Primitive Variable Formulation Research Articles

Computational analysis magnetohydrodynamics natural convection flow round vertical thermally stratified jet: Finite difference method in conjunction with primitive variable formulation

The problem of magnetohydrodynamic free convective heat transfer mechanism across a laminar, round, vertical, and thermally stratified jet is examined. The boundary layer equations are non-dimensionalized, and then, using the primitive variable formulation, the finite difference method is used to find the numerical solution of the problem. The impact of Prandtl number, magnetic parameter, and stratification parameter on velocity distribution, temperature profile, and heat transfer rate is explained graphically and in tabular form. It is found that for increasing values of thermal stratification parameter S, the velocity profile increases while temperature distribution and rate of heat transfer decrease. Similarly, with an increase in the values of Prandtl number, velocity, and temperature distribution decrease, but the rate of heat transfer increases.

Thermal radiation and soret/dufour effects on amplitude and oscillating frequency of darcian mixed convective heat and mass rate of nanofluid along porous plate

The influence of thermal radiations and Soret/Dufour significantly increases the mass and heat transport in material science, geothermal process, chemical rectors and heat exchangers due to temperature and concentration variation of nanoparticles. Main objective is to enhance the amplitude, oscillation and frequency of heat transfer using the Soret and Dufour characteristics in Darcian radiating flow of thermophoretic nanoparticles along vertical porous surface. The dimensionless analysis is performed to develop the convenient form of governing thermodynamic Darcian nanofluid formulation. To develop the programming algorithm in FORTRAN language, the primitive variable formulation is used for both time-dependent and steady models. The numerical and graphical outcomes of primitive type equations under boundary conditions are explored by using implicit finite difference approach with Gaussian elimination scheme. Various pertinent parameters are used to elaborate the steady fluid velocity, surface temperature and steady concentrated nanoparticle volume fraction. Valuable novelty of current research is to use steady results for fluctuating shear stress, fluctuation heating rate and fluctuating mass rate. It is found that the influence of Soret and Dufour parameter enhances the fluid velocity and temperature distribution with maximum variations. It is noticed that the temperature distribution enhances as thermal radiation parameter increases. The oscillation and transient stability of heating rate enhances with prominent variations as thermal radiation and Soret/Dufour parameter increases. It is depicted that amplitude in shear stress and mass rate enhances as Soret and Dufour parameter enhances.

Effect of variable viscosity on oscillatory heat and mass transfer in mixed convective flow with chemical reaction along inclined heated plate under reduced gravity

Reduced gravity refers to the actual decline in the accelerated motion of gravitation that occurs when a particular fluid comes into contact with another fluid that has different densities as a result of buoyant effects. The reduced gravity mechanism is very affective along the inclined geometries to reduce more gravitational pressure. In comparison to ordinary action of gravity, reduced gravity was reported to demonstrate less heat transfer, which helped to thicken the liquid layers. The main objective of this study is to illustrate the effects of chemical reaction and variable viscosity on oscillatory behavior of heat and mass transfer along an inclined heated plate under reduced gravity. The viscosity of fluid is assumed as a function of temperature. The mathematical model is developed in terms of partial differential equations and converted into steady and oscillating part by using oscillating Stokes conditions. The primitive variable formulation is used to convert steady and oscillating part into convenient form for smooth algorithm. The numerical results are deduced in tables and graphs with the help of FORTRAN and Tecplot 360. The velocity, temperature distribution, concentration profile, oscillatory skin friction, heat transfer and mass transfer are plotted for various governing parameters. The range of parameters has selected according as Prandtl number 0.1≤Pr≤7.0, buoyancy parameter 0<λT,λC<∞, Schmidt number 1.0≤Sc≤6.0, 0.1≤Rg≤8.0 and the choice of reaction rate parameter set the effects of chemical reaction respectively. The results show that the velocity profile increases as viscosity decreases under reduced gravity. It can be seen that temperature profile decreases as reduced gravity increases with lower viscosity. It is noticed that the heat and mass transfer increases with significant amplitude in the presence of reduced gravity and chemical reaction.

