Jian-Guo Liu - US grants

EFFICIENT NUMERICAL METHODS FOR LARGE REYNOLDS NUMBER UNSTEADY VISCOUS INCOMPRESSIBLE FLOWS Jian-Guo Liu Project Summary: The investigator proposes to develop and analyze efficient, accurate, and reliable numerical methods for unsteady viscous incompressible flows in the presence of boundaries with large Reynolds number, in hopes of simulating directly wall-generated turbulence. Efficient numerical computations of 3-D incompressible flow are currently far behind the practical needs. Indeed, even some very basic issues in formulating a numerical method are unsettled whenever boundaries are present, such as the correct numerical boundary conditions for the vorticity and efficient 3-D formulation. Some connections between different types of vorticity boundary formulations, such as local via global and the MAC scheme via Thom's formula, have already been found by this investigator and his collaborator Weinan E. With explicit treatment of the vorticity term and high-order (essentially) compact difference approach, they recently introduced a very efficient fourth order scheme in vorticity-stream function (vector) formulation, together with a complete convergence theory. The investigator proposes to continue developing efficient and accurate numerical methods for the three dimensional computation using both the vorticity-stream function and vorticity-velocity formulations. The investigator also plans to conduct a systematic study of statistical behavior and coherent structure of some 2-D and 3-D wall-bounded turbulence flows such as the driven cavity flow, backward-facing step flow, etc. and carry out a detailed numerical comparison with the homogeneous isotropic turbulence simulations. Extensive work has been done on direct simulation of isotropic turbulence in wall-free flows, much less work has involved wall-bounded turbulence. However, most turbulence is generated at the walls. Understanding turbulent flows is a grand challenge comp arable to other prominent scientific problems such as the large-scale structure of the universe and the nature of subatomic particles. In contrast to many of the other grand challenges, progress on the basic theory of turbulence translates nearly immediately into a wide range of engineering applications and technological advances that affect many aspects of everyday life. Direct numerical simulation for Navier-Stokes equation is an effective tool that complements experimental and theoretical investigation of turbulence. The investigator proposes to study two problems in the field: (1) Developing efficient, robust and reliable numerical methods for viscous incompressible flows in domains with solid boundaries, especially in the cases with large Reynolds number. (2) Using these methods to study statistical behavior and coherent structure of two and three dimensional wall-bounded turbulence via direct numerical simulations.

DMS-9805621 EFFICIENT NUMERICAL METHODS FOR VISCOUS INCOMPRESSIBLE FLOWS Jian-Guo Liu Project Summary This project will focus on continuing the development of efficient, accurate, high order finite difference and finite element methods for the unsteady viscous incompressible Navier-Stokes equations (NSE), with emphasis on finding efficient time stepping procedures and new formulations of the equations better suited to numerical computation. Some practical aspects of the methods, including extensions to flows in general 2D and 3D domains, and applications to more challenging physical problems, will be investigated. In the vorticity formulation, a new time-stepping procedure is used for high Reynolds number flows. For such flows the convection and viscous terms are treated explicitly. The stream function, and hence the velocity, is then evaluated from the vorticity via the kinematic equation. The key to the efficiency of the new time-stepping procedure is that the value of the vorticity on the boundary is obtained explicitly from the steam function without any iteration. This eliminates some traditional difficulties associated with the vorticity formulation. More akin to the primitive variable formulation, the investigator is using a new formulation of the NSE in the impulse density variable which differs from the velocity by a gauge transformation. The gauge freedom enables one to assign simple and specific boundary conditions for both the impulse and gauge fields, thus eliminating some traditional difficulties such as the pressure boundary condition. This new class of efficient numerical methods has already been used to study such real world problems as the investigation of the mechanism of drag reduction on airfoils at high velocities, as well as the development of severe storms in tropical latitudes. These methods provide an important tool that allow scientists and engineers to study related fluid problems in manufacturing and industry that were previous unsolvable with currently available numerical techniques. They represent a significant step forward in the efficient computation of solutions to such problems, and are naturally suited for implementation on high performance massively parallel computer architectures.

The investigator develops a class of efficient and accurate numerical
methods for the unsteady viscous incompressible Navier-Stokes
equations (NSE) based a new unconstrained formulation of NSE with
fully dissipation in contrast to the traditional formulation where the
Stokes operator is dissipative only in the divergence free
fields. This class of NSE solver is unconditionally stable with
explicit treatment of both pressure and convection terms. Moreover, in
this class of finite element methods, there is no requirement of the
so called inf-sup condition. The cost of solving NSE is greatly
reduced to solving a standard heat equation and a standard Poisson
equation at each time step for general three dimensional fluid problems.
The simplicity of the method also enable the PI to develop a class of
numerical methods for complex fluids such as magneto-hydrodynamics,
liquid crystal polymers, geodynamo, climate modeling, and large eddy
turbulence simulations;


Computational Fluid Dynamics has grown from a mathematical curiosity
to become an essential tool in almost every branch of fluid dynamics,
from aerospace propulsion to weather prediction and has received
extensive attention throughout the international community since the
advent of the digital computer. The accuracy and efficiency of the
proposed schemes will allow us to simulate general three-dimensional
time-dependent flows with a reasonable turn-over time. It is expected
that the proposed fast algorithms will become an important tool for
many scientists and engineers in numerous scientific and industrial
applications of current interest.

