Feature Extraction of Oscillating Flow With Vapor Condensation of Moist Air in a Sonic Nozzle

The sonic nozzle is commonly used in flow measurement. However, the nonequilibrium condensation phenomenon of moist air in the nozzle has a negative effect on the measuring accuracy. To investigate this complex phenomenon, the experiments on the oscillating condensation flow of moist air were conducted by an adjustable humidification apparatus with different relative humidities (0%–100%), temperatures (30–50°C), and carrier gas pressures (1–6 bar), where the microsize pressure measuring system was designed by Bergh–Tijdeman (B–T) model. The accurate mathematical model of nonequilibrium condensation was also built and validated by the experimental data of time-averaged pressure distribution. Then, the frequency and intensity of pressure fluctuation of oscillating flow at a wide range of operation condition were obtained combining experimental data and physical simulation model. Importantly, a new semiempirical relation of dimensionless frequency deduced from dimensionless analysis was identified accurately by experimental data. Finally, the signal nonstationarity was also observed using the continuous wavelet transform (CWT). The instantaneous frequency saltation and the energy attenuation of pressure signals were observed in the condensation flow.

addition, namely thermal choking [14], depending on the inlet stagnation condition [15], especially subcooled temperature or humidity of the moist gas. Thus, the condensation phenomenon of moist gas in the sonic nozzle should be investigated further, in order to improve the measurement accuracy and stability of mass flow rate.
For decades, the steady condensation of the moist gas has been investigated experimentally. Setoguchi et al. [16] and Matsuo et al. [17] produced a condensation flow by expanding the moist air in a supersonic nozzle with the stagnation pressure of 102 kPa and the stagnation temperature of 287 K. The static pressure distribution is acquired by moving the lower wall and the Schlieren photographs were also obtained using an optical system. Wyslouzil et al. [18] presented several experimental data of the humid nitrogen binary steady condensations in the supersonic nozzle. The inlet conditions were the total pressure of 0.6 bar, the total temperature of 286.7 K, and the maximum vapor partial pressure of 0.01 bar. Lamanna [19] and Lamanna et al. [20] also used compressed humid nitrogen to research the steady nucleation and droplet growth processes in the moist gas case. The inlet conditions were the total pressure of 0.913 bar, the total temperature of 283.1 K, and the stagnation supersaturation ratio of 0.613 (nozzle code G1 and G2). Ding et al. [13] designed a sonic nozzle, according to ISO standard 9300, to experimentally research the condensation phenomenon of the moist gas in the inlet pressure of both 4 and 8 bar, and relative humidity ranging from 42% to 63% with a constant temperature.
To predict transonic flows involving a complex nonequilibrium condensation, the mathematical computational fluid dynamics (CFD) modeling is a powerful simulation tool. The mathematic modeling of this nonequilibrium condensation flow includes the nucleation and droplet growth theory [21]. Ma et al. [22], [23] built an Eulerian multifluid turbulence model in moist air. Lv and Bai [24] built a model for nonequilibrium condensation in gaseous carrier flows. Ding et al. [25] presented a gas-liquid two-phase model for the moist gas condensation flow. Hamidi et al. [26] proposed an inviscid nonequilibrium thermodynamic model. In addition, for pure steam, Wang et al. [15], Bakhtar and Mohammadi [27], Young [28], Halama and Hric [29] and Chang et al. [30], Jabir et al. [31], and Ding et al. [32] built 2-D or 3-D mathematical models for vapor condensation.
With the latent heat addition increasing to supercritical value, the flow will occur the thermal choking and probably become unsteady. Wegener and Cagliostro [33] and Wegener 0018-9456 © 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See https://www.ieee.org/publications/rights/index.html for more information. and Mosnier [34] first noted that this unstable flow and proposed a semiempirical formula for calculating the frequency of oscillating pressure. However, this formula has a maximum error of 60% when the humidity is lower than 1 and the parameters in his formula are not independent of each other. Then, Matsuo et al. [35] observed experimentally the flow oscillation when the moist air expanded rapidly. In 1985, Frank [36] investigated experimentally the heat addition using a Laval nozzle and found that the flow becomes unsteady in the high humidity. Skillings et al. [37] also obtained the frequency and amplitude of the oscillating pressure. In 1997, Adam and Schnerr [38] presented instabilities of unsteady supersonic flows of moist gas experimentally and numerically.
