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

Modeling, simulation and mixing time calculation of stirred tank for nanofluids using partially-averaged Navier–Stokes (PANS) k u − ϵ u turbulence model

Abstract The present study deals with the numerical simulation of stirred tank in the presence of nanofluids to see the effect of different volume fraction (ϕ) of nanoparticles on the behaviour of flow characteristics and for the calculation of mixing time in the entire tank. The flow is assumed to be steady, axisymmetric, two dimensional and incompressible. For the simulation of flow inside the vessel, partially-averaged Navier–Stokes (PANS) k u − ϵ u turbulence model is used. Control volume method has been taken to descretize the governing equation along with power-law schemes. Further, semi-implicit method for pressure-linked equations revised (SIMPLER) algorithm and line-by-line tri-diagonal matrix algorithm (TDMA) have been taken to obtain the solution. The objective is to investigate the influence of ϕ on the characteristic flow variables and to calculate mixing time for different ϕ of nanofluids and for different values of f k , PANS model parameter. It is noted that with the increase in ϕ, mixing time has also been increased and it increases very fast for PANS k u − ϵ u model with the increase in filter width f k .

Development of two-dimensional hydrogen distribution model in small-scale spaces under severe accidents

Hydrogen distribution and combustion are important problems in nuclear reactor safety analysis under severe accident conditions.IPWR (Integrated pressurized water reactor) are very different from conventional nuclear power plants in design, structure and operating power. In particular, the nuclear power plant containment has a large space volume, and there are multiple compartments inside, while the small reactor containment has a small volume and a simple and compact internal layout. This makes the hydrogen in the containment behave differently in a severe accident. Thus, the hydrogen flow distribution characteristics of IPWR under severe accidents need further study.In this paper, a two-dimensional continuum model of hydrogen distribution in small-scale under severe accident conditions is established by using finite volume method and staggered grid scheme based on hydrodynamics calculation method. SIMPER (Semi-Implicit Method for Pressure Linked Equations Revised) algorithm is used to construct the pressure correction equation, alternate direction implicit algorithm is used to solve the algebraic equations, and block correction algorithm is used to accelerate the iterative convergence. Compared with the experimental study of hydrogen flow distribution characteristics in local compartments, the model can predict the change of hydrogen volume fraction distribution well with time in the experiment. After that, the flow distribution characteristics of hydrogen in the RPV (Reactorpressurevessel) and containment under the low-pressure core melt accident were analyzed with IPWR-IP200 as the research object. The results show that the coolant level in the lower head of the RPV is still about 0.8 m during hydrogen generation, so there is no hydrogen in the lower head area. The highest hydrogen volume fraction in the RPV is at the outlet of the core, which is about 23 %. The highest volume fraction of hydrogen in the containment is the outlet above the pressurizer relief valve, which is up to 17 %. Because of the high volume fraction of water vapor in the RPV, there is no risk of hydrogen combustion in the RPV. During hydrogen generation, there is a risk of hydrogen combustion within the containment, and the risk area is located in the upper space of the containment. After hydrogen production ceases, as water vapor continues to enter the containment, the hydrogen volume fraction gradually decreases, and the risk of hydrogen combustion will also decrease.The two-dimensional continuum model established in this paper can predict the hydrogen distribution under severe accidents, which can provide a reference for the hydrogen mitigation strategy of IPWR.

Thermally developing forced convection flow through porous concentric pipes annular duct

This article shows the thermally developing flow through concentric pipes annular sector duct by describing the Darcy Brinkman flow field. The cross sectional convection-diffusion terms are transformed in power law discretized form by integrating over the differential volume, whereas backward difference scheme is used in the axial direction of heat flow. With the help of semi implicit method for pressure linked equations-revised ( SIMPLE-R), we get the solution of the governing problem. The graphs of velocity profiles against R and average Nusselt number against axial distance are plotted for different values of Darcy number and geometrical configuration parameters. It has been pointed out that velocity and thermal entrance length decrease, when we decrease the value of Darcy number. By decreasing the cross section of the concentric pipes annular sector duct in the transverse direction, thermally fully developed flow region develops earlier.

