Stream Function-vorticity Formulation Research Articles

Nonconforming P1 Finite Element for the Biharmonic Problem and Its Application to Stokes Problem

The aim of this paper is to simulate the two-dimensional stationary Stokes problem. In vorticity-Stream function formulation, the Stokes problem is reduced to a biharmonic one; this approach leads to a formulation only based on the stream functions and therefore can only be applied to two-dimensional problems. The idea developed in this paper is to use the discretization of the Laplace operator by the nonconforming [Formula: see text] finite element. For the solutions which admit a regularity greater than [Formula: see text], in the general case, the convergence of the method is shown with the techniques of compactness. For solutions in [Formula: see text] an error estimate is proved, and numerical experiments are performed for the steady-driven cavity problem.

Two-Dimensional Steady Boussinesq Convection: Existence, Computation and Scaling

This research investigates laser-induced convection through a stream function-vorticity formulation. Specifically, this paper considers a solution to the steady Boussinesq Navier–Stokes equations in two dimensions with a slip boundary condition on a finite box. A fixed-point algorithm is introduced in stream function-vorticity variables, followed by a proof of the existence of steady solutions for small laser amplitudes. From this analysis, an asymptotic relationship is demonstrated between the nondimensional fluid parameters and least upper bounds for laser amplitudes that guarantee existence, which accords with numerical results implementing the algorithm in a finite difference scheme. The findings indicate that the upper bound for laser amplitude scales by O(Re−2Pe−1Ri−1) when Re≫Pe, and by O(Re−1Pe−2Ri−1) when Pe≫Re. These results suggest that the existence of steady solutions is heavily dependent on the size of the Reynolds (Re) and Peclet (Pe) numbers, as noted in previous studies. The simulations of steady solutions indicate the presence of symmetric vortex rings, which agrees with experimental results described in the literature. From these results, relevant implications to thermal blooming in laser propagation simulations are discussed.

Mixed Convection Heat Transfer and Entropy Generation in a Water-filled Square Cavity Partially Heated from Below: The Effects of Richardson and Prandtl Numbers

In the present study, fluid flow, heat transfer, and entropy generation for mixed convection inside a water-filled square cavity were investigated numerically. The sidewalls of the cavity, which move upwards, are kept at low-temperature Tc while only a part in the center of the bottom wall is kept at high-temperature Th and the remaining parts are kept adiabatic. The governing equations, in stream function–vorticity form, are discretized and solved using the finite difference method. Particular attention was paid to the influence of the Prandtl numbers of 5.534, 3.045 and 2, corresponding respectively to the water temperatures of 303.15 K, 333.15 K and 363.15 K. The numerical results are presented in the form of streamlines, isotherms, and entropy generation contours for different values of the Richardson numbers at an arbitrary Reynolds number Re=102. Besides this, the evolution of the average Nusselt number and the average entropy generation is also reported. The obtained results show interesting behaviors of the flow and thermal fields, which mainly involve stable symmetric and non-symmetric steady-state solutions, as well as unsteady regimes, depending on specific values of the Richardson and Prandtl numbers. It is additionally observed that the average Nusselt number increases and the average entropy generation decreases when both the Richardson and Prandtl numbers increase.

THE LID-DRIVEN HYDROMAGNETIC FLOW OF NEWTONIAN AND NON-NEWTONIAN LIQUIDS IN A SQUARE CAVITY

In this paper, we formulate the steady hydromagnetic lid-driven cavity problem in a stream function-vorticity form for weakly electrically conducting Newtonian and non-Newtonian liquids. Then we solve them by using the concept of pseudo time derivative. The classical benchmark results of the Newtonian liquid are recovered as a limiting case and the inhibiting influence of the magnetic field on the Newtonian and non-Newtonian liquids’ flow field is clearly depicted through graphs. We also show certain aspects of the flow for the first time in tables.

