Dynamic Simulation of Finger Seal-Rotor Interaction Using Variable Dynamic Coefficients

Hazel Marie1

Youngstown State University, Youngstown, Ohio, 44555

Compliant non-contacting finger seals with hydrostatic/hydrodynamic lift pads represent an advance in gas turbine engine sealing from the stand point of both sealing effectiveness and potential unlimited lifespan. Design optimization of the finger seal must consider optimization of the air film gap thickness between the seal and the rotor; too small a gap creates excessive viscous heating, while too large a gap causes excessive seal leakage. The work to be presented concerns the mapping of the dynamic behavior of a repetitive section of a two-layer finger seal with high- and low-pressure laminates. Dynamic stiffness coefficients in both the finger seal and the fluid film are evaluated as a function of rotor speed, finger-rotor clearance, and selected geometric features of the finger design. The fluid stiffness coefficients are determined using the solid-fluid compliant interface capabilities of the commercial package CFD-ACE. These will be compared to earlier stiffness calculations which considered a solid-fluid non-compliant interface. Finally, a third generation two degree-of-freedom dynamic analysis using a variable fluid stiffness coefficient is presented.

Nomenclature Do = seal outside diameter Di = seal inside diameter Db = finger base diameter Df = foot upper diameter Rs = stick arc radius Dcc = circle of center of the stick arc radii α = repeat angle u,v,w = fluid velocity in the Cartesian plane x-, y-, z-directions p = pressure of the fluid k = wedge taper ratio ffmin = minimum fluid film height in the radial direction ffmax = maximum fluid film height in the radial direction ω = rotor angular speed xR = displacement of the rotor mass from equilibrium xFS = displacement of the finger mass from equilibrium mREqu = equivalent rotor mass mFSEqu = equivalent finger mass kFEqu = equivalent fluid stiffness coefficient cFEqu = equivalent fluid damping coefficient kSEqu = equivalent finger stiffness coefficient cSEqu = equivalent viscous damping coefficient for Coulomb friction F = force applied to rotor Ζ = mechanical impedance Ff = friction force between equalization dam and finger sticks ωn = natural frequency of finger mass

1 Assistant Professor, Department of Mechanical and Industrial Engineering, 1 University Plaza, Youngstown, Ohio 44555, AIAA Young Professional Member.

American Institute of Aeronautics and Astronautics

1

42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit9 - 12 July 2006, Sacramento, California

AIAA 2006-4931

Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

I. Introduction Seals are an integral part of the modern gas turbine engine. They are used in many locations including between

stationary and rotating parts of the shaft, compressor and turbine; over the blade tips of the compressor and turbine; between components and throughout the internal cooling flow paths. In fact, large turbine engines can have well over fifty gas path sealing locations both at the inter-stage level and at the blade tip clearance. This reality, coupled with the fact that the engines are being required to operate at higher and higher temperatures and cycle pressure ratios, has caused the effectiveness and reliability of seals in these modern-day engines to become quite important to the enhancement of the engine performance and efficiency. With continued use and wear, the cumulative negative effect of leakage on engine thrust-to-weight ratio and specific fuel consumption (SFC) can be quite substantial. For instance the average increase in SFC of over 1% annually in large turbofan engines is attributed to the wear and erosion of seals1. Until the early 1990’s, sealing between the stationary and rotating parts in gas turbine engines was accomplished exclusively by utilizing rigid and non-compliant seals such as cylindrical, labyrinth, and honeycomb seals and/or combinations thereof. In fact, even today in air-to-air sealing for turbine engines, the labyrinth seal is still used as one of the standards by which the performance of new seal designs is measured.

These conventional seals require radial operating clearances to prevent the stationary and rotating components from rubbing, while interface surface speeds reach values as high as 1500ft/s. Basic causes for clearance deterioration can stem from the radial displacements of the rotor, differential thermal responses due to operational transients that affect both the case and the rotor, ovalization due to non-axisymmetric distribution of temperatures and/or loads, and centrifugal and gyroscopic forces born from operational maneuvers or air turbulence loads.

In the last two decades compliant shroud or shaft seals (whether outfitted with brush or finger seals) seem to have become ideal candidates for the role of passive-adaptive mitigation of these types of leakages. Today’s brush seals are in operation as shaft seals in Rolls-Royce, Pratt and Whitney, and GE jet engines. Sealing at the brush seal/shaft interface is ensured by tight packing of the bristles and interference tolerance fit between the seal inner diameter and shaft outer diameter. This configuration is much more effective than that of the labyrinth seal, even though leakage occurs both at the bristles/shaft interface and through the brush’s bristle pack. The lifespan of the brush seal is far longer and its performance degradation is delayed considerably when compared with that of a labyrinth seal. The literature detailing both experimental and numerical simulation work regarding the brush seals is very rich and, for completeness and the readers’ benefit, a comprehensive review of this literature is suggested2,3.

More recently, brush seals have found a rather challenging competitor in finger seals. Early finger seal models4-7 were comprised of a large number of flexible members fixed at one end and contacting, with the intent of sealing, a rotating surface at the other. The first numerical simulation was conducted by Hendricks et al.8 and analyzed the behavior of a finger seal using a bulk flow model. The approach was similar to numerical models used earlier to analyze brush seals9,10. This model showed good correlation when compared to experimental leakage data from an 8.2in inner diameter, two row finger seal mounted with a pre-load. In fact, the finger seal leakage performance was found to be far superior to that of labyrinth seals and on the same order as brush seals. Arora et al.11 performed comprehensive experimental testing for the finger seal assessing hysteresis characteristics, leakage performance, and life capabilities. In these tests, a three-laminate finger seal exhibited leakage that was 20% to 70% less than that of a four-knife labyrinth seal with a 0.005in initial radial clearance. The authors noted that the hysteresis was due to the frictional forces between the fingers and aft cover plate being larger than the restoring force of the fingers. To correct the hysteresis deficiency in the baseline finger seal, the authors optimized what they referred to as a “pressure balanced finger seal”12. This design added a spacer between the aft cover plate and the finger laminates to the baseline finger seal. The design of the spacer created an area for high pressure to form between the aft cover plate and the laminates, reducing the net axial force acting on the seal. The point was to keep the frictional force between the fingers and the cover plate always lower than the recovering force of the fingers. Proctor et al.13 furthered experimental testing of the pressure balanced finger seal design at rotor speeds as high as 1200 ft/s. It was found that at temperatures of 800°F and 1200°F, and pressure differentials of 10, 40 and 75 psi respectively, the difference in power loss, leakage, wear, and general performance between the brush and the finger seals were not significant. Both types of seals were mounted with a 0.003in interference with the shaft. In addition, the seal hysteresis was minimized. Arora14 and Proctor et al.15 further expanded on the early finger seal designs with a non-contacting type finger seal that includes hydrodynamic lift pads. Upon successful implementation, these self-acting hydrodynamic features will represent a significant advance of the state of the art from both the point of view of sealing effectiveness and potential unlimited lifespan. In fact, the design by Proctor et al. has been the focus of many structural, numerical, dynamic, and experimental analyses, much of which is summarized in Marie16. Specifically noticeable in her work is the need for solid-fluid interaction models due to the compliant nature of the seal. Braun et al.17 was first to use a

