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This study presents a novel investigation into the vortex dynamics of flow around a near-wall rectangular cylinder based on direct numerical simulation at 
$Re=1000$, marking the first in-depth exploration of these phenomena. By varying aspect ratios (
$L/D = 5$, 
$10$, 
$15$) and gap ratios (
$G/D = 0.1$, 
$0.3$, 
$0.9$), the study reveals the vortex dynamics influenced by the near-wall effect, considering the incoming laminar boundary layer flow. Both 
$L/D$ and 
$G/D$ significantly influence vortex dynamics, leading to behaviours not observed in previous bluff body flows. As 
$G/D$ increases, the streamwise scale of the upper leading edge (ULE) recirculation grows, delaying flow reattachment. At smaller 
$G/D$, lower leading edge (LLE) recirculation is suppressed, with upper Kelvin–Helmholtz vortices merging to form the ULE vortex, followed by instability, differing from conventional flow dynamics. Larger 
$G/D$ promotes the formation of an LLE shear layer. An intriguing finding at 
$L/D = 5$ and 
$G/D = 0.1$ is the backward flow of fluid from the downstream region to the upper side of the cylinder. At 
$G/D = 0.3$, double-trailing-edge vortices emerge for larger 
$L/D$, with two distinct flow behaviours associated with two interactions between gap flow and wall recirculation. These interactions lead to different multiple flow separations. For 
$G/D = 0.9$, the secondary vortex (SV) from the plate wall induces the formation of a tertiary vortex from the lower side of the cylinder. Double-SVs are observed at 
$L/D = 5$. Frequency locking is observed in most cases, but is suppressed at 
$L/D = 10$ and 
$G/D = 0.9$, where competing shedding modes lead to two distinct evolutions of the SV.
We explore the limiting spectral distribution of large-dimensional random permutation matrices, assuming the underlying population distribution possesses a general dependence structure. Let 
$\textbf X = (\textbf x_1,\ldots,\textbf x_n)$ 
$\in \mathbb{C} ^{m \times n}$ be an 
$m \times n$ data matrix after self-normalization (n samples and m features), where 
$\textbf x_j = (x_{1j}^{*},\ldots, x_{mj}^{*} )^{*}$. Specifically, we generate a permutation matrix 
$\textbf X_\pi$ by permuting the entries of 
$\textbf x_j$ 
$(j=1,\ldots,n)$ and demonstrate that the empirical spectral distribution of 
$\textbf {B}_n = ({m}/{n})\textbf{U} _{n} \textbf{X} _\pi \textbf{X} _\pi^{*} \textbf{U} _{n}^{*}$ weakly converges to the generalized Marčenko–Pastur distribution with probability 1, where 
$\textbf{U} _n$ is a sequence of 
$p \times m$ non-random complex matrices. The conditions we require are 
$p/n \to c >0$ and 
$m/n \to \gamma > 0$.
The numerical investigation focuses on the flow patterns around a rectangular cylinder with three aspect ratios (
$L/D=5$, 
$10$, 
$15$) at a Reynolds number of 
$1000$. The study delves into the dynamics of vortices, their associated frequencies, the evolution of the boundary layer and the decay of the wake. Kelvin–Helmholtz (KH) vortices originate from the leading edge (LE) shear layer and transform into hairpin vortices. Specifically, at 
$L/D=5$, three KH vortices merge into a single LE vortex. However, at 
$L/D=10$ and 
$15$, two KH vortices combine to form a LE vortex, with the rapid formation of hairpin vortex packets. A fractional harmonic arises due to feedback from the split LE shear layer moving upstream, triggering interaction with the reverse flow. Trailing edge (TE) vortices shed, creating a Kármán-like street in the wake. The intensity of wake oscillation at 
$L/D=5$ surpasses that in the other two cases. Boundary layer transition occurs after the saturation of disturbance energy for 
$L/D=10$ and 
$15$, but not for 
$L/D=5$. The low-frequency disturbances are selected to generate streaks inside the boundary layer. The TE vortex shedding induces the formation of a favourable pressure gradient, accelerating the flow and fostering boundary layer relaminarization. The self-similarity of the velocity defect is observed in all three wakes, accompanied by the decay of disturbance energy. Importantly, the decrease in the shedding frequency of LE (TE) vortices significantly contributes to the overall decay of disturbance energy. This comprehensive exploration provides insights into complex flow phenomena and their underlying dynamics.
