Week 11 - Simulation of Flow through a pipe in OpenFoam
1. Problem definition - In this problem we have to simulate the hagen poiseulle's flow in openfoam using a wedge shaped geometry of a pipe having `theta<5^circ` and also we have to write a code for blockmesh 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
- 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) = 0.0947625 m s^-1`
Equations icoFoam solver solves -
- Continuity equation
`nablau = 0`
- Momentum equation
`frac{del}{delt} (u) + nabla.(u u) - nabla.(nunablau) = - nablap`
4. Code -
- blockMeshDict file using matlab
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 = 2 ;
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.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
/*--------------------------------*- 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 9.99847e-3 -1.74524e-4)
(0 9.99847e-3 1.74524e-4)
(2.92 0 0)
(2.92 9.99847e-3 -1.74524e-4)
(2.92 9.99847e-3 1.74524e-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
(
);
// ************************************************************************* //- 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;
// ************************************************************************* //- 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;
// ************************************************************************* //- 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 (0.0947625 0 0);
boundaryField
{
inlet
{
type fixedValue;
value uniform (0.0947625 0 0);
}
outlet
{
type zeroGradient;
}
FrontWedge
{
type wedge;
}
BackWedge
{
type wedge;
}
noslipwalls
{
type noSlip;
}
}
// ************************************************************************* //5. Results -
- velocity
| Radial distance | Numerical_U | Analytical_U | %diff(U) |
| 0 | 0.187195 | 0.189525 | 1.22938926 |
| 1.00E-04 | 0.187042 | 0.189506053 | 1.30025044 |
| 0.000199969 | 0.186889 | 0.189449213 | 1.35139833 |
| 0.000299954 | 0.186735 | 0.18935448 | 1.38337356 |
| 0.000399939 | 0.186582 | 0.189221852 | 1.39510973 |
| 0.000499923 | 0.186429 | 0.189051333 | 1.38710126 |
| 0.000599908 | 0.186276 | 0.188842919 | 1.35928804 |
| 0.000699893 | 0.186123 | 0.188596611 | 1.31158846 |
| 0.000799878 | 0.18597 | 0.18831241 | 1.24389568 |
| 0.000899862 | 0.185816 | 0.187990318 | 1.15661182 |
| 0.000999847 | 0.185663 | 0.18763033 | 1.0485138 |
| 0.00109983 | 0.18551 | 0.187232456 | 0.91995603 |
| 0.00119982 | 0.185357 | 0.186796659 | 0.77070901 |
| 0.0012998 | 0.185204 | 0.186323013 | 0.60057689 |
| 0.00139979 | 0.185051 | 0.185811424 | 0.40924519 |
