On the stability of laminar boundary-layer flow over a flat plate with a compliant surface

Abstract

The problem has been examined using a kinematic model for wall pliability, wherein a kinematic postulation of the wall boundary conditions is made. A form of the normalized wall-displacement and its phase are used as additional parameters in an extended eigenvalue problem. Using this technique the entire gamut of possibilities regarding stability of flow past (normally) pliable walls can be examined, yet without recourse to any specific material properties for the wall. Rather, the results based on the kinematic model can be used to back-calculate the material properties corresponding to any chosen model for the dynamics of the wall. A sample back calculation is discussed herein for the Benjamin–Landahl wall model, and based on this some predictions are made regarding both stabilization of the flow and physical realizability of modes. It is believed that the kinematic model will prove useful in further understanding of the problem, and in the design of stabilizing coatings.The results show that there are three important ‘mode classes’ (distinct from ‘modes’), namely the Tollmien–Schlichting (TS), resonant (R) and Kelvin–Helmholtz (KH). Whereas the TS and R mode classes broadly agree with modes bearing similar names as found by earlier workers, the present KH mode class is difficult to classify based on earlier work. Moreover, there are also important transitional mode classes in the regions of bifurcations of the regular mode classes.Two important concepts evolve in connection with the TS and R mode classes, namely the existence of ‘stable pockets’ for the former and ‘unstable pockets’ for the latter. It is also confirmed herein that there are conflicting requirements on the damping d to stabilize TS and R modes. Considering these points it has been suggested that TS and R modes be avoided by keeping soft surfaces as compliant coatings. However, this in turn leads to instabilities from one of the transitional mode classes. It is also seen that a soft surface that is also marginally active (i.e. having a small negative value of d) could render even better stabilization.