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1

Stator clocking in SO3 compressor first stage rotor blades

Rządkowski R, Gnesin V, Kolodyazhnaya L, Szczepanik R

Zeszyty Naukowe. Cieplne Maszyny Przepływowe - Turbomachinery / Politechnika Łódzka

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2008

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nr 133

299--308

EN

Numerical calculations of the 3D transonic flow of an ideal gas through three-row compressor stage including the clocking effects are presented. The approach is based on the solution of the coupled aerodynamic-structure problem for the 3D flow through the turbine stage in which fluid and dynamic equations are integrated simultaneously in time. There has been performed the calculation for the stage of the compressor with rotor blades of 0.163 m. The aeroelastic characteristics are obtained for different position of the stator rows. The clocking effect influence the stability region of the rotor blades.

2

3D model of viscous flutter of turbine rotor blades

Rządkowski R., Gnesin V., Kolodyzhnaya L.

Transactions of the Institute of Fluid-Flow Machinery

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2007

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nr 120

3-22

EN

In this study, numerical simulations of 3D viscous flutter were performed and compared with the available experimental results. The calculations were carried out for bending oscillations or the cascade known as the Eleventh Standard Configuration. The developed numerical algorithm solves 3D Reynolds-averaged Navier-Stokes equation together with the Baldwin-Lomax, using the explicit monotonous second-order accurate Godunov-Kolgan finite-volume scheme and moving hybrid H-O structured grid. Comparison of the calculated and experimental results for the Eleventh Standard Configurations has shown good quantitative and qualitative agreement for local performances (unsteady pressure amplitude and phase distribution) at off-design conditions. Benchmark solutions are provided for various values of the inter-blade phase angle O, 180, -90, 90, 72, -72 deg.

3

The influence of steam extraction parameters on unsteady rotor forces

Rządkowski R., Gnesin V.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2007

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Vol. 11, No 3

253-261

EN

Aerodynamic unsteady forces of rotor blades in a turbine stage with steam extraction were calculated and blade failure was reported. A numerical calculation of the 3D transonic flow of an ideal gas through turbomachinery blade rows moving one in relation to another is presented. An ideal gas flow through mutually moving stator and rotor blades with periodicity on the whole annulus is described with unsteady Euler conservation equations, integrated using the explicit monotonous finite-volume difference scheme of Godunov-Kolgan and a moving hybrid H-H grid. In order to find the pressure distribution of steam parameters behind the rotor blades, calculations were performed for a steady flow through the stage and the steam extraction channels using the SPARC program and the unsteady forces of the rotor blade were calculated for four steam extraction conditions. For the maximal steam extraction the 10th harmonic of the axial force and moment was found to be close to the first natural frequency of the bladed disc with one nodal diameter. Thus, steam extraction can be the cause of the blade's failure.

4

Unsteady Forces Acting on Blades and the Shaft in the Last Stage of Steam Turbine with Exhaust Hood

Rządkowski R., Soliński M., Kwapisz L., Gnesin V.

Zeszyty Naukowe. Cieplne Maszyny Przepływowe - Turbomachinery / Politechnika Łódzka

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2005

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nr 128, vol. 2

471-478

EN

The aerodynamic unsteady forces acting on the rotor blades and the shaft in the last stage of turbine 13K215 were calculated. A numerical simulation of a 3D transonic ideal gas flow through turbomachinery blade rows moving relative to one another is presented, with a non-uniform pressure distribution of steam behind the rotor blades. In order to find the pressure distribution, numerical calculations were done for unsteady flow through the stage with the exhaust hood using the program FLUENT. The unsteady forces of the rotor blades were calculated. The natural frequencies of the bladed disc were compared with the frequencies of the flow excitations. Next, the unsteady forces acting on the shaft were calculated, taking the mistuning of the blades into consideration.

5

Three-Dimensional Viscous Flutter for a Row of Blades by Godunov-Kolgan Method

Gnesin V., Bykov Y., Rządkowski R.

Zeszyty Naukowe. Cieplne Maszyny Przepływowe - Turbomachinery / Politechnika Łódzka

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2005

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nr 128, vol. 1

209-218

EN

In this study, numerical simulations of 3D viscous flutter were performed and compared with the available experimental results. The calculations were carried out for bending oscillations of the cascade known as the Eleventh Standard Configuration. The developed numerical algorithm solves the 3D Reynolds-averaged Navier-Stokes equation together with Boldwin-Lomax and κ-ω turbulent model, using the explicit monotonous second-order accurate Godunov-Kolgan finite-volume scheme and moving hybrid H-O structured grid. Comparison of the calculated and the experimental results for the Eleventh Standard Configurations has shown good quantitative and qualitative agreement for local performances (unsteady pressure amplitude and phase distribution).

