Stable Finite Element Pair Research Articles

Finite element simulations of Herschel–Bulkley visco-plastic materials over a cylinder: Drag and lift correlation analysis

This study employs a two-dimensional and incompressible flow of Herschel–Bulkley visco-plastic materials in order to investigate the hydrodynamic forces that are acting on a barrier that is located close to the inlet of a channel. As the benchmark configuration, the flow domain that has been selected is a channel that still contains the impediment. The two important parameters of the Herschel–Bulkley Model (HBM) are the yield stress [Formula: see text] and power law index n. Obtaining special situations within the HBM, such as Newtonian, power-law, and Bingham fluids, can be accomplished by assigning certain values to these parameters at the appropriate times. Utilizing a numerical strategy grounded in the Finite Element Method (FEM), we tackle the nonlinearity of the governing equations as well as the viscosity models. As a result of this nonlinearity, FEM becomes an essential tool. The generation of a refined hybrid mesh is done in order to guarantee accuracy in the computations. The stable finite element pair ([Formula: see text]) has been selected for discretization purposes. The discretized nonlinear system is linearized with Newton’s method and subsequently, a direct linear solver PARDISO has been employed in the inner iterations. The pressure, velocity, and viscosity profiles are plotted for various values of n and Bingham number (Bn). In addition, the velocity behavior is observed along the y-direction in a channel through line graphs. Code validation is done as a special case [Formula: see text] and a good agreement is found with the results available in the literature. Finally, a correlation analysis has been performed for the drag coefficient [Formula: see text] and lift coefficient [Formula: see text].

Numerical analysis of double-diffusive natural convective flow of Ostwald-de Waele fluid in an irregular enclosure with a circular obstacle

PurposeThe current study aims to enhance the double diffusive natural convection in an Ostwald-de Waele fluid filled in a hexagonal enclosures using non-uniform magnetic field embedded with circular obstacle. The upper wall has maximum temperature Th∗ and concentration (Ch∗) whereas the lower wall is kept at minimum temperature Tc∗ and concentration Cc∗. The parameters that play a significant role in the heat and mass transfer are evaluated, including Hartmann number, power law index, Rayleigh number and buoyancy ratio. Design/Methodology/approachIn two-dimensions, the governing formulation is written as a set of balancing equations for velocities, pressures, energies and concentrations. The nonlinear governing PDEs are solved using numerical approach based on Finite Element Method (FEM). The discretization is performed using the stable finite element pair ℘2/℘1. Code validation and results are ensured by conducting a grid convergence and comparison test respectively. FindingsThe influence of flow on velocity, isotherms and iso-concentration field is examined by analyzing a broad variety of variables such as power-law index 0.7≤n≤1.2, Hartmann number 0≤Ha≤50, Rayleigh number 104≤Ra≤106, inclination angle 0°≤γ≤90° and buoyancy ratio -2≤N≤2. The findings are displayed graphically, including the flow, temperature and concentration fields. Heat and mass transfer rates are maximum for shear thinning fluids while for shear thickening fluids it reduces, whereas it is maximum for concentration dominated assistant flow and minimum for concentration dominant counter flow. ApplicationThe novel aspects of the research include its thorough examination of solar thermal power conversion and its inventive energy deficiency devices and a wide range of electrical design applications.

Flow and heat transfer analysis over a pair of heated bluff bodies in a channel: Characteristics of non-linear rheological models

In this computational study, the tribological characteristics of the flow of a visco-plastic fluid around a pair of cylinders in a channel are thoroughly analyzed. To numerically solve the governing equations, this investigation makes use of the Galerkin's Finite Element Method (FEM) in conjunction with Newton's and PARDISO solvers. The stable finite element pair (P2/P1) has been selected for the space discretization purpose. The impacts of several physically important parameters on the flow characteristics are investigated in depth, including the spacing ratio (GP), the power law index (n), the Bingham number (Bn), and the Reynolds number (Re). When using different gap spacing ratios (GP), the drag coefficient (CD) and lift coefficient CL were found to be strongly dependent on GP. The pressure, velocity, and viscosity profiles are plotted for varying n and Bn. In addition, the velocity behavior is observed along y-direction in a channel through line graphs. Code validation is done as special case when Bn=0andn=1. It has been observed that the gap between the obstacles can influence the overall flow pattern and subsequently the drag and lift forces.

