Challenge

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

SSubramanya G S12 May 2021

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

AIM:

To simulate the axis-symmetric flow by applying the Symmetry Boundary Condition. Also to simulate for the below-mentioned angles using both symmetry and wedge boundary conditions and validate them with HP equations.

1. `10^o`

2. `25^o`

3. `45^o`

THEORY:

Introduction to Internal Flow:

Liquid or gas flow through pipes or ducts is commonly used in heating and cooling applications and fluid distribution networks. The fluid in such applications is usually forced to flow by a fan or pump through a flow section. The terms pipe, duct, and conduit are usually used interchangeably for flow sections. In general, flow sections of the circular cross-section are referred to as pipes (especially when the fluid is a liquid) and flow sections of the non-circular cross-section as ducts (especially when the fluid is a gas). Small diameter pipes are usually referred to as tubes.

The fluid velocity in a pipe changes from zero at the wall because of the no-slip condition to a maximum at the pipe center. In fluid flow, it is convenient to work with an average velocity Vavg, which remains constant in incompressible flow when the cross-sectional area of the pipe is constant(refer to image 1.1)

Also, the friction between the fluid particles in a pipe does cause a slight rise in fluid temperature as a result of the mechanical energy being converted to sensible thermal energy. But this temperature rise due to frictional heating is usually too small to warrant any consideration in calculations and thus is disregarded.

Image 1.1 Fully developed laminar pipe flow.

Laminar Flow and Turbulent Flow:

Laminar flow or streamline flow in pipes (or tubes) occurs when a fluid flows in parallel layers, with no disruption between the layers. At low velocities, the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards. There are no cross-currents perpendicular to the direction of flow, nor eddies or swirls of fluids. In laminar flow, the motion of the particles of the fluid is very orderly with all particles moving in straight lines parallel to the pipe walls.

Turbulent flow is a flow regime characterized by chaotic property changes. This includes rapid variation of pressure and flows velocity in space and time. In contrast to laminar flow, the fluid no longer travels in layers, and mixing across the tube is highly efficient. Flows at Reynolds numbers larger than 4000 are typically (but not necessarily) turbulent, while those at low Reynolds numbers below 2300 usually remain laminar. Flow in the range of Reynolds numbers 2300 to 4000 and known as transition.

Reynolds Number:

The transition from laminar to turbulent flow depends on the geometry, surface roughness, flow velocity, surface temperature, and type of fluid, among other things. After exhaustive experiments in the 1880s, Osborne Reynolds discovered that the flow regime depends mainly on the ratio of inertial forces to viscous forces in the fluid. This ratio is called the Reynolds number and is expressed for internal flow in a circular pipe as

`R_e = text(Inertia Force)/text(Viscous Force) = (V_(avg)D)/nu=(rhoV_(avg)D)/mu`

where `V_(avg)` = average flow velocity `(m//s)`, `D =` characteristic length of the geometry (diameter in this case, in `m`), and `nu = mu/rho =` kinematic viscosity of the fluid `(m^2//s)`.

The Entrance Length or Hydrodynamic Length:

Consider a fluid entering a circular pipe at a uniform velocity. Because of the no-slip condition, the fluid particles in the layer in contact with the wall of the pipe come to a complete stop. This layer also causes the fluid particles in the adjacent layers to slow down gradually as a result of friction. To make up for this velocity reduction, the velocity of the fluid at the midsection of the pipe has to increase to keep the mass flow rate through the pipe constant. As a result, a velocity gradient develops along the pipe.

The region of the flow in which the effects of the viscous shearing forces caused by fluid viscosity are felt is called the velocity boundary layer or just the boundary layer. The hypothetical boundary surface divides the flow in a pipe into two regions: the boundary layer region, in which the viscous effects and the velocity changes are significant, and the irrotational (core) flow region, in which the frictional effects are negligible and the velocity remains essentially constant in the radial direction.