6

High and low frequency excitations of the last stage rotor blades and shaft of the steam turbine 13K215

Rządkowski R., Gnesin V., Soliński M.

Transactions of the Institute of Fluid-Flow Machinery

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2003

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nr 114

123-132

EN

A three-dimensional numerical analysis for aerodynamic unsteady forces of the last stage steam turbine 13K215 rotor blades and total unsteady forces acting on the shaft from considered stage, have been presented. The numerical calculations were performed for different pressure distribution behind the rotor blades in circumferential direction (p2=6000, 7500, 9000, 7500 Pa). The analysis of the 3D transonic flow of an ideal gas through turbomachinery blade rows moving relatively one to another without takong into account the blades oscillations is presented. An ideal gas flow through the mutually moving stator and rotor blades with periodicity on the whole annulus is describrd by the unsteady Euler conservation equations, which are integrated using the explicit monotonous finite-volume difference scheme of Godunov-Kolgan and moving hybrid H-H grid. The algorithm proposed allows to calculate turbine stages with an arbitrary pitch ration of stator and rotor blades. The unsteady forces acting on the rotor blades in axial, tangential and radial directions were found. Due to non-uniform pressure distribution the low frequency excitation appeared. Values of the low frequency excitations are greater than the high frequency excitation caused by the stator blades. Next, the total unsteady forces acting on the shaft from all rotor blades in considered stage were calculated. Values of unsteady forces are very small in comparison to the single blade. The low frequency excitation disappeared. The high frequency excitation is depended on the number of rotor blades.

7

Vibrations of disc with twisted blades in 3D compressible flow

Rządkowski R., Gnesin V.

Archives of Civil Engineering

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2003

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Vol. 49, nr 3

355-373

EN

Numerical calculations of the 3D transonic flow of an ideal gas through turbomachinery rotor blade row and stator, rotor blades moving relatively one to another with taking into account the blades oscillations are presented. The approach is based on the solution of the coupled aerodynamic-structure problem for the 3D flow through the turbine stage in which fluid and dynamic equations are integrated simultaneously in time. An ideal gas flow through the mutually moving stator and rotor blades with periodicity on the whole annulus is described by the unsteady Euler conservation equations, which are integrated using the explicit monotonous finite-volume difference scheme of Godunov-Kolgan and moving hybrid H-H grid. The structure analysis uses the modal approach and 3D finite element model of a blade. There has been performed the calculation for the last stage of the steam turbine with rotor blades of 0,765 m.

PL

Przedstawiono obliczenia numeryczne w 3D transonicznym przepływie gazu doskonałego poprzez łopatki wirnikowe i kierownicze, gdzie drgające łopatki wirnikowe poruszają się względem siebie. Model jest oparty na rozwiązaniu sprzężonego zagadnienia 3D mechaniczno-przepływowego poprzez stopień turbinowy. W modelu równania przepływu i dynamiki są całkowane z krokiem czasowym. Przepływ gazu doskonałego przez nieruchome łopatki kierownicze i obracające i drgające łopatki wirnikowe, z założeniem periodyczności na części obwodu, został opisany niestacjonarnymi równaniami Eulera, które zostały scałkowane przy pomocy metody objętości skończonych — Godunowa-Kolgana na ruchomych siatkach typu H-H. W modelu mechanicznym wykorzystano metodę superpozycji modalnej i 3D model MES dla łopatki. Obliczenia przeprowadzono dla ostatniego stopnia parowej turbiny o długości łopatek wirnikowych równej 0.765 [m].

8

Flutter and 3D unsteady forces acting on the last stage rotor blades with non-uniform pressure distribution behind the rotor blades of the steam turbine 13K215.

Rządkowski R., Gnesin V.

Advances in Manufacturing Science and Technology

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2003

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Vol. 27, nr 3

55-63

EN

A three-dimensional numerical analysis for aerodynamic unsteady forces and flutter parameters acting on the last stage steam turbine 13K215 rotor blades and total unsteady forces acting on the shaft have been presented. The low frequency excitation was simulated for a 94 rotating blades with 54 nozzles, it was assumed that the pressure behind tha rotor blades is changing in the circumferential direction. The flutter parameters of this stage were calculated.

