Kai Becker
email: kai.becker@kit.edu

Wolfgang Seemann

S Y Mpo

O Tatin

Stability investigations of an elastic rotor supported by actively deformed journal bearings considering the associated spectral system

Keywords:

Due to the non-linear fluid-solid interaction within journal bearings instability phenomena (o en referred to as 'oil-whirl' and/or 'oil-whip') can be observed at higher revolution speeds, which can lead to unwanted oscillations of the corresponding rotor dynamic system. To improve this behaviour, various successful methods, which are based on the idea of non-cylindrical bearing geometries, have been proposed in literature. ese investigations focus on static geometries as well as on time-varying geometries, revealing a high potential to suppress the above mentioned instability phenomena. In the present work a more elaborated model of a journal bearing with time-varying geometry is developed. e varying geometry is thereby realized by an appropriate time-dependent elastic deformation. Using the suggested bearing model as support of an elastic Jeffco rotor leads to interesting effects compared to a system with static bearing geometry. In order to ensure an efficient, comprehensive parameter study an approximation by means of an associated spectral system is derived and analysed.

INTRODUCTION

Destabilizing effects being caused by the non-linear fluidsolid interaction within journal bearings can lead to tremendously increasing oscillation amplitudes of the rotor-dynamic system and therefore represent a major disadvantage. Such phenomena are o en referred to as 'oil-whirl' and/or 'oilwhip' which are well explained in literature (cf. [ ]). In order to avoid these unwanted oscillations, various countermeasures have been developed, some of which are shortly listed below. It is thereby focused on concepts which are based on the general idea of adjustable bearing geometries, as these have shown a significant influence on the performance characteristics of the corresponding rotor system. Typical approaches from former times are non-cylindrical bearings, like multi-lobe [ , , ] or tilting pad bearings [ ], which proved themselves to have a stabilizing effect for certain parameter configurations. Nevertheless, these concepts are rather 'passive' as the current geometry is either fixed (multi-lobe) or depends on the pressure state within the bearing (tilting pad). In both cases such an 'active' intervention from outside is not possible. More recent works deal with the idea of actively adjustable journal bearings. An overview of the current state of the art can thereby be found in [ ]. Some of these concepts focus on the introduction of additional elastic or flexible elements, which lead to a change in geometry depending on the internal bearing pressure. By modifying the properties of these elements the system can successfully be influenced. E.g. C

.

[ , ] suggest a displaceable lobe which is connected to a viscoelastic element. K

. [ ] present a modification via an elastic element, whose displacement can be partly controlled by means of a hydraulic pressure unit. Other concepts with a more actively controlled change in geometry can be found as well.

P

[ ] or T ˘ . [ ] introduce the idea of a rigid cylindrical bearing which can actively be displaced by a corresponding actuation mechanism. Another approach is given by so-called active tilting pad bearings which can be found e.g. in the works of S [ ], D

. [ ] or W . [ ]. eir concept is based on the radial displacement of the pivot points of the single tilting pads in order to influence the pressure distribution. Among different other aspects the design of an appropriate controller represents the main objective in these works.

In all of the previously mentioned cases it has been shown (experimentally and/or numerically) that the stability properties of a rotor system can be significantly improved (compared to the cylindrical bearing) by using these different approaches for the journal bearings. e present work rather considers the principle effects within the bearing, without covering the question of an appropriate controller. Like captured by P

. [ ], who investigate a two-lobe bearing with harmonically changing geometry, it is thereby focused on the theoretical aspects from the mathematical/mechanical point of view. e whole rotor system should therefore be interpreted as some kind of coupled non-linear oscillators. As such systems show interesting behaviour for certain parameter configurations (cf. [ ]), this approach towards the given problem seems promising. In order to keep the idea of coupled oscillators as simple as possible in first place, a pure harmonic adjustment (similar to [ ]) of the corresponding bearing geometry is chosen which already requires an appropriate approximation strategy in order to assess a potential stabilization in a principle manner.

In the following a more elaborated model of an actively deformable journal bearing. As opposed to most of the concepts being mentioned above, the geometry adjustment is realised by an elastic deformation of an initially cylindrical bearing. As described, a harmonic variation of the bearing's geometry is implemented, aiming for an improvement of the rotor's dynamic behaviour during operation at different rotational speeds. An analytical approximation of the geometry for different degrees of deformation is derived in section . and compared to FEM results. e resulting deformation field is used to compute the pressure distribution and the corresponding bearing forces, which are relevant for the set-up of the equations of motion of a Jeffco rotor in section . . Due to the complexity of the dynamic system an associated spectral system (cf. [ ]) is derived in section . . Subsequently, a numerical analysis of the original equations of motion and the derived spectral system is carried out in section . It is thereby focused on static bearing profiles as well as on time-varying profiles. A summary and outlook for possible further investigations can be found in section .

. MODELLING e description of the total rotor system comprises the modelling of the change in the bearing's geometry due to the elastic deformation, the description of the fluid pressure distribution and the corresponding set-up of the resulting equations of motion. Focusing on an efficient solution continuation, the derivation of an associated spectral system is presented as well in the following section.

. Bearing Model

As depicted in fig. the initially circular, closed bearing of inner radius R 0 is elastically deformed by two vertically acting forces F(τ), which correspond to an appropriate actuation mechanism. e non-dimensional time τ = ωt is constructed by means of the angular frequency ω of the rotating journal. e bearing is modelled as thin, circular beam with middle radius R and rectangular cross-section with width B ≪ R and height A ≪ R, which corresponds to a rather short bearing. Further information can be obtained from fig. . e bearing is modelled as linear elastic material with Young's modulus E and it is assumed that the deformation is not influenced by the internal fluid pressure at all and that inertia terms can be neglected, which seems reasonable as a first approach. With the previously mentioned geometric properties the classical bending theory for curved beams (cf. [ ]) can be applied in order to determine the radial displacement w(ϕ, τ) of the inner bearing surface, which depends only on τ and the circumferential coordinate ϕ. As described by P

. [ ] the displacement can be determined by solving

∂ 5 w ∂ϕ 5 + 2 ∂ 3 w ∂ϕ 3 + ∂w ∂ϕ = 0 ( ) separately on the two domains D 1 ϕ = {ϕ ∈ [-π /2, π /2)} and D 2 ϕ = {ϕ ∈ [ π /2, 3π /
2)}. e following boundary conditions have to be met on D 1 ϕ (and similarly on D 2 ϕ ):

w 1 (ϕ) = w 1 (-ϕ) , ∂w 1 ∂ϕ ϕ= π 2 = 0 , π 2 ∫ 0 w 1 dϕ = 0 , E I R 3 0 ∂ 3 w 1 ∂ϕ 3 + ∂w 1 ∂ϕ = F(τ) 2 .
( )

