Artificial neural network approach to large-eddy simulation of compressible isotropic turbulence

A subgrid-scale (SGS) model for large-eddy simulation (LES) of compressible isotropic turbulence is constructed by using a data-driven framework. An artificial neural network (ANN) based on local stencil geometry is employed to predict the unclosed SGS terms. The input features are based on the first-order and second-order derivatives of filtered velocity and temperature which appear in the second-order Taylor approximation of the SGS stress and heat flux. It is shown that the proposed ANN-7 model performs better than the gradient model in the a priori test. The correlation coefficient is larger and the relative error is smaller for ANN-7 model as compared to those of the gradient model in the a priori test. In an a posteriori analysis, the performance of ANN-7 model shows advantage over the dynamic Smagorinsky model and dynamic mixed model in the prediction of spectra and structure functions of velocity and temperature, and instantaneous flow structures. Artificial neural network is a promising tool for understanding the physical fundamentals of SGS unclosed terms with further improvement.

Figure 1

Velocity spectrum from numerical simulations of solenoidally forced compressible isotropic turbulence, and the diamond represents cutoff $n=16$.

Figure 2

Schematic diagram of the ANN's network structure.

Figure 3

Contributions of $\partial {\tilde{u}}_i^{l,m,n}/\partial x_p$ and $\partial {\tilde{u}}_i^{l-1,m,n}/\partial x_p$ to ${\tau }_{11}$ for VG1 and ANN-7 models. The dashed curve denotes a parabolic fitting function for the ANN-7 model.

Figure 4

Spectra of velocity and its compressible component at $M_t=0.4$ for LES at grid resolutions of ${64}^3(h_{\mathrm{LES}}=\mathrm{\Delta })$ and ${128}^3(h_{\mathrm{LES}}=\mathrm{\Delta }/2)$ with the same filter width $\mathrm{\Delta }$. The first row shows spectra of velocity and the second row shows spectra of compressible velocity.

Figure 5

Spectra of velocity and its compressible component at $M_t=0.6$ for LES at grid resolutions of ${64}^3(h_{\mathrm{LES}}=\mathrm{\Delta })$ and ${128}^3(h_{\mathrm{LES}}=\mathrm{\Delta }/2)$ with the same filter width $\mathrm{\Delta }$. The first row shows spectra of velocity and the second row shows spectra of compressible velocity.

Figure 6

Spectra of velocity and its compressible component at $M_t=0.8$ for LES at grid resolutions of ${64}^3(h_{\mathrm{LES}}=\mathrm{\Delta })$ and ${128}^3(h_{\mathrm{LES}}=\mathrm{\Delta }/2)$ with the same filter width $\mathrm{\Delta }$. The first row shows spectra of velocity and the second row shows spectra of compressible velocity.

Figure 7

Spectra of velocity and its compressible component at $M_t=1.0$ for LES at grid resolutions of ${64}^3(h_{\mathrm{LES}}=\mathrm{\Delta })$ and ${128}^3(h_{\mathrm{LES}}=\mathrm{\Delta }/2)$ with the same filter width $\mathrm{\Delta }$. The first row shows spectra of velocity and the second row shows spectra of compressible velocity.

Figure 8

Spectrum of temperature $T$ at $M_t=0.4$, 0.6, 0.8, and 1.0 for LES at ${64}^3$ grid resolution, where $h_{\mathrm{LES}}=\mathrm{\Delta }$.

Figure 9

Spectrum of temperature $T$ at $M_t=0.4$, 0.6, 0.8, and 1.0 for LES at ${128}^3$ grid resolution, where $h_{\mathrm{LES}}=\mathrm{\Delta }/2$.

Figure 10

Spectrum of velocity at $M_t=0.4$, 0.6, 0.8, and 1.0 for LES with DMM and ANN-7 at ${128}^3$ grid resolution, where $h_{\mathrm{LES}}=\mathrm{\Delta }/2$ and DMM at ${256}^3$ grid resolution, where $h_{\mathrm{LES}}=\mathrm{\Delta }/4$.

