DOI 10.1007/s11223-015-9708-1 Strength of Materials, Vol. 47, No. 5, September, 2015

AERODYNAMIC FACTORS OF INFLUENCE ON THE RESONANCE VIBRATION OF GAS TURBINE COMPRESSOR BLADES Yu. M. Tereshchenko, E. V. Doroshenko, A. Tehrani,

UDC 629.7.035.03-036.34

and J. Abolhassanzade The paper presents results of an analysis of the gas-dynamic control of flow around the guide blades of axial compressor stage for the vibration intensity of rotor blades. It has been shown that gas-dynamic action on flow around blades equalizes the flow velocity field before the rotor and reduces the level of resonant stresses in blades in the case of their resonant excitation. Keywords: gas turbine engine, compressor, blade, vibrations, flow-around. Problem Statement. The vibration of the rotor blades of compressors and turbines is one of the most difficult and acute problems, which arise when creating aircraft gas turbine engines. This is accounted for by its complex nature since it involves the solution of problems of unsteady aerodynamics and vibrations of blades that are in elastic interaction with other rotor elements, their fatigue strength and others. All vibration modes of compressor blades, viz self-excited vibrations (flutter), resonance vibrations, acoustic resonance and rotating stall, are caused by the action of variable aerodynamic forces on them. The most common vibration mode of turbomachine blades is resonance vibrations with frequencies that are a multiple of the rotational speed of rotor [1]. They may be divided into two types: low-frequency vibrations, which are excited by the harmonics of large-scale flow nonuniformity at the compressor inlet, and high-frequency vibrations, which are caused by the intersection of rotating blades with the aerodynamic edge wakes of inlet guide blades and guide blades where there is a circumferentially periodic flow non-uniformity. In the case of cantilever design of guide blades, resonance vibrations can also arise in them in view of their reverse interaction with the rotating blades of the neighboring rotors. In the case of resonance vibrations due to the circumferentially periodic non-uniformity of air flow parameters at the compressor inlet of gas turbine engine, the mechanical stresses s in blades increase by several fold and may reach limiting values subject to the strength conditions, which results in some cases in the failure of blades. In view of this, the problem of studying and analyzing aerodynamic factors and the effect on the vibration and dynamic strength of compressor blades is important and topical. It is the purpose of this work to determine gas-dynamic effect on the structure of air flow after stator elements for the purpose of reducing its circumferential non-uniformity and decreasing thereby the level of resonant stresses in the compressor blades of gas turbine engine. Solution of the Problem and Analysis of Results. When a viscid gas flows around the inlet guide blades (IGB) and guide blades, aerodynamic edge wakes are formed, which are a source of circumferentially periodic flow non-uniformity before the rotating blade ring. In this case, the velocity and total pressure in the wake behind the stationary body are lower than in the external flow. Due to the cyclic variation of flow parameters before the blade ring, the aerodynamic loads acting on the blades vary. Their cyclic variation in value and direction is determined by the variation of the angle of attack and National Aviation University, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 5, pp. 73 – 81, September – October, 2015. Original article submitted July 6, 2015. 0039–2316/15/4705–0711 © 2015 Springer Science+Business Media New York

711

Fig. 1. Variation of flow around rotating blades and total aerodynamic force, which acts on them, in the presence of circumferentially periodic flow non-uniformity. relative flow velocity (Fig. 1). Flow non-uniformity in absolute motion after the IGBs causes a non-uniformity in relative motion, too, at the rotor wheel (RW) inlet. The flow leaves the cascades of the IGBs with the absolute velocity C1 at angle a and enters the cascades of the RW blades with the relative velocity W1 at angle b1 to their front. At the moment when rotating blade passes the aerodynamic wake region after the trailing edges of the IGBs, the absolute velocity changes its value, owing to which the relative velocity changes, too, both in value and in direction. This results in the cyclic variation of the conditions of flow around rotating blades, which manifests itself by the variation of pressure force distribution over the airfoil and the restructuring of the boundary layer, i.e. they are in a pulsating flow. Since the relative velocity W1¢ in the wake is lower than the velocity W1 after the wake boundary, the pressure force on the back of rotor blade will decrease, and that on its concave surface will increase relative to the calculated value. The dependence of the coefficient of normal force, which acts on the airfoil in the case of intersection of wakes to the blade chord, on the angle of attack is shown in Fig. 2 [2]. It can be seen that in the range of the angle of attack in question, it differs by a factor of 2–3. Thus, circumferentially periodic flow non-uniformity in axial compressor, which is caused by pulsating flow around blade rings from aerodynamic wakes after compressor stator elements, is one of important factors of the vibration excitation of rotor blades and guide blades. This vibration process may be classified as stable resonance vibrations [3], which are excited when the natural vibration frequencies of blades agree with the perturbation frequency nz, where n is rotational speed and z is the number of the blades of the cascades. It should be noted here that in terms of compressor vibration strength, the vibrations of the rotor blades of the first compressor stages are the most dangerous [1, 2, 4]. 712

