[
{
"id": "authors:9mn70-wz312",
"collection": "authors",
"collection_id": "9mn70-wz312",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170621-112302718",
"type": "book_section",
"title": "Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric Euler Equations",
"book_title": "Handbook of Mathematical Analysis in Mechanics of Viscous Fluids",
"author": [
{
"family_name": "Hou",
"given_name": "Thomas Y.",
"clpid": "Hou-T-Y"
},
{
"family_name": "Liu",
"given_name": "Pengfei",
"orcid": "0000-0002-6714-7387",
"clpid": "Liu-Pengfei"
}
],
"contributor": [
{
"family_name": "Giga",
"given_name": "Yoshikazu",
"clpid": "Giga-Yoshikazu"
},
{
"family_name": "Novotny",
"given_name": "Antonin",
"clpid": "Novotny-A"
}
],
"abstract": "Global regularity of the Euler equations in the three-dimensional (3D) setting is regarded as one of the most important open questions in mathematical fluid mechanics. In this work we consider two one-dimensional (1D) models approximating the dynamics of the 3D axisymmetric Euler equations on the solid boundary of a periodic cylinder, which are motivated by a potential finite-time singularity formation scenario proposed recently by Luo and Hou (PNAS 111(36):12968\u201312973, 2014), and numerically investigate the stability of the self-similar profiles in their singular solutions. We first review some recent existence results about the self-similar profiles for one model, and then derive the evolution equations of the spatial profiles in the singular solutions for both models through a dynamic rescaling formulation. We demonstrate the stability of the self-similar profiles by analyzing their discretized dynamics using linearization, and it is hoped that these computations can help to understand the potential singularity formation mechanism of the 3D Euler equations.",
"doi": "10.1007/978-3-319-10151-4_17-1",
"isbn": "978-3-319-10151-4",
"publisher": "Springer",
"place_of_publication": "Cham, Switzerland",
"publication_date": "2016",
"pages": "1-31"
},
{
"id": "authors:bqkrq-daq47",
"collection": "authors",
"collection_id": "bqkrq-daq47",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180808-083855888",
"type": "book_section",
"title": "Introduction to the Theory of Incompressible Inviscid Flows",
"book_title": "Nonlinear Conservation Laws, Fluid Systems and Related Topics",
"author": [
{
"family_name": "Hou",
"given_name": "Thomas Y.",
"clpid": "Hou-T-Y"
},
{
"family_name": "Yu",
"given_name": "Xinwei",
"clpid": "Yu-Xinwei"
}
],
"contributor": [
{
"family_name": "Chen",
"given_name": "Gui-Qiang",
"clpid": "Chen-Gui-Qiang"
},
{
"family_name": "Li",
"given_name": "Daqian",
"clpid": "Li-Daqian"
},
{
"family_name": "Liu",
"given_name": "Chun",
"clpid": "Liu-Chun"
}
],
"abstract": "In this chapter, we consider the 3D incompressible Euler equations. We present classical and recent results on the issue of global existence/finite time singularity. We also introduce the theories of lower dimensional model equations of the 3D Euler equations and the vortex patch problem.",
"doi": "10.1142/9789814273282_0001",
"isbn": "978-981-4273-27-5",
"publisher": "World Scientific",
"place_of_publication": "Beijing",
"publication_date": "2009-06",
"pages": "1-71"
},
{
"id": "authors:knh8q-0mp91",
"collection": "authors",
"collection_id": "knh8q-0mp91",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20100622-112319775",
"type": "book_section",
"title": "Multiscale computations for flow and transport in porous media",
"book_title": "Multiscale Phenomena in Complex Fluids: Modeling, Analysis and Numerical Simulations",
"author": [
{
"family_name": "Hou",
"given_name": "Thomas Y.",
"clpid": "Hou-T-Y"
}
],
"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 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. This manuscript is not intended to be a general survey paper on this topic. The discussion is limited by the scope of the lectures and expertise of the author.",
"isbn": "9789814273268",
"publisher": "Higher Education Press",
"place_of_publication": "Beijing",
"publication_date": "2009",
"pages": "175-285"
},
{
"id": "authors:vq7yd-gh803",
"collection": "authors",
"collection_id": "vq7yd-gh803",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:EFElnm08",
"type": "book_section",
"title": "Multiscale Computations for Flow and Transport in Heterogeneous Media",
"book_title": "Quantum Treansport: Modelling, Analysis and Asymptotics - Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, September 11\u201316, 2006",
"author": [
{
"family_name": "Efendiev",
"given_name": "Yalchin",
"orcid": "0000-0001-9626-303X",
"clpid": "Efendiev-Yalchin-R"
},
{
"family_name": "Hou",
"given_name": "Thomas Y.",
"clpid": "Hou-T-Y"
}
],
"contributor": [
{
"family_name": "Ben Abdallah",
"given_name": "N."
