# A Review: Factors Affecting Internal Two-Phase Flow-Induced Vibrations
## 4 Compilation of Existing Correlations and Models for Flow-Induced Vibrations  
Tables 1 and 2 summarize the existing database on FIV and the correlations developed to predict FIV in multiphase flow. The focus of past literature is on determining the magnitude of fluctuating force and force spectrum for different geometries and various flow regimes. Major existing works on FIV due to two-phase flow along with their operating conditions and limitations are listed in Table 1. The correlations developed over time by various researchers are listed in Table 2. These correlations were developed for different fluids, void fractions, pressure conditions, and geometries. The assumptions made to derive each of these equations are also listed for better understanding. \begin{table}
\begin{tabular}{c c c} \hline \hline
**References** & **Model** & **Remarks** \\ \hline [PERSON] et al. [48] & \(\overline{F_{RMS}}=\frac{F_{RMS}}{\rho_{1}V_{m}^{2}\left(\pi\frac{\rho_{1}^{2}}{4 }\right)}=CWe^{-0.4}\) & Two-phase flow is considered as a mixture with no slip between phases. Constant C needs to be determined for different conditions, C = 10 is the best fit, and C = 3.5 for slug flow in a 6-mm tube. \\ [PERSON] and [PERSON] [73; 98] & \(F=\left(\int_{in}PdA-\int_{out}PdA\right)-\frac{\partial}{\partial t}\int_{VC} \mu\rho dV-\int_{SC}\mu\rho\mu\cdot dV\) & \begin{tabular}{l} “Piston Flow Model”, developed for the horizontal slug flow through a pipe bend. \\ Flow regime was estimated from the liquid height without proper flow regime identification. \\ Oil and gas test section was used for the experimental database. \\ \end{tabular} \\ [PERSON] et al. [76] & \(\begin{array}{l}\text{Sluq flow, }F=\sqrt{2}\rho_{1}V_{slag}^{2}A\\ \text{Stratified and annular flow, }F=\sqrt{2}\rho_{m}V_{m}^{2}A\\ \end{array}\) & \begin{tabular}{l} Developed for a small 6-mm tube, based on consideration of momentum change alone. \\ Friction and gravity effects are not considered. \\ Taylor bubble diameter is equal to the pipe diameter. \\ \end{tabular} \\ [PERSON] et al. [66] & \(F_{FS_{s}}=-\frac{\partial}{\partial t}\iint\limits_{Q}\left(x_{f}\rho_{f}u_{ fx}+x_{g}\rho_{g}u_{gx}\right)dV\) & \begin{tabular}{l} Smooth transport of multiphase mixture through the pipe bend. \\ Elbow inlet and outlet boundary pressure and void fraction signals are required as input data. \\ \end{tabular} \\ [PERSON] et al. [66] & \(F_{FS_{s}}=-\frac{\partial}{\partial t}\iint\limits_{Q}\left(x_{f}\rho_{f}u_{ fx}+x_{g}\rho_{g}u_{gz}\right)dV\) & \begin{tabular}{l} Slug is considered as a homogenous two-phase mixture. \\ Elbow inlet and outlet boundary pressure and void fraction signals are required as input data. \\ \end{tabular} \\ [PERSON] et al. [66] & \(F_{FS_{s}}=-\frac{\partial}{\partial t}\iint\limits_{Q}\left(x_{f}\rho_{f}u_{ fx}+x_{g}\rho_{g}u_{gz}\right)dV\) & \begin{tabular}{l} Slug is considered as a homogenous two-phase mixture. \\ Elbow inlet and outlet boundary pressure and void fraction signals are required as input data. \\ \end{tabular} \\ [PERSON] et al. [66] & \(F_{FS_{s}}=-\frac{\partial}{\partial t}\iint\limits_{Q}\left(x_{f}\rho_{f}u_{ fx}+x_{g}\rho_{g}u_{gz}\right)dV\) & \
\begin{tabular}{l} The correlation was developed for a 6-inch diameter pipe, C = 25 was best fit. \\ Pressure and momentum fluctuations were considered. \\ Quasi-steady approach was used. \\ \end{tabular} \\ \hline \hline \end{tabular}
\begin{tabular}{c} \(F_{x}=\chi\,x+(p_{i,x^{\prime}}A-P_{o}A)+Mom_{x}\) \\ \(F_{y}=\chi\,y+(p_{i,x^{\prime}}A-P_{o}A)+Mom_{y}\) \\ \(\Gamma_{1}=\rho_{f}A\mu_{f}^{2},\Gamma_{2}=\alpha\rho_{g}A\mu_{g}^{2}+(1-\alpha )\rho_{f}A\mu_{f}^{2}\), \\ [PERSON] et al. & Conditions for when liquid front enters the control volume \\ \(\chi_{x}=\Gamma_{1};\chi_{y}=\Gamma_{2}\), \\ Conditions for when the mixture front enters the control volume \\ \(\chi_{x}=\Gamma_{2};\chi_{y}=\Gamma_{1}\), \\ where \(Mom_{x}\) and \(Mom_{y}\) represent the rate of change of \\ momentum in their respective coordinates. \\ \end{tabular}
\end{table}