Mixed convection of Al2O3–H2O nanoliquid in a square chamber with complicated fin

International Journal of Mechanical Sciences 165 (2020) 105192

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Mixed convection of Al2 O3 –H2 O nanoliquid in a square chamber with complicated fin Elena V. Shulepova a, Mikhail A. Sheremet a,b,∗, Hakan F. Oztop c,d, Nidal Abu-Hamdeh d a

Department of Theoretical Mechanics, Tomsk State University, 634050, Tomsk, Russian Federation Laboratory on Convective Heat and Mass Transfer, Tomsk State University, 634050, Tomsk, Russian Federation c Department of Mechanical Engineering, Technology Faculty, Firat University, Elazig, Turkey d Department of Mechanical Engineering, King Abdulaziz University, Jeddah, Saudi Arabia b

a r t i c l e

i n f o

Keywords: Adiabatic fin Differentially heated chamber Finite difference method Local adiabatic block Mixed convection Moved upper wall Nanofluid

a b s t r a c t Intensification of energy transport in different engineering devices is the challenge for many scientists and engineers. This problem can be solved by usage of various fins within the device and effective heat-transfer agent. The present study includes a combination of these opportunities for the heat transfer enhancement. Convective energy transport within a square enclosure having a mounted adiabatic fin and internal solid block under the influence of moving upper border and alumina/water nanoliquid has been studied. Vertical borders of the cavity are isothermal, namely, the left wall is heated, while the right one is cooled. Governing equations written using the non-primitive variables have been worked out by the finite difference technique. The special computational algorithm has been employed for the determination of the stream function magnitude at internal block taking into account the doubly connected region. Influences of the Rayleigh and Reynolds numbers, location of the internal adiabatic block and nanoparticles volume fraction on liquid flow and energy transport are studied. It has been revealed that the internal block location and nanoparticles concentration can control the intensity of heat transfer.

1. Introduction Control of energy transport and liquid circulation in both closed cavities and lid-driven cavities are important due to energy saving issue in engineering problems such as cooling of electronical equipments, drying process, humidification or different types of heat exchangers. Also, application of nanofluids is an important and novel subject as an alternative way to manage thermal transmission and liquid motion [1,2]. Thus, Balootaki et al. [3] worked on nanoscale LBM to make simulation for mixed convective energy transport in an air lid-driven cavity under the impact of rectangular shaped obstacle inside the cavity. They found that for Ri = 1 an intensive cell was formed in the top half of the cavity. Sheikholeslami et al. [4] illustrated a study on 3D analysis of nanofluid filled convective energy transport around elliptic obstacle in lid-driven cavity under magnetic field. They noticed that the obstacle can be employed as a managing element for mass, momentum and energy flow with magnetic influence. Ismael et al. [5] evaluated mixed convective energy transport inside a lid-driven chamber under the influence of partial slip. They found a critical condition which reflects a suppression of the convective flow and thermal transmission. Chamkha and Abu-Nada [6] numerically investigated mixed convective energy transport within

∗

single- and double-lid driven alumina-water chambers under analysis of the effect of different viscosity models. They revealed that an addition of solid nanoparticles can intensify the energy transport in the case of moderate and high Richardson numbers. Abu-Nada and Chamkha [7] simulated the mixed convection of water-based nanoliquid within a lid-driven region under the influence of bottom wavy cooled surface. It was ascertained that an inclusion of nanoparticles can enhance the thermal transmission for all Richardson numbers and bottom surface geometry parameters. Selimefendigil et al. [8] conducted a 3D numerical study on mixed convective thermal transmission. In their work, flexible bounded trapezoidal lid-driven chamber with nanoliquids was examined by using the finite element technique and found that this flexible side surface can be employed as managing factor for energy transport and liquid motion. Taghizadeh and Asaditaheri [9] performed a computational work on laminar energy transfer in an inclined lid-driven region. They inserted a circular porous cylinder into cavity to control the thermal transmission and entropy production. They observed that the proportion of the irreversibilities due to the liquid friction and energy transport varies with flow parameters. Mehmood et al. [10] made an introduction study on mixed convection in nanoliquid lid-driven enclosure with hot square obstacle under constant temperature. Gangawane

Corresponding author at: Department of Theoretical Mechanics, Tomsk State University, 36 Lenin Avenue, 634050 Tomsk, Russian Federation. E-mail address: [email protected] (M.A. Sheremet).

https://doi.org/10.1016/j.ijmecsci.2019.105192 Received 5 August 2019; Received in revised form 22 September 2019; Accepted 24 September 2019 Available online 24 September 2019 0020-7403/© 2019 Elsevier Ltd. All rights reserved.

E.V. Shulepova, M.A. Sheremet and H.F. Oztop et al.

Nomenclature Roman letters cp specific heat d dimensional distance between the wall fin and inertnal adiabatic block g gravitational acceleration h dimensional thickness of internal adiabatic block 𝑁𝑢 special functions k thermal conductivity L length and height of the square chamber l length of the internal adiabatic block Nu local Nusselt number 𝑦̄ = const average Nusselt number 𝑝̄ dimensional pressure Pr Prandtl number Ra Rayleigh number Re Reynolds number Ri Richardson number T dimensional temperature t dimensional time Tc cold wall temperature Th hot wall temperature u, v dimensionless velocity components 𝑢̄ , 𝑣̄ dimensional velocity components V0 dimensional velocity of the upper moved wall x, y dimensionless Cartesian coordinates 𝑥̄ , 𝑦̄ dimensional Cartesian coordinates Greek symbols 𝛽 thermal expansion coefficient 𝛿 dimensionless distance between the wall fin and internal adiabatic block 𝜃 dimensionless temperature 𝜇 dynamic viscosity 𝜌 density 𝜌cp heat capacitance 𝜌𝛽 buoyancy coefficient 𝜏 dimensionless time 𝛾 value of the stream function at internal block surface 𝜙 nanoparticles volume fraction 𝜓̄ dimensional stream function 𝜓 dimensionless stream function 𝜔̄ dimensional vorticity 𝜔 dimensionless vorticity Subscripts c f h nf p

