Steady natural convection flows in a square cavity with linearly heated side wall(s)

The present numerical study deals with natural convection flow in a closed square cavity when the bottom wall is uniformly heated and vertical wall(s) are linearly heated whereas the top wall is well insulated. Non-linear coupled PDEs governing the flow have been solved by penalty finite element method with bi-quadratic rectangular elements. Numerical results are obtained for various values of Rayleigh number (Ra) (10³ ⩽ Ra ⩽ 10⁵) and Prandtl number (Pr) (0.7 ⩽ Pr ⩽ 10). Results are presented in the form of streamlines, isotherm contours, local Nusselt number and the average Nusselt as a function of Rayleigh number.     

Steady natural convection flow in a square cavity filled with a porous medium for linearly heated side wall(s)

In this paper natural convection flows in a square cavity filled with a porous matrix has been investigated numerically when the bottom wall is uniformly heated and vertical wall(s) are linearly heated whereas the top wall is well insulated. Darcy–Forchheimer model without the inertia term is used to simulate the momentum transfer in the porous medium. Penalty finite element method with bi-quadratic rectangular elements is used to solve the non-dimensional governing equations. Numerical results are presented for a range of parameters (Rayleigh number Ra, 10³ ⩽ Ra ⩽ 10⁶, Darcy number Da, 10⁻⁵ ⩽ Da ⩽ 10⁻³, and Prandtl number Pr, 0.2 ⩽ Pr ⩽ 100) in terms of stream functions and isotherm contours, and local and average Nusselt numbers.     

Numerical simulation of mixed convection flows in a square lid-driven cavity partially heated from below using nanofluid

The present numerical study deals with mixed convection in a square lid-driven cavity partially heated from below and filled with water-base nanofluid containing various volume fractions of Cu, Ag, Al₂O₃ and TiO₂. Finite difference method was employed to solve the dimensionless governing equations of the problem. The effects of governing parameters, namely, Reynolds number, solid volume fraction, different values of the heat source length and different locations of the heat source on the streamlines and isotherms contours as well as Nusselt number and average Nusselt number along the heat source were considered. The present results are validated by favorable comparisons with previously published results. The results of the problem are presented in graphical and tabular forms and discussed.     

Effect of moving wall direction on mixed convection in an inclined lid-driven square cavity with sinusoidal heating

ABSTRACT

The aim of the present study is to investigate the effect of the moving wall’s direction on mixed convective flow and heat transfer in an inclined lid-driven square cavity. Sinusoidal heating is applied on the left wall while the right wall is cooled at a constant temperature. The bottom and top walls are taken to be adiabatic. The results are presented graphically in the form of streamlines, isotherms, velocity profiles, and Nusselt numbers to understand the influence of the different directions of the moving wall, Richardson number, and cavity inclination. It is observed that the flow field and temperature distribution in the cavity are affected by the moving wall’s direction. It is also observed that the heat transfer is more pronounced at low Richardson number when the wall is moving to the left.     

Numerical study of mixed convection flows in a square lid-driven cavity utilizing nanofluid

A numerical investigation of laminar mixed convection flows through a copper–water nanofluid in a square lid-driven cavity has been executed. In the present study, the top and bottom horizontal walls are insulated while the vertical walls are maintained at constant but different temperatures. The study has been carried out for the Rayleigh number 10⁴ to 10⁶, Reynolds number 1 to 100 and the solid volume fraction 0 to 0.05. The thermal conductivity and effective viscosity of nanofluid have been calculated by Patel and Brinkman models, respectively. The effects of solid volume fraction of nanofluids on hydrodynamic and thermal characteristics have been investigated and discussed. It is found that at the fixed Reynolds number, the solid concentration affects on the flow pattern and thermal behavior particularly for a higher Rayleigh number. In addition it is observed that the effect of solid concentration decreases by the increase of Reynolds number.     

