A Framework for Modeling Subgrid Effects for Two-Phase Flows in Porous Media

In this paper, we study upscaling for two-phase flows in strongly heterogeneous porous media. Upscaling a hyperbolic convection equation is known to be very difficult due to the presence of nonlocal memory effects. Even for a linear hyperbolic equation with a shear velocity field, the upscaled equation involves a nonlocal history dependent diffusion term, which is not amenable to computation. By performing a systematic multiscale analysis, we derive coupled equations for the average and the fluctuations for the two-phase flow. The homogenized equations for the coupled system are obtained by projecting the fluctuations onto a suitable subspace. This projection corresponds exactly to averaging along streamlines of the flow. Convergence of the multiscale analysis is verified numerically. Moreover, we show how to apply this multiscale analysis to upscale two-phase flows in practical applications

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

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

Multiscale analysis in Lagrangian formulation for the 2-D incompressible Euler equation

We perform a systematic multiscale analysis for the 2-D incompressible Euler equation with rapidly oscillating initial data using a Lagrangian approach. The Lagrangian formulation enables us to capture the propagation of the multiscale solution in a natural way. By making an appropriate multiscale expansion in the vorticity-stream function formulation, we derive a well-posed homogenized equation for the Euler equation. Based on the multiscale analysis in the Lagrangian formulation, we also derive the corresponding multiscale analysis in the Eulerian formulation. Moreover, our multiscale analysis reveals some interesting structure for the Reynolds stress term, which provides a theoretical base for establishing systematic multiscale modeling of 2-D incompressible flow

Multiscale computation of isotropic homogeneous turbulent flow

In this article we perform a systematic multi-scale analysis and computation for incompressible Euler equations and Navier-Stokes Equations in both 2D and 3D. The initial condition for velocity field has multiple length scales. By reparameterizing them in the Fourier space, we can formally organize the initial condition into two scales with the fast scale component being periodic. By making an appropriate multiscale expansion for the velocity field, we show that the two-scale structure is preserved dynamically. Moreover, we derive a well-posed homogenized equation for the incompressible Euler equations in the Eulerian formulations. Numerical experiments are presented to demonstrate that the homogenized equations indeed capture the correct averaged solution of the incompressible Euler and Navier Stokes equations. Moreover, our multiscale analysis reveals some interesting structure for the Reynolds stress terms, which provides a theoretical base for establishing an effective LES type of model for incompressible fluid flows

Effects of Soil Moisture Content on Germination and Physiological Characteristics of Rice Seeds with Different Specific Gravity

Soil relative water content and seed plumpness have been shown to be the key factors affecting seed germination and seedling growth of rice under direct drought cropping. It remains to be determined whether seed germination and seedling growth of water-saving and drought-resistant rice (WDR) and conventional rice with the same proportion of rice seed have the same response to soil moisture changes. The purpose of this study was to investigate the seed germination and physiological characteristics of the rice cultivars Guangliangyou 1813 (GLY-1813,indica hybrid rice) and Hanyou 73 ((HY-73), WDR) with four different specific gravities (T1, T2, T3, and T4; the rice seeds were divided into four specific gravity levels by weight using saline water, the representative specific gravities were <1.0, 1.0–1.1, 1.1–1.2 and >1.2 kg m−3, respectively), at five soil moisture content gradients (soil relative water contents of 10–20%, 20–40%, 40–60%, 60–80%, and 80–100%), under dry direct seeding conditions. The results showed that GLY-1813 had a higher germination potential, germination and seedling emergence rates, greater root dry weight, seedling dry weight, root oxidation activity, and chlorophyll content, and lower malondialdehyde (MDA) content when the soil relative water content was 20–40% or 40–60%. Cultivar HY-73 had the highest germination rate and seedling physiological activity at 20–40% relative water content; its growth vigor was better than that of GLY-1813 at the same soil moisture level. In conclusion, the soil relative water content for seed germination of HY-73 was 20–40%, which was less than that of GLY-1813. When soil relative water content was sufficient for seed germination and growth, the higher the plumpness of the rice seed, the easier it was to resist the negative effects of an adverse growth environment

Multiscale Computation of Isotropic Homogeneous Turbulent Flow

Abstract. In this article we perform a systematic multi-scale analysis and computation for incompressible Euler equations and Navier-Stokes Equations in both 2D and 3D. The initial condition for velocity field has multiple length scales. By reparameterizing them in the Fourier space, we can formally organize the initial condition into two scales with the fast scale component being periodic. By making an appropriate multiscale expansion for the velocity field, we show that the two-scale structure is preserved dynamically. Moreover, we derive a well-posed homogenized equation for the incompressible Euler equations in the Eulerian formulations. Numerical experiments are presented to demonstrate that the homogenized equations indeed capture the correct averaged solution of the incompressible Euler and Navier Stokes equations. Moreover, our multiscale analysis reveals some interesting structure for the Reynolds stress terms, which provides a theoretical base for establishing an effective LES type of model for incompressible fluid flows. 1

A FRAMEWORK FOR MODELING SUBGRID EFFECTS FOR TWO-PHASE FLOWS IN POROUS MEDIA ∗

Abstract. In this paper, we study upscaling for two-phase flows in strongly heterogeneous porous media. Upscaling a hyperbolic convection equation is known to be very difficult due to the presence of nonlocal memory effects. Even for a linear hyperbolic equation with a shear velocity field, the upscaled equation involves a nonlocal history dependent diffusion term, which is not amenable to computation. By performing a systematic multiscale analysis, we derive coupled equations for the average and the fluctuations for the two-phase flow. The homogenized equations for the coupled system are obtained by projecting the fluctuations onto a suitable subspace. This projection corresponds exactly to averaging along streamlines of the flow. Convergence of the multiscale analysis is verified numerically. Moreover, we show how to apply this multiscale analysis to upscale two-phase flows in practical applications

MULTISCALE ANALYSIS AND COMPUTATION FOR THE 3D INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Abstract. In this paper, we perform a systematic multiscale analysis for the 3D 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 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 capture 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. Key words. Multiscale analysis, turbulence modeling, 3D Navier-Stokes equations. AMS subject classifications. Primary: 76M50, 76F05, Secondary: 76F65, 76M22 1. Introduction. We develop a systematic multiscale analysis for the 3D incompressible Navier-Stoke