# A Review: Factors Affecting Internal Two-Phase Flow-Induced Vibrations
## 3 Factors Affecting Dominant Frequency  
Literature on two-phase flow-induced vibration is focused on the magnitude of flow forces as well as the spectrum of forces. Techniques, such as fast Fourier transform (FFT) and power spectral density (PSD) are used to identify the peak/dominant frequency. Dominant frequency carries the most energy and this frequency has the highest potential to cause damage. To avoid resonance with structural component's natural frequency, it is important to determine the dominant frequency. It can be observed from previously reported literature that this frequency lies in the range of 0-50 Hz. Moreover, the literature reports several factors that affect the dominant/peak frequency in two-phase flow, which is discussed here in detail. [PERSON] et al. [64] analyzed the force spectrum from their experiments on U-bend with vertically upward two-phase flow. The authors reported that for the entire range of experiments, the main/dominant frequency increases linearly with superficial mixture velocity. Moreover, it was reported that the main frequency decreases with the increasing tube diameter. For 25% void fraction, narrow peaks are observed on force spectrum and as the velocity increases and flow transitions from spherical cap bubble flow to bubbly flow, the spectrum becomes wider and shows non-harmonic multiple peaks. For 50% void fraction, two harmonic frequency peaks are observed, and they increase linearly with velocity. These peaks are attributed to the passing slugs. After 7 m/s, the slug flow transition into bubbly flow and the force spectrum decrease and become wider. For 75% void fraction, one frequency peak is observed, which increases linearly with superficial mixture velocity until 14 m/s. This frequency corresponds to the slug frequency, where only one body of slug passes within a certain period. For 95% void fraction, the spectrum is considerably wider than all the other cases and increases with velocity. This can be explained by the presence of quasi-periodic excitation, which occurs in annular flow at the interface of liquid film and gas phase. Moreover, [PERSON] and [PERSON] [96] observed a few disturbances of comparable frequencies in their void fraction signals for annular flow, which contribute to the wider spectrum in annular flow. Furthermore, [PERSON] et al. [64] reported that slug frequency can be correlated to the peak frequency of force signal, and it is possible to obtain the peak frequency of force signal from slug frequency. The authors also proposed a formula, which is shown in Equation (14), to determine the peak frequency in diameter range of 20 to 52 mm for slug, unstable slug, churn, and churn/annular flows. This formula agrees with the experimental results, except for the case of 25% void fraction, where it overestimates the value, as shown below:  
Figure 5: Relationship between void fraction with liquid and gas flowrates, and flow regimes and excitation forces. \[f_{0}=0.081\frac{V_{m}\sqrt{1-\beta}}{D} \tag{14}\]  
where \(D\) is the tube diameter, \(V_{m}\) is the mixture velocity, and \(\beta\) is the void fraction. [PERSON] et al. [65] reported the effect of superficial velocities on dominant frequency for a vertical 90\({}^{\circ}\) elbow of diameter 0.0525 m (2 inch) in upward vertical flow. The superficial gas velocity was varied from 0.5 to 9.04 m/s, and superficial liquid velocity was varied from 0.642 to 5 m/s, covering slug and churn flow regimes. It was reported that with the increase in superficial liquid velocity, the dominant frequency initially increases and drops with a further increase as the bubbly flow approaches. The peak frequency varied between 1 and 7 Hz. In the case of increasing gas velocity, the dominant frequency increases first to the highest value after the transition into slug flow and drops slightly with further increase toward the churn flow regime before increasing again in annular flow. In this case, the peak frequency was also varied between 1 and 7 Hz. [PERSON] et al.