An Unsteady Mixed Convection in a Driven Cavity Filled with Nanofluids Using an Externally Oscillating Lid ()
1. Introduction
Over the past several years, natural and mixed convection in a cavity filled with fluid has received a remarkable attention because of its wide and practical applications in real world engineering concerns [1-10]. These concerns can be observed in various fields such as cooling of electronic packages, solar energy collectors, ventilation and heating of living space and cooling of nuclear reactors (as an internal heat generator example) [11,12]. The first investigation about natural convection was probably conducted by G. De Vahl Davis et al. [13]. They studied flow and heat transfer in a cavity by considering different aspect ratios. Fusegi et al. [14] presented a numerical solution by solving natural convection governing equations at different Prandtl numbers to investigate heat transfer characteristics under various aspect ratios of a rectangular enclosure. Although many studies focus on rectangular or square enclosures, many numerical works have been done on wavy enclosures in order to investigate the effects of wavy walls on heat transfer capacity. Oztop et al. [15] numerically studied the effects of volumetric heat sources on natural convection heat transfer and flow structures in a wavy-walled enclosure. They showed that by augmenting the wave amplitude of the wavy wall, the rate of heat transfer would be higher. In mixed convection problems, the parameters that effect heat transfer are Gr, Re, and Pr (Rayleigh and Pr in natural convection). Gr and Re are usually simplified as a single parameter named Ri. It is obvious that variation in Ri number will be obtained by changing Gr and Re number either together or separately. It has been reported that the Nusselt number in a lid-driven cavity increases with augmentation of Pr while other effective parameters such as Re and Gr are kept constant [16]. For a fixed Pr, the Nusselt number increases with heightening Gr or Re while Re or Gr is kept constant respectively [17,18]. Sharifi [17] numerically studied laminar mixed heat convection in two-dimensional shallow rectangular driven cavities of aspect ratio 10. He showed that a primary vortex is observed at the right-hand side where the shear driven flow by the top moving lid is impacted on the right-hand sidewall and is forced to move downward along the sidewall. Some minor vortices are also formed in the cavity for Ri < 1. The local Nusselt number at the heated moving lid starts with a high value and decreases rapidly and monotonically to a small one towards the right side. Many cavity problems studied in literature have constant wall temperature accompanied by insulated walls (vertical or horizontal walls) [19,20]; these conditions can only be obtained if evaporation or condensation takes place on the wall’s surfaces (on constant temperature walls), but in real problem reaching these aims is a bit far from reality; in order to have a good control on the heat flux, we can consider a time-dependent driven-lid or mixed convection as a useful tool in our design’s effectual parameters. However, many recent studies of the lid-driven cavity flows are focused mainly on the problems with a constant velocity lid. Only a few works were involved with a vibrating lid. Nishimura and Kunitsugu [21] performed numerical investigation of the fluid mixing and mass transfer in two-dimensional rectangular cavities affected by the time-periodic lid velocity. Soh and Goodrich [22] outlined the basic transient flow features relevant to this problem using a time accurate finite-difference scheme. Moreover, Iwatsu et al. [23] performed a numerical investigation on the effect of external excitation on the flow structure in a square cavity. The results of their research have shown similar flow structure to steady driven-cavity flows when exerting small frequency values. A subsequent work by Iwatsu et al. [24] investigates a numerical study of the viscous flow in a heated driven-cavity under thermal stratification, where the oscillating lid was maintained at a higher temperature than the lower wall. The results of their work revealed substantial enhancement in the heat transfer rate at particular lid frequency values. Khanafer et al. [25] studied an unsteady laminar mixed convection heat transfer in a lid driven cavity. They found that the Reynolds number and Grashof number either increase or impede the energy transport process, and relatively small lid oscillation values were found to have more penetration force inside the cavity. Another concern about cavity problems relies on low thermal conductivity of fluid used in enclosures. Thus, the heat transfer capacity through such systems is relatively limited, and it is of importance to improve it from the industrial and energy concern point of view. To overcome this problem using