A Dynamic Coarse Grain Discrete Element Method for Gas-Solid Fluidized Beds by Considering Particle-Group Crushing and Polymerization

The discrete element method (DEM) coupled with computational fluid dynamics (CFD) is used extensively for the numerical simulation of gas-solid fluidized beds. In order to improve the efficiency of this approach, a coarse grain model of the DEM was proposed in the literature. In this model, a group of original particles are treated as a large-sized particle based on the initial particle distribution, and during the whole simulation process the number and components of these particle-groups remain unchanged. However, collisions between particles can lead to frequent crushing and polymerization of particle-groups. This fact has typically been ignored, so the purpose of this paper is to rationalize the coarse grain DEM-CFD model by considering the dynamic particle-group crushing and polymerization. In particular, the effective size of each particle-group is measured by a quantity called equivalent particle-group diameter, whose definition references the equivalent cluster diameter used by the energy-minimization multi-scale (EMMS) model. Then a particle-group crushing criterion is presented based on the mismatch between the equivalent diameter and actual diameter of a particle-group. As to the polymerization of two colliding particle-groups, their velocity difference after collision is chosen as a criterion. Moreover, considering the flow heterogeneity induced by the particle cluster formation, the EMMS drag force model is adopted in this work. Simulations are carried out by using a finite volume method (FVM) with non-staggered grids. For decoupling the Navier-Stokes equations, the semi-implicit method for pressure linked equations revised (SIMPLER) algorithm is used. The simulation results show that the proposed dynamic coarse grain DEM-CFD method has better performance than the original one.

Secondary Dean flow characteristics of inelastic Bird-Carreau fluids in curved microchannels

To effectively control the mixing of target materials inside microfluidic devices, the Dean flow features of generalized-Newtonian Bird-Carreau (BC) fluids in curved rectangular channels are theoretically investigated, as a passive technique. Governing equations coupled with the Cauchy momentum equation and the BC model are solved using the finite volume scheme with a semi-implicit method for pressure-linked equations-revised (SIMPLER) algorithm. The effects of the rheological parameters of BC model, such as viscosity ratio, power-law index, and relaxation time constant, on the Dean flow are systematically examined in a wide range of Dean numbers (Dn), (very low to O(102)). The entire flow characteristics of BC fluids in curved microchannels with increasing Dn are quantified using flow skewness, DnRef/DnMES, and magnitude of vorticity, resulting in two main findings of a more outward-skewed streamwise velocity profile and a more enhanced secondary Dean vortex for non-Newtonian fluids in comparison to the Newtonian case at the same Dn.

Secondary flow behavior of electrolytic viscous fluids with Bird-Carreau model in curved microchannels

A curved channel is indispensable for the lab-on-chips system because it provides a convenient scheme for increasing the channel length per unit chip area in the direction of net flow. A secondary Dean flow in curved rectangular microchannels is examined by applying a finite volume scheme with a semi-implicit method for pressure-linked equations revised (SIMPLER) algorithm for the steady flow of non-Newtonian fluids with dilute concentrations in electrolyte solution. The proposed framework is based on a theoretical model coupled with the Cauchy momentum equation of Bird-Carreau fluid, electrostatic interaction, and net charge conservation. The shear thinning effect on the secondary flow at the turn is analyzed with respect to the parameters of interest including zero-shear viscosity and power-law index. The simulation results for the realistic microchannel indicate that the streamwise axial velocity tends to shift toward the inner wall and this is caused by a stronger spanwise pressure gradient, resulting from a sufficiently low Dean number. In the case of the non-Newtonian fluid, the axial velocity becomes a blunt profile by spreading to each side of the wall, and its flow skewness directed asymmetrically to the inner wall is higher than that in the case of the Newtonian fluid. The attaining trend of higher flow skewness is changed by the competing effect of shear thinning and is well matched with the behavior of the secondary flow.