American Institute of Aeronautics and Astronautics

2

Navier-Stokes analysis to perform parametric numerical work on finger seals with lift pads. The authors visualized numerically the flow for both two- and three-dimensional distributed models of a pair of fingers arranged in-line and behind each other; the high pressure fore-finger being padless and the low pressure aft-finger padded. These models represented two limiting cases for the behavior of the padded finger seal. The 2 dimensional model limited the leakage to the area of the gap under the finger pad, while the 3 dimensional model permitted both leakage through the pad/shaft gap and unrestrained flow in between the circumferentially adjacent fingers. Braun et al.18 studied the 3 dimensional motion of various configurations of both a single padded finger and a 2 HP/1 LP finger combination with the purpose of determining the padded finger seal structural response to an outside simulated pressure stimulus. In a subsequent papers, Braun et al.19,20 mapped the thermofluid and dynamic behavior of a repetitive section of a proposed finger seal that contains four high- and four low-pressure padded fingers arranged axially in a staggered configuration, and subject to an axial pressure drop. The dynamic model introduced a simplified one-degree-of-freedom, spring-mass-damper equivalent to the complicated structure presented by the finger seal. The numerical experiments concentrated on the determination of the phase shift and displacement transmissibility. These two parameters indicated how well and under what conditions the finger would follow the rotor. It was found that under this simplified model, the geometry proposed provides satisfactory lifting capability for the fingers. The stiffness of fingers is small by comparison to that of the fluid thus causing the displacement transmissibility to be close to 1. Braun et al.21 elaborated further on the energy (temperature) generation and pressures that develop under compressible and incompressible conditions when isothermal, adiabatic and finite heat transfer coefficient cases are considered. Mass flow (leakages) and pressure forces obtained for the thermal conditions mentioned above, were studied when runner velocity, heat transfer coefficients, and axial pressure differences are varied parametrically. The authors offered a second generation two-degrees-of-freedom spring-dashpot model that incorporates the stiffness and damping of the fluid as well as the finger stiffness and the Coulomb damping caused by the finger-back plate interaction. The dynamic simulations showed that in order for the finger seal design to be successful and avoid a lag in the finger motion following the shaft, a fine balance has to be achieved between the Coulomb friction, the finger stiffness and the natural frequencies of the rotating shaft and the finger seal.

II. Scope of Work This paper continues the work started by Braun et al.17-21. The author offers solid–fluid interaction models of

both the single circumferential wedge pad and the double circumferential/axial wedge pad. The models investigated the fluid pressure build-up, fluid radial under pad force, finger lift, and fluid leakage as a function of both the rotor angular speed and the high- to low-pressure differential. In addition, the fluid radial under pad force was divided into its contribution from hydrodynamic lifting forces and hydrostatic lifting forces. These also were considered as a function of rotor angular speed while parametrically varying the high-to-low side pressure differentials. Trends in the equivalent stiffness of the seal fingers as a function of the geometry repeat angle, finger base diameter, upper foot diameter, and stick arc circle of centers are established such that the limiting values of these parameters can be determined. The equivalent fluid stiffness as a function of the wedge angle for parametric changes in the minimum film gap is established. Thus, the author offers a third generation two degree-of-freedom mass-spring-damper model that improves upon the previous models by incorporating variable fluid stiffness as a function of radial clearance between the rotor and the finger pad.

III. The Finger Seal Geometry Proctor and Steinetz15, proposed a finger seal with aerodynamic self-

acting lift pads added at the bottom of the low pressure fingers, illustrated in Figs. 1 and 2. The details of this geometry have been further clarified in Marie16. Fig. 1 presents a three-dimensional cut away view of the solid model containing the back- and front-plates and the low- and high-pressure finger laminates. The back-plate serves as an axial support against bending when the thin finger laminates are under pressure. This plate also contains a pressure equalization manifold that contributes to the mitigation of Coulomb friction between the plate and the low pressure laminate as the latter moves radially to follow the shaft orbit motion. The design configuration of the finger laminates, Fig.2, is that of an annulus; from which individual “fingers” extend to encompass the entire circumference of the rotor. An individual finger is comprised of a long thin “stick” that ends in an enlarged “foot” formation as shown. This design allows the fingers of

Figure 1. Cut Away View of Finger Seal Assembly

American Institute of Aeronautics and Astronautics

3

Solid Annulus

Finger Sticks

Feet

Finger Base

Figure 2. High Pressure Finger Seal Laminate and Exploded Detail.

the entire seal to move freely and individually like cantilever beams, exhibiting compliance to the external forces acting on them. This compliance also allows a hydrodynamic sealing action, which is responsible for the elimination of the major source of wear (friction between rotor and seal) and thus portends a potentially unlimited lifespan.

Referring to Fig. 3, the outside, Do, and inside, Di, diameters of the seal are generally determined as a consequence of the seal application. The other geometric parameters in the seal design (the finger base diameter, Db; the foot upper diameter, Df; the stick arc radius, Rs; the circle of centers of the stick arcs radii, Dcc; and the repeat angle, α) are variable and directly determine the shape of the fingers within the seal. Values for these geometric parameters are given in Table 1. In addition to the basic geometry of the finger laminate, the low pressure laminate has a pad extending from it in the axial direction, Fig. 4. The general purpose of the seal pad is to provide a self-acting lifting surface that will generate a net small hydrodynamic sealing clearance, on the order of 0.0005- to 0.001-in, between the rotor and the finger pad. Historically, conventional finger seals were padless, were mounted on a rotor with a prescribed preload, and were meant to seal through direct contact with the rotor, much like brush seals.8,11,13 The finger pad lifting surfaces provide the non-contact sealing while eliminating the inherent wear associated with a conventional preload. One of the main features of the pad is its upper side wedge and fillet that is evident in Fig. 4. As presented and discussed by Marie16, the introduction of the upper wedge/fillet combination has been essential in the reduction of the out-of-plane twisting of the finger pad in both the radial and axial directions.

A working finger seal contains at least two finger laminates. This is so that one finger laminate can be oriented in such a way to the second finger laminate, specifically staggered, so as to cover the interstices between the individual fingers. Thus, when properly assembled, the high-pressure finger sticks are centered on the interstices between the low-pressure fingers, as can be seen in Fig. 1. In

Dcc

Rs

Rs Do

Db

Di Df

a

Figure 3. Geometry and Notations for the Finger Seal

Combined Upper Wedge & Fillet

Figure 4. Pad with Upper Side Axial Wedge and Fillet

Table 1. Geometric Values of the Basic Finger Finger Geometry Symbol Value

(in) Seal Outside Diameter Do 9.666 Seal Inside Diameter Di 8.500 Finger Base Diameter Db 9.169 Stick Arc Radius Rs 4.511 Circle of Centers of Arcs Dcc 1.575 Foot Upper Diameter Df 8.600 Finger Repeat Angle a 4.444°

American Institute of Aeronautics and Astronautics

4

this regard, high pressure fingers’ primary mission is to seal the axial leakage flow that otherwise would occur between the interstices of the low pressure fingers.

IV. Numerical Implementation for Solid-Fluid Interaction Analysis The thermofluid modeling of the finger seal was performed in the context of the investigation of the under-side

shape of the pad. The desired end configuration is that of a self-lifting surface that “rides” on a thin film of air during operation while the leakage rate and the temperature rise stay acceptably low. The commercial computational fluid dynamics package CFD-ACE+ was used to evaluate the under pad force generations as a function of fluid film geometry. This allowed the calculation of the fluid stiffness for a variety of fluid film shapes. The solid-fluid interaction feature of the CFD package was also used to predict under pad radial forces, overall finger lift, and fluid leakages as functions of high-side pressure and rotor angular speed. The algorithm in the main flow module of CFD-ACE+ solves the full, compressible Navier-Stokes equations in a Cartesian system of coordinates, with rotating boundary conditions at the shaft surface. The finite-volume approach employs a conjugate gradient algebraic solver with pre-conditioning. Its adoption is especially attractive due to the fact that allows a conservative formulation for the discretized governing equations. The governing equations are numerically integrated over each of these computational cells using a cell-centered variable arrangement, where all dependent variables and material properties are stored at the cell center. The continuity, momentum and energy governing equations are expressed in the form of the generalized transport conservation equation,

{

sourcediffusionconvectiontransient

SVt φφφρρφ

+∇Γ•∇=•∇+∂

∂4342143421

r

321)()()(

(1)

where the generic scalar quantity φ becomes in turn u, v and w for the continuity and momentum equations. Integrating Eq. (1) over a control-volume cell, one obtains the following conservative form.

