Vorticity-stream Function Research Articles

Space Quasi-Periodic Steady Euler Flows Close to the Inviscid Couette Flow

We prove the existence of steady space quasi-periodic stream functions, solutions for the Euler equation in a vorticity-stream function formulation in the two dimensional channel R×[-1,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {R}}}\ imes [-1,1]$$\\end{document}. These solutions bifurcate from a prescribed shear equilibrium near the Couette flow, whose profile induces finitely many modes of oscillations in the horizontal direction for the linearized problem. Using a Nash–Moser implicit function iterative scheme, near such equilibrium we construct small amplitude, space reversible stream functions, slightly deforming the linear solutions and retaining the horizontal quasi-periodic structure. These solutions exist for most values of the parameters characterizing the shear equilibrium. As a by-product, the streamlines of the nonlinear flow exhibit Kelvin’s cat eye-like trajectories arising from the finitely many stagnation lines of the shear equilibrium.

Numerical study of the impacts of stochastic forcing on the vortex in fluid flow

This paper focuses on a numerical study about the stochastic Navier–Stokes equations. Unlike previous studies, this paper focuses on studying these equations from the perspective of vortices. The vorticity–stream function method was proposed to deal with incompressible fluid flow. And a Crank–Nicolson Fourier pseudo-spectral method was put forward to solve the formulation of stream function equation. In addition, we have conducted some numerical experiments to observe the effects of random forcing on vortices in the fluid flow by utilizing stochastic solution.

Physics-informed neural network simulation of thermal cavity flow

Physics-informed neural networks (PINNs) are an emerging technology that can be used both in place of and in conjunction with conventional simulation methods. In this paper, we used PINNs to perform a forward simulation without leveraging known data. Our simulation was of a 2D natural convection-driven cavity using the vorticity-stream function formulation of the Navier-Stokes equations. We used both 2D simulations across the x and z domains at constant Rayleigh (Ra) numbers and 3D simulations across the x, z and Ra domains. The 3D simulation was tested for a PINN’s ability to learn solutions in a higher-dimensional space than standard simulations. The results were validated against published solutions at Ra values of 103\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{3}$$\\end{document}, 104\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{4}$$\\end{document}, 105\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{5}$$\\end{document}, and 106\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{6}$$\\end{document}. Both the 2D simulations and 3D simulations successfully matched the expected results. For the 2D cases, more training iterations were needed for the model to converge at higher Ra values (105\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^5$$\\end{document} and 106\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^6$$\\end{document}) than at lower Ra (103\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^3$$\\end{document} and 104\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^4$$\\end{document}) indicating increased nonlinear fluid-thermal coupling. The 3D case was also able to converge but, but it required more training than any of the 2D cases due to the curse of dimensionality. These results showed the validity of standard simulations via PINNs and the feasibility of higher-order parameter space solutions that are not possible using conventional methods. They also showcased the additional computational demand associated with increasing the dimensionality of the learned parameter space.

Magnetohydrodynamic mixed convection of nanofluid within an annular space filled partially with a porous layer between a lid-driven wall and a discrete heater

Heat and mass transfer inside a partially porous medium is a complex coupling process in which the interface boundary conditions between free and porous space are critical for a wide range of engineering applications, including natural and engineered fractures in oil and gas extraction. In this study, we looked into the numerical simulation of magnetohydrodynamic mixed convection of nanofluid in an annular partially porous space between two coaxial cylinders with a permeable interface, in which we investigated the contribution of the interface in order to better understand the heat transfer processes. The inner cylinder is subjected to a discrete uniform heat flux, while the external cylinder is kept at a uniform cold temperature and moving at a constant speed. The base walls are designed to be impermeable and insulated. A finite difference-based vorticity-stream function method is used to solve the nonlinear coupled conservation equations using the Successive Over Relaxation approach (SOR). The acquired numerical results in terms of streamlines, isotherms, and Nusselt numbers are shown to demonstrate the effect of various control factors such as the Rayleigh number, Darcy number, Hartmann number, Reynold number, nanoparticle concentration, and the direction of the outer wall. The results of this numerical simulation show that the upward or downward direction of the wall under these control factors plays an important role in improving the rate of heat transfer or replacing the magnetic effect by limiting and controlling heat transfer

