Week 11 - Simulation of Flow through a pipe in OpenFoam

• Hagen-Poiseuille equation:

It is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through along a cylindrical pipe of constant cross-section. The Hagen-Poiseuille equation is only valid for long pipes whose length is relatively large compared to their diameter.

Hydrodynamic entrance length:

Fluid with low viscosity reaches its free stream velocities at very small distances as there is no hindrance with other molecules. But fluids with high viscosity take a large distance to reach the free stream velocity. At the entrance the velocity of fluid near the wall will have less velocity as there will be friction between the wall and the fluid towards the center velocity will be increasing and after a length, the velocity is the same at all positions. This length till which velocity varies is called hydrodynamic entrance region/length. The region after this region is a fully developed region.

Since there are no velocity gradients outside the hydrodynamic boundary layer, the shear stresses are negligible. In this undisturbed flow, the viscosity of the fluid plays no role. But within the boundary layer, the shear stresses are usually no longer negligible and have a decisive influence on the flow. Even if the boundary layer is defined very precisely, the change of the properties of the flow inside and outside of this "sharp" boundary is smooth.

• Fully Developed Laminar flow:

Flow is said to have reached a Fully developed flow if it has the same velocity profile at any cross-section within the pipe.

For the simulation of laminar flow through a pipe creating mesh for the whole pipe and running simulation may not be supported in all computers. So a wedge-shaped profile is created and simulations are performed on the wedge-shaped profile.

Plotting of shear stress:

plotting shear stress is not straight forward as plotting the velocity profile. Because in ParaView tau is not calculated directly, so it becomes relevant that we first calculate tau and then use the plot overline command to plot the shear stress.

1. We need to calculate the magnitude of vector U, this can be achieved using the mag option present in the calculator. If we provide mag(U) in the calculator we will get the magnitude of velocity.
2. Once the magnitude of velocity is computed next step is to calculate its gradient, this can be achieved by inserting the gradient of the unstructured data set, this is a computational option present in the preview. Once this option is inserted from filters we can provide the calculated velocity gradient was input form filters we can provide the calculated velocity gradient as input and hence the gradient of velocity can be calculated.
3. once the gradient is calculated, the dynamic viscosity of water at 25 c is multiplied,`mu=0.00089(kg)/(m.s)` hence tau is calculated.
4. Using a plot over the line, the probe can be placed anywhere along the radial direction and a shear plot can be obtained.