Numerical solution of steady-state turbulent flow of newtonian fluid in a two-dimensional rough fracture

Rough fractures are widely developed in rock masses and serve as the primary pathways for fluid flow. Depending on the Reynolds number, fluid flow can generally be classified as laminar and turbulent. Due to the presence of highly disordered streamlines and pronounced vortex structures, turbulent flow is inherently difficult to accurately predict. The combined effects of the complex fracture structures further complicate the flow behavior. To address these challenges, this study proposes a turbulent flow model based on Reynolds-Averaged Navier-Stokes equations (RANS) and refines the near-wall region. The model is validated through numerical simulations and can well evaluate time-averaged pressure and velocity distributions in rough fractures. The results indicate that the direction of fluctuating velocity aligns with the shear direction and decreases with increasing shear displacement. As the aperture increases, peak fluctuating velocity increases significantly along the main flow direction. As the Reynolds number increases, turbulent dissipation becomes the dominant energy loss mechanism. This phenomenon is particularly pronounced in the region slightly above the viscous sublayer, and the maximum turbulent energy loss occurs at y/h ≈ 0.03. Furthermore, the high-order nonlinear relationship between hydraulic gradient and flow rate under turbulent conditions exhibits significant deviations from the classical Forchheimer quadratic model, particularly under higher Reynolds numbers or complex fracture geometries. The proposed turbulent flow model provides a practical reference for characterizing turbulent characteristics and theoretical research in rough fractures.