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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.

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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