Projection Finite Element Method Research Articles

A consistent projection finite element method for the non-stationary incompressible thermally coupled MHD equations

This paper introduces a consistent projection finite method for the non-stationary incompressible thermally coupled MHD equations, which buoyancy affects because temperature differences in the flow cannot be neglected. It is a fully discrete projection method. The unconditional stability and error estimates of the velocity, pressure, temperature, and magnetic field under some assumptions are given. Then some numerical experiments are prescribed to support the effectiveness of the consistent projection finite element method.

Second-order rotational velocity correction projection finite element method for unsteady MHD coupled heat equation

Second-order rotational velocity correction projection finite element method for unsteady MHD coupled heat equation

Thermal Convection in a Heated-Block Duct with Periodic Boundary Conditions by Element-by-Element Treatment

The periodic nature of stream-wise flow occurs in a cooling channel so frequently due to the multiple heat sources in electronic equipment, demanding the creation of an effective technique to improve the heat-cooling convection. This work explores thermal convection enhancement in a heated-block duct for periodic boundary conditions using the element-by-element (EBE) treatment in a semi-implicit projection finite element method (FEM) through a preconditioned conjugate gradient (PCG) solver. The effects of changing the Reynolds numbers (100, 175, and 250) on rectangular cylinders installed in the channel under periodic boundary conditions were studied using time-mean Nusselt number enhancement, friction factor enhancement, and thermal performance coefficient. The results show that the rectangular cylinders installed stream-wise above an upstream block promote thermal convection in the heated-block duct due to modifying the flow of no cylinders. However, increasing the number of rectangular cylinders increases the friction factor enhancement. As a result, the case for periodic boundary conditions with a rectangular cylinder above every two blocks has the best thermal performance coefficient.

A modified characteristics projection finite element method for the non‐stationary incompressible thermally coupled MHD equations

In this paper, a modified characteristics projection finite element method is proposed for the non‐stationary incompressible thermally couple MHD equations. The modified characteristics projection finite element method combines the modified characteristics finite element method with the projection method to verify the unconditional stability and optimal error estimation of velocity, pressure, magnetic field and temperature in the non‐stationary incompressible thermally couple MHD equations. In order to show the effectiveness of the proposed method, some numerical results are presented. The numerical results are consistent with our theoretical analysis, and our method is effective.

Pressure correction projection finite element method for the 2D/3D time-dependent thermomicropolar fluid problem

Pressure correction projection finite element method for the 2D/3D time-dependent thermomicropolar fluid problem

A modified characteristic projection finite element method for the Kelvin-Voigt viscoelastic fluid equations

This paper presents a modified characteristic projection finite element method for the Kelvin-Voigt viscoelastic fluid equations. Then, the unconditional stability and optimal convergence of numerical solutions in L2-norms and H1-norms were presented. In addition, some numerical results will be given to testify the theoretical analysis. The numerical results show that the convergence orders are optimal, which show the theoretical analysis is right. The numerical results show that our method is robust for different Reynolds numbers. The numerical results imply that the numerical algorithm also converges for different retardation time. It means that our numerical method is robust for different retardation time.

A grad-div stabilized projection finite element method for a double-diffusive natural convection model

In this article, a grad-div stabilized projection finite element method is proposed for the double-diffusive natural convection model. Moreover, for the presented method, the stability analysis, and error estimates are deduced. Finally, numerical tests are provided that demonstrate the efficiency of the method. It is found that the presented method can improve the accuracy of the solution compared with the standard projection method, and compute approximations with better mass conservation.

A modified characteristics projection finite element method for unsteady incompressible Magnetohydrodynamics equations

This paper provides a modified characteristics projection finite element method for the unsteady incompressible magnetohydrodynamics(MHD) equations. In this method, modified characteristics finite element method and the projection method will be combined for solving the unsteady incompressible MHD equations. Both the stability and the optimal error estimates both in L2 and H1 norms for the modified characteristics projection finite element method will be shown. In order to demonstrate the effectiveness of our method, we will present some numerical results at the end.

A consistent projection finite element method for the incompressible MHD equations

ABSTRACT In this paper, we introduce a consistent projection finite element method for the impressible MHD (MagnetoHydroDynamics) equations. It's a fully discrete projection method, which associates the virtues of both the gauge and Uzawa methods with a variational framework. We also testify the unconditional stability and error estimates of the velocity, pressure and magnetic field by the variational method using some assumptions. Finally, we present some numerical experiments to support the validity of the consistent projection finite element method.