Numerical simulation of natural convection in a rectangular enclosure filled with porous medium saturated with magnetic nanofluid using Buongiorno's two‐component model

AbstractMotivated by studying emerging nanofluid‐based magnetic fuel cells and hybrid direct absorber solar collectors, a numerical study is presented for buoyancy‐driven flow in a vertical enclosure containing a porous medium saturated with magnetized nanofluid flow under a transverse static magnetic field. The enclosure features adiabatic side walls and vertical heat slits, ensuring consistent cold temperatures on the lower and upper bounded walls. The side walls of the regime are insulated, and the hot slits are centrally located on these walls. The finite difference method (FDM) is applied to employ the transformed dimensionless vorticity–stream function (VSF) formulation of the transport equations. The impact of pertinent parameters on isotherm, streamline, iso‐concentration, and average Nusselt and Sherwood numbers are visualized with contour plots and graphs. Increasing Darcy number values tend to amplify the isotherm magnitudes. Higher Hartmann (magnetic) number values, on the other hand, lead to a reduction in local Nusselt number profiles but do not significantly modify the local Sherwood number. The porous medium permeability, as simulated via the Darcy number, has a strong impact on streamlines, thermal contours, and iso‐concentrations. A reduction in Darcy's number suppresses local Nusselt and Sherwood numbers, whereas an elevation in Rayleigh's number enhances them. Increasing the Buongiorno nanoscale Brownian motion parameter enhances local Nusselt and Sherwood numbers at both cold walls of the enclosure. The simulations provide a deeper insight into enclosure flows involving electrically conducting nanofluids in advanced processing systems for nanomaterials and hybrid fuel cells utilizing electromagnetic and liquid fuel technologies.

Fourth-Order Conservative Non-splitting Semi-Lagrangian Hermite WENO Schemes for Kinetic and Fluid Simulations

AbstractWe present fourth-order conservative non-splitting semi-Lagrangian (SL) Hermite essentially non-oscillatory (HWENO) schemes for linear transport equations with applications for nonlinear problems including the Vlasov–Poisson system, the guiding center Vlasov model, and the incompressible Euler equations in the vorticity-stream function formulation. The proposed SL HWENO schemes combine a weak formulation of the characteristic Galerkin method with two newly constructed HWENO reconstruction methods. The new HWENO reconstructions are meticulously designed to strike a delicate balance between curbing numerical oscillation and introducing excessive dissipation. Mass conservation naturally holds due to the weak formulation of the semi-Lagrangian discontinuous Galerkin method and the design of the HWENO reconstructions. We apply a positivity-preserving limiter to maintain the positivity of numerical solutions when needed. Abundant benchmark tests are performed to verify the effectiveness of the proposed SL HWENO schemes.

On viscoelastic blood in a locally narrow artery with magnetic field: application of distributed-order time fractional Maxwell model

Purpose. The study of hemorheology of heterogeneous vessels is of importance for the therapy of cardiovascular diseases. The present paper aims to investigate the effects of the height parameter of the stenosis, the Reynolds number, the relaxation time and the Hartmann number on the blood flow by applying distributed order time fractional Maxwell model. Methodology. Distributed order time fractional Maxwell constitutive fluid model is used to simulate viscoelastic blood flowing in a narrowing blood vessel with uniform magnetic field. The continuity and momentum equations are established in two-dimensional cylindrical coordinate system for calculation. With vorticity-stream function method, the finite difference technique, combined with the fractional derivative L1 interpolation approximation, is utilized to obtain numerical solutions of velocity and wall shear stress. The effectiveness of the numerical method is verified. Outcomes. It reveals the choice of relaxation time in the governing equation has an influence on the results. The numerical results show that an appropriate increase of Reynolds number can reduce the possibility of vessel damage. Furthermore, when the height parameter of the stenosis increases, the damage to the vessel wall is getting serious. Besides, the wall shear stress along the blood vessel increases with the increase of the magnetic field, but the wall shear stress in the narrowest part of the blood vessel changes little.

