THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E. 47th St., New York, N.Y

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1 TE AMERICAN SOCIETY OF MECANICAL ENGINEERS 345 E. 47th St., New York, N.Y GT-80 The Society shall not be responsible for statements or opinions advanced in papers or discussion at meetings of the Society or of its Divisions or Sections, or printed in its publications. Discussion is printed only if the paper is published in an ASME Journal. Papers are available from ASME for 15 months after the meeting. Printed in U.S.A. Copyright 1993 by ASME A FULL NAVIER-STOKES ANALYSIS OF FLOW AND EAT TRANSFER IN STEADY TWO-DIMENSIONAL TRANSONIC CASCADES Dong yeon Kim, Joon Sik Lee, and Charn-Jung Kim Department of Mechanical Engineering Seoul National University Seoul, Korea Daesung Lee Korea Aerospace Research Institute Daeduck Science Town Daejun, Korea ABSTRACT Fluid flow and heat transfer in a turbine blade row were investigated numerically using the two-dimensional, steadystate Navier-Stokes equations and the energy equation with dissipation. The finite-volume integration approach was employed to discretize the fully elliptic governing equations. A non-staggered grid system in the boundary-fitted coordinates was used and the compressible version of the SIMPLE was employed to solve extra equations. An.0-C-' type grid system was applied owing to its advantages of easily treating the blunt trailing edge and of producing less skewness in the boundary layer region. For an accurate prediction of the heat transfer coefficient at the turbine blade, the first numerical node from the wall was placed at y +-3 so that it was embedded inside the viscous sublayer. The influence of the turbulence was analyzed with a new free-stream turbulence model which accounts for the free-stream turbulence and flow acceleration. Also the laminar-turbulent transition model was improved. Computations were performed for the low solidity Allison C3X turbine cascade. Present results showed good agreement with available experimental data in terms of the surface pressure and the heat transfer coefficient. Especially much improved distribution of the heat transfer coefficient was obtained in the vicinity of the leading and trailing edges. For practical purposes, the aerodynamic performance and the behavior of the heat transfer coefficient were analyzed by varying the inflow angle. INTRODUCTION An optimum design of the gas turbine cooling system requires a good understanding of the heat transfer characteristics in the vicinity of the turbine blade surface. Among a variety of factors that govern the internal flow through the turbine and affect the heat transfer at the blade surface, of considerable importance are the laminar-turbulent transition, the free-stream turbulence, the rapid acceleration/deceleration, the blade curvature, the ratio of the surface temperature to the free-stream temperature, the stagnation flow at the leading edge, and the secondary flow. These tremendously complicated phenomena are, however, should be taken into consideration in actual design of turbine cascades. Only a few investigations have so far included the free-stream turbulence and the transition phenomena. It has been an accepted fact that the free-stream turbulence and the acceleration of the flow substantially influence the transition process in the actual operating conditions of the turbine. It is also realized that a practical prediction of the transition behavior is a prerequisite to understand the heat transfer at the blade and to enhance the aerodynamic efficiency. In analyzing the heat transfer inside the cascade, there are a number of analytical studies that consider the laminar-turbulent transition. Wang et al. (1985) focused on the stator and carried out numerical analysis based on the boundary layer equation. They adopted the low-reynolds k-e model of Jones- Launder for the turbulence model and overcame the difficulty encountered in an earlier study of ylton et al. (1983) by properly modifying the initial conditions and the boundary layer edge conditions. Rodi (1985) also analyzed the heat transfer during the transition process by adopting boundary layer equation and the low-reynolds k-e model of Lam- Bremhorst. Meanwhile, the Navier-Stokes equations instead of the simplified boundary layer equation was first solved by Weinberg et al. (1986) who used an 0-type grid around the stator and the mixing-length model. The heat transfer in the rotor and stator is investigated by ah (1989) who used the low-reynolds k-e model of Chien. Schmidt et al. (1991) Presented at the International Gas Turbine and Aeroengine Congress and Exposition Cincinnati, Ohio May 24-27, 1993

