Week 8 - Simulation of a backward facing step in OpenFOAM
The following challenge requires the simulation of the incompressible laminar viscous flow inside a volume that has a geometry very similar to that of a backward step.
WHAT IS A LAMINAR FLOW?
It is a type of flow which is having smooth streamlines of the fluid elements, and the order of motion of such type of flow is very high. In a very contrasting way, the turbulent flow has the main characteristic of velocity fluctuations, with a high order of randomness of the fluid elements leading to the disoriented forms of the streamlines.
In nature, most of the flow patterns are of turbulence kind and in some cases, we also solve for laminar flow type, as it is also essential in certain aspects to deal with.
WHAT IS A NO-SLIP CONDITION, VISCOSITY, AND ITS SIGNIFICANCE?
A no-slip phenomenon can be said that when the relative velocity of the fluid layer with respect to the solid surface is zero(0) at the sold surface, then there is said to be a prevailing condition called a No-slip condition.
Hence at the walls, the fluid is having zero velocity.
Thus when we consider a flow through the full length of the pipe, we would eventually result in the formation of the boundary layer near the walls of the pipe which is shown in the second image below.
BOUNDARY CONDITION SPECIFICATION:
The above-shown geometry is not available directly in any form in the tutorials. Thus it would be like, we would have to work on the modification of a very similar geometry related to the above-given properties of the flow. So if we look deeply into the tutorials we would find that the most suitable type of geometry that can easily be modified as a Backward step is found under the icoFoam solver package of the incompressible flow type which has a laminar type of flow.
SETTING UP THE INITIAL CONDITIONS & FORMING THE GEOMETRY:
The initial conditions to be set are located in the 0 folder(a directory type) which is the Pressure and Velocity files(bash type of files).
The following is the sequential order of the CODE to set up the initial conditions and the geometry of the given domain.
1. First we copy the desired type of tutorial to a new dict file inside the run folder of OpenFoam by giving a different name to it.
2. Here the file we created inside the run folder is backward_step and it is being selected to get its corresponding folders edited in the corresponding steps.
3. Here, the first thing we need to do is to change the external properties that influence the fluid flow, that is, the pressure and velocity at the inlet and outlet conditions.
4. In the following steps, we would change the afore-said external factors and also the blockmesh and controlDict files.
INITIAL CONDITION SETUP:
Initially, we would be having the bash files of the pressure and velocity fields as a C++ program which can be modified according to our problem statement.
PRESSURE(Bash file):
INITIAL:
MODIFIED:
Here we assume that the pressure field is fixed to be 0 across the domain and the only thing which we would consider as a driving factor of the fluid is the velocity.
However, we make some modifications to the boundary fields by changing and adding values of pressure to the boundary terms like inlet, outlet, topAndBottom, and frontAndBack which are all representing particular types that would be represented in the blockMesh bash file along with the new geometry impartment.
VELOCITY(Bash file):
INITIAL:
MODIFIED:
- As we discussed earlier, velocity is considered to be the only flow driving factor so which we gave the value for velocity across the pipe as 1 m/s under the internalField section.
- Also, we give the same value for velocity in the inlet condition along the x-axis and indicating zero gradient value at the outlet and giving no-slip conditions to the inside top and bottom surface of the domain.
- Front and Back portions are given the type called empty which would be discussed in the blockMesh bash file setup.
GEOMETRY FORMATION:
DOMAIN SEPERATION INTO VARIOUS VOLUMES:(Assumption)
The above-seen Domain(backward step geometry), is not as it is used to simulate, but rather it is split down into 5 different rectangular control volumes and is shown in the below image.
Since it is a 3-dimensional block and has 5 volumes it will be having 22 vertices, that is, from 0 to 21 and it is pictorially represented as follows.
The Boundaries for the entire domain is represented as,
[11, 16, 5, 0 ; 17, 20, 9, 6] : [Inlet ; Outlet] ==> are designated as type patch
[11, 16, 17, 20, 21, 12 ; 0, 5, 6, 9, 10, 1] : [Front ; Back] ==> are designated as type empty
[16, 5, 6, 17 ; 11, 0, 1, 10, 9, 20, 21, 12] ; [Top ; Bottom] ==> are designated as type wall
Wherein, the type empty represents the phase in which in that particular direction the parameters are not solved and type patch and wall generally are used in (inlets and outlets) and for the phases indicating obstacles respectively.
BLOCKMESH:
The changes are made in almost every criteria namely, vertices, blocks, and boundary in accordance to the assumption made to the domain(geometry) as shown in the picture above.
INITIAL:
/*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\ / O peration | Website: https://openfoam.org
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
object blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 0.1;
vertices
(
(0 0 0)
(1 0 0)
(1 1 0)
(0 1 0)
(0 0 0.1)
(1 0 0.1)
(1 1 0.1)
(0 1 0.1)
);
blocks
(
hex (0 1 2 3 4 5 6 7) (20 20 1) simpleGrading (1 1 1)
);
edges
(
);
boundary
(
movingWall
{
type wall;
faces
(
(3 7 6 2)
);
}
fixedWalls
{
type wall;
faces
(
(0 4 7 3)
(2 6 5 1)
(1 5 4 0)
);
}
frontAndBack
{
type empty;
faces
(
(0 3 2 1)
(4 5 6 7)
);
}
);
mergePatchPairs
(
);
// ************************************************************************* //MODIFIED:
- Here, the vertices from 0 to 21, are indicated with their coordinates in a cartesian form.
