Search NASA⌕ Search

SEARCH · Search NASA

Results for “AeroAcoustics”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 487 records · Page 27
All sourcesNASA NTRSDOE OSTI
NASA NTRS1997

Computational Investigations of Noise Suppression in Subsonic Round Jets

NASA Grant NAG1-1802, originally submitted in June 1996 as a two-year proposal, was awarded one-year's funding by NASA LaRC for the period 5 Oct., 1996, through 4 Oct., 1997. Because of the inavailability (from IT at NASA ARC) of sufficient supercomputer time in fiscal 1998 to complete the computational goals of the second year of the original proposal (estimated to be at least 400 Cray C-90 CPU hours), those goals have been appropriately amended, and a new proposal has been submitted to LaRC as a follow-on to NAG1-1802. The current report documents the activities and accomplishments on NAG1-1802 during the one-year period from 5 Oct., 1996, through 4 Oct., 1997. NASA Grant NAG1-1802, and its predecessor, NAG1-1772, have been directed toward adapting the numerical tool of Large-Eddy Simulation (LES) to aeroacoustic applications, with particular focus on noise suppression in subsonic round jets. In LES, the filtered Navier-Stokes equations are solved numerically on a relatively coarse computational grid. Residual stresses, generated by scales of motion too small to be resolved on the coarse grid, are modeled. Although most LES incorporate spatial filtering, time-domain filtering affords certain conceptual and computational advantages, particularly for aeroacoustic applications. Consequently, this work has focused on the development of SubGrid-Scale (SGS) models that incorporate time- domain filters. The author is unaware of any previous attempt at purely time-filtered LES; however, Aldama and Dakhoul and Bedford have considered approaches that combine both spatial and temporal filtering. In our view, filtering in both space and time is redundant, because removal of high frequencies effects the removal of small spatial scales and vice versa.

Pruett, C. David↗
NASA NTRS1999

Development of a Linearized Unsteady Euler Analysis with Application to Wake/Blade-Row Interactions

A three-dimensional, linearized, Euler analysis is being developed to provide a comprehensive and efficient unsteady aerodynamic analysis for predicting the aeroacoustic and aeroelastic responses of axial-flow turbomachinery blading. The mathematical models needed to describe nonlinear and linearized, inviscid, unsteady flows through a blade row operating within a cylindrical annular duct are presented in this report. A numerical model for linearized inviscid unsteady flows, which couples a near-field, implicit, wave-split, finite volume analysis to far-field eigen analyses, is also described. The linearized aerodynamic and numerical models have been implemented into the three-dimensional unsteady flow code, LINFLUX. This code is applied herein to predict unsteady subsonic flows driven by wake or vortical excitations. The intent is to validate the LINFLUX analysis via numerical results for simple benchmark unsteady flows and to demonstrate this analysis via application to a realistic wake/blade-row interaction. Detailed numerical results for a three-dimensional version of the 10th Standard Cascade and a fan exit guide vane indicate that LINFLUX is becoming a reliable and useful unsteady aerodynamic prediction capability that can be applied, in the future, to assess the three-dimensional flow physics important to blade-row, aeroacoustic and aeroelastic responses.

Verdon, Joseph M.↗
NASA NTRS1999

Overview of the Testing of a Small-Scale Proprotor

This paper presents an overview of results from the wind tunnel test of a 1/4-scale V-22 proprotor in the Duits-Nederlandse Windtunnel (DNW) in The Netherlands. The small-scale proprotor was tested on the isolated rotor configuration of the Tilt Rotor Aeroacoustic Model (TRAM). The test was conducted by a joint team from NASA Ames, NASA Langley, U.S. Army Aeroflightdynamics Directorate, and The Boeing Company. The objective of the test was to acquire a benchmark database for validating aeroacoustic analyses. Representative examples of airloads, acoustics, structural loads, and performance data are provided and discussed.

