This document describes a numerical simulation of laminar and turbulent flow inside a pipe using Fluent software. The simulation models water flow inside a 1m diameter pipe that is 20m long. Two models are considered: laminar flow at a Reynolds number of 300 and turbulent flow at 8500. Theoretical equations for laminar and turbulent velocity profiles, entrance length, and Reynolds number correlations are presented. The numerical simulation sets up the models with appropriate boundary and material properties to solve the steady-state Navier-Stokes equations and compare results to experimental data.

1. SIMULATION OF LAMINAR AND TURBULENT FLOW INSIDE A PIPE
BY:
ANDI FIRDAUS SUDARMA
(432107963)
LECTURER:
DR. JAMEL ALI ORFI
SIMULATION PROJECT
NUMERICAL METHODS IN THERMOFLUIDS (ME 578)
MAGISTER PROGRAM OF MECHANICAL ENGINEERING
COLLEGE OF ENGINEERING
KING SAUD UNIVERSITY
RIYADH - KSA
FIRST SEMESTER 1433/1434 H

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ABSTRACT
The steady-state three-dimensional water flows inside a pipe are investigated by the
numerical simulation using Fluent. Both problems, laminar and turbulent, are simulated
under the same model. The solutions are compared with experimental results. The results are
illustrated in form of velocity profile and maximum velocity along the pipe.
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TABLE OF CONTENT
Abstract ...................................................................................................................................... 2
Table of Content ........................................................................................................................ 3
Nomenclature ............................................................................................................................. 4
I. Introduction ........................................................................................................................... 5
II. Theoritical Analysis ............................................................................................................. 6
III. Numerical Simulation ....................................................................................................... 11
IV. Result and Discussion ....................................................................................................... 17
V. Conclusions. ....................................................................................................................... 21
References ................................................................................................................................ 22
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NOMENCLATURE
A Area ………………………………………………….… m2
D Diameter of tube ……………………………………… m
L Channel length ……………………………………….. m
P Pressure ………………………………………………. Pa
R Radius of tube ………………………………………… m
r Radial coordinate
Re Reynolds number, ⁄
u Axial temperature m/s
V Volume m3
x Axial coordinate ……………………………………... m
Greek symbols
Dynamic viscosity …………………………………… Kg / ms
Kinematic viscosity ………………………………….. m2 / s
Density ………………………………………………. Kg / m3
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I. INTRODUCTION
Flows completely bounded by solid surfaces are called internal flows. Thus internal flows
include many important and practical flows such as those through pipes, ducts, nozzles,
diffusers, sudden contractions and expansions, valves, and fittings. The pipe networks are
common in any engineering industry. It is important to know the development of a flow at the
pipe entrance and pressure drop taking place along the pipe length. The flow of fluids in a
pipe is widely studied fluid mechanics problem. The correlations for entry length and
pressure drop are available in terms of flow Reynolds number.
Internal flows may be laminar or turbulent. Some laminar flow cases may be solved
analytically. In the case of turbulent flow, analytical solutions are not possible, and we must
rely heavily on semi-empirical theories and on experimental data. For internal flows, the flow
regime (laminar or turbulent) is primarily a function of the Reynolds number. In this project
we will only consider incompressible flows; hence we will study the flow of water inside a
smooth surface pipe.
1.1. Problem Description
The purpose of this project is to illustrate the setup and solution of a 3D turbulent and laminar
fluid flow in a pipe using Fluent. This project will consider the flow inside a pipe of diameter
1 m and a length of 20 m (Figure 1). The geometry is symmetric therefore this project will
model only half portion of the pipe. Water enters from the inlet boundary with a various
velocity (depend on Reynolds number). The flow Reynolds number is 8500 and 300 to
illustrate the turbulent and laminar flow respectively.
Inlet Outlet
𝐹𝑙𝑜𝑤
Pipe
Figure 1. Problem description
The objectives of this study are examining the results, such as velocity profile and entrance
length, compare them with experimental data and visualize the flow using animation tool.
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II. THEORITICAL ANALYSIS
The problem that will be discussed in this project is a two-dimensional single phase forced
convection flow in a pipe. To obtain the equations that govern the current problem, the
following assumption are made for the analysis;
i) Steady flow
ii) Constant transport properties of fluid
iii) Incompressible fluid flow
iv) Newtonian fluid
v) Continuum fluid
𝑢 𝑆𝑢𝑟𝑓𝑎𝑐𝑒
𝑟 𝑢
𝐹𝑙𝑜𝑤
𝑥 2𝑅
Figure 2. Schematic diagram.
2.1. Laminar Velocity Profile
In the first place we examine the flow of fluid inside the pipe set in motion. The governing
equations of this problem are continuity, momentum and energy equations. To get the
velocity profile inside the pipe, the governing equations, namely continuity, momentum and
energy equations have been derived based on the above-mentioned assumptions.
D
(Continuity equation)    V  0 (2.1)
Dt
ux 1 P
(Momentum eq. in x-direction)  V   ux    g x    2u x (2.2)
t  x
Begin by formulating two dimensional continuity equation (2.1) for conditions mentioned
above which can be written with respect to cylindrical coordinate as;
1  1  
 r ur    u    ux   0 (2.3)
r r r  x
Since ρ is constant, we will obtain V  0 . Where ur  u  0 and the velocity is not
changing with respect to x, it‟s only a function of r  u  u (r )  . An important feature of
hydrodynamic conditions in the fully developed region is the gradient of axial velocity
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component is everywhere zero. And from the assumption, there is no velocity in the r and
directions, i.e, ur  u  0 , which gives
u
0 (2.4)
x
The next step is momentum equation formulation. The flow is in the x-direction  u x  , so
ur  u  0 . Where g x  0 ,  ux  ux (r )  , ux t  0 (steady).
We can write momentum equation (2.2) as;
1 P
V   ux      2u x (2.5)
 x
Expanding the momentum equation,
ux
0
1 ux
0
ux
0
1 P  1   u  1  2u 0  2u 0