Amplitude and oscillating assessment of thermal and magnetic boundary layer flow across circular heated cylinder with heat source/sink

The effects of heat source/sink and magnetohydrodynamics on the oscillatory and periodic quantities of heat transfer and current density characteristics of viscous fluid along the magnetized and heated circular cylinder has been investigated. The governing mathematical model in terms of partial differential equations is converted into dimensionless form. The dimensionless model is again converted into steady, real and imaginary part for oscillating results. The steady, real and imaginary part is transformed into primitive form for smooth algorithm with the help of Primitive Variable Formulation (PVF). The primitive equations are reduced into system of algebraic equations by using Finite Difference Method (FDM). The heat source/sink parameter δ, the magnetic force number ξ, the buoyancy parameter λ, magneto-Prandtl factor γ, and remaining secured factors are utilized to determine computational findings of unknown quantities. Graphs are displayed for velocity, temperature distribution and magnetic profile for all pertinent parameters by FORTRAN and Teplot 360 software. The main novelty of current work is to evaluate oscillatory and periodic quantities of heat transfer and current density by using the steady solutions. It was found that prominent results in temperature distribution are deduced at each angle with the heat source effects. It was expected physically because the heat source is used as a sporting agent to compute heat transfer performance in electrically conducting fluid. The significant amplitude of oscillation in heat transfer and current density is evaluated around every position of circular cylinder. The numerical results of skin friction are compared with existing literature with excellent agreement.

A superconsistent collocation method for high Reynolds number flows

We use a novel implicit second-order projection method together with a superconsistent collocation scheme (Funaro, 1993; Fatone et al., 2005; De l’Isle and Owens, 2021) for the solution of the primitive variable formulation of the Navier–Stokes equations at very high Reynolds numbers. In particular, we apply a superconsistent collocation scheme to the convection–diffusion equation arising from one step of the projection method and this represents the first time that superconsistent collocation methods have been employed for the solution of a convection–diffusion equation having unsteady convection velocity. In order to evidence the second-order (in space and time) convergence of our scheme for both the velocity and pressure fields we choose to solve the two-dimensional unsteady Taylor–Green vortex problem (Taylor and Green, 1937). The numerical results presented for the solution of the two-dimensional square lid-driven cavity problem are in excellent agreement over the whole range of Reynolds numbers considered (5000≤Re≤1000) with some others in the literature (Ghia et al., 1982; Wang and Liu, 2019; Bruneau and Saad, 2006; Auteri et al., 2002; Pan and Glowinski, 2000).

Simulation of incompressible viscous flow using finite element method

This study focused on simulating incompressible viscous flow using the finite element method. This study used velocity and pressure as unknowns known as primitive variable formulations. Simulation of incompressible fluid flow poses numerical challenges due to the presence of nonlinear convective terms in Navier-Stokes equations and the incompressible nature of the fluid. If the connection between velocities and pressure is not discretized correctly, the stable and convergent velocities might be gained, but the obtained pressure will be oscillatory. To avoid these difficulties, continuous quadratic and additional cubic bubble functions will be used for the velocity field and linear functions for the pressure field. This kind of discretization satisfies the Ladyzhenskaya-Babuška-Brezzi (LBB) stability condition. Two cases of different Reynolds numbers were used to test the formulation's effectiveness. In the case of Reynolds number 0.12, no vortices were formed, suggesting that the flow is primarily governed by fluid friction, and fluid inertia has minimal effect. In the case of Reynolds number 120, the vortex formation, which is known as Von Kármán vortex street, appeared. These results concluded that the formulation using the finite element method is correct.

Theoretical Analysis of the Effects of Exothermic Catalytic Chemical Reaction on Transient Mixed Convection Flow along a Curved Shaped Surface.

The present problem addressed the transient behavior of convective heat and mass transfer characteristics across a curved surface under the influence of exothermic catalytic chemical reactions. The governing non-linear mathematical model wastransformed into a convenient form with the help of a primitive variable formulation. The final primitive formed model wassolved numerically by applying the finite difference method. The analysis of the above said computed numerical data in terms of oscillatory heat transfer, skin friction, and oscillatory mass transfer for various emerging parameters, such as the mixed convection parameter λT, modified mixed convection parameter λc, index parameter n, activation energy parameter E, exothermicparameter β, temperature relative parameter γ, chemical reaction parameter λ, and Schmidt number Sc is plotted in graphical form. An excellent agreement is depicted for oscillatory heat transfer behavior at the large value of activation energy E. The amplitude of heat transfer and prominent fluctuating response in mass transfer with a certain height is found at each value of the index parameter n with a good alteration. An increase in the activation energy led to an increase in the surface temperature, which yielded more transient heat transfer in the above-said mechanism. The main novelty of the current study is that first, we ensured the numerical results for the steady state heat and fluid flow and then these obtained results wereused in the unsteady part to obtain numerical results for the transient behavior of the heat and mass transfer mechanism.

