Week 12 - Symmetry vs Wedge vs HP equation
Symmetry and Wedge boundary conditions in HP equation
All units are in SI system unless specified.
| Working fluid | Water |
| Wedge angle | 4 degrees |
| Reynolds number | 2100 |
| Diameter D | 0.02 |
| Radius R | 0.01 |
| Dynamic viscosity of water at 25-degree Celcius - mu | 0.0008900 |
| Water density at 25 deg. C - rho | 997 |
| Temperature of water | 25-degree Celcius |
To solve this task help from Spoken tutorial is used.
The Hagen-Poiseuille equation describes the parabolic velocity profile of frictional, laminar pipe flows of incompressible, Newtonian fluids.
when a fluid is moving through a pipe velocity will be varying in the radial direction in a particular manner. this is because of the presence of fluid boundary friction, which may be otherwise stated as a no-slip condition. We know that as the fluid moves downstream boundary layer gets thickened and finally, at a particular location, it fills the entire pipe. Downstream of this point velocity profile will be parabolic. The entrance length can be defined as the distance between the above-mentioned point and the inlet of the pipe.
In the above figure, it's clear that the flow is developing in the Entrance region and after the entrance region is fully developed and thereafter there are no changes in the velocity profile. Viscous forces are responsible for the velocity profiles.
Entrance Length of Laminar flow pipe is calculated by the equation `((Le)/D) = (0.06*Re)`
Theoretically, we can find the following values
| Entrance length | `L_e = 0.06*Re*D` | 1.8 m |
| Total length `L_t` | 2.1 m | |
| Mean Velocity `V_(mean)` | `(Re*mu)/(rho*D)` | 0.066951 m/s |
| Maximum Velocity `V_(max)` | `V_(max) =2* V_(mean)` | 0.13390 m/s |
| Pressure Drop - `DeltaP` | `DeltaP= ( 32*mu*v_(mean)*(L_t - L_e) ) / (D^2)` | 1.42919 Pa |
| Kinematic pressure `P_("Kinematic")` | `P_("Kinematic") = (DeltaP)/rho` | 1.4335e-03 `m^2/s^2` |
A MATLAB Code is generated which will receive the wedge angle and grading parameters from the user and give a blockMesh Dict for the same as output which can be used for simulation in OpenFoam.
The MATLAB Programme is given below.
MATLAB Program to generate blockMeshDict File
%Matlab program to study flow through pipe
clear all
close all
clc
%% Input %%
Re= 1500; % Reynold's Number
D= 0.02; % Diameter of pipe
R=D/2; % Radius
Boundarytype = cstrcat("type "," ", input("Which type of boundary ? symmetry or wedge ? : ",'s'), " ;");
theta= input('Enter Wedge angle ='); % Wedge angle
mu=0.89e-3; %Dynamic viscocity @ 25 degree Celsius
rho=997; %Density of water @ 25 degree Celsius
nu= mu/rho; %Kinematic viscocity @ 25 degree Celsius
% Analytical Velocity, pressure
L_e= 0.06*Re*D; % Entrance length
L_t= L_e + 0.3; % Total length
v_avg= (Re*mu)/(rho*D); % Average velocity
v_max= 2*v_avg; % Maximum velocity
Delta_P= (32*mu*v_avg*(L_t-L_e))/(D^2); % Pressure Drop
kin_P= Delta_P/rho; % Kinematic Pressure
% Finding vertices
v0= [0 0 0];
v1= [0 R*cosd(theta/2) -R*sind(theta/2)];
v2= [0 R*cosd(theta/2) R*sind(theta/2)];
v3= [L_t 0 0];
v4= [L_t R*cosd(theta/2) -R*sind(theta/2)];
v5= [L_t R*cosd(theta/2) R*sind(theta/2)];
%blockMeshDict file.
