Constant Area Duct Research Articles

Analysis of the Oscillatory Flows of Multiple Shock Waves in a Constant Area Duct

Analysis of the Oscillatory Flows of Multiple Shock Waves in a Constant Area Duct

Assessing Transfer Matrix Models and Measurements Using Acoustic Energy Conservation Principles

Abstract Acoustic transfer matrices are widely used in the analysis of combustion dynamics of gas turbines. The reliability of the analysis thus depends on the quality of the determination of the transfer matrices of the individual acoustic elements composing the system's acoustic network. These matrices are, in some simple cases, deduced analytically using one-dimensional acoustic modeling. For more complex elements, such as swirlers, perforated plates, or injection units, the transfer matrix has to be obtained experimentally using an impedance tube setup. There are, however, uncertainties in the experimental determination of the transfer matrix coefficients and in the modeling of key elements like injection units. It is thus worth examining experimental data and assessing models using acoustic energy conservation principles. The general idea is to consider the acoustic power flow in the element represented by the transfer matrix T and compare the power input to the power output. This is best accomplished by obtaining a representation in terms of a scattering matrix S, which may be deduced from the transfer matrix T. It is first shown that standard models like those corresponding to a constant area duct or an area change comply with acoustic energy conservation. This analysis is then employed to assess the L−ζ model, widely used to describe injection unit dynamics. Acoustic conservation principles are then used to assess transfer matrices of a family of injectors determined experimentally and check that the data complies with these principles.

Importance of Preferential Segregation by Aerodynamics in Dust Rig Tests

Abstract This work studies a particle injection rig to understand how its design affects particle impingement and rebound from a target plate. The motivation behind this study is to understand how dust ingestion affects aviation gas-turbine engines. The particles are injected into a constant area duct upstream of the plate, and they exit through a converging nozzle. The major result concerns how particles respond differently to changes in the flow field based on their diameter. Near the plate, small particles follow the flow streamlines which causes them to both significantly slow down and to disperse in all directions. However, large particles move ballistically, so they impact the plate with nearly the same velocity and orientation they had at the duct exit. Reynolds-averaged Navier–Stokes (RANS) simulations are compared to large eddy simulation (LES). While RANS are capable of predicting mean particle impact statistics, they display narrower statistical variation than LES, suggesting that particle dispersion is underpredicted in RANS.

Oscillatory behaviors of multiple shock waves to upstream disturbances

The oscillatory response of multiple shock waves to upstream disturbances in a supersonic flow is studied numerically in a constant area rectangular duct. The flow is accelerated through a nozzle with an exit Mach number of 1.75 and continues in the constant area duct, where multiple shock waves are formed. To investigate the effect of upstream disturbance on shock oscillations, three parameters are varied systematically: upstream turbulent intensity, frequency of upstream pressure fluctuation, and amplitude of upstream pressure fluctuation. The wall shear stress variation along the duct length provides the location of separation and reattachment points in the flow field. The wall pressure frequency spectra were used to investigate the low-frequency unsteadiness in shock oscillations. The power spectral density of the wall static pressure and the probability density function (PDF) of shock location are analyzed, and the results suggest that as the upstream turbulent intensity is increased, the dominant frequency of oscillation is increased and the shock oscillations become more symmetrical. As the upstream disturbance frequency is increased, the shock oscillations become more symmetrical and follow the Gaussian curve closely. The shock wave oscillates with the same upstream excitation frequency when the upstream disturbance amplitude is increased. At large values of upstream disturbance amplitude, the PDF shows a large deviation from the Gaussian, and the rms amplitude of shock oscillation increases monotonously. At higher amplitudes of upstream disturbance excitation, the traces of shock train leading-edge location display path-dependence characteristics.

Dynamics of a Generalized Rayleigh System

Consider the first order differential system given by $$\begin{aligned} \begin{array}{l} {\dot{x}}= y, \qquad {\dot{y}}= -x+a(1-y^{2n})y, \end{array} \end{aligned}$$ where a is a real parameter and the dots denote derivatives with respect to the time t. Such system is known as the generalized Rayleigh system and it appears, for instance, in the modeling of diabetic chemical processes through a constant area duct, where the effect of adding or rejecting heat is considered. In this paper we characterize the global dynamics of this generalized Rayleigh system. In particular we prove the existence of a unique limit cycle when the parameter $$a\ne 0$$ .

