Singular Saddle Point Research Articles

Radial Distributions of Coronal Electron Temperatures: Specificities of the DYN Model

This paper is a follow up of the article where Lemaire and Stegen (2016) introduced their DYN method to calculate coronal temperature profiles from given radial distributions of the coronal and solar wind (SW) electron densities. Several such temperature profiles are calculated and presented corresponding to a set of given empirical density models derived from eclipse observations and in-situ measurements of the electron density and bulk velocity at 1 AU. The DYN temperature profiles obtained for the equatorial and polar regions of the corona challenge the results deduced since 1958 from singular hydrodynamical models of the SW. In these models - where the expansion velocity transits through a singular saddle point - the maximum coronal temperature is predicted to be located at the base of the corona, while in all DYN models the altitude of the maximum temperature is found at significantly higher altitudes in the mid-corona. Furthermore, the maximum of the DYN-estimated temperatures is found at much higher altitudes over the polar regions and coronal holes, than over the equator. However, at low altitudes, in the inner corona, the DYN temperatures are always smaller at high latitudes, than at low equatorial latitudes. This appears well in agreement with existing coronal hole observations. These findings have serious implications on the open questions: what is the actual source of the coronal heating, and where is the maximum energy deposited within the solar corona?

A Fast Shift-Splitting Iteration Method for Nonsymmetric Saddle Point Problems

AbstractBased on the shift-splitting technique and the idea of Hermitian and skew-Hermitian splitting, a fast shift-splitting iteration method is proposed for solving nonsingular and singular nonsymmetric saddle point problems in this paper. Convergence and semi-convergence of the proposed iteration method for nonsingular and singular cases are carefully studied, respectively. Numerical experiments are implemented to demonstrate the feasibility and effectiveness of the proposed method.

Semi-convergence analysis of Uzawa methods for singular saddle point problems

Recently, Zheng, Bai and Yang studied the parameterized Uzawa method for solving singular saddle point problems (B. Zheng, Z.-Z. Bai, X. Yang, On semi-convergence of parameterized Uzawa methods for...

Experimental investigation and physical modelling of two-phase two-component flow in a converging-diverging nozzle

Two-phase critical flow modelling methodologies are briefly reviewed. When a local knowledge of the flow behaviour is required and the influence of the geometry on the outlet conditions is to be modelled, a two-phase variable slip model is necessary. This model requires numerical integration through a singular saddle point. A simple dispersed flow model is presented and integrated numerically. Predictions of this model are compared with data taken on an air-water flow loop capable of producing high gas quality (5–100%) flows of annular dispersed character. It is shown that critical flow solutions of this model have a topological structure qualitatively similar to that of compressible gas flow. However, quantitative deviations are shown, in particular a higher upstream/downstream pressure ratio is necessary to choke the flow. Furthermore, critical flow is experimentally shown to depend on the entrained liquid fraction at the inlet.

Calculation of hysteresis and flow-rate oscillations of base pressure in supersonic annular nozzles

The interaction of the turbulent axisymmetric near wake behind the face of the central body of an annular nozzle with the supersonic annular jet discharging from this nozzle is analyzed. The flow in the monoparametric near wake is calculated by the integral method [1] while the flow in the nonviscous jet is calculated by the method of through calculation using a monotonic explicit difference system of the first order of accuracy [2]. The interaction between the nonviscous and turbulent streams is determined by the displacement thickness of the wake. The initial conditions of the wake are determined from the integral conditions of attachment with the mixing flow in the isobaric base region. The interaction flow is described by the particular solution of the equations which passes through the singular saddle point — the throat of the wake. The near wake and base pressure in different modes of discharge from an annular nozzle at the exit cross section of which the ratio of outer and inner radii is y2/y1 = 1.3 and the Mach number is M = 2.54 are calculated as an example. The region of hysteresis of the base pressure, connected with the ambiguity of the interaction flow owing to the formation of the throat of the wake within the first or second barrel of the jet, and the parameters of the low-frequency flow-rate oscillations of base pressure in this region are determined. The results of the calculations are in satisfactory agreement with experimental data.