Predictive large-eddy-simulation wall modeling via physics-informed neural networks

While data-based approaches were found to be useful for subgrid scale (SGS) modeling in Reynolds-averaged Navier-Stokes (RANS) simulations, there have not been many attempts at using machine learning techniques for wall modeling in large-eddy simulations (LESs). Large-eddy simulation differs from RANS simulation in many aspects. For one thing, LES is scale resolving. For another, LES is in and of itself a high-fidelity tool. Because data sets of higher fidelity are in general not frequently accessible or available, this poses additional challenges to data-based modeling in LES. Further, SGS modeling usually needs flow information at only large scales, in contrast with wall modeling, which needs to account for both near-wall small scales and large scales above the wall. In this work we discuss how the above-noted challenges may be addressed when taking a data-based approach for wall modeling. We also show the necessity of incorporating physical insights in model inputs, i.e., using inputs that are inspired by the vertically integrated thin-boundary-layer equations and the eddy population density scalings. We show that the inclusion of the above physics-based considerations would enhance extrapolation capabilities of a neural network to flow conditions that are not within the train data. Being cheap to evaluate and using only channel flow data at ${\mathrm{Re}}_{\tau }=1000$, the trained networks are found to capture the law of the wall at arbitrary Reynolds numbers and outperform the conventional equilibrium model in a nonequilibrium flow.

Figure 1

(a) Schematic of the setup of WMLES. The near-wall turbulence is not resolved, therefore the no-slip condition does not apply. Instead, a wall model is used. (b) Mean velocity profiles of a channel at ${\mathrm{Re}}_{\tau }\approx 180$ [64] and ${\mathrm{Re}}_{\tau }\approx 5200$ [31]. The plus superscript indicates normalization by wall units. The dashed lines are at $y^{+}=18$ and $y^{+}=520$, respectively, which are roughly the locations of the first off-wall grid of typical WMLES of the two flows.

Figure 2

Schematic of the modeled boundary-layer structure. The attached eddies are space filling and therefore the number of eddies doubles as their sizes halve.

Figure 3

Schematic of a sample neural network. The neural network is fully connected. It has two hidden layers HL1 and HL2. The input layer contains three neurons and the output layer contains one neuron.

Figure 4

Training data, network, and expected behavior. Because of symmetry, data are only shown for $x>0$ for brevity.

Figure 5

(a) Wall-shear stress as a function of the velocity at $y^{+}=50$. The mean of the velocity is removed. The LOW is Eq. (5). (b) Same as (a) but at a wall-normal height $y^{+}=10000$. In (a), all three lines are almost on top of each other. In (b), $f_{\mathrm{N}2}$ and the LOW are on top of each other.

Figure 6

Sketch of the spatial filtering of the DNS data. Both the instantaneous velocity and the instantaneous wall stress are filtered within a virtual LES computational cell of size $\mathrm{\Delta }x\times \mathrm{\Delta }y\times \mathrm{\Delta }z=h_{\text{wm}}\mathcal{R}\times h_{\text{wm}}\times h_{\text{wm}}\mathcal{R}$, where $\mathcal{R}$ varies from 1 to 12. A local coordinate system $(t,q,n)$ is such that $t$ is in the direction of the filtered velocity at $y=h_{\mathrm{wm}}$, $n$ is the wall-normal direction, and $q$ is perpendicular to both $t$ and $n$.

Figure 7

(a) The left axis gives the wall-shear stress as a function of the wall-parallel velocity at $y^{+}\approx 10$: $\times$, filtered DNS data; —, predictions of NN1; $--$, the mean velocity and the mean wall-shear stress; $--$, predictions of the zonal model. The right axis gives the PDF of the wall-parallel velocity: $--$, PDF of the filtered velocity. All quantities are normalized using wall units. (b) Same as (a) but at the wall-normal distance $y^{+}\approx 100$.

Figure 8

(a) Wall shear stress as a function of the off-wall velocity at $y^{+}\approx 10$. Different colors are used for data of different $\mathcal{R}$. The DNS data are dots and the neural network outputs are lines; WM denotes wall model. (The red and orange lines are neural network outputs for $\mathcal{R}=5$ and 9, respectively.) The thin black line shows the prediction of the zonal model. (b) Same as (a) but at $y^{+}\approx 100$.

Figure 9

(a) Crossflow wall-shear stress as a function of the crossflow pressure gradient for $h_{\text{wm}}^{+}=10$. (b) Same as (a) but for $h_{\text{wm}}^{+}=100$.

Figure 10

(a) Parameter space in terms of $h_{\mathrm{wm}}^{+}$ and $u_{\parallel }^{+}$. The contour shows the PDF of the training data. The ${\mathrm{Re}}_{\tau }={10}^{10}$ channel will cover the parameter space around the circled position. (b) Same as (a) but for parameter space in terms of $ln(h_{\mathrm{wm}}/y_0)/u_{\parallel }^{+}$ and $u_{\parallel }^{+}/h_{\mathrm{wm}}$.

Figure 11

(a) Mean velocity profiles as a function of wall-normal distance for a channel at ${\mathrm{Re}}_{\tau }=1000$. Symbols show the results of WMLES. (b) Mean velocity profiles as a function of wall-normal distance for channel flow at friction Reynolds numbers from ${\mathrm{Re}}_{\tau }={10}^3$ to ${\mathrm{Re}}_{\tau }={10}^{10}$. The symbols show the results of WMLES. The thin solid line is the logarithmic law of the wall and the bold solid line is the mean profile of the DNS channel at ${\mathrm{Re}}_{\tau }=1000$.

Figure 12

Root mean square of the streamwise velocity for the same legend as in (a) Fig. 11 and (b) Fig. 11.

Figure 13

(a) Contours of the instantaneous velocity at the wall-normal distance $y/\delta =0.047$ at $t=0$. (b) Same as (a) but at the wall-normal distance $y/\delta =0.42$. (c) Same as (a) but at a time $t=\delta /u_{\tau }$. The solid white line is the direction of the shear stress at the wall. (d) Same as (c) but at a wall-normal distance $y/\delta =0.42$. Here $u_{\tau }$ is the friction velocity of the two-dimensional channel. An underline denotes a vector and $|·|$ is the norm of the bracketed quantity.

Figure 14

Direction of the spatially averaged wall-shear stress as a function of the time. Here $\alpha$ is the angle between the spatially averaged wall-shear stress and the streamwise direction of the two-dimensional channel.

Figure 15

Spatially averaged spanwise velocity as a function of the wall-normal distance at $t{u}_{\tau }/\delta \approx 0.2$, 0.4, 0.6, 0.8, and 1.0 (from bottom to top). The WMLES results are plotted every other point for a better presentation. Velocities are normalized by the friction velocity of the two-dimensional channel flow.