Evolution of oscillatory and chaotic flows in mixed convection in porous media in the non-Darcy regime

Chaos, Solitons & Fractals Vol. 2. No. 1, pp,51-71, 1992 Printed in Great Britain

0960-0779/9255.00 + 00 (~) 1992 Pergamon Press Ltd

Evolution of Oscillatory and Chaotic Flows in Mixed Convection in Porous Media in the Non-Darcy Regime M. R. ISLAM South Dakota School of Mines, Rapid City, SD 57701, USA

Abstract--The problem of buoyancy-inducedflow in gas- or liquid-saturated porous media is of great

interest in various applications ranging from geothermal engineering to underground nuclear waste disposal. In many cases, the effect of forced and free convection may be significant. Previous studies in this regard were mainly done using the Darcy or Brinkman model. In the case of fluid flow under high temperature or pressure gradient, however, the flow regime is dictated by a Forchheimer, [P. H. Forchheimer, Z. Ver. Dt. Ing. 45, 1731-1788 (1901)] type equation. This paper investigates the time evolution of multicellular flows in a fluid-saturated porous medium in the case of mixed convection. Numerical results show the presence of periodic, quasi-periodic and chaotic behavior for increasingly high Grashof numbers.

INTRODUCTION Heat transfer in porous media is of interest in applications, such as geothermal engineering, thermal isolation, underground gas storage, nuclear waste management, etc. Convection heat transfer in fluid saturated porous media has been subject of study for more than two decades. This flow is of interest in a number of different areas, namely, geothermal engineering, thermal enhanced oil recovery, and heat loss through building insulations. The problems of geothermal and enhanced oil recovery are dominated by the effects of forced convection on the buoyancy-induced flow. This problem of combined free and forced convection in porous media has received little attention in the past [2]. Also, only a few studies have been reported on transient behavior of such systems. H o m e and O'Sullivan [3] studied the transient effects in two-dimensional natural convection in which a part of the lower boundary was heated. They observed sustained oscillatory convection in a range of different parameter values. Kimura et al. [4] studied two-dimensional, single-cell, transient convection in order to identify the route to chaos. By using a pseudo-spectral scheme, they observed periodic, quasi-periodic and non-periodic solutions in a porous medium with bottom heating. A m o n g other transient studies, two-dimensional natural convection has been studied by Poulikakos and Bejan [5]. They considered transient convection at low Rayleigh numbers in a porous medium with heated vertical walls. Inaba and Seki [6] reported both experimental and numerical results of transient natural convection in a porous medium. In their numerical study, they used a two-equation energy model leading to a rigorous treatment of finite resistance between fluid and solid matrix. They did not find any oscillatory behavior even though Chan and Banerjee [7] did find so using similar two-equation energy model in three-dimensional natural convection. Islam and Nandakumar [8] observed oscillatory and chaotic behavior in a fluid-saturated porous layer containing an internal heat source. Islam et al. [9] investigated the effect of tilt angle as well as the effect of aspect ratio on oscillatory and chaotic behavior in a porous medium with internal heat sources. 5l

52

M . R . [SLAM

All the above studies related to transient behavior have been conducted using Darcy's law in order to describe flow through porous media. There is general consensus that the Darcy's model is only valid for certain seepage velocity domain. Previous studies indicate that oscillatory and chaotic flow occur at very high Grashof n u m b e r for which inertial effects may be important. Islam and N a n d a k u m a r [10] used the generalized Forchheimer model [1] to obtain dual solution and hysteresis behavior in mixed convection in a porous medium. In this study, the same quadratic relationship between flow rate and pressure drop is used in order to observe flow transition from steady-state to periodic and quasi-periodic states in mixed convection in a porous medium.

MATHEMATICAL

FORMULATION

Let a horizontal square duct of permeability K be filled with a single fluid of density p and viscosity /t. Let a constant rate of heat transfer per unit length, Q ' and a constant gradient d p ' / d z ' be maintained in the axial flow direction. Boussinesq's approximation may be invoked as: p = p~[1 -

[J(T

-

TO].

