arXiv:2111.06646v1 [physics.flu-dyn] 12 Nov 2021

Dynamic force and torque characteristicof annular gaps - Simulation results andevaluation of the relevance of the tilt and

torque coefficientsMaximilian M. G. Kuhr*

Chair of Fluid SystemsTechnische Universität Darmstadt

[email protected]

Rainer Nordmann

Fraunhofer Institute forStructural Durability and System [email protected]

Peter F. Pelz

Chair of Fluid SystemsTechnische Universität Darmstadt

[email protected]

Abstract

We examine the dynamic force and torque characteristic of annular gaps resulting from an axial flowcomponent. First, the rotordynamic influence of annular gaps with an axial flow component is recapitulated.Until now, there is a major deficit of understanding the dynamic force and torque characteristic of annulargaps. Second, a new simulation method is presented, using a perturbed integro-differential approach incombination with a Hirs’ model and power-law ansatz functions for the velocity profiles to calculate thedynamic force and torque characteristics. Third, an extensive parameter study is carried out. To evaluatewhether or not the hydraulic tilt and torque coefficients need to be taken into account, an effective stiffnessis defined and the influence of the annulus length, an eccentrically operated shaft, the centre of rotation, amodified Reynolds number, the flow number and the pre-swirl is investigated.

I. Introduction

The reliability and performance of turboma-chinery is often limited by harmful shaft vibra-tions due to resonance effects or the responseof the system to disturbances during stationaryoperation. The dynamic behaviour of the sys-tem and therefore the mechanical vibrationsare highly influenced by the induced hydrody-namic forces and torques of the flow withinnarrow annular gaps [1, 2, 3, 4, 5, 6, 7, 8].In general, the flow in an annulus is three-dimensional. The presence of an axial pressuredifference results in an axial flow component

*Corresponding author

that is superimposed by the circumferentialflow component driven by viscous forces. Inaddition, the flow at the annulus inlet is su-perimposed by a pre-swirl due to the designparameters of the turbomachinery. The pre-swirl is then convected into the annuls by theaxial flow component. Due to the increasingdemands on flexibility, today’s turbomachineryare often operated at partial load under highlydynamic operating conditions. Therefore, in-creased vibrations occur due to flow separa-tion and recirculation areas, which pose a chal-lenge to service life and safe operation. By nowthere is a major deficit in understanding the dy-namic characteristic of the induced hydraulic

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Dynamic force and torque characteristic of annular gaps

forces and torques in annular gaps. Further-more, the existing literature mainly focuses onthe influence of hydraulic forces due to trans-lational motion whereas the influence of hy-draulic tilt and torque coefficients and theirinfluence on the stability of the rotor-annulussystems are invariably neglected. In modernturbomachinery, two essential narrow annulargaps exist, applying hydrodynamic forces andtorques on the rotating shaft [3, 9, 10]. First, an-nular seals or damper seals and second, journalbearings which are either oil or media lubri-cated. However, the usage of low viscous fluidslike water or cryogenic liquids for lubricationpurpose, as is usual for annular seals and me-dia lubricated bearings, leads to an operationat high Reynolds numbers, resulting in turbu-lent flow conditions and significant inertia ef-fects [3, 4, 5, 6, 7]. The rotordynamic influenceof those annuli is in general described by theuse of rotordynamic coefficients, namely stiff-ness K, damping C and inertia M. Here, thetilde 2 characterises dimensional variables.The generalised equation of motion includingforces and torques of the annular gap flowyields

−

FXFY

MXMY

=

KXX KXY KXα KXβ

KYX KYY KYα KYβ

KαX KαY Kαα Kαβ

KβX KβY Kβα Kββ

XYαXβY

+

+

CXX CXY CXα CXβ

CYX CYY CYα CYβ

CαX CαY Cαα Cαβ

CβX CβY Cβα Cββ

˜X˜Y

αXβY

+

+

MXX MXY MXα MXβ

MYX MYY MYα MYβ

MαX MαY Mαα Mαβ

MβX MβY Mβα Mββ

˜X˜Y

αXβY

.

(1)

Here, FX , FY and MX , MY are the induced hy-drodynamic forces and torques of the annulusacting on the rotor. X, ˜X, ˜X and Y, ˜Y, ˜Y denotesthe translational motion of the rotor and itstime derivatives. αX, αX, αX and βY, βY, βY isthe angular motion of the rotor around the

X, Y axis and its time derivatives. The 48 rotor-dynamic coefficients are in general dependenton (i) the geometry of the annulus, i.e. themean gap height ˜h, the shaft radius R, the gaplength L and the gap function h = h (ϕ, z, t)with the circumferential and axial coordinatesϕ, z; (ii) the operating parameters of the system,i.e. the static eccentric and angular position ofthe shaft inside the annulus e, α, β, the distanceof the fulcrum zT from the gap entrance, theangular velocity of the shaft Ω, the mean axialvelocity through the annulus ˜Cz and the pre-swirl velocity at the gap entrance Cϕ|z=0; (iii)the characteristics of the fluid used for lubri-cation, i.e. fluid density $ and the dynamicviscosity η, cf. figure, 1

Kij, Cij, Mij =

= f(

R, ˜h, L, e, αX , βY, zT , Ω, ˜Cz, Cϕ|z=0, $, η)

.

(2)

On dimensional ground, the dimensionlessrotordynamic coefficients are only a functionof 9 dimensionless measures: (i) the relativegap clearance ψ := ˜h/R, (ii) the dimension-less annulus length L := L/R, (iii) the relativeeccentricity ε := e/˜h, (iv) the dimensionlessfulcrum zT := zT/L, (v) the Reynolds numberin circumferential direction Reϕ := (ΩR ˜h/ν),(vi) the flow number φ := ˜Cz/(ΩR), (vii) thedimensionless pre-swirl Cϕ|z=0 := Cϕ/(ΩR)and (viii + ix) the angular displacements α :=LαX/˜h, β := LβY/˜h around the fulcrum. Kuhret al. [11] showed that the list of dimension-less measures can be further reduced becausethe relative gap clearance and Reynolds num-ber only appear as a product in the govern-ing equations, resulting in a new dimension-less measure, the modified Reynolds numberRe∗ϕ := ψRe

n fϕ . Here n f is an empirical con-

stant describing an arbitrary line within thedouble logarithmic Moody diagram. Hence,equation 2 is reduced to

Kij, Cij, Mij =

= f(

L, ε, zT , Re∗ϕ, φ, Cϕ|z=0, α, β)

.(3)

The dimensionless rotordynamic coefficients

2


෨𝑅

෨ℎ 𝜑, ǁ𝑧, ǁ𝑡

෨𝐿

ǁ𝑒

෨𝑋

ǁ𝑧

ሚҧ𝐶𝑧ሚ𝐶𝜑|z=0

෨തℎ/ ෨𝑅 ≪ 𝒪 1

෩Ω

𝑦

𝛽𝑌

ǁ𝑧𝑇

𝜚, 𝜂

𝑥 ≔ ෨𝑅𝜑

෨𝑋

𝛽𝑌

𝛼𝑋

෨𝑌 ǁ𝑒෨𝑌

Figure 1: Schematic drawing of an eccentrically operated generic annular gap with axial flow and pre-swirl at theannulus inlet.

are defined for the force and torque due totranslational and angular motion separately

KI :=2˜hKI

$Ω2R3 L, KI I :=

2˜hKI I

$Ω2R3 L2,

KI I I :=2˜hKI I I

$Ω2R3 L2, KIV :=

2˜hKIV

$Ω2R3 L3.

(4)

Here, the first two stiffness coefficients rep-resent dynamic force coefficients due to trans-lational and angular motion whereas the twolater ones represent the torque coefficients dueto translational and angular motion. The in-dices represent the corresponding sub-matricesof equation 1. The damping and inertia termsare defined accordingly

CI :=2˜hCI

$ΩR3 L, CI I :=

2˜hCI I

$ΩR3 L2,

CI I I :=2˜hCI I I

$ΩR3 L2, CIV :=

2˜hCIV

$ΩR3 L3,

MI :=2˜hMI

$R3 L, MI I :=

2˜hMI I

$R3 L2 ,

MI I I :=2˜hMI I I

$R3 L2 , MIV :=2˜hMIV

$R3 L3 .

(5)

As mentioned earlier, the vast majority ofthe existing literature, whether the studies are

of analytical, experimental or numerical na-ture, focuses only on the rotordynamic influ-ence of induced forces due to translational butnot angular motion. Furthermore, either an-nular seals operating at zero eccentricity withhigh axial pressure differences, cf. [12, 13, 14,15, 1, 2, 16, 17, 18, 19, 20, 21, 22, 23, 24] orjournal bearings operating at high eccentric-ities without an axial pressure difference, cf.[25, 26, 27, 28, 29, 30, 31, 32] are discussed.A more detailed list of the research of annularseals and journal bearings can be found in thework of Tiwari et al. [33, 34].With regard to the hydraulic torques, Childs[35] presented a bulk-flow based calculationmethod for determining skew-symmetric dy-namic force and torque coefficients for finitelength annular pressure seals operating at zeroeccentricity under fully turbulent flow condi-tions. The static and dynamic properties arecalculated by linearising the non-linear sys-tem of partial differential equations by meansof a perturbation analysis, leading to a setof linear ordinary differential equations. Fo-cusing mainly on the effect of the seal lengthon the coefficients at a constant pressure dif-ference without pre-swirl. Childs examinedthe coefficients for three different seal lengthsL/R = 0.3, 1.0, 2.0. In addition, one calculation

3


with a negative pre-swirl ratio is carried out ata seal length of L/R = 1.0 to show the effectof pre-swirl on the coefficients.Simon & Frêne [36] developed an analysis tocalculate the static and dynamic characteristicsof annular seals at eccentric shaft operation.Similar to the work of Childs [35] the methodis based on an integro-differential approach,solving the continuity and momentum equa-tion in axial and circumferential direction. Incontrast to Childs, the integrals are not solvedwith the bulk-flow approach but are modelledusing parabolic ansatz functions. The govern-ing equations are expanded using a perturba-tion method, leading to a system of non-linearpartial differential equations for the zeroth-order and a linear system of partial differentialequations for the first-order solution. Here,the zeroth-order solution gives the static char-acteristics of the flow whereas the first-ordersolution gives the dynamic properties. Bothequation systems are solved numerically bya shooting method. For the purpose of vali-dation the numeric results are first comparedto experimental and numerical data for rotor-dynamic force coefficients due to translationaldisplacement provided by Nordmann & Di-etzen [37] and Nelson & Nguyen [38]. Theresults are in good agreement with the cho-sen validation data. The paper then gives thestiffness and damping coefficients for angu-lar excitation for three relative eccentricitiesε := e/˜h = 0.0, 0.5, 0.8. Simon & Frêne [36]extend their research on the influence of theannulus length and the pre-swirl on the rotor-dynamic coefficients. In addition to positivevalues of the pre-swirl negative values are alsoinvestigated and compared to the limited datapublished by Childs [35].San Andrés [5] presents a method based on thebulk-flow approach, examining dynamic forceand torque coefficients for short annular sealsof two different lengths L := L/R = 0.4, 1.0operated at zero eccentricity. San Andres com-pares his approach to the results of Childs [35]and Simon & Frêne [39]. The method shownsa good agreement with the rotordynamic co-efficients. The paper then mainly focuses on

the influence of the fulcrum zT on the dynamicproperties. It is shown that the coefficients arehighly sensitive as to whether the centre of ro-tation is at the entrance, centre or exit of thegap. The method is later used to further inves-tigate the influence of shaft misalignment onthe dynamic properties, cf. [6].Here, the dynamic torque coefficients are de-termined for a centred annular pressure sealat high degrees of static misalignment. Thegoverning two dimensional equations are ex-panded using a perturbation method lead-ing to a non-linear partial differential equa-tion system for the zeroth-order and a lin-ear partial differential equation system for thefirst-order solution. Both equation systemsare solved numerically by using a SIMPLEC(Semi-Implicit Method for Pressure LinkedEquations-Consistent [40]) algorithm coupledto a Newton-Raphson iterative procedure. Inaddition, the analysis is enhanced by using ac-curate analytical expressions for the centredoperation. It is shown that the skew-symmetryis not longer valid for large static angles ofmisalignment which intuitively seems correct,since large angles of misalignment in parts ofthe annulus produce large static eccentricities.San Andrés [7] extends his theoretical ap-proach to the effects of journal misalignmenton the operation of turbulent flow hydrostaticbearings. The work focuses on the effect ofeccentricity and misalignment on the stiffnessand damping coefficients due to translationaland angular motions as well as the inertiaterms for translational displacement. Similarto the work of Simon & Frêne [36] no inertiaterms for angular displacements are given. Itis concluded that the presence of journal mis-alignment affects the bearing performance andneeds to be taken into account when properlydesigning turbulent flow hydrostatic bearings.Kanemori & Iwatsubo [41] present an experi-mental study of the dynamic force and torquecoefficients for long annular seals L := L/R =6.0. The test rig presented consists of a rotordriven by 2 motors to realise the spinning andwhirling motion. The induced forces were mea-sured with attached pietzo-electric load cells.

4


The corresponding torques were then calcu-lated around the fulcrum by using the geomet-ric distance and the measured loads. For thepurpose of validation the measured dynamiccoefficients were compared to the calculationmethod developed by Childs [35]. It is shownthat the dynamic coefficients for stiffness anddamping coincide well with the theory. In ad-dition, Kanemori & Iwatsubo [41] state that therotor forward whirl acts as a stabilising forceon the rotor.Kanemori & Iwatsubo [42] studied the mu-tual effects of cylindrical and conical whirlon the dynamic fluid forces and torques ona long annular seal experimentally. In addi-tion to the rotordynamic coefficients the paperpresents pressure measurements inside the an-nulus. The paper focuses mainly on the impactof phase difference at whirling motion. It isshown that the phase of the whirl acts either asa stabilising or destabilising force. Kanemori &Iwatsubo [43] extend their work to a linear sta-bility analysis using the logarithmic decrementon a real submerged motor pump includinghydrodynamic forces and torques when carry-ing out the stability analysis. The individualcontributions of the whirl and concentric mo-tion to the stability of the system is evaluated.Feng & Jiang [28] and Feng et al. [30] use theReynolds’ equation of lubrication theory to cal-culate the stiffness and damping coefficientsdue to translational and angular motions of awater lubricated hydrodynamic journal bear-ing. The influence of laminar and turbulentflow conditions is investigated as well as theinfluence of eccentricity, rotating speed and tilt-ing angle. The Reynolds’ equation is thereforemodified by using turbulent correction coeffi-cients based on the work of Frêne & Arghir[27]. Unfortunately the effects of fluid inertia,i.e. the inertia terms Mij, are neglected.

