Marker And Cell Method Research Articles

Modal analysis for incompressible fluid flow

This paper presents a numerical method for incompressible fluid flow. A difficulty in analyzing incompressible fluid flow is that the continuity equation has no time evolution term. In the marker and cell (MAC) method, Poisson’s equation is solved iteratively, which takes most of the computation time, and in the artificial compressibility method (ACM), pseudo-time iteration is necessary to solve for unsteady solutions. Here, modal analysis that uses the velocity eigenvectors corresponding to zero eigenvalues is proposed for analyzing two-dimensional incompressible fluid flow. The proposed method involves only about one third of the number of variables needed in the MAC method and the ACM, and it does not require iterative calculation of Poisson’s equation or pseudo-time iteration. Numerical results for a simple flow system and a cavity flow obtained using the proposed method are compared with those obtained using the ACM and the simplified MAC method. The results agree well, thereby validating the proposed modal analysis.

A numerical model based on the curvilinear coordinate system for the MAC method simplified

In this paper we developed a numerical methodology to study some incompressible fluid flows without free surface, using the curvilinear coordinate system and whose edge geometry is constructed via parametrized spline. First, we discussed the representation of the Navier-Stokes and continuity equations on the curvilinear coordinate system, along with the auxiliary conditions. Then, we presented the numerical method – a simplified version of MAC (Marker and Cell) method – along with the discretization of the governing equations, which is carried out using the finite differences method and the implementation of the FOU (First Order Upwind) scheme. Finally, we applied the numerical methodology to the parallel plates problem, lid-driven cavity problem and atherosclerosis problem, and then we compare the results obtained with those presented in the literature.

Thermal Radiation on Mixed Convective Flow in a Porous Cavity: Numerical Simulation

Abstract In this paper, the influence of mixed convection in a porous square enclosure under the effect of radiation is numerically examined. The top and bottom walls are maintained at uniform temperature θc while some portion of the vertical walls is partially heated with temperature θh and rest of the vertical walls are thermally insulated, with θh > θc. The non-dimensional governing equations are solved by MAC (Marker and Cell) method. The effect of various parameters (thermal Grashof number, Darcy number, Prandtl number, Reynolds number) on flow patterns and heat transfer has been presented.

Natural Convection in a Square Cavity with Uniformly Heated and/or Insulated Walls Using Marker-and-Cell Method

In this paper, a numerical Harlow–Welch MAC (Marker and Cell) method is developed to investigate the unsteady free convection in lid-driven square cavity. The constant heat flux is considered over left wall and remaining walls are kept thermally insulated. The structure of thermal convection patterns analysed via streamline, vorticity, pressure and temperature contour plots. The influence of the thermo-physical parameters, Reynolds number (Re), thermal Grashof number (Gr) Prandtl number (Pr) and Peclet number (Pe = PrRe), on these distributions are described in detail. Validation of solutions with earlier studies is included. Mesh independence is also conducted. It is observed that an increase in Prandtl number intensifies the primary circulation whereas it reduces the heat transfer rate. Increasing thermal Grashof number also decreases heat transfer rates. Furthermore, the isotherms are significantly compressed towards the left (constant flux) wall with a variation in Grashof number while Peclet number is fixed. The study is relevant to solar collector heat transfer simulations and also crystal growth technologies.

The Superconvergence Phenomenon and Proof of the MAC Scheme for the Stokes Equations on Non-uniform Rectangular Meshes

For decades, the widely used finite difference method on staggered grids, also known as the marker and cell (MAC) method, has been one of the simplest and most effective numerical schemes for solving the Stokes equations and Navier---Stokes equations. Its superconvergence on uniform meshes has been observed by Nicolaides (SIAM J Numer Anal 29(6):1579---1591, 1992), but the rigorous proof is never given. Its behavior on non-uniform grids is not well studied, since most publications only consider uniform grids. In this work, we develop the MAC scheme on non-uniform rectangular meshes, and for the first time we theoretically prove that the superconvergence phenomenon (i.e., second order convergence in the $$L^2$$L2 norm for both velocity and pressure) holds true for the MAC method on non-uniform rectangular meshes. With a careful and accurate analysis of various sources of errors, we observe that even though the local truncation errors are only first order in terms of mesh size, the global errors after summation are second order due to the amazing cancellation of local errors. This observation leads to the elegant superconvergence analysis even with non-uniform meshes. Numerical results are given to verify our theoretical analysis.

