Higher-order realizable algebraic Reynolds stress modeling based on the square root tensor

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

Figure 1

Profile of the anisotropy tensor of the Reynolds stress $b_{ij}$ against the nondimensional shear rate $\widehat{G}$.

Figure 2

Profiles of $b_{xy}$ against the nondimensional shear rate $\widehat{G}$ for ${\mathrm{\Omega }}^{\mathrm{F}}/G=0,0.25$, and $-0.25$ in the spanwise rotating shear flow.

Figure 3

Profiles of (a) the mean velocity $U_x^{+}$, (b) the turbulent energy $K^{+}$, and (c) the dissipation rate ${\varepsilon }^{+}$ in the turbulent channel flow at ${\mathrm{Re}}_{\tau }=590$. Here, $y^{+}=y{u}_{\tau }/\nu , U_x^{+}=U_x/u_{\tau }, K^{+}=K/u_{\tau }^2$, and ${\varepsilon }^{+}=\varepsilon \nu /u_{\tau }^4$.

Figure 4

Profiles of (a) the Reynolds stress and (b) the anisotropy tensor of the Reynolds stress for the present model in the turbulent channel flow at ${\mathrm{Re}}_{\tau }=590$.

Figure 5

Profile of the nondimensional shear rate $\widehat{G}$ in the turbulent channel flow at ${\mathrm{Re}}_{\tau }=590$.

Figure 6

Time evolution of (a) the turbulent energy and (b) the anisotropy tensor of the Reynolds stress in the homogeneous turbulent shear flow.

Figure 7

Profiles of (a) the mean axial velocity $U_z/U_{\mathrm{m}}$ and (b) the mean swirl velocity $U_{\theta }/U_{\theta ,\text{wall}}$ in the axially rotating turbulent pipe flow at ${\mathrm{Re}}_{\mathrm{m}}=10000$.