## 1 Introduction  
Flow measurement is crucial to flow control in various application including water supply, fuel pumps, and air conditioning systems [1, 2, 3]. A common device used to measure flow characteristics is the flow meter. However, the installation of a flow meter in large applications can be expensive. In addition, it often decreases the flow rate and changes the flow characteristics due to its invasive components. To solve this issue, non-invasive flow meters attaching to the surface of the pipeline have been developed to measure the flow rate without disturbing the fluid flow. Moreover, these devices incur lower maintenance costs, are easy to install, and do not affect the flow rate [4]. Thus far, non-invasive flow measurements using ultrasonic waves and acceleration signals have been developed [5, 6, 7, 8, 9, 10, 11]. By considering the velocity profiles across a water pipeline, the flow rate using hybrid ultrasonic flow meter was calculated [5]. To validate the accuracy of the ultrasonic flow meter, the evaluation methods of the flow rate in gas pipeline were simulated by using numerical models based on using computational fluid dynamics [6]. In another study, the velocity under two-phase flow was measured using ultrasonic signals, the corresponding flow was visualized [7, 8]. The wavenumbers of the fluid conveying pipe by using the differences of acceleration, microphone signals, and added mass were estimated to predict its flow velocity [9, 10, 11]. Wavenumbers related to the characteristics of the pipeline system are essential for measuring the flow rate. The wavenumber characteristics of acoustic silencers were predicted by using numerical model based on finite difference method [12]. Monitoring techniques that detect leakage in pipelines using the vibration signals includingwavenumber transmitted by non-invasive flow meters have been developed [13; 14; 15; 16]. However, measurements of the wavenumber at its natural frequency without accounting for the resonance of the piping system is prone to errors. To minimize the fundamental error, an improved measurement method that considers the resonance of the pipelines required.  
To accurately measure flow velocity in pipe system, well-defined boundary conditions of the system must be considered. A transfer function method based on the wave propagation analysis by considering boundary condition was proposed to precisely predict the wavenumber of complex structure for calculating dynamic properties [17; 18; 19; 20; 21]. This method is used to estimate the frequency-dependent complex modulus of a uniform beam and compliant materials by predicting its wavenumber [17; 18]. The beads on the surface of microcantilever beam were detected by calculating their position and mass [19]. The effective modulus of resonant metamaterials for longitudinal and flexural waves were estimated using experimental transfer functions [20; 21]. In this way, the mechanical properties measured by the transfer function method have a minimized error at the resonant frequency of the system.  
This study aims to predict the flow velocity of a fluid-conveying pipe by using acceleration signals at pipe surfaces as shown in Figure 1. To this end, the flexural wave propagation in a clamped-clamped pipe system was analyzed. The straight pipe clamped at both ends was modeled to the flexural wave propagation of the fluid-conveying pipe. Using this model, the characteristics of wavenumbers of pipeline with varying flow velocities were derived, and their mode shapes were investigated. To validate the proposed method, acceleration signals were replaced with numerical results based on the finite element method (FEM). This method enables the direct calculation of the parameters for the evaluation of the pipe system in a precise and consistent manner.