Symmetry vs Wedge vs HP equation
AIM: To write a program in Matlab that can generate the computational mesh automatically for any symmetry angle and grading schemes
THEORY:
Hydrodynamic Entrance Length
Many different entrance lengths exist to describe a variety of flow conditions.The Entrance length refers to the length of the entry region where effects from the pipe interior wall spread as an expanding boundary layer in to the flow. The Entrance length describes the distance travelled by the flow after entering the pipe before the flow becomes fully developed.
Hydrodynamic entrance length describes the formation of a velocity profile caused by viscous forces propagating from the pipe wall. Awareness of entrance length may be necessary for the effective placement of instrumentation, such as fluid flow meters. . After the flow becomes fully developed the flow characterstics no longer changes with the increased distance along the length of the pipe. Entrance length for the laminar flow is given as below
Entrance length = Le = 0.06 x Re x D
DATA:
Reynold\'s Number(Re) = 2100
Diameter of the pipe(D) = 0.02m
Radius of the pipe(R) - D/2 = 0.01m
Density(ρρ) = 997 kg/m^3
Kinematic Viscocity(νν) = 0.00089 N/m^2
Hagen-Poisselus Equation
Poiseuille equation, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. Poiseuille\'s Equation describes the pressure drop due to the viscosity of the fluid
blockMeshDict(For 10 degree)
/*--------------------------------*- C++ -*----------------------------------*\\
========= |
\\\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\\\ / O peration | Website: https://openfoam.org
\\\\ / A nd | Version: 7
\\\\/ M anipulation |
\\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
object blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 1;
vertices
(
(0 0 0) //1
(0 9.9619e-03 8.7155e-04) //2
(0 9.9619e-03 -8.7155e-04) //3
(2.82 0 0) //4
(2.82 9.9619e-03 8.7155e-04) //5
(2.82 9.9619e-03 -8.7155e-04) //6
);
blocks
(
hex (0 3 5 2 0 3 4 1) (500 25 1) simpleGrading (1 0.2 1)
);
edges
(
arc 1 2 (0 1e-02 0)
arc 4 5 (2.82 1e-02 0)
);
boundary
(
inlet
{
type patch;
faces
(
(0 1 2 0)
);
}
outlet
{
type patch;
faces
(
(3 5 4 3)
);
}
top
{
type wall;
faces
(
(1 4 5 2)
);
}
front
{
type symmetry;
faces
(
(0 3 4 1)
);
}
back
{
type symmetry;
faces
(
(0 2 5 3)
);
}
axis
{
type empty;
faces
(
(0 3 3 0)
);
}
);
mergePatchPairs
(
);
//*************************************************************************//blockMeshDict(For 25 degree)
/*--------------------------------*- C++ -*----------------------------------*\\
========= |
\\\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\\\ / O peration | Website: https://openfoam.org
\\\\ / A nd | Version: 7
\\\\/ M anipulation |
\\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
object blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 1;
vertices
(
(0 0 0) //1
(0 9.7629e-03 2.1643e-03) //2
(0 9.7629e-03 -2.1643e-03) //3
(2.82 0 0) //4
(2.82 9.7629e-03 2.1643e-03) //5
(2.82 9.7629e-03 -2.1643e-03) //6
);
blocks
(
hex (0 3 5 2 0 3 4 1) (500 25 1) simpleGrading (1 0.2 1)
);
edges
(
arc 1 2 (0 1e-02 0)
arc 4 5 (2.82 1e-02 0)
);
boundary
(
inlet
