To calculate difference for nodes near wall, extra nodes beyond the wall are needed. So put hypothetical nodes inside the wall like the picture above. From now, every explanations are in two dimensions for easiness.
Velocity perpendicular to the wall :
Set the same velocity with the other one across the surface of the wall. Set zero for the node right on the surface.
Velocity parallel to the wall :
Set inversely with the other one across the surface of the wall.
Solve the Navier-Stokes equation with the conditions above on the surface of the wall,
is zero if it's steady, the first of is zero because the perpendicular velocity on the surface should be zero and the representative parallel velocity on the surface should be zero also. So it will be,
Break it down for direction,
All the derivatives with should be zero because all the parallel velocity on the wall should be zero. Then discretize it,
Here, , then,
Set the pressure for hypothetical nodes by this equation.
Set them for the end side of the node sequence like below,
For example, if the last node number is 20, the calculation ends at node 18, and 19,20 will be the hypothetical nodes. Set each values of hypothetical nodes like below, it's the same calculations as the beginning of the sequence.
We don't use the pressure and the parallel velocity of node 20.
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