Solving the ordinary differential equation (ODE) ( ) on D 1 ϕ leads to

w 1 = c 1 + (c 2 + c 3 ϕ) sin ϕ + (c 4 + c 5 ϕ) cos ϕ, ( )
with integration constants c k (k = 1, .. , 5). A er imposing the boundary conditions from ( ) a final solution is given by:

w 1 = F(τ)R 3 4E I 4 π -ϕ sin ϕ -cos ϕ . ( )
By following the same principle for the second domain D 2 ϕ the deformation over the whole circumference can be stated:

w(ϕ, τ) = F(τ)R 3 4E I 4 π -ϕ sin ϕ -cos ϕ ϕ ∈ D 1 ϕ 4 π -(π -ϕ) sin ϕ + cos ϕ ϕ ∈ D 2 ϕ ( ) With w * (τ) = w ϕ = ± π 2 , τ = (8 -π 2 )F(τ)R 3 8πE I . ( )
the radial displacement ( ) can be rewri en as follows:

w = w * 2π 8 -π 2 4 π -ϕ sin ϕ -cos ϕ ϕ ∈ D 1 ϕ 4 π -(π -ϕ) sin ϕ + cos ϕ ϕ ∈ D 2 ϕ . ( )
e approximated displacement field is validated by a finiteelement (FE) calculation with C [ ] of just the upper bearing half (ϕ ∈ [0, π]). e FE-model is set up under the assumption of a plain stress field and an isotropic material with E = 210 GPa and ν = 0.3. e geometry parameters are exemplary chosen as R = 500 mm, B /R = 0.4 and different ratios for A /R. Similar boundary conditions like in ( ) are chosen and a static, radial point displacement w( π /2) /R = 1e-4 is enforced at the outer bearing radius.

0 1 2 π π -1 -0.5 0 0.5 1 •10 -4 ϕ w/R analytic A /R = 0.05 (FE) A /R = 0.1 (FE) A /R = 0.15 (FE)
Figure .

Scaled radial displacement w /R of inner bearing surface (analytic and FE) for different ratios A /R e relative radial displacement (scaled by 1 /R) of the inner bearing surface, for both the analytic approximation and the FE solution, is depicted in fig. . As the assumption of a thin, circular beam is more and more violated by increasing values of A /R the results differ obviously. In the following, only cases with sufficiently small ratios A /R are considered, for which the models show a sufficient congruency.

Motivated by the later pressure modelling the initial bearing clearance C = R 0 -R J is used to define the nondimensional displacement as W = w /C. For consistency with the thin-film approach (cf. [ ]) the elastic deformation is assumed to be of the same magnitude as the clearance C, i.e. W = O(1). As the influence of a time-varying bearing geometry should be investigated, the following harmonic approach is chosen

W * = w * C = W [1 + δ W cos(Ω W τ)] ( )
with mean value W, relative amplitude δ W < 1 and angular frequency Ω W . e special case of W = 0 corresponds to the classical circular journal bearing.

. . Pressure Distribution As the fluid-film thickness and radial displacement are considered to be of the same magnitude, the non-dimensional pressure distribution Π(ϕ, z) in the bearing can be modelled by the non-dimensional Reynolds equation

(cf. [ ]) ∂ ∂ϕ ∂Π ∂ϕ H 3 + γ 2 ∂ ∂z ∂Π ∂z H 3 = 6 ∂H ∂ϕ + 12 ∂H ∂τ ( )
with the commonly used substitutions (cf. [ ])

z = 2z B , Π = C 2 R 2 0 p µω , γ = 2R 0 B . ( )
and the non-dimensional fluid-film-thickness

H(X J , Y J , ϕ, τ) = h C = 1+W(ϕ, τ)-X J cos ϕ-Y J sin ϕ . ( )
e axial coordinate is denoted by z, the fluid pressure by p and the dynamic viscosity by µ. e coordinates X J = x J/C and Y J = y J/C correspond to the non-dimensional centre coordinates of the journal, i.e. r J = x J e x + y J e y .

e Reynolds equation ( ) has to be solved for Π, while taking the axial boundary conditions Π(z = ±1) = 0 into account. A Galerkin approach with an appropriate axial shape function of the form

Π(ϕ, z) = (1 -z 2 )g(ϕ) ( )
reduces eq. ( ) to a pure circumferential problem that is solved by a finite difference scheme on D = D 1 ϕ ∪ D 2 ϕ for periodic solutions g(ϕ) ∈ C 2 , which is an agreement with other pressure modelling approaches, e.g.

V [ ]. In total N = 121 sampling points are equally distributed over the domain D at the different points ϕ i . Applying the finite difference scheme leads to the linear system

P g = b ( )
with the discrete pressure mean values g = [g(ϕ 1 ), g(ϕ 2 ), ..] T . e linear system ( ) represents a periodic, scalar, tridiagonal system and can therefore be solved efficiently by means of an appropriate transformation. Fristly, the Sherman-Morrison-Woodbury formula is used to transform the system matrix P to a pure tridiagonal matrix. A erwards, the transformed system is solved by means of the omas algorithm. Detailed information can be found in [ ].

is procedure allows a much faster solution of these kinds of linear systems than other conventional algorithms. As a result, the semi-discrete pressure distribution is obtained:

Π(ϕ i , z) = g(ϕ i )(1 -z 2 ). ( )
. . Bearing Forces Having determined the pressure distribution, the corresponding forces which act on the journal are determined. With the given non-dimensional parameters ( ) and the approach ( ) the bearing forces are expressed by the following integrals which only take into account positive pressure values by integrating over Ω ϕ = {ϕ ∈ [π /2, 3π /2] : g(ϕ) > 0}:

F x = µBR 2 0 R J 2C 2 1 ∫ z=-1 ∫ Ω ϕ -g(ϕ)(1 -z 2 ) cos ϕ dϕdz = -F 0 S m ∫ Ω ϕ γ 2 3 g(ϕ) cos ϕ dϕ F y = µBR 2 0 R J 2C 2 1 ∫ z=-1 ∫ Ω ϕ -g(ϕ)(1 -z 2 ) sin ϕ dϕdz = -F 0 S m ∫ Ω ϕ γ 2 3 g(ϕ) sin ϕ dϕ ( )
With the discrete values g(ϕ i ) from eq. ( ) the previous integrals are approximated by means of a trapezoidal integration:

F x = -F 0 S m γ 2 3 2π N -1 N -1 i=1 max(g(ϕ i ), 0) cos ϕ i F y = -F 0 S m γ 2 3 2π N -1 N -1 i=1 max(g(ϕ i ), 0) sin ϕ i ( )
e used coefficient S m corresponds to the modified Sommerfeldnumber S m acc. to ( ) (cf. [ ]).

