Simulation flow through pipe by Wedge boundary Condition in OpenFoam

Title: Simulation flow through a pipe in openFoam

Objective:

1.To calculate length of the pipe using the entry length formula for laminar flow through a pipe

2.To show that entry length is sufficient to produce a fully developed flow.

3.To write a program in Matlab that can generate the computational mesh automatically for any wedge angle and grading schemes.

4.To show that the velocity profile is fully developed.

Theory

The main objecive of this challange is to simulate the laminar flow through a pipe by use of openFoam tool. There is use of wedge shaped domain to peroform the simulation, for that there is made Matlab code that can generate the computational mesh automatically for any wedge angle.

                For any simulation flow through pipe it is very important to calculate the velocity, pressure & shear stress distribution along the length, for that hagen poiseuille define that. When fluid is travel some distance after entering in to the pipe & that region called entrance length or hydrodynamic region. once fluid reach entire wall & in pipe there is developement of parabolic shape f fluid flow.

 Hagen poiseuille derived some equation & that equation will be use for comaprison of theoretical & computational result.

1. Entrance length of pipe

Le= 0.06* Re*D

2.Total Length

L=Le+0.3

3. Average velocity or intial velocity

`V_(avg)= (R_(e)*mu)/(rho*D)`

Re= Reynold No.

mu=Dynamic Viscosity

rho=density of water

D=Diameter of pipe

4. Maximum Velocity

`V_(max)= 2*V_(avg)`

5. Pressure distribution

`P_(1)-P_(2)= (32*mu*V_(avg)*L)/(D^2)`

5.Kinematic Pressure

`Δ_(K)= (ΔP)/(rho)`

6. Shear stress

`τ= (2*mu*V_(max))/(R^2)`

Given data for simulation

1.Re=2100 for laminar flow

2.working fluid is water

3. density of water= 997  kg/m3

4.Dynamic viscosity =0.89*10^-03

5. Diameter of pipe (D)= 0.020m (Assumed)

Calculate the following data by above formula

1.Entrance length of pipe =2.52m

2.Total length of pipw= 2.82m

3.Vavg=0.09373m/sec

4.Vmax=0.1874m/s

5.Pressure diffrence= 18.819pa

6.Kinematic pressure=0.018876m2/sec2

7.shear strees= 3.32 N/m2

8.Use of icoFoam solver.

Matlab code for prepare blockMeshDict file of openFoam

Matlab Code

close all 
clear all 
clc

% given data
Re=2100;             % Reynold No.
mu=0.89e-3;          % Dynamic viscosity
D=0.020;             % Diameter of pipe D=20mm
R=(D/2);            % Raduis of pipe
rho=997 ;            % Density of water
Le= (0.06*Re*D);     % Entry lenght of pipe
L= Le+0.3 ;          % Length of pipe
theta=4;             % Angle of wedge(less than 5 degree)
 % writing the  header part for blockMeshDict file in Matlab
h1=\'/*--------------------------------*- C++ -*----------------------------------*\\\';
h2=\'  =========                 |\';                                               
h3=\'  \\\\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox\';
h4=\'   \\\\    /   O peration     | Website:  https://openfoam.org\';
h5=\'    \\\\  /    A nd           | Version:  7\';
h6=\'     \\\\/     M anipulation  |\';
h7=\'\\*---------------------------------------------------------------------------*/\';
h8=\'FoamFile\';
h9=\'{\';
h10=\'    version     2.0;\';
h11=\'    format      ascii;\';
h12=\'    class       dictionary;\';
h13=\'    object      blockMeshDict;\';
h14=\'}\';
h15=\'// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //\';

conv=\'convertToMeters 1;\';
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 0 0];
v4=[L R*cosd(theta/2) R*sind(theta/2)];
v5=[L R*cosd(theta/2) -R*sind(theta/2)];
block=\'hex (0 3 5 2 0 3 4 1) (300 10 1) simpleGrading (1 1 1)\';
inlet=\'(0 1 2 0)\';
outlet=\'(3 5 4 3)\';
top=\'(1 4 5 2)\';
front=\'(0 3 4 1)\';
back=\'(0 2 5 3)\';
axis=\'(0 3 3 0)\';
s1=blanks(4);
s2=blanks(7);
s3=blanks(11);
s4=blanks(3);

