**1. Introduction**

When an object is immersed into a channel, a chamber, a wind tunnel, or even within our bodies, for instance, when a blood cell or clot moves through the heart or along the smallest arteries in the body, a fluid-structure interaction phenomenon occurs between the object and the surrounding fluid. This moving or static object (i.e., a cylinder, a sphere, a plate, a clot stuck in the blood vessels) will experiment a drag force in the fluid direction due to pressure and shear stress forces [1]. Drag force produced is a consequence of the velocity gradient of the fluid to the object, and it is dependent on the drag coefficient and geometry of the object.

Hydrodynamic forces are related to viscosity and inertia of the fluid. Close to the surface of the object, momentum is transferred through a layer in which viscosity takes an important role in determining the width of the layer and the way the velocity changes within it. This viscous zone is called the boundary layer [2]. Far from the solid boundary, viscous effect is less important and the inertia of the fluid particles decides the faith of the flow. The comparison between viscous and inertial forces acting on a fluid is given by the Reynolds number (Re), a dimensionless quantity.

Inertial forces are represented by the density (*ρ*) times the characteristic velocity of the fluid (*V*) times a length scale (*L*) according to the surface over which the fluid is moving. The viscous part is characterized by the dynamic viscosity of the fluid (*μ*) [2]. Thereby, *Re* can be expressed using Eq. (1):

**104**

*Technology, Science and Culture - A Global Vision*

[1] Marx K, Engels F. The Communist Manifesto. Oxford: Oxford University

[2] Hardt M, Negri A. Multitude. War and Democracy in the Age of Empire. New York: Penguin Books; 2004

[3] Beasley-Murray J. Posthegemony. Political Theory and Latin America. Minneapolis: University of Minnesota

[4] Deleuze G, Guattari F. Mil Mesetas. Capitalismo y esquizofrenia. Pre-textos:

**References**

Press; 1992

Press; 2010

Valencia; 2002

*Technology, Science and Culture - A Global Vision*

$$\mathbf{Re} = \frac{\text{Inertial Forces}}{\text{Viscous Forces}} = \frac{\rho \,\text{V}^2 \,\text{L}^2}{\mu \text{VL}} = \frac{\rho \,\text{VL}}{\mu} \tag{1}$$

For low *Re* values (below 102 ), viscous forces dominate the fluid motion having a well-defined straight path indicating that fluid moves in parallel layers, this behavior is called laminar flow. For high *Re* values (above 105 ), the fluid follows an irregular motion across the section of the channel or tube where the fluid is moving. This motion is known as turbulent flow, and when it occurs, the viscous forces are so small that they can be neglected. The stage between these ranges is called the transition regime and it always occurred when *Re* ∼3000. As it can be seen in **Figure 1**, laminar regime follows a linear behavior, and as soon as it gets closer to the transition regime and then turbulent region, it changes to a nonlinear behavior.

The relative motion of the fluid produces a hydrodynamic force in a parallel direction to the flow, also known as drag force. According to dimensionless analysis in [3], the drag force is represented by a drag coefficient *CD*, another dimensionless number, which is directly related to the Reynolds number. Depending on the *Re* observed in the object, a specific drag coefficient will occur. Hence, not only velocity is involved, but also the drag force *FD* produced by the object, its frontal area *A*, and density of the fluid *ρ*. This number is used to model all dependencies in shape, rotation, and flow conditions of the object [4] and it can be expressed by using Eq. (2):

$$\mathbf{C}\_{D} = \frac{F\_{D}}{0.5 \rho v^{2} A} \tag{2}$$

**107**

**Figure 2.**

106

, and 107

*ρ* = 1.229 kg/m3

**2.1 Calculation process**

an implicit analysis is required.

*Boundary conditions of the ICFD problem.*

*Aerodynamic Coefficient Calculation of a Sphere Using Incompressible Computational Fluid…*

(nonstationary object), a different approach is needed and an FSI simulation is required. An example of these problems is presented in [5]. For the aim of this work, the interest is focused only in stationary objects and how they affect the

The content in this chapter shows in Section 2 the process to obtain these forces and coefficients for a static sphere in an ICFD simulation. A comparison between two approaches available in the software to solve calculations, as well as the configuration for their simulations using a finite-element (FE) software is also shown. Results obtained for the ICFD using the sphere and a validation test to select the correct mesh size are presented in Section 3. Future work and conclusions of the

