A NUMERICAL STUDY OF FLOW AND HEAT TRANSFER IN A SMOOTH AND RIBBED U-DUCT WITH AND WITHOUT ROTATION

Chyu ** Department of Mechanical Engineering University of PittsburghPittsburgh, PA 15261 ABSTRACT Computations were performed to study the three-dimensional flow and heat transfer in aU

A NUMERICAL STUDY OF FLOW AND HEAT TRANSFER

IN A SMOOTH AND RIBBED U-DUCT WITH AND WITHOUT

ROTATION

Y.-L Lin + and T I-P Shih *

Department of Mechanical Engineering, Michigan State University

East Lansing, MI 48824-1226

M A Stephens #

Department of Mechanical Engineering

Carnegie Mellon UniversityPittsburgh, PA 15213-3890

M K Chyu **

Department of Mechanical Engineering

University of PittsburghPittsburgh, PA 15261

ABSTRACT

Computations were performed to study the three-dimensional flow and heat transfer in aU-shaped duct of square cross section under rotating and non-rotating conditions Theparameters investigated were two rotation numbers (0, 0.24) and smooth versus ribbed walls at aReynolds number of 25,000, a density ratio of 0.13, and an inlet Mach number of 0.05 Resultsare presented for streamlines, velocity vector fields, and contours of Mach number, pressure,temperature, and Nusselt numbers These results show how fluid flow in a U-duct evolves from

a unidirectional one to one with convoluted secondary flows because of Coriolis force,centrifugal buoyancy, staggered inclined ribs, and a 180o bend These results also show how thenature of the fluid flow affects surface heat transfer

The computations are based on the ensemble-averaged conservation equations of mass,momentum (compressible Navier-Stokes), and energy closed by the low Reynolds number SST

+ Research Associate.

* Professor Fellow ASME To whom all correspondence should be addressed (tomshih@egr.msu.edu).

# Now, Project Engineer, Pratt & Whitney, Middletown, Connecticut.

* * Leighton Orr Chair Professor and Chairman Fellow ASME.

turbulence model Solutions were generated by a cell-centered finite-volume method that usessecond-order flux-difference splitting and a diagonalized alternating-direction implicit schemewith local time stepping and V-cycle multigrid.

NOMENCLATURE

Cp constant pressure specific heat

Dh hydraulic diameter of duct

h heat transfer coefficient (h = qw/(Tw - Tb))

k turbulent kinetic energy

L length of straight portion of duct

Mi Mach number (Mi = Vi/ RTi )

Nu Nusselt number (Nu = hDh/)

Nus Nusselt number for smooth duct (Nus = 0.023 Re0.8 Pr0.4)

qw wall heat transfer rate per unit area

Ri, Ro inner and outer radius of 180o bend (Fig 1)

Rr, Rt radius from axis of rotation (Fig 1)

R gas constant for air

Re Reynolds number (Re = iViDh/)

Ro rotation number (Ro = Dh/Vi)

Tb bulk temperature defined by   p

2 r 2 2 i

Vi average velocity at duct inlet

x, y, z coordinate system rotating with duct (Fig 1)

X coordinate along the axis of the U-duct from inlet to outlet

u, v, w x-, y-, z-components of velocity relative to duct

 dissipation rate per unit k

 angular rotation speed of duct

INTRODUCTION

To improve thermal efficiency, gas-turbine stages are being designed to operate atincreasingly higher inlet temperatures This increase is enabled by advances in two areas,cooling technology and materials With cooling, inlet temperatures can far exceed allowablematerial temperatures

A widely used method for cooling vanes and blades is to bleed lower-temperature airfrom the compressor and circulate it within and around each airfoil This air, referred to as thecoolant, generally enters each airfoil from its root and exits from its tip and/or trailing edge Italso could exit from strategically placed holes for film cooling While inside each airfoil, thecoolant typically flows through a series of straight ducts connected by 180o bends with the wallsroughened with ribs or pin fins to enhance heat transfer For efficiency, effective cooling must

be accomplished with minimal cooling flow and pressure loss This need for efficiency is evenmore urgent for gas turbines with low NOx combustors, which compete for the same cooling air