cold fluid hot nanofluid (nano) particle

et al. [11] used the triangular block to control heat transfer characteristics within a lid-driven region. They observed that energy transfer is demonstrated in the form of tables for various Richardson number magnitudes. There are optimal values for triangular bodies’ location. Also, Gangawane and Manikandan [12] made a similar work to see the effects of this heated triangular block. Nasrin et al. [13] considered numerically mixed convective nanofluid energy transport within a double lid-driven region under the influence of internal power generation. They showed that a rise of the Richardson number and nanoparticles concentration have an essential impact on nanoliquid circulation and energy transport. Sheremet and Pop [14] examined numerically laminar mixed convective energy and mass transport in a square nanoliquid enclosure with mov-

International Journal of Mechanical Sciences 165 (2020) 105192

ing horizontal borders of constant temperature and thermally-insulated vertical surfaces. They found that a variation of the Brownian diffusion and thermophoresis parameters does not leads to essential modification of streamlines and temperature fields. Billah et al. [15] investigated the heat transport in a lid-driven domain including a hot cylinder. They observed that the mass and energy transfer depend on the geometrical parameters of this internal body and it makes an essential impact on the velocity field inside the region. Hussain et al. [16] used the finite element technique to study the control equations under magnetic field to simulate the mixed convective circulation and energy transport in a nanofluid filled double lid-driven enclosure. They also considered the volumetric energy production or absorption. Sheikholeslami [17] made a work on magnetic influience for forced convection of nanofluid inside a porous lid-driven region. A square obstacle is used in the study. The numerical analysis has been performed via Lattice Boltzmann Method. The obstacle can be employed as a managing characteristic for mass and energy flow as well as temperature gradient. Karbasifar et al. [18] studied the mixed convection of nanofluid in a lid-driven chamber with a hot elliptical centric cylinder and they revealed that a rise of the temperature difference between the cold borders and the cylinder surface at a constant Ri number, nanoparticles concentration and region inclination angle increases Nu and energy transport in such a way that the rising trend of Nu depends on nano-sized particle concentration, liquid velocity, and enclosure tilted angle. Shi and Khodadadi [19] performed a study by using a homemade code with finite volume method on laminar liquid circulation and energy transport in a lid-driven chamber. In their case, a thin obstacle is located inside the cavity. Solid circular cylinder is used to manage energy transport and liquid circulation in a wavy walled double-sided lid-driven region by Alsabery et al. [20]. Boraey [21] studied the different techniques for improving the mixing within a square lid-driven enclosure by proper modification of the geometric configuration by using the Multiple-Relaxation-Time Lattice Boltzmann Method (MRTLBM) and found that the position and shape of the main cavity are highly sensitive to the flow Reynolds number while two lower side vortices are not affected by the change of the Reynolds number or the presence of the obstacle compared to the standard lid-driven cavity case. Esfe et al. [22] conducted an analysis on combined convection due to lid-driven cavity having an obstacle. They investigated the effects of nanofluid variable properties. They used finite volume method and found that the thermal transmission is augmented as the size of obstacle on the bottom border increases. Golkarfard et al. [23] calculated a problem for sedimentation of solid particles in a lid-driven chamber with inner hot blocks. Their obtained results prooved that increasing of obstacles distance, then, deposition increases and the size of blocks has a huge influence on particle sedimentation. Other interesting and useful results on convective circulation and energy transport can be found in [24–29]. Hammami et al. [30] performed a computational study on the twosided lid-driven 3D chamber due to the cylinder at the cavity center. They observed that as the Reynolds number rises up to 1500, the moving parallel lids produce a vortex in the rear surfaces of the chamber back the cylinder. Selimefendigil [31] inserted an elliptic cylinder into a lid-driven nanoliquid region. He found that the energy transport is intensified essentially which is about 120.20% for single wall carbon nanotube-water nanoliquid at 𝜙 = 0.06. Munshi et al. [32] conducted a computational study on mixed convection in a lid-driven enclosure including elliptic shaped hot obstacle. They also considered the magnetic influence on this system. Kareem and Gao [33] investigated the turbulent mixed convection in a 3D lid-driven enclosure having a rotating cylinder. They indicated that only the LES technique can forecast the structure elements of the secondary vortices that have strong effects on the energy transport in the domain. Selimefendigil and Oztop [34] solved the natural convection problem in a nanofluid filled cavity with different shaped obstacles installed under the influence of a uniform magnetic field and uniform heat generation. Some interesting and

E.V. Shulepova, M.A. Sheremet and H.F. Oztop et al.

useful results for convective energy transport within lid-driven cavities can be found in [35–40]. Mehrizi et al. [41] worked on the mixed convection in a ventilated cavity with hot obstacle. Doustdar and Yekani [42] made a study on mixed convection of nanofluid in a lid-driven cavity. They inserted a heated body into cavity. They found that the parameters of the heated obstacle affect the heat transfer and fluid flow. Also, fundamentals and detailed applications of the lid-driven cavity in engineering are demonstrated by Khulmann and Romano [43]. Oztop et al. [44] performed a solution to observe the fluid flow due to combined convection in lid-driven enclosure in the presence of a circular body to control heat transfer. The objective of this study is a mathematical modeling of convective energy transport in an enclosure with moving upper adiabatic wall and vertical isothermal walls under the impacts of alumina-water nanoliquid and complicated fins (intermittent fins) within the cavity. We employed the relations for the nanoliquid properties based on the experimental study. To our best of knowledge there are no any papers published before on square nanofluid region with discrete complex fins. Therefore the novelty of the present work is an analysis of convective energy transport within a chamber with internal complicated fin that consists on the wall-mounted block and internal adiabatic body. Such approach allows to split the convective circulation inside the chamber and to intensify the thermal transmission.

