Steady-state Advection-diffusion Equation Research Articles

Physics-Informed Neural Networks for 2nd order ODEs with sharp gradients

In this work, four different methods based on Physics-Informed Neural Networks (PINNs) for solving Differential Equations (DE) are compared: Classic-PINN that makes use of Deep Neural Networks (DNNs) to approximate the DE solution;Deep-TFC improves the efficiency of classic-PINN by employing the constrained expression from the Theory of Functional Connections (TFC) so to analytically satisfy the DE constraints;PIELM that improves the accuracy of classic-PINN by employing a single-layer NN trained via Extreme Learning Machine (ELM) algorithm;X-TFC, which makes use of both constrained expression and ELM. The last has been recently introduced to solve challenging problems affected by discontinuity, learning solutions in cases where the other three methods fail. The four methods are compared by solving the boundary value problem arising from the 1D Steady-State Advection–Diffusion Equation for different values of the diffusion coefficient. The solutions of the DEs exhibit steep gradients as the value of the diffusion coefficient decreases, increasing the challenge of the problem.

Studying the Effect of Two Analytical Solutions of Advection-Diffusion Equation on Experimental Data

In this article, two proposed analytical model solutions of the steady-state advection-diffusion equation were carried out using the technique of advection diffusion multilayer method (ADMM), variable separation technique, Fourier transform, square complement method, general integrated transport technique (GITT) and Laplace transform. This work considers the wind speed u, crosswind eddy diffusivity {k}_{y} and vertical eddy diffusivity {k}_{z} as functions of power law in vertical distance "z". This consideration was applied to the two analytical models. The predicted concentrations were calculated for neutral, stable and unstable conditions. The calculated concentrations for unstable conditions were compared with already existing experimental data measured for radioactive iodine-135 (I135) at an Egyptian Atomic Energy Authority test at Inshas. Also, the calculated concentrations for stable and neutral conditions were compared with already existing experimental data on iodine I-131 (I131) released from the research reactor. A comparison of the values of the proposed concentrations and the previous works is included in this article. It was found that the second predicted model was in good agreement with observed data in unstable, stable and neutral conditions compared with the first predicted model.

Brownian sieving enhancement of microcapillary hydrodynamic chromatography. Analysis of the separation performance based on Brenner’s macro-transport theory

In a recent article [Analytical Chemistry, 93(17), 6808–6816 (2021)], an unconventional device configuration enforcing a Brownian sieving mechanism was proposed as proof of concept for the efficient implementation of microcapillary hydrodynamic chromatography (MHDC). In this article, we perform a thorough analysis of the device geometry and of operating conditions, in order to single out the optimal configuration maximizing separation resolution. Brenner’s macro-transport theory provides the technical picklock to perform the search for the optimum over a wide choice of device geometries and for a range of values of the particle Péclet number covering most conditions encountered in practical implementations of MHDC. Specifically, effective transport coefficients defining the dynamics of the suspended phase are obtained by the solution of a two-dimensional steady-state advection-diffusion equation defined onto the channel cross-section. The eigenvalue/eigenfunction structure of the associated transient problem is exploited in order to quantify the timescale for reaching the macro-transport regime conditions. Based on this timescale and on the effective transport parameters, an estimate of the column length necessary to achieve a prescribed level of separation resolution is obtained. We identify device geometry and operating conditions where the capillary length is shrunk down by a factor above ten compared to the standard MHDC configuration. Lagrangian stochastic statistics of particle ensembles are used to validate the results obtained through Brenner’s macro-transport approach. The method proposed can be readily generalized to other classes of device geometries enforcing the same Brownian sieving mechanism.

Instability of large-scale riverbed patterns

In this paper, we explore the instability of large-scale riverbed patterns, performing a linear stability analysis of a zero-pressure gradient free-surface flow in a wide straight channel with an erodible bed. The local depth-averaged turbulence state is governed by two key parameters: namely, the turbulent kinetic energy (TKE) and its dissipation rate. A depth-averaged flow model coupled with the transport equations of the TKE and its dissipation rate and the particle transport model are developed to examine the formation of large-scale patterns. Both the modes of particle transport as bedload and suspended load are considered herein, allowing for the extension of the conventional theories to cover from hydraulically smooth to transitional flow regimes. The classical Exner equation of the bed evolution is modified in the presence of suspended particles, whose concentration is coupled with the steady-state advection–diffusion equation. Applying a standard linearization technique, the periodic perturbations in both streamwise and spanwise directions are imposed on the bed to find the dispersion relationship. The stability maps for the growth rate of large-scale patterns are obtained as a function of streamwise and spanwise wavenumbers and of key parameters associated with the flow and particles.