. Rotor Model

e classical Jeffco rotor model [ ] is used like depicted in fig. . e rotor of mass M is mounted on an elastic sha with stiffness c s and is exposed to a vertically acting external force F R as well as to an external viscous damping d a . e rotor is considered to be perfectly balanced and tilting is not permi ed, such that the rotor's centre of mass coincides with the centre of the sha at r R = x R e x + y R e y + r Rz e z .

e sha is supported by two journal bearings of the above described type, wherby additional masses m ≪ M are allocated to each journal at the two bearing positions.

ω 2 X ′′ R + d a ωX ′ R + X R -X J Γ = 0 , ω 2 Y ′′ R + d a ωY ′ R + Y R -Y J Γ + f R = 0 , ηω 2 X ′′ J + X J -X R Γ -2σω f x (X J , Y J , X ′ J , Y ′ J ,τ) = 0 , ηω 2 Y ′′ J + Y J -Y R Γ -2σω f y (X J , Y J , X ′ J , Y ′ J ,τ) = 0, ( )
with the following substitutions:

X R/J = x R/J C , Y R/J = y R/J C , ω 2 = MC F 0 ω 2 , S m = B 3 µR J ω 2C 2 F 0 ,σ = S m ω ,η = 2m M , Γ = F 0 c s C , d a = C MF 0 d a , f R = F R F 0 .
( ) e parameter Γ represents the sha compliance, d a characterizes the viscous damping and η describes the mass ratio of the rotor and the allocated masses. e modified Sommerfeldnumber S m is used for the definition of the bearing load parameter σ, which is independent of the rotational speed. e external vertical load F R is represented by f R , while F 0 describes a characteristic force, used for the non-dimensional representation. e rotational speed-or rpm-parameter ω is considered to be essential for the following parameter studies and stability investigations.

e support reactions from the fluid-solid interaction within the bearings is represented by f x and f y respectively and can be obtained from ( ) via:

f x = F x S m F 0 , f y = F y S m F 0 . ( )
e system ( ) is alternatively represented by the following first-order system with state space vector z :

d dτ z = f z , τ , f : R 8 × R → R 8 with z = X J , X ′ J , X R , X ′ R , Y J , Y ′ J , Y R , Y ′ R T . ( )

. Spectral System

As the time-varying bearing deformation ( ) enters the equations of motion in ( ) as a parameter, the system is exposed to both, parametric and self-excitation, which can lead to quasi-periodic behaviour (cf. [ ]). erefore, an associated spectral system is derived according to the suggested method of S

. [ ] in order to easily describe and analyse quasi-periodic system trajectories.

e derivation of the spectral system is shortly explained while detailed information can be found in [ ] and [ ].

. . Invariance Equation

It is assumed that a torus-manifold with two base frequencies Ω W and ω 2 exists, on which the stationary system trajectories are located. It is searched for an appropriate torus function u (θ 1 , θ 2 ) with torus coordinates θ 1 and θ 2 , such that a stationary solution z * of system ( ) can be expressed by

z * = u (Ω W t, ω 2 t) .
( ) e frequencies Ω W and ω 2 thereby refer to the torus coordinates via θ ′ 1 = Ω W and θ ′ 2 = ω 2 . Under these assumptions an invariance equation for the torus function can be set up

Ω W ∂u ∂θ 1 + ω 2 ∂u ∂θ 2 = g u (θ 1 , θ 2 ), θ 1 Ω W , ( )
whereby the corresponding periodicity conditions u (θ 1 , θ 2 ) = u (θ 1 + 2π, θ 2 ) = u (θ 1 , θ 2 + 2π) have to be met. Each solution to this partial differential equation (PDE) can correspond to a quasi-periodic solution of the original system ( ).

. . Approximation via Galerkin Approach As the base frequency Ω W of the parametric excitation is known, the PDE ( ) can be approximately solved by means of a Galerkin approach with the Ansatz-function

u ⋆ = U (θ 2 ) [1, sin θ 1 , cos θ 1 ] T with U ∈ R 8×3 ( )
and the unknown coefficients U (θ 2 ). A er defining an appropriate scalar product

x, y = 1 2π 2π ∫ 0

x(θ 1 )y(θ 1 )dθ 1 ( ) the residual of PDE ( ) can be projected onto the different base functions [ψ 1 , ψ 2 , ψ 3 ] = [1, sin θ 1 , cos θ 1 ]:

Ω W ∂u ⋆ ∂θ 1 + ω 2 ∂u ⋆ ∂θ 2 -f u ⋆ , θ 1 Ω W , ψ k = 0 for k = 1, 2, 3. ( )
is results in a system of 8 × 3 = 24 ordinary differential equations for the unknown coefficients U (θ 2 ) :

ω 2 dU dθ 2 = G U (θ 2 ) ( )
With the transformation θ 2 = ω 2 θ and Z θ = U ω 2 θ the dependence on the still unknown frequency ω 2 can be eliminated:

dZ d θ = G Z ( θ) ( ) According to M .
[ ] an equilibrium solution of this spectral system can correspond to a periodic solution of the original system with period 2π /Ω W , whereas a periodic solution (with any period T θ ) of the spectral system can correspond to a quasi-periodic solution of the original one. In accordance with the previous assumptions the base frequencies of this solution are given by Ω W and ω 2 = 2π /T θ .

. RESULTS AND DISCUSSION

e two derived systems (original ( ) and spectral ( )) are investigated for their dynamic behaviour in dependence of the rpm-parameter ω by means of a bifurcation analysis via MATCONT [ ]. It is searched for configurations which may represent a system state without any negative influence of the above mentioned instability phenomena. Although the spectral system is suitable for the investigation of quasi-periodic system trajectories, it is only focused on the continuation of its stable/unstable equilibrium solutions, as they can reveal a potential stabilization caused by the introduced geometry variation. A continuation of periodic solutions of the spectral system will be part of future works.

Besides the continuation parameter ω, a standard set of parameters for the considered rotor system is taken as follows:

σ = 1, γ = 3, η = 0.001, Γ = 0.01, d a = 1, f R = 1.
( ) e investigations are divided into two parts. Firstly, an analysis of a static non-circular geometry (δ W = 0) is performed, focusing mainly on the original system. A erwards, the spectral system is used to investigate the influence of a time-varying bearing geometry (δ W 0, Ω W ).