Variational principle of a shock stationed in a duct

This paper establishes a variational principle of a normal shock stationed in ducts of the various quasi-one-dimensional (quasi-1D) steady non-conservative flow systems. It is found that the locational stability of a shock inside these ducts, namely, the stationary property of the shock, can be judged by analyzing the second-order variation of the potential energy functional of flow impulse with respect to the locational function of the shock. It proves that, for a control volume containing a shock inside a duct, the real stationary location of the shock among all the possible locations satisfying the determined inlet and outlet boundary conditions of the duct is equivalent to that the potential energy of the cross-sectional flow impulse integrated through the entire duct is a minimal. First, the shock location in general duct flows is analyzed by a momentum relaxation method. Then, based on this method, this paper's variational principle is established referring to the principle of minimum potential energy and the principle of virtual displacement. Further, this principle is applied to the flows through a quasi-1D variable-area duct, a one-dimensional (1D) frictional constant-area duct, a 1D heat-exchange constant-area duct, and a 1D mass-additional constant-area duct, which verified the generality of the principle. At last, relevant examples are provided. This variational principle affords a unified and concise theoretical criterion to analyze the stationary property of a normal shock.

Low-frequency shock train oscillation control in a constant area duct

The self-excited shock train oscillation control using partial removal of boundary layer flow in a constant area duct is studied numerically using unsteady Reynolds Averaged Navier–Stokes simulation. The effect of varying the suction flow ratio on the shock train oscillatory characteristics is analyzed using steady and unsteady statistics, space–time contour, power spectra, and cross correlation analysis. For the present study, a constant area duct of height 0.032 mm, and the aspect ratio of 25, at freestream Mach number of 2.0 and back pressure ratio of 0.52 is considered. The removal of the boundary layer through an angled suction slot with three different suction flow ratios is performed. Numerical results indicate that the removal boundary layer restricts the bifurcation process of the shock train and appears to be a single curved normal shock at a higher suction flow ratio. Also, the transition of regular reflection to the Mach reflection type is noted. The suction flow from the top and bottom slot tends to initiate a lateral oscillation that forms a wavy mixing flow region. The power spectral density contour suggests that the increase in suction flow ratio will tend to increase the dominant frequency ranges from (0.034 to 0.094). The cross correlation indicates the presence of downstream pressure disturbance that moves toward the upstream direction. With suction flow, a disturbance that emerges from the suction slot moves in the opposite direction and dominates at a lower suction flow ratio and these disturbances disappear at a large suction flow ratio.

Analysis and design optimization of an ejector integrated into an organic Rankine cycle

The organic Rankine cycle (ORC) is a promising technology to convert low-grade heat to electric power. In this decade, the improvement of the ORC performance is attracting more attention. It has been proven in previous studies that integrating an ejector into an organic Rankine cycle increases the power output capacity. In this study, the impact of both ejector geometry and working characteristics on cycle performance, in particular, the power output capacity is investigated. The ejector functions in the double-choking regime. To this aim, a parameter sensibility study is conducted using some important indicators. Results show that the net power output rises with increasing ejector area ratio or secondary mass flow rate; it falls with increasing throat diameter, primary inlet pressure, or primary inlet temperature. The constant area duct diameter is one of the most important geometrical parameters influencing the power generation and the cycle performance. Power output capacity is found to be independent of the ejector outlet parameters when the ejector operates in the double-choking region. The paper also presents and discusses various energy and exergy metrics as a means of evaluating the cycle performance.

Thermoacoustic Instability Model With Porous Media: Linear Stability Analysis and the Impact of Porous Media

Lean premixed (LPM) combustion systems are susceptible to thermoacoustic instability, which occurs when acoustic pressure oscillations are in phase with the unsteady heat release rates. Porous media has inherent acoustic damping properties and has been shown to mitigate thermoacoustic instability; however, theoretical models for predicting thermoacoustic instability with porous media do not exist. In the present study, a one-dimensional (1D) model has been developed for the linear stability analysis of the longitudinal modes for a series of constant cross-sectional area ducts with porous media using a n-Tau flame transfer function (FTF). By studying the linear regime, the prediction of acoustic growth rates and subsequently the stability of the system is possible. A transfer matrix approach is used to solve for acoustic perturbations of pressure and velocity, stability growth rate, and frequency shift without and with porous media. The Galerkin approximation is used to approximate the stability growth rate and frequency shift, and it is compared to the numerical solution of the governing equations. Porous media is modeled using the following properties: porosity, flow resistivity, effective bulk modulus, and structure factor. The properties of porous media are systematically varied to determine the impact on the eigenfrequencies and stability growth rates. Porous media is shown to increase the stability domain for a range of time delays (Tau) compared to similar cases without porous media.

Analytical and numerical study of normal shock response in a uniform duct

The response of an inviscid shock to external pressure perturbations in a constant area duct is analyzed in terms of fundamental processes like perturbation propagation and its interaction with shock. The results of these elementary processes are formulated analytically and with a Riemann wave tracking method to enable the prediction of shock movement for both upstream and downstream perturbations. The predictions thus obtained are compared with the finite-volume based numerical simulations of the Euler equations. This study shows that the shock responds nonlinearly to perturbations and the nonlinearity has a cumulative effect. Contact surfaces generated during the interaction of normal shock with perturbations, which was ignored in previous investigations, are shown to be important in order to capture this cumulative nonlinearity. The nonlinearity alters the positive and negative duty cycles, which results in a net displacement of shock after responding to one full cycle of sinusoidal perturbation. This drift in shock location is pronounced at low supersonic Mach numbers (1.2–3) but is also present at higher Mach numbers. Furthermore, it is demonstrated that the duty cycle variations are higher for perturbations originating downstream of shock than those originating upstream. The variations in frequency and amplitude are found to merely scale the response and do not introduce any new physics.

