The channel with a moving wall is considered to describe aeroelastic oscillations induced by gas flow. One of the channel walls has two degrees of freedom and it is supported by springs and dampers. The multi-fields investigation method is based on simulation transient gas flow in the channel to calculate aerodynamic forces acting on the wall. Corresponding rigidity and damping gas flow parameters obtained from these loads are included into the wall oscillations model for the stability analysis. The models are developed for two channel types: with smooth and finned wall. Aeroelastic stability boundary is shown for both channels. An effect of structural parameters on the realization of convergent oscillation and self-oscillation modes is shown too. A paradox of system destabilization with the increasing damping is observed for a certain parameter set

Alexey Selivanov

Joury Temis

English

Journal of Vibroengineering

  VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 2012. VOLUME 14, ISSUE 2. ISSN 1392-8716 
483 
770. Aeroelastic self-oscillations of gas seal wall 
  
Joury Temis, Alexey Selivanov 
Central Institute of Aviation Motors, Aviamotornaya str., 2, Moscow, 111116, Russia 
E-mail: tejoum@ciam.ru 
 
(Received 10 September 2011; accepted 14 May 2012) 
 
Abstract. The channel with a moving wall is considered to describe aeroelastic oscillations 
induced by gas flow. One of the channel walls has two degrees of freedom and it is supported by 
springs and dampers. The multi-fields investigation method is based on simulation transient gas 
flow in the channel to calculate aerodynamic forces acting on the wall. Corresponding rigidity 
and damping gas flow parameters obtained from these loads are included into the wall 
oscillations model for the stability analysis. 
The models are developed for two channel types: with smooth and finned wall. Aeroelastic 
stability boundary is shown for both channels. An effect of structural parameters on the 
realization of convergent oscillation and self-oscillation modes is shown too. A paradox of 
system destabilization with the increasing damping is observed for a certain parameter set. 
 
Keywords: aeroelastic stability, oscillations, gas flow, seal. 
 
Introduction 
 
Gas flow ducts and channels are the important parts of different technical devices. Non-
stationary processes, which take place in these channels, may influence significantly on the 
device performance and service life. It’s true both for simple devices and for extremely complex 
devices as gas turbine engines. For example modern aircraft engine have more than 100 small 
channels (seals) between rotor and stator. Some of them have smooth walls (annular seals), other 
have finned walls (labyrinth seals). The aim of the seal is to reduce gas leakage from high 
pressure area to low pressure one. Thus we have the flow induced by the pressure drop in the 
channel. In some cases energy of this flow may lead to seal wall self-oscillations and fatigue 
failure of the seal. Such problems were fixed in real engines. 
Aeroelastic analysis is a generally multi-field problem. To solve this problem we should 
combine the gas flow model and the wall dynamic model (Fig. 1). In this work we use “local” 
channel models, but in a general case the seal wall may have some “external” quasi-stationary 
motion induced by centrifugal or thermal growth of the rotor, rotor deflections and thermal 
growth of the stator. These external motions lead to a clearance change and thus may have an 
influence on gas flow and seal stability too. For the real seal initially we need to find the external 
stator/rotor motions and thereafter use the local models. To solve the external task a whole 
engine hydraulical net model and modules of thermostressed analysis can be used. These 
external models are not discussed in this work. 
 
 
Fig. 1. The aeroelastic analysis algorithm 
 
Fig. 1 shows overall algorithm of aeroelastic analysis. Gas channel with rigid moving wall is 
considered to describe seal aeroelastic oscillations induced by transient gas-dynamic loads. One 
 770. AEROELASTIC SELF-OSCILLATIONS OF GAS SEAL WALL. 
JOURY TEMIS, ALEXEY SELIVANOV 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 2012. VOLUME 14, ISSUE 2. ISSN 1392-8716 
484 
 