The equations of motion and energy are given as:

(1)

~ ' • t)' = 0

/2 V' p ' - p g = 3T o ~t- + (v'. V')T

(2)

KF(Iq'I)o '

(3)

= o( V ' T 2

where o is the heat capacity ratio given by ~[pCp]f + (1 - e)[pCp]~ O=

[pCp]f

Also, 1tl K p

F = 1 +

Iq'l.

Considering two-dimensional flow, stream functions can be defined as: Dq/ u' -

Dy'

3~p' , v' -

(4)

Dx' "

The above equations are scaled as follows: u = u'/OCa),

~ = ~'/O'/a),w

y = y'/oq 0 =

= w'(v/~),

(T -

~, = ~,'/,~, x = x ' / ~ ,

T,~) --, q = q ' / ( v / a ) , Q '/k

r = t/(oa2/cr).

Eliminating pressure from flow equations and using dimensionless forms, one obtains Gr

DO

~[ ~ ( u q )

D-f =v2q' + ~[ ay

3(vq) ]

ax

1

(5)

Evolution of oscillatory and chaotic flows

53

where Gr is the modified G r a s h o f number, given by

Gr-

o~fiKgQ'

V2/¢

The dimensionless velocity is given by, q = (u 2 + 02 + w2) 1/2. The axial flow equation becomes: -

w-

~wq

= ~.

(6)

Finally, the energy equation (3), subject to the condition of uniform axial flux (i.e. dT/dz' = Q'(pCp (w') A'), becomes SO = V2 O + Pr u - ~x- - v

-

(w)A"

(7)

The energy and flow equations are solved along with the following boundary conditions: ~p = 0, 0 = 0 around the boundary. The average Nusselt n u m b e r is calculated from:

Nu-

Y 1 (1 + r) 2 0b"

(8)

NUMERICAL SOLUTION Dimensionless flow and energy equations were discretized as follows. The convection term in equation (7) was discretized using the A r a k a w a [11] scheme and the diffusion term was discretized by the D u f o r t - F r a n k e l [12] scheme. The A r a k a w a scheme has a formal truncation error of o(At 2, Ax 2, Ay2). The flow equation (5) was discretized using central difference approximations on all internal grid points, while forward and backward differences were used at the boundaries. They were solved by the point iterative relaxation method. The variables were updated in the following order O-~p-w. As soon as the stream function is updated the secondary velocities and the local speed q, were calculated and subsequently used in updating w and 0 values. For an aspect ratio of 1, a spatial grid of 21 × 21 was used throughout this study. Sensitivity study indicated that results were only slightly affected by increasing the spatial grid to 41 x 41. For instance, for a Grashof n u m b e r of 1000, Nusselt n u m b e r changed only 0.32% when grid size was refined from 21 x 21 to 41 × 41. Time step size of 0.001 was used for smaller Gr and 0.0005 was used for higher Gr values. RESULTS AND DISCUSSION

Inertial parameter, ~ = 0.01 Initial series of runs was conducted for ~ = 0.01, ~ = 30, and Pr = 0.73. Figure 1 shows the range of Nusselt n u m b e r values obtained as a function of Gr. As can be seen from this figure, stable steady-state solutions (SS) are obtained for Gr of up to 8290. Singlefrequency solutions appear starting at a Gr of 8290. The frequency grows rapidly with increasing Gr. The amplitude in Nu remains very small (in the order of 0.05) close to the bifurcation point. There is a sudden increase in amplitude beginning at Gr of 8300. As this

54

M . R . ISLAM 17

16

15 i

ua

Id I I

i

I I!l!'

13

z ,-1 r./'j

11

z

10

• single valued Nu -

9 t

range of Nu

•

g i

7

•

~-

:

--I

0

2

I -'i 6

8

I -- ~ 10

12

14

16

18

(Thousands)

GRASHOF

NUMBER

Fig. 1. Nusselt number vs Grashof number for Pr = 0 . 7 3 , ~ = 0.01.

trend continues, frequency increases sharply as well. Figure 2 shows variations of Nu with time and the power spectrum with frequency for a Gr of 8375. The existence of a single frequency is confirmed by plotting stream function and the average Nusselt number in the phase space over seven cycles of integration corresponding to the range of dimensionless time as shown in Fig. 3. The existence of single-frequency solutions continue until a Gr of 11 250 is reached. This transition takes place by the introduction of a subfrequency f2 in addition to the primary frequency, fl. It appears that f l in the double-frequency regime is a continuous extension of f l in the single-frequency regime whereas the frequency, f2 is

d2)