The importance of efficient and effective cooling has led many investigators to study theflow and heat transfer in internal coolant passages and to develop and evaluate design concepts.Most experimental studies on internal coolant passages have focused on non-rotating ducts,which are relevant to vanes See, for example, Han, et al (1992), Chyu, et al (1995), Liou, et al.(1998), Iacovides, et al (1999), and the references cited there Experimental studies on rotatingducts, which are relevant to blades, have been less numerous Wagner, et al (1991a,b), Morris

& Salemi (1992), Han, et al (1994), and Cheah, et al (1996) investigated rotating ducts withsmooth walls Taslim, et al (1991), Wagner, et al (1992), Zhang, et al (1993), Johnson, et al.(1994), Zhang, et al (1995), Tse (1995), and Kuo & Hwang (1996) reported studies on rotatingducts with ribbed walls

Most of the earlier computational studies on internal coolant passages have been dimensional In recent years, a number of three-dimensional (3-D) studies have been reported.3-D studies are needed if there are ribs, 180o bends, and/or rotation Besserman & Tanrikut(1991), Wang & Chyu (1994), and Rigby, et al (1996) studied non-rotating smooth ducts with

two-180o bends Iacovides, et al (1991, 1996), Medwell, et al (1991), Tekriwal (1991), Dutta, et al.(1994), Tolpaldi (1994), Stephens, et al (1996a), Hwang, et al (1997), Stephens & Shih (1999),and Chen, et al (1999) studied rotating smooth ducts Prakash & Zerkle (1992, 1995), Abuaf &Kercher (1994), Stephens, et al (1995a), Rigby, et al (1997), Rigby (1998), and Bohn, et al.(1999) studied ducts with normal ribs

Very few investigators performed computational studies on ducts with inclined ribs,which are used in advanced designs Stephens, et al (1995b, 1996b) studied inclined ribs in astraight duct under non-rotating conditions Bonhoff, et al (1996) studied inclined ribs in a non-rotating U-duct (i.e., a duct with two straight sections and a 180o bend) More recently, Stephens

& Shih (1997), Bonhoff, et al (1997), and Shih, et al (1998, 2000) studied inclined ribs in ducts under rotating conditions In the study by Bonhoff, et al (1996, 1997), a Reynolds stressequations model (RSM) with wall functions was used In the studies by Stephens, et al (1996b),Stephens & Shih (1997), and Shih, et al (1998, 2000), a low-Reynolds-number SST turbulencemodel was used (i.e., integration is to the wall so that wall functions are not needed).

U-When computing U-ducts with ribs, it is important for the geometry of the ribs to becaptured correctly This is because they exert considerable influence on the flow and heattransfer When wall functions are used, the rib geometry is compromised This is becauseboundary conditions are applied at one grid point or cell away from the boundary, and thelocation of that grid point or cell boundary is typically at a y+ of 30 to 200 Also, existing wallfunctions cannot account for flow physics occurring in the low-Reynolds-number region next toribs such as density variations from temperature gradients, strong pressure gradients,impingement flows, and flow separation

The objective of this study is to use a low-Reynolds number two-equation turbulencemodel that can account for the near-wall effects to investigate the flow and heat transfer in arotating and a non-rotating U-duct with smooth and ribbed walls Wall functions will not beused The focus is to examine the nature of the flow induced by inclined ribs, a 180o bend, androtation and how that flow affects surface heat transfer, especially in the region around the bend.The bend region is of interest because it is generally smooth though there are ribs upstream anddownstream of it Also, the turning of the bend is typically very tight (i.e., the radius ofcurvature for the convex wall is much less than the duct hydraulic diameter) so that there is alarge separated region, which further complicates the flow