International Journal of Mechanical Sciences 165 (2020) 105192

Fig. 1. A sketch of the frame.

Table 1 Physical parameters of the host liquid and solid particles [35–38,46]. Physical properties

Host liquid (water)

Al2 O3

cp (J × kg−1 × K−1 ) 𝜌 (kg × m−3 ) k (W × m−1 × K−1 ) 𝛽 × 10−5 (K−1 )

4179 997.1 0.613 20.7

765 3970 40 0.846

2. Mathematical description The physical pattern of energy transport in a square chamber filled with Al2 O3 –H2 O nanoliquid is presented in Fig. 1. The considered region is the nanoliquid-filled differentially heated chamber (see Fig. 1) with an adiabatic protuberance on the bottom wall and an adiabatic block within the cavity under the effect of moving upper wall. Borders 𝑢 =

𝜕𝜓 𝜕𝑣 𝜕𝑢 , 𝑣 = − 𝜕𝜓 , 𝜔 = 𝜕𝑥 − 𝜕𝑦 are considered to be adiabatic, while the left 𝜕𝑦 𝜕𝑥 boundary is kept at constant high temperature Th and the right border is kept at constant low temperature Tc . The nanoliquid is Newtonian and the Boussinesq approach is employed. The chemical properties of the host liquid and the material of particles are presented in Table 1.

Fig. 2. Streamlines 𝜓 and isotherms 𝜃 for Ra = 105 , 𝛾 = 0.5: a – numerical data of Karki et al. [50], b – obtained results.

E.V. Shulepova, M.A. Sheremet and H.F. Oztop et al.

International Journal of Mechanical Sciences 165 (2020) 105192

Fig. 3. Streamlines 𝜓 and isotherms 𝜃 for Ra = 106 , 𝛾 = 0.5: a – numerical data of Karki et al. [50], b – obtained results.

Using these conditions the managing equations are [35–38,45] 𝜕 𝑢̄ 𝜕 𝑣̄ + =0 𝜕 𝑥̄ 𝜕 𝑦̄ ( 𝜌𝑛𝑓 ( 𝜌𝑛𝑓

(1)

𝜕 𝑢̄ 𝜕 𝑢̄ 𝜕 𝑢̄ + 𝑢̄ + 𝑣̄ 𝜕𝑡 𝜕 𝑥̄ 𝜕 𝑦̄ 𝜕 𝑣̄ 𝜕 𝑣̄ 𝜕 𝑣̄ + 𝑢̄ + 𝑣̄ 𝜕𝑡 𝜕 𝑥̄ 𝜕 𝑦̄

( ) 𝜌𝑐𝑝 𝑛𝑓

(

) =− ) =−

𝜕𝑇 𝜕𝑇 𝜕𝑇 + 𝑢̄ + 𝑣̄ 𝜕𝑡 𝜕 𝑥̄ 𝜕 𝑦̄

𝜕 𝑝̄ + 𝜇𝑛𝑓 𝜕 𝑥̄ 𝜕 𝑝̄ + 𝜇𝑛𝑓 𝜕 𝑦̄

)

( = 𝑘𝑛𝑓

(

(

𝜕 2 𝑢̄ 𝜕 2 𝑢̄ + 𝜕 𝑥̄ 2 𝜕 𝑦̄2 𝜕 2 𝑣̄ 𝜕 2 𝑣̄ + 𝜕 𝑥̄ 2 𝜕 𝑦̄2

𝜕2 𝑇 𝜕2 𝑇 + 𝜕 𝑥̄ 2 𝜕 𝑦̄2

) (2) )

( ) + (𝜌𝛽)𝑛𝑓 𝑔 𝑇 − 𝑇𝑐

𝑢̄ = 𝑉0 , 𝑣̄ = 0, 𝑢̄ = 𝑣̄ = 0,

𝜕𝑇 𝜕 𝑛̄

𝜕𝑇 𝜕 𝑦̄

for 1% ≤ 𝜙 ≤ 4%. The nanosuspension viscosity is [35–38,47]: ( ) 𝜇𝑛𝑓 = 𝜇𝑓 1 + 4.93𝜙 + 222.4𝜙2

(9)

(10)

Including the non-dimensional parameters (3)

𝑥 = 𝑥̄ ∕𝐿, 𝑦 = 𝑦̄∕𝐿, 𝜏 = 𝑉0 𝑡∕𝐿, 𝑢 = 𝑢̄ ∕𝑉0 , 𝑣 = 𝑣̄ ∕𝑉0 , ( ) ( ) ( ) 𝜃 = 𝑇 − 𝑇0 ∕ 𝑇ℎ − 𝑇𝑐 , 𝜓 = 𝜓̄ ∕ 𝑉0 𝐿 , 𝜔 = 𝜔̄ 𝐿∕𝑉0

(4)

and new non-primitive functions: Δ =

)

|𝑁𝑢𝑖×𝑗 −𝑁𝑢200×200 | 𝑁𝑢𝑖×𝑗

trol Eqs. (1)–(4) will be rewritten as

with initial and boundary conditions 𝑡 = 0∶𝑢̄ = 𝑣̄ = 0, 𝑇 = 𝑇𝑐 at 0 ≤ 𝑥̄ ≤ 𝐿, 0 ≤ 𝑦̄ ≤ 𝐿; 𝑡 > 0∶𝑢̄ = 𝑣̄ = 0, 𝑇 = 𝑇𝑐 at 𝑥̄ = 0, 0 ≤ 𝑦̄ ≤ 𝐿; 𝑢̄ = 𝑣̄ = 0, 𝑇 = 𝑇ℎ at 𝑥̄ = 𝐿, 0 ≤ 𝑦̄ ≤ 𝐿; 𝑢̄ = 𝑣̄ = 0, 𝜕𝑇 = 0 at 𝑦̄ = 0, 0 ≤ 𝑥̄ ≤ 𝐿; 𝜕 𝑦̄

