Semi-implicit Method For Pressure-linked Equations Algorithm Research Articles

Conjugate MHD natural convection in a square cavity with a non-uniform heat source thick solid partition

This paper aims to investigate the effect of a non-uniform internal volumetric energy generation on the conjugate MHD natural convection in a square cavity filled by an electrically conducting fluid. The uniform magnetic field is applied horizontally; the two horizontal and vertical left walls are thermally insulated, whereas the right vertical wall is kept at constant cold temperature. The governing conservation equations are solved numerically using an in house program based on the finite-volume method and the SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm. The effects of internal Rayleigh, Hartmann and Prandtl numbers, thermal conductivity ratio, inclination angle of the cavity, and the thickness of heat source partition are discussed. The obtained results show that regardless of Kr , α and Δ, an almost pure conduction state is observed for low RaI without MHD and for high RaI with MHD at high magnetic field intensity.

A numerical study of mixed convection heat transfer in a lid-driven cavity using Al2O3-water nanofluid

This study aims a numerical investigation of steady, laminar mixed convection heat transfer in a two-dimensional cavity by employing a finite volume method with a fourth-order approximation of convective terms, when nanoparticles are present. With the aim of solving two-dimensional momentum and energy conservation equations, a finite volume method on a non-uniform staggered grid is utilized. Second-order central differences are utilized to approximate diffusion terms in momentum and energy equations, while the development of a non-uniform four-point fourth-order interpolation (FPFOI) scheme is performed for the convective terms. Continuity and momentum equations are solved using the SIMPLE (Semi-Implicit Method for Pressure-Linked Equation) algorithm. In order to evaluate heat transfer enhancement, various viscosity and thermal conductivity models were employed. Numerical solution results were obtained in different models in cases where Gr number is between 103 and 105, Re number is 10-100-1000 and nanoparticle volumetric fraction is 0-5%.

A segregated spectral element method for the 2D transient incompressible Navier-Stokes equations

In this paper, a spectral element method for the solution of the two-dimensional transient incompressible Navier-Stokes equations is introduced, which combines the segregated velocity-pressure equal-order formulation originated from the SIMPLE (semi-implicit method for pressure linked equations) algorithm. In contrast to previous segregated finite element method based on the SIMPLE algorithm, the pressure equation is derived from the momentum and continuity equations using the element matrices to ensure the convergence of the pressure. High-order element basis functions are adopted to construct the discrete form that presents arbitrary order or accuracy. The validation test with analytical solution verifies the high accuracy of the method. The flow in a lid-driven cavity with different inclination angles and the flow over a backward-facing step are investigated to further illustrate the property of the scheme. The computed results are in excellent agreement with the benchmark solutions, even if fewer grid points are used. The almost periodic solution for the flow of Re=10000 in a lid-driven square cavity is also captured by the present scheme. The numerical results indicate that the proposed high-order method can significantly reduce the number of grid points used in the calculation.

An Efficient Parallel Implementation of the SIMPLE Algorithm Based on a Multigrid Method

This paper deals with a parallel implementation of the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm to numerically solve the Navier–Stokes system of equations for viscous incompressible flows. A mechanism of interprocess communication using a mesh decomposition with virtual cells and an algebraic multigrid method is proposed. A method of distributed matrix storage and an algorithm for matrix-vector operations reducing the number of interprocess communications are presented. The results of a series of numerical experiments on structured and unstructured grids (including a problem of external aerodynamics) are presented. Based on the results obtained, an analysis of the influence of multigrid solver settings on the overall efficiency of the algorithm is made. It is shown that the parallel algorithm based on an algebraic multigrid technique proposed for the SIMPLE method makes it possible to efficiently calculate problems on hundreds of processors.

Combined Heat and Mass Transfer of Fluid Flowing through Horizontal Channel by Turbulent Forced Convection

In the present paper, we report a numerical study of dynamic and thermal behavior of the incompressible turbulent air flow by forced convection in a two-dimensional horizontal channel. This one contains the complicated form of the deflector which has been studied by varying the inclination angle from φ = 40°, φ = 55° to φ = 65°. The baffles are mounted on lower and upper walls of the channel. The walls are maintained at a constant temperature (375 K), the inlet velocity of air is Uint = 7.8 m/s, and the Reynolds number Re = 8.73 × 104. A specifically developed numerical model was based on the finite-volume method to solve the coupled governing equations and the SIMPLE (Semi Implicit Method for Pressure Linked Equation) algorithm for the treatment of velocity-pressure coupling. For Pr = 0.71, the results obtained show that (i) the streamlines and isotherms are strongly affected by the inclinations angles at Re = 8.73 × 104, (ii) the friction coefficient near the baffles increases under the angle exchange effect, and (iii) for a constant Re, the local Nusselt number at the walls of the channel varies with increasing the inclination angle of the deflector. Furthermore, the deflectors are generally used to change the direction of the structure of flow and also to increase the turbulence levels. We can conclude that the contribution of inclined baffles improves the increase of heat and mass transfer in which the Nusselt number at a certain angle increases noticeably.

