## Abstract

This paper presents an update of the model derived by Corral and Vega (2018, “Conceptual Flutter Analysis of Labyrinth Seal Using Analytical Models. Part I—Theoretical Support,” *ASME J. Turbomach., 140*(12), p. 121006) for labyrinth seal flutter stability, providing a method of accounting for the effect of dissimilar gaps. The original Corral and Vega (CV) model was intended as a conceptual model for understanding the effect of different geometric parameters on the seal stability comprehensively, providing qualitative trends for seal flutter stability. However, the quantitative evaluation of seal flutter and the comparison of the CV model with detailed unsteady numerical simulations or experimental data require including additional physics. The kinetic energy generated in the inlet gap is not dissipated entirely in the inter-fin cavity of straight-through labyrinth seals, and part is recovered in the downstream knife. This mechanism needs to be retained in the seal flutter model. It is concluded that when the theoretical gaps are identical, the impact of the recovery factor on the seal stability can be high. The sensitivity of the seal stability to large changes in the outlet to inlet gap ratio is high as well. It is concluded that fin variations due to rubbing or wearing inducing inlet gaps more open than the exit gaps lead to an additional loss of stability concerning the case of identical gaps. The agreement between the updated model and 3D linearized Navier–Stokes simulations is excellent when the model is informed with data coming from steady Reynolds-averaged Navier–Stokes simulations of the seal.

## 1 Introduction

Labyrinth seals are the most commonly used seal type in both gas and steam turbines due to their reliability to control leakage flows. They are made of non-contacting components consisting of a series of cavities connected by small clearances. The fluid is repeatedly forced to pass through these gaps generating kinetic energy that is dissipated in the downstream inter-fin cavity. This process creates pressure losses that reduce the leakage flow through the seal.

Even though most of the studies regarding labyrinth seals are focused on sealing effectiveness (see Ref. [1] for a thorough review), it has been shown that seals are also a source of aeroelastic instabilities [2–4], though most of the seal failures are never disclosed by engine makers. Experimental analysis and post-mortem observations [5] show that seals are prone to flutter, and they represent a critical component of modern aero-engines. Ehrich [6] was one of the first authors to highlight the importance of fin clearance on seal stability. He proposed a simple analytical model neglecting the circumferential variations for a single seal cavity accounting for the effect of the geometry and the torsion center (TC) location concerning the seal center. Also, Lewis et al. [5] described the high sensitivity of seal stability to the knife-edge clearance, in particular to the most upstream one. Abbot [7] introduced the concept of acoustic circumferential resonances within the seal cavity giving rise to the well-known Abbot’s criterion. However, he did not provide any theoretical model. One decade ago, Mare et al. [8] introduced the systematic use of numerical simulations for the analysis of vibrating labyrinth seals.

Recently, a new comprehensive physical-based seal flutter model has been proposed by Corral and Vega [9]. The Corral and Vega (CV) model reconciliates the classical stability criteria of Ehrich [6] and Abbot [7] and provides more accurate and generic stability limits, including new dimensionless parameters. The model was further extended to stepped-seals [10] and applied to tip-shroud seals [11]. More recently, the formulation was updated, including the effect of non-isentropic unsteady perturbations and verified partially using CFD [12,13]. Simultaneously, Miura and Sakai [14] have released a full set of experimental data obtained in a rotating rig.

All these seal flutter models and studies assume that the clearance of each fin is constant, but modern preliminary CFD studies [13] have shown that the effect of dissimilar gaps in seal stability can be of paramount relevance. Usually, labyrinth seals are designed to operate with equal nominal closures. However, due to the difficulties in controlling seal closures because of the disk and blade thermal excursions, in practice, the geometric closure of the different seal knives is never the same due to its tilting and leaning during operation. The demands for even higher efficiencies and performances in modern engines have led to more complex seal designs with very tight clearances, and therefore their relative displacements during operation with respect to the nominal gaps are currently higher than in the past.

Moreover, even under the assumption of geometrically equal gaps, the effective or fluid dynamic gaps of the seal are never exactly the same because either the effective areas, i.e., the discharge coefficients, or the fraction of the wall-jet kinetic energy that is not dissipated in the inter-fin cavity and is carried-over to the downstream fin are different from knife to knife. This latter effect can also be translated into an equivalent effective area as well. In this context, a qualitatively correct leakage model for each fin is essential.

The original CV model already accounted for the effect of dissimilar gaps [15] but the implications on seal stability were never discussed.

This work first presents a method to estimate the effective flow passage area of the seal. The effective clearance of each gap is included in a new version of the CV flutter model. Then, the impact of dissimilar effective gaps on the seal stability is studied, comparing the prediction of the new formulation with the case of nominally identical gaps. The robustness of the seal to small perturbations of the gaps is discussed, and several conclusions are drawn. Finally, the updated model is compared to 3D linearized Navier–Stokes simulations.

## 2 Leakage Flow Model

*i*th fin of the inter-fin cavity of the seal, assuming that the flow is isentropic and adiabatic, can be written as

*π*

_{i}=

*P*

_{0,i}/

*p*

_{i}is the pressure ratio across the fin. The baseline CV model [9] makes the hypothesis that the kinetic energy of the incoming jet is fully dissipated in the cavity, the velocity in the inter-fin cavity is null, and then the total pressure of the cavity equals the static pressure. This hypothesis is used to compute the mass flow through the downstream fin of the cavity. Nevertheless, in practice, for straight-through seal configurations, there is a significant amount of the wall-jet kinetic energy that is carried-over to the downstream gap. This is especially true when the distance between two consecutive teeth is short and the inter-fin cavity cannot mix out the jet. The so-called kinetic energy carry-over coefficient,

*χ*, measures the cavity capacity to dissipate the jet kinetic energy. For stepped or staggered configurations, the wall-jet is not aligned with the downstream gap, therefore the assumption that the kinetic energy of the jet is fully dissipated is fulfilled frequently.

