## Abstract

This article discusses the mechanism for prediction of aeroacoustic resonance in cavities of hole-pattern stator seals. A Reynolds-averaged Navier–Stokes (RANS) solver developed in-house was used to simulate grazing channel flow past a cavity located in a channel. The numerical results generated with the RANS solver showed good agreement with those obtained using a commercial large eddy simulation code. In addition, the numerical results agreed well with the experimental data. Rossiter’s formula, a popular semi-empirical model used to predict frequencies of holetone acoustic instabilities caused by grazing fluid flow past open cavities, was modified using the RANS solver results to allow for its application to channel flows. This was done by modifying the empirical constant, the ratio of vortex velocity, and the freestream velocity. The RANS solver accurately captured the salient features of the flow/acoustic interaction and predicted well the dominant acoustic frequencies measured in an experimental investigation. The flow solver also provided detailed physical insight into the cavity flow instability mechanism.

## Introduction

Fluid instabilities in hole-pattern and honeycomb stator seals have been extensively investigated in recent years[1,3] The results of this body of research indicate that rotor shaft vibrations are caused by cross-coupled forces between the rotor shaft and the annular seal. Usually, these forces stabilize the rotor. However, under certain operating conditions, a destabilizing effect can been caused by an unexpected increase in friction factor with respect to an increased rotor-seal clearance. Childs et al.[1] have linked this increase in friction factor to acoustic fluid instabilities noticed in the hole-patterns of the annular seals. Unsteady pressure measurements at the base of one of the cells showed that flow instabilities in the honeycomb or hole-pattern cells cause this increase in friction factor. The oscillatory unstable cavity flow was large enough to interfere with the through flow.

The fluid instabilities in hole-pattern stator seals are produced when the flow between the rotor shaft and annular seals passes over the hole pattern of the annular seals. The holes in the annular seals act as cavities trapping pockets of recirculating fluid. The grazing flow generates a flow-acoustic feed-back loop within each cavity which in turn causes acoustic waves to emanate from the cavities. This phenomenon is similar to the bomb bay instability that occurs at certain Mach numbers when the aircraft bomb bay door is opened. An overview of past studies of cavity flow instabilities was compiled by Grace.[4]

## Physics of Cavity Flows

Fluid instabilities generated by the grazing flow past an open cavity occur due to the interaction of shear layer oscillations within the cavity, vortices within the shear layer and acoustic waves radiating from the cavity. As shown in Fig. 1, a boundary layer forms along the wall upstream of the cavity and separates from the wall as it reaches the leading edge of the cavity forming a shear layer across the top of the cavity. The faster grazing flow in the channel passing over the slower recirculation in the cavity causes Kelvin-Helmholtz oscillations in the shear layer. These oscillations cause vortices to be shed from the leading edge of the cavity and also propagate them along the shear layer. The vortices propagate downstream and impinge on the trailing edge. The interaction between the vortices and the shear layer warps the shear layer causing its reattachment point to momentarily move down the cavity wall below the lip. The shear layer stagnates just below the trailing edge causing a brief period of higher pressure. As the flow accelerates past the trailing edge of the cavity, an area of low pressure is momentarily formed along the wall just downstream of the cavity edge. As a result, an acoustic dipole is generated at the trading edge, which radiates acoustic waves in all directions. The acoustic waves that propagate upstream excite the shear layer at the leading edge of the cavity, which in turn causes the shedding of additional vortices. This vortex-acoustic interaction forms a feed-back loop which selectively amplifies a dominant frequency.

Features of cavity flow.

$f=UL⋅m−γMcoct+1/κ$

Rossiter[5] derived a semi-empirical equation that predicts the dominant frequency of the cavity flow instability, f, where U is the freestream flow velocity, L is the length of the cavity, M is the Mach number, co and cτ are the speed of sound outside and inside the cavity respectively, γ and κ are empirical terms, and m is the mode of the oscillation. Using experimental measurements, Rossiter determined that the empirical terms are γ = 0.25 and κ = 0.66. Subsequent studies have reported κ values as low as 0.57 while the value for γ has remained consistent.[6]

Rossiter, as well as the majority of researchers utilizing Rossiter’s formula, studied cavities open to the freestream with no influence of an opposing wall. For flow in holepattern stator seals the cavities are located within a channel where the opposing wall is situated in the proximity of the opening of the cavities. Consequently, Rossiter’s formula must be modified to account for the presence of the opposing wall. This was done using numerical results that were validated by experimental data.

## RANS, LES and Experimental Results

For the numerical simulations the model for the hole-pattern seal was simplified to a single, rectangular, two-dimensional cavity with a length and depth of 3.175 mm. The cavity was located in a channel with a height of 0.7112 mm, which extended 22 mm (6.9 cavity lengths) in front of and behind the cavity. The geometry was a simplified version of the experimental setup used at the Turbomachinery Laboratory at Texas A&M University. The experiments were conducted for a hole pattern containing several hundred cavities. The simplified domain, with a single cavity, allowed for reasonable computation times while still capturing the necessary flow features.

