We apply linear stability analysis to nearly incompressible flow over a two-dimensional zero-pressure-gradient boundary layer in the presence of a forward-facing step (FFS), with step height to local displacement thickness ratios between $0<h/{\delta }_s^{*}<3.2$. We employ a nominally sharp FFS with a 1 $\mu \mathrm{m}$ radius to closely resemble a prior wind tunnel experiment, but we also examine the impact of various step corner radii and slopes. The Navier-Stokes equations are solved on curvilinear grids to obtain the laminar basic states. After that, we use a combination of the harmonic linearized Navier-Stokes equations and the parabolized stability equations to compute the linear amplification of Tollmien-Schlichting instabilities. The $N$-factor envelopes for the FFS reveal crucial differences with respect to those obtained previously for backward-facing steps (BFS) with $0<h/{\delta }_s^{*}<1.3$, for which accurate predictions of transition onset could be obtained via $N$-factor correlations. As the boundary-layer flow recovers to its unperturbed state in the far wake of the BFS, the slope of the $N$-factor envelope is nearly the same as that of the unperturbed flat-plate boundary layer. In contrast, the $N$-factor envelope downstream of the FFS becomes flatter as the step height increases, i.e., $\mathrm{d}N$/$\mathrm{d}x$ becomes nearly 0, which causes the $N$-factor method to become sensitive to the prescribed value of the correlating $N$-factor value. In spite of this difficulty, the $N$-factor method still captures the overall trend associated with a progressively upstream shift in the transition location as the FFS height becomes larger. However, the critical step height corresponding to the onset of a significant upstream shift in transition location is not very well predicted when compared with the measured data. Within the framework of a variable $N$-factor method, the predicted reduction in the transition $N$-factor is given by $\mathrm{\Delta }N=1.89\arctan [(h/{\delta }_s^{*}-1.29)/0.179]-2.73$.

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• Aerodynamics
• Boundary layer receptivity, stability & separation
• Flow instability