Steady Advection-diffusion Equation Research Articles

Two-level Arrow–Hurwicz iteration methods for the steady bio-convection flows

To avoid solving a saddle-point system, in this paper, we study two-level Arrow–Hurwicz finite element methods for the steady bio-convection flows problem which is coupled by the steady Navier–Stokes equations and the steady advection–diffusion equation. Using the mini element to approximate the velocity, pressure, and the piecewise linear element to approximate the concentration, we use the linearized Arrow–Hurwicz iteration scheme to obtain the coarse mesh solution and use three different one-step Stokes/Oseen/Newton linearized scheme to obtain the fine mesh solution. The optimal error estimate O(h+H2+χm/2) of the velocity and concentration in the H1-norm and the pressure in the L2-norm are derived, where h and H are fine and coarse mesh sizes, respectively, and χm/2 denotes the iteration error with 0<χ<1. Numerical results are given to support the theoretical analysis and confirm the efficiency of the proposed two-level methods.

Penguin Huddling: A Continuum Model

Penguins huddling in a cold wind are represented by a two-dimensional, continuum model. The huddle boundary evolves due to heat loss to the huddle exterior and through the reorganisation of penguins as they seek to regulate their heat production within the huddle. These two heat transfer mechanisms, along with area, or penguin number, conservation, gives a free boundary problem whose dynamics depend on both the dynamics interior and exterior to the huddle. Assuming the huddle shape evolves slowly compared to the advective timescale of the exterior wind, the interior temperature is governed by a Poisson equation and the exterior temperature by the steady advection-diffusion equation. The exterior, advective wind velocity is the gradient of a harmonic, scalar field. The conformal invariance of the exterior governing equations is used to convert the system to a Polubarinova-Galin type equation, with forcing depending on both the interior and exterior temperature gradients at the huddle boundary. The interior Poisson equation is not conformally invariant, so the interior temperature gradient is found numerically using a combined adaptive Antoulas-Anderson and least squares algorithm. The results show that, irrespective of the starting shape, penguin huddles evolve into an egg-like steady shape. This shape is dependent on the wind strength, parameterised by the Péclet number Pe, and a parameter beta which effectively measures the strength of the interior self-generation of heat by the penguins. The numerical method developed is applicable to a further five free boundary problems.

Asymptotic analysis of the steady advection-diffusion problem in axial domains

We present the asymptotic analysis of the steady advection-diffusion equation in a thin tube. The problem is modeled in a mixed-type variational formulation, in order to separate the phenomenon in the axial direction and a transverse one. Such formulation makes visible the natural separation of scales within the problem and permits a successful asymptotic analysis, delivering a limiting form, free from the initial geometric singularity and suitable for approximating the original one. Furthermore, it is shown that the limiting problem can be simplified to a significantly simpler structure.

Dissolution of plane surfaces by sources in potential flow

The two-dimensional free boundary problem for a surface dissolving in potential flow owing to concentrated sources of dissolving agent c is formulated and solved. The surface boundary evolves quasi-steadily, and the resulting steady advection–diffusion equation for c, and Laplace’s equation for the velocity potential form a coupled pair of conformally invariant PDEs. The dynamics of the coupled fluid flow and evolving surface depends on the Péclet number: Pe=UL/D, where U and L are typical velocity and lengthscales respectively, and D is the diffusivity of c. The conformally invariant property is exploited in finding an equation of the Polubarinova–Galin class for the conformal map from the unit ζ-disk to the evolving domain in physical space. The equation is solved numerically and the time-evolution of the dissolving surface determined. For a single concentrated source with flow initially parallel to a flat surface, a Pe-dependent scallop-like surface shape typically develops. The problem involving a periodic array of concentrated sources aligned parallel to an initially flat surface is also solved.

Fingering instability in wildfire fronts

A two-dimensional model for the evolution of the fire line – the interface between burned and unburned regions of a wildfire – is formulated. The fire line normal velocity has three contributions: (i) a constant rate of spread representing convection and radiation effects; (ii) a curvature term that smooths the fire line; and (iii) a Stefan-like term in the direction of the oxygen gradient. While the first two effects are geometrical, (iii) is dynamical and requires the solution of the steady advection–diffusion equation for oxygen, with advection owing to a self-induced ‘fire wind’, modelled by the gradient of a harmonic potential field. The conformal invariance of this coupled pair of partial differential equations, which has the Péclet number $\textit {Pe}$ as its only parameter, is exploited to compute numerically the evolution of both radial and infinitely long periodic fire lines. A linear stability analysis shows that fire line instability is possible, dependent on the ratio of curvature to oxygen effects. Unstable fire lines develop finger-like protrusions into the unburned region; the geometry of these fingers is varied and depends on the relative magnitudes of (i)–(iii). It is argued that for radial fires, the fire wind strength scales with the fire's effective radius, meaning that $\textit {Pe}$ increases in time, so all fire lines eventually become unstable. For periodic fire lines, $\textit {Pe}$ remains constant, so fire line stability is possible. The results of this study provide a possible explanation for the formation of fire fingers observed in wildfires.

