Transonic swirling flow in axisymmetric nozzles

Summary The transonic flow in axisymmetric choked nozzles is computed in the case of a radial distribution of tangential velocity. The flow configuration is obtained by means of a time-dependent technique. The swirling flow is achieved through a particular surface located at the inlet of the nozzle. The pressure distribution and the sonic line are presented for choked flows without or with swirling.

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Sommario Viene calcolato il flusso transonico in ugelli assialsimmetrici in condizione critica, nel caso di distribuzione radiale di velocità tangenziale.La soluzione è ottenuta numericamente tramite una tecnica instazionaria. Il flusso vorticoso viene creato attraverso una opportuna superficie all'ingresso dell'ugello. Vengono presentati i campi di pressione e la posizione della linea sonica per flussi critici, con e senza vortice.

     

Turbulent flow through bifurcated nozzles

A finite-element model has been used to study steady-state turbulent flow through bifurcated submerged-entry nozzles with oversized ports typical of those used in the continuous casting of steel. Both 2D and 3D simulations have been performed with the commercial code FIDAP, using the standard K–? turbulence model. Predicted velocities from 3D simulations compare reasonably with experimental measurements using a hot-wire anemometer conducted in a physical water model, where severe turbulent fluctuations are present. Results show that a 2D simulation can also capture the main flow characteristics of the jet existing the nozzle and requires two orders of magnitude less computer time than the 3D simulation. A model combining the nozzle and mould was set up to study the effect of the outlet boundary conditions of the nozzle on the jet characteristics. This modelling technique will assist in the design of submerged-entry nozzles, especially as applied to enhance steel quality in the continuous casting process. Further, the model will provide appropriate inlet boundary conditions for a separate numerical model of the mould.     

Compressible air flow through a collapsing liquid cavity

We present a multiscale approach to simulate the impact of a solid object on a liquid surface: upon impact a thin liquid sheet is thrown upwards all around the rim of the impactor while in its wake a large surface cavity forms. Under the influence of hydrostatic pressure the cavity immediately starts to collapse and eventually closes in a single point from which a thin, needle‐like jet is ejected. The existing numerical treatments of liquid impact either consider the surrounding air as an incompressible fluid or neglect air effects altogether. In contrast, our approach couples a boundary‐integral method for the liquid with a Roe scheme for the gas domain and is thus able to handle the fully compressible gas stream that is pushed out of the collapsing impact cavity. Taking into account that air compressibility is crucial, since, as we show in this work, the impact crater collapses so violently that the air flow through the cavity neck attains supersonic velocities already at cavity diameters larger than 1 mm. Our computational results are validated through corresponding experimental data. Copyright © 2010 John Wiley & Sons, Ltd.     

Small amplitude pressure wave in subsonic and supersonic bubble flows in convergent-divergent nozzles

This paper makes a theoretical analysis of the propagation phenomena of the small amplitude pressure wave in the subsonic and supersonic bubble flow with a velocity slip between bubble and liquid in the convergent-divergent nozzle. From an analysis of the time-mean flow, the nondimensional parameter $\text{m = {u}\text{2}\text{G}\text{·α(1 ? α)ρ}\text{l}\text{β(2 ? 1/S)/P·[αβS + (1 ? α)βS}{}^2\text{+ α(1 ? α)]}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ corresponds to Mach number is gasdynamics where u_G is the gas velocity, α: the void fraction, ρ_L: the liquid density, P: the pressure, S: the velocity ratio of the gas and liquid flows and β: the proportional constant for the virtual mass. From a theoretical analysis of the small disturbance field, it is clarified that the parameter m also plays an essential and important role as Mach number, although the propagation performance of the disturbance is very complicated compared with that in gasdynamics. It is also shown that the pressure waves are divided into four groups depending on the velocity ratio S. Two of them are rather realistic, but the other two are required of a further investigation in future.     

Experimental investigation of rarefied gas flow through rectangular slits and nozzles

The flow of a rarefied gas through rectangular configuratons of different geometries has been experimentally studied to determine their discharge coefficient characteristics. The configurations used are a set of sharp-edged slit orifices, a smooth converging nozzle and a tube. The range of the Reynolds number based on the throat conditions varied from 0.01 to 100. The equivalent Knudsen number range based on the upstream conditions and inlet diameter varied from 0.0521 to 2.521. The results for the smooth nozzle are compared with calculations using a numerical method with one-dimensional stream tube approximation based on integrated boundary layer equations. The slit and the tube results are compared with the experimental results of Sreekanth and Davis [1988].     