Moreover, the throttling process through each seal fin leads to a contraction of the fluid, as it can be seen in Fig. 1. As a consequence, the effective passage area of the flow through the fin is smaller than the geometric seal area.

### 2.1 Kinetic Energy Carry-Over Coefficient.

*χ*. He assumed that the jet flow expanded conically into the cavity, departing from the upstream tooth, with a small angle,

*β*(see Fig. 2(a)). The rationale of the idea is that the longer the dimensionless distance from the upstream to the downstream knife,

*H*/

*L*, the higher the dissipation of the wall-jet and, the lesser energy is recovered. The angle

*β*is the spreading rate of the wall-jet what is directly related to the dissipated energy. Hodkinson [16] related the fraction of the dynamic head which is carried-over to the downstream tooth of a seal with identical gaps with the geometry and

*β*as

*H*is the nominal seal clearance. In the case of a seal with dissimilar gaps, the proper choice of

*H*in the expression (2) is the inlet gap of the cavity,

*H*=

*H*

_{1}. The main idea is that the sensitivity of boundary layer type flows to downstream information is negligible because the problem is parabolic. Therefore, it is deemed that

*χ*does not depend on the downstream gap,

*H*

_{2}. The effective total pressure of the cavity,

*P*

_{c0}, seen by the downstream fin is

*χ*= 0, the effective total pressure of the inter-fin cavity coincides with the static pressure of the cavity whereas if

*χ*= 1, the effective total pressure driving the jet flow through the downstream fin is

*P*

_{0}. Figure 2(b) compares the carry-over coefficient obtained with a Reynolds-averaged Navier–Stokes (RANS) analysis [17] using a fair resolution for the gap. The CFD results were post-processed in two different manners. First,

*χ*was derived from the CFD using Eq. (3) where

*P*

_{c0}was obtained from the middle point of the downstream gap, which is outside of the boundary layer region (□ symbols in Fig. 2(b)). This form of deriving

*χ*from the CFD is more robust than determining the angle

*β*of the dividing line between the wall-jet and the cavity recirculating flow and using Eq. (2) (○ symbols in Fig. 2(b)). In this case, the angle

*β*= 0.033 rad is nearly constant for all the simulations. The agreement in the trend of

*χ*with

*L*/

*H*

_{1}is reasonably good considering that no especial effort has been made to tune the results. The offset between the curves is not relevant since is finally removed from the model when the

*χ*derived from the CFD is used. The dashed line in Fig. 2(b) is obtained using Eq. (2) and selecting

*β*=

*β*

_{opt}to reproduce the

*χ*obtained from proper post-processing of the CFD results (Eq. (3)). The dependence of

*χ*with the pressure ratio has been checked using CFD and is weak and in line with the works of other authors [18]. The trend of Eq. (2) is believed to be good enough for this work.

It will be seen that Eq. (2) is only used to derive the functional dependence of *P*_{c0} with the geometry. The actual value of *β* is removed from the formulation at the end, and only the carry-over coefficient derived from the CFD is used.

### 2.2 Discharge Coefficient.

*A*

_{f}, and the geometric area,

*A*

_{g}, or in the case of a straight labyrinth seal, the ratio between the fluid dynamic,

*H*

_{f}, and geometric gap,

*H*

_{g}(see Fig. 1)

_{g}.

The implicit assumption for the existence of the *C*_{d} is that the flow is uniform upstream. This is usually the case for the inlet fin since it ingests air coming from plenum-like conditions, but the situation is somewhat more complex for the outlet fin where a wall-jet impinges directly in the exit gap. To evaluate the *C*_{d}, the actual mass flowrate, $m\u02d9$, is extracted from either experiments or numerical simulations while the ideal mass flow, $m\u02d9id$, is calculated using Eq. (1) using an appropriate upstream total pressure.

The dependence of the leakage of a labyrinth seal upon the seal geometry and flow conditions was studied by Suryanarayanan and Morrison [18]. They provided a leakage correlation that was validated against prior experiments. The discharge coefficient resulted to be a function of the Reynolds number, the clearance to pitch ratio, and the clearance to fin thickness ratio, *C*_{d} = *C*_{d}(Re, *H*/*L*, *H*/*t*). Szymanski et al. [19] showed experimentally that the discharge coefficient of the seal depended as well on the pressure ratio.

*C*

_{d1}≠

*C*

_{d2}a seal with nominally equal clearances behaves in practice as a seal with different inlet and outlet gaps. It will be shown that the sensitivity of the seal stability to the gap ratio,

*H*

_{2,f}/

*H*

_{1,f}, can be high, hence its relevance.

### 2.3 Impact on the Steady-State Flow.

The governing equations and the general expression for the cavity pressure in the steady-state were presented in Ref. [9], and therefore they will not be repeated here for the sake of brevity. Only those parts which are modified by the carry-over coefficient and the discharge coefficients are highlighted here.