Layer-adapted meshes for solute dispersion in a steady flow through an annulus with wall absorption: Application to a catheterized artery

This paper describes the longitudinal dispersion of passive tracer molecules injected in a steady, fully developed, viscous, incompressible, laminar flow through an annular pipe with a first order heterogeneous boundary absorption at the outer wall, numerically using layer-adapted meshes. The model is based on steady advection-diffusion equation with Dirichlet and Robin boundary conditions. The solutions are discussed in the form of iso-concentration contours of the tracer molecules in the vertical plane. An artanh transformation is used to convert the infinite domain into a finite one. A combination of central finite difference and 2-point upwind scheme is adopted to solve the governing advection-diffusion equation. It is shown that how the mixing of tracers is affected by the shear flow, aspect ratio and the first-order boundary absorption. When the flow becomes convection dominated, the monotone finite difference on a uniform mesh does not work properly, so a layer-adapted mesh, namely a “Shishkin” mesh, is used to capture the layer phenomena at the different downstream stations. The present results are compared with existing experimental and numerical data and we have earned an excellent agreement with them. It is observed that, due to the use of layer adapted mesh, we have achieved a better agreement with the experimental data than some other previous results available in the literature, especially in the closest downstream location. The results of this study are likely to be of interest to understand the basic mechanism of dispersion process of solute in blood through a catheterized artery with an absorptive arterial wall.

On dispersion of solute in steady flow through a channel with absorption boundary: an application to sewage dispersion

The paper describes the longitudinal dispersion of passive tracer materials released into an incompressible viscous fluid, flowing through a channel with walls having first-order reaction. Its model is based on a steady advection–diffusion equation with Dirichlet’s and mixed boundary conditions, and whose solution represents the concentration of the tracers in different downstream stations. For imposing the boundary conditions properly, artanh transformation is used to convert the infinite solution space to a finite one. A finite difference implicit scheme is used to solve the advection–diffusion equation in the computational region, and an inverse transformation is employed for the solution in the physical region. It is shown how the mixing of the tracer molecule influenced by the shear flow and due to the action of the absorption parameter at both the walls of the channel. For convection-dominated flow, uniform mesh is failed to capture the layer phenomena along the different downstream stations and a piecewise uniform mesh; namely, Shishkin mesh is used. The results are compared with existing experimental and numerical data available in the literature, and we have achieved an excellent agreement with them. The study plays a significant role to understand the basic mechanisms of sewage dispersion.

単調性保存スキームI : 定常移流・拡散方程式の局所厳密解に基づく有限可変差分法

単調性保存スキームI : 定常移流・拡散方程式の局所厳密解に基づく有限可変差分法

A multi-layer model for pollutant dispersion with dry deposition to the ground

This paper presents a semi-analytical solution for the steady advection–diffusion equation that allows simulating the vertical turbulent dispersion of air pollution with deposition to the ground. The performances of the solution, with a proper parameterization of the vertical profiles of wind and eddy diffusivity, were evaluated against Hanford diffusion experiment dataset using two tracers ( Doran and Horst, 1985): a non-depositing gas (SF 6) and depositing particles (ZnS). Results show that the dispersion model with the K-parameterization included produces a good fitting of the measured ground-level concentration data and there are no big differences between the parameterizations taken from literature. A comparison with other models was shown and discussed.

Enforcement of constraints and maximum principles in the variational multiscale method

We present a new theoretical framework for the enforcement of constraints in variational multiscale (VMS) analysis. The theory is first presented in an abstract operator format and subsequently specialized for the steady advection–diffusion equation. The approach borrows heavily from results in constrained and convex optimization. An exact expression for the fine-scales is derived in terms of variational derivatives of the constraints, Lagrange multipliers, and a fine-scale Green’s function. The methodology described enables the development of numerical methods which satisfy predefined attributes. A practical and effective procedure for solving the steady advection–diffusion equation is presented based on a VMS-inspired stabilized method, weakly enforced Dirichlet boundary conditions, and enforcement of a maximum principle and conservation constraint.

NUMERICAL ANALYSIS OF THE PSI SOLUTION OF ADVECTION–DIFFUSION PROBLEMS THROUGH A PETROV–GALERKIN FORMULATION

In this paper we introduce an analysis technique for the solution of the steady advection–diffusion equation by the PSI (Positive Streamwise Implicit) method. We formulate this approximation as a nonlinear finite element Petrov–Galerkin scheme, and use tools of functional analysis to perform a convergence, error and maximum principle analysis. We prove that the scheme is first-order accurate in H1 norm, and well-balanced up to second order for convection-dominated flows. We give some numerical evidence that the scheme is only first-order accurate in L2 norm. Our analysis also holds for other nonlinear Fluctuation Splitting schemes that can be built from first-order monotone schemes by the Abgrall and Mezine's technique.

Least square-based finite volumes for solving the advection–diffusion of contaminants in porous media

A second-order cell-centered finite-volume method is proposed to solve the steady advection–diffusion equation for contaminant transport in porous media. This method is based on a linear least square reconstruction that maintains the approximate cell-average values. The reconstruction is combined with an appropriate slope limiter to prevent the formation of spurious oscillations in the convection-dominated case. The theoretical convergence rate is investigated on a boundary layer problem, and the preliminar results show that the method is promising in the numerical simulation of more complex groundwater flow problems.

