Obtaining the mean fields with known Reynolds stresses at steady state

With the rising of modern data science, data--driven turbulence modeling with the aid of machine learning algorithms is becoming a new promising field. Many approaches are able to achieve better Reynolds stress prediction, with much lower modeling error ($\epsilon_M$), than traditional RANS models but they still suffer from numerical error and stability issues when the mean velocity fields are estimated using RANS equations with the predicted Reynolds stresses, illustrating that the error of solving the RANS equations ($\epsilon_P$) is also very important. In the present work, the error $\epsilon_P$ is studied separately by using the Reynolds stresses obtained from direct numerical simulation and we derive the sources of $\epsilon_P$. For the implementations with known Reynolds stresses solely, we suggest to run an adjoint RANS simulation to make first guess on $\nu_t^*$ and $S_{ij}^0$. With around 10 iterations, the error could be reduced by about one-order of magnitude in flow over periodic hills. The present work not only provides one robust approach to minimize $\epsilon_P$, which may be very useful for the data-driven turbulence models, but also shows the importance of the nonlinear part of the Reynolds stresses in flow problems with flow separations.