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

Abstract

The influence of unit Reynolds number (Re 1 = 17.5 × 10 6-80 × 10 6 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 (Re 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 Re tr with Re 1. By a systematic variation of M, Re 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, Re 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 ≈ 0.8.

Experimental analysis of suction on step-induced boundary-layer transition

“…. Similar to earlier studies ([25]), transition wasdetected at up to 100 positions on each side (port and starboard) of the model, marked by individual red dots in figure 3b (appearing as red line). Excluded from the eval-195 uation are areas of turbulent wedges or areas on the outer edges of the model which are effected by side wall effects.…”

“…In general, higher Hartree parameters β H lead to decreased critical N-factors. For low Mach numbers a higher critical N-factor was calculated meaning higher amplification ratios of TS-waves are required to trigger transition[25]. This corresponds to the phenomena described in section 4.2 as of why the expected stabilizing effect of compressibility is not observed in this investigation: initial amplitudes of TS-waves are decreased for lower Mach numbers[22].…”

“…[22][23][24]. [28]. The chord length of the model for this study will be stated as c = 200 mm despite the extension with the aft part to be able to compare results to earlier studies.…”

Experimental study of bump effects on boundary-layer transition in compressible high Reynolds number flow

“…Also the local variations in the pressure distribution induced by the small bump were strengthened by an increase in the Mach number (see Figure 15c), but this effect was clearly less pronounced than that observed in the presence of the big bump. Although the more marked adverse pressure gradient on the downstream side of the small bump probably contributed to the upstream movement of the transition location, the most important variation leading to this result is expected to be the increased level of external disturbances in DNW-KRG at larger Mach numbers (see Koch (2004), Costantini (2016), Risius et al (2018) and Risius (2018)): earlier transition was likely induced by the related increase (via the receptivity process) in the initial amplitude of the boundary-layer disturbances. The increased wall temperature ratio T w /T aw at larger Mach numbers (see Sec.…”

“…In the case of the small bump, the more marked adverse pressure gradient on the downstream side of the bump found at larger Mach numbers may have also contributed to the observed trend. Spectra of total pressure fluctuations p 0 ' in DNW-KRG were measured in Koch (2004) for different Mach numbers; the variation of p 0 ' as a function of the frequency f and of the Mach number M was approximated in Risius et al (2018) and Risius (2018) by the following relation:…”