Numerical and Experimental Study of Flapping Foils with Dynamic Wall Effect

The existence of a wall near a foil body in a freestream influences the hydrodynamic forces exerted on the body. In this research, numerical simulations are performed to investigate the hydrodynamics of a NACA 0012 foil with wall effects using open source software, a 2-D Navier-Stokes solver based on the Boundary Data Immersion Method (BDIM). The inherent difference between the 2-D and 3-D dynamic wall effects is shown by comparing experimentally obtained measurements of vortex shedding behind a flapping foil with a numerical simulation using the same kinematics. The differences are significant in terms of the thrust force coefficient profiles and the vortex development. Particle image velocimetry (PIV), using data from an experiment, shows that the vortex at the tip significantly influences the formation and phasing of the wake.

influences the formation and phasing of the wake.

Introduction
Ground effect produces a significant change in force exerted a lifting surface compared with the freestream. This can be seen in operation when in close proximity to the ground . This phenomenon has been researched extensively, especially in aerodynamics, since it can contribute to lift efficiency. In contrast with static ground effect, the airfoil lift and circulation around the foil becomes time variant in dynamic ground effect and the behavior generates an unsteady fluid wake around the foil .
This paper investigates the difference between 2-D and 3-D wall effects of heaving and pitching foils through numerical simulation. We call it the 2-D dynamic wall effect when it occurs with a lifting surface that has an infinite span or high aspect ratio.
In previous 2-D numerical studies, similar foil motions were used with various methods for different applications, such as thrust propulsion of underwater vehicles or micro aerial vehicles Wu et al., 2014b; and power extraction from tidal currents (Wu et al., 2014a;.   pitching-motion-activated flapping foil was also analyzed near a solid wall and between parallel walls using the Immersed Boundary-Lattice Boltzmann Method (Wu et al., 2014a). It was shown that, for a given amplitude and frequency, as the clearance decreases the net power extraction efficiency improves. They also observed that the leading edge vortex (LEV) from the foil between the parallel walls interacts with a wall vortex, enhancing lift.    and  as shown in figure 1.1. The pitch and angle of attack are sinusoidal functions: Heave velocity is determined by the instantaneous pitch angle and angle of attack as expressed in the following equation: Heave position is determined through integration of equation (1.3): where V ( x, t) is prescribed velocity of the body. Both equations can also be written as different from by integration over a time step δt Maertens and Weymouth (2015)  the meta-equation where µ 0 and µ 1 are the zeroth and first moments of the one-dimensional kernel over Ω f , respectively. This meta equation is solved in LilyPad for flow velocity and pressure. Using a defined kernel and the signed distance d( x) from x to the fluid-body boundary, the moments in the equation 1.10 are given by From the 2-D numerical simulation, the instantaneous thrust (C T ) and lift coefficients (C L ) are obtained and analyzed. They are expressed as the equations 1.12 and 1.13: where F x and F y are the instantaneous force in x and y direction, respectively.
ρ is the fluid density, U is the uniform flow velocity, and L is the body resolution (chord length). To simplify, the time-averaged mean force coefficients are also calculated by the equations 1.14 and 1.15. (1.14) where n c is the number of cycles and T is the period. phase average points of the profiles was calculated as: where n s is the number of sample points andŷ is the force coefficient value at the highest resolution(n = 2048). Figure 1.5 shows the RMSEs for the thrust and lift coefficient profiles. The grid size 0.00098, which equals n = 1024, was chosen for subsequent numerical experiments with the RM SE < 2%.     Table 1.5. It shows that the flapping foil motion can be considered far enough away from the boundary with the domain size where the distance from the mean heave position of the foil body is greater than 6.4L.      In this research, all numerical experiments are conducted with the settings as shown in table 1.7. The n is the number of grid cells for each direction and the body resolution L is a non-dimensional parameter that is defined as the number of grid points along the body.

Validation
The validation of simulation results was conducted by comparing them with

Results
The instantaneous force coefficient profiles during a full cycle from the 2-

Peak location in thrust coefficient profile
The magnitude of force coefficients is asymmetric at the 1st and the 2nd half phase when the wall exists (H * = 1.33) as shown in Figure 1.14. The experiment (Mivehchi, 2016) has consistent maximum peak position in the thrust coefficient.
Otherwise, the thrust coefficients in the 2-D numerical simulation have different peak positions according to Strouhal number and maximum angle of attack.    by the CFD code . The followings are the major code files written and used in the simulations. Some classes are modified for this study, thus the original class file in Lily Pad may not be compatible with this simulation code.
• LilyPad.pde : The main code for simulation • BDIM.pde : The class to solve the Boundary Data Immersed Method equation for velocity and pressure • Body.pde : The class to define the bodies in flow (modified) • NACA.pde : The extension of Body class to create NACA foil body (modified) • BodyUnion.pde : The class to unify the multiple bodies created • FloodPlot.pde : The class for flow visualization.
• Other classes is not modified and same with those in original Lily Pad code package.
All simulations in this study are run in Microsoft Windows 7 system. To execute the simulations readily, the simulation code is exported to an application that can be run with an input file of test parameters. The application of each simulation case consists of the followings.
• LilyPad.exe : The executable file to start the simulation • data : The folder including the input parameter file (See section A.2) • source : The folder including source codes