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Hydrostatic pressure effects

For cases that the hydrostatic pressure contribution

ρ(g ∙ h) \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−ρ(g ∙ h). p' = p - \rho ( \vec{g} \dprod \vec{h} ).

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

∂∂t(ρu)+∇ ∙ (ρu⊗u)−∇ ∙ (μeff∇u)=−∇p+ρg \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′=p−ρ(g ∙ h). p' = p - \rho ( \vec{g} \dprod \vec{h} ).

After the following substitutions:

−p=−p′−ρ(g ∙ h)−∇p=−∇(p′)−∇(ρ(g ∙ h))=−∇(p′)−ρg ∙ ∇h−h ∙ ∇(ρg)=−∇(p′)−ρg ∙ I−g ∙ h∇(ρ)−\cancelto0ρh ∙ ∇(g)=−∇(p′)−ρg−g ∙ h∇ρ\begin{aligned} - 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{aligned}

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

∂∂t(ρu)+∇ ∙ (ρu⊗u)−∇ ∙ (μeff∇u)=−∇p′−g ∙ h∇ρ \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

∂∂t(ρu)+∇ ∙ (ρu⊗u)−∇ ∙ (μeff∇u)=−∇p′ \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: