A time marching strategy for solving parabolic and elliptic equations with Neumann boundary conditions

Abstract

In the community of computational fluid dynamics, pressure Poisson equation with Neumann boundary condition is usually encountered when solving the incompressible Navier–Stokes equations in a segregated approach such as SIMPLE, PISO, and projection methods. To deal with Neumann boundary conditions more naturally and to retain high order spatial accuracy as well, a sixth-order accurate combined compact difference scheme developed on staggered grids (NSCCD6) is adopted to solve the parabolic and elliptic equations subject to Neumann boundary conditions. The staggered grid system is usually used when solving the incompressible Navier–Stokes equations. By adopting the combined compact difference concept, there is no need to discretize Neumann boundary conditions with one-sided discretization scheme which is of lower accuracy order. The conventional Crank–Nicolson scheme is applied in this study for temporal discretization. For two-dimensional cases, D’yakonov alternating direction implicit scheme is adopted. A newly proposed time step changing strategy is adopted to improve convergence rate when solving the steady state solutions of the parabolic equation. High accuracy order of the currently proposed NSCCD6 scheme for one- and two-dimensional cases are shown in this article.     

Local least–squares element differential method for solving heat conduction problems in composite structures

Abstract

In this article, a completely new numerical method called the Local Least-Squares Element Differential Method (LSEDM), is proposed for solving general engineering problems governed by second order partial differential equations. The method is a type of strong-form finite element method. In this method, a set of differential formulations of the isoparametric elements with respect to global coordinates are employed to collocate the governing differential equations and Neumann boundary conditions of the considered problem to generate the system of equations for internal nodes and boundary nodes of the collocation element. For each outer boundary or element interface, one equation is generated using the Neumann boundary condition and thus a number of equations can be generated for each node associated with a number of element interfaces. The least-squares technique is used to cast these interface equations into one equation by optimizing the local physical variable at the least-squares formulation. Thus, the solution system has as many equations as the total number of nodes of the present heat conduction problem. The proposed LSEDM can ultimately guarantee the conservativeness of the heat flux across element surfaces and can effectively improve the solution stability of the element differential method in solving problems with hugely different material properties, which is a challenging issue in meshfree methods. Numerical examples on two- and three-dimensional heat conduction problems are given to demonstrate the stability and efficiency of the proposed method.     

Implementation of boundary conditions and global mass conservation in pressure-based finite volume method on unstructured grids for fluid flow and heat transfer simulations

Proper boundary condition implementation can be critical for efficient, accurate numerical simulations when a pressure-based finite volume methodology is used to solve fluid flow and heat transfer problems, especially when an unstructured mesh is used for domain discretization. This paper systematically addresses the relationships between the flow boundary conditions for the discretized momentum equations and the corresponding conditions for the (Poisson type) pressure-linked equation in the context of a cell-centred finite volume formulation employing unstructured grids and a collocated variable arrangement. Special attention is paid to the treatment of the outflow boundary where flow conditions are either known to be fully developed (if a long enough channel is used) or unknown prior to solution (if a truncated channel is used). In the latter case, due to the singularity of the coefficient matrix of the pressure-linked equation, no pressure or pressure corrector solution can exist without explicitly enforcing global mass conservation (GMC) during each iteration. After evaluating published methods designed to ensure GMC, two new methods are proposed to correct for global mass imbalance. The validity of the overall methodology is demonstrated by solving the evolving flow between two parallel plates and the laminar flow over a backward-facing step on progressively truncated domains. In the latter case, our methodology is shown to handle situations where the outflow boundary passes through a recirculation zone.     

Discretized pressure Poisson algorithm for the steady incompressible flow on a nonstaggered grid

In the aspect of numerical methods for incompressible flow problems, there are two different algorithms: semi-implicit method for pressure-linked equations (SIMPLE) series algorithms and the pressure Poisson algorithm. This paper introduced a new discretized pressure Poisson algorithm for the steady incompressible flow based on a nonstaggered grid. Compared with the SIMPLE series algorithms, this paper did not introduce three correction variables. So, there is no need to implement the guess-and-correct procedure for the calculation of pressure and velocity. Compared with the pressure Poisson algorithm, there is no need to calculate unsteady Navier–Stokes equations for steady problems in the new discretized pressure Poisson algorithm. Meanwhile, as the finite volume method and cell-centered grid are used, the governing equation for pressure is obtained from the continuity equation and the boundary conditions for pressure are easily obtained. This new discretized pressure Poisson algorithm was tested at the lid-driven cavity flow problem on a nonstaggered grid and the results are also reliable.     

