4.1. Flow rheology

The mass of the PAM required to prepare the desired solution concentration was measured using a digital scale (Explorer; Ohaus Corporation) with a 0.1 mg resolution. The additives were gradually added to a 0.5 L glass beaker filled with water and mixed using a stirrer (1103; Jenway) at a moderate constant rotational speed. The stirring bar size was selected to cover most of the container's surface to mitigate the adhesion of polymers to the container walls. Each master solution was stirred for two hours, and the resultant uniform, highly concentrated solution was then gradually added to the circulating water in the flow loop. The solution was allowed to circulate at a moderate pump speed for one hour, with the channel line bypassed, before doing any tests. Samples were obtained from the flow line using a sampling valve installed close to the channel's inlet (see figure 1). Later, the solution was undergone the most extreme cavitation scenario, and another sample was taken to explore the effect of cavitation on the rheology of the solution.

A short time after taking the samples, the rheological characteristics of the solutions were determined using a rotational rheometer (Kinexus lab+; NETZSCH-Gerätebau GmbH) with a double gap geometry (DG25). The bob's internal and external diameters were 23 and 25 mm, respectively, with an internal height of 57.5 mm. The cup had a diameter of 26.25 mm, and an insert diameter of 22 mm. Three independent measurements were performed for each sample at 20 $^{\circ }$C with the shear rate controlled during the tests. The average measurement values, with corresponding standard deviations, are reported here.

Figure 4(a) shows the variation of the solution viscosities versus the strain rate. The results indicate that the selected PAM solutions in water have shear-thinning behaviour. Such a fluid behaves as a Newtonian fluid at very low, $\mu ( \dot {\gamma } \ll 1) \rightarrow \mu _{0}$, and very high, $\mu ( \dot {\gamma } \gg 1) \rightarrow \mu _{\infty }$, shear rates (Bird, Armstrong & Hassager Reference Bird, Armstrong and Hassager1987). Shear-thinning fluids follow a power law relation for intermediate $\dot {\gamma }$ values, i.e. $\mu = K {\dot {\gamma }}^n$ for $1 < {\dot {\gamma }} < \infty$ (Bird et al. Reference Bird, Armstrong and Hassager1987). Here, $K$ and $n$ are correlation constants. The standard deviation of the viscosity measurements were more than $10\,\%$ for $\dot {\gamma } < 10\,{\rm s}^{-1}$ and $\dot {\gamma } > 500\,{\rm s}^{-1}$ and hence were discarded. This discrepancy is caused by surface tension that produces artefact shear-thickening at low shear rates and the development of unstable secondary flows for higher shears, where viscosity shows a sudden increase (Morrison Reference Morrison1998). The Correau–Yasuda (CY) model (Yasuda, Armstrong & Cohen Reference Yasuda, Armstrong and Cohen1981) was fitted to the measurement data to determine the solution viscosities at higher strain rates, and the results are plotted in figure 4(a) as dashed lines. The CY model is formulated as

(4.1)\begin{equation} \mu({\dot{\gamma}}) = \mu_{\infty} +(\mu_0 - \mu_{\infty}) {\left[ 1+( \lambda_{t} {\dot{\gamma}} )^a \right] } ^{(n-1)/a}, \end{equation}

where $\mu _0$ and $\mu _{\infty }$ are the zero and infinite shear viscosities. Here, $\lambda _{t}$ is a time constant, $n$ is the power-law index and $a$ is a constant parameter suggested by Yasuda et al. (Reference Yasuda, Armstrong and Cohen1981). As discussed by Syed Mustapha et al. (Reference Syed Mustapha, Phillips, Price, Moseley and Jones1999) and later by Escudier et al. (Reference Escudier, Gouldson, Pereira, Pinho and Poole2001), the Pearson correlation coefficient $r$ should be more than 0.9975 for a rheological model to be valid and reliable. Here, $r$ is (Escudier et al. Reference Escudier, Gouldson, Pereira, Pinho and Poole2001)

(4.2)\begin{equation} r = \frac{\displaystyle N \sum_{i=1}^{N} {\mu_{{M},i} \mu_{{CY},i}} - \sum_{i=1}^{N}{\mu_{{M},i}} \sum_{i=1}^{N}{\mu_{{CY},i}} } {\displaystyle \sqrt{{\left[ N \sum_{i=1}^{N}{{\mu}^2_{{M},i}} - {\left(\sum_{i=1}^{N}{\mu_{{M},i}} \right)}^2 \right]} {\left[ N \sum_{i=1}^{N}{{\mu}^2_{{CY},i}} - {\left(\sum_{i=1}^{N}{\mu_{{CY},i}} \right)}^2 \right] } } }, \end{equation}

where $N$ denotes the total number of viscosity measurements and the subscript ‘$M$’ means ‘measured’. The coefficient of determination $R$ is also calculated for each solution to evaluate the accuracy of the utilized model. Table 3 lists the obtained coefficients of the CY model for each PAM solution, with their associated $r$ and $R$ coefficients. For all cases, $r > 0.9992$ and $R > 0.9985$; hence, the obtained correlations are valid for estimating the viscosity values at higher strain rates.

Figure 4. (a) Variation of the dynamic viscosity of the pure water and PAM solutions versus the shear rate. A dashed line depicts the fitted CY model for each solution with a concentration of more than 100 p.p.m. For pure water and the 100 p.p.m. solution, dashed lines show the average values in the span of the measured shear rates. The error bars indicate the standard deviations of three independent measurements. Here, ‘*’ signifies that the solution has experienced the cavitation process. (b) Changes of the numerically determined (see Appendix A) wall shear stress in the streamwise direction $\tau _{{{w}},x} (x)$ on the midspan of the channel in the pure water flow, plotted for three different ${ {\textit {Re}}}_{th}$. The inserted thick black curve illustrates the profile of the lower nozzle wall. A pale-green rectangle highlights the region of extreme shear stress.

Table 3. List of coefficients of the CY model for the tested PAM solutions, with their corresponding $r$ and $R$ coefficients. Here, ‘*’ signifies that the solution has experienced the cavitation process.

As figure 4(a) illustrates, the viscosity of the 400 p.p.m. solution decreases for the entire range of the tested shear strain rates after experiencing violent cavitation conditions. While still showing shear thinning behaviour at relatively lower shear strain rates, the results suggest that the cavitation process weakens this behaviour. As given in table 3, the solution has an infinite viscosity of ${\approx }18\,\%$ lower than the 200 p.p.m. solution that was not undergone the cavitation process. This result confirms that the polymers used in this work degrade mechanically due to the molecular scission at relatively large shear strain rates.

