Transp Porous Med DOI 10.1007/s11242-015-0558-x
Unsteady Natural Convection with Temperature-Dependent Viscosity in a Square Cavity Filled with a Porous Medium M. S. Astanina1 · M. A. Sheremet1,2 · J. C. Umavathi3
Received: 21 June 2015 / Accepted: 3 August 2015 © Springer Science+Business Media Dordrecht 2015
Abstract A numerical investigation is implemented on the unsteady natural convection with a temperature-dependent viscosity inside a square porous cavity. The vertical walls of the cavity are kept at constant but different temperatures, while the horizontal walls are adiabatic. The mathematical model formulated in dimensionless stream function, vorticity and temperature variables is solved using implicit finite difference schemes of the second order. The governing parameters are the Rayleigh number, Darcy number, viscosity variation parameter and dimensionless time. The effects of these parameters on the average Nusselt number along the hot wall as well as on the streamlines and isotherms are analyzed. The results show an intensification of convective flow and heat transfer with an increase in the viscosity variation parameter for the porous media, while in the case of pure fluid, the effect is opposite. Keywords Natural convection · Unsteady regimes · Temperature-dependent viscosity · Porous media · Square cavity · Numerical results · Finite difference method
1 Introduction The study of convective flow and heat transfer of fluids with temperature-dependent properties is of considerable interest in the modern fluid systems of practical importance (see Alim et al. 2014; Bondareva et al. 2013; Hirayama and Takaki 1993; Hyun and Lee 1988; Jiji 2006; Juarez et al. 2011; Martynenko and Khramtsov 2005; Parveen and Alim 2011; Sivasankaran and Ho 2008; Umavathi and Ojjela 2015). It is well known that in the case of the moderate
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M. A. Sheremet
[email protected]
1
Department of Theoretical Mechanics, Tomsk State University, Tomsk 634050, Russia
2
Department of Nuclear and Thermal Power Plants, Tomsk Polytechnic University, Tomsk 634050, Russia
3
Department of Mathematics, Gulbarga University, Gulbarga, Karnataka 585 106, India
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temperature differences, an influence of the temperature-dependent fluid properties in cavities on the fluid structure and heat transfer becomes more essential. Studies in this field can be found in the books by Martynenko and Khramtsov (2005), and Jiji (2006) and in the papers such as, for example, Hyun and Lee (1988), Hirayama and Takaki (1993), Juarez et al. (2011), Umavathi and Ojjela (2015). In spite of wide applications, there are few papers concerning an analysis of convective heat transfer in fluids of variable viscosity. For example, Hyun and Lee (1988) have numerically analyzed the unsteady natural convection in a square differentially heated cavity of a fluid with a temperature-dependent viscosity. The authors used the exponential form for the viscosity. It has been found that for a variable-viscosity fluid, thermal convection is more intensive close to the hot wall and during the transient phase, the heat input to the cavity from the hot vertical wall exceeds the heat output to the cold vertical wall. Hirayama and Takaki (1993) have conducted a numerical study of two-dimensional Benard convection of a fluid with temperature-dependent viscosity of exponential type using the Boussinesq approximation. It has been shown that for low values of the viscosity variation parameter, the convection flow spreads over the whole region, while in the case of the higher values of this parameter, the center of the convective roll shifts to the lower region and the flow almost stops near the upper boundary. Such results characterize that the layer of intensive convection is confined to the lower hot region with smaller mean viscosity. Juarez et al. (2011) have investigated the natural convective heat transfer in a square open cavity with a vertical isothermal wall and remaining adiabatic walls considering variable fluid properties. The authors used nonBoussinesq approximation for an analysis. It has been found that the comparison with the Nusselt numbers obtained with the Boussinesq model indicates that the large differences correspond to the Boussinesq number (T /T∞ = 1.6) with values between 19.9 % for Ra = 107 and 40.2 % for Ra = 104 . Moreover, when the Boussinesq number is less than 0.3, the differences are kept below 10 %. Umavathi and Ojjela (2015) have studied the heat transfer and fluid flow inside the vertical rectangular duct filled with a Newtonian variable-viscosity fluid. The results show that the negative values of the viscosity variation parameter characterize the intense velocity contour in the left