Mass conservation approximation
Mass conservation approximation#
First, we have to choose how to approximate the conservation of mass: \(\nabla \cdot (\rho \mathbf u) = 0\) (see Equation (2)).
We provide the following options, which can be selected in the parameter file in the subsection Formulation/Mass conservation
(see also Parameter name: Mass conservation):
“incompressible”:
“isothermal compression”:
\[\nabla \cdot \textbf{u} = -\rho \beta \textbf{g} \cdot \textbf{u},\]where \(\beta = \frac{1}{\rho} \frac{\partial \rho}{\partial p}\) is the compressibility, and is defined in the material model. Despite the name, this approximation can be used either for isothermal compression (where \(\beta = \beta_T\)) or isentropic compression (where \(\beta = \beta_S\)). The material model determines which compressibility is used. This is an explicit compressible mass equation where the velocity \(\textbf{u}\) on the right-hand side is an extrapolated velocity from the last timesteps.
“hydrostatic compression”:
\[\nabla \cdot \textbf{u} = - \left( \frac{1}{\rho} \left( \frac{\partial \rho}{\partial p} \right)_{T} \rho \textbf{g} + \frac{1}{\rho} \left( \frac{\partial \rho}{\partial T} \right)_{p} \nabla T \right) \cdot \textbf{u} = - \left( \beta_T \rho \textbf{g} - \alpha \nabla T \right) \cdot \textbf{u}\]where \(\beta_T = \frac{1}{\rho} \left(\frac{\partial \rho}{\partial p} \right)_{T}\) is the isothermal compressibility, \(\alpha = - \frac{1}{\rho} \left(\frac{\partial \rho}{\partial T} \right)_{p}\) is the thermal expansion coefficient, and both are defined in the material model. The approximation made here is that \(\nabla p = \rho \textbf{g}\).
“reference density profile”:
\[\nabla \cdot \textbf{u} = -\frac{1}{\bar{\rho}} \frac{\partial \bar{\rho}}{\partial z} \frac{\textbf{g}}{\|\textbf{g}\|} \cdot \textbf{u},\]where the reference profiles for the density \(\bar{\rho}\) and the density gradient \(\frac{\partial \bar{\rho}}{\partial z}\) provided by the adiabatic conditions model (Initial conditions and the adiabatic pressure/temperature) are used. Note that the gravity is assumed to point downwards in depth direction. This is the explicit mass equation where the velocity \(\textbf{u}\) on the right-hand side is an extrapolated velocity from the last timesteps.
“implicit reference density profile”:
\[\nabla \cdot \textbf{u} + \frac{1}{\bar{\rho}} \frac{\partial \bar{\rho}}{\partial z} \frac{\textbf{g}}{\|\textbf{g}\|} \cdot \textbf{u} = 0,\]which uses the same approximation for the density as “reference density profile,” but implements this term on the left-hand side instead of the right-hand side of the mass conservation equation. This effectively uses the current velocity \(\textbf{u}\) instead of an explicitly extrapolated velocity from the last timesteps.
“ask material model,” which uses “isothermal compression” if the material model reports that it is compressible and “incompressible” otherwise.
The stress tensor approximation
If a medium is incompressible, that is, if the mass conservation equation reads \(\nabla \cdot \textbf{u} = 0\), then the shear stress in the momentum and temperature equation simplifies from
to