% Writing blockMeshDict file
f1=fopen(\'blockMeshDict.txt\',\'w\');
fprintf(f1,\'%s\\n\',h1);
fprintf(f1,\'%s\\n\',h2);
fprintf(f1,\'%s\\n\',h3);
fprintf(f1,\'%s\\n\',h4);
fprintf(f1,\'%s\\n\',h5);
fprintf(f1,\'%s\\n\',h6);
fprintf(f1,\'%s\\n\',h7);
fprintf(f1,\'%s\\n\',h8);
fprintf(f1,\'%s\\n\',h9);
fprintf(f1,\'%s\\n\',h10);
fprintf(f1,\'%s\\n\',h11);
fprintf(f1,\'%s\\n\',h12);
fprintf(f1,\'%s\\n\',h13);
fprintf(f1,\'%s\\n\',h14);
fprintf(f1,\'%s\\n\',h15);
fprintf(f1,\'\\n\');
fprintf(f1,\'%s\\n\',conv);
fprintf(f1,\'\\n\');
fprintf(f1,\'%s\\n\',\'vertices\');
fprintf(f1,\'%s\\n\',\'(\');
fprintf(f1,\'%s(%d %d %d)\\n\',s1,v0(1),v0(2),v0(3));
fprintf(f1,\'%s(%d %d %d)\\n\',s1,v1(1),v1(2),v1(3));
fprintf(f1,\'%s(%d %d %d)\\n\',s1,v2(1),v2(2),v2(3));
fprintf(f1,\'%s(%d %d %d)\\n\',s1,v3(1),v3(2),v3(3));
fprintf(f1,\'%s(%d %d %d)\\n\',s1,v4(1),v4(2),v4(3));
fprintf(f1,\'%s(%d %d %d)\\n\',s1,v5(1),v5(2),v5(3));
fprintf(f1,\'%s\\n\',\');\');
fprintf(f1,\'\\n\');
fprintf(f1, \'%s\\n\',\'blocks\');
fprintf(f1,\'%s\\n\',\'(\');
fprintf(f1,\'%s %s\\n\',s4,block);
fprintf(f1,\'\\n\');
fprintf(f1,\'%s\\n\',\');\');
fprintf(f1,\'\\n\');
fprintf(f1,\'%s\\n\',\'edges\');
fprintf(f1,\'%s\\n\',\'(\');
fprintf(f1,\'%s %s %d %d (%d %d %d)\\n\',s4,\'arc\',1,2,0,R,0);
fprintf(f1,\'%s %s %d %d (%d %d %d)\\n\',s4,\'arc\',4,5,L,R,0);
fprintf(f1,\'%s\\n\',\');\');
fprintf(f1,\'\\n\');
fprintf(f1,\'%s\\n\',\'boundary\');
fprintf(f1,\'%s\\n\',\'(\');
fprintf(f1,\'%s %s\\n\',s4,\'inlet\');
fprintf(f1,\'%s %s\\n\',s4,\'{\');
fprintf(f1,\'%s %s\\n\',s2,\'type patch;\')
fprintf(f1,\'%s %s\\n\',s2,\'faces\')
fprintf(f1,\'%s %s\\n\',s2,\'(\');
fprintf(f1,\'%s (%d %d %d %d)\\n\',s3,0,1,2,0);
fprintf(f1,\'%s %s\\n\',s2,\');\');
fprintf(f1,\'%s %s\\n\',s4,\'}\');
fprintf(f1,\'%s %s\\n\',s4,\'outlet\');
fprintf(f1,\'%s %s\\n\',s4,\'{\');
fprintf(f1,\'%s %s\\n\',s2,\'type patch;\')
fprintf(f1,\'%s %s\\n\',s2,\'faces\')
fprintf(f1,\'%s %s\\n\',s2,\'(\');
fprintf(f1,\'%s (%d %d %d %d)\\n\',s3,3,5,4,3);
fprintf(f1,\'%s %s\\n\',s2,\');\');
fprintf(f1,\'%s %s\\n\',s4,\'}\');
fprintf(f1,\'%s %s\\n\',s4,\'top\');
fprintf(f1,\'%s %s\\n\',s4,\'{\');
fprintf(f1,\'%s %s\\n\',s2,\'type wall;\')
fprintf(f1,\'%s %s\\n\',s2,\'faces\')
fprintf(f1,\'%s %s\\n\',s2,\'(\');
fprintf(f1,\'%s (%d %d %d %d)\\n\',s3,1,4,5,2);
fprintf(f1,\'%s %s\\n\',s2,\');\');
fprintf(f1,\'%s %s\\n\',s4,\'}\');
fprintf(f1,\'%s %s\\n\',s4,\'front\');
fprintf(f1,\'%s %s\\n\',s4,\'{\');
fprintf(f1,\'%s %s\\n\',s2,\'type wedge;\')
fprintf(f1,\'%s %s\\n\',s2,\'faces\')
fprintf(f1,\'%s %s\\n\',s2,\'(\');
fprintf(f1,\'%s (%d %d %d %d)\\n\',s3,0,3,4,1);
fprintf(f1,\'%s %s\\n\',s2,\');\');
fprintf(f1,\'%s %s\\n\',s4,\'}\');
fprintf(f1,\'%s %s\\n\',s4,\'back\');
fprintf(f1,\'%s %s\\n\',s4,\'{\');
fprintf(f1,\'%s %s\\n\',s2,\'type wedge;\')
fprintf(f1,\'%s %s\\n\',s2,\'faces\')
fprintf(f1,\'%s %s\\n\',s2,\'(\');
fprintf(f1,\'%s (%d %d %d %d)\\n\',s3,0,2,5,3);
fprintf(f1,\'%s %s\\n\',s2,\');\');
fprintf(f1,\'%s %s\\n\',s4,\'}\');
fprintf(f1,\'%s %s\\n\',s4,\'axis\');
fprintf(f1,\'%s %s\\n\',s4,\'{\');
fprintf(f1,\'%s %s\\n\',s2,\'type empty;\')
fprintf(f1,\'%s %s\\n\',s2,\'faces\')
fprintf(f1,\'%s %s\\n\',s2,\'(\');
fprintf(f1,\'%s (%d %d %d %d)\\n\',s3,0,3,3,0);
fprintf(f1,\'%s %s\\n\',s2,\');\');
fprintf(f1,\'%s %s\\n\',s4,\'}\');
fprintf(f1,\'%s\\n\',\');\');
fprintf(f1,\'\\n\');
fprintf(f1,\'%s\\n\',\'mergePatchPairs\');
fprintf(f1,\'%s\\n\',\'(\');
fprintf(f1,\'%s\\n\',\');\');
h16=\'// ************************************************************************* //\';
fprintf(f1,\'%s\\n\',h16);
fclose(f1);

