Optical quality of flows being formed by axisymmetric nozzle configurations

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 94–98, July–August, 1989.     

Analysis of choked two-phase flows of gas and particle in a C-D nozzle

Particle–gas two-phase flows show significantly different behaviors compared to single gas flow through a convergent–divergent nozzle. Non-equilibrium effects, thermal and velocity lag results to the inefficiency of nozzle performance. In the present studies, theoretical analysis and numerical simulations were carried out to investigate particle–gas flows in a C-D nozzle. Homogeneous equilibrium model that no lag in velocity and temperature occurs between particles and gas phase was used to derive mass flow rate and sound speed of multiphase flows. Two-phase flows are regarded as isentropic flows that isentropic relations can be used for homogeneous equilibrium model. Discrete phase model(DPM) where interaction with continuous phase and discrete random walk model were considered was used to calculate particle–gas flows. Particle mass loadings were varied to investigate their effects on choking phenomena of particle–gas flows. Mass flow rate and sound speed of mixture flows were theoretically calculated by homogeneous equilibrium model and compared with numerical results. Shock wave structure and particle number density were also obtained to be different at different particle mass loading and operating pressure conditions.     

Separated two-phase flow in a nozzle

This work was performed to extend and further test the method of handling separated two-phase flow by studying each phase separately and, particularly, by placing emphasis on the study of the gas phase with interface transport expressions showing the influence of the liquid phase on it. A one-dimensional flow model for accelerating flows was used in conjunction with experimental data to obtain the pressure distribution and velocity distribution in a converging nozzle for several values of flow quality and nozzle inlet stagnation pressure. The results tend to support the use of the model (which includes the assumption that the gas is in critical flow when the two-phase mixture is in critical flow) and give some insight regarding the nature of the liquid distribution near the nozzle throat.     

Lie symmetry solutions for two-phase mass flows

We apply Lie symmetry method to a set of non-linear partial differential equations, which describes a two-phase rapid gravity mass flow as a mixture of solid particles and viscous fluid down a slope (Pudasaini, J. Geophys. Res. 117 (2012) F03010, 28 pp [1]). In order to systematically explore the mathematical structure and underlying physics of the two-phase mixture flow, we generate several similarity forms in general form and construct self-similar solutions. Our analysis generalizes the results, obtained by applying the Lie symmetry method to relatively simple single-phase pressure-driven gravity mass flows, to the two-phase mass flows that include several dominant driving forces and strong phase-interactions. Analytical and numerical solutions are presented for the symmetry-reduced homogeneous and non-homogeneous systems of equations. Analytical and numerical results show that the new models presented here can adequately describe the dynamics of two-phase debris flows, and produce observable phenomena that are consistent with the physics of the flow. The solutions are strongly dependent on the choice of the symmetry-reduced model, as characterized by the group parameters, and the physical parameters of the flows. These solutions reveal strong non-linear and distinct dynamic evolutions, and phase-interactions between the solid and fluid phases, namely the phase-heights and phase-velocities.     

Equations for two-phase flows: A primer

The main goal of these notes is to give a review of the equations for two phase flow problems with an interface between the two phases in a self-contained way, and, in particular, to properly include surface tension into the interface balance equations.     

Averaged topological equations for dispersed two-phase flows

A general relationship between the volume fraction and the specific interfacial area for averaged dispersed two-phase flows is proposed. This relationship, expressed as a basic set of two scalar evolution equations and two vectorial non-uniformity state equations, is an analytical result obtained by a systematic approach using the derivatives of some generalized functions and a local volume-averaging technique. The proposed set of equations was expressed for measurable macroscopic parameters of the system and has the same generality as the averaged transport equations of two-phase flows. By combination of the basic set of equations, called the averaged topological equations (ATEs), second-order ATEs for the volume fraction were found. The second-order ATEs were expressed both by a Lagrangian formulation and by a Eulerian formulation. The importance and physical meaning of the ATEs developed in this study were clarified within the framework of the theory of kinematic waves.     

