A Numerical Study on Turbulent Couette Flow and Heat Transfer in Concentric Annuli by Means of Reynolds Stress Turbulence Model

A numerical study is performed to investigate fluid flow and heat transfer characteristics in a concentric annulus with a slightly heated inner core moving in the flow direction and a stationary, insulated outer cylinder. Emphasis is placed on the 
effect of inner core movement on the flow structures, i.e. the normal components of the Reynolds stress and its off-diagonal one. A Reynolds stress turbulence model is employed to obtain these turbulence quantities. The governing boundary-layer equations are discretized by means of a control volume finite-difference technique and numerically solved using the marching procedure. It is found from the study that (i) the streamwise movement of the inner core causes an attenuation in the normal Reynolds stresses, although the inherent anisotropy is maintained and the appreciable turbulence remains, (ii) 
the Reynolds stress in the inner wall region is substantially diminished due to the inner core movement, resulting in a decrease in the heat transfer performance, and (iii) an increase in the velocity ratio of the moving inner core of the fluid flow induces a decrease in the Nusselt number as well as the Reynolds stress in the region near the inner core.


INTRODUCTION
The problems of heat transfer and fluid flow in concen- tric annuli can be classified three categories: (i) station- ary cylinder case, (ii) parallel Couette flow case and (iii) circular Couette flow case.The present study is focused on turbulent transport phenomenon in the parallel Couette flow, which refers to a flow in a concentric annulus with one surface moving in the flow direction and the other remaining stationary (or both surface moving in the flow direction at different velocities).Barrow and Pope [1987] conducted a simple analysis simulating flow and heat transfer in railway tunnels (such as the 54-km long Seikan tunnel in Japan and the Channel tunnel between England and France).Shigechi et al. [1990] obtained analytical solutions for the friction factor and Nusselt number for turbulent fluid flow and heat transfer in concentric annuli with moving inner cores, using a modified mixing length model, originally proposed by van Driest [1956].It was disclosed that as the relative velocity (i.e. the velocity ratio of a moving inner core to the fluid flow) is increased, the friction factor is diminished while the Nusselt number is en- hanced.The same numerical study was performed by Torii and Yang [1994], who employed the existing k-e turbulence models.It was found that: (i), the streamwise movement of the inner wall causes an attenuation in the turbulent kinetic energy, resulting in a reduction in the heat transfer rate and (ii), an increase in the relative velocity causes a decrease in both the friction factor and the Nusselt number as well as a reduction in the turbulent kinetic energy in the wall inner region.Since the two- equation k-e model basically assumes isotropic turbu- lence structure, it cannot precisely reproduce the anisot- ropy of turbulence caused by the inner core moving in the flow direction.In order to obtain the detailed infor- mation pertinent to the flow structures, the higher order closure model, i.e. a Reynolds stress turbulence model is employed.
Hanjalic and Launder [1976] and Prud'homme and Elghobashi [1986] proposed a low Reynolds number version of a Reynolds stress turbulence model, which can predict turbulence quantities in the vicinity of the wall as well as in the region far from the wall.However, it was found from the preliminary calculation that both models were unable to reproduce the inherent anisotropy in the near-wall region of isothermal circular tube flows.Better accuracy in this region was achieved by the Reynolds stress model of Launder and Shima [1989] based on a full second-momentum closure expressed in terms of the turbulent Reynolds number and independent Reynolds stress invariants.
The purpose of the present study is to investigate turbulent flow and heat transfer characteristics in con- centric annuli with the inner core moving in the flow direction.The Reynolds stress turbulence model pro- posed by Launder and Shima is employed to shed light on the mechanism of the transport phenomena.Emphasis is placed on the effects of core movement on the flow structures, i.e. the three normal components of the Reynolds stress and its off-diagonal one.

GOVERNING EQUATION AND SOLUTION PROCEDURE
Consideration is given to a steady turbulent annular flow in which a slightly heated inner core moves in the flow direction and an insulated outer cylinder is held station- ary.The physical configuration and the cylinder coordi- nate system are shown in Fig. 1.Under the assumptions of constant fluid properties and negligible viscous dissi- pation, the boundary layer approximation yields the governing equations as: Continuity equation: The turbulent thermal diffisivity o in Eq. ( 4) is given by l)

OL
Prt, where 0.9 is adapted as the value of Pr t.The turbulent viscosity vt can be represented through Boussinesq's approximation, as
The governing equations are subject to the following boundary conditions: x 0 (start of heating): hydrodynamically fully devel- oped annular flow without an inner core movement . OE 0.35fR2 (1 0.3A2) 2.5 134 S. TORII AND W.-J. YANG r Rin (inner wall): u U (velocity of a inner cylinder), . O.

OT qw
Or k r Rou (outer wall): u u 2 v 2 w 2 uv .

