CC BY
7Статья поступила в редакцию 12.08.10. Ред. рег. № 861
The article has entered in publishing office 12.08.10. Ed. reg. No. 861
УДК 517.95
ТЕПЛОВАЯ КОНВЕКЦИЯ ВНУТРИ ПРЯМОУГОЛЬНОЙ ПОЛОСТИ, ЗАПОЛНЕННОЙ НЕНЬЮТОНОВСКИМИ СТЕПЕННЫМИ ЖИДКОСТЯМИ И ДИФФЕРЕНЦИАЛЬНО НАГРЕТЫМИ ОДНОРОДНЫМИ ТЕПЛОВЫМИ ПОТОКАМИ
1 2 2 3
М. Каддири , М. Найми , А. Рахи , М. Хаснауи
Университет султана Мулай Слимана, Научно-технический факультет, Отдел машиностроения 2Физический отдел, Лаборатория моделирования потоков и перемещений (LAMET) п.я. 523, Бени-Меллал, Марокко Тел.: (212) 5 23 48 51 12/22/82; факс: (212) 5 23 48 52 01. E-mail: [email protected], [email protected] 3Университет Кади Айяда, научный факультет, департамент физики, лаборатория жидкостной механики и энергетики (LMFE),n^. 2390,
Марракеш, Марокко
Заключение совета рецензентов: 22.08.10 Заключение совета экспертов: 30.08.10 Принято к публикации: 05.09.10
Двумерные стационарные плавающие потоки неньютоновских степенных жидкостей, удерживаемых в прямоугольной полости, нагреваемых и охлаждаемых с вертикальных сторон однородными тепловыми потоками, численно рассчитываются методом конечных разностей. Число Релея и индекс режима потока являются параметрами, определяющими проблему. В работе анализируется влияние этих параметров на структуру потока и характеристики переноса тепла.
Ключевые слова: перенос тепла, естественная конвекция; неньютоновские жидкости, численные исследования, кубическая полость.
THERMAL CONVECTION WITHIN A SQUARE CAVITY FILLED WITH NON-NEWTONIAN POWER-LAW FLUIDS AND DIFFERENTIALLY HEATED
WITH UNIFORM HEAT FLUXES
M. Kaddiri1, M. Naimi2, A. Raji2, M. Hasnaoui3
'Sultan Moulay Slimane University, Faculty of Sciences and Technologies, Mechanical Engineering Department 2Physics Department, Laboratory of Flows and Transfers Modeling (LAMET) B.P. 523, Beni-Mellal, Morocco Tel.: (212) 5 23 48 51 12/22/82; Fax: (212) 5 23 48 52 01, E-mail: [email protected], [email protected] 3Cadi Ayyad University, Faculty of Sciences Semlalia, Physics Department, Laboratory of Fluids Mechanics and Energetics (LMFE), B.P. 2390,
Marrakech, Morocco
Referred: 22.08.10 Expertise: 30.08.10 Accepted: 05.09.10
Two-dimensional steady-state buoyancy driven flows of non-Newtonian power-law fluids confined in a square cavity, heated and cooled from its vertical sides with uniform heat fluxes, is conducted numerically using a finite difference technique. The parameters governing the problem are the Rayleigh number and the flow behavior index. The effects of these parameters on the flow structure and heat transfer characteristics are analyzed.
Keywords: heat transfer, natural convection, non-Newtonian fluids, numerical study, square cavity.
Vita: National doctorate (INPL, Nancy, France, 1989) State doctorate (Cadi Ayyad University, Marrakech, Morocco, 2001)
Professor of Mechanical Engineering (Sultan Moulay Slimane University, Faculty of Sciences and Technologies, Physics Department, Beni-Mellal, Morocco) Director of Flows and Transfers Modelling Laboratory (LAMET)
Research Topics
- Natural convection in non-Newtonian fluids;
- Thermosolutal convection in non-Newtonian fluids.
Mohamed Naimi
Nomenclature
g acceleration due to gravity (m/s2)
H' height or length of the enclosure (m)
k consistency index for a power-law fluid at the reference temperature (Pa-in)
n flow behavior index for a power-law fluid at the reference temperature
Nu local Nusselt number, Eq. (11)
Nu mean Nusselt number, Eq. (12)
Pr generalized Prandtl number, Eq. (10)
q' constant density of heat flux (W/m2)
Ra generalized Rayleigh number, Eq. (10)
T dimensionless temperature, (=(T' - T0')/AT *)
T0' reference temperature at the geometric centre of the enclosure (K)
AT * characteristic temperature (= q'H'/X) (K)
(u, v) - dimensionless horizontal and vertical velocities (= (u', v')/(a/H')) (x, y) - dimensionless horizontal and vertical coordinates (= (x', y')/H')
Greek symbols
a thermal diffusivity of fluid at the reference temperature (m/s2)
P thermal expansion coefficient of fluid at the reference temperature (1/K)
St dimensionless time step
X thermal conductivity of fluid at the reference temperature (W/m.°C)
dynamic viscosity for a Newtonian fluid at the reference temperature (Pa.s)
\i.a dimensionless apparent viscosity of fluid, Eq. (6)
p density of fluid at the reference temperature (kg/m3)
Q dimensionless vorticity, (= Q'/(a/H'2))
y dimensionless stream function, (= ^/a)
a apparent variable
max maximum value
0 reference value
' dimensional variables
Subscripts and superscript
Introduction
Thermal natural convection phenomenon results from density variations within a non-isothermal fluid under the influence of gravity. It is frequently encountered in nature and in many industrial applications. Its importance in various domains attracted many worldwide researchers, through the decades, to investigate the buoyant heat transfer in many flow configurations, submitted to various boundary conditions. Useful literature review relative to buoyant induced flows was documented in the article and book by Ostrach [1] and Gebhart et al. [2], respectively.