Fig. 2. Dependence of the coefficient of normal force on the angle of attack for stationary blade (1) and vibrating blade (2). Let us consider the main factors determining the vibration intensity of blades. If it is assumed according to [3] that the variation of aerodynamic force in time is of the form of rectangular pulses, then the amplitude of change in the pressure forces of the ith harmonic is given by Pi = Pb

2 Dt S 2 + C 2 = Pb Ai S 2 + C 2 , sin pi T pi

where S = sin

2pt 1 2pt n i + ... + sin i, T T

C = 1 + cos

2pt n 2pDt i + ... + cos i, T T

Dt takes into account the width of aerodynamic wake and S 2 + C 2 the number of inlet guide T blades or other stator elements before the vibrating rotor blades. The total aerodynamic force that acts on the blade in the plane of vibrations is given by

the quantity sin pi

Pb = Pa cos q + Pu sin q, b

b

where q is the angle between the direction of vibration and a normal to the front of cascades (in this particular case, it is assumed that q = a , Fig. 1) [3]. The value of the forces Pa and Pu is determined from the momentum b

b

equation, written in projections onto the abscissa and ordinate axes, the direction of which coincide with the direction of the mainstream flow and the line of the front of the cascades: Pa = b

F1 + F2 r1 ( p - 1), 2Z b

Ð u = r1 H t C1aU r2 b

F1 . rav Z b

In the case of absence of gas-dynamic effect on the aerodynamic wakes, the variation of the total aerodynamic force due to circumferentially periodic flow non-uniformity is defined by the relation: DPi = f ( DPa ; DPu ), where DPa and DPu are changes in total aerodynamic force in the case of periodic flow non-uniformity [5]. 713

The total variable aerodynamic force, which acts in resonance vibrations, will be DPi = f ( DPa ; DPu ), where P j is variable aerodynamic force arising from change in circumferential blade velocity U b in vibrations with the relative velocity W1 . The work done by variable aerodynamic force in blade vibrations during the vibration period is given by T 1

A = ò ò [ d1 ( DPi ) + d 2 ( DP j )] xdt = Ae + Ad , 0d

where Ae corresponds to the aerodynamic excitation of blade vibrations and Ad to their aerodynamic damping [2]. The energy balance in vibrations will manifest itself by the fact that the work of variable aerodynamic force A will be equal to energy dissipation in the material of blade and in its lock joint to the disk. We can determine from this the vibration amplitude of blade in its resonance vibrations and hence resonant stresses, too, which arise in it. From the energy balance we can derive a formula for the determination of the stress s r in the root section of blades [6] as the most stressed one:

sr =

dPi pI 1

1 k

ìï bk C k2 bk C k5 I ù üï 1 l 2 é Dt Dt ö æ d + 1 d P P B 3 - 19. 8 f 1 2 B m + Ñ 3¢ Å ç ÷ íÑm¢ Å 0 pt ú ý av T ø l I 1 KC k EC k êë T è l2 û ïþ ïî

,

(1)

where

Âm =

1æ

0.164 æ a ç 1 + 0. 4 k ç Ck è 1

I 1 = ò xdl , 0

ö ÷ ÷ ø

2

a

2

ò çç 1 + C 2 0è

1

I 2 = ò x 2 dl , 0

ö æ C ÷b ç ÷ ç Ck ø è

3

ö ÷÷ X ¢¢dl, ø

B3 =

æ a K = 0. 5 X k¢¢ çç 1 + 0. 4 k Ck è

é æa 0. 0067 ê1 + çç k ê è Ck ë æ a ç 1 + 0. 4 k ç Ck è ö ÷, ÷ ø

ö ÷ ÷ ø ö ÷ ÷ ø

2ù

2

2

ú ú û ,

K = 1.4–2.0.

