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Model setting without an explicit use of $ \vec{g}$ in the momentum equation

Let us take a look at the equation ([*]) as if its right hand side were known. Then we can introduce an alternative quantity, denoted by $ p_{-\varrho gh}$, to the static pressure $ p$ img243
By means of ([*]) we can formally substitute equation ([*]) by the following one img244
for the right hand side takes the same values. This is the consequence of a fact that field ([*]) has potential and thus a change in potential energy of an arbitrary element of a fluid is not dependent on its path, but on its initial and final position only. By introducing $ p_{-\varrho gh}$ we formally drop $ \vec{g}$ out from the momentum equation. However, in order to obtain the same solution as in the previous settings, it is necessary to alter the presrciption ([*]) by adding $ \varrho \vec{g} \cdot (\vec{r}_{Outlet} - \vec{r}_{Inlet})$ to its right hand side8.8. By doing this we obtain a new prescription, but this time for the quantity $ p_{-\varrho gh}$ img246
If we calculate the mean value of ([*]), we obtain img247img248 img249 img250 img251
Prescription at the outlet surface remains formally the same as ([*]), but this time for the quantity $ p_{-\varrho gh}$ img252
and hence its mean value is zero. We can see that in this setting there is no need to know a position of a turbine with respect to the water level and there is also no need to even take the measurement of $ h_{IO}$. It only suffice to know $ h$, the head.