∫∫∫∫ +∇Γ•∇=•∇+

∂∂

ϑφϑϑ

ϑ

ϑϑφϑφρϑρφ dSddVdt

)()()( r

(2)

Equation (2) is at the basis of the discretization formulation used by the CFD-ACE+ algorithm. CFD-ACE+ uses absolute convergence criteria, which for the pressure field usually requires convergence of the residual of the order of 1.0E-4. For the cases considered here the author used a 1.0E-6 convergence criterion for each primitive variable (u, v, w, and p). Application of the computational fluid dynamics analysis to the finger seal required the discretization of the fluid film into a grid comprised of 8-noded, 3 dimensional, hexahedral cells. As the nodes are positioned closer to each other, the volumes of the cells become smaller, the mesh becomes finer, and the error in the discrete results reduces. However, this scenario also increases computer processing time, sometimes prohibitively so. Thus, grid convergence simulations were conducted in order to determine an acceptable grid size that balances accuracy with practicality.

For grid convergence evaluation, the fluid film under one finger pad with a circumferential wedge was considered, Fig. 5(a). The circumferential lengths and the axial widths of the fluid film were the same as those used in the actual model simulations, as were the wedge taper ratio of k = 3 and the minimum fluid film height of ffmin=

ffmin

ffmin

3·ffmin

3·ffmin

HP Side

LP Side

Rotor Direction

r

a c

ffmin

2·ffmin

3·ffmin

3·ffmin

HP Side

LP Side

Rotor Direction

a c

r

(a) (b)

Figure 5. Pictorials of Under Pad Fluid Films, ffmin= 0.00025-in: (a) Single-Wedged (b) Double-Wedged

American Institute of Aeronautics and Astronautics

5

0.00025in. Four grids, summarized in Table 2, were considered for the grid convergence analysis. The models were run with a high side pressure equal to 25-psig and the low-side, the heel side, and the toe side pressures equal to 0-psig (high- to low-pressure differential Δp= 25-psi). The upper boundary, corresponding to the under side of the pad, was a stationary wall, while the lower boundary, corresponding to the rotor surface moved at u= 741-ft/s. This was equivalent to a rotor angular velocity of ω= 20,000-rpm.

The comparison of the grids to that of the “ultra-fine grid” was done in two ways: (1)the pressure contours of the four were compared and (2)output parameters from the numerical analysis of each were summarized for comparison: run time, iterations to convergence, mass flow out the LP side of the fluid film, the maximum pressure on the upper boundary, and the resultant axial force on the upper boundary. The pressure contours showed that there was virtually no difference in any of the four grids, both quantitatively in terms of pressure build-up and qualitatively in terms of pressure distribution under the pad. Additionally for comparison, Table 3 summarizes the output parameters. Inspection of this table revealed that the four grids chosen for comparison produced quite similar results, with even the “course grid” which offered results differing from the “ultra-fine grid” by less than 3%. Results using Grid 1 and Grid 2 differed from the “ultra-fine grid” by at most 1.54% and 0.81% respectively. However, Grid 1 decreased the computer processing time by over 95% when compared to the “ultra-fine grid” while Grid 2 only decreased the time by 11.5%. Thus, the characteristics of Grid 1 were used in all of the final models fluid zone areas, reference Table 2.

Table 2. Geometry and Grid Information for Grid Convergence Coarse (C) GRID 1 (G1) GRID 2 (G2) Ultra-Fine (UF)

Radial .00025-.00075 .00025-.00075 .00025-.00075 .00025-.00075Circumferencial 0.315 0.315 0.315 0.315

Axial 0.25 0.25 0.25 0.25Radial 5 5 5 10

Circumferencial 50 63 126 630Axial 25 50 100 500

Radial 0.0001 0.0001 0.0001 0.00005Circumferencial 0.0063 0.005 0.0025 0.0005

Axial 0.01 0.005 0.0025 0.0005

Average Aspect Ratio

Radial to Cicumferencial to

Axial1 to 63 to100 1 to 50 to 50 1 to 25 to 25 1 to 10 to 10

Geometry Dimensions

Number of Grid Points

Average Length b/t

Grid Points

Table 3. Numerical Analysis Comparison for Grid Convergence Coarse (C) GRID 1 (G1) GRID 2 (G2) Ultra-Fine (UF)

# of Nodes 6,250 15,750 63,000 150,000# of Faces 15,580 39,938 161,771 386,855# of Cells 4,704 12,152 49,500 118,604

Run Time (min) 0.14 2.05 38.27 43.22# of Iterations 81 220 551 613

Mass flow out LP side (kg/s)

3.97E-06 (2.27%)

4.02E-06 (0.98%)

4.05E-06 (0.29%)

4.06E-06 ---

Axial force on Pad (lbf)

1.077 (2.62%)

1.089 (1.54%)

1.097 (0.81%)

1.106 ---

Maximum Fluid Pressure (psig)

26.875 (0.43%)

26.919 (0.27%)

26.956 (0.13%)

26.992 ---

Final Gridding

Computer Usage

Results Comparison Numerical Value

(% Difference from UF model)

In this research, structural analysis was coupled with the flow module of CFD-ACE for an interdisciplinary analysis of the finger seal. The stress module used for the solid/fluid interaction analysis is a finite element based structural analysis module of CFD-ACE+. The stress module solves the structural mechanics equations, in finite element form, derived from the principle of virtual work. The global solver equation for linear and geometrically non-linear elasticity analysis is as follows.

{ } [ ] { }ggg FKU 1−=

(3) { } [ ] [ ] { } { }∑∑∑

===

===e

ege

ege

eg fFkKuU111

;;nnn

where represents all the nodal displacements for the entire structure, gU [ ]gK represents the global stiffness matrix, and represents all the nodal forces for the entire structure. { }gF

The two-way implicit coupling was accomplished with the aid of the grid deformation module. The grid deformation module of the CFD-ACE+ solver is necessary for moving/deforming grid problems. In this regard, pressures were sent to the stress module, where solid finger deformation and stresses were calculated. The geometry change required grid remeshing to be performed in the fluid region. The automatic remeshing option was chosen

American Institute of Aeronautics and Astronautics

6

such that the grid deformation module would automatically remesh the interiors of all structured grid fluid volumes whose boundaries were moving. The automatic remeshing feature used a standard transfinite interpolation scheme22 to determine the interior node distribution based on the motion of the boundary nodes. Then, these deformations were sent back to the flow module where the solution was recalculated on the newly deformed geometry. Information is continually transferred between the different modules as iterations were performed until convergence was obtained. For the two-way coupling to be successfully implemented, a one-to-one matching of the nodal points was required at the solid/fluid interfaces.

A. Solid – Fluid Model The solid-fluid model, Fig. 6, included both the low-pressure padded solid finger and the fluid film underneath it

in a full fluid-solid interaction simulation. The solid finger portion of the model consisted of a “straightened” finger which, for simplicity, removed the curve of the stick and the fillets and round on the foot and pad. For the upper fluid boundary of the solid-fluid interaction model, the curvature in the finger under pad was also removed. The models however, retained the desired taper ratios between the heel area and the toe area for both the circumferential wedge and the combined axial-circumferential wedge, Figs. 5(a) and (b), respectively. For the lower fluid boundary, the curvature in the rotor was also unwrapped and replaced with a straight surface. The fluid film grid generation in the solid finger–fluid film interaction model involved using a hexahedral structured grid for the fluid film that

embodied the same grid lengths and grid aspect ratios of Grid 1, Table 2. The 3-dimensional grid was obtained by an extrusion of the 2-dimensional radial-circumferential grid into the axial direction. The 2-dimensional grid was composed of quadrilateral elements, which upon extrusion became hexahedral elements.