. Static, Non-Circular Bearing Geometry

In this section the variation of the geometry is neglected (by pu ing δ W = 0) and only the influence of a non-circular bearing (controlled by W) is investigated. By increasing the mean value W the geometry becomes more and more 'elliptic-like', while W = 0 corresponds to a circular profile. In this case it is sufficient to just consider the original equations of motion

W = 0 W = 0.1 W = 0.2 5 10 15 20 -0.2 -0.1 0 0.1 ω equilibria of Y R [-]
Figure . Stable (dense) and unstable (dashed) equilibrium solutions for different W with parameter set from ( ) As the diagrams indicate, the stability of the equilibrium solution is lost in all three cases with an increasing value of the rpm-parameter via a Hopf-Bifurcation (cf. [ ]). A more elliptic bearing shape (increasing W) thereby shi s the bifurcation point to even higher rpms.

is phenomena seems reasonable, as for an elliptic bearing (in contrast to a circular one) two converging fluid-film gaps are present, resulting in a be er support of the rotating journal. A er passing the respective Hopf-Bifurcation, the system leaves its former equilibrium position and performs limitcycle oscillations (cf. [ ]), which are not depicted in fig. .  e stabilizing effect of the non-circular geometry becomes obvious. Nevertheless, these kinds of geometries cause large anisotropies in the associated bearing stiffness-matrix, which represents a disadvantage for certain applications like de-scribed by F

. [ ].

.

Time-Varying Bearing Geometry

While limiting the mean value of the bearing deformation to a certain degree, the occurring oscillations in the unstable rpm-range should now be reduced by using the previously described parametric excitation, which is caused by a comparatively small harmonic geometry variation (cf. eq. ( )).

As described e.g. in [ ], [ ],[ ] such small variations may have a positive effect onto the rotor system. In order to emphasize the advantage of the previously de- As it can be seen in fig. , the original system tends to small limit-cycle oscillations caused by the parametric excitation with frequency Ω W . Instead, the spectral system in fig. , which is freed from this frequency, approaches an equilibrium solution. e spectral components SC(. , 1) can thereby be interpreted as mean values, whereas the other two components SC(. , sin θ 1 ) and SC(. , cos θ 1 )) describe the oscillations around this mean value.

In the context a systematic solution continuation, this behaviour is much more desirable than the limit-cycle characteristic. Instead of continuing limit-cycle solutions, which will loose their stability via a Neimark-Sacker-Bifurcation (cf. [ ]), the equilibria of the spectral system are continued, for which ta Neimark-Sacker-Bifurcation corresponds to a normal Hopf-Bifurcation.

On the other hand it has to be mentioned that the increased system dimension ( ODEs) has of course a negative influence on the computation time compared to the original system ( ODEs).

In fig. an equilibrium continuation of the spectral system

θ spectral components (SC) of X R SC(X R , 1) SC(X R , sin θ 1 ) SC(X R , cos θ 1 ) 0 10 20 30 40 50 -0.15 -0.1 -0.05 0 0.05 θ spectral components (SC) of Y R SC(Y R , 1) SC(Y R , sin θ 1 ) SC(Y R , cos θ 1 )
Figure .

Transient results of spectral system for W = 0.2 and ω = 1 with geometry variation Ω W = 2.5, δ W = 0.1 and parameter set from ( ) can be found for W = 0.2. A er comparing these results to the previous ones with static geometry from fig. , the advantage of the geometry variation becomes obvious. e equilibrium solution of the system with static geometry looses its stability and stays unstable for the remaining rpm-range.

As opposed to this in the case of a time-varying geometry, the equilibrium solution of the spectral system shows a regain of stability at higher rpms at approximately ω ≈ 17. It has to be mentioned, that the original system behaviour would correspond in this case to a limit-cycle. But as the spectral components SC(. , sin θ 1 ) and SC(. , cos θ 1 )) are almost vanishing on this re-stabilized branch, the oscillation amplitudes are comparatively small. Performing a transient simulation of the original system with geometry variation for a parameter set, for which the stabilization in the spectral system has been detected, validates the obtained results: e original system tends to comparatively small limit-cycle oscillations, whereas without geometry variation, large amplitudes can be detected.

Of course a correct value for the parametric excitation frequency Ω W has to be chosen in order to achieve the above described stabilization effect. A detailed analysis of this phenomena will be part of future work. Like investigated for similar systems the stabilization might result from parametric anti-resonances as describe by P . [ ].

5 10 15 20

0 0.1 0.2 0.3 ω equilibria of X R [-] SC(X R , 1) SC(X R , sin θ 1 ) SC(X R , cos θ 1 ) 5 10 15 20 -2 -1 0 1 2 •10 -2 ω equilibria of Y R [-] SC(Y R , 1) SC(Y R , sin θ 1 ) SC(Y R , cos θ 1 )
Figure .

Equilibrium continuation of spectral system for W = 0.2 with geometry variation Ω W = 2.5, δ W = 0.1 and parameter set from ( )

. SUMMARY AND OUTLOOK

An elaborated model of an actively deformable journal bearing has been developed. It has been implemented into a Jeffco rotor system and the corresponding equations of motion as well as an associated spectral system have been set up. e results of the rotor system with static bearing geometries have shown a significant influence of a deformed bearing onto the stability properties of the respective rotor dynamic system. Transient simulations for both, the original as well as the spectral system, have been performed, emphasizing the advantage of the spectral system in the context of a possible solution continuation. An equilibrium search of the associated spectral system in dependence of the rpm-parameter ω has revealed a potential stabilization effect, caused by the time-varying geometry of the bearing. e dependence of this stabilizing effect on the parametric excitation parameters Ω W and δ W will be part of future work. Besides comprehensive parameter studies which also take into account the other system parameters, an extension of the presented model by several other effects is in progress.

e neglected influence of the pressure field onto the elastic deformation should be considered in a future model, as well as effects of a static imbalance of the rotor. Furthermore, an even be er Galerkin approach for the solution approximation of the invariance equation should be implemented, taking into account also higher harmonics in the Ansatz-function.

A proper validation of the purely theoretical results is still outstanding, whereas a comparison of different modelling strategies is in progress. Considering the stabilizing effects of other non-circular bearing profiles ([ ],[ ], [ ] etc.) the obtained qualitative results seem reasonable. Nevertheless, an experimental validation would be useful.

LIST OF SYMBOLS

W non-dimensional radial deformation (= w /C) w * bearing displacement at ϕ = ± π /2 W * non-dimensional bearing displacement at ϕ = ± π /2 W mean value of W * x J , y J centre coordinates of journal X J , Y J non-dimensional centre coordinates of journal x R , y R centre coordinates of rotor X R , Y R non-dimensional

Figure .

 . Figure . Actively deformed journal bearing

Figure .

 . Figure . Schematic sketch of Jeffco rotor

  for different static bearing geometries is shown in fig. . Stable solutions are indicated with dense lines, whereas a dashed line style is used for unstable solutions instead.

Figure .

