Numerical Simulation of Film Cooling With Auxiliary Holes

Transcription

1 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition January 2012, Nashville, Tennessee AIAA Numerical Simulation of Film Cooling With Auxiliary Holes Leandro M. Fernandes, Rômulo B. Freitas and Marcio T. Mendonca Instituto Tecnológico de Aeronáutica, São José dos Campos, SP, , Brazil Ricardo B. Flatschart and Aluísio V. Pantaleão Vale Soluções em Energia, São José dos Campos, SP, , Brazil The present study investigates the performance of film cooling in gas turbine applications when two cooling jets are used. The cooling air issuing from one hole interferes with the jet from a second hole and the resulting flow field at first may either degrade cooling performance or improve it. Care must be taken to ensure that the typical counter rotating vortices associated with jets into a main stream do not interact to uplift the cooling film. Preliminary results published else where for normal jets into a cross stream are presented here in order to illustrate the main results of jet interaction. The present investigation extends those preliminary results for jets injected at an angle of 30 degree, typical of gas turbine cooling applications. C p specific heat at constant pressure e specific internal energy k turbulent kinetic energy I turbulent intensity ṁ mass flow p pressure, N/m 2 p 0 total pressure, N/m 2 P r Prandtl number Re Reynolds number T 0 total temperature, K u velocity, m/s 2 x spatial coordinate, m t time, s δ ij Kronecker delta ɛ turbulent energy dissipation rate γ specific heat ratio µ dynamic viscosity, Ns/m 2 ρ density, kg/m 3 τ viscous tensor Subscript t turbulent variable 0 total (stagnation) state Nomenclature Grad. Student, Department of Mechanical Engineering, ITA. Grad. Student, Department of Mechanical Engineering, ITA. Research Engineer, Department of Mechanical Engineering, ITA. AIAA Senior Member CFD Research Eng., Vale Soluções em Energia. CFD Research Eng., Vale Soluções em Energia. 1 of 15 Copyright 2012 by the, Inc. All rights reserved.

2 I. Introduction In gas turbines higher specific power and lower specific fuel consumption can be achieved increasing the turbine entry temperature (TET). Higher TET may result in significant savings in fuel which translates into savings of thousands of Dollars per year. That justifies the investment in the development of new high temperature resistant alloys and improved cooling systems. through the years the TET has increased from around 1200 K in the early 60s to as high as 1850 K nowadays. The first cooling methods were based on convective cooling, but the limitations associated with relative low values of convective cooling coefficients and high blade surface temperature result in TET limitation of around 1600 K. A much more efficient technique would result if the blade surface is isolated from the hot gases by a cooling film, which is ejected over the blade through film cooling holes. Film cooling is a complex technique that depends on the interaction between the main hot stream and the cooling jet. A cooling effectiveness parameter η is used as the measure of how much the cooling flow is mixed with the main stream and varies from η = 1 close to the cooling hole and η = 0 downstream. The effectiveness depends on a large number of parameters, most of theme associated with the flow topology of a jet into a cross flow. A jet issuing from a wall into a main stream is bent in the direction of the flow and develop into a pair of counter rotating vortices that pump low temperature fluid away from the wall and high temperature fluid close to the wall. Most of the research in cooling configurations are associated with ways to reduce the development of these counter rotating vortices such as trenches, shaped holes, compound angle holes and secondary antivortex holes. The present investigation build on previous research on film cooling effectiveness. A system consisting of two holes place in line with the main stream is investigated in order to access the effect of their interaction on the temperature distribution downstream. The development of counter rotating vortices in the upstream hole changes the flow field due to the downstream hole, which may have an adverse effect on the film cooling effectiveness. The present investigation builds on these previous results by considering injection at 30 degree relative to the main stream, which is usually associated with the most effective injection angle for film cooling. The effect of blowing ratio and mass flow split between rows on the cooling effectiveness and cooling mass flow are evaluated numerically. II. Bibliographic Review The literature on gas turbine film cooling is vast and would take much more room to discuss than available for the present article. The bibliographic listing up to 1996 alone takes 10 pages. 1 Only the studies relevant to the present investigation are presented in this introduction due to space limitations. Initially the research on turbine blade film cooling addressed issues such as mass flow ratio, density ratio, temperature ratio, etc. In search of higher gain in cooling effectiveness, nowadays the investigations on film cooling are addressing the details of flow topology which recognizes the importance of the counter-rotating vortices resulting from the interaction of the cooling jets and the main stream. Detailed measurements taken by Bernsdorf 2 show the development of the vortical structures associated with the cooling jet. Latter the results presented by Bernsdorf 2 were used to calibrate a computational model based on the RANS equations. 3 Based on the investigations on the detailed flow topology new propositions are presented in the literature aiming at reducing the development of streamwise counter-rotating vortices. These investigations consider shaped holes and compound angle holes. An alternative for that end is to use anti-vortex holes 4 which are easier to manufacture. Recent experimental results regarding the anti-vortex hole positioning and orientation have been presented by Dhungel. 5 They concluded that the anti-vortex hole position has a significant impact on the cooling performance. A Detail numerical analysis was recently performed by Renze 6 addressing the issue of cooling jet topology in rows of holes using large eddy simulation (LSE). The results show that the mixing a the downstream rows have a different nature. The velocity gradients are lower downstream of the first row and the turbulence production is reduced. As a result the cooling effectiveness is significantly improved. The superior performance of LES in capturing the flow physics show strong variations in the turbulent Prandtl number. Recently Elgabry 7 discussed the penetration characteristics of film cooling jets at high blowing ratio using numerical methods. They compared the results with experimental data and discussed the shortcomings of numerical simulations for film cooling analysis. As other numerical simulations the cooling effectiveness is unpredicted due to the complexity of the problem of a jet into a cross flow stream. The main difficulty is the 2 of 15

