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

Obtaining the mean fields with known Reynolds stresses at steady state