Artykuły w czasopismach: „Transonic Cascade Flutter Study”

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Lu, P. J., and S. K. Chen. "Evaluation of Acoustic Flutter Suppression for Cascade in Transonic Flow." Journal of Engineering for Gas Turbines and Power 124, no. 1 (2000): 209–19. http://dx.doi.org/10.1115/1.1365933.

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Flutter suppression via actively excited acoustic waves is a new idea proposed recently. The high flutter frequency (typically 50–500 Hz for a fan blade) and stringent space constraint make conventional mechanical type flutter suppression devices difficult to implement for turbomachines. Acoustic means arises as a new alternative which avoids the difficulties associated with the mechanical methods. The objective of this work is to evaluate numerically the transonic flutter suppression concept based on the application of sound waves to two-dimensional cascade configuration. This is performed using a high-resolution Euler code based on a dynamic mesh system. The concept has been tested to determine the effectiveness and limitations of this acoustic method. In a generic bending-torsion flutter study, trailing edge is found to be the optimal forcing location and the control gain phase is crucial for an effective suppression. The P&W fan rig cascade was used as the model to evaluate the acoustic flutter suppression technique. With an appropriate selection of the control logic the flutter margin can be enlarged. Analogous to what were concluded in the isolated airfoil study, for internal excitation, trailing-edge forcing was shown to be optimal since the trailing-edge receptivity still works as the dominant mechanism for generating the acoustically induced airloads.

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Kobayashi, H. "Annular Cascade Study of Low Back-Pressure Supersonic Fan Blade Flutter." Journal of Turbomachinery 112, no. 4 (1990): 768–77. http://dx.doi.org/10.1115/1.2927720.

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Low back-pressure supersonic fan blade flutter in the torsional mode was examined using a controlled-oscillating annular cascade test facility. Precise data of unsteady aerodynamic forces generated by shock wave movement, due to blade oscillation, and the previously measured data of chordwise distributions of unsteady aerodynamic forces acting on an oscillating blade, were joined and, then, the nature of cascade flutter was evaluated. These unsteady aerodynamic forces were measured by direct and indirect pressure measuring methods. Our experiments covered a range of reduced frequencies based on a semichord from 0.0375 to 0.547, six interblade phase angles, and inlet flow velocities from subsonic to supersonic flow. The occurrence of unstalled cascade flutter in relation to reduced frequency, interblade phase angle, and inlet flow velocity was clarified, including the role of unsteady aerodynamic blade surface forces on flutter. Reduced frequency of the flutter boundary increased greatly when the blade suction surface flow became transonic flow. Interblade phase angles that caused flutter were in the range from 40 to 160 deg for flow fields ranging from high subsonic to supersonic. Shock wave movement due to blade oscillation generated markedly large unsteady aerodynamic forces which stimulated blade oscillation.

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Ott, P., A. Bo¨lcs, and T. H. Fransson. "Experimental and Numerical Study of the Time-Dependent Pressure Response of a Shock Wave Oscillating in a Nozzle." Journal of Turbomachinery 117, no. 1 (1995): 106–14. http://dx.doi.org/10.1115/1.2835625.

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Investigations of flutter in transonic turbine cascades have shown that the movement of unsteady normal shocks has an important effect on the excitation of blades. In order to predict this phenomenon correctly, detailed studies concerning the response of unsteady blade pressures versus different parameters of an oscillating shock wave should be performed, if possible isolated from other flow effects in cascades. In the present investigation the correlation between an oscillating normal shock wave and the response of wall-mounted time-dependent pressure transducers was studied experimentally in a nozzle with fluctuating back pressure. Excitation frequencies between 0 Hz and 180 Hz were investigated. For the measurements, various measuring techniques were employed. The determination of the unsteady shock position was made by a line scan camera using the Schlieren flow visualization technique. This allowed the simultaneous use of unsteady pressure transducers to evaluate the behavior of the pressure under the moving shock. A numerical code, based on the fully unsteady Euler equations in conservative form, was developed to simulate the behavior of the shock and the pressures. The main results of this work were: (1) The boundary layer over an unsteady pressure transducer has a quasi-steady behavior with respect to the phase lag. The pressure amplitude depends on the frequency of the back pressure. (2) For the geometry investigated the shock amplitude decreased with increasing excitation frequency. (3) The pressure transducer sensed the arriving shock before the shock had reached the position of the pressure transducer. (4) The computed unsteady phenomena agree well with the results of the measurements.

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Lepicovsky, J., R. V. Chima, E. R. McFarland, and J. R. Wood. "On Flowfield Periodicity in the NASA Transonic Flutter Cascade." Journal of Turbomachinery 123, no. 3 (2000): 501–9. http://dx.doi.org/10.1115/1.1378300.

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A combined experimental and numerical program was carried out to improve the flow uniformity and periodicity in the NASA transonic flutter cascade. The objectives of the program were to improve the periodicity of the cascade and to resolve discrepancies between measured and computed flow incidence angles and exit pressures. Previous experimental data and some of the discrepancies with computations are discussed. In the present work surface pressure taps, boundary layer probes, shadowgraphs, and pressure-sensitive paints were used to measure the effects of boundary layer bleed and tailboard settings on flowfield periodicity. These measurements are described in detail. Two numerical methods were used to analyze the cascade. A multibody panel code was used to analyze the entire cascade and a quasi-three-dimensional viscous code was used to analyze the isolated blades. The codes are described and the results are compared to the measurements. The measurements and computations both showed that the operation of the cascade was heavily dependent on the endwall configuration. The endwalls were redesigned to approximate the midpassage streamlines predicted using the viscous code, and the measurements were repeated. The results of the program were that: (1) Boundary layer bleed does not improve the cascade flow periodicity. (2) Tunnel endwalls must be shaped like predicted cascade streamlines. (3) The actual flow incidence must be measured for each cascade configuration rather than using the tunnel geometry. (4) The redesigned cascade exhibits excellent periodicity over six of the nine blades.

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Kobayashi, H. "Unsteady Aerodynamic Damping Measurement of Annular Turbine Cascade With High Deflection in Transonic Flow." Journal of Turbomachinery 112, no. 4 (1990): 732–40. http://dx.doi.org/10.1115/1.2927716.

