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Week 12 - Symmetry vs Wedge vs HP equation

RRupesh Piraji Hunkeri7 Oct 2020

1. Problem definition - In this problem we have to simulate the hagen poiseulle's flow in openfoam using a small sector of a pipe and compare the results with wedge and symmetry boundary conditions for `theta = 10 , 25 , 45` and also we have to write a code for blockmeshDict file in matlab and use it in openfoam

  • Hagen Poiseulle's Equation

`DeltaP = frac{8muLV}{R^2}`

where `DeltaP`= pressure drop

`mu` = dynamic viscosity

V = Velocity

R = Radius of pipe

2. Procedure -

  • First we will copy the cavity tutorial from tutorial-incompressible-icoFoam-cavity to $FOAM_RUN with a different name and perform our simulation on this new file
  • We will make 6 copy of these files, 3 for wedge BCs and 3 for symmetry BCs
  • Then we will edit the blockMeshDict file according to our geometry, number of cells, mesh grading factor and will specify our boundary for the geometry
  • We will also create the blockmesh file using matlab and use it in openFoam
  • Then we will specify the start time, end time and time steps in controlDict file inside system directory
  • Then we will specify the boundary condition for p and U inside the 0 directory
  • After doing all this we will run the blockMesh in command terminal
  • Then we will use icoFoam solver for running our simulation
  • Lastly we will use paraFoam to view our resuls and compare numerical and analytical values

3. Theory -

In openFoam we have to convert our geometry into number of blocks and specify its vertices as the input. For our particular problem we will take a small wedge of the pipe and perform simulation on it

Mesh grading factor - It makes the mesh more dense or less dense at the wall depending on the value of the grading factor.

plot over line - we will calculate how our velocity or pressure is varying at particular point in the geometry with respect to time steps

Boundary layer - A layer inside which a relative motion between fluid particle exists and inside our boundary layer velocity gradient exist

Fully developed flow - It implies that velocity profile does not change in fluid flow direction and we will calculate the variables in fully developed region and also we will calculate shear stress along y - direction and pressure drop along x - direction

`L_p=L_e+0.4`

where `L_p` = length of the pipe

`L_e` = hydrodynamic length

`L_e/D=0.06*Re`

where D = Diameter of pipe

Re = Reynolds number

`Re = frac{rho*V*D}{mu}`

`2100 = frac{V*0.02}{0.9025*10^-6}`

`therefore V_{INLET} = 0.0947625ms^-1`

Wedge and Symmetry Boundary conditions - Wedge boundary condition is used for axi-symmetric problem like for cylinder. The flow must be axi-symmetric along the centre line such that all variables such as velocity and shear stress of the flow must have same values at a given radius for all angles. wedge BCs is applied only for 2D flow. Wedge boundary condition applies the Navier - stokes equation in cylindrical co-ordinates whereas symmetry boundary condition applies the N-S equation in cartesian co-ordinates

Equations icoFoam solver solves -

  • Continuity equation

`nabla u =0`

  • Momentum equation

`del/(deltat)(u) + nabla*(u*u) - nabla*(nunablau)=-nablap`

4. Code -

  • blockMeshDict file using matlab for wedge boundary conditions and `theta=10^o`
clear all
close all
clc

x1 = '/*--------------------------------*- C++ -*----------------------------------*\';
x2 = '| =========                 |                                                 |';
x3 = '| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |';
x4 = '|  \\    /   O peration     | Version:  4.0                                   |';
x5 = '|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |';
x6 = '|    \\/     M anipulation  |                                                 |';
x7 = '\*---------------------------------------------------------------------------*/';

x8 = '// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //';

%specifying the inputs
D = 0.02 ;
R = D/2 ;
theta = 10 ;
Re = 2100 ;
L_hydrodynamic = 0.06*Re*D ;
L_pipe = L_hydrodynamic + 0.4 ;


%co-ordinates of 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_pipe 0 0];
v4 = [L_pipe R*cosd(theta/2) -R*sind(theta/2)];
v5 = [L_pipe R*cosd(theta/2)  R*sind(theta/2)];

%number of cells along x,y and z axis 
nx = 20 ;
ny = 1 ;
nz = 300 ;

%grading 
gx = 0.2 ;
gy = 1 ;
gz = 1 ;


%writing to a file
f1 = fopen('blockMeshDict_wedge10.txt','w')

%creating header
fprintf(f1,'%s\n',x1);
fprintf(f1,'%s\n',x2);
fprintf(f1,'%s\n',x3);
fprintf(f1,'%s\n',x4);
fprintf(f1,'%s\n',x5);
fprintf(f1,'%s\n',x6);
fprintf(f1,'%s\n',x7);

fprintf(f1 ,'FoamFile\n');
fprintf(f1,'{\n');
fprintf(f1,'%16s\n %f\n' ,'version 2.0;');
fprintf(f1,'%16s\n  %12s\n','format ascii;');
fprintf(f1,'%16s\n  %17s\n','class dictionary;');
fprintf(f1,'%16s\n  %20s\n','object blockMeshDict;');
fprintf(f1,'}\n');

fprintf(f1,x8);
fprintf(f1,'\n\n');
fprintf(f1, '%s\n\n %f\n\n','convertToMeters 1;');

fprintf(f1,'vertices\n(\n');
fprintf(f1, '(%d %d %d)\n' , v0);
fprintf(f1, '(%d %d %d)\n' , v1);
fprintf(f1, '(%d %d %d)\n' , v2);
fprintf(f1, '(%d %d %d)\n' , v3);
fprintf(f1, '(%d %d %d)\n' , v4);
fprintf(f1, '(%d %d %d)\n);\n\n' , v5);

fprintf(f1,'blocks\n(\n');
fprintf(f1, '%20s (%d %d %d) %s (%d %d %d)\n', 'hex (0 1 2 0 3 4 5 3)',nx,ny,nz,'simpleGrading',gx,gy,gz);
fprintf(f1,');\n\n');

fprintf(f1, '%s\n','edges');
fprintf(f1, '(\n);\n\n');

fprintf(f1,'%s\n', 'boundary');
fprintf(f1,'(\n');

fprintf(f1,'%10s\n','inlet');
fprintf(f1,'%6s\n','{');
fprintf(f1,'%18s\n %18s\n','type patch;');
fprintf(f1,'%11s\n','faces');
fprintf(f1,'%8s\n','(');
fprintf(f1,'%20s\n','(0 2 1 0)');
fprintf(f1,'%9s\n',');');
fprintf(f1,'%6s\n','}');

fprintf(f1,'%11s\n','outlet');
fprintf(f1,'%6s\n','{');
fprintf(f1,'%18s\n %18s\n','type patch;');
fprintf(f1,'%11s\n','faces');
fprintf(f1,'%8s\n','(');
fprintf(f1,'%20s\n','(3 4 5 3)');
fprintf(f1,'%9s\n',');');
fprintf(f1,'%6s\n','}');

fprintf(f1,'%15s\n','FrontWedge');
fprintf(f1,'%6s\n','{');
fprintf(f1,'%18s\n %18s\n','type wedge;');
fprintf(f1,'%11s\n','faces');
fprintf(f1,'%8s\n','(');
fprintf(f1,'%20s\n','(0 3 5 2)');
fprintf(f1,'%9s\n',');');
fprintf(f1,'%6s\n','}');


fprintf(f1,'%14s\n','BackWedge');
fprintf(f1,'%6s\n','{');
fprintf(f1,'%18s\n %18s\n','type wedge;');
fprintf(f1,'%11s\n','faces');
fprintf(f1,'%8s\n','(');
fprintf(f1,'%20s\n','(0 1 4 3)');
fprintf(f1,'%9s\n',');');
fprintf(f1,'%6s\n','}');

fprintf(f1,'%16s\n','noslipwalls');
fprintf(f1,'%6s\n','{');
fprintf(f1,'%18s\n %18s\n','type wall;');
fprintf(f1,'%12s\n','faces');
fprintf(f1,'%8s\n','(');
fprintf(f1,'%20s\n','(1 2 5 4)');
fprintf(f1,'%9s\n',');');
fprintf(f1,'%6s\n','}');

fprintf(f1,');\n\n');

fprintf(f1,'%s\n','mergePatchPairs');
fprintf(f1,'(\n');
fprintf(f1,');\n\n');
fprintf(f1,x8);

fclose(f1)
  • blockMeshDict file in openFoam for wedge boundary conditions and `theta=10^o`
/*--------------------------------*- C++ -*----------------------------------*\
| =========                 |                                                 |
| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
|  \\    /   O peration     | Version:  4.0                                   |
|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |
|    \\/     M anipulation  |                                                 |
\*---------------------------------------------------------------------------*/
FoamFile
{
    version     2.0;
    format      ascii;
    class       dictionary;
    object      blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

convertToMeters 1;

vertices
(
    (0 0 0)
    (0 9.96194e-3 -8.71557e-4)
    (0 9.96194e-3  8.71557e-4)
    (2.92 0 0)
    (2.92 9.96194e-3 -8.71557e-4)
    (2.92 9.96194e-3  8.71557e-4)
);

blocks
(
    hex (0 1 2 0 3 4 5 3) (20 1 300) simpleGrading (0.2 1 1)
);

edges
(
);

boundary
(
    inlet
    {
        type patch;
        faces
        (
            (0 2 1 0)
        );
    }
    outlet
    {
        type patch;
        faces
        (
            (3 4 5 3)
        );
    }
    FrontWedge
    {
        type wedge;
        faces
        (
            (0 3 5 2)
        );
    }
    BackWedge
    {
        type wedge;
        faces
        (
            (0 1 4 3)
        );
    }
    noslipwalls
    {
        type wall;
        faces
        (
            (1 2 5 4)
        );
    }
);

mergePatchPairs
(
);

// ************************************************************************* //
  • blockMeshDict file using matlab for wedge boundary conditions and `theta = 25^o`
clear all
close all
clc

x1 = '/*--------------------------------*- C++ -*----------------------------------*\';
x2 = '| =========                 |                                                 |';
x3 = '| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |';
x4 = '|  \\    /   O peration     | Version:  4.0                                   |';
x5 = '|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |';
x6 = '|    \\/     M anipulation  |                                                 |';
x7 = '\*---------------------------------------------------------------------------*/';

x8 = '// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //';

%specifying the inputs
D = 0.02 ;
R = D/2 ;
theta = 25 ;
Re = 2100 ;
L_hydrodynamic = 0.06*Re*D ;
L_pipe = L_hydrodynamic + 0.4 ;


%co-ordinates of 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_pipe 0 0];
v4 = [L_pipe R*cosd(theta/2) -R*sind(theta/2)];
v5 = [L_pipe R*cosd(theta/2)  R*sind(theta/2)];

%number of cells along x,y and z axis 
nx = 20 ;
ny = 1 ;
nz = 300 ;

%grading 
gx = 0.2 ;
gy = 1 ;
gz = 1 ;


%writing to a file
f1 = fopen('blockMeshDict_wedge25.txt','w')

%creating header
fprintf(f1,'%s\n',x1);
fprintf(f1,'%s\n',x2);
fprintf(f1,'%s\n',x3);
fprintf(f1,'%s\n',x4);
fprintf(f1,'%s\n',x5);
fprintf(f1,'%s\n',x6);
fprintf(f1,'%s\n',x7);

fprintf(f1 ,'FoamFile\n');
fprintf(f1,'{\n');
fprintf(f1,'%16s\n %f\n' ,'version 2.0;');
fprintf(f1,'%16s\n  %12s\n','format ascii;');
fprintf(f1,'%16s\n  %17s\n','class dictionary;');
fprintf(f1,'%16s\n  %20s\n','object blockMeshDict;');
fprintf(f1,'}\n');

fprintf(f1,x8);
fprintf(f1,'\n\n');
fprintf(f1, '%s\n\n %f\n\n','convertToMeters 1;');

fprintf(f1,'vertices\n(\n');
fprintf(f1, '(%d %d %d)\n' , v0);
fprintf(f1, '(%d %d %d)\n' , v1);
fprintf(f1, '(%d %d %d)\n' , v2);
fprintf(f1, '(%d %d %d)\n' , v3);
fprintf(f1, '(%d %d %d)\n' , v4);
fprintf(f1, '(%d %d %d)\n);\n\n' , v5);

fprintf(f1,'blocks\n(\n');
fprintf(f1, '%20s (%d %d %d) %s (%d %d %d)\n', 'hex (0 1 2 0 3 4 5 3)',nx,ny,nz,'simpleGrading',gx,gy,gz);
fprintf(f1,');\n\n');

fprintf(f1, '%s\n','edges');
fprintf(f1, '(\n);\n\n');

fprintf(f1,'%s\n', 'boundary');
fprintf(f1,'(\n');

fprintf(f1,'%10s\n','inlet');
fprintf(f1,'%6s\n','{');
fprintf(f1,'%18s\n %18s\n','type patch;');
fprintf(f1,'%11s\n','faces');
fprintf(f1,'%8s\n','(');
fprintf(f1,'%20s\n','(0 2 1 0)');
fprintf(f1,'%9s\n',');');
fprintf(f1,'%6s\n','}');

fprintf(f1,'%11s\n','outlet');
fprintf(f1,'%6s\n','{');
fprintf(f1,'%18s\n %18s\n','type patch;');
fprintf(f1,'%11s\n','faces');
fprintf(f1,'%8s\n','(');
fprintf(f1,'%20s\n','(3 4 5 3)');
fprintf(f1,'%9s\n',');');
fprintf(f1,'%6s\n','}');

fprintf(f1,'%15s\n','FrontWedge');
fprintf(f1,'%6s\n','{');
fprintf(f1,'%18s\n %18s\n','type wedge;');
fprintf(f1,'%11s\n','faces');
fprintf(f1,'%8s\n','(');
fprintf(f1,'%20s\n','(0 3 5 2)');
fprintf(f1,'%9s\n',');');
fprintf(f1,'%6s\n','}');


fprintf(f1,'%14s\n','BackWedge');
fprintf(f1,'%6s\n','{');
fprintf(f1,'%18s\n %18s\n','type wedge;');
fprintf(f1,'%11s\n','faces');
fprintf(f1,'%8s\n','(');
fprintf(f1,'%20s\n','(0 1 4 3)');
fprintf(f1,'%9s\n',');');
fprintf(f1,'%6s\n','}');

fprintf(f1,'%16s\n','noslipwalls');
fprintf(f1,'%6s\n','{');
fprintf(f1,'%18s\n %18s\n','type wall;');
fprintf(f1,'%12s\n','faces');
fprintf(f1,'%8s\n','(');
fprintf(f1,'%20s\n','(1 2 5 4)');
fprintf(f1,'%9s\n',');');
fprintf(f1,'%6s\n','}');

fprintf(f1,');\n\n');

fprintf(f1,'%s\n','mergePatchPairs');
fprintf(f1,'(\n');
fprintf(f1,');\n\n');
fprintf(f1,x8);

fclose(f1)
  • blockMeshDict file in openFoam using wedge boundary conditions and `theta=25^o`
/*--------------------------------*- C++ -*----------------------------------*\
| =========                 |                                                 |
| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
|  \\    /   O peration     | Version:  4.0                                   |
|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |
|    \\/     M anipulation  |                                                 |
\*---------------------------------------------------------------------------*/
FoamFile
{
    version     2.0;
    format      ascii;
    class       dictionary;
    object      blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

convertToMeters 1;

vertices
(
    (0 0 0)
    (0 9.76296e-3 -2.16439e-3)
    (0 9.76296e-3  2.16439e-3)
    (2.92 0 0)
    (2.92 9.76296e-3 -2.16439e-3)
    (2.92 9.76296e-3  2.16439e-3)
);

blocks
(
    hex (0 1 2 0 3 4 5 3) (20 1 300) simpleGrading (0.2 1 1)
);

edges
(
);

boundary
(
    inlet
    {
        type patch;
        faces
        (
            (0 2 1 0)
        );
    }
    outlet
    {
        type patch;
        faces
        (
            (3 4 5 3)
        );
    }
    FrontWedge
    {
        type wedge;
        faces
        (
            (0 3 5 2)
        );
    }
    BackWedge
    {
        type wedge;
        faces
        (
            (0 1 4 3)
        );
    }
    noslipwalls
    {
        type wall;
        faces
        (
            (1 2 5 4)
        );
    }
);

mergePatchPairs
(
);

// ************************************************************************* //
  • blockMeshDict file using matlab for wedge boundary conditions and `theta = 45^o`
clear all
close all
clc

x1 = '/*--------------------------------*- C++ -*----------------------------------*\';
x2 = '| =========                 |                                                 |';
x3 = '| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |';
x4 = '|  \\    /   O peration     | Version:  4.0                                   |';
x5 = '|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |';
x6 = '|    \\/     M anipulation  |                                                 |';
x7 = '\*---------------------------------------------------------------------------*/';

x8 = '// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //';

%specifying the inputs
D = 0.02 ;
R = D/2 ;
theta = 45 ;
Re = 2100 ;
L_hydrodynamic = 0.06*Re*D ;
L_pipe = L_hydrodynamic + 0.4 ;


%co-ordinates of 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_pipe 0 0];
v4 = [L_pipe R*cosd(theta/2) -R*sind(theta/2)];
v5 = [L_pipe R*cosd(theta/2)  R*sind(theta/2)];

%number of cells along x,y and z axis 
nx = 20 ;
ny = 1 ;
nz = 300 ;

%grading 
gx = 0.2 ;
gy = 1 ;
gz = 1 ;


%writing to a file
f1 = fopen('blockMeshDict_wedge45.txt','w')

%creating header
fprintf(f1,'%s\n',x1);
fprintf(f1,'%s\n',x2);
fprintf(f1,'%s\n',x3);
fprintf(f1,'%s\n',x4);
fprintf(f1,'%s\n',x5);
fprintf(f1,'%s\n',x6);
fprintf(f1,'%s\n',x7);

fprintf(f1 ,'FoamFile\n');
fprintf(f1,'{\n');
fprintf(f1,'%16s\n %f\n' ,'version 2.0;');
fprintf(f1,'%16s\n  %12s\n','format ascii;');
fprintf(f1,'%16s\n  %17s\n','class dictionary;');
fprintf(f1,'%16s\n  %20s\n','object blockMeshDict;');
fprintf(f1,'}\n');

fprintf(f1,x8);
fprintf(f1,'\n\n');
fprintf(f1, '%s\n\n %f\n\n','convertToMeters 1;');

fprintf(f1,'vertices\n(\n');
fprintf(f1, '(%d %d %d)\n' , v0);
fprintf(f1, '(%d %d %d)\n' , v1);
fprintf(f1, '(%d %d %d)\n' , v2);
fprintf(f1, '(%d %d %d)\n' , v3);
fprintf(f1, '(%d %d %d)\n' , v4);
fprintf(f1, '(%d %d %d)\n);\n\n' , v5);

fprintf(f1,'blocks\n(\n');
fprintf(f1, '%20s (%d %d %d) %s (%d %d %d)\n', 'hex (0 1 2 0 3 4 5 3)',nx,ny,nz,'simpleGrading',gx,gy,gz);
fprintf(f1,');\n\n');

fprintf(f1, '%s\n','edges');
fprintf(f1, '(\n);\n\n');

fprintf(f1,'%s\n', 'boundary');
fprintf(f1,'(\n');

fprintf(f1,'%10s\n','inlet');
fprintf(f1,'%6s\n','{');
fprintf(f1,'%18s\n %18s\n','type patch;');
fprintf(f1,'%11s\n','faces');
fprintf(f1,'%8s\n','(');
fprintf(f1,'%20s\n','(0 2 1 0)');
fprintf(f1,'%9s\n',');');
fprintf(f1,'%6s\n','}');

fprintf(f1,'%11s\n','outlet');
fprintf(f1,'%6s\n','{');
fprintf(f1,'%18s\n %18s\n','type patch;');
fprintf(f1,'%11s\n','faces');
fprintf(f1,'%8s\n','(');
fprintf(f1,'%20s\n','(3 4 5 3)');
fprintf(f1,'%9s\n',');');
fprintf(f1,'%6s\n','}');

fprintf(f1,'%15s\n','FrontWedge');
fprintf(f1,'%6s\n','{');
fprintf(f1,'%18s\n %18s\n','type wedge;');
fprintf(f1,'%11s\n','faces');
fprintf(f1,'%8s\n','(');
fprintf(f1,'%20s\n','(0 3 5 2)');
fprintf(f1,'%9s\n',');');
fprintf(f1,'%6s\n','}');


fprintf(f1,'%14s\n','BackWedge');
fprintf(f1,'%6s\n','{');
fprintf(f1,'%18s\n %18s\n','type wedge;');
fprintf(f1,'%11s\n','faces');
fprintf(f1,'%8s\n','(');
fprintf(f1,'%20s\n','(0 1 4 3)');
fprintf(f1,'%9s\n',');');
fprintf(f1,'%6s\n','}');

fprintf(f1,'%16s\n','noslipwalls');
fprintf(f1,'%6s\n','{');
fprintf(f1,'%18s\n %18s\n','type wall;');
fprintf(f1,'%12s\n','faces');
fprintf(f1,'%8s\n','(');
fprintf(f1,'%20s\n','(1 2 5 4)');
fprintf(f1,'%9s\n',');');
fprintf(f1,'%6s\n','}');

fprintf(f1,');\n\n');

fprintf(f1,'%s\n','mergePatchPairs');
fprintf(f1,'(\n');
fprintf(f1,');\n\n');
fprintf(f1,x8);

fclose(f1)
  • blockMeshDict file in openFoam for wedge boundary conditions and `theta=45^o`
/*--------------------------------*- C++ -*----------------------------------*\
| =========                 |                                                 |
| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
|  \\    /   O peration     | Version:  4.0                                   |
|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |
|    \\/     M anipulation  |                                                 |
\*---------------------------------------------------------------------------*/
FoamFile
{
    version     2.0;
    format      ascii;
    class       dictionary;
    object      blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

convertToMeters 1;

vertices
(
    (0 0 0)
    (0 9.23879e-3 -3.82683e-3)
    (0 9.23879e-3  3.82683e-3)
    (2.92 0 0)
    (2.92 9.23879e-3 -3.82683e-3)
    (2.92 9.23879e-3  3.82683e-3)
);

blocks
(
    hex (0 1 2 0 3 4 5 3) (20 1 300) simpleGrading (0.2 1 1)
);

edges
(
);

boundary
(
    inlet
    {
        type patch;
        faces
        (
            (0 2 1 0)
        );
    }
    outlet
    {
        type patch;
        faces
        (
            (3 4 5 3)
        );
    }
    FrontWedge
    {
        type wedge;
        faces
        (
            (0 3 5 2)
        );
    }
    BackWedge
    {
        type wedge;
        faces
        (
            (0 1 4 3)
        );
    }
    noslipwalls
    {
        type wall;
        faces
        (
            (1 2 5 4)
        );
    }
);

mergePatchPairs
(
);

// ************************************************************************* //

For writing blockMeshDict file for symmetry boundary conditions using matlab, all we have to do is change the word wedge to symmetry

  • blockMeshDict file in openFoam for symmetry boundary conditions and `theta=10^o`
/*--------------------------------*- C++ -*----------------------------------*\
| =========                 |                                                 |
| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
|  \\    /   O peration     | Version:  4.0                                   |
|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |
|    \\/     M anipulation  |                                                 |
\*---------------------------------------------------------------------------*/
FoamFile
{
    version     2.0;
    format      ascii;
    class       dictionary;
    object      blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

convertToMeters 1;

vertices
(
    (0 0 0)
    (0 9.96194e-3 -8.71557e-4)
    (0 9.96194e-3  8.71557e-4)
    (2.92 0 0)
    (2.92 9.96194e-3 -8.71557e-4)
    (2.92 9.96194e-3  8.71557e-4)
);

blocks
(
    hex (0 1 2 0 3 4 5 3) (20 1 300) simpleGrading (0.2 1 1)
);

edges
(
);

boundary
(
    inlet
    {
        type patch;
        faces
        (
            (0 2 1 0)
        );
    }
    outlet
    {
        type patch;
        faces
        (
            (3 4 5 3)
        );
    }
    FrontWedge
    {
        type symmetry;
        faces
        (
            (0 3 5 2)
        );
    }
    BackWedge
    {
        type symmetry;
        faces
        (
            (0 1 4 3)
        );
    }
    noslipwalls
    {
        type wall;
        faces
        (
            (1 2 5 4)
        );
    }
);

mergePatchPairs
(
);

// ************************************************************************* //
  • blockMeshDict file in openFoam for symmetry boundary conditions and `theta=25^o`
/*--------------------------------*- C++ -*----------------------------------*\
| =========                 |                                                 |
| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
|  \\    /   O peration     | Version:  4.0                                   |
|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |
|    \\/     M anipulation  |                                                 |
\*---------------------------------------------------------------------------*/
FoamFile
{
    version     2.0;
    format      ascii;
    class       dictionary;
    object      blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

convertToMeters 1;

vertices
(
    (0 0 0)
    (0 9.76296e-3 -2.16439e-3)
    (0 9.76296e-3  2.16439e-3)
    (2.92 0 0)
    (2.92 9.76296e-3 -2.16439e-3)
    (2.92 9.76296e-3  2.16439e-3)
);

blocks
(
    hex (0 1 2 0 3 4 5 3) (20 1 300) simpleGrading (0.2 1 1)
);

edges
(
);

boundary
(
    inlet
    {
        type patch;
        faces
        (
            (0 2 1 0)
        );
    }
    outlet
    {
        type patch;
        faces
        (
            (3 4 5 3)
        );
    }
    FrontWedge
    {
        type symmetry;
        faces
        (
            (0 3 5 2)
        );
    }
    BackWedge
    {
        type symmetry;
        faces
        (
            (0 1 4 3)
        );
    }
    noslipwalls
    {
        type wall;
        faces
        (
            (1 2 5 4)
        );
    }
);

mergePatchPairs
(
);

// ************************************************************************* //
  • blockMeshDict file in openFoam for symmetry boundary conditions and `theta=45^o`
/*--------------------------------*- C++ -*----------------------------------*\
| =========                 |                                                 |
| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
|  \\    /   O peration     | Version:  4.0                                   |
|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |
|    \\/     M anipulation  |                                                 |
\*---------------------------------------------------------------------------*/
FoamFile
{
    version     2.0;
    format      ascii;
    class       dictionary;
    object      blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

convertToMeters 1;

vertices
(
    (0 0 0)
    (0 9.23879e-3 -3.82683e-3)
    (0 9.23879e-3  3.82683e-3)
    (2.92 0 0)
    (2.92 9.23879e-3 -3.82683e-3)
    (2.92 9.23879e-3  3.82683e-3)
);

blocks
(
    hex (0 1 2 0 3 4 5 3) (20 1 300) simpleGrading (0.2 1 1)
);

edges
(
);

boundary
(
    inlet
    {
        type patch;
        faces
        (
            (0 2 1 0)
        );
    }
    outlet
    {
        type patch;
        faces
        (
            (3 4 5 3)
        );
    }
    FrontWedge
    {
        type symmetry;
        faces
        (
            (0 3 5 2)
        );
    }
    BackWedge
    {
        type symmetry;
        faces
        (
            (0 1 4 3)
        );
    }
    noslipwalls
    {
        type wall;
        faces
        (
            (1 2 5 4)
        );
    }
);

mergePatchPairs
(
);

// ************************************************************************* //
  • p file
/*--------------------------------*- C++ -*----------------------------------*\
| =========                 |                                                 |
| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
|  \\    /   O peration     | Version:  4.0                                   |
|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |
|    \\/     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;
    }

    FrontWedge
    {
        type            wedge;
    }

    BackWedge
    {
        type            wedge;
    }

    noslipwalls
    {
        type            zeroGradient;
    }
}

// ************************************************************************* //
  • U file
/*--------------------------------*- C++ -*----------------------------------*\
| =========                 |                                                 |
| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
|  \\    /   O peration     | Version:  4.0                                   |
|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |
|    \\/     M anipulation  |                                                 |
\*---------------------------------------------------------------------------*/
FoamFile
{
    version     2.0;
    format      ascii;
    class       volVectorField;
    object      U;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

dimensions      [0 1 -1 0 0 0 0];

internalField   uniform (0 0 0);

boundaryField
{
    inlet
    {
        type            fixedValue;
        value           uniform (0.0947625 0 0);
    }

    outlet
    {
        type            zeroGradient;
    }

    FrontWedge
    {
        type            wedge;
    }

    BackWedge
    {
        type            wedge;
    }

    noslipwalls
    {
        type            noSlip;
    }
}

// ************************************************************************* //

For symmetry boundary conditions for p and U ,all we have to do is change the word wedge to symmetry

  • transportProperties file
/*--------------------------------*- C++ -*----------------------------------*\
| =========                 |                                                 |
| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
|  \\    /   O peration     | Version:  4.0                                   |
|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |
|    \\/     M anipulation  |                                                 |
\*---------------------------------------------------------------------------*/
FoamFile
{
    version     2.0;
    format      ascii;
    class       dictionary;
    location    "constant";
    object      transportProperties;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

nu              [0 2 -1 0 0 0 0] 0.9025e-6;


// ************************************************************************* //
  • controlDict file
/*--------------------------------*- C++ -*----------------------------------*\
| =========                 |                                                 |
| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |
|  \\    /   O peration     | Version:  4.0                                   |
|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |
|    \\/     M anipulation  |                                                 |
\*---------------------------------------------------------------------------*/
FoamFile
{
    version     2.0;
    format      ascii;
    class       dictionary;
    location    "system";
    object      controlDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

application     icoFoam;

startFrom       startTime;

startTime       0;

stopAt          endTime;

endTime         20;

deltaT          0.01;

writeControl    timeStep;

writeInterval   100;

purgeWrite      0;

writeFormat     ascii;

writePrecision  6;

writeCompression off;

timeFormat      general;

timePrecision   6;

runTimeModifiable true;


// ************************************************************************* //

5. Results - We will calculate the resuluts in the fully developed flow

  • BCs = wedge and `theta = 10^o`
radial distance tau_X tau_Y tau_Z num_wedge_tau num_wedge_U analy_wedge_U analy_wedge_tau
0 2.64E-07 -0.0013789 0.000120637 0.001384157 0.187231 1.90E-01 1.70E-04
9.96E-05 2.62E-07 -0.0015448 0 0.00154483 0.187078 1.90E-01 5.11E-04
0.000199239 2.61E-07 -0.0017002 0 0.00170021 0.186924 1.89E-01 8.52E-04
0.000298858 2.59E-07 -0.0018556 0 0.00185559 0.186771 1.89E-01 1.19E-03
0.000398478 2.57E-07 -0.002011 0 0.00201097 0.186617 1.89E-01 1.53E-03
0.000498097 2.56E-07 -0.0021664 0 0.00216636 0.186464 1.89E-01 1.87E-03
0.000597716 2.54E-07 -0.0023217 0 0.00232174 0.186311 1.89E-01 2.22E-03
0.000697336 2.52E-07 -0.0024771 0 0.00247712 0.186157 1.89E-01 2.56E-03
0.000796955 2.51E-07 -0.0026325 0 0.00263251 0.186004 1.88E-01 2.90E-03
0.000896575 2.49E-07 -0.0027879 0 0.00278789 0.185851 1.88E-01 3.24E-03
0.000996194 2.47E-07 -0.0029433 0 0.00294327 0.185697 1.88E-01 3.58E-03
0.00109581 2.46E-07 -0.0030987 0 0.00309865 0.185544 1.87E-01 3.92E-03
0.00119543 2.44E-07 -0.003254 0 0.00325404 0.18539 1.87E-01 4.26E-03
0.00129505 2.43E-07 -0.0034094 0 0.00340942 0.185237 1.86E-01 4.60E-03
0.00139467 2.41E-07 -0.0035648 0 0.0035648 0.185084 1.86E-01 4.94E-03
0.00149429 2.21E-07 -0.0046475 2.67E-20 0.00464749 0.183207 1.85E-01 5.28E-03
0.00159391 2.16E-07 -0.0049865 3.19E-20 0.00498651 0.182712 1.85E-01 5.62E-03
0.00169353 2.10E-07 -0.0053255 3.72E-20 0.00532553 0.182217 1.84E-01 5.96E-03
0.00179315 2.05E-07 -0.0056646 4.25E-20 0.00566455 0.181723 1.83E-01 6.30E-03
0.00189277 2.00E-07 -0.0060036 4.78E-20 0.00600357 0.181228 1.83E-01 6.65E-03
0.00199239 1.92E-07 -0.0063532 2.30E-20 0.00635315 0.180422 1.82E-01 6.99E-03
0.00209201 1.84E-07 -0.0067038 -4.91E-21 0.0067038 0.179585 1.81E-01 7.33E-03
0.00219163 1.76E-07 -0.0070545 -3.28E-20 0.00705445 0.178747 1.80E-01 7.67E-03
0.00229125 1.68E-07 -0.0074051 -6.07E-20 0.00740511 0.177909 1.80E-01 8.01E-03
0.00239087 1.60E-07 -0.0077558 -8.85E-20 0.00775576 0.177072 1.79E-01 8.35E-03
0.00249049 1.52E-07 -0.0081064 -1.16E-19 0.00810642 0.176234 1.78E-01 8.69E-03
0.0025901 1.44E-07 -0.0084571 -1.44E-19 0.00845707 0.175397 1.77E-01 9.03E-03
0.00268972 1.36E-07 -0.0088077 -1.72E-19 0.00880772 0.174559 1.76E-01 9.37E-03
0.00278934 1.28E-07 -0.0091564 -2.33E-19 0.00915639 0.173565 1.75E-01 9.71E-03
0.00288896 1.21E-07 -0.0095031 -3.26E-19 0.00950313 0.17242 1.74E-01 1.01E-02
0.00298858 1.13E-07 -0.0098499 -4.19E-19 0.00984987 0.171275 1.73E-01 1.04E-02
0.0030882 1.06E-07 -0.0101966 -5.12E-19 0.0101966 0.170131 1.71E-01 1.07E-02
0.00318782 9.80E-08 -0.0105434 -6.05E-19 0.0105434 0.168986 1.70E-01 1.11E-02
0.00328744 9.04E-08 -0.0108901 -6.98E-19 0.0108901 0.167841 1.69E-01 1.14E-02
0.00338706 8.28E-08 -0.0112368 -7.91E-19 0.0112368 0.166696 1.68E-01 1.18E-02
0.00348668 7.52E-08 -0.0115836 -8.84E-19 0.0115836 0.165551 1.66E-01 1.21E-02
0.0035863 7.05E-08 -0.0119317 -7.92E-19 0.0119317 0.164182 1.65E-01 1.24E-02
0.00368592 6.67E-08 -0.0122802 -6.52E-19 0.0122802 0.162754 1.64E-01 1.28E-02
0.00378554 6.28E-08 -0.0126287 -5.12E-19 0.0126287 0.161325 1.62E-01 1.31E-02
0.00388516 5.90E-08 -0.0129772 -3.72E-19 0.0129772 0.159897 1.61E-01 1.35E-02
0.00398478 5.51E-08 -0.0133257 -2.32E-19 0.0133257 0.158469 1.59E-01 1.38E-02
0.0040844 5.13E-08 -0.0136742 -9.17E-20 0.0136742 0.15704 1.58E-01 1.41E-02
0.00418402 4.74E-08 -0.0140227 4.84E-20 0.0140227 0.155612 1.56E-01 1.45E-02
0.00428363 4.35E-08 -0.0143728 1.24E-19 0.0143728 0.154 1.55E-01 1.48E-02
0.00438325 3.96E-08 -0.0147235 1.71E-19 0.0147235 0.15231 1.53E-01 1.52E-02
0.00448287 3.57E-08 -0.0150742 2.19E-19 0.0150742 0.150619 1.51E-01 1.55E-02
0.00458249 3.17E-08 -0.0154249 2.66E-19 0.0154249 0.148929 1.50E-01 1.58E-02
0.00468211 2.78E-08 -0.0157756 3.14E-19 0.0157756 0.147238 1.48E-01 1.62E-02
0.00478173 2.39E-08 -0.0161263 3.62E-19 0.0161263 0.145548 1.46E-01 1.65E-02
0.00488135 1.97E-08 -0.0164773 3.90E-19 0.0164773 0.143813 1.44E-01 1.69E-02
0.00498097 1.43E-08 -0.0168299 3.32E-19 0.0168299 0.14188 1.43E-01 1.72E-02
0.00508059 8.84E-09 -0.0171825 2.75E-19 0.0171825 0.139947 1.41E-01 1.76E-02
0.00518021 3.43E-09 -0.0175351 2.18E-19 0.0175351 0.138014 1.39E-01 1.79E-02
0.00527983 -1.99E-09 -0.0178876 1.60E-19 0.0178876 0.136082 1.37E-01 1.82E-02
0.00537945 -7.41E-09 -0.0182402 1.03E-19 0.0182402 0.134149 1.35E-01 1.86E-02
0.00547907 -1.26E-08 -0.0185931 5.52E-20 0.0185931 0.132172 1.33E-01 1.89E-02
0.00557869 -1.67E-08 -0.018947 4.48E-20 0.018947 0.130015 1.31E-01 1.93E-02
0.00567831 -2.09E-08 -0.0193009 3.44E-20 0.0193009 0.127859 1.28E-01 1.96E-02
0.00577793 -2.50E-08 -0.0196548 2.40E-20 0.0196548 0.125702 1.26E-01 1.99E-02
0.00587754 -2.92E-08 -0.0200087 1.35E-20 0.0200087 0.123546 1.24E-01 2.03E-02
0.00597716 -3.33E-08 -0.0203627 3.10E-21 0.0203627 0.12139 1.22E-01 2.06E-02
0.00607678 -3.62E-08 -0.0207171 -2.71E-20 0.0207171 0.119089 1.20E-01 2.10E-02
0.0061764 -3.85E-08 -0.0210718 -6.57E-20 0.0210718 0.116727 1.17E-01 2.13E-02
0.00627602 -4.09E-08 -0.0214265 -1.04E-19 0.0214265 0.114364 1.15E-01 2.16E-02
0.00637564 -4.32E-08 -0.0217812 -1.43E-19 0.0217812 0.112002 1.12E-01 2.20E-02
0.00647526 -4.55E-08 -0.0221359 -1.82E-19 0.0221359 0.10964 1.10E-01 2.23E-02
0.00657488 -4.60E-08 -0.0224907 -1.68E-19 0.0224907 0.107154 1.08E-01 2.27E-02
0.0066745 -4.54E-08 -0.0228456 -1.26E-19 0.0228456 0.104602 1.05E-01 2.30E-02
0.00677412 -4.49E-08 -0.0232005 -8.39E-20 0.0232005 0.10205 1.03E-01 2.33E-02
0.00687374 -4.43E-08 -0.0235555 -4.18E-20 0.0235555 0.0994984 1.00E-01 2.37E-02
0.00697336 -4.38E-08 -0.0239104 3.52E-22 0.0239104 0.0969456 9.74E-02 2.40E-02
0.00707298 -4.31E-08 -0.0242651 7.48E-20 0.0242651 0.0942194 9.47E-02 2.44E-02
0.0071726 -4.23E-08 -0.0246198 1.49E-19 0.0246198 0.0914933 9.20E-02 2.47E-02
0.00727222 -4.16E-08 -0.0249746 2.24E-19 0.0249746 0.0887671 8.93E-02 2.50E-02
0.00737184 -4.09E-08 -0.0253293 2.98E-19 0.0253293 0.0860409 8.65E-02 2.54E-02
0.00747146 -4.07E-08 -0.0256837 2.58E-19 0.0256837 0.0831966 8.37E-02 2.57E-02
0.00757107 -4.06E-08 -0.026038 1.77E-19 0.026038 0.0803105 8.09E-02 2.61E-02
0.00767069 -4.05E-08 -0.0263922 9.55E-20 0.0263922 0.0774245 7.80E-02 2.64E-02
0.00777031 -4.04E-08 -0.0267465 1.45E-20 0.0267465 0.0745385 7.51E-02 2.68E-02
0.00786993 -4.04E-08 -0.0271002 -1.74E-22 0.0271002 0.0715322 7.21E-02 2.71E-02
0.00796955 -4.03E-08 -0.0274538 -2.67E-22 0.0274538 0.0684995 6.92E-02 2.74E-02
0.00806917 -4.03E-08 -0.0278074 -3.59E-22 0.0278074 0.0654668 6.61E-02 2.78E-02
0.00816879 -4.01E-08 -0.0281608 -3.59E-20 0.0281608 0.0624046 6.31E-02 2.81E-02
0.00826841 -3.94E-08 -0.0285136 -1.97E-19 0.0285136 0.0592375 6.00E-02 2.85E-02
0.00836803 -3.87E-08 -0.0288664 -3.59E-19 0.0288664 0.0560704 5.68E-02 2.88E-02
0.00846765 -3.81E-08 -0.0292192 -5.20E-19 0.0292192 0.0529034 5.36E-02 2.91E-02
0.00856727 -3.66E-08 -0.0295712 -8.09E-20 0.0295712 0.0496239 5.04E-02 2.95E-02
0.00866689 -3.51E-08 -0.0299231 4.17E-19 0.0299231 0.0463336 4.72E-02 2.98E-02
0.00876651 -3.36E-08 -0.030275 9.14E-19 0.030275 0.0430433 4.39E-02 3.02E-02
0.00886613 -3.21E-08 -0.0306261 6.73E-19 0.0306261 0.0396545 4.05E-02 3.05E-02
0.00896575 -3.07E-08 -0.030977 3.22E-19 0.030977 0.0362513 3.72E-02 3.08E-02
0.00906537 -2.93E-08 -0.031328 0 0.031328 0.0328399 3.38E-02 3.12E-02
0.00916499 -2.69E-08 -0.0316781 0 0.0316781 0.0293332 3.03E-02 3.15E-02
0.0092646 -2.46E-08 -0.0320282 0 0.0320282 0.0258265 2.69E-02 3.19E-02
0.00936422 -2.19E-08 -0.0323779 0 0.0323779 0.0222711 2.33E-02 3.22E-02
0.00946384 -1.90E-08 -0.0327271 0 0.0327271 0.0186695 1.98E-02 3.25E-02
0.00956346 -1.59E-08 -0.0327879 4.83E-20 0.0327879 0.0150544 1.62E-02 3.29E-02
0.00966308 -1.27E-08 -0.0312889 3.58E-19 0.0312889 0.011366 1.26E-02 3.32E-02
0.0097627 -9.44E-09 -0.0297899 6.68E-19 0.0297899 0.00767751 8.89E-03 3.36E-02
0.00986232 -5.96E-09 -0.0279639 3.37E-19 0.0279639 0.00478476 5.18E-03 3.39E-02
0.00996194 0 0 0 0 0 1.44E-03 -1.30E-04

