# Assessment of Two Streamline Curvature Correction Methods for an Elliptic Blending Turbulence Model
## 1 Introduction  
Turbulence is a ubiquitous phenomenon in nature, and, as such, determining and understanding turbulence is important in many industrial applications. Computational fluid dynamics is a useful tool for understanding turbulence. Even though computational power has grown rapidly in recent decades, at present, the Reynolds-averaged Navier-Stokes (RANS) method is still the mainstream in industrial applications, as the computing resources required for turbulence-resolved approaches--such as direct numerical simulation (DNS), large eddy simulation (LES), and hybrid LES-RANS approaches--are still unaffordable. Some widely used RANS approaches are based on linear eddy viscosity models (LEVMs). However, conventional LEVMs generally have a serious weakness: they have no capability of capturing the streamline curvature effect, which may play an important role in some industrial applications. To enable the LEVMs to have the ability to capture the effect of streamline curvature, these turbulence models need to be modified and corrected. Much progress has been made since researchers started focusing on this issue. Due to space limitations, only some representative research results are described here. Successful streamline curvature corrections to turbulence models can be divided into different types, including corrections for turbulent kinetic energy production term, corrections in the formulation of turbulent viscosity, corrections for turbulent dissipation, and so on. [PERSON] and [PERSON] [1] have proposed an efficient approach in which a curvature function is introduced to correct the turbulence production term. This approach was first applied to the one-equation Spalart-Allmaras (S-A) turbulence model. Numerical tests showed that the modified S-A model was more accurate than the original S-A model [1, 2]. Based on thiswork [1], [PERSON] and [PERSON] [3] have successfully sensitized the shear stress transport (SST) \(k-\omega\) model to rotation and curvature. [PERSON] et al. [4] have introduced [PERSON] and [PERSON]'s correction to the standard \(k-\varepsilon\) model. The modified model was applied to calculate the cavitating cloud flows, and good results were obtained. To avoid calculating the Lagrangian derivative of the strain rate tensor--which may introduce instability due to the second-order derivatives of velocity--[PERSON] and [PERSON] [5] constructed a simpler curvature correction function, using the Richardson number to replace the factor accounting for the streamline curvature effects. Their results showed that the modified model not only can predict the curvature effect as accurately as the original model but also is stable and efficient. [PERSON] et al. [6] have developed a bifurcation approach for the \(v^{2}\)-\(f\) turbulence model. The constant in the formulation of the turbulent viscosity was replaced by a functional expression, which accounts for the rotation and streamline curvature effects. This bifurcation approach has been applied in the \(k-\varepsilon\) model by [PERSON] and [PERSON] [7]. Based on the anisotropy tensor developed by [PERSON] and [PERSON] [8] for an explicit algebraic stress model, [PERSON] et al. [9] have designed a new eddy viscosity formulation to correct the streamline curvature and rotation effect and successfully applied it to the \(k-\varepsilon\) model. [PERSON] and [PERSON] [10] have introduced the eddy viscosity formulation of [PERSON] et al. [9] into the SST \(k\)-\(\omega\) model and validated its accuracy. [PERSON] et al. [11] have improved upon the formulation of [PERSON] et al. [9] by incorporating an exact calculation of the Spalart-Shur tensor, for which an approximation was used in the formulation of [PERSON] et al. [9]. After having re-defined the Richardson number and proposed a new curvature correction function, [PERSON] [12] sensitized the SST \(k\)-\(\omega\) model to streamline curvature effects by modifying the dissipation term in the \(\omega\)-equation. [PERSON] et al.