A 3-D vortex-boundary element method for the simulation of unsteady, high Reynolds number flows

Thesis (Sc. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 1995.Includes bibliographical references (p. 266-271).by Adrin Gharakhani.Sc.D

A Regularized Galerkin Boundary Element Method (RGBEM) for Simulating Potential Flow About Zero Thickness Bodies

The prediction of potential flow about zero thickness membranes by the boundary element method constitutes an integral component of the Lagrangian vortex-boundary element simulation of flow about parachutes. To this end, the vortex loop (or the panel) method has been used, for some time now, in the aerospace industry with relative success [1, 2]. Vortex loops (with constant circulation) are equivalent to boundary elements with piecewise constant variation of the potential jump. In this case, extending the analysis in [3], the near field potential velocity evaluations can be shown to be {Omicron}(1). The accurate evaluation of the potential velocity field very near the parachute surface is particularly critical to the overall accuracy and stability of the vortex-boundary element simulations. As we will demonstrate in Section 3, the boundary integral singularities, which arise due to the application of low order boundary elements, may lead to severely spiked potential velocities at vortex element centers that are near the boundary. The spikes in turn cause the erratic motion of the vortex elements, and the eventual loss of smoothness of the vorticity field and possible numerical blow up. In light of the arguments above, the application of boundary elements with (at least) a linear variation of the potential jump--or, equivalently, piecewise constant vortex sheets--would appear to be more appropriate for vortex-boundary element simulations. For this case, two strategies are possible for obtaining the potential flow field. The first option is to solve the integral equations for the (unknown) strengths of the surface vortex sheets. As we will discuss in Section 2.1, the challenge in this case is to devise a consistent system of equations that imposes the solenoidality of the locally 2-D vortex sheets. The second approach is to solve for the unknown potential jump distribution. In this case, for commonly used C{sup o} shape functions, the boundary integral is singular at the collocation points. Unfortunately, the development of elements with C{sup 1} continuity for the potential jumps is quite complicated in 3-D. To this end, the application of Galerkin ''smoothing'' to the boundary integral equations removes the singularity at the collocation points; thus allowing the use of C{sup o} elements and potential jump distributions [4]. Successful implementations of the Galerkin Boundary Element Method to 2-D conduction [4] and elastostatic [5] problems have been reported in the literature. Thus far, the singularity removal algorithms have been based on a posterior and mathematically complex reasoning, which have required Taylor series expansion and limit processes. The application of these strategies to 3-D is expected to be significantly more complicated. In this report, we develop the formulation for a ''Regularized'' Galerkin Boundary Element Method (RGBEM). The regularization procedure involves simple manipulations using vector calculus to reduce the singularity of the hypersingular boundary integral equation by two orders for C{sup o} elements. For the case of linear potential jump distributions over plane triangles the regularized integral is simplified considerably to a double surface integral of the Green function. This is the case implemented and tested in this report. Using the example problem of flow normal to a square flat plate, the linear RGBEM predictions are demonstrated here to be more accurate, to converge faster, and to be significantly less spiked than the solutions obtained by the vortex loop method

Massively parallel implementation of a 3D vortex-boundary element method

Vortex-boundary element simulation of three-dimensional wall-bounded flows is performed on a massively parallel architecture using the data parallel paradigm. With proper optimization, implementation on the Thinking Machines CM5 is shown to yield over an order of magnitude speed-up over the Cray C90, using 128 processors. This makes the direct evaluation of the flow eld represented by up to 100 000 vortex elements feasible. In this paper, the CPU time and memory requirements for the direct evaluation of the vortical eld using three parallelization algorithms are compared. In addition, performance results for the evaluation of the vortical and potential velocities, and their gradients, are presented as a function of the number of vortex elements and the number of processing nodes. Finally, to demonstrate the capability of the developed method, preliminary results from the case of flow around a stationary, idealized trailer truck near the ground level at Re = 500 are presented

3D vortex simulation of flow in an opposed-piston engine

A 3-D Lagrangian random vortex-boundary element method is extended to the case of compressible flow at low Mach numbers. The simulations utilize the equation for the transport of (vorticity/density), which is identical in form to the vorticity transport equation for incompressible flow, but with two modifications. First, diffusion involves a time-varying, spatially homogeneous diffusivity. This is implemented by appropriately modifying the diffusion time scale, so that the diffusivity is time-invariant. Second, the continuity equation includes a spatially uniform, but time-dependent density. This effect is accounted for in the potential component of the velocity field via a Poisson equation with a spatially-uniform, time-dependent volumetric source term. The latter is converted to a source on the boundary of the domain, which allows the grid-free evaluation of the potential velocity field using the boundary element method. As a result, grid-free simulation of flow in the complex geometry of engines during the entire intake and compression strokes is made possible. In this paper, the formulation for the method and preliminary results from the simulation of the swirling flow inside a typical two-stroke opposed-piston engine are presented

Simulation of three-dimensional internal flows by the random vortex and boundary element methods

A vortex boundary element method is developed for the grid-free simulation of time-dependent, incompressible, viscous flow in three dimensional configurations. The numerical scheme is based on a combination of the Lagrangian vortex method to capture the convection and stretch of the vortical field, the random walk method to describe the diffusion process, and the boundary element method to impose the normal flux boundary condition on the boundary surfaces. The no-slip boundary condition is satisfied by an extended vortex tile generation mechanism. A new boundary condition is devised to impose the fully developed flow properties at the exit plane. The formulation of the numerical scheme is presented, followed by a parametric study of the accuracy of the method using the model problem of flow in a duct with square cross-section at Re=100. Additionally, results from an example of piston driven flow in a cylinder with square cross-section and an offset port at Re=350 (based on the piston side and maximum speed) are presented