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CFD Simulation of Laminar Flow through the Pipe

Introduction to laminar flow through Pipe

Laminar flow is streamlined flow which occurs in pipe when a fluid flows in parallel layers, with no disruption between the layers
In laminar flow, viscous forces are dominant.
At low velocity flow, the fluid moves in ducts or channel without lateral mixing. Any lateral mixing (mixing of fluid normal to the flow direction) occurs by the molecular diffusion of liquid
There are no cross-flow perpendicular in the direction of fluid flow and we cannot see eddies or swirling motion if fluid
In Fluid mechanics, there is a critical Reynolds number to decide laminar and turbulent flow. For pipe flow, the critical Reynolds number is 2300.

In laminar flow, the motion of the particles of the fluid is orderly and all fluid particles streamline parallel to the pipe wall
Diffusion mixing at molecular level is very slow for large sized pipe diameter. But it can be very significant for small sized pipe

For axis symmetric flow, the variation in circumferential direction is zero

Pipe Geometry and Computational Domain

Axis-symmetric computational domain consists of a 2D plane. All flow variables are function of radial and axial coordinates, L=1 m

Geometry and Computational domain for Pipe Flow Problem

Computational Mesh Model

Summary of meshes used for for simulation of axis symmetric pipe flow

Various Mesh Models of used in this study is shown below

Grid1 (Ny=5, Nx=50):

Grid2 (Ny=10, Nx=100):

Grid3 (Ny=20, Nx=200):

$E:\IIT_Delhi_Coursework\CHL869\Grid1\Grid0_ny20_meshmodel.jpg$

Physical Properties

The physical properties for water are assumed to be constant with density (ρ) 1000kg/m³ and dynamic viscosity 0.001 kg/m-s.

Governing Equation and Boundary Condition

In the present study, the flow is assumed to be steady, two-dimensional and incompressible
The effects of body forces and viscous dissipation are neglected in this study. The thermo-physical properties are assumed to be constant
The governing equations for laminar flow in cylindrical co-ordinates system for axis symmetric problem are given below, in two dimensional cylindrical co-ordinates, all derivatives with respect to circumferential directions are zero.

Continuity Equation:

Momentum Equation:

Conservation of momentum equation is solved in in cylindriccal coo-ordinate system considering axis symmetric flow

In above, the shear stress in r-z corindate:

Where z is the axial coordinate, r is the radial co-ordinate; u and v are velocity components in z and r directions respectively.

Boundary conditions

Inlet: A uniform velocity at the inlet boundary: u = U_∞,v = 0 (6)
Outlet: Outflow condition is used at the outlet. It is a zero gradinet boundary conditions
Wall: The pipe surface is considered as wall with no slip condition: u = 0, v= 0
Axis (Symmetry for Pipe):
- In cylindrical co-ordinate system, axis-symmetry implies that there is no change in anything in the θ direction, i.e. ∂/∂θ(anything) =0
- Circumferential variation of velocity and p is the pressure are zero

Analytical Velocity distribution and Pressure drop

Velocity Profile for laminar flow through pipe is a function of maximum velocity and radius

Pressure drop in for fully developed laminar flow through the pipe is a function of flow rate, length and diameter

CFD Solver Set up in FLUENT

Numerical Procedure

ANSYS GAMBIT used for modelling and meshing, the numerical simulation was carried out by using a commercial FVM solver, ANSYS FLUENT.

Solver Setting– Steady Implicit Axis Symmetric Solver
Physical Model: Viscous-laminar Model used for numerical simulation
Boundary Conditions in ANSYS FLUENT:
- For inlet a fixed uniform velocity of 0.0005 m/s used such that,Reynold number. Based on velocity (U = 0.0005 m/s) and diameter (D = 0.2 m)n (Re =ρ U D/μ ) is 100
- The operating pressure 101325 Pa used at inlet: P_absolute = P_gauge +P _operating , assumed zero gauge pressure at inlet, P_total = P_static + 1/2ρu² for incompressible flow
- For periodic BCs at inlet and oulet, the mass flow rate is defined per 2 π radian , (ρ .u. π r²/ 2π) = 1000*0.0005*0.1²/2) =0.0025 kg/s specified. (As mentioned in ANSYS FLUENT manual for axis-symmetric problems)

