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In this thesis, we develop and apply numerical algorithms for simulation the interaction between fluid and thin flexible structures with sharp and diffused interface immersed boundary (IB) methods, respectively. With the sharp interface IB method, we simulate the interaction between air-water two-phase flows and thin flexible structures. In this algorithm, we propose a new method to avoid the unphysical fluxes across the thin structures, which effectively mitigates the issue of flow spurious penetration. In the proposed method, an explicit pressure boundary condition on the immersed boundary is applied. A new scheme for categorizing fluid, solid, and forcing nodes in the IB method is developed for avoiding extra pressure interpolation at the velocity projection step of the fractional-step method. Benchmark cases are tested and good agreement with the results in the literature is obtained. With the diffused interface IB method, we perform large-eddy simulation to investigate the dynamics of flow and canopy motions and the energy transfer in turbulent canopy flows. Different from the traditional approach that models the canopy as a continuous medium with a drag coefficient prescribed a priori, the IB method together with a beam model can explicitly capture the dynamics of individual stems. The periodic waving motion of the canopy, a phenomenon known as monami, is resolved. The simulation cases cover a broad range of stem flexibilities from rigid stems to oscillatory stems to stems yielding to the flow. For flexible canopies, as the stem flexibility reaches a high Cauchy number, the stem fluctuation amplitude decreases, and thus, the canopy behaves more like a rigid canopy, a phenomenon we call 'high flexibility-induced rigidity', which is used to explain the similarities of the flow features between rigid and highly flexible canopies. Analyses of the turbulent kinetic energy (TKE) budget are performed. Inside the canopy, the wake production term associated with the canopy-induced spatial inhomogeneity is more pronounced than the shear production term. Near the canopy top in the flexible canopy cases, the waving term associated with the canopy drag-flow velocity correlation can be as large as one-half of the shear production term. Spectral TKE budget analyses further reveal dominant effects at two characteristic scales: the monami scale associated with the coherent structures in the mixing layer and the wake scale associated with the interval between adjacent stems. The waving term converts mean kinetic energy into dispersive kinetic energy and TKE. For the TKE in flexible canopies, the waving term is found to play an important role in the inter-scale and wall-normal transport terms. Our LES data show that the spectral shortcut mechanism proposed by previous studies is caused by the waving term, not the inter-scale transport term. The underlying mechanism responsible for the waving term is further elucidated through quadrant analyses of the correlation between the drag and velocity fluctuations of flexible canopy stems.