Viscoelastic blood flow through arterial stenosis—Effect of variable viscosity

Current theoretical investigation deals with mathematical model of unsteady non-Newtonian flow of blood through a stenosed artery. The flowing blood is considered as a viscoelastic fluid having shear-thinning rheology and characterized by generalised Oldroyd-B model. The arterial wall is considered to be rigid having cosine shaped stenosis in its lumen. The governing equations of motion accompanied by appropriate choice of the initial and boundary conditions are solved numerically by MAC (Marker and Cell) method and the results are checked for numerical stability with desired degree of accuracy. The quantitative analysis has been carried out finally which includes the respective profiles of the flow-field. The key factors like the wall shear stress and flow separation are also examined for further qualitative insight into the flow through arterial stenosis. The present results show quite consistency with several existing results in the literature which substantiate sufficiently to validate the applicability of the model under consideration.

UNSTEADY ANALYSIS OF VISCOELASTIC BLOOD FLOW THROUGH ARTERIAL STENOSIS

A mathematical model of unsteady non-Newtonian blood flow in an artery under stenotic condition has been developed. The flowing blood is considered to be a viscoelastic fluid characterized by the Oldroyd-B model and the arterial wall is considered to be rigid, having cosine-shaped stenosis. The governing equations of motion accompanied by appropriate choice of the initial and boundary conditions are solved numerically by the MAC (marker and cell) method, and the results are checked, for numerical stability with desired degree of accuracy. The key factors like the wall shear stress, resistive impedance, and the other viscoelastic parameters are also examined for further qualitative insight into the flow through arterial stenosis. Comparison of the results reveals that dimensionless pressure drop for the viscoelastic model increases while it diminishes for the shear-thinning power law model over that of the Newtonian model. Moreover, the possibility of flow separation increases with increasing relaxation time (Deborah number), and in case of Newtonian fluid, delayed separation is observed. The grid independence study has also been performed successfully in order to validate the applicability of the methodology as well as the model used under consideration. Special emphasis has duly been made to compare the present theoretical results with the existing ones, and good agreement between them has been achieved both qualitatively and quantitatively.

A numerical simulation of unsteady blood flow through multi-irregular arterial stenoses

An unsteady mathematical model to study the characteristics of blood flowing through an arterial segment in the presence of a couple of stenoses with surface irregularities is developed. The flow is treated to be axisymmetric, with an outline of the stenoses obtained from a three dimensional casting of a mildly stenosed artery [1], so that the flow effectively becomes two-dimensional. The governing equations of motion accompanied by appropriate choice of boundary and initial conditions are solved numerically by MAC (Marker and Cell) method in cylindrical polar coordinate system in staggered grids and checked numerical stability with desired degree of accuracy. The pressure-Poisson equation has been solved by successive-over-relaxation (SOR) method and the pressure–velocity correction formulae have been derived. The flexibility of the arterial wall has also been accounted for in the present investigation. Further, in-depth study in the flow pattern reveals that the separation Reynolds number for the multi-irregular stenoses is lower than those for cosine-shaped stenoses and a long single irregular stenosis. The present results predict the excess pressure drop across the cosine stenoses than the irregular ones and show quite consistency with several existing results in the literature which substantiate sufficiently to validate the applicability of the model under consideration.

Numerical simulation of non-Darcian flow through a porous medium

This work reports on fluid flow in a fluid-saturated porous medium, accounting for the boundary and inertial effects in the momentum equation. The flow is simulated by Brinkman–Forchheimer–extended Darcy formulation (DFB), using MAC (Marker And Cell) and Chorin pressure iteration method. The method is validated by comparison with analytic results. The effect of Reynolds number, Darcy number, porosity and viscosity ratio on velocity is investigated. As a result, it is found that Darcy number has a decisive influence on pressure as well as velocity, and the effect of viscosity ratio on velocity is very strong given the Darcy number. Additional key findings include unreasonable choice of effective viscosity can involve loss of important physical information.

Mass transfer to blood flowing through arterial stenosis

The present investigation deals with a mathematical model representing the mass transfer to blood streaming through the arteries under stenotic condition. The mass transport refers to the movement of atherogenic molecules, that is, blood-borne components, such as oxygen and low-density lipoproteins from flowing blood into the arterial walls or vice versa. The blood flowing through the artery is treated to be Newtonian and the arterial wall is considered to be rigid having differently shaped stenoses in its lumen arising from various types of abnormal growth or plaque formation. The nonlinear unsteady pulsatile flow phenomenon unaffected by concentration-field of the macromolecules is governed by the Navier–Stokes equations together with the equation of continuity while that of mass transfer is controlled by the convection-diffusion equation. The governing equations of motion accompanied by appropriate choice of the boundary conditions are solved numerically by MAC(Marker and Cell) method and checked numerical stability with desired degree of accuracy. The quantitative analysis carried out finally includes the respective profiles of the flow-field and concentration along with their distributions over the entire arterial segment as well. The key factors like the wall shear stress and Sherwood number are also examined for further qualitative insight into the flow and mass transport phenomena through arterial stenosis. The present results show quite consistency with several existing results in the literature which substantiate sufficiently to validate the applicability of the model under consideration.