DESCRIPTION OF PROBLEM

A schematic diagram of the U-duct investigated is shown in Figs 1 and 2 It has a squarecross section and is made up of two straight ducts and a 180o bend The geometry of the straightducts is the same as that reported by Wagner, et al (1991a,b) The geometry of the bend issomewhat different, and is taken from the current experimental setup at United TechnologiesResearch Center (Wagner & Steuber (1994)) The dimensions of this U-duct are as follows (seeFig 1): The duct hydraulic diameter is Dh = 1.27 cm (0.5 in) The radial position relative to theaxis of rotation is Rr/Dh = 41.85 and Rt/Dh = 56.15 The length of the straight ducts is L/Dh =

14.3 The inner and outer radii of the 180o bend are Ri/Dh = 0.22 and Ro/Dh =1.44.

Two variations of the U-duct were investigated, one with smooth walls and another withribs When ribbed, there are ten ribs in each straight duct, five on the leading wall and five onthe trailing wall, all with the same pitch The ribs on those two walls are staggered relative toeach other with the ribs on the leading wall offset from those on the trailing wall by a half pitch(p) The ribs are located just upstream or downstream of the 180o bend All ribs are inclinedwith respect to the flow at an angle () of 45o The cross section of the rounded ribs (Fig 2) ismade up of three circular arcs of radius R, where R equals 0.0635 cm (0.025 in) so that the ribheight (e) is 0.127 cm (0.05 in) and the rib-height to hydraulic-diameter (e/Dh) is 0.1 The pitch-to-height ratio (p/e) is five (same as the UTRC experiments)

All four walls of the U-duct including rib surfaces are maintained at a constanttemperature of Tw = 344.83K At the duct inlet, the coolant air has a uniform temperature of Ti =300K at the inlet, which gives an inlet coolant-to-wall temperature ratio of 0.87 and an inletdensity ratio of / = 0.13 Unlike the temperature profile, the velocity profile at the inletshould not be uniform because of the extensive flow passages upstream of it Since fullydeveloped velocity profiles do not exist for compressible flows, the velocity profile used is theone at the exit of a non-rotating straight duct of length 150Dh with adiabatic walls and the samecross section and flow conditions as the U-duct studied here The Reynolds and rotationnumbers at the duct inlet are Re = 25,000 and Ro = 0.24, respectively To completely define thisproblem, either the inlet pressure or the inlet Mach number must be specified Here, the inletMach number is specified at Mi = 0.05, which gives rise to a rotational speed of 3132 rpm for Ro

= 0.24. A summary of the cases studied is given in Table 1

PROBLEM FORMULATION

The flow and heat transfer in the U-duct are modeled by the ensemble-averagedconservation equations of mass (continuity), momentum (compressible Navier-Stokes), and totalenergy for a thermally and calorically perfect gas with Sutherland's model for thermalconductivity These equations are written in a coordinate system that rotates with the duct sothat steady-state solutions with respect to the duct can be computed (Steinthorsson, et al (1991)and Prakash & Zerkle (1995)) The continuity equation is

0 z

w y

v x

zy yy

* xy

zx yx xx

*

Pz

Py

P

where

k 3

2 P

v 2 0

y

x

2 2

z y

x

(2c)

The first and second terms on the right-hand-side of Eq (2c) represent the centripetal and theCoriolis force, respectively In Eq (2c), the rotation is about the z-axis (see Fig 1) The energyequation is given by

z v P eˆ y u P eˆ x t

w v

dependence on freestream k and has a limiter to control overshoot in k with adverse pressuregradients so that separation is predicted more accurately.

