International Journal of Rotating Machinery 1998, Vol. 4, No. 1, pp. 35-48 Reprints available directly from the publisher Photocopying permitted by license only (C) 1998 OPA (Overseas Publishers Association) Amsterdam B.V. Published under license under the Gordon and Breach Science 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 The convective heat transfer in turbulent swirling flows is often encountered in chemical and mechan- ical mixing and separation devices, electrical and turbo-machinery, combustion chambers, pollution control devices, swirl nozzles, rocketry, and fusion reactors. In these flow fields, the heat transport phenomena in connection with the flow and the turbulence properties are substantially influenced by the centrifugal force induced by the swirl. In other words, the turbulent transfer of heat and momentum is suppressed or promoted by the interaction between turbulence and centrifugal force associated with the swirl. Murakami and Kikuyama (1980) measured the velocity profile and hydraulic loss in a hydrodynam- ically fully-developed flow region of a rotating pipe. It was disclosed that both turbulence and hydraulic loss were remarkably reduced due to pipe Corresponding author. Tel.: 099-285-8245. Fax: 099-285-8246. E-mail: [email protected]. 35
Transcript
International Journal of Rotating Machinery1998, Vol. 4, No. 1, pp. 35-48
Reprints available directly from the publisherPhotocopying permitted by license only
(C) 1998 OPA (Overseas Publishers Association)Amsterdam B.V. Published under license
under the Gordon and Breach SciencePublishers imprint.
Printed in India.
Turbulent Flow and Heat Transfer in Circular CouetteFlows 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 ofMechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, MI 48109, USA
(Received in finalform 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 heatedrotating inner cylinder and a stationary insulated outer cylinder. Emphasis is placed on theeffect of rotation of an inner core on the flow structure and the thermal field. A Reynoldsstress turbulence model is employed to determine three normal components of the Reynoldsstress and its off-diagonal one. The turbulent heat flux is expressed by Boussinesqapproximation in which the eddy diffusivity.for heat is given as functions of the temperaturevariance 7 and the dissipation rate of temperate fluctuations ct. The governing boundary-layer equations are discretized by means of a control volume finite-difference techniqueand numerically solved using the marching procedure. An inner core rotation causesan amplification of the three normal components of the Reynolds stress over the wholecross section, resulting in a substantial enhancement in the Nusselt number.
Keywords." Circular Couette flow, Reynolds stress model, Taylor number, Two-equation heattransfer model
INTRODUCTION
The convective heat transfer in turbulent swirlingflows is often encountered in chemical and mechan-ical mixing and separation devices, electrical andturbo-machinery, combustion chambers, pollutioncontrol devices, swirl nozzles, rocketry, and fusionreactors. In these flow fields, the heat transportphenomena in connection with the flow and theturbulence properties are substantially influenced
by the centrifugal force induced by the swirl. Inother words, the turbulent transfer of heat andmomentum is suppressed or promoted by theinteraction between turbulence and centrifugalforce associated with the swirl.Murakami and Kikuyama (1980) measured the
velocity profile and hydraulic loss in a hydrodynam-ically fully-developed flow region of a rotatingpipe. It was disclosed that both turbulence andhydraulic loss were remarkably reduced due to pipe
rotation and that streamwise velocity profilegradually deformed into a parabolic form with anincrease in its speed. Kikuyama et al. (1983)analyzed the variation of streamwise velocityprofiles using a modified mixing length theoryproposed by Bradshaw (1969). A combined experi-mental and theoretical study was performed byHirai and Takagi (1988) using the Reynolds stressmodel to determine the effects of pipe rotation onfluid flow and heat transfer in a thermally andhydrodynamically fully developed flow region. Itwas found that an increase in the rotation rateresulted in a decrease in heat transfer performancewith the Nusselt number asymptotically approach-ing that of a laminar pipe flow. Fluid. flow and heattransfer characteristics in the thermally and hydro-dynamically developing and fully-developed regi-ons of an axially rotating pipe were investigated byTorii and Yang (1995a,b) using the existing k-cturbulence models in which they are modified toinclude the swirling effect.On the contrary, an effect of promoting the