2 employ the boundary layer equation and use the low-reynolds k-e model of Lam-Bremhorst and Jones-Launder. They include the onset, length and path of the laminar-turbulent transition by modifying the production term in the kinetic energy equation. Boyle (1991) employs the thin-layer Navier- Stokes equation in the C-type grid with the flow-wise diffusion neglected. owever, the real configuration of the blunt trailing edge of the blade is approximated by a cuspshape. e tested the zero-equation models such as the Baldwin- Lomax model for several cascades with various flow conditions and suggested a free-stream turbulence model which is actually the same as the one by Smith and Kuethe (1966) except a constant factor. is model was successful in predicting the heat transfer coefficient of the various types of stator and rotor. The current trend of the heat transfer research on the gas turbine shows two main streams; one attempts at improving the simple zero-equation model to predict well both the transition behavior and the turbulent heat transfer, and the other is directed to include the transition effect in the multi-equation model. In order to predict the transition behavior, the transition model should properly account for the variation of the flow condition by considering the onset of the transition, its length and path all together. The present study employs the non-staggered method by Peric (1985) and extends it to account for the compressible flow. Numerical analysis is performed without transfiguration of the blade geometry such as the blunt leading edge and the thick trailing edge. This was possible by generating the 0-C- type grid arrangement. In order to resolve the dominance of the free-stream turbulence and the rapid acceleration, a new free-stream turbulence model is developed and an existing transition model is improved. For verification, the results are compared with the experimental data by ylton et al. (1983) for the low solidity Allison C3X cascade. The effects of attack angle on the aerodynamic efficiency and the heat transfer are also investigated. GOVERNING EQUATIONS AND BOUNDARY CONDITIONS Governing Equations Two-dimensional, steady-state, Reynolds-averaged Navier- Stokes equations are expressed with respect to the Cartesian coordinate system in the following tensor form; Continuity equation: Momentum equation: (PUP j)= r _Z du,) dx ; x, 3-7Ck Energy equation: d \ (pu,)=7 dx` 1., p7- {±1 _ piw} a du, &I, {u[,ux 2 auk + cry, ( =-+T3 Si dxk a (2) (3) where p is the density; U, the time-averaged velocity; u, the fluctuation velocity; the mean value of the total enthalpy; and h' the fluctuation of the static enthalpy. In the case of the high-velocity flow, the energy equation is frequently solved in terms of the total enthalpy defined as 1, =h+- V/,' + L/22 ) 2 (4) h=cpt (5) where the specific heat is assumed constant. The density is evaluated from the state equation p=p/(rt) (6) with the assumption of the ideal gas behavior. The closure condition is derived from the isotropic eddy viscosity. Then, the turbulent shear stress and the turbulent heat flux are expressed as --7[dU, au 2 6 du, Pu:u1= P ` dx, + dx, 3 '' dx k _ (7) pd,[ h pu,v= Pc dx, (8) The value of the turbulent Prandtl number is assumed here to be a constant of 0.9. The key role of the turbulence model can be critically assessed from the appropriateness of the kt, value. The thermophysical properties are determined from the standard gas table by Kreith (1990). (1) Turbulence Model The prediction of the heat transfer strongly relies on the prescription of the effective viscosity pa, and the effective Prandtl number Pr at.: = p +11, = P + YTJLTh (9) 2

3 Transition Model. The laminar-turbulent transition model 1+ P eff used in the analysis is as follows. Pro err = = (10) (K/c ) 1 _L X 1 (a) Transition Origin Model.When the free-stream P eff - Pr 1./ Pr, turbulence and the pressure gradient are taken into consideration, the onset of the transition can be determined from the model of Dunham (1972): Specification of the Free-Stream Turbulence Model. Re, The free-stream turbulence model used in this study is = [( exp(-80Tu))1[ (1 E)-1 (14) developed to predict the influence of the free-stream turbulence by utilizing the concepts of the two-layer algebraic eddy where Re, is the momentum Reynolds number evaluated at the viscosity and the mixing length. Following the suggestions by transition origin. The expression for E is known as Spalding (1969), a dimensional