13.3

13.5 Z

13 129

Z

12 B 52.7 o

5

~

15

2

2.5

3

3 5

4

4.5

5

i 35

i 40

L 45

J 50

Dimensionless Time 4 pJ'Q: £ 3 5

E

O_ CO

3

2

g_ 0 5

0

= 5

~0

~ 15

i 2o

i 25

, 3o

i

Frequency Fig. 2. T r a n s i e n t Nusselt number and its power spectrum for G r = 8375, ~ = 0 . 0 l , P r = 0 . 7 3 .

Evolution of oscillatory and chaotic flows

12.52

~ m

55

m

0 12.48 II

•mm

12.44

12.4

t,-i 12.36 Oq 12..32 12.28 12.24 12.2

Gr=8375 - - r ~ 12.7

i 12.8

T ~

I

12.9

I

Nusselt

113

I --

"1

I 13. I

132

Number

Fig. 3. Trajectory in the phase space for Gr = 8375, ~ = 0.01, Pr = 0.73.

less than a half of f l . Figure 4 shows the existence of dual fundamental frequency on the power spectrum plot for a Gr of 11250. Also shown is the oscillation in Nu as a function of dimensionless time. Similar to what was observed in the single-frequency regime, the amplitude in Nu does not vary much during the earlier stages of the transition. The amplitude in Nu oscillates around 0.85 until a Gr of 13000 was reached. At this value of Gr, the amplitude Nu increases considerably. This marks a different type of oscillation, as shown in the Nusselt number variation in Fig. 5. For this particular case, f l value actually

t4.6 ~.4.4

~

14.2

Z M ~.1 13.a

O9

~ Z

13.5 13.4

i

~

i

5

10

15

i

i

20

25

30

DIMENSIONLESS TIME

Z~

0.05

I~

0.04

[,~

0.03

[-, r,.)

o.o2

0

ool

5

~.0

15

20

FREQUENCY Fig. 4. Transient Nusselt number and its power spectrum for Gr = 11250, ~ = 0.01, Pr = 0.73.

25

56

M.R. ISLAM 15.2

14.8

Z f"-'

14.6

:14.4

z

14.2 i4 0

5

10

15

20

25

30

DIMENSIONLESS TIME 0.i

i

]

0.08 rJ

0.06 0.04

I

©

0.02

0

1/, 5

10

15

20

25

FREQUENCY F i g . 5. T r a n s i e n t

Nusselt number

a n d its p o w e r s p e c t r u m

f o r Gr = 13 100, ~ = 0 . 0 1 , Pr = (t.73.

drops even though G r is increased. However, the amplitude in N u increases and the simple harmonic values are higher. This trend is intensified at a G r of 14000 when a third transition occurs, and a third subfrequency of motion arises in the flow. Figure 6 shows the variation in N u as well as power spectrum for G r of 14000. At this point, a large frequency, f2 ( = 6.42), is accompanied by subfrequencies, f 2 = 3.69, and f 3 = 2.16. Figure 7 shows the trajectory of minimum stream function and N u in the phase space. A fourth transition takes place at a G r of 18750 when a fourth frequency of motion evolves. Figure 8 shows variation in N u and power spectrum for this Gr value. A final transition occurs at a G r of 19000 when more than four frequencies evolve. Kimura et al. [4] called such a flow to be chaotic. Figure 9 shows variation in N u along with the power spectrum. Note that the power spectrum of the time series shows broad band of background noises. Following Gollub and Benson [13] and Kimura et al. [4], these solutions are referred as non-periodic even though these solutions show relatively sharp spectral peaks. We found that the solutions in the periodic and quasi-periodic regimes exhibit hysteresis, The solutions presented here were obtained by using stable solutions before the first bifurcation point as initial conditions. At least two more different routes were identified. The first one was by using cold initial conditions (0 = 0 at r -- 0 throughout) for each run irrespective of how large the G r value is. This route varied somewhat from the one described earlier. It was found that non-periodic solutions (more than 4 frequencies, as per the nomenclature of Kimura et al. [4]) were reached at 11 500 through a series of period doubling. Beyond a Gr of 12000, however, the n u m b e r of fundamental frequencies decreased through period halving until a single-frequency solution was obtained. A third route was obtained by using unstable solutions at a higher Gr as initial conditions for obtaining solutions at lower Gr values. Each case beyond the single frequency solutions