Figure 11

Spectrum of compressible velocity component at $M_t=0.4$, 0.6, 0.8, and 1.0 for LES with DMM and ANN-7 at ${128}^3$ grid resolution, where $h_{\mathrm{LES}}=\mathrm{\Delta }/2$ and DMM at ${256}^3$ grid resolution, where $h_{\mathrm{LES}}=\mathrm{\Delta }/4$.

Figure 12

Spectrum of temperature $T$ at $M_t=0.4$, 0.6, 0.8, and 1.0 for LES with DMM and ANN-7 at ${128}^3$ grid resolution, where $h_{\mathrm{LES}}=\mathrm{\Delta }/2$ and DMM at ${256}^3$ grid resolution, where $h_{\mathrm{LES}}=\mathrm{\Delta }/4$.

Figure 13

Structure functions of the velocity and its compressible component at $M_t=0.4$, where the first row shows the structure functions of the velocity and the second row shows the structure functions of the compressible velocity component for LES at ${64}^3$ grid resolution, where $h_{\mathrm{LES}}=\mathrm{\Delta }$.

Figure 14

PDF of the normalized SGS flux $\mathrm{\Pi }/{\varepsilon }_T$ at $M_t=0.4$, 0.6, 0.8, and 1.0 for LES at ${64}^3$ grid resolution, where $h_{\mathrm{LES}}=\mathrm{\Delta }$.

Figure 15

PDF of the normalized SGS flux $\mathrm{\Pi }/{\varepsilon }_T$ at $M_t=0.4$, 0.6, 0.8, and 1.0 for LES at ${128}^3$ grid resolution, where $h_{\mathrm{LES}}=\mathrm{\Delta }/2$.

Figure 16

PDF of the normalized SGS flux $\mathrm{\Pi }/{\varepsilon }_T$ at $M_t=0.4$, 0.6, 0.8, and 1.0 for LES with DMM and ANN-7 at ${128}^3$ grid resolution, where $h_{\mathrm{LES}}=\mathrm{\Delta }/2$ and DMM at ${256}^3$ grid resolution, where $h_{\mathrm{LES}}=\mathrm{\Delta }/4$.

Figure 17

Contours of the normalized velocity divergence $\tilde{\theta }/{\tilde{\theta }}_{\mathrm{fDNS}}^{\mathrm{rms}}$ on an arbitrarily selected $x\text{−}y$ slice at $M_t=0.4$ and $t/\tau =3.37$. (a) fDNS, (b) DSM, (c) DMM, and (d) ANN-7 for LES at ${64}^3$ grid resolution, where $h_{\mathrm{LES}}=\mathrm{\Delta }$.

Figure 18

Contours of the normalized velocity divergence $\tilde{\theta }/{\tilde{\theta }}_{\mathrm{fDNS}}^{\mathrm{rms}}$ on an arbitrarily selected $x\text{−}y$ slice at $M_t=0.4$ and $t/\tau =3.37$. (a) fDNS, (b) DSM, (c) DMM, and (d) ANN-7 for LES at ${128}^3$ grid resolution, where $h_{\mathrm{LES}}=\mathrm{\Delta }/2$.

Figure 19

Contours of the normalized velocity divergence $\tilde{\theta }/{\tilde{\theta }}_{\mathrm{fDNS}}^{\mathrm{rms}}$ on an arbitrarily selected $x\text{−}y$ slice at $M_t=1.0$ and $t/\tau =3.21$. (a) fDNS, (b) DSM, (c) DMM, and (d) ANN-7 for LES at ${64}^3$ grid resolution, where $h_{\mathrm{LES}}=\mathrm{\Delta }$.

Figure 20

Contours of the normalized velocity divergence $\tilde{\theta }/{\tilde{\theta }}_{\mathrm{fDNS}}^{\mathrm{rms}}$ on an arbitrarily selected $x\text{−}y$ slice at $M_t=1.0$ and $t/\tau =3.21$. (a) fDNS, (b) DSM, (c) DMM, and (d) ANN-7 for LES at ${128}^3$ grid resolution, where $h_{\mathrm{LES}}=\mathrm{\Delta }/2$.