In this study, the mass flow rate was chosen as the independent parameter and was varied for each fluid to produce a range of flow conditions. Hence, ${ {\textit {Re}}}_{th}$ was matched between different solution flows to investigate the effect of PAM additives on the cavitation process. As discussed, the PAM solutions are shear-thinning, and their local rheological behaviour in the flow domain depends on the strain rate they experience and the relaxation time of the solution. Therefore, to determine ${ {\textit {Re}}}_{th}$ for PAM solutions, it is crucial to estimate the wall viscosity at the shear rate experienced by the throat wall, i.e. $\mu _{w,th} ({\dot {\gamma }}_{w,th})$. It was not feasible to resolve the viscous sublayer at the throat using the optical settings described in § 2.2. Therefore, ${\dot {\gamma }}_{w,th}$ was calculated using (1) the Prandtl–Kármán (PK) friction factor correlation (White Reference White2011) for turbulent water flow in smooth tubes and (2) a three-dimensional numerical simulation of the turbulent water flow. Details of the numerical set-up are given in Appendix A. It is important to note that ${\dot {\gamma }}_{w,th}$ might change in a viscoelastic flow (Naseri et al. Reference Naseri, Koukouvinis, Malgarinos and Gavaises2018), and more accurate simulations based on a non-Newtonian flow model are required to elucidate this conjecture. Nevertheless, the effect of this difference is negligible on ${ {\textit {Re}}}_{th}$. As an example, in the 400 p.p.m. solution, $\mu _{w,th} (5.0 \times 10^{4}\,{\rm s}^{-1})=1.627$ mPa s and $\mu _{w,th} (2.5\times 10^{4}\,{\rm s}^{-1})=1.667$ mPa s. For this solution, at a constant flow rate, a 50 % reduction in the shear strain rate increases the wall shear viscosity and decreases ${ {\textit {Re}}}_{th}$ by only 2.5 % in the highest PAM concentration tested. Hence, numerically obtained values of ${\dot {\gamma }}_{w,th}$ in water flow were used to approximate ${ {\textit {Re}}}_{th}$ of PAM solutions prior to tests and to match them with ${ {\textit {Re}}}_{th}$ of pure water flow.

Figure 4(b) depicts the variation of the wall shear stress $\tau _{w} (x)$ of pure water flow in the streamwise direction $x$ at the midspan of the channel, obtained numerically for three different inlet flow rates. Water flow experiences extreme shear stress in the throat region, which increases for higher flow rates. Numerical wall shear stress values were averaged over the bottom flat plate of the throat region and were used to determine the average shear rate at the throat wall $(\dot {\gamma }_{w,th})_{CFD}$. Table 4 lists the calculated average shear stresses and their associated shear strain rates for three different ${ {\textit {Re}}}_{th}$ of the pure water flow. The estimated values based on the PK friction factor are in the range of ${\pm }10\,\%$ of their numerical counterpart. Nevertheless, only the numerical values are used for further calculations.

Table 4. Calculated average wall shear stresses for three different throat velocities in pure water flow using the PK friction factor and three-dimensional numerical simulation. Here, $\langle \rho _{L}\rangle =1000\,{\rm kg}\,{\rm m}^{-3}$ and $\mu _{wat}=0.91$ mPa s, the average measured water density and viscosity, were used in the calculations.

A numerical parametric study was conducted with a similar set-up explained in Appendix A for five different inlet conditions in the range of $2 \times 10^{4}<{ {\textit {Re}}}_{th}<6\times 10^{4}$ for pure water flow. The results were used to estimate the shear rate and viscosity at the throat wall. Table 5 lists the determined viscosities of different solutions at three throat velocities and equal ${\dot {\gamma }}_{{w,th}}$, with their associated ${ {\textit {Re}}}_{th}$. At relatively high shear rates experienced by the flow in this work, the largest $\mu _{w,th}$ is $\approx$1.56 mPa s, which is $1.7 \times$ larger than the pure water viscosity and belongs to the 400 p.p.m. solution at $\bar {\bar {u}}_{th}=13.34\,{\rm m}\,{\rm s}^{-1}$. All the reported Reynolds numbers in this study were calculated based on the average wall viscosity at that particular cross-section.

Table 5. Estimated average viscosities at the throat's wall and their corresponding ${{ {\textit {Re}}}_{th}}$ for different flow rates, based on the numerical results of the pure water flow simulation. Here, ‘*’ signifies that the solution has experienced the cavitation process.

A series of oscillation frequency sweep tests using PAM solutions with concentrations of 50 to 400 p.p.m. were conducted and the resultant elastic (storage) modulus $G^{\prime } (\omega )$ and viscous (loss) modulus $G^{{\prime {\prime }}} (\omega )$ are plotted in figure 5(a,b) as functions of angular frequency $\omega$. The strain was kept constant at 1 % in all tests. As shown in figure 5(a), the elastic moduli of all tested solutions show a plateau for the range of $0.1\lessapprox \omega \lessapprox 10$ rad s$^{-1}$ and start to increase as $\omega$ increases. An increase in the polymer concentration increases the plateau $G^{\prime }$. As figure 5(b) illustrates, the viscous moduli changes almost linearly for $\omega \gtrapprox 10$ rad s$^{-1}$, which highlights the Newtonian behaviour of the fluid in this frequency range, i.e. $G^{{\prime {\prime }}} = \mu _{0} \omega$, where the viscosity $\mu _{0}$ shows an increase as the concentration increases. Oscillation tests were also conducted for higher frequencies to obtain the crossover points of $G^{\prime } (\omega )$ and $G^{{\prime {\prime }}} (\omega )$ profiles. The crossover point correlates with the relaxation time $t_{\rm R}$ of the solution. For 200 and 400 p.p.m. solutions, $t_{R} \approx 14.5$ ms and $t_{\rm R} \approx 12.5$ ms were obtained as the average of three independent tests. Due to inertial effects, no consistent crossover points were obtained for lower concentrations at higher frequencies.

Figure 5. Changes of the (a) elastic (storage) modulus $G^{\prime }$ and (b) viscous (loss) modulus $G^{\prime {\prime }}$ of different PAM solutions with angular frequency $\omega$. The strain rate was kept constant at 1 % and temperature at 20 $^\circ$C in all tests. The plotted profiles are the average of three independent measurements and the error bars show the standard deviation of the measurements at each frequency.

4.2. Drag reduction in non-cavitating tube flow

Dimensionless pressure gradient or the Fanning friction factor in a straight tube, such as the one shown in figure 1, is defined as $c_{f} = 2 \tau _{w} / \langle \rho _{L} \rangle {\langle \bar {\bar {u}}_{tu} \rangle }^2$, where the wall mean shear stress $\tau _{w}$ is a function of the time-averaged measured pressure drop $\langle {\Delta p}_{tu} \rangle$ over length $L_{tu}=0.92$ m of the tube with $D_{tu}=19$ mm. Therefore, $\tau _{w} = D_{tu} \langle {\Delta p}_{tu} \rangle / 4 L_{tu}$, and the Fanning friction factor can then be defined as

(4.3)\begin{equation} c_{f} = \frac{D_{tu}}{2 L_{tu}}\frac{\langle {\Delta p}_{tu} \rangle}{\langle {\rho}_{L} \rangle {\langle \bar{\bar{u}}_{tu} \rangle}^2}. \end{equation}

Drag reduction (DR) is the relative reduction in the pressure drop of an additive solution over a known section of a flow path with a constant cross-sectional area compared with the pressure drop of the pure solvent flow at a similar flow rate (Lumley Reference Lumley1973; Virk Reference Virk1975). Therefore, the percentage of DR can be defined as