This award supports participation in the International Conference on Hyperbolic Problems: Theory, Numerics & Applications, to be held at the University of Maryland, College Park, in June 2008.

The conference, the twelfth meeting in a bi-annual series, brings together researchers, students, and practitioners with interest in the theoretical, computational, and applied aspects of hyperbolic time-dependent problems. Topics of the conference include the study of nonlinear wave patterns in multiple dimensions, passage from microscopic to macroscopic in models for particle dynamics, theory and simulation of interfaces, and application of transport theory in complex environments including scattering in random media, biological applications, and traffic flow.

The organizers encourage participation by junior researchers and members of groups underrepresented in the mathematical sciences. The meeting provides an excellent opportunity for junior and senior participants to exchange ideas and to learn about recent research in emerging areas of theory, application, and numerical simulation connected with nonlinear hyperbolic evolution equations. Conference proceedings will be published.

Conference web site: http://hyp2008.umd.edu

In many mathematical models of physical reality, coherent structures are formed and maintained by a balance of competing influences. On the one hand, focusing, concentration, or aggregation effects are typically produced by nonlinear mechanisms. These effects are often counterbalanced by other processes that disperse, defocus, fragment, or spread things out in some way. This proposal aims to develop several novel and useful mathematical tools for analyzing how such competing effects achieve dynamic balance. The particular models of aggregation and dispersion to be studied arise specifically in studies of: animal ecology, crowd dynamics, shape matching, hydrodynamics, and mass transportation. The mathematical lessons learned are expected to be fundamental, and contribute to a body of understanding that promises to be useful to researchers across a range of disciplines. Further, the investigators plan to be substantially engaged in training and interacting with students and young researchers, at summer schools, lecture series, and disseminating results at conferences, workshops, and seminars.

The proposed research focuses on the study of dynamic behavior in four areas strongly motivated by applications and the theory of partial differential equations. The first area considers a fundamental coagulation-fragmentation model without detailed balance, coming from Niwa's scaling analysis of a large body of empirical data on fish school size in the mid-ocean. The second area will develop metrics and geodesics for crowd-configuration paths and related hydrodynamic problems for shape distances proposed by image analysts. The third area focuses on the long-time dynamics and gradient structure in a new model of nonlocal dispersion and nonlinear concentration, related to fixed-point equations for solitary wave profiles. The final area considers random sticky particle dynamics, seeking to build on recent advances in PDE theory that tie sticky particle dynamics to singular solutions of conservation laws, and on related progress for random shock clustering. Real-world applications include the fields of animal ecology, image analysis, fluid dynamics, and stochastic interacting particle systems.

The mathematical structure of numerous important models of dynamic behavior in science and data analysis is fundamentally related with optimality. Fluid motions optimize kinetic energy over time. Many systems in physical and information sciences tend to maximize entropy. Deep learning algorithms are trained by optimizing parameters for clustering and classifying big data sets. This project will improve our mathematical understanding of optimality principles and dynamics in several models of substantial current interest to researchers in a number of disciplines. These range from fluid dynamics and network routing to statistical sampling and data science to aerosol physics and animal ecology. Optimal transport theory will be used in a novel way to model fluid mixture dynamics and understand how fluid surface singularities can form. Gradient descent techniques will be investigated to analyze and improve the convergence of high-dimensional statistical sampling and wave-shape computations. Novel dynamical phenomena in merging-splitting models of clustering will be sought in models relevant to aerosol particle growth in atmospheric dynamics and the sharing of information in financial markets. These investigations will stimulate young researchers and students to participate, and will lead to results to be disseminated at conferences, research institutes, seminars, and lecture series.

In particular, this project's research will focus on bringing ideas from variational analysis to bear upon several specific topics of current interest: (1) modeling how incompressible fluids may optimally mix through an entropy-regularized multi-marginal optimal transport formulation, which ought to make numerical computations feasible and may enable a precise characterization of optimal dynamic pathways; (2) demonstrating the formation of singularities on the surface of incompressible fluids in a scenario involving expansion from a corner, through a novel perturbation analysis of a simple geodesic flow; (3) establishing convergence of gradient-like flows to explain coherent-state formation and improve statistical sampling, by developing the use of Lojasiewicz estimates in infinite-dimensional nonlocal models; (4) identifying metastable states and nontrivial temporal dynamics in kinetic models of aggregation and breakup that lack a detailed-balance structure that would drive the syst em to equilibrium.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.