II. Governing equations

In contrast to the majority of the literaturewe specialise neither on annular seals norjournal bearings, We rather develope a methodbased on the generic annulus geometry under

turbulent flow conditions, cf. figure 1. Hence,our method is applicable to annular sealsand journal bearings Kuhr et al. [11] presentthe Clearance Averaged Pressure Model(CAPM), a method determining the staticcharacteristics of generic annuli. Similar tothe bulk-flow approach, the model uses anintegro-differential approach but the velocityintegrals are treated by using power lawansatz functions. The model is experimentallyvalidated by a specifically designed testrig using active magnetic bearings. In thefollowing, the model is expanded, usinga perturbation expansion to determine thedynamic force and torque characteristics.

The dimensionless gap function for an eccen-tric and misaligned shaft reads

h = 1− ε cos ϕ− (z− zT)Lψ

tan β. (6)

Here, ε and β are time dependant. The timedependant dimensionless continuity and mo-mentum equation in circumferential and axialdirection yields

∂

∂th +

∂

∂ϕh∫ 1

0cϕ dy +

φ

L∂

∂zh∫ 1

0cz dy = 0,

∂

∂th∫ 1

0cϕ dy +

∂

∂ϕh∫ 1

0c2

ϕ dy +

+φ

L∂

∂zh∫ 1

0cϕcz dy = −h

2∂p∂ϕ

+1

2ψτyx|10,

φ∂

∂th∫ 1

0cz dy + φ

∂

∂ϕh∫ 1

0cϕcz dy +

+φ2

L∂

∂zh∫ 1

0c2

z dy = − h2L

∂p∂z

+1

2ψτyz|10.

(7)

By using power law ansatz functions of theform c = C(2y)1/n with the centreline veloci-ties Cϕ, Cz at half gap height and the exponentsnϕ, nz, the integrals for the continuity and mo-

5


mentum equation yields

∫ 1

0cϕ dy =

nϕ

nϕ + 1Cϕ +

12(nϕ + 1

) ,

∫ 1

0cz dy =

nz

nz + 1Cz,∫ 1

0c2

ϕ dy =nϕ(

nϕ + 2) (

nϕ + 1)Cϕ+

+nϕ

nϕ + 2C2

ϕ +1(

nϕ + 2) (

nϕ + 1) ,

∫ 1

0cϕcz dy =

nϕnz

nϕnz + nϕ + nzCzCϕ+

+n2

z2(nϕnz + nϕ + nz

)(nz + 1)

Cz,

∫ 1

0c2

z dy =nz

nz + 2C2

z .

(8)

The wall shear stresses τyi|10 = are modelledaccording to the bulk-flow theory for turbulentfilm flows using Hirs’ [44] approach by meansof the Fanning friction factor

τyi|10 = τyi,S − τyi,R. (9)

The directional wall shear stresses τyi,S andτyi,R reads,

τyi,R = fRCRCi,R,

τyi,S = fSCSCi,S.(10)

with the Fanning friction factor fi (i = R, S),the dimensionless effective relative velocity be-tween the wall (rotor R, stator S) and the fluidCi :=

√C2

ϕ,i + φ2C2z,i, i = R, S. The compo-

nents Cϕ,i and Cz,i are boundary layer averagedvelocities between wall and the correspondingboundary layer thickness δ assuming fully de-veloped boundary layers throughout the annu-lus. The boundary layer averaged velocities

yields

Cϕ,S :=1δ

∫ δ

0cϕ dy =

nϕ

nϕ + 1Cϕ,

Cϕ,R :=1δ

∫ δ

0

(cϕ − 1

)dy =

nϕ

nϕ + 1(Cϕ − 1

),

Cz,S :=1δ

∫ δ

0cz dy =

nz

nz + 1Cz,

Cz,R :=1δ

∫ δ

0cz dy =

nz

nz + 1Cz.

(11)

The Fanning friction factor is given by

fi = m f

(h2

CiReϕ

)−n f

. (12)

Here, the friction factor is modelled usingthe empirical constants n f and m f as well asthe corresponding velocity at the rotor and sta-tor Ci, (i = R, S) and the Reynolds number Reϕ.Arbitrary lines within the double logarithmicMoody diagram give the empirical constantsn f and m f . In order to solve the system of equa-tions, boundary conditions have to be specified.The pressure loss at the gap entrance is mod-elled by applying Bernoulli’s equation. Thepressure boundary conditions at the gap inletreads,

p|z=0 = ∆p− (1 + ζ)(

φ2C2z + C2

ϕ

), (13)

Here, ∆p is the overall axial pressure differ-ence, whereas ζ is the entrance pressure losscoefficient. A Dirichlet boundary condition forthe pressure is applied at the annulus exit:

p|z=1 = 0. (14)

In addition, the pre-swirl, i.e. the circumfer-ential velocity at the annulus inlet, is applied.This yields,

Cϕ|z=0 ∈ R. (15)

i. Perturbation analysis

For the calculation of the dynamic forces andtorque characteristics a perturbation analysis isused. Assuming small harmonic disturbances

6


∆ around the static equilibrium position, thecorresponding variables Cϕ, Cz, p and h are de-veloped using a first-order perturbation expan-sion. In the perturbation ansatz, the zeroth-order variables are independent of time whilethe first-order variables are time dependant

Cϕ (t, ϕ, z) = Cϕ,0 (ϕ, z) + ∆Cϕ,1 (t, ϕ, z) ,

Cz (t, ϕ, z) = Cz,0 (ϕ, z) + ∆Cz,1 (t, ϕ, z) ,

p (t, ϕ, z) = p0 (ϕ, z) + ∆p1 (t, ϕ, z) ,

h (t, ϕ, z) = h0 (ϕ, z) + ∆h1 (t, ϕ, z) .

(16)

By inserting the perturbation expansion intothe equations as well as into the boundary con-ditions, a set of partial differential equations forthe zeroth- and first-order is derived. The solu-tion of the zeroth-order equation system yieldsthe static characteristics of the annulus, i.e. thestatic induced forces and torques and the corre-sponding attitude angle, whereas the solutionof the first-order equation system gives thedynamic properties of the system. The zeroth-order equations read

∂

∂ϕh0

∫ 1

0cϕ,0 dy +

φ

L∂

∂zh0

∫ 1

0cz,0 dy = 0,

∂

∂ϕh0

∫ 1

0c2

ϕ,0 dy +φ

L∂

∂zh0

∫ 1

0cϕ,0cz,0 dy =

= −h0

2∂p0

∂ϕ+

12ψ

τyx,0|10,

φ∂

∂ϕh0

∫ 1

0cϕ,0cz,0 dy +

φ2

L∂

∂zh0

∫ 1

0c2

z,0 dy =

= − h0

2L∂p0

∂z+

12ψ

τyz,0|10.

(17)

By comparing the zeroth-order approximateto the initial equations, it becomes clear thatonly the time dependant terms within the con-tinuity and momentum equations vanish. Thesame holds for the integrals, the modelling ofthe shear stresses and the boundary conditions.Therefore, the zeroth-order equations are equalto the non-linear partial differential equationsystem presented by Kuhr et al. [11].

The first-order continuity equation yields

∂

∂th1 +

∂

∂ϕh0

∫ 1

0cϕ,1 dy +

∂

∂ϕh1

∫ 1

0cϕ,0 dy +

+φ

L∂

∂zh0

∫ 1

0cz,1 dy +

φ

L∂

∂zh1

∫ 1

0cz,0 dy = 0.

(18)

The first-order approximation of the momen-tum equation read for circumferential direction

∂

∂th0

∫ 1

0cϕ,1 dy +

∂

∂th1

∫ 1

0cϕ,0 dy+

+ 2∂

∂ϕh0

∫ 1

0cϕ,0cϕ,1 dy +

∂

∂ϕh1

∫ 1

0c2

ϕ,0 dy+

+φ

L∂

∂zh0

∫ 1

0cϕ,0cz,1 dy +

φ

L∂

∂zh0

∫ 1

0cϕ,1cz,0 dy+

+φ

L∂

∂zh1

∫ 1

0cϕ,0cz,0 dy =

= −h0

2∂p1

∂ϕ− h1

2∂p0

∂ϕ+

12ψ

τyx,1|10,

(19)

and axial direction

φ∂

∂th0

∫ 1

0cz,1 dy + φ

∂

∂th1

∫ 1

0cz,0 dy+

+ φ∂

∂ϕh0

∫ 1

0cϕ,0cz,1 dy + φ

∂

∂ϕh0

∫ 1

0cϕ,1cz,0 dy+

+ φ∂

∂ϕh1

∫ 1

0cϕ,0cz,0 dy + 2

φ2

L∂

∂zh0

∫ 1

0cz,0cz,1 dy+

+φ2

L∂

∂zh1

∫ 1

0c2

z,0 dy =

= − h0

2L∂p1

∂z− h1

2L∂p0

∂z+

12ψ

τyz,1|10.

(20)

The perturbation analysis is also applied tothe integrals as well as the shear stresses andthe boundary conditions. The perturbed inte-grals for the continuity and momentum equa-

7


tions yield∫ 1

0cϕ,1 dy =

nϕ

nϕ + 1Cϕ,1,

∫ 1

0cz,1 dy =

nz

nz + 1Cz,1,∫ 1

0cϕ,0cϕ,1 dy =

nϕ

nϕ + 2Cϕ,0Cϕ,1+

+nϕ

2(nϕ + 1

) (nϕ + 2

)Cϕ,1,

∫ 1

0cϕ,0cz,1 dy =

nϕnz

nϕ + nz(nϕ + 1

)Cϕ,0Cz,1+

+n2

z2 (nz + 1)

[nϕ + nz

(nϕ + 1

)]Cz,1,

∫ 1

0cϕ,1cz,0 dy =

nznϕ

nϕ + nz(nϕ + 1

)Cz,0Cϕ,1,

∫ 1

0cz,0cz,1 dy =

nz

nz + 2Cz,0Cz,1.

(21)

Separating the wall shear stresses τyi,1|10 intotheir corresponding directional part yields

τyi,1|10 = τyi,stat,1 − τyi,rot,1. (22)

By applying the binominal approxima-tion (1 + x)a ≈ 1 + ax with |x| 1, the per-turbed directional wall shear stresses τyi,stat,1and τyi,rot,1 read

τyi,stat,1 = τyi,stat,0

[Cstat,1

Cstat,0+

Ci,stat,1

Ci,stat,0+

+ n f

(h1

h0+

Cstat,1

Cstat,0

) ],

τyi,rot,1 = τyi,rot,0

[Crot,1

Crot,0+

Ci,rot,1

Ci,rot,0+

+ n f

(h1

h0+

Crot,1

Crot,0

) ].

(23)

Here Cstat,1, Crot,1 are the perturbed effectiverelative velocities between the stator, rotor andthe fluid

Cstat,1 =Cϕ,stat,0Cϕ,stat,1

Cstat,1+ φ2 Cz,stat,0Cz,stat,1

Cstat,1,

Crot,1 =Cϕ,rot,0Cϕ,rot,1

Crot,1+ φ2 Cz,rot,0Cz,rot,1

Crot,1,

(24)

and Ci,stat,1, Ci,rot,1 are the corresponding per-turbed boundary layer averaged velocities

Cϕ,stat,1 =nϕ

nϕ + 1Cϕ,1, Cϕ,rot,1 =

nϕ

nϕ + 1Cϕ,1,

Cz,stat,1 =nz

nz + 1Cz,1, Cz,rot,1 =

nz

nz + 1Cz,1.

(25)

To solve the perturbed linearised partial dif-ferential equation system the perturbation anal-ysis is applied on the boundary conditions. Theperturbed pressure boundary conditions at thegap inlet and outlet read

p1|z=0 = −2 (1 + ζ)(

φ2Cz,0Cz,1

)− 2ζCϕ,0Cϕ,1,

p1|z=1 = 0.(26)

Furthermore, it is assumed that theperturbed pre-swirl can be neglected, i.e.Cϕ,1|z=0 = 0.

As stated above, the perturbation is assumedto be harmonic in time. Thus, the perturbedvariables Cϕ,1, Cz,1 and p1 are also harmonic.This yields

Cϕ,1 = Cϕ,1,cos cos ωt + Cϕ,1,sin sin ωt,

Cz,1 = Cz,1,cos cos ωt + Cz,1,sin sin ωt,

p1 = p1,cos cos ωt + p1,sin sin ωt.

(27)

The perturbed gap function is given for lat-eral and angular perturbed movements

h1 = − (cos ϕ cos ωt + sin ϕ sin ωt) ,

h1 = − (z− zT)Lψ(cos ϕ cos ωt− sin ϕ sin ωt) .

(28)Assuming a harmonic perturbation, the time

dependant terms within the continuity andmomentum equation vanish and the first-orderlinearised partial differential equation systemcan be solved. To solve the equation systemthe same SIMPLE-C algorithm as for solvingthe zeroth-order equation system is used, cf.[11]. By integrating the first-order pressure dis-tribution the corresponding forces and torques

8


acting on the rotor are determined by

FX,1 = −∫ 1

0

∫ 2π

0p1 cos ϕ dϕ dz,

FY,1 = −∫ 1

0

∫ 2π

0p1 sin ϕ dϕ dz,

MX,1 =∫ 1

0

∫ 2π

0p1 (z− zT) sin ϕ dϕ dz,

MY,1 = −∫ 1

0

∫ 2π

0p1 (z− zT) cos ϕ dϕ dz.

