Boundary-layer receptivity for a parabolic leading edge

The effect of the nose radius of a body on boundary-layer receptivity is analysed for the case of a symmetric mean flow past a body with a parabolic leading edge. Asymptotic methods based on large Reynolds number are used, supplemented by numerical results. The Mach number is assumed small, and acoustic free-stream disturbances are considered. The case of free-stream acoustic waves, propagating obliquely to the symmetric mean flow is considered. The body nose radius, rn, enters the theory through a Strouhal number, S = ?rn/U, where ? is the frequency of the acoustic wave and U is the mean flow speed. The finite nose radius dramatically reduces the receptivity level compared to that for a flat plate, the amplitude of the instability waves in the boundary layer being decreased by an order of magnitude when S = 0.3. Oblique acoustic waves produce much higher receptivity levels than acoustic waves propagating parallel to the body chord

A principle of similarity for nonlinear vibration absorbers

This paper develops a principle of similarity for the design of a nonlinear absorber, the nonlinear tuned vibration absorber (NLTVA), attached to a nonlinear primary system. Specifically, for effective vibration mitigation, we show that the NLTVA should feature a nonlinearity possessing the same mathematical form as that of the primary system. A compact analytical formula for the nonlinear coefficient of the absorber is then derived. The formula, valid for any polynomial nonlinearity in the primary system, is found to depend only on the mass ratio and on the nonlinear coefficient of the primary system. When the primary system comprises several polynomial nonlinearities, we demonstrate that the NLTVA obeys a principle of additivity, i.e., each nonlinear coefficient can be calculated independently of the other nonlinear coefficients using the proposed formula

H∞H_\infty optimization of multiple tuned mass dampers for multimodal vibration control

In this paper, a new computational method for the purpose of multimodal vibration mitigation using multiple tuned mass dampers is proposed. Classically, the minimization of the maximum amplitude is carried out using direct $H_\infty$ optimization. However, as shall be shown in the paper, this approach is prone to being trapped in local minima, in view of the nonsmooth character of the problem at hand. This is why this paper presents an original alternative to this approach through norm-homotopy optimization. This approach, combined with an efficient technique to compute the structural response, is shown to outperform direct $H_\infty$ optimization in terms of speed and performance. Essentially, the outcome of the algorithm leads to the concept of all-equal-peak design for which all the controlled peaks are equal in amplitude. This unique design is new with respect to the existing body of knowledge.Comment: This is a new version of a preprint previously named "All-equal-peak design of multiple tuned mass dampers using norm-homotopy optimization

Leading-edge receptivity for blunt-nose bodies

This research program investigates boundary-layer receptivity in the leading-edge region for bodies with blunt leading edges. Receptivity theory provides the link between the unsteady distrubance environment in the free stream and the initial amplitudes of the instability waves in the boundary layer. This is a critical problem which must be addressed in order to develop more accurate prediction methods for boundary-layer transition. The first phase of this project examines the effects of leading-edge bluntness and aerodynamic loading for low Mach number flows. In the second phase of the project, the investigation is extended to supersonic Mach numbers. Singular perturbation techniques are utilized to develop an asymptotic theory for high Reynolds numbers. In the first year, the asymptotic theory was developed for leading-edge receptivity in low Mach number flows. The case of a parabolic nose is considered. Substantial progress was made on the Navier-Sotkes computations. Analytical solutions for the steady and unsteady potential flow fields were incorporated into the code, greatly expanding the types of free-stream disturbances that can be considered while also significantly reducing the the computational requirements. The time-stepping algorithm was modified so that the potential flow perturbations induced by the unsteady pressure field are directly introduced throughout the computational domain, avoiding an artificial 'numerical diffusion' of these from the outer boundary. In addition, the start-up process was modified by introducing the transient Stokes wave solution into the downstream boundary conditions

Boundary-layer receptivity for a parabolic leading edge. Part 2. The small-Strouhal-number limit

In Hammerton & Kerschen (1996), the effect of the nose radius of a body on boundary-layer receptivity was analysed for the case of a symmetric mean flow past a two-dimensional body with a parabolic leading edge. A low-Mach-number two-dimensional flow was considered. The radius of curvature of the leading edge, rn, enters the theory through a Strouhal number, S=?rn/U, where ? is the frequency of the unsteady free-stream disturbance and U is the mean flow speed. Numerical results revealed that the variation of receptivity for small S was very different for free-stream acoustic waves propagating parallel to the mean flow and those free-stream waves propagating at an angle to the mean flow. In this paper the small-S asymptotic theory is presented. For free-stream acoustic waves propagating parallel to the symmetric mean flow, the receptivity is found to vary linearly with S, giving a small increase in the amplitude of the receptivity coefficient for small S compared to the flat-plate value. In contrast, for oblique free-stream acoustic waves, the receptivity varies with S1/2, leading to a sharp decrease in the amplitude of the receptivity coefficient relative to the flat-plate value. Comparison of the asymptotic theory with numerical results obtained in the earlier paper confirms the asymptotic results but reveals that the numerical results diverge from the asymptotic result for unexpectedly small values of S

