thermal convection in an oscillating cube at various

7/28/2019 Thermal Convection in an Oscillating Cube at Various

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THERMAL CONVECTION IN AN OSCILLATING CUBE AT VARIOUS

FREQUENCIES AND AMPLITUDES

AKI OKAJI

Department of Mechanical Engineering, Doshisha UniversityKyoto 610-0321, Japan

[email protected]

HIROCHIKA TANIGAWA

Department of Mechanical Engineering, Maizuru National College of TechnologyMaizuru 625-8511, [email protected]

JIRO FUNAKI

Department of Mechanical Engineering, Doshisha UniversityKyoto 610-0321, Japan

[email protected]

KATSUYA HIRATADepartment of Mechanical Engineering, Doshisha UniversityKyoto 610-0321, [email protected]

Abstract

In this study, the authors numerically investigate the forced-oscillation-frequency responses

on the three-dimensional thermal convection in a cubic cavity heated below in the

gravitational field, concerning flow characteristics such as the flow structure and a global

quantity the spatially-averaged kinetic energy K . The authors assume incompressible fluid

with a Prandtl number Pr = 7.1 (water) and a Rayleigh number Ra = 1.0×104

and

4.0×104. The direction of a forced sinusoidal oscillation is parallel to the terrestrial gravity.

The authors especially focus upon the influences of the forced oscillation amplitude η and

frequency ω in non-dimensional forms, whose test ranges are 1.5 ≤ η ≤ 15 and 10 ≤ ω ≤ 103.

The obtained results are as follows. As well as Ra = 1.0×104(Tanigawa et al., 2009), for Ra

= 4.0×104 , we can observe the optimum frequency maxK

where the amplitude of K attains

the maximum at each η. And, for both Ra’s,maxK

ω becomes the minimum at η = 1.5 – 2.0.

Especially for Ra = 4.0×104 ,

maxK ω

is affected by the initial conditions. For both Ra’s,

maxK ω

uniquely exists at each η , when η < 1.5. On the other hand, we can observe not single but

plural peak frequencies with locally-maximum ’s at each η , when η ≥ 1.5. It is revealed

that the existence of such plural frequencies are related with the appearances of various flow

structures such as S1, S2, S4, S5, S6 and S8. This correspondence is also affected by the

initial conditions. In addition, the details of a new flow structure S8 is shown.

Keywords Heat Transfer , Computation, Flow Structure, Forced-Oscillation, Cubic Cavity

INTRODUCTION

Thermal convection is a key phenomenon for mass and heat transfers/diffusions and materialmixings in many appliances. So, a lot of researches have been conducted about thermal

K



convection. Also, thermal convection becomes often important in various fields, such asgeology, meteorology, power-plant engineering and manufacturing industries.

So, many researchers have tackled thermal convection since the Benard’s experiment in 1900[1]. In the present study, we consider a cubic cavity heated below as one of the simplestmodels to realise thermal convection. A lot of researchers have studied the cubic cavity. For

example, Janssen et al. [2] numerically studied bifurcations from steady flow to periodicalflow in a cubic cavity. Pallares et al. [3, 4] numerically showed the three-dimensionalconvection in a cubic cavity at low and moderate Rayleigh numbers Ra’s < 6.0×104. Theyreported seven different steady-flow structures; namely, four kinds of single-roll structures(S1, S2, S3 and S7), two kinds of four-roll structures (S5 and S6), and one kind of a toroidal-roll structure (S4).

However, most of those researches are concerned with thermal convection in a stationarycontainer without forced oscillations. On the other hand, some researchers have beeninterested in the effects of forced oscillations to thermal convection (see Gresho & Sani [5];Gershuni & Zhukhovitskiy [6]; Bringen & Danabasoglu [7]; Fu & Shieh [8]; Miyanishi et al.[9]; Tanigawa et al. [10]; Hirata et al. [11]). For instance, Fu & Shieh reported that the flow is

classified into five regions depending on the forced-oscillation frequency. Miyanishi et al. [9]and Tanigawa et al. [10] studied the frequency response on a two-dimensional cavity,respectively, and found the optimum frequency where the frequency response becomes themaximum. Such studies are concerning two-dimensional thermal convection. Concerningthree-dimensional thermal convection, Biringen & Peltier [12] studied randomly- orsinusoidally-modulated gravitational fields. Their results suggest possibilities of theenhancement or the control of mass and heat transfers/diffusions and material mixings.Recently, Hirata et al. [13] have studied the frequency response on a three-dimensionalcavity, and found the optimum frequency. And Tanigawa et al. [14] have studied theinfluence of the forced-oscillation amplitude upon the optimum frequency on a three-dimensional cavity at Ra = 1.0×104.

