In a previously published paper, a model for the nonlinear acoustic response of an area contraction including bias flow was presented. The model's prediction for the zero-driving resistance grew progressively worse as the steady-flow Mach number increased. This trend suggests that the forward loss coefficients should depend on the steady Mach number. This letter provides an empirical fitting of this Mach number dependence, along with additional validation data for the model. These additional validation data corroborate the model's prediction that the nonlinear impedance is frequency independent. This letter additionally provides an experimental methodology for determining the characteristic length with sample results.

## 1. Introduction

Direct acoustic response measurements in combustion devices such as solid and liquid rocket engines remain the best way to identify thermo-acoustic instabilities. However, the high heat and corrosive gasses contained within combustion chambers make them inhospitable to pressure transducers; thus, sense-lines are often used to offset the sensor from this harsh environment. By employing a sense-line for pressure measurements, an area contraction is created at the connection point between the sense-line and the combustion chamber. This area contraction has an associated complex acoustic impedance to acoustic driving, thereby modifying both the amplitude and phase of the pressure response measured by the sensors in the sense-line. In fact, the acoustic impedance, which is defined as the ratio of the pressure to particle velocity,^{1} behaves nonlinearly with acoustic velocity. Additionally, purge flow of an inert gas is often introduced into the sense-line to further protect the sensors from high temperatures, introducing additional complications toward understanding the measured pressure.

The foundation of this work expands on the knowledge base of the acoustically compact area contraction's response to acoustic forcing at more realistic engine operating conditions. At present, these data are only sparsely available in the literature and are often not pertinent to the combustion systems under consideration.^{2–4} Furthermore, previous work has demonstrated that experimental measurements of nonlinear acoustic elements differ significantly from predictions made with nonlinear models.^{5}

In Kawell *et al.*,^{6} a model was presented to calculate the nonlinear impedance of an area contraction. This follow-up letter centers on investigating the role of steady mean flow in the acoustic response of area contractions at various frequencies by varying the length of the EXT, thereby changing the frequency of the $14$ wave mode. By understanding the acoustic response of a compact area contraction in the lab and by measuring the pressure in the EXT, an accurate estimate of the effect of the area contraction can be obtained. In addition, this letter presents an experimental methodology for determining the length correction, which also exposes the frequency dependent nature of the length correction, an important quantity for the model.

The nonlinear impedance model enhanced in this paper can be further implemented in other reduced order models to determine complex system acoustic response. These reduced order models are used extensively by engineers to help mitigate acoustic instabilities that arise in combustors. Alternatively, they can be used in the initial design phase to avoid instabilities altogether.

## 2. Experimental methods

The objective of this study was to measure the nonlinear acoustic response of area contractions over a wide range of frequencies, acoustic velocities, and steady mean flow rates to compare with the previously published model. A multiple-microphone impedance tube, shown in Fig. 1, was used to determine the acoustic impedance of the area contraction. The convention in this paper is to call the larger tube the standing wave tube (SWT) and the smaller tube the extension tube (EXT). The area ratio, labeled in this paper as *η*, is defined as the SWT cross-sectional area divided by the EXT cross-sectional area.

### 2.1 Impedance tube setup

In this investigation, eight Kistler 211B5 sensors were used as dynamic pressure transducers inside the chamber. The sensors were placed as shown in Fig. 1, with four transducers in the SWT and four transducers in the EXT. The nearest sensors to the area contraction were placed 2.4 and 5.4 tube diameters from the sample for the SWT and EXT sections, respectively, which exceeds the 2 diameters outlined in ASTM E1050-19.^{7} This configuration allows for subtraction of the influence of the downstream components, thereby isolating the effects of the area contraction, as described in Kawell *et al.*^{6} Impedance tube measurements are highly sensitive to errors in phase between the microphones; therefore, a calibration procedure was implemented as outlined in ASTM E1050-19.^{7} An absolute static pressure transducer and thermocouple were used for measuring the pressure and temperature inside the SWT to determine the air density, which is especially critical in the cases with flow. This density is used for calculation of the area contraction Mach number and is not expected to vary significantly within the test-section it is measured in.