Young, Larry A.↗
NASA NTRS1998

Joint Institute for Advancement of Flight Sciences

The Aeroacoustics Research Program is an integral part of the Joint Institute for Advancement of Flight Sciences at The George Washington University. It is affiliated with many civil, mechanical, and environmental engineering courses, particularly those that stress theory and numerical or other analytic methods in engineering. This report lists the courses presented, the names of graduate research assistants, and bibliographic information regarding publications and presentations. Three graduate degrees were awarded and the abstracts of each dissertation is included. The dissertations were as follows: "A Numerical Investigation of Thermoacoustic Oscillations", which discusses advances in the study of acoustic phenomena through the use of computational aeroacoustics. "Computation of Vortex Shedding and Radiated Sound for a Circular Cylinder: Subcritical to Transcritical Reynolds Numbers", which discusses predicting tonal noise generated by vortex shedding from a circular cylinder. And finally, "The Radiated Field Generated by a Monopole Source in a Short, Rigid, Rectangular Duct", which develops a method for modeling the acoustic field generated by a monopole source placed in a moving rectangular duct.

Source record↗
NASA NTRS2000

Prediction of Sound Waves Propagating Through a Nozzle Without/With a Shock Wave Using the Space-Time CE/SE Method

The benchmark problems in Category 1 (Internal Propagation) of the third Computational Aeroacoustics (CAA) Work-shop sponsored by NASA Glenn Research Center are solved using the space-time conservation element and solution element (CE/SE) method. The first problem addresses the propagation of sound waves through a nearly choked transonic nozzle. The second one concerns shock-sound interaction in a supersonic nozzle. A quasi one-dimension CE/SE Euler solver for a nonuniform mesh is developed and employed to solve both problems. Numerical solutions are compared with the analytical solution for both problems. It is demonstrated that the CE/SE method is capable of solving aeroacoustic problems with/without shock waves in a simple way. Furthermore, the simple nonreflecting boundary condition used in the CE/SE method which is not based on the characteristic theory works very well.

Wang, Xiao-Yen↗
NASA NTRS1999

Current Background Noise Sources and Levels in the NASA Ames 40- by 80-Foot Wind Tunnel: A Status Report

Background noise measurements were made of the acoustic environment in the National Full-Scale Aerodynamics Complex 40- by 80-Foot Wind Tunnel (40x80) at NASA Ames Research Center. The measurements were acquired subsequent to the 40x80 Aeroacoustic Modernization Project, which was undertaken to improve the anechoic characteristics of the 40x80's closed test section as well as reduce the levels of background noise in the facility. The resulting 40x80 anechoic environment was described by Soderman et. al., and the current paper describes the resulting 40x80 background noise, discusses the sources of the noise, and draws comparisons to previous 40x80 background noise levels measurements. At low wind speeds or low frequencies, the 40x80 background noise is dominated by the fan drive system. To obtain the lowest fan drive noise for a given tunnel condition, it is possible in the 40x80 to reduce the fans' rotational speed and adjust the fans' blade pitch, as described by Schmidtz et. al. This idea is not new, but has now been operationally implemented with modifications for increased power at low rotational speeds. At low to mid-frequencies and at higher wind speeds, the dominant noise mechanism was thought to be caused by the surface interface of the previous test section floor acoustic lining. In order to reduce this noise mechanism, the new test section floor lining was designed to resist the pumping of flow in and out of the space between the grating slats required to support heavy equipment. In addition, the lining/flow interface over the entire test section was designed to be smoother and quieter than the previous design. At high wind speeds or high frequencies, the dominant source of background noise in the 40x80 is believed to be caused by the response of the in-flow microphone probes (required by the nature of the closed test section) to the fluctuations in the freestream flow. The resulting background noise levels are also different for probes of various diameters and types. The inflow microphone support strut is also a source of background noise but this source's impact may be minimized by careful design of the strut. In the present paper, the mechanisms mentioned above are discussed in detail. Their frequency and velocity ranges of dominance are defined and the differences between past and current facility background noise levels are presented. This paper gives valuable information for those wishing to make acoustic measurements in the 40x80. With this report and an estimate of the noise levels produced by the noise source of interest, it should be possible to determine the signal-to-noise ratios and measurement locations to successfully perform aeroacoustic testing in the NASA Ames Research Center's 40- by 80-Foot Wind Tunnel.