half region of the duct, while the positive values of viscosity variation parameter characterize the intense velocity contour in the right half region of the duct. Moreover, the velocity and temperature increase with the Grashof and Brinkman numbers in the upward direction. Alim et al. (2014) have investigated the combined influence of volumetric heat source and variable viscosity on natural convection flow along a wavy wall. It has been found that for the higher values of the viscosity variation parameter, one can find an intensification of the friction force at the wall and attenuation of the heat transfer rate. Parveen and Alim (2011) have analyzed an influence of the temperature-dependent viscosity on MHD natural convection along the hot vertical wavy wall. The authors have found that the local skin friction parameter and local heat transfer rate decrease with the increasing dimensionless magnetic parameter. Sivasankaran and Ho (2008) have investigated the effects of the temperature-dependent viscosity and conductivity of water near its density maximum in the presence of uniform magnetic field on the fluid flow and heat transfer. They found that the velocity and temperature profiles are distorted in the case of the influence of the temperature-dependent properties. Bondareva et al. (2013) have analyzed numerically and experimentally unsteady fluid flow and heat transfer of high-temperature silicate melt of temperature-dependent viscosity and heat capacity inside the melting furnace. Mathematical model formulated in the dimensionless stream function, vorticity and temperature has been carried out on the basis of finite difference schemes of the second order. It has been found that the formation of the recirculation zone near the inlet cross section at the initial stage is
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reflected by more qualitative melting of material supplied into the furnace and an increase in the heating time can be achieved by an extension of material stay in the furnace cavity. On the other hand, it should be pointed out that analysis of fluid flow and heat transfer in fluid-saturated porous media has a number of important applications in different industrial fields, for example thermal insulation buildings, nuclear energy systems, geothermal energy systems, solar power collectors, and many others (see Bejan 2013; Nield and Bejan 2013; Ingham and Pop 2005). There are quite few papers concerning the effect of the variable properties on fluid flow and heat transfer inside the porous cavities (see Blythe and Simpkins 1981; Guo and Zhao 2005; Hooman and Gurgenci 2008; Mehta and Sood 1992; Umavathi 2015; Chou et al. 2015). For example, Blythe and Simpkins (1981) have studied free convection inside a fluid-saturated porous layer using the integral relations in the case of the temperature-dependent viscosity. It has been found that an increase in the viscosity law parameter leads to an increase in the heat transfer rate and convective flow rate. Mehta and Sood (1992) have analyzed an influence of the temperature-dependent viscosity on the heat transfer and fluid flow along the hot vertical wall inside the porous medium using the Karman-Pohlhausen integral method. The authors have found that a decrease in viscosity increases the heat transfer rate. Temperature-dependent viscosity variation effect on Benard convection in a porous cavity has been investigated numerically by Hooman and Gurgenci (2008). The authors have used the Forchheimer–Brinkman–extended Darcy model and the second-order accurate central difference schemes. It has been shown that the reference temperature is a decreasing function of the Darcy number and is approximately independent of the other governing parameters. Guo and Zhao (2005) have studied natural convection in a differentially heated porous cavity saturated with a variable-viscosity fluid. Using the lattice Boltzmann method, they have realized the Forchheimer–Brinkman–extended Darcy model. It has been found that the flow structure is asymmetric for the variable-viscosity case and heat transfer is significantly enhanced in comparison with the case of the constant viscosity. However, these above-mentioned papers did not take into account unsteady regimes of natural convection inside the porous cavities using the Brinkman–extended Darcy model. The main purpose of the present paper is a numerical analysis of transient natural convection in a differentially heated porous cavity saturated with a fluid of temperature-dependent viscosity. It should be noted that the present work is an extension to the porous cavity of previous published paper by Hyun and Lee (1988) where the effect of variable viscosity on unsteady free convection in a square cavity filled with a pure fluid has been studied.
