Local Ratio | ScienceGate

The general problem studied is the propagation of an oblique shock wave through a two-dimensional, steady, non-uniform oncoming flow. A higher-order theory is developed to treat the refraction of the incident oblique shock wave by irrotational or rotational disturbances of arbitrary amplitude provided the flow is supersonic behind the shock. A unique feature of the analysis is the formulation of the flow equations on the downstream side of the shock wave. It is shown that the cumulative effect of the downstream wave interactions on the propagation of the shock wave can be accounted for exactly by a single parameter Φ, the local ratio of the pressure gradients along the Mach wave characteristic directions at the rear of the shock front. The general shock refraction problem is then reduced to a single non-linear differential equation for the local shock turning angle θ as a function of upstream conditions and an unknown wave interaction parameter Φ. To lowest order in the expansion variable θΦ, this equation is equivalent to Whitham's (1958) approximate characteristic rule for the propagation of shock waves in non-uniform flow. While some further insight into the accuracy of Whitham's rule does emerge, the theory is not a selfcontained rational approach, since some knowledge of the wave interaction parameter Φ must be assumed. Analytical and numerical solutions to the basic shock refraction relation are presented for a broad range of flows in which the principal interaction occurs with disturbances generated upstream of the shock. These solutions include the passage of a weak oblique shock wave through: a supersonic shear layer, a converging or diverging flow, a pure pressure disturbance, Prandtl–-Meyer expansions of the same and opposite family, an isentropic non-simple wave region, and a constant pressure rotational flow. The comparison between analytic and numerical results is very satisfactory.

Download Full-text

Loss Mechanisms of Interplatform Steps in a 1.5-Stage Axial Flow Turbine

Journal of Turbomachinery ◽

10.1115/1.4034848 ◽

2016 ◽

Vol 139 (3) ◽

Cited By ~ 1

Author(s):

Robert Kluxen ◽

Stephan Behre ◽

Peter Jeschke ◽

Yavuz Guendogdu

Keyword(s):

Entropy Production ◽

Axial Flow ◽

Suction Side ◽

Step Height ◽

Transitional Flow ◽

Local Flow ◽

Flow Capacity ◽

Local Ratio ◽

Stator Vane ◽

Recirculation Zones

In this paper, the detailed steady and unsteady numerical investigations of a 1.5-stage axial flow turbine are conducted to determine the specific influence of interplatform steps in the first stator—as caused by deviations in manufacturing or assembly. A basic first stator design and a design consisting of a bow and endwall contours are compared. Apart from step height, the position and geometry of the interplatform border are varied for the basic design. To create the steps, every third stator vane was elevated, together with its platforms at hub and shroud, such that the flow capacity is only little affected. The results show that the effects of steps on the platform borders in front and aft of the first stator can be decoupled from those occurring on the interplatform steps. For the latter, being the main contributor to the additional loss, the intensity of recirculation zones and losses increase substantially when the platform border is located close to the suction side. Using a relative step height of 1.82% span, the entropy production doubles when compared to a position close to the pressure side, which can be explained by differences in local flow velocity level. Regarding a circular-arc-shaped platform, the losses can be more than halved—mainly due to lower included angles between step and endwall flow streamlines. The findings can be explained by a nondimensional relation of the local entropy production using local values for step height and characteristic flow quantities. Furthermore, a reduction in step height leads to an attenuation of the otherwise linear relationship between step height and entropy production, which is mainly due to lower local ratio of step height and boundary layer thickness. In the case of laminar or transitional flow regions on the endwall, typical for turbine rigs with low inlet turbulence and low-pressure turbines under cruise conditions, the steps lead to immediate local flow transition and thus substantially different results.

Download Full-text

Geometric cues stabilise long-axis polarisation of PAR protein patterns in C. elegans

10.1101/451880 ◽

2018 ◽

Cited By ~ 3

Author(s):

Raphaela Geßele ◽

Jacob Halatek ◽

Laeschkir Würthner ◽

Erwin Frey

Keyword(s):

Membrane Surface ◽

Reaction Diffusion ◽

Protein Patterns ◽

C Elegans ◽

Reaction Diffusion Model ◽

Local Ratio ◽

Diffusive Fluxes ◽

Anterior Posterior ◽

Selection Of ◽

Anterior Posterior Axis

AbstractIn the Caenorhabditis elegans zygote, PAR protein patterns, driven by mutual anatagonism, determine the anterior-posterior axis and facilitate the redistribution of proteins for the first cell division. Yet, the factors that determine the selection of the polarity axis remain unclear. We present a reaction-diffusion model in realistic cell geometry, based on biomolecular reactions and accounting for the coupling between membrane and cytosolic dynamics. We find that the kinetics of the phosphorylation-dephosphorylation cycle of PARs and the diffusive protein fluxes from the cytosol towards the membrane are crucial for the robust selection of the anterior-posterior axis for polarisation. The local ratio of membrane surface to cytosolic volume is the main geometric cue that initiates pattern formation, while the choice of the long-axis for polarisation is largely determined by the length of the aPAR-pPAR interface, and mediated by processes that minimise the diffusive fluxes of PAR proteins between cytosol and membrane.

Download Full-text

Morphology, Stellar Kinematics and Dynamics of Barred Galaxies

Symposium - International Astronomical Union ◽

10.1017/s0074180900032800 ◽

1983 ◽

Vol 100 ◽

pp. 193-196

Author(s):

John Kormendy

Keyword(s):

Velocity Dispersion ◽

Rotation Velocity ◽

Stellar Kinematics ◽

Barred Galaxies ◽

Circular Velocity ◽

Local Ratio ◽

The Galaxy ◽

Stellar Velocity ◽

Random Motions ◽

Velocity Dispersions

A brief review is given of the morphology of barred galaxies, following Kormendy (1981, 1982). The features illustrated include bulges, bars, disks, lenses, and inner and outer rings.Most of the paper is devoted to a detailed discussion of the absorption-line velocity field of the prototypical SBO galaxy NGC 936. The stars in the bar region show systematic non-circular streaming motions, with average orbits which are elongated parallel to the bar. Beyond the end of the bar, the data are consistent with circular orbits. The bar region also shows large random motions: the velocity dispersion at one-half of the radius of the bar is 1/2–2/3 as large as the maximum circular velocity. The observed kinematics are qualitatively and quantitatively similar to the behavior of n-body models by Miller and Smith (1979) and by Hohl and Zang (1979). The galaxy and the models show similar radial dependences of simple dimensionless parameters that characterize the dynamics. These include the local ratio of rotation velocity to velocity dispersion, which measures the relative importance of the ordered and random motions discussed above. Also similar are the residual streaming motions (relative to the circular velocity) in a frame of reference rotating with the bar. Circulation is in the same direction as rotation in all galaxies studied to date. Thus, except for the fact that NGC 936 has a slightly larger velocity dispersion, both n-body models are good first-order approximations to bars. Thus bars are different from elliptical galaxies, which in general are also triaxial, but which rotate slowly. This study of NGC 936 will be published in Kormendy (1983).A brief discussion is given of the kinematics of lens components. In both barred and unbarred galaxies, the velocity dispersions in the inner parts of lenses are large. The ratio of rotational to random kinetic energy is ∼ 1/2 at 1/3–1/2 of the radius of the lens. This ratio then decreases to small values at the rim of the lens. Thus at least some kinds of disk components have large stellar velocity dispersions, even in unbarred galaxies.

Download Full-text