f1= fopen('blockMeshDict.txt','w'); % create blockmesh file
fprintf(f1,'%s \n','/*--------------------------------*- C++ -*----------------------------------*\');
fprintf(f1,'%s \n',' ========= |');
fprintf(f1,'%s \n',' \\ / F ield | OpenFOAM: The Open Source CFD Toolbox');
fprintf(f1,'%s \n',' \\ / O peration | Website: https://openfoam.org');
fprintf(f1,'%s \n',' \\ / A nd | Version: 8');
fprintf(f1,'%s \n',' \\/ M anipulation |');
fprintf(f1,'%s \n','\*---------------------------------------------------------------------------*/');
fprintf(f1,'%s \n','FoamFile');
fprintf(f1,'%s \n','{');
fprintf(f1,'%s \n',' version 2.0;');
fprintf(f1,'%s \n',' format ascii;');
fprintf(f1,'%s \n',' class dictionary;');
fprintf(f1,'%s \n',' object blockMeshDict;');
fprintf(f1,'%s \n','}');
fprintf(f1,'%s \n','// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //');
fprintf(f1,'%s \n','convertToMeters 1;');
b1=blanks(5);
b2=blanks(6);
b3=blanks(7);
b4=blanks(4);
% Finding vertices
fprintf(f1,'vertices \n');
fprintf(f1,'(\n');
fprintf(f1,b4,'%s \n','(');
v_0=fprintf(f1,'(%f %f %f) \n',v0(1),v0(2),v0(3));
fprintf(f1,b4,'%s','(');
v_1=fprintf(f1,'(%f %f %f)\n',v1(1),v1(2),v1(3));
fprintf(f1,b4,'%s','(');
v_2=fprintf(f1,'(%f %f %f)\n',v2(1),v2(2),v2(3));
fprintf(f1,b4,'%s','(');
v_3=fprintf(f1,'(%f %f %f)\n',v3(1),v3(2),v3(3));
fprintf(f1,b4,'%s','(');
v_4=fprintf(f1,'(%f %f %f)\n',v4(1),v4(2),v4(3));
fprintf(f1,b4,'%s','(');
v_5=fprintf(f1,'(%f %f %f)\n',v5(1),v5(2),v5(3));
fprintf(f1,'\n);\n');
% blocks creation
fprintf(f1,'\nblocks\n');
fprintf(f1,'(\n');
fprintf(f1,b4,'%s','(');
fprintf(f1,'%s','hex (0 1 2 0 3 4 5 3)');
% mesh generation
m1= input('Enter no. of mesh element along X-axis:');
m2= input('Enter no. of mesh element along Y-axis:');
m3= input('Enter no. of mesh element along Z-axis:');
fprintf(f1,blanks(1),'%s','(');
fprintf(f1,'(%2.0f %1.0f %3.0f)',m1,m2,m3);
fprintf(f1,'%s',' simpleGrading (1 1 1)');
fprintf(f1,'\n);');
% edges creation
fprintf(f1,'\nedges\n');
fprintf(f1,'(\n');
fprintf(f1,'%s arc 1 2 (0 %f 0)',blanks(1),R);
fprintf(f1,'\n%s arc 4 5 (%f %f 0)',blanks(1),L_t,R);
fprintf(f1,'\n);\n\n');
% Apply boundary conditions
% axis
fprintf(f1,'boundary\n(\n');
fprintf(f1,'%s',' axis');
fprintf(f1,'\n {\n');
fprintf(f1,'%s',blanks(10),'type empty;');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(10),'faces');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(10),'(');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(13),'(0 3 3 0)');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(10),');');
fprintf(f1,'\n }\n');
% fixedwalls
fprintf(f1,'%s',' fixedWalls');
fprintf(f1,'\n {\n');
fprintf(f1,'%s',blanks(10),'type wall;');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(10),'faces');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(10),'(');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(13),'(2 1 4 5)');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(10),');');
fprintf(f1,'\n }\n');
% front
fprintf(f1,'%s',' front');
fprintf(f1,'\n {\n');
fprintf(f1,'%s',blanks(10),Boundarytype);
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(10),'faces');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(10),'(');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(13),'(1 0 3 4)');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(10),');');
fprintf(f1,'\n }\n');
% back
fprintf(f1,'%s',' back');
fprintf(f1,'\n {\n');
fprintf(f1,'%s',blanks(10),Boundarytype);
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(10),'faces');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(10),'(');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(13),'(2 5 3 0)');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(10),');');
