Fluid Flow and Heat Transfer in an Internal Coolant Passage

Computations were performed to study the three-dimensional flow and heat transfer in a U-shaped duct of square cross section with inclined ribs on two opposite walls under rotating and non-rotating conditions. Two extreme limits in the Reynolds number (25,000 and 350,000) were investigated. The rotation numbers investigated are 0, 0.24, and 0.039. Results show rotation and the bend to reinforce secondary flows that align with it and to retard those that do not. Rotation was found to affect significantly the flow and heat transfer in the bend even at a very high Reynolds number of 350,000 and a very low Rotation number of 0:039. When there is no rotation, the flow and heat transfer in the bend were dominated by rib-induced secondary flows at the high Reynolds number limit and by bend-induced pressure-gradients at the low Reynolds number limit. Long streaks of reduced surface heat transfer occur in the bend at locations where streamlines from two contiguous secondary flows merge and then flow away from the surface. The location and size of these streaks varied markedly with Reynolds and rotation numbers.


INTRODUCTION
To increase thermal efficiency and specific thrust, advanced gas turbine stages are being designed to operate at gas temperatures that exceed acceptable material temperatures.Even with the development of thermal-barrier coatings, cooling is needed for all surfaces that come in contact with the hot gases, especially in the first stage.
*Corresponding author.Tel.: 517 432-3658, Fax: 517 353-1750, e-mail: tomshih@egr.msu.eduOne effective and widely used method of cooling is internal cooling.With internal cooling, lower- temperature air is extracted from the compressor and circulated through passages embedded inside components such as blades and vanes.Efficiency considerations demand effective cooling with minimal cooling air.This need for efficiency has led numerous investigators to study fluid flow and heat transfer processes inside internal coolant passages and to develop and evaluate design concepts that enhance heat transfer with minimal drag.
Experimental studies on internal coolant pas- sages have focused mostly on non-rotating ducts, which are relevant to vanes.See, for example, Han et al. (1992); Chyu et al. (1995); Liou et al. (1998), and references cited there.Experimental studies on rotating ducts, which are relevant to blades, have been less numerous.Wagner et al. (1991a, b); Morris and Salemi (1992); Han et   Tse (1995)  and  Kuo  and Hwang (1996) reported studies on rotating ducts with ribbed walls.
Of the computational studies, only Shih et al.   (1998) studied the flow and heat transfer in a ribbed U-duct under typical engine conditions (i.e., not only are the geometry, Reynolds number, and Rotation number typical, but also the engine speed, the coolant temperature and pressure, and the wall temperatures).They focused on industrial gas turbines and investigated an extreme situation in which the Reynolds number is 350,000 under both rotating and non-rotating conditions.When there is rotation, the duct rotated at 3,600rpm.They compared their results for a high Reynoldsnumber limit with that of Stephens and Shih (1997) for a low Reynolds-number limit at 25,000.However, that comparison was incomplete for two reasons.First, Stephens and Shih (1997) only had results for the rotating case but not for the non- rotating case.Second, only results on secondary flow and surface heat transfer were given.The objective of this study is to present a more com- plete investigation of the flow and heat transfer in a ribbed U-duct involving Reynolds numbers at two extreme limits under both rotating and non-rotat- ing conditions.