Remind that most of the fluids, considered in previous theoretical and experimental investigations on natural convection, are of Newtonian behavior. However, many fluids encountered in industrial applications, such as papermaking, oil drilling, slurry transporting, food processing, polymer engineering and so on, exhibit non-Newtonian behaviors. According to
Skelland [3], non-Newtonian fluids can be classified into three main groups, which are purely viscous, viscoelastic and time-dependent fluids. Purely viscous non-Newtonian fluids can be divided into two categories: shear-thinning (or pseudo-plastic) fluids and shear-thickening (or dilatant) fluids. For the former, the viscosity is a decreasing function of the shear rate. This property is specific to some complex solutions like ketchup, whipped cream, blood, paint, and nail polish. It is also a common property of polymer solutions and molten polymers. Pseudo-plasticity can be highlighted for example by shaking a bottle of ketchup, which leads to an unpredictable change in the viscosity of the content. The viscous force causes it to go from being thick like honey to flowing like water. For the latter, the viscosity increases with the shear rate. The dilatant effect can readily be seen with a mixture of cornstarch and water, which acts in a counterintuitive way when thrown against a surface.
International Scientific Journal for Alternative Energy and Ecology № 8 (88) 2010
© Scientific Technical Centre «TATA», 2010
Engineers, in general, are faced with the (often considerable) practical difficulties of modeling a variety of industrial processes involving the flow and heat transfer in some of these materials. Consequently, much works has been done with this in mind. In this respect, the investigations carried out by Tien et al. [4] and Ozoe and Churchill [5] count among the earliest ones that considered the rheological behavior in the case of Rayleigh-Benard convection inside a horizontal rectangular channel. Tow decades later, a numerical modeling of natural and mixed convection was conducted by Turki [6] in the case of a rectangular cavity differentially heated. Recently, the case of a shallow horizontal cavity heated and cooled with uniform heat fluxes from its vertical sides was studied analytically and numerically by Lamsaadi et al. [7]. Finally, in a more recent paper, Santra et al. [8] presented numerical results on heat transfer augmentation in a square cavity filled with nanofluids. It must be pointed out that, in all these contributions, the power-law model was adopted to approach the rheological behavior of the considered fluids. In other works, the square cavity was used and the rheological behavior of fluids was not restricted only to the power-law, such as in Santra et al. [8]. For instance, in the investigations by Demir et al. [9] and Demir [10], Criminale-Erickson-Filbey elastic fluids were considered. In a recent study by Vola et al. [11], attention was focused on the case of a Bingham plastic fluid.
Except the reference by Lamsaadi et al. [7], which is concerned by Newmann thermal boundary conditions, in all the previous references, thermal boundary conditions of Dirichlet type (imposed constant temperatures) were used. To the best of the authors' knowledge, there is no investigation dealing with natural convection in a square cavity, confining purely viscous non-Newtonian fluids, subject to thermal boundary conditions of Newmann type (imposed uniform heat fluxes). This manifest lack of works on such problems motivates the present study, which consists to analyze the conjugate effects of buoyancy and rheology on the dynamical and thermal behaviors of the studied fluids and also on the resulting heat transfer.
The square enclosure considered here is heated and cooled with uniform heat fluxes from its vertical walls while its horizontal boundaries are kept adiabatic. The power-law model, suggested originally by Ostwald-de Waele, has been used to characterize the non-Newtonian fluid behavior. A numerical solution of the full governing equations has been obtained for wide ranges of two main dimensionless parameters, which are the power-law index and the Rayleigh number. Results obtained are illustrated in terms of streamlines, isotherms and thermo-convective characteristics for various values of the governing parameters. These results are relevant to a better understanding of the general flow and heat transfer of non-Newtonian fluid flows in confined clear media.
Mathematical formulation
Problem statement The geometry under consideration is sketched in Fig. 1. It consists of a two-dimensional square enclosure of size H 'xH'. The horizontal walls are thermally insulated while the vertical sides are subjected to uniform density of heat flux, q'.