For the flowing cases, the inlet utilized an Omega FMA5500 mass flow meter to provide metered airflow through a sintered disk into the speaker block of the SWT. The Mach number through the EXT was controlled by changing the choked orifice size at the exit, which was rigidly clamped as shown in Fig. 1. The combination of this inlet and outlet was found to well-approximate a hard-walled acoustic boundary condition in comparison with flat-plate validation tests. In the non-flowing cases, the EXT was terminated with a solid plate, and the sintered disk was replaced with a solid plug.

Data were collected using a National Instruments (Austin, TX) cDAQ-9137 data acquisition system. Discrete frequency forcing was provided by two 100-watt Federal Signal (Oak Brook, IL) TS100-N speakers, producing sound pressure levels (SPLs) ranging up to 183 dB, referenced to 20 μPa. Tests were conducted using the following procedures. First, a standing wave was established in the impedance tube for 0.5 s, which was deemed sufficient to ensure that evanescent effects have subsided. Once the standing wave was established, data were acquired for 10 s at a rate of 5.12 kHz. When this process was complete, the forcing frequency was incremented as specified by the given test, and this process was repeated for all frequencies of interest for the test. Once the frequencies of interest were tested, the amplitude was incremented, and the process was repeated until the maximum amplitude achievable by the driving setup was reached. Table 1 shows the test matrix conducted for this investigation. It should be noted that due to small upstream pressure variations, the Mach number tested for each sense-line varies slightly, and therefore the 296 Hz case was chosen to be the nominal Mach number used for labeling subsequent plots.

Test series 1 . | ||||||
---|---|---|---|---|---|---|

Area ratios | M_{SL}^{a} | |||||

η = 64 | 0 | |||||

η = 16 | 0 | |||||

η = 6.6 | 0 | |||||

η = 4.0 | 0 |

Test series 1 . | ||||||
---|---|---|---|---|---|---|

Area ratios | M_{SL}^{a} | |||||

η = 64 | 0 | |||||

η = 16 | 0 | |||||

η = 6.6 | 0 | |||||

η = 4.0 | 0 |

Test series 2 . | |||||||
---|---|---|---|---|---|---|---|

Area ratio and frequency . | M
. _{SL} | Area ratio and frequency . | M
. _{SL} | Area ratio and frequency . | M
. _{SL} | ||

η = 16 f = 296 Hz | 0.010 | η = 16 f = 210 Hz | 0.010 | η = 16 f = 178 Hz | 0.011 | ||

0.025 | 0.026 | 0.023 | |||||

0.041 | 0.045 | 0.043 | |||||

0.062 | 0.064 | 0.061 | |||||

0.077 | 0.082 | 0.084 | |||||

0.089 | 0.094 | 0.098 |

Test series 2 . | |||||||
---|---|---|---|---|---|---|---|

Area ratio and frequency . | M
. _{SL} | Area ratio and frequency . | M
. _{SL} | Area ratio and frequency . | M
. _{SL} | ||

η = 16 f = 296 Hz | 0.010 | η = 16 f = 210 Hz | 0.010 | η = 16 f = 178 Hz | 0.011 | ||

0.025 | 0.026 | 0.023 | |||||

0.041 | 0.045 | 0.043 | |||||

0.062 | 0.064 | 0.061 | |||||

0.077 | 0.082 | 0.084 | |||||

0.089 | 0.094 | 0.098 |

^{a}

Sense-line Mach number (*M _{SL}*).

### 2.2 Experimental impedance calculation

The method for extracting the acoustic impedance of the area contraction is similar to that of Fujimori *et al.*^{8} utilizing the multi-point least squares method with four microphones in each section. The Fourier transformed acoustic pressure-history was calculated, yielding the complex acoustic pressure amplitudes measured by each transducer. Each set of four transducers was used to calculate the incident and reflected acoustic wave pressure and velocity in its respective section using the plane wave decomposition model presented in Kawell *et al.*^{6}

In general, the acoustic reactance, the imaginary portion of the acoustic impedance, behaves linearly and is very small. This is evidenced in the length correction portion of this letter (Sec. 4.3), where the reactance is utilized to compute the length correction. Therefore, the remainder of the analysis shown will show the acoustic resistance, the real portion of the acoustic impedance, as it is expected to behave nonlinearly near the resonance point of the EXT.