*P*

_{c0}, is given by Eq. (3), and the fluid dynamic areas are related to the nominal gaps by

*π*

_{s}=

*P*

_{0}/

*p*

_{c}, can be obtained

*π*

_{T}=

*P*

_{0}/

*p*

_{e}is the seal pressure ratio, defined as the ratio between the inlet total pressure and the static pressure at the exit of the seal, and

*π*

_{s0}=

*P*

_{0}/

*P*

_{c0}is the ratio between the upstream total pressure and the inter-fin cavity effective total pressure. Retaining the effective area of the gaps and the kinetic energy carried-over modifies the steady pressure of the inter-fin cavity when compared with the baseline prediction of the original CV model.

*P*

_{c0}/

*p*

_{e}≤ ((

*γ*+ 1)/2)

^{γ/(γ−1)}. For higher values of the cavity pressure ratio, the flow through the last seal fin is choked and the exit mass flow is constant. Therefore, for the choked condition, the following relationship is satisfied:

*χ*, causes that the critical pressure ratio of the seal,

*π*

_{s}*, is not universal anymore, contrary to what is described in the original CV model formulation. Figure 3 describes the correction to the static pressure ratio in the inter-fin cavity as a function of the kinetic energy fraction carried-over for a seal with the same fluid dynamic passage area at the inlet and the outlet (

*A*

_{1}=

*A*

_{2}). The critical total pressure ratio across the seal

*π*

_{T}* decreases

*χ*since part of the energy of the jet is recovered whereas the pressure ratio across the inlet fin,

*π*

_{s}*, increases.

It is important to recall that the steady-state description embedded in the CV mode is simple and not critical for its use. The mean flow is taken usually from a steady CFD simulation or a more sophisticated air system model of the seal.

## 3 Carry-Over Impact on the Linearized Unsteady Model

*V*

_{c}, which represents a section of the inter-fin cavity of a rotating seal (see Fig. 4) retaining the circumferential unsteady flow perturbations created by the seal vibration motion. The model assumes that the vibration amplitude of the seal is small enough, and therefore the flow and geometric variables can be decomposed into two parts: a steady or mean background flow, plus a small and periodic unsteady perturbation. Therefore, the static pressure, the cavity volume, and the seal gap area can be written as follows:

*χ*

_{s}defined in Eq. (2), the perturbation of the kinetic energy recovery factor can be written as follows:

*h*(

*π*

_{s}) and

*J*(

*π*

_{T}/

*π*

_{s0}) are known expressions of the pressure ratio:

*h*′ and

*ψ*are nondimensional functions defined as

*h*′, which was defined in previous works [9] as

*h*′ = 1 +

*h*, has been generalized here to retain the effect of the carry-over coefficient and the fact that the seal may not be choked necessarily. Finally, combining Eqs. (12) and (17), the mass conservation equation becomes

*h*′, and to a rescaling of the fluid dynamic passage area as follows:

*A*

_{1,s}* and

*A*

_{2,s}* are used.

From here on the formulation is identical to that described in Ref. [9]. Though all the ideas are compatible with a more complex and accurate higher-order model [13], it was decided to explain the implications of the carry-over coefficient and the differential gap on the baseline model to ease the discussion. Next, a short and quick rationale of the model derivation is included for the sake of completeness. Emphasis is put to highlight the presence of dissimilar gaps.

*r*from its center (see Fig. 4). The time variations of the inlet and outlet areas,

*A*

_{1,2}(

*t*,

*z*), and the control volume,

*V*

_{c,2D}(

*t*,

*z*), can be expressed in nondimensional form as

*L*and

*s*are the inter-fin cavity length and height, respectively.

The model considers the existence of different clearances for the inlet and outlet teeth, *H*_{1}* and *H*_{2}*, respectively. To avoid overloading of the nomenclature, we will use from now on *H*_{j} = *H*_{j}* in the understanding that the actual expressions are the ones given in Eq. (20). However, when the general formulation was introduced in Ref. [9], from the very beginning, it was assumed that both clearances were identical to simplify the formulation, reduce the number of dimensionless parameters, and ease the discussion of the new model. The effect of the existence of dissimilar gaps was reduced to a brief comment in Eq. (12) of the second part of the paper [15].

*θ*= Δ

*θ*sin

*τ*, and the forcing term on the right-hand side of Eq. (21) can be expressed as follows:

*H*

_{1,eff}, defined as

*r*/

*L*is large but positive and

*η*< 1, the effective distance to the TC can change sign. This effectively means that a seal supported physically in the low-pressure side (LPS) can behave in practice like a seal supported in the high-pressure side (HPS). Keeping this in mind, Eq. (21) becomes

*p*

_{c,s}, and therefore the model requires that

*ε*≪ 1.

*δ*=

*r*Δ

*θ*is the seal torsion displacement, and

*S*= 2

*πRL*is the surface of the seal land. All the parameters involved in the previous expression are sketched in Fig. 4. The dimensionless out-of-phase component of the unsteady pressure outlined in the original CV model is redefined using the new expression for the nondimensional seal clearance, $e~eff$, and the function

*h*′ as

The new definition of the seal nondimensional effective clearance or TC accounts for the contribution of the differential gapping on the stability of the seal. The difference can be due to either the geometry or the fluid dynamic behavior. The fluid dynamic difference between nominally identical inlet and outlet gaps has its origin in the inlet contraction due to the flow separation in the seal teeth (see Fig. 1) and in the partial dissipation of the kinetic energy in the inter-fin cavity which create an asymmetry between the inlet and outlet gaps. Next section analyses the impact of the gap ratio on the effective nondimensional clearance or TC $e~eff$.