The nanosuspension thermal conductivity is [35–38,47]: ( ) 𝑘𝑛𝑓 = 𝑘𝑓 1 + 2.944𝜙 + 19.672𝜙2

× 100% the con-

𝜕2 𝜓 𝜕2 𝜓 + = −𝜔 𝜕 𝑥2 𝜕 𝑦2 (5)

= 0 at 𝑦̄ = 𝐿, 0 ≤ 𝑥̄ ≤ 𝐿;

(11)

𝐻 (𝜙) 𝜕𝜔 𝜕𝜔 𝜕𝜔 +𝑣 = 1 +𝑢 𝜕𝜏 𝜕𝑥 𝜕𝑦 Re

= 0 at internal block

The nanosuspension density, heat capacity and thermal expansion parameter are calculated on the basis of the following relations [35– 38,46] 𝜌𝑛𝑓 = (1 − 𝜙)𝜌𝑓 + 𝜙𝜌𝑝

(6)

( ) ( ) ( ) 𝜌𝑐𝑝 𝑛𝑓 = (1 − 𝜙) 𝜌𝑐𝑝 𝑓 + 𝜙 𝜌𝑐𝑝 𝑝

(7)

(𝜌𝛽)𝑛𝑓 = (1 − 𝜙)(𝜌𝛽)𝑓 + 𝜙(𝜌𝛽)𝑝

(8)

𝐻 (𝜙) 𝜕𝜃 𝜕𝜃 𝜕𝜃 +𝑣 = 3 +𝑢 𝜕𝜏 𝜕𝑥 𝜕𝑦 Re ⋅ Pr

(

(

𝜕2 𝜔 𝜕2 𝜔 + 𝜕 𝑥2 𝜕 𝑦2

𝜕2 𝜃 𝜕2 𝜃 + 𝜕 𝑥2 𝜕 𝑦2

) + 𝐻2 (𝜙)

𝑅𝑎 𝜕𝜃 Pr ⋅Re2 𝜕𝑥

(12)

) (13)

with initial and boundary conditions 𝜏 = 0∶𝜓 = 0, 𝜔 = 0, 𝜃 = 0.5 at 0 ≤ 𝑥 ≤ 1, 0 ≤ 𝑦 ≤ 1; 𝜏 > 0∶𝜓 = 0, 𝜕𝜓 = 0, 𝜃 = 0 at 𝑥 = 0, 0 ≤ 𝑦 ≤ 1; 𝜕𝑥 𝜓 = 0, 𝜕𝜓 = 0, 𝜃 = 1 at 𝑥 = 1, 0 ≤ 𝑦 ≤ 1; 𝜕𝑥 𝜓 = 0, 𝜕𝜓 = 0, 𝜕𝜃 = 0 at 𝑦 = 0, 0 ≤ 𝑥 ≤ 1; 𝜕𝑦 𝜕𝑦 𝜓 = 0,

𝜓 = 𝛾,

𝜕𝜓 𝜕𝑦 𝜕𝜓 𝜕𝑛

= 1,

= 0,

𝜕𝜃 𝜕𝑦 𝜕𝜃 𝜕𝑛

= 0 at 𝑦 = 1, 0 ≤ 𝑥 ≤ 1; = 0 at internal block

(14)

E.V. Shulepova, M.A. Sheremet and H.F. Oztop et al.

International Journal of Mechanical Sciences 165 (2020) 105192

Fig. 4. Streamlines for Ra = 6.39⋅104 , Pr = 0.71: a – obtained results, b – numerical data [52], c – experimental data [53].

The non-dimensional characteristics appearing in the Eqs. (11)–(13) are (

Pr =

) 𝜇𝑐𝑝 𝑓 , 𝑅𝑎 𝑘𝑓 (𝜌𝛽)𝑛𝑓

𝐻2 (𝜙) =

𝜌𝑛𝑓 𝛽𝑓

=

=

( ) 𝑔 (𝜌𝛽)𝑓 𝜌𝑐𝑝 𝑓 (𝑇ℎ −𝑇𝑐 )𝐿3 𝜇 𝜌 .93𝜙+222.4𝜙2 , 𝐻1 (𝜙) = 𝜇𝑛𝑓 𝜌 𝑓 = 1+4 , 𝜇𝑓 𝑘 𝑓 1−𝜙+𝜙𝜌𝑝 ∕𝜌𝑓 ( 𝑓) 𝑛𝑓 1−𝜙+𝜙(𝜌𝛽)𝑝 ∕(𝜌𝛽)𝑓 𝑘𝑛𝑓 𝜌𝑐𝑝 𝑓 1+2.944𝜙+19.672𝜙2 , 𝐻3 (𝜙) = 𝑘 (𝜌𝑐 ) = 1−𝜙+𝜙(𝜌𝑐 ) ∕(𝜌𝑐 ) 1−𝜙+𝜙𝜌 ∕𝜌 𝑝

𝑓

𝑓

𝑝 𝑛𝑓

𝑝 𝑝

𝑝 𝑓

Table 3 Dependence of 𝑁𝑢 on the uniform mesh parameters. Uniform mesh

𝑁𝑢

Δ=

100 × 100 200 × 200 300 × 300

6.532 6.415 6.347

1.8 – 1.1

|𝑁𝑢𝑖×𝑗 −𝑁𝑢200×200 | 𝑁𝑢𝑖×𝑗

× 100%

(15) For analysis of the total energy transfer intensity the local Nusselt number was defined 𝑁𝑢 = −

𝑘𝑛𝑓 𝜕𝜃 | | 𝑘𝑓 𝜕𝑥 ||𝑥=0

(16)

and the average Nusselt number can be 1

𝑁𝑢 =

∫

𝑁𝑢𝑑𝑦

(17)

0

3. Numerical technique The control Eqs. (11)–(13) with additional conditions (14) were solved by the finite difference techniques [35]. For definition of the

stream function value at the internal block surface the special procedure was employed [48,49]. The developed computational code was validated using the computational data of Karki et al. [50] for thermal transmission in a domain with isothermal vertical borders with a centered adiabatic block. Figs. 2 and 3 demonstrate a good agreement for streamlines and isotherms between the obtained data and calculated results of Karki et al. [50] for various Rayleigh numbers. In the case of alumina-water nanofluid natural convection within the differentially heated square cavity the developed numerical code was verified using experimental data of Ho et al. [47] and numerical data of Saghir et al. [51]. Table 2 shows a very good agreement between the obtained results.