The k and  transport equations in the SST model are as follows:

k w y

k v x

k u

t

k

k k

y

ky

x

kx

1

k

t k

t k

k y y

k x x

k F

1 2

P z

w y

v x

u

t

2 2

k

2 t

2

2 2

k y y

k x x

k 2

max

where W is vorticity, and  is the normal distance from the solid wall The constants in Eqs (4a)

to (4e) are calculated by the following weighted formula:

In the above equation,  is a constant such as k, , k, and g that is being sought by a weightedaverage between 1 and 2 The 1 and 2 terms corresponding to constant such as k, , k, g1,and g2 are as follows:

g

2 1 1 1

g

2 2 2

by Wilcox (1993) for hydraulically smooth surfaces In that BC,  equals to 3/40, and y is thenormal distance of the first grid point from the wall The first grid point from the wall must bewithin a y+ of unity

Other BCs needed are as follows At the duct entrance, a developed profile is specifiedfor velocity, but the temperature profile is taken to be uniform (see previous section for details).Turbulence quantities (k and ) are specified in a manner that is consistent with the velocityprofile (average turbulent intensity was 5%) Only pressure is extrapolated At the duct exit, anaverage back pressure is imposed but the pressure gradients in the two spanwise directions areextrapolated This is because secondary flows induced by inclined ribs, the bend, andcentripetal/Coriolis forces cause pressure variations in the spanwise directions Density andvelocity are extrapolated

Though only solutions steady with respect to the duct are of interest, initial conditionswere needed because the unsteady form of the conservation equations was used The initialconditions used are the solutions of the steady, one-dimensional, inviscid equations, namely,

0 x

NUMERICAL METHOD OF SOLUTION

Solutions to the governing equations just described were obtained by using a researchcode, called CFL3D (Thomas, et al (1990) and Rumsey & Vatsa (1993)) In this study, theCFL3D code (Version 4.1) was modified so that it can account for steady-state solutions in arotating frame of reference by adding source terms that represent centripetal and Coriolis forces

in the momentum and energy equations (see Eqs (2) and (3)) The modified CFL3D code hasbeen validated for flow in a non-rotating duct with square and rounded ribs (Stephens, et al.(1995a, 1996a)) and flow in a rotating duct with smooth walls (Stephens, et al (1996b) andStephens & Shih (1999))

This code is based on a cell-centered finite-volume method All inviscid terms areapproximated by the second-order accurate flux-difference splitting of Roe (1981) All diffusionterms are approximated conservatively by differencing derivatives at cell faces Since onlysteady-state solutions are of interest, time derivatives are approximated by the Euler implicitformula The system of nonlinear equations that results from the aforementioned approximations

to the space- and time-derivatives are analyzed by using a diagonalized alternating-directionscheme (Pulliam & Chaussee (1981)) with local time-stepping and three-level V-cycle multigrid(Anderson, et al (1988))

The domains of the smooth and ribbed ducts are replaced by H-H structured grids (Fig.3) The number of grid points in the streamwise direction from inlet to outlet is 257 for thesmooth duct and 761 for the ribbed duct Whether smooth or ribbed, the number of grid points inthe cross-stream plane is 65 x 65 The number of grid points and their distribution were obtained

by satisfying a set of rules such as aligning the grid with the flow direction as much as possible,keeping grid aspect ratio near unity in regions with recirculating flow, and having at least 5 gridpoints within a y+ of 5 (see Stephens, et al (1996b)) As a further test, the aforementioned gridsystems were refined by a factor of 25%, first in the streamwise and then in the cross-streamdirections This grid independence study showed the predicted surface heat transfer coefficient

to vary by less than 2%

On the Cray C-90 computer, where all solutions were generated, the memory and CPUtime requirements for each run are 55 megawords (MWs) and 16 hours for the smooth duct and

155 MWs and 40 hours for the ribbed duct The CPU time given is for a converged steady-state

solution (steady in a frame relative to the duct), which typically involved 3,000 iterations.

RESULTS

The results of this study are presented in Figs 4 to 14 Note that the scales in Figs 7 to 9are not same from plot to plot in order to highlight key features in each plot In discussing theseresults, the four walls of the U-duct are referred to as leading, trailing, outer, and inner, whetherthere is rotation or not When there is no rotation, leading and trailing are the same for thesmooth case and nearly the same for the ribbed case, so only results for one wall are given Theinner and outer walls refer to the U-shaped ones at Ri and Ro, respectively (Fig 1)

Nature of Fluid Flow

The nature of the flow in the U-duct is complicated In the following, the complexity ofthis flow is examined in a step-by-step manner, adding one complicating feature at a time

Non-Rotating Smooth Duct For a smooth non-rotating duct (Case C1 in Table 1), Figs.