},
{
"family_name": "Frosali",
"given_name": "G."
}
],
"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 due to disparity of scales. From an engineering perspective, it is often sufficient to predict 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 (finite 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 four parts. In part 1, we review some homogenization theory for elliptic and hyperbolic equations. This homogenization theory provides a guideline for designing effective multiscale methods. In part 2, we review some recent developments of multiscale finite element (finite 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.",
"doi": "10.1007/978-3-540-79574-2_4",
"isbn": "978-3-540-79573-5",
"publisher": "Springer",
"place_of_publication": "New York, NY",
"publication_date": "2008-07-03",
"pages": "169-248"
},
{
"id": "authors:zt72s-9p019",
"collection": "authors",
"collection_id": "zt72s-9p019",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200127-092428769",
"type": "book_section",
"title": "Numerical Study of Nearly Singular Solutions of the 3-D Incompressible Euler Equations",
"book_title": "Mathematics and Computation, a Contemporary View",
"author": [
{
"family_name": "Hou",
"given_name": "Thomas Y.",
"clpid": "Hou-T-Y"
},
{
"family_name": "Li",
"given_name": "Ruo",
"clpid": "Li-Ruo"
}
],
"contributor": [
{
"family_name": "Munthe-Kaas",
"given_name": "Hans",
"clpid": "Munthe-Kaas-H"
},
{
"family_name": "Owren",
"given_name": "Brynjulf",
"clpid": "Owren-B"
}
],
"abstract": "In this paper, we perform a careful numerical study of nearly singular solutions of the 3D incompressible Euler equations with smooth initial data. We consider the interaction of two perturbed antiparallel vortex tubes which was previously investigated by Kerr in [16, 19]. In our numerical study, we use both the pseudo-spectral method with the 2/3 dealiasing rule and the pseudo-spectral method with a high order Fourier smoothing. Moreover, we perform a careful resolution study with grid points as large as 1,536 \u00d7 1,024 \u00d7 3,072 to demonstrate the convergence of both numerical methods. Our computational results show that the maximum vorticity does not grow faster than doubly exponential in time while the velocity field remains bounded up to T = 19, beyond the singularity time T = 18.7 reported by Kerr in [16, 19]. The local geometric regularity of vortex lines near the region of maximum vorticity seems to play an important role in depleting the nonlinear vortex stretching dynamically.",
"doi": "10.1007/978-3-540-68850-1_3",
"isbn": "978-3-540-68848-8",
"publisher": "Springer",
"place_of_publication": "Berlin",
"publication_date": "2008",
"pages": "39-66"
},
{
"id": "authors:sy24r-ap695",
"collection": "authors",
"collection_id": "sy24r-ap695",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190820-155926497",
"type": "book_section",
"title": "A Relay-Zone Technique for Computing Dynamic Dislocations",
"book_title": "Computational Mechanics",
"author": [
{
"family_name": "Tang",
"given_name": "S. Q.",
"clpid": "Tang-S-Q"
},
{
"family_name": "Liu",
"given_name": "W. K.",
"clpid": "Liu-Wing-Kam"
},
{
"family_name": "Karpov",
"given_name": "E. G.",
"clpid": "Karpov-E-G"
},
{
"family_name": "Hou",
"given_name": "T. Y.",
"clpid": "Hou-T-Y"
}
],
"contributor": [
{
"family_name": "Yao",
"given_name": "Z. H.",
"clpid": "Yao-Z-H"
},
{
"family_name": "Yuan",
"given_name": "M. W.",
"clpid": "Yuan-M-W"
}
],
"abstract": "We propose a multiscale method for simulating solids with moving dislocations. Away from atomistic subdomains where the atomistic dynamics are fully resolved, a dislocation is represented by a localized jump profile, superposed on a defect-free field. We assign a thin relay zone around an atomistic subdomain to detect the dislocation profile and its propagation speed at a selected relay time. The detection technique utilizes a lattice time history integral treatment. After the relay, an atomistic computation is performed only for the defect-free field. The method allows one to effectively absorb the fine scale fluctuations and the dynamic dislocations at the interface between the atomistic and continuum domains. In the surrounding region, a coarse grid computation is adequate. \n\nWe illustrate the algorithm for a 1D Frenkel-Kontorova model at finite temperature. By comparison of the numerical results in the following figure, the reflection is absorbed by the proposed relay-zone technique.",