  • BCs = wedge and `theta = 25^o`
radial distance tau_X tau_Y tau_Z num_wedge_U num_wedge_tau analy_wedge_U analy_wedge_tau
0 2.36E-07 -0.00136 0.000301 0.187411 1.39E-03 1.90E-01 1.67E-04
9.76E-05 2.35E-07 -0.0015 0.000333 0.187264 1.54E-03 1.90E-01 5.01E-04
0.000195259 2.34E-07 -0.00174 0 0.187103 1.74E-03 1.89E-01 8.35E-04
0.000292889 2.33E-07 -0.0019 0 0.186948 1.90E-03 1.89E-01 1.17E-03
0.000390518 2.32E-07 -0.00206 0 0.186794 2.06E-03 1.89E-01 1.50E-03
0.000488148 2.30E-07 -0.00222 0 0.18664 2.22E-03 1.89E-01 1.84E-03
0.000585778 2.29E-07 -0.00238 0 0.186486 2.38E-03 1.89E-01 2.17E-03
0.000683407 2.28E-07 -0.00254 0 0.186331 2.54E-03 1.89E-01 2.50E-03
0.000781037 2.27E-07 -0.0027 0 0.186177 2.70E-03 1.88E-01 2.84E-03
0.000878666 2.25E-07 -0.00286 0 0.186023 2.86E-03 1.88E-01 3.17E-03
0.000976296 2.24E-07 -0.00302 0 0.185869 3.02E-03 1.88E-01 3.51E-03
0.00107393 2.23E-07 -0.00318 0 0.185715 3.18E-03 1.87E-01 3.84E-03
0.00117156 2.22E-07 -0.00334 0 0.18556 3.34E-03 1.87E-01 4.17E-03
0.00126918 2.20E-07 -0.0035 0 0.185406 3.50E-03 1.86E-01 4.51E-03
0.00136681 2.19E-07 -0.00366 0 0.185252 3.66E-03 1.86E-01 4.84E-03
0.00146444 2.18E-07 -0.00382 0 0.185098 3.82E-03 1.85E-01 5.18E-03
0.00156207 1.98E-07 -0.00511 0 0.182867 5.11E-03 1.85E-01 5.51E-03
0.0016597 1.94E-07 -0.00546 0 0.18237 5.46E-03 1.84E-01 5.84E-03
0.00175733 1.90E-07 -0.00581 0 0.181873 5.81E-03 1.84E-01 6.18E-03
0.00185496 1.86E-07 -0.00616 0 0.181375 6.16E-03 1.83E-01 6.51E-03
0.00195259 1.79E-07 -0.00651 -2.46E-20 0.180566 6.51E-03 1.82E-01 6.85E-03
0.00205022 1.72E-07 -0.00687 -5.18E-20 0.179724 6.87E-03 1.82E-01 7.18E-03
0.00214785 1.65E-07 -0.00723 -7.89E-20 0.178882 7.23E-03 1.81E-01 7.51E-03
0.00224548 1.58E-07 -0.00759 -1.06E-19 0.178041 7.59E-03 1.80E-01 7.85E-03
0.00234311 1.52E-07 -0.00795 -1.33E-19 0.177199 7.95E-03 1.79E-01 8.18E-03
0.00244074 1.45E-07 -0.00831 -1.60E-19 0.176357 8.31E-03 1.78E-01 8.52E-03
0.00253837 1.38E-07 -0.00867 -1.87E-19 0.175516 8.67E-03 1.77E-01 8.85E-03
0.002636 1.31E-07 -0.00903 -2.15E-19 0.174674 9.03E-03 1.76E-01 9.18E-03
0.00273363 1.24E-07 -0.00938 -2.23E-19 0.173676 9.38E-03 1.75E-01 9.52E-03
0.00283126 1.17E-07 -0.00974 -2.13E-19 0.172526 9.74E-03 1.74E-01 9.85E-03
0.00292889 1.10E-07 -0.01009 -2.03E-19 0.171377 1.01E-02 1.73E-01 1.02E-02
0.00302652 1.02E-07 -0.01045 -1.94E-19 0.170227 1.04E-02 1.72E-01 1.05E-02
0.00312415 9.51E-08 -0.0108 -1.84E-19 0.169077 1.08E-02 1.71E-01 1.09E-02
0.00322178 8.79E-08 -0.01116 -1.74E-19 0.167928 1.12E-02 1.70E-01 1.12E-02
0.00331941 8.06E-08 -0.01151 -1.64E-19 0.166778 1.15E-02 1.69E-01 1.15E-02
0.00341704 7.34E-08 -0.01186 -1.54E-19 0.165629 1.19E-02 1.67E-01 1.19E-02
0.00351467 6.76E-08 -0.01222 -1.44E-19 0.164255 1.22E-02 1.66E-01 1.22E-02
0.0036123 6.21E-08 -0.01257 -1.33E-19 0.162822 1.26E-02 1.65E-01 1.25E-02
0.00370992 5.66E-08 -0.01293 -1.23E-19 0.161388 1.29E-02 1.63E-01 1.29E-02
0.00380755 5.11E-08 -0.01329 -1.12E-19 0.159955 1.33E-02 1.62E-01 1.32E-02
0.00390518 4.56E-08 -0.01364 -1.02E-19 0.158522 1.36E-02 1.61E-01 1.35E-02
0.00400281 4.01E-08 -0.014 -9.12E-20 0.157089 1.40E-02 1.59E-01 1.39E-02
0.00410044 3.46E-08 -0.01435 -8.07E-20 0.155656 1.44E-02 1.58E-01 1.42E-02
0.00419807 3.05E-08 -0.01471 3.15E-20 0.15404 1.47E-02 1.56E-01 1.45E-02
0.0042957 2.69E-08 -0.01507 1.87E-19 0.152345 1.51E-02 1.55E-01 1.49E-02
0.00439333 2.32E-08 -0.01542 3.43E-19 0.15065 1.54E-02 1.53E-01 1.52E-02
0.00449096 1.96E-08 -0.01578 4.99E-19 0.148955 1.58E-02 1.51E-01 1.55E-02
0.00458859 1.60E-08 -0.01614 6.55E-19 0.14726 1.61E-02 1.50E-01 1.59E-02
0.00468622 1.24E-08 -0.01649 8.11E-19 0.145565 1.65E-02 1.48E-01 1.62E-02
0.00478385 8.77E-09 -0.01685 9.09E-19 0.143826 1.69E-02 1.46E-01 1.65E-02
0.00488148 4.84E-09 -0.01721 7.52E-19 0.141889 1.72E-02 1.44E-01 1.69E-02
0.00497911 9.17E-10 -0.01757 5.95E-19 0.139953 1.76E-02 1.43E-01 1.72E-02
0.00507674 -3.01E-09 -0.01793 4.39E-19 0.138016 1.79E-02 1.41E-01 1.75E-02
0.00517437 -6.93E-09 -0.01829 2.82E-19 0.136079 1.83E-02 1.39E-01 1.79E-02
0.005272 -1.09E-08 -0.01864 1.25E-19 0.134143 1.86E-02 1.37E-01 1.82E-02
0.00536963 -1.48E-08 -0.019 -2.57E-20 0.132162 1.90E-02 1.35E-01 1.85E-02
0.00546726 -1.90E-08 -0.01936 -1.55E-19 0.130003 1.94E-02 1.33E-01 1.89E-02
0.00556489 -2.31E-08 -0.01972 -2.83E-19 0.127843 1.97E-02 1.31E-01 1.92E-02
0.00566252 -2.73E-08 -0.02008 -4.12E-19 0.125684 2.01E-02 1.29E-01 1.95E-02
0.00576015 -3.14E-08 -0.02044 -5.41E-19 0.123525 2.04E-02 1.27E-01 1.99E-02
0.00585778 -3.56E-08 -0.0208 -6.70E-19 0.121365 2.08E-02 1.24E-01 2.02E-02
0.00595541 -3.82E-08 -0.02116 -6.10E-19 0.119062 2.12E-02 1.22E-01 2.05E-02
0.00605304 -4.01E-08 -0.02152 -4.70E-19 0.116698 2.15E-02 1.20E-01 2.09E-02
0.00615066 -4.21E-08 -0.02188 -3.30E-19 0.114333 2.19E-02 1.18E-01 2.12E-02
0.00624829 -4.40E-08 -0.02224 -1.90E-19 0.111969 2.22E-02 1.16E-01 2.15E-02
0.00634592 -4.59E-08 -0.0226 -4.95E-20 0.109604 2.26E-02 1.13E-01 2.19E-02
0.00644355 -4.63E-08 -0.02296 0 0.107118 2.30E-02 1.11E-01 2.22E-02
0.00654118 -4.57E-08 -0.02332 0 0.104564 2.33E-02 1.08E-01 2.25E-02
0.00663881 -4.52E-08 -0.02368 0 0.102011 2.37E-02 1.06E-01 2.29E-02
0.00673644 -4.47E-08 -0.02405 0 0.0994578 2.40E-02 1.04E-01 2.32E-02
0.00683407 -4.41E-08 -0.02441 6.93E-22 0.0969037 2.44E-02 1.01E-01 2.35E-02
0.0069317 -4.36E-08 -0.02477 1.47E-19 0.0941771 2.48E-02 9.85E-02 2.39E-02
0.00702933 -4.31E-08 -0.02513 2.94E-19 0.0914504 2.51E-02 9.59E-02 2.42E-02
0.00712696 -4.26E-08 -0.02549 4.41E-19 0.0887238 2.55E-02 9.33E-02 2.45E-02
0.00722459 -4.20E-08 -0.02585 5.88E-19 0.0859972 2.58E-02 9.06E-02 2.49E-02
0.00732222 -4.17E-08 -0.02621 5.06E-19 0.0831529 2.62E-02 8.79E-02 2.52E-02
0.00741985 -4.15E-08 -0.02657 3.44E-19 0.0802671 2.66E-02 8.52E-02 2.55E-02
0.00751748 -4.12E-08 -0.02693 1.82E-19 0.0773813 2.69E-02 8.24E-02 2.59E-02
0.00761511 -4.09E-08 -0.02729 1.98E-20 0.0744955 2.73E-02 7.96E-02 2.62E-02
0.00771274 -4.07E-08 -0.02765 -3.20E-19 0.0714899 2.76E-02 7.68E-02 2.66E-02
0.00781037 -4.05E-08 -0.02801 -7.00E-19 0.0684581 2.80E-02 7.39E-02 2.69E-02
0.007908 -4.03E-08 -0.02836 -1.08E-18 0.0654264 2.84E-02 7.10E-02 2.72E-02
0.00800563 -3.99E-08 -0.02872 -1.34E-18 0.0623651 2.87E-02 6.81E-02 2.76E-02
0.00810326 -3.87E-08 -0.02908 -1.20E-18 0.0591994 2.91E-02 6.51E-02 2.79E-02
0.00820089 -3.76E-08 -0.02944 -1.07E-18 0.0560337 2.94E-02 6.21E-02 2.82E-02
0.00829852 -3.65E-08 -0.0298 -9.26E-19 0.052868 2.98E-02 5.90E-02 2.86E-02
0.00839614 -3.50E-08 -0.03016 -6.95E-19 0.0495903 3.02E-02 5.59E-02 2.89E-02
0.00849377 -3.36E-08 -0.03052 -4.56E-19 0.0463018 3.05E-02 5.28E-02 2.92E-02
0.0085914 -3.21E-08 -0.03087 -2.16E-19 0.0430133 3.09E-02 4.96E-02 2.96E-02
0.00868903 -3.11E-08 -0.03123 -3.75E-19 0.0396266 3.12E-02 4.64E-02 2.99E-02
0.00878666 -3.03E-08 -0.03159 -5.93E-19 0.0362255 3.16E-02 4.32E-02 3.02E-02
0.00888429 -2.93E-08 -0.03195 -7.56E-19 0.0328162 3.19E-02 3.99E-02 3.06E-02
0.00898192 -2.69E-08 -0.0323 -2.90E-19 0.0293119 3.23E-02 3.66E-02 3.09E-02
0.00907955 -2.46E-08 -0.03266 1.75E-19 0.0258076 3.27E-02 3.33E-02 3.12E-02
0.00917718 -2.19E-08 -0.03301 7.48E-20 0.0222547 3.30E-02 2.99E-02 3.16E-02
0.00927481 -1.90E-08 -0.03337 -5.63E-19 0.0186558 3.34E-02 2.65E-02 3.19E-02
0.00937244 -1.59E-08 -0.03342 -1.03E-18 0.0150433 3.34E-02 2.30E-02 3.22E-02
0.00947007 -1.26E-08 -0.03182 -5.69E-19 0.0113575 3.18E-02 1.96E-02 3.26E-02
0.0095677 -9.34E-09 -0.03021 -1.09E-19 0.00767177 3.02E-02 1.60E-02 3.29E-02
0.00966533 -5.89E-09 -0.02827 -5.44E-19 0.00481602 2.83E-02 1.25E-02 3.32E-02
0.00976296 0 0 0 0 0.00E+00 8.88E-03 -8.21E-04