2 Basic Equations The physical model of transient natural convection in a porous cavity and the coordinate system are schematically shown in Fig. 1. It is assumed that the dimension in z direction is much longer that the other two and the end effects on the flow and heat transfer are negligible, i.e., fluid flow and heat transfer are two dimensional. The left and right surfaces of the cavity are isothermal walls having constant temperatures Th and Tc < Th , respectively, while top and bottom horizontal surfaces are adiabatic walls. Therefore, the domain of interest is a square differentially heated fluid-saturated porous cavity. It is assumed in the analysis that the viscosity of the fluid is varied with temperature (see Hyun and Lee 1988; Umavathi and Ojjela 2015), and the flow is laminar. The fluid is viscous, heat-conducting and Newtonian, and the Boussinesq approximation is valid. Further, it is assumed that the temperature of the fluid phase is equal to the temperature of the solid
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structure everywhere in the porous region, and local thermal equilibrium model is applicable in the present investigation. It is known that in the Darcy–Forchheimer model, a velocity square term in the momentum equations describing the inertia effect is more important for non-Darcy effect on the convective boundary layer flow over the surface of a body embedded in a high porosity media. However, we have neglected this term in the present study because we are dealing with the natural convection flow in a cavity filled with a porous medium (see Aleshkova and Sheremet 2010; Bejan 2013; Nield and Bejan 2013; Sheremet and Trifonova 2014a, b). Under these assumptions and with the Forchheimer’s inertia term neglected the governing equations for unsteady two-dimensional natural convection flow in the porous cavity using conservation of mass, momentum and energy can be written‘ as follows: ∂ u¯ ∂ v¯ + =0 ∂ x¯ ∂ y¯ 1 ∂ u¯ u¯ ∂ u¯ v¯ ∂ u¯ 2 ∂ ∂p ∂ u¯ ρ + 2 + 2 + =− μ¯ (T ) ε ∂t ε ∂ x¯ ε ∂ y¯ ∂ x¯ ε ∂ x¯ ∂ x¯ ∂ v¯ ∂ u¯ μ¯ (T ) 1 ∂ + u¯ μ¯ (T ) − + ε ∂ y¯ ∂ y¯ ∂ x¯ K 1 ∂ v¯ u¯ ∂ v¯ v¯ ∂ v¯ 2 ∂ ∂p ∂ v¯ ρ + 2 + 2 + =− μ¯ (T ) ε ∂t ε ∂ x¯ ε ∂ y¯ ∂ y¯ ε ∂ y¯ ∂ y¯ 1 ∂ ∂ v¯ μ¯ (T ) ∂ u¯ + + v¯ + ρgβ (T − T0 ) μ¯ (T ) − ε ∂ x¯ ∂ y¯ ∂ x¯ K 2 ∂T ∂T ∂ T ∂2T ∂T = km ρC p m + + ρC p f u¯ + v¯ ∂t ∂ x¯ ∂ y¯ ∂ x¯ 2 ∂ y¯ 2
(1)
(2)
(3) (4)
where x¯ and y¯ are the dimensional Cartesian coordinates; u¯ and v¯ are the velocity components along x¯ and y¯ directions, respectively; ρ is the fluid density; t is the dimensional time; p is the T −T0 pressure; μ¯ (T ) = μ0 · exp −C Th −Tc is the temperature-dependent dimensional dynamic viscosity; T is the dimensional temperature; T0 = 0.5 (Th + Tc ) is the initial temperature
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of the cavity; μ0 is the dynamic viscosity at initial temperature; K is the porous medium permeability; ε is the porosity of the porous medium; g is the gravitational acceleration; β is the thermal expansion coefficient; C p is the specific heat at a constant pressure; ρC p m is the overall heat capacity of the porous medium; ρC p f is the heat capacity of the fluid; and km is the overall thermal conductivity of the porous medium. As the purpose of the present work is the analysis of a thermal state of object of research, transformation of the formulated governing equations of mathematical physics (1)–(4) to the form eliminating direct search of the pressure field is represented to the most expedient. For this purpose, variables such as stream we ¯formulate mathematical model in dimensionless ∂ψ ∂ ψ¯ ∂ v¯ ∂ u¯ function u¯ = ∂ y¯ , v¯ = − ∂ x¯ , vorticity ω¯ = ∂ x¯ − ∂ y¯ and temperature. By using L as
2 the length scale, L/gβε 2 (Th − Tc ) as the
time scale, gβε (Th − Tc ) L as the velocity 2 3 scale, (Th − Tc ) as the temperature scale, gβε (Th − Tc ) L as the stream function scale,
