Molecular Fluctuations Inhibit Intermittency in Compressible Turbulence

TL;DR

This work demonstrates that molecular fluctuations, incorporated via nonlinear fluctuating hydrodynamics (FHD) with stochastic fluxes, significantly alter compressible turbulence statistics across the entire dissipation range. The authors show that these fluctuations generate a thermal energy spectrum that competes with turbulent energy beyond a crossover wavenumber $k_ ext{th}$ (roughly $k_ ext{th}\approx k_{\eta}/3$) and, crucially, suppress spatio-temporal intermittency, yielding near-Gaussian statistics in the near- and far-dissipation ranges. By performing large-scale DNS of FHD for nitrogen at STP, they reveal that vorticity and divergence statistics become Gaussian and that high-pass measurements of the velocity gradient lose intermittency in the presence of fluctuations. These findings suggest that compressible Navier–Stokes models should be augmented with molecular fluctuations for accurate dissipation-range turbulence predictions, with broad implications for astrophysical, reactive, and hypersonic flows and for turbulence closures.

Abstract

In the standard picture of fully-developed turbulence, highly intermittent hydrodynamic fields are nonlinearly coupled across scales, where local energy cascades from large scales into dissipative vortices and large density gradients. Microscopically, however, constituent fluid molecules are in constant thermal (Brownian) motion, but the role of molecular fluctuations on large-scale turbulence is largely unknown, and with rare exceptions, it has historically been considered irrelevant at scales larger than the molecular mean free path. Recent theoretical and computational investigations have shown that molecular fluctuations can impact energy cascade at Kolmogorov length scales. Here we show that molecular fluctuations not only modify energy spectrum at wavelengths larger than the Kolmogorov length in compressible turbulence, but they also significantly inhibit spatio-temporal intermittency across the entire dissipation range. Using large-scale direct numerical simulations of computational fluctuating hydrodynamics, we demonstrate that the extreme intermittency characteristic of turbulence models is replaced by nearly-Gaussian statistics in the dissipation range. These results demonstrate that the compressible Navier-Stokes equations should be augmented with molecular fluctuations to accurately predict turbulence statistics across the dissipation range. Our findings have significant consequences for turbulence modeling in applications such as astrophysics, reactive flows, and hypersonic aerodynamics, where dissipation-range turbulence is approximated by closure models.

Figures (7)

• Figure 1: (a) Probability distribution function (PDF) of local vorticity $\omega$ normalized by their ensemble standard deviation $\sigma_{\omega}$ averaged over at least $8\tau_{\lambda}$, where $\tau_{\lambda}$ is the eddy turnover time for deterministic and fluctuating hydrodynamics (FHD) simulations. The PDF from an FHD simulation at thermodynamic equilibrium without turbulent forcing, FHD (eq.), is also plotted. $3$D visualization of local vorticity magnitude $|\omega|$ in a (b) deterministic and an (c) FHD simulation. Here, $|\omega|$ is normalized by the standard deviation of vorticity fluctuations at thermodynamic equilibrium $\sigma_{\omega}^{\text{eq}}\approx5\times10^6\text{s}^{-1}$; the standard deviation of vorticity fluctuations $\sigma_{\omega}\approx7.3\times10^6\text{s}^{-1}$ and $\sigma_{\omega}\approx6.3\times10^6\text{s}^{-1}$ for deterministic and FHD simulations respectively.
• Figure 2: (a) PDF of local divergence $\mathcal{D}$ normalized by its ensemble standard deviation $\sigma_{\mathcal{D}}$ for deterministic and fluctuating hydrodynamics (FHD) simulations. The inset in (a) shows the PDF of local Mach number $\hbox{Ma}$ in FHD (orange) and deterministic (blue) simulations. $3$D visualization of local divergence in a (b) deterministic and an (c) FHD simulation. Here, $\mathcal{D}$ is normalized by the standard deviation of divergence fluctuations that are $\sigma_{\mathcal{D}}\approx3.1\times10^5\text{s}^{-1}$ and $\sigma_{\mathcal{D}}\approx8.7\times10^6\text{s}^{-1}$ for deterministic and FHD simulations respectively.
• Figure 3: Mean low-pass filtered dissipation rate $\langle \epsilon ^{<}(k)\rangle$ as a function of the wavenumber $k$ computed from the mean mean low-pass filtered enstrophy in Equation \ref{['eq:enstrophy']} for deterministic Navier-Stokes and FHD simulations of compressible turbulence.
• Figure 4: (a) Comparison of the total kinetic energy spectrum $\langle E(k) \rangle$ in FHD vs. deterministic simulations. Three approximate ranges of length scales are highlighted: inertial sub-range (ISR in blue), near-dissipation range (NDR in pink) and far-dissipation range (FDR in green). In FHD simulations the thermal spectrum $E_{\text{th}}(k) = \frac{3k_B \langle T \rangle}{2\langle \rho \rangle} 4\pi k^{2}$ (red dashed-dot line) dominates for wavenumbers larger than the thermal crossover scale $k_{\text{th}}$, where $k_B$ is the Boltzmann constant. (b) Standard deviation in total kinetic energy spectrum $\delta E(k) = \langle \left(E(k) - \langle E(k)\rangle \right)^{2} \rangle^{1/2}$ normalized by $\langle E(k) \rangle$.
• Figure 5: (a) Comparison of dilatational kinetic energy $\langle E_d(k) \rangle$ in FHD vs. deterministic simulations. The FHD simulations transitions over to the thermal energy spectrum is $E_{d,\text{th}}(k)=(1/3)E_{\text{th}}(k)$ (red dashed-dot line) at $k_{\text{th}}$. (b) Standard deviation in the dilatational kinetic energy spectrum $\delta E_d(k) = \langle \left(E_d(k) - \langle E_d(k)\rangle \right)^{2} \rangle^{1/2}$ normalized by $\langle E_d(k) \rangle$.