# 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): ## 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: Pabsolute = Pgauge +P operating , assumed zero gauge pressure at inlet, Ptotal = Pstatic + 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 : 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 •   Residuals for Periodic BC Problem ## 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 ReD=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 ReD=100, the parabolic velocity profile is at the outlet of the pipe  ### Pressure Distribution (PStatic)

• Pressure fields in laminar flow through the pipe at ReD=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