Velocity interpolation

Linear

The default interpolation scheme is a bi-linear (2D) or tri-linear (3D) interpolant.

LinP

Velocity interpolation from Pusok et al. 2017. The velocity at the m-th particle is given by

\[u_m = A u_L + (1-A) u_P\]

where $u_L$ is the bi or tri-linear interpolation from the velocity nodes to the particle, $u_P$ is the bi or tri-linear interpolation from the pressure nodes to the particle, and $A=2/3$ is an empirical coefficient.

Modified Quadratic Spline MQS

Velocity interpolation from Gerya et al. 2021. The scheme guarantee bi-linear interpolation of $\partial u_i/\partial x_i$ from pressure nodes where they are defined by solving (in)compressible continuity equation.

Example for the $u_x$ component in 2D:

Step 1 Compute the normalized distances between particle and left-bottom corner of the cell:

\[t_{x} = \frac{x_m - xc_j}{\Delta x}\]

\[t_{y} = \frac{y_m - yc_j}{\Delta y}\]

Step 2 Compute $u_x$ velocity with bi-linear scheme for the bottom and top

\[u_{m}^{(13)} = u_{i,j} t_x + u_{i,j+1} t_x\]

\[u_{m}^{(23)} = u_{i+1,j} t_x + u_{i+1,j+1} t_x\]

Step 3 Compute $u_x$ of the marker with bi-linear scheme in vertical direction

\[u_{m}^{(13)} = u_{m}^{(13)} + \frac{1}{2} (t_x-\frac{1}{2})^2 (u_{i,j-1}-2u_{i,j}+u_{i,j-1})\]

\[u_{m}^{(24)} = u_{m}^{(24)} + \frac{1}{2} (t_x-\frac{1}{2})^2 (u_{i+1,j-1}-2u_{i+1,j}+u_{i+1,j-1})\]

Step 4 Compute $u_x$ vx of the marker with bi-linear scheme in vertical direction

\[u_{m} = (1-t_y) u_{m}^{(13)}+(t_y) u_{m}^{(24)}\]