## 4 Impact of Differential Gapping on the Effective Torsion Center

As it has already been mentioned, even if the geometrical gaps are identical, i.e., *H*_{1,g} = *H*_{2,g}, the fluid dynamic gaps, *H*_{1} and *H*_{2}, can be different because either the discharge coefficients of the upstream and downstream gaps are different, *C*_{d,1} ≠ *C*_{d,2}, or the carry-over coefficient is not zero (see Eq. (20)).

The *C*_{d} variation between the inlet and outlet fins can be about 5–10%, while the carry-over coefficient can be up to 20% or even larger, depending on the seal geometry. As a result, the variation of the geometric gap ratio, *η*_{g} = *H*_{2,g}/*H*_{1,g}, due to fluid dynamic effects can be up to 30% easily. In this work, it is considered that the analysis of the problem in the range of 0.5 < *η* < 2 suffices since when the gap ratio is either very small or very large, the seal behaves as a single-fin seal which is a completely different problem.

*η*. For

*η*= 1, $e~eff=e~$ and a straight line is obtained. However, if

*η*< 1 (line with circle symbols), the seal is supported in the LPS and the TC is located sufficiently far away (recall that $e~=\gamma Hr/(sL)$), $e~eff$ becomes negative if

Figure 6 displays the effective nondimensional seal clearance parameter, $e~eff$, as a function of the gap ratio, *η*, for four different characteristic scenarios defined by the position of the TC (Eq. (29)).

*r*/

*L*| > 0.5. In this case, $e~eff$ has a singularity when

*η** = (2

*r*/

*L*− 1)/(2

*r*/

*L*+ 1), and a change in its sign. Another way of looking at the problem is for given gap ratio, the dimensionless radius of the TC, (

*r*/

*L*)*, at which the effective TC or radius, $e~eff$, changes sign is:

*C*

_{d}variation, at $|r/L|*\u22737$ the behavior of the seal is opposite as expected. Figure 7 displays this critical radius as a function of

*η*. The asymptotic value of $e~eff$ when

*η*→ ∞ is

*η*are not very realistic in seal design, as it has been previously commented.

Another interesting observation is that when the TC is located at the exit of the seal cavity, *r*/*L* = 0.5, the effective dimensionless TC is independent of the gap ratio, i.e., $e~eff=e~$, and the sensitivity of the seal to dissimilar gaps is small.

On the other hand, if the TC is located at the inlet of the seal cavity, *r*/*L* = −0.5, the effective dimensionless TC is $e~eff=\u2212((\gamma H1)/2s)\eta $ and the effective TC is always negative, independently of *η*.

Finally, it must be said that an accurate quantification of the effective clearances is of major relevance since a wrong estimation of the outlet to inlet gap ratio can lead to misleading conclusions regarding seal stability.

### 4.1 Carry-Over Coefficient Impact on Gap Ratio.

The effect of the carry-over coefficient on the model gap ratio is evaluated next. The seal fins are assumed to have the same geometric closure, *H*_{1,g} = *H*_{2,g}, and the same discharge coefficients, *C*_{d1} = *C*_{d2}, and therefore, the fluid dynamic gaps are the same. However, the kinetic energy recovered in the downstream gap has an impact on the gaps of the model which is retained by using expressions (18) and (20). Even in this simplified case, the ratio between the effective gap ratio, *η*, and the geometric gap ratio, *η*_{g}, is a function of the kinetic carry-over coefficient and the pressure ratio, *η*/*η*_{g} = *η*(*χ*, *π*_{T}).

Figure 8 shows that the impact of *χ* on the gap ratio is always a slight reduction that peaks around *χ* = 0.5. Typical values of *χ* oscillate between 10% and 20% and reductions in the gap ratio of 5–10% are easily seen. The impact is larger at low-pressure ratios what is somehow surprising. Though this impact seems to be small, it will be shown that these small variations can induce significant changes in seal stability. It is worth mentioning that the choked curve corresponds to different pressure ratios since the choking condition depends on the recovery factor.

## 5 Stability Limit Correction

The stability analysis of the seal could be simply conducted estimating first the effective gap or TC, $e~eff$, as a function of the gap ratio, *η*, the dimensionless position of the TC, *r*/*L*, and the original TC parameter of any of the versions of the CV model [9,10,13] using the effective gaps formulated in Eq. (20). In the second place, use the new definition of *h*′ (see Eq. (18)) taking into account the seal pressure ratio and the carry-over coefficient. Finally, use the expression of the work-per-cycle of the models or their stability criteria with all the information in place.

However, following this process, little can be learned about the dependence of the seal stability trends with the design parameters. It can be anticipated that when the effective inlet and outlet gaps of the model are different for whatever reason, the understanding of the stability trends with the dimensionless parameters is much more complex because not only a new parameter, *η*, appears but the original parameter $e~=\gamma rH/(sL)$ is split into two, *r*/*L* and *γH*/*s*. This means that in practice the stability depends on two more dimensionless parameters and some of them are functions of the *C*_{d} and *χ* coefficients. Therefore, although the model is analytic and based on algebraic expressions, this simplicity does not translate directly in a clear understanding of the problem. In this section, we will convey just a summary of the most relevant conclusions.

