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 HP equation
Abstract: The simulation of flow through a pipe in OpenFOAM is the case of the internal flow. The flow is governed by the Hagen-Poiseuille equation.
The assumptions in the Hagen-Poiseuille equation are as under
- The flow is steady, incompressible, laminar.
- Reynolds number for laminar flow is `2100`
- There is no acceleration of the fluid since it is steady.
- There is no velocity in the radial direction.
- Only the wedge shape of the pipe is considered because the flow is axis-symmetric.
Hagen-Poiseuille equation
The fluid enters into the inlet section of the pipe. The velocity of the fluid molecules near the wall is zero because of the no-slip condition. As such the entire responsibility hangs on the shoulder of the fluid molecules present in the center of the pipe to produce a velocity profile as shown in the figure. The region where the velocity is fully developed is the hydrodynamically fully developed region, the velocity gradient and shear stress at the wall remain unchanged.
- Hydrodynamically fully developed region=`frac{delu(r,x)}{delx}=0` `u=u(r)`
- Hydrodynamically entry length `frac{L}{D}=Re**0.06` `2.52m`
- Dynamic viscosity of water `mu=0.00089` `Pa-sec`
- Density of water `rho=997frac{kg}{m^3}`
- Kinematic viscosity of water `nu=frac{mu}{rho}` `= 8.9268e-07frac{m^2}{sec}`
- Average velocity=`frac{mu**Re}{rho**D}` `0.0937frac{m}{s}`
- The velocity profile `u(r)=2**v_(avg)(1-frac{r^2}{R^2})`
- `u_(max)=2**v_(avg)` `0.1875frac{m}{s}`
- Pressure difference `trianglep=frac{32*mu*v_(avg)**L}{D^2}` `0.0169kPa`
- Shear stress `tau=frac{2**mu**u_(max)**r}{R^2}`
The simulation is being performed for different wedge angles `10^@,25^@,45^@`, and the difference between symmetry, wedge boundary condition, and analytical Hagen-Poiseuille equation is studied.
Plotting wall shear stress for all the profiles.
- The wall shear stress is maximum at the wall but is minimum at the center of pipe.
- In Newtonian fluid dynamic viscosity is constant ,so `taupropfrac{du}{dr}`
Applying wedge boundary condition
Please note that p, U, controlDict, Transportproperties remains the same for wedge angle `25^@,45^@` respectively.
BlockMeshDict file for wedge angle ten degrees.
/*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox\n
\\ / O peration | Website: https://openfoam.org\n
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
object blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 1;
vertices
(
(0.000000 0.000000 0.000000)
(0.000000 0.009962 -0.000872)
(0.000000 0.009962 0.000872)
(2.520000 0.000000 0.000000)
(2.520000 0.009962 -0.000872)
(2.520000 0.009962 0.000872)
);
blocks
(
hex (0 1 2 0 3 4 5 3) (50 1 200) simpleGrading (0.4 1 1)
);
edges
(
arc 1 2 (0.000000 0.010000 0.000000)
arc 4 5 (2.520000 0.010000 0.000000)
);
boundary
(
axis
{
type empty;
faces
(
(0 3 3 0)
);
}
inlet
{
type patch;
faces
(
(0 1 2 0)
);
}
pipe_surface
{
type wall;
faces
(
(1 4 5 2)
);
}
outlet
{
type patch
faces
(
(3 4 5 3)
);
}
front
{
type wedge
faces
(
(0 3 5 2)
);
}
back
{
type wedge
faces
(
(0 1 4 3)
);
}
);
mergePatchPairs
(
);
// ************************************************************************* //Pressure
/*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\ / O peration | Website: https://openfoam.org
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class volScalarField;
object p;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
dimensions [0 2 -2 0 0 0 0];
internalField uniform 0;
boundaryField
{
axis
{
type empty;
}
inlet
{
type zeroGradient;
}
pipe_surface
{
type zeroGradient;
}
outlet
{
type fixedValue;
value uniform 0.0169;
}
front
{
type wedge;
}
back
{
type wedge;
}
}
// ************************************************************************* //Velocity
*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\ / O peration | Website: https://openfoam.org
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class volVectorField;
object U;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
dimensions [0 1 -1 0 0 0 0];
internalField uniform (0.0937 0 0);
boundaryField
{
axis
{
type empty;
}
inlet
{
type fixedValue;
value uniform (0.0937 0 0);
}
pipe_surface
{
type fixedValue;
value uniform (0 0 0);
}
outlet
{
type zeroGradient;
}
front
{
type wedge;
}
back
{
type wedge;
}
}
// ************************************************************************* //
Transport properties
/*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\ / O peration | Website: https://openfoam.org
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
location "constant";
object transportProperties;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
nu [0 2 -1 0 0 0 0] 8.9268e-07;
// ************************************************************************* //Control Dict file
*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\ / O peration | Website: https://openfoam.org
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
location "system";
object controlDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
application icoFoam;
startFrom startTime;
startTime 0;
stopAt endTime;
endTime 10;
deltaT 0.01;
writeControl timeStep;
writeInterval 10;
purgeWrite 0;
writeFormat ascii;
writePrecision 6;
writeCompression off;
timeFormat general;
timePrecision 6;
runTimeModifiable true;
Geometry profile for wedge angle `10^@`
Velocity profile at the inlet of the pipe.
