Scalar Advection-diffusion Equation Research Articles

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.

Axial motion and scalar transport in stretched spiral vortices

We consider the dynamics of axial velocity and of scalar transport in the stretched-spiral vortex model of turbulent fine scales. A large-time asymptotic solution to the scalar advection-diffusion equation, with an azimuthal swirling velocity field provided by the stretched spiral vortex, is used together with appropriate stretching transformations to determine the evolution of both the axial velocity and a passive scalar. This allows calculation of the shell-integrated three-dimensional spectra of these quantities for the spiral-vortex flow. The dominant term in the velocity (energy) spectrum contributed by the axial velocity is found to be produced by the stirring of the initial distribution of axial velocity by the axisymmetric component of the azimuthal velocity. This gives a k−7/3 spectrum at large wave numbers, compared to the k−5/3 component for the azimuthal velocity itself. The spectrum of a passive scalar being mixed by the vortex velocity field is the sum of two power laws. The first is a k−1 Batchelor spectrum for wave numbers up to the inverse Batchelor scale. This is produced by the axisymmetric component of the axial vorticity but is independent of the detailed radial velocity profile. The second is a k−5/3 Obukov–Corrsin spectrum for wave numbers less than the inverse Kolmogorov scale. This is generated by the nonaxisymmetric axial vorticity and depends on initial correlations between this vorticity and the initial scalar field. The one-dimensional scalar spectrum for the composite model is in satisfactory agreement with experimental measurement.

Lagrangian Methods for the Tensor-Diffusivity Subgrid Model

A subgrid-scale model based on a truncated exact series expansion for Gaussian filtered products is considered for the incompressible scalar advection–diffusion equation. This model can be interpreted as a tensor diffusivity term proportional to the rate-of-strain tensor of the large-scale filtered velocity field. To control negative diffusion in the stretching directions, a Lagrangian method is used. The scalar field is represented in terms of a collection of anisotropic or axisymmetric Gaussian particles. An expansion in Hermite polynomials leads to equations of motion for particle velocity and shape based on a weighted average. A new accurate remeshing method, taking advantage of the properties of the subgrid model, is proposed and tested. A stagnation flow is used to demonstrate several theoretical and numerical aspects of the model. Better agreement with filtered DNS data is obtained than with the Smagorinsky subgrid model for a 2D time-dependent sinusoidal flow, which yields chaotic advection. The use of anisotropic particles leads to slightly more accurate results than the use of axisymmetric particles. Computational efficiency, however, makes the latter therefore the preferred choice.

The corrected operator splitting approach applied to a nonlinear advection-diffusion problem

So-called corrected operator splitting methods are applied to a 1-D scalar advection-diffusion equation of Buckley-Leverett type with general initial data. Front tracking and a 2nd order Godunov method are used to advance the solution in time. Diffusion is modelled by piece wise linear finite elements at each new time level. To obtain correct structure of shock fronts independently of the size of the time step, a dynamically defined residual flux term is grouped with diffusion. Different test problems are considered, and the methods are compared with respect to accuracy and runtime. Finally, we extend the corrected operator splitting to 2-D equations by means of dimensional splitting, and we apply it to a Buckley-Leverett type problem including gravitational effects.

Shadowing and the role of small diffusivity in the chaotic advection of scalars

Using techniques from shadowing theory, the solution of the scalar advection–diffusion equation is studied. It is shown that, under certain circumstances, the effect of small scalar diffusivity is to smooth the zero-diffusivity solution by averaging local fine-scaled structure against a Gaussian. The method of study depends on shadowing and thus fails for nonuniformly stretching systems, its failure suggesting the ways in which the effects of asymptotically small molecular diffusion can become nonlocal in chaotic fluid flows.

A particle method for a scalar advection diffusion equation

A particle method for a scalar advection diffusion equation

A new finite element formulation for computational fluid dynamics: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems

A discontinuity-capturing operator is developed for the ‘streamline’ formulation of advective-diffusive systems extending previous work on the scalar advection-diffusion equation. The operator provides a mechanism for exerting control over strong gradients in the discrete solution which appear, for example, in boundary and interior layers.

A new finite element formulation for computational fluid dynamics: II. Beyond SUPG

A discontinuity-capturing term is added to the streamline-upwind/Petrov-Galerkin weighting function for the scalar advection-diffusion equation. The additional term enhances the ability of the method to produce smooth yet crisp approximations to internal and boundary layers.

MULTIGRID TECHNIQUES FOR THE SOLUTION OF THE PASSIVE SCALAR ADVECTION-DIFFUSION EQUATION

The solution of elliptic passive scalar advection-diffusion equations is required in the analysis of many turbulent flow and convective heat transfer problems. The accuracy of the solution may be affected by the presence of regions containing large gradients of the dependent variables. The multigrid concept of local grid refinement is a method for improving the accuracy of the calculations in these problems. In combination with the multilevel acceleration techniques, an accurate and efficient computational procedure is developed. In addition, a robust implementation of the QUICK finite-difference scheme is described. Calculations of a test problem are presented to quantitatively demonstrate the advantages of the multilevel-multigrid method.