3 jet wake region, which is not well captured by the Reynolds average Navier-Stokes formulation with eddy vorticity closing. Sakai 8 studied the reliability of detached eddy simulation (DES) and RANS turbulence models for the prediction of film cooling effectiveness. They compared four different turbulent models and showed adequate agreement for the laterally averaged film cooling effectiveness. The RANS model underpredicted the effectiveness lateral spreading. The present investigation extends the results presented recently by Fernandes et. al. 9 They used a model configuration to study the interaction between rows of holes when the cooling fluid is injected normal to the main stream direction. Preliminary conclusions on film cooling effectiveness are presented and for the initial investigated configuration a negative effect on the cooling effectiveness is observed. The upstream hole vorticity uplift the downstream jet away from the wall resulting in higher wall temperatures. For small mass flow on the upstream hole this effect is not strong due to the dominance of the downstream jet vorticity which traps the upstream jet close to the wall. Fernandes et. al note that another benefit of the small mass flow upstream hole is the better cooling in the region between holes. In the present investigation instead of injecting cooling flow normal to the main stream direction, as in Fernandes et. al, 9 the cooling flow is injected at an angle of 30 degree, which is more typical of blade cooling applications. As in Fernandes et. al, different mass flow split between the two holes is investigated in order to access which ratio gives the best cooling performance. The resulting flow field is studied in search for ways to reduce the strength of the counter rotating streamwise vortices. III. Methodology The research presented in this paper was performed using OpenFoam, an open source C++ library for computational fluid dynamics. The compressible Navier-Stokes equations are solved numerically based on a finite volume scheme. The turbulent stress terms were closed using a two equation turbulence model based on the Boussinesq turbulent viscosity hypothesis. III.A. Mean flow equations The compressible Navier-Stokes equations are decomposed using the Favre decomposition. φ φ + φ, φ ρφ ρ, such that ρφ = 0; ρ φ = ρ φ = ρφ (1) where the overbar (e.g. ρφ) represents the Reynolds decomposition temporal average and ρ is the density. The Favre decomposition is introduced in the governing equations and a temporal average is taken. For the pressure, internal energy and density the Reynolds decomposition is applied. The resulting equations are ρ t + [ρũ j ] = 0, (2) x j t (ρũ i) + [ ] tot ρũ i ũ j + pδ ij τ ij = 0, (3) x j t (ρẽ 0) + [ ] ρũ j ẽ 0 + ũ j p + q tot tot j ũ i τ ij = 0. (4) x j where t is the time, u the velocity, x the spatial coordinate, p is the pressure and δ ij the Kronecker delta. The turbulent and laminar stresses are ( τ tot ij = τ lam ij + τ turb ij, τ lam ũi ij = µ + ũ j 2 ) ũ k δ ij, (5) x j x i 3 x k ( τ turb ij = ρu i u j = µ ũi t + ũ j 2 x j x i 3 ũ k δ ij x k ) 2 3 ρkδ ij. (6) k = 1/2u i u i is the turbulent kinetic energy, and µ t is the turbulent viscosity, which is evaluated with a turbulence model. 3 of 15