Heat and mass transfer under MHD mixed convection in a four-sided lid-driven square cavity

This paper investigates the heat and mass transfer under magnetohydrodynamic mixed convection flow of a binary gas mixture in a four-sided lid-driven square cavity. The enclosure's left wall is sinusoidally heated and acts as a source term, while the right wall functions as a sink. The cavity's horizontal walls are adiabatic and impermeable to mass transfer. The governing equations under Boussinesq approximation and stream function-vorticity formulation are solved using the alternating-direction-implicit scheme, a finite-difference method. The numerical scheme's consistency and stability are demonstrated using the matrix method. The MATLAB code is written, validated against some existing studies, and used to perform numerical simulations. The numerical solutions are graphically examined by visualizing the streamline, isotherm, and concentration contours for nondimensional parameters, such as Hartmann number $\left(0\le Ha\le 100\right)$, heat absorption or generation coefficient $\left(-2\le \varphi \le 2\right)$, Richardson number $\left(0.01\le Ri\le 100\right)$, and buoyancy ratio $\left(-6\le N\le 6\right)$. The magnetic field modifies the temperature and concentration distribution in the cavity, depending on the convection mode. The magnetic field forces the fluid to stagnate in different regions of the cavity, depending on the mode of convection. It was found that the difference between the maximum and minimum temperature and concentration at the cavity's midpoint increases up to 13 and 10 times, respectively, in the natural convection compared with the forced convection. The average Nusselt number on the vertical walls of the cavity is maximum in natural convection in the absence of a magnetic field but reaches a minimum value at $Ha=100$ in forced and mixed convection. The average Sherwood number on the cavity's vertical walls decreases with the magnetic field in mixed and natural convection.     

MHD mixed convection in a lid-driven cavity with corner heater

Laminar mixed convection flow in the presence of magnetic field in a top sided lid-driven cavity heated by a corner heater was considered. The corner heater is under isothermal boundary conditions with different length in bottom and right vertical walls. Finite volume technique was used to solve governing equations. The temperature of the lid is lower than that of heater. The study is performed for different Grashof and Hartmann numbers at Re = 100. The obtained results showed some very interesting results.     

Effects of moving lid direction on MHD mixed convection in a linearly heated cavity

Effects of moving lid-direction on MHD mixed convection in a cavity with the bottom wall being linearly heated are analyzed using a numerical technique. Vertical walls of the enclosure are adiabatic and the sliding wall at the top has constant temperature. The lid moves in the negative and positive x-direction. Finite volume method has been used to solve the governing equations. Results are presented for different values of Hartmann number (0 ? Ha ? 30), Reynolds number (100 ? Re ? 1000) and Grashof number (10⁴ ? Gr ? 10⁶). It is found that direction of lid is more effective on heat transfer and fluid flow in the case of mixed convection than it is the case in forced convection. Heat transfer is also decreased with increasing of magnetic field for all studied parameters.     

A Peclet number based analysis of mixed convection for lid-driven porous square cavities with various heating of bottom wall

Mixed convection flows in a lid-driven square cavity filled with porous medium are studied numerically using penalty finite element analysis for uniform and non-uniform heating of bottom wall. The relevant parameters in the present study are Darcy number (Da = 10^− 5− 10^− 5), Grashof number (Gr = 10³− 10⁵), Prandtl number (Pr = 0.026−10) and Reynolds number (Re = 1−10²). The influence of convection is analyzed with Peclet number (Pe = Re.Pr). It is observed that the temperature profiles are symmetric for low values of Pe or Pr even in the presence of asymmetric flow fields irrespective of Da. The flow distribution affects significantly temperature distributions at high Pe irrespective of Da. Effect of Peclet numbers have been further investigated for both natural convection and forced convection dominant regimes at high Da. Strong coupling between flow fields and temperature are observed at high Pe. It is interesting to observe that large isothermal mixing zone at Pr = 10 reduces the overall flow strength compared to Pr = 0.026 case. Local Nusselt numbers show almost uniform and low values for low Peclet numbers and localized enhanced heat transfer rates are observed for high Peclet numbers at Da = 10^− 3.     