Allen, Christopher S.↗
NASA NTRS2000

Advanced Methods for Aircraft Engine Thrust and Noise Benefits: Nozzle-Inlet Flow Analysis

In this project, we continue to develop the previous joint research between the Fluid Mechanics and Acoustics Laboratory (FM&AL) at Hampton University (HU) and the Jet Noise Team (JNT) at the NASA Langley Research Center (NASA LaRC). The FM&AL was established at Hampton University in June of 1996 and has conducted research under two NASA grants: NAG-1-1835 (1996-99), and NAG-1-1936 (1997-00). In addition, the FM&AL has jointly conducted research with the Central AeroHydrodynamics Institute (TsAGI, Moscow) in Russia under a Civilian Research and Development Foundation (CRDF) grant #RE2-136 (1996-99). The goals of the FM&AL programs are twofold: (1) to improve the working efficiency of the FM&AUs team in generating new innovative ideas and in conducting research in the field of fluid dynamics and acoustics, basically for improvement of supersonic and subsonic aircraft engines, and (2) to attract promising minority students to this research and training and, in cooperation with other HU departments, to teach them basic knowledge in Aerodynamics, Gas Dynamics, and Theoretical and Experimental Methods in Aeroacoustics and Computational Fluid Dynamics (CFD). The research at the HU FM&AL supports reduction schemes associated with the emission of engine pollutants for commercial aircraft and concepts for reduction of observables for military aircraft. These research endeavors relate to the goals of the NASA Strategic Enterprise in Aeronautics concerning the development of environmentally acceptable aircraft. It is in this precise area, where the US aircraft industry, academia, and Government are in great need of trained professionals and which is a high priority goal of the Minority University Research and Education (MUREP) Program, that the HU FM&AL can make its most important contribution. The main achievements for the reporting period in the development of concepts for noise reduction and improvement in efficiency for jet exhaust nozzles and inlets for aircraft engines are as follows: (1) Publications- The AIAA Paper #99-1924 has been presented at the 5th AIAA/CEAS Aeroacoustics Conference, May 10-12, 1999, Seattle, WA; the AIAA Paper #00-3315 has been accepted for the 36th AIAA/ASME/ SAE/ASEE Joint Propulsion Conference, 17-19 July, 2000, Huntsville, AL; and another paper has been accepted for the International Environmental Congress, 14-16 June, 2000, St.-Petersburg, Russia. (2) Two patents were granted on July 20, 1999, and January 12, 2000. (3) Three reports/presentations at the NASA LaRC and GRC (06/22199, 09/26/ 99, and 06/25/00). (4) Grants and Proposals: Four proposals were submitted to the NASA and CRDF; a NASA Faculty Award was granted on January, 2000. A CRDF Young Investigator Program Award was granted for a 3 months visit of the Russian scientist to the HU FM&AL (03/99-05/99). (5) Theory and Numerical Simulations- Analytical theory, numerical simulation, comparison of theoretical with experimental results, and modification of theoretical approaches, models, grids etc. have been conducted for several complicated 2D and 3D nozzle and inlet designs using NASA codes based on full Euler and Navier-Stokes solvers: CFL3D, CRAFT, GODUNOV, and others. New approach for environmental monitoring via infrasound. (6) Experimental Tests: Experimental acoustic tests at the TsAGI, Moscow, with nozzles having Screwdriver or Axisymmetric Plug and Permeable Shells. A small scale working model of the NASA Low Speed Wind Tunnel (LSWT) has been installed in the Experimental Hall of the HU FM&AL (June, 1999). Preliminary preparations for experimental tests were made. (7) Students Research Activity: Involvement of the two graduate students as research assistants in the current research project.

Gilinsky, Mikhail↗
NASA NTRS2001

Linearized Unsteady Aerodynamic Analysis of the Acoustic Response to Wake/Blade-Row Interaction

The three-dimensional, linearized Euler analysis, LINFLUX, is being developed to provide a comprehensive and efficient unsteady aerodynamic scheme for predicting the aeroacoustic and aeroelastic responses of axial-flow turbomachinery blading. LINFLUX couples a near-field, implicit, wave-split, finite-volume solution to far-field acoustic eigensolutions, to predict the aerodynamic responses of a blade row to prescribed structural and aerodynamic excitations. It is applied herein to predict the acoustic responses of a fan exit guide vane (FEGV) to rotor wake excitations. The intent is to demonstrate and assess the LINFLUX analysis via application to realistic wake/blade-row interactions. Numerical results are given for the unsteady pressure responses of the FEGV, including the modal pressure responses at inlet and exit. In addition, predictions for the modal and total acoustic power levels at the FEGV exit are compared with measurements. The present results indicate that the LINFLUX analysis should be useful in the aeroacoustic design process, and for understanding the three-dimensional flow physics relevant to blade-row noise generation and propagation.