0, (insulation)
The governing equations are discretized by means of a control volume method [1980].Since all turbulence quantities as well as the time-averaged streamwise ve- locity vary rapidly in the near-wall region, nonuniform cross-stream grids are used in which the size is increased in a geometric ratio with the maximum size being kept within 3 % of the hydrodynamic radius.The number of typical control volumes used in the radial direction are 45 at Re 10,000 and 52 at 46,000.In order to ensure the accuracy of calculated results, at least two control volumes are always located in the viscous sublayer.The resulting equations are solved from the inlet proceeding in the downstream direction by means of the marching procedure, because the equations are parabolic.The axial control volume size is constant and five times the minimum radial size at the wall.At each axial location, the pressure gradient dP/dx in Eq. ( 2) is corrected at every iteration to conserve the total mass flow rate.The procedure is repeated until a convergence criterion is satisfied.In the present study, it is set at ferent grid spacing.The calculation conditions are sum- marized in Table 2, including the Reynolds number, wall heat flux, relative velocity and tube size.
It is necessary to verify both the turbulence model employed here and the reliability of the computer code by comparing numerical predictions with experimental results for the flow fields.The model is applied to a flow in an annulus with a stationary, slightly heated inner core.Numerical results for the streamwise velocity, three components of Reynolds stress, its off-diagonal one and Nusselt number are obtained at a location 200 tube diameter downstream from the inlet, where thermally and hydrodynamically developed flow is realized.The experimental data of Brighton and Jones [1964] are used for comparison.Figure 2 depicts the radial distributions of the time-averaged streamwise velocity (dimensionless velocity u / versus y/) at Re 46,000.Figure 2(a) and (b) correspond to the distributions from the inner and outer walls to the location of the maximum stremwise velocity, respectively.It is observed that the model yields a better agreement with the experimental data, and predict the velocity profile with well-known characteris- tics of the logarithmic region, i.e. the universal wall law. Figure 3 shows the radial distributions of three normal components of the Reynolds stress tensor at Re 46,000.The numerical results are normalized by the friction velocity, U*out, on the outer wall.The model predicts an inherent anisotropy of the annular flow, although its accuracy is somewhat inferior near the inner and outer walls than in the center region.Figure 4 illustrates the radial distribution of the calculated Reynolds stress at Re 46,000.The ordinate in the figure is normalized by the square of the friction velocity (U*out) 2 on the outer wall.The predicted results are in good agreement with the experimental data.Figure 5  (max (12) for all the variables (b (u, //2, 122, W2, UP, T, and e).The superscripts M and M in Eq. ( 12) indicate two successive iterations, while the subscript "max" refers to a maximum value over the entire fields of iterations.Throughout numerical calculations, the number of con- trol volumes is properly selected between 45 and 92 to ensure validation of the numerical procedures and to obtain grid-independent solutions.It results in no appre- ciable difference between the numerical results with dif-

NUMERICAL RESULTS AND DISCUSSION
Numerical results of the Nusselt number are illustrated in Fig. 6 with the velocity ratio of a moving inner core to a fluid flow, U*, as the parameter.It is observed that the Nusselt number diminishes with an increase in the dimensionless relative velocity.A similar trend is re- ported by Torii and Yang [1994], who employ the existing k-turbulence models.It is disclosed that the substantial reduction in the Nusselt number is attributed to the inner core movement in the flow direction. .0 )'IRe-'22 (14) 0 respectively.The two equations are superimposed in Fig. 5 as solid straight lines.Note that Fig. 5 is under the temperature ratio of the inner wall to the inlet fluid, Twin/Tinlet, of unity, and the radial ratio, Rin/Rout, of 0.56.
The calculated Nusselt number and friction factor are in 0 0.2 0.4 0.6 0.8 1.0 good agreement with the correlations, Eqs. ( 13)and ( 14). (r-Rin)/(Rout-Rin) The validity of the computer code and the accuracy for FIGURE 4 Radial distribution of predicted Reynolds stress in a the model employed here are thus confirmed, stationary concentric annulus for Re 46,000 and Rin/Rou 0.56.
In summary, a decrease in the Nusselt number, as seen in Fig. 6, is caused by the streamwise movement of the inner core.This trend is amplified with an increase in the relative velocity.The mechanism is that: in the region near the inner core, a reduction in the velocity gradient induced by its streamwise movement suppresses the Reynolds stress, resulting in a decrease in the heat transfer performance.

SUMMARY
A Reynolds stress turbulence model by Launder and  Shima has employed to investigate transport phenomena in concentric annulus with a slightly heated inner core moving in the flow direction.Consideration is given to the influence of relative velocity on the flow structure.The results are summarized as follows: A Reynolds stress turbulence model predicts a reduc- tion in the Nusselt number with an increase in the relative velocity.It is also disclosed that: (i) an inner core movement causes a decrease in the velocity gradient at the inner wall, a substantial deformation of the velocity profile and an attenuation in the three normal components of the Reynolds stress in the inner wall, and (i) however, an appreciable turbulence remains and its inherent anisotropy is maintained.Consequently, the streamwise movement of the inner core suppresses the " U*=O 0 ."7U*=0.5 U*=l.0 -1.0 0 0.2 0.4 0.6 0.8 1.0 (r-Rin)/(Rout-Rin) FIGURE 9 Variation of Reynolds stress profiles with dimensionless relative velocity for Re 50,000 and Rin/Rou 0.5.
Reynolds stress in the vicinity of the inner wall, resulting in the deterioration of heat transfer performance.

)
present both the Nusselt number and the friction factor as a

FIGURE 2
FIGURE 2 Distribution of predicted time-averaged streamwise ve- locity in a stationary concentric annulus for Re 46,000 and RnfRo,t 0.56, (a) inner side and (b) outer side.

TABLE 2 .
Radial distribution of predicted normal Reynolds stresses in a stationary concentric annulus for Re 46,000 and Rio,,t 0.56.