DESCRIPTION OF PROBLEM
A schematic diagram of the problem investigated in this study is shown in Figure 1.It involves flow in a U-shaped duct of square cross section, made up of two straight sections and a 180 bend.The geometry of the straight section is the same as that reported in Wagner et al. (1991a, b).The geometry of the bend is somewhat different, and is taken from the current experimental setup at United Technologies Research Center (Wagner and   Steuber, 1994).The dimensions of the duct are as follows (see Fig. 1).The duct hydraulic diameter is Dh 1.27 cm (0.5 in).The radial position of the duct relative to the axis of rotation is given by Rr/Dh--41.85and Rt/Dh 56.15 so that the mean radius is Rm/DI (Rr q--Rt)/2D 49.The length of the duct's straight section is L/Dh 14.3.The curvature of the 180 bend is given by Ri/Dh 0.22 and Ro/Dh 1.44.
The leading and trailing walls of the straight sections are each roughened with five equally spaced ribs.The ribs on those two walls are staggered relative to each other with ribs on the leading wall offset upstream from those on the trailing wall by a half pitch (p).The ribs are located just upstream or downstream of the 180 bend.All ribs are inclined with respect to the flow at an angle of 45 The cross section of the rounded ribs is made up of three circular arcs of radius 0.0635cm (0.025 in) so that the rib height (e) is 0.127cm (0.05 in) and the rib-height to hydraulic-diameter (e/Dh) is 0.1.The pitch-to- height ratio (p/e) was set to 5 (same as UTRC experiments).With only five ribs on each face, a considerable length of the duct has smooth walls.See Stephens and Shih (1997) for details on the ribs.
For this problem, four different cases were studied, representing different limits in Reynolds and rotation numbers.A summary of all cases studied is given in Table I.For Cases and 2, the four walls of the duct including the rib surfaces were maintained at a constant temperature of Tw=800K.The coolant air entering the duct had a uniform temperature of Ti= 550 K and an average static pressure of Pi= 10atm.The Rey- nolds number at the inlet was Re--350,000.For Cases 3 and 4, Pi-1.78 atm, Mi=0.05 and Ap/p=O.13.
The angular speed of the rotating duct was 3,600rpm (376.99 radians per second).These conditions correspond to a rotation number of Ro=0.039; an inlet Mach number (relative to duct) of Mi=0.26; and an inlet density ratio of Ap/p=0.3125.For Cases 3 and 4, which repre- sent another limit, the duct wall temperature was Tw 344.83 K, and the coolant tempera- ture, pressure, and Reynolds number at the inlet were Ti 300 K, Pi 1.78 atm, and Re 25,000.
These gave rise to an inlet Mach number of Mi 0.05 and an inlet density ratio of Ap/p O. 13.
When there is rotation, the angular speed was 3,132rpm, giving rise to a rotation number of Ro 0.24.
Unlike the temperature profile, the velocity profile at the inlet was not uniform because of the extensive flow passages upstream of it.Since fully developed velocity profiles do not exist for compressible flows, the velocity profile used is the one at the exit of a non-rotating straight duct of length 150Dh with adiabatic walls and the same cross section and flow conditions as the U-duct studied here.
Of the four cases studied, Cases and 2 are fairly typical of industrial gas turbines except that the Reynolds number is near the upper limit.Cases 3 and 4 are less typical in terms of the pres- sure and temperature conditions, but the rotation number is fairly typical of the lower Reynolds num- ber limit.Note that for industrial gas turbines, the engine speed is either 3,000 or 3,600rpm so that for a given geometry and inlet conditions, the high- er the Reynolds number, the lower is the rota- tion number and vice versa.

PROBLEM FORMULATION
The problem described in the above section and depicted in Figure is modeled by the ensembleaveraged conservation equations of mass (continuity), momentum (compressible Navier- Stokes), and total energy for a thermally and calorically perfect gas with Sutherland's model for dynamic viscosity and a constant Prandtl number.These equations are written in a coordinate system that rotates with the duct so that steady-state solutions with respect to the duct can be com- puted.The form of these equations is well known (see, e.g., Steinthorsson et al., 1991; Prakash and  Zerkle, 1995 and Stephens and Shih, 1999).In this study, the ensemble-averaged equations were closed by a low-Reynolds-number two-equation k-co model known as the shear-stress-transport (SST) model (Menter, 1993 and Menter and  Rumsey, 1994).The SST model was selected over the standard k-co model of Wilcox (1993) because it eliminates dependence on freestream k and co, and has a limiter to control overshoot in k with adverse pressure gradients so that separation is predicted more accurately.
Note that by being a low-Reynolds number model, integration of the conservation equations and the turbulence model is made all the way to the wall.Thus, the wall boundary conditions (BCs) are zero velocity, constant wall temperature and zero turbulent kinetic energy.The dissipation rate per k (co) on the wall was set to O0(Ou/On)w as proposed by Wilcox (1993) for hydraulically smooth surfaces.A check, however, was imposed to ensure that the maximum value of co was less than 60u/(/3ReAy2), where Ay is the normal distance from the wall and/3 3/40, as suggested by Menter (1993).Other BCs are as follows.At the duct entrance, a developed profile was specifi- ed for velocity, but the temperature profile was taken to be uniform (see previous section for details).Turbulence quantities (k and co) were specified in a manner that is consistent with the velocity profile (average turbulent intensity was 5%).Only pressure was extrapolated.At the duct exit, an average static pressure was imposed but the pressure gradients in the two spanwise direc- tions were extrapolated.This is important because secondary flows induced by inclined ribs, the bend, and centripetal/Coriolis forces cause considerable pressure variations in the spanwise directions.Density and velocity were also extrapolated.
Though only Steady-state solutions are of inter- est in this study, initial conditions were needed because the unsteady form of the conservation equations was used.The initial conditions used are the solutions of the steady, one-dimensional, inviscid equations, namely,