{
type patch;
faces
(
(0 1 2 0)
);
}
outlet
{
type patch;
faces
(
(3 5 4 3)
);
}
top
{
type wall;
faces
(
(1 4 5 2)
);
}
front
{
type symmetry;
faces
(
(0 3 4 1)
);
}
back
{
type symmetry;
faces
(
(0 2 5 3)
);
}
axis
{
type empty;
faces
(
(0 3 3 0)
);
}
);
mergePatchPairs
(
);
//*************************************************************************//blockMeshDict (For 45 degrees)
/*--------------------------------*- C++ -*----------------------------------*\\
========= |
\\\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\\\ / O peration | Website: https://openfoam.org
\\\\ / A nd | Version: 7
\\\\/ M anipulation |
\\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
object blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 1;
vertices
(
(0 0 0) //1
(0 9.2387e-03 3.8268e-03) //2
(0 9.2387e-03 -3.8268e-03) //3
(2.82 0 0) //4
(2.82 9.2387e-03 3.8268e-03) //5
(2.82 9.2387e-03 -3.8268e-03) //6
);
blocks
(
hex (0 3 5 2 0 3 4 1) (500 25 1) simpleGrading (1 0.2 1)
);
edges
(
arc 1 2 (0 1e-02 0)
arc 4 5 (2.82 1e-02 0)
);
boundary
(
inlet
{
type patch;
faces
(
(0 1 2 0)
);
}
outlet
{
type patch;
faces
(
(3 5 4 3)
);
}
top
{
type wall;
faces
(
(1 4 5 2)
);
}
front
{
type symmetry;
faces
(
(0 3 4 1)
);
}
back
{
type symmetry;
faces
(
(0 2 5 3)
);
}
axis
{
type empty;
faces
(
(0 3 3 0)
);
}
);
mergePatchPairs
(
);
//*************************************************************************//ControlDict
/*--------------------------------*- C++ -*----------------------------------*\\
========= |
\\\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\\\ / O peration | Website: https://openfoam.org
\\\\ / A nd | Version: 7
\\\\/ M anipulation |
\\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
location \"system\";
object controlDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
application icoFoam;
startFrom startTime;
startTime 0;
stopAt endTime;
endTime 30;
deltaT 1e-2;
writeControl timeStep;
writeInterval 20;
purgeWrite 0;
writeFormat ascii;
writePrecision 6;
writeCompression off;
timeFormat general;
timePrecision 6;
runTimeModifiable true;
// ************************************************************************* //Pressure
/*--------------------------------*- C++ -*----------------------------------*\\
========= |
\\\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\\\ / O peration | Website: https://openfoam.org
\\\\ / A nd | Version: 7
\\\\/ M anipulation |
\\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class volScalarField;
object p;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
dimensions [0 2 -2 0 0 0 0];
internalField uniform 0;
boundaryField
{
inlet
{
type zeroGradient;
}
outlet
{
type fixedValue;
value uniform 0;
}
axis
{
type empty;
}
top
{
type zeroGradient;
}
front
{
type symmetry;
}
back
{
type symmetry;
}
}
// ************************************************************************* //Velocity
/*--------------------------------*- C++ -*----------------------------------*\\
========= |
\\\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\\\ / O peration | Website: https://openfoam.org
\\\\ / A nd | Version: 7
\\\\/ M anipulation |
