The damage constitutive model is of great significance to research the stress-strain relationship and damage evolution of rock under loading in engineering. In order to investigate the effect of anisotropic characteristic on the stress-strain relationship and damage evolution, a statistical damage constitutive model of anisotropic rock under true triaxial condition was developed. In this study, the plane which existed perpendicular to the coordinate axis was extracted from representative volume element (RVE) of rock. The extracted plane was assumed to be composed of abundant mesoscopic elements whose failure strength satisfied the Weibull distribution. According to the number of failure elements on the plane in each direction under loading, the anisotropic damage variable was established based on the proposed concept of areal damage. A statistical damage constitutive model of anisotropic rock was developed by using strain equivalent hypothesis and generalized Hooke constitutive model. Subsequently, the parameters in the anisotropic damage constitutive model were determined by the method of total differential. Thus, the damage evolution of anisotropic rock under various stress conditions can be conveniently evaluated by the anisotropic damage model. The model was validated based on the tests of rocks under the stress conditions of conventional triaxial and true triaxial, respectively. Moreover, for the purpose of studying the influence of parameters on the model, sensitivity analyses of mechanical parameters and model parameters were carried out. The results of statistical damage constitutive clearly demonstrate the stress-strain and damage evolution of anisotropic rock under various stress conditions.

Numerical simulations are carried out on the vortex-induced rotations of a freely rotatable rigid square cylinder in a two-dimensional uniform cross-flow. A range of Reynolds numbers between 40 and 150 and density ratios between 0.1 and 10 are considered. Results show eight different characteristic regimes, expanding the classification of Ryu & Iaccarino (J. Fluid Mech., vol. 813, 2017, pp. 482–507). New regimes include the transition and wavy rotation regimes; in the ${\rm \pi}$ -limited oscillation regime we observe multipeak subregimes. Moment-generating mechanisms of these regimes and subregimes are further elucidated. A phenomenon related to the influence of density ratio is the tooth-like shape of the ${\rm \pi} /2$ -limit oscillation regime observed in the regime map, which is explained as a result of the imbalance relation between the main frequencies of rotation response and the vortex shedding frequency. In addition, existence of multiple regimes and multistable states are discussed, indicating multiple stable attractive structures in phase space.

Numerical studies about vortex-induced rotation (VIR) have shown that six regimes exist for the dynamics of square cylinder under VIR at small Reynolds numbers; however, theoretical explanations are still rare. Aiming at stepping a little further, we select one of these regimes, that is, small-amplitude oscillation regime, to develop an analytical model. In the present model, we expand the symmetric free streamline theory to an inviscid flow pasting an inclined plate and introduce the Blasius boundary layer theory in the calculation of viscous torque on a square cylinder. This model is supported by comparing with numerical results using the immersed boundary method. Results explain the mechanism of small-amplitude oscillation regime: the main sources of the out-of-phase torque come from the imbalanced distribution of pressure on the windward face and the asymmetric distribution of pressure and viscous force on the top/bottom face. An alternative way to determine the local flow parameters by global dynamic quantities is also proposed based on this model.

Flapping wings demonstrate exceptional aerodynamic performance and the optimization of wing shape is an important issue within this field. A numerical optimization targeting wing lift maximization is conducted on a two-dimensional flapping wing with chordwise flexibility. A commonly used setup is to consider two rigid elliptical-section components connected by a torsional spring, providing chordwise flexibility. The leading body of the wing operates under specified hovering flapping kinematics, while the trailing body responds passively. Starting from this setup, the airfoil shape is parameterized using three parameters and optimized for maximum lift using the Simplex algorithm. Four configurations of the wing are evaluated, and the results indicate lift increase across all configurations, with a maximum increase in 10.6%. The optimized airfoil configuration adopts a teardrop profile, substantially narrower than the original elliptical section. According to further analysis, the lift increase can be primarily attributed to the enhancement of velocity circulation.

Bluff-body flow-energy harvesters are commonly designed around translational vibration, whereas freely rotating polygonal sections provide an alternative route that harnesses rotational oscillations. In this study, Reynolds-number-dependent energy-harvesting responses of two reconfigurable sections, a variable-angle rhombic cylinder and a variable-side-length parallelogram cylinder, are investigated using an immersed boundary method. The body rotates freely about its centroid without a linear spring or external damping. Results show that at $Re=40$, decreasing the rhombus angle improves start-up and increases the bounded-oscillation amplitude, favoring oscillation-based harvesting, while the parallelogram oscillation amplitude varies non-monotonically with side-length ratio, with $\lambda=1.1$ yielding the largest bounded-oscillation amplitude. At $Re=150$, acute rhombi sustain the largest bounded oscillations, whereas the square cylinder develops autorotation and favors rotary harvesting; the parallelogram family remains on bounded oscillatory branches, with $\lambda=1.5$ producing the largest response envelope. Phase-resolved analyses reveal that periodic near-wake reorganization maintains the oscillation of acute rhombi through favorable alignment between the hydrodynamic moment and the angular velocity. These findings provide geometry-selection guidelines for oscillatory and rotary flow-energy harvesters.