 . Figure .Transient results of original system for ω = 1, W = 0.2, Ω W = 2.5, δ W = 0.1 and parameter set from ( )

  PM. Przybylowicz. Active stabilisation of a rigid rotor by piezoelectrically controlled mobile journal bearing system. ( ): -. [ ] J. T ȗma, J. Šimek, J. Škuta, and J. Los. Active vibrations control of journal bearings with the use of piezoactuators. Mechanical Systems and Signal Processing, ( ): -, . [ ] I. Santos. Design and evaluation of two types of active tilting pad journal bearings, . [ ] DC. Deckler, RJ. Veille e, and FK. Braun, MJ .and Choy. Simulation and control of an active tilting-pad journal bearing. Tribology transactions, ( ): -, . [ ] A. Wu and M. De eiroz. A new active tilting-pad bearing: Nonlinear modeling and feedback control. Tribology Transactions, ( ): -, . [ ] B. Pfau, M. Rieken, and R. Markert. Numerische Untersuchungen eines verstellbaren Gleitlagers zur Unterdrückung von Instabilitäten mi els Parameter-Antiresonanzen. In First IFToMM DA-CH Conference, . [ ] A. Nayfeh and D. Mook. Nonlinear oscillations. John Wiley & Sons, . [ ] F. Schilder, W. Vogt, S. Schreiber, and H. Osinga. Fourier methods for quasi-periodic oscillations. International journal for numerical methods in engineering, ( ): -, . [ ] J. Presco . Applied elasticity. Longmans, Green and Co., . [ ] AB COMSOL. COMSOL Multiphysics ver. . , . [ ] A. Szeri. Fluid film lubrication: theory and design. Cambridge University Press, . [ ] B. van de Vrande. Nonlinear dynamics of elementary rotor systems with compliant plain journal bearings. Technische Universiteit, . [ ] M. Yarrow. Solving periodic block tridiagonal systems using the Sherman-Morrison-Woodbury formula. In th Computational Fluid Dynamics Conference, page , . [ ] R. Gasch, R. Nordmann, and H. Pfützner. Rotordynamik. Springer-Verlag, . ] K. Becker and W. Seemann. Approximation of quasiperiodic solutions of a rotor in two-lobe bearings with time-varying geometry. PAMM, ( ): -, . [ ] B. Marx and W. Vogt. Dynamische Systeme: eorie und Numerik. Springer-Verlag, . [ ] A. Dhooge, W. Govaerts, and Y. Kuznetsov. MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Transactions on Mathematical So ware (TOMS), ( ): -, . [ ] Y. Kuznetsov. Elements of applied bifurcation theory, volume . Springer Science & Business Media, . [ ] J. Frene, B. Degueurce, D. Berthe, and M. Godet. Hydrodynamic lubrication: bearings and thrust bearings, volume . Elsevier, . [ ] K. Becker and W. Seemann. A Journal Bearing with actively modified geometry for extending the parameterbased stability range of rotor dynamic systems. In ISRO-MAC proceedings, .

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	[ ]
					centre coordinates of rotor
					z	axial coordinate
					z	non-dimensional axial coordinate (= 2z /B)
					z	state space vector of first-order original system
					Z	state space matrix of first-order spectral system

[


Ferdinand-J Cloos

Peter F Pelz
email: peter.pelz@fst.tu-darmstadt.de

O Tatin

Swirl Boundary Layer at the Inlet of a Rotating Circular Cone

Keywords:

When a fluid enters a rotating pipe, a swirl boundary layer with thickness of δS appears at the wall and interacts with the axial momentum boundary layer with thickness of δ. e swirl is produced by the wall shear stress and not due to kinematic reasons as by a turbomachine. In the center of the pipe, the fluid is swirl-free and is accelerated due to axial boundary layer growth. Below a critical flow number ϕ < ϕ c , there is flow separation, known in the turbomachinery context as part load recirculation.

Previous work analyses the flow at the inlet of a rotating circular pipe ( R = R0 ). For a systematic approach to a turbomachine, the influence of the turbine's and pump's function, schematically fulfilled by a diffuser and a nozzle, on the evolution of the swirl and flow separation is to analyse.

e radius of the rotating pipe depends linearly on the axial coordinate, yielding a rotating circular cone. e swirl evolution depends on the Reynolds number, flow number, axial coordinate and apex angle. e influence of the la er is the paper's main task. e circumferential velocity component is measured applying D Laser Doppler Anemometry to investigate the swirl evolution.

INTRODUCTION

Turbomachines o en operate at part load due to the claim of flexibility and not at the design point where the efficiency reaches its maximum. At part load for small flow numbers ϕ := Ũ/( R0 Ω) < ϕ c , with averaged axial velocity Ũ (the superscript "∼" indicates dimensional symbols whereas quantities without this superscript are dimensionless), circumferential velocity of the pipe R0 Ω, pipe radius at the inlet R0 = R( z = 0) and axial coordinate z, losses causing flow phenomena such as cavitation, rotating stall, part load recirculation and so on appear. ese flow phenomena reduce the efficiency of a turbomachine compared to the efficiency at the design point.

e paper analyses the swirl development and distribution at the inlet of a rotating circular cone. e outcome of this work is useful to analyse and to predict swirl evolution, part load recirculation, flows in rotating gaps, e.g. secondary air flow of a gas turbine, and to clarify the inlet condition of turbo machines, e.g. shrouded turbomachines with a three dimensional boundary layer. us, this investigation supports the designers of the mentioned parts of a turbomachine. roughout this investigation, the paper nondimensionlises all velocities with the local cone circumferential velocity Ω R( z) and lengths with the cone radius at the inlet R0 . In addition to that, all radial lengths are scaled on the local cone radius R( z).

To gain more physical knowledge about swirl evolution and part load recirculation, the paper uses a generic flow situation: a flow through a rotating circular cone. By this generic model, secondary flows, e.g. tip vortices as common by blades, are avoided, yielding a clear flow situation. Previous work analyses the flow at the inlet of a rotating pipe with a constant radius R = R0 [ -]. For a systematic approach to a turbomachine, the influence of a pump's and turbine's function on the evolution of the swirl and on the flow separation is to analyse. e pump's function is to increase the pressure and this is schematically fulfilled by a diffuser. e turbine's function is to decrease the pressure, obtaining a nozzle. us, the radius of the rotating circular cone depends linearly on the axial coordinate R( z) = R0 + z tan α to accelerate or deaccelerate the flow with the apex angle α. ere, the axial velocity U(z) and the static pressure P(z) are only a function of the axial coordinate. By rotating the circular cone, a second boundary layer in circumferential direction is produced by viscosity due to non-slip condition at the wall and develops. It is the so called swirl boundary layer with thickness δ S [ -]. Inside the swirl boundary layer is a circumferential velocity component u φ (y, z), outside, the flow is swirl-free. ere is a non-negligible radial pressure distribution p(y, z) inside the swirl boundary layer for ϕ ≪ 1 due to the centrifugal force [ , ].

