Spatial development of the secondary flow in a supersonic turbulent square-to-circular transition duct

We conduct direct numerical simulations to study supersonic turbulent flow in a square-to-circular transition duct at Mach 2.7, focusing on the variation of turbulence-driven secondary flows during boundary-layer development. The circumferential pressure gradient induced by cross-sectional shape variation reshapes the distribution of secondary flows, which first move away from the wall and then reattach to it. As the cross-sectional contour gradually approaches a circular shape, the morphology of the secondary flows in the cross section transitions to a circular form. Meanwhile, the cores of the secondary flows continuously migrate away from the corners following a linear law during flow development. Under the development of the secondary flows in the duct, the instantaneous velocity iso-surfaces near the central plane in the square-to-circular transition duct elevate, resulting in a distinct low-speed feature. In the mean flow field, the square-to-circular transition duct exhibits a smaller low-momentum region area in the corner, fuller streamwise velocity profiles and an increased velocity gradient along the corner bisector, which results in higher friction coefficients at the corners compared to the square duct. Additionally, the expansion of the central cross section during the contour transition accelerates boundary-layer growth, resulting in a thicker boundary layer in the transition duct than in the square duct. These analyses provide theoretical insights into the dynamic evolution of secondary flows and the regulation of complex flows within square-to-circular transition duct in the specific flow regime.