turbulent transport of heat and momentum by thecentrifugal force occurs in a concentric annuluswith an inner cylinder rotating around the axis, inwhich Taylor vortices appear (Kuzay and Scott,1975; 1976). Such swirl flow is referred to ascircular Couette flow, which implies a flow withone surface rotating and the other stationary (orboth surfaces rotating in the same direction atdifferent angular velocities). Hirai et al. (1987)conducted an experimental study on an effectof inner core rotation on turbulent transport ofmomentum by using a two-color laser Dopplervelocimeter. It was disclosed that the Reynoldsstresses increase due to the swirl. Torii and Yang(1994) analyzed heat transfer mechanism in annuliwith an inner core rotation by means of severaldifferent two-equation k-c turbulence models. Itwas found that (i) in the entrace region, the axialrotation of the inner cylinder induces a thermaldevelopment and causes an increase in both theNusselt number and the turbulent kinetic energy inthe inner cylinder wall region, and (ii) in the fully-developed region, an increase in the Taylor number
causes an amplification of the turbulent kineticenergy over the whole cross section, resulting in asubstantial enhancement in the Nusselt number.A k-e model, which is one of the two-equation
models of turbulence, is popular in computationalanalyses of turbulent flow. However, in the case ofan annular duct flow, the importance of taking theeffect of the term, -uv utOU/Or, into account isrecognized. This is because the time-averaged localshear stress, -uv, does not go to zero at the radiallocation where the time-averaged streamwise veloc-ity has its maximum value, i.e. a radial gradient ofzero (Rehme, 1974). Furthermore, since the two-equation k-e model basically assumes isotropicturbulence structure, it cannot precisely reproducethe anisotropy of turbulence caused by the innercore rotation. In order to obtain the detailedinformation pertinent to the flow structures, it isnecessary to employ the higher order closuremodel, i.e. a Reynolds stress turbulence model.Throughout the numerical simulations for the
above turbulent heat transport problems in the swir-ling flows, the turbulent heat flux in the energy equa-tion is modeled by using the classical Boussinesqapproximation. The unknown turbulent thermaldiffusivity Oz is obtained from the definition of theknown turbulent viscosity ut and the turbulentPrandtl number Prt as Ot--- ut/Prt. In this formu-lation, an analogy between eddy diffusivities ofmomentum and heat is implicitly assumed.However, shear flow measurements (Hishida et al.,1986) and direct simulation data (Kasagi et al.,1992) reveal that its analogy, as represented by theturbulent Prandtl nu.mber, cannot adequatelyreflect the physical phenomenon of heat transportand there are no universal values of turbulentPrandtl number even in simple flows. If the aboveturbulent Prandtl number assumption is employedto determine the turbulent thermal diffusivity inannuli, it yields negative along the radial direction.This is because the turbulent viscosity, ut, becomesnegative in the radial region between velocitygradient of zero and Reynolds stress of zero, as
mentioned previously. In order to solve thisproblem and to obtain detailed information on
CIRCULAR COUETTE FLOW 37
the heat transport phenomena, two-equationmodel for thermal field and turbulent heat fluxequation model are considered to be employed.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 littleused for practical applications (for example, Laiand So, 1990). The results are not as satisfactory as
initially expected, mainly due to a few unreasonablehypotheses in the model (Nagano and Tagawa,1988). Thus, the turbulent heat flux model shouldwait until the Reynolds stress models are tested andwell developed, preferably with the near-wallmodeling, since the major source of error in heattransfer predictions is that in calculating thevelocity field (Kasagi and Myong, 1989).
This paper treats the thermal transport phenom-ena in a concentric annulus, in which a slightlyheated inner core rotates around the axis and aninsulated outer cylinder is held stationary. In orderto shed light on the mechanism of the transportphenomena, the Reynolds stress turbulence modelproposed by Launder and Shima (1989) is employed,because this model is able to reproduce the inherentanisotropy in the near-wall region of isothermalflows. The turbulent thermal diffusivity Yt is deter-mined using the two-equation model for heat tran-sport proposed by Yousseff et al. (1992). Emphasis
is placed on the mechanism of an augmentation ofheat transfer performance due to the inner core
rotation.