Fig. 1. Sketch of the geometry and coordinates system Рис. 1. Эскиз геометрии и систем координат
The non-Newtonian fluids considered here are those whose rheological behavior can be approached by the power-law or Ostwald-de Waele model, which is characterized by a relationship between the shear stress and the shear rate. In terms of laminar apparent viscosity, this one can be written here as follows:
К = k
r 12JdvvL
dx 'J + Uy '
2
V v
2 1
du' dv'
+ 1 — + — dy' dx'
n-1 ,2 11
. (1)
The two empirical constants n and k, appearing in Eq. (1), are the flow behavior and consistency indices, respectively. Note that for n = 1, Eq. (1) recovers the Newton law simply by setting k = ^(Newtonian viscosity). Thus, the deviation of n from the unity characterizes the degree of non-Newtonian behavior of the fluid. Specifically, when 0 < n < 1, the fluid is said to be pseudo-plastic (or shear-thinning), and the viscosity is found to decrease by increasing the rate of strain. On the other hand, when n > 1, the fluid is called dilatant (or shear-thickening), and the viscosity increases by increasing the shear rate. Dilatant fluids are in general much less widespread than pseudo-plastic ones. Though the power-law model does not converge to Newtonian behavior in the limit of zero and maximum shear rates, it presents the advantage of being simple and mathematically tractable. In addition, the rheological behavior of many substances can be adequately represented by this model for a relatively large ranges of shear rates (or shear stresses), which makes it useful, at least for engineering purpose, and justifies its use in most theoretical investigations of fluids having pseudoplastic or dilatant behaviors. For the mathematical formulation, the following assumptions have been adopted:
- The fluid velocities are small enough to consider the flow as laminar. In fact, in most buoyancy driven motions, the fluid circulation is slow due to moderate temperature gradients [12].
- The fluid is incompressible. For pressures close to atmospheric pressure, liquids can be considered as incompressible fluids with a good approximation.
- The viscous dissipation is negligible. According to Turki (1990), this approximation is valid for polymer solutions, weakly or moderately concentrated, such as aqueous solutions of carboxymethylcellulose (CMC). For polymer solutions highly concentrated or polymer melts, the viscous frictions are so significant that it becomes not obvious to generate the fluid motion simply by buoyancy effects.
- The physical properties are temperature-independent except for the density in the buoyancy term, which obeys the Boussinesq approximation. The derivation of the latter from the equations of compressible fluid dynamics is quite tedious and requires careful ordering of several limiting processes [13].
- The third dimension of the cavity is large enough so that the problem can be considered two-dimensional. This assumption is relatively well satisfied in general and provides insight into the more complicated three-dimensional flows [12].
Governing equations and boundary conditions
On the basis of the assumptions adopted and using the characteristic scales H', H'2/a, a/H, a/H2, q'H'/X and a, which correspond respectively to length, time, velocity, vorticity, temperature and stream function, the dimensionless governing equations (whose derivations from basic principles can be found in references [5] and [6]), written in terms of vorticity, Q, temperature, T, and stream function, are as follows:
dO d(uO) d(vO) _ dt dx dy
= Pr
V 2O + 2
dO + Эца dO
dx dx dy dy
dT + d(uT) + ЭК) = V2T .
dt dx
dy
V2у = -О .
+ So ; (2)
(3)
(4)
Where
Эу ; ду ; о dv du ;
dy ' dx ' dx dy '
VduY (dvY" du dv 2
Ka = 2 1 — 1 +1 — 1 + --1--
Ux J Idy J_ dy dx
(5)
(6)
and
= Pr
d4 d4
dx dy
du + dv dy dx
dT dx
- 2
d2 Ha
dxdy
du dx
dv
'dy.
+ PrRa
(7)
For the present problem, the appropriate non-dimensional boundary conditions are:
dT
u = v = y = — +1 = 0 for x = 0 and 1; (8) dx
dT
u = v = y = — = 0 for y = 0 and 1. (9)
dy
Note that the introduction of O, T and y, as secondary variables, presents the advantage of reducing the number of dependent variables and equations, by eliminating the pressure as a primitive variable, which is without interest in the considered situation. Such a formulation is generally more appropriate for two-dimensional and axisymmetric flows, but its major disadvantage lies in the fact that O is unknown at the boundaries. To overcome such a difficulty, the Woods formulation has been adopted for stability and accuracy reasons. Roache [14].
In addition to the flow behavior index, n, two other dimensionless parameters appear in the above equations. These are the generalized Prandtl and Rayleigh numbers defined, respectively, as
Pr =
(k / p) H '
and Ra =
a
g$H ' 2n+2 q ' (k / p)an X
(10)
and whose Newtonian expressions can be recovered by simply setting and replacing k by the Newtonian viscosity
On the other hand, it was reported in past studies Lamsaadi et al. [7], Ng et al. [15] and Mamou et al. [16] that the convection is rather insensitive to Pr variations, provided that this parameter is large enough as it is the case for the non-Newtonian fluids and for a large category of fluids having a Newtonian behavior. Therefore, the Prandtl number is not considered as an influencing parameter in this study and the simulations have been conducted with Pr ^ «>, i.e. by neglecting the inertia terms on the left hand side of Eq. (2), owing to their negligible contribution. Hence, n and Ra remain the only governing parameters for the problem under consideration.
