Experimental study and numerical sizing model for cavitation zone characterisation in cavitating venturis

Abstract

Cavitating venturi is popular as an elegant passive flow control device used in varied engineering applications. Cavitating venturi can be sized appropriately for operating with an anchored flow rate for various industrial applications. In the current work, we present the experimental results of the cavitation zone lengths in five planar venturis with different throat widths and divergent angles obtained for a pressure ratio range of 0.4 to 0.95 and an inlet Reynolds number range of 8.0$\times$10$^4$ to 2.25$\times$10$^5$. The cavitation zone lengths are obtained for quasi-steady conditions through high-speed imaging. The extracted lengths for the venturis indicate a transition in the cavitation zone behaviour and have a dependence on the divergence angle and cavitation intensity. The extracted data forms the primary dataset for validating the numerical model which we present in the subsequent part of the work. The model is a one-dimensional homogeneous two-phase model with a two-step Euler integration of the Rayleigh-Plesset equation as closure to handle the bubble dynamics. The model shows a prediction of the experimentally obtained cavitation lengths within ±10$\%$ (for small divergent angles) and ±25$\%$ (for large divergent angles) specifically at high cavitation intensities when the cavitation zone fills the divergent portion. Finally, we demonstrate the applicability of the model as a typical engineering sizing tool to predict the operating pressure ratios. The model could predict the experimentally obtained critical pressure ratios and the minimum pressure ratios within ±12$\%$ and ±20.7$\%$ respectively for the planar venturis. Specifically, if the interest is in containing the cavitation zone within the divergent portion, this model could definitely be an aid in sizing a cavitating venturi for varied engineering applications.

Appendices

Appendix A

Experimental Uncertainties: The uncertainties of different instruments are given in table 5. The systematic errors indicated in the table for all the instruments are as provided by the manufacturer. The wall pressures measured in the cavitation zone have higher uncertainties due to the cavity zone oscillations. The percentage uncertainties in pressure ratio, cavitation number, and the inlet Reynolds number quoted in section 2.3 are obtained by the rule of propagation of the measured errors in pressure, mass flow rate and venturi dimensions in lines with [46].

Table 5 $\%$ Uncertainties of various instruments used in the experiment

Appendix B

Discretisation of Model Equations: The algorithm uses the following discretised equations.

$$\begin{aligned} & \rho _{i+1}=1-\alpha _{i+1} \end{aligned}$$

(A1)

$$\begin{aligned} & x_{i+1}=\frac{1}{1+\dfrac{\rho _l}{\rho _v}\dfrac{1-\alpha _{i+1}}{\alpha _{i+1}}}\end{aligned}$$

(A2)

$$\begin{aligned} & \phi ^2_{lo,i+1}=1.0+1.2\left( \frac{\rho _l}{\rho _v}-1\right) x^{0.824}_{i+1}\end{aligned}$$

(A3)

$$\begin{aligned} & n_1=n_0(1-\alpha )\end{aligned}$$

(A4)

$$\begin{aligned} & R_{i+1}=\left[ \frac{\alpha _{i+1}}{4/3\pi n_1}\right] ^{1/3} \end{aligned}$$

(A5)

Discretised form of continuity equation is

$$\begin{aligned} u_{i+1}=\frac{u_i A_i \rho _i}{A_{i+1} \rho _{i+1}} \end{aligned}$$

(A6)

Discretised form of momentum equation is

$$\begin{aligned} & \frac{\rho _{i+1} u^2_{i+1} A_{i+1}-\rho _{i} u^2_{i} A_{i}}{z_{i+1}-z_i} =\frac{-1}{A_{avg}}\frac{P_{i+1}-P_{i}}{z_{i+1}-z_i}\nonumber \\ & \quad -f_{lo,i+1}\rho ^2_{i+1} u^2_{i+1} r_{h,i+1} \phi ^2_{lo,i+1} \end{aligned}$$

(A7)

$$\begin{aligned} A_{avg}= \frac{A_{i+1}+A_i}{2} \end{aligned}$$

(A8)

The closures are discretised as

$$\begin{aligned} & P_{b,i+1}=P_{b,i}\frac{R^3_{i}}{R^3_{i+1}} \end{aligned}$$

(A9)

$$\begin{aligned} & \left[ \frac{du_b}{dt}\right] _{i+1}=\frac{P_{b,i}-P_{i}}{R_i}-\frac{3}{2}\frac{u^2_{b,i}}{R_i} \end{aligned}$$

(A10)

The local time scale is calculated as

$$\begin{aligned} dt_{i+1}=\frac{z_{i+1}-z_i}{0.5(u_{i+1}+u_i)} \end{aligned}$$

(A11)

The new bubble wall velocity and the bubble radius are found using Euler integration as

$$\begin{aligned} & u_{b,i+1}=u_{b,i}+dt_{i+1}\left[ \frac{du_b}{dt}\right] _{i+1} \end{aligned}$$

(A12)

$$\begin{aligned} & R_{i+1}=R_{i}+dt_{i+1}u_{b,i+1} \end{aligned}$$

(A13)