Investigations on thermo-mechanical modeling of abradable ...isromac-isimet.univ-lille1.fr/upload_dir/final... · to improve the aerodynamical performances of the engine consists

Investigations on thermo-mechanical modeling ofabradable coating in the context of rotor/stator interactionsFlorence Nyssen1*, Alain Batailly1

SYM

POSI

A

ON ROTATING MACHIN

ERY

ISROMAC 2017

InternationalSymposium on

Transport Phenomenaand

Dynamics of RotatingMachinery

Maui, Hawaii

December 16-21, 2017

AbstractIn modern turbomachine designs, the nominal clearances between rotating bladed-disks and thesurrounding casings are reduced to improve aerodynamical performances. But the reduction ofnominal clearances signicantly increases the risk of occurrence of contacts between static androtating components and may lead to hazardous interaction phenomena. A common technicalsolution to mitigate this issue consists in the addition of an abradable coating on the casing innersurface. Even so, contact interactions between the blade tips and this abradable coating mayyield unexpected abradable wear removal phenomena. For that reason, recent researches havefocused on the numerical simulation of rotor/stator interactions with wear removal mechanism ofoperating clearances while neglecting thermal eects. However, high temperature areas due tocontact occurrence have been observed experimentally. ese high temperatures are suscepted togenerate a self-excitation of the system due the abradable dilatation. Accordingly, the aim of thiswork is to investigate the numerical modeling of thermal eects in the abradable coating due tocontact interactions.Keywordsrotor/stator interactions — thermal eects — abradable coating modeling1Departement de Genie Mecanique, Ecole Polytechnique de Montreal, Montreal, QC, Canada*Corresponding author: [email protected]

INTRODUCTION

Improving engine performances while lowering operatingcosts is a major issue for turbomachine manufacturers [1].Also, environmental constraints are playing a prevailing rolein modern engine designs, leading to an emphasis on re-ducing fuel consumptions. Modern technologies have beendeveloped, allowing for: (1) the increase of operating tem-peratures, (2) the increase of casing conicity which enhancescompression rates, (3) the use of lighter materials, as wellas (4) more aerodynamically ecient designs. A solutionto improve the aerodynamical performances of the engineconsists in the reduction of the nominal clearances betweenthe rotating bladed components (rotor) and the surround-ing casing (stator). However, the reduction of the clearancebetween the tip of the blades and the stator increases therisk of occurrence of unilateral contacts between static androtating components that may lead to hazardous interactionphenomena. Depositing a sacricial abradable coating alongthe casing contact surface is a common technical solutionto avoid these interactions [2, 3, 4]. But contact interactionsbetween the blade-tips and the abradable coating may leadto unexpected abradable wear removal phenomena throughvarious mechanisms, such as thermal gradient in the casing,coincidence of vibration modes, rotor imbalance resultingfrom design uncertainties, or whirl motions following maneu-ver loads [5]. For that reason, dierent researches have beencarried on the blade dynamics in the context of unilateraland frictional contacts [6, 7, 8]. In particular, recent workshave focused on the numerical simulation of rotor/stator

interactions with wear removal mechanism [3, 9, 10, 5], ne-glecting thermal eects. But high temperature areas due tocontact occurrences have been observed experimentally [11].It is assumed that these high temperatures could generate aself-excitation of the system due to the abradable dilatation.

e aim of the proposed work is to investigate the numer-ical modeling of thermal eects in the abradable coating dueto contact interactions. e rst section of the article focuseson the theoretical description of the numerical models of thedierent system components (the blade, the abradable layerwith wear and thermal modeling, and the casing), the contactmodel and the time integration algorithm. Space and timediscretizations for the resolution of the thermal problem arealso detailed, with emphasis on the reduction of computationtimes. In the second section, the proposed approach is ap-plied on an industrial nite element model of a low-pressurecompressor blade, and convergence analyses are carried out.More particularly, the eects of the spatial discretization andtime step are studied for the thermal model. In the last sectionof the article, the obtained results are analysed.