ur   ux    r
x
 2
x
 x

r r  x  x  r r  r  r 
2
x 2

 
Using continuity equation (2.5) and assumption (iii), where ux  u , we can write above
equation as follows;
1   u  1 P
r   (2.6)
r r  r   x
The momentum equation can be solved analytically to be used in the energy equation.
Multiplying energy equation (2.7) by r and integrating it twice with respect to r,
u r 2 P
r   c1 (2.7)
r 2 x
r 2 P
u  c1 ln r  c2 (2.8)
4 x
The integration constants may be determined by invoking the boundary conditions
u
For  0 will give the result c1  0 (2.9)
r r 0
And for ur  R  us  0 (no slip flow condition), will give the result
R 2 P (2.10)
c2  
4 x
Substituting equation (2.9) and (2.10) into equation (2.8) will gives
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1 P 2
u
4 x
 r  R2  (2.11)
The initial and boundary conditions for constant wall temperature problem are;
Boundary Condition 1 at r  0 , u r  0 (2.12)
Boundary Condition 2 at r  R , T  Ts for all x0 (2.13)
Then, we formulate the dimensionless form of velocity. Where um  Q A and
R R
Q   u dA   u (2 r ) dr (2.14)
0 0
Substitute equation (2.11) into equation (2.14).
1 P 2
R
Q
4 x
 r  R2 dr (2.15)
0
And substitute equation (2.14) into um  Q A
R 2 P
um  U   (2.16)
4 x
Substitute equation (2.12) and (2.15) to obtain dimensionless variable ⁄
2
u r
 1   (2.17)
um R
Equation (2.17) can be used to obtain laminar velocity profile inside the pipe.
2.2. Turbulent Velocity Profile
Except for flows of very viscous fluids in small diameter ducts, internal flows generally are
turbulent. As noted in the relation of shear stress distribution in fully developed pipe flow, in
turbulent flow there is no universal relationship between the stress field and the mean velocity
field. Thus, for turbulent flows we are forced to rely on experimental data.
The velocity profile for turbulent flow through a smooth pipe may also be approximated by
the empirical power-law equation
1
u  r n
 1   (2.18)
U  R
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Where the exponent (n) is varies with the Reynolds number. Data from Hinze suggest that the
variation of power-law exponent n with Reynolds number (based on pipe diameter, D, and
centerline velocity, U) for fully developed flow in smooth pipes is given by,
n  1.7  1.8log Reu (2.19)
For 2
Velocity profiles for n = 6 and n = 10 are shown in Figure 3. The parabolic profile for fully
developed laminar flow is included for comparison. It is clear that the turbulent profile has a
much steeper slope near the wall.
Figure 3. Velocity profiles for fully developed flow.
2.1. Reynolds Number Correlation
As discussed previously in introduction, the pipe flow regime (laminar or turbulent) is
determined by the Reynolds number, where;
UD
Re  (2.20)

