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Immersed Interface Methods For Stokes Flow With Elastic Boundaries Or Surface Tension
 SIAM J. Sci. Comput
"... . A second order accurate interface tracking method for the solution of incompressible Stokes flow problems with moving interfaces on a uniform Cartesian grid is presented. The interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids. The interf ..."
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Cited by 119 (14 self)
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. A second order accurate interface tracking method for the solution of incompressible Stokes flow problems with moving interfaces on a uniform Cartesian grid is presented. The interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids. The interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed. The Stokes equations are then discretized using the second order accurate finite difference methods for elliptic equations with singular sources developed in a previous paper (SIAM J. Numer. Anal., 31(1994), pp. 10191044). The resulting velocities are interpolated to the interface to determine the motion of the interface. An implicit quasiNewton method is developed that allows reasonable time steps to be used. Key words. Stokes flow, creeping flow, interface tracking, discontinuous coefficients, immersed interface methods, Cartesian grids, bubbles. AMS subject clas...
A phase field model for the mixture of two incompressible fluids and its approximation by a Fourierspectral method
 PHYSICA D
, 2003
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Balance and the Slow Quasimanifold: Some Explicit Results
 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57
, 1998
"... The ultimate limitations of the balance, slowmanifold, and potential vorticity inversion concepts are investigated. These limitations are associated with the weak but nonvanishing spontaneousadjustment emission, or Lighthill radiation, of inertia–gravity waves by unsteady, twodimensional or layer ..."
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Cited by 67 (7 self)
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The ultimate limitations of the balance, slowmanifold, and potential vorticity inversion concepts are investigated. These limitations are associated with the weak but nonvanishing spontaneousadjustment emission, or Lighthill radiation, of inertia–gravity waves by unsteady, twodimensional or layerwisetwodimensional vortical flow (the wave emission mechanism sometimes being called "geostrophic" adjustment even though it need not take the flow toward geostrophic balance). Spontaneousadjustment emission is studied in detail for the case of unbounded fplane shallowwater flow, in which the potential vorticity anomalies are confined to a finitesized region, but whose distribution within the region is otherwise completely general. The approach assumes that the Froude number F and Rossby number R satisfy F K 1 and R � 1 (implying, incidentally, that any balance would have to include gradient wind and other ageostrophic contributions). The method of matched asymptotic expansions is used to obtain a general mathematical description of spontaneousadjustment emission in this parameter regime. Expansions are carried out to O(F4), which is a high enough order to describe not only the weakly emitted waves but also, explicitly, the correspondingly weak radiation reaction upon the vortical flow, accounting for the loss of vortical energy. Exact evolution on a slow manifold, in its usual strict sense, would be incompatible with the arrow of time introduced by this radiation reaction and energy loss. The magnitude O(F4) of the radiation reaction may thus be taken to measure the degree of ‘‘fuzziness’ ’ of the entity that must exist in place of the strict slow manifold. That entity must, presumably, be not a simple invariant manifold, but rather an O(F4)thin, multileaved, fractal "stochastic layer" like those known for analogous but loworder coupled oscillator systems. It could more appropriately be called the "slow quasimanifold."
Benchmark spectral results on the liddriven cavity flow
 Comp. Fluids
, 1998
"... Abstract—Highlyaccurate solutions for the liddriven cavity flow are computed by a Chebyshev collocation method. Accuracy of the solution is achieved by using a substraction method of the leading terms of the asymptotic expansion of the solution of the Navier–Stokes equations in the vicinity of th ..."
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Cited by 65 (0 self)
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Abstract—Highlyaccurate solutions for the liddriven cavity flow are computed by a Chebyshev collocation method. Accuracy of the solution is achieved by using a substraction method of the leading terms of the asymptotic expansion of the solution of the Navier–Stokes equations in the vicinity of the corners, where the velocity is discontinuous. Critical comparison with former numerical experiments confirms the highaccuracy of the method, and extensive results for the flow at Reynolds number
The zero surface tension limit of twodimensional water waves
, 2005
"... We consider twodimensional water waves of infinite depth, periodic in the horizontal direction. It has been proven by Wu (in the slightly different nonperiodic setting) that solutions to this initial value problem exist in the absence of surface tension. Recently Ambrose has proven that solutions e ..."
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Cited by 62 (8 self)
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We consider twodimensional water waves of infinite depth, periodic in the horizontal direction. It has been proven by Wu (in the slightly different nonperiodic setting) that solutions to this initial value problem exist in the absence of surface tension. Recently Ambrose has proven that solutions exist when surface tension is taken into account. In this paper, we provide a shorter, more elementary proof of existence of solutions to the water wave initial value problem both with and without surface tension. Our proof requires estimating the growth of geometric quantities using a renormalized arc length parametrization of the free surface and using physical quantities related to the tangential velocity of the free surface. Using this formulation, we find that as surface tension goes to 0, the water wave without surface tension is the limit of the water wave with surface tension. Far from being a simple adaptation of previous works, our method requires a very original choice of variables; these variables turn out to be physical and well adapted to both cases.
