Large eddy simulation of transitional channel flow using a machine learning classifier to distinguish laminar and turbulent regions

While wall modeling enables significant reduction in computational cost compared to wall-resolved large eddy simulations (LESs), it often fails to capture laminar-to-turbulence transition processes realistically. This issue arises in part because wall models typically assume that the near-wall flow is in a statistically quasiequilibrium turbulent state and hence incorrectly prescribe turbulent wall stresses in regions that are still laminar during transition. In this work we propose an approach in which the application of the wall model is retained within the turbulent regions of transitional flow where even nascent spots exhibit high-Reynolds-number characteristics, but the wall model is not applied in laminar regions. The local distinction between turbulent and laminar regions is performed using a self-organized map (SOM) [see Z. Wu et al., Phys. Rev. Fluids 4, 023902 (2019)], an unsupervised machine learning classifier. We demonstrate the capability of wall-modeled LES with SOM-based turbulent/nonturbulent classification (WMSOM) in predicting both bypass and orderly transitions in channel flow at target Reynolds numbers of ${\text{Re}}_{\tau }=130$ and 200, respectively. Predictions of bypass transition initiated from localized initial disturbances agree well with direct numerical simulation. For orderly transition, we simulate K- and H-type transitions, due to the interaction of two- and three-dimensional instability waves. We show good predictions for both scenarios, with a slight delay in the transition time. The WMSOM approach offers a significant reduction in computational cost compared to wall-resolved LES.

Figure 1

Streamwise velocity $u$ contour from DNS training data with the T/NT interface at $y=0.53$.

Figure 2

Schematic of the computational domain with the base flow, and the initial perturbation visualized using isosurfaces of streamwise vorticity ${\tilde{\omega }}_x$. The isosurfaces shown correspond to ${\tilde{\omega }}_x=+0.003$ (red) and ${\tilde{\omega }}_x=-0.003$ (blue).

Figure 3

Large-amplitude ($\epsilon =0.2097$) localized perturbation contours of (a) $v^{\prime }$ and (b) $w^{\prime }$ with $\theta ={20}^{\circ }$ at $y=0.25$. The contours of (c) $v^{\prime }$ and (d) $w^{\prime }$ at $z=12.96$ are from the initial perturbation with $\theta =0^{\circ }$.

Figure 4

Evolution of small-amplitude ($\epsilon =1.5\times {10}^{-4}$ and $\theta =0^{\circ }$) (a) $v^{\prime }$ and (b) $u^{\prime }$ perturbation at $y=0.25$ and $t=10,20,30,40$.

Figure 5

Total perturbation kinetic energy for small-amplitude localized disturbance ($\epsilon =1.5\times {10}^{-4}$): (a) energy decay for different initial orientations and (b) comparison between the WMSOM (red lines) and DNS (black markers) [16]. Here $E_0=E(t=0)$ is initial energy of the perturbation.

Figure 6

Contours of wall stress ${\tau }_w$ at $y=0$ and $t=20,80,110,300$ from the (a) WRLES, (b) WMLES, and (c) WMSOM (black solid lines represent the T/NT interface). (d) Time evolution of ${\text{Re}}_{\tau }$ from LES and DNS [37]. Vertical dotted lines correspond to the time at which the contours are plotted.

Figure 7

(a) Contours of $|u^{\prime }|$ with the T/NT interface (black line) at $y=0.25$ and $t=20,80,110,300$ and (b) time evolution of ${\text{Re}}_{\tau }$ from the WMSOM for different initial orientations $\theta =0^{\circ },{10}^{\circ },{20}^{\circ },{45}^{\circ }$. Vertical dotted lines correspond to the time at which the contours are plotted. The contours of $|v^{\prime }|+|w^{\prime }|$ at $t=0$ with axes are plotted on the left side of $t=20$ contours. Grayscale and hot colormaps represent nonturbulent (NT) and turbulent (T) regions, respectively.

Figure 8

Time evolution of (a) total perturbation kinetic energy for large-amplitude disturbance with $\theta =0^{\circ },{10}^{\circ },{20}^{\circ },{45}^{\circ }$ and (b) comparison of kinetic energy evolution at initial times between large- (red) and small- (blue) amplitude perturbation.

Figure 9

Evolution of streamwise averaged streamwise energy spectra $[{\overline{E}}_u\left(n_z\right)=\frac{1}{2}{\langle {\widehat{u}}^{\prime 2}(x,y=0.25,n_z)\rangle }_x]$ at $t=0,10,20,30,40$ as a function of spanwise integer wave number $n_z=k_z{L}_z/2\pi$ for initial perturbation with (a) $\theta =0^{\circ }$ and (b) $\theta ={45}^{\circ }$.