(4.4)\begin{equation} DR\% = 1 - \frac{c_{f,s}}{c_{f,0}}, \end{equation}

where subscripts ‘$s$’ and ‘0’ stand for the additive solution and pure solvent, respectively. Here, $c_{f}$ was calculated from the measured pressure drops in the turbulent flow of pure water and four different concentrations of PAM solutions in the range of $2 \times 10^{4} \leq { {\textit {Re}}_{tu}} \leq 3 \times 10^{4}$. Each test was repeated three times with a new solution, and the mean values are plotted in figure 6. The maximum standard deviation of the calculated friction factors was 7 % of its mean value. The resultant $c_{f} ({ {\textit {Re}}_{tu}})$ profile of pure water agrees with the PK equation (White Reference White2011), shown with a dashed line in figure 6. Virk's asymptote for maximum drag reduction (MDR) (Virk, Mickley & Smith Reference Virk, Mickley and Smith1970) in dilute polymer solutions in turbulent pipe flow is indicated by a thick black line in figure 6. The MDR was not obtained for any of the tested solution concentrations. The 50 p.p.m. solution shows a higher $DR$ ($\approx$10 %–20 % more) than the 100 and 200 p.p.m. solutions. Increasing the concentration to 400 p.p.m. did not enhance pressure reduction, and a $DR\,\%$ comparable to the 50 p.p.m. solution was obtained for the tested flow rates. All examined concentrations resulted in a similar $DR \approx 50\,\%$ at the maximum tested flow rate (${ {\textit {Re}}_{tu}} = 3\times 10^{4}$).

Figure 6. Variation of the Fanning friction factor, $c_{f}$, as a function of the straight tube's Reynolds number, $ {\textit {Re}}_{tu}$, in PAM solutions of various concentrations. The thick dashed line indicates the PK friction factor (White Reference White2011) for turbulent water flow in smooth tubes. Virk's MDR asymptote (Virk et al. Reference Virk, Mickley and Smith1970) for turbulent flow of dilute polymer solutions in smooth tubes is shown by the solid black line. Each point on the plot denotes the mean of three independent measurements, with a maximum standard deviation of 7 % of the mean value in all tests.

The percentage ratio of instantaneous drag reduction, $DR (t)$ to the initial steady drag reduction of the system, $DR_{0}$, i.e. $(DR (t) / DR_{0}) \times 100\,\%$, was probed for a period of 500 s for different PAM solutions at $ {\textit {Re}}_{tu}=3.0\times 10^{4}$ to investigate the degradation level of the solutions. The results are plotted in figure 7. As shown, after $\approx$8 min, $DR (t)$ drops only by ${\approx }5\,\%$ at maximum relative to its corresponding $DR_{0}$. The data acquisition for each test took nearly 10 s, and each solution was used only for four to five tests. The solution was renewed if further tests were required. A pale-green rectangle highlights the maximum data acquisition period with each fresh solution batch ($\approx$50 s).

Figure 7. Degradation of the PAM solutions in time defined as the percentage ratio of the instantaneous drag reduction $DR (t)$ to the steady drag reduction of the flow system $DR_{0}$. Tests were conducted at $ {\textit {Re}}_{th}=3.0\times 10^{4}$.

4.3. Effect of polymer additives on the pressure drop over the nozzle

As illustrated in figure 1, pressure drop was measured over the nozzle channel section using a differential pressure transducer. The pressure drops were measured for four different concentrations of PAM solution and pure water in the range of $2\times 10^{4} < {{ {\textit {Re}}}_{th}} < 6\times 10^{4}$ to investigate the effect of PAM additives on nozzle flow. The results are plotted in figure 8(a–d) as $\sigma ( {\textit {Re}}_{th})$ profiles.

Figure 8. Variation of the cavitation number $\sigma$ versus the throat Reynolds number ${ {\textit {Re}}}_{th}$ for ramp-up (increasing ${ {\textit {Re}}}_{th}$) and ramp-down (decreasing ${ {\textit {Re}}}_{th}$) tests for (a) 50 p.p.m., (b) 100 p.p.m., (c) 200 p.p.m. and (d) 400 p.p.m. PAM concentrations. The cavitation onset points are shown by vertical solid lines coloured according to the colour code used for each solution concentration in the plots.

For pure water profiles in figure 8(a–d), $\sigma$ decreases exponentially as the flow rate increases. After the onset of cavitation, $\sigma$ drops with a smaller slope and approaches a value of $\sigma \approx 3.2$ for higher ${ {\textit {Re}}}_{th}$. The gradual decrease of the water flow rate during the ramp-down test results in similar pressure drops over the nozzle. Figure 8(a) shows that the 50 p.p.m. solution results in a higher $\sigma$ than the pure water flow at an equal ${ {\textit {Re}}}_{th}$. This difference decreases as the flow rate increases and cavitation intensifies. In the 50 p.p.m. solution, cavitation occurs at a flow rate ${\approx }20\,\%$ higher than the pure water flow. As shown in figure 8(b–d), an increase in the solution concentration shifts $\sigma$ to even higher values, and cavitation onset occurs at higher flow rates than the pure water flow.

For the tested concentrations, solution flows show higher levels of $\sigma$ during a ramp-up test than in a ramp-down test. As figure 7 illustrates, it is possible that polymers experienced mild degradation while recirculating in the flow loop. Relatively, under cavitating flow conditions, it is highly probable that the rate of polymer degradation increased significantly due to the presence of extreme shear stress fields in cavitating regions. At the end of the ramp-up test, the solutions experience relatively high shear stresses during an intense cavitation process, which may cause elongated polymer molecules to degrade or entangled molecules to straighten. As figure 4(a) and figure 8(a–d) illustrate, the transmutation of the polymer molecules due to violent cavitation and extreme shears causes a local reduction in viscosity and pressure. Hence, the ramp-down tests show lower pressure drops than the ramp-up measurements.

Figure 9(a) compares the variation of $\sigma$ of the solutions examined in this study during the ramp-down test. The inception Reynolds number ${ {\textit {Re}}}_{th,i}$ increases as the PAM concentration increases. For the maximum tested flow rate, corresponding to ${ {\textit {Re}}}_{th}\approx 5.4\times 10^{4}$, pressure drop over the nozzle increases by ${\approx }9\,\%$, ${\approx }13\,\%$, ${\approx }20\,\%$ and ${\approx }50\,\%$, respectively, in the 50, 100, 200 and 400 p.p.m. solution flows relative to the pure water flow. At the inlet of the nozzle channel, the wall shear stress is O($10^{2}$ Pa) (see figure 4b), and its corresponding apparent shear viscosity in a PAM solution is higher than the pure water. Therefore, non-Newtonian PAM solutions have the potential to produce higher local pressures at low shear regions in the flow, such as the inlet of the nozzle in this study. This result agrees with the ${\Delta p}_{ch}$ measurements in figure 8(a–d). The PAM concentration increases ${\Delta p}_{ch}$ and accordingly inlet pressure $p_{in}$ increases. It is important to note that $p_{in}$ was intentionally chosen as the characteristic pressure in the definition of the cavitation number given in (3.2) to demonstrate the physical increase of the pressure drop over the curved nozzle. Yet, most of the previous studies on cavitation in non-Newtonian viscoelastic flows have used a constant reference pressure such as a free stream or outlet pressure to define the cavitation number (Hoyt Reference Hoyt1976; Ōba et al. Reference Ōba, ItŌ and Uranishi1978; Hasegawa et al. Reference Hasegawa, Ushida and Narumi2009) and reported reductions in the incipient cavitation number $\sigma _{i}$ of viscoelastic solutions relative to pure water. Therefore, as figure 9(b) shows, if the constant outlet pressure $\langle p_{out} \rangle$ is used in (3.1), $\sigma _{i}$ demonstrates a reduction as the concentration increases.