THEORETICAL ANALYSIS
Governing Equations
Consider a steady turbulent flow through a
concentric annulus consisting of the insulatedstationary outer cylinder and the slightly heatedinner cylinder rotating around the axis, in whichthe boundary layer is developing both thermallyand hydrodynamically. The physical configurationand the cylindrical coordinate system are shown inFig. 1. Some approximations are deduced that: (i)viscous heating is negligible; (ii) the axial conduc-tion term in the energy equation is neglected forPe > 1; and (iii) the viscous dissipation term in theenergy equation is neglected. An order-of-magni-tude analysis indicates all second-derivative termsto be negligible in the streamwise and tangentialdirections. The simplified governing equations readas follows:Continuity equation
FIGURE A schematic of physical system and coordinate.
38 S. TORII AND WEN-JEI YANG
Momentum equations."x direction."
OU vOU dPU-Ox + Or p dx
+ ru rVf (2)
r direction."
OP Or2 P (W2 2Or -P--r + -r + w--- ); (3)
0 direction."
OW vOW vwUx + -ffF +-r
o(w/r)0 r3 ur2 Or Or r2--}. (4)
Energy equation."
uOT OT O ( OT )+ v o-; -r or r (5)
The Reynolds stress turbulence model proposedby Launder and Shima (1989) is employed toevaluate -uv in Eq. (2) and -vw in Eq. (4). Thetransport equations can be expressed as
U2:
Ox + V 0----=-/ 0--- r u + Cs-- -67
OU e 22(1 --fz)UV--r--fl--+-j(fl
c 2+flwfxV2---j(f +f2wfx)
OU OW W w2 V)x gf-r + vW-r VW--r --rW
(6)
(7)
W2]
uOW2 Ow2
Ox+v 0---;-
=-/05 r u+Cs-Tg Or j+2 u+Csw2
blV:
Or
r Orr + Cs v -r ’ + Csw2
r2c 3 (l 3)OU-A- -Awf- -A + f2wL or
( 3 )WlO(k )+ 2 f2 + jf2f uw G uvr FOr
k Ow-G-vw, (9)
CIRCULAR COUETTE FLOW 39
vvg:
(10)
The turbulent heat flux -v-7 in Eq. (5) is
expressed through Boussinesq’s approximation, as
OTV--’- ge
Or" (13)
Youssef et al. (1992) obtain the turbulent thermaldiffusivity, ct, using the temperature variance, 2,and the dissipation rate of temperature fluctuations
Ct together with k and c, as
ct CAfk (14)
where CA is the model constant andf is the modelfunction. Both transport equations, and Ct inEq. (14), are written in the tensor form as
b/W: Ot2 0
and
ot Ot22ujt
OT
(15)
0
Ox-1- O
OXj JCp1 fe, - ujt OX
Ct OUiC2fp - uiu;
Cm fo2 k(16)
Turbulent energy dissipation rate c is determinedfrom
(12)
In the present study, the Reynolds stress turbulencemodel is employed in place of a k-c model. Thus a
slight modification is made to the turbulentdiffusion terms in Eqs. (15) and (16). Originalturbulent diffusion terms in both equations are
written as -ujt2 and -ujc, respectively. Both termsare modeled using a gradient-type representationand are expressed as
kOt2 k__Octujt2 Cst- uiuj and ujc Cs- uiuj
e e Oxi(17)
The empirical constants and model functions inEqs. (6)-(12) are summarized in Table I.
respectively (Jones and Musonge, 1988; Sommeret 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=2.5Cs 0.22 C 0.18 CI 1.45 C2 1.9
respectively. The empirical constants and modelfunctions in Eqs. (14), (18) and (19) are summar-
ized in Table II.The boundary conditions at both the inner and
outer walls in the annulus are specified as,
CIRCULAR COUETTE FLOW 41
r--rin (inner tube wall)"
U-- V-- bl2 V2 W2 btV VW tAW-- O,
W- Ww, c
O2 02(t-2/2) OT qwO, Ct 0--Or Or2 Of" Aw
r-rout (outer tube wall)"