Fast Multipole Boundary Element Method (FMM/BEM) for the solution of the Navier-Stokes in primitive variables based on the Burton and Miller formulation in two-dimensions

In the present work the fast multipole boundary element method (FMM/BEM) is presented for the solution of incompressible Navier-Stokes equations in 2 dimensions. This is a velocity-pressure primitive variable formulation abbreviated as uv-p. The Stokes-let fundamental solution is used as a weight function in the momentum equation eliminating the pressure from free variable. This way, the LBB conditions are bypassed removing any restriction in the type of grid, placement of nodes, or interpolation. Most of the numerical methods cannot use constant elements for all fields to solve the Navier-Stokes. BEM is the only known so far semi-analytical method capable of such an achievement. The continuity is enforced with the correction of velocity, using a scalar potential, the gradient of which is computed directly with hypersingular equations. The stress hypersingular equation is added to the boundary nodes rendering the well known Burton and Miller formulation and recommendation about the coupling constant is given. Accurate results for cavity Re = 1000 with constant elements and Re=10000 with linear quadrilaterals are obtained and presented. The reached degrees of freedom (dof) are beyond 1 million. Two more problems with mixed BCs are solved. The flow past cylinder is a kind of problem in which both singular and hypersingular equations must be used to ensure convergence. The main advantage of the method is the direct computation of the wall normal stresses or tractions, which is of cornerstone importance in many realistic applications. The tractions are obtained directly from the solution of the system of equations and can be integrated on the surface of a bluff body, to obtain forces and drag and lift coefficients. The pressure is computed in a post-processing phase with hypersingular equation derived in the framework of the present method. Smooth pressure field is obtained in all cases, even when the problem is solved with unstructured grids and constant elements. The method permits the use of non-conforming grids, and it can easily be combined with other numerical methods to obtain wall tractions and stresses via hypersingular equations. Moments for pressure and convective terms are derived.

The analysis of amplitude and phase angle of periodic mixed convective fluid flow across a non-conducting horizontal circular cylinder

The present contribution deals with the study of amplitude and phase angle of periodical mixed convective fluid flow along the non-conducting circular cylinder. The surface of cylinder is thermally and electrically conducting by keeping surface temperature as a time dependent variable. The developed dimensionless equations are transformed into analytical expressions for flow characteristics by using primitive variable formulation which converts nonlinear PDE’s into primitive form for numerical data. The graphical representations of physical properties phase angle and amplitude are obtained for various values of Prandtl number Pr, mixed-convection parameter λ , magnetic-Prandtl number γ and magnetic force parameter ξ respectively. To demonstrate important properties of the solution, a parametric study is conducted for governing parameters and a set of graphical and numerical outcomes is analyzed. The amplitude of time dependent skin friction τ w , heat transfer q w and current density j w has been observed on suitable positions of the non-conducting cylinder π / 6 , π / 4 , π / 2 , 3 π / 4 , 5 π / 6 and π numerically. It is noticed that the maximum amplitude of time dependent skin friction τ w , heat transfer q w and current density j w is noted at position α = π / 2 by increasing the various amount of physical pertinent parameters.

Magneto-Exothermic Catalytic Chemical Reaction along a Curved Surface

In the current study, the physical behavior of the boundary layer flows along a curved surface owing exothermic catalytic chemical reaction, and the magnetic field is investigated. The mathematical model comprised of a part of momentum, energy, and mass equations, which are solved using a finite difference method along with primitive variable formulation. Numerical solutions, using the method of quantitative differentiation, are made with the appropriate choice of dimensionless parameters. Analysis of the results obtained shows that the field temperature and flow of fluids are strongly influenced by the combined effects of catalytic chemical reactions and the magnetic field. The effects of skin friction, heat transfer, mass transfer, mass concentration, and temperature distribution along the curved surface are illustrated in the plots and in the form of tables. By setting the controlling parameters at the boundaries, the boundary conditions at the surface and away from the surface are determined in each graph. With a larger range of body shape parameter n, skin friction and heat transfer are improved, but mass transfer is reduced. Due to the increasing values of the exothermic parameter, the fluid velocity and mass concentration are decreased gradually and the temperature distribution is increased dramatically.