1. MODELINGe proposed work deals with a single rotating blade and thecasing on which an abradable layer is deposited to acceptphysical contact events. Figure 1 summarizes the dierentcomponents involved in the modeling. Because of the pos-sibly large size of the nite element model of the bladedstructure, a reduced order model (ROM) of the blade is usedhere, embedding centrifugal eects. Only a reduced number

Investigations on thermo-mechanical modeling of abradable coating in the context of rotor/stator interactions — 2/9

casing

abradable thermal mesh

abradable mechanical mesh

impacted abradable elements blade boundary nodes

blade heat ux

z

Ω

blade

Figure 1. Representation of the thermal mesh and blade heat ux

of boundary nodes are kept in the ROM at the contact inter-face along the blade tip. e abradable layer is modeled usingone-dimensional two-node rod elements, which are mechan-ically independent of their adjacent neighbors. e modelof the material removing for the abradable layer consist in apiecewise linear plastic constitutive law [5]. In addition tothe mechanical mesh used to determine wear evolution, athermal nite element mesh is added here to compute thetemperature evolution in the abradable. When contact in-teractions occur, a heat ux is transmied from the blade tothe abradable, increasing the temperature of the impactedabradable elements. en, the heat is conducted inside theabradable layer. e temperature variations induce dilatationof the abradable elements. Finally, the surrounding casing ismodeled as a rigid component that remains insensitive to thecontact interaction with the blades. A detailed description ofeach numerical model is given in this section.

Blade modelinge governing equation of motion is given by:

Mq + Dq + K (Ω) q + Fc (q) = Fe (1)

where Ω is the angular speed, q stores the displacement ofblade degrees of freedom, M, D and K denote respectivelythe mass, damping and stiness matrices, Fc contains thecontact forces and Fe the external forces.

In operating conditions, the blade tip will move due tocentrifugal loadings. ese centrifugal eects aect the com-putation of the stiness matrix. As a rst approximation,since the angular speed Ω is constant, the stiness matrix

can be wrien as a polynomial expansion over the rotationalspeed such as [12]:

K (Ω) = K0 +Ω2K1 +Ω

4K2 (2)

where the matrices K0, K1 and K2 are obtained by computingEq. (2) for three values of the angular speed Ω, for instance,withΩ = 0,Ωmax/2 andΩmax. In this case, the three matricescan be expressed as [12, 13]:

K0 = K(0) (3)

K1 =1

3Ω2max

(16K

(Ωmax

2

)− K(Ωmax) − 15K(0)

)(4)

K2 =4

3Ω4max

(K

(Ωmax

2

)− 4K(Ωmax) + 3K(0)

)(5)

Because of the large size of the nite element model of thebladed structure, ROMs are used for explicit time integrationsimulations. e transformation matrixΦ enables to reducethe size of the model:

Φ =

[I 0

ΦR (0) Ψ

](6)

with:

Ψ =

ΦR

(Ωm

2

)?ΦR (Ωm)?

ΦL (0)ΦL

(Ωm

2

)ΦL (Ωm)

ᵀ

(7)


in which ΦR (Ω) and ΦL (Ω) stand respectively for nc con-straint modes and η xed interface modes computed for arotation speed Ω. e superscript ? indicates that matrixΦR (0) is substracted. An orthonormalization of the matrixΨ is performed to avoid potential rank-deciency due to sim-ilarities between the constraint modes. e maximum sizeof the reduced-order model is therefore equal to 3nc + 3η.

Contact interactionse contact force vector Fc (

q) is computed using Lagrange

multipliers [14, 15]. e blade is considered as the mastersurface Γmc , and the abradable layer the slave surface Γsc . Forany material point of the blade x ∈ Γmc (restricted here tothe interface nodes at the blade tip), it is possible to nd theclosest counterpart y on the the abradable slave surface:

y = arg miny∈Γse‖x − y‖ (8)

Using these notations, the discretized clearance betweenthe blade and the abradable can be wrien as:

g (x) = g0 (x) +(um (x) − u

(y(x)

))· n (9)

in which g0 (x) is the initial positive gap, n is the outwardnormal to the slave surface Γsc . Regarding to the Kuhn-Tuckeroptimality conditions, the contact conditions are:

∀x ∈ Γsc, tN ≥ 0, g(x) ≥ 0, tNg(x) = 0 (10)

in which tN denotes the discretized contact pressure whichis positive when acting on the contact interface.

