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

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.