e evolution of the boundary layers depend on the Reynolds number Re := 2 R2 0 Ω/ ν with the kinematic viscosity ν, the flow number ϕ and the relative, averaged surface roughness R z := Rz / R0 . e paper's main task is to quantify the influence of the apex angle on the swirl velocity profile and its turbulence intensity as well as on the swirl boundary layer thickness. erefore, D Laser Doppler Anemometry (LDA) is applied to measure the circumferential velocity profile. By doing so, two different inlet conditions are used: a thin laminar (configuration I) and a fully developed turbulent (configuration II) axial boundary layer at the inlet of the rotating circular cone.

is paper is structured as follows: First, it gives a literature review concerning the influence of swirl on velocity profiles and separation in the following section. §2 describes the experimental set-up. e gained results with this set-up presents §3. Finally, the main outcomes of this work are recapped in the closing of this paper.

. LITERATURE REVIEW

e flow in a rotating pipe has been much more investigated at the fully developed region than at the inlet region. At the fully developed region, the boundary layers reach the pipe centre and the velocity profiles are independent of the axial coordinate. e boundary layers reach the centre for z > 10 2 for a turbulent flow, depending on Reynolds number and flow number [ ].

An interaction of the swirl and the axial momentum is observed by a complex transformation of the axial velocity profile at the inlet of a rotating pipe. Due to turbulence damping, the axial velocity profile transforms continuously from a turbulent into a laminar profile in axial direction when a fully developed, turbulent flow enters the rotating pipe [ ]. is effect is called "laminarization" and has been observed in many investigations [ , , ]. e "laminarized" profile reaches the fully developed region for a small flow number. For a higher flow number, the axial velocity profile is retransformed into the turbulent one [ ]. An analytical approach of this transformation by Weigand and Beer [ ] meets the experimental results qualitatively, but the influence of the swirl is overestimated.

e swirl boundary layer with its velocity profile defines the evolution of the swirl. e circumferential velocity profile is parabolic u φ = r 2 , when a turbulent flow enters the rotating pipe [ , -] and for a laminar flow it follows solid body rotation u φ = r [ ] at the fully developed region. Oberlack [ ] derived the profiles by Lie group analysis. In the transition region, the circumferential velocity profile is between the parabolic and the linear profile [ ]. When a thin laminar boundary layer enters a rotating pipe for ϕ > 0.71, the circumferential velocity profile transforms and both boundary layers are thickened at the inlet of a rotating pipe [ ]. At the inlet for a smaller flow number ϕ ≤ 1 with a fully developed turbulent or a thin laminar or turbulent axial boundary layer, the circumferential velocity profile follows

u φ = 1 - y δ S 2 , ( )
for regime I and an a ached flow [ -]. For the fully developed axial turbulent flow, the measured swirl boundary layer thickness follows For a hydraulically rough flow, the circumferential velocity profile follows more or less

δ S02 = CRe m R e ϕ m ϕ z m z , ( )
with
u φ = (1 -y/δ S ) 7 [ ].
Equation for a thin axial boundary layer is confirmed by our analytical approach [ , ]. By this approach, we use the integral method of boundary layer theory and generalize the von Kármán momentum equation taking the influence of swirl by a radial pressure distribution into account. A strong influence of the swirl, i.e. the centrifugal force, on the axial momentum balance is observable for small flow numbers ϕ ≪ 1.

e centrifugal force causes flow separation for a small flow number and a measured stability map for part load recirculation is given [ , -]. When the flow separates, the circumferential velocity profile differs from the parabolic one [ , , ]. Stratford's criteria [ ] is applied to derive the critical flow number for incipient separation analytically and the results are validated by experiments [ ]. For small Reynolds number, i.e. laminar flow, flow separation is investigated by Lavan and others in a rotating pipe [ -].

For high Reynolds number as well as high flow number Eqn. is not longer valid because a second transition of the swirl boundary layer occurs in the turbulent regime [ -]. At this transition (Re, ϕ, z) > (Re, ϕ, z) t (the subscript t indicates the point of transition from regime I to II), the circumferential velocity profile transforms from the parabolic profile Eqn. of regime I into

u φ = - κ 2 log y δ S + A + B ( )
of regime II with the constants A = 0.007, B = 0.06 and

κ = 0.4 [ , ]
. Furthermore, the swirl boundary layer thickens and the turbulence intensity Tu φ := ũ′ φ,rms /( R0 Ω) increases and reaches a new self-similarity. e swirl velocity profile transformation is applied to indicate both regimes and the transition inlet length z t of regime II. When and where the transition occurs depending on the inlet condition is given by the transition inlet length and the stability map [ , ].

e flow through a non-rotating circular cone [ -] is more investigated for swirl-free flow than a flow with swirl decay [ -]. For the first case, the main task was to optimize the flow and to prevent flow separation. ere are stability maps for circular cones with a positive (diffuser) and negative (nozzle) apex angle [ ]. Less investigations analyse the swirl decay and vortex breakdown at the inlet of a rotating circular cone [ -]. Hence, up to now, the influence of the apex angle on the evolution of swirl is unknown.

.

EXPERIMENTAL SET-UP

A test-rig is designed to measure the evolution of the swirl boundary layer and the circumferential velocity component in a rotating cone applying D LDA and is used by previsous investigations [ -]. is accessibility is used to measure the circumferential velocity component by a D LDA with frequency shi and a wave length of 514.5 nm. e probe, i.e. sender and receiver, is located downstream of the rotating cone with angle of 12 • to the centreline and has a focus of mm. e measurement volume has a length of < 1.6% R0 , a diameter of < 2 R0 and is moveable with a two dimensional plane by using a traverse table. By doing so, the laser beams do not have to light through any solid material and are not deflected. An aerosol of silicon oil as tracer particles is added to the air to enable LDA measurements; see Fig. . e LDA and the experimental set-up is described in more detail by [ , , ].

e positioning of the LDA measuring volume has a systematic measuring error of ∆ zsys ≈ ±8 R0 and ∆ ỹsys ≈ ±4 R0 . e repetitive accuracy in both directions of the traverse table is 0.8 R0 . One LDA data point is measured over more than 30 s including more than 1000 bursts.

e transit time of each passing tracer particle is used to weight the measured velocity.

e maximum systematic measuring error of Reynolds number is less than ∆Re sys /Re ≤ ±4 % and of the flow number it is less than ∆ϕ sys /ϕ ≤ ±5.5 % for ϕ = 0.35 and log(Re) = 4.1. e systematic measuring errors of the Reynolds number and the flow number decrease with increasing Reynolds number and flow number, respectively. Both quantities include data points, obtaining a negligible precision error compared to the systematic error since the standard deviation is small. Nevertheless, both measuring errors are taken into account for the total measuring error. In the following, all presented data points consider the total measuring error visualized by error bars. Usually, the error bars are smaller than the marker, thus they are not visible.

e precision error of ũ′ φ,rms is unknown.