Temporal features of thermal flows over a rotating cylinder in a channel: Multigrid based simulations

The purpose of this study is to simulate thermal flows via quasi-Newtonian modeling by employing Finite Element Methods (FEM), as well as Newton Multigrid Solvers (NMS). A heated rotating cylinder is exposed to an inflowing fluid stream in a channel to facilitate the thermal energy transfer in the fluid. The NMS may be more easily parallelized as a result of an excessive number of degrees of freedom. The discretization of a dimensionless system of partial differential equations that are to be solved over an extensive computational domain is accomplished by using the higher-order stable finite element pair. Temporal discretization is performed using a 2nd order Crank-Nicolson scheme. By applying the Newton method, the discretized nonlinear system of algebraic equations is transformed into a linear form. To compute the linearized subproblems that arise from each nonlinear sweep, a geometric multigrid approach has been put in place. The whole framework has been implemented in an open-source finite element software Featflow. It has been deduced that in a clockwise rotation of the cylinder, more fluid will move from above the cylinder, and when it is rotated in an anticlockwise direction, more fluid will move from bottom side of the cylinder. In addition to this, there is a correlation between an increase in the rotational parameter ω and a reduction in both the lift coefficient CL and drag coefficient CD. The benchmark values of CD and CL fits very well with the available data for certain special cases of this study.

A fully discrete finite element method for a constrained transport model of the incompressible MHD equations

In this paper, we propose and analyze a fully discrete finite element method for a constrained transport (CT) model of the incompressible magnetohydrodynamic (MHD) equations. The spatial discretization is based on mixed finite elements, where the hydrodynamic unknowns are approximated by stable finite element pairs, the magnetic field and magnetic vector potential are discretized by H(curl)-conforming edge element. The time marching is combining a backward Euler scheme and some subtle implicit-explicit treatments for nonlinear and coupling terms. With these treatments, the fully discrete scheme is linear in the implementation and the computation of the magnetic vector potential is decoupled from the whole coupled system. The most attractive feature of this scheme that it can yield the exactly divergence-free magnetic field and current density on the discrete level. The unique solvability and unconditional stability of the scheme are also proved rigorously. By utilizing the energy argument, error estimates for the velocity, magnetic field and magnetic vector potential are further demonstrated under the low regularity hypothesis for the exact solutions. Numerical results are provided to verify the theoretical analysis and to show the effectiveness of the proposed scheme.

Unfitted finite element methods based on correction functions for Stokes flows with singular forces of low regularity

This paper presents an unfitted finite element method based on correction functions for solving stationary Stokes flows with singular forces acting on an immersed interface. It has been shown that the singular force is equivalent to a nonhomogeneous jump condition on the interface. In this paper, we consider the case that the jump has a low regularity so that it is impossible to use pointwise values on the interface to construct correction functions, as done in Guzmán et al. (2016). The natural way to deal with the problem is to use mean values of the jump on the parts of the interface cut by elements, instead of using pointwise values. However, we show that it may cause instability and the constant in the error estimate may depend on the interface location relative to the mesh. Inspired by Guo et al. (2019), we use a larger fictitious circle to overcome these issues. Associated with the correction functions, we consider two stable finite element pairs: the Mini element and the P2−P0 element, including the cases of continuous and discontinuous pressures. The optimal approximation capabilities of the correction functions and optimal error estimates of the finite element methods are both derived with a hidden constant independent of the interface location relative to the mesh. Numerical examples are provided to validate the theoretical results.

Research on three kinds of splitting finite element schemes for 2D/3D unsteady incompressible thermomicropolar fluid equations

AbstractIn this article, three kinds of splitting finite element schemes are proposed for the 2D/3D unsteady incompressible thermomicropolar fluid equations, which are named as the standard splitting scheme, the consistent splitting schemes (I and II) and the splitting Oseen scheme. The unconditionally energy stable of all the schemes and error estimation of the standard splitting scheme are proved. Furthermore, numerical experiments are carried out on the last two kinds of schemes based on different selection of finite element pairs, numerical results show that the presented schemes are reliable, especially some not inf‐sup stable finite element pairs can also be applied for the consistent splitting scheme II. In addition, for moderate Reynolds number problems, the splitting Oseen scheme exhibit the better performance of numerical stability than the consistent splitting schemes.

