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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