- Then, what we do is, represent the assumed volumes in terms of their vertices, in order of their Cartesian directions, that is, from Origin(0) ==> Vertex point @ x-axis ==> Vertex point @ y-axis ==> Vertex point @ z-axis(to the other side of the volume)
- Then, inside the boundaries, we would just indicate the individual phases in terms of vertices, but this time it is represented in a form like it obeys the right-hand-thumb rule, which is represented below.
where the normal to the area must be pointing outwards so that we can figure out the order of vertices forming the phases either clockwise or counterclockwise.
- Thus we write the phases representing their respective types based on the above rules.
- Here, we use a type of mesh called Hexahedral mesh.
- In this case, we do not involve any modification in mesh grading factors.
- In accordance with the challenge, we give the number of meshes in x,y,z axes as, [200,20,1]
/*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\ / O peration | Website: https://openfoam.org
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
object blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 0.01;
vertices
(
(0 0 0) //0//
(8 0 0) //1//
(8 0.5 0) //2//
(0 0.5 0) //3//
(8 1 0) //4//
(0 1 0) //5//
(20 1 0) //6//
(20 0.5 0) //7//
(20 0 0) //8//
(20 -1 0) //9//
(8 -1 0) //10//
(0 0 0.5) //11//
(8 0 0.5) //12//
(8 0.5 0.5) //13//
(0 0.5 0.5) //14//
(8 1 0.5) //15//
(0 1 0.5) //16//
(20 1 0.5) //17//
(20 0.5 0.5) //18//
(20 0 0.5) //19//
(20 -1 0.5) //20//
(8 -1 0.5) //21//
);
blocks
(
hex (0 1 2 3 11 12 13 14) (200 20 1) simpleGrading (1 1 1)
hex (3 2 4 5 14 13 15 16) (200 20 1) simpleGrading (1 1 1)
hex (2 7 6 4 13 18 17 15) (200 20 1) simpleGrading (1 1 1)
hex (1 8 7 2 12 19 18 13) (200 20 1) simpleGrading (1 1 1)
hex (10 9 8 1 21 20 19 12) (200 20 1) simpleGrading (1 1 1)
);
edges
(
);
boundary
(
inlet
{
type patch;
faces
(
(3 0 11 14)
(5 3 14 16)
);
}
outlet
{
type patch;
faces
(
(6 17 18 7)
(7 18 19 8)
(8 19 20 9)
);
}
topAndBottom
{
type wall;
faces
(
(4 5 16 15)
(4 15 17 6)
(11 0 1 12)
(12 1 10 21)
(21 10 9 20)
);
}
frontAndBack
{
type empty;
faces
(
(0 3 2 1)
(3 5 4 2)
(2 4 6 7)
(1 2 7 8)
(10 1 8 9)
(11 12 13 14)
(14 13 15 16)
(13 18 17 15)
(12 19 18 13)
(21 20 19 12)
);
}
);
mergePatchPairs
(
);
// ************************************************************************* //CONTROLDICTS:
In this part, we would be controlling the Courant number limitation by altering the time step(deltaT), and also by changing the simulation time required.
INITIAL:
/*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\ / O peration | Website: https://openfoam.org
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
location "system";
object controlDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
application icoFoam;
startFrom startTime;
startTime 0;
stopAt endTime;
endTime 0.5;
deltaT 0.005;
writeControl timeStep;
writeInterval 20;
purgeWrite 0;
writeFormat ascii;
writePrecision 6;
writeCompression off;
timeFormat general;
timePrecision 6;
runTimeModifiable true;
// ************************************************************************* //MODIFIED:
- Here, it was observed from the trial and error method that, only the value of 1e-5 for deltaT(time step), gives the converged result rather than other time step values lesser than it.
- Thus for this value of time step, naturally it would be a lengthier simulation. Thus we assumed a small value of the end time of simulation as 0.2 seconds for the earlier result generation.
- We also order here to write the report for simulation with a time interval of every 100 simulation steps, that is, (0.2 * 1e5)/100, and thus we yield 200 steps of simulation during visual perception through Paraview.
- We also give the maximum limit for Courant value as 2 as in some cases CFD avoids the error even for the resulting greater courant numbers.
- Then, we also give the value of the maximum limit of deltaT(time step) as 1e-4, because, fixing only with 1e-5 makes the solution process a lengthier process, and thus we give an allowance for OenFoam to oscillate between 1e-5 and 1e-4 and select its own time step for earlier possible results to be written.
