Citation
Saha, S. and Awad, M.M. (2024), "Editorial: Comment on “MHD-mixed convection flow in a lid-driven trapezoidal cavity under uniformly/non-uniformly heated bottom wall”", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 34 No. 4, pp. 1521-1527. https://doi.org/10.1108/HFF-04-2024-942
Publisher
:Emerald Publishing Limited
Copyright © 2024, Emerald Publishing Limited
1. Comments
Javed et al. (2017) performed a numerical investigation of MHD-mixed convection heat transfer inside a lid-driven trapezoidal cavity with uniform or non-uniform heating conditions. During their simulation, they varied the governing dimensionless parameters such as Rayleigh number (Ra) and Prandtl number (Pr). For a mixed convective problem, the associated heat and fluid flow characteristics due to the combined effect of natural and forced convection mechanisms are mathematically described by the combination of two crucial governing parameters, namely, Grashof number (Gr = Ra/Pr, which signifies the dominance of natural convection) and Reynolds number (Re, which indicates the dominance of forced convection). Hence, the characteristic parameter for mixed convection heat transfer is called the Richardson number (Ri = Gr/Re2), which should be 0.1 ≤ Ri ≤ 10 within the mixed convection regime (Lukose and Basak, 2021). Forced convection becomes dominant when Ri > 0.1, while the dominance of natural convection takes into play at Ri > 10. When Ri = 1, it is commonly known as pure mixed convection (Rahman et al., 2010; Hasib et al., 2015). Unfortunately, Javed et al. (2017) did not consider either Richardson or Reynolds number in their investigation while performing an extensive analysis of the mixed convection problem. They also mentioned that they varied the Hartmann number within 50 ≤ Ha ≤ 1,000 to show the MHD effect but presented all results only at Ha = 50. Besides, they mentioned the investigated range of the Prandtl number twice in their paper as 0.026 ≤ Pr ≤ 100 and 0.026 ≤ Pr ≤ 1,000, respectively, but only considered Pr = 0.026 and 10.
Javed et al. (2017) considered a lid-driven trapezoidal cavity in their problem, where the top lid was moving at a fixed speed U0. Hence, they mentioned the velocity boundary condition of the top wall as u(x, L, where L is the height of the trapezoidal enclosure) = U0 in equation (5) of their paper, which represents the sliding motion of the lid in the positive x-direction. However, while drawing the schematic diagram of their problem [see Figure 1(a)], they showed the direction of the lid movement in the negative x-direction, which was wrong as per the boundary condition described in equation (5). Moreover, they neither showed the size of the base wall of the trapezoidal chamber in Figure 1(a) nor mentioned it anywhere in their paper. By analyzing Figure 9 of Javed et al. (2017), it can be confirmed that the base length of the trapezoidal cavity was L. The corrected physical domain is now shown in Figure 1(b), along with the corrected direction of the lid movement. In the caption of Figure 1 of Javed et al. (2017), it was mentioned that ϕ = 0° represented the square cavity, where ϕ was the inclination angle of the side walls. However, the physical models developed using the physical dimensions of three different cases (ϕ = 0°, 30° and 45°) do not match with the domain shown in various qualitative plots (Figures 3–8) of Javed et al. (2017). A comparison of the difference between these physical models is presented in Figure 2. Besides, Javed et al. (2017) incorrectly mentioned the profiles of inclined side walls of the trapezoidal cavity in equation (5) as xsinϕ + ycosϕ = 0 (left side wall) and xsinϕ − ycosϕ = Lcosϕ (right side wall). It should be correctly expressed as xcosϕ + ysinϕ = 0 and xcosϕ − ysinϕ = Lsinϕ, respectively.
