In turbulent flows, kinetic energy is transferred from the largest scales to progressively smaller scales, until it is ultimately converted into heat. The Navier-Stokes equations are almost universally used to study this process. Here, by comparing with molecular-gas-dynamics simulations, we show that the Navier-Stokes equations do not describe turbulent gas flows in the dissipation range because they neglect thermal fluctuations. We investigate decaying turbulence produced by the Taylor-Green vortex and find that in the dissipation range the molecular-gas-dynamics spectra grow quadratically with wave number due to thermal fluctuations, in agreement with previous predictions, while the Navier-Stokes spectra decay exponentially. Furthermore, the transition to quadratic growth occurs at a length scale much larger than the gas molecular mean free path, namely in a regime that the Navier-Stokes equations are widely believed to describe. In fact, our results suggest that the Navier-Stokes equations are not guaranteed to describe the smallest scales of gas turbulence for any positive Knudsen number.
We report flow statistics and visualizations from gas-kinetic simulations using the Direct Simulation Monte Carlo (DSMC) method of compressible turbulent Couette flow over a porous substrate composed of an array of circular cylinders for which the Knudsen number is O(10-1). Comparisons are made with both smooth-wall DSMC simulations and direct numerical simulations of the Navier-Stokes equations for the same conditions. Roughness, permeability, and noncontinuum effects are assessed.
The Direct Simulation Monte Carlo (DSMC) method has been used for more than 50 years to simulate rarefied gases. The advent of modern supercomputers has brought higher-density near-continuum flows within range. This in turn has revived the debate as to whether the Boltzmann equation, which assumes molecular chaos, can be used to simulate continuum flows when they become turbulent. In an effort to settle this debate, two canonical turbulent flows are examined, and the results are compared to available continuum theoretical and numerical results for the Navier-Stokes equations.
The gold-standard definition of the Direct Simulation Monte Carlo (DSMC) method is given in the 1994 book by Bird [Molecular Gas Dynamics and the Direct Simulation of Gas Flows (Clarendon Press, Oxford, UK, 1994)], which refined his pioneering earlier papers in which he first formulated the method. In the intervening 25 years, DSMC has become the method of choice for modeling rarefied gas dynamics in a variety of scenarios. The chief barrier to applying DSMC to more dense or even continuum flows is its computational expense compared to continuum computational fluid dynamics methods. The dramatic (nearly billion-fold) increase in speed of the largest supercomputers over the last 30 years has thus been a key enabling factor in using DSMC to model a richer variety of flows, due to the method's inherent parallelism. We have developed the open-source SPARTA DSMC code with the goal of running DSMC efficiently on the largest machines, both current and future. It is largely an implementation of Bird's 1994 formulation. Here, we describe algorithms used in SPARTA to enable DSMC to operate in parallel at the scale of many billions of particles or grid cells, or with billions of surface elements. We give a few examples of the kinds of fundamental physics questions and engineering applications that DSMC can address at these scales.
We provide a demonstration that gas-kinetic methods incorporating molecular chaos can simulate the sustained turbulence that occurs in wall-bounded turbulent shear flows. The direct simulation Monte Carlo method, a gas-kinetic molecular method that enforces molecular chaos for gas-molecule collisions, is used to simulate the minimal Couette flow at Re=500. The resulting law of the wall, the average wall shear stress, the average kinetic energy, and the continually regenerating coherent structures all agree closely with corresponding results from direct numerical simulation of the Navier-Stokes equations. These results indicate that molecular chaos for collisions in gas-kinetic methods does not prevent development of molecular-scale long-range correlations required to form hydrodynamic-scale turbulent coherent structures.
We provide the first demonstration that molecular-level methods based on gas kinetic theory and molecular chaos can simulate turbulence and its decay. The direct simulation Monte Carlo (DSMC) method, a molecular-level technique for simulating gas flows that resolves phenomena from molecular to hydrodynamic (continuum) length scales, is applied to simulate the Taylor-Green vortex flow. The DSMC simulations reproduce the Kolmogorov -5/3 law and agree well with the turbulent kinetic energy and energy dissipation rate obtained from direct numerical simulation of the Navier-Stokes equations using a spectral method. This agreement provides strong evidence that molecular-level methods for gases can be used to investigate turbulent flows quantitatively.
The Rayleigh-Taylor instability (RTI) is investigated using the Direct Simulation Monte Carlo (DSMC) method of molecular gas dynamics. Here, two-dimensional and three-dimensional DSMC RTI simulations are performed to quantify the growth of flat and single-mode-perturbed interfaces between two atmospheric-pressure monatomic gases. The DSMC simulations reproduce all qualitative features of the RTI and are in reasonable quantitative agreement with existing theoretical and empirical models in the linear, nonlinear, and self-similar regimes. At late times, the instability is seen to exhibit a self-similar behavior, in agreement with experimental observations. For the conditions simulated diffusion can influence the initial instability growth significantly.
In this paper, the Rayleigh-Taylor instability (RTI) is investigated using the direct simulation Monte Carlo (DSMC) method of molecular gas dynamics. Here, fully resolved two-dimensional DSMC RTI simulations are performed to quantify the growth of flat and single-mode perturbed interfaces between two atmospheric-pressure monatomic gases as a function of the Atwood number and the gravitational acceleration. The DSMC simulations reproduce many qualitative features of the growth of the mixing layer and are in reasonable quantitative agreement with theoretical and empirical models in the linear, nonlinear, and self-similar regimes. In some of the simulations at late times, the instability enters the self-similar regime, in agreement with experimental observations. Finally, for the conditions simulated, diffusion can influence the initial instability growth significantly.