4 The laminar and turbulent heat fluxes are The average perfect gas relation results q lam µ j = C T p = γ µ Pr x j 1 γ Pr x j µ t q turb j = C p ρu j T = C T p = γ µ t Pr t x j 1 γ Pr t x j p = (γ 1) ρ ( ) p, (7) ρ ( ) p, (8) ρ ( ẽ 0 1 ) 2ũkũ k k. (9) where µ is the dynamic viscosity λ is the thermal conductivity, T is the static temperature, C p is the specific heat at constant pressure and Pr = C p µ/λ is the laminar Prandtl number, R is the gas constant, γ is the specific heat ratio, C v is the specific heat at constant volume, and e is the internal energy. The total energy is e 0 = e + 1/2u k u k. γ, C p, C v e R are assumed constant. III.B. Turbulence modeling For the Reynolds stress term the Boussinesq hypothesis is applied. The SST k ω turbulence model is used. 10 This model uses the k ω formulation in the inner boundary layer, which integrates up to the viscous sub-layer region, and switches to a k ɛ turbulence model away from the wall. The model is know to give reasonable results in adverse pressure gradient and separating flows. This model gives somewhat large values of turbulence close to stagnation regions, but this effect is less pronounced than with a k ɛ model. The k ω model implemented on OpenFoam have been used for turbomachinery heat transfer problems and showed good performance. 11 ρu i u j = 2µ ts ij 2 3 δ ij where k is the turbulent kinetic energy, k = u i u i /2 and ( ) u k µ t + ρk, S ij = 1 ( ui + u ) j. (10) x k 2 x j x i ν t = a 1 k max (a 1 ω, SF 2 ) (11) F 2 and a 1 are closure coefficients. The transport equations for k and ω as well as other details of the model can be found in Menter. 10 III.C. Numerical method The governing equations are solved numerically based on a cell centered unstructured finite volume scheme implemented on an open source code (OpenFoam). The computational domain is composed of a rectangular domain with a flat plate at the lower surface where the cooling holes are placed as shown in Fig. 1. The upstream hole is a square hole D D where D = m and the downstream hole has twice the area of the upstream hole with sides 2D. The holes are distant 2D apart and the upstream hole is 5D from the leading edge. The total plate length is 45D The solution methodology is base on a segregated, compressible version, pressure based SIMPLE algorithm and a steady state solver. For pressure and velocity an algebraic multigrid solver with preconditioning is used and for the turbulent quantities and the energy equation a preconditioned, bi-conjugate gradient solver is used. The vector field is interpolated using a Gauss linear schemes or a combination of Gauss linear scheme and first order Gauss upwind scheme. III.D. Boundary conditions The following boundary conditions are imposed: at the top zero pressure gradient and temperature T = T = 1000 K are specified, At the inlet zero pressure gradient, a 1/7 turbulent boundary layer power law for the streamwise velocity component and T = T are imposed. At the outlet zero gradient of all variables is imposed and the pressure is p = p atm = Pa. At solid boundaries on the plate or on the cooling 4 of 15

