Scalar Advection-diffusion Equation Research Articles

Numerical visualization of wind turbine wakes using passive scalar advection–diffusion equation and its application for wake management

To visualize and characterize the unsteady properties of the wake trailing behind a wind turbine, we propose a novel numerical methodology in this study. Through a wind-tunnel experiment using a smoke generator, we succeeded in visualizing a compact-type small-scale wind turbine wake using continuous scalar plumes from a single point source. Next, to simulate the above wind-tunnel experiment, we proposed a method using the passive scalar advection–diffusion equation based on a high-fidelity large-eddy simulation (LES). The actuator-line method (ALM) was adopted for the wind turbine model. We succeeded in qualitatively reproducing the wake visualization experiment in the wind tunnel described above and verified the effectiveness of the proposed numerical visualization method. To evaluate the characteristics of wakes generated by wind turbines in more detail, we conducted a quantitative comparison using a disk-shaped volume source with the same size as the swept area, set behind the wind turbine model. The results indicated that the non-dimensional time-averaged passive scalar profile in the near- and far-wake regions qualitatively matched the shape of the stream-wise velocity profile, despite having opposite signs. In other words, it was shown that if a disk-shaped volume source with the same size as the swept area is placed just behind the wind turbine, it is possible to accurately predict the behavior of the wind turbine wake. Finally, to validate the proposed method for wind farm wake management, we performed a visualization of wake flows in a virtual offshore wind farm consisting of 12 wind turbines with short separation distances. Through detailed comparison of two types of numerical results with different wind directions, we showed that the proposed method can effectively demonstrate the range of spatial influence of the wakes formed by the wind turbine with special attention.

On the Construction of Conservative Semi-Lagrangian IMEX Advection Schemes for Multiscale Time Dependent PDEs

This article is devoted to the construction of a new class of semi-Lagrangian (SL) schemes with implicit-explicit (IMEX) Runge-Kutta (RK) time stepping for PDEs involving multiple space-time scales. The semi-Lagrangian (SL) approach fully couples the space and time discretization, thus making the use of RK strategies particularly difficult to be combined with. First, a simple scalar advection-diffusion equation is considered as a prototype PDE for the development of a high order formulation of the semi-Lagrangian IMEX algorithms. The advection part of the PDE is discretized explicitly at the aid of a SL technique, while an implicit discretization is employed for the diffusion terms. In this way, an unconditionally stable numerical scheme is obtained, that does not suffer any CFL-type stability restriction on the maximum admissible time step. Second, the SL-IMEX approach is extended to deal with hyperbolic systems with multiple scales, including balance laws, that involve shock waves and other discontinuities. A conservative scheme is then crucial to properly capture the wave propagation speed and thus to locate the discontinuity and the plateau exhibited by the solution. A novel SL technique is proposed, which is based on the integration of the governing equations over the space-time control volume which arises from the motion of each grid point. High order of accuracy is ensured by the usage of IMEX RK schemes combined with a Cauchy–Kowalevskaya procedure that provides a predictor solution within each space-time element. The one-dimensional shallow water equations (SWE) are chosen to validate the new conservative SL-IMEX schemes, where convection and pressure fluxes are treated explicitly and implicitly, respectively. The asymptotic-preserving (AP) property of the novel schemes is also studied considering a relaxation PDE system for the SWE. A large suite of convergence studies for both the non-conservative and the conservative version of the novel class of methods demonstrates that the formal order of accuracy is achieved and numerical evidences about the conservation property are shown. The AP property for the corresponding relaxation system is also investigated.

Numerical visualization of wind turbine wakes using passive scalar advection-diffusion equation and its application to wake management

Numerical visualization of wind turbine wakes using passive scalar advection-diffusion equation and its application to wake management

On relaxation systems and their relation to discrete velocity Boltzmann models for scalar advection-diffusion equations.

The connection of relaxation systems and discrete velocity models is essential to the progress of stability as well as convergence results for lattice Boltzmann methods. In the present study we propose a formal perturbation ansatz starting from a scalar one-dimensional target equation, which yields a relaxation system specifically constructed for its equivalence to a discrete velocity Boltzmann model as commonly found in lattice Boltzmann methods. Further, the investigation of stability structures for the discrete velocity Boltzmann equation allows for algebraic characterizations of the equilibrium and collision operator. The methods introduced and summarized here are tailored for scalar, linear advection-diffusion equations, which can be used as a foundation for the constructive design of discrete velocity Boltzmann models and lattice Boltzmann methods to approximate different types of partial differential equations. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Numerical analysis of Krylov multigrid methods for stationary advection-diffusion equation