Liquid-vapor flows in heated ducts: Reference solutions

Analytical one-dimensional steady state solutions in a heated duct of constant area are determined for two-phase liquid-gas and liquid-vapor mixtures. First, the exact solution for single-phase flow is recalled in the different flow regimes (subsonic and supersonic) for given inflow conditions. These solutions are extended in the context of duct connected to an upstream tank and to a downstream pressure outflow condition. Then, reference solutions for various two-phase flow models are developed for compressible two-phase flows. They are applied in the context of ideal gas and stiffened gas equation of state for liquid-gas and liquid-vapor flows. These solutions corresponds to limit situations of partial equilibrium among the phases. The first limit situation corresponds to a two-phase flow model where both phases evolve in mechanical equilibrium only. The second one corresponds to a two-phase model where the phases evolve in both mechanical and thermal equilibria. The last one corresponds to a model describing a liquid-vapor mixture in thermodynamical equilibrium. The three limit situations are compared to show the impact given the choice of one or another equilibrium on fluid state and in particular on mass flow rate in the duct.

Effects of component polytropic efficiencies on the dimensions of monophasic ejectors

An inappropriate choice of isentropic efficiencies for the sizing of the component parts of monophasic ejectors is one of the greatest sources of error in their design. The present study shows that the replacement of isentropic efficiencies with polytropic efficiencies within 1D ejector models provides more accurate results. Polytropic efficiencies access more precisely the effects of the pressure ratio variation on the irreversibilities of the acceleration and deceleration processes. It is demonstrated that the deviation in the constant area duct length, calculated by using an isentropic efficiency from the real length is three times greater than the one calculated by using a polytropic efficiency. The polytropic efficiencies are extracted from the CFD modeling of two types of ejectors, a constant area mixing (CAM) ejector and a constant pressure mixing (CPM) ejector. A 1D thermodynamic model for CAM and CPM ejectors, based on the polytropic efficiency concept, is proposed and validated against experimental data. Parametric studies based on this model were completed. They reveal that the variation in polytropic efficiency of the primary nozzle has the most important impact on the constant area duct length. New empirical correlations to estimate the polytropic efficiencies are also provided.

Two-dimensional quantitative visualization of isolator shock trains by rainbow schlieren deflectometry

The rainbow schlieren deflectometry is effective in studying quantitatively the density fields in shock-containing free jets at high precision and high spatial resolution. However, there has been no practical application of rainbow schlieren deflectometry for shock trains in a confined duct. Therefore, in the present study, the rainbow schlieren deflectometry is applied to the flow field including a shock train in a constant-area duct where just upstream of the shock train the freestream Mach number is 1.34, the unit Reynolds number is 5.39 × 107 m-1, and the boundary layer displacement thickness is 0.149 mm. As a result, a two- dimensional density field of the shock train is for the first time quantitatively displayed and the fine structure of the shock train is illustrated as a color gradation representation.

Investigation of corner separation and suction control in constant area duct

To control shock-wave/boundary-layer interaction with suction, numerical simulations in a constant-area rectangular isolator with asymmetry nozzle are conducted to understand the complex flow phenomena. Results show that three-dimensional effect significantly weakens isolator resistance to backpressure, which indirectly emphasizes enormous role of viscous effect and geometry to the dynamic behavior of shock train or pseudo-shock wave. The large-scale separated flow only locates at the corner of the upper wall. The mechanism analysis indicates that the flow is accelerated more rapidly along the bottom wall, thereby providing more momentum at the nozzle exit. The displacement and momentum thicknesses and their increasing rate on the upper wall are all greater than that of the bottom wall, which means that upper corner flow is subjected easily to separation. The vortex structure of different cross sections in the pseudo-shock region is formed in different ways and the large-scale vortex still exists after the airflow mixing, which contributes to the deceleration and pressurization. Through the control effect analysis of suction slot, the suction slot in spanwise direction on the side wall can shorten effectively the length of shock train, which indicates that removing the low-momentum fluid near the corners can control significantly the shock-wave/boundary-layer interactions.