The thickness of this boundary layer increases in the flow direction until the boundary layer reaches the pipe center and thus fills the entire pipe, as shown in Image 1.2, and the velocity becomes fully developed a little farther downstream. The region from the pipe inlet to the point at which the velocity profile is fully developed is called the hydrodynamic entrance region, and the length of this region is called the hydrodynamic entry length `L_h`. Flow in the entrance region is called hydrodynamically developing flow since this is the region where the velocity profile develops. The region beyond the entrance region in which the velocity profile is fully developed and remains unchanged is called the hydrodynamically fully developed region. The flow is said to be fully developed when the normalized temperature profile remains unchanged as well.

Image 1.2 The development of the velocity boundary layer in a pipe.

Entry Length:

The hydrodynamic entry length(refer the image 1.3) is usually taken to be the distance from the pipe entrance to where the wall shear stress (and thus the friction factor) reaches within about 2 percent of the fully developed value. In laminar flow, the nondimensional hydrodynamic entry length is given approximately as

`L_(h, text(laminar))/D approx 0.05R_e`

Image 1.3 The variation of wall shear stress in the flow direction for flow in a pipe from the entrance region into the fully developed region.

Laminar Flow Through Pipe:

Consider the steady, laminar, incompressible flow of fluid with constant properties in the fully developed region of a straight circular pipe. The momentum equation obtained by applying a momentum balance to a differential volume element, and also the velocity profile is obtained by solving it. Then it is used to obtain a relation for the friction factor. An important aspect of the analysis here is that it is one of the few available for viscous flow.

In fully developed laminar flow, each fluid particle moves at a constant axial velocity along a streamline and the velocity profile u(r) remains unchanged in the flow direction. There is no motion in the radial direction, and thus the velocity component in the direction normal to the pipe axis is everywhere zero. There is no acceleration since the flow is steady and fully developed.

The velocity profile is fully developed laminar flow in a pipe is parabolic with a maximum at the centerline and a minimum (zero) at the pipe wall and is given by the equation as a function of radial distance from the center.

`u(r) = 2V_(avg)(1-r^2/R^2)`

The pressure drop due to viscous effects over the length of the pipe is shown below(refer the image 1.4)

`triangleP = P_1 - P_2 = (8muLV_(avg))/R^2`

Image 1.4 The pressure drop due to viscous effects

PROCEDURE:

For this simulation of Flow-Through axis-symmetric, all the above formulae are used to simulate the flow. Since the model is axis-symmetric only a small wedge is modeled such that this wedge is a symmetric part of the Cylinder(refer to the below image).

Image 2.1 Reduced Model of the Axis Symmetric Model.

In the above-reduced figure of the axis-symmetric model, the block consists of 6 vertices namely 0,1,2,3,4,5 where the face 012 is Inlet, 345 is the outlet, 1254 is the wall of the cylinder and the 0253 and 0341 is the wedge patch or symmetry patch.

About the Solver: icoFoam solver

icoFoam solves the incompressible laminar Navier-Stokes equations using the PISO algorithm. The code is inherently transient, requiring an initial condition (such as zero velocity) and boundary conditions. The icoFOAM code can take mesh non-orthogonality into account with successive non-orthogonality iterations. The number of PISO corrections and non-orthogonality corrections are controlled through user input.

Given Details:

Reynold's Number, `R_e` = 2100.

Assuming the value of Diameter,

Diameter, d = 0.01m.(r = 0.005m)

Fluid = Water at 303K.

Kinamatic Viscosity at 303K = `0.801xx10^-6 m^2//s`

Density of the Water at 303K =` 995kg//m^3`

Dynamic Viscosity at 303K = 0.000797Pa s

Wedge Angle = `3^o`(less than `5^o`)

The theoritical hydrodynamic length

`L_(h,text(laminar)) = 0.05xx2100xx0.01`

`L_(h,text(laminar)) = 1.05m`

Since the Hydrodynamic Length is 1.05m, let the length of the pipe be 1.2m so that the fully developed flow can be seen after 1.05m.

From the Reynolds Number fromula

`2100 = (V xx 1.2)/(0.801xx10^-6`

` V = 0.1682m//s`

Simulation time is the time taken by the fluid to travel the length of the pipe.

Simulation Time = `l/V`sec

`therefore` `t = 7.13`

To see the fully developed flow the simulation is run for more time and hence it is simulated for 10 seconds.

