final project for advanced fluid mechanics

7/23/2019 Final Project for Advanced Fluid Mechanics

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2-D channel Flow with/without thermal field

Muzammal Arif

Graduate Student, Department of Mechanical Engineering

King Fahd University of Petroleum and Minerals, (KFUPM)

Dhahran, Saudi Arabia

Email: [email protected]

Dr. Shahzada Zaman Shuja

Professor Mechanical Engineering DepartmentKFUPM, Dhahran, Saudi Arabia

Department of Mechanical Engineering

King Fahd University of Petroleum and Minerals

mailto:[email protected]





G201409640 MUZAMMAL ARIF AFM-I TERM PAPER (ME-532)

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Table of Contents

Introduction----------------------------------------------------------------------------------(3)

Relevance to Engineering and Industry--------------------------------------------------(3)

Literature Review --------------------------------------------------------------------------(4)

Viscous open-channel flow in laminar regime------------------------------------------(5)

Method for implementing the proposed solution along with improvements. -------(7)

Velocity Distribution in 2D channel with varying angle (θ) -------------------------(10)Velocity Distribution in pipe with varying angle (θ) ---------------------------------(11)

References/Bibliography-----------------------------------------------------------------(15)




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1. Introduction

Channel flow is defined as flow in one direction within a conduit. It may be closed channel

flow or open channel flow depending upon whether the fluid flowing is entirely in contact with

rigid boundaries or it has one of its surface free from solid boundary i.e. open to atmosphere.

Pipe flow is an example of closed channel flow while flow in rivers or canals are examples ofopen channel flow. In both closed and open channel flow, the flow can be either uniform or

non-uniform, steady or unsteady flow. In this article we will focus our attention on two

dimensional flow in a closed conduit. The flow will be laminar, uniform, steady and fully

developed and our objective will be to find distribution of different variable like velocity,

temperature and shear stress with or without the application of thermal field. We will utilize

generalized form of Navier-Stokes equations.

2. Relevance to Engineering and Industry

Channel flow is of great importance in engineering and Industry specially flow in a pipe

which is a form of closed channel flow is used extensively in Boilers, heat exchangersand condensers etc. Closed channel flow in rectangular duct that we are going to deal

with hold its application in different field of engineering like flow in ducts of heating,

ventilation and air-conditioning systems. Flow in Plate type heat exchangers etc.




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3. Literature Review

In paper [1] Simulating 2D open-channel flows through an SPH model by I. Federico, S. Marrone,

A. Colagrossi, F. Aristodemo, M. Antuono, two dimensional model of open channel flow is given.They developed an appropriate algorithm to enforce different upstream and downstream flow

conditions and simulate uniform, non-uniform and unsteady flows. The main focus was in velocity

field, pressure forces, water depths etc. Comparisons between numerical results, theory and

experimental data is provide.The governing equations were

Boundary Conditions:

Free surface

Solid boundary

In/Out flow boundary

The free surface boundary conditions can be easily handled by the SPH model. The implicit

enforcement of the free surface dynamic boundary condition is one of the main advantages of the

SPH method in comparison to other solvers where this boundary condition has to be forceddirectly.




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The channel bottom is modelled through the fixed ghost particle technique. In contrast to the

classical ghost particles that are instantaneous mirrors of the fluid particles with respect to the bodysurface, the fixed ghost particles are associated to interpolation points internal to the fluid. The

main advantage of using the fixed ghost particles instead of the classical ghost technique is that

their distribution is always uniform and does not depend on the fluid particle positions.

In total, four sets of particles are used: fluid (f), fixed ghost (s), inflow (i) and outflow (o) particles.

Similarly to the fixed ghost particles, the in/out-flow particles affect the fluid particles but not viceversa. At the inflow, the desired velocity and pressure conditions are assigned to the inflow

particles. When inflow particles cross the inflow threshold, they become fluid particles and they

evolve in accordance with the SPH equations. As concerns the outflow particles, it is possible toimpose specific outflow conditions (similarly to the inflow case) or open boundary conditions. In

the latter case, the fluid particles that cross the outflow threshold become outflow particles. Their

physical variables are frozen. Three different test cases were considered while I am going to

discuss only the first one i.e. viscous open-channel flow in laminar regime.

Viscous open-channel flow in laminar regime:

A uniform, steady and laminar flow in a free-surface channel is a special case of the Poiseuille

flow. The distribution of velocity u (z) for two-dimensional channel flow is given by a second-

order equation given by

Figure 2: Initial sketch of the computational domain




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Where g is the gravity acceleration, h is the total water depth, S 0 is the bottom slope and z is the

vertical abscissa whose origin is located at the channel bottom. The Reynold number Re =

is evaluated using the average horizontal velocity U = ∫ . The fluid domain has length L

= 2h and slope S0 = 0.001. The initial and inflow boundary conditions are imposed and elementary

fluid domain is sketched as follows

Figure 6: Particle distribution at Re=200

Figure 5: Particle distribution at Re=100Figure 4: Particle distribution at Re=10




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4. Method for implementing the proposed solution along with improvements.We will make analytical solution of the two dimensional laminar fluid flowing due to gravity

as a function of its slant height i.e. slope and visualize the distribution of velocities with the

changing of slope as well as Reynold number. Additionally we are going to replace the 2 Dchannel with a pipe and visualize the distribution of parameters like velocity as a function of

slope as well as a function of Reynolds number.