Malek et al. [39], Yu et al. [40], and Ding et al. [41] investigated the oscillation in nonequilibrium two-phase flow. However, most of researches did not provide an accurate empirical formal for oscillation frequency. In addition, there is no sufficient information for the intensity of the pressure fluctuation (PF), which is crucial for evaluating the effect of condensation on the metrology of flow rate of the sonic nozzle. Thus, the characteristics of frequency and intensity of oscillating condensation flow in the sonic nozzle at different inlet humidities, temperatures, and carrier gas pressures should be investigated further. In this article, the experiments on the unsteady nonequilibrium condensation flow of the moist air in the sonic nozzle were conducted using an adjustable humidification apparatus. The accurate mathematical model of vapor condensation was built and validated by experiment result. Then, the frequency and intensity of PF signal of self-excited oscillation were extracted combining experiment and physical simulation model. A more accurate semiempirical formula of dimensionless frequency was obtained as a function of the inlet relative humidity and saturated mass fraction of water vapor. At last, the nonstationary features were also discovered by continuous wavelet transform (CWT) that increases the complexity of fluid flow.

II. INSTABILITY AND PERIODICITY OF THE TRANSONIC CONDENSATION FLOW
Nonequilibrium condensation phenomenon in sonic nozzle can be qualitatively described, as shown in Fig. 1(a) and (b).
An unsaturated water vapor air mixture enters into the nozzle, both gas pressure and temperature drop and reach the saturation state near nozzle throat [see region I shown in Fig. 1(b)].
Owing to the absence of condensation surface, the water vapor becomes supersaturated and several condensation nuclei are formed (region II). The number is small at first and no condensation appears in this regime [42]. Then, the measurable condensation occurs at Wilson point and visible fog appears (region III). These tiny drops will continue to grow gradually along with a pressure jump, which is affected by inlet humidity, temperature, and carrier gas pressure. The dimensionless expression of critical heat addition Q cr [43] leading to the thermal choking is expressed as where q cr is critical heat addition, c p is specific heat capacity, T is temperature, M is Mach number, γ is isentropic exponent, and 0 < Q cr < 1.0417 for 1 < M < ∞. The subscripts 1 and 2 denote the initial and final states of control surface at heat addition region. When the heat addition is larger than a critical value, the condensation will be choked and an induced shock wave might appear.
When the heat addition increases, the flow will not be steady and there will occur periodic self-excited oscillation with a particular frequency and intensity. The unsteady flow frequency of condensation f 0 for a 2-D nozzle was gained by Wegener and Cagliostro [33] with a semiempirical equation as follows: where f is the frequency of PF of self-excited oscillation, 0 is the relative humidity, w 0 is the mass fraction of water vapor, l is the characteristic length, and a cr is the critical sound speed. However, when 0 ≤ 100%, it is difficult to obtain the frequency of flow oscillation accurately from (2) of which the calculating error will reach 60%. What is worse, the intensity of pressure oscillation, which implies the tendency of condensing process, is not provided in (2).

A. Adjustable Humidification Apparatus
The experiments were conducted based on an adjustable humidification apparatus, as shown in Fig. 2(a) and (b), by which the inlet humidity, temperature, and carrier gas pressure are adjusted synchronously. The compressed air is filtered and purified by degreaser and dryer. Then, the pure air is stored in the gas holder. The gas pressure is controlled by the electric control valve from 1 to 6 bar. The temperature is adjusted by a tubular pipe heater. For controlling the relative humidity, the micrometer droplets are generated by the high-pressure microfog generator, in which the liquid flowrate (Q mL = 0-17 L/h) and the droplet size (maximum diameter of 50 μm) are adjusted by the hydraulic diaphragm metering pump (HDMP) and a high-pressure (HP, P L = 0-12.4 MPa) microsprayer nozzle. Then, the small droplets mix with the heated gas flow in the evaporator and then completely evaporate into the moist air, which can ensure that no droplet exists at the upstream of the nozzle. The humidity and temperature of the moist air can be accurately controlled in the range of 0%-100% and 30-50 • C, respectively. A 3-D toroidal-throat sonic nozzle of which the throat diameter is 10 mm and curvature radius is 870 mm was designed, as plotted in Fig. 2(b). The measurement instruments used in the adjustable humidification apparatus are shown in Table I.