5 Figure 1. Topology and grid distribution holes no slip condition is imposed for the velocity components, zero gradient for pressure and adiabatic wall for the energy equation. At the cooling holes inlet uniform temperature T = 300 K and mass flow are imposed. In the spanwise direction periodic boundary conditions are enforced. The initial field is given by the free-stream conditions for all variables. The mass flow on the cooling ducts are specified such that the main stream to cooling jet velocity ratio is 0.5, for a freestream velocity of 11 m/s. Initial and boundary values for the turbulence quantities are crucial for the correct simulation of turbulent flows. They are a source of uncertainties and are usually unknown from experimental conditions. Given the turbulent intensity I, a turbulent length scale L and the ratio µ t /µ, the kinetic energy and the dissipation rate ɛ may be calculated: k = 3 2 (ui)2, where I u u, ɛ = C k 3 2 µ L, = C ρk 2 µ µ ( ) 1 µt. (12) µ where C µ = 0.09 is a fitting constant and the length scale is given as a percentage of the problem characteristic length. The turbulent intensity used for the present problem is equal to 1,2%. The characteristic length scale is L = 0.07D. IV. Previous Results From Fernandes et. al 9 In the experiment of Fernandes et. al, the main controlling parameter was the relation between the mass flow in the upstream hole and the downstream hole. Four configurations were studied, the first for a single cooling hole, the second for 30% of the total mass flow of the first configuration coming out of the upstream hole, the third for 10% and the last for 5%. The single hole case serves as a reference condition. For a jet coming out of a cooling hole normal to the main stream direction the jet is bent in the direction of the main flow and the interaction between the two streams results in the development of counter-rotating vortices with an upwash region located in the center line of the plate. The flow structure is shown in figure 2 where plate temperature and streamlines are presented. Figure 2. Temperature distribution and streamlines. Flow in a single hole 5 of 15

6 When 30% of the cooling flow comes out of the upstream hole one observes that the counter rotating vortices generated by it uplift the jet coming out of the downstream hole (Fig. 3). The resulting effect is that higher temperatures are observe over the entire plate and a stronger heating results in the region about 2D of the downstream hole and closer to the sides of the plate. The downwash effect of the downstream vortices is reinforced by the upstream vortices. The downstream jet is located exactly inside the upwash region of the first jet. This flow topology is significant when one considers rows of cooling holes located close to each other. Nether the near hole region is better cooled, nor the downstream region for the configuration tested. The streamlines resulting from both jets move away from the wall as a result of the upwash at the center line. Figure 3. Temperature distribution and streamlines. 30% mass flow in the upstream hole. The initial idea for the investigation was to have the upstream hole serve as a barrier for the downstream hole. The upstream jet would deflect the main flow from the downstream jet. The intended result would be to reduce the counter rotating vorticity of the downstream hole and enhance the plate cooling. In order to test this idea the third test case considers a weaker upstream jet in order to avoid the effect observed in the previous case where the upstream hole uplifted the main downstream cooling jet. The results presented in Fig. 4 show that a weaker jet upstream mixes with the downstream jet without lifting it up. A better cooling is observed in the plate between the two jets and significant cooling is also observed downstream. Nevertheless the waist line temperature about 2D from the downstream jet is stronger than that observed for the single jet. Figure 4. Temperature distribution and streamlines. 10% mass flow in the upstream hole. The streamlines presented in Figs. 3 and 4 show flow separation close to the jets. In order to compare the strength of this flow separation for the different configurations the following figure (Fig. 5) show the velocity magnitude on a (x, y) plane along the center line. Flow separation is observed for all configurations upstream of the main cooling hole and the least severe separation is that of the single hole configuration. The presence of the upstream hole results in a stronger region of separated flow downstream of the main cooling jet. The separation region is due to the blockage effect of the jet. 6 of 15

7 Figure 5. Velocity magnitude in the center line plane. V. The Present Investigation The summary of Fernandes et. al 9 results presented above leads to two main conclusions. One regarding the interaction between the longitudinal vortices produced by the two jets, which interact with each other and uplift the cooling flow away from the wall. The other regarding flow separation behind the cooling jets. Both effects are expected to be weaker if the jets are directed somewhat tangent to the main stream. The usual angle between the main stream and the jet in gas turbine applications is 30 degrees. The grid for this new configuration is presented in Fig. 6. To ensure that the results are grid independent four grids were tested. The temperature distribution at the centerline for these four grids is presented in Fig. 7. All grids show very similar results and the one with 288 thousand cells were used for all the following computations. This grid has an average y + = 1.76, consistent with the SST turbulent model used. The simulation was run until the pressure residue was of the order of The residue for the other dependent variables were lower than the pressure residue. Figure 6. Surface grid and topology for the inclined holes. Figure 7. Grid refinement study. As with the 90 o injection, five different conditions were tested, all with holes positioned at 30 o. One condition considering a single hole (the second hole), and four conditions considering the same total mass flow but with 5%, 10%, 30% and 70% of the total mass flow coming out of the first hole. V.A. Flow field and Temperature Distribution For the chosen test cases the resulting plate temperature distributions along the center line are presented in Fig. 8. One can see that there is no significant improvement in cooling when compared to the result from a jet coming out of a single hole. The case with 30% mass flow out of the upstream hole has a gain of about 30 K close to the exit plane, but a worse performance than the single hole in the region near the downstream 7 of 15

