Nonlinear distortions and short-wavelength secondary instability directly induced by distributed roughness in three-dimensional boundary layers

Surface roughness of fairly small (micron-sized) height is known to influence significantly three-dimensional boundary-layer transition. In this paper, we investigate this sensitive effect from the viewpoint that roughness alters the base flow thereby inducing new instabilities. We consider distributed roughness in the form of a wavy wall with its height being taken to be of $\mathit{O} (R^{-1/3 } \delta ^{\ast })$ , where the Reynolds number $R$ is defined using the local boundary-layer thickness $\delta ^{\ast }$ . Despite having a height much smaller than $\delta ^{\ast }$ , the roughness is high enough to induce nonlinear responses. The roughness-distorted boundary-layer flow is characterised by a wall layer (WL) – a thin layer adjacent to the surface – the main layer and a critical layer (CL) – the vicinity of a special position at which a singularity of the Rayleigh equation occurs. The widths of both the WL and CL are of $\mathit{O} (R^{-1/3} \delta ^{\ast })$ . Surface roughness alters the base flow significantly, leading to $\mathit{O} (1)$ vorticity distortions in these layers. We show for the first time that the nonlinearly distorted flows in these layers support small-scale local instabilities due to the roughness-induced $\mathit{O} (1)$ vorticities. Two types of modes, CL and WL modes, are identified. The CL modes have short wavelengths and high frequencies, with the spatial and temporal instabilities being governed by essentially the same equation. Thus, we focus on the former, which can be formulated as a linear generalised eigenvalue problem. The WL modes have short wavelengths but $\mathit{O} (1)$ frequencies. The temporal WL mode is governed by a linear eigenvalue problem similar to that for the CL modes, while the spatial WL mode is described by a nonlinear eigenvalue problem. The onset of these small-scale fluctuations could form a crucial step in the transition to turbulence.