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

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 nondimensionalization or global ones that characterize the configuration. We showcase the procedure on the periodic hill configuration, for which a rich direct numerical simulation database is available in the literature: in particular, the 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.

Figure 1

Computational domain for the flow over the baseline geometry of periodic hills.

Figure 2

Mean streamwise velocity field for the case with ${\text{Re}}_b=2800$. (a) DNS from Gloerfelt and Cinnella [27] and (b) RANS simulation using the original SA model.

Figure 3

A map of ${\rho }_{\text{RS}}$ for the DNS of case PH-2800.

Figure 4

Objective function convergence during data assimilation with L-BFGS. Case PH-2800 (solid line), PH-5600 (dotted line), and PH-10595 (dashed line).

Figure 5

Mean velocity field for the case with ${\text{Re}}_b=2800$. (a) Streamwise velocity field and (b) vertical velocity field. Contour plots: DA solution. Lines: DNS solution.

Figure 6

Dominant component of force by turbulent fluctuations for the case with ${\text{Re}}_b=2800$. Red indicates $-0.1$ and blue $+0.1$. (a) Total force computed from the DNS: $f_1^{\mathrm{rs},\mathrm{DNS}}$, (b) Total force computed from RANS: $f_1^{\mathrm{rs},\mathrm{pred}}$, (c) Component given by the SA model: $f_1^{{\nu }_t}$, and (d) Correction term given by the DA: $f_1^c$.

Figure 7

Mean velocity profiles for case PH-5600. Streamwise velocity (top) and vertical velocity (bottom). DNS (solid lines), DA (dashed lines), SA-RANS (dotted lines), and NN-RANS based on Scenario I (dash-dotted lines).

Figure 8

Skin friction distribution for case PH-5600: SA-RANS (dotted line), NN-RANS (dash-dotted line), and separation and reattachment locations (experiment from Breuer et al. [36], dashed lines).

Figure 9

Mean velocity fields for case PH-5600. (a) DNS, (b) SA-RANS, and (c) NN-RANS based on Scenario I.

Figure 10

Mean force in the streamwise direction for case PH-5600. DNS (solid lines), DA (dashed lines), and NN-RANS based on Scenario I (dash-dotted lines).

Figure 11

Ratio between turbulent and laminar viscosities for case PH-5600. (a) SA-RANS, (b) DA, and (c) NN-RANS based on Scenario I.

Figure 12

Mean velocity profiles for case PH-19000. Streamwise velocity (top) and vertical velocity (bottom). DNS (solid lines), DA (dashed lines), SA-RANS (dotted lines), and NN-RANS based on Scenario II (dash-dotted lines).

Figure 13

Skin friction distribution for case PH-5600: SA-RANS (dotted line), NN-RANS (dash-dotted line), and LES from Gloerfelt and Cinnella [27] (solid lines).

Figure 14

New geometries by systematically varying the steepness of the hill. Solid line: original geometry ($a=1$); dashed line: $a=0.8$; and dash-dotted line: $a=1.2$.

Figure 15

Mean velocity profiles for case PH-5600-0.8. Streamwise velocity (top) and vertical velocity (bottom). DNS (solid lines), SA-RANS (dotted lines), NN-RANS based on Scenario I (dash-dotted lines), and NN-RANS based on Scenario II (dashed lines).

Figure 16

Mean velocity profiles for case PH-5600-1.2. Streamwise velocity (top) and vertical velocity (bottom). DNS (solid lines), SA-RANS (dotted lines), NN-RANS based on Scenario I (dash-dotted lines), and NN-RANS based on Scenario II (dashed lines).

Figure 17

Mean profiles for case PH-19000. DNS (solid lines), SA-RANS (dotted lines), original NN model using set 1 (dashed lines), and NN-RANS (dash-dotted lines). (a) Streamwise velocity and (b) skin friction coefficient.