This paper outlines the problem of applying multigrid methods for the stationary scalar advection-diffusion equation for the given advection filed. We assume that all functions are smooth enough to be represented by discretization methods. Two main discretization are considered: finite difference method for Dirichlet boundary conditions and pseudo-spectral method for the periodic boundary. We test the following smoothers for multigrid methods: Jacobi smoother, Gauss-Seidel smoother, Krylov subspace smoother (GMRES method with and without preconditioners). The analysis is performed in the space formed by the cross product of discretization parameters, diffusion coefficient values, multigrid levels and smoothers.We demonstrate that the most efficient strategy depends on parameter value and given velocity field. Best variants include Gauss-Seidel smoothers which is optimal for advection-dominated problem while multigrid method is used as a preconditioner for a Krylov method. Such methods can be used for spectral or pseudo-spectral methods where explicit dense matrix storage is impossible.

An Approximate Riemann Solver for Advection–Diffusion Based on the Generalized Riemann Problem

We construct an approximate Riemann solver for scalar advection–diffusion equations with piecewise polynomial initial data. The objective is to handle advection and diffusion simultaneously to reduce the inherent numerical diffusion produced by the usual advection flux calculations. The approximate solution is based on the weak formulation of the Riemann problem and is solved within a space–time discontinuous Galerkin approach with two subregions. The novel generalized Riemann solver produces piecewise polynomial solutions of the Riemann problem. In conjunction with a recovery polynomial, the Riemann solver is then applied to define the numerical flux within a finite volume method. Numerical results for a piecewise linear and a piecewise parabolic approximation are shown. These results indicate a reduction in numerical dissipation compared with the conventional separated flux calculation of advection and diffusion. Also, it is shown that using the proposed solver only in the vicinity of discontinuities gives way to an accurate and efficient finite volume scheme.

Flow and passive transport in planar multipolar flows

We study the flow and transport of heat or mass, modelled as passive scalars, within a basic geometrical unit of a three-dimensional multipolar flow – a triangular prism – characterised by a side length$L$, a normalised thickness$0.01\leqslant \unicode[STIX]{x1D700}\leqslant 0.1$and an apex angle$0<\unicode[STIX]{x1D6FC}<\unicode[STIX]{x03C0}$, and connected to inlet and outlet pipes of equal normalised radius$0.01\leqslant \unicode[STIX]{x1D6FF}\leqslant 0.1$perpendicularly to the plane of the flow. The flow and scalar fields are investigated over the range$0.1\leqslant Re_{p}\leqslant 10$and$0.1\leqslant Pe_{p}\leqslant 1000$, where$Re_{p}$and$Pe_{p}$are respectively the Reynolds and Péclet numbers imposed at the inlet pipe when either a Dirichlet ($\text{D}$) or a Neumann ($\text{N}$) scalar boundary condition is imposed at the wall unattached to the inlets and outlets. A scalar no-flux boundary condition is imposed at all the other walls. An axisymmetric model is applied to understand the flow and scalar transport in the inlet and outlet regions, which consist of a turning region close to the pipe centreline and a channel region away from it. A separate two-dimensional model is then developed for the channel region by solving the integral form of the momentum and scalar advection–diffusion equations. Analytical relations between geometrical, flow and scalar transport parameters based on similarity and integral methods are generated and agree closely with numerical solutions. Finally, three-dimensional numerical calculations are undertaken to test the validity of the axisymmetric and depth-averaged analyses. Dominant flow and scalar transport features vary dramatically across the flow domain. In the turning region, the flow is a largely irrotational straining flow when$\unicode[STIX]{x1D700}\geqslant \unicode[STIX]{x1D6FF}$and a dominantly viscous straining flow when$\unicode[STIX]{x1D700}\ll \unicode[STIX]{x1D6FF}$. The thickness of the scalar boundary layer scales to the local Péclet number to the power$1/3$. The diffusive flux$j_{d}$and the scalar$C_{s}$at the wall where ($\text{D}$) or ($\text{N}$) is imposed, respectively, are constant. In the channel region, the flow is parabolic and dominated by a source flow near the inlet and an irrotational straining flow away from it. When$(\text{D})$is imposed the scalar decreases exponentially with distance from the inlet and the normalised scalar transfer coefficient converges to$\unicode[STIX]{x1D6EC}_{\infty }=2.5694$. When$(\text{N})$is imposed,$C_{s}$varies proportionally to surface area. Transport in the straining region downstream of the inlet is diffusion-limited, and$j_{d}$and$C_{s}$are functions of the geometrical parameters$L$,$\unicode[STIX]{x1D700}$,$\unicode[STIX]{x1D6FC}$and$\unicode[STIX]{x1D6FF}$. In addition to describing the fundamental properties of the flow and passive transport in multipolar configurations, the present work demonstrates how geometrical and flow parameters should be set to control transfers in the different regions of the flow domain.