Heat source and sink effects on periodic mixed convection flow along the electrically conducting cone inserted in porous medium.

The system of partial differential equations governing the unsteady hydromagnetic boundary-layer flow along an electrically conducting cone embedded in porous medium in the presence of thermal buoyancy, magnetic field, heat source and sink effects are formulated. These equations are solved numerically by using an implicit Finite-Difference Method. The effects of the various parameters that are source/sink parameter, porous medium parameter, Prandtl number, mixed convection parameter and magnetic Prandtl number on the velocity, temperature profiles, transverse magnetic field are predicted. The effects of heat source and sink parameter on the time-mean value as well as on transient skin friction; heat transfer and current density rate are delineated especially in each plot. The extensive results reveal the existence of periodicity and show that periodicity becomes more distinctive for source and sink in the case of the electrically conducting cone. As the source and sink contrast increases, the periodic convective motion is invigorated to the amplitude and phase angle as reflect in the each plot. The dimensionless forms of the set of partial differential equations is transform into primitive form by using primitive variable formulation and then are solved numerically by using Finite Difference Scheme which has given in literature frequently. Physical interpretations of the overall flow and heat transfer along with current density are highlighted with detail in results and discussion section. The main novelty of the obtained numerical results is that first we retain numerical results for steady part and then used in unsteady part to obtain transient skin friction, rate of heat transfer and current density. The intensity of velocity profile is increased for increasing values of porosity parameter Ω, the temperature and mass concentration intensities are reduced due heat source effects.

A higher-order numerical analysis to study the flow physics and to optimize the design of a short-dwell blade coaters for higher efficiency

Blade coaters are most commonly used for coating of paper and paperboard with higher efficiency. The efficiency of short-dwell blades coaters depends on many factors such as the properties of the coating material, design of the coating reservoir, the types of flow behaviour taking place inside the reservoir, etc. In this work, we have proposed an optimal design of the reservoir to improve the efficiency of short-dwell coaters. The reservoir has been modeled as flow inside a two-dimensional rectangular cavity. Incompressible Navier-Stokes equations in primitive variable formulation have been solved to obtain the flow fields inside the cavity. Spatial derivatives present in the momentum, and continuity equations are evaluated using a sixth-order accurate compact scheme whereas the temporal derivatives are calculated using the fourth-order Runge-Kutta method. The actual rate of convergence of the numerical scheme has been discussed in detail. In addition, the accuracy and stability of the used numerical method are also analysed in the spectral plane with the help of amplification factor and group velocity contour plot. The obtained numerical solutions have been validated with the existing literature. Four different aspect ratio cases (L/H = 3/4,4/3,4/5 and 5/4) have been considered for the simulations including the case of square cavity. It has been observed that L/H = 5/4 case provides best results among all others.

Analysis of the Physical Behavior of the Periodic Mixed-Convection Flow around a Nonconducting Horizontal Circular Cylinder Embedded in a Porous Medium

An oscillatory mixed-convection fluid flow mechanism across a nonconducting horizontal circular cylinder embedded in a porous medium has been computed. For this purpose, a model in the form of partial differential equations is formulated, and then, the governing equations of the dimensionless model are transformed into the primitive form for integration by using primitive variable formulation. The impact of emerging parameters such as porous medium parameter Ω , Richardson number λ , magnetic force parameter ξ , and Prandtl number Pr on skin friction, heat transfer, and current density is interpreted graphically. It is demonstrated that accurate numerical results can be obtained by the present method by treating nonoscillating and oscillating parts of coupled partial differential equations simultaneously. In this study, it is well established that the transient convective heat transfer, skin friction, and current density depend on amplitude and phase angle. One of the objects of the present study is to predict the mechanism of heat and fluid flow around different angles of a nonconducting horizontal circular cylinder embedded in a porous medium.