A novel analytical approach for advection diffusion equation for radionuclide release from an area source

Abstract The method of the Laplace transform has been used to obtain an analytical solution of the three-dimensional steady state advection diffusion equation for the airborne radionuclide release from any nuclear installation such as the power reactor in an area source. The present treatment takes into account the removal of the pollutants through the nuclear reaction. We assume that the pollutants are emitted as a constant rate from the area source. This physical consideration is achieved by assuming that the vertical eddy diffusivity coefficient should be a constant. The prevailing wind speed is a constant in x − direction and a linear function of the vertical height z. The present model calculations are compared with the other models and the available data of the atmospheric dispersion experiments that were carried out in the nuclear power plant of Angra dos Reis (Brazil). The results show that the present treatment performs well as the analytical dispersion model and there is a good agreement between the values computed by our model and the observed data.

Different Numerical Inversion Algorithms of the Laplace Transform for the Solution of the Advection-Diffusion Equation with Non-local Closure in Air Pollution Modeling

In this paper, a three-dimensional solution of the steady-state advection-diffusion equation is obtained applying the Generalized Integral Advection Diffusion Multilayer Technique (GIADMT), considering non-local closure for turbulent flow. Two different parameterizations were considering for the countergradient term and different methods of numerical inversion for inverse Laplace transform. The results were compared with the experimental data of Copenhagen experiment by an evaluation of statistical indices to analyse the solution of the equation through the methods of numerical inversion. Differents parameterizations for the vertical turbulent eddy diffusivity and wind profile were utilized. The results show a good agreement with the experiment and the methods of numerical inversion for inverse Laplace transform show same efficacy.

Non-unified Compact Residual-Distribution Methods for Scalar Advection–Diffusion Problems

This paper solves the advection–diffusion equation by treating both advection and diffusion residuals in a separate (non-unified) manner. An alternative residual distribution (RD) method combined with the Galerkin method is proposed to solve the advection–diffusion problem. This Flux-Difference RD method maintains a compact-stencil and the whole process of solving advection–diffusion does not require additional equations to be solved. A general mathematical analysis reveals that the new RD method is linearity preserving on arbitrary grids for the steady-state advection–diffusion equation. The numerical results show that the flux difference RD method preserves second-order accuracy on various unstructured grids including highly randomized anisotropic grids on both the linear and nonlinear scalar advection–diffusion cases.

High-order finite-volume solutions of the steady-state advection–diffusion equation with nonlinear Robin boundary conditions

We propose high-order finite-volume schemes for numerically solving the steady-state advection–diffusion equation with nonlinear Robin boundary conditions. Although the original motivation comes from a mathematical model of blood clotting, the nonlinear boundary conditions may also apply to other scientific problems. The main contribution of this work is a generic algorithm for generating third-order, fourth-order, and even higher-order explicit ghost-filling formulas to enforce nonlinear Robin boundary conditions in multiple dimensions. Under the framework of finite volume methods, this appears to be the first algorithm of its kind. Numerical experiments on boundary value problems show that the proposed fourth-order formula can be much more accurate and efficient than a simple second-order formula. Furthermore, the proposed ghost-filling formulas may also be useful for solving other partial differential equations.

仮想的な物理モデルに基づく幾何学的制約付きトポロジー最適化（型成形及びフライス加工のための幾何学的制約法）

This paper proposes a scheme for imposing geometrical constraints in topology optimization for molding and milling so that optimal configurations that guarantee manufacturability can be obtained, based on the fictitious physical model. First, a level set-based topology optimization method is briefly described, and geometrical requirements for molding and milling are clarified. In molding, molded products must embody certain geometrical features so that mold parts can be separated, and milling cannot proceed unless the desired shape allows tool cutting faces to reach the workpiece. A fictitious physical model described by a steady-state advection-diffusion equation is then constructed based on the requirements. In the fictitious physical model, material domains are represented as virtual heat sources and an advection direction is aligned with a prescribed direction, along which mold parts are moved, or attitude in the case of a milling tool. Void regions, where the value of the fictitious physical field is high, represent either undercut geometries which would prevent the mold from being parted, interior voids that cannot be manufactured, or regions that a milling tool cannot reach. Next, a geometrical constraint is formulated based on the fictitious physical model. An optimization algorithm is then constructed. Finally, in the numerical examples, the proposed method yields manufacturable optimal configurations, confirming the validity and the utility of the proposed method.

