Transonic Nozzle Flow Research Articles

Non-Intrusive Frequency Measurements With Laser Shadowgraphy And Event-Based Schlieren Imaging Of Bluff Body Wakes In Compressible Flow

Event-based imaging is an emerging technology in thermodynamics and fluid dynamics. In this work, we are leveraging an event-based camera with a conventional schlieren set-up to investigate the vortex shedding dynamics behind a blunt central injector in a transonic nozzle flow. We offer different methods to evaluate the data stream of the event-based camera. Standard Fourier methods are not directly applicable to this data due to a non-constant time step. Therefore, the data needs to be treated with special care. Clusters of on- and off-events alternate and show the periodic nature of the wake. Using a clustering algorithm on the event groups also provides an estimation of vortex shedding frequency. These results are compared with laser shadowgraphy, a second novel method to measure the vortex shedding frequency in the wake. Both approaches to measuring the vortex shedding frequency are agreeing with each other within 1%. They each offer a simple set-up with comparatively low cost for their high temporal resolution.

Analytical and Numerical Modifications of Transonic Nozzle Flows

Classical transonic hodograph-based design methods are employed and revitalized using modern numerical tools to illustrate the design of a symmetrical accelerating-decelerating nozzle throat design. The concept of Elliptic Continuation is applied to solve transonic boundary value problems avoiding the inherently nonlinear nature of the basic equations and obtaining transonic flow examples using the Method of Characteristics in an inverse mode. Purpose of the present paper, besides describing a new special flow example, is to keep these classical methods alive for education of a new generation of creative engineers.

Application of toluene LIF to transonic nozzle flows to identify zones of incomplete molecular mixing.

Toluene laser-induced fluorescence (LIF) has been applied to image the mixing deficit on the molecular level in the transonic wake of two different blunt-body injectors in a compressible accelerated nozzle flow. A single-color excitation and two-color detection scheme is employed to measure the signal red-shift caused by the quenching effect of molecular oxygen on the fluorescence of toluene, which reduces and red-shifts the LIF signal if both substances interact on a molecular level. To this end, toluene is injected alternatingly with O2-contaning and O2-free carrier gas into the air main flow. Differences of both signals mark regions where mixing on molecular level is incomplete. A zone of molecular mixing deficit extending several millimeters in stream-wise direction is identified. The effect of local variations in temperature on the sensitivity of this technique is discussed using photo-physical data measured in a stationary low-temperature cell.

Vanishing Viscosity Approach to the Compressible Euler Equations for Transonic Nozzle and Spherically Symmetric Flows

We are concerned with globally defined entropy solutions to the Euler equations for compressible fluid flows in transonic nozzles with general cross-sectional areas. Such nozzles include the de Laval nozzles and other more general nozzles whose cross-sectional area functions are allowed at the nozzle ends to be either zero (closed ends) or infinity (unbounded ends). To achieve this, in this paper, we develop a vanishing viscosity method to construct globally defined approximate solutions and then establish essential uniform estimates in weighted L p norms for the whole range of physical adiabatic exponents $${\gamma\in (1, \infty)}$$ , so that the viscosity approximate solutions satisfy the general L p compensated compactness framework. The viscosity method is designed to incorporate artificial viscosity terms with the natural Dirichlet boundary conditions to ensure the uniform estimates. Then such estimates lead to both the convergence of the approximate solutions and the existence theory of globally defined finite-energy entropy solutions to the Euler equations for transonic flows that may have different end-states in the class of nozzles with general cross-sectional areas for all $${\gamma\in (1, \infty)}$$ . The approach and techniques developed here apply to other problems with similar difficulties. In particular, we successfully apply them to construct globally defined spherically symmetric entropy solutions to the Euler equations for all $${\gamma\in (1, \infty)}$$ .

Accuracy of non-resonant laser-induced thermal acoustics (LITA) in a convergent–divergent nozzle flow

Non-resonant laser-induced thermal acoustics (LITA) was applied to measure Mach number, temperature and turbulence level along the centerline of a transonic nozzle flow. The accuracy of the measurement results was systematically studied regarding misalignment of the interrogation beam and frequency analysis of the LITA signals. 2D steady-state Reynolds-averaged Navier–Stokes (RANS) simulations were performed for reference. The simulations were conducted using ANSYS CFX 18 employing the shear-stress transport turbulence model. Post-processing of the LITA signals is performed by applying a discrete Fourier transformation (DFT) to determine the beat frequencies. It is shown that the systematical error of the DFT, which depends on the number of oscillations, signal chirp, and damping rate, is less than $$1.5\%$$ for our experiments resulting in an average error of $$1.9\%$$ for Mach number. Further, the maximum calibration error is investigated for a worst-case scenario involving maximum in situ readjustment of the interrogation beam within the limits of constructive interference. It is shown that the signal intensity becomes zero if the interrogation angle is altered by $$2\%$$ . This, together with the accuracy of frequency analysis, results in an error of about $$5.4\%$$ for temperature throughout the nozzle. Comparison with numerical results shows good agreement within the error bars.

