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Hydrostatic pressure effects

For cases that the hydrostatic pressure contribution

$\rho ( \vec{g} \dprod \vec{h} )$

is important, e.g. for buoyant and multiphase cases, it is numerically convenient to solve for an alternative pressure defined by

$p' = p - \rho ( \vec{g} \dprod \vec{h} ).$

In OpenFOAM solver applications the $$p'$$ pressure term is named p_rgh. The momentum equation

$\ddt{\rho \u} + \div ( \rho \u \otimes \u ) - \div ( \mu_{\eff} \grad \u ) = - \grad p + \rho \vec{g}$

is transformed to use $$p'$$:

$p' = p - \rho ( \vec{g} \dprod \vec{h} ).$

After the following substititions:

\begin{align} - p & = - p' - \rho ( \vec{g} \dprod \vec{h} ) \\ - \grad p & = - \grad( p') - \grad ( \rho ( \vec{g} \dprod \vec{h} ) ) \\ & = - \grad( p') - \rho \vec{g} \dprod \grad \vec{h} - \vec{h} \dprod \grad(\rho \vec{g}) \\ & = - \grad( p') - \rho \vec{g} \dprod \tensor{I} - \vec{g} \dprod \vec{h} \grad (\rho) - \cancelto{0}{\rho \vec{h} \dprod \grad (\vec{g})} \\ & = - \grad( p') - \rho \vec{g} - \vec{g} \dprod \vec{h} \grad \rho \end{align}

where, for CFD meshes the term $$\grad \vec{h}$$ is given by the gradient of the cell centres, which equates to the tensor $$\tensor{I}$$, the momentum equation becomes:

$\ddt{\rho \u} + \div ( \rho \u \otimes \u ) - \div ( \mu_{\eff} \grad \u ) = - \grad p' - \vec{g} \dprod \vec{h} \grad \rho$

For constant density applications this can be further simplified to

$\ddt{\rho \u} + \div ( \rho \u \otimes \u ) - \div ( \mu_{\eff} \grad \u ) = - \grad p'$

For examples of the use of this variable transformation, see:

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