The use of the Reynolds force vector in a physics informed machine learning approach for predictive turbulence modeling

Conditioning and accurate solutions of Reynolds average Navier–Stokes equations with data-driven turbulence closures

“…As emphasised by Duraisamy et al (2019), this problem imposes remarkable challenges to the field of data-driven turbulence modelling because if small errors in the RST field lead to significant errors in the mean velocity field, a robust ML procedure that has the RST as a target cannot avoid large propagated errors. In this regard, Wang et al (2017) and Cruz et al (2019) reported small errors in the prediction of the RST but large values of the error propagation in the mean velocity field.…”

“…The first one, EV DNS -I, is based on the works of Wu et al (2018Wu et al ( , 2019 and was already analysed in the last subsection. The second one is based on the RFV presented by Cruz et al (2019), referred to as RFV-E. The third, is a new method proposed in the present work, RFV-EV DNS -I that combines aspects of the two previous ones, as described in § 4.…”

“…In order to circumvent the problem described above with respect to data-driven turbulence modelling, Wu et al (2018) included the optimised eddy viscosity as a target for the ML procedure, addressing the linear part of the RST. Cruz et al (2019) proposed another strategy, where the Reynolds force vector (RFV, the divergence of the RST) was elected as the target for the ML technique. Both initiatives succeeded in providing better predictions for the mean velocity field.…”

“…Although the conditioning increase obtained by Wu et al (2019) could be attributed to the procedure for treating the linear part of the RST implicitly, a second ingredient of the approach, namely the use of the mean velocity field information proceeding from the DNS source, can play an important role. This fact may place their strategy in the group of approaches that produced a much lower injected error in the RST (or RFV) than the usual one available in DNS databases (Vreman & Kuerten 2014b;Andrade et al 2018;Cruz et al 2019). In other words, that procedure can be operating on the reduction of the original error, and not only on the error-amplification character of the RANS equations that is propagated to the velocity field.…”

“…4.7. RFV with implicit treatment using optimised eddy viscosity This method combines the strong points of the procedures proposed by Wu et al (2018) and Cruz et al (2019). In order to do this, the same decomposition of R into a linear and a nonlinear part with respect to S is adopted and the optimal turbulent viscosity is also obtained from (4.6).…”