A Quasi-Implicit Time-Advancing Scheme for Flow in a Three-Dimensional Curved Duct

A quasi-implicit time-marching scheme for solving unsteady incompressible three-dimensional flows on cell-centered unstructured meshes is developed. The finite-volume formulation is used for the spatial derivatives, and the flow variables at the cell face are obtained using the pressure correction. The nonlinear equations resulting from the fully implicit scheme are linearized without deterioration of the overall super-linear time accuracy. The system matrices are solved using the CG iterative method, known as the P-BiCGSTAB method for the momentum equation and the P-CG method for the pressure Poisson equation. The model is applied to simulate fully developed laminar flow in both a 90° curved 3-D circular duct and a 90° curved 3-D square duct. Steady solution is obtained in an unsteady time-marching manner. Computed results compare well with experimental data and other numerical results. It is demonstrated that the present method can be applied to unsteady incompressible laminar 3-D flow with a complex geometry on the unstructured grid system.     

Implementation of boundary conditions in pressure-based finite volume methods on unstructured grids

In this study, the implementation of boundary conditions for the Navier–Stokes and the energy equations, including the pressure and pressure correction equations, are presented in the context of finite volume formulation on cell-centered, colocated unstructured grids. The implementation of boundary conditions is formulated in terms of the contribution of boundary face of a cell to the coefficients of the discretized equation for either Dirichlet- or Neumann-type boundary conditions. Open boundaries through which the flow is not fully developed are also considered. In this case, a data reconstruction method is proposed for finding the boundary values of the variables at the correction stage. The validity of implementations is checked by comparing the results with some well-known benchmark problems.     

On the quasi-instantaneous aerodynamic load and pressure field measurements on turbines by non-intrusive PIV

An experimental technique is presented to non-intrusively measure the quasi-instantaneous aerodynamic loads and surrounding pressure field for a turbine by using particle image velocimetry (PIV). The PIV measurements provide the velocity flow field needed to calculate the pressure field around the turbine using three different methods. In the first method, the quasi-instantaneous and mean pressure fields are obtained by solving the Poisson equation and by calculating the boundary conditions from the Navier–Stokes equations. In the second method, the pressure at the boundaries is determined by spatial integration of the pressure gradient. In the third method, the pressure is calculated using the Bernoulli equation. The experimental results are compared to aerodynamic load theoretical predictions from the Blade Element Momentum theory (BEM). An analysis of the experimental results showed the importance of the local acceleration, convective and pressure terms when calculating the forces and the pressure field in a stationary reference frame. Only the Poisson method includes all these terms, and had a small standard deviation between the calculated instantaneous forces. Furthermore, the Poisson method results are independent of the control volume size investigated while the other two experimental methods are affected. This experimental technique could be used to simultaneously replace instrumentation such as force balance and pressure taps while providing for the first time quasi-instantaneous information about the surrounding flow in any turbine immersed in an incompressible flow. In addition, it could be applied to evaluate unsteady wind loads and aerodynamic stall and also provide much needed information for validating computational studies.     

An implicit lattice Boltzmann model for heat conduction with phase change

In this work, a new variation of the lattice Boltzmann method (LBM) was developed to solve the heat conduction problem with phase change. In contrast to previous explicit algorithms, the latent heat source term was treated implicitly in the energy equation, avoiding iteration steps and improving the formulation stability and efficiency. The Bhatnagar–Gross–Krook (BGK) approximation with a D2Q9 lattice was applied and different boundary conditions including Dirichlet and Neumann boundary conditions were considered. The developed model was tested by solving a one-dimensional melting problem for a pure metal, and one and two-dimensional solidification problems for a binary alloy. The results of the LBM solution were compared with analytical and finite element solutions and a good consistency was observed. Considering the special capabilities that LBM offers, like local characteristic, and inherent parallel structure, the developed model is an interesting alternative to traditional continuum models.     