Velocity profile for the fully developed flow
Pressure profile of the wedge
Velocity and pressure profile at a distance of `0.01m` from the inlet of a pipe.
Velocity and pressure profile at a distance of `0.5m` from the inlet of a pipe.
Velocity and pressure profile at a distance of `1m` from the inlet of a pipe.
Velocity and pressure profile at a distance of `2m` from the inlet of a pipe.
Shear stress plots
BlockMeshDict file for wedge angle twenty-five degrees.
/*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox\n
\\ / O peration | Website: https://openfoam.org\n
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
object blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 1;
vertices
(
(0.000000 0.000000 0.000000)
(0.000000 0.009763 -0.002164)
(0.000000 0.009763 0.002164)
(2.520000 0.000000 0.000000)
(2.520000 0.009763 -0.002164)
(2.520000 0.009763 0.002164)
);
blocks
(
hex (0 1 2 0 3 4 5 3) (50 1 200) simpleGrading (0.4 1 1)
);
edges
(
arc 1 2 (0.000000 0.010000 0.000000)
arc 4 5 (2.520000 0.010000 0.000000)
);
boundary
(
axis
{
type empty;
faces
(
(0 3 3 0)
);
}
inlet
{
type patch;
faces
(
(0 1 2 0)
);
}
pipe_surface
{
type wall;
faces
(
(1 4 5 2)
);
}
outlet
{
type patch
faces
(
(3 4 5 3)
);
}
front
{
type wedge
faces
(
(0 3 5 2)
);
}
back
{
type wedge
faces
(
(0 1 4 3)
);
}
);
mergePatchPairs
(
);
// ************************************************************************* //Geometry for wedge angle `25^@`
Velocity profile at the inlet of a pipe.
Velocity profile for a fully developed flow
Velocity and pressure profile at `0.01m`from the inlet of the pipe.
Velocity and pressure profile `0.5m` from the inlet of the pipe.
Velocity and pressure profile at `1m` from the inlet of the pipe
Velocity and pressure profile at `2m` from the inlet of the pipe`
Shear stress plots
BlockMeshDict file for wedge angle forty-five degrees.
/*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox\n
\\ / O peration | Website: https://openfoam.org\n
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
object blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 1;
vertices
(
(0.000000 0.000000 0.000000)
(0.000000 0.009239 -0.003827)
(0.000000 0.009239 0.003827)
(2.520000 0.000000 0.000000)
(2.520000 0.009239 -0.003827)
(2.520000 0.009239 0.003827)
);
blocks
(
hex (0 1 2 0 3 4 5 3) (50 1 200) simpleGrading (0.4 1 1)
);
edges
(
arc 1 2 (0.000000 0.010000 0.000000)
arc 4 5 (2.520000 0.010000 0.000000)
);
boundary
(
axis
{
type empty;
faces
(
(0 3 3 0)
);
}
inlet
{
type patch;
faces
(
(0 1 2 0)
);
}
pipe_surface
{
type wall;
faces
(
(1 4 5 2)
);
}
outlet
{
type patch
faces
(
(3 4 5 3)
);
}
front
{
type wedge
faces
(
(0 3 5 2)
);
}
back
{
type wedge
faces
(
(0 1 4 3)
);
}
);
mergePatchPairs
(
);
// ************************************************************************* //Geometry for wedge angle `45^@`
Velocity profile at the inlet of the pipe.
