Milestone 05

Lid-driven cavity

Introduction

In the previous milestone you implemented a system with periodic boundary conditions. Now we disable periodicity and instead introduce hard walls.

Figure 1: Schematic illustration of the lid-driven cavity. The top wall is sliding at a constant velocity of \(v\), the other walls are at rest. All boundaries are subject to no-slip conditions. Each box represents a discrete element of the simulation domain which contains the populations \(\vec{f}\)(x,y).

Boundary conditions

Imagine that we are given a quadratic box with a sliding lid, as shown in Fig. Figure 1. Use equilibrium initial conditions for the discrete Boltzmann transport equations. Choose the initial values of \(\rho(0)=1.0\) and \(\mathbf{u}(0)=0\) at time \(t=0\).

Suppose the 2D quadratic box has a lid, the top boundary drawn in red (see Fig. Figure 1). This lid moves with a given velocity in the direction channel 1 is pointing to.

Tasks

• Apply bounce-back boundary conditions on the black boundaries and the prescribed wall velocity boundary conditions on the red wall. Bounce-back (done appropriately) will introduce a no-slip condition at the wall.
• Calculate the velocity field in the steady state. You need to run the simulation for quite a while to reach a steady-state situation.
• Observe how the generated flow field looks like and show it in a graphical representation either with plotting the velocity vectors or with streamlines.