In this study, as well as Hirata et al. and Tanigawa et al., we numerically investigate theforced-oscillation-frequency responses on the three-dimensional thermal convection in acubic cavity heated below in the gravitational field, concerning flow characteristics such asthe flow structure and a global quantity the spatially-averaged kinetic energy K . We assumeincompressible fluid with a Prandtl number Pr = 7.1 (water) and a Rayleigh number Ra =1.0×104 and 4.0×104. The direction of a forced sinusoidal oscillation is parallel to that of theterrestrial gravity. We should note that Hirata et al. have investigated the effects of theforced-oscillation frequency at a small forced-oscillation amplitude in a wide range of Ra =5.0×103 – 7.0×104, and that Tanigawa et al. have investigated the optimum frequency over awide range of the forced-oscillation amplitude at Ra = 1.0×104. In both the studies, the maininterest in frequencies adjacent to the optimum frequency. So in the present study, we

especially focus upon frequencies not only adjacent to but also far from the optimumfrequency over wide ranges of the forced-oscillation amplitude and frequency, whose non-dimensional-form ranges are 1.5 ≤ η ≤ 15 and 10 ≤ ω ≤ 103, respectively.

METHOD

Model and Governing Equation

Fig. 1 shows the present model, namely, the computational domain to be analysed, together



with the present coordinate system and boundary conditions. The model is a cubic cavity witha length scale H

*. We analyse the thermal convection assuming incompressible fluid with aconstant Prandtl number Pr = 7.1 (water) in the gravitational field. Tested Rayleigh number

Ra is fixed to 1.0×104 and 4.0×104. Both bottom and top walls are taken to be isothermal, andthe bottom temperature is greater than the top one. Four vertical sidewalls are conductive.Under these conditions, the flow attains a steady state with such a flow structure as the S2 for

Ra = 1.0×104

, or one of two flow structures as the S1 and S5 for Ra = 4.0×104

, after enoughtime (see later).

The dimensionless Navier-Stokes equations with the Boussinesq approximation and anenergy equation are as follows;

0=⋅∇ u , (1)

zPrT RaPr pt

euu

η +∆+−∇=

D

D, (2)

and

T t

T ∆=

DD . (3)

Here, the vibrational Rayleigh numberη

Ra based on the acceleration amplitude η is defined

by( ) Raωt η Ra sin1+=

η , (4)

where u, t , p, T , e z and ω denote velocity vector, time, pressure, temperature, the unit vectorin the z direction and (angular) frequency, respectively.

Independent variables are non-dimensionalised as follows;

=

=

*

*

*

*

1

z

y x

H z

y x

x , and*

*

*

α

2 H

t t = . (5)

As well, dependent variables are non-dimensionalised as follows;

=

=

*

*

*

*

*

w

v

u H

w

v

u

α u ,

**

**

c h

c

T T

T T T

−

−= , and

2

2

**

**

α ρ

H p p = , (6)

where *α , *cT , *

hT , and * ρ denote the coefficient of thermal diffusivity, a cold top-walltemperature, a hot bottom-wall temperature and mean density, respectively. A superscript “*”represents to be dimensional.

Governing Parameter

Non-dimensional governing parameters are as follows.

