Week 12 - Validation studies of Symmetry BC vs Wedge BC in OpenFOAM vs Analytical H.P equation

Aim:Validation studies of Symmetry BC vs Wedge BC in OpenFOAM vs Analytical H.P equation.

Objectives:

1. Validate Hydro-dynamic length with the numerical result
2. Validate the fully developed flow velocity profile with its analytical profile
3. Validate maximum velocity and pressured drop for fully developed flow
4. Post-process Shear stress and validate wall shear stress for fully developed flow
5. Highligh the differences in both boundary conditions in terms of their usage and application

Angle to test:

• `10^0`
• `25^0`
• `45^0`

Assumptions:

1. The steady laminar and incompressible flow of a fluid is taken.
2. Reynolds number for the laminar flow is 2100.
3. The wedge shape pipe is considered for simulation because the flow is axis- symmetric.
4. The equation governing the fluid flow is described by the Hagen Poiseuille's equation.

Hagen Poiseuille's equation :

The assumptions of the equations are the fluid is incompressible and Newtonian, the flow is laminar through a pipe of a constant cross section and its length is longer than the diameter of pipe.

Observations based on Hagen Poiseuille's equation :

• Diameter of pipe D=0.02 m
• Hydrodynamic entrance length =`0.06*R_e*D`=2.52 m
• Dynamic viscosity of water `mu=0.89e^-3 pascal-sec`
• Density of water `rho`=997 `(kg)/m^3`
• Kinematic Viscosity `nu`=`mu/rho=``(0.89e^-3)/997`=`8.9268e^-07 m^2/sec`
• Average velocity `v_(avg)`=`(mu*R_e)/(rho*D)=0.0937 m/sec`
• Velocity profile `u(r)=``2*v_(avg)*(1-r^2/R^2)`
• The velocity profile is maximum at the center of the pipe and becomes zero at the pipe wall due to no slip condition at the pipe wall.
• Maximum velocity `u_max=2*v_(avg)=0.1875 m/sec`
• Pressure drop for the fluid in laminar flow is `DeltaP=(32*mu*v_(avg)*L_c)/(D^2)`=0.0188 kPa
• where `L_c` is the total length of pipe we assume `L_c=L_e+0.3`=2.82m
• shear stress `tau =``(2*mu*u_max*r)/(R^2)`
• where r= Radial distance 0 to R , R = Radius of pipe ,

The simulation being performed for different wedge angles `10^0,25^0,45^0` by using the boundary conditions as wedge boundary condition and Symmetry boundary condition and validate the studies with analytical HP equation.

First we apply wedge boundary condition for wedge angle `10^0`

Please note that the pressure (p) file ,velocity (U) file,controlDict file,Transport properties file remains the same for wedge angle `25^0,45^0` respectively.

BlockMeshDict file for wedge angle `10^0` :

Control Dict file :

Transport properties file:

Pressure file:

Velocity File:

Geometric Profile for wedge angle `10^0`

Velocity profile after applying the initial and boundary condition:

Velocity profile at a distance 2.52 m from the inlet of pipe (Fully developed flow region )

Pressure profile [Graphical representation] :

Shear stress plot:

BlockMeshDict file for wedge angle `25^0` (Wedge boundary condition) :

Geometric Profile for wedge angle `25^0`

Velocity profile after applying the initial and boundary condition:

Velocity profile at a distance 2.52 m from the inlet of pipe (Fully developed flow region )

Pressure profile [Graphical representation] :

Shear stress plot:

BlockMesh Dict file for wedge angle `45^0`

Geometric Profile for wedge angle `45^0`

Velocity profile after applying the initial and boundary condition:

Velocity profile at a distance 2.52 m from the inlet of pipe (Fully developed flow region )

Pressure profile [Graphical representation]:

Shear stress plot:

Applying the Symmetry boundary condition: [For wedge angle `10^0`]

• The p,U,controlDict ,Transport properties remains the same as below for wedge angle `25^0,45^0`
• Please note that the geometry of the profile is remains the same ,therefore the behavior of the parameters is shown below.

BlockMesh Dict file for wedge angle `10^0`:

Pressure File:

Velocity File:

Control Dict file:

Transport property file [ Constant file]:

Velocity profile at a distance 2.52 m from the inlet of pipe (Fully developed flow region )

Pressure Profile [Graphical representation] :

Shear stress plot:

BlockMesh Dict file for wedge angle `25^0` :

Velocity profile at a distance 2.52 m from the inlet of pipe (Fully developed flow region )

Pressure profile:[Graphical representation] :

Shear stress plot:

BlockMesh Dict file for wedge angle `45^0`

Velocity profile at a distance 2.52 m from the inlet of pipe (Fully developed flow region ) :

Pressure profile:[Graphical representation] :

Shear stress plot:

Wall shear stress plot based on Hagen Poiseuille's equation :

Conclusion:

• From the above observations we can conclude that there is no much change in maximum velocities compared with analytical result.
• The only difference observed is execution time,the execution time required for symmetry boundary condition is more as compared with wedge boundary condition.
• But there is not much difference between the analytical and numerical result .In symmetry boundary condition the courant number obtained is slightly increasing if increasing the wedge angle.
• We can say that the symmetry boundary condition plays an important role in terms of accuracy of the result.