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Model setting without an explicit use of $\color{white} \vec{g}$ 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 $\color{white} p_{-\varrho gh}$ , to the static pressure $\color{white} p$

$\displaystyle \color{white} p_{-\varrho gh} = {}'' p - \varrho g h {}'' = p - \varrho \vec{g} \cdot \vec{r} = p + \varphi \quad .$

(3.36)

By means of () we can formally substitute equation () by the following one

$\displaystyle \color{white} \vec{U} \cdot \nabla \vec{U} - \nu \Delta \vec{U} = - \tfrac{1}{\varrho} \nabla p_{-\varrho g h} \quad ,$

(3.37)

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 $\color{white} p_{-\varrho gh}$ we formally drop 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 $\color{white} \varrho \vec{g} \cdot (\vec{r}_{Outlet} - \vec{r}_{Inlet})$ to its right hand side^3.7. By doing this we obtain a new prescription, but this time for the quantity $\color{white} p_{-\varrho gh}$

$\displaystyle \color{white} p_{-\varrho g h, tot, In} (\vec{r}) = \varrho \vec{g} \cdot (\vec{r} - \vec{r}_{0,H} + \vec{r}_{0,L} - \vec{r}_{Inlet}) \quad .$

(3.38)

If we calculate the mean value of (), we obtain

$\displaystyle \color{white} \overline{p_{-\varrho g h, tot, In}}$	$\displaystyle \color{white}= \tfrac{1}{S_{Inlet}} \iint_{Inlet} \varrho \vec{g} \cdot (\vec{r} - \vec{r}_{0,H} + \vec{r}_{0,L} - \vec{r}_{Inlet}) \mathrm{d} S$	(3.39)
$\displaystyle \color{white}$	$\displaystyle \color{white}= \tfrac{1}{S_{Inlet}} \iint_{Inlet} \varrho \vec{g} \cdot \vec{r} \mathrm{d} S + \varrho \vec{g} \cdot (- \vec{r}_{0,H} + \vec{r}_{0,L} - \vec{r}_{Inlet})$	(3.40)
$\displaystyle \color{white}$	$\displaystyle \color{white}= \varrho \vec{g} \cdot (\vec{r}_{0,L} - \vec{r}_{0,H})$	(3.41)
$\displaystyle \color{white}$	$\displaystyle \color{white}= \varrho g h \quad .$	(3.42)

Prescription at the outlet surface remains formally the same as (), but this time for the quantity $\color{white} p_{-\varrho gh}$

$\displaystyle \color{white} p_{-\varrho gh, Out} (\vec{r}) = \varrho \vec{g} \cdot (\vec{r} - \vec{r}_{Outlet}) \quad ,$

(3.43)

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 $\color{white} h_{IO}$ . It only suffice to know $\color{white} h$ , the head.

Next: Model setting without an Up: Alternative formulation Previous: Alternative formulation Contents Index

Model setting without an explicit use of in the momentum equation

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