Mixed-Eularian-Lagrangian method

In last post I talked wrote about a time integration for NS equation. As a special case, time update for potential flow is actually easier, due to fact that

1. the governing equation, i.e. the Laplace equation is linear.

2. the solution dependes only on the boundary condition, in which only the free surface boundary condition is the tricky part.

3. Even though the field equation is in Eularian representation, the free surface can be described using Lagrangian representation.

For free surface potential flow, the time updating of the problem is twofold: update the (free surface) geometry and update the boundary conditioin at the free surface. Without mentioning the boundary condition at other part of the domain boundary, we are solving the problem characterized by

Field equation: $\nabla^2 \phi=0,\quad \forall x\in \sigma$

Free surface kinetic boundary condition(FSKBC): $D\vec{r}/Dt=\nabla \phi, \quad \forall x\in \gamma_0$

Free surface dynamic boundary condition(FSDBC): $\phi_t+1/2 (\nabla\phi)^2+gz+p/\rho=0,\quad \forall x\in \gamma_0$

The first boundary condition says that the particle velocity for material particle on the free surface, i.e. the velocity of free surface, equals to the flow velocity defined by velocity potential. The second boundary condition is the  Bernoulli equation, with constant on RHS assigned to 0, which could be looked as consumed by $\phi_t$. Remembering that

$\phi_t=D\phi/Dt-\vec{u}\cdot \nabla \phi=D\phi/Dt-(\nabla \phi)^2$

FSDBC can be converted to

$D\phi/Dt=1/2(\nabla \phi)^2-g\eta-p/\rho,\quad \forall x\in \gamma_0$

The plan of mixed-Eularian-Lagrangian(MEL) method, is to solve $\phi^n$ at time $t_n$, then use the BC equations to update the boundary condition. One should remember, the LHS of BC equations use material particle as variable, i.e. in Lagrangian representation. Denote particle on the free surface by $\zeta$, MEL is characterized by two steps:

1. Solve field equation with BCs with variable at $t_n$, i.e. $\phi^n, \vec{r}^n$;

2. Update the BCs at the free surface using

$(\vec{r}^{n+1}(\zeta)-\vec{r}^n(\zeta))/\Delta t=\nabla \phi^n(\zeta)$

$(\phi^{n+1}(\zeta)-\phi^n(\zeta))/\Delta t=1/2(\nabla \phi^n(\zeta))^2-g\eta-p/\rho$

As mentioned at the end of last post, usually the computing cost is characterized by the first step, while the second step concerns the stability and convergence.