**1. Introduction**

30 Will-be-set-by-IN-TECH

410 Hydrodynamics – Advanced Topics

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The flow in cavities studies the dynamics of motion of a viscous fluid confined within a cavity in which the lower wall has a horizontal motion at constant speed. There exist two important reasons which motivate the study of cavity flows. First is the use of this particular geometry as a benchmark to verify the formulation and implementation of numerical methods and second the study of the dynamics of the flow inside the cavity which become very particular as the Reynolds (Re) number is increased, i.e. decreasing the fluid viscosity.

Most of the studies, concerning flow dynamics inside the cavity, focus their efforts on the steady state, but very few study the mechanisms of evolution or transients until the steady state is achieved (Gustafson, 1991). Own to the latter aproach it was considered interesting to understand the mechanisms associated with the flow evolution until the steady state is reached and the steady state per se, since for different Re numbers (1,000 and 10,000) steady states are "similar" but the transients to reach them are completely different.

In order to study the flow dynamics and the evolution mechanisms to steady state the Lattice Boltzmann Method (LBM) was chosen to solve the dynamic system. The LBM was created in the late 90's as a derivation of the Lattice Gas Automata (LGA). The idea that governs the method is to build simple mesoscale kinetic models that replicate macroscopic physics and after recovering the macro-level (continuum) it obeys the equations that governs it i.e. the Navier Stokes (NS) equations. The motivation for using LBM lies in a computational reason: Is easier to simulate fluid dynamics through a microscopic approach, more general than the continuum approach (Texeira, 1998) and the computational cost is lower than other NS equations solvers. Also is worth to mention that the prime characteristic of the present study and the method itself was that the primitive variables were the vorticity-stream function not as the usual pressure-velocity variables. It was intended, by chosing this approach, to understand in a better way the fluid dynamics because what characterizes the cavity flow is the lower wall movement which creates itself an impulse of vorticiy which is transported within the cavity by diffusion and advection. This transport and the vorticity itself create the different vortex within the cavity and are responsible for its interaction.

In the next sections steady states, periodic flows and feeding mechanisms for different Re numbers are going to be studied within square and deep cavities.

of Lid-Driven Cavities 3

Flow Evolution Mechanisms of Lid-Driven Cavities 413

Consider a set of particles that moves in a bidimensional lattice and each particle with a finite number of movements. Now a vorticity distribution function *gi*(*x*, *t*) will be asigned to each

particle with unitary velocity *ei* giving to it a dynamic consistent with two principles:

2. Vorticity variation in a node own to particle collision

Fig. 2. D2Q5 Model.2 dimensions and 5 possible directions of moving

*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>

*g eq <sup>k</sup>* <sup>=</sup> *<sup>w</sup>* 5 

and *τ*, the dimensionless relaxation time, is determined by Re number

where *ek* are the posible directions where the vorticity can be transported as shown in Fig.2. *c* = Δ*x*/Δ*t* is the fluid particle speed, Δ*x* and Δ*t* the lattice grid spacing and the time step respectively and *τ* the dimensionless relaxation time. Clearly Eq.(5) is divided in two parts, the first one emulates the advective term of (1) and the collision term, which is in square

> <sup>1</sup> <sup>+</sup> 2.5 *ek* · *<sup>u</sup> c*

<sup>2</sup>*c*2(*<sup>τ</sup>* − 0.5)

<sup>1</sup> The evolution equations were taken from (Chen et al., 2008) and (Chen, 2009). Is strongly recomended to consult the latter references for a deeper understanding of the evolution equations and parameter

*w* = ∑ *k*≥0

*Re* <sup>=</sup> <sup>5</sup>

*<sup>τ</sup>* [*gk*(*<sup>x</sup>*, *<sup>t</sup>*) <sup>−</sup> *<sup>g</sup>*

*eq*

*<sup>k</sup>* (*<sup>x</sup>*, *<sup>t</sup>*)]<sup>1</sup> (5)

. (6)

. (8)

*gk* (7)

**Observation 0.2.** *The method only considers binary particle collisions.*

The evolution equation is discribed by

The equilibrium function is calculed by

The vorticity is calculed as

calculations.

brackets, emulates the diffusive term of equation (1).

**3.1 Numerical method**

1. Vorticity transport