In order to compare simulations with experimental results, it is necessary to have the drag curve of a sphere. According to the establish in [6], the principle of similarity simplifies the use of many variables (velocity, density, dynamic viscosity, and a dimension of the body, e.g., diameter of the sphere) to a single variable, which is the Reynolds number. For different diameters, velocities, and fluids, a characteristic

Once the curve is obtained, an ICFD analysis is conducted through a sphere immerse in a constant flow to obtain different drag and lift coefficients through diverse *Re* ranges. Simulation is configured using LS-DYNA Software and the boundary conditions of the fluid domain (see **Figure 2**) are stated as follows: **Inlet Wall** representing the entrance for the fluid, **Outlet Wall** where the fluid goes away and the pressure reaches zero, **Nonslip Condition Wall** representing that the fluid has zero velocity relative to the boundary. For an ICFD, the object is also configured as a

, which are the ones present in **Figure 1**. The fluid used is air which density

, viscosity *μ* = 0.0000173 Pa s. Speaking of the objects, the sphere has a

, 103 , 104 , 105 ,

/4).

(where the area *A* = *πd*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.83691*

chapter are listed in Section 4 and 5, respectively.

drag curve can be obtained using *Re* and is shown in **Figure 1**.

nonslip condition object and it will behave as an obstacle for the fluid. The test is conducted for seven levels of Reynold numbers: 1, 10, 102

Velocities are obtained using Eq. (1), where *L* is the diameter *d* of the sphere.

LS-DYNA has two different approaches to solve a problem: shared memory parallel (SMP) processing and massively parallel processing (MPP) [7]. SMP processing allows to distribute the model, solving process over multiple processors on the same computer. MPP capabilities allow to run the problem over a cluster of machines or use multiple processors on a single computer and it is fundamental if

frontal area (perpendicular to the flow) of 0.000314159 m2

surrounding fluid.

**2. Methodology**

The way to obtain these coefficients in a simulation is through the use of ICFD analysis. Nevertheless, these cases represent a simple calculation because the object interacting with the fluid is always stationary and only behaves as an obstacle for the fluid. However, when the object also presents a motion caused by the fluid

#### **Figure 1.**

*Characteristic drag curve acting in a sphere for different Reynolds number ranges. Transition of the boundary layer to turbulent region is represented by the dip at the last part.*

*Aerodynamic Coefficient Calculation of a Sphere Using Incompressible Computational Fluid… DOI: http://dx.doi.org/10.5772/intechopen.83691*

(nonstationary object), a different approach is needed and an FSI simulation is required. An example of these problems is presented in [5]. For the aim of this work, the interest is focused only in stationary objects and how they affect the surrounding fluid.

The content in this chapter shows in Section 2 the process to obtain these forces and coefficients for a static sphere in an ICFD simulation. A comparison between two approaches available in the software to solve calculations, as well as the configuration for their simulations using a finite-element (FE) software is also shown. Results obtained for the ICFD using the sphere and a validation test to select the correct mesh size are presented in Section 3. Future work and conclusions of the chapter are listed in Section 4 and 5, respectively.

## **2. Methodology**

*Technology, Science and Culture - A Global Vision*

For low *Re* values (below 102

Re = \_\_\_\_\_\_\_\_\_\_\_\_ *Inertial Forces*

behavior is called laminar flow. For high *Re* values (above 105

*CD* <sup>=</sup> \_\_\_\_\_\_\_ *FD*

*Viscous Forces* <sup>=</sup> <sup>ρ</sup>*V*<sup>2</sup>*L*<sup>2</sup>

ing a well-defined straight path indicating that fluid moves in parallel layers, this

irregular motion across the section of the channel or tube where the fluid is moving. This motion is known as turbulent flow, and when it occurs, the viscous forces are so small that they can be neglected. The stage between these ranges is called the transition regime and it always occurred when *Re* ∼3000. As it can be seen in **Figure 1**, laminar regime follows a linear behavior, and as soon as it gets closer to the transition regime and then turbulent region, it changes to a nonlinear behavior. The relative motion of the fluid produces a hydrodynamic force in a parallel direction to the flow, also known as drag force. According to dimensionless analysis in [3], the drag force is represented by a drag coefficient *CD*, another dimensionless number, which is directly related to the Reynolds number. Depending on the *Re* observed in the object, a specific drag coefficient will occur. Hence, not only velocity is involved, but also the drag force *FD* produced by the object, its frontal area *A*, and density of the fluid *ρ*. This number is used to model all dependencies in shape, rotation, and flow conditions of the object [4] and it can be expressed by using Eq. (2):