Equivalent Clearance Model for Solving Thermohydrodynamic Lubrication of Slider Bearings With Steps

This study focuses on the thermohydrodynamic lubrication (THD) analysis of fluid film bearings with steps on the bearing surface, such as Rayleigh step. In general, the Reynolds equation does not satisfy the continuity of fluid velocity components at steps. This discontinuity results in the difficulty to solve the energy equation for the lubricants by finite differential method (FDM), because the energy equation needs the velocity components explicitly. The authors have solved this issue by introducing the equivalent clearance height and the equivalent gradient of the clearance height at steps. These parameters remove the discontinuity of velocity components, and the Reynolds equations can be solved for any bearing surfaces with step regions by FDM. Moreover, this method results in pseudocontinuous velocity components, which enables the energy equation to be solved as well. This paper describes this method with one-dimensional and equal grids model. The numerical results of pressure and temperature distributions by the proposed method for an infinite width Rayleigh step bearing agree well with the results obtained by solving full Navier–Stokes equations with semi-implicit method for pressure-linked equations revised (SIMPLER) method.

Viscoelastic flow past a confined cylinder: Instability and velocity inflection

The flow of the Oldroyd-B fluid past a circular cylinder in a channel is simulated by a parallelized finite volume method (FVM) using fine unstructured meshes. Convergent solutions are obtained up to a high Deborah number of De = 1.6 with a high aspect ratio mesh, while it can only be obtained for De = 0.6 with a more or less equi-lateral mesh. The governing equations are re-written using the discrete elastic viscous split stress formulation together with an independent interpolation of the vorticity (DEVSS- ω ) . The numerical procedure is based on the semi-implicit method for pressure-linked equations revised (SIMPLER) algorithm, using a collocated FVM discretization. To enhance the numerical accuracy, mixed meshes (structured and unstructured) are used and adapted to the geometry and the flow physics. The results show that an instability featuring stress oscillations on the top of the cylinder is produced at a critical De number ( De ≈ 0.6 ) . This critical value is in agreement with the experiments in the literature. The oscillation generated in the shear layer on the cylinder surface at high De number is convected downstream into the wake behind the cylinder. This feature of flow oscillation agrees well with experiments reported in the literature. The mechanism of these phenomena can be explained reasonably by a boundary layer analysis. The key finding is that the instability is due to an inflectional velocity profile, near the cylinder, generated by normal stress on the cylinder surface at high De number, which can only be captured with fine meshes. According to a newly proposed theory (so called energy gradient theory), inflectional velocity profile leads to flow instability and consequently allows the convection of the oscillatory flow within the shear layer downstream of the wake and results in the flow pulsation in the spanwise direction. Therefore, the origin of the instability for the flow around a cylinder is in the shear layer on the cylinder and not in the wake itself. In addition, it seems that the increasing rate of numerical perturbation is related to the mesh aspect ratio. Computing using thin-long meshes could get convergence even if the flow is oscillatory, while the computing is prohibited by the inflectional instability when equi-lateral triangular meshes are used.

Optimization of the nonlinear convergence rate of the SIMPLER Picard algorithm for steady incompressible internal flows through enforcement of continuity projections

We discuss the performance optimization of the Semi-Implicit Method for Pressure-Linked Equations—Revised (SIMPLER) Picard algorithm for steady incompressible internal flows. We discuss the nonlinear convergence of the Picard iteration as a function of the pressure and scalar potential continuity projections stemming from the SIMPLER algorithm, for three example problems. In particular, we discuss the choice of under-relaxation method, and choice of under-relaxation factors; the choice of the projection algorithm; and the required tolerance for the linear solve of the generalized Poisson equations for the pressure and scalar potential equations that arise from the projection operations. We conclude that the convergence of the nonlinear Picard iteration can be effectively controlled by the optimal enforcement of the continuity projections. Copyright © 2005 John Wiley & Sons, Ltd.