B. Microsize Pressure Measuring System
Time-averaged pressure signals along the axis of nozzle and the PF signals at a specific point near nozzle wall are measured using the pressure probes. The accuracy is 0.2% and the pressure range is 0-8 bar. For the installation of the pressure probes, there are 12 holes drilled in nozzle wall of which the diameter is only 1.0 mm and they are numbered 1-12, respectively, as shown in Fig. 2(b). The locations of holes are shown in Table II, where x = 0 mm is the location of throat. The pressure probe array consisting of 12 pressure probes is used to measure time-averaged pressure values at different positions, as shown in Fig. 3(a).
PF signals are measured at position of No.10, where the condensation intensity is the strongest in the most cases [13].  According to (2), the frequency range of self-excited oscillation in this experiment is estimated to be 1336-3774 Hz. To measure this high-frequency pressure accurately, it is necessary to consider the frequency response characteristic of the pressure measuring system. Limited by the space and also to minimize the disturbance to flow field, a microsize pressure probe should be designed optimally in this experiment. However, a small diameter probe has adverse effect on the frequency band of the pressure measuring system. Thus, in order to balance the relationship of them and improve the amplitude-frequency characteristic, the model of pressure measuring system is analyzed first. A pressure measuring system consisting of a probe, a cavity and a pressure sensor are shown in Fig. 4, where r , l, and V are the radium and length of the probe, and the volume of the cavity, respectively. One end of the probe is inserted in the flow field and the other end is contacted to the pressure sensor; p i and p o are the actual and measured pressure values, respectively. The frequency response characteristic of the pressure measuring system is calculated by the following equation from the B-T model [44]: where V t = πr 2 l is the probe volume, k is polytropic constant for the volumes, σ is dimensionless increase in transducer volume due to diaphragm deflection, and n and φ are expressed, respectively, as follows: where γ is specific heat ratio, J 0 and J 2 are Bessel function of first kind of order 0 and 2, α is shear wavenumber, a measure of the wall shearing effects, Pr is Prandtl number, ω is angular frequency, and a 0 is the mean velocity of sound where p s is mean pressure, ρ s is mean density, μ is absolute fluid viscosity, and i is imaginary unit. According to the B-T model, the pressure measuring system is designed to be r = 0.5 mm, l = 11 mm, and V = 3.98 mm 3 . The amplitude-frequency characteristic curve obtained is shown in Fig. 5. It is shown that the cutoff frequency (where amplitude ratio drops to −3 dB) is 7337 Hz and is larger than the maximum estimated frequency. Thus, the designed pressure measuring system meets the measurement requirement. The Kulite MEMS quickly response pressure sensor with accuracy of 0.2% is applied to measure the PF signals, as shown in Fig. 3(b).
The raw signal was magnified by a voltage amplifier and processed by the tenth-order low-pass filter for the purpose of anti-interference, of which the cutoff frequency is 10 kHz. The pressure signals were sampled by NI-USB-6251 and LabVIEW. The sampling rate can be 1.25 MS/s for single channel and the average rate can be 1.00 MS/s for multiple channels. The sampling frequency is 40 kHz.

A. Signal Processing Method
The time-averaged distribution and fluctuation information of the pressure are analyzed from several raw signals. Amplitude-frequency characteristic curve of pressure measuring system.
For time-averaged distribution, the pressure signal obtained from the pressure measuring system is filtered and averaged in a period of time. Then, the time-averaged pressure is used to obtain the pressure distribution along the axis of nozzle and validate the mathematical model during the nonequilibrium condensation process under different conditions of humidity, temperature, and carrier gas pressure.