\\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class volVectorField;
object U;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
dimensions [0 1 -1 0 0 0 0];
internalField uniform (0.09373 0 0);
boundaryField
{
inlet
{
type fixedValue;
value uniform (0.09373 0 0);
}
outlet
{
type zeroGradient;
}
axis
{
type empty;
}
top
{
type noSlip;
}
front
{
type symmetry;
}
back
{
type symmetry;
}
}
// ************************************************************************* //
Transport Properties
/*--------------------------------*- C++ -*----------------------------------*\\
========= |
\\\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\\\ / O peration | Website: https://openfoam.org
\\\\ / A nd | Version: 7
\\\\/ M anipulation |
\\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
location \"constant\";
object transportProperties;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
nu [0 2 -1 0 0 0 0] 8.9256e-07;
// ************************************************************************* //Matlab Code
clear all
close all
clc
% Reynolds Number
Re = 2100
% Diameter of Pipe
D = 0.02
% Entry length of Pipe
L = 0.06*Re*D
% Total length of pipe
L_p = L + 0.3
% Radius of pipe
r = D/2
% Angle of wedge
theta = 10
%theta = 25
%theta = 45
% Dynamic Viscocity
mu = 0.89e-3
% Density
rho = 997
% Hagen Poissuel flow equation
v_avg = (Re*mu)/(rho*D)
v_max = 2*(v_avg)
del_p = (8*mu*L_p*v_avg)/(r^2)
kinematic_p = del_p/rho
% calculating shear stress
R = linspace(0,(D/2),1000)
for i = 1:length(R)
tau(i) = 2*mu*v_max*(abs(R(i)))/r^2
end
plot(R,tau)
xlabel(\'Radius\')
ylabel(\'Shear stress\')
grid on
legend(\'Shear stress\')
header1=\'/*--------------------------------*- C++ -*----------------------------------*\\\'
header2=\' ========= | \'
header3=\' \\\\ / F ield | OpenFOAM: The Open Source CFD Toolbox \'
header4=\' \\\\ / O peration | Website: https://openfoam.org \'
header5=\' \\\\ / A nd | Version: 7 \'
header6=\' \\\\/ M anipulation | \'
header7=\'\\*---------------------------------------------------------------------------*/\'
f1 = fopen(\'blockMeshDict\',\'w\')
fprintf(f1, \'%s\\n\', header1)
fprintf(f1, \'%s\\n\', header2)
fprintf(f1, \'%s\\n\', header3)
fprintf(f1, \'%s\\n\', header4)
fprintf(f1, \'%s\\n\', header5)
fprintf(f1, \'%s\\n\', header6)
fprintf(f1, \'%s\\n\', header7)
fprintf(f1, \'FoamFile\\n{\\n\')
fprintf(f1, \'%s \\t 2.0;\\n\',\'version\')
fprintf(f1, \'%s \\t ascii;\\n\',\'format\')
fprintf(f1, \'%s \\t dictionary;\\n\',\'class\')
fprintf(f1, \'%s \\t blockMeshDict;\\n\',\'object\')
fprintf(f1, \'}\\n\')
header8 =\'// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //\'
fprintf(f1, \'%s\\n\\n\',header8)
fprintf(f1, \'convertToMeters 1;\\n\\n\')
fprintf(f1,\'vertices\\n\')
fprintf(f1, \'(\\n\')
fprintf(f1, \'\\t (%d %d %d)\\n\',0,0,0)
fprintf(f1, \'\\t (%d %d %d)\\n\',0,r*cosd(theta/2),r*sind(theta/2))
fprintf(f1, \'\\t (%d %d %d)\\n\',0,r*cosd(theta/2),-r*sind(theta/2))
fprintf(f1, \'\\t (%d %d %d)\\n\',L_p,0,0)
fprintf(f1, \'\\t (%d %d %d)\\n\',L_p,r*cosd(theta/2),r*sind(theta/2))
fprintf(f1, \'\\t (%d %d %d)\\n\',L_p,r*cosd(theta/2),-r*sind(theta/2))
fprintf(f1, \');\\n\\n\')
fprintf(f1,\'blocks\\n\')