Abradable coating modelingA weak thermo-mechanical coupling is assumed for theabradable coating modeling, meaning that thermal eectsaect the system mechanics, but the mechanical deformationof the element has no eect on temperatures. Since mechan-ics has no eect on thermics, the conduction problem canbe rst solved separately. Weak coupling is well appropri-ated in the case of rapid dynamics using small time stepsand explicit resolution schemes [16]. Moreover, only heattransfer by conduction is considered in this work, convectionand radiation eects can be neglected because of their lowercontribution to heat transfer in the studied case. e detaileddescription of the mechanical and thermal model is given inthis section.

Mechanical modelinge abradable coating is modeled using one-dimentional two-node rod elements as illustrated in Figure 2. Each element ismechanically independent of its adjacent neighbors. Over thethickness of the abradable layer, a single element is used sincecalculations are performed in a quasi-static framework. It isassumed that the abradable elements undergo both elasticεe and plastic deformations εp following the contact withthe blade such as ε = εe + εp . e plastic constitutive law,illustrated in Figure 3, enables to compute the normal forcesexerted by the blade onto the abradable coating elements

and the evolution of the abradable coating prole throughpermanent plastic deformation. e set of admissible stressesEσ is given by:

Eσ = (σ, α) ∈ (R,R) | f (σ, α) ≤ 0 (11)

with α the internal hardening variable and f the yield func-tion. e relation between elastic strains εe and stresses σis assumed linear, such as σ = Eεe with E the abradableYoung’s modulus. e hardening is assumed isotropic, whichcorresponds to a uniform expansion of the initial yield sur-face without translation. e center of the yield surface doesnot move. e yield function is dened as:

f (σ, α) = σ − (σY + Kα) (12)

in which σY is the elastic limit and K is the plastic modulus.

bar elementsabradable prole

casingboundary node

blade prole

Figure 2. Abradable coating modeling

ε

σ

E

EKE+K

σY

εp

Figure 3. Plasticity constitutive law of the abradablemechanical elements

Thermal modelingIn addition to the mechanical mesh used to determine thetime evolution of the displacement eld, a thermal niteelement mesh is associated in this work to the abradablecoating. Each node of the thermal mesh has one degree offreedom, the temperature of the element. e temperatureeld in the abradable coating T(x, y, z, t) reads:

T(x, y, z, t) = N(x, y, z) T(t) (13)


where N(x, y, z) is the interpolation function matrix and T(t)contains the nodal temperatures. e heat equation is givenby:

CT T + KTT = FT (14)

in which:

• the thermal capacity matrix CT (in J·K−1) reads:

CT =

∫V

ρcPNTNdV (15)

with ρ the abradable density (in kg·m−3) and cP thethermal capacity (in J·kg−1·K−1);

• the thermal conductibility matrix KT (in W·K−1) isgiven by:

KT =

∫V

BT λBdV (16)

with

B =

∂N1∂x . . . ∂Nn

∂x∂N1∂y . . . ∂Nn

∂y∂N1∂z . . . ∂Nn

∂z

(17)

and λ the conductivity matrix, which reads, for anisotropic material (λ is the thermal conductivity coef-cient of the abradable coating, in W·m−1·K−1):

λ = λ

1 0 00 1 00 0 1

(18)

• the nodal heat ux vector FT (in W) is given by:

FT = Q +∫V

NTqdV (19)

with Q the punctual nodal heat ux (in W ) and q thevolumetric heat ux (in W·m−3). By convention, theheat ux received by the solid is counted as positive.Only punctual nodal forces are considered in the fol-lowing.

To reduce computation times, a coarser spatial discretiza-tion is used for the thermal mesh comparing to the mechan-ical one. As illustrated in Figure 4, there is one thermalelement for Rs mechanical elements. erefore, an impacton the Rs mechanical elements will provide a heat ux forthe considered thermal element.

