3D Convection Diffusion Equation Research Articles

Integrated framework for solving the convection diffusion equation on 2D Quad mesh relying on internal boundaries

The aim of this research is the design and implementation of an integrated framework to solve numerically the two-dimensional convection diffusion equations on non-rectangular grids formed only by quadrilaterals honoring the internal structures of an oil reservoir. The framework is composed of (1) a preprocessor of the internal boundaries of a 2D oil reservoir structure, (2) a 2D structural quadrilateral grid generator, (3) a solver of the 2D convection diffusion equation (CDE) on the grid, and (4) a visualization module to display the reservoir properties, wells and the simulation results on the grid. An example of typical structures corresponding to an areal view of a hydrocarbon reservoir is presented. Different scenarios were considered varying boundary conditions, source term, and diffusion constant fluid velocity. All the results are consistent with the physical interpretation of each configuration.

变系数2D对流扩散方程的高阶迭代算子分裂方法

本文针对变系数2D对流扩散方程，呈现了一种新颖的高阶迭代算子分裂方法。该方法结合了经典迭代格式和Zassenhaus乘积公式。傅立叶谱方法和维数分裂格式用于空间算子。数值实验验证了所提出的方法通过加权方法可以达到高阶精度。此外，新方法不仅可以减少误差而且能够节省大量的CPU时间。 In this paper, a novel higher order iterative operator splitting method is presented for the 2D convection diffusion equation with the variable coefficient. The proposed scheme combines the classical iterative scheme and Zassenhaus product formula for the temporal discretization. And Fourier pseudo spectral method and dimensional splitting scheme are applied for the spatial operators. The numerical results verified that the proposed method can get second order accuracy by weighted iterative scheme. Besides, the new method not only can reduce numerical error but also save a lot of CPU time than the classical iterative method.

Fourth order compact schemes for variable coefficient parabolic problems with mixed derivatives

A fourth order compact numerical scheme for variable coefficient parabolic problems with mixed derivatives is outlined. The finite difference scheme, presented here shows good wave resolution property and is stable. Implicit time discretization endows second order temporal accurateness to the scheme. Compact scheme for generalized parabolic 2D convection-diffusion equations is not available in the literature and this work addresses the same. The method is also suitable for computing incompressible flow in arbitrary domains. In this work we have successfully used the method to tackle flows, governed by the incompressible 2D unsteady Navier-Stokes (N-S) equations, in regions beyond rectangular. Further the proposed compact scheme has been found to be proficient in conjunction with numerically generated grids.

3D CFD modelling of velocities and sediment transport in the Iffezheim hydropower reservoir

The simulation of sediment transport by three-dimensional (3D) modelling is linked with the question of how accurate such models are. The current paper provides a test where detailed field measurements of velocities and bed elevation changes over a 3-month period from a prototype reservoir are compared with simulation results. The SSIIM software was used to compute water flow and sediment transport in the Iffezheim reservoir. The numerical model solved the Navier–Stokes equations on a 3D unstructured grid with dominantly hexahedral cells. The k-epsilon turbulence model was used, together with the SIMPLE method to find the pressure. The 3D convection–diffusion equation for suspended sediments was solved for nine sediment fractions. The computed velocity pattern showed good correspondence with the measurements. Grid sensitivity tests showed that the main flow features were computed in different grid sizes, but more accurately so with the finest grid. The sediment deposits were reasonably well computed in location and magnitude. A sensitivity test revealed that the computed bed elevation changes were most sensitive to the fall velocities of the finest cohesive sediment particles and sediment cohesion. The sediment deposition computations were also to some degree sensitive to the sediment discharge formula, bed roughness, discretization scheme and the grid resolution.