(29)

The rotordynamic coefficients are obtainedby calculating the induced forces and torquesat different percessional frequencies ω and per-forming a least mean square identification pro-cedure. Here, the lateral and angular move-ments are treated separately. As an example,the force FX and torque MX generated by lat-eral displacements are

− FX = KXXX + KXYY+

+ CXXX + CXYY + MXXX + MXYY,

−MX = KαXX + KαYY+

+ CαXX + CαYY + MαXX + MαYY.(30)

Evaluating the perturbed forces and torquesat ωt = 0 and ωt = π/2 the cos, sin termsvanish and the coefficients can be extracted.

− FX,1 (ωt = 0) = KXX + CXYω−MXXω2,

− FX,1 (ωt = π/2) = KXY − CXXω−MXYω2,

−MX,1 (ωt = 0) = KαX + CαYω−MαXω2,

−MX,1 (ωt = π/2) = KαY − CαXω−MαYω2.(31)

III. Validation

For the purpose of validation, the rotordy-namic coefficients for translational displace-ments are compared to the numerical resultspublished by Nordmann & Dietzen [37], Nel-son & Nguyen [38] and Simon & Frêne [36].Here, the influence of eccentricity on the ro-tordynamic coefficients is investigated for anannulus with length L = 0.5, a modified

Reynolds number Re∗ϕ = 0.043, a pressure dif-ference ∆p := 2∆ p/

($Ω2R2) = 1.78 and a

pre-swirl Cϕ|z=0 = 0.3. Nordmann & Dietzen[37] use a three- dimensional finite-differencemethod, solving the Navier-Stokes and conti-nuity equation in combination with a k-ε tur-bulence model, whereas Nelson & Nguyen [38]present a method using fast Fourier transformsto integrate the governing equation system re-sulting in hydrodynamic forces and rotordy-namic coefficients. The results by Simon &Frêne [36] are generated by a similar approachto the one presented here. The main differenceis the treatment of the integrals of the partialdifferential equation system. While the herepresented method uses ansatz functions to de-scribe the velocity profile before integration,Simon & Frêne [36] uses parabolic functionsfor the integral itself.

For angular displacements, the model is com-pared to data by Childs [35] and San Andés [5].The authors investigate the influence of gaplength on the rotordynamic torque coefficientsdue to angular displacement for a concentricannulus with the modified Reynolds numberRe∗ϕ = 0.029, a pressure difference ∆p = 8.38and no pre-swirl Cϕ|z=0 = 0. Childs [35] usesa bulk-flow approach for centred finite-lengthseals, whereas San Andés [5] presents a two-dimensional calculation method consideringfully developed flow.

Figure 2 shows the comparison of the influ-ence of eccentricity on the rotordynamic coeffi-cients for translational displacement. Here, thelines are the numeric results by Nordmann &Dietzen [37], Nelson & Nguyen [38] and Simon& Frêne , whereas the markers represent theClearance-Averaged Pressure Model. The fig-ure compares (I) the direct stiffness KXX , KYY,(II) the cross-coupled stiffness KXY, KYX, (III)the direct damping CXX, CYY, (IV) the cross-coupled damping CXY, CYX and (V) the directinertia MXX , MYY coefficients. In addition, theinfluence of eccentricity on (VI) the flow num-ber is given.

It exhibits a good agreement of the CAPMwith the data obtained from the literature. Ma-jor differences in the calculation methods are

9


𝐾𝑋𝑋,𝐾

𝑌𝑌

𝐾𝑋𝑌,𝐾

𝑌𝑋

𝐶𝑋𝑋,𝐶

𝑌𝑌

𝐶𝑋𝑌,𝐶

𝑌𝑋

(𝐼) (𝐼𝐼)

(𝐼𝐼𝐼) (𝐼𝑉)

𝐿 = 0.5; 𝑧𝑇 = 0.5; 𝑅𝑒𝜑∗ = 0.043

Δ𝑝 = 1.78; 𝐶𝜑 ቚ𝑧=0

= 0.30

RELATIVE ECCENTRICITY 𝜀RELATIVE ECCENTRICITY 𝜀

𝑀𝑋𝑋,𝑀

𝑌𝑌

𝜙

𝐾𝑋𝑋

𝐾𝑌𝑌𝐾𝑌𝑋

𝐾𝑋𝑌

NELSON & NGUYENNORDMAN & DIETZEN

SIMON & FRENE

𝐶𝑋𝑋

𝐶𝑌𝑌

𝐶𝑌𝑋𝐶𝑋𝑌

𝑀𝑌𝑌

𝑀𝑋𝑋

(𝑉) (𝑉𝐼)

Figure 2: Rotordynamic coefficients determined by the CAPM for translational displacement compared to the numericresults by Nordmann & Dietzen [37], Nelson & Nguyen [38] and Simon & Frêne.

10


only apparent for the direct stiffness KXX , KYY.Here, the results presented by Nordman & Di-etzen [37] and Simon & Frêne [36] are in goodagreement with the data obtained by the pre-sented method. However, the results of Nelson& Nguyen [38] differ at higher eccentricities.The predicted decrease of the curves starts atmuch lower eccentricities and is far more se-vere than the results predicted by the other au-thors as well the CAPM. In addition, the eccen-tricity influence on the flow number is given.The flow number increases with increasing ec-centricity. This is due to the altered frictionlosses within the annulus at eccentric opera-tion conditions and a constant axial pressuredifference ∆p.

Figure 3 shows the comparison of the influ-ence of the annulus length on the rotordynamiccoefficients for angular displacement. The fig-ure compares (I) the direct and cross-coupledstiffness |Kαα|, Kαβ, (II) the direct and cross-coupled damping Cαα, Cαβ and (III) the directand cross-coupled inertia Mαα, |Mαβ|. In ad-dition, the influence of annulus length on (IV)the flow number is given.

Again it exhibits a good agreement of theCAPM with the data obtained from the litera-ture. Minor differences in the calculation meth-ods are only apparent for the direct dampingCαα. Here, the results presented by San An-drés [5] are in good agreement with the dataobtained by the presented method. However,the results of Childs [35] differ at an annuluslength L = 0.33. The value is much higher thanpredicted by San Andrés [5] and the presentedmethod CAPM. Furthermore, the predictedvalue seems not to follow the overall trend, i.e.decreased direct damping Cαα at decreasinggap length. Compared to the results presentedhere and the results obtained by San Andrés[5], the calculation seems to be somewhat in-consistent. In addition, the length influence onthe flow number is given. The flow numberdecreased with increasing length. This is dueto the increased friction losses within the an-nulus at concentric operation conditions andconstant axial pressure difference ∆p.

IV. Parameter study

Considering the good agreement of the pre-sented model with the data from the literature,an extensive parameter study is carried out, fo-cusing on the influence of the annulus length,the eccentricity, the centre of rotation, the mod-ified Reynolds number, the flow number andthe pre-swirl on the force and the torque char-acteristics. In the following, only the resultsfor the influence of the annulus length, themodified Reynolds number and the flow num-ber are given. It is shown that the annuluslength, the modified Reynolds number and theflow number are crucial when determining therelevance of the hydraulic tilt and torque coef-ficients. The remaining results, the influenceof eccentricity, the centre of rotation and thepre-swirl are given in the appendix V.

i. Influence of the annulus length

In the following, the influence of the annu-lus length is investigated. Figures 4 to 7show the rotordynamic force and torque co-efficients for translational and angular excita-tion. The examined annulus is operated atconcentric conditions, i.e. ε = 0 at a modifiedReynolds number Reϕ = 0.031 and flow num-ber φ = 0.7. The pre-swirl is Cϕ|z=0 = 0.5,whereas the fulcrum lies in the centre of theannular gap, i.e. zT = 0.5. All four submatri-ces are skew-symmetric for the chosen concen-tric operation point, i.e KT

I..IV , CTI..IV , MT

I..IV =−KI..IV ,−CI..IV ,−MI..IV .Therefore, it is sufficient to focus on one of thedirect and one of the cross-coupled stiffnesscoefficients for each matrix. However, for thesake of completeness all coefficients are shownin the figures.

Figure 4 shows the influence of the annuluslength on the direct and cross-coupled stiffnessdue to translational excitation by the hydraulicforces (I), the direct and cross-coupled stiffnessdue to angular excitation caused by the hy-draulic forces (II), the direct and cross-coupledstiffness due to translational excitation by

11


(𝐼) (𝐼𝐼)


ANNULUS LENGTH 𝐿ANNULUS LENGTH 𝐿

𝐾𝛼𝛼,𝐾

𝛼𝛽

𝐶𝛼𝛼,𝐶

𝛼𝛽

𝑀𝛼𝛼,𝑀

𝛼𝛽

𝜙

𝜀 = 0; 𝑧𝑇 = 0.5; 𝑅𝑒𝜑∗ = 0.029

Δ𝑝 = 8.38; 𝐶𝜑 ቚ𝑧=0

= 0

|𝐾𝛼𝛼|

|𝐾𝛼𝛽|

|𝐶𝛼𝛼|

|𝐶𝛼𝛽|

𝑀𝛼𝛼

𝑀𝛼𝛽

SAN ANDRESCHILDS

Figure 3: Rotordynamic coefficients determined by the CAPM for angular displacement compared to the numericresults by Childs [35] and San Andés [5].

12


𝐾𝑋𝑋,𝐾

𝑋𝑌,𝐾

𝑌𝑋,𝐾

𝑌𝑌

𝐾𝑋𝛼,𝐾

𝑋𝛽,𝐾

𝑌𝛼,𝐾

𝑌𝛽

𝐾𝛼𝑋,𝐾

𝛼𝑌,𝐾

𝛽𝑋,𝐾

𝛽𝑌

𝐾𝛼𝛼,𝐾

𝛼𝛽,𝐾

𝛽𝛼,𝐾

𝛽𝛽

(𝐼) (𝐼𝐼)

(𝐼𝐼𝐼) (𝐼𝑉)𝜀 = 0; 𝑧𝑇 = 0.5; 𝑅𝑒𝜑

∗ = 0.031

𝜙 = 0.70; 𝐶𝜑 ቚ𝑧=0

= 0.50


𝐾𝑋𝑋, 𝐾𝑌𝑌

𝐾𝑋𝑌

𝐾𝑌𝑋 𝐾𝑌𝛼

𝐾𝛼𝑋, 𝐾𝛽𝑌

𝐾𝛼𝑌𝐾𝛼𝛽

𝐾𝛽𝛼

𝐾𝑋𝛽

𝐾𝑋𝛼 , 𝐾𝑌𝛽

𝐾𝛼𝛼 , 𝐾𝛽𝛽𝐾𝛽𝑋

Figure 4: Influence of annulus length on the stiffness due to translational and angular excitation. (I) Stiffness dueto translational excitation by the hydraulic forces. (II) Stiffness due to angular excitation by the hydraulicforces. (III) Stiffness due to translational excitation by the hydraulic torques. (IV) Stiffness due to angularexcitation by the hydraulic torques.

13


: 0.03 + 6.55

ANNULUS LENGTH 𝐿

1.79

= 5.5 ∗ 10−4

1.34

0.69= 6.0 ∗ 10−4

𝛼𝑋𝑌

= 1.4 ∗ 10−3

= 1.0 ∗ 10−3

∝ exp(1.4𝐿)

−(𝐾

𝑌𝛼𝐿𝛼𝑋)/(𝐾

𝑌𝑌𝜓𝑌)

Figure 5: Influence of the annulus length and the ratioof angular to translational excitation αX/Yon the ratio of tilt to translational stiffnesscoefficients.

the hydraulic torques (III) and the direct andcross-coupled stiffness due to angular excita-tion by the hydraulic torques (IV). Focusingon submatrix (I) the cross-coupled stiffnessKXY, |KYX | increase linearly with increasingannulus length, whereas the direct stiffness co-efficients KXX , KYY initially increase, reachinga maximum at an annulus length of L ≈ 0.6.By further increasing the annulus length,the direct stiffness decreases. It should benoted that as the gap becomes longer, thecross-coupled stiffness becomes larger than thedirect stiffness. This is particularly interesting,as it can be crucial to stability analysis. Incontrast to the the direct and cross-coupledstiffness of submatrix (I), the direct andcross-coupled stiffness of submatrix (II) showan asymptotic behaviour when increasingthe annulus length. Fist, focusing on thedirect stiffness, i.e KXα, KYβ, the stiffnessdecreases with increasing gap length. Thecurve continues to flatten reaching a valueof KXα, KYβ = −0.41 at L = 2.5. Second,examining the cross-coupled coefficients KXβ,|KYα|, the stiffness increases with increasingannulus length up to an annulus length ofL = 1.5. By further increasing the length, thecurves show an asymptotic behaviour for avalue KXβ, |KYα| = 3.40. It should be notedthat the cross-coupled coefficients of submatrix

(II) are in the same order of magnitude asthe cross- coupled stiffness of submatrix (I).This is of particular interest when examiningthe relevance of the hydraulic tilt and torquecoefficients, cf. figure 5. The direct andcross-coupled stiffness of submatrix (III)behave similarly to the ones of submatrix (II).The direct stiffness KαX, KβY, being one orderof magnitude smaller than direct stiffness KXα,KYβ, decrease with increasing annulus length,whereas the cross-coupled stiffness KαY, |KβX |first increase with increasing length up toL = 0.75. For annulus lengths longer thanL > 0.75, the cross-coupled stiffness stagnatesand slightly decreases. Focusing on the fourthsubmatrix (IV) the direct Kαα, Kββ as well asthe cross-coupled stiffness |Kβα|, Kαβ increasewhen increasing the annulus length.

As mentioned before, the relevance of thetilt and torque coefficients can be determinedby focusing on the cross-coupled stiffness ofsubmatrix (II). [3] states that the additional co-efficients become relevant at an annulus lengthgreater than L = 1.5. When determining therelevance of tilt and torque coefficients the fo-cus lies on the Y component of the inducedforces while translational motion in X-directionas well as angular motion around the Y-axis isprohibited. This yields

−FY = KYYY + KYαα. (32)

By defining an effective stiffness

Keff := KYY

(1 +

KYαα

KYYY

)=

= KYY

(1 +

KYα

KYY

Lψ

αXY

) (33)

the relevance of the additional coefficientscan be studied. If the quotient of KYαα/ (KYYY)is small, only the forces due to translationalmotion are relevant as it is for small annuli.By examining equation 33 it becomes clearthat an overall threshold for the relevance ofthe additional rotordynamic coefficients is astrong simplification. Instead, the quotient isinversely dependant on the slenderness of the

14


gap, i.e. L/ψ, and the ratio of the angular andtranslational excitation αX/Y. Childs uses aconstant ratio of αX/Y = 5.5 ∗ 10−4 to calculatethe overall length threshold of L/R = 1.5.This corresponds to an excitation angle ofαX = 3.10 ∗ 10−3 degree and a translationalexcitation of 36 ¯m, cf. [3]. Furthermore,any additional influence besides the annuluslength is neglected.