of the channel walls has two degrees of freedom and it is supported by springs and dampers. The 
investigation method is based on simulation transient gas flow in the channel to calculate 
aerodynamic forces. These forces are included into the wall oscillations model for the analysis 
of stability. 
Obviously this algorithm can be used for various gas ducts and seals, including new high-
efficient finger seals with flexible fingers. The finger seal (see Fig. 2) is comprised of a stack of 
plates with special cuts which form the fingers. The downstream fingers have lift pads on one of 
their ends and act as cantilever beams, flexing away from the rotor during gap decrease through 
centrifugal or thermal growth of the rotor or during rotordynamic deflections [1]. The problem 
of flexible finger self-oscillation, induced by gas loads acting on their pads, may be significant 
for these seals. Some feature of this analysis will be discussed further. 
 
 
Fig. 2. Compliant finger seal model: 1 – downstream finger with lift pad; 2 – upstream finger 
 
Aeroelastic self-oscillations of plane channel wall 
 
The main generalized models with two degrees of freedom are shown in the Fig. 3. These 
models are plane and walls are considered to be absolutely rigid. Springs and dampers with 
stiffness ( 0k , 1k ) and damping ( 1c ) coefficients imitate seal structure characteristics. The seal 
clearance δ  is much smaller than its length L. The stability analysis for these models can be 
carried out by standard mathematical methods, but first we must know non-stationary gas flow 
pressure distribution in the channel. At the same time this pressure distribution is depended on 
boundary conditions and actual wall position. Thus we have direct correlation between wall 
position and values of aeroelastic forces. 
 
 
(a)    (b) 
Fig. 3. Plane channel model with (a) smooth and (b) finned wall 
 
If ϕ  is sufficiently small, then wall oscillations (with two degrees of freedom: ϕ  and h) are 
described by equations: 
 770. AEROELASTIC SELF-OSCILLATIONS OF GAS SEAL WALL. 
JOURY TEMIS, ALEXEY SELIVANOV 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 2012. VOLUME 14, ISSUE 2. ISSN 1392-8716 
485 
 
1 0 1
2
1 1
( ) ( ) ,
2
( ) ( ) ,
2 3
mL
mh c h L k h k h L F
mL mLh c L h L k L h L M
φ φ φ
φ φ φ

+ + + + + + = ∆

 + + + + + = ∆

ɺɺ ɺɺɺ ɺ
ɺɺ ɺɺɺ ɺ
 (1) 
 
here m is the wall mass; h and φ  are deviations from the static equilibrium position; F∆  and 
M∆  are the aeroelastic force and moment deviations from their values at the static equilibrium. 
These deviations can be represented as follows: 
 
ϕ
ϕ
ϕ
ϕ
ɺ
ɺ
ɺ
ɺ ∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
=∆
Fh
h
FFh
h
FF , ϕ
ϕ
ϕ
ϕ
ɺ
ɺ
ɺ
ɺ ∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
=∆
Mh
h
MMh
h
MM , (2) 
 
and they are called stiffness and damping gas flow coefficients. 
The transient gas flow models for smooth and finned channels are developed to obtain 
pressure distribution p(x, t) and to calculate aerodynamic force F and moment M which are 
acting on the seal walls: 
 
∫∫ ==
LL
dxtxxptMdxtxptF
00
),()(,),()( . 
 
In a general case turbulent gas flow is described by a system of partial derivatives differential 
equations. This system consists of continuity equation, momentum equations, and energy 
equation. And it also contains some differential equations, used to describe the turbulence 
model. At the same time, there is a lot of experimental data that allows us to define friction 
coefficients, depending on Reynolds number for such small gas channels. 
Thus, the gas flow model can be built with one dimensional approximation that reduces 
calculation time. We can write continuity equation (3), momentum equation (4), energy equation 
(5), and state equation. If wall deviations are small, this gas-dynamic system can be linearized: 
 
0)()( =
∂
∂
+
∂
∂
x
u
t
δρρδ
, (3) 
 