Analysis of this expression shows that among many factors that characterize stress level in resonance vibrations of blades, the degree of flow non-uniformity dPi affects directly the quantity s r . In view of this, it is obvious that the level of resonant in compressor rotor blades can be reduced by reducing the degree of circumferentially periodic flow non-uniformity before the blade ring. We shall estimate the degree of periodic non-uniformity by the quantity: a=

v max , C0

where C 0 is constant velocity in the flow core between wakes, and v max is the maximum additional velocity in the wake. 714

The intensity of control of flow-around is estimated by the injection momentum coefficient [7]: Cm =

r b C b2 b 1 , h 2 t s sin g r 0C 0

where r 0 is density of the mainstream flow, C b and r b are the velocity and density of air that is blown out of the airfoil slots into the boundary layer, hs is relative slot height, hs = hs b, and b t is cascade solidity. The objective of the study is to find a functional relation between the degree of periodic flow non-uniformity and the intensity of control of aerodynamic wakes in the form: a = f (x, Cm ). Analytical calculations are based on the finding of the momentum thickness d ** S after the trailing edge of blades in the cascade, a quantity that determines the intensity of aerodynamic wakes. Let us write a momentum equation for the boundary layer on the surface in the form t dd ** W ¢ + (2d ** + d * ) = . dx W rW 2

(2)

After the use of the functions: f¢=

W ¢ d ** X, W

x¢ =

t rW 2

X

(3)

Eq. (2) is rearranged in the form: df ¢ W ¢ W ¢¢ {(1 + m)x ¢ - [3 + m + (1 + m) H ¢ ] f ¢} + = f ¢, dx W W¢ where m=

R d** X ¢ X

,

R d** =

Wd ** , v

X = f ( R d** ).

We shall take into account the effect of pressure gradient on the flow in the wall layers through the ratio W¢d W

**

and the change in the Reynolds number Re with the aid of the function X , which we represent as

X =

rW 2 . t

Assuming for laminar flow that x = 0. 219, X = R d** , and m =1,, we have F1 ( f ) = (1 + m)x - [3 + m + (1 + m) H ] f = 0. 44 - 5. 8 f .

(4)

For a developed turbulent boundary layer: F ( f ) = 1. 25 - 4. 8 f .

(5) 715

Analysis of expressions (4) and (5) shows that for different flow conditions, the function F ( f ) is of linear nature and can be approximated by the relation: (6) F ( f ) = a + bf . In view of (6), we reduce Eq. (2) to a linear equation df W ¢ W¢ ö æ W ¢¢ -b = a +ç ÷ f ¢, dx W W ø è W¢ the solution of which we write in the form: f =

aW ¢ Wb

x

òW

b -1

dx.

(7)

0

From expressions (7) and (3) we find x

Wd ** a W b -1 dx. = b- 2 ò v vXW 0 Writing X 1 = XR d** , we get X =

x

a vW

b- 2

òW

b -1

dx.

0

Let us write expressions for the determination of momentum thickness: d ** S =

2v W

X

for laminar flow, and d ** S =

2v é X ù W êë153. 2 úû

6/ 7

for turbulent flow. d* d ** , we can obtain a functional dependence of the =2 a a degree of periodic flow non-uniformity on the distance to the trailing edge of blades: Using the approximate relationship x app = 2

éæ t d ** S a = K1 êç 2 sin b 2 ç b a êëè

ö bù ÷ ú ÷ xú ø û

0. 5

.

