International Journal of Rotating Machinery (C) 1998 OPA (Overseas Publishers Association) 1998, Vol. 4, No. 1, pp. 35-48 Amsterdam B.V. Published under license Reprints available directly from the publisher under the Gordon and Breach Science Photocopying permitted by license only Publishers imprint. Printed in India.

Turbulent Flow and Heat Transfer in Circular Couette Flows in Concentric Annulus

SHUICHI TORII a,. and WEN-JEI YANG b

Department of Mechanical Engineering, Kagoshima University, 1-21-40 Korimoto, Kagoshima 890, Japan," Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, MI 48109, USA

(Received in final form 5 September 1996)

A numerical study is performed to investigate heat transfer and fluid flow in the hydro- dynamically and thermally fully-developed region of an annulus, consisting of a heated rotating inner cylinder and a stationary insulated outer cylinder. Emphasis is placed on the effect of rotation of an inner core on the flow structure and the thermal field. A Reynolds stress turbulence model is employed to determine three normal components of the Reynolds stress and its off-diagonal one. The turbulent heat flux is expressed by Boussinesq approximation in which the eddy diffusivity.for heat is given as functions of the temperature variance 7 and the dissipation rate of temperate fluctuations ct. The governing boundary- layer equations are discretized by means of a control volume finite-difference technique and numerically solved using the marching procedure. An inner core rotation causes an amplification of the three normal components of the Reynolds stress over the whole cross section, resulting in a substantial enhancement in the Nusselt number.

Keywords." Circular Couette flow, Reynolds stress model, Taylor number, Two-equation heat transfer model

INTRODUCTION by the centrifugal force induced by the swirl. In other words, the turbulent transfer of heat and The convective heat transfer in turbulent swirling momentum is suppressed or promoted by the flows is often encountered in chemical and mechan- interaction between turbulence and centrifugal ical mixing and separation devices, electrical and force associated with the swirl. turbo-machinery, combustion chambers, pollution Murakami and Kikuyama (1980) measured the control devices, swirl nozzles, rocketry, and fusion velocity profile and hydraulic loss in a hydrodynam- reactors. In these flow fields, the heat transport ically fully-developed flow region of a rotating phenomena in connection with the flow and the pipe. It was disclosed that both turbulence and turbulence properties are substantially influenced hydraulic loss were remarkably reduced due to pipe

