Week 12 - Validation studies of Symmetry BC vs Wedge BC in OpenFOAM vs Analytical H.P equation
AIM
To simulate the same problem as in week 1 but by applying Symmetry Boundary Condition.Do the simulation for the below mentioned angles using both symmetry and wedge boundary conditions and validate them with HP equations like in week 11 challenge.
Angles to test:
- 10°
- 25°
- 45°
OBJECTIVE
- Validate Hydro-dynamic length with the numerical result
- Validate the fully developed flow velocity profile with its analytical profile
- Validate maximum velocity and pressured drop for fully developed flow
- Post-process Shear stress and validate wall shear stress for fully developed flow
- Highligh the differences in both boundary conditions in terms of their usage and aplication
THEORY
HAGEN-POISEUILLE EQUATION
In nonideal fluid dynamics, the Hagen–Poiseuille equation, also known as the Hagen–Poiseuille law, Poiseuille law or Poiseuille equation, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section.
ASSUMPTIONS
- Fluid is incompressible and Newtonian.
- Flow is laminar through pipe of constant circular cross section area having length more than its Diameter
- No acceleration of fluid in pipe
- For velocities and pipe diameters above a threshold, actual fluid flow is not laminar but turbulent, leading to larger pressure drops than calculated by the Hagen–Poiseuille equation.
Poiseuille's equation describes the pressure drop due to the viscosity of the fluid; Other types of pressure drops may still occur in a fluid (see a demonstration here). For example, the pressure needed to drive a viscous fluid up against gravity would contain both that as needed in Poiseuille's law plus that as needed in Bernoulli's equation, such that any point in the flow would have a pressure greater than zero (otherwise no flow would happen).
In standard fluid-kinetics notation:
PRESSURE DIFFERENCE
HYDRODYNAMIC ENTRY LENGTH
-
LE`=0.06*D*Re
SHEAR STRESS(NEWTON'S LAW OF VISCOSITY)
VELOCITY PROFILE
from the above equation, velocity of flow increases parabolically from wall to center of pipe
where,
- Δp is the pressure difference between the two ends,
- L is the length of pipe,
- μ is the dynamic viscosity,
- Q is the volumetric flow rate,
- R is the pipe radius,
- A is the cross section of pipe.
The equation does not hold close to the pipe entrance.
The ratio of length to radius of a pipe should be greater than one forty-eighth of the Reynolds number for the Hagen–Poiseuille law to be valid.
The Hagen–Poiseuille equation can be derived from the Navier–Stokes equations. The laminar flow through a pipe of uniform (circular) cross-section is known as Hagen–Poiseuille flow. The equations governing the Hagen–Poiseuille flow can be derived directly from the Navier–Stokes momentum equations in 3D cylindrical coordinates (r,θ,x) by making the following set of assumptions:
- The flow is steady ( ∂(...)/∂t=0 ).
- The radial and azimuthal components of the fluid velocity are zero ( ur=uθ=0 ).
- The flow is axisymmetric ( ∂(...)/∂θ=0 ).
- The flow is fully developed (∂ux/∂x=0 ). Here However, this can be proved via mass conservation, and the above assumptions.
CONSIDERATIONS
Working temperature of water=25°C
∴density(ρ)=997`(kg)/m^3`
dynamic viscosity(μ)=8.90×10-4Pa.s
Diameter (D)=0.02m
Length of the pipe(L)=2.5m
PARAMETERS
Re=2100 ......................(given in the problem statement)
MATLAB CODE
- As the value of vertices can be calculated by resolving the components of radius.