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Unsteady aerodynamic forces acting on oscillating blades of a transonic annular turbine cascade were investigated in both aerodynamic stable and unstable domains, using a Freon gas annular cascade test facility. In the facility, whole blades composing the cascade were oscillated in the torsional mode by a high-speed mechanical drive system. In the experiment, the reduced frequency K was changed from 0.056 to 0.915 with a range of outlet Mach number M2 from 0.68 to 1.39, and at a constant interblade phase angle. Unsteady aerodynamic moments obtained by two measuring methods agreed well. Through the moment data the phenomenon of unstalled transonic cascade flutter was clarified as well as the significance of K and M2 for the flutter. The variation of flutter occurrence with outlet flow velocity in the experiments showed a very good agreement with theoretical analysis.

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Lepicovsky, Jan, David Šimurda, Jindřich Hála, Petr Šidlof, and Martin Štěpán. "Blade pressure loading and torque measurement in a transonic linear cascade." Journal of Physics: Conference Series 2511, no. 1 (2023): 012030. http://dx.doi.org/10.1088/1742-6596/2511/1/012030.

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Abstract Experimental results of a transonic compressor blade pressure loadings and blade shaft torque measurements are presented in this paper. Data were acquired for the cascade middle blade being set to a number of incidence angle offsets to simulate phases of a blade flutter oscillatory motion. This paper should be viewed as a progress report on the ongoing larger research effort on blade flutter in transonic flow regimes.

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Bakhle, Milind A., T. S. R. Reddy, and Theo G. Keith. "Subsonic/Transonic Cascade Flutter Using a Full-Potential Solver." AIAA Journal 31, no. 7 (1993): 1347–49. http://dx.doi.org/10.2514/3.49072.

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Kobayashi, H. "Effects of Shock Waves on Aerodynamic Instability of Annular Cascade Oscillation in a Transonic Flow." Journal of Turbomachinery 111, no. 3 (1989): 222–30. http://dx.doi.org/10.1115/1.3262259.

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The effects of shock waves on the aerodynamic instability of annular cascade oscillation were examined for rows of both turbine and compressor blades, using a controlled-oscillating annular cascade test facility and a method for accurately measuring time-variant pressures on blade surfaces. The nature of the effects and blade surface extent affected by shock waves were clarified over a wide range of Mach number, reduced frequency, and interblade phase angle. Significant unsteady aerodynamic forces were found generated by shock wave movement, which markedly affected the occurrence of compressor cascade flutter as well as turbine cascade flutter. For the turbine cascade, the interblade phase angle significantly controlled the effect of force, while for the compressor cascade the reduced frequency controlled it. The chordwise extent of blade surface affected by shock movement was estimated to be approximately 6 percent chord length.

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McBean, Ivan, Kerry Hourigan, Mark Thompson, and Feng Liu. "Prediction of Flutter of Turbine Blades in a Transonic Annular Cascade." Journal of Fluids Engineering 127, no. 6 (2005): 1053–58. http://dx.doi.org/10.1115/1.2060731.

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A parallel multiblock Navier-Stokes solver with the k‐ω turbulence model is used to solve the unsteady flow through an annular turbine cascade, the transonic Standard Test Case 4, Test 628. Computations are performed on a two- and three-dimensional model of the blade row with either the Euler or the Navier-Stokes flow models. Results are compared to the experimental measurements. Comparisons of the unsteady surface pressure and the aerodynamic damping are made between the three-dimensional, two-dimensional, inviscid, viscous simulations, and experimental data. Differences are found between the stability predictions by the two- and three-dimensional computations, and the Euler and Navier-Stokes computations due to three-dimensionality of the cascade model and the presence of a boundary layer separation, respectively.

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Cinnella, P., P. De Palma, G. Pascazio, and M. Napolitano. "A Numerical Method for Turbomachinery Aeroelasticity." Journal of Turbomachinery 126, no. 2 (2004): 310–16. http://dx.doi.org/10.1115/1.1738122.

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This work provides an accurate and efficient numerical method for turbomachinery flutter. The unsteady Euler or Reynolds-averaged Navier-Stokes equations are solved in integral form, the blade passages being discretised using a background fixed C-grid and a body-fitted C-grid moving with the blade. In the overlapping region data are exchanged between the two grids at every time step, using bilinear interpolation. The method employs Roe’s second-order-accurate flux difference splitting scheme for the inviscid fluxes, a standard second-order discretisation of the viscous terms, and a three-level backward difference formula for the time derivatives. The dual-time-stepping technique is used to evaluate the nonlinear residual at each time step. The state-of-the-art second-order accuracy of unsteady transonic flow solvers is thus carried over to flutter computations. The code is proven to be accurate and efficient by computing the 4th Aeroelastic Standard Configuration, namely, the subsonic flow through a turbine cascade with flutter instability in the first bending mode, where viscous effect are found practically negligible. Then, for the very severe 11th Aeroelastic Standard Configuration, namely, transonic flow through a turbine cascade at off-design conditions, benchmark solutions are provided for various values of the inter-blade phase angle.

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KAZAWA, Junichi, and Toshinori WATANABE. "Active Control of Cascade Flutter with Piezo Electric Device in Transonic Flow." Proceedings of the JSME annual meeting 2004.3 (2004): 347–48. http://dx.doi.org/10.1299/jsmemecjo.2004.3.0_347.

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Šidlof, Petr, Pavel Šidlof, Martin Štěpán, David Šimurda, and Jan Lepicovsky. "Laser triangulation measurement of blade oscillation in a transonic compressor cascade." Journal of Physics: Conference Series 2511, no. 1 (2023): 012010. http://dx.doi.org/10.1088/1742-6596/2511/1/012010.

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Abstract The paper describes a method for optical measurement of oscillation of blades in a linear cascade designed for flutter research. The method uses laser triangulation, which measures the distance between the sensor and the oscillation object. The distance can be recalculated to angular displacement of the blade. However, the relation is nonlinear due to the curvature of the blade. The nonlinear dependence between the distance and angular displacement is derived and quantified analytically. The paper further analyzes the influence of light refraction in the optical window, surface quality and sensor inclination. A procedure for the nonlinear calibration is proposed and used during wind tunnel measurements in a blade cascade with forced oscillation of the middle blade.

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Isomura, K., and M. B. Giles. "A Numerical Study of Flutter in a Transonic Fan." Journal of Turbomachinery 120, no. 3 (1998): 500–507. http://dx.doi.org/10.1115/1.2841746.