The structural domain (finger stick and pad) of the solid–fluid model however, required much less grid resolution than that required for the fluid domain. The number of elements would have become excessive if the fine grid size of the fluid zone were used throughout the solid finger region. Therefore, it was decided to use unstructured prismatic elements in the solid in order to transition from the fine grid at the solid–fluid interface to a coarser grid away from the interface. Again, the 3-dimensional solid grid was obtained by an extrusion of the 2-dimensional radial-circumferential grid into the axial direction. For the solid region, however, the 2-dimensional grid was composed of triangular elements that shared the fluid zone nodes at the solid–fluid interface. Upon extrusion in the axial direction, the triangular elements became prismatic elements. One side of the prisms positioned at the solid–fluid

Rotor Direction

r

a c Circum. Inlet

Δp = 0-psig

Axial Inlet Δp = HP-psig

Axial Exit Δp = LP-psig

Circum. Exit Δp = 0-psig

Figure 6. CFD Model for Solid-Fluid Interaction Analysis Showing Gridding and Boundary Conditions

American Institute of Aeronautics and Astronautics

7

interface shared a side of the fluid hexahedrals located at this interface. The axial extrusion of the solid finger was achieved in two steps. The stick and the pad regions were both extruded 0.030-in first to account for the thickness of the stick. The pad region was then extruded an additional 0.22-in to account for the total 0.25-in axial length of the finger pad.

B. Boundary and Initial Conditions For the fluid flow analysis, three types of boundary conditions were used: (1) wall, (2) inlet, and (3) outlet. For

the solid finger structure analysis, three different boundary conditions were used: (4) free, (5) fixed, and (6) implicit pressure loaded. In addition to the boundary conditions used on the models, iteration starting conditions were specified in the fluid to help shorten the time to convergence: (7) pressure and fluid velocity initial conditions. The fluid-fluid and solid-solid interfaces present in this model were computationally contiguous and mass, momentum and energy were conserved.

1. Wall Boundary Conditions. The under pad fluid film in the solid-fluid interaction model was bound on the bottom by a wall representing the

rotor. This wall was given a linear velocity that corresponded to the tangential velocity of the rotor surface. The fluid film was bound on the top by the under pad surface of the finger solid model. This boundary was stationary while the flow module was active. It moved with the finger, however, when the stress module was active.

2. Inlet Boundary Conditions. In the solid–fluid interaction model, the axial high-pressure side of the fluid film was designated as an inlet, as

was the heel side of the under pad fluid film. The inlet boundary conditions included a fixed pressure and fixed temperature. The pressure at the high-pressure side boundary was parametrically varied in the numerical simulations. The pressure at the heel side inlet was set at atmospheric pressure (= 0-psig). The temperature at the inlet boundaries was set to room temperature (= 75°F). The solver used the appropriate Navier-Stokes and continuity relationships to calculate the fluid velocity magnitudes and directions.

3. Outlet Boundary Conditions. In the solid–fluid interaction model, the axial low-pressure side of the fluid film was designated as an outlet, as

was the toe side of the under pad fluid film. The outlet boundary condition included only a fixed pressure. All other properties were allowed to float as they may such that the flow developed to satisfy the combination of pressure and mass flow continuity conditions across the seal. A temperature was specified (room temperature = 75°F) to be used only in the event that there was actually inflow at the outlet boundary. The outlet boundaries were set to atmospheric pressure (= 0-psig).

4. Free Solid Boundary Condition. This represented the surfaces of the solid in the solid-fluid interaction model that were free of any type of

loading and any type of constraint. All but two surfaces of the solid finger portion of the model were considered free surfaces. The entire stick, with the exception of the very end, was a free surface. In addition, all of the pad surfaces, with the exception of the solid–fluid interface, were designated as free surfaces.

5. Fixed Solid Boundary Condition. The very end of the stick was constrained from moving in the radial, axial, and circumferential directions. The

boundary condition situated at this location made the finger a slanted cantilevered beam. This fixed constraint at the end of the stick represented a finger seal that was pinned from moving at a point approximately half-way up the finger, causing a mid-stick pivot point.

6. Implicit Pressure Loaded Boundary Condition. The solid fluid interface surface could also be loaded with pressures, forces, etc. or its motion could be

constrained. The implicit pressure load boundary condition placed on this surface allowed it to be pressure loaded with the fluid pressures at that surface when the solver iterated to the finite element analysis of the solid finger. Further this boundary condition updated the changed pressure load (because of the change in the fluid film properties) with each fluid solver to solid solver iteration.

7. Fluid Pressure and Velocity Iteration Starting Conditions. The pressure field given to the fluid to start the iterations was simply the average between the specified high

pressure and atmospheric pressure, Δp/2. Likewise the beginning fluid velocity condition in the circumferential direction was the average between the specified rotor boundary velocity and the pad under surface stationary velocity. The beginning fluid velocity condition in the axial direction varied with different combinations of rotor boundary speed and high-side pressure. As a rule of thumb, 100-ft/s was specified for models with the high-side pressure over 15-psi and 50-ft/s for models with the high-side pressure less than 15-psi.

American Institute of Aeronautics and Astronautics

8

V. Two Degree-of-Freedom Dynamics Model with Variable Fluid Stiffness The two degree-of freedom-model presented

here is an extension of the two degree-of-freedom model with Coulomb friction effects21 previously presented. This model assesses more realistically the interaction between the finger seal and the rotor as the mass of the rotor is subject to a forcing function that simulates the rotor orbit. Specifically, the variable nature of the fluid stiffness as a function of the gap between the rotor and the finger pad is incorporated. For each time step of the vibration analysis, the fluid film shape between the rotor mass and the finger seal mass is considered. The variable fluid stiffness associated with that shape is compared to the stiffness used. A fluid stiffness iterative scheme, with a 1.0E-4 convergence criterion, is employed to find the correct position of the finger seal mass for each time step.

5

4

3

2

1 kS5

kF5

cF5

mHP5

Rotor Mass

Annular Rim

Finger Stick

Fluid Film

Finger Mass

cS1

Stick-Pressure Dam Interface Friction

Figure 8. Mass-Spring-Damper Representation for 3 High- and 2 Low-Pressure Fingers for 2 DOF Model

1

3 4 5

2

1

2

3 4

5

Figure 7. Characteristic Finger Cell of 3 High- and 2 Low-Pressure Fingers Used for Finger Motion Analysis

As before, the analysis is applied to a “basic cell” of the finger seal instead of to just one high-pressure or one low-pressure finger. This was done because each finger’s motion, though compliant, is directly and sometimes significantly affected by the fingers adjacent to it and by the fingers behind or in front of it. Thus, it was decided to use a basic cell consisting of two low-pressure padded fingers, one high-pressure unpadded finger, and two high-pressure unpadded fingers each with half a foot, Fig. 7. A sketch of the mass-spring-damper model for the five fingers is shown in Fig. 8. The assumptions underlying this analysis are: (a) all five fingers of the model act in unison as one mass, (b) the curvature of the fingers is neglected, (c) the harmonic motion of the rotor and the response motion of the fingers’ mass act in the same direction, (d) the cross coupling stiffness and damping are negligible, (e) there is no relative motion between the low- and high-pressure fingers, and (f) relative motion and the associated Coulomb friction occurs only between the stationary back plate and the low-pressure finger sticks.

A. Mathematical Model for Coupled and Uncoupled Rotor-Finger Interaction The feet and pads of the finger seal cell as described above are

used to represent the finger seal mass. During coupled seal/rotor interaction, Fig. 9, this lumped finger mass is connected to another lumped mass that represents the portion of the rotor interacting with the fingers of the basic cell. These two masses are connected with a spring and damper representing the fluid stiffness and damping respectively. The stick is represented by a spring which was attached on one end to the finger mass and fixed on the other end, and its inertia effects were accounted for in the lumped mass. The fixed end of the stick spring represented the absence of vibratory motion in the annular rim. The Coulomb damping was modeled with an equivalent viscous damper in parallel with the stick spring during the coupled rotor/finger mass motion. ROTOR M ASS

FLUIDDAM PING

FINGERSTIFFNESS

FS M ASS x1

x2

FLUIDFNESS

F1

LOM BM PING

kSEqu

cFEqu kFEqu

mFSEqu

cSEqu

xR

xFS

mREqu

FAppl Figure 9. Equivalent Two DOF Dynamic Model During Seal/Rotor Interaction

During uncoupled seal/rotor interaction, that is when the rotor mass contracted or lowered radially at a faster rate than the free motion model of the finger, the rotor mass actually pulled away from the finger mass. Since there is no actual physical spring and

American Institute of Aeronautics and Astronautics

9

damper connection that pulled the finger back down with the rotor, the finger mass lowered somewhat independently of rotor mass motion. Thus during this instance the motion of the finger mass, including the effects of the Coulomb damping, was modeled as a free vibration system with an initial displacement.

x+FS MASS

FINGERSTIFFNESS

BACKPLATE

kSEqu

mFingerEqu

Figure 10. Seal Dynamic Model During Uncoupled Seal/Rotor Motion

The basic finger seal cell, Fig. 7, was also used for the uncoupled finger mass motion analysis, Fig. 10. The sticks of the finger were again represented by an equivalent spring which was attached on one end to the lumped finger mass and fixed on the other end. The Coulomb damping, however, was modeled as the Coulomb friction force that develops between the back plate and the finger during relative motion. The friction force was applied in the direction opposite the motion of the lumped finger mass.