analysis yields E =min(21a 100Tu, 0.75) (15) p, =p (length scale). (velocity scale) (11) The favorable pressure gradient of a strong free-stream entails the laminar flow near the stagnation point of the cylinder even though the main free-stream is of a high turbulence. The effect of the free-stream turbulence is to augment the laminar heat transfer. The conventional free-stream turbulence models (for the case of the crossflow around a cylinder) has been used to predict the laminar heat transfer near the stagnation point. But in this study no particular treatment is considered for the laminar region near the leading edge and instead the model for the turbulent region is applied. We recommend the following free-stream turbulence model pn = p -C, D- I.Tu U e (12) where I is the length scale; Tu U, the velocity scale; Tu the intensity of the inlet turbulence; Ue the edge velocity of the boundary layer parallel to the wall. Also the mixing-length is defined here as / = min(try, 28*) (13) where A = (9 2 v)(du, / ds) and 0 is determined by using the Twaite's method (White,1974): 02(s ) _ 0.45v ru:ds W(s) 0 (16) In the above, s, designates the arc length. The origin is taken to be the geometric stagnation point. (b) Transition Length Model. The total length over which the laminar-turbulence transition persists is determined from the Dhawan and Narasimha (1958) model: = Re.+ Re, (17) Re = C Re 8 (18) where Re, means the Reynolds number based on the arc length. The subscripts t, 1, and e stand for the onset of the transition, the transition length and the termination of the transition, respectively. For 0 99% transition length, C = (c) Transition Path Model. The transition from laminar to turbulent is simulated by using the intermittency factor yn, (Abu- Ghannam and Shaw, 1980): By designating y to be the normal distance from the wall, the damping function D becomes D =1 exp( y'l K = 0.41, A. = 86, C1 = 1 and A' =26. 0 laminar zone T. In the outer = region, we 0 decided to < use the displacement YT. <1 transition zone (19) thickness 6 instead of the widely used boundary layer 1 turbulent zone thickness 8 in the mixing length model. The rationale for the choice of lies in that with a relatively coarse grid system the value of 6 can be more reliably evaluated than that of 8 (note that the number of grid points over the length of the boundary YTS = 1 (20) layer thickness is limited when the entire region between two Re, Re where 03 =( blades is to be analyzed as in our study). Re. Re. The free-stream turbulence model described as above is The exponent m is adjusted in this study such that developed for our specific use in predicting the fluid flow and {0.75 heat transfer in the gas turbine and thus should not be Tu <7% rn= interpreted as a universal one. Since the local pressure near the 1.5 Tu>7% (21) surface is a function of the local curvature, the present turbulence model indirectly accounts for the effect of the local The reason for the above modification is that when the curvature via the edge velocity Il e which depends on the local originally suggested value of m = 3 was used, the predicted heat transfer coefficient severely underestimated the C3X pressure. experimental data. 3

4 OMIIIIMI = M1 11 i LEADING EDGE TRAILING EDGE FIG.1 0-C- TYPE GRID SYSTEM FOR C3X CASCADE, 83X57 GRID. Coordinate Transformation The success of numerical analysis is heavily dependent on the choice of the coordinate system. As such, the boundaryfitted coordinate is employed to handle the complex geometry. In a general curvilinear coordinate (,7-h, governing equations (1) (3) are transformed in the following general form: id( idr o_id[rmar WO it-7777w G 2 I J q2o77)] + 1 d [r q20c1-q30,7)]+ (22) J J where G is the quantity associated with the contravariant velocity components such that G, U,x, (23) G2 =LI2x 4 U,y4 (24) and 0 = U1, U2, ]; q,= + yn2 ; q2 =xx,i + y4 yri ; q3 = x +4; J = ; and S(,t1) is the source term. The metric coefficients 4., 4, rl, are determined from the mapping and satisfy the following relations: Jy n, 4 y = rh= n y = fx, (25) Boundary Conditions (a) Inlet Boundary. At the inlet boundary, the value of the total pressure Pn, the total temperature Tr1 and the flow incident angle a are specified from the available experimental data. The static pressure is extrapolated from the pressure field in the internal region. The velocities are corrected in an implicit manner from the corrected pressure obtained through the isentropic relation. With the velocity and pressure values TABLE 1 C3X CASCADE GEOMETRY (YLTON ET AL., 1983). Blade setting angle Throat mm Vane