Gravity

Forces due to gravity often play an important role in the momentum balance for fluids and solids. The body forces due to gravity are added to the momentum equations as a source term.

Figure 1. EQUATION_DISPLAY

f_{g} = ρ g

(860)

where $g$ is the gravity vector.

When the fluid density $ρ$ varies with temperature, the gravity vector acts on the density and induces a natural convection flow.

The buoyancy forces due to a non-uniform density field are considered significant when the ratio of the Grashof (Gr) and the Reynolds (Re) numbers approaches or exceeds unity:

Figure 2. EQUATION_DISPLAY

\frac{G r}{{Re}^{2}} = \frac{ρ β Δ T | g | L}{ρ v^{2}}

(861)

where $β$ is the thermal expansion coefficient.

For natural convection flows, you can use the Boussinesq model to simulate buoyancy effects instead of using a temperature-dependent density when the temperature differences are small. The density used in the Boussinesq model is constant in all solved equations and only the buoyancy source term varies.

Working Pressure

When the Gravity model is active, the Simcenter STAR-CCM+ working pressure becomes the piezometric pressure, in which the reference altitude (which you set) is taken into account. The piezometric pressure is given as:

Figure 3. EQUATION_DISPLAY

p_{p i e z o} = p_{s t a t i c} - ρ_{r e f} g \cdot (x- x_{0})

(862)

For both variable and constant density flows it is necessary to set the reference altitude $x_{0}$ , which is used in the calculation of piezometric pressure as shown above. The reference altitude is the point at which the piezometric and static pressure are considered equal.

When setting initial or boundary pressures in simulations for which the Gravity model is active, you are setting the piezometric pressure value. Simcenter STAR-CCM+ calculates the corresponding static pressure.

Variable Density Flows

For variable density flows, the buoyancy source term is:

Figure 4. EQUATION_DISPLAY

f_{g} = (ρ - ρ_{ref}) g

(863)

where $ρ_{ref}$ is the reference density.

For VOF multiphase flows the density variation could be very significant (several orders of magnitude), and result in numerical instabilities. To obtain a robust and stable numerical solution in such cases, set the reference density to zero or to the density of the lightest phase.

Boussinesq Model

You can enhance convergence of natural convection simulations by using the Boussinesq model. If the constant density

ρ_{r e f}

is used in all terms of the governing equations by eliminating

ρ

through the Boussinesq approximation

ρ = ρ_{r e f} (1 - β Δ T)

from the buoyancy term, the buoyancy source term is approximated as:

Figure 5. EQUATION_DISPLAY

f_{g} = ρ_{r e f} g β (T_{ref} - T)

(864)

where $T_{ref}$ is the operating temperature.

This approximation is valid only for small temperature differences and consequently for small density variations. The Boussinesq approximation is valid if $β (T - T_{0}) ≪ 1$ .