Figure 5 shows the ratio of tilt totranslational stiffness coefficients, i.e.|KYαL αX/ (KYYψY) | versus the annuluslength for different αX/Y. It exhibits increas-ing influence of the additional rotordynamiccoefficients with increasing annulus length.Here, the relevance of the additional tiltand torque coefficients increase proportion-ally ∝ exp (1.4L). Furthermore, it shows astrong influence on the ratio of angular totranslational excitation. Focusing on theratio chosen by Childs, the cross-coupledstiffness of submatrix (II) KYα is 0.69 of thethe direct stiffness KYY at an annulus lengthof L = 1.5. This means that the stiffnessdue to the tilt accounts for approximately41 % of the total stiffness. By increasing theratio of angular to translational excitation toαX/Y = 1.0 ∗ 10−3 the cross-coupled stiffnessof submatrix (II) KYα is 1.34 times as greatas the direct stiffness KYY, increasing thecontribution of the tilt coefficients to theoverall stiffness to 57 %. It should be notedthat this corresponds to an increased excitationangle of αX = 6.0 ∗ 10−3 degree. Therefore, anoverall threshold of L/R = 1.5 to describe therelevance of the additional rotordynamic co-efficients is insufficient. Rather, the operatingconditions of the turbomachinery need to betaken into account.

Figure 6 shows the influence of the annu-lus length on the direct and cross-coupleddamping due to translational excitation bythe hydraulic forces (I), the direct and cross-coupled damping due to angular excitationby the hydraulic forces (II), the direct andcross-coupled damping due to translational

excitation by the hydraulic torques (III)and the direct and cross-coupled dampingdue to angular excitation by the hydraulictorques (IV). It exhibits that the direct andcross-coupled damping due to translationalexcitation by the hydraulic forces are one orderof magnitude higher than the other dampingcoefficients in the sub-matrices (II) to (IV).First, focusing on the damping coefficients ofsubmatrix (I) both, the direct CXX, CYY andcross-coupled damping CXY, |CYX | linearlydepend on the annulus length. Here, thedirect coefficients increase faster than thecross-coupled coefficients. It is noted that thedirect damping coefficients are of particularinterest when evaluating the stability of theflow inside the annulus. Second, focusing onthe direct and cross-coupled damping coeffi-cients of submatrix (II), the direct dampingcoefficients CXα, CYβ are not influenced byan increasing annulus length, whereas thecross-coupled damping coefficients CXβ, |CYα|slightly increase with increasing length. Thedamping coefficients of submatrix (III) are inthe same order of magnitude as the coefficientsof submatrix (II). Similar to the direct dampingCXα , CYβ, the direct damping coefficientsCαX, CβY are almost independent of theannulus length. In contrast, the cross-coupleddamping coefficients CαY, |CβX | increase withincreasing gap length. Finally, the directand cross-coupled damping coefficients ofsubmatrix (IV) are investigated. The directdamping coefficients Cαα, Cββ as well as thecross-coupled damping coefficients Cαβ, |Cβα|increase with increasing annulus length. Here,the direct coefficients are greater in value thanthe cross-coupled coefficients.

Figure 7 shows the influence of the annuluslength on the direct and cross-coupled inertiadue to translational excitation by the hydraulicforces (I), the direct and cross-coupled inertiadue to angular excitation by the hydraulicforces (II), the direct and cross-coupled inertiadue to translational excitation by the hydraulictorques (III) and the direct and cross-coupledinertia due to angular excitation by the

15


𝐶𝑋𝑋,𝐶

𝑋𝑌,𝐶

𝑌𝑋,𝐶

𝑌𝑌

𝐶𝑋𝛼,𝐶

𝑋𝛽,𝐶

𝑌𝛼,𝐶

𝑌𝛽

𝐶𝛼𝑋,𝐶

𝛼𝑌,𝐶

𝛽𝑋,𝐶

𝛽𝑌

𝐶𝛼𝛼,𝐶

𝛼𝛽,𝐶

𝛽𝛼,𝐶

𝛽𝛽

(𝐼) (𝐼𝐼)


𝜀 = 0; 𝑧𝑇 = 0.5; 𝑅𝑒𝜑∗ = 0.031

𝜙 = 0.70; 𝐶𝜑 ቚ𝑧=0

= 0.50


𝐶𝑋𝑌

𝐶𝑋𝑋, 𝐶𝑌𝑌

𝐶𝑌𝑋

𝐶𝑋𝛽

𝐶𝛼𝑋, 𝐶𝛽𝑌

𝐶𝛼𝑌

𝐶𝛼𝛼 , 𝐶𝛽𝛽

𝐶𝛼𝛽

𝐶𝑌𝛼

𝐶𝑋𝛼 ,𝐶𝑌𝛽

𝐶𝛽𝑋

𝐶𝛽𝛼

Figure 6: Influence of annulus length on the damping due to translational and angular excitation. (I) Damping dueto translational excitation by the hydraulic forces. (II) Damping due to angular excitation by the hydraulicforces. (III) Damping due to translational excitation by the hydraulic torques. (IV) Damping due to angularexcitation by the hydraulic torques.

16


hydraulic torques (IV). First, focusing on theinertia coefficients of submatrix (I), both, thedirect and cross-coupled inertia coefficientslinearly increase with the annulus length.Here, the cross-coupled inertia coefficientsare two orders of magnitude smaller than thedirect inertia coefficients. As a result, mostof the literature neglects the cross-coupledinertia coefficients compared to the directcoefficients. Second, focusing on the directand cross-coupled inertia coefficients ofsubmatrix (II), the coefficients are in the sameorder of magnitude as the cross-coupledinertia coefficients of submatrix (I). Therefore,no significant trend can be observed whenincreasing the annulus length. The inertiacoefficients of submatrix (III) are in the sameorder of magnitude as the coefficients ofsubmatrix (II). The direct inertia MαX, MβYdecreases slightly with increasing annuluslength, whereas the cross-coupled inertia MαY,|MβX | increases with the length. Finally, thedirect and cross-coupled inertia coefficientsof submatrix (IV) are investigated. The directinertia coefficients Mαα, Mββ as well as the thecross-coupled coefficients |Mαβ|, Mβα increasewith increasing annulus length. Here, thedirect coefficients are one order of magnitudegreater than the cross-coupled coefficients.

In summary, the following statements can bemade:

• The cross-coupled stiffness, the directdamping and the direct inertia of subma-trix (I) increase linearly with the annuluslength.

• The cross-coupled stiffness of submatrix(II) shows an asymptotic behaviour.

• The relevance of the hydraulic tilt andtorque coefficients increases proportion-ally ∝ exp (1.4L).

ii. Influence of the modified Reynoldsnumber

In the following the influence of the modifiedReynolds number on the rotordynamic coeffi-cients is investigated. Figures 8 to 11 give the

force and torque coefficients for translationaland angular excitation. The annulus of lengthL = 1.3 is operated at concentric conditions, i.e.ε = 0 with a flow number φ = 0.7. The pre-swirl before the annulus is set to Cϕ|z=0 = 0.5and the fulcrum lies in the centre of the annu-lar gap, i.e. zT = 0.5.

Figure 8 shows the influence of the modi-fied Reynolds number on the direct and cross-coupled stiffness due to translational excita-tion by the hydraulic forces (I), the direct andcross-coupled stiffness due to angular excita-tion by the hydraulic forces (II), the direct andcross-coupled stiffness due to translational ex-citation by the hydraulic torques (III) and thedirect and cross-coupled stiffness due to an-gular excitation by the hydraulic torques (IV).First, focusing on the stiffness coefficients ofthe first submatrix (I), the direct KXX, KYY aswell as the cross-coupled stiffness KXY, |KYX |decreases with increasing modified Reynoldsnumber. Here, the cross-coupled stiffness KXY,|KYX | is proportional to ∝ 1/Re∗ϕ. Second, fo-cusing on the stiffness coefficients of submatrix(II), the direct stiffness KXα, KYβ is almost in-dependent on the modified Reynolds number.However, similar to the coefficients of subma-trix (II), the cross-coupled stiffness is decreas-ing with KXβ, |KYα| ∝ 1/Re∗ϕ, cf. submatrix(I). Focusing on the stiffness of submatrix (III),the direct stiffness coefficients KαX , KβY are al-most independent of the modified Reynoldsnumber. The cross-coupled stiffness KαY, |KβX |decrease with increasing modified Reynoldsnumber. Finally, the stiffness coefficients ofsubmatrix (IV) are examined. Here, the directstiffness coefficients Kαα, Kββ are independentof the modified Reynolds number, whereas thecross-coupled stiffness Kαβ, |Kβα| decrease withincreasing modified Reynolds number.

Similar to the consideration of the influenceof the annulus length on the relevance of thetilt and torque coefficients figure 9 shows theinfluence of the modified Reynolds number onthe ratio of tilt to translational stiffness coeffi-cients, cf. equation 33. It exhibits the influence

17


𝑀𝑋𝑋,𝑀

𝑋𝑌,𝑀

𝑌𝑋,𝑀

𝑌𝑌

𝑀𝑋𝛼,𝑀

𝑋𝛽,𝑀

𝑌𝛼,𝑀

𝑌𝛽

𝑀𝛼𝑋,𝑀

𝛼𝑌,𝑀

𝛽𝑋,𝑀

𝛽𝑌

𝑀𝛼𝛼,𝑀

𝛼𝛽,𝑀

𝛽𝛼,𝑀

𝛽𝛽

(𝐼) (𝐼𝐼)


𝜀 = 0; 𝑧𝑇 = 0.5; 𝑅𝑒𝜑∗ = 0.031

𝜙 = 0.70; 𝐶𝜑 ቚ𝑧=0

= 0.50

ANNULUS LENGTH 𝐿

𝑀𝑋𝑋,𝑀𝑌𝑌

𝑀𝑋𝑌,𝑀𝑌𝑋

𝑀𝛼𝑋, 𝑀𝛽𝑌

𝑀𝛽𝑋

𝑀𝛼𝑌

𝑀𝛼𝛼 ,𝑀𝛽𝛽

𝑀𝛽𝛼

𝑀𝑋𝛼 ,𝑀𝑋𝛽 ,𝑀𝑌𝛼 ,𝑀𝑌𝛽

𝑀𝛼𝛽

ANNULUS LENGTH 𝐿

Figure 7: Influence of annulus length on the inertia due to translational and angular excitation. (I) Inertia due totranslational excitation by the hydraulic forces. (II) Inertia due to angular excitation by the hydraulic forces.(III) Inertia due to translational excitation by the hydraulic torques. (IV) Inertia due to angular excitation bythe hydraulic torques.

18


𝐾𝑋𝑋,𝐾

𝑋𝑌,𝐾

𝑌𝑋,𝐾

𝑌𝑌

𝐾𝑋𝛼,𝐾

𝑋𝛽,𝐾

𝑌𝛼,𝐾

𝑌𝛽

𝐾𝛼𝑋,𝐾

𝛼𝑌,𝐾

𝛽𝑋,𝐾

𝛽𝑌

𝐾𝛼𝛼,𝐾

𝛼𝛽,𝐾

𝛽𝛼,𝐾

𝛽𝛽

(𝐼) (𝐼𝐼)


𝐿 = 1.3; 𝜀 = 0; 𝑧𝑇 = 0.5;

𝜙 = 0.70; 𝐶𝜑 ቚ𝑧=0

= 0.50

MODIFIED REYNOLDS NUMBER 𝑅𝑒𝜑∗MODIFIED REYNOLDS NUMBER 𝑅𝑒𝜑

∗

𝐾𝑋𝑋, 𝐾𝑌𝑌𝐾𝑌𝑋


𝐾𝛼𝑌

𝐾𝛽𝛼

𝐾𝑌𝛼


𝐾𝛼𝛼 , 𝐾𝛽𝛽

𝐾𝑋𝑌𝐾𝑋𝛽

𝐾𝛽X

𝐾𝛼𝛽

Figure 8: Influence of the modified Reynolds number on the stiffness due to translational and angular excitation. (I)Stiffness due to translational excitation by the hydraulic forces. (II) Stiffness due to angular excitation by thehydraulic forces. (III) Stiffness due to translational excitation by the hydraulic torques. (IV) Stiffness due toangular excitation by the hydraulic torques.

19


−(𝐾


𝑌𝑌𝜓𝑌)

: 0.03 + 6.55

MODIFIED REYNOLDS NUMBER 𝑅𝑒𝜑∗

= 5.5 ∗ 10−4= 6.0 ∗ 10−4

𝛼𝑋𝑌= 1.4 ∗ 10−3

= 1.0 ∗ 10−3

∝1

𝑅𝑒𝜑∗ 1.8

Figure 9: Influence of the modified Reynolds numberand the ratio of angular to translational excita-tion αX/Y on the ratio of tilt to translationalstiffness coefficients.

the modified Reynolds number has on theratio of tilt to translational stiffness coefficients.This reinforces the statement that a sole de-pendence on the annulus length is insufficient.It can be shown that relevance of the tilt andtorque coefficients is proportional ∝ 1/Re∗ϕ

1.8.Therefore, the additional rotordynamic coeffi-cients become more relevant with decreasingmodified Reynolds number, i.e. a decreas-ing gap clearance ψ and Reynolds number Reϕ.