A novel pressure-correction projection finite element method for incompressible natural convection problem with variable density

This work introduces a novel pressure-correction (PC) projection finite element method for solving the incompressible natural convection problem with variable density. The main novel ideas of this work are to overcome the stability issue due to the nonlinear inertial term and the hyperbolic term for conventional finite element methods. We choose the conforming finite element pair to approximate the density, the velocity, the pressure, and the temperature field, respectively. Moreover, the stability analysis of the first-order PC projection finite element method is given. Finally, ample numerical results are presented to demonstrate the efficacy and accuracy of the proposed method.

Modified characteristics projection finite element method for time-dependent conduction-convection problems

In this paper, we give a modified characteristics projection finite element method for the time-dependent conduction-convection problems, which is gotten by combining the modified characteristics finite element method and the projection method. The stability and the error analysis shows that our method is stable and has optimal convergence order. In order to show the effect of our method, some numerical results are presented. From the numerical results, we can see that the modified characteristics projection finite element method can simulate the fluid field, temperature field, and pressure field very well.

An adaptive residual local projection finite element method for the Navier–Stokes equations

This work proposes and analyses an adaptive finite element scheme for the fully non-linear incompressible Navier-Stokes equations. A residual a posteriori error estimator is shown to be effective and reliable. The error estimator relies on a Residual Local Projection (RELP) finite element method for which we prove well-posedness under mild conditions. Several well-established numerical tests assess the theoretical results.

Convergence Analysis of a Residual Local Projection Finite Element Method for the Navier–Stokes Equations

This work presents and analyzes a new residual local projection stabilized finite element method (RELP) for the nonlinear incompressible Navier-Stokes equations. Stokes problems defined elementwise drive the construction of the residual-based terms which make the present method stable for the finite element pairs $\mathbb{P}_1/\mathbb{P}_l$, $l=0,1$. Numerical upwinding is incorporated through an extra control on the advective derivative and on the residual of the divergence equation. Well-posedness of the discrete problem as well as optimal error estimates in natural norms are proved under standard assumptions. Next, a divergence-free velocity field is provided by a simple postprocessing of the computed velocity and pressure using the lowest order Raviart-Thomas basis functions. This updated velocity is proved to converge optimally to the exact solution. Numerics assess the theoretical results and validate the RELP method.

Mixed Convective Heat Transfer Past a Heated Square Porous Cylinder in a Horizontal Channel With Varying Channel Height

The laminar mixed convection flow across the porous square cylinder with the heated cylinder bottom at the axis in the channel has been carried out numerically in this paper using a semi-implicit projection finite element method. The governing equations with the Brinkman–Forcheimer-extended Darcy model for the region of square porous cylinder were solved. The parameter studies including Grashof number, Darcy number, and channel-to-cylinder height ratio on heat transfer performance have been explored in detail. The results indicate that the heat transfer is augmented as the Darcy number and channel-to-cylinder height ratio increase. The buoyancy effect on the local Nusselt number is clearer for B/H=0.1 than for B/H=0.3 and B/H=0.5.

Numerical methods for improved efficiency in macrosegregation modeling

AbstractNew algorithms developed for the simulation of macrosegregation in directionally solidified alloys are presented. The work has been motivated by the inefficiency of models that have been used in the past. Three different developments are reported, the use of a fractional step projection finite element method to solve the momentum equations, the use of mesh‐less element free Galerkin formulations, and the combination of the Galerkin finite element method with adaptive mesh refinement. The formulation of the different algorithms is given and a two‐dimensional example of application in which the efficiency of the methods is compared is presented. The results of a three‐dimensional calculation using the Galerkin finite element method show that very significant gains are effected when compared with previously published simulations. Copyright © 2010 John Wiley & Sons, Ltd.

Numerical analysis of vortex shedding behind a porous square cylinder

Vortex shedding behind a porous square cylinder is studied by a semi‐implicit projection finite element method in this paper. Flow fields over the porous square cylinder with various porosity and Darcy number were calculated to investigate the effects of porous media on Strouhal number, lift and drag coefficients at Re=100, 200 and 250. The global quantities of the shedding flow appear evident variation for Da varied from 10−5 to 10−4. In particular, Strouhal number displays an obvious jump for Da=10−4 at Re=250, due to the variation in flow structure near the cylinder. Larger Darcy number postpones the occurrence of flow fluctuation, and the oscillation flow presents stronger amplitude. However, the flow field and the characteristic values of vortex shedding for Da≤10−6 are almost not changed. For different porosity, time histories of CD and CL look similar. In conclusion, Darcy number demonstrates more influence on the flow field than porosity does.