1. Specify the initial values of U, V, U2, 12, W2,uv, vw, uw, and e, and assign a constant axialpressure gradient.
2. Solve the equations of U, V, u2, v2, w2, uv, vw,uw, and e.
3. Repeat step 2 until the criterion of convergenceis satisfied, which is set at
max M 0M-1 < 10-4 (20)
U- V- W- u2 v2 w2 uv vw uw O,
(0- 2Ot2
e-2"\--OTrJ Or
OT0 (insulation).Or
O,02(t2/2)
t OOr2
Numerical Method
A set of governing equations employed are
discretized using a control volume finite-differenceprocedure proposed by Patankar (1980). Since allturbulence quantities as well as the time-averagedaxial and tangential velocities vary rapidly in thenear-wall region, two control volumes are alwayslocated within the viscous sublayer, y+-5. Theradial mesh size is increased from a minimum valueadjacent to the wall towards the turbulent core
region in geometrical proportion, and the maxi-mum control volume size in the turbulent core
region is always kept within 3% of D/2. Mean-while, the axial control volume size is constant atfive times the minimum radial size for the wall.Throughout numerical calculations, the number ofcontrol volumes in the radial direction was prop-erly selected between 70 and 92 to ensure validationof the numerical procedures as well as to obtaingrid-independent solutions. The maximum relativeerror over all dependent variables within thischange of grid spacing was kept within 1%. Sincethe governing equations are essentially parabolic,calculation is performed from the inlet in thedownstream direction by means of the marchingprocedure. The computations are processed in thefollowing order"
10.
11.
for all the variables 05. The superscripts M andM-1 in Eq. (20) indicate two successive itera-
tions, while the subscript "max" refers to a
maximum value over the entire fields of itera-tions.Calculate new values of U, V, u2, v2, w2, uv, vw,uw, and e with a corrected new axial pressuregradient.Repeat steps 2-4 until the conservation of thetotal mass flow rate is satisfied under thecriterion:
f y gcpO dr f f ginletO dr
f y SinletO dr_< 10-5, (21)
followed by evaluating convergent values of U,V, u2, v2, w2, uP, lw, uw, and e. Ucp is the axialvelocity under the correction process and Uinleis that at the inlet of the annulus.Repeat steps 2-5 until a hydrodynamicallyfully-developed annular flow in the absence ofrotation is realized.Start both axial rotation and heating of an
inner cylinder.Solve the equations of U, V, W, u2, v2, w2, uv,vw, uw, , T, 2, and gt"
Repeat step 8 until the criterion of conver-
gence, Eq. (20), is satisfied.Calculate new values of U, V, W, u2, v2, w2, uv,vw, uw, , T, 2, and gt with a corrected newaxial pressure gradient.Repeat steps 8-10 until the conservation of thetotal mass flow rate is satisfied under thecriterion, i.e. Eq. (21), followed by evaluating
42 S. TORII AND WEN-JEI YANG
convergent values of U, V, W,uw, , T, 2, and et.
12. Repeat steps 8-11 until x reaches the desig-nated length (200D), where thermally andhydrodynamically fully-developed flow condi-tion prevails.
The ranges ofthe parameters for the present studyare Reynolds number Re 6000-10,000; Taylornumbers Ta-0 and 5000; radius ratio r*-0.8;Prandtl number Pr- 0.7 (air); and heat flux at theinner wall qw 200 W/m2. The CPU time required incompleting the above scheme was about 50-100 hon a NEC personal computer (32 bit), depending onthe number of control volumes used.