Javed et al. (2017) used α/L as the reference velocity and ρα2/L2 as the reference pressure in equation (6) to obtain the non-dimensional governing equations [equations (7)–(10)] of their problem. These reference velocity and pressure are usually used for natural convection problems. Hence, the non-dimensional mathematical model mentioned in equations (7)–(10) of their paper is commonly used for analyzing natural convection problems. The appropriate reference velocity and pressure of their problem should be U0 and
Using the correct reference velocity and pressure, the dimensionless momentum and heat energy equations for the MHD-mixed convection problem of Javed et al. (2017) can be written as follows:
With the reference velocity considered by Javed et al. (2017), the corresponding non-dimensional velocity boundary condition at the top wall becomes U(X, 1) = U0L/α = RePr. However, they mentioned U(X, 1) = 1 in equation (11) of their paper. With this setting of velocity boundary condition U(X, 1) = RePr = 1, the unknown governing parameter, Reynolds number can be calculated as Re = 1/Pr. Since Javed et al. (2017) considered Pr = 0.026, Pr = 10 and 103 ≤ Ra ≤ 106 in their simulation, the corresponding values of Richardson number (Ri = Ra/PrRe2) can be evaluated as listed in Table 1. It is found from Table 1 that the range of Richardson number considered in the problem of Javed et al. (2017) is much higher than the usual range of mixed convection regime (0.1 ≤ Ri ≤ 10). Hence, it can be concluded that Javed et al. (2017) did not analyze mixed convection phenomena in their paper. Instead, they examined the natural convective flow inside the cavity.
The non-dimensional thermal boundary conditions mentioned in equation (11) by Javed et al. (2017) were incorrectly expressed using the dimensional temperature variable. It should be corrected as follows:
bottom wall: θ(X, 0) = 1 or sinπX;
top wall: ∂θ(X, 1)/∂Y = 0;
left side wall: θ(X, Y) = 0 at Xcosϕ + Ysinϕ = 0 and 0 ≤ Y ≤ 1; and
right side wall: θ(X, Y) = 0 at Xcosϕ − Ysinϕ = sinϕ and 0 ≤ Y ≤ 1.
The figure captions of Figures 5 and 8 in the paper of Javed et al. (2017) incorrectly displayed the value of the Prandtl number as Pr = 0.026. It should be Pr = 10, which was clearly mentioned in the Results and Discussion section of Javed et al. (2017).
Javed et al. (2017) considered uniform and non-uniform heating conditions for three different inclination angles of the side walls of the trapezoidal cavity to evaluate the mean Nusselt number along the heated bottom wall, presented in Figure 10 of their paper. However, they did not mention the formula for calculating the mean Nusselt number at the heated bottom wall. For the uniform isothermal heating condition, the mean Nusselt number (
On the other hand, for a non-uniform (sinusoidally) heated surface, the correct expression of the mean Nusselt number along the bottom wall can be expressed following the derivation of the recent publications (Shuvo et al., 2023; Deb and Saha, 2024) as follows:
Unfortunately, in their paper, Javed et al. (2017) did not use the correct expression of the mean Nusselt number at the heated bottom surface. Hence, a correction in their results is required, as illustrated in Figure 3. The data from the present simulation is generated using a Galerkin finite element method-based simulation CFD software “COMSOL Multiphysics 6.2”. It is clearly observed that the results of Javed et al. (2017) and the current simulation match quite well for the uniform heating condition when ϕ = 0° and 45° following the correct formulation of equation (5). However, when ϕ = 30°, the data from Javed et al. (2017) underpredict the value of mean Nusselt number compared to the present simulation case. It is unacceptable to have such a variation of the mean Nusselt number for the case of ϕ = 30° without any dramatic change in flow and thermal fields, which Javed et al. (2017) failed to explain clearly. Hence, it can be undoubtedly confirmed that Javed et al. (2017) computed the incorrect mean Nusselt numbers for the case of ϕ = 30°. Moreover, the case of non-uniform (sinusoidal) heating shows a significant deviation between the present computation [based on the correct formulation of equation (6)] and the data given in Javed et al. (2017). The pattern of the mean Nusselt number has an inconsistent trend with the increase of the inclination angle of the side walls. In contrast, the present simulation displays a consistent alteration of the Nusselt number with Ra for all magnitudes of ϕ. Because Javed et al. (2017) did not mention any formula for the mean Nusselt number, it can be confirmed that the deviation of these two sets of data for non-uniform heating is due to the usage of the wrong formulation of Nusselt number calculation by Javed et al. (2017).
Figures

Figure 1.
(a) Schematic illustration of the physical model considered by Javed et al. (2017), and (b) the corrected physical model of the problem by Javed et al. (2017) showing the corrected direction of wall movement (color online)

Figure 2.
Comparison of the physical model between Javed et al. (2017) and the present analysis (obtained from the corrected dimensions) for various inclination angles of the side walls (color online)

Figure 3.