- For this we can use file parsing method to write the blockMeshDict file in OpenFoam by writing a code in matlab,'fprintf' command is used to print as parameters like vertices,blocks,boundary(inlet,outlet,front,back,pipewall,axis)
Main code
- Input variables are assigned like
>reynolds number(given in the problem statement)
>Apex angle of wedge
>Diameter of the pipe
>Radius of the pipe
>density of fluid
>dynamic viscosity
>Total length of the pipe
>entrance length of the pipe
- Function code'blockMesh' is being called out
- After running the simulation on OpenFoam and getting its result from paraView .Now we have to compare the analytical and numerical values and compare its result in plot
- for this the analytical values are being calculated
- average velocity of pipe is calculated
- velocity at different points from centre of pipe is calculated
- pressure drop across length of pipe is calculated
- Wall shear stress is calculated
- now the values obtained in paraView is being imported to matlab by opening the csv file in the current folder
- And 'fopen' command is used in the code to open the csv file
- the data is for 1000 points in the csv file, so a for loop is created from 1 to 1000 to get the thousand valus of velocity
- B variable is used to command 'fgetl' that Read line(s) from file, discard newline character
- C variable is used to spilt B into rows having ','between two value
- U_mag is in the 13th column ,v_n(numerical value) is assigned the 13th column of C for every change in value of i
- now vn variable is updated by using the command 'str2double(vn)' converts an array of character vectors into a numeric array
- now to compare the results vr ,vn is plotted on Y-axis and radius on X-axis
clear all
close all
clc
%%Variables
R=2100; %Reynold's number
theta=input('Enter the wedge angle = '); %wedge angle(in degrees)
D=0.02; %Diameter of the pipe (m)
r=D/2; %Radius of the pipe(m)
rho=997; %Density(kg/m^3)
mu = 8.90e-4; %Dynamic Viscosity
L=2.5; %Total length of the pipe(m)
Le=0.058*R*D; %Entrance length
nx = 20; %cells along x
ny = 1; %cells along y
nz = 500; %cells along z
%%BlockMeshDict
[f1]=blockMesh(r,theta,L,nx,ny,nz);
%%analytical values
v_avg=(R*mu)/(rho*D); %average velocity
x=linspace(-r,r,1000); %distance from center of pipe to the circumference
%velocity at different radius in fully developed region
vr=2*v_avg*(1-(x.^2/r^2));
v_max=2*v_avg; %velocity at center of pipe
L_p=linspace(0,L,1000); %length of pipe
%pressure drop across the length of pipe
P_d=(32*mu.*L_p*v_avg)/D^2;
P_d=fliplr(P_d);
f=64/R;%darcy friction factor
tau_a=(f/4)*((rho*v_avg^2)/2);%%Wall Shear Stress
%Reading and Extracting Velocity Data From ParaView
f2=fopen('vel_ss.csv','r');
A=fgetl(f2);
for i=1:1000
B=fgetl(f2);
C=strsplit(B,',');
v_n(i)=C(13);
end
v_n=str2double(v_n);
%%[1]-Plotting - Velocity comparison from analytical method and numerical method
figure(1)
plot(x,vr,'r')
hold on
plot(x,v_n,'--','color','k','linewidth',2)
xlabel('Radius of pipe in m')
ylabel('Velocity in m/s')
legend('Numerical method',' Analytical method')
grid on
%Error
E_max=max(max(abs(vr-v_n)));
function code for blockMeshDict file
- Since the pipe is a axi-symmetric,there is no need to simulate the entire pipe.wedge having apex angle less than 5 degrees can be used making no changes in the final result.