Novel enhanced thermal capabilities of Cu-Al2O3/water hybrid nanofluid in an oblique U-shape cavity: On the molecular behaviour through vorticity analysis

This paper presents the numerical investigation of natural convection heat transfer of copper-alumina/pure water hybrid nanofluid in an oblique U-shaped enclosure. The dimensionless governing equation is formed in the stream function-vorticity formulation using dimensionless variables. The Galerkin weighted residual finite element method with the damped Newton-Raphson iteration method is applied to the problem. The heat transfer performance for the system is investigated by varying the Rayleigh numbers, nanoparticle volume fractions and their ratios, and the oblique angles of the enclosure. The fluid flow, heat transfer, and vorticity analysis are conducted extensively for the Rayleigh number up to 106. It was found that the oblique enclosure dwindled the overall heat transfer performance. Correspondingly, the maximum vorticity is at the lowest with zero obliquenesses. The results are used to gain a deeper understanding of the fluid flow and thermal behaviour of the system to achieve more sustainable and greener energy management.

Convective thermal losses of long-term underground hot water storage

A case of underground long-term hot water storage is investigated numerically. The study is based on the unsteady two-dimensional Navier-Stokes equations in Boussinesq approximation applied to a closed cavern with time-dependent temperature boundary conditions on the walls. The problem formulated in a vorticity-stream function statement is solved by finite difference method (FDM) for high values of the Rayleigh number and for the Prandtl number of water. Streamlines, velocity and temperature fields are presented graphically for given moments of time. The evolution of the thermocline thickness in the mid-section of the cavern is slow and illustrates that the hot water zone occupies more than the half of the cavern even after 6 months period. The Nusselt number on the walls shows that the convective thermal losses are small and after certain period of time tend to decrease due to the diminished temperature difference at the walls. The influence of the fluid convection on the thermal losses is evaluated quantitatively, showing high seasonal thermal efficiency of the insulated hot water storage.

Analyzing non-isothermal phase transition problems with natural convection using peridynamic differential operator

In this study, a developed model for non-isothermal phase transition with natural convection is proposed by using peridynamic differential operator (PDDO). The dimensionless governing equations of heat source approach and vorticity-stream function approach are reconstructed into the non-local integral form. The Euler forward difference is used for time integration. The application of the developed PDDO model extends to simulations involving pure phase transition, pure natural convection, and non-isothermal phase transition coupled with natural convection. The validity and accuracy of the developed non-isothermal phase transition model are verified by comparing the PDDO simulation results with results in published literature. This research provides a new alternative approach for simulating phase transition and convection problems.

Combustion of a nanoparticles-laden chemical in a vented cavity

Mixed convective characteristics of the combustion of a nanoparticles-laden fuel (n-butanol nanofluid) in a vented cavity are investigated. The nanofluid and the oxidizer enter the cavity through the inlets on the left and right vertical walls, respectively. However, the resulting product produced from the oxidation process of the fuel exits the cavity through the outlet at the bottom wall. Heat generated from the oxidation process causes natural convection within the cavity. The conjugate effect of natural and forced convection finally gives rise to mixed convection phenomena. In this regard, a mathematical model for mixed convection flow in a vented cavity is formulated with no-slip and isothermal boundary conditions. Having transformed the model into a dimensionless form, the stream function-vorticity formulation is used. The resulting equations are then solved numerically using the finite difference method. Numerical results are illustrated with the streamlines, isotherms, and isolines of fuel and oxidizer concentrations. The maximum values of the stream function (ψmax) and the temperature (θmax) are found to increase with an increase in the Frank–Kamenetskii number (Λ), volume fraction of nanoparticles (φ), and stoichiometric ratio (χ). On the contrary, they decrease with the increase in the Reynolds number (Re). When the Grashof number (Gr) is increased, ψmax increases and θmax decreases. The remaining concentrations of fuel, (CF)min, and oxidizer, (CO)min, are higher for an increase in Gr, whereas the opposite is recognized for increasing Λ. With the increase in Gr and Λ, the steady-state flow in the cavity tends to be oscillating and then chaotic.