Ejector design and performance prediction

This paper proposes a 1-D thermodynamic model for determining the critical pressure ratio, the mixing efficiency and all the dimensions of an optimum ejector providing the highest possible compression ratio for fixed inlet conditions and mass flowrates of the motive and suction fluids. The maximization of the back pressure is obtained subject to constraints imposed by the 2nd law of thermodynamics and the requirements that the flow must be subsonic at the diffuser entrance, that the mixing efficiency must be positive but smaller than one and that the length to diameter ratio for the constant area duct must be between fixed limits recommended in previous studies. The paper also describes a method for determining the off-design performance of a fixed geometry ejector which reproduces the experimental relations between the entrainment ratio, the compression ratio and the inlet conditions of the two fluids. The model uses a fixed polytropic efficiency (rather than the fixed isentropic efficiency used in previous studies) to simulate the acceleration and deceleration processes thus taking into account the effect of the pressure ratio during off-design operation. Examples of its application for isentropic and irreversible acceleration/deceleration of a perfect gas are provided and their results are analysed and compared.

Thermal and inertial parameters of the flow field of a scramjet engine

Heat flux and pressure were measured on the internal walls of a scramjet engine at a hypersonic Mach number of 8 in a shock tunnel. Temperature-time history was recorded on model walls using fast-response thermocouples and the wall heat flux was deduced thereon from the temperature records. Pressure was measured at the same locations as that of heat flux using high-frequency pressure transducers. The surface measurements were used to analyze the internal flow field of the engine. The heat flux measurements revealed the regions of peak heating that may require additional thermal protection during the operation; the regions being the beginning of the constant area duct and the nozzle. The pressure measurements and the pertinent analysis indicated an optimum compression at intake through shock waves with a viable flow separation in the combustion chamber. The experimental data indicated an optimum design of the engine for a possible fuelled operation in future.

Nonequilibrium Supersonic Freestream Studied Using Coherent Anti-Stokes Raman Spectroscopy

Measurements were conducted at the University of Virginia Supersonic Combustion Facility of the flow in a constant-area duct downstream of a Mach 2 nozzle. The airflow was heated to approximately 1200 K in the facility heater upstream of the nozzle. Dual-pump coherent anti-Stokes Raman spectroscopy was used to measure the rotational and vibrational temperatures of and at two planes in the duct. The expectation was that the vibrational temperature would be in equilibrium, because most scramjet facilities are vitiated air facilities and are in vibrational equilibrium. However, with a flow of clean air, the vibrational temperature of along a streamline remains approximately constant between the measurement plane and the facility heater, the vibrational temperature of in the duct is about 1000 K, and the rotational temperature is consistent with the isentropic flow. The measurements of vibrational temperature enabled cross-stream nonuniformities in the temperature exiting the facility heater to be documented. The measurements are in agreement with computational fluid dynamics models employing separate lumped vibrational and translational/rotational temperatures. Measurements and computations are also reported for a few percent steam addition to the air. The effect of the steam is to bring the flow to thermal equilibrium, also in agreement with the computational fluid dynamics.

Study of Physical Aspects of Rarefied Gas Flow Through Micro/Nano Scale Channels Using DSMC

Rarefied gas flow through micro/nano electro mechanical systems does not perform exactly as that in macro-scale devices. The main goal of this study was to investigate mixed subsonic–supersonic flows in micro/nano channels and to provide physical descriptions on their behaviours. We use DSMC method as a reliable numerical tool to extend our simulation because it provides accurate solution for the Boltzmann equations over the entire range of rarefied flow regime or Knudsen numbers. As is known, the appearance of oblique/normal shocks at the inlet of a channel adds to the complexity of internal flow field analyses. We found some very unique physical aspects of micro/nano flows including mixed supersonic–subsonic flow regimes in constant area ducts and the attenuation of emitted shocks, which are attributed to the strong viscous forces and dominant rarefaction effects at micro/nano scales.

Direct simulation Monte Carlo investigation of mixed supersonic–subsonic flow through micro-/nano-scale channels

The main goal in this study is to provide deep physical descriptions on the behaviour of supersonic micro-/nano-channel flow using the direct simulation Monte Carlo method. We found some unique physical aspects of micro-/nano-flows including mixed supersonic–subsonic regimes in constant-area ducts. This mixed regime is due to the formation of entrance shocks and their reflection from the thick boundary layer developed in a rarefied medium. We studied the effects of Mach number, channel aspect ratio and flow rarefaction on the channel inflow condition. The effects of wall thermal state on flow behaviour and shock wave structures were also investigated.

A note on the generalized Rayleigh equation: limit cycles and stability

The aim of the present paper is to give an analytical proof on the existence and stability of the limit cycles in the generalized Rayleigh equation, which models diabetic chemical processes through a constant area duct where the effect of heat addition or rejection is considered, \({\frac{d^{2}x}{dt^{2}}+x = \varepsilon(1-(\frac{dx}{dt}) ^{2n})\,\frac{dx}{dt}}\) where n is a positive integer and e a small real parameter. The main tool used for it is the averaging theory.