Numerical investigation on the flow and power of small-sized multi-bladed straight Darrieus wind turbine

Straight Darrieus wind turbine has attractive characteristics such as the ability to accept wind from random direction and easy installation and maintenance. But its aerodynamic performance is very complicated, especially for the existence of dynamic stall. How to get better aerodynamic performance arouses lots of interests in the design procedure of a straight Darrieus wind turbine. In this paper, mainly the effects of number of blades and tip speed ratio are discussed. Based on the numerical investigation, an assumed asymmetric straight Darrieus wind turbine is proposed to improve the averaged power coefficient. As to the numerical method, the flow around the turbine is simulated by solving the 2D unsteady Navier-Stokes equation combined with continuous equation. The time marching method on a body-fitted coordinate system based on MAC (Marker-and-Cell) method is used. O-type grid is generated for the whole calculation domain. The characteristics of tangential and normal force are discussed related with dynamic stall of the blade. Averaged power coefficient per period of rotating is calculated to evaluate the eligibility of the turbine.

A Numerical Technique for Solving the Maxwell Model for Free Surface Flows

This work is concerned with the development of a numerical technique for solving free surface flows of a Maxwell fluid. The governing equations for the flow of a Maxwell type fluid together with appropriate boundary conditions are given. The free surface stress conditions are treated in details. A novel formulation for calculating the extra stress components on rigid boundaries is given. The numerical technique presented in this work employs the finite difference method on a staggered grid and employs the ideas of the MAC (Marker-and-Cell) method. Numerical results demonstrating that this numerical technique can solve viscoelastic flows governed by the Maxwell model are presented. Moreover, validation results are presented.

Numerical simulation of forced convective evaporation system for tannery effluent

A numerical technique for evaluating the performance of forced convective evaporation system for tannery effluent is presented. The paper describes a mathematical model, which treats the airflow to be two dimensional and liquid flow to be one-dimensional. The turbulence in airflow is modeled with low Reynolds number k– ε model. The interface heat and mass transfer are represented by Reynolds flux model. The governing equations are solved using a modified MAC (Marker and Cell) method. The model is embodied into a computer code, which is applicable to forced convective evaporation system as well as mechanical draft cooling towers of both counter flow and cross flow types. Predicted results compare well with the available data in the literature and with that of experimental data and give closer results than the existing models. Distributions of air velocities, pressure, moisture fraction and air and water temperatures have been presented. The model can be used as design as well as optimization tool.

An upwinding cell-centered method with piecewise constant velocity over covolumes

We analyze as a covolume method an upwinding cell-centered method derived from a mixed formulation of the convection{diffusion equation. The covolume method uses primal and dual partitions of the problem domain to assign degrees of freedom and is a natural generalization of the well known MAC (Marker and Cell) method. The concentration is cell-centered (piecewise constant with respect to the primal partition), and the velocity is covolume- or co-cell centered (piecewise constant with respect to the dual partition). Convergence results are demonstrated in theL 2 norm. c 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 49{?? , 1999

Experimental and Computational Study of Two-Dimensional Roughness Effects on the Instability of the Flat Plate Boundary Layer.

Experimental and computational study on the linear instability region downstream of a two-dimensional roughness element(a square rod)placed in a laminar boundary layer on a flat plate is described. The numerical solutions of the two-dimensional N-S equations were obtained by the MAC(marker-and-cell)method. Hot wire measurements were conducted in the boundary layer on a flat plate in a low-turbulence wind tunnel. A sinusoidal fluctuation, which grows exponentially, is observed behind the roughness element. The characteristics of sinusoidal fluctuation(linear region)depend on κ / δ*κ, where κ is the height of the roughness element and δ*κ is the displacement thickness at the roughness element. At κ / δ*κ=1.0, the predominant frequency of naturally excited sinusoidal fluctuation lies in the unstable zone, which is predicted using the theoretical calculation based on the Orr-Sommerfeld equation applied for Lin's profile. The response characteristics of the linear instability region to artificial external excitation were investigated in detail using sound as an exciting agent. Also, flow fields with forced time-dependent disturbances are studied by the numerical method. The effects of the artificial external excitation become maximum when the frequency of disturbance coincides with the natural frequency. The numerical results for the linear instability region(frequency, phase velocity and amplification rates)agree with the experimental measurements

