--- On Mon, 23/5/11, ec.hff@btinternet.com <ec.hff@btinternet.com> wrote: From: ec.hff@btinternet.com <ec.hff@btinternet.com> Subject: International Journal of Numerical Methods for Heat and Fluid Flow - Decision on Manuscript ID HFF-Jan-2011-0005.R1 To: popm.ioan@yahoo.co.uk Date: Monday, 23 May, 2011, 10:12

23-May-2011 Dear Prof. Pop

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1 

 

Effects of inclination angle on natural convection in inclined open porous cavity with non-isothermally heated wall

Hakan F. Oztop

Department of Mechanical Engineering, Technology Faculty, Firat University, 23119 Elazig, Turkey

Department of Mechanical Engineering, College of Engineering, King Saud University, Riyadh, Saudi Arabia

Khaled Al-Salem

Department of Mechanical Engineering, College of Engineering, King Saud University, Riyadh, Saudi Arabia

Yasin Varol

Department of Mechanical Engineering, Technology Faculty, Firat University, 23119 Elazig, Turkey

Ioan Pop

Faculty of Mathematics, University of Cluj, CP 253, 3400 Cluj, Romania

Mujdat Fırat Department of Mechanical Education, Fırat University, 23119, Elazig, Turkey

Abstract

Purpose – The main purpose of this numerical study is to investigate the effects of

inclination angle and non-isothermal wall boundary conditions in a partially open cavity

filled with a porous medium.

Design/methodology/approach – In this study, the governing dimensionless equations

were written using Brinkman-Forchheimer model. They are numerically solved by using

finite volume method with SIMPLE solution algorithm by applying open boundary

conditions in one side. The opposed side of the open cavity is under non-isothermal

boundary conditions.

Findings – Results are presented by streamlines, isotherms, velocity and temperature

profiles as well as the local and mean Nusselt numbers for different values of the

governing parameters such as Grashof numbers, porosity, amplitude of sinusoidal

2 

 

function and inclination angle of the cavity. It is found that inclination angle is the most

important parameter on the temperature and flow field.

Research limitations/implications – The analysis is valid for laminar, two-dimensional

and steady natural convection in an open-ended square enclosure. An extension to three

dimensional and unsteady flow cases is left for future study.

Practical implications – It can be used for building materials or solar air collectors.

Originality/value – The originality of this study is the open sided enclosure filled with

porous media and non-isothermal wall.

Keywords Natural Convection, Open cavity, Inclination angle, Porous media

Paper type Research paper

Nomenclature

FC Forchheimer constant

Da Darcy number

g gravitational acceleration

Gr Grashof number

h length of the heated wall

H length of the square cavity

K permeability of the porous medium

n normal to the wall

xNu local Nusselt number

mNu is the mean Nusselt number

p dimensional pressure

P dimensionless pressure

Pr Prandtl number

T fluid temperature

vu , dimensional velocity components along x and y axes

3 

 

VU , dimensionless velocity components along X and Y -axes

yx, dimensional Cartesian coordinates

Y ,X dimensionless coordinates

Greek symbols

thermal diffusivity

thermal expansion coefficient

inclination angle

porosity

dimensionless temperature

kinematic viscosity

Subscript

c cold

h hot

int interior

out outlet

1. Introduction

Heat and fluid flow in porous media filled enclosures are well-known natural

phenomenon and have attracted interest of many researchers due to its many practical

situations. Among these insulation materials, geophysics applications, building heating

and cooling operations, underground heat pump systems, solar engineering and material

science can be listed. These are reviewed in several books: Pop and Ingham (2001),

Bejan et al. (2004), Ingham and Pop (2005), Nield and Bejan (2006), Vafai (2005,2010),

Vadasz (2008) and in the papers: Oztop (2007) and Varol et al. (2008).

