Scalar Advection Research Articles

Eulerian statistically preserved structures in passive scalar advection.

We analyze numerically the time-dependent linear operators that govern the dynamics of Eulerian correlation functions of a decaying passive scalar advected by a stationary, forced two-dimensional Navier-Stokes turbulence. We show how to naturally discuss the dynamics in terms of effective compact operators that display Eulerian statistically preserved structures which determine the anomalous scaling of the correlation functions. In passing we point out a bonus of the present approach, in providing analytic predictions for the time-dependent correlation functions in decaying turbulent transport.

Airflows and turbulent flux measurements in mountainous terrain: Part 1. Canopy and local effects

We have studied the effects of local topography and canopy structure on turbulent flux measurements at a site located in mountainous terrain within a subalpine, coniferous forest. Our primary aim was to determine whether the complex terrain of the site affects the accuracy of eddy flux measurements from a practical perspective. We observed displacement heights, roughness lengths, spectral peaks, turbulent length scales, and profiles of turbulent intensities that were comparable in magnitude and pattern to those reported for forest canopies in simpler terrain. We conclude that in many of these statistical measures, the local canopy exerts considerably more influence than does topographical complexity. Lack of vertical flux divergence and modeling suggests that the flux footprints for the site are within the standards acceptable for the application of flux statistics. We investigated three different methods of coordinate rotation: double rotation (DR), triple rotation (TR), and planar-fit rotation (PF). Significant variability in rotation angles at low wind speeds was encountered with the commonly used DR and TR methods, as opposed to the PF method, causing some overestimation of the fluxes. However, these differences in fluxes were small when applied to large datasets involving sensible heat and CO 2 fluxes. We observed evidence of frequent drainage flows near the ground during stable, stratified conditions at night. Concurrent with the appearance of these flows, we observed a positive bias in the mean vertical wind speed, presumably due to subtle topographic variations inducing a flow convergence below the measurement sensors. In the presence of such drainage flows, advection of scalars and non-zero bias in the mean vertical wind speed can complicate closure of the mass conservation budget at the site.

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.

Diffusion-limited scalar cascades

An important process at the heart of many fluid phenomena in the physical andengineering sciences is the advection and diffusion of a passive scalar. One of thecanonical idealizations of this process is the problem first broached by Zeldovich(1937), in which the scalar is advected inside a closed container by an incompressiblefluid with an arbitrary velocity field. There is no source or sink of the scalar withinthe fluid, and its concentration is maintained in some fixed pattern on parts of theboundary, whilst other areas of the boundary are impermeable.The mathematical formulation of Zeldovich’s problem is the advection–diffusionequation for the scalar field,

Chaotic mixing in a torus map.

The advection and diffusion of a passive scalar is investigated for a map of the 2-torus. The map is chaotic, and the limit of almost-uniform stretching is considered. This allows an analytic understanding of the transition from a phase of constant scalar variance (for short times) to exponential decay (for long times). This transition is embodied in a short superexponential phase of decay. The asymptotic state in the exponential phase is an eigenfunction of the advection-diffusion operator, in which most of the scalar variance is concentrated at small scales, even though a large-scale mode sets the decay rate. The duration of the superexponential phase is proportional to the logarithm of the exponential decay rate; if the decay is slow enough then there is no superexponential phase at all.

Construction of second order accurate monotone and stable residual distribution schemes for unsteady flow problems

The aim of this paper is to construct upwind residual distribution schemes for the time accurate solution of hyperbolic conservation laws. To do so, we evaluate a space–time fluctuation based on a space–time approximation of the solution and develop new residual distribution schemes which are extensions of classical steady upwind residual distribution schemes. This method has been applied to the solution of scalar advection equation and to the solution of the compressible Euler equations both in two space dimensions. The first version of the scheme is shown to be, at least in its first order version, unconditionally energy stable and possibly conditionally monotonicity preserving. Using an idea of Csik et al. [Space–time residual distribution schemes for hyperbolic conservation laws, 15th AIAA Computational Fluid Dynamics Conference, Anahein, CA, USA, AIAA 2001-2617, June 2001], we modify the formulation to end up with a scheme that is unconditionally energy stable and unconditionally monotonicity preserving. Several numerical examples are shown to demonstrate the stability and accuracy of the method.

