Nonlinear distortions and short-wavelength secondary instability directly induced by distributed roughness in three-dimensional boundary layers
Author(s)
Yuan, Bo
Wu, Xuesong
Type
Journal Article
Abstract
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 O(R−1/3δ∗), where the Reynolds number R is defined using the local boundary-layer thickness δ∗. Despite having a height much smaller than δ∗, 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 O(R−1/3δ∗). Surface roughness alters the base flow significantly, leading to 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 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 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.
instabilities. We consider distributed roughness in the form of a wavy wall with its height being taken to be of O(R−1/3δ∗), where the Reynolds number R is defined using the local boundary-layer thickness δ∗. Despite having a height much smaller than δ∗, 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 O(R−1/3δ∗). Surface roughness alters the base flow significantly, leading to 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 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 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.
Date Issued
2025-11-10
Date Acceptance
2025-09-26
Citation
Journal of Fluid Mechanics, 2025, 1022
ISSN
0022-1120
Publisher
Cambridge University Press
Journal / Book Title
Journal of Fluid Mechanics
Volume
1022
Copyright Statement
© The Author(s), 2025. Published by Cambridge University Press This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
License URL
Subjects
boundary layer stability
critical layers
CROSS-FLOW VORTICES
DISTURBANCES
GENERATION
Mechanics
MECHANISMS
Physical Sciences
Physics
Physics, Fluids & Plasmas
RECEPTIVITY
SCATTERING
Science & Technology
STABILITY
Technology
TOLLMIEN-SCHLICHTING WAVES
TRANSITION
transition to turbulence
Publication Status
Published
Article Number
A33
Date Publish Online
2025-11-03