The way to obtain these coefficients in a simulation is through the use of ICFD analysis. Nevertheless, these cases represent a simple calculation because the object interacting with the fluid is always stationary and only behaves as an obstacle for the fluid. However, when the object also presents a motion caused by the fluid

*Characteristic drag curve acting in a sphere for different Reynolds number ranges. Transition of the boundary* 

*layer to turbulent region is represented by the dip at the last part.*

\_\_\_\_\_ *<sup>μ</sup>*VL <sup>=</sup> <sup>ρ</sup>VL \_\_\_\_

), viscous forces dominate the fluid motion hav-

0.5<sup>ρ</sup> *<sup>v</sup>*<sup>2</sup>*<sup>A</sup>* (2)

*<sup>μ</sup>* (1)

), the fluid follows an

**106**

**Figure 1.**

In order to compare simulations with experimental results, it is necessary to have the drag curve of a sphere. According to the establish in [6], the principle of similarity simplifies the use of many variables (velocity, density, dynamic viscosity, and a dimension of the body, e.g., diameter of the sphere) to a single variable, which is the Reynolds number. For different diameters, velocities, and fluids, a characteristic drag curve can be obtained using *Re* and is shown in **Figure 1**.

Once the curve is obtained, an ICFD analysis is conducted through a sphere immerse in a constant flow to obtain different drag and lift coefficients through diverse *Re* ranges. Simulation is configured using LS-DYNA Software and the boundary conditions of the fluid domain (see **Figure 2**) are stated as follows: **Inlet Wall** representing the entrance for the fluid, **Outlet Wall** where the fluid goes away and the pressure reaches zero, **Nonslip Condition Wall** representing that the fluid has zero velocity relative to the boundary. For an ICFD, the object is also configured as a nonslip condition object and it will behave as an obstacle for the fluid.

The test is conducted for seven levels of Reynold numbers: 1, 10, 102 , 103 , 104 , 105 , 106 , and 107 , which are the ones present in **Figure 1**. The fluid used is air which density *ρ* = 1.229 kg/m3 , viscosity *μ* = 0.0000173 Pa s. Speaking of the objects, the sphere has a frontal area (perpendicular to the flow) of 0.000314159 m2 (where the area *A* = *πd*<sup>2</sup> /4). Velocities are obtained using Eq. (1), where *L* is the diameter *d* of the sphere.

#### **2.1 Calculation process**

LS-DYNA has two different approaches to solve a problem: shared memory parallel (SMP) processing and massively parallel processing (MPP) [7]. SMP processing allows to distribute the model, solving process over multiple processors on the same computer. MPP capabilities allow to run the problem over a cluster of machines or use multiple processors on a single computer and it is fundamental if an implicit analysis is required.

**Figure 2.** *Boundary conditions of the ICFD problem.*

**Figure 3.**

*Comparing two different setups. It is observable that using MPP takes less computational time than SMP even with a finer mesh. Scalability also helps to reduce time.*

In **Figure 3**, an example of SMP vs. MPP is shown. The model used is a 3D wind tunnel with an object placed at the center as an obstacle. For SMP, a mesh size of 0.008 is configured; for MPP, a finer mesh size of 0.0055 is used. Four different random velocities are chosen for the simulation and the number of CPUs used for SMP and MPP are 4 and 8, respectively. Results demonstrated that MPP finished calculations in less time than SMP for all cases, even when MPP uses a finer mesh than SMP. Due to this reason, MPP is the method selected for further simulations in order to reduce computational time. Resultant values obtained with the two methods are almost identical.

## **3. Results**

#### **3.1 Validation test**

Before obtaining the coefficients for different Reynolds numbers, it is necessary to prepare a mesh study to select the most adequate size having an equilibrium between computational time and accuracy of results. Having a low-quality mesh yields inaccuracies in results; hence, different configurations are tested for comparison (see **Figure 4**). The lift coefficient for a sphere is expected to be *CL* ≈ 0. An initial mesh size is used to compare drag and lift coefficients for an *Re* = 1000. However, configuration "X" is not suitable because results in *CL* are far from expected. This configuration is selected by using default parameters allowed by the software and then varying them in order to obtain the expected *CD*. Even when results in *CD* obtained using configuration "X" are close to the desirable values, *CL* is far from the expected, and simulation takes almost 3 h to finish calculations, and for this reason, other five options are configured for an *Re* = 1000, a running time of 3 s and a constant velocity of 0.704 m/s in search of obtaining the desirable values in *CD* and *CL*.