8 hole up to 30D. The case with 70% mass flow out of the upstream hole has a poor performance due to the strong separation associated with the large jet velocity upstream as will become clear later on. On the other hand, on the side plane the 30% case show a significant improvement, showing that the upstream hole combined with the main downstream hole result in a better lateral spreading of the cooling fluid. Figure 8. Temperature distribution along the centerline. Figure 9. Temperature distribution along the side. The characteristic mushroom structure resulting from the counter rotating vortices due to the interaction of the jet with the main flow is shown in Figs 10 and 11, at the streamwise position 13D. At this streamwise position the plate temperature distributions for the different cases are not significantly different, but at 30% mass flow upstream one can see that the top of the mushroom is somewhat more flat than in the other cases. This is the result of the mass flow of the upstream hole riding on top of the downstream jet. On the other hand the 70% mass flow on the upstream hole lift up the mass flow out of the downstream hole. Figure 10. Temperature distribution in the streamwise plane x = 13D downstream. Single hole These interaction effects can be observed on Figs 12 and 13. The 5% and 10% mass flow upstream are to week to have a positive effect on the resulting temperature distribution. The case with 30% mass flow upstream has a higher strength but enough to result in a better flow distribution and film performance. While 70% mass flow upstream is to strong and strongly interact with the main flow resulting in a strong counter rotating vortices. The upstream vorticity uplift the flow out of the downstream hole, which is relatively weak. The top view for the temperature distribution on the plate shown in Figs. 14 and 15 clearly show that the 30% mass flow upstream results in the better spanwise spreading presented first in Figs. 8 and 9. Similar protection levels are observed for the region closer to the injection hole for the single hole and for the 30% mass flow upstream, but better protection is observed further downstream for the 30% case. 8 of 15

9 5% mass flow on the upstream hole. 10% mass flow on the upstream hole. 30% mass flow on the upstream hole. 70% mass flow on the upstream hole. Figure 11. Temperature distribution in the streamwise plane x = 13D downstream. Figure 12. Temperature distribution and streamlines. Single hole. 9 of 15

10 5% mass flow on the upstream hole. 10% mass flow on the upstream hole. 30% mass flow on the upstream hole. 70% mass flow on the upstream hole. Figure 13. Temperature distribution and streamlines. Figure 14. Temperature distribution at the plate. Single hole. 10 of 15

11 5% mass flow on the upstream hole. 10% mass flow on the upstream hole. 30% mass flow on the upstream hole. 70% mass flow on the upstream hole. Figure 15. Temperature distribution at the plate. Other view angles reinforce the conclusions about the interaction between the two jets and the resulting lateral spreading as shown in Figs. 16 through 19. The side view shows that comparing the single hole streamlines with the 30% mass flow streamlines, the injection split between two holes results in a cooling flow that approaches the wall further downstream do to the upwash/downwash interaction of the two longitudinal vortical structures. On the other hand the 70% mass flow upstream, again show the cooling mass flow lifting up away from the wall. Figure 16. Streamline distribution looking from upstream. Single hole. 11 of 15

12 5% mass flow on the upstream hole. 10% mass flow on the upstream hole. 30% mass flow on the upstream hole. 70% mass flow on the upstream hole. Figure 17. Streamline distribution looking from upstream. Single hole. Figure 18. Streamline distribution looking from the side. Single hole. 12 of 15