E.V. Shulepova, M.A. Sheremet and H.F. Oztop et al.

International Journal of Mechanical Sciences 165 (2020) 105192

Fig. 5. Isolines of stream function 𝜓 and temperature 𝜃 for Ra = 104 , 𝛿 = 0.3: a – Re = 50, b – Re = 100, c – Re = 200, d – Re = 500.

Table 2 Average Nu at heated cavity surface compared with results of other authors.

7

Ra = 8.663 × 10 , Pr = 7.002

𝜙

Obtained data

Results of Ho et al. [47]

Results of Saghir et al. [51]

0.01 0.02 0.03

31.6043 31.2538 30.829

32.2037 31.0905 29.0769

30.657 30.503 30.205

E.V. Shulepova, M.A. Sheremet and H.F. Oztop et al.

International Journal of Mechanical Sciences 165 (2020) 105192

Fig. 6. Isolines of stream function 𝜓 and temperature 𝜃 for Ra = 105 , 𝛿 = 0.3: a – Re = 50, b – Re = 100, c – Re = 200, d – Re = 500.

In the case of air free convection in a square cavity cooled from the isothermal vertical walls and heated from the rectangular block placed on the bottom adiabatic wall, the developed computational code was validated successfully using the numerical data of AlAmiri et al. [52] and experimental data of Corvaro and Paroncini [53]. Fig. 4 shows a good agreement for streamlines between the obtained data, numerical results of AlAmiri et al. [52] and exper-

imental results of Corvaro and Paroncini [53] for different location of heated block at Ra = 6.39⋅104 . A grid sensitivity analysis was conducted using three various mesh sizes (100 × 100, 200 × 200, and 300 × 300) for Ra = 105 , Re = 100, Pr = 6.82, 𝜙 = 0.02, 𝛿 = 0.2. As it can be observed from Table 3 the deviations of 𝑁𝑢 for 200 × 200 and 300 × 300 are insignificant. As a result, a uniform mesh of 200 × 200 points was chosen for analysis.

E.V. Shulepova, M.A. Sheremet and H.F. Oztop et al.

International Journal of Mechanical Sciences 165 (2020) 105192

Fig. 7. Isolines of stream function 𝜓 and temperature 𝜃 for Ra = 106 , 𝛿 = 0.3: a – Re = 50, b – Re = 100, c – Re = 200, d – Re = 500.

4. Results and discussion In the considered work, we investigate mixed convection of Al2 O3 – H2 O nanoliquid within a differentially-heated square region having complicated fins and moved upper border. The impacts of the Rayleigh number (Ra = 104 –106 ), Reynolds number (Re = 50–200), nanoparticles volume fraction (𝜙 = 0.0–0.04) and fins location (𝛿 = 0.1–0.5) on the liquid motion and energy transfer are investigated for Pr = 6.82, l/L = 0.3, h/L = 0.1. The results are shown using isolines of stream function and

temperature with profiles of average Nusselt number at heated border and presented in Figs. 5–11. Figs. 5–7 demonstrate streamlines and isotherms within the considered cavity for 𝛿 = 0.3 and various magnitudes of Rayleigh and Reynolds numbers and nanoparticles concentration. In the case of low Ra magnitude (Ra = 104 in Fig. 5) one can find that regardless of the Reynolds number magnitude the major convective cell and two minor ones are formed within the cavity. The main circulation defines the clockwise motion of the medium due to the positive motion of the upper border

E.V. Shulepova, M.A. Sheremet and H.F. Oztop et al.

International Journal of Mechanical Sciences 165 (2020) 105192

Fig. 8. Profiles of the average Nu with Reynolds number for 𝛿 = 0.3 and various magnitudes of Ra and nanoparticles concentration (a) and with nanoparticles volume fraction for 𝛿 = 0.3 and various magnitudes of the Rayleigh and Reynolds numbers (b).

Fig. 9. Time profiles of 𝑁𝑢 at 𝛿 = 0.3 and 𝜙 = 0.04 for various Ra and Re.

and location of isothermal walls with heating from the left and cooling from the right. The internal block can be considered as an obstacle where the liquid surrounds this body. An appearance of the minor circulations can be explained by the location of the mounted fin and as a result liquid circulation forms such recirculation zones in the stagnant regions. An increment of the Reynolds number reflects a growth of these zones, while the left one increases essentially in comparison with right one. The later can be explained by the difference in the ascending and descending flows. It is worth noting an appearance of the additional recirculation over the internal block at Re = 100 (Fig. 5b). Such flow structure is formed due to the separation of the boundary layer near the solid obstacle. Isotherms illustrate a development of ascending thermal boundary layer close to the left heated wall and descending one close to the right cooling border. Simultaneously, the energy transport mechanism within the recirculation zones is heat conduction. A rise of the Reynolds number characterizes a reduction of the boundary layers thickness and diminution of average temperature in the chamber center. A development of the thermal plume from the right cold wall illustrates the clockwise circulation of the fluid. An inclusion of alumina nanoparticles reflects essential differences in streamlines for Re = 50 (Fig. 5a) where the recirculation zone from the left side of the mounted fin is more essential in comparison with clear water. While difference in temperature fields can be found for various Re magnitudes.