Adaptive mesh and algorithm refinement using direct simulation Monte
, 1999
"... Adaptive mesh and algorithm refinement (AMAR) embeds a particle method within a continuum method at the finest level of an adaptive mesh refinement (AMR) hierarchy. The coupling between the particle region and the overlaying continuum grid is algorithmically equivalent to that between the fine and c ..."
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Cited by 53 (6 self)
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Adaptive mesh and algorithm refinement (AMAR) embeds a particle method within a continuum method at the finest level of an adaptive mesh refinement (AMR) hierarchy. The coupling between the particle region and the overlaying continuum grid is algorithmically equivalent to that between the fine and coarse levels of AMR. Direct simulation Monte Carlo (DSMC) is used as the particle algorithm embedded within a Godunovtype compressible Navier–Stokes solver. Several examples are presented and compared with purely continuum calculations. c ° 1999 Academic Press 1.
Polygonal Eyewalls, asymmetric eye contraction, and potential vorticity mixing in hurricanes
 J. Atmos. Sci
, 1999
"... Hurricane eyewalls are often observed to be nearly circular structures, but they are occasionally observed to take on distinctly polygonal shapes. The shapes range from triangles to hexagons and, while they are often incomplete, straight line segments can be identified. Other observations implicate ..."
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Cited by 50 (19 self)
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Hurricane eyewalls are often observed to be nearly circular structures, but they are occasionally observed to take on distinctly polygonal shapes. The shapes range from triangles to hexagons and, while they are often incomplete, straight line segments can be identified. Other observations implicate the existence of intense mesovortices within or near the eye region. Is there a relation between polygonal eyewalls and hurricane mesovortices? Are these phenomena just curiosities of the hurricane’s innercore circulation, or are they snapshots of an intrinsic mixing process within or near the eye that serves to determine the circulation and thermal structure of the eye? As a first step toward understanding the asymmetric vorticity dynamics of the hurricane’s eye and eyewall region, these issues are examined within the framework of an unforced barotropic nondivergent model. Polygonal eyewalls are shown to form as a result of barotropic instability near the radius of maximum winds. After reviewing linear theory, simulations with a highresolution pseudospectral numerical model are presented to follow the instabilities into their nonlinear regime. When the instabilities grow to finite amplitude, the vorticity of the eyewall region pools into discrete areas, creating the appearance of polygonal eyewalls. The circulations associated with these pools of vorticity suggest a connection to hurricane mesovortices. At later times the vorticity
Predictivecorrective incompressible sph
 In ACM SIGGRAPH 2009 papers, ACM
"... Copyright Notice Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profi t or direct commercial advantage and that copies show this notice on the fi rst page or initial scree ..."
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Cited by 49 (3 self)
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Copyright Notice Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profi t or direct commercial advantage and that copies show this notice on the fi rst page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specifi c permission and/or a fee. Permissions may be
Model reduction for compressible flows using POD and Galerkin projection,
 Physica D: Nonlinear Phenomena
, 2004
"... Abstract We present a framework for applying the method of Proper Orthogonal Decomposition (POD) and Galerkin projection to compressible fluids. For incompressible flows, only the kinematic variables are important, and the techniques are well known. In a compressible flow, both the kinematic and th ..."
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Cited by 45 (0 self)
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Abstract We present a framework for applying the method of Proper Orthogonal Decomposition (POD) and Galerkin projection to compressible fluids. For incompressible flows, only the kinematic variables are important, and the techniques are well known. In a compressible flow, both the kinematic and thermodynamic variables are dynamically important, and must be included in the configuration space. We introduce an energybased inner product which may be used to obtain POD modes for this configuration space. We then obtain an approximate version of the NavierStokes equations, valid for cold flows at moderate Mach number, and project these equations onto a POD basis. The resulting equations of motion are quadratic, and are much simpler than projections of the full compressible NavierStokes equations.
Tsunami generation by submarine mass failure. I: Wavemaker models
, 2000
"... Numerical experiments are conducted for twodimensional model problems of tsunami generation by underwater slides and slumps. These problems consist of a semiellipse, translating down an incline with a prescribed center of mass motion. For each problem, this motion is derived as a function of geome ..."
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Cited by 43 (18 self)
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Numerical experiments are conducted for twodimensional model problems of tsunami generation by underwater slides and slumps. These problems consist of a semiellipse, translating down an incline with a prescribed center of mass motion. For each problem, this motion is derived as a function of geometric and material parameters, and we calculate corresponding characteristic tsunami amplitudes. Landslide deformation is shown to have negligible effect on these amplitudes. Results of numerical experiments for the model problems are converted into predictive curve fits of characteristic tsunami amplitudes as functions of nondimensional governing parameters. Such results are useful for estimating tsunami hazard or for performing case studies. A method is proposed to input free surface displacements, modeled shortly after tsunami generation, into tsunami propagation models, in essence treating such submarine mass failures exactly like coseismic displacement.