Numerics
The two-dimensional governing equations have been discretized by using the well-known second order central finite differences method with a regular mesh size. The integration of Eqs. (2) and (3) has been performed with the Alternating Direction Implicit method (ADI), originally used for Newtonian fluids and successfully experimented for non-Newtonian power-law fluids,
International Scientific Journal for Alternative Energy and Ecology № 8 (88) 2010
© Scientific Technical Centre «TATA», 2010
n-1
Turki [6] and Lamsaadi et al. [7]. To satisfy the mass conservation, Eq. (4) has been solved by a Point Successive Over- Relaxation method (PSOR) with an optimum relaxation factor calculated by the Frankel formula Roache [14]. Table 1 shows that a grid of 81*81 is suitable for obtaining adequate results. In fact, the grid refinement to 121*121 leads to maximum differences of
2.5 % and 3.4 % in terms of max| and Nu,
respectively. At each time step, which has been varied in the range 10-7 < 8t < 10-4 (depending on the values of n
and
XI
Ra),
V* + l vk
the
"Vi,
X|V J
convergence criterion < 10-4 has been satisfied for y,
where
Vi
k
is the value of the stream function at the
node (i, j) for the kth iteration level.
With the Ostwald power-law model, the dimensionless viscosity, given by Eq. (6), tends towards infinity at the cavity corners (where the velocity gradients are nil) for 0 < n < 1, which makes impossible direct numerical computations. However, this difficulty has been surmounted by using average values for the viscosity at the corners that has led to possible and stable computations.
The steady solution has been used to calculate the local Nusselt number as
Nu(y) =
XMT '/ L' MT
(11)
and then, the mean Nusselt number has been calculated as
_ 1
Nu = J Nu (y)dy . (12)
0
As an additional check of the results accuracy, an energy balance has been systematically verified for the system at each numerical code running. Thus, the overall heat transfer, through each vertical plane, has been evaluated and compared with the quantity of heat furnished to the system through the left vertical wall (at x = 0) and leaving it through the vertical opposite wall (at x = 1). For all the results reported here, the energy balance has been satisfied within 0.1% as a maximum difference.
Moreover, the present code has been validated against the results obtained, in the case of a square cavity differentially heated, by Turki [6] for non-Newtonian power-law fluids and by Vola et al. [11] and De Vahl Davis [17] for Newtonian fluids. Comparative results are summarized in Table 2 where it is seen that the agreement is very good; the maximum difference being within 2%.
Предварительные тесты на влияние шага координатной сетки Preliminary tests on the grid size effect
Таблица 1 Table 1
Grids 61x61 81x81 121x121
n Ra lv 1 max Nu lv 1 max Nu lv 1 max Nu
0.6 102 0.217 1.007 0.216 1.007 0.216 1.008
104 5.036 3.594 5.039 3.575 5.041 3.565
106 9.820 15.870 10.108 16.886 10.357 16.339
1.0 102 0.129 1.003 0.126 1.003 0.126 1.002
104 3.157 1.984 3.160 1.982 3.164 1.980
106 6.674 6.514 6.743 6.456 6.788 6.413
1.4 102 0.102 1.002 0.102 1.002 0.102 1.002
104 1.805 1.366 1.830 1.350 1.804 1.363
106 5.222 3.639 5.249 3.621 5.244 3.616
Таблица 2
Сравнение числового кода с предварительными исследованиями для Ra = 104
Table 2
Validation of the numerical code against previous studies for Ra = 104
Present work [7] [12] [18]
Pr n lv 1 max Nu lv 1 max Nu lv 1 max Nu lv 1 max Nu
0.71 1.0 5.072 2.248 5.032 2.240 5.071 2.238
100 0.7 10.025 4.274 4.217
1.0 5.180 2.279 2.280
1.3 2.759 1.492 1.522
Results and discussion
As mentioned before, since the computations have been conducted with high Prandtl numbers, the problem under study has became governed by only tow parameters. These ones are the flow behavior index, n, varying from 0.6 to 1.4 to include shear-thinning (0 < n < 1), Newtonian (n = 1) and shear-thickening (n > 1) fluids, and the Rayleigh number, Ra, varying in the range 102 < Ra < 5-106 to cover low, moderate and relatively high convection.
In the following subsections, combined effects of n and Ra on fluid dynamics and heat transfer characteristics will be examined.
