Y. S. Kachanov

Three-dimensional receptivity of boundary layers

Abstract The paper presents a review of results of some recent (mainly experimental) studies devoted to a quantitative investigation of the problem of receptivity of the 2D and 3D boundary layers with respect to various 3D (in general) external perturbations. The paper concentrates on the mechanisms of excitation and development of stationary and travelling instability modes in a 3D boundary layer on a swept wing, as well as in 2D boundary layers including the Blasius flow and a self-similar boundary layer with an adverse pressure gradient. In particular, the following problems of the boundary-layer receptivity are discussed: (i) receptivity to localized 3D surface roughness, (ii) receptivity to localized 3D surface vibrations, (iii) acoustic receptivity in presence of 3D surface roughness, and (iv) acoustic receptivity in the presence of 3D surface vibrations. All experiments described in the paper were conducted using controlled disturbance conditions with the help of simulation of the stationary and non-stationary perturbations by means of several disturbance generators. This approach gives us the possibility to obtain quantitative results which are independent of any uncontrolled background perturbations of the flow and the experimental model. In contrast to the data obtained at “natural” environmental conditions these results can be directly compared with calculations without any significant assumptions about the physical nature of the disturbances under investigation. The complex (amplitude and phase) coefficients of the boundary-layer receptivity to external perturbations, obtained as functions of the disturbance frequency and the spanwise wavenumber (or the wave propagation angle), represent the main results of the experiments described. These results can be used for the evaluation of the initial amplitudes and phases of the instability modes generated by various external perturbations, as well as for quantitative verification of linear receptivity theories. Several examples of the comparison of experimental results with calculations are also presented in this paper. A brief analysis of the state-of-art in the field is performed and some general properties of different receptivity mechanisms are discussed.

Resemblance of K- and N-regimes of boundary-layer transition at late stages

Abstract This paper is devoted to an experimental study of late nonlinear stages of laminar–turbulent transition in a 2D flow close to the Blasius boundary-layer. The measurements were conducted at controlled disturbance conditions with excitation of a 2D large-amplitude fundamental instability wave with frequency f1 and/or a pair of low-amplitude oblique subharmonic instability waves with frequency f1/2 and values of the spanwise wavenumber ± β 1/2 . In the case with a phase shift between the fundamental mode and the pair of the subharmonics favourable for the subharmonic resonance the transition process was found to be characterised by a rapid resonance growth of the 3D subharmonic modes. This was followed by a formation of the Λ-structures within each subharmonic period in time, positioned in a staggered order in space that is typical of the N-regime (or the subharmonic regime) of the boundary layer breakdown. However, at late stages of the disturbance development the local behaviour of the perturbations in the vicinity of the Λ-structures turned out to be very similar to that typically observed in the K-regime of breakdown. This could be seen in the formation of an intensive Λ-shaped 3D high-shear layer and the coherent structures associated with spikes in the time-traces of the hot-wire signal. Sets of consecutive spikes were found to be generated in the vicinity of the tip of each Λ-structure. The arrays of spikes had also the staggered order with the streamwise and spanwise spacing characteristic of the subharmonic wave (as in the N-regime) but their local properties were found to be qualitatively the same as those typical for the K-regime. Despite the significantly different nature of the initial stages of these two scenarios of transition, described usually in terms of weakly nonlinear interactions of the instability waves, the late stages of these two types of breakdown (described usually in terms of vortices attributed to the coherent structures) have approximately the same physical nature.

Swept-wing boundary-layer receptivity to surface non-uniformities

The linear receptivity of a swept-wing three-dimensional boundary layer is studied experimentally and theoretically. Cross-flow instability normal modes are excited by means of surface vibration or roughness perturbations. The resulting disturbances are investigated, and the normal modes are linked to the source perturbations. Experiments are performed under controlled disturbance conditions with a time-harmonic source that is localized in the spanwise direction. A localized surface vibration is used to excite wave trains consisting of cross-flow instability waves. Normal oblique modes (harmonic in time and space) are obtained by Fourier decomposition of the wave trains. This provides the spatial variation of the normal modes and, in particular, the initial amplitudes and phases of the modes at the source location. The shape of the surface vibrator is measured and used to determine the complex receptivity coefficients for the normal modes (i.e. for various spanwise wavenumbers, wave propagation angles, and disturbance frequencies – including zero frequency). The experimental receptivity coefficients are independent of the specific shape of the surface non-uniformities and can be directly compared with calculations. The theoretical work is based on a linear approximation to the disturbance source – valid for small forcing amplitudes. Like earlier studies on roughness-induced receptivity, the basic flow is locally assumed to satisfy the parallel-flow approximation. The modal response for the cross-flow instability is determined from the residue associated with the least-stable eigenmode. A detailed quantitative comparison between the experimental and theoretical receptivity characteristics is carried out. Good agreement is found for the roughness–vibrational receptivity coefficients of the swept-wing boundary layer (especially for the most-unstable cross-flow modes) over a range of disturbance frequencies and spanwise wavenumbers. The theory correctly predicts the initial spectra for the travelling and stationary cross-flow instabilities excited by the surface vibrations and surface roughness, respectively. The good agreement between theory and experiment suggests that the linear receptivity theory can be used effectively in engineering methods for transition prediction. The experimental data can also be used for validation of other theoretical approaches to the problem.