PL

Przeprowadzono trójwymiarową analizę numeryczną niestacjonarnych sił aerodynamicznych i parametrów flateru łopatek wirnikowych ostatniego stopnia turbiny 13k215. Przeanalizowano wymuszenia niskoczęstotliwościowe dla 94 łopatek wirnikowych i 54 kierowniczych, zakładając równomierny rozkład ciśnienia za łopatkami wirnikowymi.

9

3D inviscid flutter of IV Standard Configuration. Part.I. Harmonic oscillations

Gnesin V., Rządkowski R.

Transactions of the Institute of Fluid-Flow Machinery

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2002

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nr 111

37-50

EN

A three-dimensional nonlinear time-marching method and numerical analysis for aerolastic behaviour of oscillating blade row of the IV Standard Configuration has been presented. The approach is based on the solution of the coupled fluid-structure problem in which the aerodynamic and structural equations are integrated simultaneously in time. In this formulation of a coupled problem, the interblade phase angle at which stability (or instability) would occur, is a part of the solution. The ideal gas flow through multiple interblade passage (with periodicity on the whole annulus) is described by the unsteady Euler equations in the form of conservative laws, which are integrated by use of the explicit monotonous second order accurate Godunov-Kolgan finite volume scheme and a moving hybrid H-H (or H-O) grid. The structure analysis uses the modal approach and 3D finite element model of the blade. The blade motion is assumed to be a linear combination of modes shapes with the modal coefficients depending on time. The influence of the natural frequencies on the aerodynamic for the Fourth Standard Configuration is shown. The instability regions for the first two modes shapes and the distribution of the aerodamping coefficient along blade length were shown for a harmonic oscillation with the assumed interblade phase angle.

10

3D inviscid flutter of IV Standard Configuration. Part II. Coupled fluid-structure oscilstions

Rządkowski R., Gnesin V.

Transactions of the Institute of Fluid-Flow Machinery

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2002

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nr 111

51-71

EN

A three-dimensional nonlinear time-marching method and numerical analysis for aerolastic behaviour of oscillating blade of the IV Standard Configuration has been presented. The approach is based on the solution of the coupled fluid-structure problem in which the aerodynamic and structural equations are integrated simultaneously in time. In this formulation of a coupled problem, the interblade phase angle at which stability (or instability) would occur, is a part of the solution. The ideal gas flow through multiple interblade passage (with periodicity on the whole annulus) is described by the unsteady Euler equations in the form of conservative laws, which are integrated by use of the explicit monotonous second order accurate Godunov-Kolgan finite volume scheme and moving hybrid H-H (or H-O) grid. The structure analysis uses the modal approach and 3D finite element model of the blade. The blade motion is assumed to be a linear combination of mode shapes with the modal coefficients depending on time. The influence of the natural frequencies on the aerolastic coupled oscillations for the Fourth Standard Configuration is shown. It has been shown that interaction between modes plays an important role in the aerolastic blade response. This interaction has essentially nonlinear character and leads to blade limit cycle oscillations. The sign of the aerodamping coefficient calculated for the harmonic oscillations, may be considered only as a necessary, but not a sufficient condition for self-exited oscillations.

11

Aeroelastic behaviour of the last stage steam turbine blades. Part II. Coupled fluid-structure oscillations

Gnesin V., Rządkowski R., Gnesin V., Rządkowski R.

Transactions of the Institute of Fluid-Flow Machinery

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2001

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nr 108

73-94

EN

This paper presents the partial integrated method which has been employed for aeroelasticity predictions of the long steam turbine blades. The approach is based on the solution of the coupled fluid - structure problem in which the aerodynamic and structural dynamic equations are integrated simultaneously in time, thus providing the correct formulation of a problem. The ideal gas flow around the blade row is described by the unsteady 3D Euler equations in conservative form, which are integrated by using the explicit monotonous second-order accurate Godunov-Kolgan, finite volume scheme and moving grids. In the structural analysis the modal approach is used. The natural frequencies and modal shapes of the blade were calculated by using 3D finite element model. The coupled fluid-structure oscillations were shown for 5 modes shapes separately and with taking into account the interaction of five natural modes.

12

Aeroelastic behaviour of the last stage steam turbine blades. Part I. Harmonic oscillations

Rządkowski R., Gnesin V., Gnesin V.