A novel finite volume discretization method for advection–diffusion systems on stretched meshes

This work is concerned with spatial advection and diffusion discretization technology within the field of Computational Fluid Dynamics (CFD). In this context, a novel method is proposed, which is dubbed the Enhanced Taylor Advection–Diffusion (ETAD) scheme. The model equation employed for design of the scheme is the scalar advection–diffusion equation, the industrial application being incompressible laminar and turbulent flow. Developed to be implementable into finite volume codes, ETAD places specific emphasis on improving accuracy on stretched structured and unstructured meshes while considering both advection and diffusion aspects in a holistic manner. A vertex-centered structured and unstructured finite volume scheme is used, and only data available on either side of the volume face is employed. This includes the addition of a so-called mesh stretching metric. Additionally, non-linear blending with the existing NVSF scheme was performed in the interest of robustness and stability, particularly on equispaced meshes. The developed scheme is assessed in terms of accuracy – this is done analytically and numerically, via comparison to upwind methods which include the popular QUICK and CUI techniques. Numerical tests involved the 1D scalar advection–diffusion equation, a 2D lid driven cavity and turbulent flow case. Significant improvements in accuracy were achieved, with L2 error reductions of up to 75%.

DNS of scalar transfer across an air-water interface during inception and growth of Langmuir circulation

Direct numerical simulations (DNS) of an initially quiescent coupled air-water interface driven by an air flow with free stream speed of 5 m/s have been conducted. The DNS solves a scalar advection-diffusion equation for dissolved gas (or scalar) concentration in order to determine the impact of the water-side turbulence on scalar transfer from the air side to the water side and subsequent vertical transport in the water column. Two simulations are compared: one with a freely deforming interface and a second one with a flat interface. In the first simulation, the deforming interface evolves in the form of gravity-capillary waves generating aqueous Langmuir turbulence characterized by small-scale (centimeter-scale) Langmuir cells (LCs). The second simulation is characterized by pure shear-driven turbulence in the absences of LCs as the interface is intentionally held flat. It is concluded that the Langmuir turbulence serves to enhance vertical transport of the scalar in the water side and in the process it increases scalar transfer efficiency relative to the shear-dominated turbulence in the flat interface case. Furthermore, transition to Langmuir turbulence was observed to be accompanied by a spike in scalar flux characterized by an order of magnitude increase. These episodic flux increases, if linked to gusts and overall unsteadiness in the wind field, are expected to be an important contributor in determining the long-term average of the air-sea gas fluxes. The effectiveness of popular transfer velocity models, namely the small eddy model and the surface divergence model, in predicting this spike is evaluated via the DNS.

Formulation and Analysis of a Parameter-Free Stabilized Finite Element Method

We present and study a parameter-free stabilized finite element method for the scalar advection-diffusion equation. Edge element lifting of diffusive edge fluxes defines the stabilization. The amount of edge diffusion varies with the edge Péclet number and adapts to solution features without mesh-dependent parameters. We prove that the method is first-order accurate in the advective limit. Numerical studies confirm the theoretical estimates.

Direct numerical simulation of turbulent scalar transport across a flat surface

AbstractTo elucidate the physical mechanisms that play a role in the interfacial transfer of atmospheric gases into water, a series of direct numerical simulations of mass transfer across the air–water interface driven by isotropic turbulence diffusing from below has been carried out for various turbulent Reynolds numbers ($R_T=84,195,507$). To allow a direct (unbiased) comparison of the instantaneous effects of scalar diffusivity, in each of the DNS up to six scalar advection–diffusion equations with different Schmidt numbers were solved simultaneously. As far as the authors are aware this is the first simulation that is capable to accurately resolve the realistic Schmidt number, $\mathit{Sc}=500$, that is typical for the transport of atmospheric gases such as oxygen in water. For the range of turbulent Reynolds numbers and Schmidt numbers considered, the normalized transfer velocity $K_L$ was found to scale with $R_T^{-{1/2}}$ and $\mathit{Sc}^{-{1/2}}$, which indicates that the largest eddies present in the isotropic turbulent flow introduced at the bottom of the computational domain tend to determine the mass transfer. The $K_L$ results were also found to be in good agreement with the surface divergence model of McCready, Vassiliadou & Hanratty (AIChE J., vol. 32, 1986, pp. 1108–1115) when using a constant of proportionality of 0.525. Although close to the surface large eddies are responsible for the bulk of the gas transfer, it was also observed that for higher $R_T$ the influence of smaller eddies becomes more important.