### 5.1 Zero Nodal Diameter Stability Criterion.

*St*→ ∞ in the baseline model [9]. The dimensionless work-per-cycle for this particular case can be written as

*St*→ ∞. This expression is very similar to the one derived in Ref. [9], with the main difference that now the seal stability depends on two additional parameters, $e~effh\u2032=f(e~h\u2032,\eta ,s\xaf)$ (see Eq. (29)). This new dependence directly implies that the $W~cyc$ can change its sign depending on the gap ratio, as it can be seen in Fig. 9, which shows the stability map for the 0th ND as a function of the gap ratio,

*η*.

According to the CV model, when *η* = 1, the 0th ND of the seal is unstable only if $\u22121<e~h\u2032<0$. The new formulation reflects that if the clearances are different (*η* ≠ 1), there are two key parameters that control the seal stability namely the dimensionless gap, $s\xaf=s/(\gamma Hh\u2032)$, and the gap ratio, *η*. It has been decided to retain the original parameter $e~h\u2032$ of the CV model among the three controlling parameters for the sake of continuity with previous work, although it must be emphasized that this single parameter does not collapse all the dependences anymore.

*η*= 1), the stability criterion of the classical CV model [9] is recovered. The LPS is always stable whereas the HPS is stable if $e~h\u2032<\u22121$. Second, it can be noticed that if the exit gap is larger than the inlet one (

*η*> 1), the seal is more stable in the HPS but even for

*H*

_{2}≫

*H*

_{1}there is always a residual unstable region in the HPS. However, if the exit gap is smaller than the inlet one (

*η*< 1), the seal is less stable, and the LPS can become unstable. This trend is general and can be observed for other NDs as well. The kinetic energy carried-over to the downstream fin entails that

*η*<

*η*

_{g}, and hence a reduction of the stability margin.

It is important to notice that the dimensionless height, $s\xaf$, is always high for most of the seals since *s*/*H*_{1} is large necessarily. The general stability criterion for arbitrary values of $s\xaf$ depicted in Fig. 9 can then be simplified. Figure 10 shows the stability criterion for the 0th ND when $s\xaf\u226b1$. It can be seen that the stability condition reduces in this case to $\eta \u22731$ approximately. The range in which the problem is stable if for some reason the outlet gap is slightly smaller than the inlet one is very small. In other words, the 0th ND tends to be always unstable in practice, independently of the TC, though this instability tends to be weak.

### 5.2 General Stability Criterion.

*η*and $s\xaf.$

Figure 11 displays the stability regions as a function of the gap ratio *η* for clarity. Three cases are presented, two for *η* < 1 and one for *η* > 1.

Figure 11(b) sketches the stability criterion in the range $(s\xaf\u22121)/s\xaf<\eta <1$ which contains the baseline case with equal gaps. The curves in this case are alike those found in the baseline case (*η* = 1). The effect of decreasing the exit gap slightly, and therefore *η*, is to create a region on the LPS ($e~h\u2032>(\eta +1)/2s(1\u2212\eta $)) which is stable irrespectively of the vibration-to acoustic frequency ratio, *St*. However, in the HPS, the region of unstable TCs is enlarged. The asymptote that separates the stable from the unstable regions on the HPS, and that is located at $e~h\u2032=(1/2)(\eta +1)/(s\xaf(\eta \u22121)+1)$, moves to the left. As it has been already mentioned, often $s\xaf\u226b1$ and the range of validity of this solution is a small region around *η* ≃ 1, i.e., ($(1\u2212s\xaf\u22121)<\eta <1$). This type of situation can be observed in straight seals with nominally identical gaps.

Figure 11(a) represents a situation in which the exit gap has been intentionally reduced, or the inlet knife has been damaged and shortened, and $\eta <(s\xaf\u22121)/s\xaf$. In this situation, the asymptote moves to the LPS and the HPS will be always unstable since in practice the high NDs associated with high frequencies will become unstable. However, the low-frequency region is stable. On the LPS, if the frequency is low enough, there is an unstable region close to the origin, which becomes small for $s\xaf\u226b1$, exactly as in the case of equal gaps. Nevertheless for TCs located slightly far away from the inter-fin cavity center, the seal becomes unstable again for high frequencies. This instability cannot be avoided increasing the frequency of the seal but its mode-shape. This situation in which the inlet gap is increased in an uncontrolled manner, due to rubbing or contacts for instance, leads the seal towards a more unstable condition and is to be avoided.

The case in which the outlet gap, *H*_{2}, is larger than the inlet gap is more benign since a stable region on the LPS for high enough frequencies of the seal always exists (see Fig. 11(c)). Furthermore, when the support is located on HPS, the unstable region outlined by the classical stability criterion is smaller. Conversely, for low frequencies and TCs relatively far away from the seal, a new unstable region is observed. This situation is more robust from a design point of view.

Figure 12 displays the maps of the dimensionless work-per-cycle for large values of the dimensionless seal height ($s\xaf=10$). Three types of seals corresponding to the cases *η* = 0.5, *η* = 1, and *η* = 2 are displayed in Figs. 12(a), 12(b), and 12(c) respectively. The same ideas sketched in Fig. 11 are actually computed here with the model. It can be observed that when the outlet gap is smaller than the inlet one (Fig. 12(a)), the seal is always unstable for high frequencies, except in a small region in the LPS close to the origin. However, if the outlet gap is bigger than the inlet one (Fig. 12(c)), the seal is stable for vibration-to-acoustic frequency ratios larger than one, except in a small region in the HPS in the vicinity of the origin.

