In this study, an improved SPH (smoothed particle hydrodynamics) method coupling the κ-ε turbulence model and the wall function is proposed to simulate wall-bounded turbulent flows at medium and high Reynolds numbers. The second-order partial derivative term of the composite function containing the turbulent viscosity coefficient is decomposed into the sum of two terms, which helps SPH to avoid numerical errors and difficulties in dealing with boundary conditions in the nested approximation of first-order partial derivatives. Particle shifting technique, δ-SPH method, and graphics processing unit parallel technology are used in the simulations to ensure uniform particles, smooth pressure field, and high computational efficiency, respectively. The SPH method with the κ-ε turbulence model and the wall function is tested and validated by simulating four classical wall-bounded turbulent flows at medium and high Reynolds numbers, which are the turbulent flat plate boundary layer at Re = 4.2 × 106 and Re = 107, backward step separation flow at ReH = 3025, the flow around the airfoil at Re = 3 × 106, and the flow around the pitching airfoil at Re = 1.35 × 105, respectively. The simulation results are consistent with the references, validating the suitability of the current SPH method for simulating wall-bounded turbulent flows at medium and high Reynolds numbers. Finally, the airfoil motion combining the pitching and deforming at Re = 3 × 106 is investigated by the current SPH method. The results show that the deformation of the airfoil's relative thickness affects the lift coefficients of the pitching airfoil. Furthermore, the increase in the relative thickness increment reduces the amplitude of the lift coefficient, while the increase in the deformation period has little effect on the amplitude of the lift coefficient.

The special aerodynamic characteristics of insects have attracted the interest of biologists and engineers. In this paper, aerodynamic modeling methods for flapping flight are systematically reviewed in detail, especially those methods developed in the past ten years. The differences among kinds of methods, the development of each type of methods, and their applications for different flight conditions are discussed in detail. First, steady-state and several representative models are presented. The applicability of this simple model decreases when it is applied to predict the loads on small insects. Next, this paper provides a detailed description of quasi-steady (QS) models and divides these models into three groups: Osborne, Walker, and Dickinson models. Osborne models are suitable for cases with a low flight speed and flapping amplitude. Walker and Dickinson models rely on experimental and numerical data to improve the QS models for predicting nonlinear aerodynamic forces. The total forces in Walker models are divided into circulatory and non-circulatory parts. Dickinson models are established according to different high-lift mechanisms. A representative Dickinson model consists of translational, rotational, added-mass, and wake-capture components. These models provide reasonable predictions, except that their accuracy depends on empirical constants. Finally, unsteady models based on the traditional theory are examined, and several representative models are addressed. The assumption of Kutta–Joukowski conditions may not be suitable for high stroke amplitudes and flapping frequencies. Further challenges to improve aerodynamic modeling methods are mainly due to the present limited understanding of the flow complexities of various insects at low Reynold numbers.