Dynamical and thermal structures
Рис. 2. Линии потока (слева) и изотермы (справа)
для Ra = 102 и а - n = 0.6
= 0.216,Nu = 1.007
b - n = 1.0
= 0.126,Nu = 1.003) и
■=14 (k
= 0.102,Nu = 1.002)
Fig. 2. Streamlines (left) and isotherms (right) for Ra = 10 and
a - n = 0.6 k I = 0.216,Nu = 1.007),
b - n = 1.0 ((J = 0.126,Nu = 1.003)
Typical streamlines (left) and isotherms (right), are presented in Figs. 2-7, for various values of n and Ra. All these figures are characterized by a centro-symmetry (symmetry with respect to the center of the cavity) resulting from the symmetrical aspect of the geometry and the boundary conditions imposed to the system. The conjugate effect of n and Ra is seen to depend on the values attributed to these parameters.
- Low convection: For Ra = 102, Fig. 2 shows that, globally, the unicellular nature of the flow does not undergo important qualitative and quantitative changes when n is varied in its range, due to the fact that the buoyancy forces are not yet sufficient to promote the flow intensity (the pseudo-conductive regime is still dominant). This observation is well corroborated by the isotherms, which remain nearly parallel to the vertical boundaries. An increase of Ra to 103 does not change the qualitative aspect of the flow configuration but the impact of this increase is more visible on the isotherms (Fig. 3); the distortion of the latter becomes more pronounced by decreasing n.
and c - n = 1.4
= 0.102,Nu = 1.002)
Рис. 3. Линии потока (слева) и изотермы (справа) для
Ra = 103 и а - n = 0.6 k I = 2,Nu = 1.7), b- n = 1.0
\ | т max | " /
((max | = 1,NU = 1.3) C- n = Ы (¥max| = 0.5,NU = 1.1)
Fig. 3. Streamlines (left) and isotherms (right) for Ra = 103 and
a - n = 0.6 ((J = 2,Nu = 1.7), b- n = 1.0
kmax I = 1,NU = 1.3) C- n = Ы (¥max| = 0.5,NU = 1.1)
а
b
c
c
International Scientific Journal for Alternative Energy and Ecology № 8 (88) 2010
© Scientific Technical Centre «TATA», 2010
- Moderate convection: For Ra = 10 , Fig. 4 shows that the unicellular aspect of the flow structure is preserved but, qualitatively and quantitatively, the impact of n is well visible on both streamlines and isotherms and its effect is characterized by a reduction of the flow intensity and heat transfer (the streamlines tend to be less crowded near the vertical walls when n passes from 0.6 to 1.4). In addition, a close inspection of the isotherms shows that the latter change moderately by increasing n since their important distortion, observed for pseudo-plastic fluids (n = 0.6), tends to vanish for the dilatant ones (n = 1.4).
Рис. 4. Линии потока (слева) и изотермы (справа) для
Ra = 104 и a - n = 0.6 ((max| = 5.039,Nu = 3.575),
b - n = 1.0 (d = 3.160,Nu = 1.982) и с - n = 1.4 (|d| = 1.830,Nu = 1.350) Fig. 4. Streamlines (left) and isotherms (right) for Ra = 104
and a - n = 0.6 (|max| = 5.039,Nu = 3.575),
b - n = 1.0 (Id = 3.160,Nu = 1.982) and с - n = 1.4 (|d = 1.830,Nu = 1.350)
When Ra passes to 105 (Fig. 5), the effect of the flow behavior index n on the streamlines becomes more visible. The flow structure is now more complicated for the shear-thinning fluid which promotes the appearance of two
small vortices inside the main flow. The stratification of the isotherms in the central part of the enclosure is also supported by decreasing n. The tendencies observed suggest a premonition of an evolution towards more complex structures by increasing Ra.
Рис. 5. Линии потока (слева) и изотермы (справа)
для Ra = 105 и a - n = 0.6 ((max| = 7,Nu = 8),
b - n = 1.0 (|vmax | = 5,Nu = 3.5) и с - n = Ы (Id = 3,Nu = 2.4) Fig. 5. Streamlines (left) and isotherms (right) for Ra = 105
and a - n = °.6 ((m^ | = 7,Nu = 8),
Ь - n = 1.0 (|Vmaxl = 5,Nu = 3.5) and с - n = Ы (|d = 3,Nu = 2.4)
- Relatively high convection: For Ra = 106, Fig. 6 shows that the flow structure is more or less complicated depending on the flow behavior index. In fact, for n = 0.6, a breakdown is observed in the isolines located in the center of the cavity giving arise to a new arrangement and more interesting flow structure. By increasing n, the mono-cellular structure is recovered in the case of the shear-thickening fluid for which the behavior is characterized by a slowdown of the flow intensity (compare l^maxl = 10.108 and l^maxl = 5.249
a
a
b
b
c
c
corresponding respectively to n = 0.6 and 1.4). A close inspection of the isotherms leads to analogous observations inasmuch as the thermal stratification observed in the central region of the cavity and the boundary layers developed in the vicinity of the vertical sides for n = 0.6, tend to disappear for n = 1.4. This change in the behavior results from the fact that the convection effect is significantly weakened by the increase of n. It is to note that all the attempts made to increase the Rayleigh number to values as high as 107 have failed. In fact, the highest value of Ra leading to steady state solution with the numerical code was 5-106. Above this threshold, it was impossible to perform numerical calculations since all the attempts conducted with n = 0.6 leaded to a divergence of the computing code.
