Definition and Evaluation of Continuum Breakdown Parameters

In the study of hypersonic vehicles, the overlap region between the continuum and rarefied flow regimes is of particular relevance. In this transitional regime, the continuum assumption ceases to hold, and the Navier–Stokes (NS) equations lose their validity, necessitating the use of more computationally intensive approaches such as Direct Simulation Monte Carlo (DSMC) analyses. The transition does not occur uniformly throughout the flowfield. The continuum hypothesis first fails in specific regions, such as as shock waves, near-wall layers, and strong expansions. Continuum breakdown remains a relevant source of uncertainty in hypersonic flow modeling, particularly under strongly nonequilibrium conditions. Numerical cutoff values commonly employed to identify the breakdown are often defined arbitrarily and lack firm physical justification. The present study examines the applicability range of the gradient-length local Knudsen number as a breakdown parameter for hypersonic flows. Previous investigations have proposed a fixed threshold of 0.05 for non-reacting flows without slip conditions; however, this value depends on the local flow feature (e.g., shock, boundary layer, wake), and on the selected gradient parameter. DSMC and CFD simulations of hypersonic flows are performed to quantify the deviation of Navier–Stokes as a function of gradient-length local Knudsen number to establish specific ranges for the continuum validity limits. New breakdown parameters are proposed to capture different physical phenomena inside the flowfield.