Triangular elements are considered for convenience tobuild the thermal mesh. Considering an element with threenodes of coordinates (x1, y1), (x2, y2) et (x3, y3) respectively,the temperature eld in the element reads:

T (x) = N1(x, y)T1 + N2(x, y)T2 + N3(x, y)T3

= [N1(x, y) N2(x, y) N3(x, y)]

T1T2T3

(20)

casing

Rsthermal mesh

Figure 4. Space discretization of the thermal abradablemesh comparing to the mechanical one

in which (noting yab = (ya − yb), xab = (xa − xb) witha, b = 1, 2, 3):

N1(x, y) = 12Ae (y32 (x2 − x) − x32 (y2 − y))

N2(x, y) = 12Ae (y13 (x3 − x) − x13 (y3 − y))

N3(x, y) = 12Ae (y21 (x1 − x) − x21 (y1 − y))

(21)

with Ae the surface of the element, given by:

Ae =x21y31 − x31y21

2 (22)

Using these interpolation functions leads to:

B =1

2Ae

[y23 y31 y12x32 x13 x21

](23)

and:

• the elementary thermal conductibility matrix KeT reads:

KeT =

λ

4Ae

y223 + x2

32 y23y31 + x32x13 y23y12 + x32x21y31y23 + x13x32 y2

31 + x213 y31y12 + x13x21

y12y23 + x21x32 y12y31 + x21x13 y212 + x2

21

(24)

• the elementary thermal capacity matrix CeT is given

by:

CeT =

ρcP Ae

12

2 1 11 2 11 1 2

(25)

• the elementary heat vector FeT due to a volumic sourceof intensity q reads:

FeT =qAe

4

111

(26)

Matrices CT , KT and FT are then constructed by assemblingthe elementary matrices.

Denoting by subscript L the degrees of freedom of un-known temperature and by subscript P the nodes where the


temperature is known, the thermal conductivity matrix, ca-pacity matrix and heat vector can be partitionned, such as:

CT =

[CLL CLP

CPL CPP

],KT =

[KLL KLP

KPL KPP

], FT =

[FLFP

]

(27)

e temperature at a given time t + 1 can be determinedusing the following equation system [17]:

KLL∆TL = FL (t)TL (s + 1) = TL (s) + ∆TL

(28)

with:

KLL = CLL + β ∆t KLL

FL (t) = ∆t(FL (t) − CLPTP (t) − KLPTP (t) − KLLTL (s)

)(29)

and the initial conditions T (t = 0) = T0. β denotes theparameter of the integration scheme and ∆t is the time step.F contains the heat sources, in this case the heat transmiedby the blade to the abradable coating during contact phase, asillustrated in Figure 1. In this preliminary study, the heat uxis assumed proportional to the amount of removed abradable.e temperature is assumed to be unknown in all abradableelements, except at the initial time. erefore, the set P ofdegrees of freedom of known temperature is empty in thisparticular case.

Denoting ωmax the largest eigenfrequency of C−1K, onecan demontrates [17] that the integration sheme is stablefor ∆t < 2

(1−2β)ωmax, unconditionally stable for β ≥ 1

2 , andstable without oscillation for ∆t < 2

(1−β)ωmax.

Knowing the temperature evolution in the abradable coat-ing, the displacement eld can be updated by computing theelement dilatation. Assuming a linear thermo-elastic consti-tutive law leads to:

εT = αT∆T (30)

in which εT is the thermal deformation, αT is the thermalexpansion coecient and ∆T is the temperature variation.

Time integrationTo compute the displacement of the blade and the abradablewear, an explicit time integration procedure is used in thepaper, based on the central nite-dierences method. At eachiteration n, displacements xn+1 are linearly predicted usingthe following scheme:

xpn+1 = Axn + Bxn−1 (31)

with

A =[ M

h2 +D2h

]−1 [ 2Mh2 − K

](32)

B =[ M

h2 +D2h

]−1 [ D2h−

Mh2

](33)

If a contact interaction is detected at the time step n,predicted displacements are corrected such as:

xn+1 = xpn+1 + Cλ (34)

where λ denotes the contact forces and C =[Mh2 +

D+G2h

]−1.

Since the time constant of thermal eect is larger thanthe one of contact interactions, a dierent time discretizationis used to compute the evolution of abradable temperature.Instead of computing the temperature evolution at each timestep, the thermal equation is solved every Rt mechanical timesteps. e heat ux applied at each thermal time step is equalto the sum of the ux applied on the considered element alongthe Rt last mechanical time steps (as illustrated shematicallyin Figure 5). is enables to signicantly reduce computationtimes. e dilatation of the abradable mechanical elementsdue to the increase of temperature computed every Rt timestep is applied every thermal time step since the increaseof temperature are small. e dierent steps of the timeintegration algorithm is summarized in Figure 6.