Transactions of the Institute of Fluid-Flow Machinery

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2001

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nr 108

59-72

EN

This paper presents an integrated non-linear numerical model for aeroelasticity predictions of the long steam turbine blade. The approach is based on the solution of the coupled fluid - structure problem in which the aerodynamic and structural dynamic equations are integrated simultaneously in time, thus providing the correct formulation of a problem. The ideal gas flow around multiple interblade possages (with periodicity on the whole annulus) is described by the unsteady 3D Euler equations in conservative form, which are integrated by using the explicit monotonous second - order accurate Godunov-Kolgan, finite volume scheme and moving grids. In the structural analysis the modal approach is used. The natural frequencies and modal shapes of the blade were calculated using 3D finite element model. The instability regions for 5 models shapes and the distribution of the aerodamping coefficient along the blade length were shown for a harmonic oscillation with the assumed interblade phase angle.

13

3D inviscid flutter of rotor blades and stator/rotor stage

Rządkowski R., Gnesin V.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2001

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Vol. 5, No 4

567-577

EN

The approach is based on solution of the coupled aerodynamic-structure problem for 3D flow through the turbine stage in which fluid and dynamic equations are integrated simultaneously in time. An ideal gas flow through the mutually moving stator and rotor blades with periodicity on the whole annulus is described by unsteady Euler conservation equations, which are integrated using the explicit monotonous finite-volume difference scheme of Godunov-Kolgan and moving hybrid H-H grid. The structure analysis uses the modal approach and 3D finite element model of a blade. A calculation was performed for the last stage of the steam turbine with rotor blades of 0.765 m.

14

Analysis of 3D flutter of long steam turbine blades - harmonic oscillations

Rządkowski R., Gnesin V.

Zeszyty Naukowe. Cieplne Maszyny Przepływowe - Turbomachinery / Politechnika Łódzka

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2000

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nr 117, vol. 1

239-248

EN

This paper presents an integrated non-linear numerical model for aeroelasticity predictions of the long steam turbine blades. The approach is based on the solution of the coupled fluid - structure problem in which the aerodynamic and structural dynamic equations are integrated simultaneously in time, thus providing the correct formulation of a problem. The ideal gas flow around multiple interblade passages (with periodicity on the whole annulus) is described by the unsteady 3D Euler equations in conservative form, which are integrated by using the explicit monotonous second - order accurate Godunov - Kolgan finite - volume scheme and moving grids. In the structure analysis the modal approach is used. The natural frequencies and modal shapes of the blade were calculated by using 3D finite element model. The instability regions for 5 modes shapes were shown for a harmonic oscillation with the assumed interblade phase angle.

15

The numerical and experimental verification of the 3D inviscid flutter code

Rządkowski R., Gnesin V.

Transactions of the Institute of Fluid-Flow Machinery

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2000

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Nr 106

69-95

EN

In this study the numerical calculations were performed to compare the theoretical results of 3D inviscid flutter code with experimental results due to Bolcs and Fransson. The calculations were carried out for the torsional oscillations of the compressor cascade known as the First Standard Configuration, and the bending oscillations of the steam turbine cascade which has become the Fourth Standard Configuration. The developed numerical algorithm solves the Euler equation in conservative form, which is integrated by using the explicit monotonous second - order accurate Godunov - Kolgan finite - volume scheme and the moving hybrid grid. The structural model is based on the 3D and 1D models. The comparision of calculated and experimental results for the 1st and 4th Standard Configurations has shown the good quantitative and qualitative agreement for both integral performances (aerodamping coefficient) and local performances (unsteady pressure amplitude and phase distribution).

PL

Weryfiakacja numeryczna i eksperymentalna trójwymiarowego kodu obliczeniowego do obliczeń nielepkich drgań samowzbudnych. W pracy przedstawiono obliczenia numeryczne wykonane przy pomocy trójwymiarowego kodu obliczeniowego do obliczeń nielepkich drgań samowzbudnych, które porównano z danymi eksperymentalnymi Bolcs'a i Frasson'a. Obliczenia przeprowadzono dla drgań skrętnych kaskady sprężarki, znanej pod nazwą Pierwszej Konfiguracji Standardowej, oraz dla drgań giętych kaskady turbinowej parowej, którą nazwano Czwartą Konfiguracją Standardową. Opracowany algorytm numeryczny umożliwia rozwiązanie równania Eulera w formie zachowawczej, jest ono całkowane przy użyciu jawnego schematu Godunova - Kolgana drugiego rzędu dla objętościskończonych oraz przemieszczającej się siatki hybrydowej. Model strukturalny oparty jest na modelach 3D i 1D. Porównanie obliczeń z danymi eksperymentalnymi dla przypadków Pierwszej i Czwartej Konfiguracji Standardowej pokazuje dobrą zgodność ilościową oraz jakościową dla obu charakterystyk integralnych (tj. współczynnik tłumienia aerodynamicznego) oraz charakterystyk lokalnych (niestacjonarna amplituda ciśnienia i rozkład fazowy).