Mixed convection of ferrofluids in a square enclosure with obstacles: Effect of iso‐perimetric shapes

AbstractMixed convection of ferrofluid in square configurations having various iso‐perimetric shaped obstacles has been investigated for fluid flow and heat transfer characteristics. The iso‐perimetric shapes are allowed to vary in size and tested at three different vertical positions subject to adiabatic and isothermal conditions. Simulations have been carried out for various choices of controlling parameters using the higher order mixed finite element method. In particular, the higher order and stable finite element pair of cubic and quadratic () approximations on the hybrid grid is utilized to achieve the discretization of the governing non‐dimensional equations of mass, momentum and energy. The adaptive Newtons method is chosen to solve the discrete system of nonlinear equations. The present solver has been tested against a well‐known open source finite element analysis tool (FEAT2) and published results in the literature. The impact of governing parameters is investigated in the ranges as nanoparticles volume fraction ), the Richardson number , the perimeter of the obstacle , and positions of the obstacles . It is concluded that an increase in the perimeter of the inner obstacles causes a weakness in the fluid flow while a good thermal enclosure is obtained. Moreover, the isothermal obstacles enhance the average Nusselt number more than the adiabatic obstacles in all cases.

A decoupled, unconditionally energy stable and charge-conservative finite element method for inductionless magnetohydrodynamic equations

In this paper, we propose a decoupled, unconditionally energy stable and charge-conservative finite element method for the inductionless magnetohydrodynamic equations. The time marching is combined with a first order semi-implicit Euler scheme, a first order perturbation term and some delicate implicit-explicit treatments for coupling terms. The finite element discretization is based on mixed conforming elements, where the hydrodynamic unknowns are approximated by stable finite element pairs, and the current density and electric potential are discretized by divergence-conforming face elements and discontinuous volume elements. This fully discrete scheme only needs to solve a linear subsystem at each time step and yields an exactly divergence-free current density directly. We show that the proposed scheme is well-posed and unconditionally energy stable. Under a weak regularity hypothesis on the exact solution, we present the error estimates for velocity, current density and electric potential. Finally, the validity, reliability, and accuracy of the proposed algorithm are supported by several numerical experiments.

Multiphysics mixed finite element method with Nitsche's technique for Stokes‐poroelasticity problem

AbstractIn this article, we propose a multiphysics mixed finite element method with Nitsche's technique for Stokes‐poroelasticity problem. Firstly, we reformulate the poroelasticity part of the original problem by introducing two pseudo‐pressures to into a “fluid–fluid” coupled problem so that we can use the classical stable finite element pairs to deal with this problem conveniently. Then, we prove the existence and uniqueness of weak solution of the reformulated problem. And we use Nitsche's technique to approximate the coupling condition at the interface to propose a loosely‐coupled time‐stepping method to solve three subproblems at each time step–a Stokes problem, a generalized Stokes problem and a mixed diffusion problem. And the proposed method does not require any restriction on the choice of the discrete approximation spaces on each side of the interface provided that appropriate quadrature methods are adopted. Also, we give the stability analysis and error estimates of the loosely‐coupled time‐stepping method. Finally, we give the numerical tests to show that the proposed numerical method has a good stability and no “locking” phenomenon.

Numerical Computations of Non-Newtonian Fluid Flow in Hexagonal Cavity With a Square Obstacle: A Hybrid Mesh–Based Study

Thermal flow phenomena in a double lid–driven enclosure have many potential applications in numerous engineering domains. The present article theoretically investigates the heat transfer analysis of power-law fluid in a hexagonal cavity embedded with a square obstacle. The upper and lower lid walls of the cavity are considered heated along with the walls of the centered embedded square, while the rest of the cavity walls are thermally insulated. In addition, the horizontal lid walls of the cavity are uniformly moving in opposite horizontal directions. The mathematical modeling of the nonlinear fluid has been developed considering the continuity, momentum, and energy equations subjected to the appropriate boundary conditions. Due to the nonlinearity of the governing partial differential equations and of the viscosity models, we use the numerical scheme based on the finite element method The stable finite element pair (ℙ2/ℙ1) has been selected for the discretization purpose. The discretized nonlinear system is solved with the Newton method in conjunction with a direct linear solver in the inner iterations. The thermal flow features are exposed via streamlines and temperature contours for a set of governing parameters such as Reynolds number (Re), Prandtl number (Pr), and Grashof number (Gr) along with the power-law index (n). The values of local and average Nusselt numbers are calculated for the involved parameters. It is noted that the square obstacle has a strong impact for the formulation of streamlines and isotherms. Moreover, the power-law index has a strong impact on the values of the average Nusselt number and kinetic energy. The kinetic energy of the system increases with an increasing value of Gr and n, while Reynolds number has opposite effects on kinetic energy.

