Rarefaction-induced non-equilibrium characteristics in shock wave/boundary layer interaction

In recent years, trans-scale shock wave/boundary layer interactions (SWBLI) have emerged as a critical—and even bottleneck—issue in the fields of rarefied gas dynamics and non-equilibrium transport phenomena. A growing number of numerical experiments have demonstrated that many detailed aspects of SWBLI simulations cannot yield convincing results, regardless of improvements in numerical accuracy. This has prompted researchers to revisit a more fundamental question: As rarefaction effects become more significant and the temporal and spatial scales of interest shrink, are the Navier–Stokes (NS) equations—which we have trusted and relied upon for decades—still adequate and valid? At the same time, the limitations of current analysis techniques for complex physical fields have led to a serious waste of data resources. In particular, the lack of diversified or comprehensive research perspectives has resulted in inconsistencies and even contradictions in our understanding. To address these challenges, the Discrete Boltzmann Method (DBM)—a kinetic modeling and analysis method developed from the integration of kinetic theory and mean-field theory—has been proposed and further advanced.

Recently, the Jet Propulsion Laboratory from Beijing Institute of Technology (BIT), in collaboration with Prof. Aiguo Xu at the Institute of Applied Physics and Computational Mathematics (IAPCM), has extended the DBM approach for more refined kinetic studies of SWBLI. Their latest findings were published in the Chinese Journal of Aeronautics on April 12, 2025. The first author of the paper is Dr Jiahui Song, a Ph.D. candidate jointly supervised by the School of Aerospace Engineering at BIT and IAPCM. The corresponding authors are Prof. Long Miao from BIT and Prof. Aiguo Xu from IAPCM. The co-authors also include Prof. Yanbiao Gan from North China Institute of Aerospace Engineering, Prof. Feng Chen from Shandong Jiaotong University, Prof. Yugan Liao from BIT, et al.

From the perspective of the Knudsen (Kn) number, a reduction in the spatial scale of interest is effectively equivalent to an increase in rarefaction or discreteness; likewise, faster dynamic behavior (e.g., from subsonic flow to shock regions) corresponds to a higher degree of local non-equilibrium. As the degree of discreteness/rarefaction increases, more state quantities are needed to describe the system state, and more effect quantities are needed to describe the effect. This is an inevitable requirement for the physical description ability not to decrease with the increase of discreteness/rarefaction. The technical key is: how to know which physical quantities need to be increased? The core idea of the DBM is to treat the Kn=0—that is, the continuous and equilibrium state—as the expansion center of a Chapman–Enskog multiscale expansion (a generalized form of Taylor expansion). Based on the local Kn number, one can determine the required expansion order of Kn retained in the model. Then, using the hierarchical dependency between non-equilibrium distribution functions of adjacent orders, the relevant kinetic moments of the equilibrium distribution function—those needed to construct more accurate constitutive relations (such as stress and heat flux)—can be efficiently identified based solely on tensor ranks, without the need for complex equation derivations. To ensure modeling accuracy, these kinetic moments must be preserved during the discretization of velocity space. In this way, the DBM framework provides a sufficient condition for constructing accurate and physically consistent constitutive models.

A typical feature of compressible fluids is their strong coupling behavior—disturbances in one part can rapidly affect the entire system. The traditional understanding based on a single-perspective definition of the Kn is often partial, or even misleading. In SWBLI systems, both slowly varying regions (with relatively small local Kn) and rapidly varying regions such as shock fronts (with relatively large local Kn) coexist. Near any interface, different physical quantities—such as density, temperature, velocity, and pressure—often exhibit different spatial gradients, thus corresponding to different characteristic length scales and, consequently, different local Kn numbers. Early approaches that relied on a single-perspective Kn failed to capture this complexity. To address this limitation, the present study introduces the concept of a local Kn number vector, where each component represents the local Kn number from a distinct physical perspective. This allows a multi-faceted, cross-referenced characterization of local rarefaction effects within the system. The global spatial Kn number represents the averaged degree of discreteness/rarefaction across all spatial structures in the system, while the global temporal Kn number characterizes the averaged degree of non-equilibrium across all temporal variation modes. The combined use of global and local Kn numbers provides a hierarchical, multi-scale observational framework—from coarse to fine, from global overview to localized zoom-in—offering a more comprehensive understanding of the system behavior.

It is generally accepted that when the local Kn number exceeds 0.001, the validity of continuum-based descriptions begins to deteriorate. This study further confirms this understanding through direct data comparisons. In all regions where the local Kn number exceeds 0.001, the results presented in this work highlight phenomena that the NS framework either cannot capture or fails to describe accurately. For physical quantities that do exist within the NS framework—such as density, temperature, velocity, pressure, stress, and heat flux—the DBM offers more accurate representations. Moreover, non-conserved moments beyond stress and heat flux (i.e., terms related to  and their derived quantities for various physical needs lie completely outside the descriptive capability of the NS theory. As the local Kn increases, the system exhibits stronger rarefaction and non-equilibrium effects. In such cases, the physical structures of interfaces (such as shocks) and the corresponding kinetic behaviors can no longer be adequately captured by relying solely on conventional macroscopic variables like , and  Additional quantities must be introduced to maintain control and physical fidelity. This constitutes one of the core messages conveyed by this work.

The entropy generation rate is one of the key physical quantities of interest in studies such as SWBLI. A high entropy generation rate indicates not only significant thermal dissipation but also a reduced capacity for performing mechanical work. Therefore, in addition to representing a loss of useful work, it may also imply the need for enhanced thermal management measures. Moreover, entropy generation is widely used to analyze shock-induced boundary layer separation structures, as well as the associated wave drag and frictional drag arising from boundary layers. In this work, we conduct a detailed investigation into the primary mechanisms responsible for entropy generation during the SWBLI process and assess their relative significance. Several key findings are summarized as follows:

(i) The leading-edge shock wave increases the local density Kn number  behind the shock wave. As  increases, the local temperature on the degree of freedom corresponding to the shock wave () increases instead of decreasing. (ii) Behind the leading-edge shock wave, the amplitude of the nonlinear correction part  of the local temperature on the degree of freedom corresponding to the shock wave even exceeds the theoretical prediction value of its linear response ; with the help of the intersection of the curves  under different rarefaction, the area behind the leading-edge shock wave can be divided into three sections, whose characteristics are correlated with ,  and , respectively. (iii) The non-equilibrium quantities  and , as well as the viscous entropy production rate  can be used to identify the location of the separation zone, and their local minimum points correspond to the starting point and the ending point of the separation zone, respectively.

Future work will focus on developing a higher-fidelity three-dimensional DBM to further investigate the kinetic behaviors and underlying mechanisms of SWBLI under higher Mach number conditions. In addition, the formulation of kinetic boundary conditions tailored to specific physical requirements remains an open scientific challenge in the DBM community and will be an important direction for future research.

Original Source

Jiahui Song, Long Miao, Aiguo Xu, et al. Rarefaction effect on non-equilibrium characteristics of laminar shock wave/boundary layer interaction[J]. Chinese Journal of Aeronautics, 2025: 103538. https://doi.org/10.1016/j.cja.2025.103538.

About Chinese Journal of Aeronautics 

Chinese Journal of Aeronautics (CJA) is an open access, peer-reviewed international journal covering all aspects of aerospace engineering, monthly published by Elsevier. The Journal reports the scientific and technological achievements and frontiers in aeronautic engineering and astronautic engineering, in both theory and practice. CJA is indexed in SCI (IF = 5.7, Q1), EI, IAA, AJ, CSA, Scopus.