Evolution of oscillatory and chaotic flows

57

Z

r~

2; 5

~0

~5

20

25

30

DIMENSIONLESS TIME 0.05 0.04

0.03 r~ 0,02

~1~ 0 . 0 1

5

:1.0

15

20

25

FREQUENCY Fig. 6. Transient Nusselt number and its power spectrum for Gr = 1 4 0 0 0 , ~ = 0 . 0 1 , Pr = 0.73.

-14

~Z~ - 1 4 . 5

(3

o

D D

-15

Z

I~aaa

(3

[]O

a

O

(3

D O OOOO

O o

oo~

O

D O D D []

~

D

l-~ - 1 5 , 5

DO

o [3

D

D

13

[]

~O~

C]DE~jD E~ L~ [3

O

D

OO D O Oo

<

-16 O~]

~

-16.5 -17

Z

D

O

o

@

3

d~

a

-17.5

-18 0

18.5

'F

4.3

1

14.5

13 I

I"

14.7

I

I

14.9

r

T

15.1

i

T

15.,3

NUSSELT NUMBER Fig. 7. T r a j e c t o r y in the phase space for Gr = 1 4 0 0 0 , ~ = 0 . 0 1 , Pr = 0 . 7 3 .

from solutions at higher Gr leads to sustained 2-frequency solutions below the initial bifurcation point which was encountered by increasing Gr from solutions at lower Gr values. H o w e v e r , similar hysteresis was not observed for triple-frequency or non-periodic solutions.

58

M.R.

ISLAM

16.2

u~ D Z

~su

,,,-a Ua

15.6 15.4

Z 1 ~. ?

,2,

6

!0

i L?

DIMENSIONLESS

20

14

TIME

O.01E

D

e.oJ

r)

0. 000

~a

0. 0 0 6

', /'!,

0. 004

,!'1I1'

0~ 0 . 0 0 2

.-,

0

/,.

J

~22J

,

'v

,, ,'_,'d \

'~

_

~

10

,

I

/F~\ h

15

25

FREQUENCY Fig. 8. Transient Nusselt n u m b e r and its p o w e r s p e c t r u m for G r = 18750, ~ = 0.01, Pr = 1}.73,

16.4 16.2

D Z

16 15.8 15.6 J,5.4

Z 15.2

I

i

I

i

r

2

3

4

5

6

DIMENSIONLESS 10 -3

5 H

TIME

_

I

I'1 '1

I

© 0 L/'-~' 13

5

J.O

15

20

FREQUENCY Fig. 9. Transient Nusselt n u m b e r and its p o w e r spectrum for G r = 19000, ~ = 0.()1, l'r = 0.73.

25

Evolution of oscillatory and chaotic flows

59

Inertial parameter ~ = 0.02

Figure 10 shows variation in N u for various Gr values using a inertial parameter, ~ of 0.02. This series of numerical runs was conducted with ~ of 30 and Pr of 0.73. As can be seen from Fig. 10, stable steady-state solutions were obtained until Gr 3635 was reached. However, at Gr of 3625, an interesting p h e n o m e n o n occurred. Solutions of Gr = 3625 were obtained by using solutions of Gr = 3600 as initial conditions. As Gr is increased from 3600 to 3625, thermal changes led to perturbations in flow and in an abrupt manner, chaotic solutions appear. This can be seen in Fig. 11 which shows transient Nusselt numbers for a transition from Gr of 3600-3625. During initial stages of this transition, asymmetric solutions appear. However, as time progresses, symmetry grows slowly and continuously dampening oscillatory solutions appear. This trend continues until stable steady-state solutions appear beyond dimensionless time, T of 3. Similar transition was observed by Lennie et al. [14] except that they observed a transition from steady-state to a chaotic state, i.e. a deterioration in stability with time. However, they did mention the possibility of having periodic solutions for a finite time which then undergo a burst of chaos before being re-injected into the periodic orbit. That is, stabilization may occur with time. In the present work, a stabilization effect with time is observed. The trend of transition from unstable to stable flow pattern continues from a Gr of 3635. For this particular Gr, however, initially chaotic solutions stabilize into single-frequency periodic solutions. This transition is shown in Fig. 12. Figure 13 shows the transient Nusselt numbers along with the power spectrum. The power spectrum shows a sharp single-frequency of 5.7. The presence of a single frequency is further confirmed by plotting the trajectory of stream function and Nusselt number in the phase space. A closed single trajectory is shown in Fig. 14. The existence of a few scattered points is due to noises during the transition from chaotic to oscillatory regimes. As Gr is increased, another interesting phenomenon occurs. Figure 15 shows transient Nusselt numbers and maximum stream function for a Gr of 3645. Figure 16 indicates that quasi-periodic oscillations take place. This regime corresponds to periodic