Scale-Dependent Mixing for Adverse Mobility Ratio Flows in Heterogeneous Porous Media

Mixing is a scale-dependent phenomenon that has been shown to increase with increasing heterogeneity, mobility ratio, longitudinal correlation length, aspect ratio and distance traveled. The interface between miscible fluids may be dilated by velocity variations, which are promoted by certain characteristics of a porous medium, or by a high mobility ratio. Here we use a high-resolution numerical scheme to compute local dispersivities by fitting grid-block concentration histories to the 1D convection–diffusion equation. We provide new insights on the scale dependence of local dispersivity in heterogeneous porous media. We explain that irreversible mixing only increases monotonically with increasing longitudinal correlation length (layering) for unit mobility ratio displacements. The combination of an adverse mobility ratio and reservoir layering causes mixing to increase, peak and then decline with distance traveled by the injectant. We show that mixing evolves non-monotonically in layered porous media due to the effect of channeling (and not viscous fingering) at adverse mobility ratios. We also examine the effect of diffusion on local dispersivity as modeled in Eulerian simulation. The level of dispersivity is increased when diffusion is explicitly modeled, although molecular diffusion is much smaller than numerical diffusion, even in our high-resolution simulations. Diffusion is more important to consider when sharp concentration gradients exist, as is the case when mobility ratio is large and permeabilities are highly correlated in the longitudinal direction (i.e., layered). This combination gives rise to pronounced channelized flow in which a sharp concentration gradient drives diffusion over a large area of contact between fluids.

A 15-point high-order compact scheme with multigrid computation for solving 3D convection diffusion equations

We propose a method with sixth-order accuracy to solve the three-dimensional (3D) convection diffusion equation. We first use a 15-point fourth-order compact discretization scheme to obtain fourth-order solutions on both fine and coarse grids using the multigrid method. Then an iterative mesh refinement technique combined with Richardson extrapolation is used to approximate the sixth-order accurate solution on the fine grid. Numerical results are presented for a variety of test cases to demonstrate the efficiency and accuracy of the proposed method, compared with the standard fourth-order compact scheme.

A New Way to Generate an Exponential Finite Difference Scheme for 2D Convection-Diffusion Equations

The idea of direction changing and order reducing is proposed to generate an exponential difference scheme over a five-point stencil for solving two-dimensional (2D) convection-diffusion equation with source term. During the derivation process, the higher order derivatives alongy-direction are removed to the derivatives alongx-direction iteratively using information given by the original differential equation (similarly fromx-direction toy-direction) and then instead of keeping finite terms in the Taylor series expansion, infinite terms which constitute convergent series are kept on deriving the exponential coefficients of the scheme. From the construction process one may gain more insight into the relations among the stencil coefficients. The scheme is of positive type so it is unconditionally stable and the convergence rate is proved to be of second-order. Fourth-order accuracy can be obtained by applying Richardson extrapolation algorithm. Numerical results show that the scheme is accurate, stable, and especially suitable for convection-dominated problems with different kinds of boundary layers including elliptic and parabolic ones. The idea of the method can be applied to a wide variety of differential equations.

Fast and high accuracy multiscale multigrid method with multiple coarse grid updating strategy for the 3D convection–diffusion equation

An improved multiscale multigrid method with multiple coarse grid updating strategy for a three dimensional convection–diffusion equation is presented. The novelty of the proposed method lies in a fine grid updating strategy arising from the idea of multiple coarse grid computation. The new fine grid updating strategy is able to replace the iterative refinement procedure in the existing multiscale multigrid method (Wang and Zhang, 2010) [16] to obtain high order solutions. Since the proposed method needs a fourth order compact scheme with unequal-meshsize grids, a 19 point fourth order compact difference scheme with unequal meshsize in different coordinate directions is also developed for the three dimensional convection–diffusion equation. Numerical results are given to compare the computed accuracy and the computational efficiency of the multiscale multigrid method with the multiple coarse grid updating strategy against the multiscale multigrid method with the iterative refinement procedure.

An exact discretization for a transport equation with piecewise-constant coefficients and arbitrary source

The search for a robust and accurate discretization scheme for a general convection diffusion equation has been a driving force in the improvement of codes employed in the CFD community. As part of this avenue of inquiry the research in this paper was aimed at improving the accuracy of schemes implemented in an in-house RANS code by incorporating the source in the discretization stencil. To understand how to do this we started working with the equation for a transported scalar and we came up with a systematic way of transforming a source-dominant equation into a source-free equation that allowed us to obtain an exact solution, i.e., a solution whose error is within machine accuracy, for an arbitrary source and an arbitrary number of discretization nodes, as long as the coefficients are piecewise constant in the intervals. The final equation, being a source-free equation, allows the use of the exponential scheme for the coefficients which is known to be exact for the homogeneous 1D convection diffusion equation. This paper shows how to convert the exponential scheme into an exact scheme for a transport equation with piecewise constant coefficients and arbitrary source in a 1D domain.