. RESULTS

is section presents the measured swirl boundary layer thickness, the swirl velocity profile and its turbulence intensity. Hereby, both configurations are used and their influence as well as of the apex angle are discussed.

. Swirl Boundary Layer Thickness e development of the swirl boundary layer is well depictured by the isolines of u φ in figure . Between the measurement points, illustrated by the markers, the swirl velocity is interpolated. By doing so, the swirl boundary layer thickness δ S02 and δ S07 := y(u φ = 0.07) are obtained. For configuration II, the outer region of the swirl boundary layer is the duty of higher fluctuations due to the higher turbulence intensity than for configuration I; cf. § .

is is the reason for analysing the isoline δ S07 for configuration II instead of δ S02 as it is done for configuration I.

Figure and show the measured swirl boundary layer thickness for different apex angles and various cross-sections, Reynolds number and flow number for configuration I and II, respectively. Analysing the swirl boundary layer thicknesses in both figures, the swirl boundary layer thickens for increasing apex angle, especially for z 1. For a positive apex angle, the boundary layer is noticeable thicker than for a negative or a vanishing apex angle.

Analysing the slope of the boundary layer thickness in the double-logarithmic diagrams (Fig. and ) for each parameter far away from the singularity at z = 0, the thickness δ S02 and δ S07 follow the power law Eqn. . A calibration of the power law is necessary to obtain the constant C. e constants C, m Re , m ϕ and m z depend on the configuration, more or less on the apex angle but are independent of the other parameters; cf. Tab. and . e total error is ∆m i ≈ ±0.03 and of ∆C ≈ ±0.1. Constant m ϕ and C change most by varying the apex angle and increase by increasing apex angle; see Fig. c and. c. Constant m z depends on the sign of the apex angle. Hence, the apex angle has a noticeable influence on the evolution of the swirl boundary layer thickness.

. Swirl Velocity Profile

. Turbulence Intensity

Figure depicts the turbulence intensity of the swirl velocity for the same parameter combinations as the swirl velocity distribution is presented by Fig. . As it is known by the previous investigations for α = 0 [ -], the turbulence intensity is higher for configuration II than I. is is due to the higher turbulence intensity of the axial velocity at the inlet of the rotating cone; cf. § .

For configuration I and regime I, the turbulence intensity Tu φ is self-similar for the outer region y/δ S02 0.7. At the inner region y/δ S02 0.7, the turbulence intensity depends on the axial coordinate, Reynolds number and flow number but not on the apex angle; see Fig. a-d. ere, the turbulence intensity decreases with increasing axial coordinate and decreasing Reynolds number and flow number, respectively. For z 1, the turbulence intensity is self-similar for the inner and outer region.

For configuration II and regime I, the turbulence intensity is more or less self-similar and independent of the axial coordinate, Reynolds number and apex angle. For increasing flow number, the turbulence intensity increases at the inner and outer region; see Fig. . CONCLUSION is paper investigated the influence of the apex angle α on the evolution of the swirl at the inlet of a rotating circular cone. By doing so, three negative (nozzle), three positive (diffuser) and one vanishing apex angle (pipe) were applied. Furthermore, the influence of the inlet condition was investigated with two flow configurations. e measured evolution of the swirl by LDA is well described by a power law Eqn. for the swirl boundary layer thickness taking the axial coordinate, Reynolds number and flow number into account. e constants for this power law depend more or less on the apex angle; see Tab. and . e swirl boundary layer is thicker for positive than for negative apex angles for z 1.

Within the swirl boundary layer, the circumferential velocity distribution follows the well-known parabolic profile Eqn. for regime I of the swirl boundary layer; see e turbulence intensity is self-similar for the outer region. At the inner region, the turbulence intensity depends on the axial coordinate for z 1, Reynolds number and flow number but is independent of the apex angle; see Fig. .  Finally, this investigation improves the physical knowledge and understanding of swirl evolution. e gained knowledge is employable for the design of a turbomachine, especially a shrouded turbomachine and rotating flow channels of secondary air flow of a gasturbine.

Figure .

 . Figure .Flow through the inlet of a rotating circular cone.

Figure .

 . Figure . Experimental set-up with a) configuration I for a thin axial boundary layer and b) configuration II for a fully developed axial boundary layer.

Figure .

 . Figure . Laser beam path into the rotating pipe of the D Laser Doppler Anemometry.

Figure .

 . Figure . Interpolated isolines of u φ shows the development of the swirl boundary layer for configuration I. Measurement points are illustrated by the markers.

Figure . Figure . Figure .

 ... Figure . Swirl boundary layer thickness δ S02 with various apex angle versus a) axial coordinate, b) Reynolds number and c) flow number for configuration I.

Figure

  Figure shows the swirl velocity profile versus the scaled wall coordinate for various axial cross section, Reynolds number, flow number, apex angle and both configurations. For configuration I (Fig. a-d), the wall coordinate y is scaled on

  Figure depicts the turbulence intensity of the swirl velocity for the same parameter combinations as the swirl velocity distribution is presented by Fig..As it is known by the previous investigations for α = 0 [ -], the turbulence intensity is higher for configuration II than I. is is due to the higher turbulence intensity of the axial velocity at the inlet of the rotating cone; cf. § .For configuration I and regime I, the turbulence intensity Tu φ is self-similar for the outer region y/δ S02 0.7. At the inner region y/δ S02 0.7, the turbulence intensity depends on the axial coordinate, Reynolds number and flow number but not on the apex angle; see Fig.a-d. ere, the turbulence intensity decreases with increasing axial coordinate and decreasing Reynolds number and flow number, respectively. For z 1, the turbulence intensity is self-similar for the inner and outer region.For configuration II and regime I, the turbulence intensity is more or less self-similar and independent of the axial coordinate, Reynolds number and apex angle. For increasing flow number, the turbulence intensity increases at the inner and outer region; see Fig. g. For increasing Reynolds number, the turbulence intensity increases slightly at the inner region. For high Reynolds number, the transition to the turbulent regime II of the swirl boundary layer indicated by the swirl profile transformation (Fig. f) occurs. Hereby, the turbulence intensity increases strongly and yields the known self-similarity of regime II [ , ].

  Fig. .

e

  transition to regime II of the swirl boundary layer occurs as well for α 0.

Figure .