11

106 iiii ,-z

~

lO2

98

Z b-.

~

94

9

Z

•

sEngle-valued

Nu

8.6 r a n g e of Nu 8.2

78

i 05

i 15

i

i 25

T

i 35

r

r 45

(]~ousands)

GRASttOF NUMBER Fig. 10. Nusselt number variation with Grashof number for ~ = 0.02, Pr = 0.73.

60

M.R.

ISLAM

10,3

10.29

10.28

z

[-

10.27

r.,o 10.26

z 10.25

14

10.24

0

2

DIMENSIONLESS TIME F i g . 11. T r a n s i e n t N u s s e l t n u m b e r for Gr = 3 6 2 5 , ~ = 0 . 0 2 , Pr = 0 . 7 3 .

1051

,

i1

jt /~lt ,°?9

i~,i

~o28 1

..~

I ]tUl ! 1027

~9

Z

1026

l

'

L I

I'i iI

1025

.~

1024

L 16

-

-

18

T

L

20

D i m e n s i o n l e s s Time F i g . 12. T r a n s i e n t N u s s e l t n u m b e r for Gr = 3 6 3 5 , ~ = 0.{}2, Pr = 0 . 7 3 .

behavior with periodic modulation of the amplitude. Even though a few scattered noises perturbed the overall configuration slightly, the global tendency to form a large loop is evident from Fig. 15. This loop is contained by spirally-formed internal loops. This is an interesting feature which has not been discovered in fluid through porous media before, even though Lennie et al. [14] observed such oscillations in duct flow. Note that this quasi-periodic behavior is different from that observed by Kimura et al. [4] who attributed

Evolution of oscillatory and chaotic flows

61

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1.o

Z

ii"! I",, ,'

,

10294

t''¸'l

("1

I

!

~"

':

~,

~1

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I

i I,

ii L ,

.,j 10.2819

L

'L/'~ "

! I

i

i ~ :

". ,.l ',:,'

[

10 286

05

~

15

2

25

3

3

Dimensionless Time 1.2

~D_

0.8

O0

0.6

D_

02

l 5

, 10

n 1.5

, 20

, 25

, 30

, 35

, 40

, 45

i 5,~

Frequency Fig. 13. Transient Nusselt number and its power spectrum for Gr = 3635, ~ = 0.02, Pr = 0.73.

796 ,~

# 0O

•-

•

7.95 -,

alu m

i

In,

793

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792

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791

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, 10286

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10.292

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10.296

10.298

10.3

Nusse]t Number Fig. 14. Trajectory in the phase space for G r = 3635, ~ = 0.01, P r = 0.73.

the term 'quasi-periodic' to solutions with several fundamental frequencies. The quasiperiodic solutions observed at a Gr of 3645 are sustained at even higher Grashof number. Further complications occur at higher Grashof numbers. For instance, at a Gr of 4000, the number of global periodic modulations increases. For this particular case, the Nusselt number and stream function trajectories were found to be too complex to be presented clearly. When the Grashof number is increased as high as 4500, even though solutions start off as quasi-periodic, they eventually b e c o m e noisy giving rise to a chaotic behavior. Such behavior is shown in Fig. 17 for a Gr of 5000. The numerical scheme failed to converge beyond this Gr of 5000. Transitions observed for this particular inertial parameter are: SS ~

P1 ~

QP1 ~

QP2 --~ QP3 --o NP

62

M.R.