/*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\ / O peration | Website: https://openfoam.org
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
location "system";
object controlDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
application icoFoam;
startFrom startTime;
startTime 0;
stopAt endTime;
endTime 0.2;
deltaT 0.00001;
writeControl timeStep;
writeInterval 100;
purgeWrite 0;
writeFormat ascii;
writePrecision 6;
writeCompression off;
timeFormat general;
timePrecision 6;
runTimeModifiable true;
adjustTimeStep true;
maxCo 2;
maxDeltaT 0.0001;
// ************************************************************************* //Now we type the command blockMesh over here so that the simulation sets up the polyMesh file inside the constant dict file.
Hence the polyMesh dict file is written.
Now in order to solve the solution, we must ensure which solver to use. So just type sol and click enter so that the OpenFoam takes us to the solver folder and we can confirm that we use the icoFoam solver type to solve our case.
The simulation starts and ends after 0.2 seconds.
Now we would witness a certain number of reports generated over the input write interval.
RESULTS:
MESH:
The mesh which we applied is shown in the below image.
CONVERGED SOLUTION:
The solution after 0.2 seconds of flow, is given in the following image.
FINDING THE VELOCITY @ 0.085m FROM THE INLET:
FIXING THE COORDINATES:
The line position over which we are calculating the velocity is shown below(represented as an arrow).
OBTAINED PLOT RESULT:
WITH MESH GRADING FACTOR(OPTIMIZED MESH SIZE):
Here, we do the change in our previous case, that is optimizing the mesh using the mesh grading factor, thus leading us into the second case problem statement. In this problem, we would have almost everything the same, but the only things we show our interest in changing is the mesh grading factor and changing the deltaT(time step) to the order of 1e-6, thus correlating with the correct courant numbers.
The file name that we use for this case is 'backward_step_1'.
BLOCKMESH:
MODIFIED:
- Mesh Grading Factor = size of last step/ size of the first step
The above equation is applied in both x and y directions. For our case, the z- direction is not considered to be in effect. Therefore we took 0.2 as the least mesh grading factor and we know that (1/0.2)=5, therefore fixing 5 as the greatest mesh grading factor, we wrote the sizing of meshes in all the 5 volumes. The only condition is that the mesh must not have any discontinuities in terms of the mesh size. It must be like a flow of continuous similar meshes at the junctions of each volume.
/*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\ / O peration | Website: https://openfoam.org
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
object blockMeshDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 0.01;
vertices
(
(0 0 0) //0//
(8 0 0) //1//
(8 0.5 0) //2//
(0 0.5 0) //3//
(8 1 0) //4//
(0 1 0) //5//
(20 1 0) //6//
(20 0.5 0) //7//
(20 0 0) //8//
(20 -1 0) //9//
(8 -1 0) //10//
(0 0 0.5) //11//
(8 0 0.5) //12//
(8 0.5 0.5) //13//
(0 0.5 0.5) //14//
(8 1 0.5) //15//
(0 1 0.5) //16//
(20 1 0.5) //17//
(20 0.5 0.5) //18//
(20 0 0.5) //19//
(20 -1 0.5) //20//
(8 -1 0.5) //21//
);
blocks
(
hex (0 1 2 3 11 12 13 14) (200 20 1) simpleGrading (0.2 5 1)
hex (3 2 4 5 14 13 15 16) (200 20 1) simpleGrading (0.2 0.2 1)
hex (2 7 6 4 13 18 17 15) (200 20 1) simpleGrading (5 0.2 1)
hex (1 8 7 2 12 19 18 13) (200 20 1) simpleGrading (5 5 1)
hex (10 9 8 1 21 20 19 12) (200 20 1) simpleGrading (5 5 1)
);
edges
(
);
boundary
(
inlet
{
type patch;
faces
(
(3 0 11 14)
(5 3 14 16)
);
}
outlet
{
type patch;
faces
(
(6 17 18 7)
(7 18 19 8)
(8 19 20 9)
);
}
topAndBottom
{
type wall;
faces
(
(4 5 16 15)
(4 15 17 6)
(11 0 1 12)
(12 1 10 21)
(21 10 9 20)
);
}
frontAndBack
{
type empty;
faces
(
(0 3 2 1)
(3 5 4 2)
(2 4 6 7)
(1 2 7 8)
(10 1 8 9)
(11 12 13 14)
(14 13 15 16)
(13 18 17 15)
(12 19 18 13)
(21 20 19 12)
);
}
);
mergePatchPairs
(
);
// ************************************************************************* //CONTROLDICT:
MODIFIED:
As the mesh size drastically reduces to 0.2 times the previous case, it also thus affects the Courant number relating deltaT values thus allotting it a value of 1e-6 in the controlDict bash file.
/*--------------------------------*- C++ -*----------------------------------*\
========= |
\\ / F ield | OpenFOAM: The Open Source CFD Toolbox
\\ / O peration | Website: https://openfoam.org
\\ / A nd | Version: 8
\\/ M anipulation |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
location "system";
object controlDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
application icoFoam;
startFrom startTime;
startTime 0;
stopAt endTime;
endTime 0.2;
deltaT 0.000001;
writeControl timeStep;
writeInterval 100;
purgeWrite 0;
writeFormat ascii;
writePrecision 6;
writeCompression off;
timeFormat general;
timePrecision 6;
runTimeModifiable true;
adjustTimeStep true;
maxCo 2;
maxDeltaT 0.0001;
// ************************************************************************* //Hence we run the solver.
