scholarly journals Finite-Volume Multi-Stage Scheme for Advection-Diffusion Modeling in Shallow Water Flows

2011 ◽  
Vol 27 (3) ◽  
pp. 415-430 ◽  
Author(s):  
W.-D. Guo ◽  
J.-S. Lai ◽  
G.-F. Lin ◽  
F.-Z. Lee ◽  
Y.-C. Tan
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
Coupled System ◽  
High Order Accuracy ◽  
Suspended Sediment Transport ◽  
Reconstruction Method ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Multi Stage ◽  
Stage Scheme

ABSTRACTThis paper adopts the finite-volume multi-stage (FMUSTA) scheme to the two-dimensional coupled system combining the shallow water equations and the advection-diffusion equation. For the convection part, the numerical flux is estimated by adopting the FMUSTA scheme, where high order accuracy is achieved by the data reconstruction using the monotonic upstream schemes for conservation laws method. For the diffusion part, the evaluations of first-order derivatives are solved via the method of Jacobian transformation. The hydrostatic reconstruction method is employed for treatment of source terms. The overall accuracy of resulting scheme is second-order both in time and space. In addition, the scheme is non-oscillatory and conserves the pollutant mass during the transport process. For scheme validation, six advection and diffusion transport tests are simulated. The influences of the grid spacing and limiters on the numerical performance are also discussed. Furthermore, the scheme is employed in the simulation of suspended sediment transport in natural-irregular river topography. From the satisfactory agreements between the simulated results and the field measured data, it is demonstrated that the proposed FMUSTA scheme is practically suitable for hydraulic engineering applications.

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 Development of an Advection-diffusion Model Using Depth-integrated Equations Based on GPU Acceleration

2021 ◽  
Vol 21 (1) ◽  
pp. 281-289
Author(s):  
Sooncheol Hwang ◽  
Sangyoung Son
Keyword(s):  
Finite Difference ◽  
Finite Volume ◽  
Transport Model ◽  
Finite Difference Methods ◽  
Scalar Transport ◽  
Gpu Acceleration ◽  
Advection Diffusion Equation ◽  
Benchmark Tests ◽  
Advection Diffusion ◽  
Proposed Model

A scalar transport model is developed by adding a depth-averaged advection-diffusion equation to Celeris Advent, which is a Boussinesq-type numerical model that utilizes GPU acceleration. The hybrid finite volume-finite difference method is used to guarantee numerical stability along with the high accuracy of the Boussinesq equation. The advective and diffusive terms are numerically discretized using the finite volume and finite difference methods, respectively. Results of a one-dimensional scalar advection benchmark test showed that the scalar advection by the proposed model was very close to the analytical solution without any remarkable numerical diffusion. In addition, two benchmark tests using experimental data from different hydraulic experiments were numerically reproduced, and the computed results and observed data for scalar transport were found to be in good agreement. The developed model is expected to contribute to real-time disaster prediction for contaminant spills and can assist in preparing countermeasures for these types of disasters.

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