Efficient Finite Element-Based Approaches for Solving Potential Flow Problems in Fluids

Abstract

For many years, fluid flows have been modeled, starting from basic potential flow equations to full Navier-Stokes equations. The complexity of the flow increases as viscous effects, boundary layer, and flow separation are included in the fluid flow problem. However, at the preliminary design level, a simple technique to solve fluid flow problems becomes necessary for the quick assessment of 2-D aerodynamic concepts. Conventional panel methods have been popular in solving potential flow problems due to their ease of implementation for simple geometries such as circular bodies, airfoils, and 3-D applications such as wings. Nevertheless, the method becomes computationally expensive when a large number of boundary elements are required or for time-dependent problems. Moreover, due to established techniques such as panel methods, other methods go unexplored, or a smaller extent of literature is available.

The present research aims to develop a Finite Element Method (FEM) for potential flows over a range of bluff bodies like cylinders to streamlined profiles such as airfoils. In contrast to conventional panel methods, Laplace’s equation describing the potential flow was solved here for the velocity potential function using the Galerkin method. A brief discussion on edge singularities in potential flows has also been presented using a half-cylinder case study.

A novel method for implementing Kutta condition over airfoils to have a lifting flow has been investigated. Compared with other techniques such as finite difference and volume methods, the present methodology has proven to be computationally faster for airfoils with both finite angle and cusped trailing edges. The results have demonstrated excellent accuracy compared to analytical and panel methods.

The present novel Kutta condition method has been extended to quasi-unsteady flows to show its ease of adaptability for various steady and time-dependent conditions. The process of vortex shedding in the wake of an airfoil and building up of forces was studied. A case study of a sudden step change in the angle of attack was considered for quasi-unsteady flow over an airfoil, and the results were in good accordance with the panel methods.

Lastly, the application of the present FEM program was presented for a case of converting 2-D airfoil section data into 3-D wing data. 3-D wings with elliptic, rectangular, and trapezoidal planforms with tapering, sweep angle, and twist were considered. Mathematical formulas were derived from lifting-line theory, and an integration approach was used to calculate the aerodynamic coefficients. The results obtained are in good agreement with the experiments. Predicting data for 3-D wings from 2-D section airfoil using the present FEM program appears to be a very viable and cost-efficient method. Finally, the 2-D longitudinal profile of a sedan-type vehicle was considered to check the program's capability for evaluating geometries other than an airfoil. The present potential flow results for a sedan car and a modified sedan car were compared with the viscous model in ANSYS Fluent. It was observed that streamlining the sectional profile of the sedan would predict results closer to the real viscous flow due to minor flow separation.