The PF information is obtained from the unstable pressure data at the strongest condensation position No. 10. The dominant frequency is calculated in frequency domain using power spectral density (PSD) [45].
where the dominant frequency is obtained according to the spectrum line with the maximum amplitude. According to Parseval's theorem, the energy of a signal in time domain is equal to that in frequency domain. And the variance of P(t) can be given by the integral of the PSD against frequency. Thus, the intensity of PF from frequency f 1 to f 2 is where σ is the standard deviation of the p(t), p i is the i th pressure value, andp is the mean value. It reflects the energy of pressure fluctuation. The frequency and PF are calculated under different conditions of humidity, temperature, and carrier gas pressure. Considering the pressure fluctuation signal is often nonstationary and wavelet analysis is effective for the modulated and complex signal, it is analyzed based on the wavelet method.
Wavelet transform can be categorized as CWT and discrete wavelet transform (DWT) [46]. The CWT of a signal f (t) is where ψ(t) ∈ L 2 (R) is the mother wavelet, a is the scale factor, and b is the time shifting factor. The DWT is used by discretizing a and b [47] in terms of a power series. By using DWT, pressure fluctuation signals are decomposed into different frequency components. The parameter, flatness (FF) [48], [49] of each component, which can measure the sharpness of the distribution and can be used as a measure of intermittency, is calculated by To investigate the nonstationary signal in detail, CWT is used to obtain the characteristic in time-frequency domain. Generally, the wavelet coefficient can be transformed to the local wavelet energy (LWE) coefficient WL by It can enhance the energy of each frequency component, while the scale characteristic of the wavelet coefficients remains unchanged [46]. To quantitatively represent the instantaneous frequency and its intensity, wavelet ridge [50], [51] is extracted from LWE coefficient based on modulus maxima.

B. Mathematical Model Validated by Time-Averaged Pressure Distribution
At present, the mathematical CFD model provides attractive features for analyzing this unsteady nonequilibrium condensation with low cost compared with experiment. For the purpose of properly assessing the uncertainty of a mathematical model, it must be validated by sufficient experiments. As we know, in this mathematical model, the critical radius of droplet r * is calculated by r * = 2σ/(ρ d G 1 ), where ρ d is the density of dispersed phases, G 1 is bulk Gibbs free energy change, and σ is surface tension, a sensitive parameter that should be corrected by the nucleation bulk tension factor (NBTF) (σ = σ bulk × NBTF) [43]. In this model, the value of the NBTF is set as 1.02 determined by the experimental validation with accuracy of 2.0%. The details of this mathematical model can be found in [52].
The experimental time-averaged pressure distribution under different conditions of inlet humidity 0 , temperature T 0 , and carrier gas pressure P 0 is shown in Figs. 6(a) and (c), 7, and 8, where P/P 0 is the ratio of measured time-averaged pressure and inlet stagnation pressure. Meantime, the physical simulation results at the same conditions are also plotted in the figures. In addition, the latent heat addition rate under different values of 0 is shown in Fig. 6(b) and (d). The results show that the experimental results are well consistent with the physical simulation data and the pressure jump can be captured accurately.
In the conditions of P 0 = 300 kPa, T 0 = 50 • and P 0 = 400 kPa, T 0 = 50 • , respectively, Fig. 6(a) and (c) shows that with the increasing of 0 , the position of the pressure jump moves upstream and the pressure value increases. The onset point occurs at the downstream of the throat at first and then moves upstream. It means that the change of humidity has obvious effect on the condensation position and intensity. As the relative humidity is increased, the condensation is more likely to occur and the onset point occurs earlier in the nozzle with stronger intensity. Accordingly, as shown in Fig. 6(b) and (d), at low relatively humidity, the latent heat addition rate  q l is small. With 0 increasing, the release position of latent heat moves upstream at first which is consistent with the onset point, which means that the condensation is accompanied by the release of latent heat.
The condensation gets strong as the temperature increases, as shown in Fig. 7. The experiment conditions are P 0 = 300 kPa, 0 = 80% and P 0 = 500 kPa, 0 = 80%. With the increase in temperature, the position changes slightly, which means that the effect of temperature on condensation position is less than humidity.
