Homework Project 2
Potential Flow – Flow of an Incompressible Ideal Fluid around a Cylinder
Advanced Topics in Finite Elements
Summer 2015
John Connor
System Description
Potential flow around a cylinder is used to evaluated how an inviscid and incompressible fluid behaves when the flow in interrupted by a cylindrical object. The classical approach assumes that the flow far from the cylinder is unidirectional and uniform. This approach also does not account for any drag that exists within the fluid and the velocity field is irrotational. In this project we will use this methodology.
The purpose of this model is to evaluate the potential flow around multiple cylinders. The finite element method is utlized due to the complex nature of the cylinders as well as there location. By producing this finite element model the velocity around various cylinders of differing sizes and locations can be determined. Two cylinders and an ellipse was chosen for the project the dimensions and locations of these objects are detailed in Section 2.
Governing Equations, Boundary Conditions and Input Data
The method of evaluating fluid flow from a classical approach assumes an object is places in a two dimensional incompressible and inviscid flow. The governing equations assume a steady state condition where the velocity vector and pressure at the object allow the flow to remain equal to the flow far flow the object. This allows one to use a 2-D formulation of Laplace’s equation:
𝜕 2 𝑉 𝜕 2 𝑉
𝜕𝑥 2
+
𝜕𝑦 2
= 0
Where V is the potential flow. The uniform flow is then created by assuming a homogenous boundary condition applied at the edges normal to the flow path.
𝜕𝑉
= 0;
𝜕𝑉
𝜕𝑦
= 0
𝜕𝑥
A uniform flow is then simulated by applying a flux across the 2-D plane. To create the make the flow move from left to right the following boundary conditions were applied.
𝜕𝑉
𝐿𝑒𝑓𝑡: = 1;
𝜕𝑥
𝑅𝑖𝑔ℎ𝑡: 𝑉 = 0
The dimensions and locations of the fluid flow analysis are shown in Figure 1.

10m
R0.50m
4.5m
R0.75m
3m a-semiaxis: 1m b-semiaxis: 0.25m
6m
1.5m
3m
4.5m
5m
Figure 1 – Fluid Flow Analysis Input Data
The Galerkin finite element variation formulation used for the finite element analysis is:
∬ ∇ 2 𝑉𝑣𝑑𝑥𝑦 + ∬ ∇𝑉𝑣𝑑𝑥 = 0
Description of the Finite Element Model
The geometry created in COMSOL Multyphisics is shown in Figure 1. The geometry was chosen to produce a complex system for fluid interaction and to determine how the fluid velocity compares between a cylinder and an ellipse. Each shape was removed from the plate used to simulate the fluid flow. The model then has the appropriate boundary conditions applied to it such that the flow moves from left to right, as discussed in the previous section.
The element chosen for this mode is a quadratic element and is default within the COMSOL system. As the model is simple and runs, fast another element type was not needed to decrease computational resources. It was considered that the mesh density my effect the results as the shapes chosen are within close proximity however as will be discussed in the next section the mesh density had very little effect on the results given the chosen element.

Results, Discussion and Conclusion
The analysis was run with normal and extremely fine mesh densities. The results of these analyses showed no appreciable changes and as such validate that the appropriate mesh and element has been chosen. Figures 2 and 3 shown the results of the analyses.
Figure 2 – Results with a Normal Mesh Density
Figure 3 – Results with an Extremely Fine Mesh Density

The results show that the areas of high pressure and low pressure are much more distinguished around the cylinders while the ellipse shows very little change. This shows that the ellipse is a much more fluid dynamic shape and will have much less drag as compared with the cylinder.
How the velocities differ between the top cylinder and the ellipse demonstrates how each shape can affect the velocity around the bottom cylinder.
References
1.
2.
COMSOL Multiphysics. Version 4.3b: COMSOL AB, 2013. Computer Software.