Unit Reynolds number, Mach number and pressure gradient effects on laminar–turbulent transition in two-dimensional boundary layers

The influence of unit Reynolds number ($R{e}_1=17.5\times {10}^6$–$80\times {10}^6{\text{m}}^{-1}$), Mach number ($M=0.35$–0.77) and incompressible shape factor ($H_{12}=2.50$–2.66) on laminar–turbulent boundary layer transition was systematically investigated in the Cryogenic Ludwieg-Tube Göttingen (DNW-KRG). For this investigation the existing two-dimensional wind tunnel model, PaLASTra, which offers a quasi-uniform streamwise pressure gradient, was modified to reduce the size of the flow separation region at its trailing edge. The streamwise temperature distribution and the location of laminar–turbulent transition were measured by means of temperature-sensitive paint (TSP) with a higher accuracy than attained in earlier measurements. It was found that for the modified PaLASTra model the transition Reynolds number ($R{e}_{\text{tr}}$) exhibits a linear dependence on the pressure gradient, characterized by $H_{12}$. Due to this linear relation it was possible to quantify the so-called 'unit Reynolds number effect', which is an increase of $R{e}_{\text{tr}}$ with $R{e}_1$. By a systematic variation of M, $R{e}_1$ and $H_{12}$ in combination with a spectral analysis of freestream disturbances, a stabilizing effect of compressibility on boundary layer transition, as predicted by linear stability theory, was detected ('Mach number effect'). Furthermore, two expressions were derived which can be used to calculate the transition Reynolds number as a function of the amplitude of total pressure fluctuations, $R{e}_1$ and $H_{12}$. To determine critical N-factors, the measured transition locations were correlated with amplification rates, calculated by incompressible and compressible linear stability theory. By taking into account the spectral level of total pressure fluctuations at the frequency of the most amplified Tollmien–Schlichting wave at transition location, the scatter in the determined critical N-factors was reduced. Furthermore, the receptivity coefficients dependence on incidence angle of acoustic waves was used to correct the determined critical N-factors. Thereby, a found dependency of the determined critical N-factors on $H_{12}$ decreased, leading to an average critical N-factor of about 9.5 with a standard deviation of $\sigma \approx 0.8$.