mechanical time step

thermal time stepRt Rt

Figure 5. Time discretization of the thermal equationscomparing to the mechanical ones

prediction of blade displacements

computation of the blade/casingand blade/abradable distances

computation of the contactforces and abradable wear

correction of the blade displacements

mod (n, Rt ) , 0= 0

t = t + hn = n + 1

computation of nodal temperature

correction of the abradable pro-le due to thermal dilatation

Figure 6. Time integration algorithm

2. VALIDATIONONAN INDUSTRIALMODELNumerical simulations are performed on a low pressure com-pressor blade, shown in Figure 7. Eight nodes along the bladetip are considered at the contact interface in the reducedorder model (red dots in Figure 7). Ten modes are retained


in the ROM. e rotational speed of the blade is assumedconstant and equal to 450 rad·s−1. e number of mechanicalabradable element is xed to 5000. e thermal dilatationcoecient is equal to 5 · 10−6 K−1, the thermal capacity to 1J·kg−1·K−1 and the thermal conductivity to 1 W·K−1. β = 0.5for the thermal iteration scheme.

e contact between the blade and the surrounding cas-ing is initiated by applying quasi-statically a two-lobe defor-mation of the casing in order to absorb the inital clearances.e dierent simulations are conducted on 10 rounds of theblades.

Figure 7. Studied structure with the nodes retained in thereduced order model in red

undeformed casingdistorded casing

x

y

z

Figure 8. Two-lobe deformation of the casing

Spatial convergenceFigures 9a and 9b show the spatial convergence of the tem-perature prole in the abradable coating aer 10 rotationsof the blade for the leading edge and the trailing edge re-spectively. 1000, 100 and 50 elements are considered alongthe circumference of the abradable layer (this corresponds toRs = 5, 50 and 100 respectively). e number of mechanical

abradable elements is kept equal to 5000. A good conver-gence of the temperature prole is observed and 50 thermalelements along the circumference can be kept. e reductionof the number of thermal elements enables to decrease thesize of the thermal problem (Equation (28)) and ecientlyreduce computation times.

0angular position [°]

tem

pera

ture

varia

tion

[K] 1000

10050

90 180 3602700

5

10

15

20

25

(a) Leading edge


tem

pera

ture

varia

tion

[K] 1000

10050

90 180 3602700

5

10

15

20

25

(b) Trailing edge

Figure 9. Convergence of the temperature prole along theabradable circumference for dierent numbers of thermalabradable elements

Time convergenceFigures 10a and 10b show the time convergence of the tem-perature prole in the abradable coating aer 10 rotations ofthe blade for the leading edge and the trailing edge respec-tively. It is observed that when Rt increases, i.e. the timediscretisation increases, the temperature increases at the lead-ing edge and the temperature decreases at the trailing edge.is can be explained by the delay of the conduction insideof the abradable coating. A schematic illustration explainingthis eect is given in Figure 11. It is considered here that theblade enters in contact with the abradable at the mechanicaltime step 1 at the leading edge. If the thermal time step isequal to the mechanical time step, the temperature of the


abradable increases at the mechanical time step 1 at the lead-ing edge. en, the temperature progressively decreases atthe leading edge at the next time steps and the temperatureslightly increase at the trailing edge by conduction. However,if the thermal time step is equal to two times the mechan-ical time step, the heat ux generated by the blade at themechanical time step 1 will be applied at time step 2, andthe temperature of the abradable increases at the mechanicaltime step 2 at the leading edge. en, the temperature pro-gressively decreases at the leading edge at the next time stepsand the temperature slighly increases at the trailing edge byconduction. erefore, when the thermal time step increasescomparing the mechanical time step, the temperature will beslightly higher at the leading edge and slightly lower at thetrailing edge due to conduction time.

e obtained temperature prole in the abradable is there-fore more sensitive to the time discretization than to thespace discretization. In the following, the time discretizationparameter Rt is xed to 50.