Filling of a Poisson trap by a population of random intermittent searchers

We extend the continuum theory of random intermittent search processes to the case of N independent searchers looking to deliver cargo to a single hidden target located somewhere on a semi-infinite track. Each searcher randomly switches between a stationary state and either a leftward or rightward constant velocity state. We assume that all of the particles start at one end of the track and realize sample trajectories independently generated from the same underlying stochastic process. The hidden target is treated as a partially absorbing trap in which a particle can only detect the target and deliver its cargo if it is stationary and within range of the target; the particle is removed from the system after delivering its cargo. As a further generalization of previous models, we assume that up to n successive particles can find the target and deliver its cargo. Assuming that the rate of target detection scales as 1/N, we show that there exists a well-defined mean-field limit N→∞, in which the stochastic model reduces to a deterministic system of linear reaction-hyperbolic equations for the concentrations of particles in each of the internal states. These equations decouple from the stochastic process associated with filling the target with cargo. The latter can be modeled as a Poisson process in which the time-dependent rate of filling λ(t) depends on the concentration of stationary particles within the target domain. Hence, we refer to the target as a Poisson trap. We analyze the efficiency of filling the Poisson trap with n particles in terms of the waiting time density f(n)(t). The latter is determined by the integrated Poisson rate μ(t)=∫(0)(t)λ(s)ds, which in turn depends on the solution to the reaction-hyperbolic equations. We obtain an approximate solution for the particle concentrations by reducing the system of reaction-hyperbolic equations to a scalar advection-diffusion equation using a quasisteady-state analysis. We compare our analytical results for the mean-field model with Monte Carlo simulations for finite N. We thus determine how the mean first passage time (MFPT) for filling the target depends on N and n.

Multiphysics model for blood flow and drug transport with application to patient-specific coronary artery flow

A multiphysics mathematical model to simulate drug delivery in idealized and patient-specific coronary arteries is presented. Blood is modeled as an incompressible Navier–Stokes fluid, the arterial wall as a linear poroelastic medium, and the drug transport is described by a scalar advection-diffusion equation. The drug compound is released into the bloodstream, carried by the flow, deposited onto the endothelium, penetrates into the wall, and is transported within the arterial wall. NURBS-based isogeometric analysis is employed to describe the geometry and discretize the fluid-solid interaction equations.

Compact finite volume schemes on boundary-fitted grids

The paper focuses on the development of a framework for high-order compact finite volume discretization of the three-dimensional scalar advection–diffusion equation. In order to deal with irregular domains, a coordinate transformation is applied between a curvilinear, non-orthogonal grid in the physical space and the computational space. Advective fluxes are computed by the fifth-order upwind scheme introduced by Pirozzoli [S. Pirozzoli, Conservative hybrid compact-WENO schemes for shock–turbulence interaction, J. Comp. Phys. 178 (2002) 81] while the Coupled Derivative scheme [M.H. Kobayashi, On a class of Padé finite volume methods, J. Comp. Phys. 156 (1999) 137] is used for the discretization of the diffusive fluxes.Numerical tests include unsteady diffusion over a distorted grid, linear free-surface gravity waves in a irregular domain and the advection of a scalar field. The proposed methodology attains high-order formal accuracy and shows very favorable resolution characteristics for the simulation of problems with a wide range of length scales.

H-Multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

Being implicit in time, the space-time discontinuous Galerkin discretization of the compressible Navier–Stokes equations requires the solution of a non-linear system of algebraic equations at each time-step. The overall performance, therefore, highly depends on the efficiency of the solver. In this article, we solve the system of algebraic equations with a h-multigrid method using explicit Runge–Kutta relaxation. Two-level Fourier analysis of this method for the scalar advection–diffusion equation shows convergence factors between 0.5 and 0.75. This motivates its application to the 3D compressible Navier–Stokes equations where numerical experiments show that the computational effort is significantly reduced, up to a factor 10 w.r.t. single-grid iterations.