Non-uniform HOC Scheme for the 3D Convection−Diffusion Equation

In this paper, we extend the work of Kalita et al. [11] to solve the steady 3D convection-diffusion equation with variable coefficients on non-uniform grid. The approach is based on the use of Taylor series expansion, up to the fourth order terms, to approximate the derivatives appearing in the 3D convection diffusion equation. Then the original convection-diffusion equation is used again to replace the resulting higher order derivative terms. This leads to a higher order scheme on a compact stencil (HOC) of nineteen points. Effectiveness of this method is seen from the fact that it can handle the singularity perturbed problems by employing a flexible discretized grid that can be adapted to the singularity in the domain. Four difficult test cases are chosen to demonstrate the accuracy of the present scheme. Numerical results show that the fourth order accuracy is achieved even though the Reynolds number (Re) is high.

Parallel block interface domain decomposition methods for the 2D convection–diffusion equation

In this paper, a new block interface domain decomposition method (BI-DDM) with non-overlapping subdomains for the numerical solution of a two-dimensional convection–diffusion equation is presented. The block interface formulation is derived from the idea of using small groups of a certain number of mesh points where this group is treated explicitly similar to the way a single point is treated in the point method. The BI-DDM is incorporated with a correction phase which is able to economize further on the computing cost. The performance analysis of this method on several recently developed group iterative schemes implemented on a message-passing architecture are presented and discussed.

Accounting robustly for instantaneous chemical equilibriums in reactive transport: A numerical method and its application to liquid–liquid extraction modeling

Reactive transport equations are used in numerous application fields: CO2 or nuclear waste storage monitoring, separation processes in chemical engineering. We present a general method to account robustly for instantaneous chemical equilibriums in reactive transport. This method is adapted to all kinds of hydraulic transport models including 1D to 3D convection–diffusion equations. This leads to the resolution of a bound constrained system of Differential Algebraic Equations (DAEs). The algebraic constraints come from the adjunction of mass action laws related to the equilibriums, whereas the bounds account for the positivity of the computed quantities. In order to solve the numerical system associated with our method, we use an adaptation of the DASSL solver, CDASSL, that can handle the resolution of bound constrained DAE systems. We present an application of this method to liquid–liquid extraction modeling. Numerical experiments demonstrate the interest of using the CDASSL solver to ensure the bound constraints are satisfied.

Meshless Local Integral Equations Formulation for the 2D Convection-Diffusion Equations with a Nonlocal Boundary Condition

Meshless Local Integral Equations Formulation for the 2D Convection-Diffusion Equations with a Nonlocal Boundary Condition

Total variation regularization for the reconstruction of a mountain topography

The aim of this paper is to reconstruct a paleo mountain topography using a total variation (TV) regularization. A coupled system integrates the tectonic process with the surface process to simulate the evolution of a paleo mountain topography. The tectonic process and the surface process are described by a 3D convection–diffusion equation and a 2D convection–diffusion equation, respectively. We recover a piecewise smooth velocity field for the tectonic process as well as reconstruct a piecewise smooth mountain topography for the surface process using a TV regularization in an iterative fashion. The effects of the number of samples and of wavelengths on inversions are investigated. In our numerical experiments, we shall experience three difficulties: (I) recovering a large quantity of information from the limited amount of measurement data; (II) detecting sharp features; (III) choosing a properly initial guess value for a TV regularization. The numerical experiments show that a TV regularization is an efficient and stable algorithm.

Derivation of Strictly Stable High Order Difference Approximations for Variable-Coefficient PDE

A new way of deriving strictly stable high order difference operators for partial differential equations (PDE) is demonstrated for the 1D convection diffusion equation with variable coefficients. The derivation is based on a diffusion term in conservative, i.e. self-adjoint, form. Fourth order accurate difference operators are constructed by mass lumping Galerkin finite element methods so that an explicit method is achieved. The analysis of the operators is confirmed by numerical tests. The operators can be extended to multi dimensions, as we demonstrate for a 2D example. The discretizations are also relevant for the Navier---Stokes equations and other initial boundary value problems that involve up to second derivatives with variable coefficients.

