Hagen Poiseuille Flow

Overview

The aim of this test case is to validate the following parameters of incompressible steady-state laminar fluid flow through a pipe:

• Velocity
• Pressure drop

The simulation results of SimScale were compared to the analytical results shown in [1]. The mesh was created with the parametrized – hexahedralization-tool on the SimScale platform.

Geometry

A straight cylindrical pipe was chosen as the flow domain (see Fig.1.). Faces A, B and C represent the inlet, wall and outlet respectively.

Fig.1. Geometry of the cylindrical pipe

	Length	Diameter
Value [m]	1	0.01

Analysis type and Domain

A uniformly-spaced hexahedral mesh was generated on the SimScale platform using the snappyHexMesh tool (see Fig.2.). It was observed that the presence of inflation layer doesn’t not si

Mesh type	Cells in x	Cells in y	Cells in z	Number of nodes	Type
snappyHexMesh	20	1600	20	341544	3D hex

Simulation Setup

Fluid:

• Water: Dynamic viscosity (ν
  $\nu$

  ) =10−6m2s

  $={10}^{-6}{m}^{2}s$

Boundary Conditions:

	Boundary type	Velocity	Pressure
A	Inlet	Fixed Value: 0.1 ms−1 $0.1m{s}^{-1}$	Zero Gradient
B	Wall	Fixed Value: 0.0 ms−1 $0.0m{s}^{-1}$	Zero Gradient
C	Outlet	Zero Gradient	Fixed Value: 0.1 Pa $0.1Pa$

Results

The analytical solution gives us the following equations for maximum axial velocity, pressure drop and developed radial velocity profile:

uzmax=2uzavg

$$u_{z_{max}}=2u_{z_{avg}}$$

ΔP=32μLD2uzavg

$$\mathrm{\Delta }P=\frac{32\mu L}{D^2}{u}_{z_{avg}}$$

uz=−14μ∂p∂z(R2−r2)

$$u_z=-\frac{1}{4\mu }\frac{\mathrm{\partial }p}{\mathrm{\partial }z}(R^2-r^2)$$

A comparison of the velocity and pressure drop obtained with SimScale with analytical results is given in Fig.3A, 3B and 3C. Fig.3A shows the developed radial velocity profile, located 60 cm from the inlet. The variation of the axial velocity along the center-line is shown in Fig.3B, and the pressure drop along the pipe can be observed in Fig.3C.

  

Fig.3. Visualization of Axial velocity and pressure (A, B) and developed radial velocity profile (C)

References

[1]	Hagen–Poiseuille flow from the Navier–Stokes equations

Disclaimer

This offering is not approved or endorsed by OpenCFD Limited, producer and distributor of the OpenFOAM software and owner of the OPENFOAM® and OpenCFD® trade marks. OPENFOAM® is a registered trade mark of OpenCFD Limited, producer and distributor of the OpenFOAM software.

Last updated: August 7th, 2020