Development of highly accurate interpolation methodfor mesh‐free flow simulations III. Analysis of accuracy and stability

AbstractA highly accurate interpolation method, CIVA, improves the accuracy of mesh‐free and grid‐less methods by taking into consideration first‐order spatial derivatives as variables; an approach based on the same idea as that on which CIP is based. In this study, the accuracy and stability of CIVA is evaluated by analytically and numerically. First, the general formulation of CIVA for the n‐dimensional case is described. Since CIVA contains the bubble function, we consider the determination methods: constant curvature condition and utilization of another computing point. Then, the relation between the bubble function in the CIVA method and the accuracy and stability is made clear by the analysis based on the Taylor expansion. Some computations of two‐dimensional passive scalar advection and advection–diffusion problems are performed for the verification of accuracy and stability. Copyright © 2003 John Wiley & Sons, Ltd.

Discussion on: “An Optimal Control Approach for Hybrid Systems”

A finite volume approximation of the scalar hyperbolic conservation law or advection–diffusion equation is given. In the context of the method of lines, the space discretization uses weighted essentially non oscillatory (WENO) reconstructions with adaptive order (WENO-AO), and the time evolution uses implicit Runge–Kutta methods. Therefore the timestep may be larger than the CFL timestep. To reduce oscillation in the solution, ideas related to spatially partitioned Runge–Kutta methods are used. An adaptive Runge–Kutta method is developed that blends the L-stable, third order, implicit Radau IIA method with the composite backward Euler method using a weighting procedure inspired from spatial WENO methods. The weighting procedure requires a smoothness indicator, and several possibilities are considered, although one is perhaps seen to be preferred. The overall scheme is proven to maintain third order accuracy when the solution is smooth. When the solution has a discontinuity, the scheme is shown computationally to be third order accurate away from shocks, and to achieve the overall accuracy of the backward Euler method. Numerical examples show that the adaptive Runge–Kutta method reduces oscillations in the solution. Moreover, the resulting scheme is shown to be unconditionally L-stable for smooth solutions to the linear problem.

Closure Approximations for Passive Scalar Turbulence: A Comparative Study on an Exactly Solvable Model with Complex Features

Some standard closure approximations used in turbulence theory are analyzed by examining systematically the predictions these approximations produce for a passive scalar advection model consisting of a shear flow with a fluctuating cross sweep. This model has a general geometric structure of a jet flow with transverse disturbances, which occur in a number of contexts, and it encompasses a wide variety of possible spatio-temporal statistical structures for the velocity field, including strong long-range correlations. Even though the Eulerian and Lagrangian velocity statistics are not equal and the passive scalar statistics exhibit broader-than-Gaussian intermittency, this model is nevertheless simple enough so that many passive scalar statistics can be computed exactly and compared systematically with the predictions of the closure approximations. Our comparative study illustrates the strength and weaknesses of the closure approximations and points out the physical phenomena that these approximations are able or not able to describe properly. In particular it is shown that the direct interaction approximation (DIA), one of the most sophisticated closure approximations available, fails to reproduce adequately the statistical features of the scalar and may even lead to absdurd predictions, even though the equations it produces are rather complicated and difficult to analyze. Two alternative closure approximations, the Modified DIA (MDIA) and the Renormalized Lagrangian Approximation (RLA), with different levels of sophistication, both are simpler to use than the DIA and perform better. In particular, it is shown that both closure approximations always reproduce exactly the second order statistics for the scalar and that the MDIA is even able to capture intermittency effects.