A stencil of the finite-difference method for the 2D convection diffusion equation and its new iterative scheme

The paper gives the numerical stencil for the two-dimensional convection diffusion equation and the technique of elimination, and builds up the new iterative scheme to solve the implicit difference equation. The scheme's convergence and its higher rate of convergence than the Jacobi iteration are proved. And the numerical example indicates that the new scheme has the same parallelism and a higher rate of convergence than the Jacobi iteration.

A high order finite difference method with Richardsonextrapolation for 3D convection diffusion equation

In this paper, we extend the Sun and Zhang’s [24] work on high order finite difference method, which is based on the Richardson extrapolation technique and an operator interpolation scheme for the one and two dimensional steady convection diffusion equations to the three dimensional case. Firstly, we employ a fourth order compact difference scheme to get the fourth order accurate solution on the fine and the coarse grids. Then, we use the Richardson extrapolation technique by combining the two approximate solutions to get a sixth order accurate solution on coarse grid. Finally, we apply an operator interpolation scheme to achieve the sixth order accurate solution on the fine grid. During this process, we use alternating direction implicit (ADI) method to solve the resulting linear systems. Numerical experiments are conducted to verify the accuracy and effectiveness of the present method.

Analysis of a consistency recovery method for the 1D convection–diffusion equation using linear finite elements

AbstractFor residual‐based stabilization methods such as streamline‐upwind Petrov–Galerkin (SUPG) and finite calculus (FIC), the higher‐order derivatives of the residual that appear in the stabilization term vanish when simplicial elements are used. The sub‐grid scale method using orthogonal sub‐scales (OSS) attempts to recover the lost consistency by using a fine‐scale projected residual in the stabilization term. The FIC method may also be cast into an OSS form with very little manipulation using an auxiliary convective projection equation. This paper discusses the gain/loss by recovering the consistency of the discrete residual in the stabilization terms via the form that includes the convective projection (as in the OSS method). We present the von Neumann analysis of the FIC method with recovered consistency (FIC_RC) for the 1D convection–diffusion problem and we compare it with the standard Bubnov–Galerkin linear finite element method and FIC/SUPG methods. The transient analysis is done by examining the discrete dispersion relation of the stabilization methods. The spectral results for the semi‐discrete and fully discrete problem are presented with time integration done by the trapezoidal and second‐order backward differencing formula schemes. The effect of lumping the effective mass matrix T is considered relative to using a consistent form. The effect of refinement in space and time is also discussed. Finally, an optimal expression for the stabilization parameter for the FIC_RC method on a uniform grid and for the steady state is given and its performance in the transient mode is discussed. Copyright © 2008 John Wiley & Sons, Ltd.

A conformal Petrov–Galerkin method for convection‐dominated problems

AbstractIn this paper, we present the ‘conformal Petrov–Galerkin’ (CPG) method in order to solve the 2D convection–diffusion equation on meshes composed of triangular elements. By ‘conformal’ it is meant that the discrete system is obtained from the continuous weak formulation by appropriately selecting different finite‐dimensional subspaces for the shape and test functions without any additional stabilizing term. Our approach is based on searching continuous test functions that provide exact nodal values for a selected class of solutions. This method induces a stabilizing upwinding effect that removes the wiggles obtained with the Galerkin method. Copyright © 2008 John Wiley & Sons, Ltd.

Heat-Transfer Enhancement by Chaotic Advection in the Eccentric Helical Annular Flow

Chaotic advection in the eccentric helical annular heat exchanger is investigated as a means to enhance its thermal efficiency. Chaotic streak lines are generated by steadily rotating one boundary while the other is counter-rotated with a time-periodic angular velocity. The effects of the eccentricity ratio and modulation frequency on the heat-transfer rate are analyzed by numerically solving the 3D convection-diffusion equation for a broad range of parameter values. For the frequency range over which chaotic advection can be effectively promoted, the efficiency of the heat exchanger is enhanced over that obtained for steady boundary rotation. Other tools, such as stretching field calculations and streak-line plots, applicable for dissipative dynamical systems, are implemented. These tools qualitatively confirm the quantitative heat-transfer results obtained.