## 6 Numerical Verification

A thorough numerical verification of the model has been performed incrementally and systematically but its detailed presentation is outside of the scope of the paper. However, the simulations concerning just the impact of the dissimilar gaps in the stability, which are the more relevant for this work, are presented next. It is important to warn the reader that the methodology presented here is critical for a wide range of configurations and operating conditions, except for stepped seals where *χ* ≃ 0. All the analyses have been performed using a well-validated frequency-domain linearized Navier–Stokes solver [20]. The reader can find some preliminary results of the validation of the model in Refs. [12,13] where it is shown that the level of agreement between the model and the simulations is very high.

### 6.1 Numerical Model.

**V**,

**A**, and

**W**

_{c}are vectors containing, respectively, the volumes, areas, and face velocities of the cells, and

**R**is the residual. Flow unsteadiness is caused by a small periodic motion of the solid walls, in this case of the seal. Since the mesh is conformal at any instant with the vibrating boundaries, a periodic deformation of the whole mesh is created. The grid dependent geometric factors can be decomposed into a steady or mean value plus a periodic perturbation, i.e.,

**v**(

**x**,

*t*) ≪

**V**

_{0}(

**x**) and

**a**

_{0}(

**x**) ≪

**A**(

**x**,

*t*), and the mean velocity of the mesh is null,

**W**

_{0}(

**x**) = 0. The solution,

**U**(

**x**,

*t*), is decomposed as well into a mean base flow,

**U**

_{0}(

**x**), and a small perturbation,

**u**(

**x**,

*t*) ≪

**U**

_{0}, which in turn can be expressed as a Fourier series in time. If just the first harmonic of the variations is retained, any variable can be expressed as

*ω*is the angular frequency. The baseline solution is obtained solving the nonlinear problem $R(U0,A0)=0$ whereas the linear harmonic solution is obtained linearizing Eq. (42) about

**U**

_{0}to obtain the complex linear problem:

**R**/∂

**U**)

_{0}is the Jacobian of the residual evaluated with the mean solution

**U**

_{0}obtained from the nonlinear solver. The details of the solver and validation results for other type of configurations can be found in Refs. [20–22]. The model and the unsteady solver are both linear and therefore, the solutions are fully compatible since in both cases nonlinear unsteady interactions are neglected.

The frequency-domain linearized Navier–Stokes solver is spatially discretized using a MUSCL-like second-order finite volume method consistent with the matrix-valued form of the artificial viscosity. The eddy-viscosity is frozen in the linear solver and computed using the standar Wilcox 2006 *k*–*ω* model in the nonlinear counterpart of the method [17].

### 6.2 Numerical Setting.

A simplified geometry consisting in a two-fin straight-through non-rotating labyrinth seal has been used to verify the model. The computational domain includes upstream and downstream cavities that act as plenum chambers ensuring a uniform inlet and outlet pressure across the seal (see Fig. 13). The baseline seal geometry with identical geometric gaps is defined by *s*/*R* = 0.018, *s*/*H*_{g} = 50, *L*/*s* = 5/3, and *t* = 3*H*_{g}. Some more details can be found in Ref. [13].

#### 6.2.1 Mesh Description.

The meridional plane displayed in Fig. 13 is discretized using an hybrid grid with 35,000 points. The model is extruded in the circumferential direction to form a 10 deg sector containing 10 layers. Therefore, the full mesh consists of 350,000 points. The standard *k*–*ω* turbulence model is integrated to the wall and the mesh is fine enough in the whole domain to ensure that *y*^{+} ≃ 1. The seal geometry and the mesh are axisymmetric.

#### 6.2.2 Mode-Shape Definition.

The CV model assumes that the seal mode-shape is a rigid body motion around a pivot point in the meridional plane. To ease the simulation setup, a synthetic mode-shape generator has been implemented. Once the axial TC distance and the ND are selected (the radial distance is irrelevant in a straight seal since it gives rise to an axial component motion of the seal [10]), the mode displacements are applied to the seal wall nodes and then transferred to the inner nodes using a Laplacian smother. This technique allows a full control of the mode-shape for conceptual studies. Phase-shifted boundary conditions are used in the azimuthal boundaries of the mesh to simulate arbitrary NDs in the 10 deg sector.

### 6.3 Results.

The effect of the gap difference on the stability is explored numerically by varying the thickness of the first gap, keeping constant the downstream gap. The pressure ratio for three cases (*η*_{g} = 1, 4/3, and 2) has been set to *π*_{T} = 1.5. The vibration-to-acoustic frequency ratio, *St*, variation has been obtained by changing the ND keeping the frequency constant to avoid the contamination due to resonances of upstream and downstream cavities. The dimensionless TC, $e~h\u2032$, has been varied by changing the TC distance, *r*.

#### 6.3.1 Steady-State Data.

The steady nonlinear simulations are used to feed the frequency-domain Navier–Stokes linearized analyses and to derive the discharge coefficients of each fin, the carry-over coefficient, and other basic data needed to feed the model and nondimensionalize the *W*_{cyc}, such as the mean pressure of the inter-fin cavity.

