## 2 Theoretical Model
### Friction Coupling Model
#### 2.3.3 The Unconstant Friction Model  
The unconstant friction model is comprised of quasi-steady friction term and additional friction term. In view of the restrictions to the application of the two-dimensional friction model and the irreversible thermodynamic model in one-dimensional friction model, the two types are no longer discussed in this paper.  
(1) The empirical formulae model based on instantaneous acceleration  
Daily considered that the friction coupling was related to unsteady velocity and instantaneous acceleration and obtained an initial unsteady friction model. [PERSON] thought that the friction coupling was related to mean velocity of the fluid \(V\), instantaneous acceleration \(\partial V/\partial t\) and convective acceleration \(\partial V/\partial z\). Based on the Daily friction model, the classical [PERSON] model was presented and widely applied.  
The shear friction force in the model is as follows  
\[\tau_{w}=\tau_{ws}+\tau_{wu} \tag{21}\]  
\[\tau_{ws}=\frac{\rho_{f}fV|V|}{8} \tag{22}\]  
\[\tau_{wu}=\frac{\rho_{f}D}{4}.k_{3}\left(\frac{\partial V}{\partial t}-\alpha \frac{\partial V}{\partial z}\right) \tag{23}\]  
where \(\alpha\) is wave velocity of water hammer; \(k_{3}\) is the Brunone friction coefficient. In numerical simulation, \(k_{3}\) is an usually empirical value which is determined by trial or experiment, so its application is limited.  
In the Brunone model, the friction coupling of pipelines are combined with the fluid characteristics, and the influence of the convection term on friction coupling is taken into consideration. At the same time, the pressure wave velocity is included in the friction model, so the unsteady friction coupling is simulated more truly. Based on the global acceleration hypothesis, the semi-experience and semi-theory calculating formula of \(k_{3}\) in the Brunone friction model was given by [PERSON] as follows [18]  
\[k_{3}=\frac{\sqrt{7.41/Re^{\alpha_{4}}}}{2} \tag{24}\]  
\[k_{4}=\log\frac{14.3}{Re^{0.05}} \tag{25}\]  
It has been proven by many researchers that this formula can be applied to the smooth turbulent hydraulic region, and can be used directly at time of solving the friction coupling of the pipeline owing to accurate calculation results, which could provide a good analysis on friction coupling under high Reynolds number conditions.  
(2) The weight function model considering the influence of historical flow velocity and acceleration on the current flow regime  
Based on the N-S equation, [PERSON] presented the weight function friction model related to frequency. The weight function which could reflect the impact of historical flow velocity and historical acceleration on the current flow state is added to the model. The shear friction force in this model is as follows  
\[\tau_{w}=\tau_{ws}+\tau_{wu} \tag{26}\]  
\[\tau_{ws}=\frac{4\rho_{f}\
u_{f}}{R}V(t) \tag{27}\]  
\[\tau_{wu}=\frac{2\rho_{f}\
u_{f}}{R}\int_{0}^{t}\frac{\partial V}{\partial t }(t_{1})W(t-t_{1})\mathrm{d}t_{1} \tag{28}\]  
where \(\tau_{w}\) is the wall shear friction force in the unsteady flow; \(\tau_{wu}\) is the wall additional shear force term in the unsteady flow; \(W\) is the weight function of the dimensionless time \(t\), which reflects the influence of historical flow velocity and historical acceleration on the current shear friction force.  
In the Zielke model, the current velocity information is achieved by doing the integral and weighting of historical information of the local velocity. The weight function is related to the calculation frequency, and the calculation results have high accuracy in the laminar flow. So, it is generally only used for laminar flow or turbulent flow with low Reynolds number and acknowledged as an accurate model in the frequency domain.