Analysis of thermal convection in a fluid-saturated porous cavity with internal heat generation

In this paper, the thermal convection flow in a fluid-saturated porous cavity with external sidewall heating and internal heat generation is studied by a semi-implicit projection finite element method. Concerned physical phenomena are constructed by impulsively switching on a uniform internal heat generation at t = 0 in a steady state flow field established by two different sidewall temperatures. Flows in the range 103 ≤ Ra I ≤ 108 and Ra E = 105 are mainly investigated for porous media ɛ = 0.4 and 10–10 < Da < 10–1. The results include the steady state flow field and heat transfer phenomena, as well as the evolution of heat transfer. According to the results, porosity demonstrates obvious influence in heat transfer as Da ≥ 10–2, and permeability significantly affects the flow field and heat transfer rate for Da < 10–2. The decrease in permeability diminishes the strength of external heating driven circulation and provides a chance for the cell driven by internal heat generation to develop. In addition, the flow in a porous medium with smaller flow restriction more quickly reaches the steady state from the instantaneously started internal heat generation.

Numerical analysis of thermosolutal flows in a cavity with gravity modulation effects

In this work, a two-dimensional thermosolutal convection flow under a sinusoidal gravity modulation (g-jitter) field is studied to understand the effects of the periodic source on flow field, as well as heat and mass transfer mechanisms. A semi-implicit projection finite element method is adopted to solve the transient Navier–Stokes, energy and species concentration equations. The fingering regime and the diffusive regime are explored for a series of gravity modulation frequencies. Two types of flow evolution, synchronous and subharmonic responses, are obtained for different frequencies. Distribution of unstable responses for the singly unstable condition is in agreement with the literature predict. The results show that heat and mass transfer rates are affected by the response type. For a subharmonic variation flow field, the overall Nusellt number and Sherwood number exhibit larger values. In addition, the augmenting condition in combining thermally driving force and solutally driving force is analyzed for a practical crystal material and displays a different response distribution from those in the fingering regime and the diffusive regime.

Analysis of flows driven by a torsionally–oscillatory lid in a fluid-saturated porous enclosure with thermal stable stratification

The convection flow caused by a torsionally–oscillatory lid with thermal stable stratification in an enclosure filled with porous medium is studied in this paper. The governing continuity, momentum and energy transport equations are solved by a semi-implicit projection finite element method. The Grashof number and Reynolds number for ranges 10 4⩽Gr⩽10 6, 10 2⩽Re⩽10 3 are involved in this research for Darcy number 10 −2, 10 −4, 10 −6 and porosity 0.4, 0.6. The results present that the permeability of porous medium deeply dominates the flow field and diminishes the flow strength as the permeability is decreased. The increase in the value of Gr/(Re 2·Da) retards the stable stratification from the top surface to extend into the interior domain. The influence of oscillatory frequency is so serious in heat flux variation at some particular frequency corresponding to the resonant frequency. There is an evident resonant frequency at Da=10 −2 condition; however, this phenomenon is not clear for Da⩽10 −4 at different oscillatory frequencies.

AN EQUAL-ORDER VELOCITY-PRESSURE ALGORITHM FOR INCOMPRESSIBLE THERMAL FLOWS, PART 1: FORMULATION

A finite-element algorithm is presented for the solution of two- and three-dimensional incompressible laminar thermal flows. The algorithm is cast in a time-dependent form and can be classified as a projection finite-element method. However, the procedure utilized for handling Ike velocity-pressure coupling shares many features with the SIMPLER finite-difference method. In fact, given an initial or guessed velocity field, the pseudo-velocities, i.e., the velocities that would prevail in the absence of the pressure field, are found first. Then, by enforcing continuity on the pseudo-velocity field, the tentative pressure is estimated and the momentum equations are solved in sequence for the velocity components. Afterward, continuity is enforced again to find corrections that are used to modify the velocity field and the estimated pressure field. Finally, if required, the energy equation is solved before moving to the next step.