Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data

Turbulence modeling is a critical component in numerical simulations of industrial flows based on Reynolds-averaged Navier-Stokes (RANS) equations. However, after decades of efforts in the turbulence modeling community, universally applicable RANS models with predictive capabilities are still lacking. Large discrepancies in the RANS-modeled Reynolds stresses are the main source that limits the predictive accuracy of RANS models. Identifying these discrepancies is of significance to possibly improve the RANS modeling. In this work, we propose a data-driven, physics-informed machine learning approach for reconstructing discrepancies in RANS modeled Reynolds stresses. The discrepancies are formulated as functions of the mean flow features. By using a modern machine learning technique based on random forests, the discrepancy functions are trained by existing direct numerical simulation (DNS) databases and then used to predict Reynolds stress discrepancies in different flows where data are not available. The proposed method is evaluated by two classes of flows: (1) fully developed turbulent flows in a square duct at various Reynolds numbers and (2) flows with massive separations. In separated flows, two training flow scenarios of increasing difficulties are considered: (1) the flow in the same periodic hills geometry yet at a lower Reynolds number and (2) the flow in a different hill geometry with a similar recirculation zone. Excellent predictive performances were observed in both scenarios, demonstrating the merits of the proposed method.

Figure 1

The barycentric triangle that encloses all physically realizable states of Reynolds stress [31, 33]. The position within the barycentric triangle represents the anisotropy state of the Reynolds stress. Typical mapped locations of near-wall turbulence states are indicated with predictions from standard RANS models near the isotropic state (vertex 3C-I), and the actual locations are indicated near the bottom edge (2C-I). The typical RANS-predicted trend of spatial variation from the wall to shear flow and the corresponding actual trend are indicated with arrows.

Figure 2

Schematic of a simple regression tree in a two-dimensional feature space (pressure gradient along streamline $dp/ds$ and wall-distance based Reynolds number ${\text{Re}}_d$), showing (a) the stratification of feature space and (b) the corresponding regression tree built from the training data. The response is the discrepancy $\mathrm{\Delta }\eta$ in the barycentric triangle of the RANS-predicted Reynolds stress. When predicting the discrepancy for a given feature vector $\tilde{\mathbf{q}}$, the tree model in (b) is traversed to identify the leaf, and the mean of the training data is taken as the prediction $\mathrm{\Delta }\eta \left(\tilde{\mathbf{q}}\right)$.

Figure 3

Domain shape for the flow in a square duct. The $x$ coordinate represents the streamwise direction. Secondary flows induced by Reynolds stress imbalance exist in the $y-z$ plane. Panel (b) shows that the computational domain covers a quarter of the cross section of the physical domain. This is due to the symmetry of the mean flow in both $y$ and $z$ directions as shown in panel (c).

Figure 4

Barycentric map of the predicted Reynolds stress anisotropy for the test flow ($\text{R}e=3500$), learned from the training flows ($\text{Re}=2200$, 2600, and 2900). The prediction results on two streamwise locations at $y/H=0.25$ and 0.75 are compared with the corresponding baseline (RSTM) and DNS results in panels (a) and (b), respectively.

Figure 5

Profiles of normal components (a) ${\tau }_{yy}$ and (b) ${\tau }_{zz}$ of corrected Reynolds stress with the discrepancy model. The profiles are shown at four streamwise locations $y/H=0.25$, 0.5, 0.75, and 1. Corresponding DNS and baseline (RSTM) results are also plotted for comparison.

Figure 6

Computational domain and velocity field of each case in the training flow database. The velocity contours and streamlines are obtained from the baseline RANS simulations. The dimensions of each case are normalized with the respective hill heights $H$. Note that periodic boundary conditions are applied on the flows in panels (a) and (b) in the streamwise direction, but not for the flow in panel (c). (a) Periodic hills (Re = 1400 and 5600, results from latter are shown), (b) wavy channel (Re = 360), and (c) curved backward facing step (Re = 13200).

Figure 7

Magnitude (turbulence kinetic energy) of the corrected Reynolds stress for the test flow (PH10595) learned from cases with with same geometry but at different Reynolds numbers (PH1400 and PH5600). The profiles are shown at eight streamwise locations $x/H=1,2,\cdots ,8$. Corresponding baseline and DNS results are also plotted for comparison. The hill profile is vertically exaggerated by a factor of 1.3.

Figure 8

Predicted turbulence shear stress for the test flow (PH10595) learned from the flows with the same geometry but at different Reynolds numbers (PH1400 and PH5600). The profiles are shown at eight streamwise locations $x/H=1,2,\cdots ,8$. Corresponding baseline and DNS results are also plotted for comparison. The hill profile is vertically exaggerated by a factor of 1.3.

Figure 9

Magnitude (turbulence kinetic energy) of the corrected Reynolds stress for the test flow (PH10595) learned from cases with different geometries and at different Reynolds numbers (WC360 and CS13200). The profiles are shown at eight streamwise locations $x/H=1,2,\cdots ,8$. Corresponding baseline and DNS results are also plotted for comparison. The hill profile is vertically exaggerated by a factor of 2.4.

Figure 10

Predicted turbulence shear stress for the test flow (PH10595) learned from the flows with different geometries and at different Reynolds numbers (WC360 and CS13200). The profiles are shown at eight streamwise locations $x/H=1,2,\cdots ,8$. Corresponding baseline and DNS results are also plotted for comparison. The hill profile is vertically exaggerated by a factor of 2.4.

Figure 11

Feature importance of random forest regressors (a) for $\mathrm{\Delta }\eta$ and (b) for $\mathrm{\Delta }{log}_{2}k$ for scenario I (i.e., training flows in the same geometry; see Table 2). The features $q_i$ ($i=1$ to 10) are denoted by their respective abbreviations. Turb Intensity denotes the turbulence intensity (feature $q_2$), and Re_d is the wall distance based Reynolds number (feature $q_3$). For the full name list of features, see Table 1.