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35 36 S. TORII AND WEN-JEI YANG rotation and that streamwise velocity profile causes an amplification of the turbulent kinetic gradually deformed into a parabolic form with an energy over the whole cross section, resulting in a increase in its speed. Kikuyama et al. (1983) substantial enhancement in the Nusselt number. analyzed the variation of streamwise velocity A k-e model, which is one of the two-equation profiles using a modified mixing length theory models of turbulence, is popular in computational proposed by Bradshaw (1969). A combined experi- analyses of turbulent flow. However, in the case of mental and theoretical study was performed by an annular duct flow, the importance of taking the Hirai and Takagi (1988) using the Reynolds stress effect of the term, -uv utOU/Or, into account is model to determine the effects of pipe rotation on recognized. This is because the time-averaged local fluid flow and heat transfer in a thermally and shear stress, -uv, does not go to zero at the radial hydrodynamically fully developed flow region. It location where the time-averaged streamwise veloc- was found that an increase in the rotation rate ity has its maximum value, i.e. a radial gradient of resulted in a decrease in heat transfer performance zero (Rehme, 1974). Furthermore, since the two- with the Nusselt number asymptotically approach- equation k-e model basically assumes isotropic ing that of a laminar pipe flow. Fluid. flow and heat turbulence structure, it cannot precisely reproduce transfer characteristics in the thermally and hydro- the anisotropy of turbulence caused by the inner dynamically developing and fully-developed regi- core rotation. In order to obtain the detailed ons of an axially rotating pipe were investigated by information pertinent to the flow structures, it is Torii and Yang (1995a,b) using the existing k-c necessary to employ the higher order closure turbulence models in which they are modified to model, i.e. a Reynolds stress turbulence model. include the swirling effect. Throughout the numerical simulations for the On the contrary, an effect of promoting the above turbulent heat transport problems in the swir- turbulent transport of heat and momentum by the ling flows, the turbulent heat flux in the energy equa- centrifugal force occurs in a concentric annulus tion is modeled by using the classical Boussinesq with an inner cylinder rotating around the axis, in approximation. The unknown turbulent thermal which Taylor vortices appear (Kuzay and Scott, diffusivity Oz is obtained from the definition of the 1975; 1976). Such swirl flow is referred to as known turbulent viscosity ut and the turbulent circular Couette flow, which implies a flow with Prandtl number Prt as Ot--- ut/Prt. In this formu- one surface rotating and the other stationary (or lation, an analogy between eddy diffusivities of both surfaces rotating in the same direction at momentum and heat is implicitly assumed. different angular velocities). Hirai et al. (1987) However, shear flow measurements (Hishida et al., conducted an experimental study on an effect 1986) and direct simulation data (Kasagi et al., of inner core rotation on turbulent transport of 1992) reveal that its analogy, as represented by the momentum by using a two-color laser Doppler turbulent Prandtl nu.mber, cannot adequately velocimeter. It was disclosed that the Reynolds reflect the physical phenomenon of heat transport stresses increase due to the swirl. Torii and Yang and there are no universal values of turbulent (1994) analyzed heat transfer mechanism in annuli Prandtl number even in simple flows. If the above with an inner core rotation by means of several turbulent Prandtl number assumption is employed different two-equation k-c turbulence models. It to determine the turbulent thermal diffusivity in was found that (i) in the entrace region, the axial annuli, it yields negative along the radial direction. rotation of the inner cylinder induces a thermal This is because the turbulent viscosity, ut, becomes development and causes an increase in both the negative in the radial region between velocity Nusselt number and the turbulent kinetic energy in gradient of zero and Reynolds stress of zero, as the inner cylinder wall region, and (ii) in the fully- mentioned previously. In order to solve this developed region, an increase in the Taylor number problem and to obtain detailed information on CIRCULAR COUETTE FLOW 37 the heat transport phenomena, two-equation is placed on the mechanism of an augmentation of model for thermal field and turbulent heat flux heat transfer performance due to the inner core equation model are considered to be employed. rotation. The heat flux equation model ought to be more universal, at least in principle. This model, how- ever, is still in intensive development and is little THEORETICAL ANALYSIS used for practical applications (for example, Lai and So, 1990). The results are not as satisfactory as Governing Equations initially expected, mainly due to a few unreasonable Consider a steady turbulent flow through a hypotheses in the model (Nagano and Tagawa, concentric annulus consisting of the insulated Thus, the turbulent heat flux model should 1988). stationary outer cylinder and the slightly heated wait until the Reynolds stress models are tested and inner cylinder rotating around the axis, in which well developed, preferably with the near-wall the boundary layer is developing both thermally since the source of error in heat modeling, major and hydrodynamically. The physical configuration transfer predictions is that in calculating the and the cylindrical coordinate system are shown in velocity field (Kasagi and Myong, 1989). Fig. 1. Some approximations are deduced that: (i) This paper treats the thermal transport phenom- viscous heating is negligible; (ii) the axial conduc- ena in a concentric annulus, in which a slightly tion term in the energy equation is neglected for heated inner core rotates around the axis and an Pe > 1; and (iii) the viscous dissipation term in the insulated outer cylinder is held stationary. In order energy equation is neglected. An order-of-magni- to shed light on the mechanism of the transport tude analysis indicates all second-derivative terms phenomena, the Reynolds stress turbulence model to be negligible in the streamwise and tangential proposed by Launder and Shima is employed, (1989) directions. The simplified governing equations read because this model is able to reproduce the inherent as follows: anisotropy in the near-wall region of isothermal Continuity equation flows. The turbulent thermal diffusivity Yt is deter- mined using the two-equation model for heat tran- OU + OV + V o. sport proposed by Yousseff et al. (1992). Emphasis o-; -r

r, V, v outer cylinder (stationary, insulation)

,.!iiiiiii::ii::!ii!ii::iiii::iiiii::ii::ii!::iii!!i::i,!,i.:.:,::::!! ::,:::': //l inner cylinder heatln, O, W, w rotating a Ww

FIGURE A schematic of physical system and coordinate. 38 S. TORII AND WEN-JEI YANG

Momentum equations." x direction."

OU dP + vOU + ru rVf (2) U-Ox Or p dx r direction."