  • BCs = wedge and `theta = 45^o`
radial distance tau_X tau_Y tau_Z num_wedge_U num_wedge_tau analy_wedge_U analy_wedge_tau
0 2.13E-07 -0.0013 0.000539 0.18778 1.41E-03 1.90E-01 1.58E-04
9.24E-05 2.11E-07 -0.00143 0.00059 0.187646 1.54E-03 1.90E-01 4.74E-04
0.000185 2.09E-07 -0.00186 2.22E-21 0.187467 1.86E-03 1.89E-01 7.90E-04
0.000277 2.07E-07 -0.00203 3.33E-21 0.187311 2.03E-03 1.89E-01 1.11E-03
0.00037 2.05E-07 -0.00221 4.44E-21 0.187155 2.21E-03 1.89E-01 1.42E-03
0.000462 2.02E-07 -0.00238 5.55E-21 0.186999 2.38E-03 1.89E-01 1.74E-03
0.000554 2.00E-07 -0.00255 6.66E-21 0.186843 2.55E-03 1.89E-01 2.05E-03
0.000647 1.98E-07 -0.00272 7.77E-21 0.186687 2.72E-03 1.89E-01 2.37E-03
0.000739 1.96E-07 -0.00289 8.88E-21 0.186531 2.89E-03 1.88E-01 2.69E-03
0.000831 1.94E-07 -0.00306 9.99E-21 0.186375 3.06E-03 1.88E-01 3.00E-03
0.000924 1.92E-07 -0.00323 1.11E-20 0.186219 3.23E-03 1.88E-01 3.32E-03
0.001016 1.90E-07 -0.0034 1.22E-20 0.186063 3.40E-03 1.88E-01 3.63E-03
0.001109 1.88E-07 -0.00357 1.33E-20 0.185907 3.57E-03 1.87E-01 3.95E-03
0.001201 1.85E-07 -0.00374 1.44E-20 0.185751 3.74E-03 1.87E-01 4.27E-03
0.001293 1.83E-07 -0.00391 1.55E-20 0.185595 3.91E-03 1.86E-01 4.58E-03
0.001386 1.81E-07 -0.00408 1.66E-20 0.185439 4.08E-03 1.86E-01 4.90E-03
0.001478 1.79E-07 -0.00425 1.78E-20 0.185283 4.25E-03 1.85E-01 5.21E-03
0.001571 1.77E-07 -0.00442 1.89E-20 0.185127 4.42E-03 1.85E-01 5.53E-03
0.001663 1.51E-07 -0.0062 2.17E-20 0.182179 6.20E-03 1.84E-01 5.85E-03
0.001755 1.46E-07 -0.00657 2.31E-20 0.181676 6.57E-03 1.84E-01 6.16E-03
0.001848 1.41E-07 -0.00695 1.82E-20 0.180859 6.95E-03 1.83E-01 6.48E-03
0.00194 1.37E-07 -0.00734 1.27E-20 0.180009 7.34E-03 1.82E-01 6.80E-03
0.002033 1.32E-07 -0.00772 7.23E-21 0.179159 7.72E-03 1.82E-01 7.11E-03
0.002125 1.27E-07 -0.0081 1.75E-21 0.178309 8.10E-03 1.81E-01 7.43E-03
0.002217 1.23E-07 -0.00848 -3.74E-21 0.177459 8.48E-03 1.80E-01 7.74E-03
0.00231 1.18E-07 -0.00886 -9.23E-21 0.176609 8.86E-03 1.79E-01 8.06E-03
0.002402 1.13E-07 -0.00924 -1.47E-20 0.175759 9.24E-03 1.79E-01 8.38E-03
0.002494 1.09E-07 -0.00963 -2.02E-20 0.174909 9.63E-03 1.78E-01 8.69E-03
0.002587 1.04E-07 -0.01 -1.68E-20 0.173902 1.00E-02 1.77E-01 9.01E-03
0.002679 9.93E-08 -0.01038 -4.84E-21 0.172743 1.04E-02 1.76E-01 9.32E-03
0.002772 9.44E-08 -0.01075 7.14E-21 0.171583 1.08E-02 1.75E-01 9.64E-03
0.002864 8.96E-08 -0.01113 1.91E-20 0.170424 1.11E-02 1.74E-01 9.96E-03
0.002956 8.48E-08 -0.01151 3.11E-20 0.169265 1.15E-02 1.73E-01 1.03E-02
0.003049 7.99E-08 -0.01188 4.31E-20 0.168106 1.19E-02 1.72E-01 1.06E-02
0.003141 7.51E-08 -0.01226 5.51E-20 0.166946 1.23E-02 1.71E-01 1.09E-02
0.003234 7.02E-08 -0.01263 6.70E-20 0.165787 1.26E-02 1.70E-01 1.12E-02
0.003326 6.43E-08 -0.01301 6.27E-20 0.164404 1.30E-02 1.69E-01 1.15E-02
0.003418 5.81E-08 -0.01338 5.41E-20 0.162961 1.34E-02 1.67E-01 1.19E-02
0.003511 5.19E-08 -0.01376 4.55E-20 0.161518 1.38E-02 1.66E-01 1.22E-02
0.003603 4.57E-08 -0.01413 3.69E-20 0.160075 1.41E-02 1.65E-01 1.25E-02
0.003696 3.95E-08 -0.01451 2.83E-20 0.158632 1.45E-02 1.64E-01 1.28E-02
0.003788 3.33E-08 -0.01488 1.96E-20 0.157189 1.49E-02 1.62E-01 1.31E-02
0.00388 2.70E-08 -0.01526 1.10E-20 0.155746 1.53E-02 1.61E-01 1.34E-02
0.003973 2.17E-08 -0.01563 7.55E-21 0.154121 1.56E-02 1.60E-01 1.37E-02
0.004065 1.68E-08 -0.01601 6.29E-21 0.152417 1.60E-02 1.58E-01 1.41E-02
0.004157 1.19E-08 -0.01638 5.02E-21 0.150713 1.64E-02 1.57E-01 1.44E-02
0.00425 7.01E-09 -0.01676 3.75E-21 0.149009 1.68E-02 1.55E-01 1.47E-02
0.004342 2.10E-09 -0.01714 2.48E-21 0.147305 1.71E-02 1.54E-01 1.50E-02
0.004435 -2.81E-09 -0.01751 1.21E-21 0.145601 1.75E-02 1.52E-01 1.53E-02
0.004527 -7.33E-09 -0.01789 -2.93E-21 0.143853 1.79E-02 1.51E-01 1.56E-02
0.004619 -1.01E-08 -0.01827 -1.97E-20 0.141908 1.83E-02 1.49E-01 1.60E-02
0.004712 -1.29E-08 -0.01864 -3.65E-20 0.139964 1.86E-02 1.47E-01 1.63E-02
0.004804 -1.56E-08 -0.01902 -5.32E-20 0.13802 1.90E-02 1.46E-01 1.66E-02
0.004897 -1.84E-08 -0.0194 -7.00E-20 0.136075 1.94E-02 1.44E-01 1.69E-02
0.004989 -2.11E-08 -0.01977 -8.68E-20 0.134131 1.98E-02 1.42E-01 1.72E-02
0.005081 -2.33E-08 -0.02015 -1.03E-19 0.132143 2.01E-02 1.41E-01 1.75E-02
0.005174 -2.33E-08 -0.02053 -1.20E-19 0.129977 2.05E-02 1.39E-01 1.79E-02
0.005266 -2.33E-08 -0.0209 -1.36E-19 0.127812 2.09E-02 1.37E-01 1.82E-02
0.005359 -2.32E-08 -0.02128 -1.52E-19 0.125646 2.13E-02 1.35E-01 1.85E-02
0.005451 -2.32E-08 -0.02166 -1.69E-19 0.123481 2.17E-02 1.33E-01 1.88E-02
0.005543 -2.32E-08 -0.02203 -1.85E-19 0.121315 2.20E-02 1.31E-01 1.91E-02
0.005636 -2.31E-08 -0.02241 -1.53E-19 0.119008 2.24E-02 1.29E-01 1.94E-02
0.005728 -2.31E-08 -0.02279 -9.94E-20 0.116639 2.28E-02 1.27E-01 1.98E-02
0.00582 -2.31E-08 -0.02317 -4.61E-20 0.11427 2.32E-02 1.25E-01 2.01E-02
0.005913 -2.31E-08 -0.02354 7.25E-21 0.111901 2.35E-02 1.23E-01 2.04E-02
0.006005 -2.31E-08 -0.02392 6.06E-20 0.109532 2.39E-02 1.21E-01 2.07E-02
0.006098 -2.43E-08 -0.0243 6.83E-20 0.107042 2.43E-02 1.19E-01 2.10E-02
0.00619 -2.62E-08 -0.02468 5.12E-20 0.104487 2.47E-02 1.17E-01 2.13E-02
0.006282 -2.80E-08 -0.02505 3.41E-20 0.101931 2.51E-02 1.15E-01 2.16E-02
0.006375 -2.99E-08 -0.02543 1.70E-20 0.099375 2.54E-02 1.13E-01 2.20E-02
0.006467 -3.17E-08 -0.02581 -3.98E-22 0.096818 2.58E-02 1.10E-01 2.23E-02
0.00656 -3.26E-08 -0.02618 -8.46E-20 0.094091 2.62E-02 1.08E-01 2.26E-02
0.006652 -3.35E-08 -0.02656 -1.69E-19 0.091363 2.66E-02 1.06E-01 2.29E-02
0.006744 -3.44E-08 -0.02694 -2.53E-19 0.088636 2.69E-02 1.03E-01 2.32E-02
0.006837 -3.52E-08 -0.02732 -3.37E-19 0.085908 2.73E-02 1.01E-01 2.35E-02
0.006929 -3.55E-08 -0.02769 -2.26E-19 0.083064 2.77E-02 9.85E-02 2.39E-02
0.007021 -3.56E-08 -0.02807 -4.60E-20 0.080179 2.81E-02 9.61E-02 2.42E-02
0.007114 -3.57E-08 -0.02845 1.34E-19 0.077293 2.84E-02 9.36E-02 2.45E-02
0.007206 -3.57E-08 -0.02882 3.14E-19 0.074408 2.88E-02 9.11E-02 2.48E-02
0.007299 -3.54E-08 -0.0292 2.67E-19 0.071404 2.92E-02 8.86E-02 2.51E-02
0.007391 -3.50E-08 -0.02958 1.69E-19 0.068374 2.96E-02 8.60E-02 2.54E-02
0.007483 -3.46E-08 -0.02995 7.22E-20 0.065344 3.00E-02 8.34E-02 2.58E-02
0.007576 -3.41E-08 -0.03033 -3.46E-21 0.062285 3.03E-02 8.08E-02 2.61E-02
0.007668 -3.33E-08 -0.03071 -2.33E-21 0.059122 3.07E-02 7.81E-02 2.64E-02
0.007761 -3.26E-08 -0.03108 -1.20E-21 0.055959 3.11E-02 7.54E-02 2.67E-02
0.007853 -3.18E-08 -0.03146 -6.56E-23 0.052796 3.15E-02 7.26E-02 2.70E-02
0.007945 -3.07E-08 -0.03184 2.42E-23 0.049522 3.18E-02 6.99E-02 2.73E-02
0.008038 -2.97E-08 -0.03221 1.28E-23 0.046237 3.22E-02 6.71E-02 2.77E-02
0.00813 -2.86E-08 -0.03259 1.46E-24 0.042952 3.26E-02 6.43E-02 2.80E-02
0.008223 -2.70E-08 -0.03296 1.30E-19 0.03957 3.30E-02 6.14E-02 2.83E-02
0.008315 -2.54E-08 -0.03334 2.80E-19 0.036173 3.33E-02 5.85E-02 2.86E-02
0.008407 -2.37E-08 -0.03371 3.97E-19 0.032768 3.37E-02 5.56E-02 2.89E-02
0.0085 -2.22E-08 -0.03409 1.43E-19 0.029269 3.41E-02 5.26E-02 2.92E-02
0.008592 -2.07E-08 -0.03446 -1.11E-19 0.025769 3.45E-02 4.96E-02 2.96E-02
0.008684 -1.86E-08 -0.03484 -1.30E-19 0.022221 3.48E-02 4.66E-02 2.99E-02
0.008777 -1.61E-08 -0.03521 7.53E-20 0.018628 3.52E-02 4.35E-02 3.02E-02
0.008869 -1.34E-08 -0.03524 2.30E-19 0.01502 3.52E-02 4.04E-02 3.05E-02
0.008962 -9.92E-09 -0.03339 1.16E-19 0.01134 3.34E-02 3.73E-02 3.08E-02
0.009054 -6.47E-09 -0.03154 1.14E-21 0.00766 3.15E-02 3.42E-02 3.11E-02
0.009146 -4.24E-09 -0.02931 -6.24E-19 0.004879 2.93E-02 3.10E-02 3.14E-02
0.009239 0 0 0 0 0.00E+00 2.78E-02 -2.71E-03