A growth of the Rayleigh number (Ra = 105 in Fig. 6) characterizes a modification of circulation structures and temperature fields inside the analyzed domain. For example, for low Re magnitudes (Re < 200) one can find a formation of recirculation zone only over the internal block, while such recirculations are not formed near the mounted fin. The reason for this nature is low intensive circulation that is defined by the natural and mixed convection modes. Also for Re = 50 (Fig. 6a) heat transfer is weak, therefore isotherms are parallel to the vertical isothermal borders within the cavity also. A growth of Re > 200 reflects a domination of the forced convection mode where recirculation zones are formed close to the mounted fin. An addition of nanoparticles characterizes an appearance of some changes in isotherms, while streamlines have weak changes. Fig. 7 demonstrates isolines of 𝜓 and 𝜃 for high value of the Rayleigh number (Ra = 106 ). Taking into account the map of convection regimes defined by the Richardson number value, for Re < 400 we have the natural convection mode. As a result, one can find a complex flow structures in Figs. 7a–c. Such system of vortices reflect a formation of two different natural convection zones with own recirculations. One zone is located from the left side of the internal adiabatic body and another one is located from the right side. It is possible to define the temperature stratification core in these parts. An increase in Re illustrates a dissipation of the left recirculation, while the right one rises. This vortex vanishes for high Re magnitudes, when forced convection is a dominated regime. An addition of nanoparticles allows intensifying the dissipation process for the considered vortices due to a reduction of the circulation for high viscosity (effective nanofluid viscosity increases with nanoparticles concentration). Behavior of the average Nu with Ra, Re and 𝜙 is presented in Fig. 8. As it was mentioned previously, a growth of the Rayleigh and Reynolds numbers illustrates the heat transfer enhancement. It is interesting to note that for high Re (Re = 500) one can find weak difference in the average Nusselt numbers for Ra = 104 and Ra = 105 due to the development of the forced convection regime. As a result, a rise of the Reynolds number reduces the difference in the average Nu between these two Ra magnitudes. A growth of the nanoparticles volume fraction intensifies the energy transport for all considered Re at Ra = 104 and Ra = 105 . This intensification corresponds to a growth of the average Nusselt number up to 10% for Ra = 104 and Re = 500. At the same time, for Ra = 106 the energy transport degradation occurs for Re < 150 and the heat transfer enhancement is for Re > 150. Moreover, the influence of nanoparticles is profound for low and moderate Ra magnitudes. Fig. 9 demonstrates the time profiles of 𝑁𝑢 for 𝛿 = 0.3, 𝜙 = 0.04 and various magnitudes of Ra and Re numbers. The considered time 𝜏 = 300 is enough for the formation of steady state. Regardless of Re magnitudes a rise of Ra characterizes a diminution of time needed for the formation of steady state. It is interesting to note that a growth of the Reynolds

E.V. Shulepova, M.A. Sheremet and H.F. Oztop et al.

International Journal of Mechanical Sciences 165 (2020) 105192

Fig. 10. Isolines of 𝜓 and 𝜃 for Ra = 105 , Re = 200: a – 𝛿 = 0.1, b – 𝛿 = 0.2, c – 𝛿 = 0.4, d – 𝛿 = 0.5.

number reflects a significant rise of the time that is needed to reach the steady state, while for Ra = 104 and Ra = 105 at Re = 500 only small differences for the average Nusselt number can be found. Such behavior illustrates a domination of the forced convective regime. The influence of the internal block location on isolines of 𝜓 and 𝜃 is shown in Fig. 10 for Ra = 105 , Re = 200. Such values of governing parameters illustrate the formation of forced convection mode. When this block locates near the mounted fin (𝛿 ≤ 0.2) one can find a formation

of one global circulation over this internal obstacle. Further rise of 𝛿 characterizes a formation of circulation around this block. Temperature field illustrates the modification of the temperature for different modes. It is interesting to note that a location of this obstacle defines the different thickness of the temperature boundary layer. Therefore, an addition of such obstacle allows controlling the heat transfer within the cavity. This conclusion can be confirmed by results presented in Fig. 11. Here non-linear nature of 𝑁𝑢 with the location of internal block is demon-

E.V. Shulepova, M.A. Sheremet and H.F. Oztop et al.

Fig. 11. Variations of 𝑁𝑢 with the location of the internal block for 𝜙 = 0.04 and different Ra and Re.

strated. For Ra = 104 and Ra = 105 and different considered Re values the highest average Nusselt number can be achieved for 𝛿 = 0.2, while for Ra = 106 this value depends on the Reynolds number. 5. Conclusions Mixed convection of alumina–water nanoliquid in a square enclosure with a complicated fin under the influence of moving upper surface was studied. The complicated fin is a combination of bottom wall mounted fin and internal adiabatic block. Control equations written using the non-primitive variables were solved by the finite difference method. Special procedure was employed for definition of the constant magnitude of 𝜓 at the surface of internal body. Influences of the Rayleigh number, Reynolds number, internal block position and nanoparticles concentration on liquid circulation and energy transport were examined. It has been ascertained that a rise of the Rayleigh and Reynolds numbers illustrates the energy transport enhancement, while a rise of the Reynolds number reduces the difference in 𝑁𝑢 between for Ra = 104 and Ra = 105 due to the development of the forced convection regime. A rise of the nanoparticles volume fraction intensifies the energy transport for all considered Re at Ra = 104 and Ra = 105 , while for Ra = 106 the heat transfer degradation is for Re < 150 and the heat transfer enhancement is for Re > 150. Non-linear influence of the internal block location on the average Nusselt number characterizes an opportunity to manage the energy transfer. For the considered problem, in the case of Ra = 104 and Ra = 105 and different analyzed Re values the highest average Nusselt number can be achieved for 𝛿 = 0.2, while for Ra = 106 this value depends on the Reynolds number. Declaration of Competing Interest None. Acknowledgements This work was supported by the Grants Council (under the President of the Russian Federation), Grant No. MD-821.2019.8. References [1] Liang G, Mudawar I. Review of pool boiling enhancement with additives and nanofluids. Int J Heat Mass Transf 2018;124:423–53. [2] Liang G, Mudawar I. Review of single-phase and two-phase nanofluid heat transfer in macro-channels and micro-channels. Int J Heat Mass Transf 2019;136:324–54. [3] Balootaki AA, Karimipour A, Toghraie D. Nano scale lattice Boltzmann method to simulate the mixed convection heat transfer of air in a lid-driven cavity with an endothermic obstacle inside. Phys A 2018;508:681–701. [4] Sheikholeslami M, Keramati H, Shafee A, Li Z, Alawad Omer A, Tlili I. Nanofluid Mhd forced convection heat transfer around the elliptic obstacle inside a permeable lid drive 3D enclosure considering lattice Boltzmann method. Phys A 2019;523:87–104.