Prandtl Number Effects on the Entropy Generation During the Transient Mixed Convection in a Square Cavity Heated from Below

This numerical study considers the mixed convection, heat transfer and the entropy generation within a square cavity partially heated from below with moving cooled vertical sidewalls. All the other horizontal sides of the cavity are assumed adiabatic. The governing equations, in stream function–vorticity form, are discretized and solved using the finite difference method. Numerical simulations are carried out, by varying the Richardson number, to show the impact of the Prandtl number on the thermal, flow fields, and more particularly on the entropy generation. Three working fluid, generally used in practice, namely mercury (Pr = 0.0251), air (Pr = 0.7296) and water (Pr = 6.263) are investigated and compared. Predicted streamlines, isotherms, entropy generation, as well as average Nusselt numbers are presented. The obtained results reveal that the impact of the Prandtl number is relatively significant both on the heat transfer performance and on the entropy generation. The average Nusselt number increase with increasing Prandtl number. Its value varies thereabouts from 3.7 to 3.8 for mercury, from 5.5 to 13 for air and, from 12.5 to 15 for water. In addition, it is found that the total average entropy generation is significantly higher in the case of mercury (Pr«1) and water (Pr»1) than in the case of air (Pr~1). Its value varies approximately from 700 to 1100 W/m3 K for mercury, from 200 to 500 W/m3 K for water and, from 0.03 to 5 W/m3 K for air.

Stabilized leapfrog scheme run backward in time, and the explicit O(Δ t)2 stepwise computation of ill-posed time-reversed 2D Navier–Stokes equations

Richardson's leapfrog scheme is notoriously unconditionally unstable in well-posed, forward, linear dissipative evolution equations. Remarkably, that scheme can be stabilized, marched backward in time, and provide useful reconstructions in an interesting but limited class of ill-posed, time-reversed, 2D incompressible Navier–Stokes initial value problems. Stability is achieved by applying a compensating smoothing operator at each time step to quench the instability. Eventually, this leads to a distortion away from the true solution. This is the stabilization penalty. In many interesting cases, that penalty is sufficiently small to allow for useful results. Effective smoothing operators based on , with real p>2, can be efficiently synthesized using FFT algorithms. Similar stabilizing techniques were successfully applied in several other ill-posed evolution equations. The analysis of numerical stability is restricted to a related linear problem. However, as is found in leapfrog computations of well-posed meteorological and oceanic wave propagation problems, such linear stability is necessary but not sufficient in the presence of nonlinearities. Here, likewise, additional Robert–Asselin–Williams (RAW) time-domain filtering must be used to prevent characteristic leapfrog nonlinear instability unrelated to ill-posedness. Several 2D Navier–Stokes backward reconstruction examples are included, based on the stream function-vorticity formulation, and focusing on pixel images of recognizable objects. Such images, associated with non-smooth underlying intensity data, are used to create severely distorted data at time T>0. Successful backward recovery is shown to be possible at parameter values significantly exceeding expectations.

A numerical method for solution of incompressible Navier–Stokes equations in streamfunction‐vorticity formulation

In this work, we have proposed a numerical approach for solving the incompressible Navier–Stokes equations in the streamfunction-vorticity formulation. The numerical scheme is based on the diagonally implicit fractional-step θ − (DIFST) method used for the time discretization and the conforming finite element method for the spatial discretization. The accuracy and efficiency of the scheme are validated by solving some benchmark problems. The numerical simulations are carried out using the DUNE-PDELab open-source software package. The comparison of DIFST scheme with different time discretization schemes is provided in terms of CPU time. Also, different solvers with preconditioners are investigated for solving the resulting algebraic system of equations numerically.

Effects of partial magnetic field in a vented square cavity with aiding and opposing of MWCNT–water nanofluid flows

A numerical investigation for examining the effect of a uniform partial magnetic field on fluid flow and heat transfer of Multi Walled Carbon Nano Tube (MWCNT)-water nanofluid is done in a vented square cavity considering aiding and opposing flows without changing temperature boundary conditions. Modified Maron–Pierce equation for dynamic viscosity and Xue equation for thermal conductivity of nanofluid are achieved. The radial basis function (Rbf) based pseudo spectral method with cubic polyharmonic spline Rbf and nonuniform Gauss–Chebyshev–Lobatto (GCL) nodes are performed to solve the problem governed by stream function–vorticity formulation. The considered parameters are Richardson number (0.1≤Ri≤100), the center of the partial magnetic field (0.25≤lbc≤0.75), Hartmann number (0≤Ha≤100), Reynolds number (100≤Re≤400) and concentration of nanoparticles (0≤ϕ≤0.05). The results show that the partial magnetic field centered in the middle of the left wall has more reducing effect on convective heat transfer than the bottom and top centered one in both cases of flows. Convective heat transfer increases with the rise in solid volume fraction up to 0.02, but then a decrease is remarkably noticed.