13 5% mass flow on the upstream hole. 10% mass flow on the upstream hole. 30% mass flow on the upstream hole. 70% mass flow on the upstream hole. Figure 19. Streamline distribution looking from the side. The streamwise vorticity has a significant effect on the temperature distribution. But also does the flow separation downstream of the jet and the jet penetration through the boundary layer. Despite the fact that the scales in Figs. 20 and 21 do not allow direct comparison, it is clear that the 30% mass flow case has the least penetration and separation among all the test cases. This is due to the better distribution of mass flow between the to injection holes and the resulting jet velocities. Clearly the 70% mass flow has a very strong boundary layer penetration, which results in pour wall protection and strong interaction with the main flow. Figure 20. Velocity distribution at the centerline. Single hole. VI. Conclusions A comparison for film cooling configurations considering two film injection holes positioned inline was performed and compared to the film cooling for a single hole with the same total mass flow. The results show that for 30% mass flow injected there is a small gain in cooling farther downstream along the centerline. A 13 of 15

14 5% mass flow on the upstream hole. 10% mass flow on the upstream hole. 30% mass flow on the upstream hole. 70% mass flow on the upstream hole. Figure 21. Velocity distribution at the centerline. significant gain is observed considering the spanwise spreading of the cooling flow, resulting in better cooling. The main flow patterns controlling the film cooling effectiveness are the counter rotating vortices due to the interaction of the jets with the main flow and the flow separation at the jet exit. Acknowledgments The authors would like acknowledge the support from VSE Vale Soluções em Energia and from Fundação de Amparo a Pesquisa do Estado de São Paulo FAPESP. References 1 Kercher, D., A film-cooling CFD bibliography: , International Journal of Rotating Machinery, Vol. 4, No. 1, 1998, pp Bernsdorf, S., Rose, M., and Abhari, R. S., Modeling of film-cooling part I: Experimental Study of flow structure, Journal of Turbomachinery, Vol. 128, 2006, pp burdet, A., Abhari, R. S., and Rose, M. G., Modeling of film-cooling part II: Model for use in three-dimensional computational fluid dynamics, Journal of Turbomachinery, Vol. 129, 2007, pp Heidmann, J. and Ekkad, S. V., A novel antivortex turbine film cooling hole concept, Journal of Turbomachinery, Vol. 130, No. 3, 2008, pp Dhungel, A., Lu, Y., Phillips, W., Ekkad, S. V., and Heidmann, J., Film cooling from a row of holes supplemented with antivortex holes, Journal of Turbomachinery, Vol. 131, 2009, pp Renze, P., Schroder, W., and Meinke, M., large-eddy simulation of interacting film cooling jets, Proceedings of the ASME Turbo Expo 2009, Vol. GT , Orlando, Florida, USA, El-Gabry, L., Heidmann, J., and Ameri, A., Penetration Characteristics of film cooling jets at high blowing ratio, AIAA Journal, Vol. 48, No. 5, 2010, pp Sakai, E., Takahashi, T., Funazaki, K., Salleh, H. B., and Watanabe, K., large-eddy simulation of interacting film cooling jets, Proceedings of the ASME Turbo Expo 2009, Vol. GT , Orlando, Florida, USA, Fernandes, L. M., Freitas, R. B., Mendonça, M. T., Flatschart, R. B., and ao, A. V. P., Study of Secondary Hole Effect 14 of 15

15 on Film Cooling, Proceedings of COBEM 2011, 21st International Brazilian Congress of Mechanical Engineering, Brazilian Association of Mechanical and Engineering Sciences, ABCM, Natal, RN, 2011, pp Menter, F. R., Two Equation Eddy-Viscosity Turbulence Models for Engineering Applications, AIAA Journal, Vol. 32, No. 8, 1994, pp Mangani, L., Bianchini, C., Andreini, A., and facchini, B., Development and Validation of a C++ Object Oriented CFD Code Heat Transfer Analysis, Proceedings of the ASME Thermal Engineering and Summer Heat Transfer Conference, Vol. AJ-1266, Vancouver, Canada, Floryan, J. M., On the Görtler Instability of Boundary Layers, Progress in Aerospace Sciences, Vol. 28, 1991, pp Saric, W. S., Görtler Vortices, Annual Review of Fluid Mechanics, Vol. 26, 1994, pp of 15