Figure 10 shows the influence of themodified Reynolds number on the direct andcross-coupled damping due to translationalexcitation by the hydraulic forces (I), the directand cross-coupled damping due to angularexcitation by the hydraulic forces (II), thedirect and cross-coupled damping due to trans-lational excitation by the hydraulic torques(III) and the direct and cross-coupled dampingdue to angular excitation by the hydraulictorques (IV). First, focusing on the dampingcoefficients of submatrix (I), it exhibits de-creasing direct coefficients with increasingmodified Reynolds number. Similar to thedirect stiffness, the direct damping coefficientsCXX , CYY are proportional ∝ 1/Re∗ϕ. However,the cross-coupled damping coefficients CXY,CYX are almost independent of the modifiedReynolds number. Second, focusing on the

damping coefficients of submatrix (II) thedirect and cross-coupled coefficients do notdepend on the modified Reynolds number.In contrast to that, the damping coefficientsof submatrix (III) show a slight dependenceon the modified Reynolds number. Here, thedirect coefficients CαX, CβY as well as thecross-coupled coefficients CαY, |CβX | decreasesslightly with increasing modified Reynoldsnumber. Finally, the damping coefficients ofsubmatrix (IV) are examined. It is shownthat the cross-coupled damping Cαβ, |Cβα|slightly increases with increasing modifiedReynolds number. In accordance with thedirect damping coefficients of submatrix(I), the direct damping Cαα, Cββ decreasesproportionally ∝ 1/Re∗ϕ, whereas the cross-coupled damping Cαβ, |Cβα| show a lineardependence on the modified Reynolds number.

Figure 11 shows the influence of themodified Reynolds number on the directand cross-coupled inertia due to translationalexcitation by the hydraulic forces (I), thedirect and cross-coupled inertia due to angularexcitation by the hydraulic forces (II), the directand cross-coupled inertia due to translationalexcitation by the hydraulic torques (III) andthe direct and cross-coupled inertia due toangular excitation by the hydraulic torques(IV). First, focusing on the inertia coefficientsof the first submatrix (I), it exhibits slightlyincreasing inertia coefficients MXX, MYYwith increasing modified Reynolds number.The cross-coupled inertia coefficients MXY,|MYX | are one to two order of magnitudessmaller than the direct ones, exhibiting a slightincrease with increasing modified Reynoldsnumber. Second, focusing on the inertiacoefficients of submatrix (II) the direct andcross-coupled inertia are almost independentof the modified Reynolds number. It is notedthat the inertia coefficients of submatrix (II) aretwo orders of magnitude smaller than the onesof submatrix (I). Next, considering the inertiacoefficients of submatrix (III), a decreasingdirect inertia MαX, MβY is exhibited withincreasing modified Reynolds number. In

20


𝐶𝑋𝑋,𝐶

𝑋𝑌,𝐶

𝑌𝑋,𝐶

𝑌𝑌

𝐶𝑋𝛼,𝐶

𝑋𝛽,𝐶

𝑌𝛼,𝐶

𝑌𝛽

𝐶𝛼𝑋,𝐶

𝛼𝑌,𝐶

𝛽𝑋,𝐶

𝛽𝑌

𝐶𝛼𝛼,𝐶

𝛼𝛽,𝐶

𝛽𝛼,𝐶

𝛽𝛽

(𝐼) (𝐼𝐼)


𝐿 = 1.3; 𝜀 = 0; 𝑧𝑇 = 0.5;

𝜙 = 0.70; 𝐶𝜑 ቚ𝑧=0

= 0.50

𝐶𝑌𝑋

𝐶𝛼𝑌

𝐶𝛽𝑋

𝐶𝑋𝛽

𝐶𝛽𝛼𝐶𝛼𝑋, 𝐶𝛽𝑌


𝐶𝛼𝛽

𝐶𝑋𝑌


∗


𝐶𝑋𝛼 ,𝐶𝑋𝛽𝐶𝑌𝛼

Figure 10: Influence of the modified Reynolds number on the damping due to translational and angular excitation. (I)Damping due to translational excitation by the hydraulic forces. (II) Damping due to angular excitation bythe hydraulic forces. (III) Damping due to translational excitation by the hydraulic torques. (IV) Dampingdue to angular excitation by the hydraulic torques.

21


𝑀𝑋𝑋,𝑀

𝑋𝑌,𝑀

𝑌𝑋,𝑀

𝑌𝑌

𝑀𝑋𝛼,𝑀

𝑋𝛽,𝑀

𝑌𝛼,𝑀

𝑌𝛽

𝑀𝛼𝑋,𝑀

𝛼𝑌,𝑀

𝛽𝑋,𝑀

𝛽𝑌

𝑀𝛼𝛼,𝑀

𝛼𝛽,𝑀

𝛽𝛼,𝑀

𝛽𝛽

(𝐼) (𝐼𝐼)


𝐿 = 1.3; 𝜀 = 0; 𝑧𝑇 = 0.5

𝜙 = 0.70; 𝐶𝜑 ቚ𝑧=0

= 0.50𝑀𝑋𝑋,𝑀𝑌𝑌

𝑀𝑋𝑌

𝑀𝛼𝑋, 𝑀𝛽Y

𝑀𝛼𝑌

𝑀𝑋𝛼,𝑀𝑋𝛽 , 𝑀𝑌𝛼 ,𝑀𝑌𝛽

𝑀𝑌𝑋


𝑀𝛼𝛽


∗

𝑀𝛽𝑋

𝑀𝛽𝛼

Figure 11: Influence of the modified Reynolds number on the inertia due to translational and angular excitation. (I)Inertia due to translational excitation by the hydraulic forces. (II) Inertia due to angular excitation by thehydraulic forces. (III) Inertia due to translational excitation by the hydraulic torques. (IV) Inertia due toangular excitation by the hydraulic torques.

22


contrast to that, the cross-coupled inertia MαY,|MβX | increases with an increasing modifiedReynolds number. Finally, focusing on theinertia coefficients of submatrix (IV), the directinertia coefficients Mαα, Mββ show a parabolicbehaviour, whereas the cross-coupled inertia|Mαβ|, Mβα linearly increases with increasingmodified Reynolds number.


• The cross-coupled stiffness and directdamping of submatrix (I) as well as thecross-coupled stiffness of submatrix (II)and the direct damping of submatrix (IV)decrease proportionally ∝ 1/Re∗ϕ.

• The remaining coefficients only showa marginal influence of the modifiedReynolds number.

• The relevance of the tilt and torque coeffi-cients decreases proportionally ∝ 1/Re∗ϕ

1.8

iii. Influence of the flow number

Figures 12 to 15 give the influence of the flownumber on the rotordynamic force and torquecoefficients for translational and angular excita-tion. The annulus with length L = 1.3 is oper-ated at concentric conditions, i.e. ε = 0 with amodified Reynolds number Reϕ = 0.031 and apre-swirl before the annulus Cϕ|z=0 = 0.5 Thefulcrum lies in the centre of the annular gap,i.e. zT = 0.5.

Figure 12 shows the influence of the flownumber on the direct and cross-coupledstiffness due to translational excitation bythe hydraulic forces (I), the direct and cross-coupled stiffness due to angular excitationby the hydraulic forces (II), the direct andcross-coupled stiffness due to translationalexcitation by the hydraulic torques (III) andthe direct and cross-coupled stiffness due toangular excitation by the hydraulic torques(IV). First, focusing on the stiffness coefficientsof the first submatrix (I), it exhibits directstiffness coefficients KXX, KYY proportionally

increasing with ∝ φ1.9, whereas the cross-coupled stiffness KXY, |KYX | only increasesproportionally ∝ φ0.9. The difference inthe exponent is due to an increase in flownumber, resulting in an increased Lomakineffect. This effect mainly affects the directstiffness due to the altered axial pressure field.Second, focusing on the stiffness coefficients ofsubmatrix (II), the direct stiffness coefficientsKXα, KYβ are almost independent of theflow number compared to the cross-coupledcoefficients. Similar to the direct stiffnessof submatrix (I), the cross-coupled stiffnessKXβ, |KYα| increases proportionally ∝ φ1.9.It is noted that the cross-coupled stiffnesscoefficients of submatrix (II) are of the sameorder of magnitude as the direct coefficientsof submatrix (I). Again, this is of particularinterest when evaluating the relevance of thetilt and torque coefficients. Focusing on thestiffness of submatrix (III), the direct stiffnesscoefficients slightly decrease with increasingflow number, whereas the cross-coupled stiff-ness KαY, |KβX | increases proportionally ∝ φ1.8.Finally, focusing on the stiffness coefficientsof submatrix (IV), the cross-coupled stiffnesscoefficients are almost independent of the flownumber. The direct stiffness Kαα, Kββ, however,decreases proportionally ∝ −φ2.

Similar to the previous consideration ofthe tilt an torque coefficients, figure 13shows the influence of the flow numberon the ratio of tilt to translational stiffnesscoefficients, cf. equation 33. In contrast tothe influence regarding the annulus length,the influence of the flow number decreaseswith increasing flow number. Here, the ratioof tilt to translational stiffness coefficientsis proportional ∝ 1/φ1.5. Therefore, theadditional rotordynamic coefficients becomemore relevant with decreasing flow number,i.e. a decreasing axial flow component. Thisis due to the fact that the direct stiffness ofsubmatrix (I) KYY decreases faster than thecross-coupled stiffness KYα, becoming KYY = 0at vanishing flow component, i.e. φ = 0.

Figure 14 shows the influence of the flow

23


𝐾𝑋𝑋,𝐾

𝑋𝑌,𝐾

𝑌𝑋,𝐾

𝑌𝑌

𝐾𝑋𝛼,𝐾

𝑋𝛽,𝐾

𝑌𝛼,𝐾

𝑌𝛽

𝐾𝛼𝑋,𝐾

𝛼𝑌,𝐾

𝛽𝑋,𝐾

𝛽𝑌

𝐾𝛼𝛼,𝐾

𝛼𝛽,𝐾

𝛽𝛼,𝐾

𝛽𝛽

(𝐼) (𝐼𝐼)


𝐿 = 1.3; 𝜀 = 0; 𝑧𝑇 = 0.5;

𝑅𝑒𝜑∗ = 0.031; 𝐶𝜑 ቚ

𝑧=0= 0.50

FLOW NUMBER 𝜙FLOW NUMBER 𝜙

𝐾𝑌𝑋


𝐾𝛼𝑌

𝐾𝑋𝛽𝐾𝑋𝛼 , 𝐾𝑌𝛽

𝐾𝑌𝛼

𝐾𝛽X

𝐾𝛼𝛽 , 𝐾𝛽𝛼



𝐾𝑌𝑋

Figure 12: Influence of the flow number on the stiffness due to translational and angular excitation. (I) Stiffness due totranslational excitation by the hydraulic forces. (II) Stiffness due to angular excitation by the hydraulicforces. (III) Stiffness due to translational excitation by the hydraulic torques. (IV) Stiffness due to angularexcitation by the hydraulic torques.

24


= 5.5 ∗ 10−4

: 0.03 + 6.55

FLOW NUMBER 𝜙

−(𝐾


𝑌𝑌𝜓𝑌)

= 6.0 ∗ 10−4

𝛼𝑋𝑌= 1.4 ∗ 10−3

= 1.0 ∗ 10−3

∝1

𝜙1.5

Figure 13: Influence of the flow number and the ratioof angular to translational excitation αX/Yon the ratio of tilt to translational stiffnesscoefficients.

number on the direct and cross-coupleddamping due to translational excitation bythe hydraulic forces (I), the direct and cross-coupled damping due to angular excitationby the hydraulic forces (II), the direct andcross-coupled damping due to translationalexcitation by the hydraulic torques (III) andthe direct and cross-coupled damping due toangular excitation by the hydraulic torques(IV).

First, focusing on the damping coefficientsof the first submatrix (I), it exhibits the valuesof cross-coupled damping coefficients CXY,|CYX | being independent of the flow number.However, the direct damping CXX , CYY showsa linear dependence on the flow number.Second, focusing on the damping coefficientsof submatrix (II) the direct damping coeffi-cients CXα, CYβ are almost independent ofthe flow number, whereas the cross-coupleddamping CXβ, |CYα| linearly increases withthe flow number. The same dependencycan be seen when considering the dampingof submatrix (III). Being in the same orderof magnitude as the damping of submatrix(II), the direct damping coefficients CαX, CβYare almost independent of the flow number,whereas the cross-coupled damping |CβX |,CαY increases linearly with the flow number.

Finally, focusing on the damping of submatrix(IV) the cross-coupled damping coefficients|Cαβ|, Cβα are independent of the flow number,whereas the direct damping Cαα, Cββ increaseslinearly.

Figure 15 shows the influence of the flownumber on the direct and cross-coupled inertiadue to translational excitation by the hydraulicforces (I), the direct and cross-coupled inertiadue to angular excitation by the hydraulicforces (II), the direct and cross-coupled inertiadue to translational excitation by the hydraulictorques (III) and the direct and cross-coupledinertia due to angular excitation by thehydraulic torques (IV).First, focusing on the inertia coefficientsof the first submatrix (I), it exhibits inertiacoefficients almost independent of the flownumber. Merely the direct inertia MXX, MYYshows an influence at low flow numbers. Here,the direct inertia increases with increasing flownumber. This is due to the fact that the inertiacoefficients originate from a displacementof the fluid inside the annulus. Here, theaxial flow component is almost negligible,resulting only in small changes due to anincreasing flow number. Second, focusing onthe inertia coefficients of submatrix (II), thedirect coefficients MXα, MYβ are independentof the flow number, whereas the cross-coupledcoefficients MXβ, |MYα| slightly increase. It isnoted that the inertia coefficients of submatrix(II) are one to two orders of magnitude smallerthan the ones of submatrix (I). Focusing onthe inertia of submatrix (III), the cross-coupledinertia coefficients exhibit a dependenceo the flow number. Here, the coefficientsMαY, |MβX | increase with increasing flownumber, whereas the direct inertia changedinsignificantly. Finally, focusing on the inertiaof submatrix (IV) the coefficients behave in asimilar manner as the coefficients of submatrix(I). Here, the direct inertia Mαα, Mββ increaseswith increasing flow number, whereas thecross-coupled coefficients |Mαβ|, Mβα increaseat first, reaching a maximum at φ = 0.8.By further increasing the flow number the

25


𝐶𝑋𝑋,𝐶

𝑋𝑌,𝐶

𝑌𝑋,𝐶

𝑌𝑌

𝐶𝑋𝛼,𝐶

𝑋𝛽,𝐶

𝑌𝛼,𝐶

𝑌𝛽

𝐶𝛼𝑋,𝐶

𝛼𝑌,𝐶

𝛽𝑋,𝐶

𝛽𝑌

𝐶𝛼𝛼,𝐶

𝛼𝛽,𝐶

𝛽𝛼,𝐶

𝛽𝛽

(𝐼) (𝐼𝐼)𝐿 = 1.3; 𝜀 = 0; 𝑧𝑇 = 0.5;

𝑅𝑒𝜑∗ = 0.031; 𝐶𝜑 ቚ

𝑧=0= 0.50

𝐶𝑌𝑋

𝐶𝛽𝑋

𝐶𝛼𝑌

𝐶𝑌𝛼

𝐶𝛼𝛽, 𝐶𝛽𝛼𝐶𝛼𝑋, 𝐶𝛽𝑌


𝐶𝑋𝑌


𝐶𝑋𝛽




Figure 14: Influence of the flow number on the damping due to translational and angular excitation. (I) Damping dueto translational excitation by the hydraulic forces. (II) Damping due to angular excitation by the hydraulicforces. (III) Damping due to translational excitation by the hydraulic torques. (IV) Damping due to angularexcitation by the hydraulic torques.