The results are written for around 2000 timesteps.
bilal@bilal-VirtualBox:~/OpenFOAM/bilal-8/run/backward_step_1$ ls
0 0.0078 0.0156 0.0234 0.0312 0.039 0.0468 0.0546 0.0624 0.0702 0.078 0.0858 0.0936 0.1014 0.1092 0.117 0.1248 0.1326 0.1404 0.1482 0.156 0.1638 0.1716 0.1794 0.1872 0.195
0.0001 0.0079 0.0157 0.0235 0.0313 0.0391 0.0469 0.0547 0.0625 0.0703 0.0781 0.0859 0.0937 0.1015 0.1093 0.1171 0.1249 0.1327 0.1405 0.1483 0.1561 0.1639 0.1717 0.1795 0.1873 0.1951
0.0002 0.008 0.0158 0.0236 0.0314 0.0392 0.047 0.0548 0.0626 0.0704 0.0782 0.086 0.0938 0.1016 0.1094 0.1172 0.125 0.1328 0.1406 0.1484 0.1562 0.164 0.1718 0.1796 0.1874 0.1952
0.0003 0.0081 0.0159 0.0237 0.0315 0.0393 0.0471 0.0549 0.0627 0.0705 0.0783 0.0861 0.0939 0.1017 0.1095 0.1173 0.1251 0.1329 0.1407 0.1485 0.1563 0.1641 0.1719 0.1797 0.1875 0.1953
0.0004 0.0082 0.016 0.0238 0.0316 0.0394 0.0472 0.055 0.0628 0.0706 0.0784 0.0862 0.094 0.1018 0.1096 0.1174 0.1252 0.133 0.1408 0.1486 0.1564 0.1642 0.172 0.1798 0.1876 0.1954
0.0005 0.0083 0.0161 0.0239 0.0317 0.0395 0.0473 0.0551 0.0629 0.0707 0.0785 0.0863 0.0941 0.1019 0.1097 0.1175 0.1253 0.1331 0.1409 0.1487 0.1565 0.1643 0.1721 0.1799 0.1877 0.1955
0.0006 0.0084 0.0162 0.024 0.0318 0.0396 0.0474 0.0552 0.063 0.0708 0.0786 0.0864 0.0942 0.102 0.1098 0.1176 0.1254 0.1332 0.141 0.1488 0.1566 0.1644 0.1722 0.18 0.1878 0.1956
0.0007 0.0085 0.0163 0.0241 0.0319 0.0397 0.0475 0.0553 0.0631 0.0709 0.0787 0.0865 0.0943 0.1021 0.1099 0.1177 0.1255 0.1333 0.1411 0.1489 0.1567 0.1645 0.1723 0.1801 0.1879 0.1957
0.0008 0.0086 0.0164 0.0242 0.032 0.0398 0.0476 0.0554 0.0632 0.071 0.0788 0.0866 0.0944 0.1022 0.11 0.1178 0.1256 0.1334 0.1412 0.149 0.1568 0.1646 0.1724 0.1802 0.188 0.1958
0.0009 0.0087 0.0165 0.0243 0.0321 0.0399 0.0477 0.0555 0.0633 0.0711 0.0789 0.0867 0.0945 0.1023 0.1101 0.1179 0.1257 0.1335 0.1413 0.1491 0.1569 0.1647 0.1725 0.1803 0.1881 0.1959
0.001 0.0088 0.0166 0.0244 0.0322 0.04 0.0478 0.0556 0.0634 0.0712 0.079 0.0868 0.0946 0.1024 0.1102 0.118 0.1258 0.1336 0.1414 0.1492 0.157 0.1648 0.1726 0.1804 0.1882 0.196
0.0011 0.0089 0.0167 0.0245 0.0323 0.0401 0.0479 0.0557 0.0635 0.0713 0.0791 0.0869 0.0947 0.1025 0.1103 0.1181 0.1259 0.1337 0.1415 0.1493 0.1571 0.1649 0.1727 0.1805 0.1883 0.1961
0.0012 0.009 0.0168 0.0246 0.0324 0.0402 0.048 0.0558 0.0636 0.0714 0.0792 0.087 0.0948 0.1026 0.1104 0.1182 0.126 0.1338 0.1416 0.1494 0.1572 0.165 0.1728 0.1806 0.1884 0.1962
0.0013 0.0091 0.0169 0.0247 0.0325 0.0403 0.0481 0.0559 0.0637 0.0715 0.0793 0.0871 0.0949 0.1027 0.1105 0.1183 0.1261 0.1339 0.1417 0.1495 0.1573 0.1651 0.1729 0.1807 0.1885 0.1963
0.0014 0.0092 0.017 0.0248 0.0326 0.0404 0.0482 0.056 0.0638 0.0716 0.0794 0.0872 0.095 0.1028 0.1106 0.1184 0.1262 0.134 0.1418 0.1496 0.1574 0.1652 0.173 0.1808 0.1886 0.1964
0.0015 0.0093 0.0171 0.0249 0.0327 0.0405 0.0483 0.0561 0.0639 0.0717 0.0795 0.0873 0.0951 0.1029 0.1107 0.1185 0.1263 0.1341 0.1419 0.1497 0.1575 0.1653 0.1731 0.1809 0.1887 0.1965