The Mathematical Treatment for Air Pollutant Diffusion Using Laplace Adomain Decomposition Method with Ground Surface

The method of Laplace Adomain Decomposition has been used to obtain a semi-analytical solution of the threedimensional steady state advection diffusion equation for dispersion of air pollutant from a point source. The present treatment takes into account a realistic boundary condition which considers the ground surface as an absorberreflector surface for the pollutant, simultaneously. This physical consideration is achieved by assuming that the vertical eddy diffusivity coefficient should be non-zero at the ground surface for vertical diffusion to be possible. The wind prevailing speed is parameterized in terms of vertical height using the power law profile. An upper boundary condition assuming capping inversion is considered which means that pollutant is subjected to a boundary Condition of zero flux. The present model calculations are compared with the available data of the atmospheric dispersion experiments that were carried out in the Copenhagen area (Denmark) and the semi-empirical model for Gaussian plume model with the same input data. In both comparison tasks, the results are reasonably good which indicates that the present treatment performs well as a simple analytical dispersion model.

A Generalized Analytical Model for Crosswind-Integrated Concentrations with Ground-Level Deposition in the Atmospheric Boundary Layer

A generalized mathematical model describing the crosswind-integrated concentrations is presented for dispersion of pollutants emitted from a continuous source in the atmospheric boundary layer with deposition to the ground surface. The model is based on a solution of the resulting two-dimensional steady state advection-diffusion equation with deposition to the ground surface. It considers the horizontal wind speed as a generalized function of vertical height above the ground surface and eddy diffusivity as a function of both downwind distance from the source and vertical height. Various special cases of the model are deduced. A sensitivity analysis of the model prediction of the ground-level crosswind-integrated concentrations with deposition velocity is performed. Various issues and limitations associated with this work are discussed. The model is evaluated with the observations of a depositing tracer obtained from Hanford dual diffusion experiment in stable conditions. The statistical measures show that the present model, by considering deposition, is performing well with the observations. The model is giving an over-predicting trend with the observations and predicts 100 % cases within a factor of two. On the other hand, the consideration of a non-depositing condition for a depositing tracer yields the severe over-prediction and thus, it introduces the significant errors in the model prediction. The selection of the lower boundary condition for a depositing tracer at the height of deposition surface gives better prediction than those at a height of surface roughness length.

Minimal stencil finite volume scheme with the discrete maximum principle

We propose a cell-centered finite volume (FV) scheme with the minimal stencil formed by the closest neighbouring cells. The discrete solution satisfies the discrete maximum principle and approximates the exact solution with second-order accuracy. The coefficients in the FV stencil depend on the solution; therefore, the FV scheme is nonlinear. The scheme is applied to a steady state advection-diffusion equation discretized on a general polygonal mesh.

A monotone finite volume method for advection–diffusion equations on unstructured polygonal meshes

We present a new second-order accurate monotone finite volume (FV) method for the steady-state advection–diffusion equation. The method uses a nonlinear approximation for both diffusive and advective fluxes and guarantees solution non-negativity. The interpolation-free approximation of the diffusive flux uses the nonlinear two-point stencil proposed in Lipnikov [23]. Approximation of the advective flux is based on the second-order upwind method with a specially designed minimal nonlinear correction. The second-order convergence rate and monotonicity are verified with numerical experiments.

An analytical model for dispersion of pollutants from a continuous source in the atmospheric boundary layer

For the dispersion of a pollutant released from a continuous source in the atmospheric boundary layer (ABL), a generalized analytical model describing the crosswind-integrated concentrations is presented. An analytical scheme is described to solve the resulting two-dimensional steady-state advection–diffusion equation for horizontal wind speed as a generalized function of vertical height above the ground and eddy diffusivity as a function of both downwind distance from the source and vertical height. Special cases of this model are deduced and an extensive analysis is carried out to compare the model with the known analytical models by taking the particular forms of wind speed and vertical eddy diffusivity. The proposed model is evaluated with the observations obtained from Copenhagen diffusion experiments in unstable conditions and Hanford and Prairie Grass experiments in stable conditions. In evaluation of the model, a recently proposed formulation for the wind speed in the entire ABL is used. It is concluded that the present model is performing well with the observations and can be used to predict the short-range dispersion from a continuous release. Further, it is shown that the accurate parameterizations of wind speed and eddy diffusivity provide a significant improvement in the agreement between computed and observed concentrations.