Calculation of Coordinates of the Verticies of the Wedge

[0] = (0,0,0)

[1] = (0,`rcos(alpha/2)`,`rsin(alpha/2)`)

[2] = (0,`rcos(alpha/2)`,`-rsin(alpha/2)`)

[3] = (1.2,0,0)

[4] = (1.2,`rcos(alpha/2)`,`rsin(alpha/2)`)

[5] = (1.2,`rcos(alpha/2)`,`-rsin(alpha/2)`)

The coordinates are calculated from the principles of trigonometry.

1. Simulation with wedge angle as 10 degree

For this simulation, the octave code is used to create the blockMeshDict file of a wedge model with a wedge angle of 10 degrees. The same file is copied to the system folder and blockMesh code is executed. After that code, the simulation is ran using the icoFoam code.

Image 2.1.1 10 degrees wedge.

A. Boundary condition as 'wedge'

At first, the boundary condition for the wedge faces is kept as 'wedge'. The simulation ran without any errors and the result is as shown below.

Using the plot over line tool Velocity Profile is plotted. The below image shows the velocity profile of the wedge model with 'wedge' patch.

Image 2.1.2 Velocity Profile at a distance of 1.1m from the inlet.

Now the data is exported to excel and using the 'Export Scene' tool. In Exel using the velocity profile formula, the analytical solution is calculated in the excel file and is plotted in the excel. The below image shows the plot of the velocity profile obtained from simulation and from the analytical formula. The maximum velocity was found to be 0.336m/s from the simulation and 0.3299m/s from the analytical formula.

Image 2.1.3 Comparison of Velocity Profile.

Velocity: The maximum velocity was found to be 0.329m/s from the simulation and 0.336m/s from the Analytical formula.

Pressure Difference:

The pressure difference for a fully developed flow i.e at distance from 1.1m to 1.2m was found to be 4.364 Pa from the simulation and 4.2897 from the analytical solution i.e from Hagen- Poiseuille's Equation.

`trianglep = (8muLV_(avg))/R^2`

`trianglep =( 8xx0.000797xx0.1xx0.1682)/0.005^2`

`trianglep = 4.2897Pa`

Shear Force: The maximum shear force was found to be 0.103N.

B. Boundary condition as 'symmetry'.

Using the plot over line tool Velocity Profile is plotted. The below image shows the velocity profile of the wedge model with the 'wedge' patch.

Image 2.1.4 Velocity Profile at a distance of 1.1m from the inlet.

Image 2.1.5 Comparison of Velocity Profile.

Velocity: The maximum velocity was found to be 0.330m/s from the simulation and 0.336m/s from the Analytical formula.

Pressure Difference: The maximum pressure difference when the flow is fully developed after the entrance length for a distance of 0.1 m was found to be 4.391Pa.

Shear Force: The maximum shear force was found to be 0.103N.

2. Simulation with wedge angle as 25 degree

For this simulation, the octave code is used to create the blockMeshDict file of a wedge model with a wedge angle of 25 degrees. The same file is copied to the system folder and blockMesh code is executed. After that code, the simulation is ran using the icoFoam code.

Image 2.2.1 45 degrees Wedge

A. Boundary condition as 'wedge'

At first, the boundary condition for the wedge faces is kept as 'wedge'. The simulation ran without any errors and the result is as shown below.
Using the plot over line tool Velocity Profile is plotted. The below image shows the velocity profile of the wedge model with the 'wedge' patch.

Image 2.2.2 Velocity Profile at a distance of 1.1m from the inlet.

Image 2.2.3 Comparison of Velocity Profile.

Velocity: The maximum velocity was found to be 0.330m/s from the simulation and 0.336m/s from the Analytical formula.

Pressure Difference: The maximum pressure difference when the flow is fully developed after the entrance length for a distance of 0.1 m was found to be 4.55Pa.

Shear Force: The maximum shear force was found to be 0.105N.

B. Boundary condition as 'symmetry'.

For this simulation, the patch for the wedge face is changed from 'wedge' to 'symmetry'. After changing the patch it is run in Ubuntu. The simulation ran without error and the model is as shown below.
Using the plot over line tool Velocity Profile is plotted. The below image shows the velocity profile of the wedge model with the 'wedge' patch.