021 =+
∂
∂
+
∂
∂
+
∂
∂
ρδ
τ
ρ x
p
x
u
u
t
u
, (4) 
 
01
**
=
∂
∂
−
∂
∂
+
∂
∂
t
p
cx
T
u
t
T
pρ
, (5) 
 
here ρ  is the gas density, u is the flow velocity, and *T  is the stagnation temperature. The shear 
stress τ  is equal to 2 uuf ρτ = , where the wall friction factor f for a turbulent flow is equal 
to 333.0Re187.0 −⋅=
x
f . 
One-dimensional transient gas flow simulation is carried out using finite difference method 
with implicit scheme. And in some cases two-dimensional transient gas flow in the channel with 
moving wall is simulated also to test 1D model results. The analysis is carried out using 
commercial software. The difference between 1D and 2D models for aerodynamic force F(t) and 
 770. AEROELASTIC SELF-OSCILLATIONS OF GAS SEAL WALL. 
JOURY TEMIS, ALEXEY SELIVANOV 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 2012. VOLUME 14, ISSUE 2. ISSN 1392-8716 
486 
 
moment M(t) is less than 2 %. Therefore our fast 1D model (with linear approximation) can be 
used for such research. 
For the finned channel we add some equations for narrowing/widening flow interaction near 
the fins (teeth). Gas flow in this case can be described by “one-volume” model [2]: 
 
( ) ,..1,0)()( 22 102 12110 NippRTppRTSpt iiiiiiiiii ==−−−+∂
∂
−+++
δµµδµµ  (6) 
 
where N is the number of cavities between the fins, iS  is the cross-section cavity area, iµ  is the 
discharge coefficient: 
 
2252 ii
i
ββpi
pi
µ
+−+
= , where 
( 1)/
11 .ii
i
p
p
γ γ
β
−
−
 
= − +  
 
 
 
For detailed analysis of the finger seal we can use 2D gas flow model with circumferential 
coordinate s (see Fig. 2). This model is constructed on the Reynolds equation: 
 
( ) ( )






∂
∂
+
∂
∂
=





∂
∂
∂
∂
+





∂
∂
∂
∂
s
pR
t
p
x
pp
xs
pp
s
δ
ω
δ
δ
η
δ
η
δ 26
33
. (7) 
 
Reynolds approach is more effective for the 2D transient gas flow modeling, but anyway for 
first approximation can be used 1D gas flow model. Finger stiffness beamk  can be evaluated by 
3D stress analysis of fingers assembly by using commercial software (ANSYS and etc.). In all 
other aspects of the research method is similar to represented here. 
Stiffness and damping gas layer coefficients obtained from the aerodynamic loads are 
included into the wall dynamic model Eq. (1) for analysis of the stability of the aeroelastic 
oscillations. Let us find the solution Eq. (1) in the following form: 
 



Φ=
=
,
,
ti
ti
e
Heh
ω
ω
ϕ
 (8) 
 
where ω  is the self-oscillations frequency, H and Φ are complex amplitudes. Combining 
Eq. (1), (2), (8) and writing nontrivial solution existence condition, we can determine parameters 
of self-excitation oscillations. 
Fig. 4 represents safe operating area 1, unstable area 2, and boundary of stability 3 (curve of 
harmonic self-oscillations) for the smooth channel under some operating conditions. It must be 
noted, that in this case linear damping 1c  increase may cause growth of oscillations and seal 
instability. This effect is similar to Mansour`s anomaly and can be explained as follows: there is 
no direct coupling between damping coefficient increase and damping forces work increase for 
such systems with two degrees of freedom [3]. 
Let us consider the behavior of characteristic equation roots λ  with damping coefficient 1c  
vary. As a result of characteristic equation numerical solution, two pairs of complex conjugate 
roots ( 1 λ , 2 λ ) and ( 3 λ , 4 λ ) are obtained. Fig. 5a shows 1 λ  and 3 λ  versus 1c  on the complex 
plane (increasing 1c  is indicated by arrows). For Re( λ ) < 0 oscillations decrease. The 
intersection points of curves λ  with ordinate axis Re( λ ) = 0 correspond to harmonic self-
oscillations, and the system is unstable in the area Re( λ ) > 0. 
 770. AEROELASTIC SELF-OSCILLATIONS OF GAS SEAL WALL. 
JOURY TEMIS, ALEXEY SELIVANOV 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 2012. VOLUME 14, ISSUE 2. ISSN 1392-8716 
487 
 