At low flow rates (low Reynolds numbers) the flow is laminar and at high rates the flow is
transition into or turbulent. Laminar flow in a pipe may be only for Reynolds numbers less
than 2300.
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Figure 4. Flow in the entrance region of a pipe
The length of the tube between the start and the point where the fully developed flow begins
is called the Entrance Length, denoted by Le. The entrance length is a function of the
Reynolds Number Re of the flow.
LeLa min ar 0.06 Re D (2.21)
Where D is the tube diameter.
To restore a turbulent flow to parabolic flow, the entrance length is by approximation:
LeTurbulent 4.4D Re1/ 6 (2.22)
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III. NUMERICAL SIMULATION
The grid (mesh) that used in this project is already included in Fluent Tutorial-4. Using the
same mesh to generate 2 model, that is;
Model A. Laminar flow with Re = 300
Model B. Turbulent flow with Re = 8500
Reynolds number approximation is based on expectation that fully developed region will be
occurring before the flow reach pipe outlet.
Figure 5. Grid display
The problem is solved in steady state using pressure based solver. Definition of viscous
model are shown in figure (6), where the laminar and k-epsilon (2 eqn.) selected for laminar
and turbulent problem respectively.
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Figure 6. Setting of viscous model
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The fluid that flow inside the pipe is water. The properties of water are obtained from Fluent
database.
Figure 7. Material properties
The models are made with boundary conditions at inlet (at X = 0 m) and outlet (at X = 20 m)
is „Inlet‟ and „Outlet‟ respectively. The boundary condition for surface and axis of the pipe is
„Wall‟ and „Axis‟ respectively. „Axis‟ boundary condition acts like „Symmetry‟ boundary
condition but it is used for axisymmetric problem such as flow in a pipe.
The velocity inlet is obtained by using equation (2.20). The velocity is 0.0003 and 0.0085 m/s
for model A and model B, respectively. Where Turbulent Intensity can be calculated as;
T .I .  0.16  Re1/8 (2.23)
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The CFD calculation is carried out using the SIMPLE algorithm for pressure-velocity
coupling and the second order upwind differencing scheme for momentum equation and
turbulent term. These settings are shown in solution controls window figure (8).
Model B
Model A
Figure 8. Settings of algorithm for pressure-velocity coupling and spatial discretization
The convergence data are plotted to represent the fully developed velocity profile at outlet
and maximum velocity at centerline.
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Model A
0.5
Model A
0.4
0.3
0.2
0.1
Radius (m)
0.0
0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006
-0.1
-0.2
-0.3
-0.4
-0.5
Velocity (m/s)
Figure 9. Fully developed velocity profile at outlet for laminar
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Model B
0.5
Model B
0.4
0.3
0.2
0.1
Radius (m)
0.0
0 0.002 0.004 0.006 0.008 0.01 0.012
-0.1
-0.2
-0.3
-0.4
-0.5
Velocity (m/s)
Figure 10. Fully developed velocity profile at outlet for turbulent
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IV. RESULT AND DISCUSSION
The maximum velocities at centerline are presented in the chart below. Where fully
developed region will occur after the flow reaching entrance length (Le).
0.0006
0.00055
Maximum Velocity (m/s)
0.0005
0.00045
0.0004
0.00035
0.0003
0 2 4 6 8 10 12 14 16 18 20
Pipe Length (m)
Figure 11. Maximum velocity of laminar flow
The entrance length of laminar flow can be calculated using equation (2.21). For Re=300, the
entrance length may as long as 18 m. Comparing with the result obtained from simulation
(figure 11), at length of the pipe above 18 m there velocity is still developing with margin of
increment 0.052 percent.
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0.0105
0.01
Maximum Velocity (m/s)
0.0095
0.009
0.0085
0.008
0 2 4 6 8 10 12 14 16 18 20
Pipe Length (m)
Figure 12. Maximum velocity of turbulent flow
For turbulent flow, entrance length can be approximated using equation (2.22). Where at
Re=8500, the flow approximated will be fully developed at 18 m length of pipe. Comparing
with the result obtained from simulation (figure 12), at length of the pipe above 19.8 m there
velocity is still developing with margin of increment 0.0384 percent.
Using dimensionless form of velocity profile, we comparing experimental data from equation
(2.17) for laminar and equation (2.18) for turbulent and data that obtained from the
simulation. The results are illustrated in the figure (13) for laminar and (14) for turbulent.
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1.0
0.9
0.8
0.7
0.6
rR
0.5 Numerical
Experimental
0.4
0.3
0.2
0.1
0.0
0 0.2 0.4 0.6 0.8 1
u/U
Figure 13. Velocity profile of laminar flow
1.0
0.9
0.8
0.7
0.6
rR
0.5 Numerical
Experimental
0.4
0.3
0.2
0.1
0.0
0 0.2 0.4 0.6 0.8 1
u/U
Figure 14. Velocity profile of turbulent flow
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The velocity profile comparison for laminar flow shows that the velocities obtained from the
simulation are similar with the theoretical data. But the turbulent flow chart shows that there
is unmatched data between experimental and simulations. This result happen when the
problem not simulated correctly. After evaluating the turbulent model, we found that the
turbulent intensity value was 4.8%, where it should be 0.052% base on equation (2.23).
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V. CONCLUSIONS.
In general, for the above three-dimensional with two boundary conditions stated, Reynolds
number affects the velocity profile. When Reynolds number is increasing, the entrance length
will also increased. This situation is valid for both cases, laminar and turbulent.
The velocity profile of laminar flow is similar with parabolic curve, and at turbulent flow
there is extreme different between internal flow with the flow near the wall.
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REFERENCES
[1] Fluent Inc., “Tutorial 4. Simulation of Flow Development in a Pipe”, 2006
[2] Fox, R. W., McDonald, A. T., Pritchard, P. J., “Introduction to Fluid Mechanics”, 6 th ed.,
John Wiley & Sons, New York, 2003.
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