nanosized particles able to dissolve in base flow has attracted tremendous attentions recently [26-28]. The way these particles enhance heat transfer capacity is that metallic nano-sized particles have high thermal conductivity comparing to base flow, so the mixture of nano-sized particles and base flow has high convective and conductive capacity. Therefore it is enriched with both solid and fluid properties. According to what stated about nanofluid advantages, many studies and investigations both numerically and experimentally have been done in recent years considering nanofluids. The first study concerning natural convection of a nanofluid confined in a differentially heated enclosure seems to be done by Khanafer et al. [29]. A comparative study of different models based on the thermophysical properties of copper-water nanofluid was developed and investigated. Results obtained from their numerical investigation indicate that the suspended nanoparticles significantly augment the heat transfer rate at any given Grashof number. A heat transfer correlation of the average Nusselt number for various Grashof numbers and volume fractions was proposed by the authors. Mahmoudi et al. [30] applied a model to simulate the natural convection utilized by two separate heat sources in an open cavity subjected to the copperwater nanofluid. They use a model developed by Brinkman [31] to determine mixture viscosity. They showed that the Nusselt number increases due to augmentation in Rayleigh number, the distance between two heat sources, and spacing from the wall. They also showed that the average Nusselt number increases linearly with the solid volume fraction increment of nanoparticles.
The main objective of the present paper is to investigate flow and heat transfer characteristics of an unsteady laminar mixed convection heat transfer in a cavity filled with nanofluid that is insulated from below and top walls and heated and cooled by exerting hot and cold constant temperature on vertical walls. The paper is organized as follows. In the next sections, the mathematical formulation of the problem is presented. Numerical solution and algorithm validation are then proposed. The effects of varying the Reynolds number, Grashof number, the lid frequency, and solid volume fraction on streamlines, isotherms, and temporal average Nusselt number will be illustrated and discussed. The last section contains some concluding remarks.
2. Mathematical Formulation
An unsteady, periodic, laminar, incompressible mixed convection in a two-dimensional square cavity filled with Newtonian nanofluids is considered for the present study. Physical dimensions are given in Figure 1. Both top and bottom walls are assumed to be adiabatic meanwhile other two vertical sidewalls are kept at constant temperature (hot temperature at the left wall and cold temperature at the right one). The top wall of the cavity is allowed to oscillate with a velocity u = u0 sinωt in its own place longitudinally, where u0 is the maximum
Figure 1. A schematic diagram of the physical model.
oscillating speed of the lid and ω is the oscillating frequency.
The thermo-physical properties of the nanofluids, considered in this study, given in Table 1, are assumed constant except for the density in the buoyancy force which varies linearly with temperature as
according to Boussinesq approximation, where 
is the thermal expansion coefficient of nanofluid and subscript 0 denotes the reference state. It is also assumed that both the base flow and nanoparticles are in thermal equilibrium and no slip occurs between them. The shape and size of solid particles are assumed to be uniform and the diameter of them to be equal to 100 nm. The no-slip condition is imposed at the solid walls. It is further assumed that viscous dissipation is neglected in this study. Based on the abovementioned considerations, the unsteady governing equations for conservations of mass, momentum and energy are expressed in the following format:
(1)
(2)
(3)
(4)
Table 1. Thermophysical properties of water and copper nanoparticles.
The effective density of nanofluid can be determined as:
(5)
where 
and 
are nanofluid density, density of base fluid, density of nanoparticles, and solid volume fraction of the nanoparticles, respectively.
The thermal diffusivity of the nanofluid is:
(6)
where the heat capacitance of the nanofluid can be given as:
(7)
nanofluid thermal expansion coefficient can be determined by:
(8)
The effective thermal conductivity of nanofluid was suggested by Patel et al. [32] as follows:
(9)
where c is constant and must be determined experimentally, according to Hemanth [33] 
and Pe are defined as:
(10)
(11)
where dp is the diameter of nano-particles (in this study 100 nm) and df is the molecular size of base fluid (2
for water). up is the Brownian motion of particles given by [34]:
(12)
where kb is the Boltzmann constant.
