Step 10: 2D Poisson Equation¶

We're very close now to building the final Navier-Stokes simulation that brought us here in the first place but before that, let's take a quick look at the Navier-Stokes equations for an incompresssible fluid, where $\vec{v}$ represents the velocity field:

$$ \nabla \cdot \vec{v} = 0 $$$$ \frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \nabla) \vec{v} = - \frac{1}{\rho} \nabla p + \nu \nabla^2 \vec{v} $$

The first equation is a representation of mass conservation at constant density. The second equation is the conservation of momentum. Yet, we have a problem: the continuity equation for incompressible flow does not have a dominant variable and there is no obvious way to couple the velocity and the pressure. In the case of compressible flow, in contranst, mass continuity would provide an evolution equation for the density $rho$, which is couple with an equation of state relating $rho$ and $p$.

In incompressible flow, the continuity equation $\nabla \cdot \vec{v} = 0$ provides a kinematic constraint that requires the pressure field to evolve so that the rate of expansion $\nabla \cdot \vec{v} $ should vanish everywhere. A way out of this difficulty is to construct a pressure field that guarantees continuity is satisfied; such relation can be obtained by taking the divergence of the momentum equation. In that process, a Poisson equation for the pressure shows up!

We can obtain Poisson's equation from adding a source term to the right hand side of Laplace's equation:

$$ \frac{\partial^2 p}{\partial x^2} + \frac{\partial^2 p}{\partial y^2} = b $$

This leads to a different behaviour within the field where there is some finite value inside the field that affects the solution. Poisson's equation acts to "relax" the initial sources in the field.

Discretization¶

In discretized formed this looks pretty much exactly the same as in Step 9:

$$ \frac{p^n_{i+1,j} - 2p^n_{i,j} + p^n_{i-1,j}}{\Delta x^2} + \frac{p^n_{i,j+1} - 2p^n_{i,j} + p^n_{i,j-1}}{\Delta y^2} = b^n_{i,j} $$

Rearranging the discretized equation we can get:

$$ p^n_{i,j} = \frac{\Delta y^2 (p^n_{i+1,j} + p^n_{i-1,j})+ \Delta x^2 (p^n_{i,j+1} + p^n_{i,j-1}) - b^n_{i,j} \Delta x^2 \Delta y^2}{2(\Delta x^2 + \Delta y^2 )} $$

Initial and Boundary Conditions¶

We solve this equation by assuming an initial state of p = 0 everywhere. Then we add the following boundary conditions.

$p = 0 \ \text{at} \ x = 0 , 2 \ \text{and} \ y = 0,1 $

The source term consists of two initial spikes inside the domain, as follow:

$ b_{i,j} = 100 \ \text{at} \ i = nx / 4, j = ny / 4 $

$ b_{i,j} = -100 \ \text{at} \ i = nx * 3 / 4, j = \frac{3}{4} ny $

$ b_{i,j} = 0 \ \text{everywhere else} $

The iterations will advance in pseudo time to relax the initial spikes. The relaxation under Posisson's equation get slower as time progresses however.

Libraries and Variable declarations¶

Lib import¶

Function Declarations¶

Variable Declarations¶

Solving in 2D¶

Results¶

Clearly shows the sink and the source. Let's try to plot the behavior over time to get a better idea.

Animating the simulation¶

Conclusion¶

With this, we have all the necessary tools to tackle the real challenge. The navier-stokes equations. Next step we shall build a simple cavity flow simulation. Let's do this1