Effect of compliant walls on secondary instabilities in boundary-layer transition

Abstract

For aerodynamic and hydrodynamic vehicles, it is highly desirable to reduce drag and noise levels. A reduction in drag leads to fuel savings. In particular for submersible vehicles, a decrease in noise levels inhibits detection. A suggested means to obtain these reduction goals is by delaying the transition from laminar to turbulent flow in external boundary layers. For hydrodynamic applications, a passive device which shows promise for transition delays is the compliant coating. In previous studies with a simple mechanical model representing the compliant wall, coatings were found that provided transition delays as predicted from the semi-empirical e^n method. Those studies were concerned with the linear stage of transition where the instability of concern is referred to as the primary instability. For the flat-plate boundary layer, the Tollmien-Schlichting (TS) wave is the primary instability. In one of those studies, it was shown that three-dimensional (3-D) primary instabilities, or oblique waves, could dominate transition over the coatings considered. From the primary instability, the stretching and tilting of vorticity in the shear flow leads to a secondary instability mechanism. This has been theoretical described by Herbert based on Floquet theory. In the present study, Herbert''s theory is used to predict the development of secondary instabilities over isotropic and non-isotropic compliant walls. Since oblique waves may be dominant over compliant walls, a secondary theory extention is made to allow for these 3-D primary instabilities. The effect of variations in primary amplitude, spanwise wavenumber, and Reynolds number on the secondary instabilities are examined. As in the rigid wall case, over compliant walls the subharmonic mode of secondary instability dominates for low- amplitude primary disturbances. Both isotropic and non-isotropic compliant walls lead to reduced secondary growth rates compared to the rigid wall results. For high frequencies, the non-isotropic wall suppresses the amplification of the secondary instabilities, while instabilities over the isotropic wall may grow with an explosive rate similar to the rigid wall results. For the more important lower frequencies, both isotropic and non- isotropic compliant walls suppress the amplification of secondary instabilities compared to the rigid wall results. The twofold major discovery and demonstration of the present investigation are: (1) the use of passive devices, such as compliant walls, can lead to significant reductions in the secondary instability growth rates and amplification; (2) suppressing the primary growth rates and subsequent amplification enable delays in the growth of the explosive secondary instability mechanism.