Third- and fifth-order finite volume schemes for advection-diffusion equation with variable coefficients in semi-infinite domain

2017 ◽  
Vol 31 (2) ◽  
pp. 184-193 ◽  
Author(s):  
Amin Gharehbaghi
Keyword(s):  
Diffusion Equation ◽  
Finite Volume ◽  
Variable Coefficients ◽  
Finite Volume Schemes ◽  
Infinite Domain ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Fifth Order

Two-Dimensional Legendre Wavelets for Solving Variable-Order Fractional Nonlinear Advection-Diffusion Equation with Variable Coefficients

2018 ◽  
Vol 19 (7-8) ◽  
pp. 793-802 ◽  
Author(s):  
M. Hosseininia ◽  
M. H. Heydari ◽  
Z. Avazzadeh ◽  
F. M. Maalek Ghaini
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Main Idea ◽  
Variable Coefficients ◽  
Variable Order ◽  
Two Dimensional ◽  
Legendre Wavelets ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Nonlinear Advection

AbstractThis article studies a numerical scheme for solving two-dimensional variable-order time fractional nonlinear advection-diffusion equation with variable coefficients, where the variable-order fractional derivative is in the Caputo type. The main idea is expanding the solution in terms of the 2D Legendre wavelets (2D LWs) where the variable-order time fractional derivative is discretized. We describe the method using the matrix operators and then implement it for solving various types of fractional advection-diffusion equations. The experimental results show the computational efficiency of the new approach.

Streamwise dispersion and mixing in quasi-two-dimensional steady turbulent jets

2012 ◽  
Vol 711 ◽  
pp. 212-258 ◽  
Author(s):  
Julien R. Landel ◽  
C. P. Caulfield ◽  
Andrew W. Woods
Keyword(s):  
Diffusion Equation ◽  
Finite Volume ◽  
Turbulent Jets ◽  
Time Dependent ◽  
Two Dimensional ◽  
Mean Velocity ◽  
Constant Flux ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Passive Tracers

AbstractWe investigate experimentally and theoretically the streamwise transport and dispersion properties of steady quasi-two-dimensional plane turbulent jets discharged vertically from a slot of width $d$ into a fluid confined between two relatively close rigid boundaries with gap $W\ensuremath{\sim} O(d)$. We model the evolution in time and space of the concentration of passive tracers released in these jets using a one-dimensional time-dependent effective advection–diffusion equation. We make a mixing length hypothesis to model the streamwise turbulent eddy diffusivity such that it scales like $b(z){ \overline{w} }_{m} (z)$, where $z$ is the streamwise coordinate, $b$ is the jet width, ${ \overline{w} }_{m} $ is the maximum time-averaged vertical velocity. Under these assumptions, the effective advection–diffusion equation for $\phi (z, t)$, the horizontal integral of the ensemble-averaged concentration, is of the form ${\partial }_{t} \phi + {K}_{a} {\text{} {M}_{0} \text{} }^{1/ 2} {\partial }_{z} \left(\phi / {z}^{1/ 2} \right)= {K}_{d} {\text{} {M}_{0} \text{} }^{1/ 2} {\partial }_{z} \left({z}^{1/ 2} {\partial }_{z} \phi \right)$, where $t$ is time, ${K}_{a} $ (the advection parameter) and ${K}_{d} $ (the dispersion parameter) are empirical dimensionless parameters which quantify the importance of advection and dispersion, respectively, and ${M}_{0} $ is the source momentum flux. We find analytical solutions to this equation for $\phi $ in the cases of a constant-flux release and an instantaneous finite-volume release. We also give an integral formulation for the more general case of a time-dependent release, which we solve analytically when tracers are released at a constant flux over a finite period of time. From our experimental results, whose concentration distributions agree with the model, we find that ${K}_{a} = 1. 65\pm 0. 10$ and ${K}_{d} = 0. 09\pm 0. 02$, for both finite-volume releases and constant-flux releases using either dye or virtual passive tracers. The experiments also show that streamwise dispersion increases in time as ${t}^{2/ 3} $. As a result, in the case of finite-volume releases more than 50 % of the total volume of tracers is transported ahead of the purely advective front (i.e. the front location of the tracer distribution if all dispersion mechanisms are ignored and considering a ‘top-hat’ mean velocity profile in the jet); and in the case of constant-flux releases, at each instant in time, approximately 10 % of the total volume of tracers is transported ahead of the advective front.

scholarly journals FINITE DIFFERENCE SOLUTION OF TWO-DIMENSIONAL SOLUTE TRANSPORT WITH PERIODIC FLOW IN HOMOGENOUS POROUS MEDIA

2017 ◽  
Vol 8 (2) ◽  
Author(s):  
Alexandar Djordjevich ◽  
Svetislav Savović ◽  
Aco Janićijević
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Diffusion Equation ◽  
Variable Coefficients ◽  
Two Dimensional ◽  
Seepage Velocity ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Explicit Finite Difference ◽  
Explicit Finite Difference Method

Two-dimensional advection-diffusion equation with variable coefficients is solved by the explicit finite-difference method for the transport of solutes through a homogeneous, finite, porous, two-dimensional, domain. Retardation by adsorption, periodic seepage velocity, and a dispersion coefficient proportional to this velocity are permitted. The transport is from a pulse-type point source (that ceases after a period of activity). Included are the first-order decay and zero-order production parameters proportional to the seepage velocity, periodic boundary conditions at the origin and the end of the domain. Results are compared to analytical solutions reported in the literature for special cases and a good agreement was found. The solute concentration profile is greatly influenced by the periodic velocity fluctuations. Solutions for a variety of combinations of unsteadiness of the coefficients in the advection-diffusion equation are obtainable as particular cases of the one demonstrated here. This further attests to the effectiveness of the explicit finite difference method for solving two-dimensional advection-diffusion equation with variable coefficients in a finite media, which is especially important when arbitrary initial and boundary conditions are required.

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