Figure 9. Changes of $\sigma$ as a function of ${ {\textit {Re}}}_{th}$ for different concentrations of PAM solutions for the ramp-down tests when (a) inlet pressure $\langle p_{in} \rangle$ and (b) constant outlet pressure $\langle p_{out} \rangle$ is selected as the reference pressure in (3.1). Cyan circles highlight the cavitation onset and $\sigma _{i}$ denote the incipient cavitation number.

4.4. Temporal evolution of cavitation structures in pure water

The cavitation onset is followed by a complex spatial and temporal evolution of cavitation structures, which expand in size as the pressure drop increases over the nozzle. Figure 10(a–c) illustrates four instantaneous sample snapshots of vapour ratio fields of cavitating pure water flow for three different flow conditions. These fields are projected on the channel's midspan plane and cover a FOV from approximately the end of the throat region to the vicinity of the end limit of the divergence region (see figure 2). For the current test conditions, the incipient cavitation number of pure water was $\sigma _{i} \approx 4.71$, which was associated with the emergence of tiny cavitation bubbles, detectable by the optical system described in § 2.2.

Figure 10. Snapshots of instantaneous cavitating pure water structures at (a) ${{ {\textit {Re}}}_{th}} = 2.9 \times 10^4$, $\sigma = 4.23$, (b) ${{ {\textit {Re}}}_{th}} = 3.8 \times 10^4$, $\sigma = 3.61$ and (c) ${{ {\textit {Re}}}_{th}} = 4.5 \times 10^4$, $\sigma = 3.37$, projected on the channel's midspan (plane $z = 0$). The white colour ($\alpha _{G}=1$) shows the cavitation areas, and the black colour ($\alpha _{G}=0$) demonstrates the bulk liquid. Sample growing and collapsing structures are annotated using green arrows. For related videos of (a–c), refer to movies 1–3 available at https://doi.org/10.1017/jfm.2022.910.

Figure 10(a) shows the vapour area ratio snapshots at $\sigma \approx 4.23$, with a flow rate of $\approx$16 % higher than the inception. Cavitation bubbles are produced cyclically from the top or bottom throat wall surfaces at this flow condition. As cavities elongate on the upper wall, relatively smaller jet clouds are produced on the lower wall and vice versa. Cavities detach from the end of the throat when they reach a length of $\approx$2$h_{th}$ and flow downstream as they rapidly disintegrate and collapse into micron-sized bubbles. Most tiny micro cavitation bubbles collapse entirely, and any remnants are carried downstream by the bulk liquid flow. The cavity developed from one wall sporadically interacts with the passing jet cloud originating from the other and generates circular rolling clusters of cavitation bubbles with an equivalent diameter of ${\approx }h_{th}$. As the bubble clusters flow downstream, they collapse in several microseconds and vanish entirely. Cavity clouds rebound several times during the collapse process and produce radially propagating pressure waves. Interaction of the pressure waves with the growing upstream cavities generates pockets of collapsing bubbles at the front of the cavities, which transmutes the cavity into an unstable transient structure.

Further increasing the flow rate enhances the cavitation process by causing larger pressure drops over the nozzle. Figure 10(b) illustrates snapshots of the cavitation structures at $\sigma =3.61$ for the pure water flow. Here, developing cavities gain lengths up to $\approx$5$h_{th}$ before they depart from their origins in the throat. Detached cavities decelerate as they flow downstream and are followed by the next generation of developing cavities. With $p(\boldsymbol {x}, t) \ll p_{sat}$ at almost any position close to the throat, the new-born cavities grow fast and finally bond with the just detached cavities. This bonding further fluctuates the large cavitation structure and may break it off into patches of collapsing streaky clusters of bubbles. These streaky structures exist only for several milliseconds before they entirely vanish or fall apart into micron-sized cavitation bubbles. The collapse of tilted streaky bubble pockets generates a series of shockwaves that propagate upstream and break the large structures of agglomerated packets of bubbles by modifying their local pressures. This process repeats itself with a particular frequency in time. The pressure field is highly unstable at the throat's downstream region and locally fluctuates around $p_{sat}$. As a result, smaller scales of bubble pockets emerge and instantly vanish at random positions in the downstream flow field. Chains of bullet-like bubbles are produced at the throat at irregular intervals, separated by liquid slugs of widths of $\approx$0.2$h_{th}$. Thin films of lubricating liquid were also observed between the cavitation bullet-like bubbles and the channel's walls. As these bubbles travel downstream, they collide with their preceding decelerated bubbles. As a result, they accumulate in this region and extend the cavitation body from its rear as its front collapses.

Figure 10(c) is associated with the instantaneous vapour ratio fields of the cavitating water flow with $\sigma = 3.37$. Huge pressure drops over the nozzle generate cavitation superstructures with oscillating lengths of $\approx$12$h_{th}$ in the flow field. Even in this extreme condition, growing wall cavities cannot entirely occupy the core of the throat's exit region, which is dominated by a high-speed flow of the bulk liquid. Wall cavities expand, merge at the entrance of the divergence region and transform into cavitation superstructures. Interaction of the travelling shockwaves with the front of these huge clouds breaks them off into scattered clouds of cavitation bubbles. At this flow condition, the cloud cavitation collapse is chaotic and occurs at random positions in the flow field. As the clouds approach the maximum cross-sectional area, higher pressure levels ($p(\boldsymbol {x}, t) \gg p_{sat}$) shatter the bubble networks, and the fragmented bubbles vanish as they shrink in size and leave the channel. The chaotic cavitation escalates the dynamic pressure fluctuations, which intensifies the cavitation, and this cycle repeats itself as the upstream energy source is present.

As listed in table 2, a series of experiments using varying concentrations of PAM solutions in water were conducted to investigate the effect of PAM additives on the cavitation mechanism and explore the feasibility of the CR effect of these additives. Snapshots of instantaneous vapour ratio fields of the 200 p.p.m. solution on the channel's midspan are plotted in figure 11(a–c) for low, medium, and high-intensity cavitation conditions to illustrate the effect of PAM additives on the temporal evolution of cavitation structures. The cavitation process initiates at a ${ {\textit {Re}}}_{th}$, ${\approx }32\,\%$ higher than the cavitation onset of the water flow.

Figure 11. Snapshots of instantaneous cavitating structures in 200 p.p.m. PAM solution flow at (a) ${{ {\textit {Re}}}_{th}} = 3.5 \times 10^4$, $\sigma = 4.48$, (b) ${{ {\textit {Re}}}_{th}} = 3.6 \times 10^4$, $\sigma = 4.37$ and (c) ${{ {\textit {Re}}}_{th}} = 4.3 \times 10^4$, $\sigma = 4.00$, projected on the channel's midspan (plane $z = 0$). The white colour ($\alpha _{G}=1$) shows the cavitation areas, and the black colour ($\alpha _{G}=0$) demonstrates the bulk liquid. Sample growing and collapsing cavitation structures are annotated using green arrows. For related videos of (a–c), refer to movies 4–6.