B. Governing Equations 1. Coupled rotor mass/finger mass motion.

Referring to Fig. 9, if the lumped rotor and finger masses displacements are and , respectively, the equation of motion can be written in matrix form as

)(txR ( )txFS

(4) ⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡+−

−+

⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡+−

−+

⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

000 Appl

FS

R

SEquFEquFEqu

FEquFEqu

FS

R

SEquFEquFEqu

FEquFEqu

FS

R

FSEqu

REqu Fxx

kkkkk

xx

ccccc

xx

mm

&

&

&&

&&

Note that in Eq. (4), cSEqu which accounts for the Coulomb damping, is modeled as an equivalent viscous damper. Also, it is important to realize that the fluid stiffness, kFEqu, is variable and not constant. When an input harmonic forcing function of amplitude F0 and frequency ω is applied in the form of , the mechanical

impedance for the 2-DOF system can be described as

tieFtF ω0)( =

( ) jkjkjkjk kcimiZ ++−= ωωω 2 . As a result, Eq. (4) can be expressed as

( )[ ] 0FXiZrr

=ω (5)

where

( )[ ] ( ) ( )( ) ( )

( )( ) ( )( ) ( ) ( SEquFEquSEquFEquFSEqu

FEquFEqu

FEquFEquREqu

o

FS

R

kkccimiZ

kciiZiZ

kcimiZ

FF

XX

X

iZiZiZiZ

iZ

++++−=

−−==

++−=

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

=

⎥⎦

⎤⎢⎣

⎡=

ωωω

ωωω

ωωω

ωωωω

ω

222

2112

211

0

2221

1111

0;

vv

)

(6)

Solving Eq. (5), one obtains an expression for the amplitudes of the rotor and finger masses.

tiFS

tiR

eiZiZiZ

FiZtX

eiZiZiZ

FiZtX

ω

ω

ωωω

ωωωω

ω

)()()(

)()(

)()()(

)()(

2122211

012

2122211

022

−

−=

−=

(7)

American Institute of Aeronautics and Astronautics

10

2. Uncoupled rotor mass/finger mass motion. Referring to Fig. 10, this portion of the dynamic analysis considers the normal forces and the friction coefficient

constant. The finger mass equation of motion is given by fSEquFS Fxkxm ±=+&& . The negative sign is associated with the upward motion of the finger while the positive sign characterizes its downward motion. The total solution to this second order non-homogeneous equation is simply

( )nn

itiωπ

ωπ

≤≤−1

( ) ( ) ( )( )

SEqu

fin

SEqu

f

kF

tk

Fixtx 1

0 1cos12 −−+⎥⎥⎦

⎤

⎢⎢⎣

⎡−−= ω

(8)

The solution can be written for , where is the value of that causes finalii ...,,2,1= finali iSEqufi kFx ≤ . For this

model, is given by the following, where i is indicative of each successive ½ cycle. ix

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−=

SEqu

fii k

iFxx

21 0

(9)

C. Calculations of Finger and Fluid Dynamic Coefficients For both solutions of the rotor-finger motion, Eqs. (7) and (8), the equivalent finger assembly mass, the

equivalent spring coefficient for the stick, the equivalent stiffness and damping coefficients for the fluid, and the equivalent damping coefficient for the Coulomb friction were determined with the same definitions and methods as outlined for earlier generation dynamics models19,21. However, this work includes the evaluation of the equivalent stiffness of the stick as a function of geometric design. Additionally, the change in the stiffness of the fluid is analyzed as a function of the heel-to-toe fluid film ratio, k, for parametric changes in the fluid film thickness at the toe, ffmin. 1. Finger Stiffness as a function of Geometry

Referring to Fig. 3, the repeat angle (α), the stick arcs circle of centers (Dcc), the finger base diameter (Db) and the foot upper diameter (Df) are all varied from their original values, Table 1, and equivalent stiffness coefficients are found. The computational engine used for the stiffness evaluation of the finger was the commercial finite element software package ALGOR. Extensive information on the algorithm utilized in ALGOR as well as the assumptions and approximations of this analysis can be found in previous research by the author16.

To determine the equivalent stiffness coefficients of the finger sticks, a force was imposed on the seal foot and the resulting deflection was calculated and tabulated. Thus, the stiffness coefficient variation was determined by simply taking the applied force and dividing it by the corresponding tabulated displacement,

radial

appliedSEqu disp

Fk =

(10)

In order to accomplish this, a linear analysis was performed. The finger was fixed at the annular rim but otherwise allowed to move freely and a total force of 0.1-lbf was imposed on the foot at its centroid. The position of the force corresponded to the position of the lumped mass in the dynamic analysis. The repeat angle variation found the equivalent stick stiffness coefficients for designs with α = 4, 6, and 9°, which correspond to seals with 90, 60, and 40 fingers, respectively. The stick arc circle of centers variation found the stiffness coefficients for designs with Dcc = 2, 3, and 4in. The finger base diameter found the coefficients for designs with Db = 8.9, 9.1, and 9.3in, while the foot upper diameter found the coefficients for Df = 8.65, 8.75, 8.80in. 2. Fluid Film Stiffness as a function of Gap Size

Referring to Fig 5(a), the thickness of the fluid film under the toe (ffmin) as well as the ratio of the fluid film under the heel to the fluid film under the toe (k = ffmax/ffmin) are varied from the original values (ffmin = 0.00025 and k = 3) and the equivalent fluid stiffness coefficients are determined. For calculation of the equivalent fluid stiffness CFD-ACE+ was used. The boundary of the fluid corresponding to the rotor surface was moved up and down while the boundary corresponding to the pad under side was held stationary. The rotor surface was moved ±2.5% of the ffmin for each of the cases considered. The high-and low-pressure boundaries were kept constant at 25- and 0-psig, respectively. Previous research19 showed that for the same fluid film geometry the under pad forces changed with

American Institute of Aeronautics and Astronautics

11

increasing pressure, but the resulting change in the fluid stiffness coefficient was insignificant. The moving solid boundary was given a linear velocity of 741-ft/s, which corresponds to a rotor angular velocity of 20,000-rpm. The air was considered isothermal and incompressible since the calculated Mach numbers were smaller than 0.5. CFD-ACE+ integrates the nodal pressures over the area to find the equivalent forces exerted on the under pad from the fluid in the radial and circumferential directions. The direct dynamic stiffness coefficient of the fluid was found simply from the slopes of force versus ffmin graphs.

radial

radialFEqu x

Fk

ΔΔ

= (11)

The stiffness of the fluid was found for three fluid film radial heights at the toe area, ffmin= 0.00015-, 0.00025-, and 0.00035-in, with the wedge taper angle varying up to k=6.