Compressible flow hot-wire calibration

 A resistance thermal anemometer operated in compressible flow is sensitive to density, velocity, and total temperature fluctuations. To address these effects, a relatively inexpensive compressible flow calibration facility has been developed and tested. Characteristics of a hot-film anemometer in compressible flow are determined and a correction technique to account for differences in mean density between the calibration facility and the flow under study is proposed. Received: 19 October 1995 / Accepted: 18 October 1996     

Compressible and confined vortex flow

The internal compressible flow of a thin vortex chamber was investigated experimentally by measuring the radial distribution of temperature and pressure, from which the velocity field was calculated. The bulk of the internal vortex was found to be described by u_θr^0.69 = constant. The total resistance of the vortex chamber to the flow was also investigated in the context of fluidic vortex diode behavior under conditions of compressible and choked flow. It was found that the vortex chamber choked at an upstream-to-downstream pressure ratio of about 6 and in doing so passed a mass flow rate of 28% of the equivalent one-dimensional ideal nozzle. The resistance of vortex chambers is known to be strongly influenced by the presence of reversed flow in the exit due to vortex breakdown. Schlieren photography of the swirling exhaust flow was used to show that, while vortex breakdown does occur, it can only do so after the flow has become subsonic downstream of the exit and cannot therefore influence the vortex chamber resistance.     

Jet characteristics in confined swirling flow

Jets in confined swirling flow are investigated in a facility where the swirling flow in the tube is produced by a vane-type swirler. The jet is located centrally in the swirler, and the diameter ratio of the tube to the jet is 14. Both the jet and the swirling flow are fully turbulent. Results show that the confined jet is highly dissipative in nature. Consequently, the flow in the tube does not resemble a free jet with axial pressure gradient. The presence of swirl increases the rate of dissipation and the jet decays even faster. A fairly isotropic turbulence field is observed in the confined swirling flow. However, the introduction of the jet does not significantly affect this behavior and near isotropy of the turbulence field is again observed at 30 jet diameters downstream.     

Spatial instability of a swirling flow

The instability of a swirling flow of an inviscidand incompressible fluid is studied on the assumption that the wavenumber k=k_r + ik_i of the disturbance is complex while its frequency ω is real. This implies that the disturbance grows with distance along the axis of the swirling flow, but it does not grow with time. The occurrence of such disturbance is called spatial instability, in contrast to the temporal instability, in which k is a real number and ω=ω_r + iω_i is a complex one. The results show that spatial instability analysis is a useful tool for the comprehensive understanding of the instability behaviours of a swirling flow.     

Convective heat transfer in a superimposed streaming and swirling flow through a cylindrical duct

Theoretical studies have been carried out to investigate the convective heat transfer coefficients at different locations in the entrance region of a cylindrical duct with combined axial and tangential entry of time-independent power-law fluids. Investigations have been performed with uniform heat flux and uniform wall temperature boundary conditions. Theoretical model uses integral approach of hydrodynamic and thermal boundary layer theory to establish a functional relationship of local Nusselt number (Nu _z ) with the pertinent input parameters such as generalised Reynolds number based on tangential velocity of injection , generalised Prandtl number based on inlet tangential velocity (Pr _G ), the ratio of axial-to-tangential velocity at the inlet to the duct (V _R), the flow behaviour index of the fluid (n) and the ratio of axial-distance-to-duct-diameter (z/D).

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Konvektive Wärmeübertragung bei einer Drallströmung in einem zylindrischen Rohr

Zusammenfassung Es werden theoretische Untersuchungen des lokalen konvektiven Wärmeüberganges im Eintrittsbereich eines zylindrischen Rohres mit kombiniertem axialen und tangentialen Eintritt eines Fluids mit zeitunabhängigem Verhalten nach dem Potenzgesetz vorgestellt. Randbedingungen waren dabei konstanter Wärmestrom und konstante Wandtemperatur. Das theoretische Modell verwendet eine integrale Näherung der hydrodynamischen und thermischen Grenzschichttheorie. Es folgt eine funktionale Beziehung zwischen der lokalen Nusseltzahl (Nu _z ) und den relevanten Eingangsparametern wie der verallgemeinerten Reynoldszahl und der verallgemeinerten Prandtlzahl (Pr_G), die mit der tangentialen Eintrittsgeschwindigkeit gebildet werden sowie dem Verhältnis der axialen und tangentialen Geschwindigkeiten am Rohreintritt (V_R), der Kennzahl des Strömungsverhaltens des Fluids (n) und dem Verhältnis von axialer Länge zu Rohrdurchmesser (z/D).