ISLAM

804

o=

802 8 7 98

B

796 794

g,] 792 79 788

Z~

786 784 1027

10.28

10.29

103

Nusselt

10 51

Number

Fig. 15. Trajectory in the phase space for Gr = 3645, ~ = 0.02, Pr = 0.73.

10.32

T

I

+

I

l

I

I

1

10.31

E -1 z

10.3

10.29 z I0.28

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1

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I

1

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t

t.5

2

25

3

35

4

4,5

Dimensionless Time Fig. 16. Transient Nusselt n u m b e r for Gr = 3645, _~ = 0.02, Pr = 0.73.

J. 0 . 7 4

pa

10.73 Z ,-]

Z

~0.72 ~ 10.7~, ~0.7 10.6E

t

0

0.5

1

£

:1 . 5

L

I

2

DIMENSIONLESS

25

3

3.5

TIME

Fig. 17. T r a n s i e n t Nusselt n u m b e r variation for Gr = 5000, ~ = 0.02, Pr = 0.73.

Evolution of oscillatory and chaotic flows

63

where SS, P1, and NP indicate steady-state, single-frequency periodic, and non-periodic (or chaotic), respectively; and QP1, QP2, and QP3 indicate quasi-periodic solutions with orders of 1, 2, and 3, respectively. Effect o f the Prandtl number

Figure 18 shows the variation in Nusselt number with Grashof numbers for a Prandtl number of 5. For this series of runs, ~ of 0.01 and ~ of 30 were used. For a Pr of 5.0, single-frequency solutions were observed at a Gr of 1220. Figure 19 and 20 confirm this observation. At a Gr of 1250, period doubling occurs. Figure 21 shows the presence of two distinct fundamental frequencies. This observation is confirmed in Fig. 22 which shows the existence of two orbits in the closed trajectories of the stream functions and Nusself numbers. This corresponds to two fundamental frequencies for the periodic solutions. These solutions with double frequency continued to appear until a Gr of 1300 was reached. For this particular Grashof number, single-frequency solutions reappeared. Figures 23 and 24 confirm this observation. Single-frequency solutions continued to appear until a Gr of 1800 was reached. For this Gr, double-frequency solutions appeared. Figure 25 shows transient Nusselt numbers for this Grashof number. Figure 26 shows the trajectories of Nusselt number and stream functions. The existence of two helical orbits are apparent from this figure. For a Gr of 2000, which continued to show double frequency solutions, the evolution in cellular pattern of the stream function was observed. Figure 27 shows Nu variations for this Gr. Figure 28 (a)-(i) shows the evolution in stream function contours as function of time for equal dimensionless time interval of 0.025. These points (a)-(i) are indicated in Fig. 27. Figure 28a represents one of the lowest Nusselt number values (16.05). However, a four cell solution is observed for this case. This four cell solution is most likely due to hysteresis effects when Nusselt numbers decline sharply from a previous high value. This observation is supported by the fact that as the Nusselt number climbs up to 16.71 (Fig. 28b), the strength of the stream function decreases. This hysteresis phenomenon has been previously explained by Islam and Nandakumar [2] in the context of Darcy flow in a mixed convection problem. From this point, the strength increases slightly as more heat is 18

17

!liJ

16

15

i

14

Z [-, r.,,/3 r./3

Z

13

12

11

10

•

single-valued

--

range

Nu

of Nu

9

8

i

~ 04

i

i 0.8

~

= 1.2

i

i 16

T 2

I

(Thousands)

GRASttOF NUMBER

Fig. 18. Nusselt number variation with Grashof number for ~ = 0.01, Pr = 5.

I 24

64

M . R . ISLAM

J4.8

! 1

t~

:m Z

[-, ,..4

us

[

14

6

i4

il

i

t

4 i

++~++'

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,

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6

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14

12

iO

DIMENSIONLESS

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~8

20

TIME

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H U

I.t.1 0.01

US

0.005

0

+ ,, I

!

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;

b

25

20

J5

I0

FREQUENCY Fig. 19. Transient Nusselt number and its power spectrum for

-1.7

Z o

.

.

.

.

.

.

.

<

.

.

.

.

.

1220, ~ = 0.0l,

.

1.8

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Z

.