Verdon, Joseph M.↗
NASA NTRS2001

Entropy Splitting for High Order Numerical Simulation of Vortex Sound at Low Mach Numbers

A method of minimizing numerical errors, and improving nonlinear stability and accuracy associated with low Mach number computational aeroacoustics (CAA) is proposed. The method consists of two levels. From the governing equation level, we condition the Euler equations in two steps. The first step is to split the inviscid flux derivatives into a conservative and a non-conservative portion that satisfies a so called generalized energy estimate. This involves the symmetrization of the Euler equations via a transformation of variables that are functions of the physical entropy. Owing to the large disparity of acoustic and stagnation quantities in low Mach number aeroacoustics, the second step is to reformulate the split Euler equations in perturbation form with the new unknowns as the small changes of the conservative variables with respect to their large stagnation values. From the numerical scheme level, a stable sixth-order central interior scheme with a third-order boundary schemes that satisfies the discrete analogue of the integration-by-parts procedure used in the continuous energy estimate (summation-by-parts property) is employed.

Mueller, B.↗
NASA NTRS2000

Numerical Solutions to the Third CAA Workshop Benchmark Problems

This paper presents numerical solutions to the problems of propagation of sound waves through a transonic nozzle, shock-sound interactions, and automobile noise involving feedback of the third NASA Computational Aeroacoustics (CAA) Workshop on benchmark problems. The numerical algorithm is based on a dual time scheme for temporal discretization and a third-order finite volume scheme for spatial discretization. The aims of this study are to apply a dual time stepping scheme to treat aeroacoustic problems of sound propagation and to validate our CAA solver with the benchmark problems for developing a numerical tool for noise analysis and control.

Loh, Roy H.↗
NASA NTRS2003

Application of Circulation Control Technology to Airframe Noise Reduction

This report is a summary of the work performed by Georgia Tech Research Institute (GTRI) under NASA Langley Grant NAG-1-2146, which was awarded as a part of NASA's Breakthrough Innovative Technologies (BIT) initiative. This was a three-year program, with a one-year no-cost extension. Each year's study has been an integrated effort consisting of computational fluid dynamics, experimental aerodynamics, and detailed noise and flow measurements. Year I effort examined the feasibility of reducing airframe noise by replacing the conventional wing systems with a Circulation Control Wing (CCW), where steady blowing was used through the trailing edge of the wing over a Coanda surface. It was shown that the wing lift increases with CCW blowing and indeed for the same lift, a CCW wing was shown to produce less noise. Year 2 effort dealt with a similar study on the role of pulsed blowing on airframe noise. The main objective of this portion of the study was to assess whether pulse blowing from the trailing edge of a CCW resulted in more, less, or the same amount of radiated noise to the farfield. Results show that a reduction in farfield noise of up to 5 dB is measured when pulse flow is compared with steady flow for an equivalent lift configuration. This reduction is in the spectral region associated with the trailing edge jet noise. This result is due to the unique advantage that pulsed flow has over steady flow. For a range of frequencies, more lift is experienced with the same mass flow as the steady case. Thus, for an equivalent lift and slot height, the pulsed system can operate at lower jet velocities, and hence lower jet noise. The computational analysis showed that for a given time-averaged mass flow rate, pulsed jets give a higher value of C(sub l) and a higher L/D than equivalent steady jets. This benefit is attributable to higher instantaneous jet velocities, and higher instantaneous C(sub mu) values for the pulsed jet. Pulsed jet benefits increase at higher frequencies. However, these advantages are somewhat offset by the unsteadiness in the loads, which will cause structural vibrations and fatigue. Additional studies must be done, perhaps with multiple jets on the upper and lower surfaces, to smooth out the fluctuations in lift while retaining the benefits. The rest of the effort was devoted to examining ways of reducing flap edge noise by blowing air through a Coanda nozzle over a rounded tip of the flap. In this case, we were successful in moving the tip vortex away from the tip, but the device producing the blowing was noisy and we were unable to examine the noise benefits, although we believe that the movement of the tip vortex far from the tip should provide noise benefits. It should be noted that in an effort to understand the fluid dynamics and the aeroacoustics of a jet blowing over a Coanda surface, we also carried out a very extensive study of the high aspect ratio slot jets. A first-ever set of far-field noise spectra were measured for jets exhausting from slots with aspect ratios in the range 100 to 3000. Parallel measurements of velocity profiles, length scales and convection velocities were measured to understand the noise generation of high aspect ratio jets. Attempts were also made to develop jet noise prediction schemes for such jets. Much of the work done under this effort has been described in five conference papers and two doctoral theses. The first year s work on the use of steady blowing was described in two AIAA papers presented at the 2001 AIAA Aerospace Sciences Meeting in Reno. Subsequent work was presented at the 9th AIMCEAS Aeroacoustics Conference and Exhibit held at Hilton Head May 12-13. Another paper is to be presented at the 2004 AIAA Aerospace Sciences Meeting in Reno in January 2004. All six papers are included with this report as Appendices. The bulk of the experimental work done in an effort to produce a pulsed flow that is free of upstream noise is also attached as an Appendix.

