5:15 PM - 6:45 PM
[MIS21-P07] Numerical calculation of flows with phase changes: Solving slow flow fast Part 4
Keywords:phase change, permeable flow, numerical method
We have applied the explicit method and accelerated the explicit method using the Runge-Kutta method (Kawada, 2020, 2023, JpGU). In this presentation, we apply this method to a system comprising of both a bare fluid described by the Navier-Stokes equation and a permeable flow described by the Darcy's law. As an example, we consider a system with time-varying phase change. This includes a wide range of applied problems: the solidification of magma oceans during the early stages of the Earth's formation, the growth of the Earth's central core, the solidification of magma chambers, and the formation of sulphide chimneys in submarine hydrothermal vents.
First of all, let us confirm that it is possible to relate the Navier-Stokes and Darcy equations. The difference is that the drag in the Darcy equation is proportional to the velocity, while the viscous drag in the Navier-Stokes equation is proportional to the Laplacian of the velocity. If the boundary condition for the Navier-Stokes equation is free slip, the velocity and pressure fields of the Navier-Stokes flow can become the velocity and pressure fields of the Darcy flow (however, since the Fourier components are different, the steady-state solution for one is not for the other in general). From this property, if the drag for the velocity field is given as a linear combination of the Darcy and viscous drags, and the weight is assumed to vary according to the porosity of the Darcy flow (or they are switched according to some criterion), it is possible to create a system in which the two types of flow coexist. The coupling of the viscous and Darcy drags is similar to the classical treatment of phase changes in mantle convection calculations (e.g. Christensen and Yuen, 1985).
As an example, we consider the solidification of a two-phase eutectic system that simulates the solidification of the Earth's core (e.g. Worster, 1991; in the core, the upper part of the inner core solidifies, and light elements are released into the liquid outer core). Consider that a two-component fluid with a certain composition is cooled from below. The composition (density) of the liquid phase in the solid-liquid mixture is assumed to become less concentrated (lighter) towards the lower temperature side (or geometrically downwards). For simplicity, local thermodynamic equilibrium is asumed to hold (the composition of the liquid phase can only be determined from the phase diagram). In the solid-liquid mixture, the liquid phase is lighter towards the bottom, and compositional convection can be driven in this system. Furthermore, if a lighter and thinner fluid in solid-liquid mixture rises, it may dissolve the solid phase according to the phase diagram. Conversely, if a heavier and denser fluid descends, it may precipitate the solid phase. This forms a feedback effect where the porosity becomes large in an upward flow and become clogged in a downward flow.
We have performed numerical calculations for systems in which the porosity changes via the explicit method. The Darcy drag, which works when solid and liquid phases coexist, depends on the permeability (which is proportional to the reciprocal of resistance) and the porosity. It may give a great resistance when the porosity becomes small. It is difficult to solve this system using an implicit method, where the Darcy drug changes by many orders of magnitude in terms of the change of porosity. The change in the permeability due to Darcy resistance is similar to the change in viscosity in the Navier-Stokes system. In fact, when this system is solved by the SIMPLE method, the calculations proceed easily as long as the solid phase is small, but the convergence becomes extremely poor as the solid phase increases. This is because the explicit method is required. On the other hand, it has been confirmed by Navier-Stokes that the explicit method can be applied to systems where the viscosity changes by several orders of magnitude (Takeyama et al., 2017), and it is also applicable to the above-mentioned systems.
First of all, let us confirm that it is possible to relate the Navier-Stokes and Darcy equations. The difference is that the drag in the Darcy equation is proportional to the velocity, while the viscous drag in the Navier-Stokes equation is proportional to the Laplacian of the velocity. If the boundary condition for the Navier-Stokes equation is free slip, the velocity and pressure fields of the Navier-Stokes flow can become the velocity and pressure fields of the Darcy flow (however, since the Fourier components are different, the steady-state solution for one is not for the other in general). From this property, if the drag for the velocity field is given as a linear combination of the Darcy and viscous drags, and the weight is assumed to vary according to the porosity of the Darcy flow (or they are switched according to some criterion), it is possible to create a system in which the two types of flow coexist. The coupling of the viscous and Darcy drags is similar to the classical treatment of phase changes in mantle convection calculations (e.g. Christensen and Yuen, 1985).
As an example, we consider the solidification of a two-phase eutectic system that simulates the solidification of the Earth's core (e.g. Worster, 1991; in the core, the upper part of the inner core solidifies, and light elements are released into the liquid outer core). Consider that a two-component fluid with a certain composition is cooled from below. The composition (density) of the liquid phase in the solid-liquid mixture is assumed to become less concentrated (lighter) towards the lower temperature side (or geometrically downwards). For simplicity, local thermodynamic equilibrium is asumed to hold (the composition of the liquid phase can only be determined from the phase diagram). In the solid-liquid mixture, the liquid phase is lighter towards the bottom, and compositional convection can be driven in this system. Furthermore, if a lighter and thinner fluid in solid-liquid mixture rises, it may dissolve the solid phase according to the phase diagram. Conversely, if a heavier and denser fluid descends, it may precipitate the solid phase. This forms a feedback effect where the porosity becomes large in an upward flow and become clogged in a downward flow.
We have performed numerical calculations for systems in which the porosity changes via the explicit method. The Darcy drag, which works when solid and liquid phases coexist, depends on the permeability (which is proportional to the reciprocal of resistance) and the porosity. It may give a great resistance when the porosity becomes small. It is difficult to solve this system using an implicit method, where the Darcy drug changes by many orders of magnitude in terms of the change of porosity. The change in the permeability due to Darcy resistance is similar to the change in viscosity in the Navier-Stokes system. In fact, when this system is solved by the SIMPLE method, the calculations proceed easily as long as the solid phase is small, but the convergence becomes extremely poor as the solid phase increases. This is because the explicit method is required. On the other hand, it has been confirmed by Navier-Stokes that the explicit method can be applied to systems where the viscosity changes by several orders of magnitude (Takeyama et al., 2017), and it is also applicable to the above-mentioned systems.