## 3 Theoretical Modeling
### Dynamic Model Considering Friction Coupling of the BHP
#### 3.1.2 Establishment and Analysis of Friction Term of BHP  
In the FSI 14-equation for the BHP, Equations (6) and (7) are the incompressible viscous fluid flow continuity equation (N-S equation) [28], which includes the friction term for the BHP. The specific form is:  
\[f_{\rm f}(t)=2\pi r_{\rm r}\tau_{0}(t) \tag{20}\]  
\(\tau_{0}\) is the shear friction force of the fluid on the inner wall of the pipeline. The Laplace transform of Equation (20) is:  
\[F_{\rm f}(s)=2\pi r_{\rm r}\overset{\sim}{\tau}_{0}(s) \tag{21}\]  
According to the Newton friction formula, [PERSON] [29] established the wall shear force \(\tau_{0}\) in Equation (20) and obtained the relationship between the instantaneous laminar flow wall shear force \(\tau_{0}\) and the instantaneous average velocity and the weighted historical velocity. The specific expression is as follows:  
\[\tau_{\rm z}(t)=\frac{4\rho_{1}v_{1}}{r_{\rm r}}V(t)+\frac{4\rho_{1}v_{1}}{r_{ \rm r}}\int_{0}^{t}\frac{\partial V}{\partial t}(u)W(t-u)\mathrm{d}u \tag{22}\]  
where \(W\) is the weighting function of dimensionless time \(\tau\), \(\tau=v_{1}t/r_{\rm r}\). The expression of \(W(\tau)\) is:  
\[W(\tau)=\left\{\begin{array}{ll}\sum\limits_{i=1}^{5}e^{-m_{i}\tau},\,\tau \geq 0.02\\ \sum\limits_{i=1}^{6}n_{i}e^{(i-2)/2},\,\tau<0.02\end{array}\right. \tag{23}\]  
where the values for the weighted coefficients \(m_{i}\) and \(n_{i}\) are shown in Tables 2 and 3.  
Based on the theory proposed by [PERSON], [PERSON] proposed a friction model for the turbulent flow. He believes that the wall shear force is related to the average flow velocity \(V\), instantaneous acceleration \(\partial V/\partial t\), and convection acceleration \(\partial V/\partial z\)[30]. This model is widely used in the calculation of wall shear force in a turbulent flow. The expression of wall shear force in the [PERSON] model is:  
\[\tau_{B}=\frac{\rho_{1}fV|V|}{8}+\frac{\rho_{1}D}{4}.k_{3}\left(\frac{ \partial V}{\partial t}-\alpha\frac{\partial V}{\partial z}\right) \tag{24}\]  
Among them,\(f\) is the friction coefficient under turbulent flow; \(\alpha\) is the wave velocity of the water hammer; \(K_{3}\) is the Brunson friction coefficient; this numerical value generally is an empirical value, which is determined by test or trial calculation.  
Based on the [PERSON] friction model, [PERSON] established a semi-empirical and semi-theoretical formula for \(K_{3}\) by introducing the assumption of global acceleration [31]:  
\[k_{3}=\sqrt{7.41/Re^{k4}}/2 \tag{25}\]  
\[k_{4}=\log\Bigl{(}14.3/Re^{0.05}\Bigr{)} \tag{26}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \(m_{1}\) & \(m_{2}\) & \(m_{3}\) & \(m_{4}\) & \(m_{5}\) \\ \hline \(-26.3744\) & \(-70.80493\) & \(-135\) & \(-218.9216\) & \(-322.5544\) \\ \hline \end{tabular}
\end{table}
Table 2: Weighting coefficient \(m_{i}\).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \(n_{1}\) & \(n_{2}\) & \(n_{3}\) & \(n_{4}\) & \(n_{5}\) & \(n_{6}\) \\ \hline \(0.2821\) & \(-1.25\) & \(1.0579\) & \(0.9375\) & \(0.3967\) & \(0.3516\) \\ \hline \end{tabular}
\end{table}
Table 3: Weighting coefficient \(n_{i}\).  
The weighted function model by [PERSON] does not depend on empirical parameters, thus, it is more consistent with experimental test results on laminar flow and low Reynolds number turbulence calculations. In this paper, the fluid state in the BHP is the laminar flow, so the Zielke friction model is used as the friction term in the FSI 14-equation.