 . Figure . Turbulence intensity of the swirl velocity for various axial coordinate, Reynolds number, flow number and apex angle for a)-d) configuration I, a thin axial laminar boundary layer, and for e)-h) configuration II, a fully developed axial turbulent boundary layer.

  δ S02 := y(u φ = 0.02), C ≈ 4.43, m Re ≈ -0.45, m ϕ ≈ -0.46 and m z ≈ 0.47. For a thin axial boundary layer, the constant C is approximately . with m Re ≈ -0.46, m ϕ ≈ -0.49 and m z ≈ 0.44 [ , ]. e swirl boundary layer becomes independent of the Reynolds number for a hydraulically rough flow but still depends on the flow number [ , ].

Table .

 . Constants for the power law Eqn. for δ S02 and configuration I.

	α	C	m Re	m ϕ	m z
	-1.83 • 4.19 -0.46 -0.54 0.44
	-1.08 • 4.25 -0.45 -0.43 0.43
	-0.61 • 4.55 -0.46 -0.45 0.44
	±0.00 • 4.64 -0.46 -0.49 0.44
	+0.47 • 6.45 -0.47 -0.35 0.37
	+1.03 • 6.63 -0.50 -0.32 0.35
	+1.55 • 10.3 -0.47 -0.22 0.35
	Table . Constants for the power law Eqn. for δ S07 and
	configuration II.
	α	C	m Re	m ϕ	m z
	-1.83 • 2.47 -0.41 -0.46 0.41
	-1.08 • 3.51 -0.45 -0.47 0.44
	-0.61 • 4.78 -0.48 -0.44 0.45
	±0.00 • 3.16 -0.43 -0.40 0.41
	+0.47 • 3.99 -0.44 -0.38 0.35
	+1.03 • 5.07 -0.46 -0.31 0.34
	+1.55

• 5.86 -0.47 -0.28 0.35

  [ ] G. Reich. Strömung und Wärmeübertragung in einem axial rotierenden Rohr. PhD thesis, Technische Hochschule Darmstadt, . ] S. Imao, Q. Zhang, and Y. Yamada. e laminar flow in the developing region of a rotating pipe. Bulletin of JSME, : -, . [ ] C. Crane and D. Burley. Numerical studies of laminar flow in ducts and pipes. J. Comp. and appl. Mathe., : -, . [ ] G. Jungnitz. Rechnerische Untersuchung von Diffusoren. Forsch. im Ingenieurw., ( ): -, . [ ] W. Szablewski. Turbulente Strömungen in konvergenten Kanälen. Arch. Appl. Mech., ( ): -, . [ ] W Szablewski. Turbulente Strömungen in divergenten Kanälen. Arch. Appl. Mech., ( ): -, . [ ] H. Schlichting and K. Gersten. Berechnung der Strömung in rotationssymmetrischen Diffusoren mit Hilfe der Grenzschich heorie. Z. Flugwiss, ( / ): -, . [ ] G.-G. Börger. Optimierung von Windkanaldüsen für den Unterschallbereich. PhD thesis, Ruhr-Universität Bochum, . [ ] K. Gersten and H. Herwig. Strömungsmechanik: Grundlagen der Impuls-, Wärme-und Stoffübertragung aus asymptotischer Sicht. Vieweg, . [ ] F. Seifert. Berechnung inkompressibler reibungsbeha eter Ringdiffusorströmungen nach der Schlankkanaltheorie. PhD thesis, Ruhr-Universität Bochum, . [ ] G.I. Taylor. e boundary layer in the converging nozzle of a swirl atomizer. e arterly J. Mech. and Appl. Mathe., ( ): -, . [ ] A.M. Binnie and D.P. Harris. e application of boundary-layer theory to swirling liquid flow through a nozzle. e arterly J. Mech. and Appl. Mathe., ( ): -, . [ ] A.N. Broujerdi and A. Kerbriaee. Pressure loss of turbulent swirling flow in convergent nozzle. In Proceedings of th ISME, .

[ ] M. Oberlack. Similarity in non-rotating and rotating turbulent pipe flows. J. Fluid Mech., : -, .

[ ] B.S. Stratford. e prediction of separation of the turbulent boundary layer. J. Fluid Mech., : -, .

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[


Onur Tasar
email: onur.tasar@inria.fr

Yuliya Tarabalka

Pierre Alliez

Alain Giros

Sébastien Clerc

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From Satellite Images to Vector Representations

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

One of the most popular challenges in the field of remote sensing is to generate vector representations from satellite images to be included into various Geographic Information System (GIS) applications. One way to generate the aforementioned vector representations is to split this task into two consecutive sub-tasks, where the first one consists in generating pixel-wise maps in raster format using advanced machine learning techniques, and the second one aims at vectorizing the obtained raster map by applying computational geometry methods. We propose novel methods for both sub-tasks.

Conclusions

Incremental Learning: To the best of our knowledge, we propose the first work on incremental learning for semantic segmentation. Domain Adaptation: The proposed approach is architecturally simple but powerful. We achieve a better performance than the existing approaches in a much shorter period of time.

Mesh Approximation: The approach yields a better accuracy than the simplification algorithms in common GIS softwares, although our method generates significantly less number of vertices.

  Vectorizing the raster map using a triangle mesh with edge flip, half edge collapse, and vertex relocation operators

Incremental Learning

• When new classes are added to the already trained model, the current approaches suffer from catastrophic forgetting (significant performance drop for the previous classes).

• The training phase of our framework [START_REF] Tasar | Incremental learning for semantic segmentation of large-scale remote sensing data[END_REF][START_REF] Tasar | Continual learning for dense labeling of satellite images[END_REF] has two stages: adaptation and remembering.

• During the adaptation, we use new ground-truth to learn new classes on the current data, and we use the output of the previously trained network to learn old classes from the current data.

• In the remembering phase, we store only a small portion of the previous training data, and we use it to remind the network the learned information for old classes from the previous data.

Domain Adaptation

• When there is a large shift between the spectral bands of training and test images, the existing machine learning models fail to segment the test data.

• We generate fake training images that are semantically exactly the same as the original training images, but whose spectral distribution as close as possible to test images by ColorMapGAN [START_REF] Tasar | ColorMapGAN: Unsupervised domain adaptation for semantic segmentation using color mapping generative adversarial networks[END_REF].

• We then use the fake training data to fine-tune the model trained on the original training data.

Mesh Approximation

• We transform the initial mesh to the optimal mesh by minimizing an objective function with certain operators [START_REF] Tasar | Polygonization of binary classification maps using mesh approximation with right angle regularity[END_REF].

• Our objective function balances between fidelity to the classification map in 1 norm sense, right angle regularity for polygonized buildings, and the final mesh complexity.

• The operators: edge flip, half edge collapse, vertex relocation.