It is necessary to verify both the turbulencemodels of heat and momentum employed here andthe reliability of the computer code by comparingnumerical predictions with experimental results forthe flow field. The model is applied to a flow in anannulus with a stationary, slightly heated innercore. Numerical results for the thermally andhydrodynamically fully-developed annular flow ata location 200 tube diameter downstream from theinlet are compared with the experimental data(Re-46,000 and r*-0.56) of Brighton and Jones(1964). Figure 2 presents the Nusselt number as a
function of the Reynolds number. Dalle Donneand Meerwald (1966) derived the following corre-lation for the Nusselt number at the inner wall ofthe annulus as
Re8 Pro.4 (22)k. rin / Tinlet/
This equation is superimposed in Fig. 2 as a solidstraight line. It should be noted that Fig. 2 is underthe temperature ratio of the inner wall to the inletfluid, Twin/Tinlet, of unity, and the radius ratio, r*,of 0.56. The calculated Nusselt number is in goodagreement with the correlation, Eq. (22). Figure 3depicts the radial distributions of the time-aver-aged streamwise velocity (dimensionless velocityu + versus y +). (a) and (b) of Fig. 3 correspond tothe distributions from the inner and outer walls tothe location of the maximum streamwise velocity,
102
10
Turbulent ’oNu=O.O203Re’8..Pr’4
o -2
II ,,,,L
104 105ReFIGURE 2 A comparison of Predicted Nusselt number withtest results (Dalle Donne and Meerwald, 1966) for the fully-developed turbulent annular flow with a stationary inner cyl-inder for r*= 0.56 and Re 46,000.
respectively. It is observed that the model yields a
better agreement with the experimental data, andpredicts the velocity profile with the well-knowncharacteristics of the logarithmic region, i.e. theuniversal wall law. Figure 4 illustrates the radialdistributions of three normal components of theReynolds stress tensor. The numerical results arenormalized by the friction velocity, U;ut, on theouter wall. The model predicts an inherent aniso-tropy of the annular flow, although its accuracy issomewhat inferior near the inner and outer wallsthan in the center region. The predicted radialdistribution of the time-averaged temperature inthe inner wall side is illustrated in Fig. 5 in the formof Ti+n versus Yi+n It is observed that the two-equation heat transfer model reproduces the law ofthe wall for a thermal boundary layer. Through theabove comparisons, the validity of the computercode and the accuracy for the turbulence models ofheat and momentum employed here are confirmed.
RESULTS AND DISCUSSION
Figure 6, for r*-0.8, illustrates numerical resultsof the Nusselt number with the Taylor number, Ta,
2O
10
CIRCULAR COUETTE FLOW
u+=5.5+2.51ny+
Experiment
f Brighton&Jones
f Predi cti on
u+=Y+
10 10
Y+in(a) inner side
43
g [.u+=_3.05+5.001ny,fL / Experimenta o Brighton & Jones
l J Prediction
u+_y+0
1 10 102 103
Y+out(b) outer side
FIGURE 3 Dimensionless time-averaged streamwise velocity distribution in a stationary concentric annulus for r*-0.56 andRe 46,000; (a) inner side and (b) outer side.
44 S. TORII AND WEN-JEI YANG
Experimentf-’/u t’J/;u’""/-/u"’
Brighton & Jones
_Pred cti on --//0
O
u........0 0.2 0.4 0.6 0.8 1.0
FIGURE 4 A comparison of numerical and experimentalresults for radial distribution of normal Reynolds stresses in astationary concentric annulus for r*= 0.56 and Re 46,000.
IO0Turbulent
Nu=O.0189Re’SPr’4
o Ta=OTo=5000
Re105OOO 10000
FIGURE 6 Variation of predicted Nusselt numbers in a cir-cular Couette flow at Ta 0 and 5000 for r*= 0.8.
+[---I 0 [ T.+=Pry+
10 Y+in02 103
FIGURE 5 Dimensionless time-averaged temperature distri-bution in a stationary concentric annulus for r*=0.56 andRe 46,000.
