  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.
