Influence of backward-facing steps on laminar-turbulent transition in two-dimensional boundary layers at subsonic Mach numbers

Abstract

Backward-facing steps (BFS) can have a detrimental impact on laminar flow lengths because of their strong effect on boundary layer transition. BFS with normalized step heights in the range of \(h/\delta _1 \approx\) 0.1 to 0.6 (corresponding to height-based Reynolds numbers of \(\hbox{Re}_h = (U_\infty h / \nu ) \approx\) 230 to 2430) were installed in a two-dimensional wind tunnel model and tested in the Cryogenic Ludwieg-Tube Göttingen, a blow-down wind tunnel with good flow quality. The influence of BFS on the location of laminar-turbulent transition was investigated over a wide range of unit Reynolds numbers from \(\hbox{Re}_1 = {17.5\times 10^{6}\,{\text {m}^{-1}}}\) to \(80\times 10^{6}\,\hbox{m}^{-1}\), three Mach numbers, \(M= 0.35\), 0.50 and 0.65, and various streamwise pressure gradients. The measurement of the transition locations was accomplished non-intrusively by means of temperature-sensitive paint. Transition Reynolds numbers, calculated with the flow length up to the location of laminar-turbulent transition \(x_{T}\), ranged from \(\hbox{Re}_{\rm tr}\approx\) 1 × 10⁶ to 11 × 10⁶, and were measured as a function of step height, pressure gradient, Reynolds and Mach numbers. Incompressible linear stability analysis was used to calculate amplification rates of Tollmien–Schlichting waves; transition N-factors were determined by correlation with the measured transition locations. In parallel to earlier investigations with a similar setup, this systematic approach was used to identify functional relations between non-dimensional step parameters (\(h/\delta _1\) and \(\hbox{Re}_h\)) and the relative change of the transition location. Furthermore, the change of the transition N-factor \(\Delta N\) due to the installation of the steps was investigated. It was found that the installation of backward-facing steps with \(h/\delta _1 \lesssim 0.15\) and \(\hbox{Re}_h \lesssim 300\) does not lead to a reduction of \(\hbox{Re}_{\rm tr}\) and to \(\Delta N > 0\). However, increasing the step size results in a decreasing laminar flow length and thus an increasing \(\Delta N\). The reported results are in general agreement with earlier investigations at significantly lower Mach and Reynolds numbers.