Rayleigh number:( )

**

3**c

*h

**

αν

H T T β g Ra

−= . (7)



Prandtl number: *

*

α

νPr = . (8)

Non-dimensional (angular) frequency: *

**

α

H 2

ω ω = . (9)

And,

non-dimensional acceleration amplitude: *

*

g

ηη = . (10)

Here, β *, ν*, η* and g* denote thermal expansion coefficient, kinetic viscosity, acceleration

amplitude and the gravitational acceleration, respectively. In the present study, Ra is fixed to1.0×104 or 4.0×104. For Ra = 1.0×104, η = 0.1, 1, 1.5, 2, 4, 10, 15. And, for Ra = 4.0 × 104, η = 0.1, 1, 1.5, 2, 10. The present test range of ω is 10 ≤ ω ≤ 103.

Boundary Condition

Boundary conditions for flow velocity are as follows.

.wvu 0=== (on all walls) (11)

Boundary conditions for pressure are as follows.

2

2

x

uPr

x

p

∂

∂=

∂

∂, (at x = 0 and x = 1) (12)

2

2

y

vPr

y

p

∂

∂=

∂

∂, (at y = 0 and y = 1) (13)

and

T Pr Ra z

wPr

z

pη

+∂

∂=

∂

∂

2

2

. (at z = 0 and z = 1) (14)

Boundary conditions for temperature are as follows. ,T 1= (at z = 0) (15) ,T 0= (at z = 1) (16)

and

1−=∂

∂

z

T . (at x = 0, x = 1, y = 0 and y = 1) (17)

In equation (17), we consider thermally conductive conditions on four sidewalls.

The governing equations (1) – (3) are solved by a finite difference method based on the MAC

scheme with a staggered computational grid with a size of 813

.

Global Indicator

As a global indicator, we consider such a physical quantity as spatially-averaged kineticenergy K . This is defined as follows.

∫∫∫= KdxdydzV

K 1

. (18)

Here, K is the kinetic energy defined as



( )222

2

1wvuK ++= , (19)

and V is the volume of the cavity.

RESULTS AND DISCUSSION

Flow Structure in No-Forced-Oscillation State [13]

As shown in Fig. 2, Hirata et al. [13] have revealed the flow structures for Pr = 7.1 (water) asa function of Ra. Note that this result is for η = 0, which represents a no-forced-oscillationstate. The upper row in the figure corresponds to decreasing Ra, and the lower rowcorresponds to increasing Ra. Specifically speaking, the upper row indicates the flowstructure solved using the solution at a higher Ra as an initial condition, and the lower row isvi-ce versa.

In Fig. 2, we can observe three kinds of steady flow structures; namely, two single-rollstructures and one four rolls structure. The formers are called as S1 and S2, the latter is called

as S5, according to Pallares et al. [3, 4]. More specifically, the S1 has a horizontal axisperpendicular to a pair of opposite sidewalls, and the S2 has a horizontal axis on a verticaldiagonal section. The S5 has the four rolls whose axises are perpendicular to oppositesidewalls.

Optimum Frequency and Its Amplitude Effect for Ra = 1.0×104

[14]

For Ra = 1.0×104, Tanigawa et al. [14] have confirmed that there always exists an optimumfrequency where the forced-oscillation-frequency response attains the maximum for eachforced-oscillation amplitude η in a range of η = 0.1 – 15. Figure 3 shows an example; namely,the frequency response of the amplitude K of spatially-averaged kinetic energy K for η = 1.Strictly speaking, K is defined as the difference between the maximum K and the

minimum K , which are obtained from the result in fully-periodic state enough time later thebeginning of computation. We can see that K shows a peak at ω = 27. We refer to such apeak frequency with the maximum K as the optimum frequency maxK

ω . In addition to theabove result for η = 1, all the frequency responses for Ra = 1.0×104 reveal that there alwaysexists an optimum frequency maxK ω for each η ≤ 15.

Figure 4 summarises the effect of η upon maxK ω for Ra = 1.0×104. From this figure, we might

expect that a change in flow occurs at η = 1.5. This is because we can see an abrupt change of η ω dd

maxK at η = 1.5. Specifically speaking, maxK ω decreases monotonically, as η increases

from zero to 1.5. On the other hand, at η ≥ 1.5, maxK ω increases with increasing η. Thus,

maxK ω shows the minimum at η = 1.5. In fact, at η < 1.5, the flow structure is approximatelythe same as the S2, which appears at η = 0 as shown in Fig. 2. On the other hand, at η = 1.5 –15, the flow structure is characterised by the S4.