Effective property models for homogeneous two-phase flows

Using an analogy between thermal conductivity of porous media and viscosity in two-phase flow, new definitions for two-phase viscosity are proposed. These new definitions satisfy the following two conditions: namely (i) the two-phase viscosity is equal to the liquid viscosity at the mass quality = 0% and (ii) the two-phase viscosity is equal to the gas viscosity at the mass quality = 100%. These new definitions can be used to compute the two-phase frictional pressure gradient using the homogeneous modeling approach. These new models are assessed using published experimental data of two-phase frictional pressure gradient in circular pipes, minichannels and microchannels in the form of Fanning friction factor (f_m) versus Reynolds number (Re_m). The published data include different working fluids such as R-12, R-22, argon (R740), R717, R134a, R410A and propane (R290) at different diameters and different saturation temperatures. Models are assessed on the basis minimizing the root mean square error (eRMS). It is shown that these new definitions of two-phase viscosity can be used to analyze the experimental data of two-phase frictional pressure gradient in circular pipes, minichannels and microchannels using simple friction models.     

Numerical investigation of overexpanded nozzle flows

The flow in a planar overexpanded nozzle with a slope discontinuity is studied numerically by means of two- (2D) and three-dimensional (3D) Reynolds-averaged Navier–Stokes simulations and is compared to experimental results. The nozzle pressure ratios (NPR) vary from 1.6 to 10. A good agreement is found between experimental and numerical results and two configurations are observed: under a certain critical NPR, the flow is shown to be asymmetrical with respect to the nozzle axis, while it is perfectly symmetrical for higher NPRs. The value of the critical NPR is found to be very dependent on the turbulence model. Finally, an hysteresis phenomenon is evidenced since the NPR at which the change of flow configuration occurs is different whether the NPR is increasing or decreasing in the nozzle.     

Shock structure in separated nozzle flows

In the case of high overexpansion, the exhaust jet of the supersonic nozzle of rocket engines separates from nozzle wall because of the large adverse pressure gradient. Correspondingly, to match the pressure of the separated flow region, an oblique shock is generated which evolves through the supersonic jet starting approximately at the separation point. This shock reflects on the nozzle axis with a Mach reflection. Thus, a peculiar Mach reflection takes place whose features depend on the upstream flow conditions, which are usually not uniform. The expected features of Mach reflection may become much difficult to predict, depending on the nozzle shape and the position of the separation point along the divergent section of the nozzle.      

Calculation of two-phase swirling flows

The results of calculating the diffusion of a dispersed admixture in turbulent swirling jet flows using the model of momentum transfer in a turbulent gas—dispersion flow proposed by the authors are presented. These results are compared with experimental data and with calculations based on various mathematical models. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 71–78, January–February, 1994.     

Extremal nozzle contours for gas flows with particle lag

The determination of the extremal nozzle contour for gas flow without foreign particles has been carried out in several studies [1–6], based on the calculation of the flow field using the method of characteristics.In [7, 8] the equations are derived for the characteristics and the relations along the streamlines which are required for calculating two-dimensional gas flow with foreign particles. The variational problem for two-phase flow in the two-dimensional formulation may be solved by the method of Guderley and Armitage [9] with the use of equations given in [7] or [8]; however this method is very tedious, even with the use of high-speed computers.In [10, 11] studies are made of two-phase one-dimensional flows by expanding the unknown functions in series in a small parameter, defined by the particle dimensions. In [12] a solution is given for the variational problem (in the one-dimensional formulation) of designing the contour of a nozzle with maximal impulse. However that study does not take account of the static term appearing in the impulse and the solution is obtained in relative cumbersome form. Moreover, the question of account for the losses due to nonparallelism and nonuniformity of the discharge was not considered.The present paper considers in the one-dimensional formulation the flow of a two-phase medium in a Laval nozzle with small particle lags (in velocity and temperature). The variational problem of determining the maximal nozzle impulse is formulated along the nozzle contour for fixed geometric expansion ratio. The impulse losses due to nonparallelism of the discharge are simulated by a function which depends on the ordinates which are variable along the contour and on the slope of the tangent to the contour.The author wishes to thank Yu. D. Shmyglevskii and A. N. Kraiko for helpful discussions and V. K. Starkov for carrying out the calculations on the computer.     

