# 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.

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## 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 significantly improve the solution.

Tool Type : OPENFOAM®

Analysis Type : Incompressible Steady-state (Laminar)

Mesh and Element types :

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

Fig.2. Mesh used for the SimScale case

## Simulation Setup¶

Fluid:

• Water: Dynamic viscosity ($$\nu$$) $$= 10^{-6} m^2s$$

Boundary Conditions:

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

## Results¶

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

$u_{z_{max}} = 2u_{z_{avg}}$
$\Delta P = \frac{32\mu L}{D^2}u_{z_{avg}}$
$u_z = -\frac{1}{4\mu}\frac{\partial p}{\partial z}(R^2-r^2)$

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)

## 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.