Pressure gradient effect at distributed excitation of 3D TS waves by freestream and wall disturbances

The present work is a continuation of previous experiments (carried out in the Blasius boundary layer) and devoted to quantitative investigation of influence of an adverse pressure gradient on two efficient mechanisms of excitation of 3D TS instability waves due to a distributed boundary layer receptivity to free-stream vortices. These mechanisms are associated with distributed scattering of 3D amplified free-stream vortices both on the natural boundary-layer nonuniformity (on smooth surface) and on 2D surface nonuniformities (waviness). The corresponding detailed hotwire measurements were carried out in a self-similar boundary layer with Hartree parameter H = –0.115 in a wide range of the problem parameters. Complex values of quantitative characteristics of the physical phenomenon under study (the distributed receptivity coefficients) are evaluated by based on the obtained experimental data. It is found that the adverse pressure gradient leads to reduction of efficiency of the investigated vortexroughness receptivity mechanism.


Introduction
The present experiments are devoted to quantitative investigation of adverse-pressuregradient effects on the mechanisms of excitation of three-dimensional Tollmien-Schlichting (TS) waves due to distributed (in the streamwise direction) boundary layer receptivity to free-stream vortices. These mechanisms are associated with distributed scattering of threedimensional unsteady free-stream vortices both on the natural boundary layer nonuniformity (smooth surface) and on 2D surface nonuniformities (waviness). The receptivity and the boundary-layer instability mechanisms affect simultaneously on downstream evolution of TS-waves and provide the possibility of much faster growth of the latter compared to the growth rates associated with the linear instability mechanism.
The majority of previous studies were theoretical and they dealt with 2D problems. The first experimental estimations of the distributed vortex receptivity coefficients were carried out in [1] for a two-dimensional problem. Experiments [2] were devoted to investigation of 3D vortex receptivity problem in the Blasius boundary layer. These experiments gave for the first time systematic information on the distributed excitation of 3D TS-waves by freestream vortices in presence of 2D surface roughness. The values of the distributed receptivity coefficients were obtained there in a broad range of the problem parameters. The present work is a continuation of experiments [2] and devoted to a systematic quantitative investigation of influence of an adverse pressure gradient on the distributed vortical receptivity mechanisms mentioned above. The present experiments were performed in a self-similar boundary layer with Hartree parameter H = -0.115.

Experiment setup and evolution of exited TS-modes
The measurements were carried out in a low-turbulence wind tunnel T-324 of ITAM SB RAS in a boundary layer of a high-precision experimental model consisted of a flat plate (1485×1000×10 mm) and a wall bump of a special adjustable shape mounted on the windtunnel ceiling just above the plate. The shape of the wall bump provided formation on the flat plate in the region of main measurements of a self-similar flow with Hartree parameter H = -0.115. The boundary-layer edge velocity Ue varied from 8.9 to 8.6 m/s in the region of the streamwise coordinate x = 350 to 620 mm having its origin at the plate leading edge. The corresponding Reynolds numbers Re * = Ue1/ varied between 837 and 1174 (here 1 is the boundary layer displacement thickness and  is the air kinematic viscosity). The 3D freestream vortices were generated by a vibrating wire mounted normally to the flat-plate surface upstream of its leading edge. The 2D surface nonuniformities of sinusoidal shape were created by special thin-film patches applied onto the plate surface. These patches were manufactured by a special high-precision technology and had rated parameters: the streamwise wavelength sx and the amplitude hs. Thorough measurements of the boundary layer and freestream disturbances were carried out by a single hot-wire probe in several regimes (see the Table 1). In some of these regimes, the problem parameters (such as the disturbance frequency f, the streamwise surface-waviness wavelength, and the propagation angle  of the excited TS-waves) were chosen in a way to provide the satisfaction of conditions of the resonance of streamwise wavenumbers of the excited TS-modes, freestream vortices, and surface nonuniformities [2]. The characteristics of the TS-modes' linear evolution on the wavy surfaces were measured in additional series of experiments in all main studied regimes in absence of freestream disturbances. These measurements were necessary for obtaining quantitative characteristics of the studied mechanism of excitation of the instability modes -the vortex/roughness distributed receptivity coefficients. In these additional experiments the wave-trains of controlled 3D TS-waves were excited by special point disturbance source (of blowing/suction type) mounted in the beginning of the experimental model.
The experiments have shown that a controlled antisymmetric vortex street is produced in the wake of the vibrating wire and leads to a rather efficient excitation of wave trains consisted of 3D TS-modes. The effective excitation of 3D TS-waves by the point disturbance source was also achieved in the additional (stability) experiments. In the two experiments, a number of spanwise profiles of amplitudes and phases of excited boundary layer disturbances were measured at various streamwise positions. Streamwise distributions 2 EPJ Web of Conferences 159, 00005 (2017) DOI: 10.1051/epjconf/201715900005 AVTFG2016 of amplitudes and phases of the excited TS-modes were decomposed into the spanwisewavenumber spectra. It was found that due to the action of the distributed receptivity mechanisms, the amplitudes of the excited TS-waves are able to grow much faster then those generated in the case of action of the linear instability mechanism only. Fig. 1 illustrates a typical picture of comparison of streamwise evolutions of the boundary layer disturbances (aamplitudes, b -phases) in the main (receptivity) and in the additional (stability) experiments. Qualitatively the same results were obtained in all studied regimes in a wide range of the spanwise wavenumbers. These data were used then for subsequent processing and obtaining the amplitudes and phases of the distributed vortex-receptivity coefficients.

Procedure of obtaining receptivity coefficients and main results
The distributed receptivity coefficients are main quantitative characteristics of the distributed receptivity mechanism under study. They were determined in the same manner as in previous work [2] -as the corresponding coefficients of a differential equation that describes the evolution of the distributedly excited TS-modes. An analytical solution of this equation can be written in a general case as: Here d B is the complex-valued amplitude of the boundary layer disturbance; ) (x  is the TS-mode complex-valued wavenumber (obtained from processing of data of the additional, stability experiments); v B is the free-stream vortex complex-valued amplitude (measured at the boundary layer edge); H is the surface roughness complex-valued amplitude; H is its complex conjugate; ) (x G d vs are the complex-valued distributed vortex-(on smooth surface) and vortex-roughness receptivity functions, respectively; d B 0 is the complex-valued 'initial' amplitude of the excited boundary layer disturbances and x * is the