Velocity profile for the fully developed flow
Velocity and pressure profile at `0.01m` from the inlet of the pipe.
Velocity and pressure profile at `0.5m`from the inlet of the pipe.
Velocity and pressure profile at `1m`from the inlet
Velocity and pressure profile at `2m` from the inlet of the pipe.
Shear stress plots.
Applying symmetry boundary condition
- The p, U, controlDict, transportProperties remains the same as above for wedge angle `25^@,45^@` respectively.
- Please note that the geometry of the profile remains the same, therefore the behavior of parameters is shown below.
BlockMeshDict file for wedge angle ten degrees.
/*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox\n
\\ / O peration | Website: https://openfoam.org\n
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
object blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 1;
vertices
(
(0.000000 0.000000 0.000000)
(0.000000 0.009962 -0.000872)
(0.000000 0.009962 0.000872)
(2.520000 0.000000 0.000000)
(2.520000 0.009962 -0.000872)
(2.520000 0.009962 0.000872)
);
blocks
(
hex (0 1 2 0 3 4 5 3) (50 1 200) simpleGrading (0.4 1 1)
);
edges
(
arc 1 2 (0.000000 0.010000 0.000000)
arc 4 5 (2.520000 0.010000 0.000000)
);
boundary
(
axis
{
type empty;
faces
(
(0 3 3 0)
);
}
inlet
{
type patch;
faces
(
(0 1 2 0)
);
}
pipe_surface
{
type wall;
faces
(
(1 4 5 2)
);
}
outlet
{
type patch
faces
(
(3 4 5 3)
);
}
front
{
type symmetry
faces
(
(0 3 5 2)
);
}
back
{
type symmetry
faces
(
(0 1 4 3)
);
}
);
mergePatchPairs
(
);
// ************************************************************************* //Pressure
/*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\ / O peration | Website: https://openfoam.org
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class volScalarField;
object p;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
dimensions [0 2 -2 0 0 0 0];
internalField uniform 0;
boundaryField
{
axis
{
type empty;
}
inlet
{
type zeroGradient;
}
pipe_surface
{
type zeroGradient;
}
outlet
{
type fixedValue;
value uniform 0.0169;
}
front
{
type symmetry;
}
back
{
type symmetry;
}
}
// ************************************************************************* //Velocity profile
/*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\ / O peration | Website: https://openfoam.org
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class volVectorField;
object U;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
dimensions [0 1 -1 0 0 0 0];
internalField uniform (0.0937 0 0);
boundaryField
{
axis
{
type empty;
}
inlet
{
type fixedValue;
value uniform (0.0937 0 0);
}
pipe_surface
{
type fixedValue;
value uniform (0 0 0);
}
outlet
{
type zeroGradient;
}
front
{
type symmetry;
}
back
{
type symmetry;
}
}
// ************************************************************************* //
ControlDict
*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\ / O peration | Website: https://openfoam.org
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
location "system";
object controlDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
application icoFoam;
startFrom startTime;
startTime 0;
stopAt endTime;
endTime 10;
deltaT 0.01;
writeControl timeStep;
writeInterval 10;
purgeWrite 0;
writeFormat ascii;
writePrecision 6;
writeCompression off;
timeFormat general;
timePrecision 6;
runTimeModifiable true;
Transportproperties
/*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\ / O peration | Website: https://openfoam.org
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
location "constant";
object transportProperties;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
nu [0 2 -1 0 0 0 0] 8.9268e-07;
// ************************************************************************* //Velocity and pressure profile at `1m` from the inlet of the pipe.
Velocity and pressure profile at `2m` from the inlet of the pipe.
Shear stress plots
BlockMeshDict file for wedge angle twenty-five degrees.