| 0.00149977 | 0.183176 | 0.185261995 | 1.12597034 |
| 0.00159976 | 0.182681 | 0.184674615 | 1.07952868 |
| 0.00169974 | 0.182187 | 0.184049403 | 1.01190373 |
| 0.00179972 | 0.181693 | 0.1833863 | 0.92335156 |
| 0.00189971 | 0.181199 | 0.182685236 | 0.81355009 |
| 0.00199969 | 0.180394 | 0.18194635 | 0.85319102 |
| 0.00209968 | 0.179557 | 0.181169495 | 0.89004748 |
| 0.00219966 | 0.17872 | 0.180354825 | 0.90644931 |
| 0.00229965 | 0.177883 | 0.179502179 | 0.90203842 |
| 0.00239963 | 0.177047 | 0.178611726 | 0.8760487 |
| 0.00249962 | 0.17621 | 0.177683288 | 0.82916532 |
| 0.0025996 | 0.175373 | 0.176717052 | 0.76056713 |
| 0.00269959 | 0.174536 | 0.175712823 | 0.66974239 |
| 0.00279957 | 0.173543 | 0.174670803 | 0.64567368 |
| 0.00289956 | 0.172399 | 0.173590784 | 0.68654786 |
| 0.00299954 | 0.171255 | 0.17247298 | 0.70618626 |
| 0.00309953 | 0.170111 | 0.17131717 | 0.7040566 |
| 0.00319951 | 0.168967 | 0.170123583 | 0.67984875 |
| 0.00329949 | 0.167823 | 0.168892106 | 0.63301148 |
| 0.00339948 | 0.166679 | 0.167622611 | 0.56293783 |
| 0.00349946 | 0.165535 | 0.166315351 | 0.46919962 |
| 0.00359945 | 0.164167 | 0.164970065 | 0.48679415 |
| 0.00369943 | 0.16274 | 0.163587021 | 0.5177801 |
| 0.00379942 | 0.161313 | 0.162165944 | 0.52596964 |
| 0.0038994 | 0.159885 | 0.160707117 | 0.51156203 |
| 0.00399939 | 0.158458 | 0.159210248 | 0.47248724 |
| 0.00409937 | 0.15703 | 0.157675638 | 0.40947201 |
| 0.00419936 | 0.155603 | 0.156102978 | 0.32028735 |
| 0.00429934 | 0.153992 | 0.154492584 | 0.32401822 |
| 0.00439933 | 0.152303 | 0.152844134 | 0.35404273 |
| 0.00449931 | 0.150613 | 0.151157956 | 0.36052095 |
| 0.0045993 | 0.148923 | 0.149433714 | 0.34176657 |
| 0.00469928 | 0.147234 | 0.147671754 | 0.2964369 |
| 0.00479927 | 0.145544 | 0.145871721 | 0.22466377 |
| 0.00489925 | 0.14381 | 0.144033977 | 0.15550256 |
| 0.00499923 | 0.141878 | 0.142158342 | 0.19720426 |
| 0.00509922 | 0.139946 | 0.140244625 | 0.21293148 |
| 0.0051992 | 0.138014 | 0.138293207 | 0.20189514 |
| 0.00529919 | 0.136082 | 0.136303699 | 0.16265067 |
| 0.00539917 | 0.13415 | 0.134276498 | 0.09420689 |
| 0.00549916 | 0.132173 | 0.132211198 | 0.02889186 |
| 0.00559914 | 0.130018 | 0.130108214 | 0.0693374 |
| 0.00569913 | 0.127862 | 0.127967123 | 0.08214856 |
| 0.00579911 | 0.125706 | 0.125788355 | 0.06547113 |
| 0.0058991 | 0.12355 | 0.123571474 | 0.01737741 |
| 0.00599908 | 0.121394 | 0.121316922 | -0.0635345 |
| 0.00609907 | 0.119094 | 0.119024249 | -0.058602 |
| 0.00619905 | 0.116732 | 0.116693914 | -0.0326372 |
