Finite Element Analysis of
Fluid Flow Around Objects
MANE 6960 – ADVANCED TOPICS IN FINITE ELEMENT
ANALYSIS
THOMAS PROVENCHER
1
Introduction:
Fluid flow often involves complex mathematics which requires the raw computing power of
advanced computers running powerful computational fluid analysis software, a specialized
version of Finite Element Analysis (FEA). The study of fluid flow has advanced greatly over the
last few decades as knowledge and experience grew along with the ever increasing capabilities
and speed of computers. Much of this advanced, real world complexity can be simplified
through the use of some assumptions and limiting parameters. The following analysis will
assume laminar, non-viscous fluid is flowing around the objects placed in the flow path. The
velocity potential and pressure will be evaluated using this simplified FEA approach. Figure 1
shows the flow lines of a model car placed in a water tunnel and the complex interactions the
fluid has with the model.
Figure 1: Flow lines over a model car in a water tunnel
2
Formulation and Solution:
The model created for this analysis will simulate the flow over several differently shaped objects
placed in the path of a fluid using a purely theoretical mathematical Coefficient form PDE model
within COMSOL Multiphysics. This was done by creating a rectangle to limit the fluid flow
path and a smaller rectangle, triangle, oval, and circle were placed within the primary rectangle.
The surface areas of these shapes were removed from the primary rectangle to restrict where the
fluid was not permitted to travel. The left side of the primary rectangle was given a Flux/Source
boundary condition which simulates a fluid flow source. The right edge of the model was given
a Dirichlet boundary condition which simulates a place for the fluid to exit the flow path. The
top and bottom of the primary rectangle and all of the edges of the interior shapes were set have
zero flux to prevent any fluid flow from crossing their boundaries. This model can be seen in
Figure 2.
Figure 2: Flow path and obstructions
3
Three different meshes were chosen for this analysis; all were derived from COMSOL’s physics
based mesh creator and were constructed of Lagrange quadratic elements. Extra Coarse,
Normal, and Extra Fine meshes were chosen, the Normal mesh can be seen in Figure 3.
Figure 3: Normal mesh density of the flow model
The variational formulation is shown below:
(𝑢′ , 𝜐′) = (𝑓, 𝜐)
The COMSOl model was run three times, once for each mesh density to verify the mesh was
sufficiently dense. Figure 4 shows the general flow direction and relative velocities of the fluid
and Figures 5 and 6 show the velocity potential and fluid pressure contour curves, respectively,
of the fluid flow as it traveled around the objects on the normal mesh model. Table 1 shows the
maximum values provided by contour curves for all three mesh densities.
4
Figure 4: Normal mesh flow direction and relative velocities
Figure 5: Normal mesh contour plot representation of the velocity potential
5
Figure 6: Normal mesh contour plot representation of fluid pressure
Table 1: Velocity potential and fluid pressure maximum values
Maximum velocity potential and fluid pressure values
Mesh Density
Velocity Potential Fluid Pressure
Extra Coarse
5.0094
-79.889
Normal
5.0206
-82.387
Extra Fine
5.0276
-82.889
The data provided in Table 1 shows that quadratic Lagrage elements are able to provide accurate
results when the mesh density is at least set to Normal. As the mesh density increased the
maximum velocity potential and fluid pressure converged to approximately 5.027 and -82.88,
respectively.
6
Conclusions:
Fluid flow can be extremely complex to model and approximate unless significant assumptions
are made such as non-viscous fluid and laminar flow. These assumptions, while effective at
reducing the modeling complexity, also greatly reduce the range of problems that can be solved.
This purely theoretical modeling example used the same non-viscous and laminar flow
assumptions as above to analyze the flow direction and relative velocity, the velocity potential,
and fluid pressure around objects placed in a flow path. It was shown that both the normal and
extra fine mesh densities provided by COMSOL Multiphysics yielded very similar results
indicating that the model’s analysis results had converged.
References:
Bulmahn, Robert. "Aerodynamics of Model Car." Wikimedia Commons. Web. 23 June 2015.
<https://commons.wikimedia.org/wiki/File:Aerodynamics_of_model_car.jpg>.
7