Optimum Frequency and Its Amplitude Effect for Ra = 4.0×104

For Ra = 1.0×104, we have not observed any hysteresis effects. That is, the S2 always appearsin no-forced-oscillation state (at η = 0), being independent of initial conditions (see Fig. 2).So, we now examine frequency responses for Ra = 4.0×104, where either of the S1 and the S5appears depending upon initial conditions. To conclude, for Ra = 4.0×104 as well as 1.0×104,there always exists an optimum frequency maxK

ω for each η in a range of η = 0.1 – 10.



Then, we next examine the amplitude effect upon

maxK ω . As well as Fig. 4, Fig. 5

summarises the effect of η upon maxK ω for Ra = 4.0×104. In this figure, figure (a) denotes theresult with the S1 as an initial flow structure, and figure (b) denotes that with the S5 as aninitial flow structure. From both the figures, we might expect that a change in flow occurs atη ≈ 1.5 – 2.0. This is because we can see an abrupt change of η ω dd

maxK at η ≈ 1.5 – 2.0.

Specifically speaking, in each figure, maxK ω decreases monotonically, η increases from zeroto 1.5 (in Fig. 5(a)) or 2.0 (in Fig. 5(b)). On the other hand, at η > 1.5 (in Fig. 5(a)) or 2.0 (inFig. 5(b)), maxK ω increases with increasing η. Then, maxK ω

shows the minimum at η ≈ 1.5 –

2.0. As will be shown in Fig. 8, at η < 1.5, the flow structure is approximately the same as theS1 and the S5, which appears at η = 0 in Figs. 5(a) and 5(b), respectively. On the other hand,at η > 1.5, the flow structure is characterised at ω ≈ maxK ω by the S4 in both Figs. 5(a) and5(b).

Supplementary speaking, if we compare both Figs. 5(a) and 5(b), we find a small differencein the value of maxK ω at each η from a quantitative viewpoint. At η < 1.5, this differenceseems reasonable, because the flow structure is not identical between both the figures.However, it is interesting that, even at η = 2 – 5 where the flow structure is the same as the S4in both figures, the difference is not negligible yet.

Broad-Band Frequency Response covering the Optimum Frequency

Next, we examine a more broad-band frequency response covering the optimum frequency

maxK ω . To conclude, at η ≥ 1.5, we can observe other peak frequencies with locally-maximum values of K in addition to the maximum peak at ω =

maxK ω , for Ra = 1.0×104

and 4.0×104.

Figure 6 shows an example; namely, the frequency response of K for Ra = 1.0×104 at η =1.5. We can see that K shows a peak with the maximum K at ω = 22, which is defined asthe optimum frequency

maxK

ω . Moreover, we can observe another peak with a smaller K atω = 80. Plural peak frequencies with locally-maximum K ’s at ω ≠ maxK

ω suggest adiversity in flow depending upon ω even at the same η (≥ 1.5).

Flow structure

Finally, in order to confirm such a suggested diversity in last Subsection, we investigate theflow structure over a wide range of ω, both for Ra = 1.0×104 at η = 0.1 – 15 and for Ra =4.0×104 at η = 0.1 – 10.

Figure 7 summaries the flow structures for Ra = 1.0×104, as a function of ω and η. As well,Fig. 8 summaries the flow structures for Ra = 4.0×104. Fig. 8(a) and Fig. 8(b) show theresults with the S1 and the S5 as initial flow structures, respectively. In both Figs. 7 and 8, a

cross symbol represents the transitional state among stable states with such flow structures asthe S1, the S2, the S4, the S5, the S6 and the S8 (see later for the S8). And, a thick solid curvedenotes the optimum frequency

maxK ω .

At first, we consider Fig. 7. We can see that a stability boundary between the S2 and the S4exists at η ≈ 1.5 – 4 in a wide range of ω. Below the boundary, namely, at a smaller η, theflow structure is always the S2, being independent of ω. Above the boundary, namely, at alarger η, the flow structure is the S4 in the vicinity of

maxK ω . This is, of course, in



accordance with Tanigawa et al. However, far from the vicinity, we can observe a diversity inflow. That is, the flow structure is unstable and transitional at ω < 15, and is occasionally theS2 at ω > 100.