Inclined, porous media filled and non-isothermally heated open cavities are very

complex structure from the heat transfer and fluid flow point of view. Holzbecher (2004)

studied the free convection in open-top enclosures filled with a porous medium and

4 

 

heated from below. In his work, convection cells are different: while for the classical

setup partially open and closed convection cells emerge, for the more reasonable mixed

boundary condition the cells are open over the entire length of the top boundary. Mixed

convection in a ventilated rectangular cavity with a horizontal strip occupied by two

media of different permeability is studied by using the finite-volume method by Moraga

et al. (2010). They observed that both Reynolds number and thermal conductivity ratio

are the most effective parameters on heat and fluid flow. Haghshenas et al. (2010) made a

numerical work on open square cavity filled with porous media by using lattice-Boltzman

technique. The open cavity problem is also studied for viscous fluid filled enclosure

under constant temperature boundary conditions by different authors as Penot (1982),

Abib and Jaluria (1988), Elsayed et al. (1999), Polat and Bilgen (2002), and Bilgen and

Oztop (2005).

Effects of non-isothermal boundary conditions on natural convection in porous or

viscous fluid filled enclosures are studied in literature with different techniques. Among

these studies Roy and Basak (2005) studied the natural convection flow in a square cavity

under non-uniformly heated wall by using finite element technique. Saeid (2005) applied

the sinusoidally varying temperature boundary condition onto bottom wall of a cavity

filled with porous medium. Oztop et al. (2009) studied the effects of

magnetohydrodynamic buoyancy-induced flow in a non-isothermally heated square

enclosure. Varol et al. (2009) investigated the entropy generation due to natural

convection in non-uniformly heated porous isosceles triangular enclosures. They found

that non-uniform boundary condition makes different effects inside the enclosure

according to uniform boundary conditions. Other applications on non-uniform heating

can be found in Storolesten and Pop (1996), Basak et al. (2006), and Bilgen and Yedder

(2007).

Based on authors’ knowledge and above literature survey, there is no study on

inclined one side open cavities. Thus, this work is important to obtain results for

literature.

5 

 

2. Considered model

The considered physical model is plotted in Figure 1 with coordinates. One side of the

cavity is fully open and other side has non-uniformly heated boundary condition. It is a

square cavity ( LH ) that two opposite impermeable walls are adiabatic. The cavity is

filled with fluid porous saturated media. The cavity has an inclination angle which

defines by .

3. Basic equations

To write the governing equations, following assumptions are made as

The convective incompressible and viscous fluid flow is described by

Brinkman-Forchheimer model.

The Boussinesq approximation is valid.

It is also assumed that the gravity acts in downwards vertical direction

The fluid properties are constant.

Radiation mode of heat transfer is neglected according to other modes of heat

transfer.

Under above assumptions, the continuity, momentum and energy equations can be

written in dimensionless form as follow

0

Y

V

X

U (1)

CosPrGrY

U

Y

UPr

U

Da

VUCU

Da

Pr

X

P

Y

UV

X

UU

22

2

2

2

2/3

2/122

F22

(2)

6 

 

SinPrGrY

V

Y

VPr

V

Da

VUCV

Da

Pr

Y

P

Y

VV

X

VU

22

2

2

2

2/3

2/122

F22

(3)

2

2

2

2

YXYV

XU

(4)

where the following dimensionless variables have been used

ch

c

TT

TT

H

pP

H

vV

H

uU

H

yY

H

xX

,/

,,,,2

(5)

Here U and V are the dimensionless velocity components along X and Y axes, T

is the temperature of the fluid-saturated porous medium, is the dimensionless

temperature, porosity and the mining of the other quantities is explained in the

Nomenclature. We assume that the boundary conditions of Eqs. (1)-(4) are

on all solid boundaries

0,0

n

PVU (6a)

on the opening wall

0

XX

V

X

U

(6b)

on the adiabatic boundaries

0

n

(6c)

on the partial heater x2sina (6d)

Equations (1)-(4) contain the Darcy number Da , the Grashof number Gr and the Prandtl

number Pr , which are defined as

7 

 

Pr,)(

,2

3

2

HTTgGr

H

KDa ch (7)

where K is the permeability of the porous medium. The permeability K is based on

Ergun’s emprical expression (Ergun, 1952), which may be used for the packed beds that

may be closely modeled as spherical beads of diameter d and is given by

2

23

)1(150

dK . (8)

The physical quantity of interest is the mean Nusselt number mNu , calculated from the

left partially cooled wall and is given by

H

xm dyNuNu0

(9)

where xNu is the local Nusselt number, which is defined as

0

X

x XNu (10)