gβε 2 (Th − Tc )/L as the vorticity scale and μ0 as the dynamic viscosity scale, the following dimensionless variables have been introduced: x = x/L ¯ , y = y¯ /L , τ = t gβε 2 (Th − Tc )/L, θ = (T − T0 )/(Th − Tc ), u = u/ ¯ gβε 2 (Th − Tc ) L, v = v/ ¯ gβε 2 (Th − Tc ) L, ¯ ¯ 0 ψ = ψ/ gβε 2 (Th − Tc ) L 3 , ω = ω¯ L/gβε 2 (Th − Tc ), μ = μ/μ The governing equations of convective heat transfer in dimensionless variables stream function – vorticity become (see Aleshkova and Sheremet 2010; Bondareva et al. 2013): ∂ 2ψ ∂ 2ψ + = −ω 2 ∂x ∂ y2 Pr ∂ 2 (μω) ∂ 2 (μω) μω ∂θ ∂ω ∂ψ ∂ω ∂ψ ∂ω + − = + + − 2 2 ∂τ ∂y ∂x ∂x ∂y Ra ∂x ∂y Da ∂x 1 ∂ψ ∂μ Pr 1 ∂ψ ∂μ ∂ 2 μ ∂ 2 ψ +2 + + Ra 2Da ∂ y ∂ y 2Da ∂ x ∂ x ∂ x 2 ∂ y2 ∂ 2μ ∂ 2ψ ∂ 2μ ∂ 2ψ + −2 2 2 ∂y ∂x ∂ x∂ y ∂ x∂ y 2 ∂ θ ∂θ ∂ 2θ ∂ψ ∂θ ∂ψ ∂θ 1 + + − = √ ∂τ ∂y ∂x ∂x ∂y ∂ y2 Ra · Pr ∂ x 2
(5)
(6) (7)
Here, Ra is the Rayleigh number; Pr is the Prandtl number; Da is the Darcy number; and μ is the temperature-dependent dimensionless dynamic viscosity; which are defined as ρgβ (Th − Tc ) L 3 μ0 K , Pr = , Da = , μ = exp (−Cθ ) (8) αm μ0 ραm εL 2 Here, αm = km / ρC p f is the overall thermal diffusivity. The initial and boundary conditions for the formulated problem (5)–(7) are as follows Ra =
τ = 0 : ψ = ω = θ = 0 at 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 ∂ 2ψ τ > 0 : ψ = 0, ω = − 2 , θ = 1 at x = 0 and 0 ≤ y ≤ 1 ∂x
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∂ 2ψ , θ = 0 at x = 1 and 0 ≤ y ≤ 1 ∂x2 ∂ 2 ψ ∂θ ψ = 0, ω = − 2 , = 0 at y = 0, 1 and 0 < x < 1 ∂y ∂y ψ = 0, ω = −
(9)
The physical quantity of interest is the average Nusselt number Nuavg defined as:
Nuavg
1 ∂θ = dy ∂ x x=0
(10)
0
3 Numerical Method The partial differential equations (5)–(7) with corresponding initial and boundary conditions (9) were solved by finite difference method (see Aleshkova and Sheremet 2010; Bondareva et al. 2013; Sheremet 2010, 2012) using the uniform grid. For the approximation of the convective terms, we used the difference scheme of the second order, and for the approximation of the diffusion terms, we used the central differences. The parabolic equations were solved on the basis of Samarskii locally one-dimensional scheme. The discretized equations were solved by Thomas algorithm. The equation for the stream function (5) was discretized by means of five-point difference scheme on the basis of central differences for the second derivatives. The obtained difference equations were solved by the successive over relaxation method. Optimum value of the relaxation parameter was chosen on the basis of computing experiments. The time step used here was chosen to be τ = 10−3 , which was also used in papers by Aleshkova and Sheremet (2010), Bondareva et al. (2013), Sheremet (2010, 2012). The accuracy of the numerical code developed by the authors was checked by preparing the benchmark solutions for natural convection with temperature-dependent viscosity in a square cavity filled with a pure fluid (see Hyun and Lee 1988). This benchmark solution was obtained when Da = ∞ in the partial differential equations (5)–(7). Figures 2 and 3 show a good agreement between the obtained streamlines and isotherms for Pr = 7.0 and different values of the Rayleigh number and the results by Hyun and Lee (1988). The comparison of the maximum values of the stream function (ψmax ) between the obtained results and data presented in the paper by Hyun and Lee (1988) is shown in Table 1. The method of solution was tested on different meshes. Figure 4 shows sensitivity of the average Nusselt number at the left hot wall and maximum absolute value of the stream function for Ra = 106 , Pr = 7.0, Da = 10−3 , C = 3.0. On the ground of optimization of calculation accuracy for 0 ≤ τ ≤ 300, the computational mesh such as 100 × 100 into the cavity has been selected for an investigation.
4 Results and Discussion Numerical study has been conducted at the following values of the key parameters: Rayleigh number (104 ≤ Ra ≤ 106 ), Darcy number (10−5 ≤ Da ≤ 10−3 ), Prandtl number (Pr = 7.0), viscosity variation parameter (0 ≤ C ≤ 6) and dimensionless time (0 ≤ τ ≤ 300). Particular efforts have been focused on the effects of these parameters on the fluid flow and heat transfer. Streamlines, isotherms, average Nusselt number at the left hot wall and