A finite volume scheme for two-phase compressible flows

In gas-particle two-phase flows, when the concentration of the disperesed phase is low, certain assumptions may be made which simplify considerably the equations one has to solve. The gas and particle flows are then linked only via the interaction terms. One may therefore uncouple the full system of equations into two subsystems: one for the gas phase, whose homogeneous part reduces to the Euler equations; and a second system for the particle motion, whose homogeneous part is a degenerate hyperbolic system. The equations governing the gas phase flow may be solved using a high-resolution scheme, while the equations describing the motion of the dispersed phase are treated by a donor-cell method using the solution of a particular Riemann problem. Coupling is then achieved via the right-hand-side terms. To illustrate the capabilities of the proposed method, results are presented for a case specially chosen from among the most difficult to handle, since it involves certain geometrical difficulties, the treatment of regions in which particles are absent and the capturing of particle fronts.     

Nonlinear systems synchronization for modeling two-phase microfluidics flows

The aim of this work is to identify a class of models that can represent the two-phase microfluidic flow in different experimental conditions. The identification procedure adopted is based on the nonlinear systems synchronization theory. The experimental time series were assumed as the asymptotic behavior of a generic state variable of an unknown Master system, and this information was used to drive a second Slave system, with a known model and undefined parameters. To reach the convergence between the time evolutions of the two systems, so the flow identification, an error was evaluated and optimized by tuning the parameters of the Slave system, through genetic algorithm. The Chua’s oscillator has been chosen as a Slave model, and an optimal parameters set of Chua’s system was identified for each of the 18 experiments. As proof of concept on approach validity, the changes in the parameters set in the different experimental conditions were discussed taking into account the results of the nonlinear time series analysis. The results confirm the possibility with a single model to identify a variety of flow regimes generated in two-phase microfluidic processes, independently of how the processes have been generated, no directed relations with the input flow rate used are in the model.     

A probabilistic method for field solutions in two-phase flows

Solid-particle motion and related transport phenomena in two-phase flow are fluctuating processes in space and time. A deterministic method can describe only partially the intrinsic physics of these processes. In this paper, the fluctuations of the flow parameters are modelled by considering the spatial correlations, and a probabilistic computational method for two-phase flow is presented. The probabilistic governing equations have been discretized in space using a finite volume method, and then solved by applying the Neumann expansion method. This last method is time efficient, and its convergence can be guaranteed even for large fluctuations. A liquid-solid particle mixture flow in a circular pipe is taken as an example. Computational results illustrate the merit of the probabilistic approach for the prediction of two-phase flow phenomena.     

The fast procedure for predicting transient subcooled two-phase flows

This paper describes a method of predicting transient, two-phase flows in channels, and presents predictions for several problems. The model is based on a Lagrangian drift-flux formulation of the equations of mass and energy in which the liquid phase can be subcooled. The advantage of the present model over previous models lies in the solution technique, which yields accurate solutions very inexpensively and without problems related to stability. In the technique used, analytical solutions to the differential equations that are valid over limited time and space intervals are used to construct the global solution. The example problems include subcooled boiling, flow reversals and blowdown transients.     

Finite-particle-size effects in disperse two-phase flows

Three aspects of the finite radius of spherical particles in disperse two-phase flows are described. The first one is the relation between the exact volume fraction and the widely used approximation nv (n is the particle number density and v is the particle volume). The approximation affects the behavior of the effective equations at short wavelengths with possible consequences on stability and hyperbolicity. Secondly, the dilute theory of inviscid suspensions is corrected retaining the next leading order in the particle size and an application of this result to the linear problem is described. Thirdly, it is shown how several important properties of suspensions such as effective thermal conductivity and viscosity depend on the subtle effect of translation of the average fields over distances of the order of the particle size.This work has been supported by DOE and NSF under Grants DE-FG02-89ER14043 and CBT-8918144, respectively.     

LDA signal discrimination in two-phase flows

The results of an investigation into the application of LDA for measurement of two-phase flows are presented. It is determined that an important aspect of such measurement is the discrimination between the signals originating from different phases in the flow. It is shown that in the regions where the number density of the discrete phase is high, the error could become considerable. Existing methods of discrimination are investigated and shown to be inadequate. A novel method of signal discrimination is presented and is shown to be capable of complete removal of this error. A two-component LDA system is modified to incorporate the discriminator. Experimental data are presented in support of the method which are in good agreement with the calculated results.     

Flow-rate and thrust coefficients for two-layer flows in a convergent nozzle

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 187–189, January–February, 1990.