## 3. Impedance modeling

The results of this investigation are compared to a previously published model.^{6} This model is developed from the unsteady Bernoulli equation and is simplified for the special case of an area contraction. The resultant equation for the acoustic resistance is shown in Eq. (1),

where $p\u03021,1$ is the acoustic pressure upstream of the area contraction, $p\u03022,1$ is the acoustic pressure downstream of the area contraction, $u\u03021,1$ is the acoustic velocity calculated upstream of the area contraction, $u0$ is the free stream velocity, $Si$ is the cross-sectional area of the respective section, $Lc$ is a characteristic length, and $\alpha f$ and $\alpha r$ are the forward and reverse loss coefficients, respectively. The ratio *θ* is defined via the following:

The model predicts that there is no frequency dependence for the acoustic resistance of an area contraction, an important result of the development, which will be corroborated experimentally in Sec. 4. Within the model, the value of $\alpha f$ is dependent on both the geometry and potentially steady flow rate. The experimental validation data published previously and within this paper suggest that $\alpha f$ should be dependent on the steady Mach number. Thus, a linear function of Mach number was determined empirically based on the original data published in Ref. 6. This empirical formulation was determined by minimizing the least squares error of the calculated model for all the acoustic velocities at a given Mach number, yielding an optimal *α _{f}* for each Mach number. This was repeated for all the tested Mach numbers in the original data set, and a linear fit was applied to the calculated optimal loss coefficients, which yielded the following determination for

*α*shown in Eq. (3). This is the function utilized in the model comparison presented in Sec. 4.2,

_{f}Furthermore, as will be shown later, the model appears to slightly overestimate the acoustic impedance, particularly the effect of steady-flow Mach number. However, the model matches the slope of the nonlinear trend very well for each test along with an acceptable error for the zero-driving resistance, or the resistance extrapolated to zero acoustic velocity. Changing the model from a constant value of *α _{f}* and

*α*decreases the overprediction of the model, particularly for the zero-driving resistance at higher steady-flow Mach numbers. This empirical determination of the loss coefficients yields a much stronger agreement with experimental data than the previously presented comparison.

_{r}^{6}However, a physics-based approach for development of the relationship between the steady Mach number and the loss coefficients is highly desirable.

## 4. Results and discussion

In this section, impedance data are presented for various EXT diameters with varying flow rate conditions along with the associated acoustic end-corrections. In addition, comparisons with the model described previously are presented as appropriate.

### 4.1 Identification of EXT resonance

This section presents measurements of the acoustic power absorption coefficient and the complex acoustic impedance of EXTs for a range of frequencies and acoustic pressure amplitudes. In addition, it presents results for the calculated effective length correction for the various test conditions. The acoustic power absorption coefficient, *α*, is shown in Fig. 2 for four different EXT diameters without flow using the hard-termination configuration.

In general, it is expected that *α* will be maximized at the resonance of the EXT. For this particular configuration, the wave mode is expected. Typically, this can be calculated quite simply using

where *f* is the resonant frequency, *c* is the speed of sound, and *L* is the length of the EXT. However, due to the jetting effects such as those first identified by Ingard and Ising,^{9} a length correction term is needed to change the apparent length of the EXT so that the resonance frequency is accurately predicted. In addition, for other reduced order models, this quantity is particularly relevant for simplifying complex acoustic systems. The acoustic length correction allows for a simple determination of the complex acoustic interactions regarding shifts in the fundamental modes of an acoustic system. This length of the EXT is also a fundamental input to the model presented in Eq. (1), as seen in the $\rho \omega Lc/S1$ term. The length correction varies based on frequency, and thus the characteristic length of the model needs to be adjusted. A methodology for experimentally determining this length correction along with a more in-depth examination of how the length correction changes with the nonlinear behavior of the area contraction and flow is presented below.