4. Numerical method

Continuity, Momentum and Energy equations (1)-(4) along with the boundary conditions

(6) were solved numerically using uniform grid spacing. SIMPLE solution algorithm of

Patankar (1980) is used by modifying the Fortran code of Nakayama (2005). A uniform

grid consisting of 48 x 48 nodes was found to be sufficient to yield grid independent

results for moderate Grashof numbers. We consider that the geometric function FC

(Forchheimer constant) is equal to 0.011, Prandtl number Pr is taken as 1 and the

opening ratio ( OR ) is fixed at 0.5. The iteration process is terminated when the

following condition is satisfied:

51 10/ ji

mji

ii

mji

mji (11)

8 

 

where m denotes the iteration step and stands for either PVU ,, or . Published

experimental data are not available from the literature for the studied enclosure

configuration and boundary conditions. Thus, the validation of the obtained results

against suitable experimental data could not be performed. In this study, uniform mesh

sizes are used and 48×48 grids are chosen for the grid arrangement and it is found that

this grid solution is enough. The validation of the present code was made with the

previously published paper for differentially heated enclosure as listed in the Table I. As

can be shown from the table that results shows good agreement with the literature.

5. Results and discussion

In this numerical study, heat transfer and fluid flow results are obtained inside an open

sided inclined enclosure filled with fluid saturated porous medium. Results will be

present by streamlines, isotherms, velocity and temperature profiles, local Nusselt

numbers and mean Nusselt numbers for governing parameters such as porosity,

amplitude of sinusoidal function, inclination angle and Grashof numbers.

Figures 2 (a) to (c) present the streamlines (on the right) and isotherms (on the

left) for different Grashof numbers for values of 1a,2.0,01.0Da and o0 .

For this position of the cavity, the upper side of the cavity is fully open and the cavity is

heated from the bottom non-isothermally. As seen from the streamlines in Figures 2 (a)

and (b), two symmetric cells are formed inside the cavity and they turn in different

directions. As seen from isotherms, temperature distribution is also symmetric according

to mid-plane of the x-direction. This symmetric behavior is disturbed for the highest

value of Grashof number. The flow strength is increased with increasing of Grashof

number. A small circulation cell is formed at the right corner and thermal boundary layer

becomes thinner as illustrated from Figure 2 (c). The heated flow moves toward to left

side of the cavity.

Effects of permeability on fluid flow and temperature distribution are given in

Figures 3 (a) to (c). The values of permeability changes from 0.4 to 0.8 while other

9 

 

parameters are constant as 1,01.0 aDa and 610Gr . The figure can also be

comparable with Figure 2 (d) ( 2.0 ). As seen from Figures 2 (d) and 3, flow strength

increases with increasing of permeability. Thermal boundary layer becomes thinner with

increasing of permeability. This result is supported by Haghshenas et al. (2010). Effects

of inclination angle on streamlines (on the left column) and isotherms (on the right

column) are presented in Figures 4 (a), (b) and (c) for oo 60,90 and 30o, respectively.

These results are given for 1,2.0,01.0 aDa and 610Gr . For the case of left

side open cavity ( o90 ), a circulation cell is formed inside the cavity in clockwise

direction with min = -10. This result is can be comparable to see the difference between

constant (Haghshenas et al., 2010) and variable temperature boundary condition. For

other inclination angles, the flow rate is higher near the ceiling of the cavity. Thus, this

area is more heated and almost half of the cavity has constant cold temperature. With the

same parameters, Figure 5 is plotted to see the effects of amplitude of sinusoidal function.

In this case, amplitude value is taken as 5.0a . The lower values of flow strength are

formed due to low incoming energy into the cavity. Thermal boundary layer become

thicker according to value of 1a . Figure 6 compares the effects of porosity on

streamlines and isotherms for parameters of o60,1 a and 610Gr . It is seen that

flow inlet and outlet direction is on clockwise. The porosity affects the value of

streamfunction inside the cavity and absolute values of streamfunction increases with

increasing of porosity. However, thermal boundary layer becomes thinner with increasing

of porosity as seen from isotherms (Figure 6 (on the right column)).

