Two-dimensional Tollmien-Schlichting Research Articles

Stability analysis of a streaky boundary layer generated by miniature vortex generators

Delaying laminar-turbulent transition is an attractive method to reduce friction drag on streamlined bodies. In the case of natural transition, in which turbulence is triggered by the growth of two-dimensional Tollmien–Schlichting (TS) instabilities, this growth of TS instabilities can be attenuated by introducing steady streamwise streaks into the boundary layer. These streamwise streaks are generated by streamwise vortices, which rearrange the flow via the lift-up mechanism. Such streamwise vortices can be induced by so-called miniature vortex generators (MVGs). Although recently considerable attention has been devoted to the investigation of MVGs, and multiple studies investigated MVGs with different parameters, the previously investigated MVG configurations are likely suboptimal. This study aims at (partially) filling this knowledge gap by complementing previous studies by conducting a parametric study of rectangular MVGs and assessing them with local linear stability analysis. Three parameters are varied simultaneously: the height, the spanwise distance of the MVG pairs, and the distance between MVGs in each pair. First, the modelling approach is presented, which consist of an efficient base flow computation, and BiGlobal stability analysis. The base flow calculation is validated by comparing our results with the experiments of Shattarzadeh and Fransson (2015). Then, the streak amplitude, the growth factors of the TS-waves and the secondary shear-layer instabilities are analysed in a parametric study. It is shown that with the variation of the spanwise distances, the maximum amplitude and the streamwise extent of the streaks in the boundary layer can be controlled, reminiscent of the behaviour of the optimal vortices. The findings and the test of the methodology provide valuable insight, which may be useful for the optimisation of MVGs.

A CFD-Compatible Amplification Factor Transport Equation for Oblique Tollmien-Schlichting Waves in Supersonic Boundary Layers

Boundary layer transition is a hot research topic in fluid mechanics and aerospace engineering. In low-speed flows, two-dimensional Tollmien-Schlichting (T-S) waves always dominate the flow instability, which has been modeled by Coder and Maughmer from 2013. However, in supersonic flows, three-dimensional oblique Tollmien-Schlichting waves become dominant in flow instability. Inspired by Coder and Maughmer’s NTS amplification factor transport equation for two-dimensional Tollmien-Schlichting waves in low-speed flows and Kroo and Sturdza’s linear stability theory (LST) analysis results for oblique Tollmien-Schlichting waves in supersonic flows, a new amplification factor transport equation for oblique Tollmien-Schlichting waves has been developed based on LST. The compressible Falkner-Skan similarity equations are introduced to build the relationships between nonlocal variables and local variables so that all the variables used in the present model can be calculated using local variables. Applications of this new transport equation to the flows over supersonic flat plate, 3% thick biconvex airfoil, and one modified supersonic laminar airfoil show promising results compared with the standard LST analysis results.

Influence of localised smooth steps on the instability of a boundary layer

We consider a smooth, spanwise-uniform forward-facing step defined by a Gauss error function of height 4 %–30 % and four times the width of the local boundary layer thickness $\unicode[STIX]{x1D6FF}_{99}$. The boundary layer flow over a smooth forward-facing stepped plate is studied with particular emphasis on stabilisation and destabilisation of the two-dimensional Tollmien–Schlichting (TS) waves and subsequently on three-dimensional disturbances at transition. The interaction between TS waves at a range of frequencies and a base flow over a single or two forward-facing smooth steps is conducted by linear analysis. The results indicate that for a TS wave with a frequency ${\mathcal{F}}\in [140,160]$ (${\mathcal{F}}=\unicode[STIX]{x1D714}\unicode[STIX]{x1D708}/U_{\infty }^{2}\times 10^{6}$, where $\unicode[STIX]{x1D714}$ and $U_{\infty }$ denote the perturbation angle frequency and free-stream velocity magnitude, respectively, and $\unicode[STIX]{x1D708}$ denotes kinematic viscosity), the amplitude of the TS wave is attenuated in the unstable regime of the neutral stability curve corresponding to a flat plate boundary layer. Furthermore, it is observed that two smooth forward-facing steps lead to a more acute reduction of the amplitude of the TS wave. When the height of a step is increased to more than 20 % of the local boundary layer thickness for a fixed width parameter, the TS wave is amplified, and thereby a destabilisation effect is introduced. Therefore, the stabilisation or destabilisation effect of a smooth step is typically dependent on its shape parameters. To validate the results of the linear stability analysis, where a TS wave is damped by the forward-facing smooth steps direct numerical simulation (DNS) is performed. The results of the DNS correlate favourably with the linear analysis and show that for the investigated frequency of the TS wave, the K-type transition process is altered whereas the onset of the H-type transition is delayed. The results of the DNS suggest that for the perturbation with the non-dimensional frequency parameter ${\mathcal{F}}=150$ and in the absence of other external perturbations, two forward-facing smooth steps of height 5 % and 12 % of the boundary layer thickness delayed the H-type transition scenario and completely suppressed for the K-type transition. By considering Gaussian white noise with both fixed and random phase shifts, it is demonstrated by DNS that transition is postponed in time and space by two forward-facing smooth steps.

