LINEAR INSTABILITY OF THE ANNULAR HAGEN-POISEUILLE FLOW WITH SMALL INNER RADIUS BY A CHEBYSHEV PSEUDOSPECTRAL METHOD

Abstract

A new eigenvalue search procedure using the Chebyshev pseudospectral technique has been developed for the solution of the Orr-Sommerfeld equations. It can deal accurately and flexibly with various velocity profiles. The axisymmetric linear instability characteristics oflhe annular Hagen-Poiseuille Bow have been studied with this procedure. Neutral curves were obtained for 1 > R = R₁/R₀ > 0.06 where R_l is the inner radius and R₀ is the outer radius. When R = 1 which is the plane Poiseuille flow, the corresponding critical Reynolds number is 5772.22 α_r; = 1.0209. As R decreases to 0.06, the critical Reynolds number increases to 2.6 × 10₆ and the wave speed decreases to 0.0573. The wavenumber first decreases gradually from a, = 1.0209 at R = 1.0 to α _r, = 0.90 at R = 0.3. and then it increases rapidly to α _r, = 1.73 at R = 0.06.     

A linear stability analysis of thermal convection in spherical shells with variable radial gravity based on the Tau-Chebyshev method

The onset of thermal convection in a non-rotating spherical shell is investigated using linear theory. The Tau-Chebyshev spectral method is used to integrate the linearized equations. We investigate the onset of thermal convection by considering two cases of the radial gravitational field (i) a local acceleration, acting radially inward, that is proportional to the distance from the center r, and (ii) a radial gravitational central force that is proportional to r⁻ⁿ. The former case has been widely analyzed in the literature, because it constitutes a simplified model that is usually used, in astrophysics and geophysics, and is studied here to validate the numerical method. The latter case was analyzed since the case n = 5 has been experimentally realized (by means of the dielectrophoretic effect) under microgravity condition, in the experimental container called GeoFlow, inside the International Space Station. Our study is aimed to clarify the role of (i) a radially inward central force (either proportional to r or to r⁻ⁿ), (ii) a base conductive temperature distribution provided by either a uniform heat source or an imposed temperature difference between outer and inner spheres, and (iii) the aspect ratio η (ratio of the radii of the inner and outer spheres), on the critical Rayleigh number. In all cases the surface of the spheres has been assumed to be rigid. The results obtained with the linear theory based on the Tau-Chebyshev spectral method are compared with those of the integration of the full non-linear equations solved by using the spectral element method. By using the Tau-Chebyshev method, we were able to explore new cases that have not been previously reported in the literature.     

Instability of the liquid equilibrium state in an ice-water system with account for radiation heating

The problem of the stability of the equilibrium state of a weakly compressible fluid bounded from below by a stationary rigid wall and from above by a heat-conducting finite-thickness wall is studied. The thermal state of the system is investigated with account for volume heat sources due to radiation. It is shown that the mechanical equilibrium of the system is unstable. The neutral curves are plotted and the critical Rayleigh numbers are found.     

Measurement of the time structure of a coherent synchrotron radiation burst in NewSUBARU

The time structure of a burst of coherent synchrotron radiation (CSR) emitted from a high-peak-current beam in the electron storage ring NewSUBARU has been investigated. The source of this CSR burst is fine time structure in a bunch produced by longitudinal beam instability. This burst is unstable but very easy to obtain, so it can be used for some experimental applications with appropriate averaging of data. With an averaging period of 10 ms, the fluctuation of the integrated power was about 10%.     

Realization of contact resolving approximate Riemann solvers for strong shock and expansion flows

The Harten–Lax–van Leer contact (HLLC) and Roe schemes are good approximate Riemann solvers that have the ability to resolve shock, contact, and rarefaction waves. However, they can produce spurious solutions, called shock instabilities, in the vicinity of strong shock. In strong expansion flows, the Roe scheme can admit nonphysical solutions such as expansion shock, and it sometimes fails. We carefully examined both schemes and propose simple methods to prevent such problems. High‐order accuracy is achieved using the weighted average flux (WAF) and MUSCL‐Hancock schemes. Using the WAF scheme, the HLLC and Roe schemes can be expressed in similar form. The HLLC and Roe schemes are tested against Quirk's test problems, and shock instability appears in both schemes. To remedy shock instability, we propose a control method of flux difference across the contact and shear waves. To catch shock waves, an appropriate pressure sensing function is defined. Using the proposed method, shock instabilities are successfully controlled. For the Roe scheme, a modified Harten–Hyman entropy fix method using Harten–Lax–van Leer‐type switching is suggested. A suitable criterion for switching is established, and the modified Roe scheme works successfully with the suggested method. Copyright © 2009 John Wiley & Sons, Ltd.     

Pattern formation in flame spread over thin solid fuels

The fingering char pattern emerging on the surface of thin cellulosic sheets burning against an oxidizing wind is discussed. Employing collocation-based averaging, the assumption of diffusive-thermal equilibrium, the strong temperature dependence of the reaction rate, and the strong disparity between the densities of the solid and gaseous phases, an elementary two-dimensional free-interface model for the flame spread is formulated. It is shown that the pattern-forming dynamics is functionally akin to the well-studied cellular instability occurring in low Lewis number premixed gas flames.     

Natural-convective instability of electrochemical systems: A review

Basic theoretical and experimental results of research into natural-convective instability of electrochemical systems are considered.     

扭摆器中的微束团不稳定性

用理论和数值的方法, 首次研究了扭摆器中相干同步辐射效应诱发的微束团不稳定性. 这一不稳定性只存在于小能散的情况, 并当电子能量或峰值磁场取特定值时, 不稳定度达到最大值. 在BFEL的扭摆器中, 可能有轻微的不稳定性存在; 在CTF的扭摆器中, 则不会有不稳定性发生.