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How to get BC for Psi in the computing domain?

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Old   July 26, 2000, 12:56
Default How to get BC for Psi in the computing domain?
  #1
Pierre Forges
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July 26, 2000 Dear colleagues,

Notations ========= (x,y) Cartesian coordinates. (r,s) Curvilinear coordinates. P Stream Function

P is defined as follow: u = dP/dy = dP/dr dr/dy + dP/ds ds/dy [1] v =-dP/dx =-dP/dr dr/dx - dP/ds ds/dx [2]

Inlet Boundary condition ======================== For 0<y<ymax u(0,y)=1

v(0,y)=0

Now if I go to the computing domain, I have:

1=dP/dr dr/dy + dP/ds ds/dy 0=dP/dr dr/dx + dP/ds ds/dx

Two equations with two unknowns: dP/dr and dP/ds

Solving for dP/ds and integrating from s=0 to s>0 yields:

P(r=0,s)= Int_{0}^{y(s)}

[dr/dx] /[dr/dx ds/dy - dr/dy ds/dx] dy

I have used my 2 BC, namely u=1 and v=0, to get a Dirichlet BC for the stream function P. I think it's the way to do it. Let's see now another boundary condition.

Outlet Boundary condition ========================= For x=xmax and 0<y<ymax du/dx=0

dv/dx=0 In the computing domain I have:

du/dr dr/dx + du/ds ds/dx=0 dv/dr dr/dx + dv/ds ds/dx=0

Using eqs [1] and [2] yields:

[d^2P/dr^2 dr/dy + d^2P/drds ds/dy] dr/dx + [d^2P/drds dr/dy + d^2P/ds^2 ds/dy] ds/dx=0 [3]

[d^2P/dr^2 dr/dx + d^2P/drds ds/dx] dr/dx + [d^2P/drds dr/dx + d^2P/ds^2 ds/dx] ds/dx=0 [4]

From these two equations, how can I get my boundary condition for the sream function?

I cannot isolate dP/ds from eqs [3] and [4]!

Same remark applies when I have the following BC: du/dy=0 and v=0. It's a symmetry BC.

Thank you so much in advance for replying to my question.

My very best regards,

Dr. Pierre Forges

UAE University Mech. Eng. Dept. pforges@uaeu.ac.ae
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Old   July 26, 2000, 20:10
Default Re: How to get BC for Psi in the computing domain?
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John C. Chien
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(1). The coordinate transformation is not done UNTIL the coordinate transformation factors are determined or specified. You are the one who must specify these coordinate transformation factors. (2). So, you can arbitrarily create a new coordinate transformation (just like a structured mesh) and have all non-zero transformation factors. (3). But, who would do thing like that. (4). If your original inlet condition is nice and clean in x-y coordinate system, then keep the transformed coordinates exactly the same as the x-y coordinates by specifying the proper transformation factors in that region. In simple sentense, no stretching and rotation in the inlet region when creating the new coordinates. In this way, the new is identical to the old. But you are still using the new one, except that the transformation factors in those regions are such that nothing has changed. (5). This is the way to create the new coordinates. If you do not follow this rule, you will end up with a general distorted mesh over the inlet, and will have to use interpolation method to specify the inlet conditions in the new coordinates. So, you have to generate the mesh (or the cew coordinates) to simply the work by specifying the transformation factors.(or how the mesh should be stretched or rotated). (6). In other words, in your message, you have not even finished your coordinate transformation, because the transformation factors are not specified yet. You are the one who must supply the values of ds/dx(at i,j), ds/dy(at i,j), dr/dx(at i,j), and dr/dy(at i,j). (if s=x at i=1, and r=y at i=1, then ds/dx=1 at i=1, dr/dy=1 at i=1. ds/dy=dr/dx=0. Or you can let s=const*x, r=const2*y, the ds/dy and dr/dx are still zeros.)
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