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Simulation of Flow through a pipe in OpenFoam 1

KKanmani P6 May 2020
CFD

objective

  • Reynolds number based on pipe diameter and inlet velocity should be 2100
  • Working fluid - water
  • to calculate the length of the pipe
    • to Calculate length of the pipe using the entry length formula for laminar flow through a pipe
    • to Show that entry length is sufficient to produce a fully developed flow and to compare velocity profiles at different positions along the length of the pipe.
  • Solver - icoFOAM

Hagen–Poiseuille Equation

The Hagen–Poiseuille Equation (or Poiseuille equation) is a fluidic law to calculate flow pressure drop in a long cylindrical pipe and it was derived separately by Poiseuille and Hagen in 1838 and 1839, respectively.

Consider a steady flow of an incompressible Newtonian fluid in a long rigid pipe. Using conservation equations and boundary conditions for cylindrical coordinates, the laminar flow has a parabolic profile from zero at the wall to a maximum velocity at the centerline

Longitudinal cutting view of a pipe with a laminar flow and parabolic velocity profile.

Vz=ΔPR24μL(1−r2R2)0≤r≤R\">Vz=ΔPR24μL(1−r2R2)0≤r≤R
Vmax=ΔPR24μL\">Vmax=ΔPR24μL

Considering the definition of average velocity in cylindrical coordinates and, the average velocity in the pipe

V¯=12Vmax\">V¯=12Vmax

The volumetric flow rate (Q) equals cross-section area multiplied by average velocity (Q=V¯A)\">(Q=V¯A), so

ΔP=8μLQπR4\">ΔP=8μLQπR4

octVE program

clear all
close all
clc
%datum
Re=2100;
D=0.004;
L=0.06*Re*D
L_p=L+0.7;
r=D/2;
theta=4;
mu=0.89e-3
rho=997
%formula
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 strss
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     | Version:  2.2.1                                 |\'
header5=\' |  \\\\  /    A nd           | Web:      www.OpenFOAM.com                      |\'
header6=\'|    \\\\/     M anipulation  |                                                 |\'
header7=\'\\*---------------------------------------------------------------------------*/\'

f1=fopen(\'blockMeshDict.txt\',\'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,\'%12s\\t 2.0;\\n\',\'version\')
fprintf(f1,\'%12s\\t ascii;\\n\',\'format\')
fprintf(f1,\'%12s\\t dictionary;\\n\',\'class\')
fprintf(f1,\'%12s\\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)\',300,30,1)
fprintf(f1,\'simpleGrading(1 0.1 1)\\n\\n\')
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,\'mergePatchPairs\\n\')
fprintf(f1,\'(\\n\')
fprintf(f1,\');\\n\')
header9=\'// ************************************************************************* //\'
fprintf(f1,\'%s\\n\',header9)
fclose(f1)

blockmesh

/*--------------------------------*- 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 0.00199878 6.9799e-005)
	(0 0.00199878 -6.9799e-005)
	(1.204 0 0)
	(1.204 0.00199878 6.9799e-005)
	(1.204 0.00199878 -6.9799e-005)
);

blocks
(
	 hex (0 3 5 2 0 3 4 1) (300 30 1) simpleGrading (1 0.1 1)

);
edges
(
	 arc 1 2 (0 2e-3 0)
	 arc 4 5 (1.204 2e-3 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
(
);
// ************************************************************************* //

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 (0.4686 0 0);

boundaryField
{
   inlet
    {
        type           fixedValue;
	value 	uniform (0.4686 0 0); 
    }    
outlet
    {
        type            zeroGradient;
    }

   
    axis
    {
        type            empty;
    }
top
    {
        type            fixedValue;
	value 		uniform (0 0 0); 
    }    
front
    {
        type            wedge;
    } 
back
    {
        type            wedge;
    }    
}

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

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 1.0073;
    }

   
    axis
    {
        type            empty;
    }
top
    {
        type            zeroGradient;
    }    
front
    {
        type            wedge;
    } 
back
    {
        type            wedge;
    }    
}

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

generated mesh and velocity profile

velocity plot at 1m from entry length

velocity plot at 2m from entry length

conclusion:

the octave program was created and the blockMesh was automatically created and the simuaktion as done the plots were plotted.

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