A segregated spectral finite element method for the 2D transient incompressible Navier–Stokes equations

In this paper, the spectral element approximation and the velocity–pressuredecoupling method implementing the SIMPLE (semi-implicit method for pressure linked equations) algorithm are first combined to form a high-order segregated scheme for the solution of the two-dimensional transient incompressible Navier–Stokes equations. In contrast to previous segregated finite element methods based on the SIMPLE algorithm, the pressure equation is derived from the continuity equation using the element matrices to ensure convergence. High-order element basis functions are adopted, which greatly reduces the number of nodal points used in the calculation. The validation test that has an analytical solution demonstrates the high accuracy and convergence rate of the method. The flow in a lid-driven cavity with different inclination angles and the flow over a backward-facing step are investigated to further illustrate the performance of the scheme. The computed results are in excellent agreement with the benchmark solutions. The almost periodic solution for the flow of Re=10000 in a lid-driven square cavity is also captured by the present scheme.

LES simulation of flow field and pollutant dispersion in a street canyon under time-varying inflows with TimeVarying-SIMPLE approach

Field observation data have shown that the actual wind speed and direction in an urban built environment always vary with time. Realizing the flow field simulation under time-varying inflows can have a great significance in fully understanding the airflow and the transportation of air pollutants in the atmospheric boundary layer. A new scheme based on the standard SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm has been implemented in OpenFOAM, named as TimeVarying-SIMPLE approach, to simulate the time-series airflow field under time-varying inflows in this study. The accuracy of TimeVarying-SIMPLE approach compared with PISO (Pressure-Implicit with Splitting of Operators) algorithm and PIMPLE (merged PISO-SIMPLE) algorithm was validated by solving the fully developed laminar flow in a straight pipe under sinusoidally varying inflow. The wind field and pollutant dispersion in an ideal street canyon with an aspect ratio of 1 under time-varying inflows were calculated using the TimeVarying-SIMPLE approach. The flows show an obvious flapping of the shear layer near the roof layer and the expansion or shrinkage of the main vortex under time-varying inflows, resulting in a significant increase of the removal efficiency of pollutants inside the street canyon compared to that under steady inflow. The average time of actual high-frequency wind speed data is critical to produce the time-varying inflows for the boundary conditions of numerical simulations, which could produce different simulated removal efficiency of pollutants inside the street canyon. The appropriate average time should be estimated by the pre-multiplied power spectrum of the measured high-frequency wind-speed data.

Modifications to the SIMPLE algorithm with the MDCD approach for incompressible flow simulation

PurposeThe purpose of the paper is to propose some modifications to the SIMPLE (semi-implicit method for pressure-linked equations) algorithm. These modifications can ensure the numerical robustness and optimize computational efficiency. They remarkably promote the ability of the SIMPLE algorithm for incompressible DNS (direct numerical simulation) of multiscale problems, such as transitional flows and turbulent flows, by improving the properties of dispersion and dissipation.Design/methodology/approachThe MDCD (minimized dispersion and controllable dissipation) scheme and MMIM (modified momentum interpolation method) are introduced. Six typical test cases are used to validate the modified algorithm, including the linear convective flow, lid-driven cavity flow, laminar boundary layer, Taylor vortex and DHIT (decaying homogenous isotropic turbulence). Particularly, a highly unsteady DNS of separated-flow transition in turbomachinery is precisely predicted by the modified algorithm.FindingsThe numerical examples show the distinct superiority of the modified algorithm in both internal flows and external flows. The advantages of the MDCD scheme and MMIM make the SIMPLE algorithm a promising method for DNS.Originality/valueSome effective modifications to the SIMPLE algorithm are addressed. It is the first attempt to introduce the MDCD approach into the SIMPLE-type algorithms. The new algorithm is especially suitable for the incompressible DNS of convection-dominated flows.

"Skin-Core-Skin" Structure of Polymer Crystallization Investigated by Multiscale Simulation.