Figure 10

Evolution of spanwise averaged streamwise energy spectra $[{\overline{E}}_u\left(n_x\right)=\frac{1}{2}{\langle {\widehat{u}}^{\prime 2}(n_x,y=0.25,z)\rangle }_z]$ at $t=0,10,20,30,40$ as a function of streamwise integer wave number $n_x=k_x{L}_x/2\pi$ for initial perturbation with (a) $\theta =0^{\circ }$ and (b) $\theta ={45}^{\circ }$.

Figure 11

Evolution of streamwise perturbation $|u^{\prime }|$ at $t=0,10,20,30,40$ for the $\theta =0^{\circ },{45}^{\circ }$cases.

Figure 12

Schematic of the flow configuration with the initial perturbation velocity profiles for the orderly transition.

Figure 13

Eigenfunction profiles at $\text{Re}=3333.33$ of (a) and (c) streamwise and (b) and (d) wall-normal components of Tollmien-Schlichting and oblique waves. The plots show the real ($–.–.$), imaginary ($––$ ), and absolute ($—$) values of the eigenmodes.

Figure 14

Contours of wall stress ${\tau }_w$ at $y=0$ from the (a) WRLES, (b) WMLES, and (c) WMSOM with the T/NT interface at different times. The transition times for the WRLES and WMSOM are $t^{*}=40$ and 100, respectively. (d) Comparison of ${\text{Re}}_{\tau }$ evolution from the WRLES, WMLES, and WMSOM with that from the DNS. Circle markers correspond to the time at the which wall stress contours are plotted. Grayscale and hot colormaps represent nonturbulent and turbulent regions, respectively.

Figure 15

(a) Contours of streamwise perturbation $|u^{\prime }|$ from the WMSOM for different initial perturbations $\varphi =0,\pi /8,\pi /2,3\pi /4$ at $y=0.18$ and $t-t^{*}=-20,10,20,810$ with $t^{*}=100,290,100,80$. (b) Plot of ${\text{Re}}_{\tau }$ time evolution from DNS ($—$) and WRLES for $\varphi =0$ ($—$), $\varphi =\pi /8$ ($––$), $\varphi =\pi /4$ ($·····$), $\varphi =\pi /2$ ($‐·‐·$), and $\varphi =3\pi /4$ ($\sout{✱}$); WMLES for $\varphi =0$ ($—$), $\varphi =\pi /8$ ($✱$), $\varphi =\pi /4$ ($○$), $\varphi =\pi /2$ ($⛛$), and $\varphi =3\pi /4$ ($\square$); and the WMSOM for $\varphi =0$ ($—$), $\varphi =\pi /8$ ($––$), $\varphi =\pi /4$ ($·····$), $\varphi =\pi /2$ ($‐·$), and $\varphi =3\pi /4$ ($\sout{✱}$). Grayscale and hot colormaps represent nonturbulent and turbulent regions, respectively.

Figure 16

Plot of ${\text{Re}}_{\tau }$ time evolution from WRLES for $\varphi =0$ ($—$), WMLES for $\varphi =0$ ($—$), and the WMSOM for $\varphi =0$ ($—$), $\varphi =\pi /8$ ($––$), $\varphi =\pi /4$ ($·····$), $\varphi =\pi /2$ ($‐·‐·$), and $\varphi =3\pi /4$ ($\sout{✱}$) of perturbation with ${\epsilon }_{3\text{D}}=6\times {10}^{-3}$.

Figure 17

Plot of $|u^{\prime }|$ contours from the WMSOM $(\varphi =0,{\epsilon }_{3\text{D}}=6\times {10}^{-3})$ at $y=0.18$ and $t=25,200$ showing (a) the K-type and (b) the H-type transition to turbulence.

Figure 18

Spanwise averaged streamwise energy spectra $[{\overline{E}}_u\left(k_x\right)=\frac{1}{2}{\langle {\widehat{u}}^{\prime }{(k_x,y,z)}^2\rangle }_z]$ as a function of ${\kappa }_x=k_x{\lambda }_x/2\pi$ at $y=0.18$ and $t=25$ showing (a) the K-type and (b) the H-type transition.

Figure 19

Red circles depict the numerical solution of the mixing-length RANS equations over a wide range of ${\text{Re}}_{\mathrm{\Delta }}$ [46]. The dark solid line shows the empirical fit given by Eq. (A2).