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The bending mode Flutter of a modern transonic fan has been studied using a quasi-three-dimensional viscous unsteady CFD code. The type of flutter in this research is that of a highly loaded blade with a tip relative Mach number just above unity, commonly referred to as transonic stall flutter. This type of Flutter is often encountered in modern wide chord fans without a part span shroud. The CFD simulation uses an upwinding scheme with Roe’s third-order flux differencing, and Johnson and King’s turbulence model with the later modification due to Johnson and Coakley. A dynamic transition point model is developed using the en method and Schubauer and Klebanoff’s experimental data. The calculations of the flow in this fan reveal that the source of the flutter of IHI transonic fan is an oscillation of the passage shock, rather than a stall. As the blade loading increases, the passage shock moves forward. Just before the passage shock unstarts, the stability of the passage shock decreases, and a small blade vibration causes the shock to oscillate with a large amplitude between unstarted and started positions. The dominant component of the blade excitation force is due to the foot of the oscillating passage shock on the blade pressure surface.

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Thermann, Hans, and Reinhard Niehuis. "Unsteady Navier-Stokes Simulation of a Transonic Flutter Cascade Near-Stall Conditions Applying Algebraic Transition Models." Journal of Turbomachinery 128, no. 3 (2005): 474–83. http://dx.doi.org/10.1115/1.2183313.

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Due to the trend in the design of modern aeroengines to reduce weight and to realize high pressure ratios, fan and first-stage compressor blades are highly susceptible to flutter. At operating points with transonic flow velocities and high incidences, stall flutter might occur involving strong shock-boundary layer interactions, flow separation, and oscillating shocks. In this paper, results of unsteady Navier-Stokes flow calculations around an oscillating blade in a linear transonic compressor cascade at different operating points including near-stall conditions are presented. The nonlinear unsteady Reynolds-averaged Navier-Stokes equations are solved time accurately using implicit time integration. Different low-Reynolds-number turbulence models are used for closure. Furthermore, empirical algebraic transition models are applied to enhance the accuracy of prediction. Computations are performed two dimensionally as well as three dimensionally. It is shown that, for the steady calculations, the prediction of the boundary layer development and the blade loading can be substantially improved compared with fully turbulent computations when algebraic transition models are applied. Furthermore, it is shown that the prediction of the aerodynamic damping in the case of oscillating blades at near-stall conditions can be dependent on the applied transition models.

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Sanders, A. J., K. K. Hassan, and D. C. Rabe. "Experimental and Numerical Study of Stall Flutter in a Transonic Low-Aspect Ratio Fan Blisk." Journal of Turbomachinery 126, no. 1 (2004): 166–74. http://dx.doi.org/10.1115/1.1645532.

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Experiments are performed on a modern design transonic shroudless low-aspect ratio fan blisk that experienced both subsonic/transonic and supersonic stall-side flutter. High-response flush mounted miniature pressure transducers are utilized to measure the unsteady aerodynamic loading distribution in the tip region of the fan for both flutter regimes, with strain gages utilized to measure the vibratory response at incipient and deep flutter operating conditions. Numerical simulations are performed and compared with the benchmark data using an unsteady three-dimensional nonlinear viscous computational fluid dynamic (CFD) analysis, with the effects of tip clearance, vibration amplitude, and the number of time steps-per-cycle investigated. The benchmark data are used to guide the validation of the code and establish best practices that ensure accurate flutter predictions.

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Šidlof, Petr, David Šimurda, Jan Lepicovsky, and Martin Štěpán. "Aerodynamic and dynamic loading in a blade cascade designed for flutter research." EPJ Web of Conferences 264 (2022): 01041. http://dx.doi.org/10.1051/epjconf/202226401041.

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Flow-induced vibration of turbine and compressor blades, so called blade flutter, represents a serious problem for designers and operators of large turbomachines. The research of mechanisms leading to this dangerous aeroelastic instability, which can occur especially in modern long and slender blades, is hindered by lack of experimental data. A new experimental setup for controlled flutter testing has been designed in cooperation of the Institute of Thermomechanics of the Czech Academy of Sciences and Faculty of Mechatronics of the Technical University of Liberec. The test section consists of five planar blades placed in a transonic wind tunnel, with high-frequency torsional oscillation of the middle blade driven by an electric motor. The contribution presents the results of first measurements, namely the static pressure distribution for various inlet Mach numbers, aerodynamic moments and deformation of the middle blade due to inertial loads during high-frequency oscillation.

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Hála, Jindřich, Jan Lepičovský, Petr Šidlof, and David Šimurda. "CFD Investigation of the test facility for forced blade flutter research." EPJ Web of Conferences 264 (2022): 01017. http://dx.doi.org/10.1051/epjconf/202226401017.

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With the increasing share of renewable power resources turbomachines need to be operated under a wider range of operating conditions including highly off-design regimes. Under such regimes an undesirable phenomenon of blade flutter might occur and possibly destroy the machine. To prevent this, intensive research is conducted by research teams worldwide. Blade flutter research program at the Institute of Thermomechanics of the Czech academy of sciences (IT CAS) mainly aims to advance experimental techniques for investigation of sonic and transonic blade flutter. For this purpose, the new sophisticated test facility was designed and manufactured. As part of the design process, the CFD computations were conducted in order to investigate the flow field in the test facility. This paper presents results of these computations with detailed analysis of flow structures occurring during the air flow through the stationary blade cascade.

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Fransson, T. H., M. Jo¨cker, A. Bo¨lcs, and P. Ott. "Viscous and Inviscid Linear/Nonlinear Calculations Versus Quasi-Three-Dimensional Experimental Cascade Data for a New Aeroelastic Turbine Standard Configuration." Journal of Turbomachinery 121, no. 4 (1999): 717–25. http://dx.doi.org/10.1115/1.2836725.

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This paper presents a new International Standard Configuration to be added to an already existing set of 10 configurations for unsteady flow through vibrating axial-flow turbomachine cascades. This 11th configuration represents a turbine blade geometry with transonic design flow conditions with a normal shock positioned at 75 percent real chord on the suction side. Out of a set of test cases covering all relevant flow regimes two cases were selected for publication: A subsonic, attached flow case, and an off-design transonic case showing a separation bubble at 30 percent real chord on the suction side. The performed tests are shown to be repeatable and suitable for code validations of numerical models predicting flutter in viscous flows. The validity of the measured data of the two public cases was examined and comparisons with other tests were conducted. Sometimes a large difference in aerodynamic damping was observed on cases with similar flow conditions. This was investigated at three transonic cases with almost identical inlet flow conditions and only small variations in outlet Mach number. It was found that the differences in the global damping are due to very local changes on the blade surface in the shock region, which obtain a large influence by the integration because of the discrete measuring points. Hence it is recommended not to look at the global damping for code validations but more precisely to the local values. These show a common tendency, which is reproducible with different numerical methods. This was demonstrated with a potential model, a linear Euler model, a nonlinear Euler model, and a Navier–Stokes solver, all applied to predict flutter of each test case with a 2D/Q3D approach. This paper demonstrates both the limitations of inviscid codes to predict flutter in viscous flow regimes, and their cost advantage in attached flow calculations. The need for viscous code development and validation is pointed out. This should justify and encourage the publication of thoroughly measured test cases with viscous effects.