Vortex dynamics on a Möbius strip

We consider the dynamics of a two-dimensional incompressible perfect fluid on a Möbius strip embedded in $\mathbb {R}^{3}$ . The vorticity–stream function formulation of the Euler equations is derived from an exterior-calculus form of the momentum equation. The non-orientability of the Möbius strip and the distinction between forms and pseudo-forms this introduces lead to unusual properties: a boundary condition is provided by the conservation of circulation along the single boundary of the strip, and there is no integral conservation for the vorticity density or for any odd function thereof. A finite-difference numerical implementation is used to illustrate the Möbius-strip realisation of familiar phenomena: translation of vortices along boundaries, shear instability and decaying turbulence.

A Preliminary Study of Oxygen Transport from Capillary to Tissue using Finite Difference Method

In this study, Finite Difference Method is used to examine oxygen transport from blood capillary to tissue. Tissue is modelled as porous media and oxygen rich blood is transported throughout the tissue from the blood vessel by advection and diffusion. Uniform consumption by tissue is considered for simplicity. Steady state flow patterns and concentration contours are discussed. A simplistic 2-D model was created after reviewing the literature. The model considers interlinkage of capillaries. Stream Function-Vorticity formulation coupled with advection-diffusion equation are solved simultaneously to evaluate velocities and concentration values in the domain. This report focuses on the how various parameters affect flow and mass transfer inside a porous media. Effect of various non dimensional parameters like Darcy number and non-dimensional absorption coefficient are discussed.

A fourth-order compact difference algorithm for numerical solution of natural convection in an inclined square enclosure

In this work, a fourth-order compact finite difference (FD) algorithm on the nine-point 2-D stencil is proposed to simulate numerically natural convection in an inclined square enclosure using the vorticity-stream function form of the Naiver-Stokes equations. The key feature of the proposed algorithm is that for all Rayleigh numbers Ra of physical interest, the point-successive overrelaxation or point-successive underrelaxation (SOR) iteration can be used. The numerical capability of the presented algorithm is demonstrated by the application to natural convection in an inclined square enclosure. The angle of inclination of the cavity is varied from (heated from above) to (heated from below) in steps of Computations are performed for Rayleigh numbers equal to 103, 104, 105, 106 and 107 while the Prandtl number is kept constant (Pr = 0.71). Test results, which are presented in terms of streamlines, isotherms, isovorticities and local and average Nusselt number, indicate that the present algorithm could predict the benchmark results for temperature and flow fields on relatively coarser grids.

Spatio-temporal evolution of magnetohydrodynamic blood flow and heat dynamics through a porous medium in a wavy-walled artery

Background and objectiveIn a healthy body, the elastic wall of the arteries forms wave-like structures resulting from the continuous pumping of the heart. The systolic and diastolic phases generate a contraction and expansion pattern, which is mimicked in this study by considering a wavy-walled arterial structure. A numerical investigation of the spatio-temporal flow of blood and heat transfer through a porous medium under the action of magnetic field strength is conducted. MethodThe governing equations of the blood flow in the Darcy model are simulated by applying a vorticity-stream function formulation approach. The transformed dimensionless equations are further discretized using the finite difference method by developing the Peaceman-Rachford alternating direction implicit (P-R ADI) scheme. ResultsThe computational results for the axial velocity, temperature distribution, flow visualization using the streamlines and vorticity contours, isotherms, wall shear stress and the average Nusselt number are presented graphically for different values of the physical parameters. ConclusionsThe study shows that the axial velocity increases with an increase in the Darcy number, and a similar phenomenon is observed because of an amplitude variation in the wavy wall. Both temperature and wall shear stress decreases with an increase in the Darcy number. The average Nusselt number increases with the magnetic field strength, while it has a reducing tendency due to the permeability of the porous medium.

Cu and Cu-SWCNT Nanoparticles' Suspension in Pulsatile Casson Fluid Flow via Darcy-Forchheimer Porous Channel with Compliant Walls: A Prospective Model for Blood Flow in Stenosed Arteries.

