The law of incipient separation for turbulent flows as inferred by RANS

We have performed Reynolds-averaged Navier-Stokes (RANS) computations of incompressible turbulent boundary layer with adverse pressure gradient over two-dimensional smooth curved ramps of length $L$ for $982\leq Re_\theta \leq 3698$. First, we have validated the RANS by comparing the results with the experiments of Song \& Eaton ({\em Expts. Fluids}, 2004) for the separated flow over an arc at $Re_\theta=1100$. Then, we have investigated the effects of the ramp slope and curvature on the skin-friction and pressure coefficients. Our results show a numerical criterion of incipient flow separation determined only by the geometrical parameters of maximum slope, $|z'|_{\text{max}}$, length, $L$, and height, $h$, of the ramp, and by $Re_L$. Specifically, incipient separation occurs when $|z'|_{\text{max}}$ normalized by $\tilde{h}=h/L$ ($|\hat{z}'|_{\text{max}}=|z'|_{\text{max}}/\tilde{h}$) reaches a critical value that is determined according to the following law: $|\hat{z}'|_{\text{crit}}=\alpha \tilde{h} Re_L^{-1/10}+ \beta$, where $\alpha<0$ and $\beta>0$ are constants. Accordingly, the flow separates over a curved smooth ramp for which $|\hat{z}'|_{\text{max}}>|\hat{z}'|_{\text{crit}}$. The uncertainty of the law is also reported.