Modeling incompressible flows using a finite particle method

This paper describes the applications of a finite particle method (FPM) to modeling incompressible flow problems. FPM is a meshfree particle method in which the approximation of a field variable and its derivatives can be simultaneously obtained through solving a pointwise matrix equation. A set of basis functions is employed to obtain the coefficient matrix through a sequence of transformations. The finite particle method can be used to discretize the Navier–Stokes equation that governs fluid flows. The incompressible flows are modeled as slightly compressible via specially selected equations of state. Four numerical examples including the classic Poiseuille flow, Couette flow, shear driven cavity and a dam collapsing problem are presented with comparisons to other sources. The numerical examples demonstrate that FPM is a very attractive alternative for simulating incompressible flows, especially those with free surfaces, moving interfaces or deformable boundaries.     

On local ergodicity in hyperbolic systems with singularities

United Institute of Nuclear Research. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 27, No. 1, pp. 60–64, January–March, 1993.     

Numerical schemes for kinematic flows with discontinuous flux

Spatially one-dimensional kinematic flows arise in a series of applications including traffic flow and sedimentation. They lead to nonlinear systems of conservation law whose flux has an explicit “concentration times velocity” structure. A new family of simple numerical schemes which are adapted to this structure, and which handle fluxes that are discontinuous with respect to the space variable, is presented and in part analyzed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the search for singularities in incompressible flows

In these notes we give some examples of the interaction of mathematics with experiments and numerical simulations on the search for singularities.     

Spectral theory of some degenerate elliptic operators with local singularities

This paper is based on our previous results (Haroske and Skrzypczak (2008) [23], Haroske and Skrzypczak (in press) [25]) on compact embeddings of Muckenhoupt weighted function spaces of Besov and Triebel-Lizorkin type with example weights of polynomial growth near infinity and near some local singularity. Our approach also extends (Haroske and Triebel (1994) [21]) in various ways. We obtain eigenvalue estimates of degenerate pseudodifferential operators of type b₂○p(x,D)○b₁ where bi∈Lri(Rn,wi), wi∈A_∞, i=1,2, and , ?>0. Finally we deal with the ‘negative spectrum’ of some operator Hγ=A−γV for γ→∞, where the potential V may have singularities (in terms of Muckenhoupt weights), and A is a positive elliptic pseudodifferential operator of order ?>0, self-adjoint in L₂(Rn). This part essentially relies on the Birman-Schwinger principle. We conclude this paper with a number of examples, also comparing our results with preceding ones.     

Self-dual modules over local rings of curve singularities

Let C be a reduced curve singularity. C is called of finite self-dual type if there exist only finitely many isomorphism classes of indecomposable, self-dual, torsion-free modules over the local ring of C. In this paper it is shown that the singularities of finite self-dual type are those which dominate a simple plane singularity.     

Effects of manifolds and corner singularities on stretching in chaotic cavity flows

It has been assumed that the stretching field in chaotic flows evolves as the result of a random multiplicative process [F.J. Muzzio, C. Meneveau, P.D. Swanson and J.M. Ottino, Scaling and multifractal properties of mixing in chaotic flows, Phys. Fluids A, 4, 1439–1456, (1992); F. J. Muzzio, P.D. Swanson and J.M. Ottino, Partially mixed structures produced by multiplicative stretching in chaotic flows, Int. J. Bifurc. Chaos, 2, 37–50 (1992)]. This assumption has been used to derive an asymptotic scaling formalism of distributions of stretching values that has useful predictive capabilities. Deviations from this scaling were thought to be limited to cases with regular islands. However, as is shown in this paper for the chaotic cavity flow, deviations from the proposed scaling can also occur for globally chaotic flows as a result of the joint action of unstable manifolds of hyperbolic periodic point and of singularities at the corners of the cavity. A detailed examination of random multiplicative stretching, the conditions necessary for its validity, and the intensity of manifold interaction effects is performed here for the cavity flow.     

Diffusion-limited aggregation with a local particle drift

A simple model with a local particle drift is proposed to extend the diffusion-limited aggregation model. Here the attractive or the repulsive interaction between a flight particle and the aggregation cluster is taken into account. In our model the drift force depends on the local aggregation structure. The effects of the strength and the range of the particle-particle interaction on the fractal structure of the aggregations are elucidated from the scaling property of an integrated pair correlation function. From the present computer simulation results, it is found that while the attractive interaction tends to decrease the fractal dimension, the repulsive one increases it.     

Cyclic solutions in dynamical systems generated by local variational problems with objective-function singularities

The article considers cyclic systems generated by local variational problems with various singularities in the objective function. Conditions for the creation of cyclic solutions are examined. Translated from Nelineinaya Dinamika i Upravlenie, pp. 7–14, 1999.     