OP Or2 P W2 w 2 (3) Or -P--r + -r ( + ); 0 direction." --- OW vw (7) Ux + vOW-ffF +-r W2] 0 o(w/r) u 2 r3 (4) r Or Or r2--}. 2 2 u OW Ow Ox +v 0---;- Energy equation." r 2 =-/05 u+Cs-Tg Or u+Csw OT OT O OT j+2 u + v o-; -r or ( r) (5)

The Reynolds stress turbulence model proposed by Launder and Shima (1989) is employed to evaluate -uv in Eq. (2) and -vw in Eq. (4). The transport equations can be expressed as U2:

V r u Ox + 0----=-/ 0--- + Cs-- -67 blV:

OU e 2 2(1 --fz)UV--r--fl--+-j(fl Or r + Cs v + Csw2 c 2 r Or -r ' r2 +flwfxV2---j(f + f2wfx) c 3 -A- Awf- (l -A + f2wL3)OUor OU OW W x w2 3 gf-r + vW-r VW--r + 2 f2 + uw G uv --rV) ( jf2f)WlO(kr FOr ) W k Ow (6) -G-vw,- (9) CIRCULAR COUETTE FLOW 39 vvg: The turbulent heat flux -v-7 in Eq. (5) is expressed through Boussinesq's approximation, as OT V--'- ge Or" (13) Youssef et al. (1992) obtain the turbulent thermal diffusivity, ct, using the temperature variance, 2, and the dissipation rate of temperature fluctuations Ct together with k and c, as

ct CA fk (14)

where CA is the model constant and f is the model (10) function. Both transport equations, and Ct in Eq. (14), are written in the tensor form as b/W: Ot2 0 Ot2 OT ot 2ujt (15) and 0 -1- O Cp1 fe, ujt Ox OXj J OX Ct OUi C2 fp uiu; Cm fo2 (16) k - Turbulent energy dissipation rate c is determined In the present study, the Reynolds stress turbulence from model is employed-in place of a k-c model. Thus a slight modification is made to the turbulent diffusion terms in Eqs. (15) and (16). Original turbulent diffusion terms in both equations are written as -ujt2 and -ujc, respectively. Both terms are modeled using a gradient-type representation and are expressed as

kOt2 k__Oct ujt2 Cst- uiuj and ujc Cs- uiuj (12) e e Oxi (17)

The empirical constants and model functions in respectively (Jones and Musonge, 1988; Sommer Eqs. (6)-(12) are summarized in Table I. et al., 1992). Here Cst and Cs are the diffusion 40 S. TORII AND WEN-JEI YANG

TABLE Empirical constants and functions for a Reynolds stress turbulence model C =2.58 C2=0.75 Cw 1.67 C2w--0.5 CL=2.5 Cs 0.22 C 0.18 CI 1.45 C2 1.9

A2 + + +2 +2 +2

A3

O U O W v__w P -uv--- W-+ W A 1-Az+A3 fx k3/z/cLyg-- j -exp{- (0.0067Rt) /R2 exp{-(0.002Rt)2} fl 4- C1 fRIAA2/4 f2 C2AI/2 /lw fl +Clw .f2w {(f2 1)+ C2w + (f2 1)+ Cwl} 01 2.5A ( 1) 02 0.3fRz(1--0.3A2) coefficients. From this consideration, transport TABLE II Empirical constants and functions for a two- equation model for thermal transport equations of 2 and Ct for the cylindrical coordinate system in Fig. are expressed as Cp 1.70 Cp2=0.64 Cst=0.11 Cs-0.11 C) 2.0 CD2 =0.9 fel 1.0 UP2 1.0 5 Ot2 Ot 0 k Ot2 CA =0.10 BA 3.4 C2, 1.9 AA 26/Pr + v r Cst-)5+c -x r Or Rh (k/u)(k/c)-' (t-Y/ct) fD1 {1 exp(-y+/5.8)} 4- 2ctt 2ct (8) -r fD2 (1/CD2)(Ce2,f2, 1){1 exp(-y+/6)} and /3/4' {1 -exp(-y+/Aa)}2\l 4- BA/'t ) xOct + v Oct 0.3 exp{-(Rt/6.5) } r r Or Csc-+c 2 Ct t 0S + Cpfpt -Cpfp uv (0Z) 0r respectively. The empirical constants and model O( W/r) functions in Eqs. (14), (18) and (19) are summar- Cp2Jb2 CDI fDl e: rt { Or ) ized in Table II. CDZfD2 EEt (19) The boundary conditions at both the inner and k' outer walls in the annulus are specified as, CIRCULAR COUETTE FLOW 41 r--rin (inner tube wall)" 1. Specify the initial values of U, V, U2, 12, W2, uv, vw, uw, and e, and assign a constant axial U-- V-- bl2 V2 W2 btV VW tAW-- O, pressure gradient. 2. Solve the equations of U, V, u2, v2, w2, uv, vw, W- Ww, c uw, and e. 3. Repeat step 2 until the criterion of convergence O2 02(t-2/2) OT qw is satisfied, which is set at O, 0-- Or Ct Or2 Of" Aw max M 0M-1 < 10-4 (20) r-rout (outer tube wall)"

