**7. Periodicity in cavity flows**

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

of Lid-Driven Cavities 11

Flow Evolution Mechanisms of Lid-Driven Cavities 421

and breaking a negative vortex that has accompanied it since the beginning of evolution without qualitative form changes, only scaling the first configuration until the steady state

Fig. 8. Stream-function maps for a deep cavity with AR=1.5 and Re 8,000 where periodic flow take place. Maps were taken between 300,000 and 309,000 iterations. *White patches are vortices*

For Re 10,000 the positive vortex is created due to the lower wall movement and immediately itself creates a negative vortex coming from the right wall. Unlike Re 1,000 these two

*with high absolute vorticity*. Cavity upper right corner (100:200x100:300) nodes, see

Fig.4(e-right)

configuration is achieved in Fig.3(a).

(a) Upper right corner (nodes: 100:200 x 80:200).Taken at 80,000 iterations (b) Upper right corner (nodes: 100:200 x 80:200). Taken at 85,000 iterations

Fig. 7. *Left* Stream-function contour lines (**Green**), vorticity contour lines(**Red**), vorticity gradient(**Red**), velocity vector(**Black**).*Right* Stream-function contour lines (**Green**), vorticity contour lines(**Red**), vorticity gradient(**Red**), velocity vector(**Black**)and angle between [∇*ω*] and v(**Blue**)

perturbation of the dynamical system in order to study the equilibrium state or the lack of it. Let be considered the next dynamical system

$$\frac{d\vec{u}}{dt} = \left[M\_{\mathcal{V}}\right]\vec{u}.\tag{17}$$

The solution of Eq.(17) lies on finding the eigenvectors of the [*Mν*] operator which is in function of the fluid vicosity. Depending on the Re number the eigenvalues (and eigenvector) can be complex i.e. *λ* ∈ **C**, leading to periodic solutions(Toro, 2006) or Bifurcations. In (Auteri et al., 2002) the bifurcation for a cavity flow was located between 8017,6 and 8018,8 (Re numbers) but since 1995 (Goyon, 1995) reported the existence of particular periodic flow located in the upper left corner of a square cavity. In order to find the flow periodicity for Re 10,000 and determine if the system had reached its asymptotic state the system energy was used as a measure. A Periodic flow for a deep cavity is shown in Fig.84