- the below figure is the schematic representation of the wedge
- X-axis is bisecting the apex angle of the wedge
- vertice 0 is taken as the origin of the entire system v0=(0,0,0)
- vertice 1=(rcos(θ2),rsin(θ2),0)">=(rcos(θ2),rsin(θ2),0)=(rcos(θ2),rsin(θ2),0)
- vertice 2=(rcos(θ2),rsin(θ2),L)">(rcos(θ2),rsin(θ2),L)(rcos(θ2),rsin(θ2),L)
- vertice 3=(0,0,L)">(0,0,L)(0,0,L)
- vertice 4=(rcos(θ2),-rsin(θ2),0">rcos(θ2),−rsin(θ2),0rcos(θ2),-rsin(θ2),0)
- vertice 5=(rcos(θ2),-rsin(θ2),L)">(rcos(θ2),−rsin(θ2),L)(rcos(θ2),-rsin(θ2),L)
- function code is used for generation of blockMeshDict file
- It starts by writing 'fopen' command
- before define any vertices,boundary .,header information as comment is printed which is not executable in OpenFoam
- to have uniformity in dimensions , all type of length is converted to meters
- Vertices are defined as stated above in the figures
- Mesh of pipe includes a new feature called as the wedge(block having less than 8 vertices)
- hexahedral meshing is generated
- vertices are numbered according to right hand thumb rule
- vertice 4-1 and vertice 5-2 are curved so it it defined in edges by arc
- since there is no physical wall in (0-1-2-3) and(0-4-5-3),and there exists a patch wedge in the two faces thta indicates the presence of axi-symmetry
FUNCTION CODE FOR WEDGE BOUNDARY CONDITION
function[f1]=blockMesh_wedge(r,theta,L,nx,ny,nz)
%%File writing
line1 = '/*--------------------------------*- C++ -*----------------------------------*\';
line2 = '| ========= | |';
line3 = '| \\ / F ield | OpenFOAM: The Open Source CFD Toolbox |';
line4 = '| \\ / O peration | Version: v2012 |';
line5 = '| \\ / A nd | Website: www.openfoam.com |';
line6 = '| \\/ M anipulation | |';
line7 = '\*---------------------------------------------------------------------------*/';
%file open
f1 = fopen('blockMeshDict_w','w');
%To write header
fprintf(f1,'%s\n',line1);
fprintf(f1,'%s\n',line2);
fprintf(f1,'%s\n',line3);
fprintf(f1,'%s\n',line4);
fprintf(f1,'%s\n',line5);
fprintf(f1,'%s\n',line6);
fprintf(f1,'%s\n',line7);
%To write the dictionary
fprintf(f1,'FoamFile \n');
fprintf(f1,'{ \n');
fprintf(f1,' version 2.0;\n');
fprintf(f1,' format ascii;\n');
fprintf(f1,' class dictionary;\n');
fprintf(f1,' object blockMeshDict;\n');
fprintf(f1,'}');
fprintf(f1,'// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //');
fprintf(f1,'\n');
% Note: As all the calculated variables are in meter(m), keeping scale to
% be as one (1).
fprintf(f1,'convertToMeters 1;\n\n');
%Vertices
fprintf(f1,'vertices\n');
fprintf(f1,'(\n');
fprintf(f1,'\t (0 0 0)\n');
fprintf(f1,('\t (%f %f 0)\n'),r*cosd(theta/2),r*sind(theta/2));
fprintf(f1,('\t (%f %f %f)\n'),r*cosd(theta/2),r*sind(theta/2),L);
fprintf(f1,('\t (0 0 %f)\n'),L);
fprintf(f1,('\t (%f %f 0)\n'),r*cosd(theta/2),-r*sind(theta/2));
fprintf(f1,('\t (%f %f %f)\n);\n\n'),r*cosd(theta/2),-r*sind(theta/2),L);
%Write block
fprintf(f1,'blocks\n(\n');
fprintf(f1,('hex (0 4 1 0 3 5 2 3) (%d %d %d) simpleGrading (0.2 1 1)\n);\n\n'),nx,ny,nz);
