advection

A method to solve an advection equation of a physical quantity (for example, volume fraction of fluid in a grid cell) for an identification of the water surface is effective to reduce computational load of the interface tracking.

From this general solution, the exact pdf for the two-mode nonlinear advection equation has been found by solving Liouville's equation [solution displayed in (ES23)].

Here, we use the unsteady linear advection equation as a model equation and discretize this with a finite volume scheme and the implicit Euler method.

For instance, the FFD has significant numerical diffusion due to the linear interpolation used in the semi-Lagrangian solver for the advection equation. For simplicity, the one dimensional form of the linear interpolation is as follows:

Even if the initial value of the level-set function [PHI] (x,0) is set to be the distance function, the level set function [PHI] may not remain as a distance function at t > 0 when the advection equation, Equation (6), is solved for [PHI].

Therefore, instead we split off the advection equation for the density (3.1), which can be handled separately.

Similarly, scheme (25) can be converted to another solution interpolation scheme for the homogeneous advection equation:

The fractional volume is then computed and updated at each time step using the following advection equation

The volume fraction is advected with the flow and satisfies the advection equation as follows:

Let D = [0, 1]x[0, 1] .In the closed domain [0, T] x D consider the two-dimensional advection equation

In this method, the energy equation is decomposed into an advection equation and a diffusion equation as follows [8]

The Galerkin finite element methods are useful to solve the advection equation [6].

When the 1D linear advection equation is approximated by a numerical method, the amplification factor and relative phase error depend on only the cfl number.

In [22], a dam-break and oscillation experiments were performed to validate the 2D numerical model with the CIP method for the solution of advection equation of Navier-Stokes equation and also for the free surface treatment.

The air/skin polymer interface and skin/core polymer interface are traced by pseudo-concentration method and governed by two advection equations separately in the literatures [21, 22], At each time step, the advection equations are coupled with the governing equations of the two polymer melts, respectively.