BlockMeshDict file

/*--------------------------------*- 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)
    (0 9.993908e-03 3.489950e-04)
    (0 9.993908e-03 -3.489950e-04)
    (2.820000e+00 0 0)
    (2.820000e+00 9.993908e-03 3.489950e-04)
    (2.820000e+00 9.993908e-03 -3.489950e-04)
);

blocks
(
    hex (0 3 5 2 0 3 4 1) (300 10 1) simpleGrading (1 1 1)

);

edges
(
    arc 1 2 (0 1.000000e-02 0)
    arc 4 5 (2.820000e+00 1.000000e-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 wedge;
        faces
        (
            (0 3 4 1)
        );
    }
    back
    {
        type wedge;
        faces
        (
            (0 2 5 3)
        );
    }
    axis
    {
        type empty;
        faces
        (
            (0 3 3 0)
        );
    }
);

mergePatchPairs
(
);
// ************************************************************************* //

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] 8.92e-7;


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

Velocity file

/*--------------------------------*- 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 (0.0937 0 0);

boundaryField
{
    inlet
    {
        type            fixedValue;
        value           uniform (0.0937 0 0);
    }

    outlet
    {
        type            zeroGradient;
	
    }

    axis
    {
        type            empty;
    }
    
    top
    {
        type            noSlip;
    }

    front
    {
        type            wedge;
    }

    back
    {
        type            wedge;
    }
}

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

Preesure file

/*--------------------------------*- 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 0.018;
    }

    axis
    {
        type            empty;
    }
    
    top
    {
        type            zeroGradient;
    }

    front
    {
        type            wedge;
    }

    back
    {
        type            wedge;
    }
}

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

Result obtained for Wedge BC at 4 degree of wedge angle

Mesh

Velocity distribution

Velocity at X=0.01m

Velocity at X=0.2m

Velocity at X=0.5m

Velocity at X=0.9m

Velocity at X=1.2m

Velocity at X=2.4m

Velocity at  exit X=2.8m

From the above graph it show that ,as fluid is entered in to the pipe & it\'s travel up to entrance length or hydrodynamic length i.e up to x=2.4 m so. there is slightaly development of parabolic nature of flow but its not constant. but after x=2.5m there is velocity is fully developed &velocity is reached at maximum value at nearest to the centre point of pipe i.e Radius or y distance=0.

Pressure at x=2 m

from above plot it shows that as theres increase length x -direction pressure is constant.

Code for plot the graph of shear stress in Matlab

%Code for calcualte the shear stress graph form 0 to 2.82m length
% define input variable
mu=0.89e-3;  % dynamic viscosity
D=0.020;
R=linspace(0, (D/2), 1000);
dR=R(2)-R(1);

for i=1:length(R)
    du=u(i+1)-u(i);
    dU=abs(du);
    tau(i)=mu*(dU/dR);
    
plot(R,tau)
xlabel(\'Radius\')
ylabel(\'Shear stress\')
legend(\'Shear stress at x=2.82m \')
title(\'change in Shear stress at x=2.82m\')
grid on
end

Shear stress at X=2.82m

From the above plot we can see that  there is shear stress along y-axis i.e on vertical raduis. so , at centre point of pipe at R=0 shear stress is very low ,but it increases at rdial distance increases.

Result Table

conclusion

1. velocity is maximum at centre of pipe & minimum at wall surface

2. pressure drop is higher at entrance of pipe & its get constant after entrance length.

3. shear stress is maximum at wall surafce , while it minimum at centre of pipe

4. Verify the all result of hagen poiseuille with simulation result.