The use of experimental relations to approximate the efficient thermophysical properties of a nanofluid (NF) with Cu nanoparticles (NPs) and hybrid nanofluid (HNF) with Cu-SWCNT NPs and subsequently model the two-dimensional pulsatile Casson fluid flow under the impact of the magnetic field and thermal radiation is a novelty of the current study. Heat and mass transfer analysis of the pulsatile flow of non-Newtonian Casson HNF via a Darcy–Forchheimer porous channel with compliant walls is presented. Such a problem offers a prospective model to study the blood flow via stenosed arteries. A finite-difference flow solver is used to numerically solve the system obtained using the vorticity stream function formulation on the time-dependent governing equations. The behavior of Cu-based NF and Cu-SWCNT-based HNF on the wall shear stress (WSS), velocity, temperature, and concentration profiles are analyzed graphically. The influence of the Casson parameter, radiation parameter, Hartmann number, Darcy number, Soret number, Reynolds number, Strouhal number, and Peclet number on the flow profiles are analyzed. Furthermore, the influence of the flow parameters on the non-dimensional numbers such as the skin friction coefficient, Nusselt number, and Sherwood number is also discussed. These quantities escalate as the Reynolds number is enhanced and reduce by escalating the porosity parameter. The Peclet number shows a high impact on the microorganism’s density in a blood NF. The HNF has been shown to have superior thermal properties to the traditional one. These results could help in devising hydraulic treatments for blood flow in highly stenosed arteries, biomechanical system design, and industrial plants in which flow pulsation is essential.

NUMERICAL INVESTIGATION OF NATURAL CONVECTION WITH HEATED TUBES IN TUNNEL GREENHOUSE

In this research, a numerical study was carried out on heat transfer by natural convection, in a closed tunnel greenhouse, in the range of the Rayleigh number (103≤Ra≤106). Were considered in the study, the number of heating tubes used (1≤Nt≤7), which were equidistant inside the greenhouse volume, when the bottom at an average temperature and cold Roof. The governing equations written in a bicylindrical coordinates were discretized using the finite volume method and vorticity-stream function formulation; the resulting algebraic equations were solved using successive over relaxation method (S.O.R). First, the effect of the Rayleigh number on heat transfer was examined for a fixed number of tubes as reference (Nt = 3) and the number of tubes was varied to investigate the influence on heat transfer in the greenhouse. Finally, the results obtained were summarized in the form of isotherms and streamlines, and for the average Nusselt number profile; in addition to the horizontal and vertical velocities and temperatures. However, in the reference case, for low Rayleigh numbers, the heat transfer is dominated by pure conduction.With the increase of the Rayleigh number and the number of tubes Nt, the natural convection becomes more dominant and the heat transfer increases, and in general the heat transfer increase with the increasing number of tubes.

Numerical Investigation of MHD Pulsatile Flow of Micropolar Fluid in a Channel with Symmetrically Constricted Walls

This article presented an analysis of the pulsatile flow of non-Newtonian micropolar (MP) fluid under Lorentz force’s effect in a channel with symmetrical constrictions on the walls. The governing equations were first converted into the vorticity–stream function form, and a finite difference-based solver was used to solve it numerically on a Cartesian grid. The impacts of different flow controlling parameters, including the Hartman number, Strouhal number, Reynolds number, and MP parameter on the flow profiles, were studied. The wall shear stress (WSS), axial, and micro-rotation velocity profiles were depicted visually. The streamlines and vorticity patterns of the flow were also sketched. It is evident from the numerical results that the flow separation region near constriction as well as flattening of the axial velocity component is effectively controlled by the Hartmann number. At the maximum flow rate, the WSS attained its peak. The WSS increased in both the Hartmann number and Reynolds number, whereas it declined with the higher values of the MP parameter. The micro-rotation velocity increased in the Reynolds number, and it declined with increment in the MP parameter.

Flow past a rotating cylinder predicted by a compact Eulerian viscous vorticity method under non-inertial rotating frame

An Eulerian vorticity method, based on a vorticity-stream function formulation, initially developed for the 2-D viscous unsteady flow past a circular cylinder, is extended in the case of 2-D flow past a rotating cylinder in a rotational frame of reference. The vorticity equation is shown to have the same form as in the case of an inertial frame of reference. Thus, the general structure of the numerical algorithm remains the same as in the inertial frame of reference. The non-inertial effects only enter in the calculation of the velocity field, and the vorticity induced velocity is modified to account for the non-inertial contributions due to the frame rotation. Results from the current method for the flow past a rotating cylinder are presented and shown to agree very well with those from other numerical methods and experimental benchmark tests.