Second, we consider Fig. 8(a). As well as Fig. 7 we can see that a stability boundary betweenthe S1 and the S4 exists at η ≈ 1.5 in a range of ω < 100. However, as ω increases from 100,

we observe the S1 even at a larger η, as in addition to a small η. Thus, in a wide range of ω =10 – 103, at a small η < 1.5, the flow structure is always the S1, being independent of ω. Onthe other hand, at a large η > 1.5, we can confirm diversity in flow. That is, at ω > 100, theflow structure is often the S1 as well as at η < 1.5, and often unstable and transitional as wellas at ω < 15.

Thirdly, we consider Fig. 8(b). We can see that a stability boundary between the S5 and theothers exists at η ≈ 1.5 in a wide range of ω. Below the boundary, namely, at a smaller η ≤ 1.5, the flow structure is always the S5, being independent of ω. Above the boundary, namely,at a larger η ≥ 1.5, the flow structure is the S4 in the vicinity of

maxK ω . However, far from

the vicinity, we can see a rich diversity in flow. That is, we can observe such various flowstructures as the S1, the S2, the S6 and the S8, in addition to the S4, the S5 and transitional.

Figure 9 shows a new flow structure the S8. It is visualised by the iso-kinetic-energy surface

with K = 0.2× maxK , for Ra = 4.0×104 at η = 5 and ω = 180. —Generally, it is difficult tograsp three-dimensional structures. Distributions of velocity vectors, vorticity vectors and soon are effective to analyse the flows closely and quantitatively, but not suitable for theprimary comprehension of the three-dimensional whole flow structures. So, we haveproposed the iso-kinetic-energy surface to grasp the whole flow structures.— The colourdenotes fluid temperature; namely, red and blue represent hot and cold, respectively, asshown by a legend on the left side. We can clearly recognise a flow structure with doubletroidal rolls. We should note that there is almost no vertical-velocity component on a centerplane at z = 0.5: namely, the flow structure has almost no upwards flow on the center plane.

This suggests a possibility to suppose heat transfer between top and bottom walls.

CONCLUDING REMARKS

(1) For Ra = 4.0×104 as well as for Ra = 1.0×104, we can observe the optimum frequencywhere the amplitude of K attains the maximum at each η. the optimum frequencybecomes the minimum at η = 1.5 – 2.0, being affected by the initial conditions. At η < 1.5,the optimum frequency uniquely exists at each η. On the other hand, at η ≥ 1.5, we canobserve not single but plural optimum frequencies at each η. The plural optimumfrequencies suggest the appearances of various flow structures such as S1, S2, S4, S5, S6and S8. This correspondence is also affected by the initial conditions.

(2) For both Ra’s, if we consider a broad-band frequency response covering the optimumfrequency, a diversity in flow becomes rich at η ≥ 1.5. The diversity becomes more rich,as Ra increases together with hysteresis effects.

(3) We have found a new flow structure S8 for Ra = 4.0×104, at η = 5 and ω = 180 with S5 asan initial structure. And, we have revealed the details of the S8, which has the flowstructure with double troidal rolls.



REFERENCES

[1] M. H. Bénard, “Étude expérimentale des courants de convection dans une nappe liquide.– régime permanent: tourbillons cellulaires”, J. de Phys., 3e Série, 513 – 524 (1900). (inFrench)

[2] R. J. A. Janssen, R. A. W. M. Henkes and C. J. Hoogendoorn, “Transition to time-periodicity of a natural-convection flow in a 3D differentially heated cavity”, Int. J. HeatMass Transf., Vol. 36, pp. 2927 – 2940 (1993).

[3] J. Pallares, I. Cuesta, F. X. Grau and F. Giralt, “Natural convection in a cubical cavityheated from below at low Rayleigh numbers”, Int. J. Heat Mass Transf., Vol. 39, pp.3233 – 3247 (1996).