as the parameter. Equation (22), which is underr*=0.8, is superimposed on Fig. 6 as a solidstraight line. It is observed that the Nusselt numberincreases with an increase in the Taylor number.This trend becomes larger in the low Reynoldsnumber region. A similar result is reported by Toriiand Yang (1994), who employ the existingturbulence models. It is found that an amplificationof the Nusselt number is attributed to the axialrotation of the inner cylinder.An attempt is made to explore the mechanisms
of transport phenomena of circular Couette flowsin an annulus based on numerical results atRe 10,000 and r*= 0.8. Figures 7(a) and (b) showthe radial distributions of time-averaged stream-wise and tangential velocities, respectively. In both
figures, the velocity is divided by its maximumvalue at each Taylor number. Here, the maximumtangential velocity corresponds to the tangentialone on the inner cylinder. The streamwise velocityprofile at Ta=0 corresponds to a turbulentannular flow in the absence of rotation, as seen inFig. 7(a). The corresponding tangential velocity inFig. 7(b) is zero over the flow cross section. Oneobserves that the streamwise and tangential veloc-ity gradients increase near the inner and outer wallsdue to the inner core rotation. Figures 8(a) and (b)illustrate the radial variations of the Reynoldsstresses, - and with a change in Ta. Thenumerical results are divided by the square ofthe friction velocity, Uout, on the outer wall for theannular flow without the swirl. The Reynoldsstress, uv, near the wall regions is induced with an
increase in Ta. This trend becomes larger in thevicinity of the inner wall, as shown in Fig. 8(a).In Fig. 8(b), the Reynolds stress vw at Ta=0disappears over the whole cross section of the flow,while it is coursed by the swirl. The similar result,which is obtained in the isothermal circularCouette flow in an annulus, is reported by Hiraiet al. (1987). These behavior is in accord with thevariations of the streamwise and tangential velo-cities in Figs. 7(a) and (b). The radial profiles ofthree normal components of the Reynolds stress
CIRCULAR COUETTE FLOW 45
1.0
00 O.5 1.0
(F--rin)/(Fout--rin)(a) streamwise velocity
x
Ta=O
To=5,000
FIGURE 7 Variation of time-averaged velocity profiles in acircular Couette flow at Ta=0 and 5000 for r*=0.8 andRe 6000; (a) streamwise velocity and (b) tangential velocity.
are illustrated in Fig. 9 in the same form as Fig. 4 atTa=0 and 5000. One observes that (i) the threenormal stress levels over the annular cross sectionare intensified due to the inner core rotation, and(ii) this effect becomes larger near the wall sides.Thus this variation corresponds to the enhance-ment in the turbulent kinetic energy.The radial distribution of the predicted turbulent
heat flux is depicted in Fig. 10 as a function of Ta.
0.5 1.0(r--rin)/(rout--rin)(a) component
0.5
Ta=O... ......... Ta=5,000
0 O.5 1.0(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 andRe-6000; (a) uv component and vw component.
Here, the turbulent heat flux is normalized by theproduct of the friction temperature t* and thefriction velocity u* on the outer wall for the annularflow in the absence of the inner core rotation. Theturbulent heat flux level in the vicinity of the innerwall is substantially induced with an increase in Ta.Since the eddy diffusivity for heat is employed to
determine the turbulent heat flux, it is directly re-
lated to the turbulent kinetic energy, its dissipationrate, the temperature variance, and the dissipationrate of temperature fluctuations, through Eqs. (13)and (14). Here turbulent thermal diffusivity OZ isrewritten using the time-scale ratio, rm, as
k2
Ozt Ck f-- (2Tm)2. (23)
46 S. TORII AND WEN-JEI YANG
FIGURE 9Re 6000.
10.0
0
Uf-/U*outVU*out
o "Oo. oO,,,,o.O*’ **,,
,t: ........... ,.
To-5,000
Te-O0 O.5 1.0
io)Variation of normal Reynolds stress profiles in a circular Couette flow at Ta=0 and 5000 for r*=0.8 and
The predicted radial change in the time-scale ratiowith the inner core rotation (i.e. Ta) is depicted inFig. 11. Based on the asymptotic behavior of theturbulent quantities of velocity and thermal fieldsnear the wall, the time-scale ratio becomes infiniteunder the condition of uniform wall heat flux(Youssef et al., 1992). The numerical result atTa--0 reproduces this behavior in the vicinity ofthe inner wall. As Yi+n is increased, the predictedtime-scale ratio approaches a constant value, i.e.about 0.5, whose value is in good agreement withthat reported by B6guier et al. (1978). It isobserved that the radial profile of the time-scaleratio at Ta 5000 is affected by the inner core
rotation, with only a slight change over the entireflow cross section. Hence, a substantial enhance-ment in the three normal components of theReynolds stress, i.e. the turbulent kinetic energy isascribed to an increase in the Nusselt number, asshown in Fig. 6.