221 三次元移流問題に対する高精度時間積分法

221 三次元移流問題に対する高精度時間積分法

Energy-Conserving Simulation of Incompressible Electro-Osmotic and Pressure-Driven Flow

A numerical model for electro-osmotic flow is described. The advecting velocity field is computed by solving the incompressible Navier–Stokes equation. The method uses a semi-implicit multigrid algorithm to compute the divergence-free velocity at each grid point. The finite differences are second-order accurate and centered in space; however, the traditional second-order compact finite differencing of the Poisson equation for the pressure field is shown not to conserve energy in the inviscid limit. We have designed a non-compact finite differencing for the Laplacian in the pressure equation that allows exact energy conservation and affords second-order accuracy. The model also incorporates a new numerical method for passive scalar advection, called parcel advection, which accurately predicts the evolution of a passively traveling scalar pulse without requiring the addition of any artificial diffusion. The algorithm is used to confirm the experimentally observed asymmetric concentration profile that arises when an external pressure drop is imposed on electro-osmotic flow.

An unconditionally stable, explicit Godunov scheme for systems of conservation laws

AbstractCommon explicit, Godunov‐type schemes are subject to a stability constraint. The time‐line interpolation technique allows this constraint to be eliminated without having to make the scheme implicit or to linearize the equations. For 2×2 systems of conservation laws, a system of non‐linear equations has to be solved in the general case to determine the left and right states of the Riemann problems at the cell interfaces. However, if one cell in the domain is wide enough for the Courant number to be locally lower than unity, it is not necessary to solve a system anymore and the values at the next time step can be computed directly. The method is detailed for linear and non‐linear scalar advection, as well as for 2×2 systems of hyperbolic conservation laws. It is illustrated by an application to a simplified model for two‐phase flow in pipes, which is described using a 2×2 system of non‐linear hyperbolic equations. Copyright © 2002 John Wiley & Sons, Ltd.

Statistically preserved structures in shell models of passive scalar advection.

It was conjectured recently that statiscally preserved structures underlie the statistical physics of turbulent transport processes. We analyze here in detail the time-dependent (noncompact) linear operator that governs the dynamics of correlation functions in the case of shell models of passive scalar advection. The problem is generic in the sense that the driving velocity field is neither Gaussian nor delta correlated in time. We show how to naturally discuss the dynamics in terms of an effective compact operator that displays "zero modes," which determine the anomalous scaling of the correlation functions. Since shell models have neither a Lagrangian structure nor "shape dynamics," this example differs significantly from standard passive scalar advection. Nevertheless, with the necessary modifications, the generality and efficacy of the concept of statistically preserved structures are further exemplified. In passing we point out a bonus of the present approach, in providing analytic predictions for the time-dependent correlation functions in decaying turbulent transport.

Consistency with continuity in conservative advection schemes for free‐surface models

AbstractThe consistency of the discretization of the scalar advection equation with the discretization of the continuity equation is studied for conservative advection schemes coupled to three‐dimensional flows with a free‐surface. Consistency between the discretized free‐surface equation and the discretized scalar transport equation is shown to be necessary for preservation of constants. In addition, this property is shown to hold for a general formulation of conservative schemes. A discrete maximum principle is proven for consistent upwind schemes. Various numerical examples in idealized and realistic test cases demonstrate the practical importance of the consistency with the discretization of the continuity equation. Copyright © 2002 John Wiley & Sons, Ltd.

Multidimensional Least Squares Fluctuation Distribution Schemes with Adaptive Mesh Movement for Steady Hyperbolic Equations

Optimal meshes and solutions for steady conservation laws and systems within a finite volume fluctuation distribution framework are obtained by least squares methods incorporating mesh movement. The problem of spurious modes is alleviated through adaptive mesh movement, the least squares minimization giving an obvious way of determining the movement of the nodes and also providing a link with equidistribution. The iterations are carried out locally node by node, which yields good control of the moving mesh. For scalar equations an iteration which respects the flow of information in the problem significantly accelerates the convergence. The method is demonstrated on a scalar advection problem and a shallow water channel flow problem. For discontinuous solutions we introduce a least squares shock fitting approach which greatly improves the treatment of discontinuities at little extra expense by using degenerate triangles and moving the nodes. Examples are shown for a discontinuous shallow water channel flow and a shocked flow in gasdynamics governed by the compressible Euler equations.