Comparison of the mean Nusselt number computed using the correct formulation with those of Javed et al. (2017) at Pr = 0.026, Ha = 50
Values of differentgoverningparametersconsidered in theanalysis of Javedet al. (2017)
Pr | Re | Ra | Ri | Comment |
---|---|---|---|---|
0.026 | 38.46 | 103 ≤ Ra ≤ 106 | 26 ≤ Ri ≤ 2.6 × 104 | Natural convection regime (Ri > 10) |
10 | 0.1 | 103 ≤ Ra ≤ 106 | 104 ≤ Ri ≤ 107 | Natural convection regime (Ri > 10) |
Source: Table by authors
References
Deb, N. and Saha, S. (2024), “Comment on “Numerical simulation of magnetohydrodynamic buoyancy-induced flow in a non-isothermally heated square enclosure” [communications in nonlinear science and numerical simulation, 14 (2009) 770-778]”, Communications in Nonlinear Science and Numerical Simulation, Vol. 130, p. 107771.
Hasib, M.H., Hossen, M.S. and Saha, S. (2015), “Effect of tilt angle on pure mixed convection flow in trapezoidal cavities filled with water–Al2O3 nanofluid”, Procedia Engineering, Vol. 105, pp. 388-397.
Javed, T., Mehmood, Z. and Pop, I. (2017), “MHD-mixed convection flow in a lid-driven trapezoidal cavity under uniformly/non-uniformly heated bottom wall”, International Journal of Numerical Methods for Heat and Fluid Flow, Vol. 27 No. 6, pp. 1231-1248.
Lukose, L. and Basak, T. (2021), “A comprehensive review on mixed convection for various patterns of kinematically and thermally induced scenarios within cavities”, International Journal of Numerical Methods for Heat and Fluid Flow, Vol. 31 No. 9, pp. 2879-2939.
Moallemi, M.K. and Jang, K.S. (1992), “Prandtl number effects on laminar mixed convection heat transfer in a lid-driven cavity”, International Journal of Heat and Mass Transfer, Vol. 35 No. 8, pp. 1881-1892.
Rahman, M.M., Alim, M.A. and Sarker, M.M.A. (2010), “Numerical study on the conjugate effect of joule heating and magnato-hydrodynamics mixed convection in an obstructed lid-driven square cavity”, International Communications in Heat and Mass Transfer, Vol. 37 No. 5, pp. 524-534.
Sheremet, M.A. and Pop, I. (2015), “Mixed convection in a lid-driven square cavity filled by a nanofluid: Buongiorno’s mathematical model”, Applied Mathematics and Computation, Vol. 266, pp. 792-808.
Shuvo, M.S., Ikram, M.M., Hasan, M.N. and Saha, S. (2023), “Thermal resistance analysis on conjugate free convective flow in a thick-walled square chamber”, Case Studies in Thermal Engineering, Vol. 51, p. 103644.
Acknowledgements
The authors warmly thank CFDHT Research Group and Basic Research Grant [Grant ID 4747(45), SL#9, dated 13 April 2022] of BUET, Bangladesh, for the persistent assistance during this study.
About the authors
Dr Sumon Saha is a professor at the Department of Mechanical Engineering, Bangladesh University of Engineering and Technology (BUET), Dhaka. Recent publication (last 10 journal papers):
1. Chowdhury, S., Shuvo, M. S., and Saha, S., Comment on “Investigation of heat transfer enhancement of Cu-water nanofluid by different configurations of double rotating cylinders in a vented cavity with different inlet and outlet ports” [International Communications in Heat and Mass Transfer, 126 (2021) 105432], International Communications in Heat and Mass Transfer, 147 (2023), 106977. https://doi.org/10.1016/j.icheatmasstransfer.2023.106977.
2. Robin, M.R.H., Hossain, M. R., and Saha, S., Entropy generation of pure mixed convection from double circular cylinders rotating inside a confined channel, Case Studies in Thermal Engineering, 49 (2023), 103395. https://doi.org/10.1016/j.csite.2023.103395.
3. Islam, R., Islam, A. F., and Saha, S., Influence of time-dependent wall temperature fluctuation on conjugate mixed convection inside a chamber with an oscillating cylinder, International Communications in Heat and Mass Transfer, 147 (2023), 106988. https://doi.org/10.1016/j.icheatmasstransfer.2023.106988.
