## 3 Numerical Model for the Validation of Prediction Method  
To validate the transfer function method of predicting flow velocity, three acceleration signals were measured. These signals replace the numerical acceleration data based on the finite element method. When the transfer function method was performed with the numerical acceleration signals, the predicted flow velocity was found to be similar to the input flow velocity. The mechanical properties of the fluid and solid at zero flow velocity must be known before conducting this prediction method. Figure 3: Wavenumbers for the fluid-conveying pipe with (a) \(U=0\) and (b) \(U=60\) (m/s) at increased excitation frequency. The pipe was assumed to be the isotropic solid. The equation of motion governing a three-dimensional wave propagation in the pipe is [23]  
\[ \begin{array}{l}\rho_{s}\frac{\partial^{2}u_{x}}{\partial t^{2}}=\frac{ \partial\tau_{x}}{\partial x}+\frac{\partial\tau_{xy}}{\partial x}+\frac{ \partial\tau_{yz}}{\partial x}\\ \rho_{s}\frac{\partial^{2}u_{y}}{\partial t^{2}}=\frac{\partial\tau_{yx}}{ \partial x}+\frac{\partial\tau_{yz}}{\partial x}+\frac{\partial\tau_{yz}}{ \partial x}\\ \rho_{s}\frac{\partial^{2}u_{x}}{\partial t^{2}}=\frac{\partial\tau_{xx}}{ \partial x}+\frac{\partial\tau_{xy}}{\partial x}+\frac{\partial\tau_{z}}{ \partial x}\end{array}  \tag{10}\]  
where \(\rho_{s}\) is the density of the solid; \(u_{x}\), \(u_{y}\), and \(u_{z}\) are the displacements of the solid; \(\sigma\), \(\tau\) are the normal and shear stresses of the isotropic material expressed as  
\[\left\{ \begin{array}{l}\sigma_{x}\\ \sigma_{y}\\ \sigma_{z}\\ \tau_{xy}\\ \tau_{yz}\\ \tau_{zx}\end{array} \right\}=\frac{E}{(1+v)(1-2v)}\left[ \begin{array}{cccccc }1-v&v&v&0&0&0\\ v&1-v&v&0&0&0\\ v&v&1-v&0&0&0\\ 0&0&0&\frac{1-2v}{2}&0&0\\ 0&0&0&0&\frac{1-2v}{2}&0\\ 0&0&0&0&\frac{1-2v}{2}\end{array} \right]\left\{ \begin{array}{l}\frac{ \partial u_{x}}{\partial x}\\ \frac{\partial u_{y}}{\partial y}\\ \frac{\partial u_{z}}{\partial y}+\frac{\partial u_{y}}{\partial x}\\ \frac{\partial u_{z}}{\partial x}+\frac{\partial u_{z}}{\partial y}\\ \frac{\partial u_{z}}{\partial x}+\frac{\partial u_{z}}{\partial z}\end{array}  \right\}, \tag{11}\]  
where \(E\) is [PERSON]'s modulus; \(v\) is [PERSON]'s ratio. Substituting Equation (11) into Equation (10), the right-hand-side terms in Equation (10) as the displacements \(u_{x}\), \(u_{y}\), and \(u_{z}\) can be expressed. The equation of motion governing a three-dimensional pressure wave propagation in a fluid-conveying uniform flow is  
\[\frac{\partial^{2}p}{\partial x^{2}}+\frac{\partial^{2}p}{\partial y^{2}}+ \frac{\partial^{2}p}{\partial z^{2}}=\frac{1}{c^{2}}\bigg{(}\frac{\partial}{ \partial t}+U\frac{\partial}{\partial x}\bigg{)}^{2}p, \tag{12}\]  
where \(p\) is the pressure of the fluid propagating in the pipe; \(c\) is the speed of sound. For the fluid-structure interaction, the boundary condition used for the dynamic equilibrium at the contact surface between the pipe and fluid is  
\[\frac{\partial p}{\partial x}=-\rho_{f}\frac{\partial^{2}u_{x}}{\partial t^{2 }}u_{x},\ \frac{\partial p}{\partial y}=-\rho_{f}\frac{\partial^{2}u_{y}}{\partial t^{2 }},\ \frac{\partial p}{\partial z}=-\rho_{f}\frac{\partial^{2}u_{z}}{\partial t^{2 }}, \tag{13}\]  
where \(\rho_{f}\) is the density of the fluid. For the finite element analysis, the weak forms of Equations (10), (12), and (13) were obtained using a weighted integral. After approximating \(u_{x}\), \(u_{y}\), \(u_{z}\), and \(p\) using a shape function, the finite element equation was obtained for the vibration of the fluid-conveying pipe as  
\[\left[ \begin{array}{cc}\mathbf{M}_{s}&0\\ \rho_{f}\mathbf{S}&\mathbf{M}_{f}\end{array} \right]\left\{ \begin{array}{l} \ddot{\mathbf{d}}\\ \ddot{\mathbf{p}}\end{array} \right\}+\left[ \begin{array}{cc}0&0\\ 0&\mathbf{C}_{f}\end{array} \right]\left\{ \begin{array}{l}\dot{\mathbf{d}}\\ \ddot{\mathbf{p}}\end{array} \right\}+\left[ \begin{array}{cc}\mathbf{K}_{s}&- \mathbf{S}^{T}\\ 0&\mathbf{K}_{f}\end{array} \right]\left\{