“Skin-core-skin” structure is a typical crystal morphology in injection products. Previous numerical works have rarely focused on crystal evolution; rather, they have mostly been based on the prediction of temperature distribution or crystallization kinetics. The aim of this work was to achieve the “skin-core-skin” structure and investigate the role of external flow and temperature fields on crystal morphology. Therefore, the multiscale algorithm was extended to the simulation of polymer crystallization in a pipe flow. The multiscale algorithm contains two parts: a collocated finite volume method at the macroscopic level and a morphological Monte Carlo method at the microscopic level. The SIMPLE (semi-implicit method for pressure linked equations) algorithm was used to calculate the polymeric model at the macroscopic level, while the Monte Carlo method with stochastic birth-growth process of spherulites and shish-kebabs was used at the microscopic level. Results show that our algorithm is valid to predict “skin-core-skin” structure, and the initial melt temperature and the maximum velocity of melt at the inlet mainly affects the morphology of shish-kebabs.

Study of blood flow inside the stenosis vessel under the effect of solenoid magnetic field using ferrohydrodynamics principles

In this paper, biomagnetic blood flow in the stenosis vessel under the effect of the solenoid magnetic field is studied using the ferrohydrodynamics (FHD) model. The parabolic profile is considered at an inlet of the axisymmetric stenosis vessel. Blood is modeled as electrically non-conducting, Newtonian and homogeneous fluid. Finite volume and the SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm are utilized to discretize governing equations. The investigation is studied at different magnetic numbers ( $ Mn_{F}=164$ , 328, 1640 and 3280) and the number of the coil loops (three, five and nine loops). Results indicate an increase in heat transfer, wall shear stress and energy loss (pressure drop) with an increment in the magnetic number (ratio of Kelvin force to dynamic pressure force), arising from the FHD, and the number of solenoid loops. Furthermore, the flow pattern is affected by the magnetic field, and the temperature of blood can be decreased up to $1.48 {}^{\circ}{\rm C}$ under the effect of the solenoid magnetic field with nine loops and reference magnetic field ( $ B_{0}$ ) of 2 tesla.

Numerical study of mixed convection heat transfer of various fin arrangements in a horizontal channel

The mixed convection of a three-dimensional square duct with various arrangements of fins in both laminar and turbulent flow is numerically characterized and studied. This study focuses on the ability of fin arrangements to enhance a heat transfer while flow is incompressible and the fluid is air. In our models, the lower duct wall is defined with a constant heat flux condition while the two side walls and upper wall are insulated. The finite volume method with the SIMPLE (Semi Implicit Method for Pressure Linked Equations) algorithm is used for handling the pressure–velocity coupling. The numerical results are validated with experimental data and show good agreement. The computations confirm that fin arrangement significantly changes the temperature distribution and cooling performance of the fins. This study focuses on the new arrangements of the fins that are not presented in previous studies. These arrangements have 3 Models with two angles (30 and 45°). According to the results, heat transfer highly improves (approximately 40–50% in the laminar flow stream and 15–20% in turbulent flow in comparison to base plate heat transfer) as the simple fin arrangement is replaced by inclined fins. Moreover, the influence of limited heat flux on temperature distribution is extensively investigated. It is notable that the temperature of fins significantly reduces as the best arrangement of the fins is presented in the air flow stream. Therefore, an enhanced cooling zone occurs when the fin arrangement is adjusted according to the location of the source of the heat flux on the base plate.

Effect of Radiation on Mixed Convection Inside a Lid-Driven Square Cavity With Various Optical Thicknesses and Richardson Numbers

ABSTRACTThis work studies numerically the effect of the radiative heat transfer on the flow and thermal behaviors of the mixed convection in a lid-driven square cavity in the presence of radiatively emitting, absorbing, and isotropically scattering gray medium. The Boussinesq approximation has been used in modeling the governing equations, and the SIMPLE (semi-implicit method for pressure-linked equations) algorithm is used in coupling the velocity and pressure fields. The radiative transfer equation and the governing equations have been solved respectively by the discrete ordinates method and the finite-volume method in order to obtain the temperature, velocity, and heat flux distributions in the participating medium. The present numerical simulations are validated by comparison with several earlier studies. Then, the temperature and velocity distributions and Nusselt numbers have been analyzed in a broad range of optical thicknesses from 0 to 100 and Richardson numbers from 0.01 to 100. The results show that the radiation has a significant role on the flow and thermal behaviors in the lid-driven square cavity. As an example, we can refer to a sweep behavior that is detected in the velocity distributions of the lid-driven cavity.