  • BCs = symmetry and `theta = 10^o`
radial distance tau_X tau_Y tau_Z num_symmetry_U num_symmetry_tau analy_symmetry_U analy_symmetry_tau
0 9.31E-08 -0.00135 0.000118 0.187086 1.36E-03 1.90E-01 1.70E-04
9.96E-05 9.33E-08 -0.00152 0 0.186936 1.52E-03 1.90E-01 5.11E-04
0.000199 9.35E-08 -0.00167 0 0.186786 1.67E-03 1.89E-01 8.52E-04
0.000299 9.36E-08 -0.00183 0 0.186635 1.83E-03 1.89E-01 1.19E-03
0.000398 9.38E-08 -0.00198 0 0.186485 1.98E-03 1.89E-01 1.53E-03
0.000498 9.40E-08 -0.00213 0 0.186334 2.13E-03 1.89E-01 1.87E-03
0.000598 9.42E-08 -0.00229 0 0.186184 2.29E-03 1.89E-01 2.22E-03
0.000697 9.43E-08 -0.00244 0 0.186034 2.44E-03 1.89E-01 2.56E-03
0.000797 9.45E-08 -0.0026 0 0.185883 2.60E-03 1.88E-01 2.90E-03
0.000897 9.47E-08 -0.00275 0 0.185733 2.75E-03 1.88E-01 3.24E-03
0.000996 9.48E-08 -0.00291 0 0.185582 2.91E-03 1.88E-01 3.58E-03
0.001096 9.50E-08 -0.00306 0 0.185432 3.06E-03 1.87E-01 3.92E-03
0.001195 9.52E-08 -0.00321 0 0.185282 3.21E-03 1.87E-01 4.26E-03
0.001295 9.53E-08 -0.00337 0 0.185131 3.37E-03 1.86E-01 4.60E-03
0.001395 9.55E-08 -0.00352 0 0.184981 3.52E-03 1.86E-01 4.94E-03
0.001494 9.48E-08 -0.00461 2.08E-20 0.183119 4.61E-03 1.85E-01 5.28E-03
0.001594 9.48E-08 -0.00495 2.50E-20 0.182629 4.95E-03 1.85E-01 5.62E-03
0.001694 9.48E-08 -0.00529 2.91E-20 0.18214 5.29E-03 1.84E-01 5.96E-03
0.001793 9.48E-08 -0.00562 3.32E-20 0.18165 5.62E-03 1.83E-01 6.30E-03
0.001893 9.48E-08 -0.00596 3.73E-20 0.181161 5.96E-03 1.83E-01 6.65E-03
0.001992 9.23E-08 -0.00631 1.31E-20 0.180359 6.31E-03 1.82E-01 6.99E-03
0.002092 8.95E-08 -0.00667 -1.41E-20 0.179525 6.67E-03 1.81E-01 7.33E-03
0.002192 8.67E-08 -0.00702 -4.12E-20 0.178691 7.02E-03 1.80E-01 7.67E-03
0.002291 8.40E-08 -0.00737 -6.84E-20 0.177857 7.37E-03 1.80E-01 8.01E-03
0.002391 8.12E-08 -0.00772 -9.55E-20 0.177023 7.72E-03 1.79E-01 8.35E-03
0.00249 7.84E-08 -0.00807 -1.23E-19 0.176189 8.07E-03 1.78E-01 8.69E-03
0.00259 7.56E-08 -0.00843 -1.50E-19 0.175355 8.43E-03 1.77E-01 9.03E-03
0.00269 7.28E-08 -0.00878 -1.77E-19 0.174521 8.78E-03 1.76E-01 9.37E-03
0.002789 6.94E-08 -0.00913 -2.37E-19 0.17353 9.13E-03 1.75E-01 9.71E-03
0.002889 6.52E-08 -0.00948 -3.29E-19 0.172388 9.48E-03 1.74E-01 1.01E-02
0.002989 6.11E-08 -0.00982 -4.20E-19 0.171245 9.82E-03 1.73E-01 1.04E-02
0.003088 5.70E-08 -0.01017 -5.12E-19 0.170103 1.02E-02 1.71E-01 1.07E-02
0.003188 5.29E-08 -0.01052 -6.04E-19 0.168961 1.05E-02 1.70E-01 1.11E-02
0.003287 4.87E-08 -0.01087 -6.96E-19 0.167818 1.09E-02 1.69E-01 1.14E-02
0.003387 4.46E-08 -0.01122 -7.88E-19 0.166676 1.12E-02 1.68E-01 1.18E-02
0.003487 4.05E-08 -0.01156 -8.79E-19 0.165534 1.16E-02 1.66E-01 1.21E-02
0.003586 3.68E-08 -0.01191 -7.88E-19 0.164167 1.19E-02 1.65E-01 1.24E-02
0.003686 3.31E-08 -0.01226 -6.49E-19 0.16274 1.23E-02 1.64E-01 1.28E-02
0.003786 2.95E-08 -0.01261 -5.09E-19 0.161313 1.26E-02 1.62E-01 1.31E-02
0.003885 2.59E-08 -0.01296 -3.70E-19 0.159887 1.30E-02 1.61E-01 1.35E-02
0.003985 2.23E-08 -0.01331 -2.30E-19 0.15846 1.33E-02 1.59E-01 1.38E-02
0.004084 1.87E-08 -0.01366 -9.10E-20 0.157034 1.37E-02 1.58E-01 1.41E-02
0.004184 1.51E-08 -0.01401 4.84E-20 0.155607 1.40E-02 1.56E-01 1.45E-02
0.004284 1.25E-08 -0.01436 1.24E-19 0.153997 1.44E-02 1.55E-01 1.48E-02
0.004383 1.03E-08 -0.01471 1.71E-19 0.152307 1.47E-02 1.53E-01 1.52E-02
0.004483 8.20E-09 -0.01506 2.19E-19 0.150618 1.51E-02 1.51E-01 1.55E-02
0.004582 6.05E-09 -0.01541 2.67E-19 0.148929 1.54E-02 1.50E-01 1.58E-02
0.004682 3.90E-09 -0.01576 3.15E-19 0.14724 1.58E-02 1.48E-01 1.62E-02
0.004782 1.75E-09 -0.01612 3.62E-19 0.145551 1.61E-02 1.46E-01 1.65E-02
0.004881 -1.64E-15 -0.01647 3.91E-19 0.143816 1.65E-02 1.44E-01 1.69E-02
0.004981 -1.60E-15 -0.01682 3.33E-19 0.141885 1.68E-02 1.43E-01 1.72E-02
0.005081 -1.56E-15 -0.01717 2.76E-19 0.139953 1.72E-02 1.41E-01 1.76E-02
0.00518 -1.51E-15 -0.01753 2.19E-19 0.138021 1.75E-02 1.39E-01 1.79E-02
0.00528 -1.47E-15 -0.01788 1.61E-19 0.136089 1.79E-02 1.37E-01 1.82E-02
0.005379 -1.42E-15 -0.01823 1.04E-19 0.134157 1.82E-02 1.35E-01 1.86E-02
0.005479 -1.38E-15 -0.01859 5.60E-20 0.132181 1.86E-02 1.33E-01 1.89E-02
0.005579 -1.32E-15 -0.01894 4.55E-20 0.130025 1.89E-02 1.31E-01 1.93E-02
0.005678 -1.27E-15 -0.0193 3.49E-20 0.127869 1.93E-02 1.28E-01 1.96E-02
0.005778 -1.22E-15 -0.01965 2.43E-20 0.125713 1.97E-02 1.26E-01 1.99E-02
0.005878 -1.17E-15 -0.02 1.37E-20 0.123557 2.00E-02 1.24E-01 2.03E-02
0.005977 -1.12E-15 -0.02036 3.17E-21 0.121402 2.04E-02 1.22E-01 2.06E-02
0.006077 -1.28E-09 -0.02071 -2.70E-20 0.119101 2.07E-02 1.20E-01 2.10E-02
0.006176 -3.10E-09 -0.02107 -6.56E-20 0.116739 2.11E-02 1.17E-01 2.13E-02
0.006276 -4.92E-09 -0.02142 -1.04E-19 0.114377 2.14E-02 1.15E-01 2.16E-02
0.006376 -6.75E-09 -0.02178 -1.43E-19 0.112015 2.18E-02 1.12E-01 2.20E-02
0.006475 -8.57E-09 -0.02213 -1.81E-19 0.109653 2.21E-02 1.10E-01 2.23E-02
0.006575 -1.09E-08 -0.02249 -1.68E-19 0.107168 2.25E-02 1.08E-01 2.27E-02
0.006675 -1.35E-08 -0.02284 -1.26E-19 0.104616 2.28E-02 1.05E-01 2.30E-02
0.006774 -1.60E-08 -0.0232 -8.39E-20 0.102064 2.32E-02 1.03E-01 2.33E-02
0.006874 -1.86E-08 -0.02356 -4.18E-20 0.099512 2.36E-02 1.00E-01 2.37E-02
0.006973 -2.12E-08 -0.02391 3.53E-22 0.09696 2.39E-02 9.74E-02 2.40E-02
0.007073 -2.17E-08 -0.02427 7.49E-20 0.094233 2.43E-02 9.47E-02 2.44E-02
0.007173 -2.23E-08 -0.02462 1.49E-19 0.091507 2.46E-02 9.20E-02 2.47E-02
0.007272 -2.29E-08 -0.02498 2.24E-19 0.088781 2.50E-02 8.93E-02 2.50E-02
0.007372 -2.35E-08 -0.02533 2.99E-19 0.086055 2.53E-02 8.65E-02 2.54E-02
0.007471 -2.36E-08 -0.02569 2.58E-19 0.08321 2.57E-02 8.37E-02 2.57E-02
0.007571 -2.36E-08 -0.02604 1.77E-19 0.080324 2.60E-02 8.09E-02 2.61E-02
0.007671 -2.36E-08 -0.02639 9.57E-20 0.077438 2.64E-02 7.80E-02 2.64E-02
0.00777 -2.36E-08 -0.02675 1.45E-20 0.074551 2.67E-02 7.51E-02 2.68E-02
0.00787 -2.35E-08 -0.0271 -1.34E-22 0.071545 2.71E-02 7.21E-02 2.71E-02
0.00797 -2.35E-08 -0.02746 -2.05E-22 0.068512 2.75E-02 6.92E-02 2.74E-02
0.008069 -2.34E-08 -0.02781 -2.76E-22 0.065479 2.78E-02 6.61E-02 2.78E-02
0.008169 -2.33E-08 -0.02816 -3.58E-20 0.062416 2.82E-02 6.31E-02 2.81E-02
0.008268 -2.30E-08 -0.02852 -1.97E-19 0.059249 2.85E-02 6.00E-02 2.85E-02
0.008368 -2.27E-08 -0.02887 -3.59E-19 0.056081 2.89E-02 5.68E-02 2.88E-02
0.008468 -2.24E-08 -0.02922 -5.20E-19 0.052913 2.92E-02 5.36E-02 2.91E-02
0.008567 -2.14E-08 -0.02958 -8.08E-20 0.049634 2.96E-02 5.04E-02 2.95E-02
0.008667 -2.03E-08 -0.02993 4.17E-19 0.046343 2.99E-02 4.72E-02 2.98E-02
0.008767 -1.93E-08 -0.03028 9.15E-19 0.043052 3.03E-02 4.39E-02 3.02E-02
0.008866 -1.84E-08 -0.03063 6.73E-19 0.039662 3.06E-02 4.05E-02 3.05E-02
0.008966 -1.76E-08 -0.03098 3.22E-19 0.036259 3.10E-02 3.72E-02 3.08E-02
0.009065 -1.68E-08 -0.03133 0 0.032847 3.13E-02 3.38E-02 3.12E-02
0.009165 -1.57E-08 -0.03168 0 0.029339 3.17E-02 3.03E-02 3.15E-02
0.009265 -1.46E-08 -0.03203 0 0.025832 3.20E-02 2.69E-02 3.19E-02
0.009364 -1.33E-08 -0.03238 0 0.022276 3.24E-02 2.33E-02 3.22E-02
0.009464 -1.18E-08 -0.03273 0 0.018673 3.27E-02 1.98E-02 3.25E-02
0.009563 -1.01E-08 -0.03279 4.83E-20 0.015058 3.28E-02 1.62E-02 3.29E-02
0.009663 -8.09E-09 -0.0313 3.58E-19 0.011368 3.13E-02 1.26E-02 3.32E-02
0.009763 -6.06E-09 -0.0298 6.68E-19 0.007679 2.98E-02 8.89E-03 3.36E-02
0.009862 -3.75E-09 -0.02797 3.37E-19 0.004786 2.80E-02 5.18E-03 3.39E-02
0.009962 0 0 0 0 0.00E+00 1.44E-03 -1.30E-04