/*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox\n
\\ / O peration | Website: https://openfoam.org\n
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
object blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 1;
vertices
(
(0.000000 0.000000 0.000000)
(0.000000 0.009763 -0.002164)
(0.000000 0.009763 0.002164)
(2.520000 0.000000 0.000000)
(2.520000 0.009763 -0.002164)
(2.520000 0.009763 0.002164)
);
blocks
(
hex (0 1 2 0 3 4 5 3) (50 1 200) simpleGrading (0.4 1 1)
);
edges
(
arc 1 2 (0.000000 0.010000 0.000000)
arc 4 5 (2.520000 0.010000 0.000000)
);
boundary
(
axis
{
type empty;
faces
(
(0 3 3 0)
);
}
inlet
{
type patch;
faces
(
(0 1 2 0)
);
}
pipe_surface
{
type wall;
faces
(
(1 4 5 2)
);
}
outlet
{
type patch
faces
(
(3 4 5 3)
);
}
front
{
type symmetry
faces
(
(0 3 5 2)
);
}
back
{
type symmetry
faces
(
(0 1 4 3)
);
}
);
mergePatchPairs
(
);
// ************************************************************************* //Velocity and pressure profile at `1m` from the inlet of the pipe
Velocity and pressure profile at `2m` from the inlet of the pipe.
Shear stress plots
BlockMeshDict file for wedge angle forty-five degrees.
/*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox\n
\\ / O peration | Website: https://openfoam.org\n
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
object blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 1;
vertices
(
(0.000000 0.000000 0.000000)
(0.000000 0.009239 -0.003827)
(0.000000 0.009239 0.003827)
(2.520000 0.000000 0.000000)
(2.520000 0.009239 -0.003827)
(2.520000 0.009239 0.003827)
);
blocks
(
hex (0 1 2 0 3 4 5 3) (50 1 200) simpleGrading (0.4 1 1)
);
edges
(
arc 1 2 (0.000000 0.010000 0.000000)
arc 4 5 (2.520000 0.010000 0.000000)
);
boundary
(
axis
{
type empty;
faces
(
(0 3 3 0)
);
}
inlet
{
type patch;
faces
(
(0 1 2 0)
);
}
pipe_surface
{
type wall;
faces
(
(1 4 5 2)
);
}
outlet
{
type patch
faces
(
(3 4 5 3)
);
}
front
{
type symmetry
faces
(
(0 3 5 2)
);
}
back
{
type symmetry
faces
(
(0 1 4 3)
);
}
);
mergePatchPairs
(
);
// ************************************************************************* //Velocity and pressure profile at `1m` from the inlet of the pipe.
Velocity and pressure profile at `2m` from the inlet of the pipe.
Shear stress plots.
Conclusion
The OpenFOAM results are tabulated as under
| Output |
Hagen-Poiseuille equation |
Wedge angle `10^@` BC: Wedge |
Wedge angle `10^@` BC: Symmetry |
Wedge angle `25^@` BC: Wedge |
Wedge angle `25^@` BC: Symmetry |
Wedge angle `45^@` BC: Wedge |
Wedge angle `45^@` BC: Symmetry |
| Mean Courant number | `-` | `0.02872` | `0.0769289` | `0.0769407` | `0.07694` | `0.0769` | `0.0769692` |
| Max Courant number | `-` | `0.37947` | `0.305395` | `0.308834` | `0.3088` | `0.3181` | `0.318186` |
| Execution time | `-` | `117.74s` | `122.7s` | `116.36s` | `120.76s` | `119.69s` | `126.68s` |
| Clock time | `-` | `145s` | `148s` | `142s` | `148s` | `147s` | `153s` |
| `u_(max)` | `0.1875` | `0.17952` | `0.17952` | `0.17952` | `0.17952` | `0.17952` | `0.17952` |
| Kinematic pressure | `0.0169` | `0.0169` | `0.0169` | `0.0169` | `0.0169` | `0.0169` | `0.0169` |
| Wall shear stress | `0.034` | `0.034` | `0.034` | `0.034` | `0.034` | `0.034` | `0.034` |
From the above data, it can be concluded that simulation performed for different wedge angles produces similar analytical results.
The deviation from the Hagen-Poiseuille equation is negligible.
The execution time for symmetry boundary condition is more as compared to wedge boundary condition.