Numerical simulation for unsteady flow over marine current turbine rotors

The numerous benefits of Savonius turbine such as simple in structure, has appropriate self-start ability, relatively low operating velocity, water acceptance from any direction and low environmental impact have generated interests among researchers. However, it suffers from a lower efficiency compared to other types of water turbine. To improve its performance, parameters such flow pattern, pressure and velocity in different conditions must be analyzed. For this purpose, a detailed description on the flow field of various types of Savonius rotors is required. This article presents a numerical study on a nonlinear two-dimensional flow over a classic Savonius type rotor and a Benesh type rotor. In this experiment, sliding mesh was used for solving the motion of the bucket. The unsteady Reynolds averaged Navier-Stokes equations were solved for velocity and pressure coupling by using the SIMPLE (Semi-Implicit Method for Pressure linked Equations) algorithm. Other than that, the turbulence model using k-e standard obtained good results. This simulation demonstrated the method of the flow field characteristics, the behavior of velocity vectors and pressure distribution contours in and around the areas of the bucket.

Numerical studies of lithium-ion battery thermal management systems using phase change materials and metal foams

This article investigates thermal management systems (TMS) of lithium-ion battery made from pure octadecane, pure gallium and octadecane–Al foam composite materials by numerical simulations. Porosity of the Al foam changes from 0.97 to 0.925 and 0.88. The numerical simulation is based on SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm, staggered grid, and temperature transforming model. Three different heat fluxes of 400, 600, and 800W/m2 at the left and right boundaries of the computational domain are considered to simulate the heat released from the battery. Different TMS’s thicknesses have been studied. The time variations of battery surface temperatures are compared with different phase change materials (PCMs) to compare the effectivity of the TMS. Results show that the discharge time before the average battery surface temperature reaches above 60°C increases with an increasing thickness (between 7.5 and 15mm) of the TMS. The result with pure octadecane indicates that the discharge time is increased by 87 percent when the thickness of the TMS is increased from 7.5mm to 15mm. The surface temperature of the battery is more uniform and the discharge time is 4.7 times longer when gallium is used as the PCM, compared with those with octadecane for all thicknesses at the heat flux of 600W/m2 and thickness of 12.5mm. Adding metal matrix of 0.88 porosity to the octadecane led to 7.3 times longer discharge time compared to the pure octadecane. It is also found that adding the Al foam to octadecane remarkably increases the uniformity of the battery surface temperature, i.e. the maximum temperature difference at the surface of the battery decreased from 25°C with pure octadecane to 2×10−4°C with octadecane–Al foam composite (0.88 porosity) after the battery discharges for 1000s at 600W/m2 heat flux.

Entropy Generation and Natural Convection of CuO-Water Nanofluid in C-Shaped Cavity under Magnetic Field

This paper investigates the entropy generation and natural convection inside a C-shaped cavity filled with CuO-water nanofluid and subjected to a uniform magnetic field. The Brownian motion effect is considered in predicting the nanofluid properties. The governing equations are solved using the finite volume method with the SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm. The studied parameters are the Rayleigh number (1000 ≤ Ra ≤ 15,000), Hartman number (0 ≤ Ha ≤ 45), nanofluid volume fraction (0 ≤ φ ≤ 0.06), and the cavity aspect ratio (0.1 ≤ AR ≤ 0.7). The results have shown that the nanoparticles volume fraction enhances the natural convection but undesirably increases the entropy generation rate. It is also found that the applied magnetic field can suppress both the natural convection and the entropy generation rate, where for Ra = 1000 and φ = 0.04, the percentage reductions in total entropy generation decreases from 96.27% to 48.17% for Ha = 45 compared to zero magnetic field when the aspect ratio is increased from 0.1 to 0.7. The results of performance criterion have shown that the nanoparticles addition can be useful if a compromised magnetic field value represented by a Hartman number of 30 is applied.

Study on Laminar Two-Dimensional Lid-Driven Cavity Flow with Inclined Side Wall

In this present work, a computational code is developed to solve a laminar two-dimensional lid driven cavity flow with inclined side wall. SIMPLE (Semi-Implicit Method for Pressure-Linked Equation) algorithm based on finite volume method on staggered grid ...