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Fig. 2 Comparison of streamlines ψ and isotherms θ at Ra = 3.5 × 104 : numerical results of Hyun and Lee (1988) (a), present study (b)
maximum absolute values of the stream function inside the cavity for different values of key parameters mentioned above are illustrated in Figs. 5, 6, 7, 8 and 9. Figures 5 and 6 show transient contours of the stream function ψ and temperature θ for Ra = 106 , Da = 10−3 , C = 0 and C = 3. It should be noted that presented illustrations in Figs. 5 and 6 reflect the transient behavior of the fluid flow and heat transfer. The effect of the dimensionless time on streamlines and isotherms in the case of C = 0 where we have natural convection inside a porous cavity saturated with a fluid of constant viscosity is presented in Fig. 5. At the initial time τ ≤ 3 due to the presence of horizontal temperature difference, one can find an evolution of hot and cold temperature waves close to the vertical isothermal walls. The governing heat transfer mechanism at the initial time is a heat conduction. Such penetration of temperature waves from vertical walls into the cavity leads to a formation of a convective cell with two cores of weak intensity. An appearance of such cores is caused by the temperature differences between isothermal walls and initial temperature of the cavity where Tc < T0 < Th . Moreover, these two cores are located along the borders between the zones of initial temperature and penetrative hot and cold temperature waves. Therefore, at
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Fig. 3 Comparison of streamlines ψ and isotherms θ at Ra = 3.5 × 105 : numerical results of Hyun and Lee (1988) (a), present study (b) Table 1 Comparison of maximum values of the stream function (ψmax ) between present results and data of Hyun and Lee (1988) for C = 3 Ra Ra = 3.5 × 104
Ra = 3.5 × 105
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τ
Data of Hyun and Lee (1988)
Obtained results
0.00004
0.006
0.0059
0.003
0.18
0.18
0.07
2.0
1.9
1.5
1.2
1.2 0.053
0.00004
0.051
0.003
1.5
1.54
0.02
6.0
6.0
1.5
2.15
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Fig. 4 Variation of the average Nusselt number (a) and maximum absolute value of the stream function (b) versus the dimensionless time and the mesh parameters
τ = 3, these cores are located close to vertical walls. An increase in time up to τ = 12 (Fig. 5b) leads to a formation of hot convective wave near the left wall and cold convective wave near the right wall. The observed distortions of the isotherms close to the left upper corner and right bottom corner characterize a displacement of the convective cores to the central part of the cavity with more intensive circulation. At time τ = 12, the locations of vortices are defined by the distortion of isotherms θ = 0.1 and θ = −0.1. Further increase in the dimensionless time in the range τ ≤ 30 reflects an intensification of convective flow, heating of the left and top parts of the cavity and cooling of the right and bottom parts of the cavity. At the same time, one can find an approach of convective cores in the central part of the enclosure. That is defined by the heating and cooling of the domain of interest. At time τ = 60 (Fig. 5e), one can find a core coalescence of less violently circulation in comparison with previous time moments. Also, this time moment is characterized by a formation of single vortex in the central part of the cavity elongated along the horizontal axis. Further increase in the dimensionless time τ > 60 leads to a stratification of the flow and temperature with a slight counterclockwise rotation of the convective core. High-density distributions of isotherms close to the left bottom and right upper corners characterize a formation of thermal boundary layers along the vertical