YZβ discontinuity capturing for advection‐dominated processes with application to arterial drug delivery

AbstractThe YZβ discontinuity‐capturing operator, recently introduced in (Encyclopedia of Computational Mechanics, Vol. 3, Fluids. Wiley: New York, 2004) in the context of compressible flows, is applied to a time‐dependent, scalar advection–diffusion equation with the purpose of modelling drug delivery processes in blood vessels. The formulation is recast in a residual‐based form, which reduces to the previously proposed formulation in the limit of zero diffusion and source term. The NURBS‐based isogeometric analysis method, proposed by Hughes et al. (Comput. Methods Appl. Mech. Eng. 2005; 194:4135–4195), was used for the numerical tests. Effects of various parameters in the definition of the YZβ operator are examined on a model problem and the better performer is singled out. While for low‐order B‐spline functions discontinuity capturing is necessary to improve solution quality, we find that high‐order, high‐continuity B‐spline discretizations produce sharp, nearly monotone layers without the aid of discontinuity capturing. Finally, we successfully apply the YZβ approach to the simulation of drug delivery in patient‐specific coronary arteries. Copyright © 2007 John Wiley & Sons, Ltd.

Stability analysis of second-order time accurate schemes for ALE–FEM

In this work we will introduce and analyze the Arbitrary Lagrangian Eulerian formulation for a model problem of a scalar advection–diffusion equation defined on a moving domain. Moving from the results illustrated in our previous work [J. Num. Math. 7 (1999) 105], we will consider first and second-order time advancing schemes and analyze how the movement of the domain might affect accuracy and stability properties of the numerical schemes with respect to their counterpart on fixed domains. Theoretical and numerical results will be presented, showing that stability properties are not, in general, preserved, while accuracy is maintained.

Generalized Fourier analyses of the advection–diffusion equation—Part I: one‐dimensional domains

AbstractThis paper presents a detailed multi‐methods comparison of the spatial errors associated with finite difference, finite element and finite volume semi‐discretizations of the scalar advection–diffusion equation. The errors are reported in terms of non‐dimensional phase and group speed, discrete diffusivity, artificial diffusivity, and grid‐induced anisotropy. It is demonstrated that Fourier analysis provides an automatic process for separating the discrete advective operator into its symmetric and skew‐symmetric components and characterizing the spectral behaviour of each operator. For each of the numerical methods considered, asymptotic truncation error and resolution estimates are presented for the limiting cases of pure advection and pure diffusion. It is demonstrated that streamline upwind Petrov–Galerkin and its control‐volume finite element analogue, the streamline upwind control‐volume method, produce both an artificial diffusivity and a concomitant phase speed adjustment in addition to the usual semi‐discrete artifacts observed in the phase speed, group speed and diffusivity. The Galerkin finite element method and its streamline upwind derivatives are shown to exhibit super‐convergent behaviour in terms of phase and group speed when a consistent mass matrix is used in the formulation. In contrast, the CVFEM method and its streamline upwind derivatives yield strictly second‐order behaviour. In Part II of this paper, we consider two‐dimensional semi‐discretizations of the advection–diffusion equation and also assess the affects of grid‐induced anisotropy observed in the non‐dimensional phase speed, and the discrete and artificial diffusivities. Although this work can only be considered a first step in a comprehensive multi‐methods analysis and comparison, it serves to identify some of the relative strengths and weaknesses of multiple numerical methods in a common analysis framework. Published in 2004 by John Wiley & Sons, Ltd.

Generalized fourier analyses of the advection–diffusion equation—Part II: two‐dimensional domains

AbstractPart I of this work presents a detailed multi‐methods comparison of the spatial errors associated with the one‐dimensional finite difference, finite element and finite volume semi‐discretizations of the scalar advection–diffusion equation. In Part II we extend the analysis to two‐dimensional domains and also consider the effects of wave propagation direction and grid aspect ratio on the phase speed, and the discrete and artificial diffusivities. The observed dependence of dispersive and diffusive behaviour on propagation direction makes comparison of methods more difficult relative to the one‐dimensional results. For this reason, integrated (over propagation direction and wave number) error and anisotropy metrics are introduced to facilitate comparison among the various methods. With respect to these metrics, the consistent mass Galerkin and consistent mass control‐volume finite element methods, and their streamline upwind derivatives, exhibit comparable accuracy, and generally out‐perform their lumped mass counterparts and finite‐difference based schemes. While this work can only be considered a first step in a comprehensive multi‐methods analysis and comparison, it serves to identify some of the relative strengths and weaknesses of multiple numerical methods in a common mathematical framework. Published in 2004 by John Wiley & Sons, Ltd.

A new stabilized finite element formulation for scalar convection–diffusion problems: the streamline and approximate upwind/Petrov–Galerkin method

A new stabilized and accurate finite element formulation for convection-dominated problems is herein developed. The basis of the new formulation is the choice of a new upwind function. The upwind function chosen for the new method provokes its degeneration into the SUPG or CAU methods, depending on the approximate solution’s regularity. The accuracy and stability of the new formulation for the linear and scalar advection–diffusion equation is demonstrated in several numerical examples.