VI. Results and Discussion

A. Solid – Fluid Interaction Results This section discusses the solid-fluid interaction model as it was used to compare two under pad configurations:

(a) a single wedge (SW) under pad with a circumferential wedge only and (b) a double wedge (DW) under pad with a combined circumferential and axial wedge. A rationale for the solid-fluid interaction model is first presented by comparing a compliant model with a rigid one of the same geometry. Then CFD-ACE+ was used to evaluate the under pad pressure generations as a function of both high-side pressure and rotor angular speed for the single and double wedge pads. Results are presented that predict under pad radial forces, overall finger lift, and fluid leakages as functions of high-side pressure and rotor angular speed. 1. Solid Fluid Interaction Rationalization

A comparison was made between the single wedge solid-fluid interaction model and the exact model minus the effect of solid-fluid (S-F) interaction. This was done to show the difference in fluid and finger performance results of the two. The models were run at a rotor speed of ω= 20,000-rpm with pressure differentials up to 25-psi. The pressure contours of the non-complaint models as compared to the S-F interaction models, Figs. 11(a)-(d), showed that both the magnitude and the contour shapes of the under pad pressure varied significantly between the two models. This lead to an over-estimation of the non-compliant models under pad forces, Fig. 12, which became slightly worse as the pressure differential increased. The leakage showed the opposite trend, with the non-deforming model underestimating the amount of seal leakage. This of course was because there was no lift of the finger in the non-compliant model. The finger lift changes the fluid film shape which in turn changes the pressure profile which again changes the finger lift. Thus its importance is obvious in finding the steady state response of the finger to pressure differential and rotor speed inputs.

11.6

2.9

5.8

8.7

0

14.5

(a1) (a2)

Pressure (psig)

ca

r

HP

LP Rotor

Direction

(a) (b)

20.3

2.9

8.7

0

14.5

26.1Pressure (psig)

(d1) (d2)

(c) (d)

Figure 11. Fluid Pressure Profiles at ω= 20,000-rpm for Increasing Pressure Differentials: (a) 0-psi, Non-Compliant, (b) 0-psi, S-F Interaction, (c) 25-psi, Non-Compliant, (d) 25-psi, S-F Interaction

2. Under Pad Shape Comparison For both the single and double wedge pads, the radial under pad force as a function of rotor speed, Fig. 13(a),

showed a slight increase in force as the rotor speed was increased. Also for any given rotor speed, the radial under pad force increased with increasing pressure differential. Graphed together, one can see that the radial under pad force was more dependent on rotor speed in the single wedge than the double wedge pad. For any constant value of high- to low-side pressure differential, the double wedge pad had greater under pad force at lower rotor speeds while the single wedge pad did at higher rotor speeds. The rotor speed at which the transition happened, however,

American Institute of Aeronautics and Astronautics

12

increased with increasing pressure differential. At a pressure differential of 5-psi, the single wedge had more under pad force at rotor speeds greater than 2,000-rpm. At 15-psi, the single wedge did not have more force than the double wedge until speeds greater than 7,500-rpm; at 25-psi not until speeds greater than 18,000-rpm. From these findings, one can conclude that the single wedge design reacted more to increases in rotor speed than did the double wedge design.

The maximum finger deflection as a function of rotor speed and pressure differential increased for both pad types, Fig. 13(b) with the double wedge again showing less dependency on the rotor speed. For any given pressure differential the finger lift was greater in the double wedge at lower rotor speeds and greater in the single wedge at higher rotor speeds. The speed at which the transition occurred increased for increasing pressure differential. In fact, it occurred at the same speed as the transition speed for the radial under pad force. The leakage as a function of rotor speed, Fig. 13(c), followed in line with the radial under pad force and the maximum finger deflection, with the exception that the transition speed was at a greater rpm. For a pressure differential of 25-rpm, the leakage for the double wedge pad was greater than that of the single wedge pad all the way up to a rotor speed of 30,000-rpm where the leakage for the two was about equal. This meant that there was a range of speeds where the single wedge exhibited more radial under pad force and lift yet had less leakage than the double wedge.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 5 10 15 20 25High Side Pressure (psi)

Radi

al U

nder

Pad

For

ce (l

bf) Non-deforming Model

S-F Interaction Model

Figure 12. Radial Under Pad Force as a Function of Rotor Speed for Non-Compliant and S-F Interaction Models

B. Dynamic Simulation Results This section presents the finger seal

dynamic response through the 2 degree-of-freedom model that was introduced above. A rationale for using a variable fluid stiffness is presented. A prescribed motion is imposed on the rotor that interacts with the finger seal through the fluid film separating them; the resulting interaction between the rotor and the finger is then followed in time. This simulation serves to determine the finger seal operational response characteristics, and uses the finger mass and stiffness, the Coulomb friction and the fluid damping and variable stiffness as parameters.

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 30Rotor Speed (x1000 rpm)

Radi

al U

nder

Pad

For

ce (l

bf)

SW, 5 psi SW, 15 psi SW, 25 psi

DW, 5 psi DW, 15 psi DW, 25 psi

(a)

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

0 5 10 15 20 25 30Rotor Speed (x1000 rpm)

Max

Fin

ger

Pad

Def

lect

ion

(in)

SW, 5 psi SW, 15 psi SW, 25 psiDW, 5 psi DW, 15 psi DW, 25 psi

(b)

0.0E+00

1.0E-06

2.0E-06

3.0E-06

4.0E-06

5.0E-06

6.0E-06

7.0E-06

0 5 10 15 20 25 30Rotor Speed (x1000 rpm)

Mas

s Fl

ow O

ut L

P Si

de (k

g/s)

SW, 5 psi SW, 15 psi SW, 25 psi

DW, 5 psi DW, 15 psi DW, 25 psi

(c)

Figure 13. (a) Radial Under Pad Force, (b) Maximum Finger Pad Deflection, (c) Leakage; as a Function of Rotor Speed

1. Equivalent Finger Stiffness Coefficients As was previously stated, four geometric

parameters of the finger seal were individually

American Institute of Aeronautics and Astronautics

13

varied while the other parameters were kept constant at the original values as summarized in Table 1. Stiffness coefficients were determined from the finite element analysis and plotted as a function of the varied parameter, Fig.(a)-(d). A power equation trend line was added to the data in EXCEL with good correlation. This trend line option calculates the least squares fit through the points using the equation . The purpose of the graphs was to see the sensitivity of the finger stiffness to a change in a geometric parameter. Thus, there is no need to examine the actual power equation.

bcxy =

It is the repeat angle, α, of the finger stick arcs that determines how many individual fingers will be in the total seal. One simply can divide the full circle angle, 360°, by the desired number of fingers to calculate the correct repeat. Thus as the repeat angle increases, the number of individual fingers decreases. This means that each finger stick must be thicker16. As one would expect, the thicker finger causes an increase in the equivalent finger stiffness coefficient, Fig. 14(a). The figure shows the three stiffness coefficients values that were calculated, α = 4 , 6, and 9°. Even at a repeat angle of 9°, which corresponds to a 40 finger seal, the finger stiffness is still sufficiently low to ensure seal compliance16.

When the diameter of the circle of centers, Dcc, of the finger stick arcs is increased while all other dimensions remain the same, the sticks become thicker and point more directly toward the center of the seal. In fact, the upper limit of Dcc is at a diameter in which the finger sticks become posts pointing radially down toward the geometric center of the seal. Conversely, when the circle of centers is decreased, the sticks narrow and become more concentric to the inside diameter of the seal. The lower limit of Dcc, then, is at the diameter in which the finger sticks lengthen, narrow and ultimately collapsed into one arc. An increase in the circle of centers causes a dramatic

0

20

40

60

80

100

120

0 5 10

0

800

1600

2400

3200

4000

0 2 4

Stick Arc Circle of Centers (in)Fi

nger

Stif

fnes

s (lb

f/in) FEA Calculated

Finger Seal Repeat Angle (deg)

Fing

er S

tiffn

ess

(lbf/i

n) FEA Calculated

Power (FEACalculated)

Power (FEACalculated)

(a) (b)

0

10

20

30

40

50

60

8.8 9 9.2 9.4Finger Base Diameter (in)

Fing

er S

tiffn

ess

(lbf/i

n) FEA Calculated

Power (FEACalculated)

0

5

10

15

20

25

30

35

8.6 8.65 8.7 8.75 8.8 8.85

Foot Upper Diameter (in)

Fing

er S

tiffn

ess

(lbf/i

n) FEA Calculated

Power (FEACalculated)

(c) (d)

Figure 14. Equivalent Finger Stiffness Coefficients as a function of Finger Geometry: (a) Repeat Angle, α, (b) Stick Arc Circle of Centers, Dcc, (c) Finger Base Diameter, Db, and (d) Foot Upper Diameter, Df

American Institute of Aeronautics and Astronautics

14

increase in the equivalent stiffness coefficient, Fig. 14(b). In fact, for this seal, there is a very small acceptable range in Dcc that keeps the finger stiffness sufficiently low for compliance.