As figure 11(a) illustrates, at a condition with a flow rate of only 6 % higher than the onset, the produced cavities form streaky topologies with wiggling outlines in the 200 p.p.m. solution. The instantaneous length of the streaky bubbles increases up to ${\approx }2 h_{th}$, with a length-to-width ratio of $\approx$6. At infrequent periods, the dominant core liquid flow is discontinued instantly by the merging wall cavities at the region where the wall profiles begin to diverge. As the fronts of the cavities break off, relatively uniform structures of detached cavities are generated. They oscillate, partially rebound and collapse as lumps of densely packed cavities, which emerge as continuous cavitation structures in the recorded images. Long polymer molecules act as flexible dampers in the flow field and relax extreme pressure field oscillations adjacent to the collapsing bubbles by releasing their stored energy under tension and storing energy under compression. This effect eases the cavitation process, increases the rebounding period of cavitating bubbles, and imposes lower pressure fluctuations on the surrounding bulk flow.

For the flow condition with ${{ {\textit {Re}}}_{th}} = 3.6 \times 10^4$, $\sigma = 4.37$, shown in figure 11(b), the core liquid flow exists only to a length of ${\approx }2.5h_{th}$, the position where the growing wall cavities merge. A high-velocity thin layer of lubricating liquid flows upstream between the channel walls and the growing cavities. These continuous film flows possess high levels of momentum, and as they approach the throat, they come to a complete stop near the oncoming growing cavity. The interaction of the front of the lubricating films with the cavities transfers momentum to the cavities, lifts them in the positive wall-normal direction, and separates them from the channel walls. This phenomenon accelerates the merging of the cavities. The recirculating flow regions trap the detached collapsing cavities at irregular periods in the downstream flow field. As a result, the collapse rate is enhanced in regions with a recovered pressure, where the cavities attune to the pressure distribution patterns in the recirculating region by altering their topology.

As figure 11(c) illustrates, cavitation clouds develop into immense structures downstream at a higher cavitation level. The visualization results indicated that the developed cavities remained attached to the wall surfaces during the entire recording time. Under this condition, small-scale clouds were produced due to the break-off and chaotic collapse of the front of the developed cavities, which shed downstream. These clouds show random topologies and generate mesoscale clusters of cavitation bubbles as they collapse. In contrast to pure water flow (see figure 10c), where the collapse of shedding small-scale cavitation clouds downstream produced consistent micron-sized bubbles, the 200 p.p.m. solution flow produced mesoscale bubble networks with sizes in the range of $\approx$0.2–0.5 mm.

Instantaneous snapshots of cavitating flow structures at the convergence and throat regions of different PAM solution flows are illustrated in figure 12(a–c) at three different $ {\textit {Re}}_{th}$. As seen, cavitation structures emerge on the walls at about $x/h_{th}\approx -1$, adjacent to the midway of the throat region, oscillating on both walls cyclically as they grow downstream. As expected, increasing the PAM concentration reduces the cavitation intensity and suppresses it at higher concentrations for the tested flow conditions.

Figure 12. Snapshots of instantaneous vapour ratio fields in different PAM solution flows at the convergence and throat regions for (a) ${{ {\textit {Re}}}_{th}} = 2.6 \times 10^4$, (b) ${{ {\textit {Re}}}_{th}} = 3.0 \times 10^4$ and (c) ${{ {\textit {Re}}}_{th}} = 3.4 \times 10^4$, projected on the channel's midspan (plane $z = 0$). The white colour ($\alpha _{G}=1$) shows the cavitation areas, and the black colour ($\alpha _{G}=0$) demonstrates the bulk liquid. The flow direction is from left to right and a pale-yellow rectangle, with dashed blue borders, signifies the throat region on each figure. For sample videos, refer to movies 7–8.

Spatiotemporal maps of cavitating flow fields were evaluated to demonstrate the CR effect of PAM additives on the temporal evolution of cavitation structures. These fields were obtained by averaging each instantaneous vapour ratio field relative to the $y$-direction, i.e. $\bar {\alpha }_{G} (x,t)$, and stitching the resultant lines chronologically. The results are shown in figure 13(a,b) for an intermediate and a high flow rate of pure water flow and four different concentrations of PAM additives for 45 ms. The PAM additives demonstrate significant CR effects by strongly relaxing the chaotic unstable cavity fronts, attenuating their growth downstream, and suspending the cavitation onset to higher ${ {\textit {Re}}}_{th}$. As given in table 2, cavitation initiates at ${ {\textit {Re}}}_{th,i} = 3.7 \times 10^4$ in the 400 p.p.m. solution, which is ${\approx }50\,\%$ higher than ${ {\textit {Re}}}_{th,i}$ of pure water.

Figure 13. Spatiotemporal fields of cavitating PAM solutions for (a) ${{ {\textit {Re}}}_{th}} = 3.4 \times 10^4$, and (b) ${{ {\textit {Re}}}_{th}} = 4.4 \times 10^4$. Each row corresponds to a different concentration. From top to bottom, results for pure water, 50, 100, 200 and 400 p.p.m. solutions are illustrated, respectively. From the total imaging period of $T=0.895$ s, only 45 ms is illustrated for each flow condition.

In figure 13(a), the front of the vaporous structures in the 50 p.p.m. solution flow oscillates with a mean length similar to that in pure water. The population of the miniature cavitation bubbles swimming at the closure of the cavity in the pure water flow is strongly alleviated in the 50 p.p.m. solution. In pure water, the footprints of these tiny bubbles are visible as irregular fading regions of $\bar {\alpha }_{G}$ at the cavity closure, which are moderated in the 50 p.p.m. solution flow, as shown in figure 13(a). At ${{ {\textit {Re}}}_{th}} = 3.4 \times 10^4$, a further increase of the PAM concentration suppresses the cloud cavitation by damping the growth of the wall cavities and retarding them to sheet cavities. Cavitation is entirely suppressed for a 400 p.p.m. solution, with the only infrequent appearance of small evanescent cavities.

Under extreme cavitation conditions, as shown in figure 13(b), large-scale cavitation structures shed downstream, filling almost the entire visualized FOV. At this relatively high flow rate, the 50 p.p.m. solution displays an even more intense cavitation process than the pure water, enhancing the cavitation instead of reducing it. As the solution concentration increases to 100 p.p.m., the relatively tiny cavitation bubbles adjacent to the oscillating cloud fronts fade out. Growing cavities maintain lengths of similar scales to that in the pure water flow while oscillating periodically, with lower fluctuations relative to the average cavity length. As figure 13(b) illustrates, increasing the concentration to even 400 p.p.m. cannot suppress the vivacious cavitation structures at this flow condition and only linearize the growth rate of the cloud cavities in time. Once a cavity begins to grow, it expands at an almost constant rate to its maximum length, comparable to the entire length of the divergence region downstream, then it dwindles at a constant and relatively steeper rate, and this process repeats itself in time. Conversely, as the vaporous structures grow in the pure water flow at this high flow rate, they shrink and grow sporadically, at a time-dependent random rate, and produce significant batches of micron-sized dispersed bubbles, which swim in their vicinity and dispense in the entire downstream region.