Let us determine the momentum thickness in the wake after compressor stator with control of flow around surfaces in the form b æ ö d ** = ç d 0** - Cm r 0 Wb sin g + d x** ÷ , t è ø where d ** x is the momentum thickness on the area from the site of the slots to the trailing edge. 716

Fig. 3. Dependence of the relative level of resonant stresses in the root section of compressor blades on the intensity of control of aerodynamic wakes {curves – calculation by (8), symbols – experimental results [5]}. In consideration of the assumptions made, we have am

é æ t ö sin b 2 b æ ** b ** = K ê2 ç ÷ ç d S 0 - Cm r m Wm sin g - d S x t ë è bø a xè

öù ÷ú øû

0. 5

,

where r m and Wm are determined for the blade passage in accordance with the Zhukovskii–Stechkin theorem for cascade. Expression (1) for the determination of the level of resonant stresses in the case of resonance vibrations at the different intensity of control of aerodynamic wakes can be rearranged to the dimensionless form:

sr =

sr = s r0

é ùæ 1 u êb 0 - arccos W (1 + a m ) ú çç 1 + a 1 m ë ûè

2

ö æ ÷ - ç b 0 - arccos u ç ÷ W1 è ø 2

æ é ùæ 1 ö u u êb 0 - arccos W (1 + a ) ú ç 1 + a ÷ - çç b 0 - arccos W ø 1 1 è ë ûè

ö ÷÷ ø

ö ÷÷ ø

,

(8)

where s r is stresses at Cm = 0. 0

Figure 3 shows researh results of the effect of control of aerodynamic wakes on the level of resonant stresses in forced resonance vibrations of the blades of axial compressor stage. Thus, the results of the study showed that at the intensity of gas-dynamic control of flow-around which corresponds to the values Cm = 0.01–0.015 in the number range M 0 = 0.4–0.8, the level of resonant stresses can be reduced by 90–95% relative to the initial one, i.e., vibrations of the blades are practically avoided. At the intensity of control of aerodynamic wakes above the optimal values, an increase in resonant stresses, which is caused by vibrations of blades, is observed again. CONCLUSIONS 1. The gas-dynamic control of flow around stator elementsmakes it possible to reduce the level of resonant stresses in the compressor rotor blades in their resonance vibrations through the influence on circumferentially 717

periodic flow non-uniformity, which is caused by the aerodynamic wakes after stator elements, which are before the rotor blade ring. 2. Relations have been derived, which allow one to assess the effect of the intensity of gas-dynamic control of flow around guide blades on the level of resonant stresses in the rotating blades. 3. A comparison of the results of calculations from the obtained relations with the results of experimental studies showed their good agreement. 4. It has been shown that at the intensity of gas-dynamic control of flow around the guide blades Cm = 0.01–0.02, the level of resonant stresses in the root section of rotor blades in their resonance vibrations can be reduced by 90–95%. REFERENCES 1. 2. 3. 4. 5. 6. 7.

718

G. S. Samoilovich, Vibrations Excitation of Turbomachine Blades [in Russian], Mashinostroenie, Moscow (1975). G. S. Samoilovich, Unsteady Flow Around and Aeroelastic Vibrations of Turbomachine Cascades [in Russian], Nauka, Moscow (1969). V. A. Kulagina, “Approximate calculation of the aerodynamic excitation and damping of resonance vibrations of compressor blades,” Lopat. Mashiny Struin. Apparat., No. 4, 51–64 (1968). F. D. Heiman, “Vibrations of turbine blades arising from the action of edge wakes of nozzle apparatus blades,” Énerg. Mash. Ustan., No. 4, 1–20 (1969). Yu. M. Tereshchenko, “Investigation of methods of reducing the level of vibratory stresses in elements of turbomachines,” Strength Mater., 6, No. 10, 1263–1265 (1974). I. A. Birger, Strength Design of Blades. Manual for Strength Designer of Gas Turbine Engine [in Russian], Issue 2, Oborongiz, Moscow (1956). Yu. M. Tereshchenko, A Way of Avoiding Flow Non-Uniformity after the Axial Compressor Stator Blades [in Russian], Inventor’s Certificate No. 411232, published in the bulletin “Discoveries, Inventions, Production Prototypes, Trade Marks” (1974), No. 2.