It is expected that the acoustic power absorption will increase with increasing nonlinear damping contributed by the EXT. This is demonstrated by the two middle area contractions tested, $\eta =16.0$ and $\eta =6.6$, having the highest absorption coefficient at their peaks. The non-flowing acoustic resistance was shown in Kawell *et al.*,^{6} and since the newly determined loss coeffiicents are unchanged in the zero flow cases, these resistances are unchanged. The *η* = 16 case was chosen for further examination due to the nature of its high absorption coefficient, representing what is expected to demonstrate strong nonlinear behavior.

For the remainder of the results shown in this paper, the acoustic velocity sweeps were conducted at the absorption coefficient maximums for each EXT. In the flowing cases, the configuration is slightly longer due to the holding hardware for the exit orifice plate, and thus the same procedure was conducted prior to testing each EXT in its flowing configuration, and each respective maximum was used.

### 4.2 Area contraction acoustic resistance with flow

The normalized nondimensional resistance is an important parameter for engineers when designing acoustic systems. Often, this damping is desirable for tuning the acoustics of a system. In Sec. 4.1, the acoustic resistance of the area contraction was shown to be nonlinear near the resonance of the system without flow. Previous work has shown that the acoustic impedance of the area contraction is most likely caused by vortex shedding that occurs as the acoustic wave oscillates through the abrupt area change.^{9} The nonlinear nature of the acoustic resistance is most likely due to the flow reversal through the area contraction. Thus, as the steady mean flow rate increases, it is expected that the nonlinear nature of the acoustic resistance will diminish, as eventually there is no flow reversal through the area contraction.

In the interest of keeping the data set reasonable, a single area ratio was chosen for examination, the $\eta =16.0$ case, as it had the highest peak absorption. The three lengths of the EXT were chosen to be 21.3, 31.8, and 36.7 cm, which corresponded to nominal resonance frequencies of 296, 210, and 178 Hz, respectively. As mentioned previously, the control of the steady-flow Mach number was subject to small upstream pressure differences in the supply of air, and thus there are slight discrepancies between the Mach number for each test. This is evidenced by a slightly higher than typical difference for the 296 Hz case at the 0.025 and 0.041 Mach number tests, resulting in a lower initial resistance. The labeled Mach number is the nominal Mach number for the 296 Hz case in the EXT for each velocity sweep. The experimentally determined acoustic resistance along with the model prediction is shown in Fig. 3, with the shaded region representing the 95% confidence interval. This confidence interval was determined by using dithering to compute the sensitivity coefficients, followed by computing the regression confidence intervals from the standard error, pooling this variance across the ensmembles, and then showing the computed uncertainty of the ensemble average. It should also be noted that the acoustic response of the chosen drivers decreased with decreasing frequency, and thus the acoustic velocity available during each test diminished as the frequency decreased.

Once again, the data show that the area contraction is strongly nonlinear at low steady-flow Mach numbers. In addition, it should be noted that for all but the lowest steady-flow velocity, there is a linear region where the acoustic velocity is insufficient to overcome the steady-flow velocity. This is shown particularly well at the highest steady-flow Mach number tested, where even the highest acoustic velocity achievable by the test setup was insufficient to overcome the steady-flow velocity. The corresponding zero-driving resistance of the area contraction also increased with increasing flow rate. As the steady-flow velocity increases, the acoustic velocity required to exceed the steady-flow velocity increases, supporting the hypothesis that the flow reversal through the area contraction is what causes the nonlinearity in the acoustic resistance. The model presented in Sec. 3 matches fairly well, with the addition of a linear relationship for Mach number significantly improving the prediction of the zero-driving resistance, especially at higher Mach numbers.

Overall, this improvement in determination of the zero-driving resistance also highlights the strong agreement in prediction of the nonlinear trend, demonstrating strong agreement in the nonlinear region as well. The model presented is fundamentally frequency independent, which the data strongly corroborate as the data are highly consistent across the frequencies tested. While the empirical modeling yields a much better fit, the model still overestimates the point at which the acoustic velocity overcomes the steady-flow velocity, as indicated by the slope of the model lines beginning much later than the experimental data for all but the fully linear case. A more involved determination of *α _{f}* accounting for more of the fundamental physics may yet still enhance the impedance prediction of the model.

