The three-dimensional stability of boundary-layer flow over compliant walls

Abstract

This paper examines the linear stability of the Blasius boundary layer over compliant walls to three-dimensional (oblique) disturbance wave modes. The formulation of the eigenvalue problem is applicable to compliant walls possessing general material anisotropy. Isotropic-material walls and selected classes of anisotropic-material walls are studied. When the properties of the wall are identical with respect to all oblique wave directions, the stability eigenvalue problem for unstable three-dimensional wave modes may be reduced to an equivalent problem for two-dimensional modes. The results for isotropic-material walls show that three-dimensional Tollmien–Schlichting instability modes are more dominant than their two-dimensional counterparts when the walls are sufficiently compliant. The critical Reynolds number for Tollmien-Schlichting instability may be given by three-dimensional modes. Furthermore, for highly compliant walls, calculations based solely on two-dimensional modes are likely to underestimate the maximum disturbance growth factor needed for transition prediction and correlation. However, because the disturbance growth rates on highly compliant walls are much lower than those on a rigid wall, significant delay of transition may still be possible provided compliance-induced instabilities are properly suppressed. Walls featuring material anisotropy which have reduced stiffness to shear deformation in the transverse and oblique planes are also investigated. Such anisotropy is found to be effective in reducing the growth rates of the three-dimensional modes relative to those of the two-dimensional modes.