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14.1

14.3

NUSSELT Fig. 20. Trajectory in the phase space for

!4.5

147

NUMBER Gr

= 1220, ~ = 0.01,

Pr = 5.

Pr = 5.

Evolution of oscillatory and chaotic flows

65

15.2

I;:3 ~4.~

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[.., ~4.~ ,-1 r.~

II

II II tlt

!4.4 14 2 14

~b

O

•:

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16

!8

20

TIME

0.05

[-,., r,,)

0.03 0.02

A

z'~

A i0

i

i

15

20

25

FREQUENCY Fig. 21. Transient Nusselt number and its power spectrum for

Gr

= 1250, .~ = 0.01,

P r = 5.

-1 3

Z O

-1.9

F--, rj

[] [~] [] LlO

D

~

lIE

Z

£

D

< tr..] [-

-2.1

-2.2

~

[l L]

D D

-2.3

~b 2.4

I

14

I

14.2

C I

I

1

I

144

NUSSELT

I

14.6

I

[

14.8

NUMBER

Fig. 22. Trajectory in the phase space for

Gr

= 1250, ~ = 0.01,

P r = 5.

T 15

M . R . ISLAM

66 15.4

l.,q

I5.2 15

Z

14.8

[-, I4.6 :14.4

Z

14.2 14 5

8

10

12

J4

16

18

DIMENSIONLESS TIME

0.I2 0.I

[-., 0.08

[..r4 0.06 0.04

0

0.02 0 10

15

20

25

FREQUENCY Fig. 23. Transient Nusselt number and its power spectrum for G r = 1300, ~ = 0.01, P r = 5.

"--1.8

DEll

Z 0

-1.9

[]

m

0

o

0

D o

r,.;

~°

Z -2

0

o

D

[3

D

< [.-, r.z)

0

-2.1

D

-2.2

[]

Z -2.3

D

--2.4

i

14.1

i

14..3

D

in

Ell

14.5

i

i

i

14.7

i

14-.9

E

i

15.1

NUSSELT NUMBER Fig. 24. T r a j e c t o r y in the p h a s e s p a c e for G r = 1300, ~ = 0.01, P r = 5.

15.3

Evolution of oscillatory and chaotic flows

67

16.8

~6.6 16.4 ~6.2

,-3

~6 15.8 ~5.6 15.4

~ 6

' 2O

DIMENSIONLESS

TIME

0,04

0.035

O~ rj r.~ •~

0.03 0.025 0,02 0.0~5 0.0~

0

0.005 5

10

20

15

25

FREQUENCY Fig. 25. T r a n s i e n t N u s s e l t n u m b e r a n d its p o w e r s p e c t r u m f o r

--2.1

-

G r = 1800, ~ = 0 . 0 1 , P r = 5.

-

-o c~o txaJu.a.tud~m

Z

--2.2

q~ Z

--2.3

D° ~ ° ~ o ° ° °

2

noo P~u o . ~ 0 ~

OJ~

~ 0 O0

U

OD

~

O~

-

__oOS'

~ ~.~.jooo:po~

oOO

0

~O

O~

OOo

0 ~ 0 ~

[D

~

o~ --2.4o

<

p~

o --2.5

o o D O

--2.6

O

I:1

[D [3 [3 (D

--2.7

[] 0

D

~

s

--2.8

~OglO ODcl ~ --2.9

)

15.5

15.7

I

i

i

15.9

NUSSELT

I

i

16.1

i

~

16.3

~

16.5

NUMBER

Fig. 26. Trajectory in the phase space for

Gr

= 1800, ~ = 0.01,

P r = 5.

i

16.7

68

M.R. ISLAM 17 169 I 168

I b

,~

16,7 166

Z ,,~

165

16 2

16.1

iI i

[

t

vl~

0

",l

/

I \,J

T

02

r

04

r

I

06

-7

T--

0,8

T

10

1.2

Dimensionless Fig. 27. Transient Nusselt number for

Gr

•

14

: 16

1.8

2

Time = 2000, ~ = 0.01,

Pr = 5.

(a)

(d)

(g)

(b)

(e)

(h)

(c)

(!)