%Edges
fprintf(f1,'edges\n(\n');
fprintf(f1,'\t arc 4 1 (0 %f 0)\n',r);
fprintf(f1,'\t arc 5 2 (%f %f 0)\n);\n\n',L,r);
%Boundary
fprintf(f1,'boundary\n(\n');
%Inlet
fprintf(f1,' inlet\n');
fprintf(f1,' {\n');
fprintf(f1,' type patch;\n');
fprintf(f1,' faces\n');
fprintf(f1,' (\n');
fprintf(f1,' (0 4 1 0)\n');
fprintf(f1,' );\n');
fprintf(f1,' }\n');
%outlet
fprintf(f1,' outlet\n');
fprintf(f1,' {\n');
fprintf(f1,' type patch;\n');
fprintf(f1,' faces\n');
fprintf(f1,' (\n');
fprintf(f1,' (3 5 2 3)\n');
fprintf(f1,' );\n');
fprintf(f1,' }\n');
%Front wedge
fprintf(f1,' front\n');
fprintf(f1,' {\n');
fprintf(f1,' type wedge;\n');
fprintf(f1,' faces\n');
fprintf(f1,' (\n');
fprintf(f1,' (0 1 2 3)\n');
fprintf(f1,' );\n');
fprintf(f1,' }\n');
%Back wedge
fprintf(f1,' back\n');
fprintf(f1,' {\n');
fprintf(f1,' type wedge;\n');
fprintf(f1,' faces \n');
fprintf(f1,' (\n');
fprintf(f1,' (0 3 5 4)\n');
fprintf(f1,' );\n');
fprintf(f1,' }\n');
%Top wall
fprintf(f1,' pipewall\n');
fprintf(f1,' {\n');
fprintf(f1,' type wall;\n');
fprintf(f1,' faces\n');
fprintf(f1,' (\n');
fprintf(f1,' (1 4 5 2)\n');
fprintf(f1,' );\n');
fprintf(f1,' }\n');
%Bottom wall
fprintf(f1,' axis\n');
fprintf(f1,' {\n');
fprintf(f1',' type empty;\n');
fprintf(f1,' faces\n');
fprintf(f1,' (\n');
fprintf(f1,' (0 3 3 0)\n');
fprintf(f1,' );\n');
fprintf(f1,' }\n');
fprintf(f1,');\n\n');
fprintf(f1,'margePatchPairs\n');
fprintf(f1,'(\n');
fprintf(f1,');\n\n');
fprintf(f1,'// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //');
fclose(f1);
end
FUNCTION CODE FOR SYMMETRY BOUNDARY CONDITION
function[f1]=blockMesh_symmetry(r,theta,L,nx,ny,nz)
%%File writing
line1 = '/*--------------------------------*- C++ -*----------------------------------*\';
line2 = '| ========= | |';
line3 = '| \\ / F ield | OpenFOAM: The Open Source CFD Toolbox |';
line4 = '| \\ / O peration | Version: v2012 |';
line5 = '| \\ / A nd | Website: www.openfoam.com |';
line6 = '| \\/ M anipulation | |';
line7 = '\*---------------------------------------------------------------------------*/';
%file open
f1 = fopen('blockMeshDict_s','w');
%To write header
fprintf(f1,'%s\n',line1);
fprintf(f1,'%s\n',line2);
fprintf(f1,'%s\n',line3);
fprintf(f1,'%s\n',line4);
fprintf(f1,'%s\n',line5);
fprintf(f1,'%s\n',line6);
fprintf(f1,'%s\n',line7);
%To write the dictionary
fprintf(f1,'FoamFile \n');
fprintf(f1,'{ \n');
fprintf(f1,' version 2.0;\n');
fprintf(f1,' format ascii;\n');
fprintf(f1,' class dictionary;\n');
fprintf(f1,' object blockMeshDict;\n');
fprintf(f1,'}');
fprintf(f1,'// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //');
fprintf(f1,'\n');
% Note: As all the calculated variables are in meter(m), keeping scale to
% be as one (1).
fprintf(f1,'convertToMeters 1;\n\n');
%Vertices
fprintf(f1,'vertices\n');
fprintf(f1,'(\n');
fprintf(f1,'\t (0 0 0)\n');
fprintf(f1,('\t (%f %f 0)\n'),r*cosd(theta/2),r*sind(theta/2));
fprintf(f1,('\t (%f %f %f)\n'),r*cosd(theta/2),r*sind(theta/2),L);
fprintf(f1,('\t (0 0 %f)\n'),L);
