**8. Flow transients**

Studying vorticity and stream-function maps was found that the way to get to the *same* state in most of the flow (Fig.3(a) and Fig.3(b)), with the exception of the corners for Re 10,000 which oscillate, change significantly as the number of Re varies. In order to illustrate this "bifurcation" vorticity transients for Re 1,000 and Re 10,000 are shown in Figs.9, 10 and 11 until steady state configuration is reached.

#### **8.1 Transient description**

For Re 1,000 the positive vortex is created on the lower right corner by the bottom wall movement. Latter vortex is feeded and grows until the whole cavity is taken cornering

<sup>4</sup> A well discribed periodic flow for square cavity can be found in (Goyon, 1995).

10 Will-be-set-by-IN-TECH

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*]

*du*´

used as a measure. A Periodic flow for a deep cavity is shown in Fig.84

<sup>4</sup> A well discribed periodic flow for square cavity can be found in (Goyon, 1995).

perturbation of the dynamical system in order to study the equilibrium state or the lack of it.

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

Studying vorticity and stream-function maps was found that the way to get to the *same* state in most of the flow (Fig.3(a) and Fig.3(b)), with the exception of the corners for Re 10,000 which oscillate, change significantly as the number of Re varies. In order to illustrate this "bifurcation" vorticity transients for Re 1,000 and Re 10,000 are shown in Figs.9, 10 and 11

For Re 1,000 the positive vortex is created on the lower right corner by the bottom wall movement. Latter vortex is feeded and grows until the whole cavity is taken cornering

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

*dt* <sup>=</sup> [*Mν*] *<sup>u</sup>*´. (17)

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

Let be considered the next dynamical system

until steady state configuration is reached.

and v(**Blue**)

**8. Flow transients**

**8.1 Transient description**

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 configuration is achieved in Fig.3(a).

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 with high absolute vorticity*. Cavity upper right corner (100:200x100:300) nodes, see Fig.4(e-right)

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

of Lid-Driven Cavities 13

Flow Evolution Mechanisms of Lid-Driven Cavities 423

Fig. 10. Vorticity maps: Positive vorticiy (**Blue**), Negative vorticity(**Red**) (200x200 nodes

Fig. 11. Vorticity maps: Positive vorticiy (**Blue**), Negative vorticity(**Red**) (200x200 nodes

**Definition 0.3.** *A vorticity channel is a bondary layer, coming from a wall, that feeds and creates*

mesh). The nine maps were taken from 60,000 to 110,000 iterations.

*vortex.*

mesh). The twelve maps were taken from 10,000 to 60,000 iterations.

Fig. 9. Vorticity maps: Positive vorticiy (**Blue**), Negative vorticity(**Red**) (200x200 nodes mesh). The nine maps were taken at 10,000, 20,000, 30,000, 40,000, 50,000, 60,000, 70,000, 80,000, 100,000 and 110,000 iterations respectively.

vortices qualitatively change during evolution, changing size and shape until the stable state configuration is reached shown in Fig.3(b).

It is worth to notice that vorticity maps for Re 1,000 and Re 10,000 are topologicaly very different. For Re 1,000 no interaction between positive and negative vorticity is presented but for Re 10,000 interaction is presented since the begining of evolution until the steady state and in the steady state itself because what causes the flow periodicity is the interaction of positive and negative vortices on the corners of the cavity.

#### **9. Vortex feeding mechanisms**

Cavity flow is a phenomenon characterized by a continuos vorticity injection to the system induced by the moving wall. The vorticity arises because the no-slip condition (viscous fluid) creating an impulse of vorticity that is transported into the cavity by advection or diffusion Eq.(1). As seen since the beginning the vorticity transport equation is divided in a diffusive term *<sup>ν</sup>*∇2*<sup>ω</sup>* <sup>≈</sup> <sup>1</sup> *Re* <sup>∇</sup>2*<sup>ω</sup>* and in an advective term [∇*ω*] *<sup>v</sup>*. At the beginning of the flow evolution the vorticity input is transported from the wall purely by diffusion but as the flow evolves both terms of the vorticity transport equation start to have different weights, being the diffusive term the most sensitive to Re number variations.

12 Will-be-set-by-IN-TECH

Fig. 9. Vorticity maps: Positive vorticiy (**Blue**), Negative vorticity(**Red**) (200x200 nodes mesh). The nine maps were taken at 10,000, 20,000, 30,000, 40,000, 50,000, 60,000, 70,000,

vortices qualitatively change during evolution, changing size and shape until the stable state

It is worth to notice that vorticity maps for Re 1,000 and Re 10,000 are topologicaly very different. For Re 1,000 no interaction between positive and negative vorticity is presented but for Re 10,000 interaction is presented since the begining of evolution until the steady state and in the steady state itself because what causes the flow periodicity is the interaction of

Cavity flow is a phenomenon characterized by a continuos vorticity injection to the system induced by the moving wall. The vorticity arises because the no-slip condition (viscous fluid) creating an impulse of vorticity that is transported into the cavity by advection or diffusion Eq.(1). As seen since the beginning the vorticity transport equation is divided in a diffusive

the vorticity input is transported from the wall purely by diffusion but as the flow evolves both terms of the vorticity transport equation start to have different weights, being the diffusive

*Re* <sup>∇</sup>2*<sup>ω</sup>* and in an advective term [∇*ω*] *<sup>v</sup>*. At the beginning of the flow evolution

80,000, 100,000 and 110,000 iterations respectively.

positive and negative vortices on the corners of the cavity.

term the most sensitive to Re number variations.

configuration is reached shown in Fig.3(b).

**9. Vortex feeding mechanisms**

term *<sup>ν</sup>*∇2*<sup>ω</sup>* <sup>≈</sup> <sup>1</sup>

Fig. 10. Vorticity maps: Positive vorticiy (**Blue**), Negative vorticity(**Red**) (200x200 nodes mesh). The twelve maps were taken from 10,000 to 60,000 iterations.

Fig. 11. Vorticity maps: Positive vorticiy (**Blue**), Negative vorticity(**Red**) (200x200 nodes mesh). The nine maps were taken from 60,000 to 110,000 iterations.

**Definition 0.3.** *A vorticity channel is a bondary layer, coming from a wall, that feeds and creates vortex.*

of Lid-Driven Cavities 15

Flow Evolution Mechanisms of Lid-Driven Cavities 425

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

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

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

values of circulation that are achieved for each value of Re (Table.1).

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

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

(b) Superposition for Re 10,000 during

(b) Square cavity circulation evolution. Positive

Γ (Red) and negative Γ (Blue)

(a) Superposition for Re 1,000 during

(a) Square cavity circulation evolution. Positive

evolution.

in positve circulation diferential.

Γ (Red) and negative Γ (Blue)

10,000.

vorticity (Aquamarine).
