Nonlinear sinusoidal and varicose instability in a boundary layer

Abstract

It is well known [1] that the laminar‐turbulent transition at a low turbulence level of the free flow is associated with the development of instability waves, the so-called Tollmien‐Schlichting waves. When a twodimensional Tollmien‐Schlichting wave reaches a certain amplitude at the nonlinear stage of its development, it undergoes three-dimensional distortion and, as a result, characteristic three-dimensional Λ structures arise [1]. Owing to certain features of the appearance and development of these structures, they are not only typical for the classical laminar‐turbulent transition, but are also inevitable attributes of a transition to more complex flows, e.g., flows modulated with longitudinal streaky structures, such as Hertler vortices, transverseflow vortices on sliding wings, etc., as well as flows in the viscous sublayer of a turbulent boundary layer. In these cases, they arise in particular due to the secondary high-frequency instability of such flows and may be manifested not only as Λ structures, but also in the form of horseshoe vortices ( Ω structures), hairpin vortices, etc. A characteristic feature of the development of such structures, e.g., on a sliding wing, is the disappearance of one of the counter-rotating vortices due to the transverse flow, whereas the development of a classical Λ structure can be observed on a straight wing [1]. The high-frequency secondary instability of transition and turbulent near-wall flows in the presence of streaky structures is often attributed to so-called sinusoidal and varicose instability. Both instability modes were investigated under controlled conditions at the linear and initial stages of nonlinear development. When the transverse size of the streaky structure was larger than the thickness of the shear layer, growth of varicose instability was observed. At the same time, when the transverse size of the streaky structure was comparable to or smaller than the thickness of the shear layer, it became more instable with respect to antisymmetric (sinusoidal) modes than to symmetric (varicose) modes. The experiment reported in [2] clearly shows that the growth of the symmetric mode leads to the formation of hairpin vortices, which are a pair of counterrotating longitudinal vortices that are connected by a head, i.e., a Λ vortex, while an antisymmetric mode is developed to a train of quasi-longitudinal vortices with alternating-sign vorticity. Unfortunately, the experiments reported in [2] concerned only the initial stage of the nonlinear development of disturbances, and spatial resolution was insufficiently high to reveal the structure of the flow in more detail. In this paper, we report on our experimental investigations of the nonlinear stage of the varicose and sinusoidal instability of the streaky structure in the Blasius boundary layer. In contrast to the experiment reported in [2], the study is more detailed (thermal anemometer measurements of the longitudinal velocity component and velocity pulsations in space ( xyz ) at 5 × 10 4 points) in order to reveal the features of the dynamics of the appearance, development, and internal structure of coherent formations up to the later stages of their nonlinear development. The experiments were carried out under controlled conditions in a low-turbulent wind tunnel. A plane plate 14 mm in thickness, 1000 mm in width, and 2000 mm in length was placed in parallel in the operation part of the tunnel. The streaky structure was generated by means of a cylindrical roughness element, which had a height of 1.1 mm and a diameter of 5.8 mm and was placed in the center of the plate at a distance of x 0 = 438 mm from the fore. The velocity of the flow was equal to U ∞ = 7.8 m/s, and the turbulence level was no higher than 0.04%. In the absence of the roughness element, the laminar boundary layer was developed without any waves and the velocity profile was close to the Blasius profile. A roughness-element height of h = 1.1 mm is close to the thickness of the displacement of