Analysis of single-mode Richtmyer–Meshkov instability using high-order incompressible vorticity—streamfunction and shock-capturing simulations

Two- and three-dimensional simulation results obtained using a new high-order incompressible, variable-density vorticity–streamfunction (VS) method and data from previous ninth-order weighted essentially nonoscillatory (WENO) shock-capturing simulations [M. Latini and O. Schilling, “A comparison of two- and three-dimensional single-mode reshocked Richtmyer-Meshkov instability growth,” Physica D 401, 132201 (2020)] are used to investigate the nonlinear dynamics of single-mode Richtmyer–Meshkov instability using a model of a Mach 1.3 air(acetone)/SF6 shock tube experiment [J. W. Jacobs and V. V. Krivets, “Experiments on the late-time development of single-mode Richtmyer–Meshkov instability,” Phys. Fluids 17, 034105 (2005)]. A comparison of the density fields from both simulations with the experimental images demonstrates very good agreement in the large-scale structure with both methods but differences in the small-scale structure. The WENO method captures the small-scale disordered structure observed in the experiment, while the VS method partially captures such structure and yields a strong rotating core. The perturbation amplitude growth from the simulations generally agrees well with the experiment. The simulation bubble and spike amplitudes agree well at early times. At later times, the WENO bubble amplitude is smaller than the VS amplitude and vice versa for the spike amplitude. The predictions of nonlinear single-mode instability growth models are shown to agree with the simulation amplitudes at early-to-intermediate times but underpredict the amplitudes at later times in the nonlinear regime. Visualizations of the mass fraction and enstrophy isosurfaces, velocity and vorticity fields, and baroclinic vorticity production and vortex stretching terms from the three-dimensional simulations indicate that, with the exception of the small-scale structure within the rollups, the VS and WENO results are in good agreement.

Blowup analysis for a quasi-exact 1D model of 3D Euler and Navier–Stokes

We study the singularity formation of a quasi-exact 1D model proposed by Hou and Li (2008 Commun. Pure Appl. Math. 61 661–97). This model is based on an approximation of the axisymmetric Navier–Stokes equations in the r direction. The solution of the 1D model can be used to construct an exact solution of the original 3D Euler and Navier–Stokes equations if the initial angular velocity, angular vorticity, and angular stream function are linear in r. This model shares many intrinsic properties similar to those of the 3D Euler and Navier–Stokes equations. It captures the competition between advection and vortex stretching as in the 1D De Gregorio (De Gregorio 1990 J. Stat. Phys. 59 1251–63; De Gregorio 1996 Math. Methods Appl. Sci. 19 1233–55) model. We show that the inviscid model with weakened advection and smooth initial data or the original 1D model with Hölder continuous data develops a self-similar blowup. We also show that the viscous model with weakened advection and smooth initial data develops a finite time blowup. To obtain sharp estimates for the nonlocal terms, we perform an exact computation for the low-frequency Fourier modes and extract damping in leading order estimates for the high-frequency modes using singularly weighted norms in the energy estimates. The analysis for the viscous case is more subtle since the viscous terms produce some instability if we just use singular weights. We establish the blowup analysis for the viscous model by carefully designing an energy norm that combines a singularly weighted energy norm and a sum of high-order Sobolev norms.

A novel Lagrangian–Eulerian weighted-least squares scheme coupled with other stable techniques for multi-physical fluid flow around complex obstacle