OpU
OpU 2 OP Opuh Ox O, 0----0--+ pf2X, OX puf2X (1 where h is enthalpy.For an ideal gas, the solution to Eq. ( 1) is NUMERICAL METHOD OF SOLUTION Solutions to the ensemble-averaged conservation equations were obtained by using a cell-centered finite-volume code called CFL3D (Thomas et al.,  1990 and Rumsey and Vatsa, 1993).In this study, the CFL3D code was modified so that it can account for steady-state solutions in a rotating frame of reference by adding source terms that represent centripetal and Coriolis forces in the momentum and energy equations.
The CFL3D code contains a number of differ- ent algorithms.The one used in this study is as follows.All inviscid terms were approximated by the second-order accurate flux-difference splitting of Roe (1981).All diffusion terms were approxi- mated conservatively by differencing derivatives at cell faces.Since only steady-state solutions were of interest, time derivatives were approximated by the Euler implicit formula.The system of nonlinear equations that resulted from the aforementioned approximations to the space-and time-derivatives were analyzed by using a diagonalized alternating- direction implicit scheme (Pulliam and Chaussee, 1981) with local time-stepping (local Courant number always set to unity) and three-level Vcycle multigrid (Anderson et al., 1988).
The domain of the problem was replaced by a single-block grid system that was assembled by patching together 37 blocks of H-H structured grids with C 2 continuity at all block interfaces (see Fig. 2).For this grid system, the total number of grid points in the streamwise direction from inflow to outflow is 761.The number of grid points in the cross-stream plane is 65 x 65.From Figure 2, it can be seen that grid lines are clustered near all solid surfaces to resolve the sharp gradients there, are smooth as grid spacings change from fine to coarser, and are orthogonal everywhere except along rib surfaces.Along rib surfaces, the grid lines are aligned with the rib inclination.The grid system used satisfy a set of rules that are needed to obtain meaningful solutions such as having five grid points within y+ of 5.The rules and the process can be found in Stephens et al. (1996a), and will not be repeated here.As a further test, the aforementioned grid systems were refined by a factor of 25%, first in the streamwise and then in the cross-stream directions.This grid independence study showed the predicted surface heat transfer coefficient to vary by less than 2%.
On a Cray C-90 computer, where all computa- tions were performed, the memory requirement for each run is 155 megawords (MWs).The CPU time required for a converged steady-state solu- tion is about 40 hours and involves about 3,000 iterations.A solution is said to be converged when the following conditions are satisfied: (1) residual drops at least three orders of magnitude; (2) the second norm of the residual is less than 10-13; and (3) predicted surface heat transfer varies by less than 1%.
The code used in this study has been validated for two problems.The first problem is flow in a non-rotating square duct in which two opposite walls are lined with inclined ribs.In that problem, the geometry of the duct and rib is identical to that studied here for the up-leg portion of the U- duct.The computed heat-transfer enhancement for that problem was compared with measurements obtained by using liquid crystal thermography (see Stephens et al., 1996a).The comparisons showed that the code used predicted the qualita- tive features very well.The predicted quantitative features were also found to be reasonable (peak values are within 10 to 20%).The second problem is flow in a rotating U-duct with smooth walls.The geometry of that problem is the same as that studied here except for the absence of ribs.For that problem, the computed Nusselt numbers on the leading and trailing walls for the up-leg part of the U-duct were compared with measured one by Wagner et al. (1991a).Those comparisons, described in Stephens et al. (1996b), showed very good agreements.The good comparisons obtained in those two studies give some confidence to the code for studying the present problem.

RESULTS
As noted in the Introduction, the objective of this study is to investigate the flow and heat transfer in a ribbed U-duct for two extremes in Reynolds number under rotating and non-rotating condi- tions.To meet this objective, computations were performed for four cases summarized in Table I.The results generated are given in Figures 3 to 6.

