Hagen Poiseuille s Flow

Google drive link to all the blockMeshDict file for various angles:

https://drive.google.com/open?id=1GuYMGjgkJHC8hovIfkqA1sp4k1OdNR1t

Google drive link to all the plots and source code:

https://drive.google.com/open?id=1ntyenSwIed9JrNxDXwMKVZIeMk0-xNrk

 

Assumptions:

1. Reynolds Number = 1000 (For Laminar Flow)
2. Temperature= 20 degree celsius

at this temperature (From Saturated Water Tables Ref. Cengel and Cimbala)

Kinematic Viscosity = 1.004008e-06 m2/s
Density = 998 kg/m3

3. Length of pipe = 1 m
4. Diameter of pipe = 20 mm

 

Analytical Results:

From

1. Reynolds Number = (rho*V*D/mu)

2.Del_p=(32*mu*V*L/D2)   %Hagen Poiseuille's Equation 

We get: 

Pressure Drop = 4.024064e+00 pa
Kinematic Inlet pressure = 4.032128e-03 m2/s2
Kinematic Outlet pressure = 0 m2/s2
Max velocity = 1.004008e-01 m/s
Avg Velocity = 5.020040e-02 m/s 

Note:

1. The pressures(p) used to calculate the kinematic pressures(p/rho) were Gauge Pressures at inlet and outlet.

2. A pressure driven approach was used i.e the inlet and outlet kinematic pressures were fixed in initial conditions and the velocity was left to float.

3. The Transport Model used was Newtonian .

 

CFD Results:

1. Velocity Profile:

The Velocity profile obtained at the inlet and outlet are shown below (the plots shown are for wedge angle = 2 degrees )

The reflect filter in paraview was used to model the complete velocity profile across the diameter of pipe.

Observations:

1. We see that the profile is fully developed and is a parabola which matches with the Hagen Poiseuille's Result.

2. Also the maximum velocity obtained from the above plots was 0.09912 m/s which has only 1.27 % error i.e almost equal to the result obtained by Hagen Poiseuille equation.

 

Further comparison for Maximum Velocity and Average Velocity with changing Wedge Angle :

We see that the average velocity remains mostly unchanged with changing the wedge angle while the maximum velocity approaches towards the analytical value as a result of decreasing the wedge angle.

 

2. Pressure Drop (Loss of Head in Pipe)

The Pressure Drop is the pipe along the length is linear.

 

3. Wall Shear Stress

As the flow is laminar, so the turbulence model used to compute the Wall Shear stress was Laminar.

 

4. Accuracy vs Wedge Angle

The percentage error in maximum velocity for decreasing wedge angle was plotted and it is seen that the percentage error decreases as we decrease the wedge angle.

 

5. Simulation Time vs Wedge Angle

It is seen that the simulation time rises at first when we decrease the wedge angle but it falls subsequently after that as we further decrease the wedge angle.

 

 

Program in MATLAB to generate blockMeshDict file for any wedge angle and grading schemes:

Following is the program to generate the blockMeshDict file automatically to generate the computational mesh

clear all
close all
clc

%input
theta=2;  %In degrees %Wedge angle
grading=0.25; %Grading factor along the walls of cylinder 

length=1000;
radius=10;
b1=400; %cells along the length of cylinder
b2=12; %cells along the radius of cylinder

%vertices of the wedge
v0=[0 0 0];
v1=[length 0 0];
v2=[length radius*cosd(theta/2) -radius*sind(theta/2)];
v3=[0 radius*cosd(theta/2) -radius*sind(theta/2)];
v4=[0 radius*cosd(theta/2) radius*sind(theta/2)];
v5=[length radius*cosd(theta/2) radius*sind(theta/2)];

s1='/*--------------------------------*- C++ -*----------------------------------*\';
s2='| =========                 |                                                 |';
s3='| \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox           |';
s4='|  \\    /   O peration     | Version:  4.1                                   |';
s5='|   \\  /    A nd           | Web:      www.OpenFOAM.org                      |';
s6='|    \\/     M anipulation  |                                                 |';
s7='\*---------------------------------------------------------------------------*/';

s8='// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //';

%wedge block
% hex (0 1 2 3 0 1 5 4) (80 20 1) simpleGrading(1 1 1)

f1=fopen('blockMeshDict.txt','wt');

fprintf(f1,'%s\n',s1);
fprintf(f1,'%s\n',s2);
fprintf(f1,'%s\n',s3);
fprintf(f1,'%s\n',s4);
fprintf(f1,'%s\n',s5);
fprintf(f1,'%s\n',s6);
fprintf(f1,'%s\n\n',s7);


fprintf(f1,'FoamFile\n{\n');
fprintf(f1,'%16s \t 2.0;\n','version');
fprintf(f1,'%16s \t ascii;\n','format');
fprintf(f1,'%16s \t dictionary;\n','class');
fprintf(f1,'%16s \t blockMeshDict;\n}\n','object');
fprintf(f1,'%s\n\n',s8);

fprintf(f1,'convertToMeters 0.001;\n\n');
fprintf(f1,'vertices\n(\n');
fprintf(f1,'\t(%d %d %d)\n',v0(1),v0(2),v0(3));
fprintf(f1,'\t(%d %d %d)\n',v1(1),v1(2),v1(3));
fprintf(f1,'\t(%d %d %d)\n',v2(1),v2(2),v2(3));
fprintf(f1,'\t(%d %d %d)\n',v3(1),v3(2),v3(3));
fprintf(f1,'\t(%d %d %d)\n',v4(1),v4(2),v4(3));
fprintf(f1,'\t(%d %d %d)\n);\n\n',v5(1),v5(2),v5(3));

fprintf(f1,'blocks\n(\n');
fprintf(f1,'\t hex (0 1 2 3 0 1 5 4) (%d %d %d)\n',b1,b2,1);
fprintf(f1,'\t simpleGrading\n\t (1 %d 1)\n);\n\n',grading);

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

fprintf(f1,'boundary\n(\n');
fprintf(f1,'\t%s\n\t{\n\t\t type %s;\n\t\t faces \n\t\t (\n\t\t\t (%d %d %d %d) \n\t\t );\n\t}\n\n','inlet','patch',0,4,3,0);
fprintf(f1,'\t%s\n\t{\n\t\t type %s;\n\t\t faces \n\t\t (\n\t\t\t (%d %d %d %d) \n\t\t );\n\t}\n\n','outlet','patch',1,2,5,1);
fprintf(f1,'\t%s\n\t{\n\t\t type %s;\n\t\t faces \n\t\t (\n\t\t\t (%d %d %d %d) \n\t\t );\n\t}\n\n','Wall','wall',2,3,4,5);
fprintf(f1,'\t%s\n\t{\n\t\t type %s;\n\t\t faces \n\t\t (\n\t\t\t (%d %d %d %d) \n\t\t );\n\t}\n\n','axis','empty',0,1,1,0);
fprintf(f1,'\t%s\n\t{\n\t\t type %s;\n\t\t faces \n\t\t (\n\t\t\t (%d %d %d %d) \n\t\t );\n\t}\n\n','wedgeFront','wedge',0,1,2,3);
fprintf(f1,'\t%s\n\t{\n\t\t type %s;\n\t\t faces \n\t\t (\n\t\t\t (%d %d %d %d) \n\t\t );\n\t}\n\n','wedgeBack','wedge',0,1,5,4);
fprintf(f1,');\n\n');

fprintf(f1,'mergePatchPairs\n(\n);\n\n');
fprintf(f1,'%s',s8);

fclose(f1);

 

 


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