The effective dynamic viscosity of the nanofluid was introduced by Brinkman as follows:
(13)
We can rewrite governing equations as non-dimensional form as below:
(14)
(15)
(16)
(17)
Here, the non-dimensional variables are: U the Xcomponent velocity, V the Y-component velocity, θ the temperature, and τ the time. Gr is the Grashof number
, Re is the Reynolds number
, and Pr is the Prandtl number
. The dimensionless variables in the above equations are defined as:
(18)
For
, the stagnation state 
for temperature and flow fields is used as an initial condition for the sinusoidal-oscillating lid in this investigation. Meanwhile, the used dimensionless boundary conditions are summarized as follows:
(19)
(20)
(21)
(22)
The time history of the dimensionless heat flux is computed through calculating the average Nusselt number along both the hot and cold sidewalls using:
(23)
3. Numerical Solution
The solution of the governing equations along with the initial and boundary conditions is obtained using a control volume approach and SIMPLE algorithm with a staggered grid. The hybrid scheme is used to evaluate the flow, heat and mass fluxes across each of the control volume boundaries. In this study a non-uniform grid mesh which is finer near the boundaries is used, in order to have better accuracy in final results. An implicit scheme is employed to deal with the time differential terms. A line-by-line SOR method with relaxation is used in conjunction with iterations to solve the nonlinear discretized equations. The convergence of solutions is assumed when the relative error for each variable between consecutive iterations is recorded below the convergence criterion ε such that:
(34)
where n is the number of iteration index and
. The convergence criterion was set to 10–5.
4. Algorithm Validation
A non-uniform mesh size system of 52 × 52 is put into practice in the present study with finer mesh near the cavity walls in order to capture accurate results. Table 2 shows the number of grid points and their influence on the maximum temporal average Nusselt number while lid moves at its maximum speed. First, the present numerical solution is verified against two well-known numerical studies. Namely, the numerical solutions of natural convection in a cavity reported by De Vahl Davis [13] and N. Massarotti et al. [35]. The findings of the comparisons are written down in Table 3 for the bounds on the magnitude of the average Nusselt number predictions. Both comparisons illustrate close proximity in the predictions made between the various solutions. Another fold for validation of our code is comparison between the predicted stream function contours under steady state pure fluid mixed convection of the present work to that of F. Talebi et al. [36]. As displayed in Figure 2, the comparison strikes a superb concurrence between both studies.
5. Results and Discussion
As stated earlier, the overall objective of the current study is to survey a time dependent two-dimensional
Table 2. Influence of number of grid for a test case at Re = 100, Gr = 106, f = 0.0, Ω = π/30.
Table 3. Comparison of the average Nusselt number at the hot wall between the present solution and other solutions.
(a)
(b)
Figure 2. Comparison of the steady state mixed convection stream function between the present work (a) and Talebi [36] (b) for Re = 100, Gr = 2.1 × 105.
mixed convection in a lid-driven cavity utilized with nano-particles according to various Reynolds number
, Grashof number
, lid oscillation frequency
, and solid volume fraction
. In this part, all discussion is based on focusing on the hot wall due to the same periodic trend on the cold wall. In order to have a better comprehension of the average Nusselt number variation, average Nusselt number at the steady state mixed convection inside the cavity is written in Table 4 for both cases of u = 1, u = –1, and numerous Reynolds and Grashof numbers. According to Table 4, by increasing Grashof number, average Nusselt number tends to be equal in both cases.
Figure 3 shows the impact of variations of Reynolds number on the temporal average Nusselt number along the hot wall while Grashof number and non-dimensional lid frequency are fixed at 105(a), 106(b) and π/30 respectively.
