Fig. 14. Vortex diagram

16 Will-be-set-by-IN-TECH

Positive Γ 48.52 30.16 83.5 33 Negative Γ 23.8 3.09 60.67 2.55

positive circulation increases negative circulation increases too maintaining a circulation

It can be seen in Fig.13 that for Re 1,000 positive (negative) circulation reaches its maximum (minimum) and stabilizes around latter value, which fails to happen for Re 10,000 where circulation peaks at a "constant" rate but after reaching maximum starts decreasing. The motivation of this subsection is to explain why this change of slope took place (Fig.13(b)) and try to predict it analiticaly because it was observed that for different Re numbers the same

In order to understand this phenomena recall that the cavity has vorticity channels that feed and remove vorticity into and out the system affecting the circulation values. Having mentioned this observation and due to the low weight diffusive term has in the transport

which is when the positive vortex has taken the cavity, reflecting the "constant" increase of circulation Fig.13(b). More interesting and contradicting the assumption made was that *<sup>d</sup>*<sup>Γ</sup>

does not fall after the 100,000 iterations, situation that was expected since a slope change was observed in the Fig.13(b) after 100,000 iterations. Willing to explain this behavior the following

This unit feeds the positive vortex. The vortex is not able to accumulate more circulation, as it has reached the steady state configuration therefore this unit of vorticty has to be "passed" to each of the corner vortices, which also are not able to accumulate more circulation having to pass it to the upper wall and balancing the accounts of vorticity on the walls. Since the way of

Through the present study was seen that viscosity is who decides if vorticity can travel without diffusing itself, curl up, accumulate and form vortices. In a word is who decides how will the flow evolves. The interesting thing is that after being so influential in the flow pattern everything was in vain because the configuration of steady state regardless of the number Re (100-10,000) is very similar, a positive vortex has taken the cavity and two or three vortices were cornered. Latter observation trigger on of the most important remaining open

*dt* is based on counting how much vorticity is entering and leaving the system

*dt* is calculed according to the gradient of vorticity on the walls (18), which is the

∇*w* · *nds* (18)

*dt* was constant until 100.000 iterations,

*dt*

*dt* remains constant.

Table 1. Circulation values comparison

equation, *<sup>d</sup>*<sup>Γ</sup>

hypothesis was proposed:

**11. Discussion and open questions**

calculating the *<sup>d</sup>*<sup>Γ</sup>

differential of about 30 throughout evolution (Fig.13 a and b).

**10.1 Why does the circulation fall after rising for Re 10,000?**

After ploting Eq.(18) through time it was found that *<sup>d</sup>*<sup>Γ</sup>

Assume a unit of vorticity entering to the system Fig.14.

change in slope occures reaching different values of maximum circulation.

same as quantifying how much vorticity is entering and leaving the system.

*d*Γ *dt* <sup>=</sup> *∂*Ω

the circulation loss between vortice was not quantified, explaining why *<sup>d</sup>*<sup>Γ</sup>

Re 1,000 Re 10,000 max min max min

> question for future studies, why after so many turns, so many games, the flow reaches the same configuration?. It is believed that a study from game theory involving two players, "positive vorticity" and "negative vorticity" who fight a common good, the space of the cavity, can clarify why the positive vortex end taking the whole cavity behavior that is not achieved in the deep cavity scenario. Along with the latter question, other two remain open. First would be to answer, why the configuration of stable state coincide when the system can not store more vorticity and secondly why can not be achieved by the square cavit flow the configuration that occurs to happen in the deep cavity between the positive and negative knowing before that during the flow evolution this configuration is achieved but then lost.