High-Order Monotonicity-Preserving Compact Schemes for Linear Scalar Advection on 2-D Irregular Meshes

This paper is concerned with the numerical solution for linear scalar advection problems, the velocity field of which may be uniform or a given function of the space variable. We would like to propose the following: (1) a new family of 1-D compact explicit schemes, which preserve monotonicity while maintaining high-order accuracy in smooth regions; and (2) an extension to the 2-D case of this family of schemes, which ensures good accuracy and isotropy of the computed solution even for very distorted meshes. A few theoretical results are proven, while abundant numerical tests are shown in order to illustrate the quality of the schemes at issue.

Calculation of the anomalous exponents in the rapid-change model of passive scalar advection to order epsilon(3).

The field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar advected by the Gaussian velocity field with zero mean and correlation function approximately equal to delta(t-t('))/k(d + epsilon). Inertial-range anomalous exponents, identified with the critical dimensions of various scalar and tensor composite operators constructed of the scalar gradients, are calculated within the epsilon expansion to order epsilon(3) (three-loop approximation), including the exponents in anisotropic sectors. The main goal of the paper is to give the complete derivation of this third-order result, and to present and explain in detail the corresponding calculational techniques. The character and convergence properties of the epsilon expansion are discussed, the improved "inverse" epsilon expansion is proposed, and the comparison with the existing nonperturbative results is given.

Mixing in frozen and time-periodic two-dimensional vortical flows

In the first part of this paper, we investigate passive scalar or tracer advection–diffusion in frozen, two-dimensional, non-circular symmetric vortices. We develop an asymptotic description of the scalar field in a time range 1 [Lt ] t/T [Lt ] Pe1/3, where T is the formation time of the spiral in the vortex and Pe is a Péclet number, assumed much larger than 1. We derive the leading-order decay of the scalar variance E(t) for a singular non-circular streamline geometry,The variance decay is solely determined by a geometrical parameter μ and the exponent β describing the behaviour of the closed streamline periods. We develop a method to predict, in principle, the variance decay from snapshots of the advected scalar field by reconstructing the streamlines and their period from just two snapshots of the advected scalar field.In the second part of the paper, we investigate variance decay in a periodically moving singular vortex. We identify three different regions (core, chaotic and KAM-tori). We find fast mixing in the chaotic region and investigate a conjecture about mixing in the KAM-tori region. The conjecture enables us to use the results from the first section and relates the Kolmogorov capacity, or box-counting dimension, of the advected scalar to the decay of the scalar variance. We check our theoretical predictions against a numerical simulation of advection–diffusion of scalar in such a flow.

FETI domain decomposition methods for scalar advection–diffusion problems

In this paper, we show that iterative substructuring methods of finite element tearing and interconnecting type can be successfully employed for the solution of linear systems arising from the finite element approximation of scalar advection–diffusion problems. Using similar ideas as those of a recently developed Neumann–Neumann method, we propose a one-level algorithm and a class of two-level algorithms, obtained by suitably modifying the local problems on the subdomains. We present some numerical results for some significant test cases. Our methods appear to be optimal for flows without closed streamlines and possibly very small values of the viscosity. They also show very good performances for rotating flows and moderate Reynolds numbers. Therefore, the algorithms proposed appear to be well-suited for many convection-dominated problems of practical interest.

Erratum: Anomalous exponents to orderɛ3in the rapid-change model of passive scalar advection [Phys. Rev. E63, 025303 (2001)

Erratum: Anomalous exponents to order<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi>ɛ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>in the rapid-change model of passive scalar advection [Phys. Rev. E<b>63</b>, 025303 (2001)