16

A theoretical model of 3D flutter in subsonic, transonic and supersonic inviscid flow

Gnesin V., Rządkowski R.

Transactions of the Institute of Fluid-Flow Machinery

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2000

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Nr 106

45-68

EN

A three - dimensional nonlinear time - marching method for aeroelastic behaviour of oscilating turbine blade row has been presented. The approach is based on the solution of the coupled fluid - structure problem, where the aerodynamic and structural dynamics equations are integrated simultaneously in time, thus providing the correct formulation of a coupled problem as the interblade phase angle at which stability (instability) would occur is also a part of solution. The ideal gas flow around multiple interblade passages (with periodicity on the whole annulus) is described by the unsteady Euler equations in conservative form, which are integrated by using the explicit monotouns second - order accurate Godunov - Kolgan finite - volume scheme and moving hybrid H-O (or H-H) grid. The structural model is bades on the 3D and 1D models. In 3D model the mode shapes and natural frequencies have been obtained via standard FE analysis techniques. The 1D blade model applied here is a one - dimensional beam described by an extended beam - theory including all important effects on a rotating blade. The fluid and the structural equations are solved using the direct integration method or the modal superposition method. The fluid - structure model is also presented for a very simple two degree of freedom blade model.

PL

Teoretyczny model trójwymiarowego flatteru nielepkiego przepływu poddźwiękowego, transonicznego i ponaddźwiękowego. Przedstawiono trójwymiarową, nieliniową metodę kroczącą w czasie do badań zachowań aeroelastycznych drgającego szeregu łopatek turbiny. Metoda jest oparta na rozwiązaniu zagadnienia sprężonego płyn - wieniec turbiny, gdzie równania opisujące aerodynamikę i dynamikę konstrukcji całkowane są równocześnie w czasie i dlatego przedstawiają właściwy opis zagadnienia sprzężonego, gdyż międzyłopatkowy kąt fazowy, dla którego obserwujemy stabilność (niestabilność) jest także częścią rozwiązania. Przepływ kanałami międzyłopatkowymi gazu doskonałego (z określoną okresowością na całym pierścieniu) jest opisany niestacjonarnymi równaniami Euela, w formie zachowawczej, które są całkowane przy użyciu jawnego schematu drugiego rzędu w formie objętości skończonych Godunowa - Kolgana oraz przemieszczającej się siatki hybrydowej H-O (lub H-H). Model strukturalny jest oparty na modelach 3D i 1D. W przypadku modelu 3D częstości własne oraz postacie drgań własnych otrzymywane są przy pomocy standardowych technik elementów skończonych. Jednowymiarowy model łopatki, zastosowany w pracy, jest jednowymiarową belką opisaną przy pomocy rozszerzonej teorii belek, która umożliwia opis wszystkich ważnych efektów wirującej łopatki. Równania płynu oraz konstrukcji rozwiazywane są przy użyciu metody bezpośredniego całkowania lub superpozycji modalnej. Model płyn - konstrukcja jest także pokazany na przypadku bardzo prostego modelu łopatki o dwóch stopniach swobody.

17

The numerical study of 3D flutter in a transonic blade row

Gnesin V., Rządkowski R., Kolodyazhnaya L

Zeszyty Naukowe. Cieplne Maszyny Przepływowe - Turbomachinery / Politechnika Łódzka

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1999

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nr 115

149--158

EN

A three - dimensional nonlinear time - marching method and numerical analysis of 3D flutter for oscillating blade row has been presented. The approach is based on the solution of the coupled fluid - structure problem in which the aerodynamic and structural dynamic equations are integrated simultaneously in time, thus providing the correct formulation of a problem, as the interblade phase angle at which a stability (instability) would occur is a part of solution. The ideal gas flow around multiple interblade passages (with periodicity on the whole annulus) is described by the unsteady Euler equations in conservative form, which are integrated by using the explicit monotonous second - order accurate Godunov - Kolgan finite - volume scheme and moving grids. In he structure analysis the modal approach is used. The natural frequencies and modal shapes of the blade were calculated by using the different models: 3D finite element model; 1D blade model is of a one-dimensional beam described by an extended beam-theory including all important effects on a rotating blade.

