Vorticity Variables Research Articles

Transient natural convection with temperature-dependent viscosity in a square partially porous cavity having a heat-generating source

A numerical study is performed on the transient natural convection with a temperature-dependent viscosity inside a square partially porous cavity with a local heat-generating and heat-conducting source. The vertical walls of the cavity are kept at constant cooling temperature while the horizontal walls are adiabatic. The discrete heat-conducting and heat-generating source is located on the bottom wall. A porous layer is located under the clear fluid layer. Governing equations formulated in dimensionless stream function, vorticity and temperature variables with corresponding initial and boundary conditions are solved using implicit finite difference schemes of the second order. The control parameters are the Darcy number, Ostrogradsky number, viscosity variation parameter, height of the porous layer, and dimensionless time. The effects of these parameters on the average Nusselt number along the heat source surface, average temperature of the heater, fluid flow rate inside the cavity, as well as on the streamlines and isotherms are analyzed. The results show that porous layer thickness and viscosity of the working fluid are very good control parameters for optimization of the passive cooling system.

Free convection of copper–water nanofluid in a porous gap between hot rectangular cylinder and cold circular cylinder under the effect of inclined magnetic field

Natural convection heat transfer of copper–water nanofluid in a porous gap between hot internal rectangular cylinder and cold external circular cylinder under the effect of inclined uniform magnetic field has been investigated. Domain of interest is a porous sector, where horizontal and vertical adiabatic borders are the external circular cylinder radii. Governing equations formulated in dimensionless stream function, vorticity and temperature variables using the Brinkman-extended Darcy model for the porous medium, single-phase nanofluid model with Brinkman correlation for the nanofluid viscosity and Hamilton and Crosser model for the nanofluid thermal conductivity have been solved numerically by the control volume finite element method. Effects of the Rayleigh number, Hartmann number, Darcy number, magnetic field inclination angle, nanoparticles volume fraction, nanoparticles shape factor, nanoparticles material, nanofluid thermal conductivity and dynamic viscosity models and nanofluid electrical conductivity correlation on streamlines, isotherms, local and average Nusselt numbers have been studied. Obtained results have shown the heat transfer enhancement with the Rayleigh number, Darcy number, nanoparticles volume fraction and nanoparticles shape factor, while the heat transfer rate reduces with the Hartmann number and magnetic field inclination angle. At the same time, the average Nusselt number increases at about 16% when nanoparticles volume fraction rises from 0 till 4% for Ra = 105, Ha = 25, while for Ha = 0 one can find the heat transfer rate augmentation at about 9% for the same conditions. In the case of different nanofluid thermal conductivity and dynamic viscosity models, it has been found that KKL model reflects the heat transfer rate reduction with nanoparticles volume fraction, while for the Hamilton–Crosser–Brinkman model, the heat transfer rate increases. Comparison between the Maxwell correlation for the nanofluid electrical conductivity and the base fluid electrical conductivity illustrates an intensification of the convective heat transfer rate for high values of the Rayleigh number (Ra ≥ 104) in the case of Maxwell correlation for the nanofluid electrical conductivity. At the same time, the effect of the nanoparticles volume fraction becomes more significant when nanofluid electrical conductivity is a function of nanoparticles volume fraction.

Mixed convection with entropy generation of nanofluid in a lid-driven cavity under the effects of a heat-conducting solid wall and vertical temperature gradient

Mixed convection combined with entropy generation of an alumina–water nanofluid in a lid-driven cavity with a bottom solid wall of finite thickness and conductivity has been examined numerically. Governing equations formulated in dimensionless stream function and vorticity variables on the basis of a single-phase nanofluid model under the effect of Brownian diffusion have been solved by finite difference method of the second-order accuracy. The effects of Richardson number (Ri = 0.01–10.0), thermal conductivity ratio (1.0 ≤ K ≤ 20.0), solid wall thickness (0.1 ≤ δ ≤ 0.3) and nanoparticles volume fraction (0 ≤ ϕ ≤ 0.05) on streamlines, isotherms and isentropic lines as well as average Nusselt number at solid–fluid interface and rate of fluid flow have been analyzed. It has been found that an increase in nanoparticles volume fraction leads to the heat transfer enhancement and reduction of the average Bejan number.