NON-DARCIAN SURFACE TENSION EFFECTS ON FREE SURFACE TRANSPORT IN POROUS MEDIA

The phenomenological analysis of free surface transport through porous media using the Brinkman-Forchheimer-extended Darcy model is presented. A finite difference scheme using the marker-and-cell (MAC) method is employed to investigate the momentum and energy transport in a porous channel involving free surface transport phenomena. The interfacial tension effect at the free surface is incorporated in the analysis. The present investigation constitutes one of the first numerical investigations of the free surface momentum and energy transport through porous media using the MAC method. Fully developed velocity and temperature fields for saturated as well as unsaturated porous channels, for cases with different Darcy numbers, are compared and verified against existing analytical solutions. Temporal free surface distributions for cases with different Darcy numbers and Reynolds numbers are presented. Also explored is the effect of the free surface transport in porous media on the energy transfer. It is found that the boundary and inertial effects have a significant influence on the free surface transport through porous media and that the surface tension effects become insignificant for ReK > 1. The present work constitutes one of the first studies on Non-Darcian effects on free surface transport in porous media.

Experimental and Computational Study of the Effect of Two-Dimensional Roughness on the Mean Velocity of a Laminar Boundary Layer.

In this report an experimental and computiational study of the flow behind a two-dimensional square roughness element placed in a laminar boundary layer on a flat plate is described Numerical solutioins of the two-dimensional N-S equations were obtained by the MAC(Marker-and-Cell)method. The experiment was carried out in a low-turbulence wind tunnel. The flow behind the roughness element consisted of a separated region and an external layer. There was no mass transfer between these regions. The computational and the experimental results for these regions were compared. The extent of the separated region depends on κ / δ*κ, where κ is the height of the roughness element and δ*κ is the displacement thickness at the roughness element. When κ / δ*κ is less than a critical value, the computational results agree with the experimental results. The boundary-layer flow separates at the roughness element and returns at 70-κ to the laminar boundary layer. In this case the length of the separated region increases with κ / δ*κ. When κ / δ*κ is larger than the critical value, the length of the separated region decreases with increasing κ / δ*κ, but the computational results do not agree with the experimental ones.

Numerical modeling of flows in flotation columns

Abstract Flotation columns are widely used in the mineral processing and coal preparation industry. Conventional mathematical modeling of flotation columns has utilized axial dispersion or tanks-in-series models. However, for design and scale up purposes these models are somewhat lacking because they do not provide detailed information on the flow fields within the column. This paper presents a two-dimensional, two-phase, fluid-dynamic model with numerical solutions for some cases encountered in column flotation. The MAC (Marker And Cell) method was employed to obtain the numerical solutions. The simulated flow field reveals the presence of a circulating flow structure in flotation columns, which could be one of the primary reasons for mixing in column flotation. The effects of some parameters such as superficial gas and liquid velocities on the characteristics of the flow structure in flotation columns were also tested with the numerical model.

Droplet formation on metallic surfaces during low-fluence laser irradiation

We have developed a technique to numerically investigate laser-induced modification of metal surfaces at low laser fluences, where plasma effects and thermal sputtering can be neglected. In this contribution we focus on the technique and the model employed to solve the corresponding Navier-Stokes equations for laser irradiation with an axially symmetric intensity profile. All equations are numerically solved using the method of finite differences. Specifically, the linearized set of the Navier-Stokes equations for an inviscid and incompressible fluid is solved using a slightly modified MAC (marker and cell) method, whereas the thermal behavior of the solid-liquid system is computed using the ADI (alternating direction implicit) algorithm. Under the assumptions made, the heat conduction in the solid and the melt can be computed separately from the actual melt dynamics. The results obtained by solving the heat conduction equation, where the phase transition solid-liquid has been taken into account, are parametrically considered in the MAC module. Our results show that the surface does not remain flat in general after laser irradiation of gold targets for low laser fluences. Small asperities, whose height increases with each consecutive laser shot, are observed.

Hydrodynamic phenomena during laser irradiation: a finite difference approach

We have developed a numerical method to compute the physical behavior of metals and semiconductors, such as Au and Si, under the influence of laser irradiation at low fluences. Electronic effects and evaporation of single atoms are neglected. In the model employed, the thermal reaction of the solid-melt system is calculated by solving the two-dimensional heat conduction equation, where the phase transition solid-liquid is taken into account. Accelerations acting on the melt caused by the thermal expansion of the solid and the phase transition at the interface are also computed. All this information is then parametrically considered in our melt dynamics solver, which makes use of the so-called MAC (Marker and Cell) method. This technique describes the flow of an incompressible fluid, the boundary of which is partially confined and partially free. Our results for laser irradiation of Au and Si show the same tendency as the experimentally observed data: Au, in contrast to Si, tends to from droplets on the surface. Those experimental findings are consequently consistent with the dominance of hydrodynamic effects under low-fluence laser irradiation.