International Journal of Mechanical Sciences 165 (2020) 105192 [5] Ismael MA, Pop I, Chamkha AJ. Mixed convection in a lid-driven square cavity with partial slip. Int J Therm Sci 2014;82:47–61. [6] Chamkha AJ, Abu-Nada E. Mixed convection flow in single- and double-lid driven square cavities filled with water-Al2 O3 nanofluid: effect of viscosity models. Eur J Mech B/Fluids 2012;36:82–96. [7] Abu-Nada E, Chamkha AJ. Mixed convection flow of a nanofluid in a lid-driven cavity with a wavy wall. Int Commun Heat Mass Transf 2014;57:36–47. [8] Selimefendigil F, Öztop HF, Chamkha AJ. Analysis of mixed convection of nanofluid in a 3D lid-driven trapezoidal cavity with flexible side surfaces and inner cylinder. Int Commun Heat Mass Transf 2017;87:40–51. [9] Taghizadeh S, Asaditaheri A. Heat transfer and entropy generation of laminar mixed convection in an inclined lid driven enclosure with a circular porous cylinder. Int J Therm Sci 2018;134:242–57. [10] Mehmood K, Hussain S, Sagheer M. Mixed convection in alumina-water nanofluid filled lid-driven square cavity with an isothermally heated square blockage inside with magnetic field effect: introduction. Int J Heat Mass Transf 2017;109:397–409. [11] Gangawane KM, Oztop HF, Ali ME. Mixed convection in a lid-driven cavity containing triangular block with constant heat flux: effect of location of block. Int J Mech Sci 2019;152:492–511. [12] Gangawane KM, Manikandan B. Mixed convection characteristics in lid-driven cavity containing heated triangular block. Chin. J. Chem. Eng. 2017;25:1381–94. [13] Nasrin R, Alim MA, Chamkha AJ. Modeling of mixed convective heat transfer utilizing nanofluid in a double lid-driven chamber with internal heat generation. Int J Numer Methods Heat Fluid Flow 2014;24:36–57. [14] Sheremet MA, Pop I. Mixed convection in a lid-driven square cavity filled by a nanofluid: buongiorno’s mathematical model. Appl Math Comput 2015;266:792–808. [15] Billah MM, Rahman MM, Sharif UM, Rahim NA, Saidur R, Hasanuzzaman M. Numerical analysis of fluid flow due to mixed convection in a lid-driven cavity having a heated circular hollow cylinder. Int Commun Heat Mass Transf 2011;38:1093– 1103. [16] Hussain S, Öztop HF, Mehmood K, Abu-Hamdeh N. Effects of inclined magnetic field on mixed convection in a nanofluid filled double lid-driven cavity with volumetric heat generation or absorption using finite element method. Chin J Phys 2018;56:484–501. [17] Sheikholeslami M. Influence of magnetic field on Al2 O3 –H2 O nanofluid forced convection heat transfer in a porous lid driven cavity with hot sphere obstacle by means of LBM. J Mol Liq 2018;263:472–88. [18] Karbasifar B, Akbari M, Toghraie D. Mixed convection of Water-Aluminum oxide nanofluid in an inclined lid-driven cavity containing a hot elliptical centric cylinder. Int J Heat Mass Transf 2018;116:1237–49. [19] Shi X, Khodadadi JM. Laminar fluid flow and heat transfer in a lid-driven cavity due to a thin fin,. J Heat Transf 2002;124:1056–63. [20] Alsabery AI, Sheremet MA, Chamkha AJ, Hashim I. Impact of nonhomogeneous nanofluid model on transient mixed convection in a double lid-driven wavy cavity involving solid circular cylinder. Int J Mech Sci 2019;150:637–55. [21] Boraey MA. Characteristics of the vortical structure in a square cavity with a central obstacle at different Reynolds numbers. Am J Aerosp Eng 2018;5:39–46. [22] Esfe MH, Ghadi AZ, Esforjani SSM, Akbari M. Combined convection in a lid-driven cavity with an inside obstacle subjected to Al2 O3 -water nanofluid: effect of solid volume fraction and nanofluid variable properties. Acta Phys Pol A 2013;124:665–72. [23] Golkarfard V, Nasab SAG, Ansari AB, Bagheri GH. Numerical investigation on deposition of solid particles in a lid-driven square cavity with inner heated obstacles. Adv Powder Technol 2012;23:736–43. [24] Ben-Nakhi A, Chamkha AJ. Natural convection in inclined partitioned enclosures. Heat and Mass Transfer 2006;42:311–21. [25] Ben-Nakhi A, Chamkha AJ. Conjugate natural convection around a finned pipe in a square enclosure with internal heat generation. Int J Heat Mass Transf 2007;50:2260–71. [26] Sathiyamoorthy M, Chamkha A. Effect of magnetic field on natural convection flow in a liquid gallium filled square cavity for linearly heated side wall(s). Int J Therm Sci 2010;49:1856–65. [27] Chamkha AJ, Ismael MA. Conjugate heat transfer in a porous cavity filled with nanofluids and heated by triangular thick wall. Int J Therm Sci 2013;67:135–51. [28] Chamkha AJ, Ismael MA. Natural convection in differentially heated partially porous layered cavities filled with a nanofluid. Numer Heat Transf A 2014;65:1089– 1113. [29] Selimefendigil F, Oztop HF, Chamkha AJ. MHD mixed convection and entropy generation of nanofluid filled lid-driven cavity under the influence of inclined magnetic fields imposed to its upper and lower diagonal triangular domains. J Magn Magn Mater 2016;406:266–81. [30] Hammami F, Souayeh B, Ben-Cheikh N, Ben-Beya B. Computational analysis of fluid flow due to a two-sided lid driven cavity with a circular cylinder. Comput Fluids 2017;156:317–28. [31] Selimefendigil F. Mixed convection in a lid-driven cavity filled with single and multiple-walled carbon nanotubes nanofluid having an inner elliptic obstacle. Propul Power Res 2019;8:128–37. [32] Munshi MJH, Alim MA, Bhuiyan AH, Ali M. Hydrodynamic mixed convection in a lid- driven square cavity including elliptic shape heated block with corner heater. Procedia Eng 2017;194:442–9. [33] Kareem AK, Gao S. Mixed convection heat transfer of turbulent flow in a three-dimensional lid-driven cavity with a rotating cylinder. Int J Heat Mass Transf 2017;112:185–200. [34] Selimefendigil F, Öztop HF. Natural convection and entropy generation of nanofluid filled cavity having different shaped obstacles under the influence of magnetic field and internal heat generation. J Taiwan Inst Chem Eng 2015;56:42–56.