Ahuja, K. K.↗
NASA NTRS2001

Influence of Wake Models on Calculated Tiltrotor Aerodynamics

The tiltrotor aircraft configuration has the potential to revolutionize air transportation by providing an economical combination of vertical take-off and landing capability with efficient, high-speed cruise flight. To achieve this potential it is necessary to have validated analytical tools that will support future tiltrotor aircraft development. These analytical tools must calculate tiltrotor aeromechanical behavior, including performance, structural loads, vibration, and aeroelastic stability, with an accuracy established by correlation with measured tiltrotor data. The recent test of the Tilt Rotor Aeroacoustic Model (TRAM) with a single,l/4-scale V-22 rotor in the German-Dutch Wind Tunnel (DNW) provides an extensive set of aeroacoustic, performance, and structural loads data. This paper will examine the influence of wake models on calculated tiltrotor aerodynamics, comparing calculations of performance and airloads with TRAM DNW measurements. The calculations will be performed using the comprehensive analysis CAMRAD II.

Johnson, Wayne↗
NASA NTRS2004

Design and Evaluation of Modifications to the NASA Langley Flow Impedance Tube

The need to minimize fan noise radiation from commercial aircraft engine nacelles continues to provide an impetus for developing new acoustic liner concepts. If the full value of such concepts is to be attained, an understanding of grazing flow effects is crucial. Because of this need for improved understanding of grazing flow effects, the NASA Langley Research Center Liner Physics Group has invested a large effort over the past decade into the development of a 2-D finite element method that characterizes wave propagation through a lined duct. The original test section in the Langley Grazing IncidenceTube was used to acquire data needed for implementation of this finite element method. This test section employed a stepper motor-driven axial-traversing bar, embedded in the wall opposite the test liner, to position a flush-mounted microphone at pre-selected locations. Complex acoustic pressure data acquired with this traversing microphone were used to educe the acoustic impedance of test liners using this 2-D finite element method and a local optimization technique. Results acquired in this facility have been extensively reported, and were compared with corresponding results from various U.S. aeroacoustics laboratories in the late 1990 s. Impedance data comparisons acquired from this multi-laboratory study suggested that it would be valuable to incorporate more realistic 3-D aeroacoustic effects into the impedance eduction methodology. This paper provides a description of modifications that have been implemented to facilitate studies of 3-D effects. The two key features of the modified test section are (1) the replacement of the traversing bar and its flush-mounted microphone with an array of 95 fixed-location microphones that are flush-mounted in all four walls of the duct, and (2) the inclusion of a suction device to modify the boundary layer upstream of the lined portion of the duct. The initial results achieved with the modified test section are provided in this report, and a comparison of these results with those achieved using the original test section is used to demonstrate that the data acquisition and analysis with the new test section can be confidently used for impedance eduction.

Jones, Michael G.↗
NASA NTRS1999

Overview of the Testing of a Small-Scale Proprotor

This paper presents an overview of results from the wind tunnel test of a 1/4-scale V-22 proprotor in the Duits-Nederlandse Windtunnel (DNW) in The Netherlands. The small-scale proprotor was tested on the isolated rotor configuration of the Tilt Rotor Aeroacoustic Model (TRAM). The test was conducted by a joint team from NASA Ames, NASA Langley, U.S. Army Aeroflightdynamics Directorate, and The Boeing Company. The objective of the test was to acquire a benchmark database for validating aeroacoustic analyses. Representative examples of airloads, acoustics, structural loads, and performance data are provided and discussed.

