**3. The ANSYS FLUENT software**

solution in a closed form for this equation system. In fact, this is one of the most important reasons of the use of CFD, since it provides numerical approximations to the equations solutions, which have not been found yet using analytical methods, except for idealized cases

The mathematical character of the equations has a significant impact on CFD. First of all, it is very important that the problem is adequately posed, and it means that the solution of the problem exists and it is unique. Furthermore, this solution is only affected by initial and boundary conditions. It is also important to classify the governing equations as elliptical, parabolic, or hyperbolic. Elliptical equations have to be solved simultaneously in the com‐ plete flow domain, whereas hyperbolic and parabolic equations propagate from one posi‐

Mathematically, parabolic and hyperbolic equations are susceptible to be solved using time steps. On the other hand, for the elliptical equations, the flow variables in a given point should be solved simultaneously with the flow variables in other positions. It can be stated that the Navier-Stokes equations have, in general, a mixed nature. They are parabolic in the time

As it was mentioned before, the governing equations can be applied for laminar and turbu‐ lent flows by means of the use of additional terms to represent the influence of the fluctuat‐ ing eddies. In a basic and rigorous way, direct numerical simulation (DNS) works solving numerically these equations to a desired accuracy degree without any additional model or correlation. However, its application is still limited because of the large quantity of resour‐ ces required to solve the majority of the existent problems, which exceed the capacity of

For turbulent flows, there is a wide spectrum of time and length scales that have to be solved that increase the computational time in DNS simulation. However, from a scientific point of view, the information obtained using DNS is very valuable because it is considered a valida‐ tion means for turbulent models (with less complexity). The traditional performance of the CFD methods for turbulent flows has been focused on predicting the average influence of the main characteristics of turbulence. Very often, such influence is approached including new transport equations to replicate the effects of generation or dissipation of Reynolds stresses , which represent the transport of momentum in the streamwise direction (x-direction) caused by the turbulent eddies in the "i" and "j" directions. Frequently, average stress models adjust to semi-empirical models, which depend on a number of constants obtained using experi‐ mental correlations. These correlations are very often obtained from experiments where the working fluid is air or water, and very frequently, the experimental setup includes a flat plate. In the literature, this kind of closing model is known as a Reynolds Average Navier–Stokes (RANS) turbulence model. This kind of model is the most frequently used for engineering applications today. Unfortunately, for complex turbulent flows there is not a "universal" model that can be applied to any kind of flow, since boundary layer effects can be very different. In fact, the description of what happens within the boundary layer is the main weakness of the RANS models. For many years, RANS models have had success reproduc‐ ing turbulent zones far away from the wall for fully developed flows. However, they still have

for one-dimensional or two-dimensional flows in laminar or creeping flows.

tion to another one.

conventional computers.

domain and elliptical in the space domain [1, 3].

320 Numerical Simulation - From Brain Imaging to Turbulent Flows

The software ANSYS FLUENT is one of the most used tools to discretely solve the govern‐ ing equations in fluid dynamics. It is very useful as a tool oriented to reproduce the behav‐ ior and give information for a wide range of fluid flows (compressible or incompressible, laminar or turbulent, steady or unsteady). It also possesses a wide variety of mathematical models for transport phenomena (such as heat transfer and chemical reactions) that can be applied to both simple and more complex geometries. FLUENT can be applied to laminar or turbulent flows within devices, such as heat exchangers, nozzles, complex-shaped ducts, turbomachinery (heat transfer and aerodynamic), heat engine components, external flows, flow through compressors, pumps, fans, and multiphase flows, among others.

In order to make a generalization of the governing equations to include compressibility effects and interactions among different phases, the continuity equation used by FLUENT can be described as follows:

$$\frac{\partial \rho}{\partial t} + \nabla \cdot \left(\rho \vec{V}\right) = S\_m \tag{1.12}$$

where the source term *Sm* represents the mass added to the continuous phase by the disperse phase. Other source terms can be defined by the user.

With respect to the other quantities transported by the fluid, they are modeled based on an integral general equation, where variable *ϕ* can be substituted by different flow variables before being discretized in its matrix form. For that Eq. (1.13), represents a dissipation constant for the transported fluid, which can be, for example, the dynamic viscosity *μ* for the momen‐ tum transport equations in any direction, or the Eddie viscosity *μt* associated with the energy production–dissipation models for the turbulent kinetic energy, whose magnitude depends on secondary factors such as turbulent kinetic energy intensity and some semi-empirical correlations.

$$\frac{\partial}{\partial t} \int\_{V}^{a} \rho \phi dV + \oint\_{A} \rho \phi \vec{V} \cdot dA = \oint\_{A} \Gamma\_{\phi} \nabla \phi \cdot dA + \int\_{V}^{a} S\_{\phi} dV \tag{1.13}$$


**Table 1** shows some equivalencies associated to the general variable.

**Table 1.** Different variables used in the general equation.

**Figure 1.** The coupled and segregated solvers comparison diagram, adapted with permission from [8].

The FLUENT solvers work based on a logical sequence to solve the discretized equations in two ways, the segregated one or the coupled one [8]. The segregated mode is recommended for flow problems where it can be expected the formation of large velocity gradients. These gradients can be produced by a boundary layer formation with or without flow separation where the compressibility effects are minimal (Mach < 0.3) and with or without temperature gradients at the boundary. Some examples of the application of the segregated solver are as follows: vehicular aerodynamics problems, internal flows, centrifugal pumps velocity fields' analysis, cyclone separator flows, flows in refrigeration ducts, flow pattern in cooling fins, etc. Regarding the coupled solver, it is widely used for the solution of problems related to large compressibility effects in the fluid, such as supersonic flows, hypersonic flows within nozzles combustion chambers, the movement of projectiles, and other objects where the shock waves are produced due to large pressure gradients and density changes.

Numerically speaking, the biggest difference between both solver modes is the way the flow governing equations are solved. The segregated mode solve them sequentially, taking the velocity and pressure equations (or corrected pressure equations) as the main variables, the coupled one solves globally the equations for continuity, momentum, and species exchange. **Figure 1** shows a diagram of both solver modes' performances.