On the role of adaptivity for robust laminar flow control

AbstractIn boundary-layer flows, one may reduce skin-friction drag by delaying the onset of laminar-to-turbulent transition via the attenuation of small-amplitude Tollmien–Schlichting (TS) waves. In this work, we use numerical simulations and experiments to compare the robustness of adaptive and model-based techniques for reducing the growth of two-dimensional TS disturbances. In numerical simulations, the optimal linear quadratic Gaussian (LQG) regulator shows the best performance under the conditions it was designed for. However, it is found that the performance deteriorates linearly with the drift of the Reynolds number from its nominal value. As a result, an order-of-magnitude loss of performance is observed when applying the computation-based LQG controller in wind-tunnel experiments. In contrast, it is shown that the adaptive filtered-X least-mean-squares (FXLMS) algorithm is able to maintain an essentially constant performance for significant deviations of the nominal values of the disturbance amplitude and Reynolds number.

Numerical Simulation of Riblet Controlled Spatial Transition in a Zero-Pressure-Gradient Boundary Layer

To analyze the fundamental physical mechanism which determines the damping effect of a riblet surface on three-dimensional transition several numerical simulations of spatial transition in a flat plate zero-pressure-gradient boundary layer above a riblet wall are performed in this study. Two types of forced transition scenarios are investigated. The first type of transition is defined by K-type transition induced by a dominant two-dimensional Tollmien–Schlichting (TS) wave and a weak spanwise disturbance. The second type of transition is purely excited by two oblique waves. By a qualitative analysis of the occurring maximum wall-normal and spanwise velocity components and the Fourier modes of the disturbances the two-dimensional TS waves are found to be amplified by riblets, whereas three-dimensional structures, i.e., Λ-, hairpin, and streamwisely aligned vortices, are damped. At oblique transition the breakdown to turbulence is delayed by the riblets compared to transition on a clean surface. The investigation of the near wall flow structure reveals secondary flows induced by the riblets and reduced wall normal ejections as well as a reduced downwash.

Floquet analysis of secondary instability of boundary layers distorted by Klebanoff streaks and Tollmien–Schlichting waves

Previous studies of the interaction between boundary layer streaks and Tollmien–Schlichting (TS) waves have shown puzzling effects. Streaks were shown to reduce the growth rate of primary TS waves and, thereby, to delay transition; however, they can also promote transition by inducing a secondary instability. The outcome of the interaction depends on the spanwise wavelength and intensity of the streaks as well as on the amplitude of the TS waves. A Floquet analysis of secondary instability is able to explain many of these features. The base state is periodic in two directions: it is an Ansatz composed of a saturated TS wave (periodic in x) and steady streaks (periodic in z). Secondary instability analysis is extended to account for the doubly periodic base flow. Growth rate computations show that, indeed, the streak can either enhance or diminish the overall stability of the boundary layer. The stabilizing effect is a reduction in the growth rate of the primary two-dimensional TS wave; the destabilizing effect is a secondary instability. Secondary instability falls into two categories, depending on the spanwise spacing of the streaks. The response of one category to perturbations is dominated by fundamental and subharmonic instability; the response of the other is a detuned instability.