0.0016 0.0094 0.0172 0.025 0.0328 0.0406 0.0484 0.0562 0.064 0.0718 0.0796 0.0874 0.0952 0.103 0.1108 0.1186 0.1264 0.1342 0.142 0.1498 0.1576 0.1654 0.1732 0.181 0.1888 0.1966
0.0017 0.0095 0.0173 0.0251 0.0329 0.0407 0.0485 0.0563 0.0641 0.0719 0.0797 0.0875 0.0953 0.1031 0.1109 0.1187 0.1265 0.1343 0.1421 0.1499 0.1577 0.1655 0.1733 0.1811 0.1889 0.1967
0.0018 0.0096 0.0174 0.0252 0.033 0.0408 0.0486 0.0564 0.0642 0.072 0.0798 0.0876 0.0954 0.1032 0.111 0.1188 0.1266 0.1344 0.1422 0.15 0.1578 0.1656 0.1734 0.1812 0.189 0.1968
0.0019 0.0097 0.0175 0.0253 0.0331 0.0409 0.0487 0.0565 0.0643 0.0721 0.0799 0.0877 0.0955 0.1033 0.1111 0.1189 0.1267 0.1345 0.1423 0.1501 0.1579 0.1657 0.1735 0.1813 0.1891 0.1969
0.002 0.0098 0.0176 0.0254 0.0332 0.041 0.0488 0.0566 0.0644 0.0722 0.08 0.0878 0.0956 0.1034 0.1112 0.119 0.1268 0.1346 0.1424 0.1502 0.158 0.1658 0.1736 0.1814 0.1892 0.197
0.0021 0.0099 0.0177 0.0255 0.0333 0.0411 0.0489 0.0567 0.0645 0.0723 0.0801 0.0879 0.0957 0.1035 0.1113 0.1191 0.1269 0.1347 0.1425 0.1503 0.1581 0.1659 0.1737 0.1815 0.1893 0.1971
0.0022 0.01 0.0178 0.0256 0.0334 0.0412 0.049 0.0568 0.0646 0.0724 0.0802 0.088 0.0958 0.1036 0.1114 0.1192 0.127 0.1348 0.1426 0.1504 0.1582 0.166 0.1738 0.1816 0.1894 0.1972
0.0023 0.0101 0.0179 0.0257 0.0335 0.0413 0.0491 0.0569 0.0647 0.0725 0.0803 0.0881 0.0959 0.1037 0.1115 0.1193 0.1271 0.1349 0.1427 0.1505 0.1583 0.1661 0.1739 0.1817 0.1895 0.1973
0.0024 0.0102 0.018 0.0258 0.0336 0.0414 0.0492 0.057 0.0648 0.0726 0.0804 0.0882 0.096 0.1038 0.1116 0.1194 0.1272 0.135 0.1428 0.1506 0.1584 0.1662 0.174 0.1818 0.1896 0.1974
0.0025 0.0103 0.0181 0.0259 0.0337 0.0415 0.0493 0.0571 0.0649 0.0727 0.0805 0.0883 0.0961 0.1039 0.1117 0.1195 0.1273 0.1351 0.1429 0.1507 0.1585 0.1663 0.1741 0.1819 0.1897 0.1975
0.0026 0.0104 0.0182 0.026 0.0338 0.0416 0.0494 0.0572 0.065 0.0728 0.0806 0.0884 0.0962 0.104 0.1118 0.1196 0.1274 0.1352 0.143 0.1508 0.1586 0.1664 0.1742 0.182 0.1898 0.1976
0.0027 0.0105 0.0183 0.0261 0.0339 0.0417 0.0495 0.0573 0.0651 0.0729 0.0807 0.0885 0.0963 0.1041 0.1119 0.1197 0.1275 0.1353 0.1431 0.1509 0.1587 0.1665 0.1743 0.1821 0.1899 0.1977
0.0028 0.0106 0.0184 0.0262 0.034 0.0418 0.0496 0.0574 0.0652 0.073 0.0808 0.0886 0.0964 0.1042 0.112 0.1198 0.1276 0.1354 0.1432 0.151 0.1588 0.1666 0.1744 0.1822 0.19 0.1978
0.0029 0.0107 0.0185 0.0263 0.0341 0.0419 0.0497 0.0575 0.0653 0.0731 0.0809 0.0887 0.0965 0.1043 0.1121 0.1199 0.1277 0.1355 0.1433 0.1511 0.1589 0.1667 0.1745 0.1823 0.1901 0.1979
0.003 0.0108 0.0186 0.0264 0.0342 0.042 0.0498 0.0576 0.0654 0.0732 0.081 0.0888 0.0966 0.1044 0.1122 0.12 0.1278 0.1356 0.1434 0.1512 0.159 0.1668 0.1746 0.1824 0.1902 0.198
0.0031 0.0109 0.0187 0.0265 0.0343 0.0421 0.0499 0.0577 0.0655 0.0733 0.0811 0.0889 0.0967 0.1045 0.1123 0.1201 0.1279 0.1357 0.1435 0.1513 0.1591 0.1669 0.1747 0.1825 0.1903 0.1981
