**5.1 Deep cavities**

6 Will-be-set-by-IN-TECH

*ν<sup>e</sup>* = *ν<sup>t</sup>* + *ν*.

*τe*

*and* |*S*| = |*ω*|

[*gk*(*x*, *t*) − *g*

3.

(b) Stream-function map in steady state

for Re 10,000.

Fig. 3. Steady states. Maps were taken at 100,000 and 110,000 iterations respectively.

<sup>3</sup> Is strongly recomended to consult (Chen, 2009) for a deeper understanding of the evolution equations

*eq*

*<sup>k</sup>* (*x*, *t*)] (16)

As the transport equation has changed, the LBM evolution equation has also changed

<sup>5</sup>(*C*Δ)2|*S*<sup>|</sup> 2*c*2Δ*t*

Having a new evolution equation Eq.(16) the algorithm has to be modified adding a new step where *τ<sup>e</sup>* is calculated based on the vorticity field. After making this improvement to the method, the algorithm began to work eficiently allowing to achive higher Re numbers without

It is said that the flow has reached steady state when collisions and transport do not affect each node probability. Concerning the algorithm it was considered that the flow had reached the steady state when its energy had stabilized and when the maps of vorticity and stream

Steady state vortex configuration for Re 1,000 and Re 10,000 is shown in Fig.3. It worth to notice that both are very similar, a positive vortex that fills the cavity and two negative vortices at the corners of the cavity. This configuration was observed from Re 1,000 to Re 10,000 being a prime characteristic of cavity flows. It is also important to clarify that for Re 10,000 the steady state presents a periodicity which is located in the upper left vortex that we shall see later,

*gk*(*<sup>x</sup>* <sup>+</sup> *<sup>c</sup>ek*Δ*t*, *<sup>t</sup>* <sup>+</sup> <sup>Δ</sup>*t*) <sup>−</sup> *gk*(*<sup>x</sup>*, *<sup>t</sup>*) = <sup>−</sup> <sup>1</sup>

*τ<sup>e</sup>* = *τ* +

compromising the computer cost, justifing the use of a LBM.

**5. Steady state study for different Re numbers**

function showed no changes through time.

indeed Fig.3(b) is a "snapshot" of the flow.

(a) Stream-function map in steady state

for Re 1,000.

and parameter calculations.

where

where

Several studies have proposed to study the deep cavity geometry (Gustafson, 1991; Patil et al., 2006) but none has reached to simulate high Re numbers possibly because the mesh sizes. Due to the LBM low computational cost it was decided to present the study of a deep cavity with an aspect ratio (AR) of 1.5 for Re 8,000.