Streamline-Upwind Method.

A new treamline-upwind method is developed for predicting the velocity and the pressure in transient incompressible viscous fluid flows over the wide range from high to low Reynolds numbers. This new technique is composed of three parts : an exact solution of advection equation, an exact solution of locally-linearized advection-diffusion equation and semi-Lagrangian method. Element Green function plays an important role solving the steady advection-diffusion equation in each element. Semi-Lagrangian method is a time integration of the Navier-Stokes equation along the steamline coordinate. The essential part of this time-marching method is similar to the Crank-Nicolson method. Many methods presented up to now are included in this new method.

Some remarks about the collocation method on a modified Legendre grid

We compare the results obtained by applying the standard collocation method at the Legendre Gauss-Lobatto nodes, for a model problem simulating a steady advection-diffusion equation, with those obtained by collocating at a new set of nodes. These nodes are derived from a suitable modification of the Legendre grid, according to the magnitude of the ratio between the advective and the diffusive parts of the differential operator.

New Theory of Hybrid-Upwind Technique and Discrete Del Operator for Finite-Element Method

In this paper, we present a new hybrid-streamline-upwind finite-element method, which is based on the finite analytic method developed in the field of the finite difference method. In order to obtain an optimal shape function, we introduce an adjoint differential operator to the differential operator for a steady advection-diffusion equation. The shape functions which satisfy these differential operators are mutually dual. One of them interpolates accurately the functions appearing in convection-dominated flows, the other becomes a hybrid-streamline-upwind weighting function. Furthermore, we define a discrete del operator for the reduction of memory storage in the computer. As a result, we achieve simplicity of formulation and high-speed calculation of the finite-element method.

New Hybrid-Streamline-Upwind Finite-Element Method in Dual Space. (Fundamental Theory and DiscreteDel Operator).

In this paper, we present a new hybrid-streamline-upwind finite-element method, which is based on the finite analytic method developed in the field of the finite difference analysis. In order to obtain an optimal shape function, we introduce an adjoint differential operator to the differential operator for a steady advection-diffusion equation. The shape functions which satisfy these differential operators are mutually dual. One of them accurately interpolates the functions appearing in convection-dominated flows, and the other becomes a hybrid-streamline-upwind weight function. Furthermore, we define a discrete del operator for the reduction of memory storage in computers. As a result, we achieve simplicity of formulation and high-speed calculation in the finite-element method.

Discrete Del Operator Using Streamline-Upwind Shape Function for Mixed-Element Method.

In this paper, we have proposed the hybrid streamline-upwind finite-element method using the discrete del operator for the case of mixed elements, which has already been described for the case of a hexa element in the previous paper. Generally speaking, it is the demerit of the finite-element method that it tapes much time to construct computational grids. Therefore it is important to extend the theory to the mixed-element case in order to save time in constructing computational grids in the complex computational area. The main content of this paper is as follows. First, we define the streamline-upwind shape function of each element using a solution of a steady advection-diffusion equation Second, we have proposed the discrete del operator for the mixed-element method, which is defined as an element average of the gradient of the shape function in the discrete space.

New Hybrid-Streamline-Upwind Finite-Element Method in a Dual Space. 1st Report. Fundamental Theory and Discrete Del Operator.

In this paper, we present a new hybrid-streamline-upwind finite-element method, which is based on the finite analytic method developed in the field of the finite difference method. In order to obtain an optimal shape function, we introduce an adjoint differential operator to the differential operator for a steady advection-diffusion equation. The shape functions which satisfy these differential operators are mutually dual. One of them interpolates accurately the functions appearing in convection-dominated flows, the other becomes a hybrid-streamline-upwind weight function. Furthermore, we define a discrete del operator for the reduction of memory storage in the computer. As a result, we achieve simplicity of formulation and high-speed calculation of the finite-element method.

“Ｗｉｇｇｌｅｓ”と中心一風上混合差分

Spatial oscillations or “wiggles” have been encountered in many high-Reynolds number flow solutions when a central-difference method is used. As Roache demonstrates in his famous text book, it occurs even in a linear equation and the source of “wiggles” is simple. In this paper, it is clearly and mathematically shown based on the monotonicity of the solution that the “wiggles” must occur in a central-difference solution of a steady advection-diffusion equation if the cell Reynolds number is greater than 2. In order to get an accurate 3-point finite difference solution as well as to prevent such wiggles, central-upwind hybrid finite difference methods are derived.

An analysis of advective diffusion in branching channels

Solutions to the steady advection–diffusion equation in a branching channel are obtained for both uniform and spatially varying flow fields and for two channel geometries. An interesting feature of the solutions is that anisotropy of the dispersion coefficients in the direction of the streamlines may be accounted for. The analysis reveals that mixing is confined to a distance, b2U/π2KN, downstream of the junction in the advection-dominated case and a distance, KS/U, upstream in the diffusion-dominated situation, KS and KN being the diffusivities along and across the flow respectively, U the characteristic velocity of the flow, and b the breadth of channel downstream of the junction.