Reevaluation of the Vertex Collocated Mesh for Primitive Variable Computation of Incompressible Fluid Flows by Enforcing Continuity-Based Boundary Conditions

Complementing a previous comparative study of the accuracy of the fundamental mesh structures for primitive variable computations of incompressible fluid flows, this article considers some alternative approaches to the closure of the pressure equations in the boundary nodes of the vertex collocated mesh. In the previous study, these boundary pressures were determined directly by a discretized Poisson equation; in the present article the pressure equations at these nodes are derived from specific continuity equations, obtained by mass balance on the half-cells and by unilateral parabolic approximation of the velocity component normal to the wall. The first approach reduces the accuracy of the vertex collocated mesh to first-order, while the second approach remains second-order-accurate, except in the traditional cavity problem, and provides better results than the Poisson equation approach for rough and moderate refinements. However, in comparison with the other types of mesh, the vertex collocated mesh remains the least accurate for refined meshes.     

SPECIAL TREATMENT OF PRESSURE CORRECTION BASED ON CONTINUITY CONSERVATION IN A PRESSURE-BASED ALGORITHM

The special features of pressure-correction equations and their effects on the performance of the SIMPLE algorithm have been systematically investigated based on the concept of continuity conservation. Except for use of the same iterative method as for the momentum equations, iterative solution of the pressure-correction equation has special features in three respects: initial values, boundary conditions (BCs), and iterative procedure. First, the initial values in each outer loop are independent and should be reset as zeroes. Second, the BCs are fully reverse to that of velocity: Dirichlet velocity BCs correspond to Neumann BCs of pressure correction, and Neumann velocity BCs lead to pressure-correction Dirichlet BCs. Third, more inner iterations for the pressure-correction equation are required to better satisfy continuity conservation. Dealing properly with these features can greatly improve the efficiency of the SIMPLE algorithm. Computational results and comparisons have shown that global mass conservation BCs are favorable to convergence, but may be slowed down by the local conservation BCs. During the course of convergence, the BCs of the pressure-correction equation are vital: only correct BCs can boost convergence, incorrect BCs cannot. Increasing the inner iterations of the pressure-correction equation will significantly decrease the outer-loop iterations, and therefore effectively improve the performance of the SIMPLE algorithm.     

A Thermo-Mechanical Analysis of Isotropic and Composite Beams via Collocation with Radial Basis Functions

In this article, the mechanical behavior of beams subjected to thermal loads is investigated. The temperature field is obtained by exactly solving Fourier's heat conduction equation and it is considered as an external load within the mechanical analysis. Several higher-order beam models as well as Timoshenko's classical theory are derived thanks to a compact notation for the a priori approximation of the displacement field upon the cross-section. The governing differential equations and boundary conditions are obtained in a compact nucleal form that does not depend upon the displacements’ expansion order. The latter can be regarded as a free parameter of the formulation. A meshless strong-form solution based upon collocation with Wendland's radial basis functions is adopted. Isotropic and laminated orthotropic beams are investigated. Results are validated toward an analytical Navier-type solution and three-dimensional FEM results. It is shown that good accuracy can be obtained.     

On the Accuracy of a Finite-Difference Method for Parabolic Partial Differential Equations with Discontinuous Boundary Conditions

Although the numerical solution of parabolic partial differential equations (PDEs) is widely documented, the effect of discontinuous boundary conditions on numerical accuracy is not. This article employs the Keller box finite-difference method to study the effect of such discontinuities when solving the linear one-demensional transient heat equation. We demonstrate that this formally second-order-accurate scheme can lose accuracy, but that an analytical understanding of the behavior of the solution helps in providing an accuracy-restoring formulation. Benchmark computations are presented that will provide guidance in the numerical solution of nonlinear parabolic PDEs for which there are no closed-form analytical solutions.     

Higher-Order Numerical Scheme for the Fractional Heat Equation with Dirichlet and Neumann Boundary Conditions

In this article, we consider a higher-order numerical scheme for the fractional heat equation with Dirichlet and Neumann boundary conditions. By using a fourth-order compact finite-difference scheme for the spatial variable, we transform the fractional heat equation into a system of ordinary fractional differential equations which can be expressed in integral form. Further, the integral equation is transformed into a difference equation by a modified trapezoidal rule. Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm.     

