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Discretizing the Boundary Conditions of the Gulf of Mexico Project
Rich Affalter
Math 1110
February 18, 2001
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Goals
To write discretized mathematical expressions for the flow potential, along our boundary.
To examine the normal and velocity vectors at each node along the boundary.
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What We Know We have determined that along our
boundary we shall set =0. We know the equation for
We Know that the velocity of flow is equivalent to the gradient of the flow potential.
02
2
2
2
yx
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Three Circumstances of Node Points Along the boundary of the Gulf of
Mexico we encounter three possible circumstances for each node. 1) On Land 2) On River 3) On Open Sea
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The Boundary of the Gulf of Mexico
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Nodes On Land
For the node points that lie on land, we conclude that the normal component of the velocity points into the land.
Normal , n=0 Velocity will point orthogonal to the
normal. Thus, the velocity points along the tangent.Land
Gulf
0 vn
n
v
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Nodes On River
For the node points that lie on the river, we know that the normal component of the velocity points up the river.
Velocity will point out into the Gulf.
Not necessarily –5 but some value.
5 vn
n
v
River
Gulf
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Nodes on Open Sea
For any of the node points that lie on the open sea of our boundary, we will assume that flow potential equals zero.
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Discretizing this information into equations
We will use this knowledge of the velocity at each node along the boundary to create a solvable system involving functions of flow potential.
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5 Boundary Point Configurations
There will be five possible boundary point configurations 1) North-South with 2 known points 2) East-West with 2 known points 3) North-South with 1 known point 4) East-West with 1 known point 5) Points lying along a diagonal
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Boundary Points with 2 Known Points This means that the points are
lying either N-S or E-W with 2 known points.
These known points can be either interior or other boundary points
1
3
25
64v
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Finding the Equation for
In the picture, and are the two boundary nodes, and are two known nodes, and and are the corresponding mid points of the nodes.
Now, knowing using the definition of derivative we have
y vy y
6 5
y
y x y y x
y
) , ( ) , (
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Final Equation of East-West nodes
yvy
2
)( 4321
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Applying to North-South Nodes
This function can easily be converted to assess North-South nodes with 2 known points by substituting x for y.
It will look like this
xvx
2
)( 4321
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Equation for Boundary Points with One Known
This will apply to E-W or N-S nodes that have only one known value. This known value must be a boundary point.
1 24
5
3v
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Finding the In the picture, and are the
boundary nodes, is the one known node, and and are the corresponding mid points.
Now, using the definition of derivative we have
y vy y
) ( 25 4
y
y x y y x
y
) , ( ) , (
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The Final Equation of East
yvy
2
2 321
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Applying this to North-South Nodes This equation can also easily be
applied to N-S nodes by substituting x for y.
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Equations for Diagonal Nodes This process will apply to any
diagonal node points along the boundary.
1
23
c
x
yd
f
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Finding Flow Potential In the picture, and are the boundary
nodes,and is a known valued node. Using geometry I found the lengths of d,f,c.
y
cd
2
x
cf
2
22 yx
xyc
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Finding Flow Potential cont. Using this information we decompose the
velocity vector into x and y components.
xx
32
yy
31
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Final Flow Potential Equation for Diagonal Nodes
22
31
22
32
)()()()( yxyx
vn
22
),(
df
dfn