(i)

Fig. 28. Stream function contours for Gr = 2000, ~ = 0.01. Pr = 5 at various times.

generated ( N u = 16.98, Fig. 28c). H o w e v e r , from this point as the Nusselt number declines slowly four cells collapse gradually to form two a s y m m e t r i c cells (Fig. 28d). A s the Nusselt n u m b e r decreases further (Fig. 28e), a s y m m e t r y in flow is gradually r e m o v e d . At this point the two cell solution appears. This flow regime continues through Fig. 28f. H o w e v e r , as the Nusselt n u m b e r declines further, four cell solutions reappear as shown in Fig. 28g. N o t e that Fig. 28g and Fig. 28a are situated at similar location in the N u axis even though half

Evolution of oscillatory and chaotic flows

69

cycle separates the two points. They both show four cell solutions. In Fig. 28h, Nu climbs the second peak giving rise to stronger cells (Nu = 16.5). As Nu declines further, the similar peak (Fig. 28i) is achieved. At this point asymmetric solutions appear. They are analogous to the first peak (Fig. 28c and Fig. 28d) even though the cells change sites (they collapse on the left-hand side rather than on the right-hand side). For this particular Prandtl number, triple-frequency solutions appeared at a Gr of 2200. Figure 29 shows the existence of different frequency patterns in transient Nusselt numbers. As the Grashof numbers were increased further, chaotic solutions appeared after a Gr of 2300. Figure 30 shows the existence of chaotic nature for this Gr. For this particular Pr of 5, the following sequence was observed: SS --~ P1 (n ~ P2 m ~ P1 ~2) ~ P2 (2) ~ P3 --~ P4 --* NP where SS, P1, P2, P3, P4, and NP represent steady-state, single-frequency, doublefrequency, triple-frequency, and non-periodic (chaotic) solutions, respectively. The superscripts indicate the number of occurrence of a particular type of solution.

CONCLUSIONS

The route to chaotic solutions is observed through periodic and quasi-periodic solutions in mixed convection heat transfer in a porous medium in non-Darcy flow regime. The effect of Grashof number, Prandtl number, and inertial parameter is investigated. For a particular path in increasing Grashof number, the existence of periodic solutions of various fundamental frequencies, quasi-periodic solutions and non-periodic solutions depends largely on the inertial parameter as well as the Prandtl number.

17.8 17.4 17 2

Z m]

17

~6.8 166

Z 16.4 0

2

4

6

B

10

DIMENSIONLESS

12

14

ib

.~

TIME

0.04 0.035

~I r0

ooa O,025 OO2

o o15

~ :: 0

001

o oo~ 0 0

5

i0

I~

FREQUENCY Fig. 29. Transient Nusselt number and its power spectrum for

~' .3

Gr

= 2200, ~ = 0.01,

25

P r = 5.

70

M . R . ISt.AM 17 8

m I~

17.6

~ Z

17.4

[--d

~7

r.~

Z

2 ~7

1B.8 "-6.6 0

Z=

4

6

8

10

12

t4

16

1B

D I M E N S I O N L E S S TIME 0 05

[....,

-

-

0.04

I.~ 0 0 3 0.02 0

o.o~

10

5

15

20

25

FREQUENCY Fig. 30. Transient Nusselt n u m b e r and its power spectrum for Gr = 2300, ~ = 0.01, Pr = 5.

Acknowledgements--This Technology.

study was possible with a grant from the South Dakota

NOMENCLATURE A, A',

%

Gr, K, k, m,

Nu, Pr, p',

q, Q',

T, l), W,

x,y,z,

Y, 0, V,

p,

dimensionless cross-sectional area cross-sectional area of the porous duct specific heat Grashof n u m b e r , Q ' g f l K a / k v 2 permeability effective thermal conductivity inertial constant Nusselt n u m b e r Prandtl n u m b e r pressure local speed rate of heat transfer per unit length temperature velocity vector axial velocity coordinate axes effective thermal diffusivity coefficient of thermal expansion aspect ratio of the duct normalized pressure drop dimensionless temperature viscosity kinematic viscosity inertial parameter, m K / a density porosity stream function

School of Mines and

Evolution

of oscillatory

and chaotic

flows

71

Superscripts, subscripts, symbols b, f, s, w, ?):

dimensionless quantity bulk fluid solid wall reference average quantity

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