At each and every go out step, assess brand new pushes that boundaries enforce toward water

These forces are determined by distortion of your own elastic limitations. Additional outside pushes familiar with drive the new action of the line can also be put on the new liquid.

Pass on the latest force from the Lagrangian grid explaining the career off this new limitations into Cartesian grid regularly resolve the brand new Navier–Stokes equations.

Getting numerical benefits, i set that it container in this a slightly large occasional domain name, out-of proportions (1 meters+29 h) ? (step 1 meters+31 h),in which h=?x=?y ‘s the mesh depth of our own water grid

Utilize the the fresh new velocity career in order to revise the positioning of one’s border. The newest boundary try moved in the regional liquid speed, enforcing the no-sneak updates.

Numerical simulations

Both-dimensional mathematical simulations off airline in this report was basically developed so you’re able to be similar to the physical experiments of Dickinson and Go?tz (Dickinson and Go?tz,1993) and previous two-dimensional numerical simulations of clap and fling (Miller and Peskin,2005). Dickinson and Go?tz used an aluminium wing with a chord of 5 cm immersed in a sucrose service that have a great kinematic viscosity from 0.0000235 meters 2 s –step one (throughout the 20 times that liquids) swinging with a feature velocity throughout the directory of 0.04–0.12 m s –step 1 . The size of the new sucrose tank included in the latest physical experiment had been step one m in total because of the 0.4 m wide. A similar details as the noted because of it actual check out were utilized in every of your following mathematical experiments having several exclusions: (1) how big the new computational container was risen up to reduce wall outcomes at the lower Re and you can (2) brand new translational velocity try changed to replicate Re=10. About after the simulations, i explore an excellent computational tank that’s 1 yards ? step 1 meters in dimensions. The brand new sides of your own computational tank are manufactured from engrossed boundary issues that try connected because of the stiff springs to help you stationary target affairs. The location in four walls is known as the fresh new `computational tank’. Brand new Navier–Stokes equations was solved to your a great 1230?1230 Cartesian grid,each side are discretized to your an effective Lagrangian assortment of 120 activities. Miller and you will Peskin (Miller and you will Peskin,2005) shown a convergence research one showed that that it mesh size is in the set of overlap for the two-wing state in the Re below a hundred.

Unless otherwise stated, the motion of the flexible wings was prescribed by attaching target points to the top 1/5 of the boundary along the leading edge of the wing with springs (Fig. 2). The target points moved with the prescribed motion and applied a force to the boundary proportional to the distance between the target and corresponding boundary points. The bottom 4/5 of the wing (trailing edge) was free to bend. This has the effect of modeling a flexible wing with a rigid leading edge. In the `nearly-rigid’ case, springs were attached to target points along the entire length of the wing, which prevented any significant deformation. The dimensional stiffness coefficient of the springs that attach the boundary points to the target points was k_targ=1.44?10 5 kg s –2 ,and the stiffness coefficient for the tension or compression of the wing was also set to k_str=1.44?10 5 kg s –2 . These values were chosen to prevent any significant stretching or deformation of the wings in the rigid case.

The dimensional flexural or bending stiffness of the wing was varied from k_ray=0.125 ? to 2 ? Nm 2 , where?=5.5459?10 –6 Nm 2 . As stated above,this range of bending stiffnesses corresponds to dimensionless values ranging from 0.25 to 4. We chose this value of ? to represent the case where deformations during translation are small but bending does occur when the wings are close. This range of values was found to produce deformations that are qualitatively similar to those observed in flight videos.