Turbulence production in an APG-boundary-layer transition induced by randomized perturbations

This paper is devoted to an experimental investigation of formation and development of coherent vortical structures at late stages of a laminar-turbulent transition initiated by a harmonic, almost two-dimensional Tollmien-Schlichting (TS) wave perturbed by weak (initially) broadband disturbances. The initial base flow represented a self-similar boundary layer with an adverse pressure gradient (APG) with Hartree parameter βH = −0.115. Experiments were performed at controlled disturbance conditions with the help of the ‘deterministic noise’ method and a universal disturbance source of instability waves. The main measurements were carried out by means of a hot-wire anemometer in a broad spatial region of the flow starting with stages of quasi-sinusoidal small-amplitude instability wave and ending with final stages of transition characterized by formation of concentrated localized vortical structures. The excited perturbations were partly random (within 20 TS-wave fundamental periods) but periodical at very large time scales during which the flow passes the model several times. The detailed measurements and the experimental data processing gave us the possibility to obtain instantaneous velocity and vorticity fields in the (x, y, z, t)-space and to perform computer-aided ‘visualization’ of the instantaneous flow structure. Specific features of the turbulence production mechanism occurring at late stages of transition are studied and compared with previously reported data obtained at sinusoidal excitation. A qualitative similarity is found between essentially nonlinear stages of transition observed in the present (randomized) case and those studied previously in cases of transition initiated by a harmonic TS wave or by a TS wave packet. It is found that interaction of primary wave with a broadband ‘noise’ of 3D TS waves leads at late stages of transition to formation of Λ-vortices, intensive Λ -shaped high-shear (HS) layers, Ω -shaped vortices, ring-like vortices and associated ‘spikes’ in time-traces, and other phenomena. These structures are found to be disposed in a random order in time and space and have somewhat distorted shapes (in contrast to the periodic-excitation case studied earlier). However, their general properties are qualitatively similar. The obtained instantaneous pictures of velocity and vorticity fields seem to correspond qualitatively to those present in the near-wall region of the developed turbulent boundary layer.

A methodology for optimal laminar flow control: Application to the damping of Tollmien–Schlichting waves in a boundary layer

A methodology for determining the optimal steady suction distribution for the delay of transition in a boundary layer is presented. The flow state is obtained from the coupled system of boundary layer equations and parabolized stability equations (PSE), to account for the spatially developing nature of the flow. The wall suction is defined by an optimal control procedure based on the iterative solution of the equations for the state and the dual state; the latter is available from the adjoint boundary layer equations and the adjoint PSE. The technique is applied to the control of two-dimensional Tollmien–Schlichting (TS) waves. Results show that the onset of the instability can be significantly postponed and/or the growth rate considerably reduced by applying an appropriate suction through the whole wall length, in a wide frequency band. Control over panels of finite length completes the study and brings useful, preliminary information on the practicality of the approach in view of implementation. Finally, a simplified methodology which does not rely on the PSE is discussed, based on the minimization of the shape factor. Satisfactory results are achieved with this simpler approach which might, thus, constitute a method of choice when results are needed rapidly, i.e., during on-line control of TS waves.

Numerical investigation of the interaction of the Klebanoff-mode with a Tollmien–Schlichting wave

Direct numerical simulations (DNS) of the Navier–Stokes equations are used to investigate the role of the Klebanoff-mode in laminar–turbulent transition in a flatplate boundary layer. To model the effects of free-stream turbulence, volume forces are used to generate low-frequency streamwise vortices outside the boundary layer. A suction/blowing slot at the wall is used to generate a two-dimensional Tollmien–Schlichting (TS) wave inside the boundary layer. The characteristics of the fluctuations inside the boundary layer agree very well with those measured in experiments. It is shown how the interaction of the Klebanoff-mode with the two-dimensional TS-wave leads to the formation of three-dimensional TS-wavepackets. When the disturbance amplitudes reach a critical level, a fundamental resonance-type secondary instability causes the breakdown of the TS-wavepackets into turbulent spots.

Interaction between Görtler vortices and two-dimensional Tollmien–Schlichting waves

The nonlinear interaction between Görtler vortices and two-dimensional Tollmien–Schlichting (TS) waves is studied with a spatial, nonparallel model based on the parabolized stability equations. The effect of the TS waves on the development of the vortices is accounted for, showing that TS wave amplitudes of the same order of magnitude as the vortices result in significant nonlinear interaction. The range of governing parameters that has been studied so far is extended and the main effects of Görtler number, spanwise wave number, and initial amplitudes are identified. The study shows that the relative growth rates and initial amplitudes are the two most significant parameters.