fprintf(f1,('\t (%f %f 0)\n'),r*cosd(theta/2),-r*sind(theta/2));
fprintf(f1,('\t (%f %f %f)\n);\n\n'),r*cosd(theta/2),-r*sind(theta/2),L);
%Write block
fprintf(f1,'blocks\n(\n');
fprintf(f1,('hex (0 4 1 0 3 5 2 3) (%d %d %d) simpleGrading (0.2 1 1)\n);\n\n'),nx,ny,nz);
%Edges
fprintf(f1,'edges\n(\n');
fprintf(f1,'\t arc 4 1 (0 %f 0)\n',r);
fprintf(f1,'\t arc 5 2 (%f %f 0)\n);\n\n',L,r);
%Boundary
fprintf(f1,'boundary\n(\n');
%Inlet
fprintf(f1,' inlet\n');
fprintf(f1,' {\n');
fprintf(f1,' type patch;\n');
fprintf(f1,' faces\n');
fprintf(f1,' (\n');
fprintf(f1,' (0 4 1 0)\n');
fprintf(f1,' );\n');
fprintf(f1,' }\n');
%outlet
fprintf(f1,' outlet\n');
fprintf(f1,' {\n');
fprintf(f1,' type patch;\n');
fprintf(f1,' faces\n');
fprintf(f1,' (\n');
fprintf(f1,' (3 5 2 3)\n');
fprintf(f1,' );\n');
fprintf(f1,' }\n');
%Front wedge
fprintf(f1,' front\n');
fprintf(f1,' {\n');
fprintf(f1,' type symmetry;\n');
fprintf(f1,' faces\n');
fprintf(f1,' (\n');
fprintf(f1,' (0 1 2 3)\n');
fprintf(f1,' );\n');
fprintf(f1,' }\n');
%Back wedge
fprintf(f1,' back\n');
fprintf(f1,' {\n');
fprintf(f1,' type symmetry;\n');
fprintf(f1,' faces \n');
fprintf(f1,' (\n');
fprintf(f1,' (0 3 5 4)\n');
fprintf(f1,' );\n');
fprintf(f1,' }\n');
%Top wall
fprintf(f1,' pipewall\n');
fprintf(f1,' {\n');
fprintf(f1,' type wall;\n');
fprintf(f1,' faces\n');
fprintf(f1,' (\n');
fprintf(f1,' (1 4 5 2)\n');
fprintf(f1,' );\n');
fprintf(f1,' }\n');
%Bottom wall
fprintf(f1,' axis\n');
fprintf(f1,' {\n');
fprintf(f1',' type empty;\n');
fprintf(f1,' faces\n');
fprintf(f1,' (\n');
fprintf(f1,' (0 3 3 0)\n');
fprintf(f1,' );\n');
fprintf(f1,' }\n');
fprintf(f1,');\n\n');
fprintf(f1,'margePatchPairs\n');
fprintf(f1,'(\n');
fprintf(f1,');\n\n');
fprintf(f1,'// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //');
fclose(f1);
end
OpenFoam
controlDict
/*--------------------------------*- C++ -*----------------------------------*\
| ========= | |
| \\ / F ield | OpenFOAM: The Open Source CFD Toolbox |
| \\ / O peration | Version: v2012 |
| \\ / A nd | Website: www.openfoam.com |
| \\/ M anipulation | |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
location "system";
object controlDict;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
application icoFoam;
startFrom latestTime; // startTime, or to resume, latestTime
startTime 0;
stopAt endTime;
endTime 30;
deltaT 0.02;
writeControl runTime; // or timeStep
writeInterval 0.5; // while using timeStep, writing an oputput folder every writeInterval*deltaT or else the value specified
purgeWrite 0; // retain the last specified number of files...to retain all 0
writeFormat ascii;
writePrecision 6;
writeCompression off;
timeFormat general;
timePrecision 6;
runTimeModifiable true;
// ************************************************************************* //Initial Conditions
velocity
At the inlet of the pipe ,initial velocity in the Z directions is given.This velocity is the average velocity of the pipe that is calculated in the main code.