Figures 7-10 illustrates the variation of velocity profiles at different location

inside the cavity for different inclination angle. Figures 7(a) and (b) are plotted for

velocity profiles at different inclination angle for 01.0,2.0,104 DaGr and

1a at 5.0X and 0.75, respectively. A low negative value is formed for velocity at

o60 with a parabolic distribution. Then, the lowest velocity values are obtained at

o45 . It means that inclination angle plays a critical role on velocity profiles. Near the

heated wall, velocity reachs a maximum value for o0 as seen from Figure 7(a). Near

10 

 

the opening side, values are decreases and velocity distribution becomes effective around

the heater. For the same parameters, except 510Gr , velocity profiles are shown in

Figures 8(a) and (b). It is interesting to see that the inclination angle shows different

behavior depending on the Grashof number. In Figure 8(a), the highest velocity profiles

are obtained for o45 , contrary of Figure 7. Near the opening, the velocity profiles

have maximum values around the non-isothermally heated part and the lowest velocity is

formed for o0 . The profiles show different variations near the heated wall for

o30 and o0 for 75.0X on the contrary of 5.0X (Figure 9 (a)) due to top

wall opening as given in Figure 9 (b). Further, Figure 10 presents the variation of the

velocity profiles for different values of porosity as 4.0 to 0.8. The figure is plotted for

the parameters of o6 60,1,10 aGr and 01.0Da . Finally, Figures 11 and 12

illustrate the variation of the local Nusselt number with different parameters. These

figures are presented to understand the effects of the Grashof number on variation of the

local Nusselt numbers as given in Figures 11(a) to (d). As can be seen from these figures,

the role of the inclination angle strongly depends on the Grashof number. Namely, the

highest value for local Nusselt number is obtained for o60 at 410Gr and o45

at 510Gr . A decreasing of the local Nusselt number is shown in Figure 11(a) at o0

due to stagnation point of that area and there is a lower velocity as can be seen from

Figure 2(a). Figures 12(a) and (b) presents the effects of porosity on variation of the local

Nusselt number for different inclination angles at o6 30,10 Gr and 01.0Da for

1a and a = 0.5, respectively. The figures indicate that maximum point of the local

Nusselt number decreases with increasing of porosity. Negative values of local Nusselt

numbers stem from the direction of heat transfer due to non-linear heating even there is

no heat absorption. As seen from the figure, lower heat transfer is formed for 5.0a , as

expected.

Variation of the mean Nusselt number for different parameters is listed in Table

II. As can be seen from this table the mean Nusselt number takes different values for the

fixed value of the porosity ( 2.0 ), amplitude of non-linear temperature ( 1a ) and

11 

 

Grashof number ( 410Gr ). The maximum heat transfer is formed for this values at

o90 . Global viewing of this table indicates that heat transfer increases with increasing

of Grashof number. Also, heat transfer increases with increasing of porosity for same

parameters. However, the heat transfer decreases with decreasing of amplitude of non-

linear temperature. For the highest value of Grashof number, the best value of the heat

transfer is obtained at o30 .

6. Conclusions

Two dimensional heat and fluid flow model is developed to examine the flow and

temperature field inside an open sided enclosure with a non-isothermal temperature

boundary conditions. The following major conclusions are drawn as

With the increase of the Grashof number, heat transfer increases as

independent of other parameters due to incoming more energy into the system.

Symmetric, double cells are formed inside the cavity for low Grashof numbers

and = 90o.

Stream function values are affected from the inclination angle at the same

Grashof number.

The inclination angle is the most important control parameter for flow and

temperature field in an open ended porous media filled enclosure.

Based on amplitude of sinusoidal function of temperature, the heat transfer

decreases with decreasing of amplitude value.

Acknowledgement First author thanks to King Saud University for their support to this

study in Visiting Professor Program.

12 

 

References

Abib, A.H. and Jaluria, Y. (1988), Numerical Simulation of the buoyancy-induced flow

in a partially open enclosure, Numerical Heat Transfer, Part A, Vol. 14, pp. 235-

254.