Table 1 shows the main steady data derived from the nonlinear analysis for three different geometric gap ratios, namely, *η*_{g} =1, 4/3, and 2. The first observation is that the carry-over coefficient decreases when *η*_{g} is increased. This is due to the fact that *H*_{1}/*L* increases since *H*_{2} is kept constant (see Eq. (2) and Fig. 2(b)). The second observation is that the discharge coefficients are very high ($Cd\u22721$) in this case. This is partially due to the fact that the tip of the fin is relatively thick leading to a high *C*_{d} (see Ref. [19]). In this case, the ratio *C*_{d,1}/*C*_{d,2} ∼ 5–$8%$ and its role is small compared to that of the carry-over coefficient. In any case, the steady data obtained by the CFD are injected in the unsteady model and therefore most of the uncertainties associated with the steady validation of the code are removed from the comparison between the work-per-cycle obtained by the linearized Navier–Stokes solver and the model.

η_{g} | χ_{s} | C_{d1} | C_{d,2} | η | Re | $s\xaf$ | h′ | Ω |
---|---|---|---|---|---|---|---|---|

1 | 0.23 | 0.97 | 0.93 | 0.84 | 21.000 | 10.7 | 3.32 | 11.1 |

4/3 | 0.18 | 0.98 | 0.92 | 1.05 | 18.800 | 13.8 | 3.34 | 12.3 |

2 | 0.1 | 1 | 0.92 | 1.48 | 13.000 | 11.6 | 6.13 | 15.8 |

η_{g} | χ_{s} | C_{d1} | C_{d,2} | η | Re | $s\xaf$ | h′ | Ω |
---|---|---|---|---|---|---|---|---|

1 | 0.23 | 0.97 | 0.93 | 0.84 | 21.000 | 10.7 | 3.32 | 11.1 |

4/3 | 0.18 | 0.98 | 0.92 | 1.05 | 18.800 | 13.8 | 3.34 | 12.3 |

2 | 0.1 | 1 | 0.92 | 1.48 | 13.000 | 11.6 | 6.13 | 15.8 |

Note: The pressure ratio is kept constant, *π*_{T} = 1.5.

*η*=

*A*

_{2,s}*/

*A*

_{1,s}* which according to Eq. (20) turns out to be

*η*< 1, whereas the seal with

*η*

_{g}= 4/3 behaves approximately as a seal with identical gaps (

*η*≃ 1) (see Fig. 14). This is the reason why is so important to retain the impact of the carry-over coefficient in the formulation.

#### 6.3.2 Dimensionless Work-Per-Cycle.

The CV model predicts the work-per-cycle performed by the inner walls that define the inter-fin cavity. However, the motion injected to the 3D unsteady simulations by the artificial generator of mode-shapes includes displacements of the outer fin walls (see Fig. 13). The work associated with the motion these external walls is disregarded in the CFD analysis to make a fair comparison with the model.

Figure 14 compares the dimensionless work-per-cycle obtained with the frequency-domain linearized Navier–Stokes solver [20] (top) with the higher-order model described in Ref. [12], updated with the carry-over corrections described in this work (bottom). A simulation matrix of 15 TCs and 17 NDs is used to construct each of the plots of the top row, totaling 765 simulations.

The work-per-cycle contours have been bounded in the range $\u22120.5<\u2212W~cyc<0.5$ to enhance the visualization. Figure 14 shows that the matching of the CFD results with the prediction of the model is excellent for the three cases. The impact of the carry-over coefficient is clearly seen in the case of nominally identical gaps, *η*_{g} = 1. The recovery of kinetic energy in the exit fin makes that the effective gap ratio become *η* = 0.84. The stability pattern outlined in Ref. [9] for a seal with identical gaps changes completely and three different scenarios can be distinguished.

The first scenario is the seal with nominally identical geometrical gaps, *η*_{g} = 1, which is largely the most recurring case. The discharge coefficients of both fins are similar and their ratio is close to the unity. The analytical model and the simulations predict that even if the discharge coefficients of both gaps are nearly identical, the effective outlet to inlet gap ratio is less than one, *η* < 1, due to the effect of the kinetic energy carried-over to the downstream cavity. The map shows a large unstable region on the HPS for *St* > 1 and a narrow unstable interval on the LPS, which is characteristic of a seal with *η* < 1, as it has been described before.

The second scenario is generated by decreasing the nominal gap on the first fin to reach a geometric gap ratio of *η*_{g} = 4/3. The ratio of the discharge coefficients is very close to one (*C*_{d,2}/*C*_{d,1} ≃ 0.94) but the effect of the kinetic energy carried-over to the exit makes that the effective outlet to inlet gap ratio is *η* ≃ 1. Figure 14(b) shows in fact that the CV model stability criterion described in Ref. [9] is recovered. A seal supported on the LPS is unstable when $St\u22721$ and unstable in a bounded region when the seal support is on the HPS. It is concluded that, to recover the CV stability criterion for nominally identical gaps, a differential gapping is required to ensure that *η* ≃ 1. The matching between the CV model and the simulations is excellent.

Finally, the case in which the outlet gap doubles the first, *η*_{g} = 2, is presented. This situation is not common and usually never considered as the design intent. However, it is representative of a case in which due to rubbing and contact with the static parts, the second fin has been deteriorated or even worn out completely. This configuration is benign and robust since tends to stabilize seals supported on the HPS and to deteriorate slightly the stability of seals supported on the HPS, which are always stable if *St* > 1. There is always a narrow unstable region close to $e~effh\u2032=0$ on the HPS. The degree of matching of the analytical model with the CFD predictions is surprisingly good.