4. Deb, N., Fardin, S., Fardin, M.M., Nawal, N., Nizam, M.R. and Saha, S., Thermal management inside a discretely heated rectangular cuboid using P, PI and PID controllers, Case Studies in Thermal Engineering, 51 (2023), 103601. https://doi.org/10.1016/j.csite.2023.103601.
5. Shuvo, M.S., Ikram, M.M., Hasan, M.N. and Saha, S., Thermal resistance analysis on conjugate free convective flow in a thick-walled square chamber, Case Studies in Thermal Engineering, 51 (2023), 103644. https://doi.org/10.1016/j.csite.2023.103644.
6. Deb, N. and Saha, S., Comment on “Steady natural convection flows in a square cavity with linearly heated side wall (s)” [International Journal of Heat and Mass Transfer, 50 (2007) 766–775], International Journal of Heat and Mass Transfer, 219 (2024), 124890. https://doi.org/10.1016/j.ijheatmasstransfer.2023.124890.
7. Tasnim, S., Shuvo, M.S., Deb, N., Islam, M.S. and Saha, S., Entropy generation on magnetohydrodynamic conjugate free convection with Joule heating of heat-generating liquid and solid element inside a chamber, Case Studies in Thermal Engineering, 52 (2023), 103711. https://doi.org/10.1016/j.csite.2023.103711.
8. Shuvo, M.S., Mahmud, M.J. and Saha, S., Multi-scaling analysis of turbulent boundary layers over an isothermally heated flat plate with zero pressure gradient, Heliyon, 9(12) (2023), e22721. https://doi.org/10.1016/j.heliyon.2023.e22721.
9. Deb, N. and Saha, S., Comment on “Numerical simulation of magnetohydrodynamic buoyancy-induced flow in a non-isothermally heated square enclosure” [Communications in nonlinear science and numerical simulation, 14(2009) 770-778], Communications in Nonlinear Science and Numerical Simulation, 130 (2024), 107771. https://doi.org/10.1016/j.cnsns.2023.107771. 10. Deb, N. and Saha, S., Role of internal heat generation, magnetism and Joule heating on entropy generation and mixed convective flow in a square domain, Annals of Nuclear Energy, 198 (2024), 110324. https://doi.org/10.1016/j.anucene.2023.110324. Web: https://sumonsaha.buet.ac.bd/ Google Scholar: https://rb.gy/4huqd9
M.M. Awad is a professor of energy systems at Mechanical Power Engineering Department, Faculty of Engineering, Mansoura University, Mansoura, Egypt. He was a postdoctoral fellow in mechanical engineering in the Faculty of Engineering and Applied Science, Memorial University of Newfoundland, Canada. He received his PhD from Memorial University of Newfoundland in 2007 and his undergraduate degree and his master’s degree from Mansoura University, Egypt, in 1996 and 2000, respectively. His research focus is on the development of robust models for characterizing transport phenomena using fundamental theory. These models are validated using experimental and/or numerical results. He is the author of 14 book chapters. He has published more than 145 papers in refereed journals and conference proceedings in these areas. Presently, his research is focused on renewable energy, the modeling of complex fluid dynamics and heat transfer problems in internal flows. These include transport in porous media, compact heat exchangers, two-phase flow in oil and gas operations, microchannel flows, non-Newtonian flows and thermal design/optimization of energy systems. He is a member of the American Society of Mechanical Engineers (ASME). He was also a recipient of the ASME International Petroleum Technology Institute (IPTI) Award in 2005 and 2006. He won silver medal at 45th International Exhibition of Inventions, Geneva, Swiss, 29 March–2 April 2017. His patent was “Solar Powered Portable Apparatus for Extracting Water from Air Using Desiccant Solution.” He was representative of Mansoura University in Research and Policy Committee Members for the mega research project among MIT and three Egyptian Universities: Ain Shams University, Mansoura University and Aswan University (August 2019-October 2020).http://news.mit.edu/2019/usaid-grant-30-million-energy-challenges-egypt-0327. Also, he is a corresponding member of the International Information Center for Multiphase Flow (ICeM) in Egypt [ICeM Newsletter No. 53 (March 2020–Present)]. (http://www.jsmf.gr.jp/icem_2.shtml) (http://www.jsmf.gr.jp/file/No53.pdf). In addition, he is an Editorial Board Member of Energy Nexus (https://www.journals.elsevier.com/energy-nexus/editorial-board).