c
Рис. 6. Линии потока (слева) и изотермы (справа) для
Ra = 106 и а - n = 0.6 ((J = 10.108,Nu = 16.
b - n = 1.0 ((max| = 6.743,Nu = 6.456) и c - n = 1.4 (Id = 3.249,Nu = 3.621) Fig. 6. Streamlines (left) and isotherms (right) for Ra =106 and a - n = 0.6 (Ц = 10.108,Nu = 16.8:
b - n = 1.0 ((max| = 6.743,Nu = 6.456) and c - n = 1.4 (Ц = 3.249, Nu = 3.621)
Hence, results presented in Fig. 7 with Ra = 5-10 show that the breakdown of the inner isolines occurs for all the values of n with an increasing impact by decreasing n. It is obvious that this behavior is retarded for the shear-thickening fluids for which higher values of Ra are required to observe this break. This means that all the behaviors observed are delayed by increasing n from small to larger values. In addition, it can be seen from Fig. 7 that the width of the peripheral flow is considerably reduced and the thermal stratification is better established for the shear-thinning fluids.
c
Рис. 7. Линии потока (слева) и изотермы (справа)
для Ra = 5-106 и а - n = 0.6 (|Vmax| = 12.359,Nu = 24.846),
b - n = 1.0 (Ц = 8.160,Nu = 9.684)
и c - n = 1.4 (d = 6.120, Nu = 5.108) Fig. 7. Streamlines (left) and isotherms (right) for
Ra = 5-106 and a - n = 0.6 (|Vmax| = 12.359,Nu = 24.846),
b - n = 1.0 (Ц = 8.160,Nu = 9.684
= 6.120, Nu = 5.108
and c - n = 1.4
Stream function, velocity and temperature distributions In this subsection, the attention is mainly focused on the combined effects of Ra and n by presenting in Figs. 8-13 the profiles of the stream function, y, (top), vertical
International Scientific Journal for Alternative Energy and Ecology № 8 (88) 2010
© Scientific Technical Centre «TATA», 2010
a
b
b
velocity, v, (middle) and temperature, T, (bottom), at mi-height of the cavity.
- Low convection: For Ra = 102, it appears evident from Fig. 8 that the regime is pseudo-conductive since the stream function and velocity amplitudes are weak and the temperature distribution is quasi-linear and independent of n.
0,&
0,4-
-0,4-
-0,&
п = 0.6
» Ч 1 I О - i A \
V^^.^V Г \ J
0,0
0,5
1,0
T 0,6
0,2 0,0 -0,2
-0,6
■ п =0.6
V п=1 л=14
\
>ч a¡¡ \ "я ч
0,0
0,5
1,0
Рис. 8. Изменения потоковой функции (вверху), вертикальной скорости (в центре) и температуры (внизу) по оси x на полувысоте полости и различных значений n Fig. 8. Stream function (top), vertical velocity (middle) and temperature (bottom) variations with the x-coordinate at mid-height of the cavity (y = 1/2), for Ra = 102 and various values of n
By increasing Ra up to 103, Fig. 9 shows that the profiles shape of у and v remain qualitatively unchanged, but their amplitudes undergo noticeable
increases whose importance increases by decreasing n. The impact of the increase of Ra on the temperature profile is more pronounced for n = 0.6 since it differs now notably from the linear aspect while for Newtonian and shear-thickening fluids the deviation from the linear profile remains weak and confused, with respect to n, due probably to the fact that this parameter does not affect significantly the apparent viscosity of these fluids for relatively low values of Ra.
Рис. 9. Изменения потоковой функции (вверху), вертикальной скорости (в центре) и температуры (внизу) по оси x на полувысоте полости (у = 1/2), для Ra = 10
и различных значений n Fig. 9. Stream function (top), vertical velocity (middle) and temperature (bottom) variations with the x-coordinate at mid-height of the cavity (y = 1/2), for Ra = 103 and various values of n
¥
-1
-2
-3
-4
-5
- у Tj
1 % s/
■ 4 "X^jM / {
/ '
\ V у
X ч ^ :
/
ч r
■ [
1 •
temperature profile from the linear shape becomes more and more pronounced by decreasing n and this profile tends to exhibit a curved aspect with strong gradient zones in the vicinity of the vertical walls and weak gradient ones elsewhere. The effects observed with Ra = 104 are accentuated by increasing Ra up to 105 as shown in Fig. 11. It can be seen from this figure that the behavior of the boundary layer regime is already visible for n = 0.6.