A semi-analytical solution for the three-dimensional advection–diffusion equation considering non-local turbulence closure

Atmospheric air pollution turbulent fluxes can be assumed to be proportional to the mean concentration gradient. This assumption, along with the equation of continuity, leads to the advection–diffusion equation. Moreover, large eddies are able to mix scalar quantities in a manner that is counter to the local gradient. In this work we present a semi-analytical solution for the three-dimensional steady-state advection–diffusion equation, considering non-local turbulence closure using the Generalized Integral Advection Diffusion Multilayer Technique (GIADMT). We report some examples of applications of the new solution for two different datasets and for a water tank experiment.

An analytical solution of the advection-diffusion equation considering non-local turbulence closure

Atmospheric air pollution turbulent fluxes can be assumed to be proportional to the mean concentration gradient. This assumption, along with the equation of continuity, leads to the advection-diffusion equation. Moreover, large eddies are able to mix scalar quantities in a manner that is counter to the local gradient. We present a general solution of a two-dimension steady state advection-diffusion equation, considering non-local turbulence closure using the General Integral Laplace Transform Technique. We show some examples of applications of the new solution with different vertical diffusion parameterisations.

Semi-analytical solution of the steady three-dimensional advection-diffusion equation in the planetary boundary layer

We present a three-dimensional solution of the steady-state advection-diffusion equation considering a vertically inhomogeneous planetary boundary layer (PBL). We reach this goal applying the generalized integral transform technique (GITT), a hybrid method that had solved a wide class of direct and inverse problems mainly in the area of heat transfer and fluid mechanics. The transformed problem is solved by the advection-diffusion multilayer model (ADMM) method, a semi-analytical solution based on a discretization of the PBL in sub-layers where the advection-diffusion equation is solved by the Laplace transform technique. Numerical simulations are presented and the performances of the solution are compared against field experiments data.

A finite analytic method for solving the 2-D time-dependent advection–diffusion equation with time-invariant coefficients

Difficulty in solving the transient advection–diffusion equation (ADE) stems from the relationship between the advection derivatives and the time derivative. For a solution method to be viable, it must account for this relationship by being accurate in both space and time. This research presents a unique method for solving the time-dependent ADE that does not discretize the derivative terms but rather solves the equation analytically in the space–time domain. The method is computationally efficient and numerically accurate and addresses the common limitations of numerical dispersion and spurious oscillations that can be prevalent in other solution methods. The method is based on the improved finite analytic (IFA) solution method [Lowry TS, Li S-G. A characteristic based finite analytic method for solving the two-dimensional steady-state advection–diffusion equation. Water Resour Res 38 (7), 10.1029/2001WR000518] in space coupled with a Laplace transformation in time. In this way, the method has no Courant condition and maintains accuracy in space and time, performing well even at high Peclet numbers. The method is compared to a hybrid method of characteristics, a random walk particle tracking method, and an Eulerian–Lagrangian Localized Adjoint Method using various degrees of flow-field heterogeneity across multiple Peclet numbers. Results show the IFALT method to be computationally more efficient while producing similar or better accuracy than the other methods.

Comparison of sigma schemes for estimation of air pollutant dispersion in low winds

The dispersion coefficients are key parameters in most of the air quality models and their use in low wind speed conditions involves some degree of uncertainty. Various estimates of dispersion parameters are discussed for low wind conditions. Specifically, schemes viz., standard, split sigma, split sigma theta, segmented plume (I) and (II), short-term averaging and U min approach are studied through intercomparison. The concentration formula used in this intercomparison is obtained from the steady-state advection-diffusion equation. The results show that with hourly data of wind velocity and standard deviation of horizontal wind direction ( σ θ ), split sigma and split sigma theta schemes perform much better than the traditional standard method as they take into account the wind direction fluctuations for the horizontal dispersion. On the other hand, with high-frequency data of wind velocity and σ θ , schemes like segmented plume (I) and (II) and short-term averaging simulate the observations much better especially in terms of the multiple peak nature of the concentration distribution. The influence of wind fluctuations from instantaneous plumes is captured by the high-frequency data. Further, short-term averaging scheme has the advantage of not requiring σ θ and still performing well. The difference in these results and those obtained from the Gaussian plume solution has been found to be marginal.