Characterising Görtler vortices in supersonic turbulent flows over concave surfaces

Görtler vortices induced by concave curvature in supersonic turbulent flows are investigated using resolvent analysis and large-eddy simulations at Mach 2.95 and Reynolds number $ Re_{\delta }=63\,500$ based on the boundary-layer thickness $ \delta$ . Resolvent analysis reveals that the most amplified coherent structures manifest as streamwise counter-rotating vortices with optimal spanwise wavelength $ 2.4\delta$ at cut-off frequency $f\delta /{u}_{\infty } =0.036$ , where $ {u}_{\infty }$ is the freestream velocity. The leading spectral proper orthogonal decomposition modes with spanwise wavelength approximately $ 2\delta$ align well with the predicted coherent structures from resolvent analysis at $f\delta /{u}_{\infty } =0.036$ . These predicted and extracted coherent structures are identified as Görtler vortices, driven by the Görtler instability. The preferential spanwise scale of the Görtler vortices is further examined under varying geometric and freestream parameters. The optimal spanwise wavelength is insensitive to the total turning angle beyond a critical value, but sensitive to the concave curvature $ K$ at the same turning angle. A limit spanwise wavelength $ 1.96\delta$ , corresponding to an infinite concave curvature as $ K\rightarrow \infty$ , is identified and validated. Increasing the freestream Mach number or decreasing the ratio of wall temperature to freestream temperature reduces the optimal wavelength normalised by $ \delta$ , while variations in freestream Reynolds number have negligible impact. Additionally, a modified definition of the turbulent Görtler number $ G_{T}$ based on the peak eddy viscosity in boundary layers is proposed and employed to assess the occurrence of Görtler instability.