An elementary coordinate-dependent local resolution of singularities and applications

An elementary classical analysis resolution of singularities method is developed, extensively using explicit coordinate systems. The algorithm is designed to be applicable to subjects such as oscillatory integrals and critical integrability exponents. As one might expect, the trade-off for such an elementary method is a weaker theorem than Hironaka's work [H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero I, Ann. of Math. (2) 79 (1964) 109-203; H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero II, Ann. of Math. (2) 79 (1964) 205-326] or its subsequent simplications and extensions such as [E. Bierstone, P. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (2) (1997) 207-302; S. Encinas, O. Villamayor, Good points and constructive resolution of singularities, Acta Math. 181 (1) (1998) 109-158; J. Kollar, Resolution of singularities—Seattle lectures, preprint; A.N. Varchenko, Newton polyhedra and estimates of oscillatory integrals, Funct. Anal. Appl. 18 (3) (1976) 175-196]. Nonetheless the methods of this paper can be used to prove a variety of theorems of interest in analysis. As illustration, two consequences are given. First and most notably, a general theorem regarding the existence of critical integrability exponents are established. Secondly, a quick proof of a well-known inequality of Lojasiewicz [S. Lojasiewicz, Ensembles semi-analytiques, Inst. Hautes Études Sci., Bures-sur-Yvette, 1964] is given. The arguments here are substantially different from the general algorithms such as [H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero I, Ann. of Math. (2) 79 (1964) 109-203; H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero II, Ann. of Math. (2) 79 (1964) 205-326], or the elementary arguments of [E. Bierstone, P. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988) 5-42] and [H. Sussman, Real analytic desingularization and subanalytic sets: an elementary approach, Trans. Amer. Math. Soc. 317 (2) (1990) 417-461]. The methods here have as antecedents the earlier work of the author [M. Greenblatt, A direct resolution of singularities for functions of two variables with applications to analysis, J. Anal. Math. 92 (2004) 233-257], Phong and Stein [D.H. Phong, E.M. Stein, The Newton polyhedron and oscillatory integral operators, Acta Math. 179 (1997) 107-152], and Varchenko [A.N. Varchenko, Newton polyhedra and estimates of oscillatory integrals, Funct. Anal. Appl. 18 (3) (1976) 175-196].     

On local inflexional instability in boundary-layer flows

Summary This paper describes a linearized theory of the two- and three-dimensional incompressible viscous flows ensuing from locally unstable velocity profiles. The theory is used to propose a hypothesis for the mechanisms governing formation of the turbulence wedge behind a fixed roughness element.

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Résumé On a présenté une théorie linearizée pour l'évolution des écoulements visqueux, incompressibles, bi-et tri-dimensionels, suivant l'instabilité locale des profils de vitesse de la couche limite laminaire. Utilisant les résultats de la théorie, on a proposé une hypothèse sur les mécanismes qui régissent la formation du coin de turbulence en aval d'un élément de rugosité fixée.

     

Constancy regions of mixed multiplier ideals in two‐dimensional local rings with rational singularities

The aim of this paper is to study mixed multiplier ideals associated with a tuple of ideals in a two‐dimensional local ring with a rational singularity. We are interested in the partition of the real positive orthant given by the regions where the mixed multiplier ideals are constant. In particular we reveal which information encoded in a mixed multiplier ideal determines its corresponding jumping wall and we provide an algorithm to compute all the constancy regions, and their corresponding mixed multiplier ideals, in any desired range.     

Modeling of building evacuation problems by network flows with side constraints

In this paper we model building evacuations by network flows with side constraints. Side constraints come from variable arc capacities on some arcs which are functions of flows in incident arcs. In this context we study maximum flow, minimum cost, and minimax objectives. For some special structured networks we propose ‘greedy’ algorithms for solving these problems. For more general network structures, solution procedures are recommended which take advantage of the network structures of the problems.     

Modeling of flows over inclined wavy bottoms

We study the flow of an incompressible liquid film down a wavy incline. Starting from the Navier–Stokes equations we derive an integral boundary layer equation (IBLe) by applying a Galerkin method with a single test and ansatz function. In comparison with older models this approach has several advantages. First, a linear stability analysis for stationary solutions yields a critical Reynolds number which corresponds in the limit of a flat incline with results directly obtained by the Navier–Stokes equations. Second, the IBLe is consistent with the pertinent Benney equation. Finally, the velocity profile is not assumed to be parabolic, so that also parameter regimes with recirculations can be modelled. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A decoupled finite particle method for modeling incompressible flows with free surfaces

Smoothed particle hydrodynamics (SPH) is a meshfree Lagrangian particle method, and it has been applied to different areas in engineering and sciences. One concern of the conventional SPH is its low accuracy due to particle inconsistency, which hinders the further methodology development. The finite particle method (FPM) restores the particle consistency in the conventional SPH and thus significantly improves the computational accuracy. However, as pointwise corrective matrix inversion is necessary, FPM may encounter instability problems for highly disordered particle distribution. In this paper, through Taylor series analyses with integration approximation and assuming diagonal dominance of the resultant corrective matrix, a new meshfree particle approximation method, decoupled FPM (DFPM), is developed. DFPM is a corrective SPH method, and is flexible, cost-effective and easy in coding with better computational accuracy. It is very attractive for modeling problems with extremely disordered particle distribution as no matrix inversion is required. One- and two-dimensional numerical tests with different kernel functions, smoothing lengths and particle distributions are conducted. It is demonstrated that DFPM has much better accuracy than conventional SPH, while particle distribution and the selection of smoothing function and smoothing length have little influence on DFPM simulation results. DFPM is further applied to model incompressible flows including Poiseuille flow, Couette flow, shear cavity and liquid sloshing. It is shown that DFPM is as accurate as FPM while as flexible as SPH, and it is very attractive in modeling incompressible flows with possible free surfaces.     

Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities

We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe equation with isolated singularities to higher order equations. Our approach uses blow up analysis for local integral equations, and is unified for all critical elliptic equations of order smaller than the dimension. We also prove the existence of Fowler solutions to the global equations, and establish a sup ? inf type Harnack inequality of Schoen for integral equations.