**1. Introduction**

At the beginning of a transition in shear flow, a large number of vortex structures are formed. The formation mechanism and dynamic characteristics of vortex structures are still key issues in fluid mechanics research.

Lee presented direct comparisons of experimental results on transitions in wall-bound flows obtained by flow visualizations, hot-film measurement, and Particle Image Velocimetry (PIV), along with a brief mention of relevant theoretical progresses, based on a critical review of about 120 selected publications. Despite the somewhat different initial disturbance conditions used in the experiments, the flow structures were found to be practically the same [1]. Sharma presented a new theory of coherent structures in wall turbulence. The theory is the first to predict packets of hairpin vortices and other structures in turbulence and their dynamics, based on an analysis of the Navier–Stokes equations, under an assumption of a turbulent mean profile [2]. Wedin studied finite-amplitude coherent structures with a reflection symmetry in the span-wise direction of a parallel boundary layer flow; some states computed displayed a span-wise spacing between streaks of the same length scale as turbulence flow structures observed in experiments [3]. Wu presented a mathematical theory to describe the nonlinear dynamics of coherent structures. The formulation was based on a triple decomposition of the instantaneous flow into a mean field, coherent fluctuations, and small-scale turbulence, but with the mean-flow distortion induced by nonlinear interactions of coherent fluctuations being treated as part of the organized motion [4]. Mcmullan implemented large eddy simulations of the plane mixing layer for the purpose of reducing the stream-wise vortex structure that may exist in these flows. Both an initially laminar and initially turbulent mixing layer were considered in this study. The initially laminar flow originated from Blasius profiles with a white noise fluctuation environment, whilst the initially turbulent flow had an inflow condition obtained from an inflow turbulence generation method. Flow visualization images demonstrated that both mixing layers contained organized turbulent coherent structures, and that the structures contained rows of stream-wise vortices distributed across the span of the mixing layer [5]. Wall studied three spatially extended traveling wave exact coherent states, together with one span-wise localized state for channel flow. Two of the extended flows were derived by the homotopy method from solutions. Both these flows were asymmetric with respect to the channel center plane, and featured streaky structures in stream-wise velocity flanked by staggered vortical structures; one of these flows featured two streak/vortex systems per span-wise wavelength [6]. Kang studied the direct numerical simulation data of a wave packet in laminar turbulent transition in a Blasius boundary layer. The decomposition of this wave packet into a set of modes could be achieved in a wide variety of ways [7]. Shinneeb experimentally studied the turbulent wake generated by a vertical sharp-edged flat plate suspended in a shallow channel flow with a gap near the bed. Two different gap heights were studied which were compared with the no-gap flow case. The Reynolds number based on the water depth was 45,000. Extensive measurements of the flow field in the vertical and horizontal planes were made using a PIV system. The large vortices were exposed by analyzing the PIV velocity fields using the proper orthogonal decomposition method [8]. Lemarechal experimentally investigated the laminar turbulent transition of a Blasius boundary layer-like flow at the Institute of Aerodynamics and Gas Dynamics, University of Stuttgart. The late stage of controlled transition with K-type breakdown was investigated with the temperature-sensitive paint (TSP) method on the flat-plate surface. The test conditions enabled the TSP method to resolve the complete transition process temporally and spatially. Therefore, it was possible to detect the coherent structures occurring in the late stage of laminar–turbulent transition from the visualizations on the flat-plate surface, namely, Λ and Ω vortices [9]. The effects of isolated, cylindrical roughness elements on laminar–turbulent transition in a flat-plate boundary layer were investigated in a laminar water channel. Most predictions by global linear stability theory could be confirmed, but additional observations in the physical flow demonstrated that not all features could be captured adequately by global linear stability theory [10].

There are many kinds of small disturbance models for boundary layer transition, such as combination Tollmier–Schlichting(T–S) waves, multi-eddy structures which satisfy continuity equation, slot jets, and so on. The initial disturbance sources of other physical models except slot jets are still not clear. In this paper, vortex structures induced by local instantaneous small wall vibration combinations similar to the real physical mechanism are adopted, and the internal mechanisms of the influence of different disturbance combinations on the flow stability are analyzed.

To perform approximate simulations of the wall local forced vibration, an initial disturbance was set in the form of local single-period micro-vibration at the bottom of the wall of *y* = 0. However, the influence of physical deformation on the mesh was neglected since it is outside of the scope of this paper. Two disturbance models were implemented, namely, the positive–negative (P–N) model and the negative–positive (N–P) model. The P–N model is defined as follows: the normal disturbance velocity at mesh points in the circle region, ) (*x* − 10.5) <sup>2</sup> <sup>+</sup> *<sup>z</sup>*<sup>2</sup> <sup>&</sup>lt; 2 is supposed to be *<sup>v</sup>* <sup>=</sup> 0.02 sin(2<sup>π</sup> · *<sup>t</sup>*/15), where <sup>0</sup> <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> 15, and that in the circle region, ) (*x* − 14.7) <sup>2</sup> <sup>+</sup> *<sup>z</sup>*<sup>2</sup> <sup>&</sup>lt; 2, is supposed to be *<sup>v</sup>* <sup>=</sup> <sup>−</sup>0.02 sin(2π·*t*/15), where 0 ≤ *t* ≤ 15. Similarly, the N–P model is defined as follows: the normal disturbance velocity at mesh points in the circle region, ) (*x* − 10.5) <sup>2</sup> <sup>+</sup> *<sup>z</sup>*<sup>2</sup> <sup>&</sup>lt; 2, is supposed to be *<sup>v</sup>* <sup>=</sup> <sup>−</sup>0.02 sin(2<sup>π</sup> · *<sup>t</sup>*/15), where <sup>0</sup> <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> 15, and that in the circle region, ) (*x* − 14.7) <sup>2</sup> <sup>+</sup> *<sup>z</sup>*<sup>2</sup> <sup>&</sup>lt; 2, is supposed to be *<sup>v</sup>* <sup>=</sup> 0.02 sin(2<sup>π</sup> · *<sup>t</sup>*/15), where 0 ≤ *t* ≤ 15. The amplitude of the small disturbance velocity is commonly chosen to be about 1% of the maximum mean basic flow velocity. A relatively small amplitude of the disturbance velocity can lead to a relatively long time of disturbance evolution. The computational domain and initial disturbance location are shown in Figure 1.

**Figure 1.** Distribution of initial disturbance at the wall.

#### **2. Governing Equations and Numerical Method**