Self-excited oscillation of non-equilibrium condensation in critical flow nozzle

The unsteady flow with vapor condensation in critical flow nozzle includes many different self-excited periodic oscillation modes and might affect its flow field and mass flow-rate. The instability of condensation in transonic nozzle flow was discussed firstly. To investigate the phenomenon of self-excited oscillation and bifurcation induced by non-equilibrium condensation of the pure superheated steam flow in critical flow nozzle, an Eulerian viscous flow solver using SST k-ω turbulent model for vapor condensation was established and validated by experimental data of both steady and unsteady condensations, and algebraic formula. Three self-excited symmetric oscillation modes were classified by the oscillation amplitude and frequency of induced shock. The phenomenon of symmetric oscillation modes and asymmetric bifurcation were discussed in detail. Moreover, the effect of self-excited oscillation on its mass flow-rate was clarified. The maximum deviation of mass flow-rate is 1.2%, which is worthy of attention.

多項式カオス展開を用いた遷音速ノズル流れの不確定性解析の研究

多項式カオス展開を用いた遷音速ノズル流れの不確定性解析の研究

Numerical simulation of transonic nozzle flows in a near-critical condition

Numerical simulation of transonic nozzle flows in a near-critical condition

Experimental research on coarse water formation in steam condensing flow on a transition through the shock wave

In this work the condensation phenomena in steam turbine were discussed. The motivation for presented research was an interaction of liquid phase with shock waves existing in the transonic flow. The paper presents the experimental results of the steam condensing transonic flow in Laval nozzles. For the tests the geometries of the half arc nozzles were used. The behaviour of shock waves in the wet steam region was investigated. Due to the high back pressure, in the divergent part of the nozzle the shock wave was induced and interacting with the nozzle walls caused instability in the flow.

Non-equilibrium condensation process of water vapor in moist air expansion through a sonic nozzle

Non-equilibrium vapor condensation phenomenon through a sonic nozzle is very complicated and closely related to the flow measurement of the sonic nozzle. The gas–liquid two-phase flow Eulerian models for homogeneous and heterogeneous nucleation of moist gas in transonic nozzle flow were built to investigate the effect of vapor condensation on the mass flow rate of the sonic nozzle. Grid independence was achieved by using a solution-adaptive refinement. The CFD models were carefully validated by published experimental data and analytical results. It was shown that the flow rate of the sonic nozzle is affected by both homogeneous and heterogeneous nucleation. In comparison with the experimental data, the effects of vapor condensation on the mass flow rate of the sonic nozzle were obtained. Besides, experiments on periodic, unsteady condensation flow in the sonic nozzle were also reported. This unsteady flow will also affect the flow rate of the sonic nozzle. The results of experiments accorded well with simulations and semi-empirical formula. All the results can be used to further analyze the effect of unsteady flow induced by vapor condensation on the flow rate of the sonic nozzle.

COMPUTATIONAL AND ASYMPTOTIC METHODS IN AEROACOUSTICS WITH APPLICATIONS

In this article the computational and asymptotic methods used in aeroacoustics are reviewed. In particular, two different aeroacoustic applications are demonstrated.In the first problem we investigate the first and second order asymptotic predictions of the thickness and loading noise of a subsonic B-bladed helicopter rotor in the far field and compare the SPL noise results with those of full numerical computations. The results of the second order asymptotic formula seem to be in better agreement with full numerical computations than the first order asymptotic formula. In the second problem, the effect of acoustic wave propagation in transonic nozzle flow is investigated by solving the unsteady quasi-one-dimensional transonic nozzle equations in conservative form using high order computational aeroacoustic schemes, where a novel non-reflecting boundary condition is implemented in addition to the standard non-reflecting boundary condition using characteristics. Excellent agreement with the exact solution is obtained in each case.

A novel nonreflecting boundary condition for unsteady flow

SUMMARYA novel nonreflecting boundary condition, which converges to the specified time‐dependent boundary condition within any degree of accuracy, is introduced for the numerical simulation of hyperbolic systems and validated against the solution of two fundamental boundary value problems in fluids. First, transonic nozzle flow with backward acoustic disturbance is considered. Using high‐order aeroacoustic numerical schemes, the proposed nonreflecting boundary condition yields results that are in excellent agreement with those obtained using conventional nonreflecting boundary conditions based on the method of characteristics as well as with the results of the exact solution. The novel nonreflecting boundary condition, implemented into a semi‐analytical solution algorithm of unsteady bubbly cavitating nozzle flows, is also validated against results obtained using a Lagrangian finite volume scheme. Copyright © 2013 John Wiley & Sons, Ltd.

