The stability of an incompressible two-dimensional wake

Abstract

The growth of small disturbances in a two-dimensional incompressible wake has been investigated theoretically and experimentally. The theoretical analysis is based upon inviscid stability theory wherein small disturbances are considered from both temporal and spatial reference frames. Through a combined stability analysis, in which small disturbances are permitted to amplify in both time and space, the relationship between the disturbance characteristics for the temporal and spatial reference frames is shown. In these analyses a quasi-uniform assumption is adopted to account for the continuously varying mean-velocity profiles that occur behind flat plates and thin airfoils. It is found that the most unstable disturbances in the wake produce transverse oscillations in the mean-velocity profile and correspond to growing waves that have a minimum group velocity.Experimentally, the downstream development of the wake of a thin airfoil and the wave characteristics of naturally amplifying small disturbances are investigated in a water tank. The disturbances that develop are found to produce transverse oscillations of the mean-velocity profile in agreement with the theoretical prediction. From the comparison of the experimental results with the predictions for the characteristics of the most unstable waves via the temporal and spatial analyses, it is concluded that the stability analysis for the wake is to be considered solely from the more realistic spatial viewpoint. Undoubtedly, this conclusion is also applicable to other highly unstable flows such as jets and free shear layers.In accordance with the disturbance vorticity distribution as determined from the spatial model, a description of the initial development of a vortex street is put forth that contrasts with the description given by Sato & Kuriki (1961).

An experimental study of airfoil instability tonal noise with trailing edge serrations

“…For the S0 case, it was found that localized, non-convective normal-component velocity fluctuations are prominent inside the wake. This aerodynamic feature, which has been both predicted and measured previously [6,7,29], can be considered as a noise source. The distance of predicted by feedback Model A is observed experimentally.…”

“…For example, in an experimental study of the flow behind flat plates and cylinders, Mattingly and Criminale [29] observed a localized velocity oscillation, which can only be sustained in the absolute instability region. Koch [30] proposed a global resonance condition in which wake tones can sometimes be emitted.…”

Linear analysis of the cylinder wake mean flow

“…[3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Over the years a theory of spatially developping flows has advanced to the point of addressing fully nonlinear structures in wakes [17,18].…”

Inviscid instability characteristics of free shear layers with non-uniform density

“…A11 quantities used here are normalized with the average velocity (U,+U,)/Z, average density (p,+p,)/2 and the local layer The main result is that when the wake component is present there are two unstable branches as opposed to one in the case of the hyperbolic tangent. These two branches do not have much in common with the "sinuous" and "varicose" modes of instability calculated by Mattingly & Criminale (1972) in the case of pure wakes since the symmetry conditions that exist in the pure wake do not hold here. The stronger branch leads to the usual Kelvin-Helmholtz roll-up patterns observed in shear layers and it is referred to here as the shear layer branch.…”