Thermal visualization of Ostwald-de Waele liquid in wavy trapezoidal cavity: Effect of undulation and amplitude

The present study is concerned with the numerical simulations of Ostwald-de Waele fluid flow in a wavy trapezoidal cavity in the presence of a heated cylinder situated at the center of the cavity. The work consists in characterizing the mixed convection as a function of the intensity of heat flow. The flow behaviour and temperature distribution in a cavity are the main focus of this study. The lower wall of the cavity is fixed and heated while the wavy surface is insulated and moves with a constant speed. The sloping walls are kept cold and are subject to zero no-slip conditions for velocity components. The whole setup is modeled as a set of coupled partial differential equations and is solved by the Finite Element Method. For pressure and velocity approximations, we use the stable finite element pair P2−P1, while for temperature approximation we use the space of linear polynomial as P1. The ranges of the parameters involved in the study are the Ostwald-de Wale index (0.5≤n≤1.5), Prandlt number (1≤Pr≤10), Grashof number (100≤Gr≤1000), the number of undulation (0≤λ≤4), and the non-dimensional amplitude of the wavy surface (0.02≤A≤0.06). The major findings of the study are shown using velocity profile, streamlines, and isotherms. Moreover, the kinetic energy and average (Nuavg) Nusselt number is determined for various values of the parameters involved.

Velocity-vorticity-pressure formulation for the Oseen problem with variable viscosity

We propose and analyse an augmented mixed finite element method for the Oseen equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and homogeneous Dirichlet boundary condition for the velocity. The weak formulation includes least-squares terms arising from the constitutive equation and from the incompressibility condition, and we show that it satisfies the hypotheses of the Babu\vska-Brezzi theory. Repeating the arguments of the continuous analysis, the stability and solvability of the discrete problem are established. The method is suited for any Stokes inf-sup stable finite element pair for velocity and pressure, while for vorticity any generic discrete space (of arbitrary order) can be used. A priori and a posteriori error estimates are derived using two specific families of discrete subspaces. Finally, we provide a set of numerical tests illustrating the behaviour of the scheme, verifying the theoretical convergence rates, and showing the performance of the adaptive algorithm guided by residual a posteriori error estimation.

A full discretization for the saddle-point approach of a degenerate parabolic problem involving a moving body

This paper aims to study a degenerate parabolic problem for a solenoidal vector field in which the time derivative acts on a moving body. We propose a fully-discrete finite element scheme combined with backward Euler’s method for the saddle-point variational formulation. The convergence of this numerical scheme is proved and error estimates for some stable finite element pairs are also established.

Numerical simulations of MHD mixed convection of hybrid nanofluid flow in a horizontal channel with cavity: Impact on heat transfer and hydrodynamic forces

In this paper, we explore the characteristics of heat transfer and fluid flow of hybrid nanofluid enclosed in a horizontal channel with cavity having an obstacle. The dimensionless governing partial equations for various physical parameters are simulated via higher order and stable Galerkin based finite element method. In particular, the higher order and stable finite element pair, ¶2/¶1 is utilized for the discretization of the modelled partial differential equations. The discretised system of nonlinear algebraic equations are linearized with the help of Newton's method and the resulting linear problems are computed with the help of geometric multi-grid techniques. Effective governing parameters such as Reynolds number, cylinder rotation, aspect ratio of cavity and the impact of variance in cylinder radius on the heat transfer and fluid flow are analyzed in detail. Moreover, the computation of hydrodynamic forces such as drag and lift coefficients on the obstacle, are also analyzed in this paper. It is shown that by increasing the radius of the obstacle the heat transfer in the channel is enhanced up to 119%. Furthermore, horizontal position of cylinder only contributes a heat transfer efficacy of 2.54%.

Analysis of biomagnetic blood flow in a stenosed bifurcation artery amidst elastic walls

In recent years, fluid-structure interaction (FSI) analysis has generated remarkable interest of researchers in interdisciplinary sciences problems, for instance, mechanical engineering, biomedical engineering, i.e., incorporating elastic wall behaviour in human arteries. Here, in this paper, we considered incompressible Newtonian blood flow and the elastic bifurcated artery wall in a non-uniform magnetic field. The considered model of Biomagnetic Fluid Dynamics (BFD) describes both magnetization and electrical conductivity of blood. Moreover, an Arbitrary Lagrangian-Eulerian (ALE) formulation is used by two-way fluid-structure interaction coupling of the problem. Finally, for discretization, a stable finite element pair is employed to approximate the displacement, velocity and pressure spaces independently and the resulting nonlinear algebraic system is linearized by implementing the Newtons procedure. A quantitative analysis is made against Reynolds number Re and Hartmann number Ha on the bifurcated artery amidst elastic walls showing noticeable effects on recirculation. Also, the displacement field is plotted against Ha for different values of Re and their converse behaviour is observed. The decreasing behaviour for the wall shear stresses of the bifurcated artery are also drawn in context of Ha and Re. Finally, the conclusion is drawn at the end showing the significance of the elastic behaviour of artery walls.