  • BCs = symmetry and `theta = 25^o`
radial distance tau_X tau_Y tau_Z num_symmetry_tau num_symmetry_U analy_symmetry_U analy_symmetry_tau
0 2.18E-07 -0.00136 0.000301 0.00139 0.187402 1.90E-01 1.67E-04
9.76E-05 2.17E-07 -0.0015 0.000333 0.001539 0.187255 1.90E-01 5.01E-04
0.000195 2.17E-07 -0.00174 0 0.001743 0.187093 1.89E-01 8.35E-04
0.000293 2.17E-07 -0.0019 0 0.001902 0.186939 1.89E-01 1.17E-03
0.000391 2.17E-07 -0.00206 0 0.002061 0.186785 1.89E-01 1.50E-03
0.000488 2.16E-07 -0.00222 0 0.002221 0.186631 1.89E-01 1.84E-03
0.000586 2.16E-07 -0.00238 0 0.00238 0.186477 1.89E-01 2.17E-03
0.000683 2.16E-07 -0.00254 0 0.002539 0.186323 1.89E-01 2.50E-03
0.000781 2.16E-07 -0.0027 0 0.002699 0.186169 1.88E-01 2.84E-03
0.000879 2.15E-07 -0.00286 0 0.002858 0.186015 1.88E-01 3.17E-03
0.000976 2.15E-07 -0.00302 0 0.003017 0.185861 1.88E-01 3.51E-03
0.001074 2.15E-07 -0.00318 0 0.003177 0.185707 1.87E-01 3.84E-03
0.001172 2.15E-07 -0.00334 0 0.003336 0.185553 1.87E-01 4.17E-03
0.001269 2.15E-07 -0.0035 0 0.003495 0.185399 1.86E-01 4.51E-03
0.001367 2.14E-07 -0.00365 0 0.003654 0.185245 1.86E-01 4.84E-03
0.001464 2.14E-07 -0.00381 0 0.003814 0.185091 1.85E-01 5.18E-03
0.001562 1.98E-07 -0.00511 0 0.005112 0.182862 1.85E-01 5.51E-03
0.00166 1.96E-07 -0.00546 0 0.005459 0.182365 1.84E-01 5.84E-03
0.001757 1.93E-07 -0.00581 0 0.005807 0.181868 1.84E-01 6.18E-03
0.001855 1.90E-07 -0.00615 0 0.006154 0.181371 1.83E-01 6.51E-03
0.001953 1.82E-07 -0.00651 -2.47E-20 0.006512 0.180561 1.82E-01 6.85E-03
0.00205 1.74E-07 -0.00687 -5.19E-20 0.006871 0.17972 1.82E-01 7.18E-03
0.002148 1.66E-07 -0.00723 -7.91E-20 0.00723 0.178878 1.81E-01 7.51E-03
0.002245 1.58E-07 -0.00759 -1.06E-19 0.007589 0.178037 1.80E-01 7.85E-03
0.002343 1.50E-07 -0.00795 -1.34E-19 0.007948 0.177196 1.79E-01 8.18E-03
0.002441 1.42E-07 -0.00831 -1.61E-19 0.008307 0.176354 1.78E-01 8.52E-03
0.002538 1.34E-07 -0.00867 -1.88E-19 0.008666 0.175513 1.77E-01 8.85E-03
0.002636 1.26E-07 -0.00903 -2.15E-19 0.009025 0.174671 1.76E-01 9.18E-03
0.002734 1.18E-07 -0.00938 -2.24E-19 0.009382 0.173673 1.75E-01 9.52E-03
0.002831 1.09E-07 -0.00974 -2.14E-19 0.009736 0.172524 1.74E-01 9.85E-03
0.002929 1.00E-07 -0.01009 -2.04E-19 0.010091 0.171375 1.73E-01 1.02E-02
0.003027 9.17E-08 -0.01044 -1.94E-19 0.010445 0.170225 1.72E-01 1.05E-02
0.003124 8.30E-08 -0.0108 -1.84E-19 0.010799 0.169076 1.71E-01 1.09E-02
0.003222 7.43E-08 -0.01115 -1.74E-19 0.011154 0.167926 1.70E-01 1.12E-02
0.003319 6.57E-08 -0.01151 -1.64E-19 0.011508 0.166777 1.69E-01 1.15E-02
0.003417 5.70E-08 -0.01186 -1.54E-19 0.011862 0.165627 1.67E-01 1.19E-02
0.003515 5.08E-08 -0.01222 -1.44E-19 0.012217 0.164254 1.66E-01 1.22E-02
0.003612 4.53E-08 -0.01257 -1.34E-19 0.012573 0.162821 1.65E-01 1.25E-02
0.00371 3.98E-08 -0.01293 -1.23E-19 0.012928 0.161388 1.63E-01 1.29E-02
0.003808 3.43E-08 -0.01328 -1.13E-19 0.013284 0.159955 1.62E-01 1.32E-02
0.003905 2.88E-08 -0.01364 -1.02E-19 0.013639 0.158522 1.61E-01 1.35E-02
0.004003 2.33E-08 -0.01399 -9.15E-20 0.013995 0.157089 1.59E-01 1.39E-02
0.0041 1.78E-08 -0.01435 -8.10E-20 0.01435 0.155656 1.58E-01 1.42E-02
0.004198 1.41E-08 -0.01471 3.12E-20 0.014707 0.154039 1.56E-01 1.45E-02
0.004296 1.13E-08 -0.01506 1.87E-19 0.015064 0.152344 1.55E-01 1.49E-02
0.004393 8.50E-09 -0.01542 3.43E-19 0.015421 0.15065 1.53E-01 1.52E-02
0.004491 5.68E-09 -0.01578 4.99E-19 0.015779 0.148955 1.51E-01 1.55E-02
0.004589 2.86E-09 -0.01614 6.55E-19 0.016136 0.14726 1.50E-01 1.59E-02
0.004686 4.53E-11 -0.01649 8.10E-19 0.016493 0.145565 1.48E-01 1.62E-02
0.004784 -2.32E-09 -0.01685 9.08E-19 0.01685 0.143826 1.46E-01 1.65E-02
0.004881 -2.67E-09 -0.01721 7.52E-19 0.017209 0.141889 1.44E-01 1.69E-02
0.004979 -3.03E-09 -0.01757 5.95E-19 0.017568 0.139953 1.43E-01 1.72E-02
0.005077 -3.38E-09 -0.01793 4.39E-19 0.017926 0.138016 1.41E-01 1.75E-02
0.005174 -3.73E-09 -0.01828 2.82E-19 0.018285 0.13608 1.39E-01 1.79E-02
0.005272 -4.09E-09 -0.01864 1.25E-19 0.018644 0.134143 1.37E-01 1.82E-02
0.00537 -4.71E-09 -0.019 -2.57E-20 0.019003 0.132162 1.35E-01 1.85E-02
0.005467 -6.40E-09 -0.01936 -1.55E-19 0.019362 0.130003 1.33E-01 1.89E-02
0.005565 -8.08E-09 -0.01972 -2.83E-19 0.019722 0.127844 1.31E-01 1.92E-02
0.005663 -9.77E-09 -0.02008 -4.12E-19 0.020082 0.125685 1.29E-01 1.95E-02
0.00576 -1.15E-08 -0.02044 -5.41E-19 0.020441 0.123525 1.27E-01 1.99E-02
0.005858 -1.31E-08 -0.0208 -6.70E-19 0.020801 0.121366 1.24E-01 2.02E-02
0.005955 -1.65E-08 -0.02116 -6.10E-19 0.021161 0.119063 1.22E-01 2.05E-02
0.006053 -2.07E-08 -0.02152 -4.70E-19 0.021522 0.116699 1.20E-01 2.09E-02
0.006151 -2.48E-08 -0.02188 -3.30E-19 0.021882 0.114334 1.18E-01 2.12E-02
0.006248 -2.89E-08 -0.02224 -1.90E-19 0.022243 0.11197 1.16E-01 2.15E-02
0.006346 -3.31E-08 -0.0226 -4.95E-20 0.022603 0.109605 1.13E-01 2.19E-02
0.006444 -3.57E-08 -0.02296 0 0.022963 0.107119 1.11E-01 2.22E-02
0.006541 -3.76E-08 -0.02332 0 0.023324 0.104565 1.08E-01 2.25E-02
0.006639 -3.94E-08 -0.02368 0 0.023685 0.102012 1.06E-01 2.29E-02
0.006736 -4.13E-08 -0.02405 0 0.024045 0.099459 1.04E-01 2.32E-02
0.006834 -4.31E-08 -0.02441 6.93E-22 0.024406 0.096905 1.01E-01 2.35E-02
0.006932 -4.24E-08 -0.02477 1.47E-19 0.024766 0.094178 9.85E-02 2.39E-02
0.007029 -4.17E-08 -0.02513 2.94E-19 0.025126 0.091452 9.59E-02 2.42E-02
0.007127 -4.09E-08 -0.02549 4.41E-19 0.025487 0.088725 9.33E-02 2.45E-02
0.007225 -4.02E-08 -0.02585 5.88E-19 0.025847 0.085998 9.06E-02 2.49E-02
0.007322 -3.99E-08 -0.02621 5.06E-19 0.026207 0.083154 8.79E-02 2.52E-02
0.00742 -3.97E-08 -0.02657 3.44E-19 0.026567 0.080268 8.52E-02 2.55E-02
0.007517 -3.94E-08 -0.02693 1.82E-19 0.026927 0.077382 8.24E-02 2.59E-02
0.007615 -3.92E-08 -0.02729 1.98E-20 0.027287 0.074497 7.96E-02 2.62E-02
0.007713 -3.85E-08 -0.02765 -3.20E-19 0.027647 0.071491 7.68E-02 2.66E-02
0.00781 -3.76E-08 -0.02801 -7.00E-19 0.028006 0.068459 7.39E-02 2.69E-02
0.007908 -3.67E-08 -0.02837 -1.08E-18 0.028365 0.065427 7.10E-02 2.72E-02
0.008006 -3.58E-08 -0.02872 -1.34E-18 0.028724 0.062366 6.81E-02 2.76E-02
0.008103 -3.50E-08 -0.02908 -1.20E-18 0.029083 0.0592 6.51E-02 2.79E-02
0.008201 -3.42E-08 -0.02944 -1.07E-18 0.029442 0.056035 6.21E-02 2.82E-02
0.008299 -3.34E-08 -0.0298 -9.26E-19 0.029801 0.052869 5.90E-02 2.86E-02
0.008396 -3.26E-08 -0.03016 -6.95E-19 0.030159 0.049591 5.59E-02 2.89E-02
0.008494 -3.17E-08 -0.03052 -4.56E-19 0.030516 0.046303 5.28E-02 2.92E-02
0.008591 -3.09E-08 -0.03087 -2.16E-19 0.030874 0.043014 4.96E-02 2.96E-02
0.008689 -2.92E-08 -0.03123 -3.75E-19 0.031232 0.039627 4.64E-02 2.99E-02
0.008787 -2.75E-08 -0.03159 -5.93E-19 0.031589 0.036226 4.32E-02 3.02E-02
0.008884 -2.57E-08 -0.03195 -7.56E-19 0.031946 0.032817 3.99E-02 3.06E-02
0.008982 -2.39E-08 -0.0323 -2.90E-19 0.032302 0.029312 3.66E-02 3.09E-02
0.00908 -2.21E-08 -0.03266 1.75E-19 0.032659 0.025808 3.33E-02 3.12E-02
0.009177 -2.00E-08 -0.03301 7.48E-20 0.033015 0.022255 2.99E-02 3.16E-02
0.009275 -1.74E-08 -0.03337 -5.63E-19 0.033371 0.018656 2.65E-02 3.19E-02
0.009372 -1.48E-08 -0.03342 -1.03E-18 0.033421 0.015044 2.30E-02 3.22E-02
0.00947 -1.14E-08 -0.03182 -5.69E-19 0.031817 0.011358 1.96E-02 3.26E-02
0.009568 -8.07E-09 -0.03021 -1.09E-19 0.030213 0.007672 1.60E-02 3.29E-02
0.009665 -5.20E-09 -0.02827 -5.44E-19 0.028269 0.004816 1.25E-02 3.32E-02
0.009763 0 0 0 0 0 8.88E-03 -8.21E-04