VELOCITY FILE FOR WEDGE BOUNDARY CONDITION
/*--------------------------------*- C++ -*----------------------------------*\
| ========= | |
| \\ / F ield | OpenFOAM: The Open Source CFD Toolbox |
| \\ / O peration | Version: v2012 |
| \\ / A nd | Website: www.openfoam.com |
| \\/ M anipulation | |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class volVectorField;
object U;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
dimensions [0 1 -1 0 0 0 0];
internalField uniform (0 0 0);
boundaryField
{
inlet
{
type fixedValue;
value uniform (0 0 0.0937);
}
outlet
{
type zeroGradient;
}
front
{
type wedge;
}
back
{
type wedge;
}
axis
{
type empty;
}
pipewall
{
type noSlip;
}
}
// ************************************************************************* //VELOCITY FILE FOR SYMMETRY BOUNDARY CONDITION
/*--------------------------------*- C++ -*----------------------------------*\
| ========= | |
| \\ / F ield | OpenFOAM: The Open Source CFD Toolbox |
| \\ / O peration | Version: v2012 |
| \\ / A nd | Website: www.openfoam.com |
| \\/ M anipulation | |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class volVectorField;
object U;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
dimensions [0 1 -1 0 0 0 0];
internalField uniform (0 0 0);
boundaryField
{
inlet
{
type fixedValue;
value uniform (0 0 0.0937);
}
outlet
{
type zeroGradient;
}
front
{
type symmetry;
}
back
{
type symmetry;
}
axis
{
type empty;
}
pipewall
{
type noSlip;
}
}
// ************************************************************************* //PRESSURE FILE FOR WEDGE BOUNDARY CONDITION
/*--------------------------------*- C++ -*----------------------------------*\
| ========= | |
| \\ / F ield | OpenFOAM: The Open Source CFD Toolbox |
| \\ / O peration | Version: v2012 |
| \\ / A nd | Website: www.openfoam.com |
| \\/ 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;
}
front
{
type wedge;
}
back
{
type wedge;
}
axis
{
type empty;
}
pipewall
{
type zeroGradient;
}
}
// ************************************************************************* //
PRESSURE FILE FOR SYMMETRY BOUNDARY CONDITION
/*--------------------------------*- C++ -*----------------------------------*\
| ========= | |
| \\ / F ield | OpenFOAM: The Open Source CFD Toolbox |
| \\ / O peration | Version: v2012 |
| \\ / A nd | Website: www.openfoam.com |
| \\/ 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;
}
front
{
type symmetry;
}
back
{
type symmetry;
}
axis
{
type empty;
}
pipewall
{
type zeroGradient;
}
}
// ************************************************************************* //
transportProperties
In openFoam,every tutorial has three files namely 0,constant and system.The 'constant' file stores transport properties file in notepad. this has value ν">ννi.e,kinetic viscosity.we to convert the dynamic viscosity that we have considered for this problem statement into kinetic viscosity by using the below formula
`nu=mu/rho`
where `nu`-kinetic viscosity
μ">μ-dynamic viscosity=8.90e-4
ρ">ρ-density=997
/*--------------------------------*- C++ -*----------------------------------*\
| ========= | |
| \\ / F ield | OpenFOAM: The Open Source CFD Toolbox |
| \\ / O peration | Version: v2012 |
| \\ / A nd | Website: www.openfoam.com |
| \\/ M anipulation | |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
location "constant";
object transportProperties;
}
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
nu [0 2 -1 0 0 0 0] 8.9268e-07;
// ************************************************************************* //blockMeshDict
- Wedge
1.for 10°
/*--------------------------------*- C++ -*----------------------------------*\
| ========= | |
| \\ / F ield | OpenFOAM: The Open Source CFD Toolbox |
| \\ / O peration | Version: v2012 |
| \\ / A nd | Website: www.openfoam.com |
| \\/ M anipulation | |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
object blockMeshDict;
}// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 1;