Image 2.2.4 Velocity Profile at a distance of 1.1m from the inlet.

Now the data is exported to excel and using the 'Export Scene' tool. In Excel using the velocity profile formula, the analytical solution is calculated in the excel file and is plotted in excel. The below image shows the plot of the velocity profile obtained from simulation and from the analytical formula.

Image 2.2.5 Comparison of Velocity Profile.

Velocity: The maximum velocity was found to be 0.330m/s from the simulation and 0.336m/s from the Analytical formula.

Pressure Difference: The maximum pressure difference when the flow is fully developed after the entrance length for a distance of 0.1 m was found to be 4.391Pa.

Shear Force: The maximum shear force was found to be 0.103N.

3. Simulation with wedge angle as 45 degree

For this simulation, the octave code is used to create the blockMeshDict file of a wedge model with a wedge angle of 45 degrees. The same file is copied to the system folder and blockMesh code is executed. After that code, the simulation is ran using the icoFoam code.

Image 2.3.1 45 degrees Wedge

A. Boundary condition as 'wedge'

Image 2.3.2 Velocity Profile at a distance of 1.1m from the inlet.

Image 2.3.3 Comparison of Velocity Profile.

Velocity: The maximum velocity was found to be 0.331m/s from the simulation and 0.336m/s from the Analytical formula.

Pressure Difference: The maximum pressure difference when the flow is fully developed after the entrance length for a distance of 0.1 m was found to be 5.064Pa.

Shear Force: The maximum shear force was found to be 0.110N.

B. Boundary condition as 'symmetry'.

Image 2.3.4 Velocity Profile at a distance of 1.1m from the inlet.

Image 2.3.5 Comparison of Velocity Profile.

Velocity: The maximum velocity was found to be 0.331m/s from the simulation and 0.336m/s from the Analytical formula.

Pressure Difference: The maximum pressure difference when the flow is fully developed after the entrance length for a distance of 0.1 m was found to be 5.064Pa.

Shear Force: The maximum shear force was found to be 0.110N.

RESULTS:

The Below table shows the results from all the simulations done i.e for different wedge angles and different boundary conditions.

Wedge Angle	Boundary Condition	Analytical Maximum Velocity	Maximum Velocity	Pressure Difference Analytical(Pa)	Pressure Difference (Pa)	Maximum Shear Stress(N/m^2)
10 Degree	wedge	0.336 m/s	0.3299 m/s	4.2897	4.391	0.103
10 Degree	symmetry	0.336 m/s	0.3300 m/s	4.2897	4.391	0.103
25 Degree	wedge	0.336 m/s	0.3300m/s	4.2897	4.55	0.105
25 Degree	symmetry	0.336 m/s	0.3300 m/s	4.2897	4.55	0.105
45 Degree	wedge	0.336 m/s	0.3310 m/s	4.2897	5.064	0.110
45 Degree	symmetry	0.336 m/s	0.3310 m/s	4.2897	5.064	0.110

From the results, it is seen that in both the cases i.e for different boundary conditions, the results were almost the same. But as the Wedge angle increases the accuracy of the simulation results decreases as the distance from the axis to the wall becomes lesser than the radius of the pipe.

CONCLUSION:

The axis-symmetric flow for different wedge angles using both symmetry and wedge boundary conditions is simulated and results are validated with given analytical equations. It is seen that in both the cases i.e for different boundary conditions, the results were almost the same. But as the Wedge angle increases the accuracy of the simulation results decreases as the distance from the axis to the wall becomes lesser than the radius of the pipe.

The 'wedge' type boundary condition is normally used for the axis-symmetric models whereas the 'symmetry' boundary condition is used for the planar symmetric models. Since the given pipe model was very simple in design the results were matching. The 'wedge' patch is more likely to be used in models like flow through Pipes and symmetry boundary condition is used in ducts, or flow-through rectangular cross-section tubes, etc.

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

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Ship work like thisin your own course.

Talk to a career counsellor

Ship work like this
in your own course.