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Pátý, Marek, and Jan Halama. "ON THE USE OF A FLUX-SPLITTING SCHEME IN THE NUMERICAL FLUTTER ANALYSIS OF A LOW-PRESSURE TURBINE STAGE." Acta Polytechnica 61, SI (2021): 135–47. http://dx.doi.org/10.14311/ap.2021.61.0135.

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The endeavour to increase the power output of steam turbines results in the design of low-pressure stages with large diameters. Such designs, featuring long and thin blades, are increasingly susceptible to unfavourable aeroelastic effects. The interaction of structure and flow may induce blade vibrations, known as flutter, which act detrimentally on the operational life of the machine. The present work employs a time-marching numerical simulation to investigate the flutter behaviour of a low-pressure transonic turbine cascade. Its blades are subject to a harmonic motion based on the results of a structural analysis and its susceptibility to flutter is evaluated via the energy method. The computations are performed with an in-house Finite Volume Method code. The flow model is based on 2D Euler equations in Arbitrary Lagrangian-Eulerian formulation with the AUSM+-up scheme for inviscid flux discretization. A higher-order spatial accuracy is achieved by using a MUSCL approach, for which both the gradient reconstruction and the slope limiting are given a careful examination-by comparing the convergence and accuracy of multiple methods. The computational model is validated by experimental data on the Fourth Standard Configuration turbine cascade.

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He, L., and J. D. Denton. "Inviscid-Viscous Coupled Solution for Unsteady Flows Through Vibrating Blades: Part 2—Computational Results." Journal of Turbomachinery 115, no. 1 (1993): 101–9. http://dx.doi.org/10.1115/1.2929193.

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A quasi-three-dimensional inviscid-viscous coupled approached has been developed for unsteady flows around oscillating blades, as described in Part 1. To validate this method, calculations for several steady and unsteady flow cases with strong inviscid-viscous interactions are performed, and the results are compared with the corresponding experiments. Calculated results for unsteady flows around a biconvex cascade and a fan tip section highlight the necessity of including viscous effects in predictions of turbomachinery blade flutter at transonic flow conditions.

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Karnick, Pradeepa T., and Kartik Venkatraman. "Shock–boundary layer interaction and energetics in transonic flutter." Journal of Fluid Mechanics 832 (October 26, 2017): 212–40. http://dx.doi.org/10.1017/jfm.2017.629.

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We study the influence of shock and boundary layer interactions in transonic flutter of an aeroelastic system using a Reynolds-averaged Navier–Stokes (RANS) solver together with the Spalart–Allmaras turbulence model. We show that the transonic flutter boundary computed using a viscous flow solver can be divided into three distinct regimes: a low transonic Mach number range wherein viscosity mimics increasing airfoil thickness thereby mildly influencing the flutter boundary; an intermediate region of drastic change in the flutter boundary due to shock-induced separation; and a high transonic Mach number zone of no viscous effects when the shock moves close to the trailing edge. Inviscid transonic flutter simulations are a very good approximation of the aeroelastic system in predicting flutter in the first and third regions: that is when the shock is not strong enough to cause separation, and in regions where the shock-induced separated region is confined to a small region near the trailing edge of the airfoil. However, in the second interval of intermediate transonic Mach numbers, the power distribution on the airfoil surface is significantly influenced by shock-induced flow separation on the upper and lower surfaces leading to oscillations about a new equilibrium position. Though power contribution by viscous forces are three orders of magnitude less than the power due to pressure forces, these viscous effects manipulate the flow by influencing the strength and location of the shock such that the power contribution by pressure forces change significantly. Multiple flutter points that were part of the inviscid solution in this regime are now eliminated by viscous effects. Shock motion on the airfoil, shock reversal due to separation, and separation and reattachment of flow on the airfoil upper surface, also lead to multiple aerodynamic forcing frequencies. These flow features make the flutter boundary quantitatively sensitive to the turbulence model and numerical method adopted, but qualitatively they capture the essence of the physical phenomena.

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Schuff, Matthias, and Virginie Anne Chenaux. "Coupled Mode Flutter of a Linear Compressor Cascade in Subsonic and Transonic Flow Conditions." Journal of Physics: Conference Series 1909, no. 1 (2021): 012033. http://dx.doi.org/10.1088/1742-6596/1909/1/012033.

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Ayer, T. C., and J. M. Verdon. "Validation of a Nonlinear Unsteady Aerodynamic Simulator for Vibrating Blade Rows." Journal of Turbomachinery 120, no. 1 (1998): 112–21. http://dx.doi.org/10.1115/1.2841372.

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A time-accurate Euler/Navier–Stokes analysis is applied to predict unsteady subsonic and transonic flows through a vibrating cascade. The intent is to validate this nonlinear analysis along with an existing linearized inviscid analysis via result comparisons for unsteady flows that are representative of those associated with blade flutter. The time-accurate analysis has also been applied to determine the relative importance of nonlinear and viscous effects on blade response. The subsonic results reveal a close agreement between inviscid and viscous unsteady blade loadings. Also, the unsteady surface pressure responses are essentially linear, and predicted quite accurately using a linearized inviscid analysis. For unsteady transonic flows, shocks and their motions cause significant nonlinear contributions to the local unsteady response. Viscous displacement effects tend to diminish shock strength and impulsive unsteady shock loads. For both subsonic and transonic flows, the energy transfer between the fluid and the structure is essentially captured by the first-harmonic component of the nonlinear unsteady solutions, but in transonic flows, the nonlinear first-harmonic and the linearized inviscid responses differ significantly in the vicinity of shocks.

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Hennings, H., and W. Send. "Experimental Investigation and Theoretical Predication of Flutter Behavior of a Plane Cascade in Low Speed Flow." Journal of Engineering for Gas Turbines and Power 120, no. 4 (1998): 766–74. http://dx.doi.org/10.1115/1.2818465.