4.5. Cavitation length

The r.m.s. of the fluctuating component of the vapour ratio field ${\alpha '}_{G,rms}$ was calculated for each flow condition to quantify the cavitation intensity. A set of $\approx$17 000 images were analysed for each flow case to ensure the values converged statistically. Figure 14(a–d) show the variation of ${\alpha '}_{G,rms}$ in the streamwise direction, downstream of the throat on the $y=0$ centreline of the midspan plane, for different flow conditions and PAM solutions. Each ${\alpha '}_{G,rms}(y=0)$ profile demonstrates a peak, which is associated with the mean position of the oscillating cavitation front. The streamwise distance of this point relative to the origin at $x = 0$ characterizes an appropriate length scale for cavitation structures, denoted by $L_{ca}$ in this study. Dular et al. (Reference Dular, Bachert, Stoffel and Širok2004) and Zhang et al. (Reference Zhang, Zuo, Morch and Liu2019), among others, used a similar approach to measure the cavitation length. The loci of the peak points are highlighted by cyan circles on the plots in figure 14(a–d). There is another peak present in each ${\alpha '}_{G,rms}(y=0)$ profile in the $x < 2{h_{th}}$ region. This point corresponds to the mean streamwise position, where cavities detach from the wall surfaces. In figure 14(a–d), these detachment regions are highlighted by green dashed rectangles.

Figure 14. Variation of the cavitation intensity ${\alpha '}_{G,rms}$ on the midspan plane $z=0$ and at the channel's centreline $y = 0$, for the selected range of flow conditions with increasing ${ {\textit {Re}}}_{th}$ and decreasing $\sigma$ for (a) pure water, (b) 100 p.p.m., (c) 200 p.p.m. and (d) 400 p.p.m. solution flows. The positions of ${[{\alpha '}_{G,rms}(y=0)]}_{max}$ are highlighted with cyan circles on each figure and connected with a black dashed line to demonstrate the variation of the maxima. The regions of cavitation detachment adjacent to the throat are emphasized using rectangles coloured in pale-green with dashed dark green outlines.

Figure 14(a) shows that as the flow rate (equivalently the pressure drop) of the pure water flow increases, cloud cavitation structures develop into a wider region downstream while maintaining a maximum of ${\alpha '}_{G,rms} \approx 0.42$ until ${ {\textit {Re}}}_{th} = 3.9 \times 10^{4}$ and $\sigma = 3.2$. After this point, the maximum intensity decays linearly with increasing ${ {\textit {Re}}}_{th}$. This reduction in the maximum intensity reveals that the cavitation cloud's oscillating front is more stable at higher flow rates. For water flows with $\sigma < 3.2$, the growing cavities detach from the wall surface after a partial expansion and are transferred by the bulk liquid flow farther downstream, where they collapse due to pressure recovery. At the same time, new cavities are generated and develop from the throat when bulk liquid fills the collapse region. Therefore, $\alpha _{G}$ shows high fluctuations in this region, resulting in a higher ${\alpha '}_{G,rms}$ at the cloud's closure region. For $\sigma > 3.2$, the detachment and collapse periods are longer for larger, more violent structures. The flow region is repeatedly filled with newly growing cavitation bubbles, which statistically increases the probability of the existence of bubble pockets of a similar population in the same region. Therefore, the flow experiences lower fluctuation levels of $\alpha _{G}$ at the edge of the cavitation structures.

Similar to the pure water flow shown in figure 14(a), intensity values on the centreline of PAM solutions, given in figure 14(b–d), show two peaks: (1) in the close vicinity of the throat, where the clouds detach from the surface; and (2) at the edge of the cavitation clouds, where the cavity closure oscillates periodically. Maximum cavitation intensity is maintained for the full range of the examined ${ {\textit {Re}}}_{th}$ and $\sigma$ for PAM solutions tested, while its position is shifted farther downstream as ${ {\textit {Re}}}_{th}$ increases. As a result, the addition of PAM polymers intensifies the fluctuations of the cavitation edge at higher flow rates, while relative to the pure water, they delay the cavitation onset to larger ${ {\textit {Re}}}_{th}$, and damp the expansion of the cavitation structures in the streamwise direction. Figure 14(a) reveals that the cavitation detachment position remains almost constant for the entire range of the tested flow rates for pure water at $x / {h_{th}} \approx 0.4$. Conversely, as shown in figure 14(b–d), rising the concentration level in the solution generally shifts the detachment position downstream, respectively, to $x / {h_{th}} \approx 1.0$, $x / {h_{th}} \approx 1.2$ and $x / {h_{th}} \approx 1.8$. However, as ${ {\textit {Re}}}_{th}$ increases, the mean detachment position reverses towards upstream.

The estimated cavitation lengths based on the loci of ${[{{\alpha }'}_{G,rms}(y=0)]}_{max}$ are plotted in figure 15(a,b) for varying ${ {\textit {Re}}}_{th}$ and $\sigma$ and different solutions of PAM in water. As shown in figure 15(a), the occurrence of the minimum cavitation length $L_{ca,min} \approx 2 h_{th}$ is shifted towards higher ${ {\textit {Re}}}_{th}$ and $\sigma$ as the concentration increases. For the 400 p.p.m. solution, this happens at a ${ {\textit {Re}}}_{th}$ of ${\approx }18\,\%$ higher than pure water. For all examined solutions, $L_{ca}$ increases almost linearly with ${ {\textit {Re}}}_{th}$. At relatively higher flow rates, $3.2\times 10^{4} < {{ {\textit {Re}}}_{th}} < 4.0 \times 10^{4}$, 50 and 100 p.p.m. solutions expand to longer cavitation structures than pure water, enhancing the growth of streaky and cloud cavities. Solutions with concentrations of more than 100 p.p.m. decelerate the expansion of the cavitation clouds and shrink their size in the streamwise direction. For instance, at ${ {\textit {Re}}}_{th}=4\times 10^{4}$, $L_{ca}$ in the 400 p.p.m. solution is reduced by 30 % relative to pure water.

Figure 15. Variation of the normalized cavitation length with (a) throat Reynolds number ${ {\textit {Re}}}_{th}$ and (b) cavitation number $\sigma$ for pure water and four different concentrations of PAM solution in water.

As shown in figure 15(b), cavitation length increases almost linearly with the reduction of $\sigma$ for all examined solutions. A similar $L_{ca}$ is generated at a higher $\sigma$ in PAM solutions relative to pure water. A cavity of $L_{ca,min}\approx 2h_{th}$ occurs at $\sigma \approx 5.7$ in the 400 p.p.m. solution, which is ${\approx }34\,\%$ higher than its counterpart in pure water. There is a noticeable shift (${\approx }11\,\%$) in the $L_{ca}(\sigma )$ profile for the 50 p.p.m. solution compared with pure water. Increasing the mixture concentration from 50 to 200 p.p.m. does not indicate a significant shift in $L_{ca}(\sigma )$. As the PAM concentration increases to 400 p.p.m., $L_{ca}(\sigma )$ shifts ${\approx }23\,\%$ towards higher $\sigma$ when compared with the 50–200 p.p.m. profiles. These shifts follow the increase of the inlet pressure due to the local increase of the solution's shear viscosity discussed in § 4.1.

