Data-driven constitutive relation reveals scaling law for hydrodynamic transport coefficients

Finding extended hydrodynamics equations valid from the dense gas region to the rarefied gas region remains a great challenge. The key to success is to obtain accurate constitutive relations for stress and heat flux. Data-driven models offer a new phenomenological approach to learning constitutive relations from data. Such models enable complex constitutive relations that extend Newton's law of viscosity and Fourier's law of heat conduction by regression on higher derivatives. However, the choices of derivatives in these models are ad hoc without a clear physical explanation. We investigated data-driven models theoretically on a linear system. We argue that these models are equivalent to nonlinear length scale scaling laws of transport coefficients. The equivalence to scaling laws justified the physical plausibility and revealed the limitation of data-driven models. Our argument also points out that modeling the scaling law could avoid practical difficulties in data-driven models like derivative estimation and variable selection on noisy data. We further proposed a constitutive relation model based on scaling law and tested it on the calculation of Rayleigh scattering spectra. The result shows our data-driven model has a clear advantage over the Chapman-Enskog expansion and moment methods.

Figure 1

Rayleigh scattering of electromagnetic (EM) waves with wave vector ${\mathbf{k}}_i$ and frequency ${\omega }_i$ through gas with density fluctuation $\delta \rho$. $\widehat{\mathbf{n}}$ is a unit vector. The intensity $I\left(\omega \right)$ of scattered EM wave with frequency shift $\omega$ is the Rayleigh scattering spectra. The Rayleigh scattering spectra are proportional to and determined by the density fluctuation spectra $\langle \delta {\rho }^2\rangle$. Therefore to compute the Rayleigh scattering spectra, we need only to compute the density fluctuation spectra of gas.

Figure 2

Flow chart of our model to calculate the density fluctuation spectra $\langle {\delta \rho }^2\rangle$.

Figure 3

Comparison between spectra calculated using DSMC, the NS equation, the Grad 13 method, and our model for various Knudsen numbers. At a small Knudsen number, the spectra consist of three peaks and correspond to entropy fluctuation and pressure fluctuation. As the Knudsen number increases, these peaks disappear gradually and blur into a bell shape. The result shows that our model-calculated spectra match the DSMC result much better than the NS equation and Grad 13 method, especially in the high Knudsen number region.

Figure 4

(a) Test comparison between our model and NS equation predicted velocity fluctuation spectra with DSMC results at Knudsen number $\text{Kn}=0.15$. Though our model has never trained on the velocity fluctuation spectra, it still outperforms the NS equation. (b) The effective viscosity $\frac{\mu \left(\mathrm{Kn}\right)}{{\mu }_0}$ and its $95\%$ confidence interval (shadowed) of our model for Rayleigh scattering, compared with the NS equation and results from nonlinear microchannel flow (IP, Karniadakis). To compare microchannel flow and Rayleigh scattering results, we match their Knudsen number at effective viscosity 0.5 by multiplying a constant to the microchannel flow Knudsen number.