Validation of wall boundary conditions for simulating complex fluid flows via the Boltzmann equation: Momentum transport and skin friction

TL;DR

This study validates wall boundary conditions for simulating complex fluid flows using the Boltzmann equation, demonstrating accurate predictions of non-equilibrium phenomena and turbulence across different flow regimes with reduced computational complexity.

Contribution

The paper introduces a comprehensive validation of wall boundary conditions for the Boltzmann equation, including the first direct simulations of wall-bounded turbulence.

Findings

High spatial resolution is crucial for accurate flow instability predictions.

Accurate momentum transfer can be achieved with minimal velocity domain resolution.

Wall boundary conditions are effective across a wide range of flow regimes.

Abstract

The influence and validity of wall boundary conditions for non-equilibrium fluid flows described by the Boltzmann equation remains an open problem. The substantial computational cost of directly solving the Boltzmann equation has limited the extent of numerical validation studies to simple, often two-dimensional, flow problems. Recent algorithmic advancements for the Boltzmann--BGK equation introduced by the authors, consisting of a high-order spatial discretization augmented with a discretely-conservative velocity model, have made it feasible to accurately simulate unsteady three-dimensional flow problems across both the rarefied and continuum regimes. This work presents a comprehensive evaluation and validation of wall boundary conditions across a variety of flow regimes, primarily for the purpose of exploring their effects on momentum transfer in the low Mach limit. Results are presented for a range of steady and unsteady wall-bounded flow problems across both the rarefied and continuum regimes, from canonical two-dimensional laminar flows to unsteady three-dimensional transitional and turbulent flows, the latter of which are the first instances of wall-bounded turbulent flows computed by directly solving the Boltzmann equation. We show that approximations of the molecular gas dynamics equations can accurately predict both non-equilibrium phenomena and complex hydrodynamic flow instabilities and show how spatial and velocity domain resolution affect the accuracy. The results indicate that an accurate approximation of particle transport (i.e. high spatial resolution) is significantly more important than particle collision (i.e. high velocity domain resolution) for predicting flow instabilities and momentum transfer consistent with that predicted by the hydrodynamic equations and that these effects can be computed accurately even with very few degrees of freedom in the velocity domain.