vertices
(
(0 0 0)
(0.009962 0.000872 0)
(0.009962 0.000872 2.500000)
(0 0 2.500000)
(0.009962 -0.000872 0)
(0.009962 -0.000872 2.500000)
);
blocks
(
hex (0 4 1 0 3 5 2 3) (20 1 500) simpleGrading (0.2 1 1)
);
edges
(
arc 4 1 (0 0.010000 0)
arc 5 2 (2.500000 0.010000 0)
);
boundary
(
inlet
{
type patch;
faces
(
(0 4 1 0)
);
}
outlet
{
type patch;
faces
(
(3 5 2 3)
);
}
front
{
type wedge;
faces
(
(0 1 2 3)
);
}
back
{
type wedge;
faces
(
(0 3 5 4)
);
}
pipewall
{
type wall;
faces
(
(1 4 5 2)
);
}
axis
{
type empty;
faces
(
(0 3 3 0)
);
}
);
margePatchPairs
(
);
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //hydrodynamic length
velocity at the hydrodynamic length
pressure along the pipe
wall shear stress at the hydrodynamic length
2.for 25°
/*--------------------------------*- C++ -*----------------------------------*\
| ========= | |
| \\ / F ield | OpenFOAM: The Open Source CFD Toolbox |
| \\ / O peration | Version: v2012 |
| \\ / A nd | Website: www.openfoam.com |
| \\/ M anipulation | |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
object blockMeshDict;
}// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 1;
vertices
(
(0 0 0)
(0.009763 0.002164 0)
(0.009763 0.002164 2.500000)
(0 0 2.500000)
(0.009763 -0.002164 0)
(0.009763 -0.002164 2.500000)
);
blocks
(
hex (0 4 1 0 3 5 2 3) (20 1 500) simpleGrading (0.2 1 1)
);
edges
(
arc 4 1 (0 0.010000 0)
arc 5 2 (2.500000 0.010000 0)
);
boundary
(
inlet
{
type patch;
faces
(
(0 4 1 0)
);
}
outlet
{
type patch;
faces
(
(3 5 2 3)
);
}
front
{
type wedge;
faces
(
(0 1 2 3)
);
}
back
{
type wedge;
faces
(
(0 3 5 4)
);
}
pipewall
{
type wall;
faces
(
(1 4 5 2)
);
}
axis
{
type empty;
faces
(
(0 3 3 0)
);
}
);
margePatchPairs
(
);
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //velocity at the hydrodynamic length
pressure along the pipe
wall shear stress at the hydrodynamic length
3.for 45°
/*--------------------------------*- C++ -*----------------------------------*\
| ========= | |
| \\ / F ield | OpenFOAM: The Open Source CFD Toolbox |
| \\ / O peration | Version: v2012 |
| \\ / A nd | Website: www.openfoam.com |
| \\/ M anipulation | |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
object blockMeshDict;
}// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 1;
vertices
(
(0 0 0)
(0.009239 0.003827 0)
(0.009239 0.003827 2.500000)
(0 0 2.500000)
(0.009239 -0.003827 0)
(0.009239 -0.003827 2.500000)
);
blocks
(
hex (0 4 1 0 3 5 2 3) (20 1 500) simpleGrading (0.2 1 1)
);
edges
(
arc 4 1 (0 0.010000 0)
arc 5 2 (2.500000 0.010000 0)
);
boundary
(
inlet
{
type patch;
faces
(
(0 4 1 0)
);
}
outlet
{
type patch;
faces
(
(3 5 2 3)
);
}
front
{
type wedge;
faces
(
(0 1 2 3)
);
}
back
{
type wedge;
faces
(
(0 3 5 4)
);
}
pipewall
{
type wall;
faces
(
(1 4 5 2)
);
}
axis
{
type empty;
faces
(
(0 3 3 0)
);
}
);
margePatchPairs
(
);
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //velocity at the hydrodynamic length
pressure along the pipe
wall shear stress at the hydrodynamic length
- Symmetry
1.for 10°
/*--------------------------------*- C++ -*----------------------------------*\
| ========= | |
| \\ / F ield | OpenFOAM: The Open Source CFD Toolbox |
| \\ / O peration | Version: v2012 |
| \\ / A nd | Website: www.openfoam.com |
| \\/ M anipulation | |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
object blockMeshDict;
}// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 1;
vertices
(
(0 0 0)
(0.009962 0.000872 0)
(0.009962 0.000872 2.500000)
(0 0 2.500000)
(0.009962 -0.000872 0)
(0.009962 -0.000872 2.500000)
);
blocks
(
hex (0 4 1 0 3 5 2 3) (20 1 500) simpleGrading (0.2 1 1)
);
edges
(
arc 4 1 (0 0.010000 0)
arc 5 2 (2.500000 0.010000 0)
);
boundary
(
inlet
{
type patch;
faces
(
(0 4 1 0)
);
}
outlet
{
type patch;
faces
(
(3 5 2 3)
);
}
front
{
type symmetry;
faces
(
(0 1 2 3)
);
}
back
{
type symmetry;
faces
(
(0 3 5 4)
);
}
pipewall
{
type wall;
faces
(
(1 4 5 2)
);
}
axis
{
type empty;
faces