0.0032 0.011 0.0188 0.0266 0.0344 0.0422 0.05 0.0578 0.0656 0.0734 0.0812 0.089 0.0968 0.1046 0.1124 0.1202 0.128 0.1358 0.1436 0.1514 0.1592 0.167 0.1748 0.1826 0.1904 0.1982
0.0033 0.0111 0.0189 0.0267 0.0345 0.0423 0.0501 0.0579 0.0657 0.0735 0.0813 0.0891 0.0969 0.1047 0.1125 0.1203 0.1281 0.1359 0.1437 0.1515 0.1593 0.1671 0.1749 0.1827 0.1905 0.1983
0.0034 0.0112 0.019 0.0268 0.0346 0.0424 0.0502 0.058 0.0658 0.0736 0.0814 0.0892 0.097 0.1048 0.1126 0.1204 0.1282 0.136 0.1438 0.1516 0.1594 0.1672 0.175 0.1828 0.1906 0.1984
0.0035 0.0113 0.0191 0.0269 0.0347 0.0425 0.0503 0.0581 0.0659 0.0737 0.0815 0.0893 0.0971 0.1049 0.1127 0.1205 0.1283 0.1361 0.1439 0.1517 0.1595 0.1673 0.1751 0.1829 0.1907 0.1985
0.0036 0.0114 0.0192 0.027 0.0348 0.0426 0.0504 0.0582 0.066 0.0738 0.0816 0.0894 0.0972 0.105 0.1128 0.1206 0.1284 0.1362 0.144 0.1518 0.1596 0.1674 0.1752 0.183 0.1908 0.1986
0.0037 0.0115 0.0193 0.0271 0.0349 0.0427 0.0505 0.0583 0.0661 0.0739 0.0817 0.0895 0.0973 0.1051 0.1129 0.1207 0.1285 0.1363 0.1441 0.1519 0.1597 0.1675 0.1753 0.1831 0.1909 0.1987
0.0038 0.0116 0.0194 0.0272 0.035 0.0428 0.0506 0.0584 0.0662 0.074 0.0818 0.0896 0.0974 0.1052 0.113 0.1208 0.1286 0.1364 0.1442 0.152 0.1598 0.1676 0.1754 0.1832 0.191 0.1988
0.0039 0.0117 0.0195 0.0273 0.0351 0.0429 0.0507 0.0585 0.0663 0.0741 0.0819 0.0897 0.0975 0.1053 0.1131 0.1209 0.1287 0.1365 0.1443 0.1521 0.1599 0.1677 0.1755 0.1833 0.1911 0.1989
0.004 0.0118 0.0196 0.0274 0.0352 0.043 0.0508 0.0586 0.0664 0.0742 0.082 0.0898 0.0976 0.1054 0.1132 0.121 0.1288 0.1366 0.1444 0.1522 0.16 0.1678 0.1756 0.1834 0.1912 0.199
0.0041 0.0119 0.0197 0.0275 0.0353 0.0431 0.0509 0.0587 0.0665 0.0743 0.0821 0.0899 0.0977 0.1055 0.1133 0.1211 0.1289 0.1367 0.1445 0.1523 0.1601 0.1679 0.1757 0.1835 0.1913 0.1991
0.0042 0.012 0.0198 0.0276 0.0354 0.0432 0.051 0.0588 0.0666 0.0744 0.0822 0.09 0.0978 0.1056 0.1134 0.1212 0.129 0.1368 0.1446 0.1524 0.1602 0.168 0.1758 0.1836 0.1914 0.1992
0.0043 0.0121 0.0199 0.0277 0.0355 0.0433 0.0511 0.0589 0.0667 0.0745 0.0823 0.0901 0.0979 0.1057 0.1135 0.1213 0.1291 0.1369 0.1447 0.1525 0.1603 0.1681 0.1759 0.1837 0.1915 0.1993
0.0044 0.0122 0.02 0.0278 0.0356 0.0434 0.0512 0.059 0.0668 0.0746 0.0824 0.0902 0.098 0.1058 0.1136 0.1214 0.1292 0.137 0.1448 0.1526 0.1604 0.1682 0.176 0.1838 0.1916 0.1994
0.0045 0.0123 0.0201 0.0279 0.0357 0.0435 0.0513 0.0591 0.0669 0.0747 0.0825 0.0903 0.0981 0.1059 0.1137 0.1215 0.1293 0.1371 0.1449 0.1527 0.1605 0.1683 0.1761 0.1839 0.1917 0.1995
0.0046 0.0124 0.0202 0.028 0.0358 0.0436 0.0514 0.0592 0.067 0.0748 0.0826 0.0904 0.0982 0.106 0.1138 0.1216 0.1294 0.1372 0.145 0.1528 0.1606 0.1684 0.1762 0.184 0.1918 0.1996
0.0047 0.0125 0.0203 0.0281 0.0359 0.0437 0.0515 0.0593 0.0671 0.0749 0.0827 0.0905 0.0983 0.1061 0.1139 0.1217 0.1295 0.1373 0.1451 0.1529 0.1607 0.1685 0.1763 0.1841 0.1919 0.1997
0.0048 0.0126 0.0204 0.0282 0.036 0.0438 0.0516 0.0594 0.0672 0.075 0.0828 0.0906 0.0984 0.1062 0.114 0.1218 0.1296 0.1374 0.1452 0.153 0.1608 0.1686 0.1764 0.1842 0.192 0.1998