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The Institute of Aeroelasticity operates a test facility which enables aeroelastic investigations of plane cascades in low-speed flow. The test stand serves as a pilot facility to develop tools for analogous investigations in transonic flow. Eleven blades are elastically suspended in a windtunnel with a 1 × 0.2 m2 cross section. This paper describes the experimental method of determining the flutter boundary by extrapolation of the results measured in subcritical flow. A two-dimensional theoretical model of the 11 blades, including the windtunnel walls, permits the computation of unsteady pressures, forces, and moments in close relation to the experiment. The prediction of flutter is compared with experimental results. In the present investigation, the motion of the blades is constrained to pitch around mid-chord. The vibrating blades are mechanically uncoupled. Any interaction between the blades is effected by the air stream, leading to a sensitive dependence on the aerodynamic forces.

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Ji, Lucheng, Jia Yu, Weiwei Li, and Weilin Yi. "Study on aerodynamic optimal super/transonic turbine cascade and its geometry characteristics." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 231, no. 3 (2016): 435–43. http://dx.doi.org/10.1177/0954410016638875.

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The shock waves are important phenomena in transonic turbines, which cause lots of negative effects on the aerodynamic performance. Much of attention had been paid on reducing the strength of the shock waves via modifying turbine cascade geometry, and it is highly preferred to build experiences on the relationship between the cascade aerodynamic performance and the geometric parameters. The paper presents a numerical study on the aerodynamic optimal transonic turbine cascade and its geometry characteristics. Three typical Russia transonic turbine cascades with different design conditions are selected and optimized using adjoint method at three different back pressures, respectively. Thus, the best geometry parameters for optimum aerodynamic performance can be found. Then the key geometry parameters of optimized cascades are extracted and compared with the original ones. Results show that even the best designs by hands could be less efficient than ones by computer-aided optimizations. Some experiences on how to set the key geometry parameters for a best performance are obtained. The reduced shock profiling is applied to the thermal turbomachinery and machine dynamics transonic turbine by using the adjoint method. The performance of the thermal turbomachinery and machine dynamics transonic turbine was increased significantly.

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Lepicovsky, Jan, David Šimurda, and Petr Šidlof. "Verification tests of a new blade flutter research facility." MATEC Web of Conferences 345 (2021): 00020. http://dx.doi.org/10.1051/matecconf/202134500020.

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Long term strategic changes in power generation approaches will require more flexibility for large power generating turbines as an unavoidable consequence of the increasing share of power generated by alternative energy sources. Demanded flexibility for the power turbine output will augment undesired flow phenomena in the low-pressure turbine module, which will consequently enhance blade flutter problems of long slender blades in turbine last stages. In order to advance the understanding of blade flutter onset conditions, the Institute of Thermomechanics of the Czech Academy of Sciences instigated an advanced research program on blade flutter research in high-speed turbomachines. A new innovative test facility for Blade Forced Flutter research was designed and built in the High-Speed Laboratory of the Institute of Thermomechanics. The concept of the new test facility is based on extensive experience with an older Transonic Flutter Cascade facility operated at the NASA Glenn Research Center in Cleveland, Ohio. At present, the first phase of verification tests of the new facility is in progress. The ongoing steady-state tests are intended for exploration of a newly proposed quasi-stationary method to investigate instigating flow conditions leading to an onset of intense blade flutter. Results of some opening tests under steady flow conditions are presented in the paper. The blade drive mechanism for unsteady tests with oscillating blades has not yet been installed in the facility. The presented paper is a work-in-progress report on the ongoing research of complex blade flutter problems.

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Chiarelli, Mario Rosario, and Salvatore Bonomo. "Numerical Investigation into Flutter and Flutter-Buffet Phenomena for a Swept Wing and a Curved Planform Wing." International Journal of Aerospace Engineering 2019 (February 27, 2019): 1–19. http://dx.doi.org/10.1155/2019/8210235.

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The results of numerical studies carried out on high-aspect-ratio wings with different planforms are discussed: the transonic regime is analysed for a swept wing and a curved planform wing. The wings have similar aspect ratios and similar aerodynamic profiles. The analyses were carried out by CFD and FE techniques, and the reliability of the numerical aerodynamic results was proven by a sensitivity study. Analysing the performances of the two wings demonstrated that in transonic flight conditions, a noticeable drag reduction can be obtained by adopting a curved planform wing. In addition, for such a wing, the aeroelastic instability condition, consisting in a classical flutter, is postponed compared to a conventional swept wing, for which a flutter-buffet instability occurs. In a preliminary manner, the study shows that, for a curved planform wing, the high speed buffet is not an issue and at the same time notable fuel saving can be achieved.

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SATO, Iwataro, and Shojiro KAJI. "Study of transonic cascade airfoils by a panel method." Transactions of the Japan Society of Mechanical Engineers Series B 52, no. 484 (1986): 3880–87. http://dx.doi.org/10.1299/kikaib.52.3880.

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Kiss, Tibor, Joseph A. Schetz, and Hal L. Moses. "Experimental and numerical study of transonic turbine cascade flow." AIAA Journal 34, no. 1 (1996): 104–9. http://dx.doi.org/10.2514/3.13028.

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Kholodar, Denis B., Jeffrey P. Thomas, Earl H. Dowell, and Kenneth C. Hall. "Parametric Study of Flutter for an Airfoil in Inviscid Transonic Flow." Journal of Aircraft 40, no. 2 (2003): 303–13. http://dx.doi.org/10.2514/2.3094.

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Shelton, M. L., B. A. Gregory, R. L. Doughty, T. Kiss, and H. L. Moses. "A Statistical Approach to the Experimental Evaluation of Transonic Turbine Airfoils in a Linear Cascade." Journal of Turbomachinery 115, no. 3 (1993): 366–75. http://dx.doi.org/10.1115/1.2929263.

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In aircraft engine design (and in other applications), small improvements in turbine efficiency may be significant. Since analytical tools for predicting transonic turbine losses are still being developed, experimental efforts are required to evaluate various designs, calibrate design methods, and validate CFD analysis tools. However, these experimental efforts must be very accurate to measure the performance differences to the levels required by the highly competitive aircraft engine market. Due to the sensitivity of transonic and supersonic flow fields, it is often difficult to obtain the desired level of accuracy. In this paper, a statistical approach is applied to the experimental evaluation of transonic turbine airfoils in the VPI & SU transonic cascade facility in order to quantify the differences between three different transonic turbine airfoils. This study determines whether the measured performance differences between the three different airfoils are statistically significant. This study also assesses the degree of confidence in the transonic cascade testing process at VPI & SU.