Concerning seal deterioration, the worst-case scenario arises when the inlet seal is eroded or partially removed. In this case, stable seals supported either on the LPS or the HPS can become unstable unexpectedly. This can be seen in Fig. 12(a) where the stability map for *η* = 0.5 obtained with the model is plotted. The high-frequency seal modes are always unstable except if the TC is located in the LPS close to $e~effh\u2032=0$.

Figure 15 is intended to display a more quantitative comparison between the model and the simulations. This figure compares the $W~cyc$ obtained by the model and the linearized Navier–Stokes simulations for a constant TC and varying *St*, for the same seal and pressure ratio (*π*_{T} = 1.5). The vibration-to-acoustic ratio is changed varying the ND and keeping constant the frequency of vibration, exactly as it was done to obtain the stability maps. Different samples of Fig. 14 including cuts at the LPS and HPS have been selected. The actual physical and dimensionless positions of the cuts are given in the caption of Fig. 15. It can be seen that the actual shape of the curves varies significantly from case to case but the level of matching between the model and the simulation is good in all the cases.

It is concluded that the CV model if properly fed with the correct parameters can reproduce the results obtained using linearized Navier–Stokes simulations.

## 7 Concluding Remarks

The baseline CV model for labyrinth seal flutter has been explored in detail to investigate the impact of the effect of dissimilar gaps. The different closure of the inlet and outlet gaps can be either geometric or induced by the flow. There are two mechanisms that give rise to the effective gaps. The first is the contraction of the streamlines at the gap inlet due to flow separation in the fin seal. This phenomenon is accounted for using the classical concept of discharge coefficient. The second has to do with the partial recovery of the kinetic energy of the incoming jet on the downstream closure. This effect is retained using the so-called carry over coefficient. Unlike the discharge coefficient, the kinetic energy carry-over coefficient entails a non-negligible modification of the underlying model that finally can be recast in the same for as the original one if the proper effective gaps are used in the nondimensionalisation of the problem.

It is shown that seal stability is very sensitive to the gap ratio and that a seal supported on the LPS can behave as if it were physically supported on the HPS. The analytical analysis of the stability of the problem as a function of the dimensionless parameters is simple but the results are difficult to express simply. It was finally decided to focus on the operating conditions which are of interest for the industry including the main and clearest conclusions.

It is concluded that straight-through seals designed with nominal gaps are not robust against small perturbations of the gaps and in practice behave as seal with the first gap more open than in nominal conditions. For the typical range of the seal design parameters, perturbations of the design intent leading to values of the gap ratio slightly smaller than one ($\eta \u22721$) make the seal behave very differently than predicted by the baseline CV model. Small perturbations of the nominally equal seal gaps can lead to a more unstable seal than originally foreseen.

The motivation for modifying the formulation was the inability of matching the model with the simulations in some particular cases. A large simulation matrix of 3D frequency-domain linearized Navier–Stokes simulation has been included to support all the claims discussed in this work. The model and the simulations exhibit excellent degree of matching in the whole range of parameters. It is concluded that the model can be used to make quantitative predictions as well.

## Acknowledgment

Roque Corral and Michele Greco want to thank ITP Aero for providing access to ITP’s computing framework and its support. This research work has been supported by the European project ARIAS, H2020 research and innovation program under grant agreement no. 769346. The authors gratefully acknowledge the financial support.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The authors attest that all data for this study are included in the paper. Data provided by a third party are listed in Acknowledgment.

## Nomenclature

*h*=upstream fin nondimensional pressure function

- $m\u02d9$ =
mass flow per span-wise or circumferential length unit

*r*=TC position

*s*=cavity height

- $s\xaf$ =
*s*/(*γHh*′), nondimensional cavity height *t*=tip fin thickness

*z*=azimuthal coordinate

*A*=seal clearance area

*H*=fin clearance

*J*=downstream fin nondimensional pressure function

*L*=seal cavity length

*R*=cavity radius

*a*_{0}=sound speed in the cavity

*p*_{c}=cavity static pressure

*p*_{e}=exit static pressure

*t*_{d}=$pc,sVc.sm\u02d9sa02,$ discharge time

*v*_{θ}=circumferential velocity

*C*_{d}=$m\u02d9/m\u02d9id$, discharge coefficient

*P*_{0}=inlet total pressure

*R*_{g}=specific gas constant

*W*_{cyc}=work-per-cycle

*a*_{j}(*t*) =*j*th gap time perturbation- Re =
$m\u02d9/(2\pi R\mu )$ Reynolds number based on the gap

*St*=$\omega RNDa0$, vibration-to-acoustic frequency ratio

### Greek Symbols

*β*=jet opening angle

*γ*=heat capacity ratio

- Δ
*θ*=rotation angle

*η*=*H*_{2}/*H*_{1}, gap ratio*π*_{s}=*P*_{0}/*p*_{c}, cavity pressure ratio*π*_{T}=*P*_{0}/*p*_{e}, total pressure ratio*π*_{c}=*π*_{T}/*π*, pressure ratios relationship*ρ*=fluid density

*τ*=nondimensional time

*χ*=kinetic energy carry-over coefficient

*ω*=vibration angular frequency (rad/s)

- Ω =
*ωt*_{d}, nondimensional discharge time - $\Omega ~$ =
$\Omega ~/h\u2032$, Ω rescaling with

*π*_{T}