"doi": "10.1007/978-3-540-75999-7_28",
"isbn": "978-3-540-75998-0",
"publisher": "Springer",
"place_of_publication": "Berlin",
"publication_date": "2007",
"pages": "228-228"
},
{
"id": "authors:w0cmf-d0020",
"collection": "authors",
"collection_id": "w0cmf-d0020",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110602-094821767",
"type": "book_section",
"title": "Multiscale computation of isotropic homogeneous turbulent\n flow",
"book_title": "Inverse problems, multi-scale analysis, and effective medium theory",
"author": [
{
"family_name": "Hou",
"given_name": "Tom",
"clpid": "Hou-T-Y"
},
{
"family_name": "Yang",
"given_name": "Danping",
"clpid": "Yang-Danping"
},
{
"family_name": "Ran",
"given_name": "Hongyu",
"clpid": "Ran-Hongyu"
}
],
"contributor": [
{
"family_name": "Ammari",
"given_name": "Habib",
"clpid": "Ammari-H"
},
{
"family_name": "Kang",
"given_name": "Hyeonbae",
"clpid": "Kang-Hyeonbae"
}
],
"abstract": "In this article we perform a systematic multi-scale analysis and\ncomputation for incompressible Euler equations and Navier-Stokes Equations\nin both 2D and 3D. The initial condition for velocity field has multiple length\nscales. By reparameterizing them in the Fourier space, we can formally organize\nthe initial condition into two scales with the fast scale component being\nperiodic. By making an appropriate multiscale expansion for the velocity field,\nwe show that the two-scale structure is preserved dynamically. Moreover, we\nderive a well-posed homogenized equation for the incompressible Euler equations\nin the Eulerian formulations. Numerical experiments are presented to\ndemonstrate that the homogenized equations indeed capture the correct averaged\nsolution of the incompressible Euler and Navier Stokes equations. Moreover,\nour multiscale analysis reveals some interesting structure for the Reynolds\nstress terms, which provides a theoretical base for establishing an effective LES\ntype of model for incompressible fluid flows.",
"isbn": "0-8218-3968-3",
"publisher": "American Mathematical Society",
"place_of_publication": "Providence, RI",
"publication_date": "2006",
"pages": "111-135"
},
{
"id": "authors:d2egm-3fc19",
"collection": "authors",
"collection_id": "d2egm-3fc19",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200226-133732456",
"type": "book_section",
"title": "Coastal hydrodynamics of ocean waves on beach",
"book_title": "Advances in Applied Mechanics",
"author": [
{
"family_name": "Zhang",
"given_name": "Jin E.",
"clpid": "Zhang-Jin-E"
},
{
"family_name": "Wu",
"given_name": "Theodore Y.",
"clpid": "Wu-T-Y-T"
},
{
"family_name": "Hou",
"given_name": "Thomas Y.",
"clpid": "Hou-T-Y"
}
],
"contributor": [
{
"family_name": "van der Giesen",
"given_name": "Erik",
"clpid": "van-der-Giesen-E"
},
{
"family_name": "Wu",
"given_name": "Theodore Y.",
"clpid": "Wu-T-Y-T"
}
],
"abstract": "This chapter describes the coastal hydrodynamics of ocean waves on beach. A comprehensive study on modeling three-dimensional ocean waves coming from an open ocean of uniform depth and obliquely incident on beach with arbitrary offshore slope distribution, while evolving under balanced effects of nonlinearity and dispersion is presented. A family of beach configurations that is uniform in the long-shore direction as a first approximation for beaches with negligible long-shore curvature is considered. The beach slope variation is assumed to have such distributions that the ocean waves will evolve on beach without breaking. The overall approach adopted begins with development of a three-dimensional linear shallow-water wave theory, followed by taking, step by step, the nonlinear and dispersive effects into account. The linear theory is shown to provide a fundamental solution involving a central function, called the beach-wave function that delineates the evolution of the incoming train of simple waves during interaction with any beach belonging to this broad family of beach configurations. This linear theory can easily afford to cover such factors as oblique wave incidence, arbitrary distribution of offshore beach slope, and wavelength variations with respect to beach breadth.",
"doi": "10.1016/s0065-2156(00)80005-8",
"isbn": "9780120020379",
"publisher": "Elsevier",
"place_of_publication": "New York, NY",
"publication_date": "2001",
"pages": "89-165"
}
]