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
        Radius = Diameter/2;                       % 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)
        Graded Mesh = 0.2

        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:

NB: Since 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.