0.0049 0.0127 0.0205 0.0283 0.0361 0.0439 0.0517 0.0595 0.0673 0.0751 0.0829 0.0907 0.0985 0.1063 0.1141 0.1219 0.1297 0.1375 0.1453 0.1531 0.1609 0.1687 0.1765 0.1843 0.1921 0.1999
0.005 0.0128 0.0206 0.0284 0.0362 0.044 0.0518 0.0596 0.0674 0.0752 0.083 0.0908 0.0986 0.1064 0.1142 0.122 0.1298 0.1376 0.1454 0.1532 0.161 0.1688 0.1766 0.1844 0.1922 0.2
0.0051 0.0129 0.0207 0.0285 0.0363 0.0441 0.0519 0.0597 0.0675 0.0753 0.0831 0.0909 0.0987 0.1065 0.1143 0.1221 0.1299 0.1377 0.1455 0.1533 0.1611 0.1689 0.1767 0.1845 0.1923 constant
0.0052 0.013 0.0208 0.0286 0.0364 0.0442 0.052 0.0598 0.0676 0.0754 0.0832 0.091 0.0988 0.1066 0.1144 0.1222 0.13 0.1378 0.1456 0.1534 0.1612 0.169 0.1768 0.1846 0.1924 system
0.0053 0.0131 0.0209 0.0287 0.0365 0.0443 0.0521 0.0599 0.0677 0.0755 0.0833 0.0911 0.0989 0.1067 0.1145 0.1223 0.1301 0.1379 0.1457 0.1535 0.1613 0.1691 0.1769 0.1847 0.1925
0.0054 0.0132 0.021 0.0288 0.0366 0.0444 0.0522 0.06 0.0678 0.0756 0.0834 0.0912 0.099 0.1068 0.1146 0.1224 0.1302 0.138 0.1458 0.1536 0.1614 0.1692 0.177 0.1848 0.1926
0.0055 0.0133 0.0211 0.0289 0.0367 0.0445 0.0523 0.0601 0.0679 0.0757 0.0835 0.0913 0.0991 0.1069 0.1147 0.1225 0.1303 0.1381 0.1459 0.1537 0.1615 0.1693 0.1771 0.1849 0.1927
0.0056 0.0134 0.0212 0.029 0.0368 0.0446 0.0524 0.0602 0.068 0.0758 0.0836 0.0914 0.0992 0.107 0.1148 0.1226 0.1304 0.1382 0.146 0.1538 0.1616 0.1694 0.1772 0.185 0.1928
0.0057 0.0135 0.0213 0.0291 0.0369 0.0447 0.0525 0.0603 0.0681 0.0759 0.0837 0.0915 0.0993 0.1071 0.1149 0.1227 0.1305 0.1383 0.1461 0.1539 0.1617 0.1695 0.1773 0.1851 0.1929
0.0058 0.0136 0.0214 0.0292 0.037 0.0448 0.0526 0.0604 0.0682 0.076 0.0838 0.0916 0.0994 0.1072 0.115 0.1228 0.1306 0.1384 0.1462 0.154 0.1618 0.1696 0.1774 0.1852 0.193
0.0059 0.0137 0.0215 0.0293 0.0371 0.0449 0.0527 0.0605 0.0683 0.0761 0.0839 0.0917 0.0995 0.1073 0.1151 0.1229 0.1307 0.1385 0.1463 0.1541 0.1619 0.1697 0.1775 0.1853 0.1931
0.006 0.0138 0.0216 0.0294 0.0372 0.045 0.0528 0.0606 0.0684 0.0762 0.084 0.0918 0.0996 0.1074 0.1152 0.123 0.1308 0.1386 0.1464 0.1542 0.162 0.1698 0.1776 0.1854 0.1932
0.0061 0.0139 0.0217 0.0295 0.0373 0.0451 0.0529 0.0607 0.0685 0.0763 0.0841 0.0919 0.0997 0.1075 0.1153 0.1231 0.1309 0.1387 0.1465 0.1543 0.1621 0.1699 0.1777 0.1855 0.1933
0.0062 0.014 0.0218 0.0296 0.0374 0.0452 0.053 0.0608 0.0686 0.0764 0.0842 0.092 0.0998 0.1076 0.1154 0.1232 0.131 0.1388 0.1466 0.1544 0.1622 0.17 0.1778 0.1856 0.1934
0.0063 0.0141 0.0219 0.0297 0.0375 0.0453 0.0531 0.0609 0.0687 0.0765 0.0843 0.0921 0.0999 0.1077 0.1155 0.1233 0.1311 0.1389 0.1467 0.1545 0.1623 0.1701 0.1779 0.1857 0.1935
0.0064 0.0142 0.022 0.0298 0.0376 0.0454 0.0532 0.061 0.0688 0.0766 0.0844 0.0922 0.1 0.1078 0.1156 0.1234 0.1312 0.139 0.1468 0.1546 0.1624 0.1702 0.178 0.1858 0.1936
0.0065 0.0143 0.0221 0.0299 0.0377 0.0455 0.0533 0.0611 0.0689 0.0767 0.0845 0.0923 0.1001 0.1079 0.1157 0.1235 0.1313 0.1391 0.1469 0.1547 0.1625 0.1703 0.1781 0.1859 0.1937