E.V. Shulepova, M.A. Sheremet and H.F. Oztop et al. [35] Gibanov NS, Sheremet MA, Oztop HF, Al-Salem K. Convective heat transfer in a lid-driven cavity with a heat-conducting solid backward step under the effect of buoyancy force. Int J Heat Mass Transf 2017;112:158–68. [36] Gibanov NS, Sheremet MA, Oztop HF, Abu-Hamdeh N. Effect of uniform inclined magnetic field on mixed convection in a lid-driven cavity having a horizontal porous layer saturated with a ferrofluid. Int J Heat Mass Transf 2017;114:1086–97. [37] Gibanov NS, Sheremet MA, Oztop HF, Nusier OK. Convective heat transfer of ferrofluid in a lid-driven cavity with a heat-conducting solid backward step under the effect of a variable magnetic field. Numer Heat Transf A 2017;72:54–67. [38] Astanina MS, Sheremet MA, Oztop HF, Abu-Hamdeh N. Mixed convection of Al2 O3 -water nanofluid in a lid-driven cavity having two porous layers. Int J Heat Mass Transf 2018;118:527–37. [39] Garoosi F, Rashidi MM. Conjugate-mixed convection heat transfer in a two-sided lid-driven cavity filled with nanofluid using Manninen’s two phase model. Int J Mech Sci 2017;131-132:1026–48. [40] Ismael MA, Abu-Nada E, Chamkha AJ. Mixed convection in a square cavity filled with CuO-water nanofluid heated by corner heater. Int J Mech Sci 2017;133:42–50. [41] Mehrizi AA, Farhadi M, Afroozi HH, Sedighi K, Darz AAR. Mixed convection heat transfer in a ventilated cavity with hot obstacle: effect of nanofluid and outlet port location. Int Commun Heat Mass Transf 2012;39:1000–8. [42] Doustdar MM, Yekani MK. Numerical study of mixed convection of nano fluid in a lid-driven cavity containing hot obstacles. Aerosp Mech J 2016;12:67–78. [43] Kuhlmann HC, Romano F. The lid-driven cavity. In: Gelfgat A, editor. Computational modeling of bifurcations and instabilities in fluid dynamics. Springer; 2018. p. 233–310.

International Journal of Mechanical Sciences 165 (2020) 105192 [44] Oztop HF, Zhao Z, Yu B. Fluid flow due to combined convection in lid-driven enclosure having a circular body. Int J Heat Fluid Flow 2009;30:886–901. [45] Cho C-C. Mixed convection heat transfer and entropy generation of Cu-water nanofluid in wavy-wall lid-driven cavity in presence of inclined magnetic field. Int J Mech Sci 2019;151:703–14. [46] Al-Rashed AAAA. Investigating the effect of alumina nanoparticles on heat transfer and entropy generation inside a square enclosure equipped with two inclined blades under magnetic field. Int J Mech Sci 2019;152:312–28. [47] Ho CJ, Li WK, Chang YS, Lin CC. Natural convection heat transfer of alumina-water nanofluid in vertical square enclosures: an experimental study. Int J Therm Sci 2010;49:1345–53. [48] Adlam JH. Computation of two-dimensional time-dependent natural convection in a cavity where there are internal bodies. Comput Fluids 1986;14:141–57. [49] Lippke A, Wagner H. Numerical solution of the Navier-Stokes equations in multiply connected domains. Comput Fluids 1991;20:19–28. [50] Karki P, Yadav AK, Perumal DA. Study of adiabatic obstacles on natural convection in a square cavity using lattice Boltzmann method. J Therm Sci Eng Appl 2019;11:034502. [51] Saghir MZ, Ahadi A, Mohamad A, Srinivasan S. Water aluminum oxide nanofluid benchmark model. Int J Therm Sci 2016;109:148–58. [52] AlAmiri A, Khanafer K, Pop I. Buoyancy-induced flow and heat transfer in a partially divided square enclosure. Int J Heat Mass Transf 2009;52:3818–28. [53] Corvaro F, Paroncini M. An experimental study of natural convection in a differentially heated cavity through a 2D-PIV system. Int J Heat Mass Transf 2009;52:355–65.