# Performance Study of a Fluidic Hammer Controlled by an Output-Fed Bistable Fluidic Oscillator
## 4 Results and Discussion
### Performance Analysis Related to the Flow Behavior
The dimensions of the parameters in all cases examined in this study are listed in Table 2.  
In order to investigate the effect of operational parameter (i.e., supply flow rate) on the flow behavior in the fluidic oscillator, a series of computations were performed on the baseline fluidic hammer. Figure 6 shows the relationship between pressure recovery and flow recovery at different supply flow rate obtained by the CFD simulation. The corresponding Reynolds numbers ranged from 2.90 \(\times\) 10\({}^{6}\) to 5.22 \(\times\) 10\({}^{6}\). The diagram indicates that \(\mu_{\text{q}}\) is found as a function of \(\mu_{\text{p}}\) and the relationship between them is independent of Reynolds number (or the supply flow rate). The variations of pressure recovery versus flow recovery can be divided into two sections, corresponding to two states of the supply jet. As displayed in Figure 6, at the state of a-b, the pressure recovery changes sharply as the \begin{table}
\begin{tabular}{c c c c c} \hline
**Case No.** & \(D_{\text{p}}\) **(mm)** & \(D_{\text{r}}\) **(mm)** & \(m\) **(kg)** & \(L\) **(mm)** \\ \hline  
\hline \end{tabular}
\end{table} 
Table 2: The values of parameters of the impact body investigated in the current study. flow recovery increases. It can be thus attributed because of the delay caused by the fluid inertia and the finite time for the acoustic waves propagating in the feedback loops, the jet-switching deflection process has not finished before the impact body rebounds and moves forward. While at the state of b-c, the supply jet has already deflected and switched its attachment to the other side, and drives the impact body to move forward. As shown in Figure 6, the data shown at the section of b-c has a linear fit, made for supply flow rate Q\({}_{\text{S}}\) = 125 L/min. The value of the adjusted squared multiple correlation coefficient is 0.99388, near to 1.0, suggesting that the pressure recovery is simply negatively and linearly related with the flow recovery, and as the value of the flow recovery increases the value of the pressure recovery decreases. The point \"d\" at the fitting line, where the value of the flow recovery is 1.00, indicates that the flow rate of output flow is equal to that of the supply flow. As the flow recovery increases continuously, there will be extra liquid entrained into the output channel due to the escalating jet-pump effect caused by the diminishing fluid resistance for entering the rear chamber of the cylinder. In order to make the presentation of the CFD computation results more general and try to find some trace for designing fluidic hammers with high performance, the general form of the fitting line equation is expressed as follows:  
\[\mu_{\text{p}}=\mu_{p0}-C_{pq}\mu_{\text{q}} \tag{6}\]  
where the intercept \(\mu_{p0}\) of Equation (6) can be regarded as the value of the pressure recovery when the velocity of the impact body is 0 and thus all of the supply flow is discharged into the vents, and the slope \(C_{pq}\) represents a dimensionless characteristic coefficient between the pressure recovery and flow recovery. Pressure drop P\({}_{\text{s}}\) can be expressed as:  
\[\text{P}_{\text{s}}\text{=}\mu^{2}/2 \tag{7}\]  
Figure 6: The pressure recovery vs flow recovery dependences obtained at different Reynolds numbersin the forward movement of the impact body. Combining Equations (4)-(7), the following expression is obtained:  
\[v=\frac{\left(\mu_{p0}-\mu_{p}\right)A_{s}u}{C_{pq}A_{p}}=\frac{\left(\rho\mu_{p0 }u^{2}-2P_{o}\right)A_{s}u}{C_{pq}A_{p}\mu^{2}} \tag{8}\]  
\[v_{\max}=\frac{\left(\mu_{p0}-\mu_{p\min}\right)A_{s}u}{C_{pq}A_{p}} \tag{9}\]  
where \(v_{\max}\) is the maximum value of \(v\), and \(\mu_{p\min}\) is the minimum value of \(\mu_{p}\).