0.0066 0.0144 0.0222 0.03 0.0378 0.0456 0.0534 0.0612 0.069 0.0768 0.0846 0.0924 0.1002 0.108 0.1158 0.1236 0.1314 0.1392 0.147 0.1548 0.1626 0.1704 0.1782 0.186 0.1938
0.0067 0.0145 0.0223 0.0301 0.0379 0.0457 0.0535 0.0613 0.0691 0.0769 0.0847 0.0925 0.1003 0.1081 0.1159 0.1237 0.1315 0.1393 0.1471 0.1549 0.1627 0.1705 0.1783 0.1861 0.1939
0.0068 0.0146 0.0224 0.0302 0.038 0.0458 0.0536 0.0614 0.0692 0.077 0.0848 0.0926 0.1004 0.1082 0.116 0.1238 0.1316 0.1394 0.1472 0.155 0.1628 0.1706 0.1784 0.1862 0.194
0.0069 0.0147 0.0225 0.0303 0.0381 0.0459 0.0537 0.0615 0.0693 0.0771 0.0849 0.0927 0.1005 0.1083 0.1161 0.1239 0.1317 0.1395 0.1473 0.1551 0.1629 0.1707 0.1785 0.1863 0.1941
0.007 0.0148 0.0226 0.0304 0.0382 0.046 0.0538 0.0616 0.0694 0.0772 0.085 0.0928 0.1006 0.1084 0.1162 0.124 0.1318 0.1396 0.1474 0.1552 0.163 0.1708 0.1786 0.1864 0.1942
0.0071 0.0149 0.0227 0.0305 0.0383 0.0461 0.0539 0.0617 0.0695 0.0773 0.0851 0.0929 0.1007 0.1085 0.1163 0.1241 0.1319 0.1397 0.1475 0.1553 0.1631 0.1709 0.1787 0.1865 0.1943
0.0072 0.015 0.0228 0.0306 0.0384 0.0462 0.054 0.0618 0.0696 0.0774 0.0852 0.093 0.1008 0.1086 0.1164 0.1242 0.132 0.1398 0.1476 0.1554 0.1632 0.171 0.1788 0.1866 0.1944
0.0073 0.0151 0.0229 0.0307 0.0385 0.0463 0.0541 0.0619 0.0697 0.0775 0.0853 0.0931 0.1009 0.1087 0.1165 0.1243 0.1321 0.1399 0.1477 0.1555 0.1633 0.1711 0.1789 0.1867 0.1945
0.0074 0.0152 0.023 0.0308 0.0386 0.0464 0.0542 0.062 0.0698 0.0776 0.0854 0.0932 0.101 0.1088 0.1166 0.1244 0.1322 0.14 0.1478 0.1556 0.1634 0.1712 0.179 0.1868 0.1946
0.0075 0.0153 0.0231 0.0309 0.0387 0.0465 0.0543 0.0621 0.0699 0.0777 0.0855 0.0933 0.1011 0.1089 0.1167 0.1245 0.1323 0.1401 0.1479 0.1557 0.1635 0.1713 0.1791 0.1869 0.1947
0.0076 0.0154 0.0232 0.031 0.0388 0.0466 0.0544 0.0622 0.07 0.0778 0.0856 0.0934 0.1012 0.109 0.1168 0.1246 0.1324 0.1402 0.148 0.1558 0.1636 0.1714 0.1792 0.187 0.1948
0.0077 0.0155 0.0233 0.0311 0.0389 0.0467 0.0545 0.0623 0.0701 0.0779 0.0857 0.0935 0.1013 0.1091 0.1169 0.1247 0.1325 0.1403 0.1481 0.1559 0.1637 0.1715 0.1793 0.1871 0.1949
RESULTS:
MESH:
This is the mesh that looks right after doing the Mesh grading.
CONVERGED SOLUTION:
The solution after 0.2 seconds of flow, is given in the following image.
OBTAINED PLOT RESULT:
INFERENCE:
- The mesh looks finer in the second case but eventually, it took more time for the solver to solve it since it has very little deltaT value compared to the farther, but eventually, it would always yield an accurate result.
- Both the graphs are almost similar. This is because with the conscience of running for very little time we chose 0.2 seconds. If it was a larger time then we would have witnessed some slight changes in the graph.
- However, due to the finer mesh in the second case, we may actually witness a sudden drop in velocity at 0.01m height which is not shown in the above case, which infers that the Mesh grading gives accurate results.