Two-Phase Numerical Simulation and Box-Counting Analysis of Kelvin–Helmholtz Instabilities in Sediment-Laden Shear Flows

Kelvin–Helmholtz (K–H) instabilities play an important role in mixing and entrainment processes in stratified and sediment-laden flows, while their development can be affected by particle properties and rheological contrasts. In this study, a two-phase numerical framework is used to investigate the onset and nonlinear evolution of K–H billows in sediment-laden shear layers. The governing equations are solved using a finite-volume approach with second-order TVD discretization, and the effects of the Richardson number (Ri), sediment particle size, and viscosity ratio (W) are examined systematically. In addition to concentration, vorticity, and interphase slip-velocity fields, a box-counting dimension (BCD) diagnostic is introduced as a complementary measure to quantify the instability onset time, the effective growth-start time, and the maximum geometrical growth rate of the sediment–water interface. The results show that, within the tested parameter range, coherent instabilities develop for Ri≲0.25. Decreasing Ri leads to earlier BCD-based onset, shorter effective growth intervals, larger BCD growth rates, and stronger vortical structures. Sediment granulometry also affects the instability evolution: fine silt-like particles delay the onset and produce broader, more diffuse vortical structures, whereas fine-sand-like particles promote earlier destabilization and more compact billows. The viscosity ratio W modifies the vortex morphology, with high viscosity contrast reducing peripheral deformation and concentrating vorticity within the core region. These findings complement classical descriptions of stratified shear instabilities by highlighting the additional effects of sediment granulometry and rheological contrast. The proposed BCD-based diagnostic provides a practical quantitative tool for comparing interfacial growth in sediment-laden shear flows relevant to estuarine, coastal, and hydraulic engineering applications.