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#### What is buoyancy

What is buoyancy? The definition says it is an upward directed force, exerted by a fluid, that opposes the weight of the partially or fully immersed object and was first described by Archimedes more than 2000 years ago.

So let’s take a look at an example and assume we have a cylindrical object, completely submerged in a box of water. The water pressure increases linearly with the depth, so the water pressure at the bottom of the cylinder is larger than the water pressure at the top of the cylinder.

If we express this in terms of forces, we can say that the pressure difference will result in a net upward directed force on the cylinder, which is called the buoyant force. \[F_{buoyant}=F_{up}-F_{down}\] By taking into account the following relation between pressure and forces, \(F=p\cdot A\), the equation above can be rewritten as: \[F_{buoyant}=p_{up}A-p_{down}A=(p_{up}-p_{down})A\] And taken into account that the pressure can be expressed as \(p=\rho\cdot g \cdot h\), the buoyant force can be written as: \[F_{buoyant}=\rho_{f}g(h_{top}-h_{bottom})A\] with \(h_{top}\) and \(h_{bottom}\) the depths of the top and bottom of the cylinder respectively and \(\rho_{f}\) the density of the fluid.

The term \((h_{top}-h_{bottom}) \cdot A\) can also be written as a volume. This is the volume of displaced fluid by the submerged object and not the volume of the object itself. When an object is fully submerged, both volumes are equal. However when an object is only partly submerged the volume of the displaced fluid is smaller than the volume of the object and only the submerged volume should be taken into account.

So the final equation becomes: \[F_{buoyant}=\rho_{f} g V_{displaced fluid}\] Hence the buoyancy force depends on the density of the fluid in which it is submerged, the gravity and the volume of displaced fluid. This force is independent of the mass or the density of the submerged object. Furthermore this force is also independent of how deep the object is in the fluid as the pressure at the top and the bottom of the object increase by the same amount if you go deeper.

On every object, fully or partially submerged, there will be a buoyant force upwards. However, depending on the weight of the object \((F_{g})\) it can either:

- Sink when \(F_{g} \gt F_{buoyant}\)
- Remain in place when \(F_{g}=F_{buoyant}\)
- Rise upwards when \(F_{g} \lt F_{buoyant}\) and the object is fully submerged or floats when the object is only partially submerged.

An important example when buoyancy is important is in natural convection cases, as in such cases the fluid motion occurs by buoyancy effects. When a heat source is added to the fluid, the density will vary with the temperature. The material will become less dense with increasing temperature and cause the hot fluid to rise. Buoyancy is also present in forced convection cases, but less dominant.

#### How is buoyancy taken into account in the platform

When the density variation is limited, the density can be taken as a constant in the unsteady and convection terms and only as a variable in the gravitational term. This method, the so-called Boussinesq approximation, is implemented in our platform to account for buoyancy. The density which takes into account the thermal expansion is calculated as: \[\rho_{eff} = \rho(1-\beta(T-T_{ref}))\] With \(\rho_{eff}\) the effective driving density, \(\rho\) the density, \(T\) the temperature , \(T_{ref}\) the reference temperature and \(\beta\) the thermal expansion coefficient.

Note that the effect of buoyancy on turbulence is also taken into account in our platform. This happens fully automatically through the application of source terms on the turbulent kinetic energy \(k\) and the turbulent kinetic energy dissipation rate \(\varepsilon\) or specific dissipation rate \(\omega\) depending on the chosen turbulence model.

#### Which parameters to set in the platform

When buoyancy effects are to be included in a specific region, several parameters need to be set. These parameters are set in the region settings of the concerning region:

Parameter | Units | Explanation |
---|---|---|

\(g\) | \([m^{2}/s]\) | The directional gravity vector. |

\(\beta\) | \([1/K]\) | The thermal expansion coefficient needs to be set for the material that is assigned to the region. It can be specified as a constant value or an array with the polynomial coefficients to include the temperature dependency. |

\(T_{ref}\) | \([K]\) | Optional: The initial temperature in the specific region, serving as a reference temperature. If nothing is specified, a temperature of \(293.15\:K\) is set. |

\(p_{ref}\) | \([Pa]\) | Optional: The initial pressure in the specific region, serving as a reference pressure. If nothing is specified, a pressure of \(1\:atm\:(101325\:Pa)\) is set. |