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32

Chuang, H. A., and J. M. Verdon. "A Nonlinear Numerical Simulator for Three-Dimensional Flows Through Vibrating Blade Rows." Journal of Turbomachinery 121, no. 2 (1999): 348–57. http://dx.doi.org/10.1115/1.2841321.

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The three-dimensional, multistage, unsteady, turbomachinery analysis, TURBO, has been extended to predict the aeroelastic response of a blade row operating within a cylindrical annular duct. In particular, a blade vibration capability has been incorporated, so that the TURBO analysis can be applied over a solution domain that deforms with a vibratory blade motion. Also, unsteady far-field conditions have been implemented to render the computational inlet and exit boundaries transparent to outgoing unsteady disturbances and to allow for the prescription of incoming aerodynamic excitations. The modified TURBO analysis has been applied to predict unsteady subsonic and transonic flows. The intent is to validate this nonlinear analysis partially for blade flutter applications via numerical results for benchmark unsteady flows, and to demonstrate this analysis for a realistic fan rotor. For these purposes, we have considered unsteady subsonic flows through a three-dimensional version of the 10th Standard Cascade and unsteady transonic flows through the first-stage rotor of the NASA Lewis Rotor 67 fan. Some general correlations between aeromechanical stabilities and fan operating characteristics will be presented.

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33

Silva, Roberto G. A., Olympio A. F. Mello, and Joao L. F. Azevedo. "Navier-Stokes-Based Study into Linearity in Transonic Flow for Flutter Analysis." Journal of Aircraft 40, no. 5 (2003): 997–1000. http://dx.doi.org/10.2514/2.6886.

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Geissler, W. "Numerical study of buffet and transonic flutter on the NLR 7301 airfoil." Aerospace Science and Technology 7, no. 7 (2003): 540–50. http://dx.doi.org/10.1016/s1270-9638(03)00065-8.

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Lepicovsky, J., E. R. McFarland, R. V. Chima, V. R. Capece, and J. Hayden. "Intermittent Flow Regimes in a Transonic Fan Airfoil Cascade." International Journal of Rotating Machinery 10, no. 2 (2004): 135–44. http://dx.doi.org/10.1155/s1023621x04000144.

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A study was conducted in the NASA Glenn Research Center (NASA-GRC) linear cascade on the intermittent flow on the suction surface of an airfoil section from the tip region of a modern low aspect ratio fan blade. Experimental results revealed that, at a large incidence angle, a range of transonic inlet Mach numbers exist where the leading-edge shock-wave pattern was unstable. Flush-mounted, high-frequency response pressure transducers indicated large local jumps in the pressure in the leading edge area, which generates large intermittent loading on the blade leading edge. These measurements suggest that for an inlet Mach number between 0.9 and 1.0, the flow is bi-stable, randomly switching between subsonic and supersonic flows. Hence, it appears that the change in overall flow conditions in the transonic region is based on the rate of switching between two stable flow states rather than on the continuous increase of the flow velocity.To date, this flow behavior has only been observed in a linear transonic cascade. Further research is necessary to confirm this phenomenon occurs in actual transonic fans and is not the by-product of an endwall restricted linear cascade.

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36

Piovesan, Tommaso, Andrea Magrini, and Ernesto Benini. "Accurate 2-D Modelling of Transonic Compressor Cascade Aerodynamics." Aerospace 6, no. 5 (2019): 57. http://dx.doi.org/10.3390/aerospace6050057.

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Modern aeronautic fans are characterised by a transonic flow regime near the blade tip. Transonic cascades enable higher pressure ratios by a complex system of shockwaves arising across the blade passage, which has to be correctly reproduced in order to predict the performance and the operative range. In this paper, we present an accurate two-dimensional numerical modelling of the ARL-SL19 transonic compressor cascade. A large series of data from experimental tests in supersonic wind tunnel facilities has been used to validate a computational fluid dynamic model, in which the choice of turbulence closure resulted critical for an accurate reproduction of shockwave-boundary layer interaction. The model has been subsequently employed to carry out a parametric study in order to assess the influence of main flow variables (inlet Mach number, static pressure ratio) and geometric parameters (solidity) on the shockwave pattern and exit status. The main objectives of the present work are to perform a parametric study for investigating the effects of the abovementioned variables on the cascade performance, in terms of total-pressure loss coefficient, and on the shockwave pattern and to provide a quite large series of data useful for a preliminary design of a transonic compressor rotor section. After deriving the relation between inlet and exit quantities, peculiar to transonic compressors, exit Mach number, mean exit flow angle and total-pressure loss coefficient have been examined for a variety of boundary conditions and parametrically linked to inlet variables. Flow visualisation has been used to describe the shock-wave pattern as a function of the static pressure ratio. Finally, the influence of cascade solidity has been examined, showing a potential reduction of total-pressure loss coefficient by employing a higher solidity, due to a significant modification of shockwave system across the cascade.

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KOBAYASHI, Hiroshi. "Unsteady aerodynamic characteristics of annular cascade oscillating in transonic flow. 3rd Report. Low back-pressure supersonic compressor blade flutter." Transactions of the Japan Society of Mechanical Engineers Series B 52, no. 480 (1986): 2920–29. http://dx.doi.org/10.1299/kikaib.52.2920.

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Procházka, Pavel, Pavel Šnábl, Sony Chindada, Ondřej Bublík, and Václav Uruba. "On the stall flutter occurrence in a blade cascade set to turbine and compressor geometry." EPJ Web of Conferences 264 (2022): 01032. http://dx.doi.org/10.1051/epjconf/202226401032.

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A blade cascade allowing free pitch movement of five blades was developed in Institute of Thermomechanics. This model is introduced to study the phenomenon of stall flutter existing in rotary bladed wheels of steam turbines and other devices. This article describes how to induce the stall flutter for prescribed boundary conditions (as inlet velocity, various angles of attack, etc.) and gives survey about flow field differences around the cascade set to both the turbine and the compressor geometry. This experimental research utilized time-resolved Particle Image Velocimetry (PIV) to measure and to evaluate statistical quantities of the wake behind the NACA0010 profiles of the cascade. Also dynamical analysis was performed in the form of Fast-Fourier transform and also phase-locked measurement was applied. Gained knowledge will be utilized to design advanced model of the cascade allowing stall flutter examination without the use of ball bearings.

