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I would like to ask a question on the implementation of finite volume method on a non-uniform grid in solving Navier-Stokeq equations. I will just post the screenshot of a PhD thesis, where I found the evaluation of the derivative term difficult to understand. The screenshot is below

enter image description here

You can also see the non-uniform grid in the bottom. I can more or less understand the eq. 2.38a. But for equation 2.38b, the author simply use, for example, $(u_5+u_2)/2$ to interpolate the u velocity between $u_5$ and $u_2$. Because $\Delta y_1\ne\Delta y_2$, I'm confused by this evaluation. Is it common to do so in finite volume method? Thanks.

jengmge
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1 Answers1

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With reference to the figure below, the equations are discretized in the region enclosed within the red dotted lines. And, the blue colored line denotes the subscript 2 in your equations.

$$ \frac{\partial u^2}{\partial x}|_2 = \frac{\left({u_e^2 - u_w^2}\right)}{\frac{(\Delta x_1 + \Delta x_2)}{2}} \\ \frac{\partial uv}{\partial y}|_2 = \frac{\left({u_s v_s - u_n v_n}\right)}{\Delta y_1} $$ where $$ u_e = \frac{u_{right} + u_2}{2}, u_w = \frac{u_2 + u_1}{2} \\ u_s = \frac{u_2 + u_0}{2}, u_n = \frac{u_2 + u_5}{2} \\ v_s = \frac{v_0 + v_1}{2}, v_n = \frac{v_3 + v_4}{2} $$

As you see, the length and height of this red dotted lined-region are $\frac{(\Delta x_1 + \Delta x_2)}{2}$ and $\Delta y_1$, respectively. Hence, the same have been used in the discretized equations. enter image description here

Sthavishtha Bhopalam
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  • May I ask if u_n is on the middle point between u_5 and u_2 in your case? If it's not, how come we can use the average of u_5 and u_2 to get u_n. This is my original question. Thanks. – jengmge Jan 14 '22 at 07:03
  • Yes, u_n here lies at the mid-point of u_5 and u_2. Actually, I would not expect one to use their average to compute u_n. But the average of u_5 and u_2 is what seems to match with the results from your picture. Weighted averaging in non-uniform meshes is a good choice. – Sthavishtha Bhopalam Jan 15 '22 at 05:07
  • can you share the full text of the reference? without sharing the full text of the reference you are following, its very difficult to interpret what and why the authors are doing so. – Sthavishtha Bhopalam Jan 15 '22 at 05:09
  • In this question, a non-uniform mesh is considered in the y direction. And u_n is not located on the middle point between u_5 and u_2. This is the setting in the thesis. The reference thesis is here: https://open.bu.edu/handle/2144/15705.

    I agree with you on the weighted averaging. But this seems not to be the case followed in the thesis (where the mesh is non-uniform, but the author only takes the plain average of the two velocities). This confuses me.

     – jengmge Jan 16 '22 at 00:48
  • The discretization of the scheme in the thesis follows from the references paper1 and paper2. – Sthavishtha Bhopalam Jan 19 '22 at 05:22
  • in pg. 113 of paper1, the authors mention that discretization of the two terms you asked for is identical in non-uniform or uniform meshing spaced domains (for second order accuracy). they avoid the usage of weighted averaging as that would lead to non-conservative discrete schemes. – Sthavishtha Bhopalam Jan 19 '22 at 05:25
  • so, following eq. 4 from paper2 which they cite, the discretized equations in the thesis are correct. – Sthavishtha Bhopalam Jan 19 '22 at 05:28