The inlet total pressure was 104,190 Pa, the exit static pressure was 101,325 Pa and the total temperature was 305.4 K. The Reynolds number based on the cavity length was 14,300. Each simulation was started from an initial condition with a uniform velocity of 68 m/s parallel to the channel. These parameters were chosen to provide a flow with a Mach number of approximately 0.2.

Figure 2 shows a series of pressure contours illustrating a complete cycle of the cavity flow mechanism described above. A vortex is shed from the leading edge of the cavity and increases in magnitude as it propagates along the shear layer. As the vortex approaches the trailing edge of the cavity, a dipole is formed releasing an acoustic wave. As the original vortex impinges on the trailing edge, a second vortex is shed.

Pressure contours at time, t: (a) 0, (b) 12μs, (c) 24μs, (d) 36μs

Velocity profiles are shown in Fig. 3 for five evenly spaced locations through the cavity. The velocity profiles produced with the LES and RANS simulations are nearly identical except near the bottom of the cavity.

Simulations using the flow conditions described above predicted a dominant frequency using RANS and LES of 20.95 kHz and 21.80 kHz, respectively. This compares well with the experimental result of 21.75 kHz. This shows that, even though the flow domain was significantly simplified, both types of numerical simulations were able to accurately capture the important flow features and predict the dominant frequency. This indicates that cross flow and cavity interactions are secondary effects for the dominant frequencies. The computational time required by the RANS solver was more than one order of magnitude smaller than that of the LES solver.

Snapshot of velocity profiles generated with the LES and RANS solvers; profiles located at five evenly spaced axial locations through the cavity. Frequency, f [Hz]

## Determination of Empirical Values for Rossiter’s Formula

It was found that Rossiter’s formula (1) as reported in literature[5,6] was not able to accurately predict dominant frequencies. It was believed that the error was due to the influence of the opposing wall of the channel. Numerical simulations showed that the feed-back mechanism for cavity channel flows was the same as that used to derive Rossiter’s formula. For this reason the form of Rossiter’s formula remained valid while only the value of one empirical variable, κ, was modified.

The empirical term κ can be physically understood as the ratio between the propagation speed of vortices and the freestream velocity. Originally, Rossiter found the value for κ by fitting the equation to experimental results. Reported κ values for open cavity flow range from 0.57 by Larcheveque et al.[6] to 0.66 by Rossiter.[5] Using CFD, however, the position of the vortices can be directly tracked throughout a given cycle. The velocity is then easily calculated giving the K ratio. Using the technique described above, the authors found the K ratio for channel cavity flow to be 0.52.

Figure 4 shows pressure variation as a function of frequency for a Mach number at the cavity leading edge of 0.166. Simulations were also run for a Mach number of 0.183. The pressure variation calculated using the RANS predict dominant frequencies of 23.68 kHz for Mach 0.166 and 26.32 kHz for Mach 0.183. The dominant frequency was also calculated with Rossiter’s formula (1) using the third mode, m=3. Two values of κ constant were used: Rossiter’s κ=0.66 and the value calculated herein for channel flows, κ=0.52. Figure 4 shows a good match between the dominant frequency predicted by the RANS solver and Rossiter’s formula with κ=0.52, for both Mach numbers. A summary of the predicted dominant frequencies is shown in Table 1.

Pressure variation vs. frequency at M=0.166

Table 1

MachRANSRossiter’s formula (1)
Channel κ=0.52Open κ=0.66
0.16623.68kHz23.90 kHz29.70 kHz
0.18326.32 kHz26.14 kHz32.42 kHz
MachRANSRossiter’s formula (1)
Channel κ=0.52Open κ=0.66
0.16623.68kHz23.90 kHz29.70 kHz
0.18326.32 kHz26.14 kHz32.42 kHz

## Conclusions

A Reynolds-averaged Navier-Stokes (RANS) solver developed in-house was used to simulate grazing channel flow past a cavity in a channel. The objective of this investigation was to predict fluid instabilities in holepattern stator seals. The numerical results generated with the RANS solver showed good agreement with those obtained using a commercial Large Eddy Simulation code. In addition, the numerical results agreed well with experimental data. Rossiter’s formula, a popular semiempirical model used to predict frequencies of hole-tone acoustic instabilities caused by grazing fluid flow past open cavities, was modified using the RANS solver results to allow for its application to channel flows. This was done by modifying the empirical constant K, the ratio of vortex velocity and the freestream velocity. The corrected K value was obtained by tracking vortex position through a cycle of the flow. The dominant frequencies predicted using the Rossiter’s formula with the new K value matched well the experimental data for hole-pattern stator seals. The RANS solver accurately captured the salient features of the flow/acoustic interaction and predicted well the dominant acoustic frequencies measured in an experimental investigation. The flow solver also provided detailed physical insight into the cavity flow instability mechanism.

This work was sponsored by the Turbomachinery Research Consortium. The authors thank Professor Dara Childs and Mr. Bassem Kheireddin of the Turbomachinery Laboratory at Texas A&M University for providing the geometry, flow conditions, and experimental results for the hole-pattern stator seal.

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