Machine-learning-augmented domain decomposition method for near-wall turbulence modeling

To tackle the challenging near-wall turbulence modeling while preserving low computational cost, the near-wall nonoverlapping domain decomposition (NDD) method is proposed, incorporating the machine-learning technique. Using recently proposed implicit NDD (INDD), the solution can be calculated with a Robin-type (slip) wall boundary condition on a relatively coarse mesh and then corrected in the near-wall region at every iteration through an estimated turbulent viscosity profile obtained from a neural network. To maintain a reasonable complexity with acceptable accuracy, only the near-wall field properties, i.e., wall-normal distance, near-wall velocities, streamwise pressure gradients, and one near-wall scale parameter, are employed as the input features for the neural network. The benefit of incorporating machine learning is twofold. First, the near-wall turbulent viscosity is predicted more accurately than by the traditional algebraic functions used in the approximate approach. Second, similarly to the conventional NDD, the present simulations save one order of computational cost over the fully resolved one-block simulations. The accuracy and efficiency of the method are demonstrated on test cases with the $k\text{−}\varepsilon$ model, including turbulent flows in a channel and an asymmetric diffuser at different Reynolds numbers. Results compare favorably with the one-block benchmark solutions and show a better agreement when compared with approximate NDD predictions. In the latter case, a variation of diffuser geometry is also considered to test the model performance on engineering design tasks with another turbulent model (i.e., Spalart-Allmaras), showing good generalization capability on different turbulence closures.

Figure 1

Sketch of the data-driven implicit NDD.

Figure 2

Sketch of the solution obtained in two stages of implicit NDD.

Figure 3

Channel flow, interface position $y^{*+}=100$. Comparison of nondimensional turbulent viscosity ${\nu }_t^{+}$ (a) and streamwise velocity $u^{+}$ (b) between the results from implicit NDD (${\text{Re}}_{\tau }=590$ in the red line, 1995 in the blue line, 5180 in the green line, and 9000 in the magenta line) and the one-block LRN solution.

Figure 4

Geometry of asymmetric diffuser.

Figure 5

Diffuser flow, $\text{Re}=1.8\times {10}^4$, interface position $y^{*+}=40$. (a) Comparison of mean streamwise velocities $v_x$ between the results from experiment [60], one-block solution, explicit (approximate), and implicit NDD. (b) Enlargement of the region from the straight wall to $y/H=0.06$. (c) Enlargement of the shaded region in (a).

Figure 6

Diffuser flow, $\text{Re}=1.8\times {10}^4$, interface position $y^{*+}=40$. Comparison of spanwise velocities $v_y$ between the results from the one-block solution, explicit and implicit NDD.

Figure 7

Diffuser flow, $\text{Re}=1.8\times {10}^4$, interface position $y^{*+}=40$. Comparison of $C_f$ and $C_p$ between the results from experiment [60], one-block solution, explicit and implicit NDD. The subfigures (b), (c), and (d) share the same legend as (a).

Figure 8

Diffuser flow, $\text{Re}=3.6\times {10}^4$, interface position $y^{*+}=60$. (a) Comparison of mean stream-wise velocities $v_x$ between the results from one-block solution, explicit (approximate), and implicit NDD. (b) Comparison of velocity component $v_y$ between the results from one-block solution, explicit and implicit NDD.

Figure 9

The variation in opening angle $\theta$ of the diffuser flow cases.

Figure 10

The adopted training and predictive cases.

Figure 11

Comparison of $C_f$ and $C_p$ on the inclined wall of the diffuser flow for different cases with $Re$ and $\theta$ specified at the left bottom of respective subfigures. Subfigures (c) and (d) share the same legend as (b).

Figure 12

Diffuser flow, $\text{Re}=1.8\times {10}^4$, interface position $y^{*+}=40$. (a) Comparison of mean stream-wise velocities $v_x$ between the results from experiment [60], one-block solution, and implicit NDD with Spalart-Allmaras model. (b) Comparison of velocity component $v_y$ between the results from one-block solution and implicit NDD with Spalart-Allmaras model. (c) Enlargement of the shaded region in (a).

Figure 13

Relative computational times with the explicit (approximate) and implicit NDD for both $k\text{−}\varepsilon$ and Spalart-Allmaras models. The times are normalized by that of companion one-block solution.

Figure 14

The variations of training and validation loss with training epochs for different cases with the $k\text{−}\varepsilon$ model (a) and the Spalart-Allmaras model (b).

Figure 15

Diffuser flow, $\text{Re}=1.8\times {10}^4$, interface position $y^{*+}=40$. Comparison of $C_f$ and $C_p$ along the inclined wall between the results from the one-block solution, original INDD [20], and data-driven INDD.

Figure 16

Error evolution of the cases ($\text{Re}=1.8\times {10}^4,\theta ={10}^o$) at $y^{*+}=40$ with the $k\text{−}\varepsilon$ model and the Spalart-Allmaras model.