Different mesh sizes used for this *Re* need to have a *CD* ≈ 0.5 and *CL* ≈ 0. Configurations "A" and "E" have the coarser mesh with the lower number of elements and the finer mesh with the highest amount of elements, respectively. According to the results shown in **Table 1**, configuration "X" yields the highest computational time. From configurations "A" to "C," computational time is considerably lower than "E" and "X" but *CD* is far from expected. Configuration "D" possess a close *CD* but a higher *CL* than previous variations, whereas configuration

**109**

ficients will be.

problem.

**Figure 4.**

**Table 1.**

tion selected for the next test.

*total running time of 3 seconds.*

**3.2 Sphere drag simulation**

lent conditions are present (*Re* > 105

**Figures 5** and **6** by using configuration "E."

*Aerodynamic Coefficient Calculation of a Sphere Using Incompressible Computational Fluid…*

"E" is the one with the closest results to the expected values for *CD* ≈ 0.5 and *CL* ≈ 0 and a similar computational time than "X." For these reasons, "E" is the configura-

*Validation test for the sphere using different mesh sizes. The simulations were configured using a Re=1000 and a* 

**Drag coefficient (***CD***)**

A 19,100 7.43E-01 8.43E-02 4.9 B 36,762 6.79E-01 7.55E-02 13.5 C 39,892 3.74E-01 2.79E-02 28 D 65,446 5.54E-01 1.20E-01 35 E 78,392 4.27E-01 2.43E-02 150 X 70,730 5.67E-01 1.61E-01 162

Once the mesh is selected, the simulation is configured using the parameters listed in Section 2. Results observed during test demonstrated that Navier-Stokes equation used by LS-DYNA is not suitable to solve ICFD problems where turbu-

1 ≤ *Re* ≤ 100,000. We show the results obtained for drag and lift coefficients in

It is observed that the higher the Reynolds number, the lower drag and lift coef-

The software is capable of managing laminar zones for these cases without any

). Therefore, simulations are only trusted for

**Lift coefficient (***CL***)**

**Running time (#CPUs = 16 with MPP) (min)**

*DOI: http://dx.doi.org/10.5772/intechopen.83691*

*Configuring different mesh sizes for the sphere.*

**elements**

*Sphere meshing w/different sizes. Re = 1000, running time = 3 s.*

**Configuration No. of** 

*Aerodynamic Coefficient Calculation of a Sphere Using Incompressible Computational Fluid… DOI: http://dx.doi.org/10.5772/intechopen.83691*

**Figure 4.** *Configuring different mesh sizes for the sphere.*


#### **Table 1.**

*Technology, Science and Culture - A Global Vision*

*with a finer mesh. Scalability also helps to reduce time.*

ods are almost identical.

**3. Results**

**Figure 3.**

**3.1 Validation test**

In **Figure 3**, an example of SMP vs. MPP is shown. The model used is a 3D wind tunnel with an object placed at the center as an obstacle. For SMP, a mesh size of 0.008 is configured; for MPP, a finer mesh size of 0.0055 is used. Four different random velocities are chosen for the simulation and the number of CPUs used for SMP and MPP are 4 and 8, respectively. Results demonstrated that MPP finished calculations in less time than SMP for all cases, even when MPP uses a finer mesh than SMP. Due to this reason, MPP is the method selected for further simulations in order to reduce computational time. Resultant values obtained with the two meth-

*Comparing two different setups. It is observable that using MPP takes less computational time than SMP even* 