spacing mm True chord mm Axial chord mm Air inlet angle 0 Air exit angle determined as above, the temperature and the density are evaluated from equations (4) and (6). (b) Wall Boundary. The surface of the turbine blade is assumed impermeable and no-slip conditions are applied to it. The boundary conditions for the pressure field are of the Neumann type, and those for the temperature field are of the Dirichlet type (measured variable wall temperature). (c) Outlet Boundary. The exit flow is established by specifying the measured constant static pressure. The velocity can be derived from the pressure gradient, and for all other variables including the total enthalpy, the asymptotic boundary condition (d 20/4 2 )= 0 is applied. (d) Periodic Boundary. For the treatment of the periodic line, a subsidiary control volume is deployed along the upper and the lower periodic lines so that flow variables assume the same values at the periodic grid point and the identical mass flow rate is imposed along the periodic line. Grid Generation and Solution Procedure Grid system is generated by highly utilizing the geometric configuration of the flow passage. The currently-used turbine blade can be characterized by the rounded leading edge, the thick trailing edge and the large staggered angle. To deploy the meshes in an optimum fashion, grid generation is made by 4

5 TABLE 2 C3X CASCADE TEST CONDITIONS (YLTON ET AL., 1983). Run PTI(kpa) TT!(K) M1 Re, x10-6m 2Re, x10-6 Tu(%) (K) solving the Poisson equation. Figure 1 illustrates the 83 x 57 grid arrangement for the flow passage. No particular treatment was attempted for the control volumes at the leading/trailing edges of the blade. The 0-C- type grid (Stow, 1989) was modified such that it produces a small degree of skewness along the surface and maintains an excellent orthogonality near the blade surface; this is the main reason why the 0-C- type grid was preferred here to the usual -type grid. It has an additional merit in treating the transonic viscous flow inside turbine blade cascade with a large turning angle. A drawback of using a nonorthogonal coordinate system such as used here is that it produces unrealistic values for the heat transfer coefficient when the grid skewness is large near the regions having steep gradients of flow variables, e.g. near the blade surface, near the leading and trailing edges. Therefore, there is practical limitation to obtain grid-independent solution by increasing the number of grid points. In fact, we repeated the computation with a denser 166 x 84 grid system and observed that the local heat transfer coefficient shows an upper or lower peak at both the leading and trailing edges due to a high degree of skewness (This kind of peak was also reported in other numerical studies for turbine cascades.). owever, such an extreme variation is confined to a small region only and fortunately grid-independent solution can be observed for most of the regions. The numerical procedure to solve the governing equations is constructed within a frame of non-staggered method by Peric (1985). But in order to treat the transonic flow the method is modified to account for the compressible Navier-Stokes equation and the dissipation in the energy equation. This was done by evaluating the density at the control volume surface from.,e density biasing method (Karki, 1986), and by relating the density and pressure connections Delaney Euler Present Navier-Stokes FIG. 2 BOUNDARY LAYER EDGE VELOCITY DISTRIBUTION FOR C3X CASCADE, RUN # PT/ ylton data (#144) Present Navler-Stokes Kwon Navler-Stokes Delaney Euler RESULTS AND DISCUSSION The free-stream turbulence model developed in this study is FIG. 3(A) BLADE SURFACE STATIC PRESSURE FOR applied to analyze the internal flow and the heat transfer C3X CASCADE, P r1 /P2=1.69. through the gas turbine cascade. The available experimental data by ylton et al. (1983) for low solidity Allison C3X x/cx 5