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Nomenclature A _E Area of the tangential entry ports - a ₁ Area of the axial entry port - C _p Specific heat of fluid - D Duct diameter - E A non-dimensional parameter defined by Eq. (10) - G A non-dimensional parameter defined by Eq. (10) - K Flow consistency index of the fluid - K _t Thermal conductivity of the fluid - L Length of the duct - Axial mass flux at the inlet plane - Tangential mass flux at the inlet plane - Nu _z Local Nusselt number, Nu_z=h_z D/K_t - n Flow behaviour index of the fluid - P Static pressure - Pr Prandtl number - Pr _G Generalised Prandtl number - r _I Radius of the duct - Generalised Reynolds number - r Radial distance from the axis - r _c Radius of forced vortex core - t Temperature of fluid at any location - t ₀ Free stream temperature of the fluid - t _w Wall temperature - u ₀ Uniform axial velocity at the inlet - V _r Radial velocity component - V _z Axial velocity component - V _Ø Tangential velocity component - z Distance along the axis of the duct - Thermal diffusivity - Hydrodynamic boundary layer thickness - _t Thermal boundary layer thickness - _w Defined by Eq. (31), _w= t_w – t₀ - Dynamic viscosity - Density - Circulation constant - Angular velocity in forced vortex core     

Starting of an axisymmetric convergent-divergent nozzle in a hypersonic flow

The starting of an axisymmetric convergent-divergent nozzle, with the result that supersonic flow is formed within almost the entire channel, is modeled, as applied to the hypersonic aerodynamic setup of the Institute of Mechanics of Moscow State University. A successful starting is realized when the nozzle is thrown in a uniform supersonic air flow at a fairly high Mach number. The steady flow structure is studied. It is numerically shown that in the convergent section of the channel there arises an oblique shock wave whose interaction with the nozzle axis leads to the formation of a reflected shock and a curvilinear Mach disk with a region of unsteady subsonic flow in the vicinity of the throat. The mathematical model is based on the two-dimensional Euler equations for axisymmetric gas flows.     

Heat transfer of gas-particle flow in a supersonic convergent-divergent nozzle

Summary Heat flux, wall heat transfer coefficients, and wall pressures are determined for high velocity flow of gas-solid mixtures in a converging-diverging nozzle. Flow separation accompanied with oblique shock formation occurs in the diverging section of the nozzle. The shock strength is reduced upon the addition of solid particles. The wall pressure in the convergent section of the nozzle appears unaffected by the presence of solid particles. In the divergent section, however, the wall pressure is slightly lowered. At the maximum ratio of solid to air flow used in the experiments (3.7) increases in the heat transfer rate of up to 20 and 50 percent are obtained in the convergent and separated (divergent) regions of the nozzle, respectively. Slightly larger increases in the wall heat transfer coefficients are also obtained. It is concluded that the wall heat flux and heat transfer coefficients are influenced strongly by the presence of disturbances upstream of the nozzle inlet.Nomenclature W _a air flow rate - W _s solids flow rate - x axial distance from nozzle entrance - L axial length of nozzle - specific heat ratio of fluid - A _e exit cross section of flow - A _* throat cross section of flow - P ₀ inlet pressure - P _s wall separation pressure - P _a ambient exhaust pressure - shock wave angle - shock wave deflection angle - M ₁ Mach number upstream of shock wave - Mach number normal to shock wave - q heat flux - k _f thermal conductivity of fluid - T _wi inside wall temperature - T _wo outside wall temperature - T _ad adiabatic wall temperature - h wall heat transfer coefficient - C nozzle constant - A local cross section of flow - c _p specific heat of fluid - Pr Prandtl number - viscosity of fluid - r _c throat radius of curvature - factor accounting for variation of and Units absolute temperature °R(ankine) °F+459.7 - conductivity 1 BTU (hr ft °F)^–1 4.137×10^–3 cal (s cm °C)^–1 - specific heat 1 BTU (1b °F)^–1 1 cal (g °C)^–1 - absolute pressure 1 psia 0.0680 atm Supported in part by aid provided by the UCLA Space Science Center (Grant NsG 236-62 Libby).Listed for readers not familiar with the units adopted in this paper (editor).     