Flow Field
The flow field in the U-duct with ribs and a coolant-to-wall temperature that is less than unity can be inferred from Figures 3 to 5   normalized temperature, r/, along several planes.Figure 5 shows normalized pressure, P/Pref, on walls with velocity vectors at 0.0016Dh from walls.
Here, the coloration is in gray scale, which can be difficult to read.interested reader can contact the first author to obtain colored plots.
Before discussing the results in these figures, it is important to review the nature of the flow in a smooth U-duct under non-rotating and rotating conditions.When there is no rotation (Ro=0.0,Re=25,000), Stephens et al. (1996b) and Lin et al. (1998) showed that the velocity profile to have a maximum about the center of the ductcross-section in the up-leg portion.As the flow approaches the bend, it is decelerated near the concave part of the bend with some flow reversal near the leading/trailing walls because of the adverse pressure gradient induced by the bend.Near the convex part of the bend, the flow is accelerated instead of decelerated.As the flow goes around the bend, Dean-type secondary flows form, and these secondary flows persist in the down-leg part of duct.Around the bend, a large Streamwise separation region forms on the convex side of the bend.The Dean-type secondary flows induce another pair of secondary flows in the separated region downstream of the bend.
When there is rotation (Ro 0.24, Re 25,000), Stephens and Shih (1997, 1999) trailing.Scale is same as that in Figure 3. outward flow, this pair flows from the trailing face to the leading face along the two side walls, which transports cooler air from near the center of the duct cross section to the trailing wall.Since the thermal boundary layer starts on the trailing wall, air temperature near the trailing wall is lower than that near the leading wall.With high- er temperature and hence lower density near the leading wall, centrifugal buoyancy decelerates the flow there.With lower velocity and hence thicker boundary layer next to the leading face, the Coriolis-induced secondary flows cause the formation of additional pairs of vortices near the leading face (Stephens and Shih, 1999).With the above as a backdrop, the results of this study given in Figures 3 to 5 are described.

Non-rotating Ribbed Duct
When there is no rotation (Cases and 3, Tab.I), the flow in the U-duct is dominated by the inclin- ed ribs.The staggered inclined ribs induce two counter-rotating secondary flows with the sense of rotation dependent upon the rib inclination.In the up-leg part, the secondary flows flow from the inner side wall to the outer side wall along the two faces of the duct with ribs.Note that there is flow reversal in the region between the ribs (see Figs. 3 and 4).Because of the ribs' stag- gered arrangement, the size of these secondary flows oscillates along the duct.The magnitude of this oscillation is weaker at the higher Reynolds num- ber.These secondary flows transport cooler fluid towards the ribbed faces.
As the flow approaches the bend, the secondary flow components have higher values near the concave wall than the convex wall (Fig. 4).This is because the flow component along the duct is decelerated near the concave wall and accelerated near the convex wall.Further into the bend, the pressure gradient induced by the flow curvature reinforces secondary flows that are aligned with Dean-type ones and retards those that do not (Fig. 3).At Re-25,000, the secondary flow in the bend is asymmetric with bend-induced pressure gradient exerting considerable influence.At Re 350,000, however, the secondary flows in the bend are nearly symmetric with the effects of the bend-induced pressure gradient being minimal.Note that whenever different secondary flows meet near surfaces, the temperature reaches a local maximum.
Around the bend, there is a large separated region next to the convex surface.This separated region extends all the way to the first rib in the down-leg part of the U-duct.The size of the separation bubble is larger when Re is lower.Unlike smooth U-ducts, there was essentially no flow separation next to the concave wall of the bend.
Re 25,000  If the rib inclination was -45 instead of t45 in the up-leg part of the duct, then Dean-type secondary flows in the bend would reinforce rib- induced secondary flows.To continue this reinfor- cement in the down-leg part of the duct, the rib inclination there should remain at -+-45 Rotating Ribbed Duct With rotation (Cases 2 and 4, Tab.I), Coriolis- induced secondary flows in the smooth part of duct are quickly dominated by those induced by inclined ribs.However, at Re=25,000, Coriolis force can clearly be seen to reinforce the secondary flow that is aligned with it (i.e., turning in the same direction) and to retard the one that is opposed to it.Thus, one secondary flow (the one next to the leading face) becomes larger and stronger than the other.At Re=350,000, the effects of Coriolis diminishes but its effects are clearly not negligible by noting differences in the flow with and without rotation.
Around the bend, the curvature-induced pres- sure gradient further reinforces the secondary flow that is aligned with it and retards the one that is not.At Re 25,000, one secondary flow dominate the entire duct cross section.At Re 350,000, the two secondary flows created by inclined ribs in the up-leg part persist throughout the bend, but they are definitely affected by the pressure gradient there by becoming highly asymmetric (Fig. 3).
When there is rotation, flow reversal occurs in the bend next to the leading and trailing faces when Re=25,000, but not when Re= 350,000.Recall that when there is no rotation, flow reversal does not take place if ribs are present at both Reynolds numbers investigated.
In the down-leg part of the U-duct, the secondary flows are again reinforced or retarded by the Coriolis force, which further accentuate the asymmetry created in the up-leg part and the bend.Thus, rotation clearly has strong effects on the nature of the flow even at Re-350,000 and Ro--0.039.

