About finite volume methods for NS
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 NavierStokes equations. I have a question about numerical implementation. As NS 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 NS equation, I will have to convert u*du/dx in u(k)*[u(k)u(k1)]/deltaX where u(k) is the xspeed at the kth 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 
Re: About finite volume methods for NS
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)*fg*f(both at last itteration). so your nonlinear term f*g equals three terms but all of them are linear.

Re: About finite volume methods for NS
Ok, it means to make a Taylor linearisation around the last iteration point.
Thank you for the answer ! 
Re: About finite volume methods for NS
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 1D 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) 
Re: About finite volume methods for NS
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 1D. 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 ? 
Re: About finite volume methods for NS
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. 
Re: About finite volume methods for NS
The book "numerical heat transfer an fluid flow" of Patankar is a good source to see that.

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