Mathematical Modeling of Pressure on Cylindrical Ellipse using Side-by-Side Configuration

The application of the concept of fluid is often used to solve problems in the daily life. One of them is the problem of fluid around an elliptical cylinder. This study aims to solve the problems of the fluid around two elliptical cylinder configuration with side-by-side using the Navier-Stokes equations. Navier-Stokes equations–incompressible, viscous and unsteady-are solved using finite difference method staggered grid and SIMPLE (Semi Implicit Method for Pressure-Linked Equation) algorithms. Finite difference method is used to complete the grid arrangement, whereas the SIMPLE algorithm is used to obtain components of velocity and pressure value. Results of this study are the pressure value based on fluid flow profile and a mathematical model which received an elliptical cylinder pressure. Profile of fluid flow is simulated by varying the Reynolds number of 100, 1000, 7000, and 10000 as well as variations in the distance between the cylinder with a ratio of 2 <= S/a <= 6 where L is the distance between the cylinder and a is the minor axis of the cylinder ellipse. Then the pressure is calculated based on the value of the received cylinder pressure components. After obtaining the pressure value, then we create a mathematical model of the stresses imposed on the elliptical cylinder.

A high‐order discontinuous Galerkin solver for low Mach number flows

SummaryIn this work, we present a high‐order discontinuous Galerkin method (DGM) for simulating variable density flows at low Mach numbers. The corresponding low Mach number equations are an approximation of the compressible Navier–Stokes equations in the limit of zero Mach number. To the best of the authors'y knowledge, it is the first time that the DGM is applied to the low Mach number equations. The mixed‐order formulation is applied for spatial discretization. For steady cases, we apply the semi‐implicit method for pressure‐linked equation (SIMPLE) algorithm to solve the non‐linear system in a segregated manner. For unsteady cases, the solver is implicit in time using backward differentiation formulae, and the SIMPLE algorithm is applied to solve the non‐linear system in each time step. Numerical results for the following three test cases are shown: Couette flow with a vertical temperature gradient, natural convection in a square cavity, and unsteady natural convection in a tall cavity. Considering a fixed number of degrees of freedom, the results demonstrate the benefits of using higher approximation orders. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Numerical research on flow and thermal transport in cooling pool of electrical power station using three depth-averaged turbulence models

This paper describes a numerical simulation of thermal discharge in the cooling pool of an electrical power station, aiming to develop general-purpose computational programs for grid generation and flow/pollutant transport in the complex domains of natural and artificial waterways. Three depth-averaged two-equation closure turbulence models, {m1}, {m2}, and {m3}, were used to close the quasi three-dimensional hydrodynamic model. The {m3} model was recently established by the authors and is still in the testing process. The general-purpose computational programs and turbulence models will be involved in a software that is under development. The SIMPLE (Semi-Implicit Method for Pressure-Linked Equation) algorithm and multi-grid iterative method are used to solve the hydrodynamic fundamental governing equations, which are discretized on non-orthogonal boundary-fitted grids with a variable collocated arrangement. The results calculated with the three turbulence models were compared with one another. In addition to the steady flow and thermal transport simulation, the unsteady process of waste heat inpouring and development in the cooling pool was also investigated.

Three‐dimensional energetic and exergetic analysis of the injection orientation of DI diesel engine under different engine speeds

AbstractThree‐dimensional (3‐D) computational code was implemented to solve conservation equations based on finite volume method as to simulate 1.8 L Ford diesel engine. Velocity and pressure of each computational cell is achieved by SIMPLE (semi‐implicit method for pressure‐linked equations) algorithm. For the exergetic aspect, the initial condition is set at 0.1 MPa and 300 K. The engine modeling is performed with 130 °, 140 °, and 150 ° with respect to x‐axis under 1500 and 2500 rpm engine speeds. The results, however, indicate better air/fuel mixture (near stoichiometric equivalence ratio) for 130 ° of injection angle, albeit smaller spray droplets (lower sauter mean diameter) were introduced with 140 °. It is seen that higher soot and NOx mass fraction is attributed to 1500 rpm engine speed. The highest NOx and soot are exhausted at 130 ° and 150 ° of injection, respectively. Second law efficiency was calculated for different spray angle and engine speed schemes such that 36.62%, 30.2%, and 32.07% are associated with 130 °, 140 °, and 150 ° of injection angle under 1500 rpm, respectively. In terms of engine performance, that is, indicated mean effective pressure, indicated specific fuel consumption, and temperature, the best performance metrics are of 130 ° equal to 15.4 bar, 0.3856 kg/kW‐h, and 2074.97 K under 1500 rpm, respectively. Instant irreversibility rate is the highest amount with peak value of 17.48 J/deg for 130 deg‐1500 rpm, while 140 ° shows higher mean irreversibility rate over crank angle (CA) period.