= g sin ()

= g cos ()

v=w=0 which gives = 0

Navier Stokes equation in x-direction is

Putting t= v = w = = 0 also

= 0 (no pressure gradient along x) and

= 0 (no

variation in u along the z- direction) will give




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=

= sin θ

= sin θ

Integration gives = sin θ ∗ y + A

Again Integration gives

= sin θ ∗ y2 + Ay + (1)

Applying the BC

1. @ y = 0 u = 0 gives B = 0

2. @ y = h τ = 0 i.e. =0

0 = sin θ ∗ h + A

or sin θh = A

Put the values of A and B in equation (1) gives

= 2 sin θy

2

+

sin θhy

= sin θ2ℎ




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First we will reproduce the results shown in figure 3.

We will takesin θ = 0.001, = 1000/3 and g = 9.81m2/s, μ=8.9x10-4 kg/(ms)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.002 0.004 0.006 0.008 0.01

Graph at slope =0.001

velocity (m/s)

H e i g h t o f l i q u i d c o l u m n ( m )




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Velocity Distribution as a function of angle in 2D channel:

Now we will vary the angle theta of the 2D plane from 0 degree to 60 with an interval of 15 degree

and we will measure the distribution of velocities. Plot of velocities as a function of height as well

as a function of angle theta is given in the following diagram.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.00E+00 1.00E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 6.00E+00 7.00E+00

Velocities at different values of angle ()

velocity (m/s)

H e i g h t o f l i q u i d c o l u m n ( m )

15 deg 30 deg 45 deg 60 deg




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5. Velocity Distribution in pipe with varying angle ( ):

In our next step we will find velocity distribution of fluid flowing in a pipe having a gradient.

With u = ur = 0 and∂

∂ = 0

Continuity equation gives us∂∂z = 0

Navier-Stokes r-direction:

N-S equation in r-direction for steady flow gives

0 =

gr = gcos (

)

gz =g sin ()

+ρgr




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0 =

Navier-Stokes θ-direction:

N-S equation in θ-direction for steady flow gives

0 =

Navier-Stokes z-direction:

N-S equation in z-direction for steady flow gives

0 = 1 ( ) +

0 =

1

(

)+

1 ( ) =

+ρgθ




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( ) =

Integration w.r .t “r” gives

( ) =

2 +

= 2 +

Integrating again w.r.t “r” gives

=

4 + +

Constants A and B can be found by the B.C which are

@ r = 0 = 0 which gives A=0

= − + (2)

@ r =R uz =0 which gives

= 4

Put the value of B in equation (2) gives

= 4 + 4

= 4




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Distribution of velocities for the flow of a liquid in a pipe for the various angle “θ” is shown in the

figure. Angle is varied from. The various parameters taken are given below

= 1000/3, g = 9.81m2/s, μ=8.9x10-4 kg/(ms), R=1 m

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.00E+00 5.00E-01 1.00E+00 1.50E+00 2.00E+00 2.50E+00 3.00E+00

Velocity distribution as a function of angle ()

velocity(m/2)

p i p e r a d i u s

( m )

15 deg 30 deg 45 deg 60 deg




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References/Bibliography

1. Recurrence in 2D inviscid channel flow(Y. Charles Li)

2. Simulating 2D open-channel flows through an SPH model(I. Federico, S. Marrone, A. Colagrossi, F. Aristodemo, M. Antuono)

3. Understanding heat transfer in 2D channel flows including recirculation (M.H.

Dirkse, W.K.P. van Loon, J.D. Stigter, G.P.A. Bot)

4. CFD 2D simulation of viscous flow during ECAE through a rectangular die withparallel slants(Alexander V. Perig & Nikolai N. Golodenko)

5. Second law analysis of the 2D laminar flow of two-immiscible, incompressibleviscous fluids in a channel(Fethi Kamish, Hakan F. oztop)

6. Guidelines for Modeling a 2D Rough Wall Channel Flow(Stefano Leonardi · Paolo Orlandi · Lyazid Djenidi Robert A. Antonia)

7. Direct numerical simulation of a turbulent open channel flow with passive heattransfer(D. M. LU and G. HETSRONIt)

8. DNS of turbulent heat transfer in a channel flow with a high spatial resolution(Makoto Kozuka, Yohji Seki, Hiroshi Kawamura)

9. DNS of velocity and thermal fields in turbulent channel flow with transverse-ribroughness(Yasutaka Nagano, Hirofumi Hattori a, Tomoya Houra)

10. Passive heat transfer in a turbulent channel flow simulation using large eddy

simulation based on the lattice Boltzmann method framework (Hong Wu, Jiao Wang,Zhi Tao)

11. Use of direct numerical simulation to study the effect of Prandtl number ontemperature fields(Yang Na a, Dimitrios V. Papavassiliou b, Thomas J. Hanratty)

12. Simulating two-dimensional thermal channel flows by means of a lattice Boltzmannmethod with new boundary conditions(Annunziata D’Orazio, Sauro Succi)

13. An investigation of turbulent open channel flow with heat transfer by large eddysimulation(Lei Wang, Yu-Hong Dong, Xi-Yun Lu)

14. Flow distribution and pressure drop in 2D meshed channel circuits(Daniel Tondeur , YilinFan b, LingaiLuo)

15. Large eddy simulation of stably stratified turbulent open channel flows with low- tohigh-Prandtl number (Lei Wang, Xi-Yun Lu)

16. Heat transfer in fully developed turbulent channel flow: comparison betweenexperiment and direct numerical simulations(M. TEITEL and R. A. ANTONIA)

final project for advanced fluid mechanics

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