On the consistent similarity of equilibrium supersonic pressure-gradient turbulent boundary layers over smooth and rough walls

The consistent outer-layer similarity of equilibrium supersonic pressure-gradient turbulent boundary layers (PGTBLs) over smooth and rough walls, which has not been systematically explored for rough-wall configurations, is investigated using a unified similarity theory and direct numerical simulations (DNS). A unified similarity framework applicable to both wall conditions is developed, within which a compressible pressure-gradient parameter, upper Lamda Subscript c Λ c $\\varLambda _c$ , is identified as the governing similarity parameter. The theory predicts that smooth- and rough-wall PGTBLs are self-similar when upper Lamda Subscript c Λ c $\\varLambda _c$ constant in the streamwise direction, and that consistent similarity between the two flows is achieved when they share the same upper Lamda Subscript c Λ c $\\varLambda _c$ . To construct such equilibrium PGTBLs, a prediction–correction procedure is proposed to prescribe the boundary-layer edge velocity, ensuring a constant upper Lamda Subscript c Λ c $\\varLambda _c$ in DNS. The resulting simulations confirm collapse of outer-layer mean velocity and Reynolds-stress profiles between smooth and rough walls under matched upper Lamda Subscript c Λ c $\\varLambda _c$ condition. Combining theory and DNS, the similarity conditions expressed in terms of upper Lamda Subscript c Λ c $\\varLambda _c$ are shown to be equivalent to requiring a constant compressible Rotta–Clauser parameter beta β $\\beta$ . Furthermore, with appropriate scaling, both mean and fluctuating temperature profiles exhibit consistent similarity, demonstrating that the outer-layer similarity extends naturally to the thermal field in supersonic PGTBLs.