 
Fig. 4. Smooth channel stability region: 1 – safe operating area, 2 – unstable area, 3 – stability threshold 
 
On the boundary of stability there may be qualitative changes when stiffnesses are varied. If 
stiffnesses k are “small” values, then with damping increasing, the system turns from oscillations 
increase to oscillations decrease. With stiffness increasing, nonlinear effects appear – see       
Fig. 5b. 
 
 
(a)    (b) 
Fig. 5. (a) Characteristic equation roots λ  versus 1c ; (b) stability threshold for different stiffness 
 
Fig. 6а shows stability region for the finned channel. The effect of seal destabilization with 
increase of damping coefficient isn’t observed for selected seal parameters. 
The Eq.(1) and Eq.(6) can be joining into one nonlinear system, where 
∫ −=∆
L
hFdxtxpF
0
00 ),(),( ϕ  and ∫ −=∆
L
hMxdxtxpM
0
00 ),(),( ϕ . 
Numerical solutions of this system for different damping coefficient values confirm the 
boundary of stability. For example Fig. 6b shows h(t) and ϕ (t) behavior for small damping 
(point from unstable area). More detail gas-dynamic analysis and stability analysis for these 
channels are represented at the works [4, 5]. 
 770. AEROELASTIC SELF-OSCILLATIONS OF GAS SEAL WALL. 
JOURY TEMIS, ALEXEY SELIVANOV 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. JUNE 2012. VOLUME 14, ISSUE 2. ISSN 1392-8716 
488 
 
 
(a)    (b) 
Fig. 6. Finned channel stability region (a) and system behavior for small damping (b): 1 – safe operating 
area; 2 – unstable area; 3 – stability threshold 
 
Conclusions 
 
This research is first stage of seal/channel aeroelastic stability study. Developed 
mathematical models allow obtaining initial results on the behavior and stability of such 
systems. It is shown that the aeroelastic oscillations problem is essentially nonlinear even for our 
models with two degrees of freedom (for example a paradox of system destabilization with the 
increasing damping is shown). Obviously such problem should be solved with use of multi-
fields investigation method only. 
 
References 
 
[1] Braun M. J., Pierson H. M., Li H., Dong D. Numerical simulations and an experimental 
investigation of a Finger Seal. NASA/CP-2006-214383, Vol. 1, 2006, p. 309 – 359. 
[2] Childs D. W. Turbomachinery Rotordynamics: Phenomena, Modeling and Analysis. John Wiley & 
Sons Inc., USA, 1993, 476 p. 
[3] Mansour W. M. Quenching of limit cycles of a Van der Pol oscillator. Journal of Sound and 
Vibration, Vol. 25, No. 3, 1972, p. 395 – 404. 
[4] Kadaner J. S., Selivanov A. V., Temis J. M. Performance analysis of sealing devices in gas turbine 
engines. Proc. of the 3rd International Symposium on Stability Control of Rotating Machinery 
(ISCORMA–3), Cleveland, USA, 19-23 September, 2005, p. 593 – 604. 
[5] Temis J. M., Selivanov A. V. Aeroelastic self-oscillations of plane channel wall. Proc. of the 3rd 
International Conference on Nonlinear Dynamics, ND-KhPI-2010, Kharkov, Ukraine, 21 – 24 
September, 2010, p. 421 – 426. 


Aeroelastic self-oscillations of gas seal wall

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