4.6. Temporal variance of cavitation structures

In an instantaneous image captured from a cavitating flow, each square pixel can be fully or partially occupied by a vaporous phase, i.e. $0 \leq \alpha _{G} \leq 1$. In an Eulerian reference frame, the time difference of $\alpha _{G}$ can provide information on the local growth or collapse of the cavities. In this study, the first-order time difference of $\alpha _{G}$ field is defined as ${\delta \alpha }_{G} (x, y, t_{k-1})={\alpha }_{G} (x, y, t_{k}) - {\alpha }_{G} (x, y, t_{k-1})$, where the index $k$ indicates the current instant. Therefore, ${\delta \alpha }_{G} > 0$ corresponds to local growth of the cavitation structure, and correspondingly ${\delta \alpha }_{G} < 0$ at a position in the flow field refers to the occurrence of a collapse. A value of ${\delta \alpha }_{G} = 0$ means that a similar fluid phase is present at the same location at the instants $t_{k-1}$ and $t_{k}$; hence, it does not show any growth or collapse. Here, ${\delta \alpha }_{G} > 0$ and ${\delta \alpha }_{G} < 0$ are symbolized as ${{\delta \alpha }_{G}}^{+}$ and ${{\delta \alpha }_{G}}^{-}$, respectively. Appendix B illustrates the spatiotemporal variations of ${\delta \alpha }_{G}$, averaged in the $y$-direction at each instant, for only 5$\%$ of the total recording time of $T=0.895$ s. The process of obtaining $\overline {{\delta \alpha }}_{G} (x,t)$ is commonly called the frame difference method (Sato, Taguchi & Hayashi Reference Sato, Taguchi and Hayashi2013).

Visualization results given in § 4.4 reveal that PAM additives suppress cloud cavitation under moderate conditions and alleviate the growth of cavity fronts in the supercavitation regime. It is well understood that the propagation of high-speed pressure waves induced by the collapse of cavitation bubbles in a pure water flow intensifies the pressure fluctuations in the flow field and enhances the cavitation process (Karathanassis et al. Reference Karathanassis, Trickett, Koukouvinis, Wang, Barbour and Gavaises2018; Zhang et al. Reference Zhang, Zuo, Morch and Liu2019; Wu et al. Reference Wu, Deijlen, Bhatt, Ganesh and Ceccio2021). Therefore, it can be hypothesized that polymer additives reduce cavitation intensity by affecting the local collapse and growth process in the flow field. To verify this hypothesis, the r.m.s. of the fluctuations of the time difference fields, i.e. ${({\delta \alpha })'}_{G,rms}$, were calculated separately for the collapse and growth regions in each solution flow. The resultant r.m.s. fields were normalized by their associated time-averaged fields $\langle {\delta \alpha }_{G} \rangle$. It is assumed here that the cavitation process is statistically symmetric relative to the centreline $y=0$ and shows the most intense fluctuations in the streamwise direction and on the centreline. Here, ${({\delta \alpha })'}_{G^{-},rms} / {\langle {\delta \alpha }_{G^{-}} \rangle }$ is the defined to be the cavitation collapse level (CCL).

Figure 16(a–d) show the variation of the normalized time variance fluctuations of collapse on $y=0$, respectively, for water and 100, 200 and 400 p.p.m. solution flows. The maxima of the profiles are highlighted by cyan circles and connected via dashed black lines to demonstrate the trend of changes. Figure 16(a) illustrates that increasing the water flow rate elevates the CCL. For the highest tested throat Reynolds number, ${ {\textit {Re}}}_{th} = 4.5 \times 10^{4}$, the collapse fluctuations can be as high as $10 \times$ the mean collapse time variance, revealing the extreme violence and chaos associated with the supercavitation regime. Until ${ {\textit {Re}}}_{th} = 3.4 \times 10^{4}$, pure water flow experiences the highest collapse level at $x \approx 1.2h_{th}$. With ${\approx }10\,\%$ increase in the flow rate, the CCL shows a rapid increase, and the position of its peak shifts farther downstream to $x \approx 1.8h_{th}$. After ${ {\textit {Re}}}_{th} = 3.4 \times 10^{4}$, maximum CCL continues to elevate almost linearly with an increase in the flow rate.

Figure 16. Streamwise variation of the normalized r.m.s. of the time difference fluctuations of the collapsing vapour ratio field on the centreline $y = 0$ for (a) pure water and (b) 100 p.p.m., (c) 200 p.p.m. and (d) 400 p.p.m. PAM solutions in water at different flow conditions. The maxima positions are highlighted by cyan circles and connected by a black dashed line on each figure to elucidate their variation. The reduction of the maximum CCL at the highest flow rate in each PAM solution relative to its counterpart in pure water is also annotated in each figure.

Figure 16(b) shows that the cavitation process in the 100 p.p.m. solution occurs with relatively lower CCLs compared with the cavitating pure water flow, where the maximum CCL shifts downstream almost linearly for the entire range of the tested conditions. At the highest tested flow rate with ${ {\textit {Re}}}_{th} = 4.6 \times 10^{4}$, maximum CCL occurs at $x \approx 2.2h_{th}$ and $x \approx 3.1h_{th}$, respectively, for the pure water and the 100 p.p.m. solution flows, while the maximum CCL is relatively 15 % lower in the 100 p.p.m. solution. Also, compared with pure water, the PAM additives flatten the CCL profiles and relax the fluctuations globally. This result elucidates the CR effects of PAM additives. As figure 16(c,d) illustrate, a further increase of the solution concentration to 200 and 400 p.p.m. alleviates the fluctuations significantly. For the maximum flow rate tested, the cavitation collapse level reduces by 52 % and 62 % for the 200 and 400 p.p.m. solutions.

The dependency of the maximum values of CCL and cavitation growth level (CGL) on ${ {\textit {Re}}}_{th}$ and $\sigma$ are depicted in figure 17(a,b). Here, ${({\delta \alpha })'}_{G^{+},rms} / {\langle {\delta \alpha }_{G^{+}}} \rangle$ is denoted as CGL. Statistically, the absolute values of CCL and CGL should be identical. In the current study, the maximum relative difference between the absolute CCL and CGL is ${\approx }0.1\,\%$. Figure 17(a) shows that the cavitation collapse and growth fluctuations increase by increasing ${ {\textit {Re}}}_{th}$ for pure water. As discussed in § 4.4 and shown in figure 17(a), PAM additives elevate the maximum CCL in the 50 p.p.m. solution relative to the pure water flow. This behaviour of the 50 p.p.m. solution escalates in the supercavitation regime. For solution concentrations more than 50 p.p.m., PAM additives reveal a CR effect by relaxing the maximum CCL and CGL in the entire range of the tested ${ {\textit {Re}}}_{th}$. As figure 17(b) shows, in a similar manner, a reduction in $\sigma$ exacerbates the cavitation process by increasing CCL. This increase is abrupt for water and the 50 p.p.m. solution and relaxes for higher concentrations. As discussed in § 4.3, for a similar ${ {\textit {Re}}}_{th}$, the addition of PAM additives increases the pressure drop over the nozzle compared with pure water. Hence, CCL and CGL profiles do not collapse on the same abscissa in figure 17(b).

Figure 17. Variation of the extrema of the collapsing and growing ${({\delta \alpha })'}_{G,rms}$, normalized by its corresponding extrema of ${\langle {\delta \alpha }_{G} \rangle }$ versus (a) ${ {\textit {Re}}}_{th}$ and (b) $\sigma$, for pure water and different solutions of PAM in water. The subscript ‘ext’ stands for extrema, i.e. a maximum or a minimum.