2 2 2 U- V- W- u v w uv vw uw O, for all the variables 05. The superscripts M and 2 Ot2 02(t2/2) M-1 in Eq. (20) indicate two successive itera- (0- O, O Or t Or2 tions, while the subscript "max" refers to a e-2"\--OTrJ maximum value over the entire fields of itera- OT 0 (insulation). tions. Or Calculate new values of U, V, u2, v2, w2, uv, vw, uw, and e with a corrected new axial pressure Numerical Method gradient. A set of governing equations employed are Repeat steps 2-4 until the conservation of the discretized using a control volume finite-difference total mass flow rate is satisfied under the procedure proposed by Patankar (1980). Since all criterion: turbulence quantities as well as the time-averaged y gcpO dr ginletO dr axial and tangential velocities vary rapidly in the f f f _< 10 -5, (21) near-wall region, two control volumes are always f y SinletO dr located within the viscous sublayer, y+-5. The radial mesh size is increased from a minimum value followed by evaluating convergent values of U, adjacent to the wall towards the turbulent core V, u2, v2, w2, uP, lw, uw, and e. Ucp is the axial region in geometrical proportion, and the maxi- velocity under the correction process and Uinle mum control volume size in the turbulent core is that at the inlet of the annulus. region is always kept within 3% of D/2. Mean- Repeat steps 2-5 until a hydrodynamically while, the axial control volume size is constant at fully-developed annular flow in the absence of five times the minimum radial size for the wall. rotation is realized. Throughout numerical calculations, the number of Start both axial rotation and heating of an control volumes in the radial direction was prop- inner cylinder. erly selected between 70 and 92 to ensure validation Solve the equations of U, V, W, u2, v2, w2, uv, of the numerical procedures as well as to obtain vw, uw, , T, 2, and gt" grid-independent solutions. The maximum relative Repeat step 8 until the criterion of conver- error over all dependent variables within this gence, Eq. (20), is satisfied. change of grid spacing was kept within 1%. Since 10. Calculate new values of U, V, W, u2, v2, w2, uv, the governing equations are essentially parabolic, vw, uw, , T, 2, and gt with a corrected new calculation is performed from the inlet in the axial pressure gradient. downstream direction by means of the marching 11. Repeat steps 8-10 until the conservation of the procedure. The computations are processed in the total mass flow rate is satisfied under the following order" criterion, i.e. Eq. (21), followed by evaluating 42 S. TORII AND WEN-JEI YANG