| 0.00629904 | 0.11437 | 0.114325451 | -0.0389671 |
| 0.00639902 | 0.112008 | 0.111919332 | -0.0792247 |
| 0.00649901 | 0.109646 | 0.109475078 | -0.1561291 |
| 0.00659899 | 0.107162 | 0.106993176 | -0.1577899 |
| 0.00669897 | 0.10461 | 0.104473384 | -0.1307666 |
| 0.00679896 | 0.102058 | 0.101915444 | -0.1398764 |
| 0.00689894 | 0.0995065 | 0.099319869 | -0.1879089 |
| 0.00699893 | 0.0969539 | 0.096686139 | -0.2769387 |
| 0.00709891 | 0.0942279 | 0.09401478 | -0.2266878 |
| 0.0071989 | 0.0915018 | 0.091305258 | -0.2152576 |
| 0.00729888 | 0.0887757 | 0.088558116 | -0.245696 |
| 0.00739887 | 0.0860496 | 0.085772804 | -0.3227087 |
| 0.00749885 | 0.0832052 | 0.082949878 | -0.3078027 |
| 0.00759884 | 0.0803191 | 0.080088774 | -0.2875877 |
| 0.00769882 | 0.0774331 | 0.077190065 | -0.3148522 |
| 0.00779881 | 0.074547 | 0.074253171 | -0.3957127 |
| 0.00789879 | 0.0715405 | 0.071278678 | -0.3673215 |
| 0.00799878 | 0.0685077 | 0.068265992 | -0.3540673 |
| 0.00809876 | 0.0654748 | 0.065215716 | -0.3972718 |
| 0.00819875 | 0.0624124 | 0.06212724 | -0.4589941 |
| 0.00829873 | 0.0592451 | 0.05900118 | -0.4134153 |
| 0.00839871 | 0.0560777 | 0.055837231 | -0.4306612 |
| 0.0084987 | 0.0529103 | 0.052635069 | -0.5229036 |
| 0.00859868 | 0.0496306 | 0.049395336 | -0.476287 |
| 0.00869867 | 0.0463399 | 0.046117384 | -0.4824991 |
| 0.00879865 | 0.0430492 | 0.042801868 | -0.577854 |
| 0.00889864 | 0.03966 | 0.039448124 | -0.5370998 |
| 0.00899862 | 0.0362564 | 0.036056824 | -0.553503 |
| 0.00909861 | 0.0328446 | 0.03262729 | -0.6660379 |
| 0.00919859 | 0.0293374 | 0.029160207 | -0.6076548 |
| 0.00929858 | 0.0258302 | 0.025654881 | -0.6833747 |
| 0.00939856 | 0.0222743 | 0.022112014 | -0.7339255 |
| 0.00949855 | 0.0186723 | 0.018530898 | -0.7630626 |
| 0.00959853 | 0.0150566 | 0.014912247 | -0.9680134 |
| 0.00969852 | 0.0113676 | 0.01125534 | -0.9973953 |
| 0.0097985 | 0.00767864 | 0.007560906 | -1.5571403 |
| 0.00989848 | 0.00477836 | 0.003828583 | -24.807547 |
| 0.00999847 | 0 | 5.79902E-05 | 100 |
velocity at the fully developed flow
- shear stress
| radial distance | dr | U_analytical | dU | tau_Numerical | tau_analytical |
| 0 | 1.00E-04 | 0.189525 | 1.89467E-05 | 0.00138233 | 1.71E-04 |
| 1.00E-04 | 1.00E-04 | 0.189506053 | 5.68398E-05 | 0.0015372 | 5.13E-04 |
| 0.000199969 | 1.00E-04 | 0.189449213 | 9.47337E-05 | 0.00169186 | 8.55E-04 |
| 0.000299954 | 1.00E-04 | 0.18935448 | 0.000132627 | 0.00184652 | 1.20E-03 |
| 0.000399939 | 1.00E-04 | 0.189221852 | 0.000170519 | 0.00200118 | 1.54E-03 |
| 0.000499923 | 1.00E-04 | 0.189051333 | 0.000208414 | 0.00215584 | 1.88E-03 |
