**6. Concluding remarks**

Rayleigh number Pr *<sup>3</sup>*

turbulent flow takes place approximately for *Ra* =*10 <sup>7</sup>*

284 Numerical Simulation - From Brain Imaging to Turbulent Flows

values for the *u* and for *w* components of velocity

**Figure 7.** Isothermal lines. Ra = 105

conducted for *Ra* =*10 <sup>5</sup>*

*<sup>2</sup> , g TL Ra <sup>v</sup>* b

(laminar regime). Air is the operating fluid, for which Pr *= 0.71*. The

where *β* is the thermal expansion coefficient, *g* is the gravity acceleration and *ΔT* =*Th* −*Tc* is the temperature difference between the vertical walls. The transition between laminar and

hybrid advection scheme was used and the Boussinesq approximation was adopted. Compu‐ tations were performed in a non-uniform grid, with 82 × 82 = 6400 nodes. Reference results are reported in [17] and [18] for several Rayleigh numbers in laminar regime, comparing solutions given by several authors. Results for laminar and turbulent flow are also presented in [19].

**Figure 7(a)** and **(b)** displays isothermal lines generated using a constant value spacing between the minimum and the maximum verified within the domain. **Figure 8(a)** and **(b)** shows the flow streamlines. The flow, in the steady-state situation, is characterized by a large vortex filling the cavity, rotating in the clockwise direction. Two small vortices rotating in the same direction are located near the cavity centre. For this case, the minimum and the maximum streamline values used in the visualization do not correspond to the total amplitude of the stream function within the domain. These values were, instead, adjusted in EasyCFD to correspond to those employed in [19]. The agreement between the calculations and those reported in the literature is very good. Vahl Davis and Jones [18] present normalized maximum

a

\* \* = = *uL wL*

. (a) EasyCFD and (b) Dixit and Babu [19].

 a

occurring in the vertical and horizontal symmetry lines, respectively. **Table 1** shows the results obtained with EasyCFD, the reference values in [17] and the range of variation for the *37* contributions reported in [18]. This range does not include the minimum and maximum reported values since these clearly fall outside the general trend of the remaining contributions.

*u ;w* (87)

<sup>D</sup> <sup>=</sup> (86)

. In the present work, simulations were

The numerical simulation of fluid flow for 2D problems was addressed. The physical principles and numerical models here presented correspond to the implementation in the software package EasyCFD. Transformation of the original equations to cope with a non-orthogonal generalized mesh is described in detail, along with the coupling of momentum and continuity with an adapted SIMPLEC algorithm for non-staggered meshes. Although not addressed in the present chapter, this software package was developed entirely based on a graphical interface, aiming at an easy and intuitive utilization. With a fast learning curve, this package is very suitable for learning the principles and application methods in computational fluid dynamics and has a great value both as a didactic and an applied tool. Although, at first, the restriction to 2D situations may seem very limitative, a great number of practical situations may be addressed with this approach.