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39

Frey, Christian, Graham Ashcroft, Hans-Peter Kersken, and Daniel Schlüß. "Flutter Analysis of a Transonic Steam Turbine Blade with Frequency and Time-Domain Solvers." International Journal of Turbomachinery, Propulsion and Power 4, no. 2 (2019): 15. http://dx.doi.org/10.3390/ijtpp4020015.

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The aim of this study was to assess the capabilities of different simulation approaches to predict the flutter stability of a steam turbine rotor. The focus here was on linear and nonlinear frequency domain solvers in combination with the energy method, which is widely used for the prediction of flutter onset. Whereas a GMRES solver was used for the linear problem, the nonlinear methods employed a time-marching procedure. The solvers were applied to the flutter analysis of the first rotor bending mode of the open Durham Steam Turbine test case. This test case is representative of the last stage of modern industrial steam turbines. We compared our results to those published by other researchers in terms of aerodynamic damping and local work per cycle coefficients. Time-domain, harmonic balance, and time-linearised methods were compared to each other in terms of CPU efficiency and accuracy.

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40

SAWAKI, Yuta, Yuichi KUYA, and Keisuke SAWADA. "Study of Fast Prediction Method for Transonic Flutter Boundary Using Radial Basis Function." AEROSPACE TECHNOLOGY JAPAN, THE JAPAN SOCIETY FOR AERONAUTICAL AND SPACE SCIENCES 18 (2019): 133–41. http://dx.doi.org/10.2322/astj.jsass-d-17-00087.

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Singh, Indrajeet, R. K. Mishra, and P. S. Aswatha Narayana. "Numerical Study of a Transonic Aircraft Wing for the Prediction of Flutter Failure." Journal of Failure Analysis and Prevention 17, no. 1 (2016): 107–19. http://dx.doi.org/10.1007/s11668-016-0209-8.

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Sláma, Václav, Bartoloměj Rudas, Petr Eret, et al. "EXPERIMENTAL AND NUMERICAL STUDY OF CONTROLLED FLUTTER TESTING IN A LINEAR TURBINE BLADE CASCADE." Acta Polytechnica CTU Proceedings 20 (December 31, 2018): 98–107. http://dx.doi.org/10.14311/app.2018.20.0098.

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In this paper, experimental testing of flutter and numerical simulations using a commercial code ANSYS CFX and an in-house code TRAF are performed on an oscillating linear cascade of turbine blades installed in a subsonic test rig. Bending and torsional motions of the blades are investigated in a travelling wave mode approach. In each numerical approach, a rig geometry model with a different level of complexity is used. Good agreement between the numerical simulations and experiments is achieved using both approaches and benefits and drawbacks of each technique are commented in this paper. It is demonstrated that both used computational techniques are adequate to predict turbine blade flutter. It is concluded that validated numerical tools can provide a better insight of flutter phenomena of operationally flexible steam turbine last stage blades.

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43

Szumowski, A., J. Amecke, and J. Agocs. "Photographic study of obstacle-induced disturbations of transonic turbine cascade flow." Journal of Thermal Science 7, no. 3 (1998): 139–48. http://dx.doi.org/10.1007/s11630-998-0010-4.

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Wang, Liyue, Cong Wang, Sheng Qin, et al. "Experimental Study on Performance of Transonic Compressor Cascade with Microgroove Polyurethane Coatings." Fluids 7, no. 6 (2022): 190. http://dx.doi.org/10.3390/fluids7060190.

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Due to the harsh operating environment of aero-engines, a surface structure that provides excellent aerodynamic performance is urgently required to save energy and reduce emissions. In this study, microgroove polyurethane coatings fabricated by chemical synthesis are investigated in terms of their effect on aerodynamic performance, which is a new attempt to investigate the impact on aerodynamic performance of compressor cascade at transonic speeds. This method reduces manufacturing and maintenance cost significantly compared with traditional laser machining. Wake measurements are conducted in the high-speed linear compressor cascade wind tunnel to evaluate the performance of cascade attached with different microgroove polyurethane coatings. Compared with the Blank case, the microgroove polyurethane coatings have the characteristic of reducing flow loss, with a maximum reducing rate of 5.87% in the area-averaged total pressure loss coefficient. The mechanism of flow loss control is discussed through analyzing the correlation between the total pressure distribution and turbulence intensity distribution. The results indicate that a large quantity of energy loss in the flow field due to turbulence dissipation and the reduction in viscous drag by microgroove polyurethane coatings relates to its effect on turbulence control. This paper demonstrates a great perspective on designing micro-nano surface structure for aero-engine applications.

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45

Glodic, Nenad, Carlos Tavera Guerrero, and Mauricio Gutierrez Salas. "Blade oscillation mechanism for aerodynamic damping measurements at high reduced frequencies." E3S Web of Conferences 345 (2022): 03002. http://dx.doi.org/10.1051/e3sconf/202234503002.

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Accurate prediction of aerodynamic damping is essential for flutter and forced response analysis of turbomachinery components. Reaching a high level of confidence in numerical simulations requires that the models have been validated against the experiments. Even though a number of test cases have been established over the past decades, there is still a lack of suitable detailed test data that can be used for validation purposes in particular when it comes to aero damping at high reduced frequencies which is more relevant in the context of forced response analysis. A new transonic cascade test rig, currently undergoing commissioning at KTH, has been designed with the goal to provide detailed blade surface unsteady pressure data for compressor blades profiles oscillating at high reduced frequencies. The paper provides an overview of the blade actuation system employed in the test rig and presents the result of a series of bench tests characterizing the blade vibration amplitudes achieved with this actuation system.

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Abhari, R. S., and A. H. Epstein. "An Experimental Study of Film Cooling in a Rotating Transonic Turbine." Journal of Turbomachinery 116, no. 1 (1994): 63–70. http://dx.doi.org/10.1115/1.2928279.