Transonic nozzle flows and free boundary problems for the full Euler equations

We establish the existence and uniqueness of transonic flows with a transonic shock through a two-dimensional nozzle of slowly varying cross-sections. The transonic flow is governed by the steady, full Euler equations. Given an incoming smooth flow that is close to a constant supersonic state (i.e., smooth Cauchy data) at the entrance and the subsonic condition with nearly horizontal velocity at the exit of the nozzle, we prove that there exists a transonic flow whose downstream smooth subsonic region is separated by a smooth transonic shock from the upstream supersonic flow. This problem is approached by a one-phase free boundary problem in which the transonic shock is formulated as a free boundary. The full Euler equations are decomposed into an elliptic equation and a system of transport equations for the free boundary problem. An iteration scheme is developed and its fixed point is shown to exist, which is a solution of the free boundary problem, by combining some delicate estimates for the elliptic equation and the system of transport equations with the Schauder fixed point argument. The uniqueness of transonic nozzle flows is also established by employing the coordinate transformation of Euler–Lagrange type and detailed estimates of the solutions.

A method of moments based CFD model for polydisperse aerosol flows with strong interphase mass and heat transfer

A Computational Fluid Dynamics method is introduced to study the nucleation, coagulation, evaporation and condensation of polydisperse particles in multi-dimensional multi-species laminar and turbulent flows with strong mass and energy coupling between the phases. The model is based on the Reduced Navier–Stokes (RNS) methodology and incorporates a lognormal aerosol moment method to describe the evolution of suspended particulates. The model is validated against available analytic and numerical techniques for one and two-dimensional geometries. The sensitivity of both the bulk flow and aerosol particle properties to boundary layer effects, even in high Reynolds number flows, is demonstrated for transonic two-phase flow in wet steam de Laval nozzles. It is shown that the developed model and method are useful for the prediction of particle properties such as mass and number concentrations, geometric mean particle size and standard deviation of the particle size distribution in multi-dimensional turbulent compressible aerosol flows with strong heat and mass transfer.

Non-uniqueness and multi-shock solutions for transonic nozzle flows

Non-uniqueness for steady inviscid nozzle flows with supersonic inlet and subsonic exit conditions is studied via pseudo-arclength continuation coupled to the discrete flow equations. An interplay between geometry, boundary conditions and bifurcation behaviour is examined, including solutions with one or two shocks.

Pressure-sensitive pain measurements in planar transonic nozzle flow

Pressure-sensitive pain measurements in planar transonic nozzle flow

Shock capturing approximations to the compressible Euler equations with geometric structure and related equations

In this paper we analyze a shock capturing method, developed by Chen-Glimm [4], to construct approximate solutions of the transonic nozzle flow and the spherically symmetric flow for the compressible Euler equations and related equations. The \(L^\infty\)-stability, \(H^{-1}\)-compactness, and convergence of the approximate solutions are proved for any adiabatic exponent \(\gamma>1\) for the isentropic Euler equations. The corresponding global solutions are obtained for arbitrarily large \(L^\infty\) initial data.

Global solutions to the compressible Euler equations with geometrical structure

We prove the existence of global solutions to the Euler equations of compressible isentropic gas dynamics with geometrical structure, including transonic nozzle flow and spherically symmetric flow. Due to the presence of the geometrical source terms, the existence results themselves are new, especially as they pertain to radial flow in an unbounded region,\(|\vec x| \geqq 1,\), and to transonic nozzle flow. Arbitrary data withL ∞ bounds are allowed in these results. A shock capturing numerical scheme is introduced to compute such flows and to construct approximate solutions. The convergence and consistency of the approximate solutions generated from this scheme to the global solutions are proved with the aid of a compensated compactness framework.

Computation of unsteady three-dimensional transonic nozzle flows using k-epsilon turbulence closure

A three-dimensional Navier-Stokes solver is applied to the computation of unsteady nozzle flows resulting from fluctuating backpressure and validated by comparison with experimental measurements. The flow is modeled by the three-dimensional compressible Navier-Stokes equations using the Launder-Sharma near-wall k-e turbulence model. The mean-flow and turbulence-transport equations are integrated in time using a first-order implicit scheme with third-order MUSCL upwind-biased Van Leer flux-vector-splitting. The configuration studied is a three-dimensional nozzle, with thick sidewall boundary layers. The three-dimensional effect is produced by the shock wave-boundary layer interaction at the corners of the nozzle, where an important three-dimensional recirculating zone appears. The flow is unsteady because of backpressure fluctuation, produced experimentally by a rotating rod of elliptical cross section. The unsteady computations were run for a backpressure fluctuation frequency of 180 Hz.

A comparison of nucleation theories by the asymptotic solution of condensing nozzle flows

The classical and Dillmann-Meier nucleation rates are compared in transonic nozzle flows of moist air under atmospheric supply conditions using the asymptotic solution of Delale, Schnerr and Zierep. The comparison is made in slender nozzles using two distinct expressions for the poorly known surface tension, one fitted to the experiments of Peters and Paikert by the classical theory and the other extrapolated from room temperatures to the range of temperatures investigated. The droplet growth law is fixed by the Hertz-Knudsen formula. It is shown that the Dillmann-Meier theory predicts higher nucleation rates than the classical theory together with a delay of the onset of condensation when either of the surface tension expressions is employed.