Multiscale computation of isotropic homogeneous turbulent flow

Hou, Tom; Yang, Dinghui; Ran, Hongyu

doi:10.1090/conm/408/07690

“…The oscillatory part of θ ε in general could have order one contribution to the mean velocity of the incompressible Euler equation. In [65][66][67], Hou and Yang and co-workers have studied convection of microstructure of the 2-D and 3-D incompressible Euler equations using a new approach. They do not assume that the oscillation is propagated by the mean flow.…”

Section: Convection Of Microstructurementioning

confidence: 99%

Multiscale Computations for Flow and Transport in Porous Media

Hou

¹

2009

Series in Contemporary Applied Mathematics

View full text Add to dashboard Cite

Abstract. Many problems of fundamental and practical importance have multiple scale solutions. The direct numerical solution of multiple scale problems is difficult to obtain even with modern supercomputers. The major difficulty of direct solutions is the scale of computation. The ratio between the largest scale and the smallest scale could be as large as 10 5 in each space dimension. From an engineering perspective, it is often sufficient to predict the macroscopic properties of the multiple-scale systems, such as the effective conductivity, elastic moduli, permeability, and eddy diffusivity. Therefore, it is desirable to develop a method that captures the small scale effect on the large scales, but does not require resolving all the small scale features. This paper reviews some of the recent advances in developing systematic multiscale methods with particular emphasis on multiscale finite element methods with applications to flow and transport in heterogeneous porous media.

show abstract

“…In our recent work in [17,15,16], we analyzed the structure of the multiscale solution for 2D and 3D Euler equations from a different viewpoint. A key technique is to use a nested multiscale expansion to characterize the propagation of small scales.…”

Section: A Nested Multiscale Expansion For the 3d Euler Equationsmentioning

confidence: 99%

Multiscale Analysis and Computation for the Three-Dimensional Incompressible Navier–Stokes Equations

Hou¹,

Yang²,

Ran³

2008

Multiscale Model. Simul.

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper, we perform a systematic multiscale analysis for the three-dimensional incompressible Navier-Stokes equations with multiscale initial data. There are two main ingredients in our multiscale method. The first one is that we reparameterize the initial data in the Fourier space into a formal two-scale structure. The second one is the use of a nested multiscale expansion together with a multiscale phase function to characterize the propagation of the small-scale solution dynamically. By using these two techniques and performing a systematic multiscale analysis, we derive a multiscale model which couples the dynamics of the small-scale subgrid problem to the large-scale solution without a closure assumption or unknown parameters. Furthermore, we propose an adaptive multiscale computational method which has a complexity comparable to a dynamic Smagorinsky model. We demonstrate the accuracy of the multiscale model by comparing with direct numerical simulations for both two-and three-dimensional problems. In the two-dimensional case we consider decaying turbulence, while in the three-dimensional case we consider forced turbulence. Our numerical results show that our multiscale model not only captures the energy spectrum very accurately, it can also reproduce some of the important statistical properties that have been observed in experimental studies for fully developed turbulent flows.

show abstract

“…The oscillatory part of θ ε in general could have order one contribution to the mean velocity of the incompressible Euler equation. In [53][54][55], Hou and Yang and co-workers have studied convection of microstructure of the 2-D and 3-D incompressible Euler equations using a new approach. They do not assume that the oscillation is propagated by the mean flow.…”

Section: Convection Of Microstructurementioning

confidence: 99%

Multiscale Computations for Flow and Transport in Heterogeneous Media

Efendiev

¹

,

Hou

²

2008

Lecture Notes in Mathematics

View full text Add to dashboard Cite

Abstract. Many problems of fundamental and practical importance have multiple scale solutions. The direct numerical solution of multiple scale problems is difficult to obtain even with modern supercomputers. The major difficulty of direct solutions is the scale of computation. The ratio between the largest scale and the smallest scale could be as large as 10 5 in each space dimension. From an engineering perspective, it is often sufficient to predict the macroscopic properties of the multiple-scale systems, such as the effective conductivity, elastic moduli, permeability, and eddy diffusivity. Therefore, it is desirable to develop a method that captures the small scale effect on the large scales, but does not require resolving all the small scale features.The purpose of this lecture note is to review some recent advances in developing multiscale finite element (volume) methods for flow and transport in strongly heterogeneous porous media. Extra effort is made in developing a multiscale computational method that can be potentially used for practical multiscale for problems with a large range of nonseparable scales. Some recent theoretical and computational developments in designing global upscaling methods will be reviewed. The lectures can be roughly divided into 4 parts. In part 1, I will review some homogenization theory for elliptic and hyperbolic equations. This homogenization theory provides the critical guideline for designing effective multiscale methods. In part 2, I will review some recent developments of multiscale finite element (volume) methods. We also discuss the issue of upscaling one-phase, two-phase flows through heterogeneous porous media and the use of limited global information in multiscale finite element (volume) methods. In part 4, we will consider multiscale simulations of two-phase flow immiscible flows using a flow-based adaptive coordinate, and introduce a theoretical framework which enables us to perform global upscaling for heterogeneous media with long range connectivity.

show abstract

Multiscale computation of isotropic homogeneous turbulent flow

Cited by 3 publications

References 10 publications

Multiscale Computations for Flow and Transport in Porous Media

Multiscale Computations for Flow and Transport in Porous Media

Multiscale Analysis and Computation for the Three-Dimensional Incompressible Navier–Stokes Equations

Multiscale Computations for Flow and Transport in Heterogeneous Media

Contact Info

Product

Resources

About