**10. Circulation study for different Re numbers**

In order to understand more about what is happening with the vorticity of the system was decided to study the circulation behavior. The circulation is defined as Γ = *ωdA*. An interesting aspect of the circulation is that, although it must be constant in the system over 14 Will-be-set-by-IN-TECH

Channel creation is derived from two different phenomena: First is the energy transformation that occurs in the wall because the system continually transforms translational energy into rotational energy. Secondly a vortex whatever its sign is creates a channel of opposite sign. In oder to understand the latter suppose a positive vortex near a wall. The vortex make the particles that lie between it and the wall start spinning or *rotate*, due to viscosity, in the

There are three important features on the channels. The first and most important is that the channels transport vorticity from the walls inside the cavity and also diffuses vorticity along the route to nearby channels in proportion to the existing vorticity gradient. Secondly a positive channel always wraps a negative vortex and a negative channel always wraps a

• **Channel creation:** The transient is shown in Fig.9. Since the beginning there is a feeding channel from the right wall that grows merging in a left wall channel. It is worth noticing that the channel wraps the positive vortex during evolution (Fig.12(a)) but never interacts

• **Channel characteristics:** In Fig.9 can be observed that the feeding channels are thick. This ows to the fact the diffusive term of the transport equation is big enough to let vorticity be

Before studying the channels it is worth to clarify that in Fig.10 and 11 channels are the thin

• **Channel creation:** In the transient shown in Fig.10 can be seen since the beginning the appearance of a feeding channel coming from the right wall, but unlike the Re 1,000 transient, it begins to feed a vortex (sixth square of transient Fig.10) that grows inside the cavity. This vortex has the ability to interact in different ways (Fig.10 and 11) with the positive vortex that eventually will take the cavity. What is interesting about the vortex interaction, apart from the different forms that arise in the transient, is that the latter vortex has as many vorticity as the positive one, allowing them to interact in many ways. This interaction is able to produce a configuration seen in the deep cavity steady state where both vortices occupy the cavity without cornering each other but highly unstable ( twelfth square Fig.10). This occurs because the diffusive term of the transport equation has less weigth, allowing to concentrate vorticity without being spread across the cavity, which is the case for Re 1,000. It is also important to mention that for Re 10,000 negative channel

red "tubes" and the color patches are formed vortices which are fed by channels.

wraped positive vortex and vice versa (Fig.12(b)) as happens for Re 1,000.

• **Channel characteristics:** Unlike Re 1,000 channels the thickness of Re 10,000 channels are smaller, due to the diffusive low weight term in the vorticity transport equation.

In order to understand more about what is happening with the vorticity of the system was decided to study the circulation behavior. The circulation is defined as Γ = *ωdA*. An interesting aspect of the circulation is that, although it must be constant in the system over

opposite direction causing a vorticity input - in this case negative - to the system.

positive vortex. And finally channel thickness is function of the Re number.

spread within the fluid apart from being transported.

**10. Circulation study for different Re numbers**

**9.1 Channel creation and some other characteristics**

**9.2 Channel study for Re 1,000**

**9.3 Channel study for Re 10,000**

with it.

(a) Superposition for Re 1,000 during evolution. (b) Superposition for Re 10,000 during evolution

Fig. 12. Stream-function contour lines (blue) and vorticity maps superposition. *Left* Positive vorticity (Dark red) Negative vorticity (Light red), *right* Positive vorticity (Aqua) Negative vorticity (Aquamarine).

time according to Kelvins theorem, it can be split into positive and negative values. As seen, the prime characteristic of the flow is the positive vorticity input from the lower wall deriving in positve circulation diferential.

(a) Square cavity circulation evolution. Positive Γ (Red) and negative Γ (Blue) (b) Square cavity circulation evolution. Positive Γ (Red) and negative Γ (Blue)

Fig. 13. *Left* Square cavity circulation for Re 1,000. *Right* Square cavity circulation for Re 10,000.

In both figures can be seen that the flow reaches a maximum around the 100.000 iterations when the positive vortex has taken all the cavity (Fig.3.1 and 3.2). What is interesting are the values of circulation that are achieved for each value of Re (Table.1).

Several important things are shown in Table.1. First the circulation increase for Re 10,000 is three times bigger than Re 1,000 i.e. ΔΓ*Re*1,000 = 18.36 compared with ΔΓ*Re*10,000 = 50.5. Latter observation means that as the viscosity decreases the system is able to accumulate more circulation. Finally, system circulation is consistent whit Kelvin's theorem even though

of Lid-Driven Cavities 17

Flow Evolution Mechanisms of Lid-Driven Cavities 427

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.

Among all the results it was clearly seen the power and the preponderance of the viscosity in the evolution of cavity flows, how it affects the dynamics of vortices, transient or evolution of the flow and the accumulation or dissipation of energy. Was also observed the periodicity of steady-state flow for both cavities being the first to show a complete cycle of periodicity in the deep one. In conjunction with the above the feeding channels definition were proposed which were key to understanding the transient flow. It was also proposed a transient "Bifurcation" since they vary dramatically as the number of Re is increased. This "Bifurcation" is mainly

As for deep cavities in addition to finding the periodicity of the flow for Re 8.000 it was presented an interesting phenomenon observed in Sec.5.1.2 where a *quasi cavity* is created that replicates cavity flow transients that occur before reaching steady state in a square cavity. Finally, the numerical method implemented, based on the equations presented in (Chen, 2009; Chen et al., 2008), was a great help for the simplicity of its programming and its primitive

The authors are very greatful to Dr.Omar López for helpful discussions and advice.

Fig. 14. Vortex diagram

**12. Conclusions**

due to viscosity.

**13. Acknowledgments**

variable, vorticity, was central in the study.


Table 1. Circulation values comparison

positive circulation increases negative circulation increases too maintaining a circulation differential of about 30 throughout evolution (Fig.13 a and b).

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

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 change in slope occures reaching different values of maximum circulation.

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 equation, *<sup>d</sup>*<sup>Γ</sup> *dt* is calculed according to the gradient of vorticity on the walls (18), which is the same as quantifying how much vorticity is entering and leaving the system.

$$\frac{d\Gamma}{dt} = \int\_{\partial\Omega} \nabla w \cdot n ds \tag{18}$$

After ploting Eq.(18) through time it was found that *<sup>d</sup>*<sup>Γ</sup> *dt* was constant until 100.000 iterations, 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> *dt* 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 hypothesis was proposed:

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

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 calculating the *<sup>d</sup>*<sup>Γ</sup> *dt* is based on counting how much vorticity is entering and leaving the system the circulation loss between vortice was not quantified, explaining why *<sup>d</sup>*<sup>Γ</sup> *dt* remains constant.

#### **11. Discussion and open questions**

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