Рис. 10. Изменения потоковой функции (вверху), вертикальной скорости (в центре) и температуры (внизу) по оси x на полувысоте полости (у = 1/2), для Ra = 104, и
различных значений n Fig. 10. Stream function (top), vertical velocity (middle) and temperature (bottom) variations with the x-coordinate at mid-height of the cavity (y = 1/2), for Ra = 104 and various values of n
- Moderate convection: As can be seen from Fig. 10, an increase of Ra up to 104 modifies the profile of y, which tends to now to have a flat shape in the central part of the cavity by decreasing n (Fig. 10, top). Simultaneously, a modification in the velocity profile is observed (Fig. 10, middle). By decreasing n, the extrema of v are displaced towards the thermally active walls giving rise to the appearance of an inflexion point in the centre of the enclosure. Likewise, the deviation of the
40-
0-
-40
-80 0,0
n = 0.6
■ Д I Лч 1 4 s....... ^Sivy * 1 ■ \J
0,5
0,2-
0,1
0,0
-0,1-
-0,2-
1.4
41
v V V
V n = 0.6 V\ \ \
0,0
0,5
1,0
Fig. 11. Изменения потоковой функции (вверху), вертикальной скорости (в центре) и температуры (внизу) по оси x на полувысоте полости (у = 1/2), для Ra = 105 и
различных значений n Fig. 11. Stream function (top), vertical velocity (middle) and temperature (bottom) variations with the x-coordinate at mid-height of the cavity (y = 1/2), for Ra = 105 and various values of n
International Scientific Journal for Alternative Energy and Ecology № 8 (88) 2010
© Scientific Technical Centre «TATA», 2010
fluids (case of n = 1.4) and it is shifted away from the active walls. Similarly, the temperature variations are more important near the active boundaries indicating the development of thermal boundary layers with a decreasing thickness by decreasing the flow behavior index n. In addition, the decrease of n makes the zone outside the boundary layers increasingly flat. An augmentation of Ra to 5-106 (Fig. 13) makes increasingly complex the situation inasmuch as the phenomena observed with Ra = 106 and n = 0.6 appear strongly developed for almost all the explored values of n, confirming the findings of the previous subsection.
Fig. 12. Изменения потоковой функции (вверху), вертикальной скорости (в центре) и температуры (внизу) по оси x на полувысоте полости (у = 1/2), для Ra = 106 и
различных значений n Fig. 12. Stream function (top), vertical velocity (middle) and temperature (bottom) variations with the x-coordinate at mid-height of the cavity (y = 1/2), for Ra = 106 and various values of n
- Relatively high convection: At Ra = 106, the breakdown of the inner isolines and the boundary layer regime are now observed for Newtonian and shear-thinning fluids (Fig. 12). In fact, the deformation caused in the flat zone of the у profile consists in the appearance of two extrema while |v| increases sharply to maximum values very close to the active walls and decreases after that with the same rate towards zero not far from the vertical walls where the vertical velocity remains nil. This description is particularly valid for n = 0.6. Comparatively, the maximum of v undergoes a drastic decrease for the shear-thickening
300
100-
-100
-3000,0
T
0,10
0,0&
0,02
-0,02
-0,05
п = 0.6
•
' 1
г ÍX. .1.4
»if1
*
0- -.—
0,5
x 1,0
-0,10
0,0
1.4
Г
.....
л = 0.6 \ \ V
0,5
1,0
Fig. 13. Изменения потоковой функции (вверху), вертикальной скорости (в центре) и температуры (внизу) по оси x на полувысоте полости (у = 1/2), для Ra = 5106 и
различных значений n Fig. 13. Stream function (top), vertical velocity (middle) and temperature (bottom) variations with the x-coordinate at mid-height of the cavity (y = 1/2), for Ra = 5-106 and various values of n
Flow intensity and heat transfer rate The effect of the Rayleigh number on the maximum stream function, l^max I, and the average Nusselt number, Nu, is illustrated in Fig. 14 for various n corresponding to fluids with different rheological behaviors. At low values of Ra, l^max I and Nu tend towards their asymptotic values of pure conduction with increasing rates by decreasing n. The variations of
I^max I and Nu with Ra are in general characterized by a monotonous increase. This behavior is expected since, in general, buoyancy forces act in order to improve the convection heat transfer when the flow structure is not complicated as it is the case here (dominant monocellular structure).
Nu
10:
n = 0.§ У X
*
У 1-4
Itf
10J
104
10s
10b
10
Ra
,7
b
Fig. 14. Изменения максимальной потоковой функции (а) и средний критерий Нуссельта (b) c числом Релея
для различных значений n Fig. 14. Variations of the maximum stream function (a) and the mean Nusselt number (b) with the Rayleigh number for various values of n
Quantitatively, for a given Ra, both I ^max I and Nu decrease by increasing n. In fact, according to Eq. (6), this behavior is mainly attributed to an increase of the apparent viscosity with the behavior index and the slowing down role of the latter on the fluid motion is well known. It is to note that, the irregularities observed for n = 0.6 in the evolutions of I ^max I and Nu at Ra around 106 are attributed to the breakdown of the inner isolines marking the flow structure more complex at high Rayleigh numbers. It appears finally that the parameters n and Ra have opposite effects on the fluid flow and heat transfer characteristics. This means that convection heat transfer can be enhanced by decreasing n (for a given Ra) or increasing Ra (for a given n). The situation is however more sensitive to the change of Ra for the shear-thinning behavior (n = 0.6) compared to the shear-thickening one (n = 1.4). Analogous observations relative to the conjugate effects of n and Ra on convection heat transfer were reported in the past by Turki [6] and lately by Hadim [18] while investigating natural convection in respectively power-law non-Newtonian fluid-filled cavity and saturated porous layer differentially heated from the vertical sides (imposed constant but different temperatures on the vertical sides).