isothermal walls. These mentioned corners are the origins of ascending and descending boundary layers close to the left and right walls, respectively. Streamlines and isotherms in the case of C = 3 where we have natural convection inside a porous cavity saturated with a variable-viscosity fluid are presented in Fig. 6. These fluid flow structure and thermal fields differ essentially from ones presented in Fig. 5. At the initial time τ = 3 (Fig. 6a), one can find also a formation of two temperature waves close to the vertical walls like in Fig. 5a, but penetration zones of these waves are different. The latter leads to an appearance of a convective cell with a single core located near the left hot wall. An increase in the dimensionless time leads to more intensive evolution of the left temperature wave with a single convective core near the hot wall. The main reason for such behavior is the effect of the viscosity law [Eq. (8)]. Taking into account this law, it is possible to conclude that an increase in the temperature leads to a decrease in the viscosity (Pr = 7.0) that characterizes more intensive motion of hot fluid. At the same time, a decrease in the temperature leads to an increase in the dynamic viscosity that reflects in an attenuation of the convective flow.
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Fig. 5 Streamlines ψ and isotherms θ at Ra = 106 , Da = 10−3 , C = 0: τ = 3 (a), τ = 12 (b), τ = 21 (c), τ = 30 (d), τ = 60 (e), τ = 120 (f), τ = 300 (g)
Therefore, we have intensive flow close to the hot wall and weak flow near the cold wall. The steady-state regime (Fig. 6g) reflects a formation of convective cell with a single core close to the left hot wall elongated along the horizontal axis and slightly counterclockwise rotated. The variable-viscosity fluid essentially heats up the cavity taking into account the position of the isotherm θ = 0 in comparison with the fluid of constant viscosity (Fig. 5g). The main reason for such results is a weakening of the viscous force for the hot fluid and an enhancement of this force for the cold fluid. Figure 7 shows the effect of the viscosity variation parameter on the streamlines and isotherms for Ra = 106 , Da = 10−3 , τ = 300. Regardless of the viscosity variation parameter value, one can find inside the cavity a single-core convective cell with a stratified temperature field in the central part of the cavity. The core is located close to the left hot wall as has been mentioned above. Convective cell characterizes a formation of ascending flow close to the left wall and descending flow close to the right cold wall. An increase in the viscosity variation parameter leads to an intensification of convective flow. One can find an increase in
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Fig. 6 Streamlines ψ and isotherms θ at Ra = 106 , Da = 10−3 , C = 3: τ = 3 (a), τ = 12 (b), τ = 21 (c), τ = 30 (d), τ = 60 (e), τ = 120 (f), τ = 300 (g)
the maximum absolute value of the stream function with C presented in Fig. 8b. Moreover, an increase in C characterizes a decrease in the cold temperature wave penetration rate and as a result more intensive heating of the cavity. The former is confirmed by the location of the isotherm θ = −0.4 for different values of the viscosity variation parameter, while the latter is confirmed by the location of the isotherm θ = 0 for different values of C. It should be noted that for high values of the viscosity variation parameter (C ≥ 4), one can find a formation of local distortion of streamlines and isotherms close to the left upper corner that can be explained by a decrease in the boundary layer thickness. The effects of the dimensionless time and the viscosity variation parameter on the average Nusselt number at the hot vertical wall and the maximum absolute value of the stream function are presented in Fig. 8. An increase in the viscosity variation parameter leads to an increase in the heat transfer rate and fluid flow intensity. The effect of the dimensionless time on these integral parameters is non-monotonic due to a formation of three time intervals taking into account the distributions of the average Nusselt number. The first time interval characterizes a
