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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 img146 1 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 side3.7. 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.

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Previous: greenSolver – steady state, Up: TCFD Solvers Next: Compressible Mathematical Model
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greenDyMSolver - transient, cavitation

TCFD solver for transient, cavitating fluid flow is called greenDyMSolver. It was gradually developed during the time from the interPhaseChangeDyMFoam solver. In any matters the greenDyMSolver behaves the same way as any standard OpenFOAM solver. It is compatible with all OpenFOAM applications and libraries. Solver is modified to be more robust, limits for variables can be specified and many other changes have been done.

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