(
(0 3 3 0)
);
}
);
margePatchPairs
(
);
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //velocity at the hydrodynamic length
pressure along the pipe
wall shear stress at the hydrodynamic length
2.for 25°
/*--------------------------------*- C++ -*----------------------------------*\
| ========= | |
| \\ / F ield | OpenFOAM: The Open Source CFD Toolbox |
| \\ / O peration | Version: v2012 |
| \\ / A nd | Website: www.openfoam.com |
| \\/ M anipulation | |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
object blockMeshDict;
}// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 1;
vertices
(
(0 0 0)
(0.009763 0.002164 0)
(0.009763 0.002164 2.500000)
(0 0 2.500000)
(0.009763 -0.002164 0)
(0.009763 -0.002164 2.500000)
);
blocks
(
hex (0 4 1 0 3 5 2 3) (20 1 500) simpleGrading (0.2 1 1)
);
edges
(
arc 4 1 (0 0.010000 0)
arc 5 2 (2.500000 0.010000 0)
);
boundary
(
inlet
{
type patch;
faces
(
(0 4 1 0)
);
}
outlet
{
type patch;
faces
(
(3 5 2 3)
);
}
front
{
type symmetry;
faces
(
(0 1 2 3)
);
}
back
{
type symmetry;
faces
(
(0 3 5 4)
);
}
pipewall
{
type wall;
faces
(
(1 4 5 2)
);
}
axis
{
type empty;
faces
(
(0 3 3 0)
);
}
);
margePatchPairs
(
);
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //velocity at the hydrodynamic length
pressure along the pipe
wall shear stress at the hydrodynamic length
3.for 45°
/*--------------------------------*- C++ -*----------------------------------*\
| ========= | |
| \\ / F ield | OpenFOAM: The Open Source CFD Toolbox |
| \\ / O peration | Version: v2012 |
| \\ / A nd | Website: www.openfoam.com |
| \\/ M anipulation | |
\*---------------------------------------------------------------------------*/
FoamFile
{
version 2.0;
format ascii;
class dictionary;
object blockMeshDict;
}// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //
convertToMeters 1;
vertices
(
(0 0 0)
(0.009239 0.003827 0)
(0.009239 0.003827 2.500000)
(0 0 2.500000)
(0.009239 -0.003827 0)
(0.009239 -0.003827 2.500000)
);
blocks
(
hex (0 4 1 0 3 5 2 3) (20 1 500) simpleGrading (0.2 1 1)
);
edges
(
arc 4 1 (0 0.010000 0)
arc 5 2 (2.500000 0.010000 0)
);
boundary
(
inlet
{
type patch;
faces
(
(0 4 1 0)
);
}
outlet
{
type patch;
faces
(
(3 5 2 3)
);
}
front
{
type symmetry;
faces
(
(0 1 2 3)
);
}
back
{
type symmetry;
faces
(
(0 3 5 4)
);
}
pipewall
{
type wall;
faces
(
(1 4 5 2)
);
}
axis
{
type empty;
faces
(
(0 3 3 0)
);
}
);
margePatchPairs
(
);
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //velocity at the hydrodynamic length
pressure along the pipe
wall shear stress at the hydrodynamic length
CONCLUSION：
| angle | 10 | 25 | 45 |
| hydrodynamic length | 0.09 | 0.08 |
0.07 |
The values of pressure and velocity calculated using Hagen Poisuelles equation are given
below：
Velocity： 0.4214m/s
Pressure：0.07107(Kinematic Pressure)
It is observed that with an increase in wedge angle， the velocity and pressure near the
exact solution.
The computational time for the symmetric BC is more than that of wedge BC
The velocity dipped toO at the boundary walls testifies the no-slip BC for all the cases.
We have performed Simulation for two Boundary Conditions； those are Symmetry
Boundary Condition and the Wedge Boundary Condition(As performed in the previous
challenge) .During the Wedge Boundary condition simulation， the simulation across the arc
of the wedge is not done， but whereas in the case of Symmetry Boundary Condition， the
calculation across the arc of the wedge is done； hence the simulation time is higher in that
case.The Courant number is increasing slightly in the case of wedge Boundary Condition
and hence the Symmetry Boundary Condition wll be slightly more accurate.The Symmetry
Boundary condition considers the Meshas3D and the calculation is done along all the
three global axes.
We understand the results should not change due to the change in the Boundary
conditions， and this is because the Physics of the problem does not change and remains
the same.