Numerical investigation of the effect of temperature variation on heat transfer rate in a square cavity filled water-Al2O3 nanofluid with a hot circular cylinder in the center of the cavity

In this paper, the natural convection flow in a square cavity filled with nanofluid water-[Formula: see text] with a hot circular cylinder in the center of the cavity is numerically analyzed. All the walls are in lower temperatures than the circular cylinder. The Navier–Stokes and energy equations in the primitive variable form are discretized and solved by the finite element method (FEM). The effect of the volume fraction, the radius of the circular cylinder, the temperature and Rayleigh number is considered on the average Nusselt number. For the calculation of the viscosity coefficient and thermal conductivity coefficient of water-[Formula: see text] nanofluid, an experimental model is used which is the function of the volume fraction, temperature and nanoparticles diameter. This model is compared to the Brinkman model for viscosity and Maxwell model for thermal conductivity which are only the functions of volume fraction and are used by many researchers. The results show the experimental model leads to different results in comparison with the Brinkman model and Maxwell model, and indicate that the rate of the heat transfer can increase or decrease with the increase in volume fraction and temperature.

The meshless numerical simulation of Kelvin–Helmholtz instability during the wave growth of liquid–liquid slug flow

Kelvin–Helmholtz instability (KHI) occurs at the interface of two fluids, in which the heavier fluid flows at the bottom. In the present numerical work, the KHI was analyzed by solving the modification of two-dimensional (2-D) incompressible Navier–Stokes equations. To capture the interface, the Cahn–Hilliard equation was implemented into the Navier–Stokes equations. The phenomena around the KHI of two and three-component fluids were investigated numerically by using radial basis function (RBF) combined with the domain decomposition method (DDM) in a primitive variable formulation. Here DDM is able to solve the large scale problem. On the other hand the calculation accuracy decreases with the increase of the number of the subdomains. For the above reason, in the present works, the domain was partitioned into 15 x 15 subdomains in order to reduce the decrease in accuracy due to the domain division. Next, fractional step method was used to solve the modification of the Navier–Stokes equations.The numerical results indicate that the procedure used in the present work can easily handle the KHI problem under the variations of the interface thickness, density ratios, and initial velocity differences. Moreover, the interface evolutions obtained from the present method agree well with those of the finite difference method. The effects of the interface thickness, density ratio and magnitude of velocity difference on the KHI were also investigated. The decrease of the interface thickness produces a non-smooth concentration profile, and the increase of the interface thickness produces too much surface diffusion. It was found also that the increase of the density ratio reduces the growth of KHI. The interface rolls up are strongly affected by the initial horizontal velocity difference. Finally, the present study also shows that the RBF method is a reliable method to solve the KHI on the complex domains.

A fluid mechanic's analysis of the teacup singularity.

The mechanism for singularity formation in an inviscid wall-bounded fluid flow is investigated. The incompressible Euler equations are numerically simulated in a cylindrical container. The flow is axisymmetric with the swirl. The simulations reproduce and corroborate aspects of prior studies reporting strong evidence for a finite-time singularity. The analysis here focuses on the interplay between inertia and pressure, rather than on vorticity. The linearity of the pressure Poisson equation is exploited to decompose the pressure field into independent contributions arising from the meridional flow and from the swirl, and enforcing incompressibility and enforcing flow confinement. The key pressure field driving the blowup of velocity gradients is that confining the fluid within the cylinder walls. A model is presented based on a primitive-variables formulation of the Euler equations on the cylinder wall, with closure coming from how pressure is determined from velocity. The model captures key features in the mechanics of the blowup scenario.

MAGNETIC FIELD EFFECT ON THE HEAT TRANSFER IN A NANOFLUID FILLED LID DRIVEN CAVITY WITH JOULE HEATING

In this paper, the effects of magnetic field, Joule heating and volumetric heat generation on the heat transfer and fluid flow in a Cu-Water nanofluid filled lid driven cavity using enhanced streamfunction–velocity method are investigated. The cavity is heated by a uniform volumetric heat density and side walls have constant temperature. The top wall moves with constant velocity in +x direction, while no-slip boundary conditions are imposed on the other walls of the cavity. An inclined fixed magnetic field is applied to the left side wall of the cavity. The dimensionless governing equations are solved numerically for the stream function and temperature using finite difference method for various Richardson(Ri), Reynolds(Re), Hartmann (Ha), Eckert(Ec)numbers, magnetic field angle(α) and solid volume fraction of the nanofluid() in MATLAB software. To discretize the streamfunction-velocity formulation, a five point constant coefficient second-order compact finite difference approximation which avoids difficulties inherent in the conventional streamfunction–vorticity and primitive variable formulations is used. The stream function equation is solved using fast Poisson's equation solver on a rectangular grid (POICALC function in MATLAB) and the temperature equation is solved using Jacobi bi-conjugate gradient stabilized (BiCGSTAB) method. The heat transfer within the cavity is characterized by Nusselt number (Nu1). The results show that Nu1 is significantly increased by increasing Ri and  and increasing the Reynolds number enhances convective cooling. The heat transfer within the cavity is decreased by increasing Hartmann number which improves conduction heat transfer and reduces Nu1. Joule heating has a negative effect on the convection within the cavity and convection is decreased by increasing the value of Ec. It can be investigated that Nu1 is decreased by increasing Ec due to the strong distortion effect of Joule heating on convection current of heat transfer.

