## OpenFoam Simulation of Flow through a Pipe using Axi-symmetric Boundary Condition

Formulae involved:

a) Entry Length of a Pipe for Laminar Flow, L = 0.07*Re*D

b) Velocity = (Re*mu)/(rho*Diameter)

c) Hagen-Poiseuille's Pressure Drop, dP = -(32*mu*u_avg*L)/(D*D)
Here, u_avg = u_max/2

d) Hagen-Poiseuille's Velocity Distribution, u(r) = -[1/(4*mu)]*(dP/dX)*[(R*R)-(r*r)]
Here, r = Radius at which Velocity to be calculated

e) Shear Stress Distribution, Tau(r) = -(r/2)*(dP/dX)
Here, r = Radius at which Shear Stress to be calculated

In the above Equations,
dX = X_out-X_in, D = Diameter, R = Outer Radius, L = Length,
Re = Reynold's Number, mu = Dynamic Viscosity

INPUTS:
Re = 1900;
rho = 997;                                 % kg/m3
Diameter = 0.01;                           % meter
Wedge Angle = 4;                           % Degree
mu = 8.9e-4;                               % Pa.sec
Kinematic Outlet Pressure = 90             % m2/sec2
Inlet Velocity = 0.1696                    % m/sec
(from Reynold's No. Formula)

Solver used = icoFoam


SCREENSHOTS  of  the  ParaView:

Velocity  Contour  in  Wedged  Pipe:

Pressure  Contour:

Velocity  Profile  obtained from  Hagen-Poiseuille's  Equation:

POST  PROCESS  of  the  Simulation:

Velocity  Profile  obtained  at  the  Starting  of  the  Pipe:

Velocity  Profile  obtained  at  the  Middle  of  the  Pipe:

Velocity  Profile  obtained  at  the  End  of  the  Pipe:

Comparison  of  Velocity  Profile  at  the  End  of  the  Pipe:

Shear  Stress  Profile  obtained:

NBSince the flow was simulated for axi-symmetric BC, the above profiles are half of the Actual Profiles.

Kinematic  Pressure  Distribution  obtained  along  the  Length  of  the  Pipe:

Results:

(1)  Entry Length = 1.33 m

(2)

(3)

(4)  Pressure Drop (dP) acccording to Hagen-Poiseuille = 64.2417664 Pa

(5)  Kinematic Inlet Pressure obtained after the Simulation = 92.326 m2/sec2

(6)  Kinematic Pressure drop obtained after the Simulation = 2.326 m2/sec2

(7)  Maximum Shear Stress obtained after the Simulation = 0.00437 Pa

Result  Discussion:

If we look at the above Plots and Results, we will find that at the initial stage,flow was not fully developed.But at the Exit of the Pipe(Total Length = Entry Length = 1.33 m),velocity and its profile closely matched with that of the Hagen Poiseuillei's Equation.So we can say that the Entry Length is sufficient to produce a fully developed flow in a circular pipe. Again, Shear Stress profile also   looks fine.

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