Application and evaluation of dynamic Bradshaw coefficient in shear stress transport turbulence model for hydraulic machinery

The Bradshaw assumption, that the ratio of Reynolds shear stress to turbulence kinetic energy (TKE) is a constant roughly equal to 0.3, is introduced into eddy viscosity and TKE dissipation term of the shear stress transport turbulence model (SST). This constant is calibrated by the zero-pressure-gradient plate boundary layer test. This is not applicable to rotating separation flows in hydraulic machinery, while its dynamic effects are usually ignored. In this article, a comprehensive evaluation of the dynamic Bradshaw coefficient (DBC) in SST is conducted. First, theoretical analyses of the existing typical DBCs are carried out, and a suitable expression form driven by a single turbulence Reynolds number is adopted in view of its well-reflected dynamic effects and high robustness. According to the equation structures of SST, three dynamic strategies are proposed, including only introducing DBC into eddy viscosity (SST-M1), only introducing DBC into TKE dissipation term (SST-M2), and introducing DBC both into eddy viscosity and TKE dissipation term (SST-M1 + M2). Second, the classical case of flows around a hydrofoil is employed to evaluate the application effects of these three dynamic strategies. The results show that SST-M1 exhibits severe lift/drag oscillations at large angles of attack accompanied by the eddy viscosity fluctuations, indicating poor numerical stability and potential risk of this strategy. In contrast, both SST-M2 and SST-M1 + M2 can effectively improve the deficiencies of SST in overestimating lift and underestimating drag at large angles of attack, which is attributed to the promotion of earlier and larger flow separation. The consistency between these two strategies implies that introducing DBC into the TKE dissipation term plays a key role in enhancing the prediction of SST. Therefore, the dynamic strategy SST-M2 is recommended to extend the applicability of SST for rotating separation flows in hydraulic machinery.