4 to 6 and Fig 9 show the following In the up-leg part of the U-duct, the velocity profile has amaximum at about the center of the duct cross section The coolant is coolest near the center ofthe duct cross section with thermal boundary layers growing along all four walls of the ductalong the streamwise direction (Fig 6) The expected pairs of vortices in each of the fourcorners of the duct cross section were not predicted because the turbulence model used cannotaccount for anisotropic effects Not resolving these vortices is acceptable since their magnitudesare extremely small when compared to secondary flows induced by rotation, bend, and ribs (seeLin, et al (1998))

At about 1.25Dh to 1Dh upstream of the bend, the flow becomes affected by the bend(Figs 4, 5, 6, & 9) As the flow approaches and enters the bend, it accelerates near the inner wallbut decelerates and reverses next to the outer wall (Fig 5) Near the middle of the bend, the flowseparates on the inner wall (Fig 5) This separated region is largest about the mid x-z plane andsmallest next to the leading/trailing face With this separated region, which reduces the effectivepassage cross-section, the speed of flow near the outer wall is increased markedly

Along the bend, a rather complicated pressure gradient forms with higher pressure next tothe outer wall and lower pressure next to the inner wall (Fig 9) The nature of this gradientdepends on the curvature of the streamlines in the core of the duct where flow speed is highest.The curvature in the streamlines, however, depends on the geometry of the bend and whether or

not there are separation bubbles in the bend since they effectively change the bend geometry.The Dean-type secondary flows created by this pressure gradient can clearly be seen in Fig 4and Fig 6 (C1-P3) Because of the separation bubble in the bend, the pressure gradient and theDean-type secondary flows induce another pair of secondary flows within the separation bubble(Fig 6 C1-P4) The net effect of these secondary flows is to transport cooler fluid near thecenter of the duct cross section towards the outer wall and parts of the leading/trailing faces (Fig.6).

Downstream of the bend, only the secondary flows of the Dean type persist until the ductexit The Dean-type secondary flow coupled with the separation bubble around the bend causedthe maximum in the velocity profile in the down-leg part of the duct to be shifted towards theouter wall

Rotating Smooth Duct The effects of rotation on a smooth U-duct in which thecoolant-to-wall temperature is less than unity (Case C2 in Table 1) can be inferred from Figs 4

to 6 and Fig 9 In the up-leg part of the duct, Fig 4 shows rotation to induce secondary flows.Two symmetric counter-rotating flows are formed by the Coriolis force as early as Dh into theduct With radially outward flow, the rotation orientation is from the trailing face to the leadingface along the outer and inner walls, transporting cooler air from near the center of the duct crosssection to the trailing face first Since the thermal boundary layer starts on the trailing face, gastemperature near that face is lower than that near the leading face (Fig 6)

With higher temperature and hence lower density near the leading face, centrifugalbuoyancy tends to decelerate the flow on the leading face more so than that on the trailing face(Fig 5) With lower velocity and hence thicker boundary layer next to the leading face, theCoriolis-induced secondary flows cause the formation of additional pairs of vortices near thatface (C2 in Fig 4) This was also observed by Iacovides & Launder (1991), Stephens, et al.(1996a), Bonhoff, et al (1996), and Stephens & Shih (1999) Stephens, et al (1996b) showedthat at a rotation number of 0.48 and a density ratio of 0.13, centrifugal buoyancy causes massiveflow separation on the leading face

With rotation, the pressure gradient in the 180o bend changed considerably The pressuregradient induced by rotation in the radial direction is much stronger than the pressure gradientinduced by the bend from streamline curvature Note that when there is rotation, lines ofconstant pressure are nearly flat in the bend (contrast C1 and C2 in Fig 9) With such a pressure