26


𝑀𝑋𝑋,𝑀

𝑋𝑌,𝑀

𝑌𝑋,𝑀

𝑌𝑌

𝑀𝑋𝛼,𝑀

𝑋𝛽,𝑀

𝑌𝛼,𝑀

𝑌𝛽

𝑀𝛼𝑋,𝑀

𝛼𝑌,𝑀

𝛽𝑋,𝑀

𝛽𝑌

𝑀𝛼𝛼,𝑀

𝛼𝛽,𝑀

𝛽𝛼,𝑀

𝛽𝛽

(𝐼) (𝐼𝐼)

𝐿 = 1.3; 𝜀 = 0; 𝑧𝑇 = 0.5

𝑅𝑒𝜑∗ = 0.031; 𝐶𝜑 ቚ

𝑧=0= 0.50


𝑀𝑋𝑌


𝑀𝛼𝑌

𝑀𝑋𝛼,𝑀𝑋𝛽 ,𝑀𝑌𝛼 ,𝑀𝑌𝛽

𝑀𝑌𝑋


𝑀𝛼𝛽

𝑀𝛽𝑋



𝑀𝛽𝛼

Figure 15: Influence of the flow number on the inertia due to translational and angular excitation. (I) Inertia due totranslational excitation by the hydraulic forces. (II) Inertia due to angular excitation by the hydraulic forces.(III) Inertia due to translational excitation by the hydraulic torques. (IV) Inertia due to angular excitationby the hydraulic torques.

27


cross-coupled inertia coefficients decrease.


• The direct stiffness of submatrix (I) as wellas the cross-coupled stiffness of submatrix(II) and (IV) increase proportionally ∝ φ1.9.

• The damping coefficients exhibit a lineardependence on the flow number.

• The relevance of the tilt and torque coeffi-cients decreases proportionally ∝ 1/φ1.5.

V. Conclusions

In the presented paper we discuss the dynamicforce and torque characteristic of annular gapswith an axial flow component. First, the ro-tordynamic influence of annular gaps is dis-cussed. So far there is a severe lack of under-standing with regard to the dynamic charac-teristic including hydraulic forces and torquesof the flow inside the annulus. Second, a newcalculation method is presented, using a per-turbed integro-differential approach in combi-nation with power-law ansatz functions anda Hirs’ model to calculate the dynamic forceand torque characteristics. For validation pur-poses, the Clearance-Averaged Pressure Model(CAPM) is compared to existing literature by[37, 45, 36], exhibiting a good agreement withthe results shown therein. Third, an extensiveparameter study is carried out. The results areused to evaluate the relevance of the tilt andtorque coefficients. It is shown that the pre-conceived idea of an overall threshold depend-ing only on the annulus length is insufficient.Rather, the operating conditions of the turbo-machinery has to be taken into account. Theinfluence of the operating conditions is partic-ularly evident when considering the modifiedReynolds number, the flow number and the ra-tio of the excitation amplitudes in translationaland rotational degree of freedom.

Acknowledgements

We gratefully acknowledge the financial sup-port of the Federal Ministry for Economic Af-fairs and Energy (BMWi) due to an enactmentof the German Bundestag under Grant No.03EE5036B and KSB SE & Co. KGaA. In addi-tion, we gratefully acknowledge the financialsupport of the industrial collective researchprogramme (IGF no. 21029 N/1), supported bythe Federal Ministry for Economic Affairs andEnergy (BMWi) through the AiF (German Fed-eration of Industrial Research Associations e.V.)due to an enactment of the German Bundestag.Special gratitude is expressed to the participat-ing companies and their representatives in theaccompanying industrial committee for theiradvisory and technical support.

Declaration of competing interest

The authors declare that they have no knownpersonal relationships or competing financialinterests that could have appeared to influencethe work reported in this paper.

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[29] Yuan, X., Zhang, G., Li, B., and Miao,X., 2006, “Theoretical and experimen-tal results of water-lubricated, high-speed,short-capillary-compensated hybrid jour-nal bearings,” Proceedings of the Interna-tional Joint Tribology Conference, pp. 391–398.

[30] Feng, H., Jiang, S., and Ji, A., 2019, “In-vestigations of the static and dynamiccharacteristics of water-lubricated hydro-dynamic journal bearing considering tur-bulent, thermohydrodynamic and mis-aligned effects,” Tribology International,130, pp. 245–260.

[31] Wang, L., Pei, S., Xiong, X., and Xu,H., 2013, “Investigation of the com-bined influence of turbulence and ther-mal effects on the performance of water-lubricated hybrid bearings with circum-ferential grooves and stepped recesses,”Proceedings of the Institution of MechanicalEngineers, Part J: Journal of Engineering Tri-bology, 228(1), pp. 53–68.

[32] Dousti, S., Allaire, P. E., Dimond, T., andCao, J., 2016, “An extended reynold equa-tion applicable to high reduced reynoldsnumber operation of journal bearings,”Tribology International, 102, pp. 182–197.

[33] Tiwari, R., Manikandan, S., and Dwivedy,S. K., 2005, “A review of the experimentalestimation of the rotor dynamic param-eters of seals,” The Shock and VibrationDigest, 37(4), pp. 261–284.

[34] Tiwari, R., Lees, A. W., and Friswell, M. I.,2004, “Identification of dynamic bearingparameters: A review,” The Shock and Vi-bration Digest, 36(2), pp. 99–124.

[35] Childs, D. W., Nelson, C. C., Noyes, T.,and Dressman, J. B., 1982, “A high-reynolds-number seal test facility: Facil-ity description and preliminary test data,”NASA. Lewis Research Center Rotordyn. In-stability Probl. in High-Performance Turbo-machinery.

[36] Simon, F., and Frêne, J., 1992, “Rotordy-namic coefficients for turbulent annularmisaligned seals,” Rotating machinery - Dy-namics; Proceedings of the 3rd InternationalSymposium on Transport Phenomena and Dy-namics of Rotating Machinery (ISROMAC-3),pp. 207–222.

[37] Nordmann, R., and Dietzen, F.-J., 1988,“Finite difference analysis of rotordynamicseal coefficients for an eccentric shaftposition,” NASA, Lewis Research Center,Rotordynamic Instability Problems in High-Performance Turbomachinery, pp. 269–284.

30


[38] Nelson, C. C., and Nguyen, D. T., 1988,“Analysis of eccentric annular incompress-ible seals: Part 1—a new solution usingfast fourier transforms for determininghydrodynamic force,” Journal of Tribology,110(2), pp. 354–359.

[39] Simon, F., and Frêne, J., 1992, “Analysisfor incompressible flow in annular pres-sure seals,” Journal of Tribology, 114(3),pp. 431–438.

[40] Schäfer, M., 2006, Computational Engineer-ing: Introduction to Numerical Methods, 1 ed.Springer, Berlin and Heidelberg.

[41] Kanemori, Y., and Iwatsubo, T., 1992, “Ex-perimental study of dynamic fluid forcesand moments for a long annular seal,”Journal of Tribology, 114(4), pp. 773–778.

[42] Kanemori, Y., and Iwatsubo, T., 1994, “Ro-tordynamic analysis of submerged motorpumps : Influence of long seal on thestability of fluid machinery,” JSME in-ternational journal. Ser. C, Dynamics, control,robotics, design and manufacturing, 37(1),pp. 193–201.

[43] Kanemori, Y., and Iwatsubo, T., 1994,“Forces and moments due to combinedmotion of conical and cylindrical whirlsfor a long seal,” Journal of Tribology, 116(3),pp. 489–498.

[44] Hirs, G. G., 1973, “A bulk-flow theory forturbulence in lubricant films,” Journal ofLubrication Technology, 95(2), pp. 137–145.

[45] Nelson, C. C., and Nguyen, D. T., 1988,“Analysis of eccentric annular incompress-ible seals: Part 2—effects of eccentricityon rotordynamic coefficients,” Journal ofTribology, 110(2), pp. 361–366.

31


A. Influence of the eccentricity

In the following, the influence of the eccentric-ity on the rotordynamic coefficients is inves-tigated. Figures 16 to 19 give the force andtorque coefficients for translational and angu-lar excitation. The annulus of length L = 1.3is operated at a modified Reynolds numberReϕ = 0.031 and flow number φ = 0.7. The pre-swirl before the annulus is set to Cϕ|z=0 = 0.5and the fulcrum lies in the centre of the gap, i.e.zT = 0.5. In contrast to the skew-symmetricsub-matrices when investigating concentric op-eration conditions, i.e. ε = 0, the skew-symmetry of the sub-matrices vanishes withincreasing eccentricity.

Figure 16 shows the influence of an eccentricoperated shaft on the direct and cross-coupledstiffness due to translational excitation bythe hydraulic forces (I), the direct and cross-coupled stiffness due to angular excitation bythe hydraulic forces (II), the direct and cross-coupled stiffness due to translational excitationby the hydraulic torques (III) and the directand cross-coupled stiffness due to angularexcitation by the hydraulic torques (IV). First,focusing on the stiffness coefficients of the firstsubmatrix (I), it exhibits a skew-symmetricsubmatrix up to an eccentricity ε ≈ 0.4.This is in good agreement with the existingliterature, cf. [3]. Furthermore, the curve ofthe direct and cross-coupled stiffness becomesnon-linear when increasing the eccentricity.Here, the direct stiffness KXX, KYY slightlydecreases with increasing eccentricity, whereasthe cross-coupled stiffness KXY slightlyincreases. In contrast, the cross-coupledstiffness KYX decreases exponentially withincreasing eccentricity. Second, focusing onthe stiffness coefficients of submatrix (II) thedirect and cross-coupled stiffness are almostindependent of the eccentricity. Here, thedirect stiffness KXα slightly decreases, whereasthe direct stiffness KYβ slightly increases withincreasing eccentricity. Furthermore, thecross-coupled stiffness KYα decreases, whereasthe cross-coupled stiffness KXβ increases with

increasing eccentricity. It is noted that thestiffness coefficients of submatrix (II) are inthe same order of magnitude as the ones ofsubmatrix (I). In contrast to the coefficientsof submatrix (II) the stiffness coefficients ofsubmatrix (III) exhibit an eccentricity influence.Here, the direct stiffness KαX, KβY as well asthe cross-coupled stiffness KαY decrease withincreasing eccentricity. However, the cross-coupled stiffness KβX shows an exponentialprogression and a sign change at ε ≈ 0.75when increasing the eccentricity. Finally, focus-ing on the direct and cross-coupled stiffnessof submatrix (IV) the direct stiffness Kαα, Kββ

as well as the cross-coupled stiffness Kβα

decrease with increasing eccentricity, whereasthe cross-coupled stiffness Kαβ increasesexponentially with increasing eccentricity.

Similar to the consideration of the influenceof length on the relevance of the tilt andtorque coefficients, figure 17 shows theinfluence of the eccentricity on the ratio oftilt to translational stiffness coefficients, cf.equation 33. It is shown that, although theadditional coefficients are relevant at theselected operating point for an annulus lengthL = 1.3, the influence of the eccentricity isnegligible.

Figure 16 shows the influence of an ec-centrically operated rotor on the direct andcross-coupled damping due to translationalexcitation by the hydraulic forces (I), thedirect and cross-coupled damping due toangular excitation by the hydraulic forces (II),the direct and cross-coupled damping dueto translational excitation by the hydraulictorques (III) and the direct and cross-coupleddamping due to angular excitation by thehydraulic torques (IV). First, focusing on thedamping coefficients of the first submatrix(I), it exhibits a skew-symetric submatrix upto an eccentricity ε = 0.4. This is in goodagreement with the corresponding stiffnesscoefficients, cf. figure 16. Furthermore, thecurve of the direct and cross-coupled dampingbecomes non-linear when increasing the

32


𝐾𝑋𝑋,𝐾

𝑋𝑌,𝐾

𝑌𝑋,𝐾

𝑌𝑌

𝐾𝑋𝛼,𝐾

𝑋𝛽,𝐾

𝑌𝛼,𝐾

𝑌𝛽

𝐾𝛼𝑋,𝐾

𝛼𝑌,𝐾

𝛽𝑋,𝐾

𝛽𝑌

𝐾𝛼𝛼,𝐾

𝛼𝛽,𝐾

𝛽𝛼,𝐾

𝛽𝛽

(𝐼) (𝐼𝐼)

(𝐼𝐼𝐼) (𝐼𝑉)𝐿 = 1.3; 𝑧𝑇 = 0.5; 𝑅𝑒𝜑

∗ = 0.031

𝜙 = 0.70; 𝐶𝜑 ቚ𝑧=0

= 0.50

RELATIVE ECCENTRICITY 𝜀RELATIVE ECCENTRICITY 𝜀


𝐾𝑌𝑋

𝐾𝛽𝑌

𝐾𝛼𝑌

𝐾𝛼𝛽

𝐾𝛽𝛼

𝐾𝑌𝛼



𝐾𝑋𝑌

𝐾𝑋𝛽

𝐾𝛽X𝐾𝛼X

Figure 16: Influence of eccentricity on the stiffness due to translational and angular excitation. (I) Stiffness due totranslational excitation by the hydraulic forces. (II) Stiffness due to angular excitation by the hydraulicforces. (III) Stiffness due to translational excitation by the hydraulic torques. (IV) Stiffness due to angularexcitation by the hydraulic torques.

