Generation of secondary instability modes in the interaction of a Tollmien-Schlichting wave with isolated roughness

Abstract

The infinite plane channel flow arising from the impingement of a plane instability wave of finite amplitude on isolated three-dimensional wall roughness is considered. The problem of the transformation of perturbations developing on the roughness in growing modes of secondary instability is solved. This problem describes the development of perturbations leading to the occurrence of a turbulent wedge. Simple relations describing the flow at large distances from the roughness are obtained. From these relations it follows that the angle at the vertex of the turbulent wedge is determined by the amplitude of the impinging wave, while the value of the perturbations generated is proportional to the roughness wlume.When the free-stream turbulence is fairly low, lzmlnar-turbulent transition starts with the onset and growth of a plane Tollmien-Schlichting wave. When the amplitude of this primary wave reaches a threshold value, secondary three-dimensional perturbations begin to develop rapidly against its background. Once the latter have reached a value comparable with the fundamental wave, the higher harmonics increase very rapidly and turbulent flow develops on a section several boundary-layer thicknesses long. On the basis of this perturbation development we can assume that the position of the transition coincides, with a high degree of accuracy, with the point at which the secondary perturbation amplitude becomes equal to the fundamental wave amplitude. In order to determine this point it is necessary to know the development law of the secondary perturbations and their initial amplitude. The development of the secondary perturbations is well descn'bed by the secondary instability theory [L2], whose conclusions agree with experiment. According to this theory, three-dlmensional perturbations, which are modes of a certain eigenvalue problem, are amplified against the two-dimensional fundamental wave background. There are two types of such modes. Unsteady subharmonic perturbations belong to the first type. Modes of the second type consist of a steady component and an unsteady part with period equal to that of the fundamental wave. The development of perturbations of the first and second types leads to the subharmonic and Klebanoff transitions, respectively. The type of modes amplified in a specific flow is determined by the initial conditions for the secondary perturbations. These initial conditions must be determined from the problem of the transformation of the external three-dimensional perturbations into secondary instability modes. Up till now this problem, similar to the susceptibility problem for the primary instability wave, has been considered only qualitatively [2].In the present paper the problem is solved for the perturbations resulting from the interaction of a primary instability wave with small isolated roughness on the streamlined surface. For the sake of simplicity, we will consider plane channel flow; however, the method of solution and the main results can also be applied to boundary layer flow.1. Consid...

Investigation of Mechanisms of Excitation and Growth of Unstable Perturbations in Pulsating Flows

“…Physically, the formulation of the problem investigated in the present study is analogous to the formulations considered in [7,8]: here too unstable perturbations are generated as a result of direct scattering of the main flow on steady-state roughness, and the characteristics of the excited waves (oscillation period and shape, wavelength, and growth rate) are determined by the flow-core parameters and significantly depend on the unsteady nature of the undisturbed flow.…”