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This turbulent pipe flow validation case belongs to fluid dynamics. The aim of this test case is to validate the following parameters:

- Pressure drop between the inlet and outlet of the pipe
- Velocity distribution across the flow direction

The simulation results of SimScale were compared to the analytical results presented by Henryk Kudela in one of his lectures\(^1\).

The geometry can be seen below:

This is a cylindrical pipe with a diameter of 0.01 \(m\), and a length of 1 \(m\).

**Tool Type**: OpenFOAM®

**Analysis Type**: Incompressible steady-state analysis.

**Turblence Model**: Two turbulence models were tested, the k-epsilon and the k-omega SST.

**Mesh and Element Types**:

Two approaches were tested in this validation case: Wall functions and full resolution on the walls. For the wall functions, the desired \(y^+\) range is [30 , 300], and the generated mesh looks like this:

Full resolution on the walls requires a \(y^+\) lower than \(1\), so the final mesh has the following form:

More details about the meshes used in the three cases can be seen bellow:

Cases | Near-wall approach | Number of cells | Mesh type | Turbulence model |
---|---|---|---|---|

A | Wall functions | 176574 | Standard | k-omega SST |

B | Wall functions | 176574 | Standard | k-epsilon |

C | Full resolution | 1393776 | Standard | k-omega SST |

**Fluid**:

**Water**- \((\nu)\)
*Kinematic viscosity*= 10\(^{-6}\) \(m^2 \over \ s\) - \((\rho)\)
*Density*= 1000 \(kg \over\ m^3\)

- \((\nu)\)

**Boundary Conditions**:

- Velocity inlet of 1 \(m \over \ s\)
- Pressure outlet of 0 \(Pa\)
- No-slip walls with wall functions for cases A and B, and full resolution for case C

**Initial Conditions:**

- Turbulent kinetic energy \((k)\) of 3.84e-3 \(m^2 \over \ s^2\)
- Case A & C: Specific dissipation rate (\(ω\)) of 88.53 \(1 \over \ s\)
- Case B: Dissipation rate \((ε)\) of 3.059e-2 \(m^2 \over \ s^3\)

The velocity profile for turbulent pipe flow is approximated by the Power-law velocity profile equation \(^1\):

$$\bar{u}_y(r) = \bar{u}_{y_{max}}\left(\frac{R-r}{R}\right)^{1/n}$$

where:

- \({u}_{y_{max}}\): the maximum y-velocity of the cross-section (along the pipe axis)
- \(R\): the radius of the cylinder
- \(r\): the distance from the center of the cross-section
- \(n\): a constant that depends on the Reynolds number, estimated as 7 for this case

For turbulent flow, the ratio of \(u_{y_{max}}\) to the mean flow velocity is a function of \(Re\). In this case, this ratio is calculated to be 1.234.

The pressure drop for turbulent flow in pipes is obtained by using the Darcy-Weisbach \(^2\):

$$\Delta P = f\ \frac{\rho\ u^2 \ l}{2 \ d}$$

where:

- \(f\): is the Darcy friction factor calculated by the solution of the Colebrook equation
- \(ρ\): is the density of the fluid
- \(u\): is the average velocity of the cross section
- \(l\): is the length of the pipe
- \(d\): is the diameter of the cylinder

According to the Moody diagram (Figure 4) and for this case, the value of \(f\) is 0.0309.

For the “wall function approach” the average \(y^+\) value on the walls of the pipe is 31.95 for k-omega SST and 32.39 for k-epsilon.

Pressure drop along the pipe length can be observed below:

The following graph shows the developed velocity profile, located 60 \(cm\) from the inlet:

For “full resolution”, the average value for \(y^+\) is 0.017. The corresponding graphs are created:

The pressure drop along the pipe length:

The developed radial velocity profile, located 60 \(cm\) from the inlet:

Besides good agreement with the analytical model, results show that all approaches and turbulence models are successful in predicting the pressure drop along pipe length for the given meshes.

Note

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Last updated: July 30th, 2021

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