Three-dimensional calculations of the flow field around a turbine blade with film cooling injection near the leading edge (Q5952018)

Three-dimensional calculations of turbulent compressible flows around a high-pressure turbine cascade model with film cooling injection near the leading edge are conducted using a finite volume multi-block solver for averaged Navier-Stokes equations. The governing differential equations are integrated over arbitrary three-dimensional control volumes with the aid of Gauss theorem. The conservative form of the resulting discretization equations is further enhanced by adopting Cartesian velocity components as the principal unknowns for the momentum equation. All variables are stored at the control volume centres, and the momentum interpolation procedure of \textit{C. M. Rhie} and \textit{W. L. Chow} [AIAA J. 21, 1521--1532 (1983; Zbl 0528.76044)] is used to avoid the checker-board splitting of the pressure field. The discretized momentum and continuity equations are combined to form a pressure-correction equation following the SIMPLE algorithm of Patankar and Spalding. The resulting system of the algebraic difference equations is solved in an uncoupled manner by the strongly implicit procedure of Stone. A high level of vectorization is accomplished, allowing for fast results on fine grids. The pressure-correction method adopted in FAST-3D has been originally developed for incompressible flows. In order to allow for compressibility effects, the fluid density is computed as a function of both the fluid pressure and absolute temperature with the aid of the perfect-gas equation of state. Density corrections are then introduced into the discretized continuity equation as proposed by \textit{K. C. Karki} and \textit{S. V. Patankar} [Numer. Heat Transfer 14, No. 3, 295--307 (1988; Zbl 0662.78048); ibid., 309--321 (1988; Zbl 0662.76049)]. These density corrections are related to the pressure corrections via the equation of state, under the assumption of an isentropic transformation. As a result, additional terms arise in the final form of the pressure correction equation. The method is successfully applied to the investigation of the flow field structure in the vicinity of the showerhead cooling jets at the leading-edge of the blade. The calculations are extended to the plenum and injections pipes. The present method yields excellent agreement with experiments for the isentropic Mach number distributions on the blade surface. The standard \(k-\varepsilon\) model with wall functions showed a remarkable capability of capturing the major features of the flow, including the injection-induced secondary-flow vortices, in particular on the suction side of the blade. On the pressure side, however, the lateral jet spreading is underpredicted and somewhat exaggerated near-wall sink-flow vortices prevail. The discrepancies with the measurements on the pressure side become more evident with increasing blowing rate. This weakness is a direct consequence of employing an isotropic eddy-viscosity model for turbulence which inherently tends to underpredict the magnitude of the turbulent stresses in the lateral direction wherever the turbulence exhibits a strong anisotropy (i.e. on convex walls); the present results definitely confirm this statement. The results show also that, whenever the turbulence anisotropy is weak (i.e. on concave walls), the standard model predicts the flow remarkably well and does not require an extra correction.