**6. Vortex binding**

A particular process for Re 10,000 in square and deep cavities was found to take place through evolution. This process occurs several times throughout evolution, named Vortex Binding. In this process isolated vortices get connected forming a "massive" vortex which eventually will configure the steady state vortices distribution. A binding process that occured through evolution is shown in Fig.6 binding a positive vortex that appeared in the upper right corner with the positive vortex that came from the movement of the bottom wall.

In order to explain the binding process, which is illustrated in Fig.6, recall the vorticity transport equation Eq.(1). The transport equation is divided in two terms that dictate the transport of vorticity, the diffusive term *<sup>ν</sup>*∇2*<sup>ω</sup>* and the advective term [∇*ω*]*v*. For a high Re number flow the diffusive term can be neglected, turning the attention in the advective term. As the flow evolved it was seen that the vorticity and stream-function contour lines tended to align as shown in Fig.7(a) making the vorticity gradient vector and velocity vector orthogonal at different places (Fig.7(a)) causing [∇*ω*]*v* = 0, i.e. no vorticity transport.

As shown in Fig.7(b) vorticity contour lines started to curve, due to its own vorticity, crossing with the stream-function contour lines and making [∇*ω*]*v* �= 0. In Fig.7(b)can be seen that

of Lid-Driven Cavities 9

Flow Evolution Mechanisms of Lid-Driven Cavities 419

Fig. 5. *Left* Stream-function map for Re 8,000 in a cavity with AR=1.5 (200x300 nodes) *Right*

Fig. 6. Stream-function maps for Re 10,000 were Vortex binding process take place. Four

transport in different places which made possible the vortex binding to take place.

the vorticity gradient and the velocity vector are no longer orthogonals creating vorticity

In the study of dynamic systems, being the case of the present study the NS equations, and their solutions there exist bifurcations leading to periodic solutions. Specifically in cavity flows, when the Re number is increased, such bifurcations take place known as **Hopf Bifurcations**. Willing to understand how this Bifurcation takes place the *Sommerfelds* infinitesimal perturbation model is introduced. This perturbation model considers a small

maps were taken between 80,000 and 90,000 iterations

**7. Periodicity in cavity flows**

Stream-function map in a square cavity for Re 8,000 (200x200 nodes).

Fig. 4. Stream-function map for different times through evolution for a cavity with AR=1.5 and Re 8,000 in a 200x300 nodes mesh. *a,b,c,d and e* were taken at 20,000, 50,000, 150,000, 180,000 and 260,000-340,000 iterations.

8 Will-be-set-by-IN-TECH

Fig. 4. Stream-function map for different times through evolution for a cavity with AR=1.5 and Re 8,000 in a 200x300 nodes mesh. *a,b,c,d and e* were taken at 20,000, 50,000, 150,000,

180,000 and 260,000-340,000 iterations.

Fig. 5. *Left* Stream-function map for Re 8,000 in a cavity with AR=1.5 (200x300 nodes) *Right* Stream-function map in a square cavity for Re 8,000 (200x200 nodes).

Fig. 6. Stream-function maps for Re 10,000 were Vortex binding process take place. Four maps were taken between 80,000 and 90,000 iterations

the vorticity gradient and the velocity vector are no longer orthogonals creating vorticity transport in different places which made possible the vortex binding to take place.