Transient natural convection in a partially open trapezoidal cavity filled with a water-based nanofluid under the effects of Brownian diffusion and thermophoresis

PurposeThe purpose of this paper is to study about the natural convection of water-based nanofluid in a partially open trapezoidal cavity under the influence of Brownian diffusion and thermophoresis.Design/methodology/approachGoverning equations formulated in dimensionless stream function – vorticity variables – have been solved by finite difference method with a homemade code C++. Effects of Rayleigh number (Ra = 50-1,000), Lewis number (Le = 10-1,000), buoyancy-ratio parameter (Nr = 0.1-5.0), Brownian motion parameter (Nb = 0.1, 1.0) and thermophoresis parameter (Nt = 0.1, 1.0) on nanofluid flow and heat transfer have been studied.FindingsIt is found that high values of Rayleigh and Lewis numbers lead to the homogenization of nanoparticles distributions. For high values of Nt and Nb, heating is more essential and the cavity average temperature rises.Originality/valueThe originality of this work is to analyze natural convection in an open-sided trapezoidal cavity with Brownian diffusion and thermophoresis.

A non-primitive boundary element technique for modeling flow through non-deformable porous medium using Brinkman equation

In this study, we develop a non-primitive boundary integral equation (BIE) method for steady two-dimensional flows of an incompressible Newtonian fluids through porous medium. We assume that the porous medium is isotropic and homogeneous, and use Brinkman equation to model the fluid flow. First, we present BIE method for 2D Brinkman equation in terms of the non-primitive variables namely, stream-function and vorticity variables. Subsequently, a test problem namely, the lid-driven porous cavity over a unit square domain is presented to assert the accuracy of our BEM code. Finally, we discuss an application of our proposed method to flows through porous wavy channel, which is a problem of significant interest in the micro-fluidics, biological domains and groundwater flows. We observe that the rate of convergence ( $$R_{c}$$ ) increases with increasing Darcy number. For low Darcy number streamlines follow the curvature of the wavy-walled channel and no circulation occurs irrespective of the wave–amplitude, while for high Darcy number the flow circulation occurs near the crest of the wavy-walled channel, when the wave–amplitude is large enough.

Optimal vorticity accuracy in an efficient velocity–vorticity method for the 2D Navier–Stokes equations

We study a velocity–vorticity scheme for the 2D incompressible Navier–Stokes equations, which is based on a formulation that couples the rotation form of the momentum equation with the vorticity equation, and a temporal discretization that stably decouples the system at each time step and allows for simultaneous solving of the vorticity equation and velocity–pressure system (thus if special care is taken in its implementation, the method can have no extra cost compared to common velocity–pressure schemes). This scheme was recently shown to be unconditionally long-time $$H^1$$ stable for both velocity and vorticity, which is a property not shared by any common velocity–pressure method. Herein, we analyze the scheme’s convergence, and prove that it yields unconditional optimal accuracy for both velocity and vorticity, thus making it advantageous over common velocity–pressure schemes if the vorticity variable is of interest. Numerical experiments are given that illustrate the theory and demonstrate the scheme’s usefulness on some benchmark problems.

Transient free convection in an inclined square porous cavity filled with a nanofluid using LTNE and Buongiorno’s models

The combined effect of Brownian diffusion, thermophoresis and cavity inclination angle on natural convective heat transfer in an inclined porous enclosure has been studied numerically. Fluid containing nanoparticles of low concentration circulates inside the cavity under the effect of the buoyancy force. Governing equations with corresponding boundary conditions formulated using the non-dimensional stream function and vorticity variables have been solved by the finite difference method. An influence of the cavity inclination angle, Darcy and Nield numbers on nanofluid flow and heat transfer has been investigated. It has been found that high Nield numbers illustrate more equilibrium temperature distribution inside the porous cavity.

Natural convection in a rotating cavity partially filled with a porous medium under the effect of a local heater

The combined effect of uniform rotation and porous layer on natural convective heat transfer in a square cavity with a local heater has been studied numerically. The local thermal equilibrium model with the Brinkman-extended Darcy law has been used for description of the heat transfer and fluid flow within the porous layer. Governing equations with corresponding boundary conditions formulated using the non-dimensional stream function and vorticity variables have been solved by the finite difference method. The effects of Taylor number and porous layer thickness on fluid flow and heat transfer have been examined. It has been found a non-linear effect of the porous layer thickness on the heat transfer rate.

Effect of thermal radiation on natural convection in a square porous cavity filled with a fluid of temperature-dependent viscosity

A numerical study of the natural convection combined with thermal radiation inside a square porous cavity filled with a fluid of temperature-dependent viscosity is carried out. The side horizontal walls are assumed to be adiabatic while both the left and right vertical walls are kept at constant but different temperatures. The Rosseland diffusion approximation is used to describe the radiative heat flux in the energy equation. The governing equations formulated in dimensionless stream function, vorticity, and temperature variables are solved using finite difference method. A parametric analysis illustrating the effects of the radiation parameter (0 ? Rd ? 10), Darcy number (10?5 ? Da ? 10?2), and viscosity variation parameter (0 ? C ? 6) on fluid flow and heat transfer is implemented. The results show an essential intensification of convective flow with an increase in the radiation parameter.