Leading-edge receptivity for bodies with mean aerodynamic loading

Boundary-layer receptivity in the leading-edge region of a cambered thin airfoil is analysed for the case of a low-Mach-number flow. Acoustic free-stream disturbances are considered. Asymptotic results based on large Reynolds number ($U^2 / \omega \nu \,{\gg}\, 1$) are presented, supplemented by numerical solutions. The influence of mean aerodynamic loading enters the theory through a parameter $\mu$, which provides a measure of the flow speed variations in the leading-edge region, due to flow around the leading edge from the lower surface to the upper. A Strouhal number based on airfoil nose radius, $S\,{=}\,\omega r_n/U$, also enters the theory. The variation of the receptivity level as a function of $\mu$ and $S$ is analysed. Modest levels of aerodynamic loading are found to decrease the receptivity level for the upper surface of the airfoil, while the receptivity is increased for the lower surface. For larger angles of attack close to the critical angle for boundary layer separation, a local rise in the receptivity occurs for the upper surface, while on the lower surface the receptivity decreases. These effects are more pronounced at larger values of $S$. While the Tollmien–Schlichting wave does not emerge until a downstream distance of $O((U^2 / \omega \nu)^{1/3} U / \omega)$, the amplitude of the Tollmien–Schlichting wave is influenced by the acoustic free-stream disturbances only in a relatively small region near the leading edge, of length approximately $4 U/\omega$. The numerical receptivity coefficients calculated, together with the asymptotic eigenfunctions presented, provide all the necessary information for transition analysis from the interaction of acoustic disturbances with leading-edge geometry

Suppression of Limit Cycle Oscillations using the Nonlinear Tuned Vibration Absorber

The objective of the present study is to mitigate, or even completely eliminate, the limit cycle oscillations in mechanical systems using a passive nonlinear absorber, termed the nonlinear tuned vibration absorber (NLTVA). An unconventional aspect of the NLTVA is that the mathematical form of its restoring force is not imposed a priori, as it is the case for most existing nonlinear absorbers. The NLTVA parameters are determined analytically using stability and bifurcation analyses, and the resulting design is validated using numerical continuation. The proposed developments are illustrated using a Van der Pol-Duffing primary system

Boundary-layer receptivity and laminar-flow airfoil design

Boundary-layer receptivity examines the way in which external disturbances generate instability waves in boundary layers. Receptivity theory is complementary to stability theory, which studies the evolution of disturbances that are already present in the boundary layer. A transition prediction method which combines receptivity with linear stability theory would directly account for the influence of free-stream disturbances and also consider the characteristics of the boundary layer upstream of the neutral stability point. The current e sup N transition prediction methods require empirical correlations for the influence of environmental disturbances, and totally ignore the boundary layer characteristics upstream of the neutral stability point. The regions where boundary-layer receptivity occurs can be separated into two classes, one near the leading edges and the other at the downstream points where the boundary layer undergoes rapid streamwise adjustments. Analyses were developed for both types of regions, and parametric studies which examine the relative importance of different mechanisms were carried out. The work presented here has focused on the low Mach number case. Extensions to high subsonic and supersonic conditions are presently underway

Broadband passive targeted energy pumping from a linear dispersive rod to a lightweight essentially non-linear end attachment

We examine non-linear resonant interactions between a damped and forced dispersive linear finite rod and a lightweight essentially nonlinear end attachment. We show that these interactions may lead to passive, broadband and on-way targeted energy flow from the rod to the attachment, which acts, in essence, as non-linear energy sink (NES). The transient dynamics of this system subject to shock excitation is examined numerically using a finite element (FE) formulation. Parametric studies are performed to examine the regions in parameter space where optimal (maximal) efficiency of targeted energy pumping from the rod to the NES occurs. Signal processing of the transient time series is then performed, employing energy transfer and/or exchange measures, wavelet transforms, empirical mode decomposition and Hilbert transforms. By computing intrinsic mode functions (IMFs) of the transient responses of the NES and the edge of the rod, and examining resonance captures that occur between them, we are able to identify the non-linear resonance mechanisms that govern the (strong or weak) one-way energy transfers from the rod to the NES. The present study demonstrates the efficacy of using local lightweight non-linear attachments (NESs) as passive broadband energy absorbers of unwanted disturbances in continuous elastic structures, and investigates the dynamical mechanisms that govern the resonance interactions influencing this passive non-linear energy absorption