Semirobust analysis of an H(div)-conforming DG method with semi-implicit time-marching for the evolutionary incompressible Navier–Stokes equations

Abstract In this paper, we present a fully discrete analysis of an H(div)-conforming DG method with semi-implicit time-marching for the evolutionary incompressible Navier–Stokes equations. We use a semi-implicit time-discrete scheme in which the convection velocity is treated explicitly for the convection term. A stability analysis and a priori error estimates are given, in which the constants are independent of the negative powers of the viscosity. For inf-sup stable H(div)-conforming finite element pairs $BDM_k/P_{k-1}$ and $RT_k/P_k$, the rate of convergence $k+1/2$ is proved for the $L^2$ error of the velocity in the case of $\nu \leq C h$, where $k$ is the degree of the polynomials in the velocity approximation. In particular, for the inf-sup stable finite element pair $RT_k/P_k$, the convergence rate of the pressure is also $k+1/2$ when $\nu \leq C h$. The numerical experiments verify the analytical results.

Penalty decoupled iterative methods for the stationary natural convection equations with different Rayleigh numbers

In this paper, we propose and analyze six combined schemes in the framework of penalty method to solve the natural convection (NC) equations numerically. The key role of the penalty technique is to manage the incompressibility constraint divu=0 effectively and improve the positivity of the resulting system. For definiteness and without loss of generality, we mainly explore a inf-sup stable finite element pair P1b-P1-P1b and a inf-sup unstable one P1-P0-P1. Three efficient iterative schemes are applied to deal with the nonlinear terms. Moreover, the results of the discrete stability and optimal convergence under the exactly different stability conditions are proved for the given methods. Numerical simulations and results for three benchmark problems are explored to illustrate the theoretical findings.

Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity

AbstractThis paper proposes and analyzes a posteriori error estimator based on stress equilibration for linear elasticity with emphasis on the behavior for (nearly) incompressible materials. It is based on an H(div)‐conforming, weakly symmetric stress reconstruction from the displacement‐pressure approximation computed with a stable finite element pair. Our focus is on the Taylor‐Hood combination of continuous finite element spaces of polynomial degrees k + 1 and k for the displacement and the pressure, respectively. This weak symmetry allows us to prove that the resulting error estimator constitutes a guaranteed upper bound for the error with a constant that depends only on local constants associated with the patches and thus on the shape regularity of the triangulation. It does not involve global constants like those from Korn's in equality which may become very large depending on the location and type of the boundary conditions. Local efficiency, also uniformly in the incompressible limit, is deduced from the upper bound by the residual error estimator. Numerical results for the popular Cook's membrane test problem confirm the theoretical predictions.

A comprehensive finite element examination of Carreau Yasuda fluid model in a lid driven cavity and channel with obstacle by way of kinetic energy and drag and lift coefficient measurements

In current pagination the flow problems involving shear rate dependent nonlinear viscosity have been treated successfully in the sense of space discretization and solvers. The stable finite element pair Q2/P1disc is employed to approximate the velocity and pressure spaces independently. Discretized form of non-linear expressions is linearized by implementing Newton’s procedure and the resulting systems are solved using a geometric multigrid approach. The flow generated by way of driven cavity and by an obstacle are very important benchmarks of computational fluid dynamics. In current pagination shear rate reliant viscosity model renowned as Carreau Yasuda fluid is capitalized. The obtained results are demonstrated and analyzed with the help of velocity and viscosity plots. In addition, we have produced new reference data for kinetic energy (K.E) for driven cavity problem and drag and lift coefficients for circular obstacle problem. The obtained results for driven cavity problem reveal the fact that K.E is an increasing function of the relaxation parameter (λ) and power law exponent (n) whereas a decreasing function of the model parameter (a). For the case of flow around obstacle the drag coefficient (CD) and the lift coefficient (CL) show strong dependence on (λ) and (n), however a weak dependence on the parameter (a).