Basak, T., Roy, S., Paul, T. and Pop, I. (2006), Natural convection in a square cavity

filled with a porous medium: effects of various thermal boundary conditions,

International Journal of Heat and Mass Transfer, Vol. 49, pp.1430–1441.

Bejan, A., Dincer, I., Lorente, S., Miguel, A.F. and Reis, A.H. (2004), “Porous and

Complex Flow Structures in Modern Technologies”, Springer, New York.

Bilgen, E. and Oztop, H. (2005), Natural convection heat transfer in partially open

inclined square cavities, International Journal of Heat and Mass Transfer, Vol.

48, pp.1470-1479.

Bilgen, E. and Ben Yedder, R. (2007), Natural convection in enclosure with heating and

cooling by sinusoidal temperature profiles on one side, International Journal of

Heat and Mass Transfer, Vol. 50, pp. 139–150.

Elsayed, M.M., Al-Najem, N.M., El-Refaee, M.M. and Noor, A.A. (1999), Numerical

study of natural convection in fully open tilted cavities, Heat Transfer

Engineering, Vol. 20, pp. 73-85.

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Vol. 48, pp. 89-94.

Haghshenas, A., Nasr, M.R. and Rahimian, M.H. (2010), Numerical simulation of natural

convection in an open square cavity filled with porous medium by lattice

Boltzmann method, International Communications in Heat and Mass Transfer,

Vol. 37, pp. 1513-1519.

Holzbecher, E. (2004), Free convection in open-top enclosures filled with a porous

medium heated from below, Numerical Heat Transfer, Part A, Vol. 46, pp. 241-

254.

Ingham, D.B. and Pop, I. (eds.). 2005, “Transport Phenomena in Porous Media III”,

Elsevier, Oxford.

13 

 

Moraga, N.O., Sanchez, G.C. and Riquelme, J.A. (2010), Unsteady mixed convection in

a vented enclosure partially filled with two non-darcian porous layers, Numerical

Heat Transfer, Part A, Vol. 57, pp. 473-495.

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Press, Boca Raton, FL.

Nield, D.A. and Bejan, A. (2006), “Convection in Porous Media (3rd edition), Springer,

New York.

Oztop, H.F. (2007), Natural convection in partially cooled and inclined porous

rectangular enclosures, International Journal of Thermal Sciences, Vol. 46, pp.

149-156.

Oztop, H.F., Oztop, M. and Varol, Y. (2009), Numerical simulation of

magnetohydrodynamic buoyancy-induced flow in a non-isothermally heated

square enclosure, Communications in Nonlinear Science and Numerical

Simulation, Vol. 14, pp. 770-778.

Patankar, S.V. (1980), “Numerical Heat Transfer and Fluid Flow”, Hemisphere, New

York.

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isothermal open cavities, Numerical Heat Transfer, Part A, Vol. 5, pp. 421-437.

Polat, O. and Bilgen, E. (2002), Laminar natural convection in inclined open shallow

cavities, International Journal of Thermal Sciences, Vol. 41, pp. 360-368.

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Oxford.

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14 

 

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Springer, NewYork.

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heated triangular cavity, International Journal of Heat and Mass Transfer, Vol.

51, pp. 5040–5051.

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in non-uniformly heated porous isosceles triangular enclosures at different

positions, International Journal of Heat and Mass Transfer, Vol. 52, pp.1193-

1205.

 

Figure Captions

Fig. 1. Physical model and system of coordinates.

Fig. 2. Streamlines (on the left) and isotherms (on the right) for ,2.0,01.0 Da o0,1 a : a) 410Gr ; b) 510Gr ; c) 610Gr .

Fig. 3. Streamlines (on the left) and isotherms (on the right) for 610Gr,01.0Da ,

1a : a) 4.0 ; b) 6.0 ; c) 8.0 .

Fig. 4. Streamlines (on the left) and isotherms (on the right) for 2.0,01.0Da ,

1a , 610Gr : a) o90 , b) o60 , c) o30 .

Fig. 5. Streamlines (on the left) and isotherms (on the right) for 2.0,01.0Da ,

5.0a , 610Gr : a) o90 , b) o60 , c) o30 .

Fig. 6. Streamlines (on the left) and isotherms (on the right) for 1a , o60 , 610Gr :

a) 4.0 ; b) 8.0 .

