Jahrul M. Alam

Simultaneous space-time adaptive wavelet solution of nonlinear parabolic differential equations

Dynamically adaptive numerical methods have been developed to efficiently solve differential equations whose solutions are intermittent in both space and time. These methods combine an adjustable time step with a spatial grid that adapts to spatial intermittency at a fixed time. The same time step is used for all spatial locations and all scales: this approach clearly does not fully exploit space-time intermittency. We propose an adaptive wavelet collocation method for solving highly intermittent problems (e.g. turbulence) on a simultaneous space-time computational domain which naturally adapts both the space and time resolution to match the solution. Besides generating a near optimal grid for the full space-time solution, this approach also allows the global time integration error to be controlled. The efficiency and accuracy of the method is demonstrated by applying it to several highly intermittent (1D+t)-dimensional and (2D+t)-dimensional test problems. In particular, we found that the space-time method uses roughly 18 times fewer space-time grid points and is roughly 4 times faster than a dynamically adaptive explicit time marching method, while achieving similar global accuracy.

Scaling of space-time modes with Reynolds number in two-dimensional turbulence

is bounded by Re 4/3 for forced turbulence and Re for decaying turbulence. The usual computational estimate of the number of space–time modes required to calculated decaying two-dimensional turbulence is N∼ Re 3/2 . These bounds neglect intermittency, and it is not known how sharp they are. In this paper we use an adaptive multi-scale wavelet collocation method to estimate for the first time the number of space–time computational modes N necessary to represent two-dimensional decaying turbulence as a function of Reynolds number. We find that N∼ Re 0.9 for 1260 Re 40 400 over many eddy turn-over times, and that temporal intermittency is stronger than spatial intermittency. The spatial modes alone scale like Re 0.7 .T heβ-model then implies that the spatial fractal dimension of the active regions is 1.2, and the temporal fractal dimension is 0.3. These results suggest that the usual estimates are not sharp for adaptive numerical simulations. The relatively high compression confirms the importance of intermittency and encourages the search for reduced mathematical models of two-dimensional turbulence (e.g. in terms of coherent vortices).

A computational methodology for two‐dimensional fluid flows

SUMMARY A weighted residual collocation methodology for simulating two-dimensional shear-driven and natural convection flows has been presented. Using a dyadic mesh refinement, the methodology generates a basis and a multiresolution scheme to approximate a fluid flow. To extend the benefits of the dyadic mesh refinement approach to the field of computational fluid dynamics, this article has studied an iterative interpolation scheme for the construction and differentiation of a basis function in a two-dimensional mesh that is a finite collection of rectangular elements. We have verified that, on a given mesh, the discretization error is controlled by the order of the basis function. The potential of this novel technique has been demonstrated with some representative examples of the Poisson equation. We have also verified the technique with a dynamical core of a two-dimensional flow in primitive variables. An excellent result has been observed—on resolving a shear layer and on the conservation of the potential and the kinetic energies—with respect to previously reported benchmark simulations. In particular, the shear-driven simulation at CFL = 2.5 (Courant–Friedrichs–Lewy) and Re=1000 (Reynolds number) exhibits a linear speed up of CPU time with an increase of the time step, Δt. For the natural convection flow, the conversion of the potential energy to the kinetic energy and the conservation of total energy is resolved by the proposed method. The computed streamlines and the velocity fields have been demonstrated. Copyright

Toward a Multiscale Approach for Computational Atmospheric Modeling

AbstractAtmospheric motions are generally characterized by a wide range of multiple length and time scales, and a numerical method must use a fine grid to resolve such a wide range of scales. Furthermore, a very fine grid requires an extremely small time step in order to keep explicit time integration schemes stable. Therefore, high-resolution meteorological simulations are very expensive.A novel multiscale modeling approach is, therefore, presented for simulating atmospheric flows. In this approach, a prognostic variable representing a highly intermittent multiscale feature is decomposed into a significant and a nonsignificant part using wavelets, where the significant part is represented by a small fraction of the wavelet modes. The proposed multiscale methodology has been verified for simulating three cases: Smolarkiewicz’s deformational flow model, warm thermals in a dry atmosphere, and the dynamics of a vortex pair with ambient stable stratification. Comparisons with benchmark simulations and with a ...

A multiscale modeling study for the convective mass transfer in a subsurface aquifer

Abstract Quantitative and realistic computer simulations of mass transfer associated with

A numerical study of two-phase miscible flow through porous media with a Lagrangian model:

\hbox {CO}_2

Experimental and Numerical Investigations of Bubble Dynamics in Porous and Non-Porous Media

CO2 disposal in subsurface aquifers is a challenging endeavor. This article has proposed a novel and efficient multiscale modeling framework, and has examined its potential to study the penetrative mass transfer in a