As illustrated In Figure 3(a), at relatively high Reynolds numbers
, the average Nusselt number has two maximums and two minimums in each time period. The greater maximum occurs whilst lid is at its highest negative speed value, because at this moment thermal boundary layer produced by buoyancy force will be disturbed by the flow motivated by lid about upper
Table 4. Steady state average Nusselt number according to Reynolds and Grashof number variation for both case of lid velocity (u = ±1).
(a)
(b)
Figure 3. Variation of the average Nusselt number at different Grashof and Reynolds numbers: (a) Gr = 105, Ω = π/30, f = 0%; (b) Gr = 106, Ω = π/30, f = 0%.
half part of the hot wall. The inferior maximum takes place at maximum positive speed value of lid due to forced and free convection operation in the same direction. Although in this case these two forces operate in the same direction and enhance heat transfer on the hot wall, the effect of boundary layer disturbance caused by negative speed is more effectual that results in higher maximum Nusselt number. When lid stops in order to switch its velocity from positive to negative value and vice versa, two minimums come into sight due to lack of forced convection. As it is evident in Figure 3(a), by increasing Reynolds number, the temporal average Nusselt number will follow the same periodic behavior, but the difference between two maximums becomes more remarkable. It can be deduced that in high Reynolds numbers, thermal boundary layer disturbance turn in to be more outstanding. Figure 4 shows streamlines and isotherms for various velocities equal to 1, 0+, –1, and 0– at Re = 100 and fixed value of Gr = 105, and Ω = π/30. As it can be seen in Figure 4(a), natural convection and forced convection effects are accumulated and a big vortex is created inside the domain. This vortex makes up a narrow thermal boundary layer near upper part of the cold wall. On the contrary, thermal boundary layer stretches near upper part of the hot wall and causes a reduction of the local Nusselt number over there, but an increase in the average Nusselt number yields on the hot wall (inferior maximum) due to overall reduction of thermal boundary layer thickness along the hot wall. When the lid gradually decelerates to become stagnant, it can be observed in Figure 4(b) that natural convection is dominant inside the cavity due to lack of adequate forced convection. In
this figure streamlines and isotherms are approximately similar to the case of pure natural convection. As lid accelerates in the negative direction, the primary vortex is found to break up into two vortices and a secondary vortex appears and occupies about half of the cavity. This phenomenon disturbs the thermal boundary layer near upper parts of the hot wall and increases the average Nusselt number. A careful observation indicates that thermal boundary layer becomes thinner at upper part of the hot wall whereas the region where two vortices cannot cover (the magnitude of stream function is equal to zero), this layer will become thicker and local Nusselt number will reduce. But the overall effect is in the way of augmentation of the average Nusselt number. It should be noted that in the steady state condition the opposite behavior will be observed, in other word, the greater average Nusselt number is at u = 1 and the lesser one at u = –1 (Table 4). While negative speed value of lid becomes zero, it is expected to observe the same flow and thermal field like Figures 4(c) and (d), but Figures 4(g) and (h) shows that streamlines and isotherms are still under manipulation of negative speed value of lid due to time lapse that is needed to completely fade away the effects of the negative lid velocity inside the domain. It can be seen that the region near hot wall of the cavity can adjust itself sooner than the region near cold wall; it manifests that buoyancy force and forced convection do better and add to each other faster near hot wall.
Figure 3(b) shows the impact of variation of Reynolds number on the temporal average Nusselt number along the hot wall at Grashof number equal to 106 as well. Due to greater Grashof number, the average Nusselt number would be greater. The same behavior can be observed in Figure 3(b), but a precise observation reveals that at Re = 100, greater maximums occurs while lid has maximum positive speed value and inferior one happens at maximum negative speed value. It is true that lid negative motion disturbs thermal boundary layer near hot wall, but as Grashof number rises up, the strengthen of buoyancy force will become greater, and disturbance will be limited to rather small area near upper part of the hot wall (Figures 5(c) and (d)) and cannot heighten the Nusselt number to a great extent. In that case it can be uttered that although disturbance caused by secondary vortex can enhance heat transfer rate in general, but the vicinity impinged by this vortex is also of great importance.