fprintf(f1,'\n }\n');
% inlet
fprintf(f1,'%s',' inlet');
fprintf(f1,'\n {\n');
fprintf(f1,'%s',blanks(10),'type patch;');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(10),'faces');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(10),'(');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(13),'(0 1 2 0)');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(10),');');
fprintf(f1,'\n }\n\n');
% outlet
fprintf(f1,'%s',' outlet');
fprintf(f1,'\n {\n');
fprintf(f1,'%s',blanks(10),'type patch;');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(10),'faces');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(10),'(');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(13),'(3 5 4 3)');
fprintf(f1,'\n');
fprintf(f1,'%s',blanks(10),');');
fprintf(f1,'\n }\n\n');
fprintf(f1,'\n);\n');
fprintf(f1,'\nmergePatchPairs');
fprintf(f1,'\n(\n);\n\n');
fprintf(f1,'%s','// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //');
fclose(f1);blockMeshDict file -Wedge Condition - 4 degrees
/*--------------------------------*- 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;
object blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 1;
vertices
(
(0.000000 0.000000 0.000000)
(0.000000 0.009994 -0.000349)
(0.000000 0.009994 0.000349)
(2.100000 0.000000 0.000000)
(2.100000 0.009994 -0.000349)
(2.100000 0.009994 0.000349)
);
blocks
(
hex (0 1 2 0 3 4 5 3) (50 1 100) simpleGrading (1 1 1)
);
edges
(
arc 1 2 (0 0.010000 0)
arc 4 5 (2.100000 0.010000 0)
);
boundary
(
axis
{
type empty;
faces
(
(0 3 3 0)
);
}
fixedWalls
{
type wall;
faces
(
(2 1 4 5)
);
}
front
{
type wedge ;
faces
(
(1 0 3 4)
);
}
back
{
type wedge ;
faces
(
(2 5 3 0)
);
}
inlet
{
type patch;
faces
(
(0 1 2 0)
);
}
outlet
{
type patch;
faces
(
(3 5 4 3)
);
}
);
mergePatchPairs
(
);
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //blockMeshDict file -Symmetry Condition - 45 degrees
/*--------------------------------*- 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;
object blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 1;
vertices
(
(0.000000 0.000000 0.000000)
(0.000000 0.009239 -0.003827)
(0.000000 0.009239 0.003827)
(2.100000 0.000000 0.000000)
(2.100000 0.009239 -0.003827)
(2.100000 0.009239 0.003827)
);
blocks
(
hex (0 1 2 0 3 4 5 3) (50 1 100) simpleGrading (1 1 1)
);
edges
(
arc 1 2 (0 0.010000 0)
arc 4 5 (2.100000 0.010000 0)
);
boundary
(
axis
{
type empty;
faces
(
(0 3 3 0)
);
}
fixedWalls
{
type wall;
faces
(
(2 1 4 5)
);
}
front
{
type symmetry;
faces
(
(1 0 3 4)
);
}
back
{
type symmetry;
faces
(
(2 5 3 0)
);
}
inlet
{
type patch;
faces
(
(0 1 2 0)
);
}
outlet
{
type patch;
faces
(
(3 5 4 3)
);
}
);
mergePatchPairs
(
);
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //blockMeshDict file -Symmetry Condition - 25 degrees
/*--------------------------------*- 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;
object blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 1;
vertices
(
(0.000000 0.000000 0.000000)
(0.000000 0.009763 -0.002164)
(0.000000 0.009763 0.002164)
(2.100000 0.000000 0.000000)
(2.100000 0.009763 -0.002164)
(2.100000 0.009763 0.002164)
);
blocks
(
hex (0 1 2 0 3 4 5 3) (50 1 100) simpleGrading (1 1 1)
);
edges
(
arc 1 2 (0 0.010000 0)
arc 4 5 (2.100000 0.010000 0)
);
boundary
(
axis
{
type empty;
faces
(
(0 3 3 0)
);
}
fixedWalls
{
type wall;
faces
(
(2 1 4 5)
);
}
front
{
type symmetry;
faces
(
(1 0 3 4)
);
}
back
{
type symmetry;
faces
(
(2 5 3 0)
);
}
inlet
{
type patch;
faces
(
(0 1 2 0)
);
}
outlet
{
type patch;
faces
(
(3 5 4 3)
);
}
);
mergePatchPairs
(
);
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //blockMeshDict file -Symmetry Condition - 10 degrees
