The Cavern Problem

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Numerical solution for Navier Stokes Equation using SIMPLE (Semi-Implicit Method for Pressure Linked Equations, Patankar & Spalding, 1972) method. : Explicit Finite Volume Method

The flow of an incompressible viscous fluid in the unit square $\Omega$ is described by the non-stationary Navier-Stokes equations:

$$\begin{cases} \frac{\partial \vec{u}}{\partial t} + (\vec{u}, \nabla) \vec{u} - \nu \Delta \vec{u} + \nabla p = \vec{0} \\ (\nabla, \vec{u}) = 0 \end{cases}$$

Boundaries:

The normal velocity components at the boundary are zero. A constant speed $\mathbf{u}=(1,0)$ is set on the upper boundary of the region. At the initial time $\mathbf{u} = 0$.

Example of solution