flow inside a pipe with fluent modelling

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

Simulation Project Numerical Methods in Thermo-Fluids (ME 578)

2

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.


3

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


4

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


5

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.

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.

𝐹𝑙𝑜𝑤 Pipe

Inlet Outlet


6

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

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.

(Continuity equation) 0D

VDt

(2.1)

(Momentum eq. in x-direction) 21xx x x

u PV u g u

t x

(2.2)

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

0r xr u u ur r r x

(2.3)

Since ρ is constant, we will obtain 0V . Where 0ru u 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

𝑢𝑆𝑢𝑟𝑓𝑎𝑐𝑒

𝑢

2𝑅 𝐹𝑙𝑜𝑤

𝑟

𝑥


7

component is everywhere zero. And from the assumption, there is no velocity in the r and

directions, i.e, 0ru u , which gives

0u

x

(2.4)

The next step is momentum equation formulation. The flow is in the x-direction xu , so

0ru u . Where 0xg , ( )x xu u r , 0xu t (steady).

We can write momentum equation (2.2) as;

21x x

PV u u

x

(2.5)

Expanding the momentum equation,

xr

uu

r

0

1 xu

r

0

xx

uu

x

0 2

2 2

1 1 1x xu uPr

x r r r r

02

2

xu

x

0

Using continuity equation (2.5) and assumption (iii), where xu u , we can write above

equation as follows;

1 1u Pr

r r r x

(2.6)

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,

2

12

u r Pr c

r x

(2.7)

2

1 2ln4

r Pu c r c

x

(2.8)

The integration constants may be determined by invoking the boundary conditions

For 0

0r

u

r

will give the result 1 0c (2.9)

And for 0r R su u (no slip flow condition), will give the result

2

24

R Pc

x

(2.10)

Substituting equation (2.9) and (2.10) into equation (2.8) will gives


8

2 21

4

Pu r R

x

(2.11)

The initial and boundary conditions for constant wall temperature problem are;

Boundary Condition 1 at 0r

, 0u r (2.12)

Boundary Condition 2 at r R

, sT T for all 0x (2.13)

Then, we formulate the dimensionless form of velocity. Where mu Q A and

0 0

(2 )

R R

Q u dA u r dr (2.14)

Substitute equation (2.11) into equation (2.14).

2 2

0

1

4

RP

Q r R drx

(2.15)

And substitute equation (2.14) into mu Q A

2

4m

R Pu U

x

(2.16)

Substitute equation (2.12) and (2.15) to obtain dimensionless variable ⁄

2

1m

u r

u R

(2.17)

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

1nu r

U R

(2.18)


9

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,

1.7 1.8log Reun (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;

ReUD

(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.


10

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.

min 0.06ReLa arLe D (2.21)

Where D is the tube diameter.

To restore a turbulent flow to parabolic flow, the entrance length is by approximation:

1/64.4 ReTurbulentLe D (2.22)


11

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.


12

Figure 6. Setting of viscous model


13

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;

1/8. . 0.16 ReT I (2.23)


14

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).

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.

Model A

Model B


15

Figure 9. Fully developed velocity profile at outlet for laminar

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006

Rad

ius

(m)

Velocity (m/s)

Model A

Model A


16

Figure 10. Fully developed velocity profile at outlet for turbulent

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0 0.002 0.004 0.006 0.008 0.01 0.012

Rad

ius

(m)

Velocity (m/s)

Model B

Model B


17

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).

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.

0.0003

0.00035

0.0004

0.00045

0.0005

0.00055

0.0006

0 2 4 6 8 10 12 14 16 18 20

Max

imu

m V

elo

city

(m

/s)

Pipe Length (m)


18

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.

0.008

0.0085

0.009

0.0095

0.01

0.0105

0 2 4 6 8 10 12 14 16 18 20

Max

imu

m V

elo

city

(m

/s)

Pipe Length (m)


19

Figure 13. Velocity profile of laminar flow

Figure 14. Velocity profile of turbulent flow

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.2 0.4 0.6 0.8 1

r\R

u/U

Numerical

Experimental

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.2 0.4 0.6 0.8 1

r\R

u/U

Numerical

Experimental


20

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).


21

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.


22

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”, 6th

ed.,

John Wiley & Sons, New York, 2003.

flow inside a pipe with fluent modelling

Education