  • BCs = symmetry and `theta = 45^o`
radial distance tau_X tau_Y tau_Z num_symmtery_U num_symmetry_tau analy_symmtery_U analy_symmetry_tau
0 2.13E-07 -0.0013 0.000539 0.187778 1.41E-03 1.90E-01 1.58E-04
9.24E-05 2.11E-07 -0.00142 0.00059 0.187645 1.54E-03 1.90E-01 4.74E-04
0.000185 2.08E-07 -0.00186 2.22E-21 0.187466 1.86E-03 1.89E-01 7.90E-04
0.000277 2.06E-07 -0.00203 3.33E-21 0.18731 2.03E-03 1.89E-01 1.11E-03
0.00037 2.04E-07 -0.0022 4.44E-21 0.187154 2.20E-03 1.89E-01 1.42E-03
0.000462 2.01E-07 -0.00238 5.55E-21 0.186998 2.38E-03 1.89E-01 1.74E-03
0.000554 1.99E-07 -0.00255 6.66E-21 0.186842 2.55E-03 1.89E-01 2.05E-03
0.000647 1.97E-07 -0.00272 7.77E-21 0.186686 2.72E-03 1.89E-01 2.37E-03
0.000739 1.94E-07 -0.00289 8.88E-21 0.18653 2.89E-03 1.88E-01 2.69E-03
0.000831 1.92E-07 -0.00306 9.99E-21 0.186374 3.06E-03 1.88E-01 3.00E-03
0.000924 1.90E-07 -0.00323 1.11E-20 0.186218 3.23E-03 1.88E-01 3.32E-03
0.001016 1.87E-07 -0.0034 1.22E-20 0.186062 3.40E-03 1.88E-01 3.63E-03
0.001109 1.85E-07 -0.00357 1.33E-20 0.185906 3.57E-03 1.87E-01 3.95E-03
0.001201 1.82E-07 -0.00374 1.44E-20 0.18575 3.74E-03 1.87E-01 4.27E-03
0.001293 1.80E-07 -0.00391 1.55E-20 0.185594 3.91E-03 1.86E-01 4.58E-03
0.001386 1.78E-07 -0.00408 1.66E-20 0.185438 4.08E-03 1.86E-01 4.90E-03
0.001478 1.75E-07 -0.00425 1.78E-20 0.185282 4.25E-03 1.85E-01 5.21E-03
0.001571 1.73E-07 -0.00442 1.89E-20 0.185126 4.42E-03 1.85E-01 5.53E-03
0.001663 1.58E-07 -0.0062 2.25E-20 0.182178 6.20E-03 1.84E-01 5.85E-03
0.001755 1.54E-07 -0.00657 2.39E-20 0.181676 6.57E-03 1.84E-01 6.16E-03
0.001848 1.48E-07 -0.00695 1.90E-20 0.180858 6.95E-03 1.83E-01 6.48E-03
0.00194 1.43E-07 -0.00734 1.34E-20 0.180008 7.34E-03 1.82E-01 6.80E-03
0.002033 1.37E-07 -0.00772 7.82E-21 0.179158 7.72E-03 1.82E-01 7.11E-03
0.002125 1.32E-07 -0.0081 2.23E-21 0.178308 8.10E-03 1.81E-01 7.43E-03
0.002217 1.26E-07 -0.00848 -3.35E-21 0.177458 8.48E-03 1.80E-01 7.74E-03
0.00231 1.20E-07 -0.00886 -8.94E-21 0.176608 8.86E-03 1.79E-01 8.06E-03
0.002402 1.15E-07 -0.00924 -1.45E-20 0.175758 9.24E-03 1.79E-01 8.38E-03
0.002494 1.09E-07 -0.00963 -2.01E-20 0.174909 9.63E-03 1.78E-01 8.69E-03
0.002587 1.04E-07 -0.01 -1.68E-20 0.173902 1.00E-02 1.77E-01 9.01E-03
0.002679 9.75E-08 -0.01038 -4.81E-21 0.172742 1.04E-02 1.76E-01 9.32E-03
0.002772 9.15E-08 -0.01075 7.17E-21 0.171583 1.08E-02 1.75E-01 9.64E-03
0.002864 8.55E-08 -0.01113 1.91E-20 0.170424 1.11E-02 1.74E-01 9.96E-03
0.002956 7.95E-08 -0.01151 3.11E-20 0.169265 1.15E-02 1.73E-01 1.03E-02
0.003049 7.34E-08 -0.01188 4.31E-20 0.168105 1.19E-02 1.72E-01 1.06E-02
0.003141 6.74E-08 -0.01226 5.51E-20 0.166946 1.23E-02 1.71E-01 1.09E-02
0.003234 6.14E-08 -0.01263 6.70E-20 0.165787 1.26E-02 1.70E-01 1.12E-02
0.003326 5.63E-08 -0.01301 6.27E-20 0.164403 1.30E-02 1.69E-01 1.15E-02
0.003418 5.14E-08 -0.01338 5.41E-20 0.16296 1.34E-02 1.67E-01 1.19E-02
0.003511 4.66E-08 -0.01376 4.55E-20 0.161518 1.38E-02 1.66E-01 1.22E-02
0.003603 4.17E-08 -0.01413 3.69E-20 0.160075 1.41E-02 1.65E-01 1.25E-02
0.003696 3.68E-08 -0.01451 2.83E-20 0.158632 1.45E-02 1.64E-01 1.28E-02
0.003788 3.19E-08 -0.01488 1.97E-20 0.157189 1.49E-02 1.62E-01 1.31E-02
0.00388 2.71E-08 -0.01526 1.10E-20 0.155746 1.53E-02 1.61E-01 1.34E-02
0.003973 2.21E-08 -0.01563 7.57E-21 0.15412 1.56E-02 1.60E-01 1.37E-02
0.004065 1.71E-08 -0.01601 6.30E-21 0.152417 1.60E-02 1.58E-01 1.41E-02
0.004157 1.22E-08 -0.01638 5.03E-21 0.150713 1.64E-02 1.57E-01 1.44E-02
0.00425 7.20E-09 -0.01676 3.76E-21 0.149009 1.68E-02 1.55E-01 1.47E-02
0.004342 2.22E-09 -0.01714 2.48E-21 0.147305 1.71E-02 1.54E-01 1.50E-02
0.004435 -2.76E-09 -0.01751 1.21E-21 0.145601 1.75E-02 1.52E-01 1.53E-02
0.004527 -7.33E-09 -0.01789 -2.93E-21 0.143853 1.79E-02 1.51E-01 1.56E-02
0.004619 -1.01E-08 -0.01827 -1.97E-20 0.141908 1.83E-02 1.49E-01 1.60E-02
0.004712 -1.29E-08 -0.01864 -3.65E-20 0.139964 1.86E-02 1.47E-01 1.63E-02
0.004804 -1.56E-08 -0.01902 -5.32E-20 0.13802 1.90E-02 1.46E-01 1.66E-02
0.004897 -1.84E-08 -0.0194 -7.00E-20 0.136075 1.94E-02 1.44E-01 1.69E-02
0.004989 -2.11E-08 -0.01977 -8.68E-20 0.134131 1.98E-02 1.42E-01 1.72E-02
0.005081 -2.34E-08 -0.02015 -1.03E-19 0.132143 2.01E-02 1.41E-01 1.75E-02
0.005174 -2.38E-08 -0.02053 -1.20E-19 0.129977 2.05E-02 1.39E-01 1.79E-02
0.005266 -2.41E-08 -0.0209 -1.36E-19 0.127812 2.09E-02 1.37E-01 1.82E-02
0.005359 -2.45E-08 -0.02128 -1.52E-19 0.125646 2.13E-02 1.35E-01 1.85E-02
0.005451 -2.49E-08 -0.02166 -1.69E-19 0.123481 2.17E-02 1.33E-01 1.88E-02
0.005543 -2.52E-08 -0.02203 -1.85E-19 0.121315 2.20E-02 1.31E-01 1.91E-02
0.005636 -2.63E-08 -0.02241 -1.53E-19 0.119008 2.24E-02 1.29E-01 1.94E-02
0.005728 -2.77E-08 -0.02279 -9.94E-20 0.116639 2.28E-02 1.27E-01 1.98E-02
0.00582 -2.91E-08 -0.02317 -4.61E-20 0.11427 2.32E-02 1.25E-01 2.01E-02
0.005913 -3.05E-08 -0.02354 7.24E-21 0.111901 2.35E-02 1.23E-01 2.04E-02
0.006005 -3.19E-08 -0.02392 6.06E-20 0.109532 2.39E-02 1.21E-01 2.07E-02
0.006098 -3.33E-08 -0.0243 6.83E-20 0.107043 2.43E-02 1.19E-01 2.10E-02
0.00619 -3.48E-08 -0.02468 5.12E-20 0.104487 2.47E-02 1.17E-01 2.13E-02
0.006282 -3.63E-08 -0.02505 3.41E-20 0.101931 2.51E-02 1.15E-01 2.16E-02
0.006375 -3.78E-08 -0.02543 1.70E-20 0.099375 2.54E-02 1.13E-01 2.20E-02
0.006467 -3.93E-08 -0.02581 -3.98E-22 0.096819 2.58E-02 1.10E-01 2.23E-02
0.00656 -3.85E-08 -0.02618 -8.46E-20 0.094091 2.62E-02 1.08E-01 2.26E-02
0.006652 -3.78E-08 -0.02656 -1.69E-19 0.091363 2.66E-02 1.06E-01 2.29E-02
0.006744 -3.71E-08 -0.02694 -2.53E-19 0.088636 2.69E-02 1.03E-01 2.32E-02
0.006837 -3.63E-08 -0.02732 -3.37E-19 0.085908 2.73E-02 1.01E-01 2.35E-02
0.006929 -3.57E-08 -0.02769 -2.26E-19 0.083064 2.77E-02 9.85E-02 2.39E-02
0.007021 -3.52E-08 -0.02807 -4.60E-20 0.080179 2.81E-02 9.61E-02 2.42E-02
0.007114 -3.46E-08 -0.02845 1.34E-19 0.077294 2.84E-02 9.36E-02 2.45E-02
0.007206 -3.41E-08 -0.02882 3.14E-19 0.074408 2.88E-02 9.11E-02 2.48E-02
0.007299 -3.38E-08 -0.0292 2.67E-19 0.071404 2.92E-02 8.86E-02 2.51E-02
0.007391 -3.35E-08 -0.02958 1.69E-19 0.068374 2.96E-02 8.60E-02 2.54E-02
0.007483 -3.33E-08 -0.02995 7.22E-20 0.065344 3.00E-02 8.34E-02 2.58E-02
0.007576 -3.30E-08 -0.03033 -3.45E-21 0.062285 3.03E-02 8.08E-02 2.61E-02
0.007668 -3.23E-08 -0.03071 -2.33E-21 0.059122 3.07E-02 7.81E-02 2.64E-02
0.007761 -3.15E-08 -0.03108 -1.20E-21 0.055959 3.11E-02 7.54E-02 2.67E-02
0.007853 -3.08E-08 -0.03146 -6.53E-23 0.052796 3.15E-02 7.26E-02 2.70E-02
0.007945 -2.97E-08 -0.03184 2.42E-23 0.049522 3.18E-02 6.99E-02 2.73E-02
0.008038 -2.86E-08 -0.03221 1.29E-23 0.046237 3.22E-02 6.71E-02 2.77E-02
0.00813 -2.74E-08 -0.03259 1.46E-24 0.042952 3.26E-02 6.43E-02 2.80E-02
0.008223 -2.62E-08 -0.03296 1.30E-19 0.03957 3.30E-02 6.14E-02 2.83E-02
0.008315 -2.50E-08 -0.03334 2.80E-19 0.036173 3.33E-02 5.85E-02 2.86E-02
0.008407 -2.37E-08 -0.03371 3.97E-19 0.032768 3.37E-02 5.56E-02 2.89E-02
0.0085 -2.13E-08 -0.03409 1.43E-19 0.029269 3.41E-02 5.26E-02 2.92E-02
0.008592 -1.90E-08 -0.03446 -1.11E-19 0.025769 3.45E-02 4.96E-02 2.96E-02
0.008684 -1.66E-08 -0.03484 -1.30E-19 0.022221 3.48E-02 4.66E-02 2.99E-02
0.008777 -1.41E-08 -0.03521 7.53E-20 0.018628 3.52E-02 4.35E-02 3.02E-02
0.008869 -1.16E-08 -0.03524 2.30E-19 0.01502 3.52E-02 4.04E-02 3.05E-02
0.008962 -9.14E-09 -0.03339 1.16E-19 0.01134 3.34E-02 3.73E-02 3.08E-02
0.009054 -6.66E-09 -0.03154 1.14E-21 0.00766 3.15E-02 3.42E-02 3.11E-02
0.009146 -4.32E-09 -0.02931 -6.24E-19 0.004879 2.93E-02 3.10E-02 3.14E-02
0.009239 0 0 0 0 0.00E+00 2.78E-02 -2.71E-03

  • pressure values for different angles and boundary conditions and also comparison between analytical and numerical values
Boundary conditions theta `P_(max)` `P_(min)` `DeltaP_{Numerical}` `DeltaP_{Analytical}`
wedge 10 0.00187018 0.000138345 1.731835 1.71046
25 0.00194545 0.000143928 1.801522 1.71046
45 0.00216826 0.000160469 2.007791 1.71046
symmetry 10 0.00187017 0.00013839 1.73178 1.71046
25 0.00194545 0.000143936 1.801514 1.71046
45 0.00216822 0.000160467 2.007753 1.71046

6. Conclusion -

  • The shear stress is zero at the centre and maximum at the wall because there is friction at the wall
  • As we increase theta we are going away from our analytical/accurate results as seen from above plots
  • We get same results for both wedge and symmetry boundary conditions for particular value of theta
  • If we increase the number of points along y direction for calculating velocity and shear stress , the plot will be same
  • The max velocity and max shear stress values for both anaytical and numerical are as follows
Boundary conditions theta U_numerical U_analytical tau_numerical tau_analytical
wedge 10 0.187231 0.19 0.032789 0.0339
25 0.187411 0.19 0.0334 0.0332
45 0.18778 0.19 0.0352 0.0314
symmetry 10 0.187086 0.19 0.0328 0.0339
25 0.187402 0.19 0.0334206 0.0332
45 0.187778 0.19 0.0352 0.0314
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