Machine learning-augmented turbulence modeling for RANS simulations of massively separated flows

Abstract

Most widely used Reynolds-Averaged Navier-Stokes (RANS) models employ the Boussinesq approximation, which assumes a linear relationship between the turbulent Reynolds stresses and the mean-velocity gradient tensor. This assumption, which can be very stringent, is more suited for simple shear-flows and is regarded as an important shortcoming for the improvement of the representation of turbulence in complex geometries. Correction of the local turbulence length scales, as achieved for example by the introduction of a correction term in the eddy-viscosity equation of a Spalart-Allmaras model, then indeed only allows relatively small corrections to the base-line model and may not be sufficiently flexible to adapt to common situations where the Reynolds stresses are not aligned with the velocity gradient tensor. For the variational data-assimilation step, we consider a vectorial source correction term in the momentum equations (output quantity) together with the Spalart-Allmaras model, which allows full flexibility to adapt to any mean-flow topology. We show how the machine-learning framework should be adapted, in particular with respect to the vectorial nature of the correction term and given invariance properties. As for the input quantities, we discuss the impact of considering either local quantities for the non-dimensionalization or global ones that characterize the configuration. We showcase the procedure on the periodic hill configuration, for which a rich DNS database is available in the literature: in particular, availability of the full mean-flow solution for a range of Reynolds numbers and geometries, ensures identification of "physical" correction terms and allows learning of turbulence models that accurately extrapolate to new Reynolds numbers and geometries.

Development and Validation of a Machine Learned Turbulence Model

“…The studies that focused on RANS model augmentation can be further subdivided into three categories: (i) direct estimation of the turbulent eddy viscosity ( 𝑇 ) for linear models [47][48][49][50][51][52][53][54][55], (ii) correction terms for the linear models [56][57][58][59][60][61][62][63], and (iii) enhancement of the accuracy of the turbulence transport equations used in linear models [64][65][66][67][68]. Studies in category (i) have been applied for both incompressible [47][48][49][50] and compressible flows [51][52][53][54].…”

“…Studies in category (ii) involving correction of terms for linear models have either proposed to use corrections to shear stresses (∆𝜏) over the linear model prediction [56][57][58][60][61] or to use corrections to turbulent eddy viscosity (∆ 𝑇 ) [62][63]. The models have been developed for both k-/k- and SA turbulence models and have been mostly developed for incompressible separated flows, except for Wang et al [58] which extended the Wang et al [56] model for compressible flows, and He et al [63] performed simulation for shock induced separation.…”

Deep Reinforcement Learning-Augmented Spalart–Allmaras Turbulence Model: Application to a Turbulent Round Jet Flow

“…This first step of retrieving an optimal or best-fit closure field can be considered as collecting a priori information. Based on the information collected, datadriven approaches typically apply supervised learning algorithms in a subsequent step to determine an improved closure model [17,18]. The quality of the identified models can then be evaluated in an a posteriori analysis.…”

Research on grid‐dependence of neural network turbulence model

“…On the basis of RANS framework, Volpiani et al 25 used SA turbulence model combined with variational fraction data assimilation to correct the source term of momentum equation and achieve better flexibility correction. The fully predicted Reynolds stress term of the corrected RANS model reads…”

“…On the basis of RANS framework, Volpiani et al 25 used SA turbulence model combined with variational fraction data assimilation to correct the source term of momentum equation and achieve better flexibility correction. The fully predicted Reynolds stress term of the corrected RANS model reads $$$$ {f}_i^{RS, pred}={f}_i^{\upsilon_t}+{f}_i^c $$$$ where $$$ {f}_i^{\upsilon_t} $$$ are the Reynolds stresses due to the eddy‐viscosity; $$$ {f}_i^c $$$ is an additional vectorial source term.…”