Linear instability mechanisms on airfoils at low Reynolds numbermassive Separation, Wingtip Vortex Formation and the Trailing Vortex System

Resumo

Abstract Two- and three-dimensional modal and non-modal instability mechanisms of steady spanwise homogeneous laminar separated flows over airfoil profiles, placed at large angles of attack against the oncoming flow, have been investigated using linear global theory. Three NACA profiles of distinct thickness and camber were considered, in order to assess geometry effects on the laminar-turbulent transition paths discussed. At the conditions investigated large-scale steady separation occurs, such that Tollmien-Schlichting and crossflow mechanisms were not considered. It is found that the leading modal instability on all three airfoils is that associated with the Kelvin-Helmholtz (KH) mechanism, taking the form of the eigenmodes known from analysis of generic bluff bodies. The three-dimensional stationary eigenmode of the two-dimensional laminar separation bubble, associated in earlier analyses with the formation on the airfoil surface of large-scale separation patterns akin to stall-cell, is shown to be stronger damped than the KH mode. Non-modal instability analysis reveals the potential of the flows to sustain transient growth which becomes stronger with increasing angle of attack and Reynolds number. Optimal initial conditions were computed and were found to be analogous to those on a cascade of Low Pressure Turbine blades. By changing the time-horizon of the analysis these linear optimal initial conditions are found to evolve into the KH mode. Steady Navier-Stokes equations for a high angle of attack and Reynolds number flow are achieved through selective damping frequency method, and its modal analysis is performed. The most unstable mode is oscillating after the airfoil and dominates about O(10) chord length. The sub-leading mode is a KH type as appeared in the low Reynolds number steady flow. The stationary mode starts immediately behind the airfoil and then decays into the wake. The time-periodic base flows ensuing linear amplification of the KH mode are analyzed here via temporal Floquet theory. Two amplified modes are discovered, having characteristic spanwise wavelengths of approximately 0:6 and 2 chord lengths, respectively. Unlike secondary instabilities on the circular cylinder, three-dimensional short-wavelength perturbations are the first to become linearly unstable on all airfoils. Long-wavelength perturbations are quasi-periodic, standing or traveling-wave perturbation that also become unstable as the Reynolds number is increased further. The dominant short-wavelength instability gives rise to spanwise periodic wall-shear patterns, akin to the separation cells encountered on airfoils at low angles of attack and the stall cells found in flight at conditions close to stall. Thickness and camber have quantitative but not qualitative effect on the secondary instability analysis results obtained. The previous analysis assums an idealistic wing flow which has a homogeneous boundary conditions in the spanwise direction. A generalized wingtip developed downstream should be taken into account. To this purpose, a finite wing laminar flow has been performed. The flow field over the airfoil and in its wake was computed by full three-dimensional direct numerical simulation at a chord Reynolds number of Rec = 1750 and two angles of attack, AoA = 0 and 5 degrees. The spatial eigenvalue problem governing linear global small-amplitude perturbations superposed upon this base flow has been solved and results were used to initialize a linear PSE-3D marching procedure without any simplifying assumptions regarding the form of the trailing vortex system, other than weak dependence of all flow quantities on the axial spatial direction. Two classes of linearly unstable perturbations were identified, namely stronger-amplified symmetric modes and weaker-amplified antisymmetric disturbances, both peaking at the vortex sheet which connects the trailing vortices. The amplitude functions of both classes of modes were documented and N-factor predictions for potential laminar breakdown have been computed.