6 PTI ylton data (#143) Present Navier-Stokes Kwon Navier-Stokes Delaney Euler FIG. 3(B) BLADE SURFACE STATIC PRESSURE FOR C3X CASCADE, Pn/P2=2.01. FIG. 5 VELOCITY VECTOR CONTOUR FOR C3X CASCADE, RUN #144. (A) STATIC PRESSURE (B) MAC NUMBER FIG. 4 CALCULATED STATIC PRESSURE AND MAC NUMBER CONTOURS FOR C3X CASCADE, RUN #144. turbine cascade are compared with our results to validate the proposed model. Their experimental data have served as the "bench-mark" data in many other works (Wang, 1985; Weinberg, 1986; Kwon, 1988; Boyle, 1991). The geometric configuration of the cascade used in their experiments is shown in Table 1. The flow has an incident angle of 0 and exits at an angle of Table 2 summarizes the boundary conditions used in the analysis where Tv denotes the mean wall temperature. For computation, the inlet is placed at axial chord upstream from the leading edge and the uniform profiles are prescribed for the flow variables. The exit is placed at downstream from the trailing edge and the exit static pressure is specified by assuming the fully-developed flow. It is found that, when the first node from the wall is placed at y+ 3 (i.e., inside the viscous sublayer), the present free-stream turbulence model well predicts the heat transfer coefficient. The suction surface is treated with the laminar-turbulent transition model and the pressure surface with the fully turbulent model. The boundary layer edge is determined as follows. The normalized vorticity is first defined such that w = (du,, / dr) (Li, / U, where L p is the pitch length. Next, check how many times co changes its sign from the wall to the midpoint between the suction and pressure surfaces. If co changes its sign only once, find a point y, at which co 100. Otherwise, find a point y, at which o.) changes its sign the second time. Then starting from the above-determined point y, to the wall, find a point y2 where 1(01 coc.. Finally, search for the maximum velocity (U, ) m,, over a range from the wall to y 2. The boundary edge is identified to be a point y 3 where U,, = In this study, we used coca, =1. Figure 2 displays the boundary edge velocity along the arc length for both the pressure and suction surfaces in the case of Run #145. For comparison, the results from the Euler equation of Delaney (see ylton et al.) are also shown in the figure. A 6

7 o ylton data Present Navier-Stokes =1135 W/m 2 /K Tu=6.5% o -0.5 suction side 0.5 pressure side FIG. 6 EAT TRANSFER COEFFICIENT FOR C3X CASCADE, Tu=6.5%, RUN #144, RUN #143. o ylton data =1135 W/m2/K Present Navier-Stokes 0 Tu=8.3% suction side 0.5 pressure side FIG. 7 EAT TRANSFER COEFFICIENT FOR C3X CASCADE, Tu=8.3% RUN #159, RUN #111. satisfactory agreement can be recognized for the pressure side, whereas for the suction side the present study overestimates the data up to 30% of the true chord and underestimates over the 30-50% interval. Note that the present results show an earlier acceleration in the front portion of the suction surface. Figure 3(a) shows the distribution of the surface static pressure and compares with other works. The abscissa is the distance along the axial direction starting from the geometric stagnation point. The results agree to a satisfactory extent with the data, especially for the pressure side. For the suction side, the present results clearly reveal the rapid acceleration behavior appearing around 50% axial chord and the subsequent deceleration. Note that such a clarity is not so distinguishable in other works as in our study. Figure 3(a) also shows that the pressure gradient from this study is the largest up to 30% axial chord of the suction side, and that the results from the Euler equation (Delaney) underestimate for 30-50% axial chord of the suction side. The constant pressure lines and the iso-mach lines in the internal passage of the turbine are plotted in Fig.4 for the case of Run #144. It can be seen that the flow rapidly accelerates toward the geometry throat, as indicated by an increase in the 7

8 ylton data Present Navier-Stokes Wang boundary layer o FIG. 8 EAT TRANSFER COEFFICIENT FOR C3X CASCADE, Tu=6.5%, RUN #145, RUN #149. I o ylton data Present Navier-Stokes =1135 W/m 2 /K Wang boundary layer o Tu=8.3% e I o 0 - FIG. 9 EAT TRANSFER COEFFICIENT FOR C3X CASCADE, Tu=8.3% RUN #157, RUN #159. pressure gradient. owever, the pressure gradient begins to decrease downstream away the throat. The maximum Mach number of unity is observed and the choking occurs at the geometric throat. On the contrary, the Navier-Stokes solutions by Kwon predicted maximum Mach number which is lower than our case. Figure 5 presents the velocity vector distribution near the trailing edge. It shows two asymmetric cells resulting from the secondary motion which is due to a larger velocity on the pressure side than that on the suction side. Figure 3(b) shows the distribution of the surface static pressure for Run #143. The major difference between Run #143 and Run #144 is the cascade exit Mach number. The exit Mach numbers are 5 for Run #143 and 0.9 for Run #144 (see Table 2). As shown there, the predicted static pressure at the wall agrees fairly well with the experimental data. The results by Delaney show dual acceleration and deceleration around 50% axial chord in view of the pressure gradient. The results by Kwon weakly generate the procedure of acceleration/deceleration. owever, the present Navier-Stokesequation-based results give rise to a quite slow deceleration and an obviously remarkable rapid-acceleration. The 8