convergent values of U, V, W, 102 uw, , T, 2, and et. Turbulent 12. Repeat steps 8-11 until x reaches the desig- 'o nated length (200D), where thermally and Nu=O.O203Re'8..Pr'4 hydrodynamically fully-developed flow condi- o -2 tion prevails. The ranges of the parameters for the present study are Reynolds number Re 6000-10,000; Taylor numbers Ta-0 and 5000; radius ratio r*-0.8; Prandtl number Pr- 0.7 (air); and heat flux at the inner wall qw 200 W/m2. The CPU time required in II ,,,,L completing the above scheme was about 50-100 h 10 4 5 on a NEC personal computer (32 bit), depending on 10 Re 10 the number of control volumes used. It is necessary to verify both the turbulence FIGURE 2 A comparison of Predicted Nusselt number with test results (Dalle Donne and Meerwald, 1966) for the fully- models of heat and momentum employed here and developed turbulent annular flow with a stationary inner cyl- the reliability of the computer code by comparing inder for r*= 0.56 and Re 46,000. numerical predictions with experimental results for the flow field. The model is applied to a flow in an respectively. It is observed that the model yields a annulus with a stationary, slightly heated inner better agreement with the experimental data, and core. Numerical results for the thermally and predicts the velocity profile with the well-known hydrodynamically fully-developed annular flow at characteristics of the logarithmic region, i.e. the a location 200 tube diameter downstream from the universal wall law. Figure 4 illustrates the radial inlet are compared with the experimental data distributions of three normal components of the (Re-46,000 and r*-0.56) of Brighton and Jones Reynolds stress tensor. The numerical results are (1964). Figure 2 presents the Nusselt number as a normalized by the friction velocity, on the function of the Reynolds number. Dalle Donne U;ut, outer wall. The model predicts an inherent aniso- and Meerwald (1966) derived the following corre- tropy of the annular flow, although its accuracy is lation for the Nusselt number at the inner wall of somewhat inferior near the inner and outer walls the annulus as than in the center region. The predicted radial distribution of the time-averaged temperature in 8 o.4 Re Pr (22) the inner wall side is illustrated in Fig. 5 in the form k. rin / Tinlet/ of Ti+n versus Yi+n It is observed that the two- This equation is superimposed in Fig. 2 as a solid equation heat transfer model reproduces the law of straight line. It should be noted that Fig. 2 is under the wall for a thermal boundary layer. Through the the temperature ratio of the inner wall to the inlet above comparisons, the validity of the computer fluid, Twin/Tinlet, of unity, and the radius ratio, r*, code and the accuracy for the turbulence models of of 0.56. The calculated Nusselt number is in good heat and momentum employed here are confirmed. agreement with the correlation, Eq. (22). Figure 3 depicts the radial distributions of the time-aver- aged streamwise velocity (dimensionless velocity RESULTS AND DISCUSSION u + versus y +). (a) and (b) of Fig. 3 correspond to the distributions from the inner and outer walls to Figure 6, for r*-0.8, illustrates numerical results the location of the maximum streamwise velocity, of the Nusselt number with the Taylor number, Ta, CIRCULAR COUETTE FLOW 43

u+=5.5+2.51ny+ 2O

10 Experiment f Brighton&Jones f Predi cti on u+=Y+

10 10 Y+in (a) inner side

g [.u+=_3.05+5.001ny,f L / Experiment a o Brighton & Jones Prediction l J u+_y+ 0 102 103 1 10 Y+out (b) outer side

FIGURE 3 Dimensionless time-averaged streamwise velocity distribution in a stationary concentric annulus for r*-0.56 and Re 46,000; (a) inner side and (b) outer side. 44 S. TORII AND WEN-JEI YANG IO0 f-'/u t 'J/;u'""/-/u"' Experiment Brighton & Jones Turbulent Nu=O.0189Re'SPr'4 _Pred cti on --// 0 O o Ta=O To=5000 u......

0 0.2 0.4 0.6 0.8 1.0 10 5OOO Re 10000 FIGURE 4 A comparison of numerical and experimental results for radial distribution of normal Reynolds stresses in a FIGURE 6 Variation of predicted Nusselt numbers in a cir- stationary concentric annulus for r*= 0.56 and Re 46,000. cular Couette flow at Ta 0 and 5000 for r*= 0.8.