Before obtaining the coefficients for different Reynolds numbers, it is necessary to prepare a mesh study to select the most adequate size having an equilibrium between computational time and accuracy of results. Having a low-quality mesh yields inaccuracies in results; hence, different configurations are tested for comparison (see **Figure 4**). The lift coefficient for a sphere is expected to be *CL* ≈ 0. An initial mesh size is used to compare drag and lift coefficients for an *Re* = 1000. However, configuration "X" is not suitable because results in *CL* are far from expected. This configuration is selected by using default parameters allowed by the software and then varying them in order to obtain the expected *CD*. Even when results in *CD* obtained using configuration "X" are close to the desirable values, *CL* is far from the expected, and simulation takes almost 3 h to finish calculations, and for this reason, other five options are configured for an *Re* = 1000, a running time of 3 s and a constant velocity of 0.704 m/s in search of obtaining the desirable values in *CD* and *CL*. Different mesh sizes used for this *Re* need to have a *CD* ≈ 0.5 and *CL* ≈ 0. Configurations "A" and "E" have the coarser mesh with the lower number of elements and the finer mesh with the highest amount of elements, respectively. According to the results shown in **Table 1**, configuration "X" yields the highest computational time. From configurations "A" to "C," computational time is considerably lower than "E" and "X" but *CD* is far from expected. Configuration "D" possess a close *CD* but a higher *CL* than previous variations, whereas configuration

**108**

*Validation test for the sphere using different mesh sizes. The simulations were configured using a Re=1000 and a total running time of 3 seconds.*

"E" is the one with the closest results to the expected values for *CD* ≈ 0.5 and *CL* ≈ 0 and a similar computational time than "X." For these reasons, "E" is the configuration selected for the next test.

#### **3.2 Sphere drag simulation**

Once the mesh is selected, the simulation is configured using the parameters listed in Section 2. Results observed during test demonstrated that Navier-Stokes equation used by LS-DYNA is not suitable to solve ICFD problems where turbulent conditions are present (*Re* > 105 ). Therefore, simulations are only trusted for 1 ≤ *Re* ≤ 100,000. We show the results obtained for drag and lift coefficients in **Figures 5** and **6** by using configuration "E."

It is observed that the higher the Reynolds number, the lower drag and lift coefficients will be.

The software is capable of managing laminar zones for these cases without any problem.

**Figure 5.**

*The curve obtained during simulations is presented in red. The turbulent part is not accurate as expected due to the Navier-Stokes equation used by the software.*

**Figure 6.**

*Lift coefficients of the sphere are expected to be approximately zero. All results are between 0 and 0.05.*

**111**

provided the original work is properly cited.

and gibran.etcheverry@udlap.mx

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: jose.duranhz@udlap.mx; rene.ledesma@udlap.mx

*Aerodynamic Coefficient Calculation of a Sphere Using Incompressible Computational Fluid…*

it is only present as an obstacle. Future investigations are focused in configuring objects as being nonstationary, i.e., using a fluid-structure interaction (FSI) problem, the objective of this is to understand how other geometries will affect the fluid when they have their own motion. Simulating artificial valves is the main interest of future approaches, especially the ones shown in [5]. Future work includes not only an FSI simulation of a cardiac valve but also a parametric study where characteristic velocity and frequency of the fluid and the elastic modulus of the material are varied to create different situations. This will allow us to create a data matrix that will be focused in determining possible control parameters. Once these factors are settled, a modification to the electric analogy of the cardiac cycle represented by the Windkessel model [8–10] will be conducted in order to replicate

The aim of this work is to understand how an object interacts with a fluid when

The aim of this research is only focused in low-velocity fields; hence, inaccuracies observed in turbulent regions can be neglected because they only occurred for

Using the MPP software approach is imperative for these kinds of simulations where an implicit analysis is needed. Parallelizing the workload helps to decrease computational time in almost 80% compared to the use of the SMP approach.

Conducting a validation test is essential in order to select a proper mesh quality

ICFD analysis is a useful tool to understand hydrodynamic problems, providing the user with additional information to that obtained when using experimental particle image velocimetry method. The main advantage is the possibility to change

\*, Gibran Etcheverry1

\*

(resolution) and this radically impacts the veracity and accuracy of results.

geometry in an easier and faster manner, saving a large amount of time.

\*, Rene Ledesma-Alonso2

1 Department of Computing, Electronics and Mechatronics, Universidad de las

2 Department of Industrial and Mechanical Engineering, Universidad de las

Américas Puebla, San Andrés Cholula, Puebla, Mexico

Américas Puebla, San Andrés Cholula, Puebla, Mexico

*DOI: http://dx.doi.org/10.5772/intechopen.83691*

**4. Future work**

the behavior of a valve.

**5. Conclusions**

high-velocity fields.

**Author details**

Carlos Duran-Hernandez1

and Rogelio Perez-Santiago2

*Aerodynamic Coefficient Calculation of a Sphere Using Incompressible Computational Fluid… DOI: http://dx.doi.org/10.5772/intechopen.83691*
