  In this study, realizable algebraic Reynolds stress modeling based on the
square root tensor [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 an algebraic Reynolds
stress 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 algebraic
Reynolds stress 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 algebraic
Reynolds stress 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.