255075100150200


tem

pera

ture

varia

tion

[K]

90 180 3602700

5

10

15

20

25

(a) Leading edge

255075100150200


tem

pera

ture

varia

tion

[K]

90 180 360270

0

5

10

15

20

25

(b) Trailing edge

Figure 10. Time convergence of the temperature prolealong the abradable circumference for dierent values of Rt

0 1 2 3 4 5 6 7 8 9 10Mechanical time step [-]

LEte

mpe

ratu

re[K

]

TLEmax

0

(a) Leading edge

0 1 2 3 4 5 6 7 8 9 10Mechanical time step [-]

TEte

mpe

ratu

re[K

]

TTEmax

0

(b) Trailing edge

Figure 11. Schematic explanation of the time convergenceat the leading edge (LE) and trailing edge (TE). Black curve:Rt = 1, gray curve: Rt = 2.

ResultsFigures 12 and 13 show respectively the time evolution ofwear and temperature over the circumference of the abrad-able at the leading and trailing edges. In the rst graph, a pla-nar representation is given. e horizontal axis correspondsto the mechanical time step, the vertical axis correspondsto the angular position of the abradable element. e colorscale gives the normalized wear amplitude (Figure 12) andtemperature (Figure 13). e second graph gives the normal-ized wear prole in polar coordinates for dierent time steps.Lighter lines correspond to rst time steps and darker curvesto last mechanical time steps.

At the leading edge, two contact areas are observed, lead-ing to two wear lobes. For the last time steps, a small contri-bution of four wear lobes is observed. At the trailing edge,four contact lobes are obtained. ese contact areas bringheat from the blades to the abradable layer. Two temperaturepeaks areas are observed at the leading edge, while four tem-perature peaks areas can be seen at the trealing edge. eheat is then transmied in the abradable layer by conduction.Along time, a uniform temperature distribution is observed atboth leading and trailing edges, and four higher temperatureareas are observed on both edges.


0

time step [-]

angu

larp

ositi

on[°]

90

180

360

270

0 2 4 6 8 10 12 0

1

×104

wea

r

0 1 2wear

024

681012

time

step

[-]×104

0

270

90

angular180

position [°]

(a) Leading edge

0

time step [-]

angu

larp

ositi

on[°]

90

180

360

270

0 2 4 6 8 10 12 0

1

×104

wea

r

0 1 2wear

024

681012

time

step

[-]

×104

0

270

90

angular180

position [°]

(b) Trailing edge

Figure 12. Time evolution of the wear over thecircumference of the abradable at the leading and trailingedges

0

time step [-]

angu

larp

ositi

on[°]

90

180

360

270

0 2 4 6 8 10 12295

×104

300

310

315

320

325

tem

pera

ture

[K]

295 320 345temperature [K]

024

681012

time

step

[-]

×104

0

270

90

angular180

position [°]

(a) Leading edge

0

time step [-]

angu

larp

ositi

on[°]

90

180

360

270

0 2 4 6 8 10 12295

×104

300

310

315

320

325

tem

pera

ture

[K]

295 320 345temperature [K]

024

6

81012

time

step

[-]

×104

0

270

90

angular180

position [°]

(b) Trailing edge

Figure 13. Time evolution of the temperature over thecircumference of the abradable at the leading and trailingedges


3. CONCLUSIONis paper investigates the numerical modeling of thermal ef-fects in the abradable coating. A mechanical mesh consistingin independent two-node rod elements with elasto-plasticconstitutive law is considered to compute the abradable wear.A thermal mesh of the abradable layer has been added to com-pute the evolution of the temperature in the abradable dueto blade contacts. To reduce computation times of the tem-perature simulation, a dierent space and time discretizationhas been used. A convergence analysis has been performedwhich shows that the space discretization has low eect onthe obtained results. e reduction of computation time canbe done by considering a low number of abradable thermalelements. However, the time discretization has a larger eectdue to conduction inside of the abradable.

In future works, the numerical model can be improved byconsidering temperature diusion inside of the carter, provid-ing carter dilatation. e two-dimensional model (tangential-axial) has to be extended to three dimensions (tangential-axial-radial). e thermal ux of the blade should also becalibrated using experiments to obtain a representative evo-lution of temperatures during rotor/stator interactions in realoperating conditions. Moreover, a more accurate predictionof the blade heat ux could be to compute it as proportionalto the friction between the blade and the abradable. Finally,a dependence of the mechanical property of the abradablecoating with temperature should be included.

ACKNOWLEDGMENTSis research was supported by the Natural Sciences and En-gineering Research Council of Canada (NSERC). e authorsare also grateful to the industrial partner for supporting thisproject, Safran Aircra Engines.

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