Higher-order realizable turbulence modeling based on the square root tensor of the Reynolds stress.

In this study, realizable turbulence modeling based on the square root tensor of the Reynolds stress [Phys. Rev. E \textbf{92}, 053010 (2015)] is further developed for extending its applicability to more complex flows. In conventional methods, it was difficult to construct a turbulence model satisfying the realizability conditions when the model involves higher-order nonlinear terms on the mean velocity gradient. Such higher-order nonlinear terms are required to predict turbulent flows with three-dimensional mean velocity. The present modeling based on the square root tensor enables us to make the model always satisfy the realizability conditions, even when it involves higher-order nonlinearity. To construct a realizable model applicable to turbulent flows with three-dimensional mean velocity, a quartic-nonlinear eddy-viscosity model is proposed. The performance of the model is numerically verified in a turbulent channel flow, a homogeneous turbulent shear flow, and an axially rotating turbulent pipe flow. The present model gives a good result in each turbulent flow. Note that the mean swirl flow in an axially rotating turbulent pipe flow is reproduced because the present model involves cubic nonlinearity. Such a higher-order realizable turbulence model, involving quartic nonlinearity on the mean velocity, is expected to be useful in numerically stable predictions of turbulent flows with three-dimensional mean velocity.