In this paper, a stable novel meshless coupled method is proposed to simulate the non-isothermal magnetohydrodynamics (MHD) flow problems (multi-physics quantities) inside a lid-driven cavity around complex obstacle. The proposed method is mainly motivated by a Lagrangian–Eulerian (L–E) weighted-least squares (WLS) scheme combined with a stream function-vorticity (SFV) and other stable techniques, and it is further to investigate the non-isothermal MHD flow around an airfoil obstacle at large Hartmann (Ha) or Reynolds (Re) number, for the first time. In the present meshless coupled approach (named L–E WLS–SFV), the traditional MHD equations are derived as another form with an SFV method under divergence-free constraint, which can avoid the tedious treatment of pressure on complex irregular obstacle. Then, a stable L–E WLS coupled algorithm is proposed to approximate the space derivatives of multi-physical quantities (velocity, magnetic, temperature, etc.), in which a corrected particle shifting technique is employed to improve the tensile instability among Lagrangian particles moving inside the domain and a second-order upwind scheme is adopted to stabilize large Re number problem in Eulerian fixed nodes near the boundary. Several benchmarks are simulated to show the numerical accuracy and convergence rates of the proposed WLS scheme for MHD flow at different parameters. Subsequently, the case of the non-isothermal MHD flow around a square obstacle under large parameters is simulated by the proposed L–E WLS–SFV method and compared with other numerical results to demonstrate the validity and capacity of the proposed method for multi-physical flow and the necessity of imposing the above two stable techniques. Finally, the case of non-isothermal MHD flow around the circular or airfoil obstacle is numerically investigated, and the important effects of the Hartmann, Rayleigh, and Reynolds numbers on the multi-physical quantities (stream function, vorticity, temperature, and magnetic field) are discussed. The advantages of the proposed method for the muti-physical flow around irregular obstacles are also exemplified. All the numerical results show that the proposed L–E WLS–SVF method is robust and accurate to simulate the multi-physical fluid flow around complex obstacles.

Magnetohydrodynamic and Ferrohydrodynamic Fluid Flow Using the Finite Volume Method

Many problems in fluid mechanics describe the change in the flow under the effect of electromagnetic forces. The present study explores the behaviour of an electric conducting, Newtonian fluid flow applying the magnetohydrodynamics (MHD) and ferrohydrodynamics (FHD) principles. The physical problems for such flows are formulated by the Navier–Stokes equations with the conservation of mass and energy equations, which constitute a coupled non-linear system of partial differential equations subject to analogous boundary conditions. The numerical solution of such physical problems is not a trivial task due to the electromagnetic forces which may cause severe disturbances in the flow field. In the present study, a numerical algorithm based on a finite volume method is developed for the solution of such problems. The basic characteristics of the method are, the set of equations is solved using a simultaneous direct approach, the discretization is achieved using the finite volume method, and the solution is attained solving an implicit non-linear system of algebraic equations with intense source terms created by the non-uniform magnetic field. For the validation of the overall algorithm, comparisons are made with previously published results concerning MHD and FHD flows. The advantages of the proposed methodology are that it is direct and the governing equations are not manipulated like other methods such as the stream function vorticity formulation. Moreover, it is relatively easily extended for the study of three-dimensional problems. This study examines the Hartmann flow and the fluid flow with FHD principles, that formulate MHD and FHD flows, respectively. The major component of the Hartmann flow is the Hartmann number, which increases in value the stronger the Lorentz forces are, thus the fluid decelerates. In the case of FHD fluid flow, the major finding is the creation of vortices close to the external magnetic field source, and the stronger the magnetic field of the source, the larger the vortices are.

Effect of discrete heating on the key parameters and entropy generation in a corrugated enclosure filled with hybrid nanofluid

This research work aims to study the effect of discrete heating on natural convective heat transfer, fluid flow, and entropy generation within a corrugated enclosure filled with water-based molybdenum disulfide (MoS2)–silicon dioxide (SiO2) hybrid nanofluid. In the study, various heights of the discrete heaters in terms of aspect ratios ( AR = 0.2 , 0.4 , 0.6 , 0.8 ) are considered for various Rayleigh numbers ( Ra = 10 3 − 10 6 ) with varying nanoparticle volume fractions ( ϕ = 0 % − 4 % ). Stream function-vorticity formulations of two-dimensional (2D) Navier-Stokes equations and energy equation are used as the governing equations, and a higher-order compact (HOC) finite difference scheme is employed to discretize those equations. The results obtained from the current developed code are first validated with the existing numerical and experimental data. Finally, the present computed results are studied, both qualitatively and quantitatively, in terms of streamlines, isotherms, entropy generations, average Nusselt numbers, and Bejan numbers. At Ra = 10 6 and the aspect ratio of heater length AR = 0.2 , a different and interesting flow and heat transfer phenomenon is observed. The results show that an increase in the aspect ratio of heater length from 0.2 to 0.8 decreases the average Nusselt number and total entropy generation by 57.49% and 13.76%, respectively, for Ra = 10 6 and ϕ = 4 % . In addition, the effect of nanoparticle concentration on total entropy generation increases up to 73.40% when the aspect ratio of heater length increases to 0.8.