As shown in Fig. 8, with the increase in carrier gas pressure (T 0 = 50 • , 0 = 80% and T 0 = 30 • , 0 = 89%), the position of pressure jump moves upstream, while the relative intensity (the change of the pressure ratio) becomes weak. The reason is that the increase in carrier gas pressure, which makes the change of the pressure ratio, decreases.  In a word, the position of condensation moves upstream with 0 , T 0 , and P 0 increasing. And the increase of 0 and T 0 enhances the relative intensity of the condensation, but the increase of P 0 reduces it. In addition, according to Figs. 6(a) and (c), 7, and 8, the condensation onset closest to the inlet is at x = −2.6 mm with P 0 = 400 kPa, T 0 = 50 • C, and 0 = 97%.
As shown in Fig. 1, there is a pressure jump during condensation process. Corresponding to Figs. 6, 7, and 8, the position x p of pressure jump, as shown in Fig. 6(a) and its quantity P j , which is the pressure increment comparing with the pressure of frozen flow at the same time, are listed in Table III. It shows that the maximum pressure jump quantity is 35.5 kPa at x = 10.2 mm with P 0 = 500 kPa, T 0 = 50 • C, and 0 = 80%.
From the above results shown in Figs. 6, 7, and 8, it can be seen that the NBTF value of 1.02 is appropriate for present mathematical model, which can ensure the adaptability of the model. Thus, the present model is accurate enough to apply to the further research of pressure fluctuation.

C. Feature Extraction and Semiempirical Relation for Pressure Fluctuation
The frequency and intensity information are extracted and analyzed in detail. First of all, the signal is filtered by DWT.
In order to calculate the DWT coefficients, Daubechies 4 was chosen as the mother wavelet and an eight-level DWT was carried out on the raw signal. According to the estimated frequency range of the signal, D3 (2500-5000 Hz) and D4 (1250-2500 Hz) were chosen to be reconstructed.
The filtered signal in time and frequency domain is shown in Fig. 9. Calculated by (8) and (9), the main frequency is 2705 Hz, where the power spectrum P f is 0.17 W/Hz, as shown in Fig 9(b).
The frequency and intensity of pressure fluctuation signals are calculated and shown in Figs. 10-12. The results of the experiment and mathematical model are also compared. It can be seen that this physical simulation model has an excellent predictive perform. The fit curves in these figures also show the change tendency of frequency and intensity of pressure fluctuation.
For P 0 = 250 kPa and T 0 = 30 • , Fig. 10 shows that the frequency increases with 0 increasing, which is corresponding to (2). The unsteady flow occurs at 0 ≥ 60%. Then, along with 0 increasing, the intensity of PF will increase rapidly at first and then start to go down generally. That is because the condensation is more likely to occur and the position moves upstream with humidity increasing. The larger released latent heat and lower critical heat addition jointly result in stronger oscillating state; thus, the frequency of self-excited oscillation is higher. In higher humidity, the oscillating condensation shock moves from divergent section to the convergent section, and the supersaturation at condensation region becomes weak leading to oscillation intensity decreasing.
At P 0 = 360 kPa and 0 = 87%, as shown in Fig. 11, the temperature limit for unsteady flow is 22 • C. With temperature increasing, the frequency also increases, especially at the lower temperature region. The intensity value of PF in the whole temperature range is changing, which is closed to zero at low temperature and keeps rising when temperature is higher. For the flow with the same Mach number, it is easier to reach the critical heat addition at high-temperature condition. Thus, more released heat leads to a high-frequency oscillation. Due to the effect of temperature on condensation position is less than humidity, the frequency changes more gently with temperature increasing.  As shown in Fig. 12, when T 0 = 40 • and 0 = 81%, the unsteady flow occurs at P 0 ≤ 620 kPa. With inlet carrier gas pressure increasing, both frequency and intensity PF of the pressure signal increase at first and then go down at high pressure. When the pressure is low, the released speed of the latent heat increases with the carrier gas pressure increasing. Thus, the frequency increases while the intensity enhances. In high-pressure region, the released speed is constant, but the critical heat addition continues to increase rapidly leading to a weaker oscillation flow with lower frequency and intensity. The unsteady flow will switch to the stable state with inlet pressure further increasing.