HIGHER-ORDER COMPACT SCHEMES FOR NUMERICAL SIMULATION OF INCOMPRESSIBLE FLOWS,PART I: THEORETICAL DEVELOPMENT

A higher-order-accurate numerical procedure has been developed for solving incompressible Navier-Stokes equations for fluid flow problems. It is based on low-storage Runge-Kutta schemes for temporal discretization and fourth- and sixth-order compact finite-difference schemes for spatial discretization. New insights are presented on the elimination of the odd-even decoupling problem in the solution of the pressure Poisson equation. For consistent global accuracy, it is necessary to employ the same order of accuracy in the discretization of the Poisson equation. Accuracy and robustness issues are addressed by application to several pertinent benchmark problems in Part II.     

INCOMPRESSIBLE COMPUTATIONAL FLUID DYNAMICS AND THE CONTINUITY CONSTRAINT METHOD FOR THE THREE-DIMENSIONAL NAVIER-STOKES EQUATIONS

Abstract

As the field of computational fluid dynamics (CFD;) continues to mature, algorithms are required to exploit the most recent advances in approximation theory, numerical mathematics, computing architectures, and hardware. Meeting this requirement is particularly challenging in incompressible fluid mechanics, where primitive-variable CFD formulations that are robust, while also accurate and efficient in three dimensions, remain an elusive goal. This monograph asserts that one key to accomplishing this goal is recognition of the dual role assumed by the pressure, i.e., a mechanism for instantaneously enforcing conservation of mass and a force in the mechanical balance law for conservation of momentum. Proving this assertion has motivated the development of a new, primitive-variable, incompressible, CFD algorithm called the continuity constraint method (CCM;). The theoretical basis for the CCM consists of a finite-element spatial semidiscretization of a Galerkin weak statement, equal-order interpolation for all state variables, a 6-implicit time-integration scheme, and a quasi-Newton iterative procedure extended by a Taylor weak statement;(TWS) formulation for dispersion error control. This monograph presents: (I) the formulation of the unsteady evolution of the divergence error, (2) an investigation of the role of nonsmoothness in the discretized continuity-constraint function, (3;) the development of a uniformly H’ Galerkin weak statement for the Reynolds-averaged Navier-Stokes pressure Poisson equation, and(4;) a derivation of physically and numerically well-posed boundary conditions. In contrast to the general family of ‘pressure-relaxation’ incompressible CFD algorithms, the CCM does not use the pressure as merely a mathematical device to constrain the velocity distribution to conserve mass. Rather, the mathematically smooth and physically motivated genuine pressure is an underlying replacement for the nonsmooth continuity-constraint function to control inherent dispersive-error mechanisms. The genuine pressure is calculated by the diagnostic pressure Poisson equation, evaluated using the verified solenoidal velocity field. This new separation of tasks also produces a genuinely clear view of the totally distinct boundary conditions required for the continuity constraint function and genuine pressure.     

Development of a Moving and Stationary Mixed Particle Method for Solving the Incompressible Navier-Stokes Equations at High Reynolds Numbers

In the current particle method, we propose a new semi-implicit particle method for more effectively solving the incompressible Navier-Stokes equations at a high Reynolds number. Within the Lagrangian framework, the convective terms in the equations of motion are eliminated, without the problem of convective numerical instability. Also, the crosswind diffusion error generated normally in the case of a large angle difference between the velocity vector and the coordinate line disappears. Only the Laplacian operator for the velocity components and the gradient operator for the pressure need to be approximated on the basis of particle interaction through the currently proposed kernel function. As the key to getting better predicted accuracy, the kernel function is derived subject to theoretical constraint conditions. In the conventional moving-particle method, it is almost impossible to get convergent solution at a high Reynolds number. To overcome this simulation difficulty so that the moving-particle method is applicable to a wider range of flow simulations, a new solution algorithm is proposed for solving the elliptic-parabolic set of partial differential equations. In the momentum equations, calculation of the velocity components is carried out in the particle-moving sense. Unlike the traditional moving-particle semi-implicit method, the pressure values are not calculated at the particle locations being advected along the flowfield. After updating the fluid particle locations within the Lagrangian framework, we interpolate the velocities at uniformly distributed pressure locations. In the current semi-implicit solution algorithm, pressure is governed by the elliptic differential equation with the source term being contributed entirely to the velocity gradient terms. The distribution of particle locations can become highly nonuniform in cases involving a high Reynolds number and under conditions having an apparently vortical flow. As a result, the elliptic nature of the pressure can be considerably destroyed in the course of Lagrangian motion. To retain the embedded ellipticity in the incompressible viscous flow equations, the Poisson equation adopted for the calculation of pressure is solved in a mathematically more plausible fixed uniform mesh so as to get not only fourth-order accuracy for the pressure but also to enhance ellipticity in the pressure Poisson equation. Moreover, the velocity–pressure coupling can be more enhanced in the semi-implicit solution algorithm. The proposed moving and stationary mixed particle semi-implicit solution algorithm and the particle kernel will be demonstrated to be suitable to simulate high-Reynolds number fluid flows by investigating the lid-driven cavity flow problem at Re = 100 and Re = 1,000. Besides the validation of the proposed semi-implicit particle method in the fixed domain, the broken-dam problem is also solved to demonstrate the ability of accurately capturing the time-evolving free surface using the proposed semi-implicit particle method.     