Fig. 7. Velocity profiles at different inclination angles for 410Gr , 1a , 01.0Da ,

2.0 : a) 5.0X , b) 75.0X .

Fig. 8. Velocity profiles at different inclination angles for 510Gr , 1a , 01.0Da ,

2.0 : a) 5.0X , b) 75.0X .

Fig. 9. Velocity profiles at different inclination angles for 610Gr , 1a , 01.0Da ,

2.0 : a) 5.0X , b) 75.0X .

Fig. 10. Velocity profiles at different porosities for 610Gr , 1a , 01.0Da , o60 :

a) 5.0X , b) 75.0X .

Fig. 11. Variation of local Nusselt number along the heated surfaces at different

inclination angles for 1,2.0,01.0 aDa : a) 410Gr , b) 510Gr ,

c) 610Gr .

Fig. 12. Effect of porosity on local Nusselt number for 610Gr , 01.0Da , o30

a) 1a , b) 5.0a .

Fig. 1. Physical model and system of coordinates.

          

0.16 -0.16

0.05

0.1

0.15

0.2

0.25

0.3

0.350.40.45

0.55

0.7

0.8

a)  

3 -3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

 

b)  

-22

 

0.1

0.150.250.30.450.550.6

0.650.7

c)   

Fig. 2

 

-32

0.6

0.5

0.350.3

0.150.1

0.750.85  

a)  

-38

0.1

0.150.250.350.450.550.65 0.8 0.85

b)  

-45

   

0.1

0.20.30.40.55

0.7

c)  

Fig. 3

-10

0.05

0.15

0.25

0.450.

55

0.65

a)

-17

0.0

5

0.1

50.250.350.45

0.6

b)

-16

0.05

0.150.250.35

0.4

0.50.65

c)

Fig. 4

-8

0.4

0.3

0.2

0.1

a)

-13

0.05

0.15

0.25

0.350.45

b)

-12

0.05

0.15

0.250.3

0.4

c)

Fig. 5

-20

0.05

0.10.15

0.2

0.3

0.4

a)

-30

0.05

0.10.15

0.2

0.3

0.4

b)

Fig. 6 

 

V

-120 -100 -80 -60 -40 -20 0

Y

0.0

0.2

0.4

0.6

0.8

1.0

 

a)  

V

-30 -25 -20 -15 -10 -5 0

Y

0.0

0.2

0.4

0.6

0.8

1.0

 

b)

Fig. 7  

V

-40 -30 -20 -10 0

Y

0.0

0.2

0.4

0.6

0.8

1.0

 

a)   

V

-12 -10 -8 -6 -4 -2 0

Y

0.0

0.2

0.4

0.6

0.8

1.0

 

b)

Fig. 8

V

-40 -30 -20 -10 0

Y

0.0

0.2

0.4

0.6

0.8

1.0

 

a)

V

-6 -5 -4 -3 -2 -1 0

Y

0.0

0.2

0.4

0.6

0.8

1.0

 

b)

Fig. 9

V

-80 -60 -40 -20 0

Y

0.0

0.2

0.4

0.6

0.8

1.0

=0.4

=0.6

=0.8

 

a)

V

-20 -15 -10 -5 0

Y

0.0

0.2

0.4

0.6

0.8

1.0

=0.4

=0.6

=0.8

 

b) 

Fig. 10

X

0.0 0.2 0.4 0.6 0.8 1.0

Nu L

-4

-2

0

2

4

6

8

10

12

 

 a)

         X

0.0 0.2 0.4 0.6 0.8 1.0

Nu L

-4

-2

0

2

4

6

8

10

12

 

b)

 

X

0.0 0.2 0.4 0.6 0.8 1.0

Nu L

-4

-2

0

2

4

6

8

10

 

c)

Fig. 11

 

X

0.0 0.2 0.4 0.6 0.8 1.0

Nu L

-4

-2

0

2

4

6

8

10

12

14

=0.4

=0.6

=0.8

 

a)

X

0.0 0.2 0.4 0.6 0.8 1.0

Nu L

-2

0

2

4

6

8

=0.4

=0.6

=0.8

 

b)

Fig. 12