b. Closed-end

There are the following equations when the right-traveling wave d*p*<sup>R</sup> reaches the closed end of the right end.

$$\begin{cases} \mathbf{d}p\_{\mathbb{R}} = a\rho \mathbf{d}u\_{\mathbb{R}} \\ \mathbf{d}p\_{\mathbb{L}} = -a\rho \mathbf{d}u\_{\mathbb{L}} \\ \mathbf{d}p = \mathbf{d}p\_{\mathbb{R}} + \mathbf{d}p\_{\mathbb{L}} \\ \mathbf{d}u = \mathbf{d}u\_{\mathbb{R}} + \mathbf{d}u\_{\mathbb{L}} = \mathbf{0} \end{cases} \tag{7}$$

Thus, d*p*<sup>L</sup> = d*p*R, d*u*<sup>L</sup> = �d*u*R, d*p* = 2d*p*R, d*v* = 0.

It can be seen that the amplitude of the incident pressure wave and reflected pressure wave is the same after the pressure wave propagates to the nozzle when the needle is closed. However, the velocity disturbance value is the opposite. The velocity change at the nozzle is zero and the pressure variation is twice the amplitude of the incident pressure wave. The reflection with the property of incident wave and reflected wave is the same, that is, an expansion wave or a compression wave at the same time, and the amplitude of pressure disturbance is the same is called complete positive reflection. Therefore, the pressure wave reflection when the needle valve is closed is a complete positive reflection.

c. Orifice flow end

As shown in **Figure 2**, the fuel flow from the C-C boundary surface of the pipeline with section *F* through the orifice of section *Ft* to the space where the backpressure is *pc*. Assuming the initial state of fuel is *p*<sup>0</sup> = *pc* and *u*<sup>0</sup> = 0, the Bernoulli equation and continuity equation from C-C boundary surface to the minimum throat section t-t can be established as follows.

$$\begin{cases} p + \frac{1}{2}\rho u^2 = p\_c + \frac{1}{2}\rho u\_t^2\\ Fu = a\_F F\_t u\_t \end{cases} \tag{8}$$

**Figure 2.** *Schematic of the orifice flow end.*

where *p* and *u* are the fuel pressure and flow rate at C-C boundary surface, *α<sup>F</sup>* is the orifice flow coefficient, and *ut* is the fuel flow velocity at t-t of minimum throat section.

Thus,

$$p = \frac{1}{2}\rho u^2 \frac{1 - \phi\_F^2}{\phi\_F^2} + p\_c \tag{9}$$

where, *φ<sup>F</sup>* = *αFFt*/*F*, which is the effective flow section ratio at the orifice end.

The above equation is the boundary condition equation of the orifice outflow of the high-pressure common rail fuel injection system. It can be seen from the above equation that when *φ<sup>F</sup>* = 0, *Ft* = 0, is a closed-end, and when *φ<sup>F</sup>* = 1, *Ft* = *F*, is a outlet flow opening end. However, the reflection at the boundary surface transitions from complete positive reflection to complete negative reflection when *φ<sup>F</sup>* changes from 0 to 1.