Pressure Velocity Coupling :
- Semi –implicit Method for Pressure Linked Equation (SIMPLE) used for steady state problem
Discretization Scheme–

- Pressure : Standard
- Momentum: QUICK (Quadratic Upwind interpolation for Convective Kinetics)
Convergence Criteria: Convergence criteria for all flow variable kept 10^e-7

Convergence Histories of flow variables

Residuals for Velocity inlet BC Problem

$E:\IIT_Delhi_Coursework\CHL869\Grid2_ny10\Mesh2_residual.jpg$

Residuals for Periodic BC Problem

$E:\IIT_Delhi_Coursework\CHL869\periodicBC\pipepPBCNy20_residuaks.jpg$

Grid Independence Study

Pipe Flow Simulation with Velocity Inlet:

Grid Sensitive studies for Velocities Profiles in laminar flow of the pipe
Velocity Profile at Outlet

The variation of axial velocity profile computed in the Centre of pipe for different mesh densities

Static Pressure drop variation across the circular pipe Axis checked for different mesh densities for the laminar flow at Re=100

Based on the above a grid-independent study Grid2 with mesh size 1000 is considered as an optimistic mesh model for the present CFD study.

Validation of Numerical Results with Analytical Solution

The analytical solution of velocity profile and pressure drop is calculated using the equations (10) and (11) for L=1 m, r=0.1 m, D=0.2m, U =0.0005 m/s,ρ=10000 kg/m³ and µ=0.0001 kg/m-s by assuming laminar fully developed flow through the pipe
Pressure Drop is same for both analytical and CFD simulation
- Analytical Pressure Drop:-4.00E-04 Pa
- Numerical (CFD) Pressure Drop: -4.00E-04 Pa

Pressure Drop (ΔP)	Pascal
Analytical Pressure Drop	-4.00E-04
Numerical (CFD) Pressure Drop	-4.08E-04

CFD Results for Laminar Flow

Fully Developed Velocity profile

Comparison of Numerical Velocities Profiles with analytical solutions for fully developed laminar flow through the pipe at Re=100

Velocity Development for various axial and radial positions

Radial Velocity profile at various cross-sections of pipe is presented in the following figure

Axial velocity Profile at different radii need to be compared

Velocity Distribution in Pipe

Velocity Contours in laminar flow through the pipe at Re_D=100
Fully developed velocity profile is noted at the outlet. The maximum velocity for fully developed laminar flow is double of average velocity.

Streamlines colored with velocity magnitude shows regions of different velocity magnitude

Velocity Vector

Vectors with Velocity magnitude are shown in below
Due to flow development, the magnitude of velocity changes in the flow transition region
As per, the velocity Vectors in laminar flow through the pipe at Re_D=100, the parabolic velocity profile is at the outlet of the pipe

Pressure Distribution (P_Static)

Pressure fields in laminar flow through the pipe at Re_D=100
High fluid pressure at the inlet and low at outlet

Velocity along the Axis of Pipe

Axial Velocity in laminar flow through is shown in the Centre of pipe at R=0.0

Pipe Flow Simulation with Periodic Conditions

Velocity fields in laminar flow through the pipe for periodic flow

Pressure Distribution:

Pressure fields in laminar flow through the pipe for periodic flow

Velocity Profile for Periodic BC in laminar flow through the Pipe:

Radial velocity profile at different axial location X=0.1m, 0.3m, 0.4m, 0.9m for line-01, line-03, line-4, line-9 respectively
Axial Velocity Profile is linear at the Centre of pipe
Radial and Axial Velocity in laminar flow through the pipe for periodic flow

Conclusion

CFD simulation of laminar flow through the pipe is carried out for steady-state axis-symmetric flow
SIMPLE algorithm used for Pressure – Velocity coupling
Axial variation of variables is negligible if model laminar flow through a 3D pipe

CFD Modelling of Laminar Flow Through Pipe