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Time-resolved measurements of heat transfer on a fully cooled transonic turbine stage have been taken in a short duration turbine test facility, which simulates full engine nondimensional conditions. The time average of this data is compared to uncooled rotor data and cooled linear cascade measurements made on the same profile. The film cooling reduces the time-averaged heat transfer compared to the uncooled rotor on the blade suction surface by as much as 60 percent, but has relatively little effect on the pressure surface. The suction surface rotor heat transfer is lower than that measured in the cascade. The results are similar over the central 3/4 of the span, implying that the flow here is mainly two dimensional. The film cooling is shown to be much less effective at high blowing ratios than at low ones. Time-resolved measurements reveal that the cooling, when effective, both reduced the dc level of heat transfer and changed the shape of the unsteady waveform. Unsteady blowing is shown to be a principal driver of film cooling fluctuations, and a linear model is shown to do a good job in predicting the unsteady heat transfer. The unsteadiness results in a 12 percent decrease in heat transfer on the suction surface and a 5 percent increase on the pressure surface.

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47

Abhari, R. S., and M. Giles. "A Navier–Stokes Analysis of Airfoils in Oscillating Transonic Cascades for the Prediction of Aerodynamic Damping." Journal of Turbomachinery 119, no. 1 (1997): 77–84. http://dx.doi.org/10.1115/1.2841013.

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An unsteady, compressible, two-dimensional, thin shear layer Navier–Stokes solver is modified to predict the motion-dependent unsteady flow around oscillating airfoils in a cascade. A quasi-three-dimensional formulations is used to account for the stream-wise variation of streamtube height. The code uses Ni’s Lax–Wendroff algorithm in the outer region, an implicit ADI method in the inner region, conservative coupling at the interface, and the Baldwin–Lomax turbulence model. The computational mesh consists of an O-grid around each blade plus an unstructured outer grid of quadrilateral or triangular cells. The unstructured computational grid was adapted to the flow to better resolve shocks and wakes. Motion of each airfoil was simulated at each time step by stretching and compressing the mesh within the O-grid. This imposed motion consists of harmonic solid body translation in two directions and rotation, combined with the correct interblade phase angles. The validity of the code is illustrated by comparing its predictions to a number of test cases, including an axially oscillating flat plate in laminar flow, the Aeroelasticity of Turbomachines Symposium Fourth Standard Configuration (a transonic turbine cascade), and the Seventh Standard Configuration (a transonic compressor cascade). The overall comparison between the predictions and the test data is reasonably good. A numerical study on a generic transonic compressor rotor was performed in which the impact of varying the amplitude of the airfoil oscillation on the normalized predicted magnitude and phase of the unsteady pressure around the airfoil was studied. It was observed that for this transonic compressor, the nondimensional aerodynamic damping was influenced by the amplitude of the oscillation.

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48

Schäfer, Dominik. "Influence of Fluid Viscosity and Compressibility on Nonlinearities in Generalized Aerodynamic Forces for T-Tail Flutter." Aerospace 9, no. 5 (2022): 256. http://dx.doi.org/10.3390/aerospace9050256.

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The numerical assessment of T-tail flutter requires a nonlinear description of the structural deformations when the unsteady aerodynamic forces comprise terms from lifting surface roll motion. For linear flutter, a linear deformation description of the vertical tail plane (VTP) out-of-plane bending results in a spurious stiffening proportional to the steady lift forces, which is corrected by incorporating second-order deformation terms in the equations of motion. While the effect of these nonlinear deformation components on the stiffness of the VTP out-of-plane bending mode shape is known from the literature, their impact on the aerodynamic coupling terms involved in T-tail flutter has not been studied so far, especially regarding amplitude-dependent characteristics. This term affects numerical results targeting common flutter analysis, as well as the study of amplitude-dependent dynamic aeroelastic stability phenomena, e.g., Limit Cycle Oscillations (LCOs). As LCOs might occur below the linear flutter boundary, fundamental knowledge about the structural and aerodynamic nonlinearities occurring in the dynamical system is essential. This paper gives an insight into the aerodynamic nonlinearities for representative structural deformations usually encountered in T-tail flutter mechanisms using a CFD approach in the time domain. It further outlines the impact of geometrically nonlinear deformations on the aerodynamic nonlinearities. For this, the horizontal tail plane (HTP) is considered in isolated form to exclude aerodynamic interference effects from the studies and subjected to rigid body roll and yaw motion as an approximation to the structural mode shapes. The complexity of the aerodynamics is increased successively from subsonic inviscid flow to transonic viscous flow. At a subsonic Mach number, a distinct aerodynamic nonlinearity in stiffness and damping in the aerodynamic coupling term HTP roll on yaw is shown. Geometric nonlinearities result in an almost entire cancellation of the stiffness nonlinearity and an increase in damping nonlinearity. The viscous forces result in a stiffness offset with respect to the inviscid results, but do not alter the observed nonlinearities, as well as the impact of geometric nonlinearities. At a transonic Mach number, the aerodynamic stiffness nonlinearity is amplified further and the damping nonlinearity is reduced considerably. Here, the geometrically nonlinear motion description reduces the aerodynamic stiffness nonlinearity as well, but does not cancel it.

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Irina Carmen, ANDREI. "Numerical Study of Transonic Axial Flow Rotating Cascade Aerodynamics – Part 1: 2D Case." INCAS BULLETIN 6, no. 2 (2014): 3–13. http://dx.doi.org/10.13111/2066-8201.2014.6.2.1.

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Noorsalehi, Mohammad Hossein, Mahdi Nili-Ahmadabadi, Seyed Hossein Nasrazadani, and Kyung Chun Kim. "Aerodynamic Inverse Design of Transonic Compressor Cascades with Stabilizing Elastic Surface Algorithm." Applied Sciences 11, no. 11 (2021): 4845. http://dx.doi.org/10.3390/app11114845.

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The upgraded elastic surface algorithm (UESA) is a physical inverse design method that was recently developed for a compressor cascade with double-circular-arc blades. In this method, the blade walls are modeled as elastic Timoshenko beams that smoothly deform because of the difference between the target and current pressure distributions. Nevertheless, the UESA is completely unstable for a compressor cascade with an intense normal shock, which causes a divergence due to the high pressure difference near the shock and the displacement of shock during the geometry corrections. In this study, the UESA was stabilized for the inverse design of a compressor cascade with normal shock, with no geometrical filtration. In the new version of this method, a distribution for the elastic modulus along the Timoshenko beam was chosen to increase its stiffness near the normal shock and to control the high deformations and oscillations in this region. Furthermore, to prevent surface oscillations, nodes need to be constrained to move perpendicularly to the chord line. With these modifications, the instability and oscillation were removed through the shape modification process. Two design cases were examined to evaluate the method for a transonic cascade with normal shock. The method was also capable of finding a physical pressure distribution that was nearest to the target one.

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