18

Comparison of the 2d and 3d models of flutter of a palisade in an inviscid flow

Rządkowski R., Gnesin V., Kovalyov A

Zeszyty Naukowe. Cieplne Maszyny Przepływowe - Turbomachinery / Politechnika Łódzka

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1999

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nr 115

349--356

EN

In recent years the works of the coupled fluid-structure problems appeared. The computational method used to solve this problem was based on a time-marching algorithm, so it was natural to consider a time domain flutter analysis method. The time domain method of flutter analysis is based on the simultaneous integration in time of the equation of motion for the structure and the fluid. In this work the comparison of the 2D and 3D Flutter results for the turbine cascade (IV Configuration) is shown. It was observed that the negative aerodamping coefficient calculated for the harmonic oscillation are not sufficient condition for growing oscillation during fluid-structure interaction.

19

Two - dimensional flutter of a bladed disc

Grządkowski R., Gnesin V., Kovalyov A.

Prace Instytutu Lotnictwa

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1998

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nr 4 (155)

101-110

EN

In the present study, the simultaneous time domain method is used to determine the aeroelastic stability of a bladed disc. The unsteady equations of motion for the structure and the fluid are integrated simultaneously in time starting with a steady flowfiel and a small initial disturbances or the assumed unsteady forces, applied to the airfoil. Each blade is allowed to move independently, and the motion of all blades is analyzed to determine the aeroelastic stability of a palisade. The flow model is capable of representing 2D-flows over a wide Mach number range from low subsonic to supersonic, including transonic flows. The aerodinamic model fully accounts for blade thickness and camber and theangle-of-attack effects. The unsteady Euler 2D equations are integrated using an explicit monotonous second-order accuracy Godunov's type difference scheme. The blade model applied here is a one-dimensional beam described by an extended beam-theory including all important effects on a rotating blade. The disc is modelled by a moderately thick plate theory. Assuming rigidly fixed blades on the disk rim, the displacements of any particle of the bladed disc are found. In the time domain method, the equation of motion was integrated in time, by a method of constant average acceleration (Wilson method). The initial conditions are the steady flowfield and an assumed unsteady forces applied to the blades. In order to verfied the numerical program written on the basis of algorithms presented above numerical results weren compared to experimental and theoretical results presented by Bolcs and Fransson. The First Standard Configuration (compressor cascade in low subsonic flow) were used. The good agreement between the experimental and numerical results was observed.

PL

W pracy przedstawiono metodę bezpośredniego całkowania równań ruchu do wyznaczenia aeroelastycznej stabilności tarczy z łopatkami. Równania opisujące drgania modelu mechanicznego i przepływającego czynnika scałkowano wykorzystując jako warunki początkowe przepływ ustalony oraz małe dowolne wymuszenie jednej lub wszystkich łopatek. W przedstawionym modelu każda łopatka może poruszać się w sposób wynikający ze sprzężonych równań ruchu cieczy i palisady, ruch łopatki jest analizowany do wyznaczenia aeroelastycznej stabilności palisady. Model przepływowy opisuje dwuwymiarowy przepływ w szerokim zakresie liczb Macha od przepływu poddźwiękowego, transonicznego do naddźwiękowego. W modelu tym uwzględniono grubość i wygięcie oraz ich wpływ na lokalny kąt natarcia. Niestacjonarne dwuwymiarowe równania Eulera scałkowano wykorzystując metodę objętości skończonych Godunova-Kolgana, drugiego rzędu dokładności. Łopatkę zamodelowano przy pomocy modelu belkowego jednowymiarowego, uwzględniającego wstępne skręcene, niesymetryczne przekroje poprzeczne i kąt ustawienia łopatek na tarczy oraz uwzględniono wpływ ruchu obrotowego. Tarczę zamodelowano płytą średniej grubości. Zakładając sztywne zamocowanie łopatek na tarczy znaleziono przemieszczenia dowolnego elementu tarczy z łopatkami. W przedstawionej metodzie (time-domain) równania ruchu scałkowano wykorzystując metodę Wilsona. W celu zweryfikowania programu numerycznego otrzymane wyniki zostały porównane z wynikami eksperymentalnymi i teoretycznymi przedstawionymi przez Bolcsa i Franssona. Do testowania wykorzystano pierwszą standardową konfigurację (palisada sprężarkowa dla poddźwiękowego przepływu z małymi liczbami Macha). Uzyskano dobrą zbieżność wyników eksperymentalnych i numerycznych.