The interaction of disturbance modes behind an isolated cylindrical roughness element in a laminar boundary layer is investigated by means of hot-film anemometry and particle image velocimetry in a low-turbulence laminar water channel. Both sinuous and varicose disturbance modes are found in the wake of a roughness with unit aspect ratio (diameter/height $=$ 1). Interestingly, the frequency of the varicose mode synchronizes with the first harmonic of the sinuous mode when the critical Reynolds number from three-dimensional global linear stability theory is exceeded. The coupled motion of sinuous and varicose modes is explained by frequency lock-in. This mechanism is of great importance in many aspects of nature, but has not yet received sufficient attention in the field of boundary-layer theory. A Fourier mode decomposition provides detailed analyses of sinuous and varicose modes. The observation is confirmed by a second experiment with the same aspect ratio at a different position in the laminar boundary layer. When the aspect ratio is increased, the flow is fully governed by the varicose mode. Thus, no frequency lock-in can be observed in this case. The significance of this work is to explain how sinuous and varicose modes can co-exist behind a roughness and to propose a mechanism which is well established in physics but not encountered often in boundary-layer theory.
Experimental observation of frequency lock-in of roughness-induced instabilities in a laminar boundary layer