system, rotor-stator system.

CONCLUDING REMARKS
The Prandtl number effects on buoyancy-influenced non-isothermal ttow between coaxial disks have been studied by using a similarity model.The results reveal that the change in Pr leads to a variation of temperature distribution, which, in turn, modifies the velocity fields through a coupling of the thermal and hydrodynamic natures linked by the rotation-induced buoyancy effects.
From the present theoretical analysis, it can be disclosed that, for the two-disk problems, the Prandtl number presents significant impact on the flow and heat transfer characteristics in either free-convection and mixed con- vection.In the typical cases of mixed convection flow and heat transfer with the same measure of buoyancy, IBI 0.05, the Pr-effect in buoyancy-opposed flows (B 0.05) is more pronounced than that in buoyancy-assisted ones (B -0.05).
FIGUREPhysical model of two co-axially rotating infinite disks.

FFIGURE 3 FIGURE 5
FIGURE 3 Pr-effects on solutions of free convection between co- rotating disks.
FIGURE:7 Friction factors in mixed convection flows in rotor-stator system.