When the finger base diameter, Db, changes, the finger stick length changes. In fact, this variation only alters the geometric length of the stick; as the diameter increases, so does the length. Further, as the length increases, the equivalent stiffness coefficient decreases, Fig. 14(c). Thus, once all other geometric finger parameters are decided upon, one can use the finger base diameter to tweak the finger stiffness to the optimum.

Increasing the upper foot diameter, Df shortens the effective stick length, which in turn causes only a slight increase in the stiffness coefficient, Fig. 14(d). However, it also dramatically increases the mass of the foot. One must keep in mind that altering the mass has an impact on the finger dynamics. An increase or decrease in mass may also be desirable to achieve optimum dynamic performance; however, this will not be discussed currently. 2. Variable Fluid Stiffness Coefficient

The equivalent fluid stiffness coefficient is determined for a number of fluid film shapes so as to determine the level of dependence of the stiffness to changes in the relative position of the finger pad to the rotor. The stiffness of the fluid is also dependent on the angular speed of the rotor; however, this is not considered in the current work. The dynamic simulation looks at the steady state case with the rotor turning at 20,000 rpm. The fluid stiffness as a function of the heel-to-toe wedge taper ratio has a very similar shape for all the minimum fluid film heights considered, with the major variation occurring in the magnitude, Fig. 15(a). This can be seen more clearly by considering the fluid stiffness as a function of the minimum fluid film height for increasing taper ratios, Fig. 15(b). Past dynamic analyses19,21 has considered the equilibrium shape of fluid film to be ffmin = 0.00025in with k=3 (ffmax = 0.00075in). This was the shape of the fluid film about which the dynamic simulation occurred and was therefore the shape used for the determination of the previous constant equivalent fluid stiffness, kFEqu. The current work will consider the equilibrium shape of the fluid to have the same ffmin, but with k=2.5. Further, a simplifying assumption will be made that considers that the fluid film will retain this ratio during the dynamic simulation with the only change occurring in the minimum fluid film height. Simply stated as the relative position, of the finger pads with respect to the rotor changes, the film height will change but the film wedge ratio will remain a constant k=2.5. This approximation works well because the stiffness changes very little between taper ratios of 2 and 3, for any given minimum fluid film height, Fig. 15(b). The power trend lines (with R2 values > 0.9998) produced the following equation for stiffness as a function of minimum fluid film.

(12) 058.3

minmin

863.2minmin

))(084()(:3k

))(072()(:2k−

−

−==

−==

ffEffk

ffEffk

FEqu

FEqu

0

5000

10000

15000

20000

25000

0 1 2 3 4 5 6k (ffmax/ffmin)

Flui

d St

iffne

ss (l

bf/in

)

7

ffmin=0.00015

ffmin=0.00025

ffmin=0.00035

(a)

0

5000

10000

15000

20000

25000

0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004

ffmin (in)

Flui

d St

k=2

ess

(lbf/i

n) k=3k=4k=5k=6

iffn

Pow er (k=3)Pow er (k=2)

(b)

Figure 15. Equivalent Fluid Stiffness Coefficient as a function of (a) wedge taper ratio for increasing values of minimum fluid film, (b) the minimum fluid film for increasing values of taper ratio.

American Institute of Aeronautics and Astronautics

15

Since the dynamic equations of motion for the rotor and the finger mass, Eqs. 7 and 8, are from their equilibrium positions, the minimum fluid film height at any given time is determined as follows.

00025.0)(min +−= RFS xxff (13)

3. Rotor and Finger Motion Simulation The full cycle dynamic

simulation showing the relationship between the motion of the rotor mass and the finger mass was explored with the rotor speed held constant at 20,000-rpm, while the stiffness of the stick, kSEqu, was varied from 11.72- to 70.32-lbf/in. The net pressure drop through the equalization holes was considered constant at 5.0psi. The dynamic values used in the simulation are summarized in Tables 4 and 5. This simulation was run for both constant fluid stiffness (6832 lbf/in) and variable fluid stiffness (~1450-21000 lbf/in), Figs. 16(a) and (b), respectively. Note that the results show the time = 0.0-s when the rotor is crossing its equilibrium position and beginning a new cycle. This was arbitrary since the simulation looks at the steady state response, and was chosen so that the period of excitation and thus the frequency could be easily determined.

Both graphs exhibited many similar trends in the motion of the finger mass. First, at an angular input speed of 20,000 rpm, the frequency response of the rotor is such that it is faster than the natural frequency response of the finger mass. Thus the fingers do not “keep up” with the rotor as it pulls away from them. In fact, the rotor begins its upward motion and thus forcing function

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

0 0.0004 0.0008 0.0012 0.0016 0.002 0.0024 0.0028 0.0032 0.0036

Time (s)

Disp

lace

men

t fro

m E

quili

briu

m (i

n)Rotor

FS k=11.72 lbf/in

FS k=17.58 lbf/in

FS k=23.44 lbf/in

FS k=35.16 lbf/in

FS k=46.88 lbf/in

FS k=58.60 lbf/in

FS k=70.32 lbf/in

(a)

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

0 0.0004 0.0008 0.0012 0.0016 0.002 0.0024 0.0028 0.0032 0.0036

Time (s)

Disp

lace

men

t fro

m E

quili

briu

m (i

n)

Rotor

FS k=11.72 lbf/in

FS k=17.58 lbf/in

FS k=23.44 lbf/in

FS k=35.16 lbf/in

FS k=46.88 lbf/in

FS k=58.60 lbf/in

FS k=70.32 lbf/in

(b) Figure 16. Rotor and Finger Response for Increasing Values of Stick Stiffness, k with = 5-psi: (a) Constant Fluid Stiffness, (b) Variable Fluid Stiffness

SEqu netP

Table 4. Coefficient and Coulomb Friction Values for the 2 DOF Coupled/Uncoupled Dynamic Response

Mass (lbm)Stick

Stiffness (lbf/in)

Stick Damping (lbf·s/in)

Fluid Stiffness (lbf/in)

Fluid Damping (lbf·s/in)

3.711 x 10-3 Varies 0.114 Varies 0.15

mPnet

(psi)Acontact

(in2)Ff

(lbf)

0.3 5 0.0213 0.032

American Institute of Aeronautics and Astronautics

16

interaction on the fingers as they are still returning downward to equilibrium. This is indicative of the situation in which the air gap between the fingers and the rotor is larger than the initial gap. Also, both simulations showed that finger stiffness values above 58.60-lbf/in do not work favorably with the rotor motion. Under the given simulation conditions, there is too much potential energy in the finger stick that is converted to kinetic energy, and “chatter” of the finger mass is seen. Finally, both simulations show that the lowest stick stiffness values also do not work favorably with the rotor because of the sluggish motion of the finger mass compared to the rotor mass.

There are, however, key differences in the two simulations. The most notable difference occurs when the rotor mass is moving radially outward and consequently squeezing the fluid film. The constant fluid stiffness simulation shows the rotor and finger masses virtually touching. In reality, as the fluid is squeezed, its stiffness coefficient value increases significantly, Figs. 15(a) and (b) and Eq. 12. Also, as the ratio of fluid stiffness to finger stiffness increases, the displacement transmissibility tends to one for the given amount of Coulomb and fluid damping16. Thus the variable stiffness simulation showing a continuous gap between the finger mass and the rotor mass is more realistic. Another difference occurs when the rotor has reached its peak position radially. The constant fluid stiffness simulation shows an enlarged gap between the two masses. This again is not realistic, because there would not be enough stiffness in the fluid at that film height to sustain the finger mass distance from the rotor. In the variable stiffness simulation, the gap remains fairly constant although the peak of the finger mass is still slightly delayed due to the Coulomb damping in the finger. Finally, as the rotor mass pulls away from the finger mass, the finger mass descends back at a slightly slower rate in the variable stiffness simulation than in the constant stiffness simulation. This is due to the fact that the finger mass did not have as large of an initial displacement from equilibrium before the uncoupled portion of the dynamic response.

