An analysis is performed to examine the equilibrium states and the stability of modeled Reynolds stress equations for homogeneous turbulent flows. For a homogeneous flow, the system of governing equations consists of several ordinary differential equations. Depending on the coefficients used in a turbulence model, this system may produce undesirable solutions. Therefore, we must find the condition to secure that all other states except the realistic one should not be stable.
For the flows under relaxation process, the turbulence structure is governed by the two coupled ordinary differential equations for $Ⅱ_b$ and $Ⅲ_b$. The slow part of the pressure-strain is responsible for this flow. For several models (2nd, 3th, 5th order, Fu et al., Craft and Launder), all the possible equilibrium states as well as the isotropic states are found analytically. Then acceptable bounds of the coefficients are obtained by examining the stability of each states. It is found that all models adopt the coefficients that are all in their respective bounds. However, experimental data imply that the coefficient bounds for most models are too narrow, except for the 5th order model.
For the homogeneous shear flows, the system of the governing equations consists of four coupled ordinary differential equations for $b_11$, $b_22$, $b_12$ and ε/Sk. The behavior of solutions are much complicated in this case. Depending on the coefficients adopted, the solution of this system may converge, diverge, oscillate, or the equilibrium solution may not exist. It is shown that the rapid part of the pressure-strain correlation is most responsible for such different behaviors. Constraints for the model constants in various Reynolds stress models are obtained by stability conditions for the equilibrium states as well as by their physically realizable bounds. It is observed that quadratic or cubic models often oscillate and converge onto unrealistic states, especially under high shear conditions.
Next, it is shown that the model of Shih and Lumley (1985), the simplest form of quadratic model, can be adjusted to produce stable and realistic behavior. Employing the 5th order model as a model for the slow part, this model is modified to set coefficients to A = 0.928, B =-1.26. Performance test reveals that the present model overcomes the defects of the original SL model.