Numerical simulation of flow and heat transfer in a pipe with nail-shaped hurdles

We investigate fluid flow and heat transfer in a channel obstructed by nail-shaped hurdles under forced convection using the Finite Difference Method. Due to the absence of an analytical solution, numerical techniques are employed. The channel is geometrically transformed into a rectangular configuration using curvilinear coordinates. Elliptic grid generation is used to discretize the domain. The Navier–Stokes equations are reformulated through the vorticity stream function formulation, and the Finite Difference Method incorporates a coordinate transformation approach. Effects of Reynolds numbers between (20≤Re≤600), Prandtl numbers (0.71,17.1), and nail spans (0.1≤h≤0.7) on velocity, temperature profiles is studied. Local and Average Nusselt numbers on Nail-shaped hurdle are also calculated. The observed maximum Nusselt numbers are 13.49,27.83, and 11.86 at Ra=600,Pr=7.1, and h=0.1, respectively Results show that higher Reynolds and Prandtl numbers increase Nusselt numbers, enhancing heat transfer. Narrower nail spans improve heat transfer efficiency.

Non-orthogonal multiple-relaxation-time lattice Boltzmann method for vorticity-streamfunction formulation

In this work, we propose a non-orthogonal multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) for the two-dimensional (2D) incompressible Navier-Stokes equations in vorticity-stream function formulation. In this method, the vorticity and the streamfunction equations are calculated by two separate LB equations, which are constructed by using the non-orthogonal transformation matrix. Most importantly, to eliminate the residual term that may appear in the recovered vorticity equation, the convection term is handled as a source term in the LB equation rather than being directly recovered by the evolution equation. In addition, due to the convection term in our method is computed with a local scheme, it retains the parallel feature of the standard LBM. We then test the accuracy and the stability of the present LBM by considering various benchmark examples. The results show that the present method has a second-order convergence rate in space. Our numerical results indicate that the present LBM is a good candidate for solving streamfunction-vorticity formulation.

Natural convection of nanofluid flow in a porous medium in a right-angle trapezoidal enclosure: a Tiwari and Das’ nanofluid model

The study uses Tiwari and Das’ nanofluid model to present a study on free convective flows in a right-angled trapezoidal cavity that is saturated with a porous bed and filled with Cu-water nanofluid material. This investigation aims to enhance the characteristics of hybrid fuel cells and energy depository devices by analysing the cavity's heat expansion and fluid flow properties. In trapezoidal enclosures, the inclined and right wall portions are maintained at different isothermal temperatures at all times. Finite difference-based stream function-vorticity numerical simulations are employed to carry out this analysis. The outcomes have been presented for isotherms, streamlines, and Nusselt numbers concerning nanoparticle volume fraction, Darcy number, and various Rayleigh numbers. It is found that the inverse relationship between the thermal Rayleigh number and the nanofluid’s volume has a significant impact on the average Nusselt number.

Numerical study of heating transfer by natural convection in an inclined elliptical cylinder charged with nanofluid

In this paper, thermal transfer with natural convection in a tilted annular cylinder with a Cu-water nanofluid has been numerically studied. The hot interior and cold exterior elliptical surfaces of the enclosure were maintained at constant temperatures Th and Tc , respectively. The governing equations were solved by the stream function-vorticity approach. The finite volume approach was utilized to discretise the controlling equations. The volume fraction range of the nanoparticles and the Rayleigh number was as follows: 0<ϕ<0.08 and 10^4<Ra<10^6, respectively. The inclination angles were γ=30°,45°,and 60°. Results were given as isotherm contours, streamlines, average and local Nusselt numbers. The results indicate that the thermal transfer ratio increases with an increase in the tilt angle, regardless of the nanoparticle size values. and the impact of the inclination angle on the heating transfer rate is more important the higher the Rayleigh number and the more convection there is.