We include Smagorinsky's algebraic eddy viscosity approach into the multiple-relaxation-time (MRT) lattice Boltzmann equation (LBE) for large-eddy simulations (LES) of turbulent flows. The main advantage of the MRT-LBE model over the popular lattice BGK model is a significant improvement of numerical stability which leads to a substantial reduction of oscillations in the pressure field, especially for turbulent flow simulations near the numerical stability limit. The MRT-LBE model for LES is validated with a benchmark case of a surface mounted cube in a channel at Re = 40 000. Our preliminary results agree well with experimental data.

Large-Eddy Simulations with a Multiple-Relaxation-Time LBE Model

The purpose of this paper is to extend Himmelberg's fixed point theorem replacing the usual convexity in topological vector spaces by an abstract topological notion of convexity which generalizes classical convexity as well as several metric convexity structures found in the literature. We prove the existence, under weak hypothese, of a fixed point for a compact approachable map and we provide sufficient conditions under which this result applies to maps whose values are convex in the abstract sense mentionned above.

Abstract Convexity and Fixed Points

We study spontaneous breakdown of chiral symmetry during the nonlinear evolution of the Tayler instability. We start with an initial steady state of zero helicity. Within linearized perturbation calculations, helical perturbations of this initial state have the same growth rate for either sign of helicity. Direct numerical simulations (DNS) of the fully nonlinear equations, however, show that an infinitesimal excess of one sign of helicity in the initial perturbation gives rise to a saturated helical state. We further show that this symmetry breaking can be described by weakly nonlinear finite-amplitude equations with undetermined coefficients which can be deduced solely from symmetry consideration. By fitting solutions of the amplitude equations to data from DNS, we further determine the coefficients of the amplitude equations.

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Breakdown of chiral symmetry during saturation of the Tayler instability.

A radially expanding liquid sheet of a finite radius can be formed by impacting a liquid jet on a circular disk [Savart, Ann. Chim. Phys. 54, 55 (1833); Ann. Chim. 54, 113 (1833)] or by impinging two jet heads on against each other [Savart, Ann. Chim. Phys. 55, 257 (1833)]. The breakup of the liquid sheet first observed by Savart at the outer rim of the sheet is explained from the point of view of absolute and convective instability. Whether the sheet is convectively or absolutely unstable depends on the local Weber number We=ρU2H/S, where ρ and S are, respectively, the liquid density and the surface tension, and H and U are, respectively, the local half-sheet thickness and the local average liquid velocity. It is shown that absolute instability occurs at We=1, and convective instability occurs in the region where We>1. In the radially expanding liquid sheet, H decreases inversely with the radial distance from the center of the sheet, but U remains constant. Thus, the local Weber number that is greater th...

Absolute and convective instability of a radially expanding liquid sheet

The effect of thickness anisotropy on the degenerate elliptical resonance modes of micro-hemispherical shell resonators (μHSRs) created using the thermal oxidation process is investigated. This anisotropy arises from the variation in wet thermal oxide growth according to the exposed crystal planes of the single-crystal-silicon hemispherical mold used to generate the μHSRs. It is shown that, despite the presence of thickness anisotropy, the degenerate resonance modes of oxide μHSRs can exhibit zero intrinsic frequency split depending on the particular resonance mode and symmetry of the thickness anisotropy imparted from the underlying silicon wafer. Measured results verified by simultaneous electrical excitation on the 0° and 45° axes demonstrate less than 94 Hz intrinsic m=3 frequency split for a 1240 μm oxide μHSR (limited by measurement conditions), which is to the authors' knowledge the smallest as-fabricated frequency split reported to date for any μHSR.

Effect of thickness anisotropy on degenerate modes in oxide micro-hemispherical shell resonators

This paper investigates the dynamics of a TCP system described by a first- order nonlinear delay differential equation. By analyzing the associated characteristic transcendental equation, it is shown that a Hopf bifurcation sequence occurs at the pos- itive equilibrium as the delay passes through a sequence of critical values. The explicit algorithms for determining the Hopf bifurcation direction and the stability of the bifur- cating periodic solutions are derived with the normal form theory and the center manifold theory. The global existence of periodic solutions is also established with the method of Wu (Wu, J. H. Symmetric functional differential equations and neural networks with mem- ory. Transactions of the American Mathematical Society 350(12), 4799-4838 (1998)).

Bin Han

Papers

Applied Mathematics and Mechanics