Computational Study of the Coupled Mechanism of Thermophoretic Transportation and Mixed Convection Flow around the Surface of a Sphere.

The main goal of the current work was to study the coupled mechanism of thermophoretic transportation and mixed convection flow around the surface of the sphere. To analyze the characteristics of heat and fluid flow in the presence of thermophoretic transportation, a mathematical model in terms of non-linear coupled partial differential equations obeying the laws of conservation was formulated. Moreover, the mathematical model of the proposed phenomena was approximated by implementing the finite difference scheme and boundary value problem of fourth order code BVP4C built-in scheme. The novelty point of this paper is that the primitive variable formulation is introduced to transform the system of partial differential equations into a primitive form to make the line of the algorithm smooth. Secondly, the term thermophoretic transportation in the mass equation is introduced in the mass equation and thus the effect of thermophoretic transportation can be calculated at different positions of the sphere. Basically, in this study, some favorite positions around the sphere were located, where the velocity field, temperature distribution, mass concentration, skin friction, and rate of heat transfer can be calculated simultaneously without any separation in flow around the surface of the sphere.

Magneto-thermo analysis of oscillatory flow around a non-conducting horizontal circular cylinder

The present study is based on oscillatory mixed-convection flow of electrically conducting fluid around a non-conducting horizontal circular cylinder. By following the statement of the problem, a mathematical model is constituted, and then, the formulated model is transformed into convenient form for integration by using primitive-variable formulation. The behavior of oscillatory skin friction, heat transfer and magnetic flux for pertinent parameters involved in the flow model such as mixed-convection parameter $$ \lambda $$ , magnetic force parameter $$ \xi $$ , the magnetic Prandtl number $$ \gamma $$ and the Prandtl number Pr is discussed numerically. For this purpose, first velocity profile, temperature distribution and magnetic-field profile at various positions $$ \alpha = \pi /6, \;\pi /3 $$ and $$ \pi $$ of the non-conducting horizontal circular cylinder for steady state are secured and then are used to calculate oscillatory skin friction, heat transfer and current density. It is predicted that oscillatory behavior of skin friction, heat transfer and current density is increased at position $$ \alpha = \pi /4 $$ for increasing values of the various physical pertinent parameters. It is pertinent to mention here that the convective heat transfer is practically associated with oscillatory flow behavior.

Meshless numerical model based on radial basis function (RBF) method to simulate the Rayleigh–Taylor instability (RTI)

The Rayleigh-Taylor instability (RTI) is the instability at the interface between two fluids when a heavier fluid is placed on top of lighter fluid in a gravitational field. In the present work, the RTI was studied numerically by using a meshless radial basis function (RBF) method. The present manuscript describes the development of the meshless RBF method to solve the RTI problem in an incompressible viscous two-phase immiscible fluid. This method can address the difficulty of the classical base method which often requires much computing time for the generation of the computational mesh. Moreover, the meshless RBF method does not require connectivity information among the nodes. Consequently, the present manuscript provides a new numerical procedure in the solution of the RTI problem by the combination of meshless RBF and Cahn-Hilliard equations. In the present numerical study, the RBF method was combined with the domain decomposition method (DDM) to solve the large scale problem. The problem was governed by the Navier-Stokes and Cahn-Hilliard equations in a primitive variable formulation. The Cahn-Hilliard equations were used to capture the interface between two fluids systems. The RBF method was used for spatial discretization and the Euler implicit method was implemented for time discretization. The fractional step scheme was used to solve the pressure velocity coupling. Here, the effects of Atwood numbers as representing the density ratio on the RTI were investigated. As a result, it was found that the position of the rising bubble and falling spike during RTI conforms well to the results from the previous works.