Determination of mass flow characteristics of supersonic axisymmetric nozzle diaphragms in modeling variable duties of low-consumption turbines

В работе приводятся обобщающие зависимости коэффициентов расхода сопловых аппаратов со сверхзвуковыми осесимметричными соплами в широком диапазоне изменения определяющих геометрических и режимных параметров. Предложена двухпараметрическая функция, учитывающая влияние расположения сопел в сопловом аппарате и степени конфузорности дозвуковой части осесимметричного сопла на коэффициент расхода. Показано слабое влияние на коэффициент расхода относительного радиуса закругления стенки в узкой части сопла и относительной длины дозвуковой части сопла в области их оптимальных значений определенных по минимуму потерь кинетической энергии. Переменные режимы работы сопла учитываются зависимостью относительного коэффициента расхода в функции от числа Рейнольдса в критическом сечении сопла. Полученные в работе эмпирические зависимости позволяют использовать их при моделировании переменных режимов и многорежимной оптимизации малорасходных турбин. The research presents generalizing dependences of mass flow rates in supersonic axisymmetric nozzle diaphragms n a wide range of variation of the governing geometric and operating parameters. A two-parameter function is proposed that takes into account the influence of the location of the nozzles in the nozzle apparatus and the degree of compression of the flow of the subsonic part of the nozzle on the mass flow rate. It is shown that the relative radius of rounding of the nozzle wall in the vicinity of the throat section and the relative length of the subsonic part of the nozzle in the region of their optimal values determined by the minimum of kinetic energy losses have a weak effect on the flow rate. Variable duties of nozzle operation are taken into account by the dependence of the relative flow rate as a function of the Reynolds number in the throat of the nozzle. The empirical dependencies obtained in this work make it possible to use them in modeling variable modes and multi-mode optimization of low-consumption turbines.

Aerodynamic Design of a Laval Nozzle for Real Gas Using Hodograph Method

We propose an approach for the design of the subsonic part of plane and axisymmetric Laval nozzles for real gases. The proposed approach is based on the hodograph method and allows one to solve the inverse design problem directly. Real gas effects are taken into consideration using the chemical equilibrium model. We present nozzle contours computed with the proposed method for a stoichiometric methane-air mixture. Results confirm that real gas effects have a strong influence on the nozzle shape. The described method can be used in the design of nozzles for rocket engines and for high-enthalpy wind tunnels.

Subsonic structure and optically thick winds from Wolf–Rayet stars

Mass loss by stellar wind is a key agent in the evolution and spectroscopic appearance of massive main sequence and post-main sequence stars. In Wolf–Rayet stars the winds can be so dense and so optically thick that the photosphere appears in the highly supersonic part of the outflow, veiling the underlying subsonic part of the star, and leaving the initial acceleration of the wind inaccessible to observations. Here we investigate the conditions and the structure of the subsonic part of the outflow of Galactic Wolf–Rayet stars, in particular of the WNE subclass; our focus is on the conditions at the sonic point of their winds. We compute 1D hydrodynamic stellar structure models for massive helium stars adopting outer boundaries at the sonic point. We find that the outflows of our models are accelerated to supersonic velocities by the radiative force from opacity bumps either at temperatures of the order of 200 kK by the iron opacity bump or of the order of 50 kK by the helium-II opacity bump. For a given mass-loss rate, the diffusion approximation for radiative energy transport allows us to define the temperature gradient based purely on the local thermodynamic conditions. For a given mass-loss rate, this implies that the conditions in the subsonic part of the outflow are independent from the detailed physical conditions in the supersonic part. Stellar atmosphere calculations can therefore adopt our hydrodynamic models as ab initio input for the subsonic structure. The close proximity to the Eddington limit at the sonic point allows us to construct a sonic HR diagram, relating the sonic point temperature to the luminosity-to-mass ratio and the stellar mass-loss rate, thereby constraining the sonic point conditions, the subsonic structure, and the stellar wind mass-loss rates of WNE stars from observations. The minimum stellar wind mass-loss rate necessary to have the flow accelerated to supersonic velocities by the iron opacity bump is derived. A comparison of the observed parameters of Galactic WNE stars to this minimum mass-loss rate indicates that these stars have their winds launched to supersonic velocities by the radiation pressure arising from the iron opacity bump. Conversely, stellar models which do not show transonic flows from the iron opacity bump form low-density extended envelopes. We derive an analytic criterion for the appearance of envelope inflation and of a density inversion in the outer sub-photospheric layers.

Experimental and numerical investigation of an axisymmetric supersonic jet

A comprehensive study of a steady axisymmetric supersonic jet of CO2, including experiment, theory, and numerical calculation, is presented. The experimental part, based on high-sensitivity Raman spectroscopy mapping, provides absolute density and rotational temperature maps covering the significant regions of the jet: the zone of silence, barrel shock, Mach disk, and subsonic region beyond the Mach disk. The interpretation is based on the quasi-gasdynamic (QGD) system of equations, and its generalization (QGDR) considering the translational–rotational breakdown of thermal equilibrium. QGD and QGDR systems of equations are solved numerically in terms of a finite-difference algorithm with the steady state attained as the limit of a time-evolving process. Numerical results show a good global agreement with experiment, and provide information on those quantities not measured in the experiment, like velocity field, Mach numbers, and pressures. According to the calculation the subsonic part of the jet, downstream of the Mach disk, encloses a low-velocity recirculation vortex ring.

Exact solutions for flow of a perfect gas in a two-dimensional Laval nozzle

A family of exact solutions is found for the problem of steady irrotational isentropic shockfree transsonic flow of a perfect gas through a Laval nozzle in two dimensions. The hodograph method is used, whereby the position co-ordinates x , y are expressed in terms of the velocity variables; the expressions are infinite series in the subsonic part of the flow field, infinite integrals (analytic continuations of the series) in the supersonic part. An inversion is required to get the velocity as a function of position; in general, this requires detailed numerical calculations, but approximate formulae (62) are found for the neighbourhood of the sonic point on the axis.