Numerical simulation of growth of a vapor bubble during flow boiling of water in a microchannel

The present study is performed to numerically analyze the growth of a vapor bubble during flow of water through a microchannel. The complete Navier–Stokes equations, along with continuity and energy equations, are solved using the SIMPLER (semi-implicit method for pressure-linked equations revised) method. The liquid–vapor interface is captured using the level set technique. The microchannel is 200-μm square in cross-section and the bubble is placed at the center of the channel with superheated liquid around it. The results show steady initial bubble growth followed by a rapid axial expansion after the bubble fills the channel cross-section. A trapped liquid layer is observed between the bubble and the channel as it elongates. The bubble growth rate increased with the incoming liquid superheat, but decreased with Reynolds number. The formation of a vapor patch at the walls is found to be dependent on the time the bubble takes to fill up the channel. The upstream interface of the bubble is found to exhibit both forward and reverse movement during bubble growth. The results show little effect of gravity on the bubble growth rate under the specified conditions. The bubble growth features obtained from the numerical results are found to be qualitatively similar to experimental observations.

A distributed lagrange multiplier based computational method for the simulation of particulate-Stokes flow

In this paper we present a distributed lagrange multiplier (DLM) based Stokes flow algorithm for particulate flows. The entire fluid–particle domain is treated as a fluid. The “fluid” in the particle domain is ensured to move rigidly by adding a rigidity constraint. We modify the SIMPLER (Semi-Implicit Method for Pressure Linked Equations—Revised) algorithm for fluids (by Patankar [Numerical Heat Transfer and Fluid Flow, Taylor and Francis, London, 1980]), to account for the presence of the particle. The modification pertains to the presence of the rigidity constraint in the particle domain. We validate the algorithm with suitable test cases.

A comparison study of the convergence characteristics and robustness for four variants of SIMPLE‐family at fine grids

A comparative study is performed to reveal the convergence characteristics and the robustness of four variants in the semi‐implicit method for pressure‐linked equations (SIMPLE)‐family: SIMPLE, SIMPLE revised (SIMPLER), SIMPLE consistent (SIMPLEC), and SIMPLE extrapolation (SIMPLEX). The focus is concentrated in the solution at fine grid system. Four typical fluid flow and heat transfer problems are taken as the numerical examples (lid‐driven cavity flow, flow in an axisymmetric sudden expansion, flow in an annulus with inner surface rotating and the natural convection in a square enclosure). It is found that an appropriate convergence condition should include both mass conservation and momentum conservation requirements. For the four problems computed, the SIMPLEX always requires the largest computational time, the SIMPLER comes the next, and the computational time of SIMPLE and SIMPLEC are the least. As far as the robustness is concerned, the SIMPLE algorithm is the worst, the SIMPLER comes the next and the robustness of SIMPLEX and SIMPLEC are superior to the others. The SIMPLEC algorithm is then recommended, especially for the computation at a fine grid system. Brief discussion is provided to further reveal the reasons which may account for the difference of the four algorithms.

Computational analysis of surface-tension-driven coating-defect flow

Bondline readout (BLRO) is a coating defect frequently exhibited on adhesively-bonded, polymeric automotive body panels painted with high-glamor/flow clearcoats. BLRO or telegraphing results from Marangoni-type, surface-tension-driven flows. An efficient numerical code was developed to predict temperature- and concentration-induced BLRO flows to use it as an effective tool for screening and developing potential clearcoat systems. Two numerical codes were developed: a 1-D code based on the lubrication approximation; a 2-D code based on the SIMPLER (Semiimplicit method for pressure-linked equations revised) algorithm. The 1-D code produced BLRO profiles in agreement with those predicted from the 2-D code and those measured from BLRO flow experiments, thus confirming a proposed BLRO mechanism. Moreover, the 1-D code was vastly more time-efficient than the 2-D code due to a severe time-step stability limitation found with the latter.