The ranges of humidity, temperature, and carrier gas pressure are listed in Table IV. The frequency f and intensity PF obtained under these conditions are also listed in it.  The values of dimensionless frequency f 0 under the experiment conditions are calculated by semiempirical (2), as shown in Fig. 13(a), where the line denotes that the predicted result is equal to the experimental result. The results of (2) deviate from the experiment results seriously. The maximum error is 60%, indicating that the semiempirical (2) has poor predictability when applying to 3-D nozzle in this experiment. Thus, it is necessary to deduce a new semiempirical equation.
Reduced by Buckingham's theorem, the dimensionless frequency f 0 can be obtained from the frequency of the unsteady flow f [33], as shown in (13), where l is the characteristic nozzle length and l = (R * h * ) 1/2 , R * is the throat radius of curvature, h * is the throat diameter, L is the latent heat of condensation, and τ c is the characteristic time at condensation condition. The subscript v denotes vapor and ∞ denotes referring to flat surface of liquid.
The first two terms P v0 /P ∞0 and P v0 /P 0 are, respectively, proportional to the inlet relative humidity 0 and mass fraction of water vapor w 0 . Moreover, w 0 is a function of 0 , as shown in (14), where w s is the saturated mass fraction of water vapor In order to make the variables independent of each other, w s is chosen as the dependent variable. Thus, f 0 can be expressed as Due to the small interval of inlet temperature, L/(c p T 0 ) is regarded as a constant. For fixed γ and a small range of nozzle geometries h * /l, τ c a * /l is expected to show little variation. Finally, a simple semiempirical function can be obtained as Based on the above results of the experiment and mathematical model, the final identified semiempirical equation is where a 0 = 1.32, b 1 = 1.79, b 0 = −0.73, c 2 = −8.58 × 10 −6 , c 1 = 0.0013, and c 0 = 0.70. The predicted result of (17) is plotted in Fig. 13(b), in which the scatters are distributed near the line. The average error of (17) is 5.51%, the maximum error is 13.53%, the sum of squares error is 0.08, and R 2 is 0.86. It is indicated that the error is reduced significantly compared with the semiempirical equation of Wegner.

D. Nonstationarity of Pressure Fluctuation
To analyze the nonstationarity of pressure fluctuation, two pressure fluctuation signals with typical nonstationary characteristic were chosen and compared to illustrate the signal features. The experiment conditions are as follows: for Signal 1, P 0 = 360 kPa, T 0 = 40 • C, and 0 = 85%; and for Signal 2, P 0 = 300 kPa, T 0 = 25 • C, and 0 = 88%.
The characteristic of time series reconstructed by DWT is analyzed preliminarily. In time domain, probability density function (PDF) of the pressure increment P(t, t) = P(t + t) − P(t) is used to reflect the probability distribution of the pressure changing rate. Fig. 14 shows the PDF of pressure increment P for time delay t = 5 ms. Compared with Signal 2, the shape of Signal 1 is wider and shorter. The ranges of P are −10-10 and −5-5 kPa, respectively, for the two signals. The P values are dispersed in Signal 1 while concentrate in a smaller range in Signal 2. It is illustrated that there are more drastic changes in Signal 1.
Since the pressure fluctuation signals contain several frequency components, the flatness FF of A8, D8-D1 corresponding to different frequency ranges is calculated by (11). The results of Signals 1 and 2 are shown in Table V. For Gaussian distribution, the value of FF is 3 [53]. For intermittent systems, the flatness can also be seen as the ratio of time spent under quiescent conditions to the time spent under active conditions. The value of FF increases with frequency increasing, indicating that high-frequency components of these two signals are more intermittent. Especially for the value of D3 and D4, Signal 1 with higher FF is more intermittent than Signal 2.
