Momentum equation in buoyantBoussinesqSimpleFoam

CuriousFluidDynamics · September 9, 2024, 11:03

Dear all,

I am very new to OpenFOAM so my apologies in advance if this question is too basic.

For the solver of buoyantBoussinesqSimpleFoam there is this term in the momentum equation:

Code:

ghf*fvc::snGrad(rhok)

I understood it as:

$g_n \nabla ( 1 - \beta (T -T_{ref}))$

Where $\beta$ is the thermal expansion coefficient,

the temperature and

the inner product of the gravitational acceleration with the location of the face centres. However, why do we take the gradient? From the Boussinesq assumption, there does not seem to be one at first glance: https://en.wikipedia.org/wiki/Boussi...ion_(buoyancy)

Tobermory · September 10, 2024, 13:43

This term arises because OpenFOAM buoyant solvers don't solve for the static pressure, they solve for a quantity called p_rgh (for stability reasons), which is akin to an anhydrostatic pressure:

$p_{rgh} = p - \rho( \vec{g}\cdot \vec{h})$

Stick this into the momentum equation, and you will see where the $\nabla \rho$ term comes from (and where the gravity term disappears to).

CuriousFluidDynamics · September 12, 2024, 09:27

Thank Tobermory! This really helped. Finally I also found an derivation here: https://www.openfoam.com/documentati...orm-p-rgh.html

September 9, 2024, 11:03	Momentum equation in buoyantBoussinesqSimpleFoam	#1
CuriousFluidDynamics New Member Join Date: Sep 2024 Posts: 3 Rep Power: 2	Dear all, I am very new to OpenFOAM so my apologies in advance if this question is too basic. For the solver of buoyantBoussinesqSimpleFoam there is this term in the momentum equation: Code: ghf*fvc::snGrad(rhok) I understood it as: $g_n \nabla ( 1 - \beta (T -T_{ref}))$ Where $\beta$ is the thermal expansion coefficient, the temperature and the inner product of the gravitational acceleration with the location of the face centres. However, why do we take the gradient? From the Boussinesq assumption, there does not seem to be one at first glance: https://en.wikipedia.org/wiki/Boussi...ion_(buoyancy)

September 12, 2024, 09:27		#3
CuriousFluidDynamics New Member Join Date: Sep 2024 Posts: 3 Rep Power: 2	Thank Tobermory! This really helped. Finally I also found an derivation here: https://www.openfoam.com/documentati...orm-p-rgh.html Tobermory likes this.

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September 10, 2024, 13:43		#2
Tobermory Senior Member Join Date: Apr 2020 Location: UK Posts: 746 Rep Power: 14	This term arises because OpenFOAM buoyant solvers don't solve for the static pressure, they solve for a quantity called p_rgh (for stability reasons), which is akin to an anhydrostatic pressure: $p_{rgh} = p - \rho( \vec{g}\cdot \vec{h})$ Stick this into the momentum equation, and you will see where the $\nabla \rho$ term comes from (and where the gravity term disappears to).