Caltech

A series of lectures

**11h00/11:00 am
Pavillon André-Aisenstadt, salle/Room 6214**

"*Multiscale Modeling and Computation of Flow and Transport
in Strongly Heterogeneous Porous Media.
Part I*"

**16h00/4:00 pm
Pavillon André-Aisenstadt, salle/Room 6214
**

"

in Strongly Heterogeneous Porous Media.

Part II

Many problems of fundamental and practical importance contain multiple scale solutions. Composite materials, flow and transport in porous media, and turbulent flow are examples of this type. Direct numerical simulations of these multiscale problems are extremely difficult due to the wide range of length scales in the underlying physical problems. Another difficulty is the lack of scale separation in the multiscale problems. So traditional two-scale methods based on homogenization theory may not apply. In this talk, I will review some of our effort in developing multiscale methods for flow and transport in strongly heterogeneous porous media. In particular, we will describe the physical background, the need for upscaling, the mathematical challenges in reducing the modeling errors across scales, the derivation of microscopic boundary conditions for the local bases, and how to incorporate the long-range effects associated with some strongly channelized media. We will also address the challenging issue of upscaling the convection dominated transport. A new homogenization result will be presented which accounts for the memory effect of the transport equation. Finally, we will present a number of practical applications to demonstrate the robustness of the method. The examples include the multiscale computations of two-phase flow in heterogeneous media, convection dominated transport, the nonlinear Richards flow, and some recent benchmark tests, such as the SPE comparative solution project posed by the Society of Petroleum Engineering, where porous media have strong channelized structures.
Une réception suivra la seconde conférence au Salon Maurice-l'Abbé, Pavillon André-Aisenstadt
There will be a reception after the 2nd lecture in Salon Maurice-l'Abbé, Pavillon André-Aisenstadt |

**May 10 mai
**

**11h00/11:00 am
Pavillon André-Aisenstadt, salle/Room 6214
**

"

The understanding of scale interactions for the incompressible Euler and N avier-Stokes equations has been a major challenge. For high Reynolds number flows, the degrees of freedom are so high that it is almost impossible to resolve all small scales by direct numerical simulations. Deriving an effective equation for the large scale solution is very useful in engineering applications. On the other hand, the nonlinear and nonlocal nature of the Euler equations makes it difficult to perform multiscale analysis. One of the main challenges is to understand how small scales propagate in time and whether the multiscale structure in the initial data is preserved dynamically. In this lecture, we will present a systematic multiscale analysis for the 3D incompressible Euler equation with rapidly oscillating initial data. We first present a multiscale analysis based on the Lagrangian formulation. By using a Lagrangian description, we characterize the nonlinear convection of small scales exactly and turn a convection dominated transport problem into an elliptic problem for the stream function. Thus, homogenization theory for elliptic problems can be used to obtain a multiscale expansion for the stream function. At the end, we derive a coupled multiscale system for the flow map and the stream function, which can be solved uniquely. Based on our understanding in the Lagrangian formulation, we derive a similar multiscale analysis using the Eulerian formulation, which is easier to implement for computational purposes. Finally, we show how to apply our multiscale analysis to problems with continuous spectrum of scales, which is typical for high Reynolds number flows. Our multiscale analysis reveals some interesting structures of the Reynolds stress terms and provide a theoretical guidance in developing a systematic multiscale modeling of incompressible flow. Numerical results will be presented to demonstrate the accuracy and the robustness of the multiscale method. |

**16h00/4:00 pm
Pavillon André-Aisenstadt, salle/Room 6214
**

"*Geometric Properties and Non-Blowup of 3D Incompressible Euler Flow*"

Whether the 3D incompressible Euler equation can develop a finite time singularity from smooth initial data has been an outstanding open problem. It has been believed that a finite singularity of the 3D Euler equation could be the onset of turbulence. Here we review some existing computational and theoretical work on possible finite blow-up of the 3D Euler equation. Further, we show that there is a sharp relationship between the geometric properties of the vortex filament and the maximum vortex stretching. By exploring this local geometric property of the vorticity field, we have obtained a global existence of the 3D incompressible Euler equation provided that the normalized unit vorticity vector has certain mild regularity property in a very localized region containing the maximum vorticity. Our assumption on the local geometric regularity of the vorticity field seems consistent with recent numerical experiments. Further, we discuss how viscosity may help to prevent singularity formation and present a new result on the global existence of the viscous Boussinesq equations. |