figures, the velocity is divided by its maximum value at each Taylor number. Here, the maximum tangential velocity corresponds to the tangential +[---I 0 [ T.+=Pry+ one on the inner cylinder. The streamwise velocity profile at Ta=0 corresponds to a turbulent annular flow in the absence of rotation, as seen in 10 02 103 Fig. 7(a). The corresponding tangential velocity in Y+in Fig. 7(b) is zero over the flow cross section. One FIGURE 5 Dimensionless time-averaged temperature distri- observes that the streamwise and tangential veloc- bution in a stationary concentric annulus for r*=0.56 and ity gradients increase near the inner and outer walls Re 46,000. due to the inner core rotation. Figures 8(a) and (b) illustrate the radial variations of the Reynolds as the parameter. Equation (22), which is under stresses, and with a change in Ta. The r*=0.8, is superimposed on Fig. 6 as a solid numerical results are divided by the square of straight line. It is observed that the Nusselt number the friction velocity, Uout, on the outer wall for the increases with an increase in the Taylor number. annular flow- without the swirl. The Reynolds This trend becomes larger in the low Reynolds stress, uv, near the wall regions is induced with an number region. A similar result is reported by Torii increase in Ta. This trend becomes larger in the and Yang (1994), who employ the existing vicinity of the inner wall, as shown in Fig. 8(a). turbulence models. It is found that an amplification In Fig. 8(b), the Reynolds stress vw at Ta=0 of the Nusselt number is attributed to the axial disappears over the whole cross section of the flow, rotation of the inner cylinder. while it is coursed by the swirl. The similar result, An attempt is made to explore the mechanisms which is obtained in the isothermal circular of transport phenomena of circular Couette flows Couette flow in an annulus, is reported by Hirai in an annulus based on numerical results at et al. (1987). These behavior is in accord with the Re 10,000 and r*= 0.8. Figures 7(a) and (b) show variations of the streamwise and tangential velo- the radial distributions of time-averaged stream- cities in Figs. 7(a) and (b). The radial profiles of wise and tangential velocities, respectively. In both three normal components of the Reynolds stress CIRCULAR COUETTE FLOW 45 1.0

0.5 1.0 (r--rin)/(rout--rin) (a) component 0 0 O.5 1.0 (F--rin)/(Fout--rin) Ta=O ...... Ta=5,000 (a) streamwise velocity ... 0.5 Ta=O 0 O.5 1.0 x To=5,000 (r-rin)/(rout-rin) (b) component

FIGURE 8 Variation of Reynolds stress profiles in a circu- lar Couette flow at Ta=0 and 5000 for r*=0.8 and Re-6000; (a) uv component and vw component.

Here, the turbulent heat flux is normalized by the product of the friction temperature t* and the friction velocity u* on the outer wall for the annular flow in the absence of the inner core rotation. The FIGURE 7 Variation of time-averaged velocity profiles in a circular Couette flow at Ta=0 and 5000 for r*=0.8 and turbulent heat flux level in the vicinity of the inner Re 6000; (a) streamwise velocity and (b) tangential velocity. wall is substantially induced with an increase in Ta. Since the eddy diffusivity for heat is employed to are illustrated in Fig. 9 in the same form as Fig. 4 at determine the turbulent heat flux, it is directly re- Ta=0 and 5000. One observes that (i) the three lated to the turbulent kinetic energy, its dissipation normal stress levels over the annular cross section rate, the temperature variance, and the dissipation are intensified due to the inner core rotation, and rate of temperature fluctuations, through Eqs. (13) (ii) this effect becomes larger near the wall sides. and (14). Here turbulent thermal diffusivity OZ is Thus this variation corresponds to the enhance- rewritten using the time-scale ratio, rm, as ment in the turbulent kinetic energy. The radial distribution of the predicted turbulent k2 Ozt Ck f-- (2Tm) 2. (23) heat flux is depicted in Fig. 10 as a function of Ta. 46 S. TORII AND WEN-JEI YANG Uf-/U*out VU*out 10.0

o "Oo. oO,,,,o.O*' **,, ,t: ...... ,. To-5,000

0 Te-O 0 O.5 1.0 io)

FIGURE 9 Variation of normal Reynolds stress profiles in a circular Couette flow at Ta=0 and 5000 for r*=0.8 and Re 6000.

The predicted radial change in the time-scale ratio with the inner core rotation (i.e. Ta) is depicted in Fig. 11. Based on the asymptotic behavior of the turbulent quantities of velocity and thermal fields near the wall, the time-scale ratio becomes infinite under the condition of uniform wall heat flux (Youssef et al., 1992). The numerical result at Ta--0 reproduces this behavior in the vicinity of the inner wall. As Yi+n is increased, the predicted time-scale ratio approaches a constant value, i.e. about 0.5, whose value is in good agreement with that reported by B6guier et al. (1978). It is observed that the radial profile of the time-scale ratio at Ta 5000 is affected by the inner core rotation, with only a slight change over the entire 0 1 flow cross section. Hence, a substantial enhance- O.5 ment in the three normal components of the (r-Fin)/(rout-r n) Reynolds stress, i.e. the turbulent kinetic energy is FIGURE 10 Variation of turbulent heat flux profiles in a ascribed to an increase in the Nusselt number, as circular. Couette flow at Ta--0 and 5000 for r*=0.8 and shown in Fig. 6. Re 6000. CIRCULAR COUETTE FLOW 47