Numerical and Experimental Investigation of the K-Regime of Boundary-Layer Transition

One particular case of K-type transition has been investigated using hot-wire measurements and spatial direct numerical simulation (DNS). Detailed quantitative comparisons of the results of both approaches showed very good agreement of the spatial disturbance development, the disturbance spectra, the instantaneous velocity traces, and the local frequency-spanwise-wave-number spectra. Indications for a direct generation of three-dimensional modes as higher harmonics of the fundamental modes were found. A closer look at the phase speeds of these modes, however, revealed that weak-nonlinear interactions are only initially appropriate to describe the flow, they fail when local events dominate, like, for example the formation of small-scale vortices in the boundary layer. The investigation of the later stages showed that the hot-wire ‘spike’-signals are connected with small ring-like vortices.

Three-dimensional acoustic-roughness receptivity of a boundary layer on an airfoil: experiment and direct numerical simulations

We describe an experimental and numerical investigation of the problem of excitation of three-dimensional Tollmien-Schlichting (TS) waves in a boundary layer on an airfoil owing to scattering of an acoustic wave on localized microscopic surface non-uniformities. The experiments were performed at controlled disturbance conditions on a symmetric airfoil section at zero angle of attack. In each set of measurements, the acoustic wave had a fixed frequency f ac , in the range of unstable TS-waves. The three-dimensional surface non-uniformity was positioned close to the neutral stability point at branch I for the two-dimensional perturbations. To avoid experimental difficulties in the distinction of the hot-wire signals measured at the same (acoustic) frequency but having a different physical nature, the surface roughness was simulated by a quasi-stationary surface non-uniformity (a vibrator) oscillating with a low frequency f v

Study of 3D Wall Roughness Acoustic Receptivity on an Airfoil

Hot-wire measurements are perfomed on linear 3D acoustic receptivity in a two-dimensional laminar boundary layer. A localized quasi steady surface roughness serves as the receptivity element. A plane acoustic wave with frequency f ac scatters at this vibrating source and the generated TS-wave train is measured downstream in spanwise cuts and at combination frequencies (f 1,2 = f ac ∓ f v ). After Fourier decomposition linear stabiltity theory is used for upstream extrapolation to the initial amplitudes at the roughness element. The dispersion characteristic is determined and the complex receptivity function is calculated by normalization of the initial TS-spectra with the related amplitudes and phases of the surface vibrator and the acoustics. The results are compared with Direct Numerical Simulations based on a vorticity-velocity formulation of the complete Navier-Stokes equations and a new embeded wall model. Good overall agreement is achieved.

Experimental study of resonant interactions of modulated waves in a non self-similar boundary layer

The current work is devoted to the study of weakly non-linear interactions of Tollmien-Schlichting waves in an incompressible, 2D airfoil boundary layer. Selected resonant regimes are investigated with emphasis to the regimes where more than one fundamental T-S wave are present in the flow. The results show that amplification rates of the effective sub-harmonic modes are not affected by modulations in time of the fundamental wave. It is also shown that modulations of the 2D fundamental waves tend to generate additional modes at modulation frequency. Furthermore, these modes can resonate with the fundamental ones as detuned subharmonics. This mechanism seems to be responsible for the initial seeds of subharmonics in cases of ‘natural’ transition.

Systematic investigations of 3D acoustic receptivity with respect to steady and unsteady disturbances. Experiment and DNS

In the present paper, the linear 3D acoustic-roughness receptivity of the 2D boundary layer of an airfoil is investigated systematically with respect to main influence parameters by means of wind tunnel experiments and Direct Numerical Simulations (DNS). A plane acoustic wave with frequency f ac scatters at a specially designed localized roughness element, that is capable to vibrate with frequency f v . The resulting Tollmien-Schlichting (TS) waves appear at combination frequencies (f 1,2 = f ac ∓ f v ) and develop as a wave train downstream of the surface roughness. The complex receptivity function is evaluated by normalization of the initial TS-spectra with the related amplitudes and phases of the surface vibrator and the acoustics. The main influence parameters investigated are the frequency (of the acoustic and the TS wave), the pressure gradient (given by the streamwise position of the surface roughness) and the influence of surface vibrations on the acoustic receptivity. Additionaly pure vibrational receptivity is studied. The main goal of this investigation is to study and compare systematically different types of receptivity in the same base flow developing on an airfoil at a Reynolds number typical for glider applications.

Experimental Investigation on 3D Acoustic Receptivity of a Laminar Boundary Layer in the Presence of Surface Non-Uniformities

Hot-wire measurements were performed under controlled conditions in the Laminar Wind Tunnel on an airfoil with pressure gradient. A quantitatively known acoustic field was established in the test section. The receptivity of the 2-dimensional laminar boundary layer in the presence of a localized 3D surface roughness was studied. The development of the generated Tollmien-Schlichting-waves was measured at different streamwise stations as cuts in spanwise direction. The quasi steady 3D roughness was modeled by a vibrating source driven at a frequency two orders below the acoustic one. The amplitude of the TS-waves was measured at combination frequencies. Fourier decomposition in time and space leads to wave number spectra which can be compared to linear stability theory. The receptivity function was evaluated by upstream extrapolation to the initial distributions at the source position.

EXPERIMENTAL STUDY OF RESONANT INTERACTIONS OF INSTABILITYWAVES IN AN AIRFOIL BOUNDARY LAYER

The present paper is devoted to the detailed experimental study of weakly nonli- near resonant interactions of Tollmien-Schlichting waves in a specially designed 2D non self-similar boundary layer on an airfoil. The influence of the funda- mental frequency on the efficiency of the tuned subharmonic resonance is in- vestigated as well as the influence of frequency and spanwise wavenumber de- tunings. The results are compared with Direct Numerical Simulations based on a vorticity-velocity formulation of the complete Navier-Stokes equations. Good overall agreement is achieved.