9 0.15 Tu=6.5% 0 Present prediction Run #145 Run #149 suction pressure FIG. 10 WALL SEAR DISTRIBUTION FOR C3X CASCADE, Tu=6.5%. P 'Ti Tu=6.5% angle increase attack angle = O.% xic x 0 attack angle = attack angle = -1 0 ylton data (0 ) FIG. 11 EFFECT OF ATTACK ANGLE ON BLADE SURFACE STATIC PRESSURE DISTRIBUTION, RUN #107. maximum Mach number observed throughout the turbine passage is approximately 1.24 which is a little larger than 1.1 observed by Kwon. This fact implies that the flow is at a transonic stage. Comparison of the external heat transfer coefficient with the experimental data by ylton (4 cases are selected, see Table 2) is displayed in Fig. 6 for the turbulence intensity of 6.5% and in Fig. 7 for 8.3%. Excellent agreement is evident in the figure. The heat transfer coefficient at the suction side increases with an S-shaped curve during the laminar-turbulent transition. The present results accurately identify the starting point of the transition and predict well the laminar region that has heat transfer coefficients lower than those in the fully turbulent region. The heat transfer coefficient at the pressure surface increases along the arc length starting from the geometric stagnation point to the flow stagnation point, and then rapidly decreases as one moves further toward the concave surface. Due to the acceleration of the flow, the heat transfer coefficient gradually increases downstream at the pressure surface. Figures 8 and 9 compare the experimental data with the present results and those of Wang et al. in terms of the heat transfer coefficient. The turbulence intensities considered are 6.5% and 8.3%, respectively. This study produces better agreement with experimental results both in the laminar region close to the leading edge on the suction surface and in the transition region. The decreasing trend of the heat transfer coefficient in the downstream of the suction surface is manifested in our study better than in Wang et al. Meanwhile, the present study generally overestimated the heat transfer coefficient at the "pressure" surface in the downstream region. Figure 10 depicts the distribution of the wall skin friction coefficient along the arc length. Only the predicted results are shown because the experimental data are not available. Although two cases of Run #145 and Run #149 are very distinctive of the flow conditions, differences in the wall skin friction coefficients between them are relatively insignificant compared with the variation encountered in the heat transfer coefficients shown in Fig. 8. This reflects the fact that the changes in the inlet and outlet conditions, i.e. the variation of Reynolds number, hardly affect the friction coefficient. 9

10 attack angle = 0 attack angle = +1 0 attack angle = -1 0 o =1135 W/m 2 /K Tu=6.5% ylton data (0 ) FIG. 12 EFFECT OF ATTACK ANGLE ON EAT TRANSFER DISTRIBUTION, RUN #107. One of the key features of the present work is that a parametric study is carried out by varying the flow attack angle. The results cover the off-design condition ranging from -10 deg to +10 deg. For the flow condition corresponding to Run #107, Fig.11 shows the static pressure at the blade surface in response to variation of the attack angle. It can be seen that with an increasing attack angle the pressure gradient remains nearly unchanged in front of the suction surface but the acceleration point of the flow migrates toward the leading edge. owever, on the pressure surface the static pressure distribution is very insensitive to the variation of the attack angle. For the same case as in Fig.11, Fig.12 shows the response of the heat transfer coefficient. The suction surface undergoes a mild change during the transition but does no change within the fully turbulent region. In the laminar region, the heat transfer coefficient is shifting downward with an increasing attack angle. In the transition region, it rather tends to shift upward at an angle of +10 deg but remains unchanged for the angle range of -10 deg 0 deg. The heat transfer coefficient at the pressure surface, from the beginning point of the concavity to the 50% true chord, shows higher values in both the cases of ±10 deg than the case of zero angle. Subsequently, its trend becomes reversed although the magnitude of the changes is negligible. CONCLUDING REMARKS A numerical study is made of