**4. Computational fluid dynamics and fluent**

The fluid equations can be solved either in differential form or integral form. The differential form is obtained by applying conservation laws to a fluid particle in an Eulerian frame of reference. The integral form is obtained by applying conservation laws to a control volume.

The following are the differential form of fluid equations:

$$\frac{\partial u}{\partial \mathbf{x}} + \frac{\partial v}{\partial \mathbf{y}} = \mathbf{0}$$

$$\rho \left( u \frac{\partial u}{\partial \mathbf{x}} + v \frac{\partial u}{\partial \mathbf{y}} \right) = -\frac{\partial p}{\partial \mathbf{x}} + \mu \nabla^2 u$$

$$\rho \left( u \frac{\partial v}{\partial \mathbf{x}} + v \frac{\partial v}{\partial \mathbf{y}} \right) = -\frac{\partial p}{\partial \mathbf{y}} + \mu \nabla^2 v \tag{13}$$

There is also the integral form of fluid equations, as listed below:

**Figure 4.** *QBlade simulation graphs.*

$$\int\_{\mathcal{S}} (\nu.n)d\mathcal{S} = \mathbf{0}$$

$$\int\_{\mathcal{S}} \rho(\nu.\hat{n})\overrightarrow{\mathcal{V}}d\mathcal{S} = -\int\_{\mathcal{S}} \rho\hat{n}d\mathcal{S} + \overrightarrow{F}\_{\text{vic}} \tag{14}$$

The integral form of fluid equations is mainly preferred since conservation is always valid for any control volume or mesh "chunk." Conservation does not apply to each element in the differential form; hence, in industry, the integral form is preferred. Our software for CFD simulation is ANSYS Fluent, which uses the integral form for solving the problem. Both these equations of integral form are valid for any region or arbitrary shape of a control volume. However, Fluent will produce numeric solutions satisfying these equations for only a particular shape of control volume defined during the meshing step. Meshing breaks the 3D geometry into smaller "chunks" processed as individual control volumes.

The solution methodology introduces errors when solving these equations in Fluent. Two types of errors are introduced, namely, discretization and linearization errors. Discretization error occurs because we assume the value at the interface of adjoining cells is nothing but the average of each of those cell center values. **Figure 5** shows the control volumes and cell centers for a typical uniform meshing. Uniform rectangular chunks will not be the case for complex geometry, as the geometry will be broken into tetrahedrons. The averaging algorithm is also more exotic and will not be covered here.

However, on solving these equations, we end up with a set of nonlinear algebraic equations. These can be solved only by Newton-Raphson (NR) method by assuming a guess value for each cell center. NR will continually iterate until the error falls below a particular threshold. NR method leads to another source of error known as linearization error. The ultimate aim is to reduce both errors as much as possible.

**Figure 5.** *Discretization I.*

The mesh geometry plays an important role here. More "chunks" leads to less discretization error but more linearization error and vice versa. Hence, in Fluent, it is essential to hit the sweet spot for all geometry. Here, we conclude the inner working of Fluent for the translational or inertial frame of reference. In the next section, the Navier-Stokes equation will be modified for the rotational frame of reference.