33


: 0.03 + 6.55

RELATIVE ECCENTRICITY 𝜀

= 5.5 ∗ 10−4

−(𝐾


𝑌𝑌𝜓𝑌)

= 6.0 ∗ 10−4

𝛼𝑋𝑌

= 1.4 ∗ 10−3

= 1.0 ∗ 10−3

Figure 17: Influence of the eccentricity and the ratio ofangular to translational excitation αX/Y onthe ratio of tilt to translational stiffness coef-ficients.

eccentricity. Here, the direct damping CXXincreases exponentially when increasing theeccentricity, whereas CYY only increases athigh eccentricities. The cross-coupled dampingcoefficients are almost independent of theeccentricity, with CYX slightly decreasing withincreasing eccentricity. Second, focusing onthe damping coefficients of submatrix (II) thedirect and cross-coupled damping coefficientsare almost independent of the eccentricity,being one order of magnitude smaller than thecoefficients of submatrix (I). In contrast to thecoefficients of submatrix (II) the damping coef-ficients of submatrix (III) exhibit an eccentricityinfluence. Here, the direct and cross-coupleddamping coefficients CαX, CβX increase withincreasing eccentricity, whereas the direct andcross-coupled damping coefficients CβY, CαYdecrease. Finally, focusing on the direct andcross-coupled damping of submatrix (IV), thedirect damping Cββ exponentially increaseswith with increasing eccentricity, whereas thedirect and cross-coupled damping coefficientsCαα, Cαβ, Cβα are almost independent of theeccentricity.

Figure 16 shows the influence of an ec-centrically operated rotor on the direct andcross-coupled inertia due to translationalexcitation by the hydraulic forces (I), the

direct and cross-coupled inertia due to angularexcitation by the hydraulic forces (II), the directand cross-coupled inertia due to translationalexcitation by the hydraulic torques (III) and thedirect and cross-coupled inertia due to angularexcitation by the hydraulic torques (IV). First,focusing on the inertia coefficients of the firstsubmatrix (I), it exhibits a skew-symmetricsubmatrix up to an eccentricity ε = 0.4.Furthermore, the direct inertia exponentiallyincreases with increasing eccentricity. Here,the direct inertia MXX increases faster thanthe direct inertia MYY. The cross-coupledinertia coefficients are almost independent ofthe eccentricity, with MXY slightly increasingand MYX slightly decreasing with increasingeccentricity. Second, focusing on the inertiacoefficients of submatrix (II) the direct andcross-coupled inertia coefficients are almost in-dependent of the eccentricity, being two ordersof magnitude smaller than the coefficients ofsubmatrix (I). In contrast to the coefficientsof submatrix (II), the damping coefficients ofsubmatrix (III) exhibit an eccentricity influence.Here, the direct and cross-coupled dampingcoefficients MαX, MαY and MβX increasewith increasing eccentricity, whereas thedirect inertia MβY decrease with increasingeccentricity. Finally, focusing on the directand cross-coupled inertia of submatrix (IV),the direct inertia behaves similar to the directinertia of submatrix (I). Mββ exponentiallyincreases with with increasing eccentricity. Thecross-coupled inertia coefficients Mαβ, Mβα

slightly decrease with increasing eccentricity.

B. Influence of the centre of

rotation

In the following, the influence of the centreof rotation on the rotordynamic coefficients isinvestigated. Figures 20 to 23 give the force andtorque coefficients for translational and angularexcitation. The annulus of length L = 1.3 isoperated at concentric conditions, i.e. ε = 0with a modified Reynolds number Reϕ = 0.031and flow number φ = 0.7. The pre-swirl before

34


𝐶𝑋𝑋,𝐶

𝑋𝑌,𝐶

𝑌𝑋,𝐶

𝑌𝑌

𝐶𝑋𝛼,𝐶

𝑋𝛽,𝐶

𝑌𝛼,𝐶

𝑌𝛽

𝐶𝛼𝑋,𝐶

𝛼𝑌,𝐶

𝛽𝑋,𝐶

𝛽𝑌

𝐶𝛼𝛼,𝐶

𝛼𝛽,𝐶

𝛽𝛼,𝐶

𝛽𝛽

(𝐼) (𝐼𝐼)

(𝐼𝑉)

𝐿 = 1.3; 𝑧𝑇 = 0.5; 𝑅𝑒𝜑∗ = 0.031

𝜙 = 0.70; 𝐶𝜑 ቚ𝑧=0

= 0.50

𝐶𝑌𝑌

𝐶𝑌𝑋

𝐶𝛼𝑌

𝐶𝛽𝑋

𝐶𝛽𝛽

𝐶𝑋𝛼 , 𝐶𝑋𝛽 , 𝐶𝑌𝛼 , 𝐶𝑌𝛽

𝐶𝛽𝑌 𝐶𝛽𝛼


𝐶𝛼𝑋

𝐶𝛼𝛼

𝐶𝛼𝛽

𝐶𝑋𝑌

𝐶𝑋𝑋

(𝐼𝐼𝐼)


Figure 18: Influence of eccentricity on the damping due to translational and angular excitation. (I) Damping due totranslational excitation by the hydraulic forces. (II) Damping due to angular excitation by the hydraulicforces. (III) Damping due to translational excitation by the hydraulic torques. (IV) Damping due to angularexcitation by the hydraulic torques.

35


𝑀𝑋𝑋,𝑀

𝑋𝑌,𝑀

𝑌𝑋,𝑀

𝑌𝑌

𝑀𝑋𝛼,𝑀

𝑋𝛽,𝑀

𝑌𝛼,𝑀

𝑌𝛽

𝑀𝛼𝑋,𝑀

𝛼𝑌,𝑀

𝛽𝑋,𝑀

𝛽𝑌

𝑀𝛼𝛼,𝑀

𝛼𝛽,𝑀

𝛽𝛼,𝑀

𝛽𝛽

(𝐼) (𝐼𝐼)

(𝐼𝑉)

𝐿 = 1.3; 𝑧𝑇 = 0.5; 𝑅𝑒𝜑∗ = 0.031

𝜙 = 0.70; 𝐶𝜑 ቚ𝑧=0

= 0.50

𝑀𝑌𝑌

𝑀𝑋𝑌

𝑀𝛼𝑋

𝑀𝛼𝑌

𝑀𝛽𝛽

𝑀𝑋𝛼 ,𝑀𝑋𝛽 , 𝑀𝑌𝛼 ,𝑀𝑌𝛽

𝑀𝑋𝑌

𝑀𝑋𝑋

𝑀𝛽𝑌𝑀𝛽𝑋

𝑀𝛼𝛼

𝑀𝛼𝛽

𝑀𝛽𝛼


(𝐼𝐼𝐼)


Figure 19: Influence of eccentricity on the inertia due to translational and angular excitation. (I) Inertia due totranslational excitation by the hydraulic forces. (II) Inertia due to angular excitation by the hydraulic forces.(III) Inertia due to translational excitation by the hydraulic torques. (IV) Inertia due to angular excitationby the hydraulic torques.

36


the annulus is set to Cϕ|z=0 = 0.5. Due to theconcentric operation conditions, the matricesare skew-symmetric.

Figure 20 shows the influence of the centreof rotation on the direct and cross-coupledstiffness due to translational excitation bythe hydraulic forces (I), the direct and cross-coupled stiffness due to angular excitationby the hydraulic forces (II), the direct andcross-coupled stiffness due to translationalexcitation by the hydraulic torques (III)and the direct and cross-coupled stiffnessdue to angular excitation by the hydraulictorques (IV). First, focusing on the stiffnesscoefficients of the first submatrix (I), it exhibitscoefficients that are independent of the centreof rotation. This is due to the fact that thecoefficients originate from the translationalexcitation by the hydraulic forces. As willbe shown, all coefficients of submatrix (I),i.e. stiffness KI , damping CI and inertia MI ,are independent for the centre of rotation.Second, focusing on the stiffness coefficientsof submatrix (II) the direct and cross-coupledcoefficients show a linear dependence on thecentre of rotation. Here, the direct stiffnesscoefficients KXα, KYβ increase with the centreof rotation moving form the inlet to the outletof the annulus, whereas the cross-coupledstiffness coefficients KXβ, |KYα| decreasewith moving centre of rotation. It is notedthat the stiffness coefficients of submatrix(II) are of the same order of magnitude asthe ones of submatrix (I). Focusing on thestiffness coefficients of submatrix (III), theyexhibit a linear dependence on the centre ofrotation. Here, the direct stiffness KαX, KβYand the cross-coupled stiffness KβX decreasewith moving centre of rotation, whereas thecross-coupled stiffness KαY increases linearlywith the centre of rotation moving form theinlet to the outlet of the annulus. In contrast tothe linear dependence on the centre of rotationof submatrices (II) and (III), the stiffness ofsubmatrix (IV) show a parabolic behaviour.Here, the direct stiffness Kαα, Kββ stronglydecreases with the centre of rotation moving

form the annulus inlet to the annulus outlet.Furthermore, the cross-coupled stiffness Kαβ,|Kβα| first decreases with moving centre ofrotation, reaching a minimum at zT = 0.54. Bymoving the centre of rotation further towardthe outlet of the annulus, the cross-coupledstiffness increases. The fact that the minimumdoes not coincide with a centre of rotationzT = 0.5 originated from the axial flowcomponent. Due to that, a bias torque isinduced on the rotor, shifting the minimumtowards the outlet of the annulus.

Similar to the consideration of the influenceof length on the relevance of the tilt and torquecoefficients, figure 21 shows the influence ofthe centre of rotation on the ratio of tilt totranslational stiffness coefficients, cf. equation33. It is shown that although the additionalcoefficients are relevant at the selected operat-ing point for an annulus length L = 1.3, theinfluence of centre of rotation on the effectivestiffness is negligible.

Figure 22 shows the influence of the centreof rotation on the direct and cross-coupleddamping due to translational excitation bythe hydraulic forces (I), the direct and cross-coupled damping due to angular excitationby the hydraulic forces (II), the direct andcross-coupled damping due to translationalexcitation by the hydraulic torques (III) and thedirect and cross-coupled damping due to an-gular excitation by the hydraulic torques (IV).First, focusing on the damping coefficients ofthe first submatrix (I), it exhibits damping co-efficients independent of the centre of rotation.Second, focusing on the damping coefficientsof submatrix (II), the direct and cross-coupleddamping show a linear dependence on thecentre of rotation. Here, the direct dampingcoefficients CXα, CYβ increase with the centreof rotation moving from the inlet to the outletof the annulus, whereas the cross-coupleddamping CXβ, |CYα| decreases with movingcentre of rotation. Focusing on the dampingcoefficients of submatrix (III), the coefficientsalso exhibit a linear dependence on the centre

37


𝐾𝑋𝑋,𝐾

𝑋𝑌,𝐾

𝑌𝑋,𝐾

𝑌𝑌

𝐾𝑋𝛼,𝐾

𝑋𝛽,𝐾

𝑌𝛼,𝐾

𝑌𝛽

𝐾𝛼𝑋,𝐾

𝛼𝑌,𝐾

𝛽𝑋,𝐾

𝛽𝑌

𝐾𝛼𝛼,𝐾

𝛼𝛽,𝐾

𝛽𝛼,𝐾

𝛽𝛽

(𝐼) (𝐼𝐼)

(𝐼𝑉)𝐿 = 1.3; 𝜀 = 0; 𝑅𝑒𝜑

∗ = 0.031

𝜙 = 0.70; 𝐶𝜑 ቚ𝑧=0

= 0.50

CENTRE OF ROTATION 𝑧𝑇CENTRE OF ROTATION 𝑧𝑇


𝐾𝑌𝑋

𝐾𝑌𝛼


𝐾𝛼𝑌

𝐾𝛼𝛽

𝐾𝛽𝛼

𝐾𝑋𝛽



𝐾𝑋𝑌

𝐾𝛽𝑋

(𝐼𝐼𝐼)

Figure 20: Influence of the centre of rotation on the stiffness due to translational and angular excitation. (I) Stiffnessdue to translational excitation by the hydraulic forces. (II) Stiffness due to angular excitation by thehydraulic forces. (III) Stiffness due to translational excitation by the hydraulic torques. (IV) Stiffness due toangular excitation by the hydraulic torques.

38


= 5.5 ∗ 10−4

: 0.03 + 6.55

CENTRE OF ROTATION 𝑧𝑇

−(𝐾


𝑌𝑌𝜓𝑌)

= 6.0 ∗ 10−4

𝛼𝑋𝑌

= 1.4 ∗ 10−3

= 1.0 ∗ 10−3

Figure 21: Influence of the centre of rotation and theratio of angular to translational excitationαX/Y on the ratio of tilt to translational stiff-ness coefficients.

of rotation. Here, the direct damping CαX , CβYand the cross-coupled damping CβX decreasewith moving centre of rotation, whereas thecross-coupled damping CαY increases linearlywith the centre of rotation moving from theinlet to the outlet of the annulus. Similar tothe stiffness coefficients of submatrix (IV) thedamping coefficients also exhibit a parabolicdependence on the centre of rotation. Here, thedirect damping coefficients Cαα, Cββ as well asthe cross-coupled damping Cαβ, |Cβα| decreaseat first, reaching a minimum at zT = 0.54for the direct damping and zT = 0.47 for thecross-coupled damping. By further movingthe the centre of rotation towards the outletof the annulus, the corresponding coefficientsincrease.