two-dimensional, steady-state, compressible transonic flow through the passage of the gas turbine. A striking feature of the present approach lies in that the fully-elliptic Navier-Stokes equation is solved unlike more simplified set of equations used in other works. The limitation of the method by Peric to the incompressible flow is liberated in this study by extending it to account for the compressible flow and the energy dissipation term. In addition, we proposed the free-stream turbulence model mostly suitable for the problem considered here and found good agreement with the available experimental data. Also, the transition model on use is modified. These improvements made here were valid especially for the transonic flow through cascade in which the effects of the free-stream turbulence and the rapid acceleration are predominant. A tip for further accurate simulation of the heat transfer phenomena in a thin boundary layer was to place the first node from the wall at a position of y The 0-C- type grid system used here produced less skewness near the wall and was able to directly working with the thick trailing edge without simplification of the complex geometry. Additional key feature of the results is that the aerodynamic efficiency and the heat transfer characteristics in response to variation of the attack angle can be investigated even under the off-design condition. In general, the predictions agree favorably with the available experimental data better than existing analyses. ACKNOWLEDGEMENT The authors are grateful for the support provided by a grant from The Korea Science & Engineering Foundation and Turbo and Power Machinery Research Center. REFERENCES Abu-Ghannam, B. J., and Shaw, R., 1980, "Natural Transition of Boundary Layers-The Effects of Turbulence,

11 Pressure Gradients, and Flow istory," J. Mech. Engr. Science, Vol. 22, No. 5, pp Boyle, R. J., 1991, "Navier-Stokes Analysis of Turbine Blade eat Transfer," ASME Journal of Turbomachinery, Vol. 113, pp Dhawan, S., and Narasimha, R., 1958, "Some Properties of Boundary Layer Flow During Transition From Laminar to Turbulent Motion," J. Fluid Mechanics, Vol. 3, pp Dunham, J., 1972, "Predictions of Boundary Layer Transition on Turbomachinery Blades," AGARD-AG-164. Frank Kreith, P. E., 1990, eat Transfer and Fluid Flow Data Books, Genium Publishing Co., New York. ah, C., 1989, " Numerical Study of Three-Dimensional Flow and eat Transfer Near the Endwall of a Turbine Blade Row," AIAA Paper No ylton, L. D., Mihelc, M. S., Turner, E. R., Nealy, D. A., and York, R. E., 1983, "Analytical and Experimental Evaluation of the eat Transfer Distribution Over the Surfaces of Turbine Vanes," NASA CR Karki, K. C., 1986, ''A Calculation Procedure for Viscous Flows at All Speeds in Complex Geometries," Ph.D. Thesis, University of Minnesota, Minneapolis. Kwon, 0. K., 1988, "Navier-Stokes Solution for Steady Two- Dimensional Transonic Cascade Flows," ASME Journal of Turbomachinery, Vol. 110, pp Peric, M., 1985, "A Finite Volume Method for the Prediction of Three-Dimensional Fluid Flow in Complex Ducts," Ph.D. Thesis, University of London, Imperial College. Rhie, C. -M., Chow, W. L., 1983, "Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation, AIAA J., Vol. 21, No. 11, pp Rodi, W., Scheuerer, G., 1985, "Calculation of eat Transfer to Convection-Cooled Gas Turbine Blades," ASME Journal of Engineering for Gas Turbines and Power, Vol. 107, pp Schmidt, R. C., Patankar, S. V., 1991, "Simulating Boundary Layer Transition With Low-Reynolds Number k - e Turbulence Models: Part2- An Approach to Improving the Predictions," ASME Journal of Turbomachinery, Vol. 113, pp Smith, M. C., and Kuethe, A. M., 1966, "Effects of Turbulence on Laminar Skin Friction and eat transfer, " Physics of Fluids, Vol. 9, pp Spalding, D. B., 1969, "Applications of Boundary Layer Theory," Imperial College Mechanical Engineering Department Report BL/TN/A/8. Stow, P., 1989, "The Development of Advanced Computational Methods for Turbomachinery Blade Design," International Journal for Numerical Methods in Fluids, Vol. 9, pp Wang, J.., Jen, h. F., artel, E. 0., 1985, "Airfoil eat Transfer Using Low Reynolds Number Version of a Two Equation Turbulence Model," ASME Journal of Engineering for Gas Turbines and Power, Vol. 107, No. 1, pp Weinberg, B. C., Yang, R. J., McDonald,., and Shamroth, S. J., 1986, "Calculations of Two- and Three-Dimensional Transonic Cascade Flow Fields Using the Navier-Stokes Equations," ASME Journal of Engineering for Gas Turbines and Power, Vol. 108, pp White, F.M., 1974, Viscous Fluid Flow, McGraw-ill, New York. 11