| 0.000599908 | 1.00E-04 | 0.188842919 | 0.000246308 | 0.0023105 | 2.22E-03 |
| 0.000699893 | 1.00E-04 | 0.188596611 | 0.000284201 | 0.00246516 | 2.57E-03 |
| 0.000799878 | 1.00E-04 | 0.18831241 | 0.000322092 | 0.00261982 | 2.91E-03 |
| 0.000899862 | 1.00E-04 | 0.187990318 | 0.000359988 | 0.00277447 | 3.25E-03 |
| 0.000999847 | 1.00E-04 | 0.18763033 | 0.000397874 | 0.00292913 | 3.59E-03 |
| 0.00109983 | 1.00E-04 | 0.187232456 | 0.000435798 | 0.00308379 | 3.93E-03 |
| 0.00119982 | 1.00E-04 | 0.186796659 | 0.000473646 | 0.00323845 | 4.28E-03 |
| 0.0012998 | 1.00E-04 | 0.186323013 | 0.000511589 | 0.00339311 | 4.62E-03 |
| 0.00139979 | 1.00E-04 | 0.185811424 | 0.000549429 | 0.00354777 | 4.96E-03 |
| 0.00149977 | 1.00E-04 | 0.185261995 | 0.00058738 | 0.00462571 | 5.30E-03 |
| 0.00159976 | 1.00E-04 | 0.184674615 | 0.000625213 | 0.00496321 | 5.64E-03 |
| 0.00169974 | 1.00E-04 | 0.184049403 | 0.000663103 | 0.0053007 | 5.99E-03 |
| 0.00179972 | 1.00E-04 | 0.1833863 | 0.000701064 | 0.0056382 | 6.33E-03 |
| 0.00189971 | 1.00E-04 | 0.182685236 | 0.000738886 | 0.0059757 | 6.67E-03 |
| 0.00199969 | 1.00E-04 | 0.18194635 | 0.000776855 | 0.00632379 | 7.01E-03 |
| 0.00209968 | 1.00E-04 | 0.181169495 | 0.000814669 | 0.00667295 | 7.35E-03 |
| 0.00219966 | 1.00E-04 | 0.180354825 | 0.000852646 | 0.00702212 | 7.70E-03 |
| 0.00229965 | 1.00E-04 | 0.179502179 | 0.000890453 | 0.00737128 | 8.04E-03 |
| 0.00239963 | 1.00E-04 | 0.178611726 | 0.000928438 | 0.00772045 | 8.38E-03 |
| 0.00249962 | 1.00E-04 | 0.177683288 | 0.000966236 | 0.00806961 | 8.72E-03 |
| 0.0025996 | 1.00E-04 | 0.176717052 | 0.001004229 | 0.00841878 | 9.06E-03 |
| 0.00269959 | 1.00E-04 | 0.175712823 | 0.00104202 | 0.00876794 | 9.41E-03 |
| 0.00279957 | 1.00E-04 | 0.174670803 | 0.00108002 | 0.00911518 | 9.75E-03 |
| 0.00289956 | 1.00E-04 | 0.173590784 | 0.001117803 | 0.00946054 | 1.01E-02 |
| 0.00299954 | 1.00E-04 | 0.17247298 | 0.001155811 | 0.00980591 | 1.04E-02 |
| 0.00309953 | 1.00E-04 | 0.17131717 | 0.001193587 | 0.0101513 | 1.08E-02 |
| 0.00319951 | 1.00E-04 | 0.170123583 | 0.001231477 | 0.0104966 | 1.11E-02 |
| 0.00329949 | 1.00E-04 | 0.168892106 | 0.001269495 | 0.010842 | 1.15E-02 |
| 0.00339948 | 1.00E-04 | 0.167622611 | 0.00130726 | 0.0111874 | 1.18E-02 |
| 0.00349946 | 1.00E-04 | 0.166315351 | 0.001345286 | 0.0115327 | 1.21E-02 |
| 0.00359945 | 1.00E-04 | 0.164970065 | 0.001383044 | 0.0118796 | 1.25E-02 |
| 0.00369943 | 1.00E-04 | 0.163587021 | 0.001421077 | 0.0122269 | 1.28E-02 |
| 0.00379942 | 1.00E-04 | 0.162165944 | 0.001458827 | 0.0125741 | 1.32E-02 |