Theoretical investigation on the performance prediction of solar still

A theoretical investigation on the performance prediction of solar still is presented in this paper. A solar still of conventional type is considered. The mathematical model is based on time-averaged Navier-Stokes equations. The effect of the variable fluid properties is taken into consideration by using a mixture of air and vapour in the still. A steady state two-dimensional approach with constant temperature boundaries is considered. A discretization schema with finite-difference technique is adapted. The SIMPLER (Semi-Implicit Method for Pressure-Linked Equations Revised) methodology is used. The grid size independence solution is checked for convergence. The numerical results show clearly different zones of circulation with reverse velocity on the inside still glass cover. The numerical results prove the necessity to undertake a numerical investigation before the sizing of solar still. The objective of the present investigation is to present a mathematical model in order to improve the still design, also the obtained results prove to be a good tool for performance prediction according to a given geometry.

Viscoelastic flow past a cylinder: drag coefficient

The channel flow of a viscoelastic fluid past a circular cylinder is simulated using a parallelized unstructured Finite Volume Method (FVM) [H.-S. Dou, N. Phan-Thien, J. Non-Newtonian Fluid Mech. 77 (1998) 21–51] under a distributed computing environment through the Parallel Virtual Machine (PVM) libraries. The numerical method is based on the SIMPLER (Semi-Implicit Method for Pressure-Linked Equations Revised) algorithm, using the simplified Phan-Thien/Tanner (PTT) and the Upper Convective Maxwell (UCM) constitutive models, with the Elastic Viscous Split Stress (EVSS) formulation. The parallelization of the program is implemented by a domain decomposition strategy and the PVM software. It is found that the both shear thinning and normal stress contribute to the drag reduction. With the PTT model, the drag reduction is mainly due to shear thinning. With the UCM model, it is primarily due to the normal stress effect.

The flow of an Oldroyd-B fluid past a cylinder in a channel: adaptive viscosity vorticity (DAVSS- ω) formulation

A parallel unstructured finite volume method (FVM) is developed and implemented under a distributed computing environment through the parallel virtual machine (PVM) libraries, and is used to simulate the channel flow of the Oldroyd-B fluid past a circular cylinder. Differing from our previous work [11, 12], a discrete elastic viscous split stress (DEVSS) formulation together with an independent interpolation of the vorticity (DEVSS- ω) is proposed in this paper. This method has almost the same stability behavior as the elastic viscous split stress (EVSS) formulation, and is suitable for complex constitutive models. To further improve the stability at high Deborah numbers, we combine the idea of the discrete adaptive elastic viscous split stress (DAVSS) formulation [7] with the independent interpolation of the vorticity to arrive at the DAVSS- ω method. The numerical implementation is based on the unstructured FVM method and the semi-implicit method for pressure-linked equations revised (SIMPLER) algorithm. The parallelization of the program is implemented by a domain decomposition strategy and using PVM software libraries. The results are compared with those by the EVSS, DEVSS, and the plain Oldroyd-B formulation (without splitting the stress). It is found that the drag coefficient first decreases and then increases with the De number, for a channel half width to cylinder radius ratio of h/ R = 2. It is also confirmed that the drag enhancement at high Deborah number is due to the increasing extension effect in the regions near the front and the rear stagnation points.

APPLICATION OF BLOCK-IMPLICIT MULTIGRID APPROACH TO THREE-DIMENSIONAL HEAT TRANSFER PROBLEMS INVOLVING DISCRETE HEATING

In this study, the performance of the symmetrically coupled Gauss-Seidel (SCGS) based multigrid method was investigated by applying it to three-dimensional conjugate heat transfer in two different configurations with discrete heat sources. These include a natural convection cooled rectangular enclosure with a discrete heat source mounted on one vertical wall, and a mixed forced convection cooled channel with a discrete heat source on the bottom wall. The convergence rates using the semi-implicit method for pressure-linked equations revised (SIMPLER), SCGS, and multigrid SCGS (MG-SCGS) were compared for both configurations. The performance of MG-SCGS was tested for variations in Rayleigh number, Reynolds number, and boundary conditions. It is shown that faster convergence can be achieved by applying the multigrid technique with only two grid levels. With MG-SCGS, significant and consistent performance enhancement over SIMPLER and SCGS was obtained for the conjugate natural, mixed, and forced convection problems considered.