Matt Hunwardsen's Modeling Homepage.

Advisors: Dr. Cindy Wyels, and Dr. Paul Stanley

Computational Fluid Dynamics
Summary:
We will explore the flow of an compressible viscous fluid in a controlled 2-D environment ( using a boundary, we can force an object through the flow domain ) so we can predict the effects of changes to the system when using different objects at different speeds.  We want to model and observe the effects of an object moving through a fluid field. Then we will examine the effects of changing the surface of the object and changing the speed at which it moves through the field.
Assumptions:
Physical:  compressible fluid, equilibrium pressure initially, temperature does not change, conservation laws hold, boundary values remain constant.

Mathematical:  discrete data, averaging fluid motion, speed of motion, linear motion throughout, no information is lost between the cells.

How Do We Do This?
Use a spreadsheet (Microsoft Excel) to input data and formulas into a field representing our fluid.  Construct an algorithm (plan) to move an object through our field and to calculate the ramifications of this motion.  Possibly write a program to simulate this, so we can increase our computing power and the degree of our parameters (3-D?).

Here is our initial field at equilibrium

The Field

 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

This is how we determine how each cell is affected by the changes in the fluid domain.

Cell Averaging Algorithm

 1/24 1/12 1/24 1/12 1/2* 1/12 1/24 1/12 1/24

* this is the object cell.

What is our Object?
Initially we will move a plate through our field.  This is how we do it.

Each object cell in the path of our motion will have this calculation algorithm instead of the Cell Averaging Algorithm.

Position                       1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Pointer                        0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Is Occupied?                0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Was Occupied?            1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
Counter             62.0
Step                    1.0
Set/Clear             1.0

=IF(C\$3>0.5,0,IF(B\$3>0.5,D45+C45,(C44+B45+D45+C46)/12+C45/2+(B44+D44+B46+D46)/24))

9.5 9.4 9.2 9.1 9.2 9.4 9.6 9.8 10.1 10.2 10.1 10.0 10.0
9.0 8.5 8.1 7.8 7.8 7.9 8.1 8.4  8.8   9.9 10.9 10.7 10.1
8.3 7.6 6.8 6.2 5.9 5.7 5.5 5.3  4.8   4.9   0.0 21.2 11.1
7.8 6.8 5.8 5.0 4.5 4.3 4.0 3.8  3.5   3.8   0.0 22.0 11.8
7.4 6.3 5.2 4.3 3.9 3.6 3.4 3.3  3.1   3.7   0.0 22.2 11.9
7.2 6.1 4.9 4.0 3.6 3.3 3.2 3.2  3.1   3.6   0.0 22.2 12.0
7.1 5.9 4.8 3.9 3.5 3.2 3.2 3.1  3.0   3.6   0.0 22.3 12.0
7.1 5.9 4.7 3.9 3.4 3.2 3.1 3.1  3.0   3.6   0.0 22.3 12.0
7.1 5.9 4.7 3.9 3.4 3.2 3.1 3.1  3.0   3.6   0.0 22.3 12.0
7.2 6.0 4.8 3.9 3.4 3.2 3.2 3.1  3.0   3.6   0.0 22.3 12.0
7.4 6.2 5.0 4.1 3.7 3.4 3.2 3.2  3.0   3.6   0.0 22.3 12.0
7.7 6.7 5.6 4.8 4.3 4.0 3.8 3.5  3.2   3.7   0.0 22.3 12.0
8.3 7.5 6.7 6.1 5.8 5.6 5.4 5.1  4.7   4.4   0.0 22.2 11.9
8.9 8.4 8.0 7.7 7.6 7.7 7.8 8.0  8.4   8.9 10.0 11.2 10.9
9.4 9.2 9.0 8.9 8.9 9.0 9.2 9.4 9.7   9.9 10.2 10.3 10.2

We will then manipulate the plate into a wedge.

Eventually we will be able to simulate a ball moving through the field and then we will be able to modify the shape of the ball to increase the efficiency of motion.  Unfortunately, the augment our program for the ball involved too many levels of recursion for Excel.  But, to modify written cide would not be any trouble at all.

Observations:
We were able to successfully move simple objects through a fluid field and examine the results.  We discovered that our simulator is consistent with the known laws of Physics (vacuum). Microsoft Excel is a really sweet program in some regards, but its power limits us to only simple objects.  We found that Excel does not calculate all the cells simultaneously, this account for the deformation of the Wedge graph.

Conclusions
We are able to successfully model linear motion in a 2-D fluid field given sufficiently simple objects.  With Excel, we can create and implement minor changes to our algorithm and view the data in graphical form.  The anti-symmetry of the graphs is due to Excels mthod of calculation.  This can be overcome by using multiple grids, one for new values, one to act as a reference and to store old values.

Strengths:  Use of everyday tools, graphs are easily understandable, adaptability of our algorithm.

Weakness:  Block Shapes, confined to 2-D, rounding errors, computational speed, methods of calculation, and well, me.

Where Do We Go From Here?
Perhaps from here we can make generalization about the numerical approximations used in Fluid Dynamics.  We can evaluate the efficiency of different numerical methods given a particular problem, the vacuum, and how it affects motion, etc.  Refine our model to fix the weaknesses in it, eliminate the errors ( modify the Cell Averaging Algorithm, and the calculation algorithm ), and increase its efficiency.