VII. Conclusion The single and double wedge pad analysis was conducted to compare the two configurations in terms of lifting

capabilities. The single wedge pad generates more radial force and finger lift at higher rotor speeds, especially when the pressure differential is low, than does the double wedge pad. The double wedge pad generates more radial force and finger lift at higher pressure differentials when the rotor speed is low. The double wedge is designed to increase the hydrostatic lift of the single wedge at any condition. However, with this comes greater leakage that also increases at a faster rate than the radial force or the finger lift. So with the double wedge pad, there is a concern of not enough finger lift at low pressure differentials (because of reduced hydrodynamic lifting capabilities) and too much finger lift and leakage at high pressure differentials (because of enhanced hydrostatic lifting capabilities). In conclusion, the single wedge pad configuration exhibits more than adequate lift from the hydrostatic pressure differential, while still utilizing lift from the hydrodynamic rotor motion when needed.

The change in the stiffness of the finger seal as a function of various geometric parameters is significant in that it allows the designer several options in the optimization of the finger seal design. The rotor finger motion simulation showed that at a steady state operating speed of 20,000 rpm, with a 5psi pressure drop in the equalization chamber, there exists a compromise range of stiffness values, around 46.88-lbf/in that work reasonably well. However, this optimum stiffness value would change if the speed of the rotor or the equalization pressure drop were to change. Thus the seal designer must know the usual steady state operating conditions as well as the characteristics of the equalization chamber (i.e. the pressure drop going into the chamber) before deciding on optimal finger stiffness.

The comparison of the constant and variable fluid stiffness dynamic simulations revealed that the same conclusions concerning the optimum finger stiffness would have been derived from either. However, the notable inaccuracies in the constant stiffness simulation, that is the too small and too large of gaps between the finger and the rotor (where the fluid stiffness would become very large and very small, respectively), were alleviated with the variable stiffness simulation. Also, the variable simulation showed that there existed a slightly larger gap between the finger and rotor mass when the rotor pulled away from the fingers.

Finally, it is the author’s belief that the differences between the constant and variable fluid stiffness dynamic simulation would become more significant for certain different seal geometric designs as well as for different seal operating conditions. Thus the reader is cautioned in regard to concluding that a constant fluid stiffness simulation is sufficient. Further investigation into the importance of the variable simulations would include running a comparison for a variety of seal geometries, which include changes in all the dynamic parameters. Further work for an even more accurate dynamic model could find both the fluid stiffness and damping as a function of both rotor speed and minimum fluid film height. This would allow for simulations to characterize finger seal dynamics during ramp up, steady state, and coast down.

American Institute of Aeronautics and Astronautics

17

References 1Chupp, R.E. and Nelson, P., “Evaluation of Brush Seals for Limited Life Engines,” AIAA /SAE /ASME /ASEE 26th Joint

Propulsion Conference, AIAA 90-2140, Orlando, Florida, 1990. 2Lattime, S.B., Braun, M.J., and Choy, F.K., “Rotating Brush Seal,” International Journal of Rotating Machinery, Vol. 7, No.

2, 2002. 3Lattime, S.B., “A Hybrid Floating Brush Seal (HFBS) For Improved Sealing And Wear Performance In Turbomachinery

Applications,” Ph.D. Dissertation, Dept. of Mechanical Engineering, University Of Akron, Akron, OH, 2001. 4Heydrich, H., “Bi-directional Finger Seal,” U.S. Patent No. 5,031,922, July 1991. 5Mackay, C.G., “Laminated Finger Seal,” U.S. Patent No. 5,042,823, August, 1991. 6Mackay, C.G., “Laminated Finger Seal,” Patent 5,071,138, December, 1991. 7Johnson, M.C., “Laminated Finger Seal with Logarithmic Curvature,” U.S. Patent No. 5,108,116, April, 1992. 8Hendricks, R.C., O’Halloran, B., Arora, G.K., Addy, H.E., Flowers, J., Carlile, J., and Steinetz, B.M., “Advances in Contact

Sealing,” NASA CP-3282, Vol. 1, 1994, pp.363-371. 9Braun, M.J., Hendricks, R.C., and Canacci V., “Non-Intrusive Qualitative and Quantitative Flow Characterization and Bulk

Flow Model for Brush Seals,” Proceeding of Japan International Tribology Conference, Vol. III, Nagoya, Japan, 1990, pp. 1611-1616.

10Hendricks, R.C., Schlumberger, J., Braun, M.J., Choy, F.K., and Mullen, R.L., “A Bulk Flow Model of a Brush Seal System,” Proceedings of ASME International Gas Turbine and Aeroengine Congress and Exposition, Paper No. 91-GT-325, Orlando, Florida, June 1991.

11Arora, G.K., Proctor, M.P., and Steinetz, B.M., “Pressure Balanced, Low Hysteresis Finger Seal Test Results,” NASA TM 1999-209191, 1991.

12Arora, G.K. and Glick, D.L. “Pressure Balanced Finger Seal,” U.S, Patent No. 6,196,550, March, 2001. 13Proctor, M.P., Kumar, A., and Delgado, I.R., “High Speed, High-Temperature Finger Seal Test Results,” Proceedings of

AIAA/ASME/SAE/ASEE 38th Joint Propulsion Conference, AIAA-2002-3793, Indianapolis, IN, July 2002. 14 Arora, G.K. “Noncontacting Finger Seal with Hydrodynamic Foot Portion,” U.S. Patent No. 5,755,445, May 1998. 15Proctor, M.P. and Steinetz, B.M., “Noncontacting Finger Seal”, U.S. Patent No. 6,811,154, November, 2004. 16Marie, Hazel, “A Study of Non-contacting Passive Adaptive Turbine Finger Seal Performance,” Ph.D. Dissertation, Dept.

of Mechanical Engineering, University of Akron, Akron, OH, 2005. 17Braun, M.J., Kudriavtsev, V.V., Steinetz, B.M., and Proctor, M.P., “Two- and Three-Dimensional Numerical Experiments

Representing Two Limiting Cases of an In-line Pair of Finger Seal Components,” Proceeding of the 9th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, Honolulu, HI February 2002.

18Braun, M.J., Pierson, H.M., and Kudriavtsev, V.V., “Finger Seal Solid Modeling Design and Some Solid/Fluid Interaction Considerations,” Tribology Transactions, Vol. 46, No. 3, 2003, pp. 566-575.

19Braun, M.J., Pierson, H.M., Deng, D., Choy, F.K, Proctor, M..P., and Steinetz, B.M., “Structural And Dynamic Considerations Towards The Design Of A Padded Finger Seal,” Proceedings of AIAA/ASME/SAE/ASEE 39th Joint Propulsion Conference, AIAA 2003-4698, Huntsville, Alabama, July 2003.

20Braun, M.J., Deng, D.F., Pierson, H.M., Proctor, M.P., and Steinetz, B.M., “A Three Dimensional Thermofluid Analysis And Simulation Of Flow, Temperature And Pressure Patterns In A Passive-Adaptive Compliant Finger Seal,” Proceedings of 10th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery [CD-ROM], Paper 126, Honolulu, HI, March 2004.

21Braun, M.J., Pierson, H.M., and Deng, D., “Thermofluids Considerations and the Dynamic Behavior of a Finger Seal Assembly,” Tribology Transactions, (to be published).

22Walker, Marshall. “Current Experience with Transfinite Interpolation,” Computer Aided Geometric Design, Vol. 16, 1999, pp. 77-83.

American Institute of Aeronautics and Astronautics

18