| 0.0038994 | 1.00E-04 | 0.160707117 | 0.001496868 | 0.0129214 | 1.35E-02 |
| 0.00399939 | 1.00E-04 | 0.159210248 | 0.001534611 | 0.0132686 | 1.39E-02 |
| 0.00409937 | 1.00E-04 | 0.157675638 | 0.00157266 | 0.0136159 | 1.42E-02 |
| 0.00419936 | 1.00E-04 | 0.156102978 | 0.001610394 | 0.0139631 | 1.45E-02 |
| 0.00429934 | 1.00E-04 | 0.154492584 | 0.001648451 | 0.014312 | 1.49E-02 |
| 0.00439933 | 1.00E-04 | 0.152844134 | 0.001686177 | 0.0146616 | 1.52E-02 |
| 0.00449931 | 1.00E-04 | 0.151157956 | 0.001724242 | 0.0150111 | 1.56E-02 |
| 0.0045993 | 1.00E-04 | 0.149433714 | 0.001761961 | 0.0153607 | 1.59E-02 |
| 0.00469928 | 1.00E-04 | 0.147671754 | 0.001800033 | 0.0157103 | 1.62E-02 |
| 0.00479927 | 1.00E-04 | 0.145871721 | 0.001837744 | 0.0160598 | 1.66E-02 |
| 0.00489925 | 1.00E-04 | 0.144033977 | 0.001875634 | 0.0164097 | 1.69E-02 |
| 0.00499923 | 1.00E-04 | 0.142158342 | 0.001913717 | 0.0167613 | 1.73E-02 |
| 0.00509922 | 1.00E-04 | 0.140244625 | 0.001951418 | 0.0171129 | 1.76E-02 |
| 0.0051992 | 1.00E-04 | 0.138293207 | 0.001989508 | 0.0174644 | 1.80E-02 |
| 0.00529919 | 1.00E-04 | 0.136303699 | 0.002027201 | 0.017816 | 1.83E-02 |
| 0.00539917 | 1.00E-04 | 0.134276498 | 0.002065299 | 0.0181675 | 1.86E-02 |
| 0.00549916 | 1.00E-04 | 0.132211198 | 0.002102985 | 0.0185193 | 1.90E-02 |
| 0.00559914 | 1.00E-04 | 0.130108214 | 0.00214109 | 0.0188722 | 1.93E-02 |
| 0.00569913 | 1.00E-04 | 0.127967123 | 0.002178768 | 0.0192251 | 1.97E-02 |
| 0.00579911 | 1.00E-04 | 0.125788355 | 0.002216882 | 0.019578 | 2.00E-02 |
| 0.0058991 | 1.00E-04 | 0.123571474 | 0.002254552 | 0.0199309 | 2.04E-02 |
| 0.00599908 | 1.00E-04 | 0.121316922 | 0.002292673 | 0.0202838 | 2.07E-02 |
| 0.00609907 | 1.00E-04 | 0.119024249 | 0.002330335 | 0.0206372 | 2.10E-02 |
| 0.00619905 | 1.00E-04 | 0.116693914 | 0.002368464 | 0.0209908 | 2.14E-02 |
| 0.00629904 | 1.00E-04 | 0.114325451 | 0.002406119 | 0.0213444 | 2.17E-02 |
| 0.00639902 | 1.00E-04 | 0.111919332 | 0.002444255 | 0.021698 | 2.21E-02 |
| 0.00649901 | 1.00E-04 | 0.109475078 | 0.002481902 | 0.0220516 | 2.24E-02 |
| 0.00659899 | 1.00E-04 | 0.106993176 | 0.002519792 | 0.0224055 | 2.27E-02 |
| 0.00669897 | 1.00E-04 | 0.104473384 | 0.002557939 | 0.0227595 | 2.31E-02 |
| 0.00679896 | 1.00E-04 | 0.101915444 | 0.002595575 | 0.0231134 | 2.34E-02 |
| 0.00689894 | 1.00E-04 | 0.099319869 | 0.00263373 | 0.0234674 | 2.38E-02 |
| 0.00699893 | 1.00E-04 | 0.096686139 | 0.002671359 | 0.0238214 | 2.41E-02 |
| 0.00709891 | 1.00E-04 | 0.09401478 | 0.002709521 | 0.0241752 | 2.45E-02 |
