Synthetic turbulent inflow generator using machine learning

We propose a methodology for generating time-dependent turbulent inflow data with the aid of machine learning (ML), which has the possibility to replace conventional driver simulations or synthetic turbulent inflow generators. As for the ML model, we use an autoencoder-type convolutional neural network with a multilayer perceptron. For the test case, we study a fully developed turbulent channel flow at the friction Reynolds number of ${\mathrm{Re}}_{\tau }=180$ for easiness of assessment. The ML models are trained using a time series of instantaneous velocity fields in a single cross section obtained by direct numerical simulation (DNS) so as to output the cross-sectional velocity field at a specified future time instant. From the a priori test in which the output from the trained ML model are recycled to the input, the spatiotemporal evolution of cross-sectional structure is found to be reasonably well reproduced by the proposed method. The turbulence statistics obtained in the a priori test are also, in general, in reasonable agreement with the DNS data, although some deviation in the flow rate was found. It is also found that the present machine-learned inflow generator is free from the spurious periodicity, unlike the conventional driver DNS in a periodic domain. As an a posteriori test, we perform DNS of inflow-outflow turbulent channel flow with the trained ML model used as a machine-learned turbulent inflow generator (MLTG) at the inlet. It is shown that the present MLTG can maintain the turbulent channel flow for a long time period sufficient to accumulate turbulent statistics, with much lower computational cost than the corresponding driver simulation. It is also demonstrated that we can obtain accurate turbulent statistics by properly correcting the deviation in the flow rate.

Figure 1

Schematic diagram of the present machine learning and its use as an inflow generator. (a) Training stage; (b) a priori and a posteriori tests.

Figure 2

Schematic structure of the ML model using the autoencoder-type convolutional neural network with multilayer perceptron (e.g., Case 1).

Figure 3

Learning curve: (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4.

Figure 4

“Reynolds stresses” computed based on the raw output, $R_{ij}=\overline{u_i^{\prime +}{u}_j^{\prime +}}$: (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4.

Figure 5

Mean profiles of “velocity fluctuations”: (a) $\overline{u^{\prime }}$, (b) $\overline{v^{\prime }}$, (c) $\overline{w^{\prime }}$.

Figure 6

Statistics based on the corrected fluctuations: (a) $u_{\mathrm{rms}}^{+}$, (b) $v_{\mathrm{rms}}^{+}$, (c) $w_{\mathrm{rms}}^{+}$, (d) $p_{\mathrm{rms}}^{+}$, (e) $-\overline{u^{\prime \prime +}{v}^{\prime \prime +}}$, (f) ${\omega }_{x\mathrm{rms}}^{+}$.

Figure 7

Spanwise energy spectrum: (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4.

Figure 8

Cross-sectional contour of streamwise velocity fluctuations $u^{\prime }$ after recycling for 200 time steps (i.e., about 250 wall unit time): (a) DNS, (b) Case 1, (c) Case 2, (d) Case 3, (e) Case 4. Animations are available at Ref. [34].

Figure 9

Temporal statistics in the $apriori$ test: (a) temporal spectrum at $y^{+}=13.2$; (b) temporal two-point correlation coefficient at $y^{+}=13.2$; (c) temporal spectrum at $y^{+}=177.0$; (d) temporal two-point correlation coefficient at $y^{+}=177.0$.

Figure 10

Spatiotemporal development of peak RMS velocity: (a) $u_{\mathrm{MLTG}}^{\prime }/u_{\mathrm{DNS}}^{\prime }$; (b) $v_{\mathrm{MLTG}}^{\prime }/v_{\mathrm{DNS}}^{\prime }$; (c) $w_{\mathrm{MLTG}}^{\prime }/w_{\mathrm{DNS}}^{\prime }$; (d) ${\mathrm{Re}}_{\tau }$.

Figure 11

Turbulence statistics in $aposteriori$ test using MLTG: (a) mean velocity profile; (b) RMS of $u_i^{\prime }$; (c) RMS of ${\omega }_i^{\prime }$; (d) shear stress balance; (e) streamwise energy spectrum of $u^{\prime }$; (f) spanwise energy spectrum of $u^{\prime }$.