Free Convection in an Open Triangular Cavity Filled With a Nanofluid Under the Effects of Brownian Diffusion, Thermophoresis and Local Heater

Natural convection of a water-based nanofluid in a partially open triangular cavity with a local heat source of constant temperature under the effect of Brownian diffusion and thermophoresis has been analyzed numerically. Governing equations formulated in dimensionless stream function and vorticity variables on the basis of two-phase nanofluid model with corresponding initial and boundary conditions have been solved by finite difference method. Detailed study of the effect of Rayleigh number, buoyancy-ratio parameter, and local heater location on fluid flow and heat transfer has been carried out. It has been revealed that an increase in the buoyancy force magnitude leads to homogenization of nanoparticles distribution inside the cavity. A growth of a distance between the heater and the cavity corner illustrates the heat transfer enhancement.

Stokes Flow Inside Topographically Patterned Microchannel Using Boundary Element Method

Abstarct This study focuses on the investigation of two-dimensional steady Stokes flow inside topographically patterned microchannel. Boundary element method (BEM) is used to solve the Stokes equation and obtain the streamline profiles. The velocity field and pressure gradients are obtained by taking the appropriate spatial derivatives of the stream function and vorticity variables. We restrict ourselves to rectangular stepped geometries and study the effect of variation of step width, step height and step frequency. Interestingly, ‘crown-shaped’ patterns in the horizontal velocity profiles are formed when a sudden contraction is met in the flow region. Pressure gradients, together with the velocity and streamline profiles are analyzed to gain a wholesome understanding of the flow physics.

Algorithm for solving the Navier–Stokes equations for the modeling of creeping flows

We study a coupled algorithm for solving the two-dimensional Navier–Stokes equations in the stream function–vorticity variables. The algorithm is based on a finite-difference scheme in which the inertial terms in the vortex transport equation are taken from the lower time layer and the dissipative terms, from the upper time layer. In the linear approximation, we study the stability of this algorithm and use test computations to show its advantages when modeling flows at moderate Reynolds numbers.

Heatline visualization of natural convection in a thick walled open cavity filled with a nanofluid

Natural convection of an alumina-water nanofluid in a partially open rectangular cavity with a left heat-conducting solid wall of finite thickness and conductivity has been studied numerically. Governing equations formulated in dimensionless stream function and vorticity variables on the basis of a single-phase nanofluid model with corresponding boundary conditions have been solved by finite difference method. Analysis of the influence of Rayleigh number (Ra=103–106), thermal conductivity ratio (1⩽K⩽20), solid wall thickness (0.1⩽δ⩽0.3) and nanoparticles volume fraction (0⩽ϕ⩽0.05) on streamlines, heatlines and isotherms as well as average Nusselt number at solid-fluid interface and fluid flow rate has been carried out. It has been revealed that for the considered models for an effective thermal conductivity ratio and dynamic viscosity an increase in the nanoparticles volume fraction leads to the heat transfer reduction and decrease of the fluid flow rate.

Numerical simulation of a conjugate turbulent natural convection combined with surface thermal radiation in an enclosure with a heat source

A detailed numerical analysis of complex heat transfer (turbulent natural convection, conduction and surface thermal radiation) in a rectangular enclosure with a heat source has been carried out. The finite difference method for the solution of the governing equations using the dimensionless stream function, vorticity and temperature variables has been utilized. The effects of Rayleigh number in a range from 108 to 1011, thermal conductivity ratio in a wide range from 10 to 1000, as well as internal surface emissivity 0≤ε˜<1 on the fluid flow and heat transfer have been extensively explored. Detailed results including temperature fields, flow profiles, and average Nusselt numbers have been presented. In this investigation we have tried to study the main regularities of heat and mass transfer in the considered domain.

Free convection in a trapezoidal cavity filled with a micropolar fluid

The present investigation deals with the study of steady laminar natural convective flow and heat transfer of micropolar fluids in a trapezoidal cavity. The bottom wall of the cavity is kept at high constant temperature, the inclined walls is kept at low constant temperatures while the top horizontal wall is adiabatic. Governing equations formulated in dimensionless stream function and vorticity variables has been solved by finite difference method of the second order accuracy. Comprehensive verification of the utilized numerical method and mathematical model has shown a good agreement with numerical data of other authors. Computations have been carried out to analyze the effects of Rayleigh number, Prandtl number and vortex viscosity parameter both for weak and strong concentration cases. Obtained results have been presented in the form of streamlines, isotherms and vorticity profiles as well as the variation of the average Nusselt number and fluid flow rate. It has been shown that an increase in the vortex viscosity parameter leads to attenuation of the convective flow and heat transfer inside the cavity.