Numerical Simulation of the Thermally Developed Pulsatile Flow of a Hybrid Nanofluid in a Constricted Channel

Heat transfer analysis of the pulsatile flow of a hybrid nanofluid through a constricted channel under the impact of a magnetic field and thermal radiation is presented. Hybrid nanofluids form a new class of nanofluids, distinguished by the thermal properties and functional utilities for improving the heat transfer rate. The behaviors of a water-based copper nanofluid and water-based copper plus a single-wall carbon nanotube, i.e., (Cu–SWCNT/water), hybrid nanofluid over each of velocity, wall shear stress, and temperature profiles, are visualized graphically. The time-dependent governing equations of the incompressible fluid flow are transformed to the vorticity-stream function formulation and solved numerically using the finite difference method. The laminar flow simulations are carried out in 2D for simplicity as the flow profiles are assumed to vary only in the 2D plane represented by the 2D Cartesian geometry. The streamlines and vorticity contours are also shown to demonstrate the flow behviour along the channel. For comparison of the flow characteristics and heat transfer rate, the impacts of variations in Hartmann number, Strouhal number, Prandtl number, and the thermal radiation parameter are analyzed. The effects of the emerging parameters on the skin friction coefficient and Nusselt number are also examined. The hybrid nanofluid is demonstrated to have better thermal characteristics than the traditional one.

Impact of Lorentz Force in Thermally Developed Pulsatile Micropolar Fluid Flow in a Constricted Channel

This work aimed to analyze the heat transfer of micropolar fluid flow in a constricted channel influenced by thermal radiation and the Lorentz force. A finite difference-based flow solver, on a Cartesian grid, is used for the numerical solution after transforming the governing equations into the vorticity-stream function form. The impact of various emerging parameters on the wall shear stress, axial velocity, micro-rotation velocity and temperature profiles is discussed in this paper. The temperature profile is observed to have an inciting trend towards the thermal radiation, whereas it has a declining trend towards the Hartman and Prandtl numbers. The axial velocity profile has an inciting trend towards the Hartman number, whereas it has a declining trend towards the micropolar parameter and Reynolds number. The micro-rotation velocity escalates with the micropolar parameter and Hartman number, whereas it de-escalates with the Reynolds number. The Nusselt number is observed to have a direct relationship with the Prandtl and Reynolds numbers.

Analysis of Thermal Creep Effects on Fluid Flow and Heat Transfer in a Microchannel Gas Heating

Abstract Thermal creep effects on fluid flow and heat transfer in a microchannel gas flow at low velocities are studied numerically. The continuity and Navier–Stokes equations in vorticity–stream function form, coupled with the energy equation, are solved, considering the thermal creep effect due to the longitudinal temperature gradient along the channel wall in addition to the combined effects of viscous dissipation, pressure work, axial conduction, shear work, and nonequilibrium conditions at the gas–wall interface. The governing equations are also solved without thermal creep, and comparisons between the two solutions are presented to evaluate the thermal creep effect on the flow field in the slip flow regime at relatively low Reynolds numbers. The results presented show that the thermal creep effect on both velocity and temperature fields become more significant as the Reynolds number decreases. Thermal creep effect on the velocity field also extends a longer distance downstream the channel as the Reynolds number decreases, hence increasing the hydrodynamics entrance length. Thermal creep can cause high positive velocity gradients at the upper channel wall for gas heating and hence reverse the flow rotation in the fluid layers adjacent to the wall. Thermal creep also results in a higher gas temperature in the developing region and higher heat exchange between the fluid and the channel wall in the entrance region. Thermal creep effect on heat exchange between the gas and the channel wall becomes more significant as the Knudsen number decreases.

Simulation of Thermal Radiation and Turbulent Free Convection in an Enclosure with a Glass Wall and a Local Heater

In this study, a numerical modelling of thermal radiation and turbulent thermogravitational convection in a large-scale chamber containing a thermally-generating element is conducted. The lower border of the cabinet is maintained under adiabatic conditions, while on the other walls the convective boundary conditions (Robin boundary condition) are used. The managing equations with corresponding restrictions are transformed using the stream function–vorticity formulation and then solved by employing a finite difference method. The influence of both the height and wall emissivity of the heated source on fluid motion and the heat transmission in a large-scale chamber is investigated. Our results of the calculations on non-uniform grids with algebraic transformation are in excellent agreement with other available experimental and numerical outcomes for turbulent thermal convection in enclosures. The computations indicate that the average total Nusselt number is enhanced up to 2 times with an increase in the heater height. The results show that the surface emissivity of the heat source has a great influence on the total thermal transference coefficient. Furthermore, a growth of the heater surface emissivity has no significant effect on the flow structure.