fprintf(f1, \'(\\n\')
fprintf(f1, \'\\t hex (0 3 5 2 0 3 4 1) (%d %d %d)\',500,25,1)
fprintf(f1, \' SimpleGrading (1 0.2 1)\\n\\n\')
fprintf(f1, \');\\n\\n\')
fprintf(f1,\'edges\\n\')
fprintf(f1, \'(\\n\')
fprintf(f1, \'\\t arc %d %d (%d %d %d)\\n\',1,2,0,r,0)
fprintf(f1, \'\\t arc %d %d (%d %d %d)\\n\',4,5,L_p,r,0)
fprintf(f1, \');\\n\\n\')
fprintf(f1,\'boundary\\n\')
fprintf(f1, \'(\\n\')
fprintf(f1, \'\\t inlet \\n\')
fprintf(f1, \'\\t {\\n\')
fprintf(f1, \'\\t\\t type patch;\\n\')
fprintf(f1, \'\\t\\t faces \\n\')
fprintf(f1, \'\\t\\t (\\n\')
fprintf(f1, \'\\t\\t\\t(0 1 2 0)\\n\')
fprintf(f1, \'\\t\\t);\\n\')
fprintf(f1, \'\\t }\\n\')
fprintf(f1,\'\\t outlet\\n\')
fprintf(f1, \'\\t{\\n\')
fprintf(f1, \'\\t\\t type patch;\\n\')
fprintf(f1, \'\\t\\t faces \\n\')
fprintf(f1, \'\\t\\t (\\n\')
fprintf(f1, \'\\t\\t\\t(3 5 4 3)\\n\')
fprintf(f1, \'\\t\\t);\\n\')
fprintf(f1, \'\\t }\\n\')
fprintf(f1,\'\\t top\\n\')
fprintf(f1, \'\\t{\\n\')
fprintf(f1, \'\\t\\t type wall;\\n\')
fprintf(f1, \'\\t\\t faces \\n\')
fprintf(f1, \'\\t\\t (\\n\')
fprintf(f1, \'\\t\\t\\t(1 4 5 2)\\n\')
fprintf(f1, \'\\t\\t);\\n\')
fprintf(f1, \'\\t }\\n\')
fprintf(f1,\'\\t front\\n\')
fprintf(f1, \'\\t{\\n\')
fprintf(f1, \'\\t\\t type wedge;\\n\')
fprintf(f1, \'\\t\\t faces \\n\')
fprintf(f1, \'\\t\\t (\\n\')
fprintf(f1, \'\\t\\t\\t(0 3 4 1)\\n\')
fprintf(f1, \'\\t\\t);\\n\')
fprintf(f1, \'\\t }\\n\')
fprintf(f1,\'\\t back\\n\')
fprintf(f1, \'\\t{\\n\')
fprintf(f1, \'\\t\\t type wedge;\\n\')
fprintf(f1, \'\\t\\t faces \\n\')
fprintf(f1, \'\\t\\t (\\n\')
fprintf(f1, \'\\t\\t\\t(0 2 5 3)\\n\')
fprintf(f1, \'\\t\\t);\\n\')
fprintf(f1, \'\\t }\\n\')
fprintf(f1,\'\\t axis\\n\')
fprintf(f1, \'\\t{\\n\')
fprintf(f1, \'\\t\\t type empty;\\n\')
fprintf(f1, \'\\t\\t faces \\n\')
fprintf(f1, \'\\t\\t (\\n\')
fprintf(f1, \'\\t\\t\\t(0 3 3 0)\\n\')
fprintf(f1, \'\\t\\t);\\n\')
fprintf(f1, \'\\t }\\n\')
fprintf(f1, \');\\n\\n\')
fprintf(f1, \'mergePatchParis\\n\')
fprintf(f1, \'(\\n\')
fprintf(f1, \');\\n\')
header9 =\'// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //\'
fprintf(f1, \'%s\\n\',header9)
fclose(f1)Geometry for the wedge angle 10 degree
Mesh for the wedge angle 10 degree
Geometry for the wedge angle 25 degree
Mesh for the wedge angle 25 degree
Geometry for the wedge angle 45 degree
Mesh for the wedge angle 45 degree
Grapical Representation(velocity profiles)
The below plots are plotted for wedge angle 45 degree
Velocity profile at the inlet of the pipe
Velocity profile at 0.1m from the inlet of the pipe
Velocity profile at 0.3m from the inlet of the pipe
Velocity profile at 0.5m from the inlet of the pipe
Velocity profile at 1m from the inlet of the pipe
Velocity profile at 1.5m from the inlet of the pipe
Velocity profile at 2m from the inlet of the pipe
Velocity profile at 2.5m from the inlet of the pipe
Velocity profile at 2.82m from the inlet of the pipe
The below plot is plotted for wedge angle 25 degree
Velocity profile at 2.82m from the inlet of the pipe
The below plot is plotted for wedge angle 10 degree
Velocity profile at 2.82m from the inlet of the pipe
Shear Stress Graph
TABULATION:
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