Three-Dimensional Semi-Analytical Thermo-Elasticity Solution for a Functionally Graded Solid and an Annular Circular Plate

Three-dimensional analysis of a functionally graded (FG) solid and an annular circular plate subjected to thermo-mechanical load with various boundary conditions are carried out in this paper. At first, the temperature gradient can be derived by solving the heat conduction equation exactly and then by applying the differential quadrature method (DQM) along the radial direction and by using the state-space method in the thickness direction to the governing state equations results in a system of first-order differential equations, which can be solved analytically. The thermo-elastic constants of the plate are assumed to vary exponentially through the thickness, and the Poisson ratio is held to be constant. To verify the accuracy of the present work, a comparison is made with previously published results. The effects of temperature change, mechanical load, gradient index, edges conditions and thickness to radial length ratio on the behavior of the plate are examined.     

Developing a ghost fluid lattice Boltzmann method for simulation of thermal Dirichlet and Neumann conditions at curved boundaries

ABSTRACT

In this paper, a ghost fluid thermal lattice Boltzmann method is developed to simulate Dirichlet and Neumann thermal boundary conditions at curved boundaries. As such, a new formulation for both thermal boundary conditions is developed using a bilinear interpolation method. The presented method is also formulated to address the special cases that arise when the values of the macroscopic variables are interpolated at the image points surrounded by many solid nodes as well as the fluid nodes. The results of the presented method are compared to those available in the literature from conventional numerical methods, and excellent agreement is observed.     

Slip-enhanced reverse electrodialytic power generation in ion-selective nanochannels

Power generation by reverse electrodialysis in ion-selective nanochannels is numerically investigated. Especially,in the present study, the influence of hydrodynamic slip at the surface of nanochannels is investigated. The current-potential characteristics of the nanochannels are calculated by solving several governing equations:Nernst-Planck equation for the ionic concentrations, the Poisson equation for the electric potential, and the Navier-Stokes equation for the diffusioosmotic flow. Hydrodynamic slip is applied as the boundary condition at the surface of nanochannels. As the slip length increases, the diffusioosmotic flow velocity and electrical conductance of ions increase because the friction at the surface of nanochannels decreases. It is shown that the power generation is enhanced by 44% with a moderate 100nm slip length by using a nanochannel with 10nm height.     

Development of a Compact and Accurate Discretization for Incompressible Navier-Stokes Equations Based on an Equation-Solving Solution Gradient,Part III: Fluid Flow Simulations on Staggered Polygonal Grids

A compact and accuracy discretization of incompressible Navier-Stokes equations on staggered polygonal grids is presented in this article. It is a sequel to our efforts in developing a feasible solution procedure to simulate incompressible flow problems in complicated domains. By taking advantage of the discretization procedure for the convection-diffusion equation described in our previous work, difference counterparts of the Navier-Stokes equations can be obtained on staggered polygonal grids. Additional ingredients of pressure–velocity coupling and boundary conditions for velocity gradients in the solution procedure are also described. Several test problems are solved to illustrate the feasibility of the present formulation. From the numerical results obtained, it is evident that the proposed scheme is a useful tools to simulate incompressible flow field in arbitrary domains.