Unsteady natural convection with summer boundary conditions in a habitat at high Rayleigh number and at high time

A numerical study of the thermoconvective instabilities at high time in a habitat filled with Newtonian fluid is conducted. The gable roof of the habitat is subjected to a heat flux of constant density, and its side walls and floor are, respectively, adiabatic and isothermal. Based on the Boussinesq assumptions, the summer thermal and dynamic conditions are numerically studied using unsteady natural convection equations formulated with vorticity and stream-function variables. The finite volume method is used to generate the set of equations, which are solved by the iterative under-relaxation line-by-line method of Gauss–Seidel. Sudden changes in the average Nusselt number and in the extreme values of the stream functions at Ra=1×108 show that the initially unicellular flow in a pseudo-conductive regime becomes a multicellular flow with the emergence and disappearance of cells. In time and space, the change of the flow behaviour is observed more rapidly if the active walls are close to the cold horizontal wall.

Pressure Losses of Power-Law Fluid Flow through an Axisymmetric Sudden Contraction

In this paper the laminar stationary power-law fluid flow through an axisymmetric pipe contraction is investigated. The mathematical statement of the problem is formulated using stream function and vorticity variables. For obtaining a stationary solution the relaxation method with following realization of numerical algorithm based on finite difference alternative directions scheme is utilized. Implemented parametrical investigations allow obtaining the dependence of local resistance coefficients on Reynolds number, nonlinearity degree and piping contraction ratio.

Magnetic field effect on the unsteady natural convection in a wavy-walled cavity filled with a nanofluid: Buongiorno's mathematical model

A numerical investigation is performed on the unsteady natural convection of water based nanofluid within a wavy-walled cavity under the influence of a uniform inclined magnetic field using the mathematical nanofluid model proposed by Buongiorno. The left vertical wavy and right vertical flat walls of the cavity are kept at constant but different temperatures whereas the top and bottom horizontal walls are adiabatic. All boundaries are assumed to be impermeable to the base fluid and nanoparticles. The mathematical model formulated in dimensionless stream function, vorticity and temperature variables is solved using implicit finite difference schemes of the second order. The governing parameters are the Hartmann number, undulation number, wavy contraction ratio, inclination angle of the magnetic field relative to the gravity vector and dimensionless time. The effects of these parameters on the average Nusselt number along the hot wavy wall, as well as on the streamlines, isotherms and isoconcentrations are analyzed.

Numerical simulation of turbulent natural convection combined with surface thermal radiation in a square cavity

Purpose – The purpose of this paper is to present transient turbulent natural convection with surface thermal radiation in a square differentially heated enclosure using non-primitive variables like stream function and vorticity. Design/methodology/approach – The governing equations formulated in dimensionless variables “stream function, vorticity and temperature,” within the Boussinesq approach taking into account the standard two equation k-ε turbulence model with physical boundary conditions have been solved using an iterative implicit finite-difference method. Findings – It has been found that using of the presented algebraic transformation of the mesh allows to effectively conduct numerical analysis of turbulent natural convection with thermal surface radiation. It has been shown that the average convective Nusselt number increases with the Rayleigh number and decreases with the surface emissivity, while the average radiative Nusselt number is an increasing function of these key parameters. It has been shown that a presence of surface thermal radiation effect leads to an expansion of the eddy viscosity zones close to the walls. Originality/value – It should be noted that for the first time in this paper we used stream function and vorticity variables with very effective algebraic transformation of the mesh in order to create a non-uniform mesh for an analysis of turbulent flow. Such method allows to reduce the computational time essentially in comparison with using of the primitive variables. The considered method has been successfully validated on the basis of the experimental and numerical data of other authors in case of turbulent natural convection without thermal radiation. The used numerical method would benefit scientists and engineers to become familiar with the analysis of turbulent convective heat and mass transfer, and the way to predict the properties of the turbulent flow in advanced nuclear systems, in industrial sectors including transportation, power generation, chemical sectors, ventilation, air-conditioning, etc.

Study of melting of a pure gallium under influence of magnetic field in a square cavity with a local heat source

Numerical analysis of the unsteady natural convection with phase transitions inside the pure galluim under the influence of a uniform magnetic field with a heat source of constant temperature has been conducted in the presence of magnetic field. The vertical walls of the cavity are kept at low constant temperatures whereas the top and bottom horizontal walls are adiabatic with the exception of the heat source of high constant temperature. The mathematical model formulated in dimensionless stream function, vorticity and temperature variables is solved using the implicit finite difference schemes of the second order. The governing parameters are the Rayleigh and Hartmann numbers, and the dimensionless time. The effects of these parameters on the streamlines and isotherms are analyzed.