Conclusion
A numerical investigation of steady thermal convection in a square cavity filled with non-Newtonian power-law fluids is performed. The case where both vertical sides are submitted to uniform heat fluxes (Neumann type condition) is considered. The impact of the main governing parameters, which are the Rayleigh number (102 < Ra < 5-106) and the flow behavior index (0.6 < n < 1.4), on the fluid flow and heat transfer characteristics is examined. The main conclusions that emerge from the study are summarized in the following points:
- At low and moderate Rayleigh numbers, the flow structure is organized in one cell regardless of the flow behavior index.
- At relatively high Rayleigh numbers, the flow structure losses its well organized aspect since a breakdown of the inner isolines, decreasing the flow behavior index, leads to more complex structures.
- A decrease of the flow behavior index or an increase of the Rayleigh number has similar effects on the flow intensity and the heat transfer rate (opposite effects of these parameters on fluid flow and heat transfer characteristics).
References
1. Ostrach S. Natural convection in enclosures // J. Heat Transfer. 1988. 110. P. 1175-1190.
2. Gebhart B., Jaluria Y., Mahajan Hemisphere, Washington DC; 1988.
a
International Scientific Journal for Alternative Energy and Ecology № 8 (88) 2010
© Scientific Technical Centre «TATA», 2010
3. Skelland A.H.P., Non-Newtonian flow and heat transfer, Wiley, New York; 1967.
4. Tien C., Tsuei H.S., Sun Z.S. Thermal instability of a horizontal layer of non-Newtonian fluid heated from below // Int. J. Heat Mass Transfer. Shorter Comm. 1969. 12. P. 1173-1178.
5. Ozoe H., Churchill S.W. Hydrodynamic stability and natural convection in Ostwald-De Waele and Ellis fluids: the development of a numerical solution // AIChE J. 1972. 18. P. 1196-207.
6. Turki S., Contribution à l'étude numérique des transferts par convection naturelle et par convection mixte dans les fluides non-Newtoniens confinés, Thèse de Doctorat, CNAM, Paris, France; 1990.
7. Lamsaadi M., Naïmi M., Hasnaoui M. Natural convection of non-Newtonian power-law fluids in a shallow horizontal rectangular enclosure uniformly heated from the side // Energy Conversion and Management. 2006. 47. P. 2535-2551.
8. Santra A.K., Sen S., Chakraborty N. Study of heat transfer augmentation in a differentially heated square cavity using copper-water nanofluid // Int. J. Thermal Sciences. 2008. 47. P. 1113-1122.
9. Demir H., Akyyldyz F.T. Unsteady thermal convection of a non-Newtonian fluid // Int. J. Eng. Science. 2000. 38. P. 1923-1938.
10. Demir H. Thermal convection of viscoelastic fluid with Biot boundary conduction // Math. Comp. Simulation. 2001. 56. P. 277-296.
11. Vola D., Boscardin L., Latche J.C. Laminar unsteady flows of Bingham fluids: a numerical strategy and some benchmark results // J. Comp. Physics. 2003. 187. P. 441-456.
12. Siginer D.A., Valenzuela-Rendon A. On the laminar free convection and stability of grade fluids in enclosures // Int. J. Heat Mass Transfer. 2000. 43. P. 3391-405.
13. Gray D.D., Giorgini A. The validity of the Boussinesq approximation for liquids and gases // Int. J. Heat Mass Transfer; 1976. 19. P. 545-51.
14. Roache P. J., Computational fluid dynamics, Hermosa Publishers, Albuquerque, New Mexico; 1982.
15. Ng M.L., Hartnett J.P. Natural convection in power-law fluids // Int. Comm. Heat Mass Transfer. 1986. 13. P. 115-120.
16. Mamou M., Vasseur P., Hansoui M. On numerical stability analysis of double-diffusive convection in confined enclosures // J. Fluid. Mech. 2001. 433. P. 209-250.
17. De Vahl Davis G. Natural convection of air in a square cavity: a bench mark numerical solution // Int. J. Num. Method Fluids. 1983. 3. P. 249-264.
18. Hadim H. Non-Darcy natural convection of a non-Newtonian fluid in a porous cavity // Int. Comm. Heat Mass Transfer. 2006. 33 P. 1179-1189.