Supersonic flow separation in planar nozzles

We present experimental results on separation of supersonic flow inside a convergent–divergent (CD) nozzle. The study is motivated by the occurrence of mixing enhancement outside CD nozzles operated at low pressure ratio. A novel apparatus allows investigation of many nozzle geometries with large optical access and measurement of wall and centerline pressures. The nozzle area ratio ranged from 1.0 to 1.6 and the pressure ratio ranged from 1.2 to 1.8. At the low end of these ranges, the shock is nearly straight. As the area ratio and pressure ratio increase, the shock acquires two lambda feet. Towards the high end of the ranges, one lambda foot is consistently larger than the other and flow separation occurs asymmetrically. Downstream of the shock, flow accelerates to supersonic speed and then recompresses. The shock is unsteady, however, there is no evidence of resonant tones. The separation shear layer on the side of the large lambda foot exhibits intense instability that grows into large eddies near the nozzle exit. Time-resolved wall pressure measurements indicate that the shock oscillates in a piston-like manner and most of the energy of the oscillations is at low frequency.      

Radiating hydrogen flow in axisymmetric nozzles

A considerable number of studies published in recent years have been devoted to the study of gas in channels and pipes. In view of the complexity of the question and the lack of analytic techniques, individual aspects of the problem are generally considered. The determination of the radiant field characteristics in regions of simple geometric form filled with a stationary radiating-absorbing medium has been carried out in several studies. The articles [1–3] are devoted to the calculation of the radiant field and the temperature field for a given flow of a perfect inviscid nonheat-conducting radiating gas with constant absorption coefficient. The flow is assumed to be irrotational [1, 2] or nearly potential [3]. The authors investigated the accuracy of the solution obtained with the aid of various approximate methods and found that the diffusion approximation yields a small error in calculating the radiation density field and the values of the radiant thermal fluxes for a quite broad class of wall reflecting properties. We may note also [4, 5], in which a calculation is made of one-dimensional steady flow of a viscous heat-conducting radiating perfect gas with constant transport coefficients.In [1–5] the absorption coefficient is considered constant. This assumption simplifies the solution process considerably, since as the independent variables we can take the corresponding optical thicknesses. The study [3] contains a remark that the calculation method proposed there may be used with a variable absorption coefficient. However, this possibility was not used in the calculations presented.For a constant absorption coefficient these studies yield a rather complete analysis of the methods for solving two-dimensional problems in geometrically simple regions in the absence of mechanical motion and one-dimensional problems with motion. They contain results obtained for the exact integral or integrodlfferential equations and present an analysis of the approximate methods. The study [3] considers broader possibilities of solving two-dimensional problems (using the Monte-Carlo method), but the flow is assumed known ahead of time.In the following we present a method for calculating the two-dimensional equilibrium flow of an inviscid non-heat-conducting radiating gas with variable absorption coefficient. As an example, we consider the flow of radiating-absorbing hydrogen in axisymmetric nozzles. It is assumed that the radiation is gray and is in local thermodynamic equilibrium. The transport equation is considered in the diffusion approximation. The nozzles examined have a semi-infinite cylindrical inlet section. The initial gas flow in the cylindrical section is supersonic. In the solution process we determine the radiation density field and all the flow parameters within the nozzle.The author wishes to thank Yu. D. Shmyglevskii for his continued interest in this study.     

Supersonic swirling flow past a blunt body

The problem of supersonic swirling flow past a blunt body is studied numerically on the basis of the complete Navier-Stokes equations.St. Petersburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 158–160, November–December, 1994.     

Coherent structures in unsteady swirling jet flow

An LDA technique and phase-averaging analysis were used to study unsteady precessing flow in a model vortex burner. Detailed measurements were made for Re=15,000 and S=1.01. On the basis of the analysis of phase-averaged data and vortex detection by the λ₂-technique of Joeng and Hussain (1995), three precessing spiral vortex structures were identified: primary vortex (PV), inner secondary vortex (ISV), and outer secondary vortex (OSV). The PV is the primary and most powerful structure as it includes primary vorticity generated by the swirler; the ISV and OSV are considered here as secondary vortical structures. The jet breakdown zone is the conjunction of a pair of co-rotating co-winding spiral vortices, PV and ISV. The interesting new feature described is that the secondary vortices form a three-dimensional vortex dipole with a helical geometry. The effect of coupling of secondary vortices was suggested as a mechanism of enhanced stability reflected in their increased axial extent.