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 crossflow 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
 Axissymmetric 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/m^{3} and dynamic viscosity 0.001 kg/ms.
Governing Equation and Boundary Condition
 In the present study, the flow is assumed to be steady, twodimensional and incompressible
 The effects of body forces and viscous dissipation are neglected in this study. The thermophysical properties are assumed to be constant
 The governing equations for laminar flow in cylindrical coordinates system for axis symmetric problem are given below, in two dimensional cylindrical coordinates, all derivatives with respect to circumferential directions are zero.
Continuity Equation:
Momentum Equation:
Conservation of momentum equation is solved in in cylindriccal cooordinate system considering axis symmetric flow
In above, the shear stress in rz corindate:
Where z is the axial coordinate, r is the radial coordinate; 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 coordinate system, axissymmetry 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: Viscouslaminar 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} + 1/2ρu^{2} for incompressible flow
 For periodic BCs at inlet and oulet, the mass flow rate is defined per 2 π radian , (ρ .u. π r^{2}/ 2π) = 1000*0.0005*0.1^{2}/2) =0.0025 kg/s specified. (As mentioned in ANSYS FLUENT manual for axissymmetric 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^e7
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 gridindependent 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/m^{3} and µ=0.0001 kg/ms by assuming laminar fully developed flow through the pipe
 Pressure Drop is same for both analytical and CFD simulation
 Analytical Pressure Drop:4.00E04 Pa
 Numerical (CFD) Pressure Drop: 4.00E04 Pa
Pressure Drop (ΔP)  Pascal 
Analytical Pressure Drop  4.00E04 
Numerical (CFD) Pressure Drop  4.08E04 
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 crosssections 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 line01, line03, line4, line9 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 steadystate axissymmetric flow
 SIMPLE algorithm used for Pressure – Velocity coupling
 Axial variation of variables is negligible if model laminar flow through a 3D pipe