Towards the study of flying snake aerodynamics, and an analysis of the direct forcing method
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Immersed boundary methods are a class of techniques in computational fluid dynamics where the Navier-Stokes equations are simulated on a computational grid that does not conform to the interfaces in the domain of interest. This facilitates the simulation of flows with complex moving and deforming geometries without considerable effort wasted in generating the mesh. The first part of this dissertation is concerned with the aerodynamics of the cross-section of a species of flying snake, Chrysopelea paradisi (paradise tree snake). Past experiments have shown that the unique cross-section of this snake, which can be described as a lifting bluff body, produces an unusual lift curve--with a pronounced peak in lift coefficient at an angle of attack of 35 degrees for Reynolds numbers 9000 and beyond. We studied the aerodynamics of the cross-section using a 2-D immersed boundary method code. We were able to qualitatively reproduce the spike in the lift coefficient at the same angle of attack for flows beyond a Reynolds number of 2000. This phenomenon was associated with flow separation at the leading edge of the body that did not result in a stall. This produced a stronger vortex and an associated reduction in pressure on the dorsal surface of the snake cross-section, which resulted in higher lift. The second part of this work deals with the analysis of the direct forcing method, which is a popular immersed boundary method for flows with rigid boundaries. We begin with the fully discretized Navier-Stokes equations along with the appropriate boundary conditions applied at the solid boundary, and derive the fractional step method as an approximate block LU decomposition of this system. This results in an alternate formulation of the direct forcing method that takes into consideration mass conservation at the immersed boundaries and also handles the pressure boundary conditions more consistently. We demonstrate that this method is between first and second-order accurate in space when linear interpolation is used to enforce the boundary conditions on velocity. We then develop a theory for the order of accuracy of the direct forcing method with linear interpolation. For a simple 1-D case, we show that the method can converge at a range of rates for different locations of the solid body with respect to the mesh. But this effect averages out in higher dimensions and results in a scheme that has the same order of accuracy as the expected order of accuracy of the interpolation at the boundary. The discrete direct forcing method for the Navier-Stokes equations exhibits an order of accuracy between 1 and 2 because the velocities at the boundary are linearly interpolated, but the resulting boundary conditions on the pressure gradient turn out to be only first-order accurate. We recommend linearly interpolating the pressure gradient as well to make the method fully second-order accurate. We have also developed two open source codes in the course of these studies. The first, cuIBM, is a two-dimensional immersed boundary method code that runs on a single GPU. It can simulate incompressible flow around rigid bodies with prescribed motion. It is based on the general idea of a fractional step method as an approximate block LU decomposition, and can incorporate any type of immersed boundary method that can be made to fit within this framework. The second code, PetIBM, can simulate both two and three-dimensional incompressible flow and runs in parallel on multiple CPUs. Both codes have been validated using well-known test cases.