Figure 23 shows the influence of the centre ofrotation on the direct and cross-coupled inertiadue to translational excitation by the hydraulicforces (I), the direct and cross-coupled inertiadue to angular excitation by the hydraulicforces (II), the direct and cross-coupled inertiadue to translational excitation by the hydraulictorques (III) and the direct and cross-coupledinertia due to angular excitation by thehydraulic torques (IV). First, focusing on theinertia coefficients of the first submatrix (I), itexhibits inertia coefficients independent of the

centre of rotation, similar to the stiffness anddamping coefficients. Second, focusing on theinertia coefficients of submatrix (II), the directinertia coefficients MXα, MYβ are independentof the centre of rotation, whereas the cross-coupled coefficients MXβ, MYα show a lineardependency. Here, the cross-coupled inertiaMXβ decreases with the centre of rotationwhile the cross-coupled inertia MYα increases.Focusing on the inertia coefficients of subma-trix (III), the cross-coupled coefficients exhibita linear dependence on the centre of rotation.Similar to the direct inertia coefficients ofsubmatrix (II) the ones of submatrix (III) arealso independent of the centre of rotation.Here, the cross-coupled inertia MβX decreaseswith moving centre of rotation, whereas thecross-coupled inertia MαY increases linearlywith the centre of rotation moving from theinlet to the outlet of the annulus. In accordancewith the stiffness and damping coefficients ofsubmatrix (IV), the inertia coefficients show aparabolic behaviour. Here, the cross-coupledinertia |Mαβ|, Mβα slightly increases with thecentre of rotation moving from the inlet tothe outlet of the annulus, whereas the directinertia Mαα, Mββ decreases at first, reaching aminimum at zT = 0.49. By moving the centreof rotation further towards the outlet of theannulus, the direct inertia increases again.

C. Influence of the pre-swirl

Figures 24 to 27 give the influence of pre-swirlon the rotordynamic force and torque coeffi-cients for translational and angular excitation.The annulus of length L = 1.3 is operated atconcentric conditions, i.e. ε = 0 with a mod-ified Reynolds number Reϕ = 0.031 and flownumber φ = 0.7. The pre-swirl before the annu-lus is set to Cϕ|z=0 = 0.5 and the fulcrum liesin the centre of the annular gap, i.e. zT = 0.5.

Figure 24 shows the influence of the pre-swirl on the direct and cross-coupled stiffnessdue to translational excitation by the hydraulicforces (I), the direct and cross-coupled stiffness

39


𝐶𝑋𝑋,𝐶

𝑋𝑌,𝐶

𝑌𝑋,𝐶

𝑌𝑌

𝐶𝑋𝛼,𝐶

𝑋𝛽,𝐶

𝑌𝛼,𝐶

𝑌𝛽

𝐶𝛼𝑋,𝐶

𝛼𝑌,𝐶

𝛽𝑋,𝐶

𝛽𝑌

𝐶𝛼𝛼,𝐶

𝛼𝛽,𝐶

𝛽𝛼,𝐶

𝛽𝛽

(𝐼) (𝐼𝐼)

(𝐼𝑉)

𝐿 = 1.3; 𝜀 = 0; 𝑅𝑒𝜑∗ = 0.031

𝜙 = 0.70; 𝐶𝜑 ቚ𝑧=0

= 0.50

𝐶𝑋𝑌


𝐶𝑌𝑋

𝐶𝑋𝛽

𝐶𝛼𝑋, 𝐶𝛽𝑌

𝐶𝛼𝑌


𝐶𝛼𝛽

𝐶𝑌𝛼

𝐶𝑋𝛼 , 𝐶𝑌𝛽

𝐶𝛽𝑋

𝐶𝛽𝛼


(𝐼𝐼𝐼)


Figure 22: Influence of the centre of rotation on the damping due to translational and angular excitation. (I) Dampingdue to translational excitation by the hydraulic forces. (II) Damping due to angular excitation by thehydraulic forces. (III) Damping due to translational excitation by the hydraulic torques. (IV) Damping dueto angular excitation by the hydraulic torques.

40


𝑀𝑋𝑋,𝑀

𝑋𝑌,𝑀

𝑌𝑋,𝑀

𝑌𝑌

𝑀𝑋𝛼,𝑀

𝑋𝛽,𝑀

𝑌𝛼,𝑀

𝑌𝛽

𝑀𝛼𝑋,𝑀

𝛼𝑌,𝑀

𝛽𝑋,𝑀

𝛽𝑌

𝑀𝛼𝛼,𝑀

𝛼𝛽,𝑀

𝛽𝛼,𝑀

𝛽𝛽

(𝐼) (𝐼𝐼)


𝐿 = 1.3; 𝜀 = 0; 𝑅𝑒𝜑∗ = 0.031

𝜙 = 0.70; 𝐶𝜑 ቚ𝑧=0

= 0.50


𝑀𝑋𝑌

𝑀𝛼𝑋, 𝑀𝛽𝑌𝑀𝛽𝑋

𝑀𝛼𝑌


𝑀𝑋𝛽

𝑀𝛼𝛽 ,𝑀𝛽𝛼


𝑀𝑋𝑌

𝑀𝑌𝛼

𝑀𝑋𝛼,𝑀𝑌𝛽


Figure 23: Influence of the centre of rotation on the inertia due to translational and angular excitation. (I) Inertia dueto translational excitation by the hydraulic forces. (II) Inertia due to angular excitation by the hydraulicforces. (III) Inertia due to translational excitation by the hydraulic torques. (IV) Inertia due to angularexcitation by the hydraulic torques.

41


𝐾𝑋𝑋,𝐾

𝑋𝑌,𝐾

𝑌𝑋,𝐾

𝑌𝑌

𝐾𝑋𝛼,𝐾

𝑋𝛽,𝐾

𝑌𝛼,𝐾

𝑌𝛽

𝐾𝛼𝑋,𝐾

𝛼𝑌,𝐾

𝛽𝑋,𝐾

𝛽𝑌

𝐾𝛼𝛼,𝐾

𝛼𝛽,𝐾

𝛽𝛼,𝐾

𝛽𝛽

(𝐼) (𝐼𝐼)


𝐿 = 1.3; 𝜀 = 0; 𝑧𝑇 = 0.5;𝑅𝑒𝜑

∗ = 0.031;𝜙 = 0.70

PRE-SWIRL 𝐶𝜑ȁ𝑧=0PRE-SWIRL 𝐶𝜑ȁ𝑧=0

𝐾𝑌𝑋


𝐾𝑋𝛽𝐾𝑋𝛼 , 𝐾𝑌𝛽

𝐾𝑌𝛼

𝐾𝛽𝛼

𝐾𝛼𝛼 , 𝐾𝛽𝛽𝐾𝛽𝑋

𝐾𝛼𝑌


𝐾𝑋𝑌

𝐾𝛼𝛽

Figure 24: Influence of the pre-swirl on the stiffness due to translational and angular excitation. (I) Stiffness due totranslational excitation by the hydraulic forces. (II) Stiffness due to angular excitation by the hydraulicforces. (III) Stiffness due to translational excitation by the hydraulic torques. (IV) Stiffness due to angularexcitation by the hydraulic torques.

42


: 0.03 + 6.55

PRE-SWIRL 𝐶𝜑ȁ𝑍=0

= 5.5 ∗ 10−4= 6.0 ∗ 10−4

𝛼𝑋𝑌

= 1.4 ∗ 10−3

= 1.0 ∗ 10−3

−(𝐾


𝑌𝑌𝜓𝑌)

Figure 25: Influence of the pre-swirl and the ratio of an-gular to translational excitation αX/Y onthe ratio of tilt to translational stiffness coef-ficients.

due to angular excitation by the hydraulicforces (II), the direct and cross-coupledstiffness due to translational excitation by thehydraulic torques (III) and the direct and cross-coupled stiffness due to angular excitation bythe hydraulic torques (IV). First, focusing onthe stiffness coefficients of the first submatrix(I), it exhibits direct stiffness coefficients KXX,KYY independent of the pre-swirl, whereasthe cross-coupled stiffness coefficients KXY,|KYX | increase linearly with an increasingpre-swirl. This is due to the fact that anincreased pre-swirl mainly alters the tangentialforce component, i.e. FY, on the rotor. It isnoted that the cross-coupled stiffness is ofparticular interest when analysing the stabilityof the system. Here, high pre-swirl rations actin a destabilising manner. Second, focusing onthe stiffness coefficients of submatrix (II), thedirect and cross-coupled stiffness are almostindependent of the pre-swirl. Only the directstiffness coefficients slightly increase with anincreasing pre-swirl. Focusing on the stiffnessof submatrix (III), the cross-coupled stiffnesscoefficients KαY, |KβX | sightly decrease withan increasing pre-swirl, whereas the directstiffness coefficients KαX , KβY decrease linearlywith the pre-swirl ratio. Finally focusing onthe stiffness coefficients of submatrix (IV) thedirect stiffness Kαα, Kββ is almost independent

of the pre-swirl. In contrast, the cross-coupledstiffness shows a linear dependence on thepre-swirl.

Similar to the previous consideration onthe relevance of the 48 coefficients, figure 25shows the influence of the pre-swirl on theratio of tilt to translational stiffness coefficients,cf. equation 33. Similar to the influence ofthe eccentricity and the centre of rotation, thepre-swirl only slightly affects the effectivestiffness. Here, the influence only becomesapparent with large pre-swirl ratios, i.e Cϕ|z=0.

Figure 26 shows the influence of the pre-swirl on the direct and cross-coupled dampingdue to translational excitation by the hydraulicforces (I), the direct and cross-coupled damp-ing due to angular excitation by the hydraulicforces (II), the direct and cross-coupleddamping due to translational excitation bythe hydraulic torques (III) and the directand cross-coupled damping due to angularexcitation by the hydraulic torques (IV). First,focusing on the damping coefficients of thefirst submatrix (I), it exhibits direct dampingcoefficients CXX, CYY almost independent ofthe pre-swirl. However, the cross-coupleddamping coefficients CXY, |CYX | increaselinearly with the pre-swirl. Second, focusingon the damping coefficients of submatrix (II),the direct and cross-coupled damping areindependent of the pre-swirl. This is not onlytrue for the damping coefficients of submatrix(II) but also holds for the damping coefficientsof submatrix (III) and submatrix (IV). Solelythe direct damping coefficients of submatrix(III) slightly decrease with increasing pre-swirl.

Figure 27 shows the influence of the pre-swirl on the direct and cross-coupled inertiadue to translational excitation by the hydraulicforces (I), the direct and cross-coupled inertiadue to angular excitation by the hydraulicforces (II), the direct and cross-coupled inertiadue to translational excitation by the hydraulictorques (III) and the direct and cross-coupledinertia due to angular excitation by the

43


𝐶𝑋𝑋,𝐶

𝑋𝑌,𝐶

𝑌𝑋,𝐶

𝑌𝑌

𝐶𝑋𝛼,𝐶

𝑋𝛽,𝐶

𝑌𝛼,𝐶

𝑌𝛽

𝐶𝛼𝑋,𝐶

𝛼𝑌,𝐶

𝛽𝑋,𝐶

𝛽𝑌

𝐶𝛼𝛼,𝐶

𝛼𝛽,𝐶

𝛽𝛼,𝐶

𝛽𝛽

(𝐼) (𝐼𝐼)𝐿 = 1.3; 𝜀 = 0; 𝑧𝑇 = 0.5;𝑅𝑒𝜑

∗ = 0.031;𝜙 = 0.70

𝐶𝛽𝑋

𝐶𝛼𝑌

𝐶𝛽𝛼𝐶𝛼𝑋, 𝐶𝛽𝑌


𝐶𝑋𝑌


𝐶𝛼𝛼 ,𝐶𝛽𝛽


𝐶𝑌𝛼

𝐶𝛽𝛼


𝐶𝑋𝛽

𝐶𝑌𝑋

Figure 26: Influence of the pre-swirl on the damping due to translational and angular excitation. (I) Damping due totranslational excitation by the hydraulic forces. (II) Damping due to angular excitation by the hydraulicforces. (III) Damping due to translational excitation by the hydraulic torques. (IV) Damping due to angularexcitation by the hydraulic torques.

44


𝑀𝑋𝑋,𝑀

𝑋𝑌,𝑀

𝑌𝑋,𝑀

𝑌𝑌

𝑀𝑋𝛼,𝑀

𝑋𝛽,𝑀

𝑌𝛼,𝑀

𝑌𝛽

𝑀𝛼𝑋,𝑀

𝛼𝑌,𝑀

𝛽𝑋,𝑀

𝛽𝑌

𝑀𝛼𝛼,𝑀

𝛼𝛽,𝑀

𝛽𝛼,𝑀

𝛽𝛽

(𝐼) (𝐼𝐼)

𝐿 = 1.3; 𝜀 = 0; 𝑧𝑇 = 0.5𝑅𝑒𝜑

∗ = 0.031;𝜙 = 0.70𝑀𝑋𝑋,𝑀𝑌𝑌

𝑀𝑋𝑌


𝑀𝛽𝑋

𝑀𝑋𝛼, 𝑀𝑋𝛽 ,𝑀𝑌𝛼 ,𝑀𝑌𝛽

𝑀𝑌𝑋


𝑀𝛼𝛽

𝑀𝛼𝑌

𝑀𝛽𝛼



Figure 27: Influence of the pre-swirl on the inertia due to translational and angular excitation. (I) Inertia due totranslational excitation by the hydraulic forces. (II) Inertia due to angular excitation by the hydraulic forces.(III) Inertia due to translational excitation by the hydraulic torques. (IV) Inertia due to angular excitationby the hydraulic torques.

45


hydraulic torques (IV). First, focusing on theinertia coefficients of the first submatrix (I),it exhibits direct inertia coefficients MXX,MYY independent of the pre-swirl. However,the cross-coupled inertia linearly dependson the pre-swirl. Here the cross-coupledinertia coefficients experience a sign changeat negative pre-swirl ratios Cϕ|z=0 < −0.17.Second, focusing on the inertia coefficients ofsubmatrix (II), the direct and cross-coupledinertia are independent of the pre-swirl. Focus-ing on the inertia coefficients of submatrix (III),the direct inertia coefficients MαX , MβY as wellas the cross-coupled coefficient |MβX |, MαYdecrease with increasing pre-swirl. Finally,focusing on the inertia coefficients of submatrix(IV), the direct coefficients Mαα, Mββ linearlyincrease with increasing pre-swirl, whereas thecross-coupled inertia coefficients |Mαβ|, Mβα

slightly decrease with increasing pre-swirlratio.

46

Documents

arXiv:2111.06646v1 [physics.flu-dyn] 12 Nov 2021