Surface Heat Transfer
The heat transfer characteristics can be inferred from Figures 3, 4 and 6.Figures 3 and 4 show normalized temperature contours, r/, at selected planes, and Figure 6 shows surface heat transfer coefficient via a normalized Nusselt number, Nu/ Nus.From these figures, it is noted that whether there is rotation or not, high heat transfer rates occur at the ribs.Low heat transfer rates take place at the following locations on the ribbed faces: just downstream of the rib tops and towards the outer wall for the up-leg part; just downstream of the rib tops and towards the inner wall for the down-leg part; in the separated region just downstream of the bend; and in thin strips on the leading and trailing faces in the bend where different secondary flow leave the surface.Basi- cally, when two secondary flows impinge on a sur- face, heat transfer is high, and when they leave a surface, heat transfer is low.All streaks of low and high surface heat transfer in the bend can be attributed to this.Thus, the structure of the secondary flows in the bend has a significant effect on surface heat transfer.
When there is no rotation, the heat-transfer enhancement on the ribbed faces is nearly identical with the average heat transfer rate in the up-leg part higher than that in the down-leg part.In the bend, the effects of increasing Re from 25,000 to 350,000 is to shift the streak of low heat-transfer Re 25,000 Re-350,000 enhancement from near the middle of the leading and trailing faces towards the concave wall.On the concave wall, a similar increase in Re causes the two streaks of low heat-transfer enhancement to be near its middle instead of near the ribbed walls.
When there is rotation, the heat-transfer en- hancement on the trailing face is markedly higher than that on the leading face in the up-leg part.In the down-leg part, heat-transfer enhancement is higher on the trailing wall when Re 350,000 and higher on the leading wall when Re 25,000.The reason is that hotter fluid was convected to the trailing wall at .Re=25,000.Again, secondary flows affect surface heat transfer by transporting hotter or cooler fluid to the leading and trailing faces.

SUMMARY
Computations were performed by using a vali- dated CFD code to examine the flow and heat transfer in a ribbed U-shaped duct with and without rotation.The Reynolds numbers investi- gated are 25,000 and 350,000, which represent two extreme limits.Results show in detail the evolution of the secondary flows induced by inclined ribs and a smooth 180 bend under rotating and non- rotating conditions.Both rotation and the bend were found to reinforce secondary flows that align with it and to retard those that do not.Even at Re 350,000, the effects of rotation were found to be significant.The structure of the secondary flows is important because it determines how cooler fluid in the core is convected to the ribbed or leading and trailing faces.The cooler the fluid is near the surface, the higher is the heat transfer rate.Long streaks of reduce surface heat transfer in the bend were found to be due to secondary flows meeting and then leaving the surface.These streaks were a strong function of the Reynolds and rotation numbers.Nusselt number for smooth duct (Nus 0.023 Re '8 Pr'4) static pressure, pressure at inlet heat transfer rate per unit area from duct surface inner and outer radius of 180 bend (Fig. 1) radius from axis of rotation (Fig. 1) gas constant for air Reynolds number (piViDh/#) rotation number (Dh/gi) temperature, coolant inlet temp., wall temp.

FIGURE
FIGURESchematic of ribbed U-duct studied.

FIGURE 3
FIGURE 3 Normalized temperature with projected velocities at several cross-stream planes.
FIGURE 4 Normalized temperature with projected velocities at several streamwise planes.Left: leading, Middle: mid-plane, Right:

FIGURE 5
FIGURE 5 Normalized pressure on walls with velocity vectors at 0.0016Dh from walls.Left: leading, Right: trailing, (a) Non- rotating; (b)Rotating.