In order to characterize the nonstationary signal on its local frequency properties in detail, Signals 1 and 2 are transformed into time-frequency domain by CWT. The complex Morlet wavelet was chosen as the mother wavelet. As shown in Fig. 15(a) and (c), there are the LWE map derived from CWT coefficients for Signals 1 and 2, which are calculated by (12). The estimated scale is located within the range of 1000-1300, namely a frequency range is 1000-5000 Hz. In the LWE map, the dark color represents greater LWE modulus, while the light color means smaller absolute values. Quantitatively, wavelet ridge in the above frequency range is extracted to show the instantaneous frequency of the signal, as shown in Fig. 15(c) and (d).
From the result of Signal 1 [see Fig. 15(a) and (b)], it is found that on the larger scale which is corresponding to lower frequency, there are dark areas that occur alternately. Several dark lines occur occasionally with time. In the scale range of 1200-1300, there is a dark band corresponding to the frequency of 2.66 kHz shown in Fig. 15(b). At t v = 0.76 s, there occurs a frequency saltation from 2.66 to 2.81 kHz. After the saltation, the energy of the frequency of 2.66 kHz becomes weak. Meanwhile, there are also some signals with high frequency but weak energy. The LWE map of Signal 2, as shown in Fig. 15(c), is clearer than that of Signal 1. There is a clear dark band corresponding to frequency of 2.18 kHz. But the color becomes light obviously after time of t d = 1.18 s, meaning its energy attenuates.
For frequency of 2.66 and 2.18 kHz, respectively, of Signals 1 and 2, the LWE coefficients are extracted and shown in Fig. 16, and it can be seen that the coefficients decrease at t v = 0.76 s and t d = 1.18 s. By contrast, the coefficients of Signal 1 are larger than that of Signal 2 because of its higher energy. And there are more changes in Signal 1.
In addition, the specific information of wavelet ridge is listed in Table VI, where t c is the time when the signal occurs a sudden change including frequency saltation for Signal 1 and energy attenuation for Signal 2.
It is shown that the pressure fluctuation signals probably are nonstationary, both its frequency and intensity. In addition, the nonstationary features extracted from CWT are match up with the results of PDF and FF in Section IV-C. Compared with Signal 2, Signal 1 is more intermittent. These results are benefit for analyzing the effects of humidity, temperature, and carrier gas pressure on the vapor condensation of moist air in a sonic nozzle and guiding the sonic nozzle design and mass flow-rate measurement.

V. CONCLUSION
The nonequilibrium condensation of moist air in the sonic nozzle was investigated combining a series of experiments and physical simulation. The results indicated that along with 0 increasing, the frequency of pressure fluctuation increases, while the intensity PF increases rapidly at first and then goes down generally. The frequency increases with T 0 increasing and the intensity PF is closed to zero at low temperature and keeps rising when temperature is higher. With P 0 increasing, both frequency and intensity PF increase at first and then go down. The new semiempirical relation of dimensionless frequency at various conditions is f 0 = 1.32 (1.79 0 − 0.73) (−8.58 × 10 −6 w −2 s + 0.0013 w −1 s + 0.70) and the average error is reduced to 5.51%. The signal nonstationarity was also observed using CWT. For the typical nonstationary Signal 1, it occurs a frequency saltation from 2.66 to 2.81 kHz at 0.76 s. For Signal 2, its energy attenuates after 1.18 s. The results above are useful in assessing the effect of vapor condensation on the accuracy and stability of mass flow rate of the sonic nozzle.
Yiming Li was born in Hebei, China, in 1995. She is currently pursuing the master's degree with Tianjin University, Tianjin, China.
Her research interests are in the field of single-phase and multiphase flow measurement. He is currently a Marie Sklodowska-Curie Individual Fellow with the Faculty of Engineering, University of Nottingham, U.K. He focuses on the development and application of computational fluid dynamics (CFD) modeling for multiphase flow in complex systems involving supersonic flows, particles, droplets, and gas separation.

Chao
Yuhe Tian was born in Hebei, China, in 1996. She is currently pursuing the master's degree with the Tianjin University of Technology, Tianjin, China.
Her research interests are in the field of single-phase and multiphase flow measurement.