0 O.5 1(r-Fin)/(rout-r n)
FIGURE 10 Variation of turbulent heat flux profiles in acircular. Couette flow at Ta--0 and 5000 for r*=0.8 andRe 6000.
CIRCULAR COUETTE FLOW 47
Ta=5,000 /
’,,1
FIGURE ll Variation of radial profiles of time-scale ratioin a circular Couette flow at Ta 0 and 5000 for r*= 0.8 andRe 6000.
tangential velocity, (ii) the presence of the tangen-tial velocity intensifies the three normal compo-nents of the Reynolds stress, resulting in anamplification of the turbulent heat flux, and (iii)an effect of inner core rotation on the time-scaleratio, which is used to determine the turbulentthermal diffusivity in Eq. (23), is minor over theflow cross section. Consequently, the turbulentkinetic energy is induced due to the axial rotationof the inner cylinder, resulting in the enhancementof heat transfer performance.
In summary, an increase in the Nusselt number,as seen in Fig. 6, is caused by the axial rotation ofthe inner cylinder. The mechanism is that (i) aninner core rotation courses an increase in thestreamwise velocity gradient near the inner andouter walls and a presence of the tangentialvelocity, (ii) the tangential velocity induces a
production of the three normal components ofthe Reynolds stress, and (iii) it yields an enhance-ment in the turbulent thermal diffusivity, i.e. an
amplification of the turbulent heat flux, resulting inan augmentation of heat transfer performance.
CONCLUSIONS
The two-equation model for heat transfer and theReynolds stress turbulence model have beenemployed to numerically investigate the thermaltransport phenomena in concentric annulus with a
slightly heated inner core rotating around the axis.Consideration is given to the influence ofinner corerotation on the flow and thermal fields. The resultsderived from the present study are summarized asfollows.The turbulence models of heat and momentum
employed in this work predict an increase in theNusselt number due to an axially rotating inner
cylinder. It is disclosed that (i) the inner corerotation induces the streamwise velocity gradientnear the inner and outer walls and causes the
NOMENCLATURE
D
NuPPrPrtPeq
rin
routReRt
2
t*TT+
hydraulic diameter of the annulus,2(rout-tin), mheat transfer coefficient, W/m2Kturbulent kinetic energy, (u2 + v2 + w2)/2, m2/s2Nusselt number, hUm/Atime-averaged pressure, PaPrandtl numberturbulent Prandtl numberPeclet numberheat flux, W/m2
radial coordinate, mradius ratio, rin/routinner radius of the annulus, mouter radius of the annulus, mReynolds number, UmD/uturbulent Reynolds number,fluctuating temperature component, Ktemperature variance, K2
Reynolds stress, m2/s2time-averaged velocity componentsin axial, radial and tangential direc-tions, respectively, m/saxial mean velocity over tube cross
section, m/sturbulent heat flux, mK/stangential velocity on the innercylinder, m/saxial coordinate, mdistance from wall, mdimensionless distance, u*y/u
Greek Letters
Eto, o
/’,
p07-
turbulent energy dissipation rate, m2/sdissipation rate of , K/s2.molecular and turbulent thermal diffusiv-ities, m2/smolecular and turbulent viscosities, mZ/smolecular thermal conductivity, W/mKdensity of gas, Pasectangential directiontime-scale ratio, (tz/zct)/(k/c)
Subscripts
ininletmax
outw
inner sideinletmaximumouter sidewall
References
Beguier, C., Dekeyser, I. and Launder, B.E., 1978. Ratio ofScalar and Velocity Distribution Time Scales in Shear FlowTurbulence, Physics of Fluids, 21, 307-310.
Bradshaw, P., 1969. The Analogy between Streamline Curvatureand Buoyancy in Turbulent Shear Flow, J. Fluid Mech., 36,177-191.
Brighton, J.A. and Jones, J.B., 1964. Fully Developed TurbulentFlow in Annuli, Trans. ofASME, Ser. D, 835-844.
Dalle Donne, M. and Meerwald, E., 1966. Experimental LocalHeat Transfer and Average Friction Coefficients for SubsonicTurbulent Flow of Air in an Annulus at High Temperatures,Int. J. Heat Mass Transfer, 9, 1361-1376.