Young, Larry A.↗
NASA NTRS2000

Self-Scheduling Parallel Methods for Multiple Serial Codes with Application to WOPWOP

This paper presents a scheme for efficiently running a large number of serial jobs on parallel computers. Two examples are given of computer programs that run relatively quickly, but often they must be run numerous times to obtain all the results needed. It is very common in science and engineering to have codes that are not massive computing challenges in themselves, but due to the number of instances that must be run, they do become large-scale computing problems. The two examples given here represent common problems in aerospace engineering: aerodynamic panel methods and aeroacoustic integral methods. The first example simply solves many systems of linear equations. This is representative of an aerodynamic panel code where someone would like to solve for numerous angles of attack. The complete code for this first example is included in the appendix so that it can be readily used by others as a template. The second example is an aeroacoustics code (WOPWOP) that solves the Ffowcs Williams Hawkings equation to predict the far-field sound due to rotating blades. In this example, one quite often needs to compute the sound at numerous observer locations, hence parallelization is utilized to automate the noise computation for a large number of observers.

Long, Lyle N.↗
NASA NTRS2001

Proceedings of the Jet Noise Workshop

Jet noise has been a major problem for aircraft for nearly 50 years. There has been considerable research performed around the world aimed at identifying ways to reduce jet noise. This work was first intended for turbojet aircraft and later extended to low bypass ratio turbofans. Many of the people who performed this pioneering research have retired or are no longer active in aeroacoustics. After so many years of work in jet noise, it is a challenge to piece together the history of its development through existing publications due to the large volume of documents. It is possible to forget important developments from the past as new researchers tackle similar problems. Therefore, a jet noise workshop was organized by the AeroAcoustics Research Consortium (AARC) with the intent of reviewing research that has been done by experts throughout the world. The forum provided a unique opportunity for current researchers to hear the diverse views from world experts on issues related to jet noise modeling and interpretation of experimental data.

Huff, Dennis↗
NASA NTRS2004

Computation of Gust-Cascade Interaction Using the CE/SE Method

The problem 2 in Category 3 of the 4th Computational Aeroacoustic(CAA) Workshop is solved using the space-time conservation element and solution element (CE/SE) method. This problem models rotor-stator interaction in a 2D cascade. It involves complex geometries and flow physics including vortex shedding and acoustic radiation. The parallel version of the 2D nonlinear Euler solver is used with an unstructured triangular mesh to solve this problem. The Giles approach is incorporated with the CE/SE method to handle non-equal pitches of the rotor and stator. Validation on the Giles approach is performed using Problem 3.1 in the 2nd CAA Workshop. The space-time CE/SE method is a finite volume method with second-order accuracy in both space and time. The flux conservation is enforced in both space and time instead of space only. It has low numerical dissipation and dispersion errors. It uses simple non-reflecting boundary conditions and is compatible with unstructured meshes. It is simple, flexible, and generate reasonably accurate solutions. The CE/SE method has been successfully applied to solve numerous practical problems, especially aeroacoustic problems. Some preliminary numerical results of the benchmark problem 3.2 of the 4th CAA Workshop are shown. The steady-state pressure contour is plotted. The mean pressure distribution on the blade surface is compared with Turbo solution showing a good agreement. The sound pressure level versus the rotor harmonic n at the six designated positions on the blade surface, three locations at inlet plane, and three locations at the outlet plane are plotted. It can be seen that the acoustic response exists only at the excitation frequencies (n = 1,2,3). On the blade surface, the acoustic wave at n = 1 is dominant, while at the inlet and outlet planes, the sound pressure level at n = 2 becomes the largest, which is similar to the results presented. The distribution of sound pressure level at different spatial modes along the z- direction is plotted for n = 1,2,3, respectively. It shows that the spatial modes m = -32 and 22 at n = 1 exponentially decay, and the spatial modes m = 10 at n = 2, m = -42 and 12 at n = 3 propagate both upstream and downstream, which agrees with the prediction based on the linearized theory. Some oscillations are observed, which needs to be investigated further. In the final paper, the numerical results will be compared with a frequency-domain solver LINFLUX solution if it is available.

Wang, X.-Y.↗