Coarse Grid Modeling of Turbine Film Cooling Flows Using Volumetric Source Terms NASA / TM — 2001-210817 May 2001 2001 – GT – 138

The recent trend in numerical modeling of turbine film cooling flows has been toward higher fidelity grids and more complex geometries. This trend has been enabled by the rapid increase in computing power available to researchers. However, the turbine design community requires fast turnaround time in its design computations, rendering these comprehensive simulations ineffective in the design cycle. The present study describes a methodology for implementing a volumetric source term distribution in a coarse grid calculation that can model the small-scale and three-dimensional effects present in turbine film cooling flows. This model could be implemented in turbine design codes or in multistage turbomachinery codes such as APNASA, where the computational grid size may be larger than the film hole size. Detailed computations of a single row of 35 degree round holes on a flat plate have been obtained for blowing ratios of 0.5, 0.8, and 1.0, and a density ratios of 1.0 and 2.0 using a multiblock grid system to resolve the flows on both sides of the plate as well as inside the hole itself. These detailed flow fields were spatially averaged to generate a field of volumetric source terms for each conservative flow variable. Solutions were also obtained using three coarse grids having streamwise and spanwise grid spacings of 3d, 1d, and d/3. These coarse grid solutions used the integrated hole exit mass, momentum, energy, and turbulence quantities from the detailed solutions as volumetric source terms. It is shown that a uniform source term addition over a distance from the wall on the order of the hole diameter is able to predict adiabatic film effectiveness better than a near-wall source term model, while strictly enforcing correct values of integrated boundary layer quantities. NOMENCLATURE Ah hole cross-sectional area cp specific heat cfl Courant number d film cooling hole diameter DR density ratio = ρc/ρin e specific internal energy I momentum ratio = ( ρV)c/(ρV)in k turbulence kinetic energy M blowing ratio = (ρV)c/(ρV)in Pr Prandtl number Prt turbulent Prandtl number qn updated conservative variable qo old conservative variable R gas constant for air Rq residual Re Reynolds number s effective slot width Sq source term ∆t local time step T temperature u x component of velocity v y component of velocity V absolute velocity (vol)R residual volume (vol)S source term volume w z component of velocity x streamwise distance from hole leading edge y transverse coordinate from hole centerline y dimensionless grid spacing z normal coordinate from wall η film effectiveness ρ density ω dissipation per unit turbulence kinetic energy Subscripts aw adiabatic wall conditions c coolant conditions in freestream inlet conditions o stagnation conditions National Aeronautics and Space Administration 1 NASA/TMÑ—2001-210817 INTRODUCTION As gas turbine inlet temperatures increase in the drive toward higher cycle efficiencies, film cooling of high pressure turbine blades is increasingly needed to ensure their durability. However, film cooling flows are complex three-dimensional flows involving streams of different temperature, and are not easily described using low-order models. In addition, the scale of film cooling flows is much smaller than the scale over which the aerodynamic design of turbine systems is typically performed. Thus film cooling systems have generally been designed using databases of information from detailed studies. However, in order to optimize the aerodynamics and heat transfer of a film-cooled turbine blade, it would be highly desirable to have a model which would account for the detailed effects of film cooling on a coarser grid such as the type used for turbine aerodynamic design. Such an optimization would reduce the amount of cooling flow required by placing the cooling air only where it is required, and reducing overcooling. Goldstein (1971) summarizes early studies in the area of film cooling. These studies concentrated on slot flows, and film cooling effectiveness values were found to correlate well with the parameter x/Ms, where x is the downstream distance, M is the blowing ratio, and s is the slot width. This parameter has also been used for discrete hole cooling, with s defined as the effective slot width for the row of holes. However, the physics of discrete hole cooling is quite different from that of a slot. A row of discrete holes typically has a much lower spanaveraged downstream film effectiveness distribution for the same x/Ms due to the formation of vortices which allow hot gas to penetrate to the wall. These vortices are of the scale of the hole size, so if a numerical simulation has a spanwise grid spacing greater than the film hole spanwise pitch, as is typical for turbine blade aerodynamic design, their effect is lost. In essence, any such calculation is two-dimensional on the scale of the film holes. To overcome this problem, several computational studies have computed turbine blade geometries with accurate resolution of the film holes, and in some cases, of the hole pipes and plena as well. Garg and Gaugler (1997) showed the importance of film hole exit profiles. Garg and Rigby (1999) resolved the plenum and hole pipes for a three-row showerhead film cooling arrangement, and Heidmann, et al. (2000) computed the heat transfer for a realistic turbine vane with 12 rows of film cooling holes with shaped holes and plena resolved. Garg (2000) presented results of a full rotating blade with 172 film holes, resolving the film hole exits, but not the hole pipes and plena. These studies are appropriate in the research environment, but are too computationally intensive for turbine design. A simplified model is thus needed. Many published studies have discussed the physics and presented data for discrete hole film cooling in simplified geometries. The most basic geometry consists of a row of round holes in a flat plate. There is a relatively large body of experimental data for 35 deg. round holes with a pitch of 3d (Pedersen, et al., 1977; Foster and Lampard, 1980; Pietrzyk, et al., 1989; Pietrzyk, et al., 1990; Sinha, et al., 1991). This geometry allows for a study of jet lift-off behavior at various blowing ratios and is perhaps the most realistic simplified geometry for turbine film cooling. In addition, the computational study of Leylek and Zerkle (1994) uses this geometry and gives an excellent description of the vortical flows associated with this geometry. The present study will therefore use this geometry in the development of the film cooling model. A source term model for film cooling flows has been described by Hunter (1998), and implemented in the APNASA code (Adamczyk, 1985). In this model, source terms of mass, momentum, and energy are added at the blade surface to account for the flux of these quantities due to film cooling or other types of injection. The source terms are added at a single computational cell adjacent to the wall at the hole exit location. The typical distance from the wall for the cell adjacent to the wall is 0.015d for these wall function-type grids. The source terms are inferred from the known blowing ratio, hole angle, and temperature ratio. However, the flow produced by these source terms behaves as slot flow for large spanwise grid spacings, because they are applied in a constant manner across the blade span. It is thus the goal of this research effort to develop a model to account for the three-dimensional mixing of discrete hole injection on the coarse grids typical for turbine aerodynamic design. The test of this model will be its ability to predict span-averaged film effectiveness downstream of a row of discrete film holes in a coarse grid calculation. COMPUTATIONAL METHOD The initial approach of the study was to determine the source terms of mass, momentum, energy, and turbulence quantities which must be added in a two-dimensional (slot) calculation to yield the same flow solution as the span-average of a full three-dimensional calculation including the plenum and hole pipe. The geometry of choice is shown in Figure 1. The holes are aligned with the streamwise direction with an inclination angle of 35 deg. The hole spanwise pitch is 3d. The geometry is the same as that of Leylek and Zerkle (1994) and Sinha et al. (1991).