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Old   June 24, 2006, 16:07
Default About finite volume methods for N-S
  #1
Lionel BRS
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Hi everybody,

I'm a student in mechanical engineering (at Swiss Federal Institute of Technology), and I had some courses about finite volume methods.

For the moment, I didn't have any application on finite volume method to Navier-Stokes equations. I have a question about numerical implementation.

As N-S has a term:

V*grad(V) where V=(u,v) is the speed vector,

it's non linear equations.

So, if I want to make a numerical compute of N-S equation, I will have to convert

u*du/dx

in

u(k)*[u(k)-u(k-1)]/deltaX

where u(k) is the x-speed at the k-th point of discretisation (the center of a finite volume for example).

But the goal is to convert differential equations to a *linear algebraic* system. And the previous equation is NOT linear, even it is algebraic.

So I wonder how to make it linear. Maybe I could take a Taylor's developement, but It would make a strong error.

How is it done in most commercial solvers ?

Thank you in advance for your help ! I would really like to understand the numerical solving of NS.

Lionel
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Old   June 26, 2006, 01:04
Default Re: About finite volume methods for N-S
  #2
yousef
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hi I'm yousef I study aerospace at Sharif univer. in Tehran. if you have a term like "f*g" you can linearize it as below: f*g=f(at last itteration)*g+g(last itteration)*f-g*f(both at last itteration). so your non-linear term f*g equals three terms but all of them are linear.

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Old   June 26, 2006, 05:08
Default Re: About finite volume methods for N-S
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Lionel BRS
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Ok, it means to make a Taylor linearisation around the last iteration point.

Thank you for the answer !
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Old   June 27, 2006, 05:23
Default Re: About finite volume methods for N-S
  #4
Salva
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In finite volume schemes you tend avoid this terms and express everthing in conservative form I=nabla (u phi ). Once this terms is integrated over the control volume (the cell)and applying Gauss theorem.

I=( u phi)_R-( u phi)_L

where R and L are points in the surface of the cell (in 1-D sense)

Now you interpolate as you wish. You use u from the previous iteration or step and you solve for phi. The result is an algebraic equation. If phi=u you still use the same procedure the convective u is diffrenet form the "u" you are solving. This schemes are iteratives in nature (so repeat, at the end the diference between both "u's" must be very small)

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Old   June 27, 2006, 07:34
Default Re: About finite volume methods for N-S
  #5
Lionel BRS
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Hi Salva, and thank you very much for this information.

I understand the interest of conservative form, but I dont see how to integrate numerically in 2D or 3D.

Because as you said,

I=( u phi)_R-( u phi)_L

is only applicable for 1-D.

In 2D, can I just take the same formula for integration along X for example, and then multiply by the height of cell (if it's rectangular) ?

Or is there some more accruate methods to integrate on a 2D domain ?
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Old   June 29, 2006, 05:15
Default Re: About finite volume methods for N-S
  #6
Salva
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Because you are integrating over teh cell doesn't matter if the system is 1D,2D or 3D.

I=sum_k (F_k S_k)

Where F_k is the flux across face K and S_k is the area of the face. (asuming polyhedarl cells, but it doens't matter) Basically you can understand it as

I= Flux going in- flux going out

The advantage of this method is that it is conservative at cell level (no mass loss).

The numerics inteven in the fact that F_k is an averaged flux you hav eto evaluate by interpolation or wahtever you want.

Equation for I is NOT an approximation (no numerics involved), if we knew F_k exactly F_k S_k= int_k F n dS there would be no numericall error in equation I.

All comercial codes worked like this.

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Old   July 16, 2006, 03:15
Default Re: About finite volume methods for N-S
  #7
Alej
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The book "numerical heat transfer an fluid flow" of Patankar is a good source to see that.
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