[4] J. Pallares, F. X. Grau and F. Giralt, “Flow transitions in laminar Rayleigh-Benardconvection in a cubic cavity at moderate Rayleigh numbers”, Int. J. Heat Mass Transf.,Vol. 42, pp. 753 – 769 (1999).

[5] P. M. Gresho and L. Sani, “The effect of gravity modulation on the stability of a heatedfluid layer”, J. Fluid Mech., Vol. 40, pp. 783 – 806 (1970).

[6] G. Z. Gershuni and Y. M. Zhukhovitskiy, “Vibrational-induced thermal convection inweight-lessness”, Fluid Mech. - Sov. Res., Vol. 15, pp. 63 – 84 (1986).

[7] S. Biringen and G. Danabasoglu, “Computation of convective flow with gravitymodulation in rectangular cavities”, J. Thermophy., Vol. 4, pp. 357 – 365 (1990).

[8] W. S. Fu and W. J. Shieh, “A study of thermal convection in an enclosure inducedsimultaneously by gravity and vibration”, Int. J. Heat Mass Trans., Vol. 35, pp. 1695 –

1710 (1992).

[9] T. Miyanishi, K. Hirata and H. Tanigawa, “A numerical study of natural convection in acylindrical cavity with gravity modulation”, Trans. JSME, Series B, Vol. 65, pp. 118 –125 (1999). (in Japanese)

[10] H. Tanigawa, T. Miyanishi and K. Hirata, “A numerical study of natural convection in asquare cavity with gravity modulation”, Trans. JSME, Series B, Vol. 66, pp. 1053 – 1060(2000). (in Japanese)

[11] K. Hirata, T. Sasaki and H. Tanigawa, “Vibrational effects on convection in a squarecavity at zero gravity”, J. Fluid Mech., Vol. 445, pp. 327 – 344 (2001).

[12] S. Biringen and L. J. Peltier, “Numerical simulation of 3-D Benard convection withgravitational modulation”, Phys. Fluids A, Vol. 2, pp. 754 – 764 (1990).

[13] K. Hirata, R. Hakui, K. Ishihara, H. Tanigawa and J. Funaki, “On frequency response of fluid in a cubic cavity heated below”, Trans. JSME, Series B, Vol. 72, pp. 279 – 284(2006). (in Japanese)

[14] H. Tanigawa, N. Nakamura, S. Fujita, J. Funaki and K. Hirata, “On the Effect of Forced-Oscillation Amplitude upon the Flow in a Cubic Cavity Heated Below”, Trans. JSME,Series B, Vol. 75, pp. 2106 – 22114 (2009). (in Japanese)

[15] L. Valencia, J. Pallares, I. Cuesta, and F. X. Grau, “Rayleigh- Bénard convection of water in a perfectly conducting cubical cavity : effects of temperature-dependent physicalproperties in laminar and turbulent regimes”, Numerical Heat Transf., Part A, Vol. 47,pp. 333 – 352 (2005).



Figure 1: Computational domain, together with coordinate system and boundary conditions.

Figure 2: Flow structures for Pr = 7.1, at η = 0 (Hirata et al. [13]).

Figure 3: Frequency response for Ra = 1.0×104

and Pr = 7.1, at η = 1(Tanigawa et al.

[14]).

Figure 4: Amplitude effects for Ra = 1.0×104

and Pr = 7.1 (Tanigawa et al. [14]).



(a) Wtih S1 as an initial flow structure (b) With S5 as an initial flow structure

Figure 5: Amplitude effects for Ra = 4.0×104

and Pr = 7.1.

Figure 6: Frequency response for Ra = 1.0×104

and Pr = 7.1, at η = 1.5.

Figure 7: Flow structures for Ra = 1.0×104

and Pr = 7.1.



(a) S1 as an initial flow structure

(b) S5 as an initial flow structure

Figure 8: Flow structures for Ra = 4.0×104

and Pr = 7.1

Figure 9: Flow structure S8 for Ra = 4.0×104

and Pr = 7.1 at η = 5 and ω = 180 with S5 as

an initial flow structure (at ωt = 16 π + 1.83).

thermal convection in an oscillating cube at various

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