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Fig. 7 Streamlines ψ and isotherms θ at Ra = 106 , Da = 10−3 , τ = 300: C = 1 (a), C = 2 (b), C = 4 (c), C = 6 (d)
Fig. 8 Variation of the average Nusselt number (a) and maximum absolute value of the stream function (b) with dimensionless time and viscosity variation parameter for Ra = 106 , Da = 10−3
decrease in the average Nusselt number due to a conductive heating of the surrounding region close to the left vertical wall. The second time interval illustrates an increase in Nuavg due to an intensification of convective flow and after that one can find a formation of a steady-state regime. It is worth noting that for high values of the viscosity variation parameter (C ≥ 6), the average Nusselt number decreases monotonically. An increase in C also leads to both an increase in the global minimum value of Nuavg and a decrease in the dimensionless time period for an approach to the maximum value of the function |ψ|avg (τ ). Figure 9 shows the effect of the viscosity variation parameter, Rayleigh and Darcy numbers on the average Nusselt number at the left hot wall. As has been mentioned above, an increase in C and Ra leads to an increase in the heat transfer rate. Moreover, for high values of the Rayleigh number, an increase in the viscosity variation parameter leads to more intensive increment of Nuavg . In the case of the porous cavity, an increase in the Darcy number leads
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Fig. 9 Variation of the average Nusselt number with viscosity variation parameter and Rayleigh number for Da = 10−3 (a) and with viscosity variation parameter and Darcy number for Ra = 106 (b)
to an increase in the average Nusselt number. An increment of the average Nusselt number with C is more intensive for high values of Da. In the case of pure fluid (Da = ∝ in Fig. 9b), an increase in the viscosity variation parameter leads to a decrease in Nuavg .
5 Conclusions The unsteady natural convection in a differentially heated square porous cavity of a fluid with a temperature-dependent viscosity has been analyzed in dimensionless stream function, vorticity and temperature using a second-order accurate finite difference method. The enclosure has the hot left and cold right walls, and adiabatic horizontal walls. Distributions of streamlines, isotherms, average Nusselt number and maximum absolute value of the stream function in a wide range of governing parameters have been obtained. Based on the findings in this study, we conclude the following: 1. The distributions of streamlines and isotherms in dependence on the dimensionless time evidently show the transient intervals of a formation of the steady-state regime. The average Nusselt number and maximum absolute value of the stream function are the non-monotonic functions of the dimensionless time. 2. An increase in the viscosity variation parameter leads to an intensification of convective flow and heat transfer and a formation of a single-core convective cell for the porous cavity. Also an increase in C characterizes a decrease in the cold temperature wave penetration rate and as a result more intensive heating of the cavity. In the case of the pure fluid (Da = ∝), an increase in the viscosity variation parameter leads to a decrease in the heat transfer rate. 3. An increase in the Rayleigh and Darcy numbers for the porous cavity leads to an increase in the heat transfer rate. Acknowledgments This work of M.S. Astanina and M.A. Sheremet was conducted as a government task of the Ministry of Education and Science of the Russian Federation, Project Number 13.1919.2014/K. The authors also wish to express their thank to the very competent Reviewers for the valuable comments and suggestions.
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