tangential velocity, (ii) the presence of the tangen- tial velocity intensifies the three normal compo- nents of the Reynolds stress, resulting in an amplification of the turbulent heat flux, and (iii) / an effect of inner core rotation on the time-scale Ta=5,000 ratio, which is used to determine the turbulent ',,1 thermal diffusivity in Eq. (23), is minor over the flow cross section. Consequently, the turbulent kinetic energy is induced due to the axial rotation FIGURE ll Variation of radial profiles of time-scale ratio of the inner cylinder, resulting in the enhancement in a circular Couette flow at Ta 0 and 5000 for r*= 0.8 and of heat transfer performance. Re 6000.

In summary, an increase in the Nusselt number, NOMENCLATURE as seen in Fig. 6, is caused by the axial rotation of the inner cylinder. The mechanism is that (i) an D hydraulic diameter of the annulus, inner core rotation courses an increase in the 2(rout-tin), m streamwise velocity gradient near the inner and heat transfer coefficient, W/m2K 2 2 outer walls and a presence of the tangential turbulent kinetic energy, (u + v + w2)/ 2 velocity, (ii) the tangential velocity induces a 2, m2/s production of the three normal components of Nu Nusselt number, hUm/A the Reynolds stress, and (iii) it yields an enhance- P time-averaged pressure, Pa ment in the turbulent thermal diffusivity, i.e. an Pr Prandtl number amplification of the turbulent heat flux, resulting in Prt turbulent Prandtl number an augmentation of heat transfer performance. Pe Peclet number q heat flux, W/m2 radial coordinate, m CONCLUSIONS radius ratio, rin/rout rin inner radius of the annulus, m The two-equation model for heat transfer and the rout outer radius of the annulus, m Reynolds stress turbulence model have been Re Reynolds number, UmD/u employed to numerically investigate the thermal Rt turbulent Reynolds number, transport phenomena in concentric annulus with a fluctuating temperature component, K slightly heated inner core rotating around the axis. 2 temperature variance, K2 Consideration is given to the influence of inner core t* friction temperature, qw/pCpU*, K rotation on the flow and thermal fields. The results T time-averaged temperature, K derived from the present study are summarized as T+ dimensionless time-averaged tempera- follows. ture, (Tw- T)/(qw/pCpU*) The turbulence models of heat and momentum Taylor number, Ta Ww(rout-rin) /rout-rin V, tin. employed in this work predict an increase in the fluctuating velocity components n Nusselt number due to an axially rotating inner axial, radial and tangential directions, cylinder. It is disclosed that (i) the inner core respectively, m/s rotation induces the streamwise velocity gradient friction velocity, m/s near the inner and outer walls and causes the dimensionless velocity, U/u* 48 S. TORII AND WEN-JEI YANG