Analysis of mixed convection of nanofluid in a 3D lid-driven trapezoidal cavity with flexible side surfaces and inner cylinder

Numerical study of mixed convection in a lid-driven 3D flexible walled trapezoidal cavity with nanofluids was performed by using Galerkin weighted residual finite element method. Effects of various pertinent parameters such as Richardson number (between 0.05 and 50), elastic modulus of the side surfaces (between 1000 and 10⁵), side wall inclination angle (between 0° and 20°) and solid particle volume fraction (between 0 and 0.04) on the fluid flow and heat transfer characteristics in a 3D lid-driven-trapezoidal cavity were numerically examined. It was observed that these characteristics are influenced when the pertinent parameters change. Flexible side surface can be used as control element for heat transfer rate. Increment and reduction in the space which are provided by the flexible side walls result in heat transfer enhancement and deterioration for side wall inclination angle of 0° and 10°. Average Nusselt number enhances by about 9.80% when the value of the elastic modulus is increased from 1000 to 10⁵ for side wall inclination angles of θ = 0°. Adding nanoparticles to the base fluid results in linear increment of heat transfer and at the highest volume fraction, 25.30% of heat transfer enhancement is obtained. A polynomial type correlation for the average Nusselt number along the hot wall was proposed and it has a fourth order polynomial dependence upon the Richardson number and first order dependence upon the solid particle volume fraction.     

Accurate numerical simulation for non-Darcy double-diffusive mixed convection in a double lid-driven porous cavity using SEM

The non-Darcy double-diffusive mixed convection in a double lid-driven porous cavity with two thermosolutal sources is numerically investigated in this article, depicting the effects of different physical parameters on heat and mass transfer in the drying chamber. The flow is generated due to the motion of the horizontal moving lids and the buoyancy produced by the temperature and concentration gradients. The governing equations are discretized by the Legendre spectral element method (SEM) with high accuracy, and an improved time-splitting method is developed to deal with the coupled pressure and velocity in the Brinkman-Forchheimer extended Darcy model. The effects of Darcy number (Da?=?10^?5～10^?1), Richardson number (Ri?=?10^?2～10¹), and buoyancy ratio (Br = ?5?～?5) are investigated, and numerical results are analyzed by contours of streamline, isotherm, heatline, isoconcentration, and massline in detail. Results reveal the pattern of heat and mass transfer with the variation on significant parameters by the average Nusselt and Sherwood numbers on the moving lids of the cavity.     

Finite element based heatline approach to study mixed convection in a porous square cavity with various wall thermal boundary conditions

A penalty finite element method based simulation is performed to analyze the influence of various walls thermal boundary conditions on mixed convection lid driven flows in a square cavity filled with porous medium. The relevant parameters in the present study are Darcy number (Da = 10^?5 ? 10^?3), Grashof number (Gr = 10³ ? 10⁵), Prandtl number (Pr = 0.7–7.2), and Reynolds number (Re = 1–10²). Heatline approach of visualizing heat flow is implemented to gain a complete understanding of complex heat flow patterns. Patterns of heatlines and streamlines are qualitatively similar near the core for convection dominant flow for Da = 10^?3. Symmetric distribution in heatlines, similar to streamlines is observed irrespective of Da at higher Gr in natural convection dominant regime corresponding to smaller values of Re. A single circulation cell in heatlines, similar to streamlines is observed at Da = 10^?3 for forced convection dominance and heatlines are found to emanate from a large portion on the bottom wall illustrating enhanced heat flow for Re = 100. Multiple circulation cells in heatlines are observed at higher Da and Gr for Pr = 0.7 and 7.2. The heat transfer rates along the walls are illustrated by the local Nusselt number distribution based on gradients of heatfunctions. Wavy distribution in heat transfer rates is observed with Da ? 10^?4 for non-uniformly heated walls primarily in natural convection dominant regime. In general, exponential variation of average Nusselt numbers with Grashof number is found except the cases where the side walls are linearly heated. Overall, heatlines are found to be a powerful tool to analyze heat transport within the cavity and also a suitable guideline on explaining the Nusselt number variations.     

Numerical analysis of fluid flow due to mixed convection in a lid-driven cavity having a heated circular hollow cylinder

Mixed convection heat transfer in a lid-driven cavity along with a heated circular hollow cylinder positioned at the center of the cavity has been analyzed numerically. The present study simulates a realistic system such as air-cooled electronic equipment with a heat component or an oven with heater. A Galerkin weighted residual finite element method with a Newton–Raphson iterative algorithm is adopted to solve the governing equations. The computation is carried out for wide ranges of the Richardson numbers, cylinder diameter and solid fluid thermal conductivity ratio. Results are presented in the form of streamlines, isothermal lines, average Nusselt number at the heated surface and fluid temperature in the cavity for the mentioned parameters. It is found that the flow field and temperature distribution strongly depend on the cylinder diameter and also the solid–fluid thermal conductivity ratio at the three convective regimes.     

Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids

The behaviour of nanofluids is investigated numerically inside a two-sided lid-driven differentially heated square cavity to gain insight into convective recirculation and flow processes induced by a nanofluid. A model is developed to analyze the behaviour of nanofluids taking into account the solid volume fraction χ. The transport equations are solved numerically with finite volume approach using SIMPLE algorithm. Comparisons with previously published work on the basis of special cases are performed and found to be in excellent agreement. The left and the right moving walls are maintained at different constant temperatures while the upper and the bottom walls are thermally insulated. Three case were considered depending on the direction of the moving walls. Governing parameters were 0.01 < Ri < 100 but due to space constraints only the results for 0.1 < Ri < 10 are presented. It is found that both the Richardson number and the direction of the moving walls affect the fluid flow and heat transfer in the cavity. Copper–Water nanofluid is used with Pr = 6.2 and solid volume fraction χ is varied as 0.0%, 8%, 16% and 20%. Detailed results are presented for flow pattern and heat transfer curves.     

Hydromagnetic effect on mixed convection in a lid-driven cavity with sinusoidal corrugated bottom surface

The present numerical simulation is conducted to analyze the mixed convection flow and heat transfer in a lid-driven cavity with sinusoidal wavy bottom surface in presence of transverse magnetic field. The enclosure is saturated with electrically conducting fluid. The cavity vertical walls are insulated while the wavy bottom surface is maintained at a uniform temperature higher than the top lid. In addition, the transport equations are solved by using the finite element formulation based on the Galerkin method of weighted residuals. The implications of Reynolds number (Re), Hartmann number (Ha) and number of undulations (λ) on the flow structure and heat transfer characteristics are investigated in detail while, Prandtl number (Pr) and Rayleigh number (Ra) are considered fixed. The trend of the local heat transfer is found to follow a wavy pattern. The results of this investigation illustrate that the average Nusselt number (Nu) at the heated surface increases with an increase of the number of waves as well as the Reynolds number, while decreases with increasing Hartmann number.     

Mixed convection in a lid driven square cavity with an isothermally heated square blockage inside

Laminar mixed convection characteristics in a square cavity with an isothermally heated square blockage inside have been investigated numerically using the finite volume method of the ANSYS FLUENT commercial CFD code. Various different blockage sizes and concentric and eccentric placement of the blockage inside the cavity have been considered. The blockage is maintained at a hot temperature, T_h, and four surfaces of the cavity (including the lid) are maintained at a cold temperature, T_c, under all circumstances. The physical problem is represented mathematically by sets of governing conservation equations of mass, momentum, and energy. The geometrical and flow parameters for the problem are the blockage ratio (B), the blockage placement eccentricities (?_x and ?_y), the Reynolds number (Re), the Grashof number (Gr), and the Richardson number (Ri). The flow and heat transfer behavior in the cavity for a range of Richardson number (0.01–100) at a fixed Reynolds number (100) and Prandtl number (0.71) is examined comprehensively. The variations of the average and local Nusselt number at the blockage surface at various Richardson numbers for different blockage sizes and placement eccentricities are presented. From the analysis of the mixed convection process, it is found that for any size of the blockage placed anywhere in the cavity, the average Nusselt number does not change significantly with increasing Richardson number until it approaches the value of the order of 1 beyond which the average Nusselt number increases rapidly with the Richardson number. For the central placement of the blockage at any fixed Richardson number, the average Nusselt number decreases with increasing blockage ratio and reaches a minimum at around a blockage ratio of slightly larger than 1/2. For further increase of the blockage ratio, the average Nusselt number increases again and becomes independent of the Richardson number. The most preferable heat transfer (based on the average Nusselt number) is obtained when the blockage is placed around the top left and the bottom right corners of the cavity.