Physics-informed machine learning approach for augmenting turbulence models: A comprehensive framework

Reynolds-averaged Navier-Stokes (RANS) equations are widely used in engineering turbulent flow simulations. However, RANS predictions may have large discrepancies due to the uncertainties in modeled Reynolds stresses. Recently, Wang et al. demonstrated that machine learning can be used to improve the RANS modeled Reynolds stresses by leveraging data from high-fidelity simulations [J.-X. Wang et al., Phys. Rev. Fluids 2, 034603 (2017)]. However, solving for mean flows from the improved Reynolds stresses still poses significant challenges due to potential ill-conditioning of RANS equations with Reynolds stress closures. Enabling improved predictions of mean velocities is of profound practical importance, because often the velocity and its derived quantities (quantities of interest, e.g., drag, lift, and surface friction), and not the Reynolds stress itself, are of ultimate interest in RANS simulations. To this end, we present a comprehensive framework for augmenting turbulence models with physics-informed machine learning, illustrating a complete workflow from identification of input features to final prediction of mean velocities. This work has two innovations. First, we demonstrate a systematic procedure to generate mean flow features based on the integrity basis for mean flow tensors. Second, we propose using machine learning to predict linear and nonlinear parts of the Reynolds stress tensor separately. Inspired by the finite polynomial representation of tensors in classical turbulence modeling, such a decomposition is instrumental in overcoming the ill-conditioning of RANS equations. Numerical tests demonstrate the merits of the proposed framework.

Figure 1

(a) Computational domain for the flow in a square duct. The $x$ coordinate represents the streamwise direction. Secondary flows induced by Reynolds stress imbalance exist in the $y\text{−}z$ plane. (b) 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 the $y$ and $z$ directions as shown in (c).

Figure 2

Computational domain for the flow over periodic hills. The solid line indicates the configuration of the test flow and the dashed line indicates the configuration of the training flow. The hill width of the training flow is 0.8 of the hill width of the test flow. The $x,y$, and $z$ coordinates are aligned in the streamwise, wall-normal, and spanwise directions, respectively.

Figure 3

Comparison of secondary flow velocity $U_z$ by using (a) the Reynolds stress model with explicit treatment and (b) the Reynolds stress model with implicit treatment. The results obtained using the linear eddy viscosity model capture no secondary flow and are thus omitted here.

Figure 4

Solved mean velocity field for the flow over periodic hills at $\text{Re}=5600$ by using (a) the Reynolds stress model with an explicit treatment, (b) the eddy viscosity model, and (c) the Reynolds stress model with implicit treatment. The DNS data are utilized as the modeled term to represent the best possible performance among the respective classes of models.

Figure 5

Indicator of the deficiency of the linear eddy viscosity model due to the misalignment of eigenvectors between the Reynolds stress tensor and strain rate tensor.

Figure 6

Prediction of DNS Reynolds stress components of the flow in a square duct at Reynolds number $\text{Re}=3500$, including (a) ${\mathbf{\tau }}_{yy}^{\mathrm{DNS}}$ and (b) ${\mathbf{\tau }}_{zz}^{\mathrm{DNS}}$ at $y/h=0.25$, 0.5, 0.75, and 1. The training flow is at Reynolds number $\text{Re}=2200$.

Figure 7

Prediction of the linear part of Reynolds stress components of the flow in a square duct at Reynolds number $\text{Re}=3500$, including (a) ${\mathbf{\tau }}_{yy}^L$ and (b) ${\mathbf{\tau }}_{zz}^L$ at $y/h=0.25$, 0.5, 0.75, and 1. The training flow is at Reynolds number $\text{Re}=2200$.

Figure 8

Nonlinear part of the Reynolds stress ${\mathbf{\tau }}_{yy}^{\perp }$ in a square duct at Reynolds number $\text{Re}=3500$, including (a) RANS simulated results, (b) DNS data, and (c) prediction of RSMs with implicit treatment. The training flow is at Reynolds number $\text{Re}=2200$. The color of the contour denotes the value of the stress component; the light color here indicates small magnitude.

Figure 9

Optimal eddy viscosity ${\nu }_t^L$ of the flow in a square duct at Reynolds number $\text{Re}=3500$ at $y/h=0.25$, 0.5, 0.75, and 1. The training flow is at Reynolds number $\text{Re}=2200$.

Figure 10

Secondary flow fields in a square duct at Reynolds number $\text{Re}=3500$, including (a) RANS simulated results, (b) DNS data, and (c) prediction of the RSM with implicit treatment. The training flow is at Reynolds number $\text{Re}=2200$. The color of the contour denotes the magnitude of the secondary flow. The light color here indicates large magnitude.

Figure 11

Secondary flow in a square duct at Reynolds number $\text{Re}=3500$ predicted by the RSM with implicit treatment at $y/h=0.25$, 0.5, 0.75, and 1 including (a) $U_y$ and (b) $U_z$. The training flow is at Reynolds number $\text{Re}=2200$.

Figure 12

Comparison of predicted secondary velocity $U_z$ at Reynolds number $\text{Re}=1.25\times {10}^5$ with experimental data [53] (denoted by $▴)$. Comparisons are shown along (a) the vertical axis of symmetry and (b) the diagonal of the duct. The training flow is at Reynolds number $\text{Re}=2200$.

Figure 13

Prediction of (a) the shear stress component ${\mathbf{\tau }}_{xy}$ of the Reynolds stress tensor and (b) the TKE at $x/H=1,2,...,8$. The test flow is the flow over a periodic hill at $\text{Re}=5600$. The training flow is at the same Reynolds number but has a steeper hill profile, as shown in Fig. 2.

Figure 14

Prediction of the shear stress component ${\mathbf{\tau }}_{xy}$ of the linear part of Reynolds stress at $x/H=1,2,...,8$. The test flow is the flow over a periodic hill at $\text{Re}=5600$. The training flow is at the same Reynolds number but has a steeper hill profile, as shown in Fig. 2.

Figure 15

Machine-learning-predicted optimal eddy viscosity at $x/H=1,2,...,8$. The test flow is the flow over a periodic hill at $\text{Re}=5600$. The training flow is at the same Reynolds number but has a steeper hill profile, as shown in Fig. 2.

Figure 16

Streamwise velocity by the RSM with implicit treatment at $x/H=1,2,...,8$. The test flow is the flow over a periodic hill at $\text{Re}=5600$. The training flow is at the same Reynolds number but has a steeper hill profile, as shown in Fig. 2.