**3.3. Pressure–velocity coupling**

**Figure 4.** Translational periodicity in a heat exchanger array.

328 Numerical Simulation - From Brain Imaging to Turbulent Flows

axis in the fluid in studies related to turbomachinery.

Planes (**Figure 4**). Also, a mass flow can be defined.

**3.2. Some technical suggestions for an adequate solving**

Periodic boundary condition is useful to cut down the size of the studied domain, saving simulation time in cases where the geometry of the given problem and the flow pattern have characteristics that are repetitive and periodic along a certain characteristic length "L". The periodic flow behavior can be found in many applications such as heat exchanger channels, flow over pipe banks, and fully developed flow in pipes and ducts. There exist two types of periodicity, such as translational and rotational periodicity (**Figure 3**). For rotational perio‐ dicity a constant pressure is defined along the periodicity planes, also defining a rotational

Regarding translational periodicity, a finite pressure drop can be defined between two periodic

To solve adequately a CFD problem, it is recommended a series of best practices whose Application can help to obtain better convergence values for CFD simulations. The first one is to verify that the skewness value of most of the mesh elements is between 0 and 0.5, being the values closest to 0 the most desirable ones. This is to minimize the numerical dissipation for the algorithms responsible of carrying out the flow balances at the cell faces. Another important suggestion to improve initial convergence in fluid, heat transfer, and multiphase flow problems is to first obtain solutions using single precision and first-order discretization models. After that, based on these solutions, change the discretization models to second order models. This should be carried out considering that second-order schemes produce lower error values. From this point, it is recommended to activate the energy equation (if needed) or any multiphase flow or reactive flow parameters (if needed) to obtain adequate convergence. It is

*3.1.6. Periodic boundary condition*

In the Navier-Stokes Eqs. (1.7)–(1.10), calculated velocities are related to the momentum transport in two or three dimensions. These velocities should satisfy the mass conservation equation. However, continuity equation does not depend on the pressure as it does the momentum exchange. The convective and dissipative terms of the momentum conservation equation, which depends on the pressure, have second-order, nonlinear terms. When governing equations have to be solved, it is necessary to find the velocity and pressure fields for stationary and time-dependent problems. If the flow is incompressible, continuity equation can be used as a transport equation for density, energy conservation equation represents temperature transport, and the existing pressure gradients can be obtained using equations such as the ideal gas law or any valid state equation. For incompressible flow, however, density has a constant value and there is not a direct link among the governing equations for the mass and momentum transport equations. This situation provokes unpredictable oscillations in the solving process making it considerably difficult.

Regarding the semi-implicit method for pressure-linked equations (SIMPLE) algorithm proposed by Patankar and Spalding in 1972 [13], it has been used for many years as a useful and convenient solving strategy. This algorithm is depicted in **Figure 5**. Unit mass flows go through the faces of the cells, and are evaluated from an initial guessed velocity value in two or three dimensions. From these velocity and pressure fields, a momentum transport equa‐ tion solution is obtained. Then, from this solution, a corrected pressure is obtained using a series of relationships obtained from the continuity equation.

Finally, such corrected pressure values will be used for the transport and continuity equa‐ tions solving. The whole process has to be repeated in an iterative manner until a conver‐ gence criterion is reached. As it was mentioned before, to reach a solution, it is necessary to initialize the flow using arbitrary values for the velocity and pressure fields. Such values are often selected from meaningful flow quantities such as free stream velocity and the atmos‐ pheric pressure for external flows. In the case of an internal flow, the main velocity value(the one with larger magnitude when several inlets are present) and the internal pressure are used.

Nowadays, there are a series of improvements made to the SIMPLE algorithm. One of them is the SIMPLEC algorithm also known as SIMPLE-Consistent [14], which is one of the most popular. This algorithm works similarly to SIMPLE, being its main difference that the momentum conservation equations are modified in a way that allows the velocity correction equations are simplified. The result of this change is that for some cases, the convergence time decreases. This algorithm is optimal to solve laminar flow problems without temperature gradients. It requires less convergence time than the conventional SIMPLE algorithm. Nevertheless, its use is not recommendable to solve turbulent flow, species transport,

combustion problems, or any problem involving not only fluid movements, but also a more complex physical problem.

**Figure 5.** Overall stages in the SIMPLE algorithm, adapted with permission from Versteeg and Malalesekera [3].

The pressure implicit with splitting of operators (PISO) algorithm is an improved version of the SIMPLE algorithm. It was created for non-iterative calculations in the numerical solution of non-steady flows. It has been adopted for iterative solution process for steady and nonsteady flows, being especially useful for the last ones. This algorithm involves a stage where velocities and pressures are arbitrarily predicted, and two correcting stages. Therefore, it has an additional correcting stage with respect to SIMPLE and SIMPLEC. In the prediction and correction stages, the obtained pressures involve functions that contain within the continui‐ ty equation [15].

Even though the PISO algorithm requires additional memory to store the values of the added correction equations, and in general, it requires relaxation factors to stabilize the calculation process, it performs in a quick and efficient way.

In relation to periodic or transient problems, although less popular than PISO and SIMPLE, the fractional step method (FSM) algorithm is a method created as a solving alternative of noniterative nature (NITA–non-iterative time advancement). Even though its use is less popular than the SIMPLE algorithm and its modifications, due to its formulation, involving a decou‐ pling of the mass and momentum conservation equations using mathematical operators is capable of decreasing the simulation time [16].

Regarding the pressure–velocity coupling algorithm known as COUPLED, it is recommend‐ ed to solve problems involving steady-state single-phase flows. For this algorithm, the solving process of all flow components is carried out using vectors. This means using a square coefficients matrix derived from flows entering or leaving the control volume through each control surface. This matrix is then algebraically associated with a unidimensional unknown vector. The product of both matrices is then equivalent to a solution vector where boundary conditions are included. The COUPLED algorithm is an alternative to SIMPLE and its spinoffs in the segregated solver, and it is recommended for cases where the domain presents lowquality elements for non-steady problems.

Finally, before starting a solution process, it is recommended to take a careful look into the most adequate coupling schemes for the problem at hand. This can be done considering the most important flow characteristics and looking for a similar case in any classic study case where a benchmark has already been performed (e.g., jets, flow over a flat plate, internal flows).