Hirai, S. and Takagi, T., 1988. Prediction of Heat TransferDeterioration in Turbulent Swirling Pipe Flow, JSME Int. J.,Ser. II, 31(4), 694-700.
Hirai, S., Takagi, T., Tanaka, K. and Kida, K., 1987. Effect ofSwirl on the Turbulent Transport of Momentum in aConcentric Annulus with a Rotating Inner Cylinder, Trans.,JSME, 53(486), 432-437 (in Japanese).
Hishida, M., Nagano, Y. and Tagawa, M., 1986. TransportProcesses of Heat and Momentum in the Wall Region ofTurbulent Pipe Flow, Proc. Eighth Int., Heat Transfer Conf.,3, 925-930.
Jones, W.P. and Musonge, P., 1988. Closure of the ReynoldsStress and Scalar Flux Equations, Physics of Fluids, 31(12),3589-3604.
Kasagi, N. and Myong, H.K., 1989. An Outlook: Modeling ofTurbulent Heat Transport, J. Heat Transfer Society ofJapan,28(108), 4-17.
Kasagi, N., Tomita, Y. and Kuroda, A., 1992. Direct NumericalSimulation of Passive Scalar Field in a Turbulent ChannelFlow, J. Heat Transfer, 114, 598-606.
Kikuyama, K., Murakami, M., Nishibori, K. and Maeda, K.,1983. Flow in an Axially Rotating Pipe (A calculation offlow in the saturated region), Bulletin of the JSME, 26(214),506-513.
Kuzay, T.M. and Scott, C.J., 1975. Turbulent Heat TransferStudies in Annulus with Inner Cylinder Rotation, Trans. ofASME, 75-WA/HT-55, 1-11.
Kuzay, T.M. and Scott, C.J., 1976. Turbulent Prandtl Numbersfor Fully Developed Rotating Annular Axial Flow of Air,Trans. ofASME, 76-HT-36, 1-13.
Lai, Y.G. and So, R.M.C., 1990. Near-Wall Modeling ofTurbulent Heat Fluxes, Int. J. Heat Mass Transfer, 33(7),1429-1440.
Launder, B.E. and Shima, N., 1989. Second-Moment Closurefor the Near-Wall Sublayer: Development and Application,AIAA J., 27(10), 1319-1325.
Murakami, M. and Kikuyama, K., 1980. Turbulent Flow inAxially Rotating Pipes, J. Fluids Engineering, 102, 97-103.
Nagano, Y. and Tagawa, M., 1988. Statistical Characteristics ofWall Turbulence with a Passive Scalar, J. Fluid Mech., 196,157-185.
Patankar, S.V., 1980. Numerical Heat Transfer and Fluid Flow,Hemisphere Publishing, Washington, DC.
Rehme, K., 1974. Turbulent Flow in Smooth Concentric Annuliwith Small Radius Ratios, J. Fluid Mech., 64, 263-287.
Sommer, T.P., So, R.M.C. and Lai, Y.G., 1992. A Near-WallTwo-equation Model for Turbulent Heat Fluxes, Int. J. HeatMass Transfer, 35(12), 3375-3387.
Torii, S. and Yang, W.J., 1994. A Numerical Study onTurbulent Flow and Heat Transfer in-Circular CouetteFlows, Numerical Heat Transfer, Part A, 26, 231-336.
Torii, S. and Yang, W.J., 1995a. Numerical Prediction of Fully-Developed Swirling Flows in an Rotating Pipe by Means of aModified k-c Turbulence Model, Int. J. Numerical Methodsfor Heat & Fluid Flow, 5(2), 175-183.
Torii, S. and Yang, W.J., 1995b. A Numerical Analysis on Flowand Heat Transfer in the Entrance Region of an AxiallyRotating Pipe, Int. J. Rotating Machinery, 2(2), 123-129.
Youssef, M.S., Nagano, Y. and Tagawa, M., 1992. A Two-equation Heat Transfer Model for Predicting TurbulentThermal Fields under Arbitrary Wall Thermal Conditions,Int. J. Heat Mass Transfer, 35(11), 3095-3104.
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