4.7. Shedding frequency

It was discussed in § 4.6 that one mechanism through which the PAM additives relax the cavitation process is by significantly mitigating the collapse and growth fluctuations in the flow field. Tracking the temporal evolution of the cavitation structures in the PAM solutions (see figure 13) reveals that additives also alter the shedding periodicity. A spectral analysis was performed on the high-speed pressure sensor data, explained in § 2.1, to quantify this behaviour. Two approaches were applied to the time series of the pressure fluctuations ${p'}_{d} (t)$ to obtain the spectra of the dominant shedding frequencies: (1) fast Fourier transform (FFT) was utilized to obtain the power spectral density (PSD) of the data, i.e. ${\mathcal {A}}_{PSD} (f)$, where $f$ is the frequency; and (2) continuous wavelet transform (CWT) was used to obtain the time-dependent spectra of the shedding frequencies, i.e. ${\mathcal {A}}_{CWT} (f)$. The probability of the instantaneous dominant frequencies ${\mathcal {A}}_{pdf} (f)$ was obtained from the CWT results.

Figure 18(a,b) illustrate the spectral analysis results for two sample flow conditions with an intermediate and a high flow rate in pure water flow, and figure 18(c,d) show similar results for the 200 p.p.m. PAM solution. For each flow condition, left to right diagrams illustrate the temporal variation of the pressure fluctuations normalized by its maximum value, the PSD spectrum, the CWT spectrogram of the pressure signal, and the normalized probability density function (p.d.f.) percentage of the maximum CWT-based signal frequency. The maximum dominant frequencies obtained from the PSD-based and CWT-based spectral analysis were used to calculate the Strouhal number $St$, given in (3.2), for different solutions in the range of the tested flow conditions. Using the throat hydraulic diameter $D_{h,th}$ as the characteristic length scale in (3.2), the resultant $St({ {\textit {Re}}}_{th}$) and $St(\sigma$) profiles of different solutions are plotted, respectively, in figure 19(a,b). The first and second rows are associated with the PSD and CWT spectral analysis. Comparing the PSD-based and CWT-based results for $St({ {\textit {Re}}}_{th})$ and $St(\sigma )$, the maximum relative difference was less than 1.5 % in the entire range of the tested solution concentrations and flow conditions. Therefore, the following discussion is attributed to both spectral analyses.

Figure 18. Spectral analysis results of the pressure fluctuation signal at the channel downstream, ${p'}_{d}$, in pure water flow at (a) ${ {\textit {Re}}}_{th}=3.6\times 10^{4}$, $\sigma = 3.69$ and (b) ${ {\textit {Re}}}_{th}=4.8\times 10^{4}$, $\sigma = 3.31$, and in 200 p.p.m. solution flow at (c) ${ {\textit {Re}}}_{th}=3.6\times 10^{4}$, $\sigma = 4.48$ and (d) ${ {\textit {Re}}}_{th}=4.8\times 10^{4}$, $\sigma = 4.86$. Cavitation onset of pure water and 200 p.p.m. solution occur, respectively, at ${ {\textit {Re}}}_{th,i}=2.5\times 10^{4}$, $\sigma _{i}=4.71$ and ${ {\textit {Re}}}_{th,i}=3.3\times 10^{4}$, $\sigma _{i}=4.42$. In each of (a–d) subplots from left to right represent, respectively, the pressure fluctuation signal at the channel's downstream, ${p'}_{d}$, normalized by its maximum value ${p'}_{d,max}$, for a period of 2 s; the normalized PSD of ${p'}_{d}$ as a function of frequency, obtained using discrete FFT; the time-dependent behaviour of the signal's frequency, coloured by the normalized amplitude of the signal's CWT; and the normalized p.d.f. percentage of the maximum CWT-based signal frequency. Cyan circles on the PSD and p.d.f. plots highlight the maximum dominant frequencies obtained using FFT and CWT.

Figure 19. Throat hydraulic diameter $D_{h,th}$ based Strouhal number $St$ (see (3.2)) as a function of (a) the throat Reynolds number ${ {\textit {Re}}}_{th}$, and (b) cavitation number $\sigma$ in pure water and different PAM solution flows. Panels (a,b) and (c,d) display the results of PSD and CWT spectral analysis, respectively.

Figure 19(a) shows that $St$ is almost constant before the cavitation onset in the pure water flow and is followed by a rapid increase in $St$ as the flow begins to cavitate. After the onset, $St$ reduces exponentially as the flow rate increases. The PAM solutions with concentrations in the range of 50–100 p.p.m. show a similar trend of changes in the $St({ {\textit {Re}}}_{th}$) profile, with relatively smaller peaks of $St$. At a constant ${ {\textit {Re}}}_{th}>{ {\textit {Re}}}_{th,i}$, the 200 p.p.m. solution shows larger $St$ relative to the pure water and other concentrations. In contrast, the 400 p.p.m. solution demonstrates lower $St$ than the other semidilute solutions until its inception at ${ {\textit {Re}}}_{th,i}=3.7\times 10^{4}$, after which $St$ suddenly increases to values relatively higher than the other tested solutions. As figure 19(a) illustrates, the dominant shedding frequency at the inception reduces by increasing the solution concentration. Figure 19(b) shows that the $St$ of the pure water abruptly increases as the cavitation process starts and gradually decreases as $\sigma$ decreases. The PAM solutions of 50 to 200 p.p.m. show similar trends of changes in $St(\sigma$) profiles. As the flow rate increases and $\sigma \approx 6$ in the 400 p.p.m. solution, $St$ increases slightly and maintains a value of $St\approx 0.1$ until the inception point, where $St$ suddenly increases to $St\approx 0.2$, which is ${\approx }70\,\%$ smaller than the inception $St$ of the pure water. These findings suggest that another mechanism by which the viscoelastic polymer molecules damp out cavitation is by reducing the dominant inception shedding frequency.

Strouhal numbers were also calculated based on the cavitation length $L_{ca}$. The resultant profiles are illustrated in figure 20(a,b). As shown in figure 20(a), the $St$ profile of pure water flow shows a local peak at the cavitation onset and after a slight decrease starts to increase almost linearly, in contrast to its counterpart in figure 19(a). This highlights that as $ {\textit {Re}}_{th}$ increases, the increase in $L_{ca}$ dominates the decrease in $f$. The $St( {\textit {Re}}_{th})$ profiles of 50 and 100 p.p.m. solutions in figure 20(a) show similar local peaks at the cavitation onset, with a reduced amplitude relative to the pure water. After the onset, $St$ remains almost constant with an increase of $ {\textit {Re}}_{th}$ for the 50 and 100 p.p.m. solutions. The $St$ definition based on $L_{ca}$ is ill-defined for non-cavitating flow. For instance, the 200 and 400 p.p.m. solutions have $L_{ca}=0$ at $ {\textit {Re}}_{th}\lesssim 3.2 \times 10^{4}$, equivalent to $St=0$ in this range. For higher $ {\textit {Re}}_{th}$, $St$ grows linearly in both concentrations, with a smaller slope than that of pure water flow. As shown in figure 20(b), as $\sigma$ decreases, $St(\sigma )$ profiles of the solutions display similar trends as of $St( {\textit {Re}}_{th})$ for increasing $ {\textit {Re}}_{th}$.

Figure 20. Cavitation length $L_{ca}$ based Strouhal number $St$ (see (3.2)) as a function of (a) the throat Reynolds number ${ {\textit {Re}}}_{th}$, and (b) cavitation number $\sigma$ in pure water and different PAM solution flows. Panels (a,b) and (c,d) display the results of PSD and CWT spectral analysis.