2 b/F, VW, HW Reynolds stress, m2/s Hirai, S., Takagi, T., Tanaka, K. and Kida, K., 1987. Effect of U,V,W time-averaged velocity components Swirl on the Turbulent Transport of Momentum in a Concentric Annulus with a Rotating Inner Cylinder, Trans., in axial, radial and tangential direc- JSME, 53(486), 432-437 (in Japanese). tions, respectively, m/s Hishida, M., Nagano, Y. and Tagawa, M., 1986. Transport Processes of Heat and Momentum in the Wall Region of axial mean velocity over tube cross Turbulent Pipe Flow, Proc. Eighth Int., Heat Transfer Conf., section, m/s 3, 925-930. Jones, W.P. and Musonge, P., 1988. Closure of the Reynolds turbulent heat flux, mK/s Stress and Scalar Flux Equations, Physics of Fluids, 31(12), tangential velocity on the inner 3589-3604. cylinder, Kasagi, N. and Myong, H.K., 1989. An Outlook: Modeling of m/s Turbulent Heat Transport, J. Heat Transfer Society ofJapan, x axial coordinate, m 28(108), 4-17. distance from wall, m Kasagi, N., Tomita, Y. and Kuroda, A., 1992. Direct Numerical Simulation of Passive Scalar Field in a Turbulent Channel dimensionless distance, u*y/u Flow, J. Heat Transfer, 114, 598-606. Kikuyama, K., Murakami, M., Nishibori, K. and Maeda, K., 1983. Flow in an Axially Rotating Pipe (A calculation of Greek Letters flow in the saturated region), Bulletin of the JSME, 26(214), 506- 513. turbulent energy dissipation rate, m2/s Kuzay, T.M. and Scott, C.J., 1975. Turbulent Heat Transfer Et dissipation rate of , K/s 2. Studies in Annulus with Inner Cylinder Rotation, Trans. of ASME, 75-WA/HT-55, 1-11. o, o molecular and turbulent thermal diffusiv- Kuzay, T.M. and Scott, C.J., 1976. Turbulent Prandtl Numbers ities, m2/s for Fully Developed Rotating Annular Axial Flow of Air, Trans. of ASME, 76-HT-36, 1-13. /', molecular and turbulent viscosities, mZ/s Lai, Y.G. and So, R.M.C., 1990. Near-Wall Modeling of molecular thermal conductivity, W/mK Turbulent Heat Fluxes, Int. J. Heat Mass Transfer, 33(7), 1429-1440. p density of gas, Pasec Launder, B.E. and Shima, N., 1989. Second-Moment Closure 0 tangential direction for the Near-Wall Sublayer: Development and Application, AIAA J., 1319-1325. 7- time-scale ratio, 27(10), (tz/zct)/(k/c) Murakami, M. and Kikuyama, K., 1980. Turbulent Flow in Axially Rotating Pipes, J. Fluids Engineering, 102, 97-103. Subscripts Nagano, Y. and Tagawa, M., 1988. Statistical Characteristics of Wall Turbulence with a Passive Scalar, J. Fluid Mech., 196, in inner side 157-185. Patankar, S.V., 1980. Numerical Heat Transfer and Fluid Flow, inlet inlet Hemisphere Publishing, Washington, DC. max maximum Rehme, K., 1974. Turbulent Flow in Smooth Concentric Annuli with Small Radius Ratios, J. Fluid Mech., 64, 263-287. out outer side Sommer, T.P., So, R.M.C. and Lai, Y.G., 1992. A Near-Wall w wall Two-equation Model for Turbulent Heat Fluxes, Int. J. Heat Mass Transfer, 35(12), 3375-3387. Torii, S. and Yang, W.J., 1994. A Numerical Study on References Turbulent Flow and Heat Transfer in-Circular Couette Flows, Numerical Heat Transfer, Part A, 26, 231-336. Beguier, C., Dekeyser, I. and Launder, B.E., 1978. Ratio of Torii, S. and Yang, W.J., 1995a. Numerical Prediction of Fully- Scalar and Velocity Distribution Time Scales in Shear Flow Developed Swirling Flows in an Rotating Pipe by Means of a Turbulence, Physics of Fluids, 21, 307- 310. Modified k-c Turbulence Model, Int. J. Numerical Methods Bradshaw, P., 1969. The Analogy between Streamline Curvature for Heat & Fluid Flow, 5(2), 175-183. and Buoyancy in Turbulent Shear Flow, J. Fluid Mech., 36, Torii, S. and Yang, W.J., 1995b. A Numerical Analysis on Flow 177-191. and Heat Transfer in the Entrance Region of an Axially Brighton, J.A. and Jones, J.B., 1964. Fully Developed Turbulent Rotating Pipe, Int. J. Rotating Machinery, 2(2), 123-129. Flow in Annuli, Trans. of ASME, Ser. D, 835-844. Youssef, M.S., Nagano, Y. and Tagawa, M., 1992. A Two- Dalle Donne, M. and Meerwald, E., 1966. Experimental Local equation Heat Transfer Model for Predicting Turbulent Heat Transfer and Average Friction Coefficients for Subsonic Thermal Fields under Arbitrary Wall Thermal Conditions, Turbulent Flow of Air in an Annulus at High Temperatures, Int. J. Heat Mass Transfer, 35(11), 3095-3104. Int. J. Heat Mass Transfer, 9, 1361-1376. Hirai, S. and Takagi, T., 1988. Prediction of Heat Transfer Deterioration in Turbulent Swirling Pipe Flow, JSME Int. J., Ser. II, 31(4), 694-700. N F EW 2 O 0 R 0 ENERGY MATERIALS 6 Materials Science & Engineering for Energy Systems Maney Publishing on behalf of the Institute of Materials, Minerals and Mining

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