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In my control equation, there is a term $$ \frac{\partial \rho u}{\partial t}+\frac{\partial \rho u^2}{\partial x}+\cdots=\frac{\partial p}{\partial x} $$

How to discretize it in staggered grid?

For convenience, I include a picture of a staggered grid:

enter image description here

If u is replaced by a scalar variable like $p$, we can simply integrate it in the scalar CV and write $p_P-p_P^o$ in which superscript $o$ means old time value. However in staggered grid, we store speed at cell face $w,e,n,s$ of a scalar CV then how can we write it in discrete form?

Basically I don't know how should we integrate this equation because there are two types of variables involved.

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I can't believe that I forgot in SIMPLE we first treat $p$ as known.

The we should integrate this equation over the u-control volume. Because the time term no relation to $x$ and $y$, so it's simply multiplied by $\Delta V$. Thus we can write $$\rho (u_w-u_w^o)\Delta V$$

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