/*--------------------------------*- 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;
object blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 1;
vertices
(
(0.000000 0.000000 0.000000)
(0.000000 0.009962 -0.000872)
(0.000000 0.009962 0.000872)
(2.100000 0.000000 0.000000)
(2.100000 0.009962 -0.000872)
(2.100000 0.009962 0.000872)
);
blocks
(
hex (0 1 2 0 3 4 5 3) (50 1 100) simpleGrading (1 1 1)
);
edges
(
arc 1 2 (0 0.010000 0)
arc 4 5 (2.100000 0.010000 0)
);
boundary
(
axis
{
type empty;
faces
(
(0 3 3 0)
);
}
fixedWalls
{
type wall;
faces
(
(2 1 4 5)
);
}
front
{
type symmetry;
faces
(
(1 0 3 4)
);
}
back
{
type symmetry;
faces
(
(2 5 3 0)
);
}
inlet
{
type patch;
faces
(
(0 1 2 0)
);
}
outlet
{
type patch;
faces
(
(3 5 4 3)
);
}
);
mergePatchPairs
(
);
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //p files for wedge case
/*--------------------------------*- 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;
}
fixedWalls
{
type zeroGradient;
}
front
{
type wedge;
}
back
{
type wedge;
}
inlet
{
type zeroGradient;
}
outlet
{
type fixedValue;
value uniform 1;
}
}
// ************************************************************************* //p files for Symmetry case
/*--------------------------------*- 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;
}
fixedWalls
{
type zeroGradient;
}
front
{
type symmetry;
}
back
{
type symmetry;
}
inlet
{
type zeroGradient;
}
outlet
{
type fixedValue;
value uniform 1;
}
}
// ************************************************************************* //U files for wedge case
/*--------------------------------*- 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 (0.06691 0 0);
boundaryField
{
inlet
{
type fixedValue;
value uniform (0.06691 0 0);
}
fixedWalls
{
type fixedValue;
value uniform (0 0 0);
}
front
{
type wedge;
}
back
{
type wedge;
}
axis
{
type empty;
}
outlet
{
type zeroGradient;
}
}
// ************************************************************************* //U files for Symmetry case
/*--------------------------------*- 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 (0.06691 0 0);
boundaryField
{
inlet
{
type fixedValue;
value uniform (0.06691 0 0);
}
fixedWalls
{
type fixedValue;
value uniform (0 0 0);
}
front
{
type symmetry;
}
back
{
type symmetry;
}
axis
{
type empty;
}
outlet
{
type zeroGradient;
}
}
// ************************************************************************* //transportProperties 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 "constant";
object transportProperties;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
nu [0 2 -1 0 0 0 0] 8.9268e-07;
// ************************************************************************* //RESULTS
Pressure drop
Pressure drop along the fully developed region has benn plotted.
wedge case
Symmetry case -wedge angle = 10 degree
Symmetry case -wedge angle = 25 degree
Symmetry case -wedge angle = 45 degree
Maximum Velocity.
Since flow is fully developed from 1.8 m onwards at outlet velocity profile is plotted and maximum velocity is noted.
Wedge case -wedge angle = 4 degree
Symmetry case -wedge angle = 45 degree
Symmetry case -wedge angle = 25 degree
Symmetry case -wedge angle = 10 degree
Conclussion
| RESULTS | Theoretical | Wedge Boundary condition | Symmetry Boundary Condition - 10 degree wedge angle | Symmetry Boundary Condition - 25 degree wedge angle | Symmetry Boundary Condition - 45 degree wedge angle |
| Pressure drop | 1.4335e-03 | 1.69e-03 | 1.69e-03 | 1.75e-3 | 1.91e-03 |
| Maximum velocity | 0.13390 | 0.109224 | 0.10938 | 0.110393 | 0.113188 |
The accuaracy can be incresed by incresing the mesh density.
All the case files will be attache below.