### 4.3 Acoustic end-correction

The acoustic end-correction is a useful characteristic of an acoustic insertion element due to its concise summary of complicated physics. In reduced order models, the acoustic end-correction serves as a valuable way to capture the effects of complicated flow phenomena with a single parameter. Primarily, this physical effect results from evanescent waves that arise from the sharp transition in area. In addition, it is required for proper determination of the phase correction term, $i(\rho \omega Lc/S1)$ in Eq. (1), which represents the phase correction of the impedance due to these physical effects. The acoustic end-correction can be easily computed from the data collected for determination of the impedance and is presented in this section for reference in this model and others.

To determine the acoustic end-correction, the $14$ wave resonance frequency is determined experimentally by finding the *x*-intercept of the phase of the absorption coefficient, $\varphi $. Figure 4 shows a representative plot of $\varphi $, in this case the non-flowing *η* = 16 test. Once the frequency is determined, the length correction can be calculated via Eq. (5),

where $\Delta L$ is the acoustic end-correction, $d\varphi /df$ is the derivative of the phase with respect to frequency at the resonance, and *M*_{0} is steady-flow Mach number. This value is then nondimensionalized by the wave number *k* to represent the overall phase correction required by the model, shown in Eq. (6),

where *f _{res}* is the resonant frequency.

The results for the cases without flow are tabulated in Table 2. When nondimensionalized by the wavenumber, the results are consistent across the area ratios, indicating that the phase correction required for the model is independent of area ratio.

Throughout the experimental testing conducted, it was found that there was no significant trend with acoustic velocity. However, with the flow rate control scheme, small variations within each nominal Mach number were impractical to suppress. To mitigate these effects, the length correction for the remainder of the data presented is averaged across the acoustic velocities to accommodate the small variations in Mach number between each acoustic velocity tested. Table 3 shows the calculated length correction for the flowing cases.

Mach number . | Nondimensional kL
. | ||
---|---|---|---|

f = 178 Hz
. | f = 210 Hz
. | f = 296 Hz
. | |

0.010 | 0.058 | 0.117 | 0.253 |

0.025 | 0.066 | 0.086 | 0.217 |

0.041 | 0.061 | 0.096 | 0.255 |

0.062 | 0.060 | 0.093 | 0.256 |

0.077 | 0.061 | 0.090 | 0.244 |

0.089 | 0.044 | 0.094 | 0.244 |

Mach number . | Nondimensional kL
. | ||
---|---|---|---|

f = 178 Hz
. | f = 210 Hz
. | f = 296 Hz
. | |

0.010 | 0.058 | 0.117 | 0.253 |

0.025 | 0.066 | 0.086 | 0.217 |

0.041 | 0.061 | 0.096 | 0.255 |

0.062 | 0.060 | 0.093 | 0.256 |

0.077 | 0.061 | 0.090 | 0.244 |

0.089 | 0.044 | 0.094 | 0.244 |

These data show that the length correction is relatively independent of steady-flow velocity, at least in the range tested. In addition, the length correction appears to be frequency dependent, with higher frequencies corresponding to increased length corrections.

## 5. Conclusions

In this investigation, additional validation experimental results were shown for the acoustic resistance and length correction of area contractions as an acoustic element. From this analysis, it is clear that the acoustic response of the area contraction is strongly nonlinear, as predicted by the model, which is linearized by steady flow through the area contraction. In this study, steady-flow Mach numbers of approximately 0.1 in the EXT were sufficient to fully linearize the nonlinear acoustic resistance for the acoustic velocities tested. Additional lengths were tested, showing that the resistance results compared favorably with a model developed in a previous paper and its prediction that the acoustic resistance should be independent of frequency.

In addition to the acoustic resistance data, the acoustic length correction was determined for each area contraction, and its dependence on acoustic velocity as well as mean steady flow was investigated. For an area contraction without flow, it was shown that the length correction is strongly dependent on the area ratio. The acoustic length correction was also found to increase with increasing flow rate. This work provides foundational data for implementation in reduced order acoustic models and to help further the understanding of the nonlinear nature of area contractions in acoustic systems.

## Acknowledgments

The authors would like to thank Eric Van Horn and Thomas Teasley for their valuable technical discussions and help with data collection and problem solving.