A Time-Domain Harmonic Balance Unified Gas-Kinetic Scheme for Temporally Periodic Flows Across all Knudsen Regimes

TL;DR

This paper presents a novel time-domain harmonic balance unified gas-kinetic scheme (HB-UGKS) that efficiently simulates temporally periodic flows across all Knudsen regimes, capturing complex phenomena with reduced computational cost.

Contribution

The paper introduces a harmonic balance approach integrated with UGKS to efficiently simulate multiscale periodic flows across all Knudsen numbers, including nonlinear effects.

Findings

Accurately captures flow dynamics across Knudsen and Strouhal numbers.

Successfully models anti-resonance and damping phenomena.

Achieves up to 9-fold speedup over explicit methods.

Abstract

This paper introduces a time-domain harmonic balance unified gas-kinetic scheme (HB-UGKS) designed to simulate temporally periodic flows across all Knudsen regimes. The harmonic balance approach reformulates the periodic problem into a block-coupled, quasi-steady system via a time-spectral source term. This allows for pseudo-time marching, local time-stepping, and the concurrent resolution of all sub-time levels, drastically reducing wall-clock time. Coupled with the UGKS-which maintains essential transport-collision coupling in its flux evaluations--the framework ensures multiscale validity across the entire Knudsen number range. The method is validated against two representative cavity flows. For a shear-driven oscillatory cavity under small-amplitude excitation, the fundamental harmonic alone accurately resolves the flow dynamics across various Knudsen and Strouhal numbers, successfully capturing the anti-resonance phenomenon and matching hydrodynamic damping predictions from linearized Boltzmann analyses. For a thermally driven cavity with large temperature modulations, higher-order harmonics prove essential to capture strong nonlinear waveform distortions and rarefaction effects. Beyond its physical fidelity, the HB-UGKS demonstrates substantial computational efficiency over explicit time-domain methods. This advantage peaks in high-frequency regimes, achieving speedup factors of 9.0 and 8.26 for the shear-driven and thermally driven cases, respectively.