| 0.0071989 | 1.00E-04 | 0.091305258 | 0.002747142 | 0.024529 | 2.48E-02 |
| 0.00729888 | 1.00E-04 | 0.088558116 | 0.002785313 | 0.0248828 | 2.51E-02 |
| 0.00739887 | 1.00E-04 | 0.085772804 | 0.002822926 | 0.0252366 | 2.55E-02 |
| 0.00749885 | 1.00E-04 | 0.082949878 | 0.002861104 | 0.02559 | 2.58E-02 |
| 0.00759884 | 1.00E-04 | 0.080088774 | 0.002898709 | 0.0259432 | 2.62E-02 |
| 0.00769882 | 1.00E-04 | 0.077190065 | 0.002936895 | 0.0262965 | 2.65E-02 |
| 0.00779881 | 1.00E-04 | 0.074253171 | 0.002974493 | 0.0266497 | 2.69E-02 |
| 0.00789879 | 1.00E-04 | 0.071278678 | 0.003012686 | 0.0270024 | 2.72E-02 |
| 0.00799878 | 1.00E-04 | 0.068265992 | 0.003050276 | 0.027355 | 2.75E-02 |
| 0.00809876 | 1.00E-04 | 0.065215716 | 0.003088477 | 0.0277076 | 2.79E-02 |
| 0.00819875 | 1.00E-04 | 0.06212724 | 0.00312606 | 0.02806 | 2.82E-02 |
| 0.00829873 | 1.00E-04 | 0.05900118 | 0.003163949 | 0.0284117 | 2.86E-02 |
| 0.00839871 | 1.00E-04 | 0.055837231 | 0.003202161 | 0.0287635 | 2.89E-02 |
| 0.0084987 | 1.00E-04 | 0.052635069 | 0.003239733 | 0.0291153 | 2.92E-02 |
| 0.00859868 | 1.00E-04 | 0.049395336 | 0.003277952 | 0.0294662 | 2.96E-02 |
| 0.00869867 | 1.00E-04 | 0.046117384 | 0.003315516 | 0.029817 | 2.99E-02 |
| 0.00879865 | 1.00E-04 | 0.042801868 | 0.003353743 | 0.0301678 | 3.03E-02 |
| 0.00889864 | 1.00E-04 | 0.039448124 | 0.0033913 | 0.0305178 | 3.06E-02 |
| 0.00899862 | 1.00E-04 | 0.036056824 | 0.003429535 | 0.0308677 | 3.10E-02 |
| 0.00909861 | 1.00E-04 | 0.03262729 | 0.003467083 | 0.0312175 | 3.13E-02 |
| 0.00919859 | 1.00E-04 | 0.029160207 | 0.003505326 | 0.0315665 | 3.16E-02 |
| 0.00929858 | 1.00E-04 | 0.025654881 | 0.003542867 | 0.0319155 | 3.20E-02 |
| 0.00939856 | 1.00E-04 | 0.022112014 | 0.003581117 | 0.0322641 | 3.23E-02 |
| 0.00949855 | 1.00E-04 | 0.018530898 | 0.00361865 | 0.0326121 | 3.27E-02 |
| 0.00959853 | 1.00E-04 | 0.014912247 | 0.003656908 | 0.0326752 | 3.30E-02 |
| 0.00969852 | 1.00E-04 | 0.01125534 | 0.003694434 | 0.0311964 | 3.33E-02 |
| 0.0097985 | 1.00E-04 | 0.007560906 | 0.003732324 | 0.0297177 | 3.37E-02 |
| 0.00989848 | 1.00E-04 | 0.003828583 | 0.003770592 | 0.0279144 | 3.40E-02 |
| 0.00999847 | -1.00E-02 | 5.79902E-05 | 5.79902E-05 | 0 | -5.23E-06 |
shear stress at fully developed flow
6. Conclusion -
- The shear stress is zero at the centre and maximum at the wall because there is friction at the wall
- The max velocity and max shear stress values for both anaytical and numerical are as follows
| WEDGE | U_ANALYTICAL | U_NUMERICAL | TAU_ANALYTICAL | TAU_NUMERICAL |
| 2 | 0.189525 | 0.187195 | 3.27E-02 | 3.40E-02 |